Research Article Design of Optimal PID Controller with ...downloads.hindawi.com/journals/mpe/2013/582879.pdf · pole placement method is used to design the PID controller. In [ ],

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2013 Article ID 582879 22 pageshttpdxdoiorg1011552013582879

Research ArticleDesign of Optimal PID Controller with 120576-Routh Stability forDifferent Processes

XianHong Li12 HaiBin Yu1 and MingZhe Yuan1

1 Department of Information Service and Intelligent Control Shenyang Institute of Automation Chinese Academy of SciencesShenyang 110016 China

2University of Chinese Academy of Sciences (Graduate School of Chinese Academy of Sciences) Beijing 100039 China

Correspondence should be addressed to HaiBin Yu yhbsiacn

Received 30 November 2012 Revised 27 February 2013 Accepted 11 March 2013

Academic Editor Shane Xie

Copyright copy 2013 XianHong Li et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

This paper presents a design method of the optimal proportional-integral-derivative (PID) controller with 120576-Routh stability fordifferent processes through Lyapunov approach The optimal PID controller could be acquired by minimizing an augmentedintegral squared error (AISE) performance index which contains control error and at least first-order error derivative or even maycontain nth-order error derivative The optimal control problem could be transformed into a nonlinear constraint optimization(NLCO) problem via Lyapunov theoremsTherefore optimal PID controller could be obtained by solving NLCO problem throughinterior method or other optimization methods The proposed method can be applied for different processes and optimal PIDcontrollers under various control weight matrices and 120576-Routh stability are presented for different processes Control weight matrixand 120576-Routh stabilityrsquos effects on system performances are studied and different tuning methodsrsquo system performances are alsodiscussed 120576-Routh stabilityrsquos effects on disturbance rejection ability are investigated and different tuning methodsrsquo disturbancesrejection ability is studied To further illustrate the proposed method experimental results of coupled water tank system (CWTS)under different set points are presented Both simulation results and experiment results show the effectiveness and usefulness ofthe proposed method

1 Introduction

PID controllers have been widely used in various industrialprocesses due to simple structure few parameters and easyimplementation In [1ndash43] many tuning methods such asempirical methods robust methods frequency methodsnumerical methods and intelligent methods had been pre-sented to design PID controllers In [3 4] Ziegler-Nichols (Z-N) methods include continuous cycling (C-C) method andprocess reaction curve (PRC) method and they can be calledas first Z-N method and second Z-N method respectivelyThe PRC method is especialy proposed for processes withtime delay In [5] the Cohen-Coon method is presented forfirst-order plus time delay (FOPTD) processes In [6] theintegral absolute error (IAE) performance index is used todesign the optimal PID controller for FOPTD and second-order plus time delay (SOPTD) plants In [7ndash13] the immunealgorithm modified genetic algorithm modified ant colony

algorithm particle swarm algorithm (PSA) and modifiedPSA are used to design optimal PID controller for differentsystems In [14] multiple tabu search algorithm (MTSA) isapplied to design the optimal fuzzy-logic PID controller forload frequency control system In [15] some formulas aredeveloped to directly compute the optimal PID controllerfor certain processes In [16] zero-pole assignment methodis directly applied to compute optimal PID controller forcontinuous-time processes In [17] direct search algorithmis used to design the optimal PID controller In [18] theoptimal PID controller is presented for FOPTD plants viadimensional analysis andnumerical optimization techniquesIn [19] a design method is presented to design the optimalPID controller via minimizing load disturbances and solvinga constrained optimization problem in frequency domainIn [20 21] design methods are proposed to develop optimalPIPID controller via solving a nonconvex optimization prob-lem in frequency domain In [22] hybrid Taguchi genetic

2 Mathematical Problems in Engineering

algorithm (HTGA) is used to design optimal PID con-troller for pulse width modulation (PWM) feedback systemsthrough solving a constrained optimization problem In [23ndash25] linear quadratic regulator (LQR) method is proposed todesign the optimal PID controller for different systems andprocesses In [26] the loop transfer recovery technique isapplied to devise multivariable robust optimal PID controllerfor a nonminimum phase boiler system In [27] optimalPID controller with 119867

2119867infin

norm constraint is presentedfor the superheated steam temperature system via PSA In[28] a parameter space approach is proposed to design theminimum variance PID controller for linear time-invariant(LTI) plants In [29] a design method is proposed to designPID controller by considering a transient performance itemIn [30] a designmethod of multivariable fixed-structure PIDcontroller is proposed for multivariable plants by solving anoptimization problem with linear matrix inequality (LMI)constraint In [31] iterative feedback tuning (IFT) method isused to design the optimal PID controller for achieving fastresponse to set point variations In [32] a two-layer onlineautotuning algorithm is presented to design the optimal PIDcontroller via general predictive control method In [33]the LQR method is presented to design the optimal PIDcontroller for robot arms In [39] a design method of PIDcontroller is proposed for SOPTD processes via internalmodel control (IMC) framework In [40] analytic rules ofthe PID controller are presented for FOPTD and SOPTDprocesses via the IMCmethod andmodel reduction methodIn [41] a performance assessmentmethod of PID controller isproposed for integral processes via focusing on step responsesof the set point change and load disturbance In [42] a designmethod of the PID controller with feedforward compensatoris proposed for the FOPTD process In [43] a design methodof the multiloop PIPID controller is presented for squaremultiple-input and multiple-output (MIMO) process whichis decomposed into a series of equivalent effective open-loop process (EOP) with reasonable gain margins and phasemargins In [44] an on-line relay feedback approach is usedto identify assess and tune the PID controller In [45] thepole placement method is used to design the PID controllerIn [46] the multiobjective genetic algorithm is presented todesign the optimal fractional-order PID controller In [47] adesign method is presented for optimal PID controller withdynamic performance constrained

Design methods of the PID controller in [1ndash43] can beroughly divided into the following cases the empirical meth-ods intelligent algorithm methods numerical parametricmethods classic LQRLQGmethods robust designmethodsIMC methods and pole placement methods Many designmethods of the PID controller are proposed for first-orderand second-order processes In this paper a novel systematicdesign method is presented to design optimal PID controllerwith 120576-Routh stability for different processes The optimalPID controllerwith 120576-Routh stability is proposed byminimiz-ing augmented integral squared error (AISE) performanceindexThe proposed method is unlike classic optimal controldesign methods which need to solve the Riccati equation(RE) The proposed optimal control problem is equivalentlytransformed into theNLCOproblem via Lyapunov theorems

Therefore optimal controller parameters can be obtained bysolving a NLCO problem The proposed method is used todesign optimal PID controller for a third-order SOPTD andFOPTDprocesses respectivelyThe proposedmethod is usedto control the liquid level of the CWTS under different setpoints The simulation results and experiment results showeffectiveness and usefulness of the proposed method Theproposed method is convenient to design the optimal PIDcontroller which can provide high performance for controlsystem

This paper is arranged as follows in Section 2 prelimi-nary and optimal control problem statements are describedIn Section 3 CWTS model is established and systematicdesign procedures of the optimal PID controller are presentedin detail Characteristics and robustness of 120576-Routh stabilityare analyzed and the interior method is recommended tosolve the NLCO problem of the proposed optimal PIDcontroller In Section 4 optimal PID controllers and sim-ulation results are presented for different processes Thecontrolweightmatrix and 120576-Routh stabilityrsquos effects on systemperformances are studied The system performances anddisturbance rejection ability of different tuning methodsare investigated 120576-Routh stabilityrsquos effects on disturbancerejection ability are discussed as well To further validate theproposed method the proposed method is used to controlthe liquid level of the CWTS under different set points InSection 5 study contents are reviewed and some conclusionsare made

2 Preliminary and Optimal ControlProblem Statements

Preliminary Based on Lyapunov theorems a systematic tun-ing method is proposed to design the optimal PID controllerwith 120576-Routh stability The following theorems and 120576-Routhstability definition are utilized to design the optimal PIDcontroller

Theorem 1 (see [1 2]) Sufficient and necessary conditions forlinear time-invariant system = 119860119909 asymptotically stable in alarge scope are that for any given matrix 119876 = 119876

119879gt 0 there

is a matrix 119875 = 119875119879

gt 0 that satisfies the Lyapunov algebraicequation (LAE) 119860119879119875 + 119875119860 = minus119876

Theorem 2 (see [1 2]) The linear time-invariant (LTI) system = 119860119909 and 119909(0) = 119909

0is asymptotically stable for any given

matrix 119876 = 119876119879

gt 0 then the performance index J will haveequivalent relationships

119869 = int

infin

0

119909119879119876119909119889119905 = 119909

119879(0) 119875119909 (0) 119860

119879119875 + 119875119860 = minus119876

(1)

Definition of 120576-Routh stability 119863(119904) is characteristic poly-nomial of the linear time-invariant control system For anynonnegative real 120576 if all elements of the Routh arrayrsquosfirst column of characteristic polynomial 119863(119904) = 119886

0119904119899+

1198861119904119899minus1

+ sdot sdot sdot + 119886119899minus1

119904 + 119886119899can satisfy the following inequality

Mathematical Problems in Engineering 3

relationships then the linear time-invariant control system isconsidered to possess the 120576-Routh stability

11987711

(1198860 1198861 119886

119899) gt 120576 119877

21(1198860 1198861 119886

119899) gt 120576

1198771198991

(1198860 1198861 119886

119899) gt 120576 119877

119899+11(1198860 1198861 119886

119899) gt 120576

(120576 ge 0)

(2)

where 11987711

11987721

11987731

119877119899 1 and 119877

119899+11are the elements of

the Routh arrayrsquos first column It calls Routh arrayrsquos firstcolumn as the Routh column and the general computationformulas of the Routh column can be founded in [48 49]It could be known that the 120576-Routh stability degeneratesinto the Routh-Hurwitz stable when the 120576 is zero (120576 =

0) The motivation for the definition of 120576-Routh stabilitycomes from stability analysis of linear systems As knownif any one element in Routh column of polynomial 119863(119904) iszero or negative real then linear control systems may bringoscillation behaviors or unstable behaviors Each element inthe Routh column of the polynomial 119863(119904) is much closer tothe zero 0

+ at least it is more possible to bring oscillationbehaviors Consequently the definition of 120576-Routh stability ismotivated to describe how far away each Routh column fromthe zero 0

+ It can be known that 120576-Routh stability will implysystem robustness in some degreeOptimal Control Problem Statements We assume that theprocesses can be described by general transfer function119866

119901(119904)

119866119901(119904) =

1198871119904119908+ 1198872119904119908minus1

+ sdot sdot sdot + 119887119908119904 + 119887119908+1

119904119899+ 1198861119904119899minus1


119904 + 119886119899

=

sum119908+1

119895=1119887119895119904119908+1minus119895

119904119899+ sum119899

119894=1119886119894119904119899minus119894

=

119873 (119904)

119872 (119904)

(3)

where 1198871 119887

119908+1 1198861 and 119886

119899are system parameters and

119872(119904) and 119873(119904) are the denominator and numerator respec-tively The denominator and numeratorrsquos orders are thedeg119872(119904) = 119899 and deg119873(119904) = 119908 respectively For mostpractical processes the number of poles 119899 is greater thanthe number of zeros 119908 (119899 gt 119908) and 119899 and 119908 are thepositive integers It could be known that for the all-pole plants1198871 1198872 119887

119908= 0 additional for the type I plants 119886

119899= 0 and

for the type II plants 119886119899minus1

= 119886119899= 0 and so on The diagram

of the control system is shown in Figure 1 and the controlsystem can be described by the following equation

119906 (119904) = 119866119888(119904) 119890 (119904) 119866

119888(119904) =

(119896119901119904 + 119896119894+ 1198961198891199042)

119904

119903 (119904) minus 119910 (119904) ≜ 119890 (119904) 119910 (119904) = 119866119901(119904) 119906 (119904)

(4)

where 119890(119905) is control error 119903(119905) and 119910(119905) are reference com-mand and system output respectively119866

119888(119904) is PID controller

and 119896119901 119896119894 and 119896

119889are controller parameters Equation (4) can

be equally expressed as

119903 (119904) = 119910 (119904) + 119890 (119904) 119910 (119904) =

119873 (119904) 119906 (119904)

119872 (119904)

119906 (119904) =

(119896119901119904 + 119896119894+ 1198961198891199042) 119890 (119904)

119904

(5)

Therefore error transfer function can be obtained as119890 (119904)

119903 (119904)

=

1

(1 + 119866119901(119904) 119866119888(119904))

lArrrArr 119863(119904) 119890 (119904)

= 119904119872 (119904) 119903 (119904)

119863 (119904) = 119904119872 (119904) + 119873 (119904) (119896119901119904 + 119896119894+ 1198961198891199042)

deg 119863 (119904) = 119899 + 1

(6)

where 119863(119904) is the control systemrsquos characteristic polynomialThe error differential equation can be obtained by inversingLaplace transform of (6)

(120575119872 (120575) + 119873 (120575) (119896119901120575 + 119896119894+ 1198961198891205752)) 119890 (119905) = 120575119872 (120575) 119903 (119905)

(7)

where 120575 is the differential operator When reference com-mand 119903(119905) is constant signal or zero signal (119903(119905) = 0) or eventhe piecewise constant signal the error differential equationwill yield

(120575119899+1

+

119899

sum

119894=1

119886119894120575119899+1minus119894

)

+ (119896119901120575 + 119896119894+ 1198961198891205752)

119908+1

sum

119895=1

119887119895120575119908+1minus119895

119890 (119905) = 0

(8)

Considering the polynomial differential operator 119872(120575) and119873(120575) error differential equation (8) could be expressed as

120575119899+1

+

119899

sum

119894=1

119886119894120575119899+1minus119894

+

119908+1

sum

119895=1

119887119895119896119901120575119908+2minus119895

+

119908+1

sum

119895=1

119887119895119896119894120575119908+1minus119895

+

119908+1

sum

119895=1

119887119895119896119889120575119908+3minus119895

119890 (119905) = 0

(9)

Therefore (9) can be equally expressed as

119890(119905)(119899+1)

+ 1198861119890(119905)(119899)

+ 1198862119890(119905)(119899minus1)

+ sdot sdot sdot + 119886119899minus119908minus2

119890(119905)(119908+3)

+ (119886119899minus119908minus1

+ 1198871119896119889) 119890(119905)(119908+2)

+ (119886119899minus119908

+ 1198871119896119901+ 1198872119896119889) 119890(119905)(119908+1)

+

119908minus1

sum

119895=1

(119886119899minus119908+119895

+ 119887119895+1

119896119901+ 119887119895119896119894+ 119887119895+2

119896119889) 119890(119905)(119908+1minus119895)

+ (119886119899+ 119887119908+1

119896119901+ 119887119908119896119894) 119890 (119905) + 119887

119908+1119896119894119890 (119905) = 0

(10)


For the symbol simplicity the error differential equation (10)is equally expressed as

119890(119905)(119899+1)

+ 1198861119890(119905)(119899)

+ 1198862119890(119905)(119899minus1)


119890(119905)(119908+3)

+ 1198981119890(119905)(119908+2)

+ 1198982119890(119905)(119908+1)

+

119908minus1

sum

119895=1

119898119895+2

119890(119905)(119908+1minus119895)

+ 119898119908+2

119890 (119905) + 119898119908+3

119890 (119905) = 0

(11)

where 119898119894(119894 = 1 2 119908 + 3) is the decision parameter

Parameter119898119894(119894 = 1 2 119908 + 3) is obtained as

1198981= (119886119899minus119908minus1

+ 1198961198891198871) 119898

2= (119886119899minus119908

+ 1198871119896119901+ 1198872119896119889)

119898119895+2

= (119886119899minus119908+119895

+ 119887119895+1

119896119901+ 119887119895119896119894+ 119887119895+2

119896119889)

(119895 = 1 2 119908 minus 1)

119898119908+2

= (119886119899+ 119887119908+1

119896119901+ 119887119908119896119894) 119898

119908+3= 119887119908+1

119896119894

(12)

Thus the corresponding state-space model of the differentialequation (11) can be obtained

= 119860119909 (13)

where 119909 is a state vector 119860 is a state matrix 1198621times119899

is aparameter vector 119874

119899times1is a zero vector and 119868

119899times119899is an unit

matrix They are obtained as

119860 = (

119874119899times1

119868119899times119899

minus119898119908+3

1198621times119899

) 119909 = (119890 (119905) 119890 (119905) 119890 (119905) sdot sdot sdot 119890(119905)(119899)

)

119879

1198621times119899

= (minus119898119908+2

sdot sdot sdot minus 1198981minus 119886119899minus119908minus2

sdot sdot sdot minus 1198862minus 1198861)

(14)

To obtain the optimal PID controller it needs to selectsuitable optimized performance indices The performanceindices such as the integral squared error (ISE) integralabsolute error (IAE) integral time absolute error (ITAE) andintegral time squared error (ITSE) are often used for criteriawhen designing optimal PID controller The precision andsteady-state property of control system are directly reflectedby control error In this paper augmented integral squarederror (AISE) in [34 36] is used to design an optimal PIDcontroller Hence optimal PID controller not only stabilizesthe control system with 120576-Routh stability but also minimizesthe performance indexTherefore the proposed optimal con-trol problem can be stated as (i) the optimal PID controllerstabilizes the control system with 120576-Routh stability (ii) theoptimal PID controller minimizes the given performance

index Eventually the proposed optimal control problem canbe depicted by the following equation

119869lowast(119896lowast

119901 119896lowast

119894 119896lowast

119889) = min119866lowast

119888(119904)

119869 (1198981 119898

119908+3)

= minint

infin

0

119909119879(119905) 119876119909 (119905) 119889119905

= min

119899

sum

119894=0

int

infin

0

119902119894+1119894+1

(119890(119905)(119894))

2

119889119905

subject to (st)

(1) = 119860119909

(2)Re (eig (119860)) lt 0 Re 119904 | det (119904119868 minus 119860) = 0 lt 0

(3) 120576 ge 0 11987711

(1198981 119898

119908+3) gt 120576

11987721

(1198981 119898

119908+3) gt 120576

119877119899+11

(1198981 119898

119908+3) gt 120576 119877

119899+21(1198981 119898

119908+3) gt 120576

(4)1198984= 1198911(1198981 1198982 1198983) 119898

119908+3= 119891119908(1198981 1198982 1198983)

(5) 0 lt 119896119901le 1205910

119901 0 lt 119896

119894le 1205910

119894 0 lt 119896

119889le 1205910

119889

(15)

where 119876 is a control weight matrix and at least a positivesemidefinite real symmetric matrix and 119891

1 119891

119908is the

linear function of the controller parameters 119896119901 119896119894 and 119896

119889 and

119896lowast

119901 119896lowast119894 and 119896

lowast

119889are optimal controller parameters Constraint

(1) is the control systemrsquos state constraint constraint (2)

guarantees the control system stable in a large scope andconstraint (3) guarantees the control systemwith the 120576-Routhstability However there are three independent decisionvariables among decision variables119898

119894(119894 = 1 2 119908+3) the

rest decision variables are interrelated variables Constraint(4) reflects decision variablesrsquo inner relationships Constraint(5) can assure the existence of feasible solutions and canreduce the searching space Constraint (5) can also assure PIDcontroller to be physically achieved In some actual controlsystem PID controller parameters are limited in certainrange Thus the controller parameters cannot be too largeThe parameters 120591

0

119901 1205910119894 and 120591

0

119889are the scope of controller

parameters

3 CWTS Model and ProposedDesign Procedures

The proposed design method is used to design optimal PIDcontroller which is employed to control the liquid level ofthe CWTS The experiment setup is shown in Figure 2 Itcan use the experimental setup to construct different physicalprocesses which are shown in Figure 3

The physical process in Figure 3 contains the interactingbehavior because the liquid level ℎ

2of tank-2 depends on the

liquid level ℎ1of tank-1 In this paper it does not consider

nonlinearities of the coupled water tank system Thus the


Command

119903(119905) +

+

minusminus

119890(119905) Controlledplant 119866119901(119904)

Optimal tuningmechanism

119910(119905)

Output119863119906(119905)

119866lowast119888 (119904) = 119896

lowast119901 +

119896lowast119894119904+ 119896lowast119889119904

Optimal PIDcontroller

Figure 1 Feedback control system with optimal PID controller

Control valve

ValveWater tank

Tank-1

Tank-2

Figure 2 Experimental setup of CWTS

water tank system is considered as the linear process Basedon the volume balance principle the liquid level equation oftank-1 can be obtained

119876119894119888minus 1198761119888

minus 11987612

= 1198621

119889ℎ1

119889119905

(16)

where119876119894119888is the flow of control valve119876

1119888is the flow of valve-

1 11987612

is the flow of valve-3 1198621is the hydraulic capacity of

tank-1 and ℎ1is the liquid level of tank-1 Likewise the liquid

level equation of tank-2 can be obtained

11987612

minus 1198762119888

= 1198622

119889ℎ2

119889119905

(17)

where 1198762119888is the flow of valve-2 119862

2is the hydraulic capacity

of tank-2 and ℎ2is the liquid level of tank-2 The water flows

equation of valve-1 valve-2 and valve-3 can be obtained

1198761119888

=

ℎ1

1198771

1198762119888

=

ℎ2

1198772

11987612

=

ℎ1minus ℎ2

11987712

(18)

where1198771and119877

2are the liquid resistance of valve-1 and valve-

2 respectively and 11987712

is the liquid resistance of valve-3Thus the liquid level model of water tank-1 is acquired

119866ℎ(119904) =

1198671(119904)

119876119894119888(119904)

=

1198871119904 + 1198872

11988601199042+ 1198861119904 + 1198862

=

11988711199041198860+ 11988721198860

1199042+ 11988611199041198860+ 11988621198860

=

1198731(119904)

1198721(119904)

(19)

where 119866ℎ(119904) is the liquid level transfer function of tank-1

The transfer function119866ℎ(119904) shows that physical process is the

Tank-1 Tank-2

Control valve

1198771

119876121198671 + ℎ1

11986221198621

1198772

11987621198881198761119888

1198760 + 119876119894119888

11987712

1198672 + ℎ2Valve-3

Valv

e-2

Valv

e-1

Figure 3 The liquid level process of the CWTS

second-order processwith a negative zero Systemparametersare obtained as

1198871= 11987711198771211987721198622 119887

2= 1198771(11987712

+ 1198772)

1198860= 119877111986211198771211987721198622

1198861= 119877111986211198772+ 1198771119862111987712

+ 119877111987721198622+ 1198771211987721198622

(20)

1198862= 1198771+ 11987712

+ 1198772 (21)

31 Proposed Design Procedures Without losing generalitiesa third-order controlled process is considered in the unityfeedback system The process (119899 = 3 119908 = 2) and PIDcontroller have forms respectively

119866119901(119904) =

11988711199042+ 1198872119904 + 1198873

11988601199043+ 11988611199042+ 1198862119904 + 1198863

=

119873 (119904)

119872 (119904)

119866119888(119904) = 119896

119901+

119896119894

119904

+ 119896119889119904

(22)

Thus open-looped transfer function of control system yields

119866 (119904) = 119866119888(119904) 119866119901(119904) =

(119896119901119904 + 119896119894+ 1198961198891199042)119873 (119904)

119904119872 (119904)

(23)

Then the error equation and of the control system areobtained

119890 (119904)

119903 (119904)

=

1

(1 + 119866119901(119904) 119866119888(119904))

lArrrArr 119863(119904) 119890 (119904) = 119904119872 (119904) 119903 (119904)

119863 (119904) = (119904119872 (119904) + 119873 (119904) (119896119901119904 + 119896119894+ 1198961198891199042))

deg 119863 (119904) = 4

(24)

where 119863(119904) is the control systemrsquos characteristic polynomialThe error differential equation can be obtained by inversingLaplace transform of the error equation

(120575119872 (120575) + 119873 (120575) (119896119901120575 + 119896119894+ 1198961198891205752)) 119890 (119905) = 120575119872 (120575) 119903 (119905)

(25)

where 120575 is a differential operator When reference command119903(119905) is the constant signal or zero signal (119903(119905) = 0) or even


piecewise constant signal then the error differential equationwill yield

(1198860+ 1198961198891198871) 119890(119905)(4)

+ (1198861+ 1198961199011198871+ 1198961198891198872)119890 (119905)

+ (1198862+ 1198961198941198871+ 1198961199011198872+ 1198961198891198873) 119890 (119905)

+ (1198863+ 1198961198941198872+ 1198961199011198873) 119890 (119905) + 119896

1198941198873119890 (119905) = 0

(26)

For symbol simplicity error differential equation is equallyexpressed as

1198981119890(119905)(4)

+ 1198982

119890 (119905) + 119898

3119890 (119905) + 119898

4119890 (119905) + 119898

5119890 (119905) = 0 (27)

where 119898119894(119894 = 1 5) is the decision parameter The

decision parameter yields

1198981= 1198860+ 1198961198891198871 119898

2= 1198861+ 1198961199011198871+ 1198961198891198872 119898

5= 1198961198941198873

1198983= 1198862+ 1198961198941198871+ 1198961199011198872+ 1198961198891198873 119898

4= 1198863+ 1198961198941198872+ 1198961199011198873

(28)

Thus the corresponding state-space model of error differen-tial equation (27) can be obtained

= 119860119909

119909 = (119890 (119905) 119890 (119905) 119890 (119905)

119890 (119905))119879 119860 = (

1198743times1

1198683times3

minus1198981

41198621times3

)

1198621times3

= (minus1198981

3minus 1198981

2minus 1198981

1)

(29)

where 119909 is state vector 119860 is state matrix 1198981119894is the middle

decision variables and the 1198981

119894= 119898119894+1

1198981(119894 = 1 2 3 4)

Therefore the optimal control problem is formulated as


119901 119896lowast

119894 119896lowast

119889) = min119866lowast

119888

119869 (1198981 119898

5)

= minint

infin

0

119909119879(119905) 119876119909 (119905) 119889119905

= min

3

sum

119894=0

int

infin

0

119902119894+1119894+1

(119890(119905)(119894))

2

119889119905

st

(1) = 119860119909

(2)Re (eig (119860)) lt 0 or Re 119904 | det (119904119868 minus 119860) = 0 lt 0

(3) 120576 ge 0 11987711

(1198981 119898

119908+3) gt 120576

11987721

(1198981 119898

119908+3) gt 120576

11987731

(1198981 119898

119908+3) gt 120576 119877

41(1198981 119898

119908+3) gt 120576

11987731

(1198981 119898

119908+3) gt 120576 119877

41(1198981 119898

119908+3) gt 120576

11987751

(1198981 119898

119908+3) gt 120576

(4)1198984= 1198911(1198981 1198982 1198983) 1198985= 1198912(1198981 1198982 1198983)

(5) 0 lt 119896119901le 1205910

119901 0 lt 119896

119894le 1205910

119894 0 lt 119896

119889le 1205910

119889

(30)

If optimal control problem (30) is solved then the proposedoptimal PID controller could be obtained The constraint(2) assures that the control systems asymptotically stablein a large scope The determinant is det(119904119868 minus 119860) = 119904

4+

1198981

11199043+ 1198981

21199042+ 1198981

3119904 + 119898

1

4 Based on LTI stablility theory

constraint (2) is equivalently transformed into the followinginequality

Re (eig (119860)) lt 0 or Re 119904 | det (119904119868 minus 119860) = 0 lt 0

lArrrArr 1198981

1gt 0 119898

1

11198981

2minus 1198981

3gt 0

1198981

11198981

21198981

3minus (1198981

3)

2

minus 1198981

4(1198981

1)

2

gt 0 1198981

4gt 0

(31)

The constraint (3) assures that the control system possessesthe 120576-Routh stability The characteristic polynomial is119863(119904) =

11989811199044+11989821199043+11989831199042+1198984119904+1198985 According to 120576-Routh stability

definition the constraint (3) is formulated by the followinginequalities

120576 ge 0 11987711

(1198981 119898

5) gt 120576 119877

21(1198981 119898

5) gt 120576

11987731

(1198981 119898

5) gt 120576

11987741

(1198981 119898

5) gt 120576 119877

51(1198981 119898

5) gt 120576

lArrrArr 1198981gt 120576 119898

2gt 120576 119898

5gt 120576

11989811198984minus 11989821198983gt 1205761198982

(1198982)2

1198985minus 1198981(1198984)2

minus 119898211989831198984gt 120576 (119898

11198984minus 11989821198983)

(32)

The inequalities (31) and (32) assure control system asymp-totic stability and 120576-Routh stability respectively Thereforethe control system is asymptotically stable in a large scopebased on the LyapunovTheorems 1 and 2 performance index119869 is equivalently transformed into the performance index

119869lowast= minint

infin

0

119909119879119876119909119889119905 = min119909

119879(0) 119875119909 (0)

119860119879119875 + 119875119860 = minus119876

(33)

where 119875 is a positive definite real symmetric matrix andmatrix 119875 meets the LAE 119860119879119875 + 119875119860 = minus119876 The performanceindex 119869 is determined by the matrix 119875 and initial state


Therefore the proposed optimal control problem (30) can beequivalently transformed into the following NLCO problem


119901 119896lowast

119894 119896lowast

119889) = min119866lowast

119888(119904)

119869 (1198981 119898

5)

= min119909(0)119879119875119909 (0)

st

(1) 119860119879119875 + 119875119860 = minus119876

(2)1198981

1gt 0 119898

1

4gt 0 119898

1

11198981

2minus 1198981

3gt 0

1198981

11198981

21198981

3minus (1198981

3)

2

minus 1198981

4(1198981

1)

2

gt 0

(3)1198981gt 120576 119898

2gt 120576 119898

5gt 120576 119898

11198984minus 11989821198983gt 1205761198982

(1198982)2

1198985minus 1198981(1198984)2

minus 119898211989831198984gt 120576 (119898

11198984minus 11989821198983)

(4)1198984= 1198911(1198981 1198982 1198983) 1198985= 1198912(1198981 1198982 1198983)

(5) 0 lt 119896119901le 1205910

119901 0 lt 119896

119894le 1205910

119894 0 lt 119896

119889le 1205910

119889

(34)

The proposed optimal control problem (30) is equivalentlytransformed into aNLCOproblemvia applying the Lyapunovtheorems The optimal PID controller with 120576-Routh stabilitycan be obtained by solving NLCO problem (34) The designproblem of optimal PID controller with 120576-Routh stabilityis deduced into the issue that for the given control weightmatrix and initial states we pursue a suitable matrix 119875 tominimize performance index J We assume that the controlsystem is static at the beginning then a unit step command119903(119905) = 1(119905) is inputted into the system and the initial statescan be obtained 119909

1(0) = 1 119909

2(0) = 0 119909

3(0) = 0 and

1199094(0) = 0 The control weight matrix 119876 is a diagonal matrix

with 119876 = diag(1199021119902211990231199024) Apparently NLCO problem (34)

has five middle decision variables In fact it only has threeindependent variables Different optimization methods suchas the Newton method quasi-Newton method Lagrangemethod conjugate gradient method interior point methodlinear programming method particle swarm algorithmgenetic algorithm evolution algorithm and other intelligentalgorithms in [50ndash63] have been well established to solvethe NLCO problem Therefore optimal parameters 119898lowast

119894and

proposed PID controller are gotten

119866lowast

119888(119904) = 119896

lowast

119901+

119896lowast

119894

119904

+ 119896lowast

119889119904 119896

lowast

119901=

119898lowast

4minus 1198863minus 1198872119896lowast

119894

1198873

119896lowast

119894=

119898lowast

5

1198873

119896lowast

119889=

119898lowast

1minus 1198860

1198871

(35)

It is noted that (i) the proposed method is also suitablefor first-order second-order and high-order processes (ii)the proposed method is also suitable for nonunit feedbacksystem The non-unit feedback system can be transformedinto unit feedback system through transfer function equiva-lent principle (iii) the proposed method is also suitable forprocesses with time delay which can be approximated bytransfer function (iv) the proposed method can be used fordifferent processes

32 Characteristic Analysis of 120576-Routh Stability The 120576-Routhstability constraint will imply control system robustness insome degree The 120576-Routh stability constraint could avoidtoo conservative system performances and may help toimprove the robustness of control systems The closed-loopcharacteristic polynomial119863(119904) is 119904119899+1+119886

1119904119899+sdot sdot sdot+119886

119899minus119908minus2119904119908+3

+

1198981119904119908+2

+ 1198982119904119908+1

+ sdot sdot sdot + 119898119908+2

119904 + 119898119908+3

The polynomialcoefficient 119898

119896(119896 = 1 2 119908 + 3) is a linear mapping

function of the PID controller parameters 119896119901 119896119894 and 119896

119889 The

parameters 119896119901 119896119894 and 119896

119889are constrained in intervals (0 1205910

119901]

(0 1205910

119894] and (0 120591

0

119889] respectively Therefore the PID controller

parameters can be seen as the uncertain bounded parametersin some degree In this sense polynomial119863(119904) can be seen asinterval polynomial with three uncertain parameters 119896

119901 119896119894

and 119896119889 To analyze the robustness of 120576-Routh stability without

losing generality interval polynomial 119865(119904) has the generalform

119865 (119904) = 119904119899+ 1199011119904119899minus1

+ 1199012119904119899minus2


119904 + 119901119899 (36)

where 1199011 1199012 and 119901

119899are the uncertain coefficients of

the interval polynomial 119865(119904) The uncertain coefficients1199011 1199012 and 119901

119899are continuous mapping functions of

uncertain bounded parameters 1199021 1199022 and 119902

119898

1199011= 1198911(1199021 119902

119898) 119901

119899= 119891119899(1199021 119902

119898)

Ω119902= (1199021 119902

119898)1003816100381610038161003816[119902minus

1 119902+

1] times sdot sdot sdot times [119902

minus

119898 119902+

119898]

1199021isin [119902minus

1 119902+

1] 119902

119898isin [119902minus

119898 119902+

119898]

(37)

whereΩ119902is the uncertain parameter space 119902minus

119894and 119902+

119894are the

lower bound and upper bound of uncertain parameter 119902119894 and

119891119894(sdot) is continuous mapping function of polynomial 119865(119904) for

uncertain parameters Based on 120576-Routh stability definitionfor interval polynomial 119865(119904) the following inequalities canbe obtained

11987711

(1199021 119902

119898) = 1198911199031

(1199011 119901

119899) gt 120576

119877119899+11

(1199021 119902

119898) = 119891119903119899+1

(1199011 119901

119899) gt 120576

Ω119903120576

= (1199021 119902

119898) | 1198911199031

(1199011 119901

119899) gt 120576

119891119903119899+1

(1199011 119901

119899) gt 120576 (120576 ge 0)

(38)

where 119891119903119894(sdot) is the element of first column of Routh array

According to the interval polynomial stable criterion in [3664] if the disjoint set Ω

119891of feasible set Ω

119903120576and uncertain

parameters space Ω119902is not a empty set then the interval

polynomial 119865(119904) will satisfy the interval polynomial stablecriterion

min(1199021 119902119898)isinΩ119891

1198911199031

(1199011 119901

119899) 119891

119903119899+1(1199011 119901

119899) gt 0

(Ω119891= (Ω119903120576

⋂Ω119902) = 0)

(39)

Therefore the interval polynomial 119865(119904) is Routh-Hurwitzstable in the feasible set Ω

119891 and the 120576-Routh stability could


guarantee the robustness of linear systems in a certain degreeFor any positive real 120576

2gt 1205761gt 0 two inequalities for 120576-Routh

stability are obtained

I Ω1= (119902

1 119902

119898) | 1198911199031

(1199011 119901

119899) gt 1205761

119891119903119899+1

(1199011 119901

119899) gt 1205761

II Ω2= (119902

1 119902

119898) | 1198911199031

(1199011 119901

119899) gt 1205762

119891119903119899+1

(1199011 119901

119899) gt 1205762

(40)

The inequality (I) and inequality (II) have the feasible spacesΩ1andΩ

2 and the first inequality of inequalities (I) and (II)

has the feasible set

Ω1

1= (1199021 119902

119898) | 1198911199031

(1199011 119901

119899) gt 1205761

Ω1

2= (1199021 119902

119898) | 1198911199031

(1199011 119901

119899) gt 1205762

(41)

Then any point 119872lowast

1in the feasible set Ω

1

2will have the

relationship

exist119872lowast

1= (119902lowast

11 119902lowast

21 119902

lowast

1198981) 119891

1199031(119872lowast

1) = 120576lowast

2gt 1205762 (42)

Likewise any point 119872lowast2in the feasible set Ω1

1will have the

relationship

exist119872lowast

2= (119902lowast

12 119902lowast

22 119902

lowast

1198982) 119891

1199031(119872lowast

2) = 120576lowast

1gt 1205761 (43)

For the feasible set Ω11 there at least a point 119873lowast

1which will

satisfy the relationship

Case 1 exist119873lowast

1= (119902lowast

11 119902lowast

21 119902

lowast

1198981) 1205761lt 1198911199031

(119873lowast

1) lt 1205762

lArrrArr 1198911199031

(119873lowast

1) isin (1205761 1205762)

997904rArr 119873lowast

1isin Ω1

1 119873lowast

1notin Ω1

2997904rArr Ω

1

1sup Ω1

2

Case 2 1198911199031

(119873lowast

1) notin (1205761 1205762) 997904rArr Ω

1

1= Ω1

2

(44)

Therefore the feasible setsΩ1198961andΩ

119896

2will have the following

relationship

Case 1 Ω119896

1sup Ω119896

2 (forall119896 = 1 2 119899 119899 + 1)

Case 2 Ω119896

1= Ω119896

2 (forall119896 = 1 2 119899 119899 + 1)

Ω119896

1= (1199021 119902

119898) | 119891119903119896

(1199011 119901

119899) gt 1205761

Ω119896

2= (1199021 119902

119898) | 119891119903119896

(1199011 119901

119899) gt 1205762

(45)

Then the overall feasible spaces Ω1and Ω

2will have follow-

ing relationships

Ω1supe Ω2lArrrArr (Ω

1sup Ω2) cup (Ω

1= Ω2)

Ω1=

119899+1

⋂

119895=1

Ω119895

1 Ω

2=

119899+1

⋂

119895=1

Ω119895

2

(46)

The overall feasible set will be gradually condensed with the120576 increasing With 119886 real 120576 sequence 120576

119905+1gt 120576119905gt 120576119905minus1

gt sdot sdot sdot gt

1205762gt 1205761 one will get following relationships

Ω119905= (1199021 119902

119898) | 1198911199031

(1199011 119901

119899) gt 120576119905

119891119903119899+1

(1199011 119901

119899) gt 120576119905

lArrrArr Ω119905= (Ω1

119905cap sdot sdot sdot cap Ω

119899+1

119905) = (

119899+1

⋂

119895=1

Ω119895

119905)

(119905 = 1 2 3 119905 isin Z+)

Ω119896

119905= (1199021 119902

119898) | 119891119903119896

(1199011 119901

119899) gt 120576119905

(119896 = 1 2 119899 119899 + 1)

(47)

Therefore the feasible set Ω119905will has the following relation-

ship

Ω119905+1

sube Ω119905sube sdot sdot sdot sube Ω

2sube Ω1

lArrrArr (Ω119905+1

Δ119905Ω119905sub sdot sdot sdot Δ

2Ω2Δ1Ω1)

⋃ (Ω119905+1

= Ω119905= sdot sdot sdot = Ω

2= Ω1)

exist119896 Δ119896= ldquo sub10158401015840 119896 isin 1 2 119905

(48)

When the feasible setΩ119905is strictly condensedwith 120576

119905 then the

limit of feasible space will yield the following relationships

Case 1 lim120576119905rarr+infin

(1199021 119902

119898) | 1198911199031

(1199011 119901

119899) gt 120576119905

119891119903119899+1

(1199011 119901

119899) gt 120576119905

= lim120576119905rarr+infin

Ω119905= lim120576119905rarr+infin

(Ω1

119905cap Ω2


119899+1

119905)


(

119899+1

⋂

119895=1

Ω119895

119905) = Ω

lowast

= (119902lowast

10119902lowast

20 119902

lowast

1198980)



(1199021 119902

119898) | 1198911199031

(1199011 119901

119899) gt 120576119905

119891119903119899+1

(1199011 119901

119899) gt 120576119905


Ω119905= lim120576119905rarr+infin

(Ω1

119905cap Ω2


119899+1

119905)


(

119899+1

⋂

119895=1

Ω119895

119905) = 0

(49)

If the 120576119905tends to infinity then the limit of the feasible set

will be condensed into a point Ωlowast or an empty set 0 If thefeasible set is condensed into a point Ωlowast then the intervalpolynomial degenerates into a usual polynomial It reflectsthat the point Ω

lowast is a fixed point in the stable region oflinear control systems If the feasible set tends to empty set 0therefore it has a real 120576lowast

119905to satisfy the following relationships

120576119905997888rarr +infin lim

120576119905rarr+infin

Ω119905= 0 lArrrArr exist120576

lowast

119905 120576119905gt 120576lowast

119905 Ω119905= 0

0 lt 120576119905le 120576lowast

119905 Ω119905

= 0

120576lowast

119905gt 120576119905gt 120576119905minus1

gt sdot sdot sdot gt 1205762gt 1205761ge 0

Ωlowast

119905sub Ω119905sub Ω119905minus1

sub sdot sdot sdot sub Ω2sub Ω1

(50)

when the feasible set has the relationshipΩ119905+1

= Ω119905= Ω119905minus1

=

sdot sdot sdot = Ω2= Ω1 However this is a special event which can be

depicted by the probability event

119875 Ω119905+1

= Ω119905= Ω119905minus1

= sdot sdot sdot = Ω2= Ω1

lArrrArr 119875 Ω119905+1

= Ω119905 Ω119905= Ω119905minus1

Ω2= Ω1

(51)

It assumes that all the probability events are independent andidentically distributed Hence the probability can be writtenas

119875 Ω119905+1

= Ω119905 Ω119905= Ω119905minus1

Ω2= Ω1

= 119875 Ω119905+1

= Ω119905 times 119875 Ω

119905= Ω119905minus1

times sdot sdot sdot times 119875 Ω2= Ω1

=

119905+1

prod

119895=2

119875 Ω119895= Ω119895minus1

= (119875 Ω2= Ω1)119905

(119905 = 1 2 119905 isin Z+)

(52)

Considering probability event Ω2= Ω1 first the probability

will have119875 Ω1= Ω2

= 119875 (Ω1


119899+1

1) = (Ω

1


119899+1

2)

= 119875

(

119899+1

⋂

119895=1

Ω119895

1) = (

119899+1

⋂

119895=1

Ω119895

2)

lArrrArr 119875(Ω1


119899+1

2) sub Ω

119896

1

(Ω1


119899+1

1) sub Ω

119896

2

(forall119896 = 1 2 119899 119899 + 1)

(53)

The probability 119875Ω2= Ω1 for the probability event Ω

2=

Ω1 will yield

119875 (Ω1

2cap Ω2


119899

2) sub Ω

1

1 = 1205821

119875 (Ω1

2cap Ω2


119899

2) sub Ω

119899+1

1 = 1205821

119875 (Ω1

1cap Ω2


119899

1) sub Ω

1

2 = 1205822

119875 (Ω1

1cap Ω2


119899

1) sub Ω

119899+1

2 = 1205822

119875 Ω1= Ω2

= 119875

(

119899+1

⋂

119895=1

Ω119895

2) sub Ω

119896

1 (forall119896 = 1 2 119899 119899 + 1)

times 119875

(

119899+1

⋂

119895=1

Ω119895

1) sub Ω

119896

2 (forall119896 = 1 2 119899 119899 + 1)

=

119899+1

prod

119896=1

119875

(

119899+1

⋂

119895=1

Ω119895

2) sub Ω

119896

1

times

119899+1

prod

119896=1

119875

(

119899+1

⋂

119895=1

Ω119895

1) sub Ω

119896

2

= 120582119899+1

1120582119899+1

2 (0 lt 120582

1lt 1 0 lt 120582

2lt 1)

(54)

Therefore the probability of the probability event yields

119875 Ω119905+1

= Ω119905 Ω119905= Ω119905minus1

Ω2= Ω1 = 120582(119899+1)119905

1120582(119899+1)119905

2

119875 Ω119905+1

Δ119905Ω119905Δ119905minus1

Ω119905minus1

sdot sdot sdot Δ2Ω2Δ1Ω1 = 1 minus 120582

(119899+1)119905

1120582(119899+1)119905

2

(55)

33 Interior Method for NLCO Problem Without losinggenerality NLCO problem (34) of the optimal PID controllercan be written in the following general forms

min 119891 (119909)

st ℎ (119909) = 0 119892 (119909) le 0

(56)


where 119891(119909) 119877119899

rarr 119877 ℎ(119909) 119877119899

rarr 119877119898 and 119892(119909)

119877119899

rarr 119877119902 are the smooth and differentiable functions 119909 is

the decision variable and 119899 119898 and 119902 denote the numberof the decision variable equality constraint and inequalityconstraint respectively The various optimization methodsin [50ndash63] are developed to solve optimization problemsThe optimization methods such as the Newton methodsconjugate gradient methods steepest descent methods inte-rior point methods genetic algorithms and particle swarmalgorithms in [50ndash63] are well established to solve constraintoptimization problems Based on interior point methods in[56ndash61] the interior method is recommended to solve theNLCO problem Then the NLCO problem (56) could betransformed into the following form

min 119891 (119909) minus 120592

119902

sum

119894=1

ln 120575119894

st ℎ (119909) = 0 119892 (119909) + 120575 = 0

(57)

where 120592 gt 0 is the barrier parameter the slack vector 120575 =

(1205751 1205752 120575

119902)119879

gt 0 is set to be positive and 119892(119909) is anexpanded constraint inequality It introduces the Lagrangemultipliers 119910 and 119911 for the barrier problem (57)

119871 (119909 119910 119911 120575) = 119891 (119909) minus 120592

119902

sum

119894=1

ln 120575119894+ 119910119879(119892 (119909) + 120575) + 119911

119879ℎ (119909)

(58)

where 119871(119909 119910 119911 120575) is Lagrange function y = (1199101 1199102 119910

119902)119879

and z = (1199111 1199112 119911

119898)119879 are Lagrange multipliers for

constraints 119892(119909) + 120575 and ℎ(119909) respectively Based on theKarush-Kuhn-Tucker (KKT) optimality conditions [57ndash59]the optimality conditions for NLCO problem (56) can beexpressed as

nabla119909119871 (119909 119910 119911 120575) = nabla119891 (119909) + (nabla119892 (119909))

119879

119910 + (nablaℎ (119909))119879119911 = 0

nabla120575119871 (119909 119910 119911 120575) = minus120592119878

minus1

120575119890 + 119910 = 0 lArrrArr minus120592119890 + 119878

120575119884119890 = 0

nabla119910119871 (119909 119910 119911 120575) = 119892 (119909) + 120575 = 0

nabla119911119871 (119909 119910 119911 120575) = ℎ (119909) = 0

(59)

where 119878120575

= diag(1205751 1205752 120575

119902) is the diagonal matrix and

its elements are the components of the vector 120575 and e is avector of all ones nablaℎ(119909) and nabla119892(119909) are the Jacobian matricesof the vectors ℎ(119909) and 119892(119909) respectively nabla119891(119909) is the grandof function 119891(119909) and 119884 = diag(119910

1 1199102 119910

119902) is a diagonal

matrix and its elements are the components of vector y Thesystem (59) is the KKT condition of the NLCO problem (56)When in the search approach it should remain 120575 119910 gt 0To obtain the iteration direction it can make the point (119909 +

Δ119909 120575+Δ120575 119911+Δ

119911 119910+Δ119910) satisfying the KKT conditions(59)

then the following system will be obtained

(nabla2119891 (119909) + nabla

2119892(119909)119879119910 + nabla

2ℎ(119909)119879119911) Δ119909 + nabla119892(119909)

119879Δ119910

+ nablaℎ(119909)119879Δ119911 + (nabla119891 (119909) + nabla119892(119909)

119879119910 + nablaℎ(119909)

119879119911) = 0

minus120592119890 + 119878120575119884119890 + 119878

120575Δ119910 + 119884Δ120575 = 0

119892 (119909) + nabla119892 (119909) Δ119909 + 120575 + Δ120575 = 0

nablaℎ (119909) Δ119909 + ℎ (119909) = 0

(60)

System (60) is obtained by ignoring higher-order incrementaland replaces the nonlinear terms with linear approximationin system (59) The system (60) is written in the matrix form

(

119867(119909 119910 119911) 0 nablaℎ(119909)119879

nabla119892(119909)119879

0 119878minus1

120575119884 0 119868

nablaℎ (119909) 0 0 0

nabla119892 (119909) 119868 0 0

)(

Δ119909

Δ120575

Δ119910

Δ119911

)

= (

minusnabla119891 (119909) minus nabla119892(119909)119879119910 minus nablaℎ(119909)

119879119911

120592119878minus1

120575119890 minus 119910

minusℎ (119909)

minus119892 (119909) minus 120575

)

(61)

119867(119909 119910 119911) = nabla2119891 (119909) + nabla

2119892(119909)119879119910 + nabla

2ℎ(119909)119879119911 (62)

where119867(119909 119910 119911) is the Hessian matrix in system Finally thenew iterate direction is obtained via solving the system (61)which is the essential process in the interior point methodThus the new iteration point can be gotten in the nextiteration

(119909 120575 119911 119910) larr997888 (119909 120575 119911 119910) + 1205771(Δ119909 Δ120575 Δ119911 Δ119910) (63)

where 1205771is the step size Choosing the step size 120577

1holds the

120575 119910 gt 0 in search process In this paper the interior pointmethod is recommended to solve the NLCO problem Theinterior algorithm framework is depicted as follows

Interior Algorithm Framework for NLCOProblem One has thefollowing

Step 1 Choose an initial iteration point (119909(0) 120575(0) 119911(0) 119910(0))in the feasible region set = 119909 | ℎ(119909) = 0 119892(119909) le 0 andthe 120575(0) gt 0 119910

(0)gt 0 119896 = 0

Step 2 Construct current iterate we have the current iteratevalues 119909(119896) 120575(119896) 119911(119896) and 119910

(119896) of the primal variable 119909 the ofthe slack variable 120575 the multipliers 119910 and 119911 respectively

Step 3 Calculate the Hessian matrix 119867(119909 119910 119911) of theLagrange system 119871(119909 119910 119911 120575) and the Jacobian matricesnablaℎ(119909) and nabla119892(119909) are of the vectors ℎ(119909) and 119892(119909) in thecurrent iterate (119909(119896) 120575(119896) 119911(119896) 119910(119896))

Step 4 Solve the linear system (61) and construct the iteratedirection (Δ119909 Δ120575 Δ119911 Δ119910) Solving the linearmatrix equation


(61) we obtain the primal solution Δ119909 multipliers solutionΔ119911 Δ119910 and also the slack variable solution Δ120575

Step 5 Choosing the step size 1205771holds the 120575 119910 gt 0

in search process 1205771

isin [0 1] Update the iterate val-ues (119909

(119896+1) 120575(119896+1)

119911(119896+1)

119910(119896+1)

) larr (119909(119896)

120575(119896)

119911(119896)

119910(119896)

) +

1205771(Δ119909 Δ120575 Δ119911 Δ119910) 119896 larr 119896 + 1

Step 6 Check the ending conditions in regionΩ If it does notsatisfied go to Step 3 else minimum 119891min of NLCO problemis obtained in feasible regionΩ

Step 7 End

4 Simulation and Experiment Results

The proposed method is used to design the optimal PIDcontroller with 120576-Routh stability for processes 119866

1199011(119904) [7]

1198661199012(119904) [15] and 119866

1199013(119904) [6] respectively 119866

1199011(119904) is a third-

order all-pole process and 1198661199012(119904) and 119866

1199013(119904) are the SOPTD

and FOPTD processes respectively

1198661199011

(119904) =

15

(1199042+ 09119904 + 5) (119904 + 3)

1198661199012

(119904) =

119890minus05119904

(119904 + 1)2 119866

1199013(119904) =

4119890minus10119904

10119904 + 1

(64)

In order to use the proposedmethod time delay of plant2and plant3 is approximated as 119890minus05119904 asymp (1 minus 025119904)(1 + 025119904)

and 119890minus10119904

asymp (1199042minus 06119904 + 012)(119904

2+ 06119904 + 012) respectively

Thus transfer functions of 1198661199012(119904) and 119866

1199013(119904) are the (minus119904 +

4)(1199043+ 61199042+ 9119904 + 4) and (04119904

2minus 024119904 + 0048)(119904

3+ 07119904

2+

018119904 + 0012) respectively The proposed method is usedto design the optimal PID controller with 120576-Routh stabilityfor different processes and optimal PID controllers underdifferent control weight matrices and 120576-Routh stability areshown in Table 1

For the proposed method the plant1rsquos step response andsystem performances under various control weight matricesare shown in Figure 4 and Table 2 respectively The controlweight matrix has effects on system performances Figure 4and Table 2 obviously show that system performances areaffected by control weight factors 119902

1 1199022 1199023 and 119902

4 The

results show that the response velocity will be heightened byincreasing weight factor 119902

1or reducing weight factors 119902

2 1199023

and 1199024 When control weight factor 119902

1(with 119902

2 1199023 and 119902

4

fixed) is increased the overshoot is increased and the risetime delay time peak time and setting time are reducedLikewise the overshoot will be increased and the rise timedelay time and peak time will be reduced by reducing weightfactors 119902

2 1199023 and 119902

4(with 119902

1fixed) Response results show

that system performances are affected by control weightmatrix

For 119899th-order processes it can be inferred that overshootwill be heightened and the rise time delay time and peaktime will be reduced by increasing control weight factor 119902

1

(with other factors fixed) or reducing other weight factors(with 119902

1fixed) On the contrary the overshoot will be

reduced and the rise time delay time and peak time willbe increased by reducing weight factor 119902

1(with other factors

fixed) or increasing other factors (with 1199021fixed) Therefore

system performances are affected by control weight factors1199021 1199022 119902

119899+1 In all control weight matrix has effects on

system performancesFor plant2 step response results and system perfor-

mances under different 120576-Routh stability are shown inFigure 5 and Table 3 respectively The response results showthat system performances are affected by 120576-Routh stability(with control weight matrix fixed) The results show thatresponse velocity will be heightened by increasing 120576 Whenthe 120576 (with control weight matrix fixed) is increased theovershoot is increased and the rise time delay time andpeak time are reduced However setting time is graduallyincreased and then is decreased by increasing 120576 Thereforethe rise time delay time and peak time will be increased andthe overshootswill be reduced by reducing the 120576 In all systemperformances are affected by the 120576-Routh stability too

41 System Performances of Different Tuning Methods Thestep response and system performances of different tuningmethods for plant1 plant2 and plant3 are shown in Figures6 7 and 8 and Table 4 respectively For plant1 the resultsobviously show that the Z-N method IA-ISE method [7]and IA-ITSE method [7] show oscillatory behaviors butthe proposed method shows smoothly response without anyoscillation The Z-N method has small rise time delay timeand peak time compared with the proposed method IA-ISEmethod and IA-ITSEmethod However the Z-Nmethod hasthe largest overshoot and setting time among these tuningmethods The overshoot and setting time of the Z-N methodis about 1108 and 641 sec and the overshoot and settingtime of the proposed method IA-ISE method and IA-ITSEmethod are about 408 and 354 sec 424 and 313 sec and527 and 488 sec respectively Thus these tuning methodsown advantages for plant1

The plant2 is a SOPTD plant with dead time ratio 120591119879 =

05 The results in Figure 7 show that dynamic performancesof the Z-Nmethod A-Hmethod [19] gain and phasemethod[15] and proposed method have different features The Z-N method A-H method [19] gain and phase method [15]and the proposed method have small rise time delay timeand peak time setting time and overshoot respectively Theproposed method has the largest rise time delay time andpeak time and the A-H method and Z-N method have thelargest setting time and overshoot respectively The longestrise time delay time peak time and setting time are about4778 sec 1841 sec 6622 sec and 443 sec respectively andthe largest overshoot is about 4309 The gap between thelargest and the smallest items of the rise time delay timepeak time and setting time is small but only the gap betweenthe largest and the smallest overshoot is about 13 times of thesmallest onesThus the comprehensive performances of Z-Nmethod are poorer than rest methods


Table 1 Proposed optimal PID controllers of plants1ndash3 under various control weight matrices

Plants Control weight matrix Q = diag (1199021119902211990231199024) Optimal PID controller

120576-Routh stability119896lowast

119901119896lowast

119894119896lowast

119889

Plant1

diag (5 5 1 1) 013121 058138 058655

120576 = 0

diag (5 2 1 1) 001633 058138 051933diag (10 2 1 1) 023263 082219 064449diag (10 10 1 1) 047606 082219 077901diag (10 10 4 2) 015570 058138 064101

Plant2

diag (02 1 1 1) 115698 052966 086781

120576 = 05

diag (02 1 1 02) 216962 084500 182140diag (02 2 2 05) 178668 058700 159290

diag (01 08 08 02) 185901 064942 162251diag (04 05 5 1) 122806 062168 142095

Plant3

diag (004 27 158 175) 032447 001537 145256

120576 = 0


Table 2 Plant1rsquos system performances of proposed method under various control weight matrices

Control weight matrix Plant1 Rise time Delay time Peak time Setting time Peak overshootQ = diag (119902

1119902211990231199024) 119896

lowast

119901119896lowast

119894119896lowast

119889119905119903(sec) 119905

119889(sec) 119905

119901(sec) 119905

119904(sec) 119872

119901()

diag (5 5 1 1) 013121 058138 058655 4060 1685 531 354 408diag (5 2 1 1) 001633 058138 051933 3681 1801 500 607 669diag (10 2 1 1) 023263 082219 064449 2850 1318 402 535 847diag (10 10 1 1) 047606 082219 077901 3161 1140 438 2756 458diag (10 10 4 2) 015570 058138 064101 4120 1691 537 360 423

0 2 4 6 8 10 12 14

0

02

04

06

08

1

12

Syste

m re

spon

se

minus02

Time (s)

119876 = diag (5 5 1 1)119876 = diag (5 2 1 1)119876 = diag (10 2 1 1)

119876 = diag (10 10 1 1)119876 = diag (5 5 2 1) lowast 2

(a)

0 2 4 6 8 10 12 14

0

02

04

06

08

1

12

Con

trol e

rror

minus02

Time (s)

119876 = diag (5 5 1 1)119876 = diag (5 2 1 1)119876 = diag (10 2 1 1)

119876 = diag (10 10 1 1)119876 = diag (5 5 2 1) lowast 2

(b)

Figure 4 Step response of the proposed method under various control weight matrices (for plant1)


0 1 2 3 4 5 6 7 8 9 100

02

04

06

08

1

12

14

Time (s)

Syste

m re

spon

se

120576 = 05

120576 = 25

120576 = 3

120576 = 4

120576 = 5

(a)

0 1 2 3 4 5 6 7 8 9 10

0

02

04

06

08

1

Time (s)

minus02

minus04

Con

trol e

rror

120576 = 05

120576 = 25

120576 = 3

120576 = 4

120576 = 5

(b)

Figure 5 Step response of the proposed method under various 120576-stability (for plant2)

0 5 10 15 20 25 30

0

02

04

06

08

1

12

Proposed methodZ-N method

IA-ISE methodIA-ITSE method

Syste

m re

spon

se

Time (s)

minus02

(a)

0 5 10 15 20 25 30

0

02

04

06

08

1

12



Con

trol e

rror

Time (s)

minus02

(b)

Figure 6 Step response of plant1 under different tuning methods

Table 3 System performances of proposed method under different 120576 values (for plant2)

Control matrix Q and 120576 value 119896119901

119896119894

119896119889

Risetime

119905119903(sec)

Delaytime

119905119889(sec)

Peaktime

119905119901(sec)

Settingtime

119905119904(sec)

Peakovershoot119872119901()

Plants

Q = diag (02 1 1 1) and 120576 = 05 115698 052966 086781 4778 1841 6622 400 329 Plant2Q = diag (02 1 1 1) and 120576 = 25 145019 062500 130523 4201 1655 6451 720 541 Plant2Q = diag (02 1 1 1) and 120576 = 30 179987 075000 178883 3727 1488 5961 808 761 Plant2Q = diag (02 1 1 1) and 120576 = 40 231710 100000 200000 3151 1316 4201 735 1100 Plant2Q = diag (02 1 1 1) and 120576 = 50 246310 125000 100000 1795 1266 2587 397 2846 Plant2


0 2 4 6 8 10 12 14 16 180

05

1

15Sy

stem

resp

onse

Gain and phase methodA-H method


Time (s)

(a)

0 2 4 6 8 10 12 14 16 18

0

05

1


minus05


Con

trol e

rror

Time (s)

(b)


0 20 40 60 80 100 120 140 160 180 2000

02

04

06

08

1

12

14

Proposed methodZ-N method (C-C)Cohen-Coon method

Z-N method (PRC)Upper tolerance boundLower error tolerance bound

Syste

m re

spon

se

Time (s)

095

105

(a)

0 20 40 60 80 100 120 140 160 180 200

0

02

04

06

08

1

Proposed methodZ-N method (C-C)

Cohen-Coon methodZ-N method (PRC)

Con

trol e

rror

Time (s)

minus02

minus04

(b)


The plant3 is a FOPTD plant with large dead time ratio120591119879 = 10 The results in Figure 8 show that all the tuningmethods show oscillation response The overshoot of the Z-N method (C-C) and Z-N method (PRC) are about 1818and 2117 respectively The results demonstrate that theproposedmethodhas the smallest peak time 2087 sec and thesmallest overshoot 667 and the Cohen-Coon method hasthe smallest rise time 17252 sec and the smallest delay time14306 sec and the Z-Nmethod (C-C) has about the smallest

setting time 5160 sec However the Z-N method (PRC)has the largest rise time 20286 sec delay time 14635 secand peak time 2643 sec and the Cohen-Coon methodhas the largest setting time 6705 sec and largest overshoot3806 For plant3 the Cohen-Coon method shows poorersystemperformances than other tuningmethods By studyingdifferent tuning methodsrsquo dynamic performances it can beknown that the proposed method could provide good systemperformances


Time (s)0 5 10 15 20 25 30

0

02

04

06

08

1

12

0 5 10 15 20 25 30

0

02

04

06

08

1

12

Syste

m re

spon

seSy

stem

resp

onse

Con

trol e

rror

Con

trol e

rror

Time (s)

Time (s)Time (s)16 18 20 22 24 26 28

075

08

085

09

095

1

105

11



16 18 20 22 24 26 28

0

005

01

015

02

025

03



minus02 minus02

minus005

119863119906(119905) = 03119906(119905 minus 16)

Figure 9 Disturbance rejection ability of different tuning methods for plant1

Table 4 System performances of different tuning methods for plants1ndash3

Methods 119896119901

119896119894

119896119889

Rise time119905119903(sec)

Delay time119905119889(sec)

Peak time119905119901(sec)

Setting time119905119904(sec)

Peak overshoot119872119901() Plants

Proposed method 013121 058138 058655 4060 1685 5310 354 408 Plant1Z-N method 05971 10750 02687 1371 0877 1643 641 1108 Plant1IA-ISE method 04479 09994 03340 1678 0983 4015 313 424 Plant1IA-ITSE method 07377 09886 03557 1429 0845 3506 488 527 Plant1Proposed method 115698 052966 086781 4778 1841 6622 400 329 Plant2Z-N method 28130 17194 11505 1620 1195 2458 381 4309 Plant2Gain and phase method 23870 08747 09476 1938 1302 2478 334 1502 Plant2A-H method 25778 14518 12941 1763 1286 2537 443 3365 Plant2Proposed method 032447 001537 145256 19486 14306 2087 6056 667 Plant3Z-N method

Z-N method (C-C) 03060 00205 11384 19238 14367 2498 5160 1818 Plant3Z-N method (PRC) 03000 00150 00600 20286 14635 2643 5606 2117 Plant3

Cohen-Coon method 03958 00219 12178 17252 13400 2143 6705 3806 Plant3A-H method (real PID controller form) with derivative filter constant 119879119891 = 004947


16 18 20 22 24 26 28 30086

088

09

092

094

096

098

1

102

104





0 5 10 15 20 25 300

05

1

15

0 5 10 15 20 25 30

0

05

1

18 20 22 24 26 28 30

0

002

004

006

008

01

012

014

Con

trol e

rror

Con

trol e

rror

Syste

m re

spon

seSy

stem

resp

onse

minus05

Time (s) Time (s)

Time (s) Time (s)

minus002

119863119906(119905) = 03119906(119905 minus 18)


42 Disturbance Rejection Ability of Different Methods Dis-turbance response results for plant1 plant2 and plant3 areshown in Figures 9 10 and 11 respectively For plant1 theresults show that the proposed method Z-N method IA-ISEand IA-ITSE methods have good ability to reject the distur-bances and they show roughly the samedisturbance responseamplitude (DRA) and disturbance attenuation time (DAT)The proposed method has the best disturbance rejectionability among the methods The proposed method does notshow oscillatory behaviors but Z-Nmethod IA-ISEmethodand IA-ITSE method show oscillatory behaviors For plant2the disturbance response results of the proposed method Z-N method A-H method and gain and phase method areshown in Figure 10The results show that these methods havegood ability to reject disturbances and they have roughly thesame DATThe DRAs of the proposed method Z-N methodA-H method and gain and phase method are about 013010 010 and 011 respectively The proposed method haslarger DRA than the rest methods but the proposed methodshows smooth disturbance response For plant3 disturbance

response results in Figure 11 show that these tuning methodshave roughly the same ability to reject disturbances Theproposed method Z-N method (C-C) Z-N method (PRC)and Cohen-Coon method have roughly the same DRA andDAT Among these tuning methods the DRA of the Z-Nmethod (PRC) is about 0275 the proposed method Z-Nmethod (C-C) andCohen-Coonmethod have the sameDRAwhich is about 025 Overall as known from disturbanceresponse results it could be that the proposed method A-Hmethod and gain and phase method roughly have roughlythe same ability to reject disturbances

To study 120576-Routh stabilityrsquos effects on disturbance rejec-tion ability the disturbance response results under various120576 are shown in Figure 12 The disturbance response resultsin Figure 12 clearly show that the 120576 has effects on the DRAand DAT The DAT will be continuously reduced whenthe 120576 increases from 05 to 50 However the DAR will begradually reduced and then be increased when the 120576 increasesfrom 05 to 50 The DAT reflects how fast the disturbanceswill be eliminated and DRA reveals how much disturbance


200 250 300 350 40007

075

08

085

09

095

1

105

11

200 250 300 350 400

0

005

01

015

02

025

03

0 50 100 150 200 250 300 350 4000

02

04

06

08

1

12

14

0 50 100 150 200 250 300 350 400

0

02

04

06

08

1

095

105

Time (s) Time (s)

Time (s)Time (s)





minus005

Con

trol e

rror

Syste

m re

spon

se

Con

trol e

rror

Syste

m re

spon

se

minus02

minus04

119863119906(119905) = 01119906(119905 minus 180)


amplitude of the control system Thus 120576-Routh stability haseffects on the disturbance rejection ability

43 Experiment Results for the CWTS The proposed PIDcontroller is utilized to control the liquid level of a CWTSwith interactive behavior The proposed PID controllersrsquoexperimental results of the CWTS under various controlweight matrix and different set points are shown in Figures14 15 16 17 18 19 20 and 21 The theoretical model ofthe CWTS is already obtained in previous section thereforethe identification model can be acquired via open-loopexperiment The liquid levelrsquos real-time data is stored in thedisk of monitoring computer and the date can be accessedon-line for process control (OPC) databaseThe identificationmodel of CWTS is identified as ℎ

1(119904)119876119894119888(119904) = (04465119904 +

03064)(1199042+ 1340119904 + 001264) via least square method

(sample time is 1 second) when open ratio of control valveoutput valve-1 output valve-2 and connected valve-3 is setat 75 100 30 and 100 respectively The steady-state

liquid level is about 29400mm and it will take about 12minutes to reach this liquid level The actual liquid level andthe identification modelrsquos liquid level are shown in Figure 13The identification model is consistent with the actual modelof the CWTS thus the identification model describes theactual model accurately

In the closed-loop experiment the proposed methodis used to control the liquid level of the CWTS underdifferent set points Proposed PID controllers under differentcontrol weight matrices and 120576-Routh stability are used inthe experiments The proposed PID controllers (119896

119901=

10000 119896119894

= 032549 119896119889

= 0001 120576 = 0 119896119901

=

10000 119896119894

= 010118 119896119889

= 0001 120576 = 0 119896119901

=

10000 119896119894= 022986 119896

119889= 0001 120576 = 0 119896

119901= 6000 119896

119894=

013817 119896119889= 0001 and 120576 = 0) are used to control liquid

level of the CWTS with interactive behaviors The first groupexperimentsrsquo setting point of the liquid level ℎ

1is set at

30000mm and second group experimentsrsquo setting point ℎ1

is set at 16000mm The corresponding experiment results


0 5 10 15 20 25 300

02

04

06

08

1

12

14

Time (s)

16 18 20 22 24 26 28 30086

088

09

092

094

096

098

1

102

104

0 5 10 15 20 25 30

0

02

04

06

08

1

Time (s)

Time (s) Time (s)

16 18 20 22 24 26 28 30

0

002

004

006

008

01

012

014

minus02

minus04

minus002

120576 = 05

120576 = 25

120576 = 3

120576 = 4

120576 = 5

120576 = 05

120576 = 25

120576 = 3

120576 = 4

120576 = 5

Figure 12 Disturbance rejection ability of the proposed method under various 120576-stability (for plant2)

of the first group and second group experiments are shownin Figures 14ndash21 respectively In Figures 14ndash21 the solid redline is the dynamic response of the liquid level ℎ

1 the solid

magenta line is the dynamic response of the liquid level ℎ2

and the solid cyan line is the control signal In the first groupexperiments the dynamic response results of the proposedmethod shown in Figures 14ndash17 obviously reveal that theproposedmethod provides good systemperformances for theCWTSwith interactive behaviors In first group experimentsthe setting time for the CWTS is less than 7 minutes andthe overshoot of the proposed method is less than 4 undervarious control weight matrices of the proposed method

In the second group experiments the dynamic responseresults of the proposed method shown in Figures 18ndash21clearly reveal that the proposedmethodprovides good systemperformances and robustness The proposed PID controlleris designed for the liquid level of set point at 294mm Theactual model of the CWTS has been changed in the secondgroup experiments because the liquid level set point hasmadea huge change In the second group experiments the setting

time of the proposed method for the CWTS is less than 35minutes and the overshoot of the proposed method is lessthan 12 Figures 18 and 20 show that the liquid level hasa small fluctuation however overall system performancesare still accepted The system parameters of liquid modelwill show perturbations when the set point of liquid levelmakes a huge changeThe proposed method can still providegood system performances when the set point of the liquidlevel changes from 300mm to 160mm Furthermore thedynamic response results in Figures 18ndash21 also reflect that theproposed method shows good robustness to rejection modelperturbation system parameters perturbation and set pointvariation

Simulation results and experiment results reveal that theproposedmethod can provide good system performances fordifferent processes The proposed method also demonstratesgooddisturbance rejection ability and robustness for differentprocessesTherefore the proposedmethod is useful to designthe optimal PID controller In short the proposed methodis suitable for different processes and can provide good


0 200 400 600 800 1000 1200 1400 1600 1800 20000

50

100

150

200

250

300

Time (s)

Liqu

id le

vel o

f CW

TS (m

m)

Actual liquid level of CWTS (open loop)Identified modelrsquos liquid levelof CWTS (open loop)

Figure 13 Actual liquid level and identified modelrsquos liquid level(open loop)

Figure 14 Experimental response of the CWTS with the PIDcontroller 119896

lowast

119901= 10000 119896lowast

119894= 032549 119896lowast

119889= 0001 and 120576 = 0 in

set point 300mm

Figure 15 Experimental response of the CWTS with the PIDcontrollers 119896lowast

119901= 10000 119896lowast

119894= 010118 119896lowast

119889= 0001 and 120576 = 0 in

set point 300mm


119901= 10000 119896lowast

119894= 022986 119896lowast

119889= 0001 and 120576 = 0 in

set point 300mm

Figure 17 Experimental response of the CWTS with the PIDcontrollers 119896

lowast

119901= 6000 119896lowast

119894= 013817 119896lowast

119889= 0001 and 120576 = 0 in

set point 300mm

system performancesThe simulation results and experimentresults show the effectiveness and usefulness of the proposedmethod

5 Conclusions

In this paper a systematic method is presented to design theoptimal PID controller with 120576-Routh stability for differentprocesses via Lyapunov approachThe optimal PID controlleris obtained by minimizing AISE performance index whichcontains the control error and at least first-order errorderivative or evenmay contain the 119899th-order error derivativeThe proposed optimal control problem could be equivalentlytransformed into a NLCO problem via Lyapunov theoremsTherefore the optimal PID controller can be obtained bysolving the NLCO problem via the interior method orother optimization methods Optimal PID controllers underdifferent control weight matrices and 120576-Routh stability arepresented for different processes For the proposed methodcontrolweightmatrix and 120576-Routh stabilityrsquos effects on systemperformances are analyzed Different tuningmethodsrsquo systemperformances are discussed



119901= 10000 119896lowast

119894= 032549 119896lowast

119889= 0001 and 120576 = 0 in

set point 160mm


119901= 10000 119896lowast

119894= 010118 119896lowast

119889= 0001 and 120576 = 0 in

set point 160mm

The controlweightmatrix and 120576-Routh stability have largeeffects on system performances For third-order processessystemperformances and response velocity are affected by thecontrol weight factors 119902

1 1199022 1199023 and 119902

4 The peak overshoot

and response velocity will be heightened by increasing weightfactor 119902

1(with other factors fixed) Likewise the overshoot

and response velocity will be heightened by reducing weightfactors 119902

2 1199023 and 119902

4(with 119902

1fixed) On the contrary the

overshoot and response velocity will be reduced by reducingweight factor 119902

1(with other factors fixed) or increasing

weight factors 1199022 1199023 and 119902

4(with 119902

1fixed) Through the

study for 119899th-order processes the above conclusions stillhold it can be inferred that system performances are affectedby the control weight factors 119902

1 1199022 119902

119899+1 The 120576-Routh

stability (with control weight matrix fixed) has effects onsystem performances The results show that the overshootsand response velocity will be heightened by increasing the 120576Overshoots will be increased and the rise time delay timeand peak time will be reduced by increasing the 120576 On thecontrary overshoots will be reduced and the rise time delaytime and peak time will be increased by reducing the 120576

Disturbance rejection ability of different tuning methodsis also investigated and simulation results reveal that the


119901= 10000 119896lowast

119894= 022986 119896lowast

119889= 0001 and 120576 = 0 in

set point 160mm


lowast

119901= 6000 119896lowast

119894= 013817 119896lowast

119889= 0001 and 120576 = 0 in

set point 160mm

proposed method shows good disturbance rejection abilityThe 120576-Routh stabilityrsquos effects on disturbance rejection abilityare investigated The disturbance response results clearlyshow that the 120576 has effects on the DRA and DAT The DATwill be continuously reduced by increasing the 120576 Howeverthe DAR will be gradually reduced and then be increased byincreasing the 120576

To further validate the proposed method the proposedmethod is utilized to control the liquid level of CWTS withinteractive behaviors The experimental results under differ-ent set points and control weight matrices are presented indetail Both simulation results and experimental results showthe effectiveness and usefulness of the proposed method

Acknowledgments

A partial content of this paper as the previous work andearlier version paper was presented at the 2009 Interna-tional Conference on Information Engineering and Com-puter Science (ICIECS-2009) Wuhan China December 19-20 2009This work is supported by the Research Projects no2007BAF09B01 and 2009A A04Z155 The authors would liketo thank the reviewers for suggestions


References

[1] J J DrsquoAzzo andCHHoupisLinearControl SystemAnalysis andDesign McGraw-Hill New York NY USA 2nd edition 1981

[2] M D Tong Linear SystemTheory and Design University of Sci-ence and Technology of China press Hefei China (Chinese)2004

[3] J G Ziegler andN BNichols ldquoOptimum settings for automaticcontrollersrdquo ASME Transactions vol 64 pp 759ndash768 1942

[4] J G Ziegler and N B Nichols ldquoProcess lags in automaticcontrol circuitsrdquoTransactions of the ASME vol 65 pp 433ndash4441943

[5] G H Cohen and G A Coon ldquoTheoretical consideration ofrelated controlrdquo Transactions of the ASME vol 75 pp 827ndash8341953

[6] C RMadhuranthakam A Elkamel andH Budman ldquoOptimaltuning of PID controllers for FOPTD SOPTD and SOPTDwithlead processesrdquoChemical Engineering and Processing vol 47 no2 pp 251ndash264 2008

[7] D H Kim and J H Cho ldquoIntelligent tuning of PID con-troller with disturbance function using immune algorithmrdquoin Proceedings of the IEEE Annual Meeting Fuzzy InformationProcessing vol 1 pp 286ndash291 June 2004

[8] Y P Wang D R Hur H H Chung N R Watson J Arrillagaand S S Matair ldquoDesign of an optimal PID controller inAC-DC power system using modified genetic algorithmrdquo inProceedings of the 2000 International Conference on PowerSystem Technology vol 3 pp 1437ndash1442 December 2000

[9] Q Zeng and G Tan ldquoOptimal design of PID controller usingmodified ant colony system algorithmrdquo in Proceedings of the 3rdInternational Conference on Natural Computation (ICNC rsquo07)vol 5 pp 436ndash440 August 2007

[10] Z L Gaing ldquoA particle swarm optimization approach foroptimum design of PID controller in AVR systemrdquo IEEETransactions on Energy Conversion vol 19 no 2 pp 384ndash3912004

[11] H Hu Q Hu Z Lu and D Xu ldquoOptimal PID controller designin PMSM servo system via particle swarm optimizationrdquo inProceedings of the 31st Annual Conference of IEEE IndustrialElectronics Society (IECON rsquo05) pp 79ndash83 November 2005

[12] TH Kim IMaruta and T Sugie ldquoParticle swarmoptimizationbased robust PID controller tuning schemerdquo in Proceedings ofthe 46th IEEE Conference on Decision and Control (CDC rsquo07)pp 200ndash205 New Orleans La USA December 2007

[13] M I Solihin Wahyudi M A S Kamal and A LegowoldquoOptimal PID controller tuning of automatic gantry craneusing PSO algorithmrdquo in Proceedings of the 5th InternationalSymposium on Mechatronics and its Applications (ISMA rsquo08)Amman Jordan May 2008

[14] S Pothiya and I Ngamroo ldquoOptimal fuzzy logic-based PIDcontroller for load-frequency control including superconduct-ing magnetic energy storage unitsrdquo Energy Conversion andManagement vol 49 no 10 pp 2833ndash2838 2008

[15] M Zhuang and D P Atherton ldquoAutomatic tuning of optimumPID controllersrdquo IEE Proceedings D vol 140 no 3 pp 216ndash2241993

[16] G Herjdlfsson and A S Hauksddttir ldquoDirect computationof optimal PID controllersrdquo in Proceedings of the 42nd IEEEConference onDecision andControl pp 1120ndash1125MauiHawaiiUSA December 2003

[17] S Daley and G P Liu ldquoOptimal PID tuning using direct searchalgorithmsrdquoComputing andControl Engineering Journal vol 10no 2 pp 51ndash56 1999

[18] S Tavakoli andM Tavakoli ldquoOptimal tuning of PID controllersfor first order plus time delay models using dimensionalanalysisrdquo in Proceedings of the 4th International Conference onControl and Automation pp 942ndash946 Montreal Canada June2003

[19] H Panagopoulos K J Astrom and T Hagglund ldquoDesignof PID controllers based on constrained optimizationrdquo IEEProceedings vol 149 no 1 pp 32ndash40 2002

[20] C Hwang and C Y Hsiao ldquoSolution of a non-convex optimiza-tion arising in PIPID control designrdquo Automatica vol 38 no11 pp 1895ndash1904 2002

[21] Y L Xue D H Li and C D Lv ldquoOptimal design of PIDcontroller based on sensitivity constraintrdquo in Proceedings ofthe 5th World Congress on Intelligent Control and AutomationHangzhou China June 2004

[22] C H Hsieh and J H Chou ldquoDesign of optimal PID controllersfor PWM feedback systems with bilinear plantsrdquo IEEE Transac-tions on Control Systems Technology vol 15 no 6 pp 1075ndash10792007

[23] R T J OrsquoBrien and J M Howe ldquoOptimal PID controller designusing standard optimal control techniquesrdquo in Proceedings ofthe 2008 American Control Conference pp 4733ndash4738 SeattleWash USA June 2008

[24] J B He Q G Wang and T H Lee ldquoPIPID controller tuningvia LQR approachrdquo in Proceedings of the 37th IEEE Conferenceon Decision and Control pp 1177ndash1182 Tampa Fla USADecember 1998

[25] G R Yu and R C Hwang ldquoOptimal PID speed control of brushless DCmotors using LQR approachrdquo in Proceedings of the IEEEInternational Conference on SystemsMan andCybernetics (SMCrsquo04) pp 473ndash478 October 2004

[26] H Zargarzadeh M R Jahed Motlagh and M M ArefildquoMultivariable robust optimal PID controller design for a non-minimum phase boiler system using loop transfer recoverytechniquerdquo in Proceedings of the 6th Mediterranean Conferenceon Control and Automation (MED rsquo08) pp 1520ndash1525 AjaccioFrance June 2008

[27] P Han Y Huang Z Z Jia D F Wang and Y L Li ldquoMixedH2Hinfinoptimal pid control for superheated steam temperature

system based on PSO optimizationrdquo in Proceedings of theInternational Conference on Machine Learning and Cybernetics(ICMLC rsquo05) pp 960ndash964 Guangzhou China August 2005

[28] P B Dickinson andA T Shenton ldquoA parameter space approachto constrained variance PID controller designrdquoAutomatica vol45 no 3 pp 830ndash835 2009

[29] J C Basilio and S R Matos ldquoDesign of PI and PID controllerswith transient performance specificationrdquo IEEE Transactions onEducation vol 45 no 4 pp 364ndash370 2002

[30] M Saeki ldquoFixed structure PID controller design for standardHinfin

control problemrdquo Automatica vol 42 no 1 pp 93ndash1002006

[31] O Lequin M Gevers M Mossberg E Bosmans and L TriestldquoIterative feedback tuning of PID parameters comparison withclassical tuning rulesrdquo Control Engineering Practice vol 11 no9 pp 1023ndash1033 2003

[32] M Xu S Li C Qi and W Cai ldquoAuto-tuning of PID controllerparameters with supervised receding horizon optimizationrdquoISA Transactions vol 44 no 4 pp 491ndash500 2005


[33] G L Luo and G N Saridis ldquoLQ design of PID controllers forrobot armsrdquo IEEE Journal of Robotics and Automation vol RA-1 no 3 pp 152ndash159 1985

[34] X H Li H B Yu and M Z Yuan ldquoDesign of an optimalPID controller based on Lyapunov approachrdquo in Proceedingsof the International Conference on Information Engineering andComputer Science (ICIECS rsquo09)WuhanChinaDecember 2009

[35] K J Astrom and T Hagglund ldquoThe future of PID controlrdquoControl Engineering Practice vol 9 pp 1163ndash1175 2001

[36] X H Li H B Yu M Z Yuan and J Wang ldquoDesign of robustoptimal proportional-integral-derivative controller based onnew interval polynomial stability criterion and Lyapunov the-orem in the multiple parametersrsquo perturbations circumstancerdquoIET Control Theory and Applications vol 4 no 11 pp 2427ndash2440 2010

[37] H Biao and R Kadali ldquoSystem identification conventionalapproachrdquo in Dynamic Modeling Predictive Control and Per-formance Monitoring vol 374 of Lecture Notes in Control andInformation Sciences pp 9ndash29 Springer Press Berlin Germany2008

[38] F P Ti System Identification Zhe Jiang University PressHangzhou China (Chinese) 2004

[39] D E Rivera M Morari and S Skogestad ldquoInternal modelcontrol 4 PID controller designrdquo Industrial and EngineeringChemistry Process Design and Development vol 25 no 1 pp252ndash265 1986

[40] S Skogestad ldquoSimple analytic rules for model reduction andPID controller tuningrdquo Journal of Process Control vol 13 no4 pp 291ndash309 2003

[41] M Veronesi and A Visioli ldquoPerformance assessment andretuning of PID controllers for integral processesrdquo Journal ofProcess Control vol 20 no 3 pp 261ndash269 2010

[42] A Visioli ldquoA new design for a PID plus feedforward controllerrdquoJournal of Process Control vol 14 no 4 pp 457ndash463 2004

[43] H P Huang J C Jeng C H Chiang and W Pan ldquoA directmethod for multi-loop PIPID controller designrdquo Journal ofProcess Control vol 13 no 8 pp 769ndash786 2003

[44] K K Tan T H Lee and X Jiang ldquoOn-line relay identificationassessment and tuning of PID controllerrdquo Journal of ProcessControl vol 11 no 5 pp 483ndash496 2001

[45] QGWang Z Zhang K J Astrom andL S Chek ldquoGuaranteeddominant pole placement with PID controllersrdquo Journal ofProcess Control vol 19 no 2 pp 349ndash352 2009

[46] L Meng and D Xue ldquoDesign of an optimal fractional-orderPID controller using multi-objective GA optimizationrdquo inProceedings of the Chinese Control and Decision Conference(CCDC rsquo09) pp 3849ndash3853 June 2009

[47] X H Li H B Yu and M Z Yuan ldquoA new design methodof optimal PID controller with dynamic performances con-strainedrdquo Control and Intelligent Systems vol 40 no 4 pp 1ndash212012

[48] I J Nagrath and M Gopal Control Systems Engineering WileyEastern New Delhi India 1975

[49] G H Hostetter C J Savant Jr and R T Stefani Design ofFeedback Control System CBS college 1982

[50] T F Coleman and Y Li ldquoOn the convergence of interior-reflective Newton methods for nonlinear minimization subjectto boundsrdquoMathematical Programming vol 67 no 2 pp 189ndash224 1994

[51] D F Shanno ldquoConditioning of quasi-Newton methods forfunction minimizationrdquo Mathematics of Computation vol 24pp 647ndash656 1970

[52] W W Hager and H Zhang ldquoA new conjugate gradient methodwith guaranteed descent and an efficient line searchrdquo SIAMJournal on Optimization vol 16 no 1 pp 170ndash192 2005

[53] J E Dennis M El-Alem and K Williamson ldquoA trust-regionapproach to nonlinear systems of equalities and inequalitiesrdquoSIAM Journal on Optimization vol 9 no 2 pp 291ndash315 1999

[54] W W Hager and D T Phan ldquoAn ellipsoidal branch andboun d algorithm for global optimizati onrdquo SIAM Journal onOptimization vol 20 no 2 pp 740ndash758 2009

[55] N I M Gould ldquoThe generalized steepest-edge for linearprogrammingrdquo Combinatorics and Optimization TechnicalReport 83-2 University of Waterloo Waterloo Canada 1983

[56] Y Zhang ldquoSolving large-scale linear programs by interior-pointmethods under the MATLAB environmentrdquo Tech Rep TR96-01 Department of Mathematics and Statistics University ofMaryland Baltimore Md USA 1995

[57] R J Vanderbei and D F Shanno ldquoAn interior-point algorithmfor non-convex nonlinear programmingrdquo Computational Opti-mization and Applications vol 13 no 1ndash3 pp 231ndash252 1999

[58] H Y Benson and D F Shanno ldquoInterior-point methods fornonconvex nonlinear programming regularization and warm-startsrdquo Computational Optimization and Applications vol 40no 2 pp 143ndash189 2008

[59] R AWaltz J LMorales J Nocedal andD Orban ldquoAn interioralgorithm for nonlinear optimization that combines line searchand trust region stepsrdquoMathematical Programming vol 107 no3 pp 391ndash408 2006

[60] H B Richard M E Hribar and J Nocedal ldquoAn interiorpoint algorithm for large-scale nonlinear programmingrdquo SIAMJournal on Optimization vol 9 no 4 pp 877ndash900 1997

[61] A Forsgren P E Gill and M H Wright ldquoInterior methods fornonlinear optimizationrdquo SIAM Review vol 44 no 4 pp 525ndash597 2002

[62] D E GoldbergGenetic Algorithms in Search Optimization andMachine Learning Addison-Wesley Reading Mass USA 1989

[63] HHHolger andT Stutzle Stochastic Local Search Foundationsand Applications Morgan Kaufmann San Francisco CalifUSA 2004

[64] X H Li H B Yu and M Z Yuan ldquoRobust stability of intervalpolynomials and matrices for linear systemsrdquo in Proceedings ofthe IASTED International Conference Modelling Identificationand Control pp 138ndash147 Phuket Thailand November 2010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


algorithm (HTGA) is used to design optimal PID con-troller for pulse width modulation (PWM) feedback systemsthrough solving a constrained optimization problem In [23ndash25] linear quadratic regulator (LQR) method is proposed todesign the optimal PID controller for different systems andprocesses In [26] the loop transfer recovery technique isapplied to devise multivariable robust optimal PID controllerfor a nonminimum phase boiler system In [27] optimalPID controller with 119867

2119867infin

norm constraint is presentedfor the superheated steam temperature system via PSA In[28] a parameter space approach is proposed to design theminimum variance PID controller for linear time-invariant(LTI) plants In [29] a design method is proposed to designPID controller by considering a transient performance itemIn [30] a designmethod of multivariable fixed-structure PIDcontroller is proposed for multivariable plants by solving anoptimization problem with linear matrix inequality (LMI)constraint In [31] iterative feedback tuning (IFT) method isused to design the optimal PID controller for achieving fastresponse to set point variations In [32] a two-layer onlineautotuning algorithm is presented to design the optimal PIDcontroller via general predictive control method In [33]the LQR method is presented to design the optimal PIDcontroller for robot arms In [39] a design method of PIDcontroller is proposed for SOPTD processes via internalmodel control (IMC) framework In [40] analytic rules ofthe PID controller are presented for FOPTD and SOPTDprocesses via the IMCmethod andmodel reduction methodIn [41] a performance assessmentmethod of PID controller isproposed for integral processes via focusing on step responsesof the set point change and load disturbance In [42] a designmethod of the PID controller with feedforward compensatoris proposed for the FOPTD process In [43] a design methodof the multiloop PIPID controller is presented for squaremultiple-input and multiple-output (MIMO) process whichis decomposed into a series of equivalent effective open-loop process (EOP) with reasonable gain margins and phasemargins In [44] an on-line relay feedback approach is usedto identify assess and tune the PID controller In [45] thepole placement method is used to design the PID controllerIn [46] the multiobjective genetic algorithm is presented todesign the optimal fractional-order PID controller In [47] adesign method is presented for optimal PID controller withdynamic performance constrained

Design methods of the PID controller in [1ndash43] can beroughly divided into the following cases the empirical meth-ods intelligent algorithm methods numerical parametricmethods classic LQRLQGmethods robust designmethodsIMC methods and pole placement methods Many designmethods of the PID controller are proposed for first-orderand second-order processes In this paper a novel systematicdesign method is presented to design optimal PID controllerwith 120576-Routh stability for different processes The optimalPID controllerwith 120576-Routh stability is proposed byminimiz-ing augmented integral squared error (AISE) performanceindexThe proposed method is unlike classic optimal controldesign methods which need to solve the Riccati equation(RE) The proposed optimal control problem is equivalentlytransformed into theNLCOproblem via Lyapunov theorems

Therefore optimal controller parameters can be obtained bysolving a NLCO problem The proposed method is used todesign optimal PID controller for a third-order SOPTD andFOPTDprocesses respectivelyThe proposedmethod is usedto control the liquid level of the CWTS under different setpoints The simulation results and experiment results showeffectiveness and usefulness of the proposed method Theproposed method is convenient to design the optimal PIDcontroller which can provide high performance for controlsystem

This paper is arranged as follows in Section 2 prelimi-nary and optimal control problem statements are describedIn Section 3 CWTS model is established and systematicdesign procedures of the optimal PID controller are presentedin detail Characteristics and robustness of 120576-Routh stabilityare analyzed and the interior method is recommended tosolve the NLCO problem of the proposed optimal PIDcontroller In Section 4 optimal PID controllers and sim-ulation results are presented for different processes Thecontrolweightmatrix and 120576-Routh stabilityrsquos effects on systemperformances are studied The system performances anddisturbance rejection ability of different tuning methodsare investigated 120576-Routh stabilityrsquos effects on disturbancerejection ability are discussed as well To further validate theproposed method the proposed method is used to controlthe liquid level of the CWTS under different set points InSection 5 study contents are reviewed and some conclusionsare made

2 Preliminary and Optimal ControlProblem Statements

Preliminary Based on Lyapunov theorems a systematic tun-ing method is proposed to design the optimal PID controllerwith 120576-Routh stability The following theorems and 120576-Routhstability definition are utilized to design the optimal PIDcontroller

Theorem 1 (see [1 2]) Sufficient and necessary conditions forlinear time-invariant system = 119860119909 asymptotically stable in alarge scope are that for any given matrix 119876 = 119876

119879gt 0 there

is a matrix 119875 = 119875119879

gt 0 that satisfies the Lyapunov algebraicequation (LAE) 119860119879119875 + 119875119860 = minus119876

Theorem 2 (see [1 2]) The linear time-invariant (LTI) system = 119860119909 and 119909(0) = 119909

0is asymptotically stable for any given

matrix 119876 = 119876119879

gt 0 then the performance index J will haveequivalent relationships

119869 = int

infin

0

119909119879119876119909119889119905 = 119909

119879(0) 119875119909 (0) 119860

119879119875 + 119875119860 = minus119876

(1)

Definition of 120576-Routh stability 119863(119904) is characteristic poly-nomial of the linear time-invariant control system For anynonnegative real 120576 if all elements of the Routh arrayrsquosfirst column of characteristic polynomial 119863(119904) = 119886

0119904119899+

1198861119904119899minus1


119904 + 119886119899can satisfy the following inequality



11987711

(1198860 1198861 119886

119899) gt 120576 119877

21(1198860 1198861 119886

119899) gt 120576

1198771198991

(1198860 1198861 119886

119899) gt 120576 119877

119899+11(1198860 1198861 119886

119899) gt 120576

(120576 ge 0)

(2)

where 11987711

11987721

11987731

119877119899 1 and 119877






119901(119904)

119866119901(119904) =

1198871119904119908+ 1198872119904119908minus1

+ sdot sdot sdot + 119887119908119904 + 119887119908+1

119904119899+ 1198861119904119899minus1


119904 + 119886119899

=

sum119908+1

119895=1119887119895119904119908+1minus119895

119904119899+ sum119899

119894=1119886119894119904119899minus119894

=

119873 (119904)

119872 (119904)

(3)

where 1198871 119887

119908+1 1198861 and 119886




119899= 0 and




119906 (119904) = 119866119888(119904) 119890 (119904) 119866

119888(119904) =

(119896119901119904 + 119896119894+ 1198961198891199042)

119904

119903 (119904) minus 119910 (119904) ≜ 119890 (119904) 119910 (119904) = 119866119901(119904) 119906 (119904)

(4)



and 119896119901 119896119894 and 119896



119903 (119904) = 119910 (119904) + 119890 (119904) 119910 (119904) =

119873 (119904) 119906 (119904)

119872 (119904)

119906 (119904) =

(119896119901119904 + 119896119894+ 1198961198891199042) 119890 (119904)

119904

(5)


119903 (119904)

=

1

(1 + 119866119901(119904) 119866119888(119904))

lArrrArr 119863(119904) 119890 (119904)

= 119904119872 (119904) 119903 (119904)

119863 (119904) = 119904119872 (119904) + 119873 (119904) (119896119901119904 + 119896119894+ 1198961198891199042)

deg 119863 (119904) = 119899 + 1

(6)


(120575119872 (120575) + 119873 (120575) (119896119901120575 + 119896119894+ 1198961198891205752)) 119890 (119905) = 120575119872 (120575) 119903 (119905)

(7)


(120575119899+1

+

119899

sum

119894=1

119886119894120575119899+1minus119894

)

+ (119896119901120575 + 119896119894+ 1198961198891205752)

119908+1

sum

119895=1

119887119895120575119908+1minus119895

119890 (119905) = 0

(8)


120575119899+1

+

119899

sum

119894=1

119886119894120575119899+1minus119894

+

119908+1

sum

119895=1

119887119895119896119901120575119908+2minus119895

+

119908+1

sum

119895=1

119887119895119896119894120575119908+1minus119895

+

119908+1

sum

119895=1

119887119895119896119889120575119908+3minus119895

119890 (119905) = 0

(9)


119890(119905)(119899+1)

+ 1198861119890(119905)(119899)

+ 1198862119890(119905)(119899minus1)


119890(119905)(119908+3)

+ (119886119899minus119908minus1

+ 1198871119896119889) 119890(119905)(119908+2)

+ (119886119899minus119908

+ 1198871119896119901+ 1198872119896119889) 119890(119905)(119908+1)

+

119908minus1

sum

119895=1

(119886119899minus119908+119895

+ 119887119895+1

119896119901+ 119887119895119896119894+ 119887119895+2

119896119889) 119890(119905)(119908+1minus119895)

+ (119886119899+ 119887119908+1

119896119901+ 119887119908119896119894) 119890 (119905) + 119887

119908+1119896119894119890 (119905) = 0

(10)



119890(119905)(119899+1)

+ 1198861119890(119905)(119899)

+ 1198862119890(119905)(119899minus1)


119890(119905)(119908+3)

+ 1198981119890(119905)(119908+2)

+ 1198982119890(119905)(119908+1)

+

119908minus1

sum

119895=1

119898119895+2

119890(119905)(119908+1minus119895)

+ 119898119908+2

119890 (119905) + 119898119908+3

119890 (119905) = 0

(11)



1198981= (119886119899minus119908minus1

+ 1198961198891198871) 119898

2= (119886119899minus119908

+ 1198871119896119901+ 1198872119896119889)

119898119895+2

= (119886119899minus119908+119895

+ 119887119895+1

119896119901+ 119887119895119896119894+ 119887119895+2

119896119889)

(119895 = 1 2 119908 minus 1)

119898119908+2

= (119886119899+ 119887119908+1

119896119901+ 119887119908119896119894) 119898

119908+3= 119887119908+1

119896119894

(12)


= 119860119909 (13)






119860 = (

119874119899times1

119868119899times119899

minus119898119908+3

1198621times119899

) 119909 = (119890 (119905) 119890 (119905) 119890 (119905) sdot sdot sdot 119890(119905)(119899)

)

119879

1198621times119899

= (minus119898119908+2



(14)




119901 119896lowast

119894 119896lowast

119889) = min119866lowast

119888(119904)

119869 (1198981 119898

119908+3)

= minint

infin

0

119909119879(119905) 119876119909 (119905) 119889119905

= min

119899

sum

119894=0

int

infin

0

119902119894+1119894+1

(119890(119905)(119894))

2

119889119905

subject to (st)

(1) = 119860119909


(3) 120576 ge 0 11987711

(1198981 119898

119908+3) gt 120576

11987721

(1198981 119898

119908+3) gt 120576

119877119899+11

(1198981 119898

119908+3) gt 120576 119877

119899+21(1198981 119898

119908+3) gt 120576

(4)1198984= 1198911(1198981 1198982 1198983) 119898

119908+3= 119891119908(1198981 1198982 1198983)

(5) 0 lt 119896119901le 1205910

119901 0 lt 119896

119894le 1205910

119894 0 lt 119896

119889le 1205910

119889

(15)


1 119891

119908is the


119889 and

119896lowast

119901 119896lowast119894 and 119896

lowast




119894(119894 = 1 2 119908+3) the


0

119901 1205910119894 and 120591

0


parameters








Command

119903(119905) +

+

minusminus



119910(119905)

Output119863119906(119905)

119866lowast119888 (119904) = 119896

lowast119901 +

119896lowast119894119904+ 119896lowast119889119904



Control valve

ValveWater tank

Tank-1

Tank-2



119876119894119888minus 1198761119888

minus 11987612

= 1198621

119889ℎ1

119889119905

(16)



1 11987612




11987612

minus 1198762119888

= 1198622

119889ℎ2

119889119905

(17)





1198761119888

=

ℎ1

1198771

1198762119888

=

ℎ2

1198772

11987612

=

ℎ1minus ℎ2

11987712

(18)

where1198771and119877




119866ℎ(119904) =

1198671(119904)

119876119894119888(119904)

=

1198871119904 + 1198872

11988601199042+ 1198861119904 + 1198862

=

11988711199041198860+ 11988721198860

1199042+ 11988611199041198860+ 11988621198860

=

1198731(119904)

1198721(119904)

(19)



Tank-1 Tank-2

Control valve

1198771

119876121198671 + ℎ1

11986221198621

1198772

11987621198881198761119888

1198760 + 119876119894119888

11987712

1198672 + ℎ2Valve-3

Valv

e-2

Valv

e-1



1198871= 11987711198771211987721198622 119887

2= 1198771(11987712

+ 1198772)

1198860= 119877111986211198771211987721198622

1198861= 119877111986211198772+ 1198771119862111987712

+ 119877111987721198622+ 1198771211987721198622

(20)

1198862= 1198771+ 11987712

+ 1198772 (21)


119866119901(119904) =

11988711199042+ 1198872119904 + 1198873

11988601199043+ 11988611199042+ 1198862119904 + 1198863

=

119873 (119904)

119872 (119904)

119866119888(119904) = 119896

119901+

119896119894

119904

+ 119896119889119904

(22)


119866 (119904) = 119866119888(119904) 119866119901(119904) =

(119896119901119904 + 119896119894+ 1198961198891199042)119873 (119904)

119904119872 (119904)

(23)


119890 (119904)

119903 (119904)

=

1

(1 + 119866119901(119904) 119866119888(119904))

lArrrArr 119863(119904) 119890 (119904) = 119904119872 (119904) 119903 (119904)

119863 (119904) = (119904119872 (119904) + 119873 (119904) (119896119901119904 + 119896119894+ 1198961198891199042))

deg 119863 (119904) = 4

(24)


(120575119872 (120575) + 119873 (120575) (119896119901120575 + 119896119894+ 1198961198891205752)) 119890 (119905) = 120575119872 (120575) 119903 (119905)

(25)




(1198860+ 1198961198891198871) 119890(119905)(4)

+ (1198861+ 1198961199011198871+ 1198961198891198872)119890 (119905)

+ (1198862+ 1198961198941198871+ 1198961199011198872+ 1198961198891198873) 119890 (119905)

+ (1198863+ 1198961198941198872+ 1198961199011198873) 119890 (119905) + 119896

1198941198873119890 (119905) = 0

(26)


1198981119890(119905)(4)

+ 1198982

119890 (119905) + 119898

3119890 (119905) + 119898

4119890 (119905) + 119898

5119890 (119905) = 0 (27)



1198981= 1198860+ 1198961198891198871 119898

2= 1198861+ 1198961199011198871+ 1198961198891198872 119898

5= 1198961198941198873

1198983= 1198862+ 1198961198941198871+ 1198961199011198872+ 1198961198891198873 119898

4= 1198863+ 1198961198941198872+ 1198961199011198873

(28)


= 119860119909

119909 = (119890 (119905) 119890 (119905) 119890 (119905)

119890 (119905))119879 119860 = (

1198743times1

1198683times3

minus1198981

41198621times3

)

1198621times3

= (minus1198981

3minus 1198981

2minus 1198981

1)

(29)



119894= 119898119894+1

1198981(119894 = 1 2 3 4)



119901 119896lowast

119894 119896lowast

119889) = min119866lowast

119888

119869 (1198981 119898

5)

= minint

infin

0

119909119879(119905) 119876119909 (119905) 119889119905

= min

3

sum

119894=0

int

infin

0

119902119894+1119894+1

(119890(119905)(119894))

2

119889119905

st

(1) = 119860119909


(3) 120576 ge 0 11987711

(1198981 119898

119908+3) gt 120576

11987721

(1198981 119898

119908+3) gt 120576

11987731

(1198981 119898

119908+3) gt 120576 119877

41(1198981 119898

119908+3) gt 120576

11987731

(1198981 119898

119908+3) gt 120576 119877

41(1198981 119898

119908+3) gt 120576

11987751

(1198981 119898

119908+3) gt 120576

(4)1198984= 1198911(1198981 1198982 1198983) 1198985= 1198912(1198981 1198982 1198983)

(5) 0 lt 119896119901le 1205910

119901 0 lt 119896

119894le 1205910

119894 0 lt 119896

119889le 1205910

119889

(30)


4+

1198981

11199043+ 1198981

21199042+ 1198981

3119904 + 119898

1




lArrrArr 1198981

1gt 0 119898

1

11198981

2minus 1198981

3gt 0

1198981

11198981

21198981

3minus (1198981

3)

2

minus 1198981

4(1198981

1)

2

gt 0 1198981

4gt 0

(31)




120576 ge 0 11987711

(1198981 119898

5) gt 120576 119877

21(1198981 119898

5) gt 120576

11987731

(1198981 119898

5) gt 120576

11987741

(1198981 119898

5) gt 120576 119877

51(1198981 119898

5) gt 120576

lArrrArr 1198981gt 120576 119898

2gt 120576 119898

5gt 120576

11989811198984minus 11989821198983gt 1205761198982

(1198982)2

1198985minus 1198981(1198984)2

minus 119898211989831198984gt 120576 (119898

11198984minus 11989821198983)

(32)



infin

0

119909119879119876119909119889119905 = min119909

119879(0) 119875119909 (0)

119860119879119875 + 119875119860 = minus119876

(33)





119901 119896lowast

119894 119896lowast

119889) = min119866lowast

119888(119904)

119869 (1198981 119898

5)

= min119909(0)119879119875119909 (0)

st

(1) 119860119879119875 + 119875119860 = minus119876

(2)1198981

1gt 0 119898

1

4gt 0 119898

1

11198981

2minus 1198981

3gt 0

1198981

11198981

21198981

3minus (1198981

3)

2

minus 1198981

4(1198981

1)

2

gt 0

(3)1198981gt 120576 119898

2gt 120576 119898

5gt 120576 119898

11198984minus 11989821198983gt 1205761198982

(1198982)2

1198985minus 1198981(1198984)2

minus 119898211989831198984gt 120576 (119898

11198984minus 11989821198983)

(4)1198984= 1198911(1198981 1198982 1198983) 1198985= 1198912(1198981 1198982 1198983)

(5) 0 lt 119896119901le 1205910

119901 0 lt 119896

119894le 1205910

119894 0 lt 119896

119889le 1205910

119889

(34)


1(0) = 1 119909

2(0) = 0 119909

3(0) = 0 and




119894and


119866lowast

119888(119904) = 119896

lowast

119901+

119896lowast

119894

119904

+ 119896lowast

119889119904 119896

lowast

119901=

119898lowast


119894

1198873

119896lowast

119894=

119898lowast

5

1198873

119896lowast

119889=

119898lowast

1minus 1198860

1198871

(35)



1119904119899+sdot sdot sdot+119886

119899minus119908minus2119904119908+3

+

1198981119904119908+2

+ 1198982119904119908+1

+ sdot sdot sdot + 119898119908+2

119904 + 119898119908+3




119889 The

parameters 119896119901 119896119894 and 119896


119901]

(0 1205910

119894] and (0 120591

0



119901 119896119894



119865 (119904) = 119904119899+ 1199011119904119899minus1

+ 1199012119904119899minus2


119904 + 119901119899 (36)

where 1199011 1199012 and 119901





119898

1199011= 1198911(1199021 119902

119898) 119901

119899= 119891119899(1199021 119902

119898)

Ω119902= (1199021 119902

119898)1003816100381610038161003816[119902minus

1 119902+


minus

119898 119902+

119898]

1199021isin [119902minus

1 119902+

1] 119902

119898isin [119902minus

119898 119902+

119898]

(37)


119894and 119902+

119894are the




11987711

(1199021 119902

119898) = 1198911199031

(1199011 119901

119899) gt 120576

119877119899+11

(1199021 119902

119898) = 119891119903119899+1

(1199011 119901

119899) gt 120576

Ω119903120576

= (1199021 119902

119898) | 1198911199031

(1199011 119901

119899) gt 120576

119891119903119899+1

(1199011 119901

119899) gt 120576 (120576 ge 0)

(38)







min(1199021 119902119898)isinΩ119891

1198911199031

(1199011 119901

119899) 119891

119903119899+1(1199011 119901

119899) gt 0

(Ω119891= (Ω119903120576

⋂Ω119902) = 0)

(39)







I Ω1= (119902

1 119902

119898) | 1198911199031

(1199011 119901

119899) gt 1205761

119891119903119899+1

(1199011 119901

119899) gt 1205761

II Ω2= (119902

1 119902

119898) | 1198911199031

(1199011 119901

119899) gt 1205762

119891119903119899+1

(1199011 119901

119899) gt 1205762

(40)




Ω1

1= (1199021 119902

119898) | 1198911199031

(1199011 119901

119899) gt 1205761

Ω1

2= (1199021 119902

119898) | 1198911199031

(1199011 119901

119899) gt 1205762

(41)



1

2will have the

relationship

exist119872lowast

1= (119902lowast

11 119902lowast

21 119902

lowast

1198981) 119891

1199031(119872lowast

1) = 120576lowast

2gt 1205762 (42)


1will have the

relationship

exist119872lowast

2= (119902lowast

12 119902lowast

22 119902

lowast

1198982) 119891

1199031(119872lowast

2) = 120576lowast

1gt 1205761 (43)


1which will



1= (119902lowast

11 119902lowast

21 119902

lowast

1198981) 1205761lt 1198911199031

(119873lowast

1) lt 1205762

lArrrArr 1198911199031

(119873lowast

1) isin (1205761 1205762)


1isin Ω1

1 119873lowast

1notin Ω1

2997904rArr Ω

1

1sup Ω1

2

Case 2 1198911199031

(119873lowast

1) notin (1205761 1205762) 997904rArr Ω

1

1= Ω1

2

(44)


119896


relationship

Case 1 Ω119896

1sup Ω119896

2 (forall119896 = 1 2 119899 119899 + 1)

Case 2 Ω119896

1= Ω119896

2 (forall119896 = 1 2 119899 119899 + 1)

Ω119896

1= (1199021 119902

119898) | 119891119903119896

(1199011 119901

119899) gt 1205761

Ω119896

2= (1199021 119902

119898) | 119891119903119896

(1199011 119901

119899) gt 1205762

(45)


2will have follow-

ing relationships


1sup Ω2) cup (Ω

1= Ω2)

Ω1=

119899+1

⋂

119895=1

Ω119895

1 Ω

2=

119899+1

⋂

119895=1

Ω119895

2

(46)


119905+1gt 120576119905gt 120576119905minus1



Ω119905= (1199021 119902

119898) | 1198911199031

(1199011 119901

119899) gt 120576119905

119891119903119899+1

(1199011 119901

119899) gt 120576119905



119899+1

119905) = (

119899+1

⋂

119895=1

Ω119895

119905)

(119905 = 1 2 3 119905 isin Z+)

Ω119896

119905= (1199021 119902

119898) | 119891119903119896

(1199011 119901

119899) gt 120576119905

(119896 = 1 2 119899 119899 + 1)

(47)


ship

Ω119905+1


2sube Ω1



2Ω2Δ1Ω1)

⋃ (Ω119905+1


2= Ω1)


(48)


119905 then the



(1199021 119902

119898) | 1198911199031

(1199011 119901

119899) gt 120576119905

119891119903119899+1

(1199011 119901

119899) gt 120576119905


Ω119905= lim120576119905rarr+infin

(Ω1

119905cap Ω2


119899+1

119905)


(

119899+1

⋂

119895=1

Ω119895

119905) = Ω

lowast

= (119902lowast

10119902lowast

20 119902

lowast

1198980)



(1199021 119902

119898) | 1198911199031

(1199011 119901

119899) gt 120576119905

119891119903119899+1

(1199011 119901

119899) gt 120576119905


Ω119905= lim120576119905rarr+infin

(Ω1

119905cap Ω2


119899+1

119905)


(

119899+1

⋂

119895=1

Ω119895

119905) = 0

(49)





120576119905997888rarr +infin lim



lowast

119905 120576119905gt 120576lowast

119905 Ω119905= 0

0 lt 120576119905le 120576lowast

119905 Ω119905

= 0

120576lowast

119905gt 120576119905gt 120576119905minus1


Ωlowast

119905sub Ω119905sub Ω119905minus1


(50)


= Ω119905= Ω119905minus1

=



119875 Ω119905+1

= Ω119905= Ω119905minus1


lArrrArr 119875 Ω119905+1

= Ω119905 Ω119905= Ω119905minus1

Ω2= Ω1

(51)


119875 Ω119905+1

= Ω119905 Ω119905= Ω119905minus1

Ω2= Ω1

= 119875 Ω119905+1

= Ω119905 times 119875 Ω

119905= Ω119905minus1


=

119905+1

prod

119895=2

119875 Ω119895= Ω119895minus1

= (119875 Ω2= Ω1)119905

(119905 = 1 2 119905 isin Z+)

(52)



= 119875 (Ω1


119899+1

1) = (Ω

1


119899+1

2)

= 119875

(

119899+1

⋂

119895=1

Ω119895

1) = (

119899+1

⋂

119895=1

Ω119895

2)

lArrrArr 119875(Ω1


119899+1

2) sub Ω

119896

1

(Ω1


119899+1

1) sub Ω

119896

2

(forall119896 = 1 2 119899 119899 + 1)

(53)


2=

Ω1 will yield

119875 (Ω1

2cap Ω2


119899

2) sub Ω

1

1 = 1205821

119875 (Ω1

2cap Ω2


119899

2) sub Ω

119899+1

1 = 1205821

119875 (Ω1

1cap Ω2


119899

1) sub Ω

1

2 = 1205822

119875 (Ω1

1cap Ω2


119899

1) sub Ω

119899+1

2 = 1205822

119875 Ω1= Ω2

= 119875

(

119899+1

⋂

119895=1

Ω119895

2) sub Ω

119896

1 (forall119896 = 1 2 119899 119899 + 1)

times 119875

(

119899+1

⋂

119895=1

Ω119895

1) sub Ω

119896

2 (forall119896 = 1 2 119899 119899 + 1)

=

119899+1

prod

119896=1

119875

(

119899+1

⋂

119895=1

Ω119895

2) sub Ω

119896

1

times

119899+1

prod

119896=1

119875

(

119899+1

⋂

119895=1

Ω119895

1) sub Ω

119896

2

= 120582119899+1

1120582119899+1

2 (0 lt 120582

1lt 1 0 lt 120582

2lt 1)

(54)


119875 Ω119905+1

= Ω119905 Ω119905= Ω119905minus1

Ω2= Ω1 = 120582(119899+1)119905

1120582(119899+1)119905

2

119875 Ω119905+1

Δ119905Ω119905Δ119905minus1

Ω119905minus1


(119899+1)119905

1120582(119899+1)119905

2

(55)


min 119891 (119909)

st ℎ (119909) = 0 119892 (119909) le 0

(56)


where 119891(119909) 119877119899

rarr 119877 ℎ(119909) 119877119899

rarr 119877119898 and 119892(119909)

119877119899



min 119891 (119909) minus 120592

119902

sum

119894=1

ln 120575119894

st ℎ (119909) = 0 119892 (119909) + 120575 = 0

(57)


(1205751 1205752 120575

119902)119879


119871 (119909 119910 119911 120575) = 119891 (119909) minus 120592

119902

sum

119894=1

ln 120575119894+ 119910119879(119892 (119909) + 120575) + 119911

119879ℎ (119909)

(58)


119902)119879

and z = (1199111 1199112 119911




119879

119910 + (nablaℎ (119909))119879119911 = 0

nabla120575119871 (119909 119910 119911 120575) = minus120592119878

minus1

120575119890 + 119910 = 0 lArrrArr minus120592119890 + 119878

120575119884119890 = 0

nabla119910119871 (119909 119910 119911 120575) = 119892 (119909) + 120575 = 0

nabla119911119871 (119909 119910 119911 120575) = ℎ (119909) = 0

(59)

where 119878120575

= diag(1205751 1205752 120575



1 1199102 119910



Δ119909 120575+Δ120575 119911+Δ



(nabla2119891 (119909) + nabla

2119892(119909)119879119910 + nabla

2ℎ(119909)119879119911) Δ119909 + nabla119892(119909)

119879Δ119910


119879119910 + nablaℎ(119909)

119879119911) = 0

minus120592119890 + 119878120575119884119890 + 119878

120575Δ119910 + 119884Δ120575 = 0

119892 (119909) + nabla119892 (119909) Δ119909 + 120575 + Δ120575 = 0

nablaℎ (119909) Δ119909 + ℎ (119909) = 0

(60)


(

119867(119909 119910 119911) 0 nablaℎ(119909)119879

nabla119892(119909)119879

0 119878minus1

120575119884 0 119868

nablaℎ (119909) 0 0 0

nabla119892 (119909) 119868 0 0

)(

Δ119909

Δ120575

Δ119910

Δ119911

)

= (


119879119911

120592119878minus1

120575119890 minus 119910

minusℎ (119909)

minus119892 (119909) minus 120575

)

(61)

119867(119909 119910 119911) = nabla2119891 (119909) + nabla

2119892(119909)119879119910 + nabla

2ℎ(119909)119879119911 (62)


(119909 120575 119911 119910) larr997888 (119909 120575 119911 119910) + 1205771(Δ119909 Δ120575 Δ119911 Δ119910) (63)


1holds the




(0)gt 0 119896 = 0










(119896+1) 120575(119896+1)

119911(119896+1)

119910(119896+1)

) larr (119909(119896)

120575(119896)

119911(119896)

119910(119896)

) +

1205771(Δ119909 Δ120575 Δ119911 Δ119910) 119896 larr 119896 + 1


Step 7 End



1199011(119904) [7]

1198661199012(119904) [15] and 119866

1199013(119904) [6] respectively 119866

1199011(119904) is a third-


1199013(119904) are the SOPTD


1198661199011

(119904) =

15

(1199042+ 09119904 + 5) (119904 + 3)

1198661199012

(119904) =

119890minus05119904

(119904 + 1)2 119866

1199013(119904) =

4119890minus10119904

10119904 + 1

(64)


and 119890minus10119904

asymp (1199042minus 06119904 + 012)(119904

2+ 06119904 + 012) respectively


1199013(119904) are the (minus119904 +

4)(1199043+ 61199042+ 9119904 + 4) and (04119904

2minus 024119904 + 0048)(119904

3+ 07119904

2+



1 1199022 1199023 and 119902

4 The



2 1199023


1(with 119902

2 1199023 and 119902

4


2 1199023 and 119902

4(with 119902




1

















119901119896lowast

119894119896lowast

119889

Plant1

diag (5 5 1 1) 013121 058138 058655

120576 = 0


Plant2

diag (02 1 1 1) 115698 052966 086781

120576 = 05

diag (02 1 1 02) 216962 084500 182140diag (02 2 2 05) 178668 058700 159290

diag (01 08 08 02) 185901 064942 162251diag (04 05 5 1) 122806 062168 142095

Plant3

diag (004 27 158 175) 032447 001537 145256

120576 = 0




1119902211990231199024) 119896

lowast

119901119896lowast

119894119896lowast

119889119905119903(sec) 119905

119889(sec) 119905

119901(sec) 119905

119904(sec) 119872

119901()


0 2 4 6 8 10 12 14

0

02

04

06

08

1

12

Syste

m re

spon

se

minus02

Time (s)

119876 = diag (5 5 1 1)119876 = diag (5 2 1 1)119876 = diag (10 2 1 1)

119876 = diag (10 10 1 1)119876 = diag (5 5 2 1) lowast 2

(a)

0 2 4 6 8 10 12 14

0

02

04

06

08

1

12

Con

trol e

rror

minus02

Time (s)

119876 = diag (5 5 1 1)119876 = diag (5 2 1 1)119876 = diag (10 2 1 1)

119876 = diag (10 10 1 1)119876 = diag (5 5 2 1) lowast 2

(b)



0 1 2 3 4 5 6 7 8 9 100

02

04

06

08

1

12

14

Time (s)

Syste

m re

spon

se

120576 = 05

120576 = 25

120576 = 3

120576 = 4

120576 = 5

(a)

0 1 2 3 4 5 6 7 8 9 10

0

02

04

06

08

1

Time (s)

minus02

minus04

Con

trol e

rror

120576 = 05

120576 = 25

120576 = 3

120576 = 4

120576 = 5

(b)


0 5 10 15 20 25 30

0

02

04

06

08

1

12



Syste

m re

spon

se

Time (s)

minus02

(a)

0 5 10 15 20 25 30

0

02

04

06

08

1

12



Con

trol e

rror

Time (s)

minus02

(b)




119896119894

119896119889

Risetime

119905119903(sec)

Delaytime

119905119889(sec)

Peaktime

119905119901(sec)

Settingtime

119905119904(sec)


Plants



0 2 4 6 8 10 12 14 16 180

05

1

15Sy

stem

resp

onse



Time (s)

(a)

0 2 4 6 8 10 12 14 16 18

0

05

1


minus05


Con

trol e

rror

Time (s)

(b)


0 20 40 60 80 100 120 140 160 180 2000

02

04

06

08

1

12

14



Syste

m re

spon

se

Time (s)

095

105

(a)

0 20 40 60 80 100 120 140 160 180 200

0

02

04

06

08

1



Con

trol e

rror

Time (s)

minus02

minus04

(b)





Time (s)0 5 10 15 20 25 30

0

02

04

06

08

1

12

0 5 10 15 20 25 30

0

02

04

06

08

1

12

Syste

m re

spon

seSy

stem

resp

onse

Con

trol e

rror

Con

trol e

rror

Time (s)

Time (s)Time (s)16 18 20 22 24 26 28

075

08

085

09

095

1

105

11



16 18 20 22 24 26 28

0

005

01

015

02

025

03



minus02 minus02

minus005

119863119906(119905) = 03119906(119905 minus 16)



Methods 119896119901

119896119894

119896119889










16 18 20 22 24 26 28 30086

088

09

092

094

096

098

1

102

104





0 5 10 15 20 25 300

05

1

15

0 5 10 15 20 25 30

0

05

1

18 20 22 24 26 28 30

0

002

004

006

008

01

012

014

Con

trol e

rror

Con

trol e

rror

Syste

m re

spon

seSy

stem

resp

onse

minus05

Time (s) Time (s)

Time (s) Time (s)

minus002

119863119906(119905) = 03119906(119905 minus 18)






200 250 300 350 40007

075

08

085

09

095

1

105

11

200 250 300 350 400

0

005

01

015

02

025

03

0 50 100 150 200 250 300 350 4000

02

04

06

08

1

12

14

0 50 100 150 200 250 300 350 400

0

02

04

06

08

1

095

105

Time (s) Time (s)

Time (s)Time (s)





minus005

Con

trol e

rror

Syste

m re

spon

se

Con

trol e

rror

Syste

m re

spon

se

minus02

minus04

119863119906(119905) = 01119906(119905 minus 180)




1(119904)119876119894119888(119904) = (04465119904 +





119901=

10000 119896119894

= 032549 119896119889

= 0001 120576 = 0 119896119901

=

10000 119896119894

= 010118 119896119889

= 0001 120576 = 0 119896119901

=

10000 119896119894= 022986 119896

119889= 0001 120576 = 0 119896

119901= 6000 119896

119894=



1is set at




0 5 10 15 20 25 300

02

04

06

08

1

12

14

Time (s)

16 18 20 22 24 26 28 30086

088

09

092

094

096

098

1

102

104

0 5 10 15 20 25 30

0

02

04

06

08

1

Time (s)

Time (s) Time (s)

16 18 20 22 24 26 28 30

0

002

004

006

008

01

012

014

minus02

minus04

minus002

120576 = 05

120576 = 25

120576 = 3

120576 = 4

120576 = 5

120576 = 05

120576 = 25

120576 = 3

120576 = 4

120576 = 5



1 the solid







0 200 400 600 800 1000 1200 1400 1600 1800 20000

50

100

150

200

250

300

Time (s)

Liqu

id le

vel o

f CW

TS (m

m)




lowast

119901= 10000 119896lowast

119894= 032549 119896lowast

119889= 0001 and 120576 = 0 in

set point 300mm


119901= 10000 119896lowast

119894= 010118 119896lowast

119889= 0001 and 120576 = 0 in

set point 300mm


119901= 10000 119896lowast

119894= 022986 119896lowast

119889= 0001 and 120576 = 0 in

set point 300mm


lowast

119901= 6000 119896lowast

119894= 013817 119896lowast

119889= 0001 and 120576 = 0 in

set point 300mm


5 Conclusions




119901= 10000 119896lowast

119894= 032549 119896lowast

119889= 0001 and 120576 = 0 in

set point 160mm


119901= 10000 119896lowast

119894= 010118 119896lowast

119889= 0001 and 120576 = 0 in

set point 160mm


1 1199022 1199023 and 119902





2 1199023 and 119902

4(with 119902





4(with 119902

1fixed) Through the


1 1199022 119902

119899+1 The 120576-Routh




119901= 10000 119896lowast

119894= 022986 119896lowast

119889= 0001 and 120576 = 0 in

set point 160mm


lowast

119901= 6000 119896lowast

119894= 013817 119896lowast

119889= 0001 and 120576 = 0 in

set point 160mm



Acknowledgments



References











































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











11987711

(1198860 1198861 119886

119899) gt 120576 119877

21(1198860 1198861 119886

119899) gt 120576

1198771198991

(1198860 1198861 119886

119899) gt 120576 119877

119899+11(1198860 1198861 119886

119899) gt 120576

(120576 ge 0)

(2)

where 11987711

11987721

11987731

119877119899 1 and 119877






119901(119904)

119866119901(119904) =

1198871119904119908+ 1198872119904119908minus1

+ sdot sdot sdot + 119887119908119904 + 119887119908+1

119904119899+ 1198861119904119899minus1


119904 + 119886119899

=

sum119908+1

119895=1119887119895119904119908+1minus119895

119904119899+ sum119899

119894=1119886119894119904119899minus119894

=

119873 (119904)

119872 (119904)

(3)

where 1198871 119887

119908+1 1198861 and 119886




119899= 0 and




119906 (119904) = 119866119888(119904) 119890 (119904) 119866

119888(119904) =

(119896119901119904 + 119896119894+ 1198961198891199042)

119904

119903 (119904) minus 119910 (119904) ≜ 119890 (119904) 119910 (119904) = 119866119901(119904) 119906 (119904)

(4)



and 119896119901 119896119894 and 119896



119903 (119904) = 119910 (119904) + 119890 (119904) 119910 (119904) =

119873 (119904) 119906 (119904)

119872 (119904)

119906 (119904) =

(119896119901119904 + 119896119894+ 1198961198891199042) 119890 (119904)

119904

(5)


119903 (119904)

=

1

(1 + 119866119901(119904) 119866119888(119904))

lArrrArr 119863(119904) 119890 (119904)

= 119904119872 (119904) 119903 (119904)

119863 (119904) = 119904119872 (119904) + 119873 (119904) (119896119901119904 + 119896119894+ 1198961198891199042)

deg 119863 (119904) = 119899 + 1

(6)


(120575119872 (120575) + 119873 (120575) (119896119901120575 + 119896119894+ 1198961198891205752)) 119890 (119905) = 120575119872 (120575) 119903 (119905)

(7)


(120575119899+1

+

119899

sum

119894=1

119886119894120575119899+1minus119894

)

+ (119896119901120575 + 119896119894+ 1198961198891205752)

119908+1

sum

119895=1

119887119895120575119908+1minus119895

119890 (119905) = 0

(8)


120575119899+1

+

119899

sum

119894=1

119886119894120575119899+1minus119894

+

119908+1

sum

119895=1

119887119895119896119901120575119908+2minus119895

+

119908+1

sum

119895=1

119887119895119896119894120575119908+1minus119895

+

119908+1

sum

119895=1

119887119895119896119889120575119908+3minus119895

119890 (119905) = 0

(9)


119890(119905)(119899+1)

+ 1198861119890(119905)(119899)

+ 1198862119890(119905)(119899minus1)


119890(119905)(119908+3)

+ (119886119899minus119908minus1

+ 1198871119896119889) 119890(119905)(119908+2)

+ (119886119899minus119908

+ 1198871119896119901+ 1198872119896119889) 119890(119905)(119908+1)

+

119908minus1

sum

119895=1

(119886119899minus119908+119895

+ 119887119895+1

119896119901+ 119887119895119896119894+ 119887119895+2

119896119889) 119890(119905)(119908+1minus119895)

+ (119886119899+ 119887119908+1

119896119901+ 119887119908119896119894) 119890 (119905) + 119887

119908+1119896119894119890 (119905) = 0

(10)



119890(119905)(119899+1)

+ 1198861119890(119905)(119899)

+ 1198862119890(119905)(119899minus1)


119890(119905)(119908+3)

+ 1198981119890(119905)(119908+2)

+ 1198982119890(119905)(119908+1)

+

119908minus1

sum

119895=1

119898119895+2

119890(119905)(119908+1minus119895)

+ 119898119908+2

119890 (119905) + 119898119908+3

119890 (119905) = 0

(11)



1198981= (119886119899minus119908minus1

+ 1198961198891198871) 119898

2= (119886119899minus119908

+ 1198871119896119901+ 1198872119896119889)

119898119895+2

= (119886119899minus119908+119895

+ 119887119895+1

119896119901+ 119887119895119896119894+ 119887119895+2

119896119889)

(119895 = 1 2 119908 minus 1)

119898119908+2

= (119886119899+ 119887119908+1

119896119901+ 119887119908119896119894) 119898

119908+3= 119887119908+1

119896119894

(12)


= 119860119909 (13)






119860 = (

119874119899times1

119868119899times119899

minus119898119908+3

1198621times119899

) 119909 = (119890 (119905) 119890 (119905) 119890 (119905) sdot sdot sdot 119890(119905)(119899)

)

119879

1198621times119899

= (minus119898119908+2



(14)




119901 119896lowast

119894 119896lowast

119889) = min119866lowast

119888(119904)

119869 (1198981 119898

119908+3)

= minint

infin

0

119909119879(119905) 119876119909 (119905) 119889119905

= min

119899

sum

119894=0

int

infin

0

119902119894+1119894+1

(119890(119905)(119894))

2

119889119905

subject to (st)

(1) = 119860119909


(3) 120576 ge 0 11987711

(1198981 119898

119908+3) gt 120576

11987721

(1198981 119898

119908+3) gt 120576

119877119899+11

(1198981 119898

119908+3) gt 120576 119877

119899+21(1198981 119898

119908+3) gt 120576

(4)1198984= 1198911(1198981 1198982 1198983) 119898

119908+3= 119891119908(1198981 1198982 1198983)

(5) 0 lt 119896119901le 1205910

119901 0 lt 119896

119894le 1205910

119894 0 lt 119896

119889le 1205910

119889

(15)


1 119891

119908is the


119889 and

119896lowast

119901 119896lowast119894 and 119896

lowast




119894(119894 = 1 2 119908+3) the


0

119901 1205910119894 and 120591

0


parameters








Command

119903(119905) +

+

minusminus



119910(119905)

Output119863119906(119905)

119866lowast119888 (119904) = 119896

lowast119901 +

119896lowast119894119904+ 119896lowast119889119904



Control valve

ValveWater tank

Tank-1

Tank-2



119876119894119888minus 1198761119888

minus 11987612

= 1198621

119889ℎ1

119889119905

(16)



1 11987612




11987612

minus 1198762119888

= 1198622

119889ℎ2

119889119905

(17)





1198761119888

=

ℎ1

1198771

1198762119888

=

ℎ2

1198772

11987612

=

ℎ1minus ℎ2

11987712

(18)

where1198771and119877




119866ℎ(119904) =

1198671(119904)

119876119894119888(119904)

=

1198871119904 + 1198872

11988601199042+ 1198861119904 + 1198862

=

11988711199041198860+ 11988721198860

1199042+ 11988611199041198860+ 11988621198860

=

1198731(119904)

1198721(119904)

(19)



Tank-1 Tank-2

Control valve

1198771

119876121198671 + ℎ1

11986221198621

1198772

11987621198881198761119888

1198760 + 119876119894119888

11987712

1198672 + ℎ2Valve-3

Valv

e-2

Valv

e-1



1198871= 11987711198771211987721198622 119887

2= 1198771(11987712

+ 1198772)

1198860= 119877111986211198771211987721198622

1198861= 119877111986211198772+ 1198771119862111987712

+ 119877111987721198622+ 1198771211987721198622

(20)

1198862= 1198771+ 11987712

+ 1198772 (21)


119866119901(119904) =

11988711199042+ 1198872119904 + 1198873

11988601199043+ 11988611199042+ 1198862119904 + 1198863

=

119873 (119904)

119872 (119904)

119866119888(119904) = 119896

119901+

119896119894

119904

+ 119896119889119904

(22)


119866 (119904) = 119866119888(119904) 119866119901(119904) =

(119896119901119904 + 119896119894+ 1198961198891199042)119873 (119904)

119904119872 (119904)

(23)


119890 (119904)

119903 (119904)

=

1

(1 + 119866119901(119904) 119866119888(119904))

lArrrArr 119863(119904) 119890 (119904) = 119904119872 (119904) 119903 (119904)

119863 (119904) = (119904119872 (119904) + 119873 (119904) (119896119901119904 + 119896119894+ 1198961198891199042))

deg 119863 (119904) = 4

(24)


(120575119872 (120575) + 119873 (120575) (119896119901120575 + 119896119894+ 1198961198891205752)) 119890 (119905) = 120575119872 (120575) 119903 (119905)

(25)




(1198860+ 1198961198891198871) 119890(119905)(4)

+ (1198861+ 1198961199011198871+ 1198961198891198872)119890 (119905)

+ (1198862+ 1198961198941198871+ 1198961199011198872+ 1198961198891198873) 119890 (119905)

+ (1198863+ 1198961198941198872+ 1198961199011198873) 119890 (119905) + 119896

1198941198873119890 (119905) = 0

(26)


1198981119890(119905)(4)

+ 1198982

119890 (119905) + 119898

3119890 (119905) + 119898

4119890 (119905) + 119898

5119890 (119905) = 0 (27)



1198981= 1198860+ 1198961198891198871 119898

2= 1198861+ 1198961199011198871+ 1198961198891198872 119898

5= 1198961198941198873

1198983= 1198862+ 1198961198941198871+ 1198961199011198872+ 1198961198891198873 119898

4= 1198863+ 1198961198941198872+ 1198961199011198873

(28)


= 119860119909

119909 = (119890 (119905) 119890 (119905) 119890 (119905)

119890 (119905))119879 119860 = (

1198743times1

1198683times3

minus1198981

41198621times3

)

1198621times3

= (minus1198981

3minus 1198981

2minus 1198981

1)

(29)



119894= 119898119894+1

1198981(119894 = 1 2 3 4)



119901 119896lowast

119894 119896lowast

119889) = min119866lowast

119888

119869 (1198981 119898

5)

= minint

infin

0

119909119879(119905) 119876119909 (119905) 119889119905

= min

3

sum

119894=0

int

infin

0

119902119894+1119894+1

(119890(119905)(119894))

2

119889119905

st

(1) = 119860119909


(3) 120576 ge 0 11987711

(1198981 119898

119908+3) gt 120576

11987721

(1198981 119898

119908+3) gt 120576

11987731

(1198981 119898

119908+3) gt 120576 119877

41(1198981 119898

119908+3) gt 120576

11987731

(1198981 119898

119908+3) gt 120576 119877

41(1198981 119898

119908+3) gt 120576

11987751

(1198981 119898

119908+3) gt 120576

(4)1198984= 1198911(1198981 1198982 1198983) 1198985= 1198912(1198981 1198982 1198983)

(5) 0 lt 119896119901le 1205910

119901 0 lt 119896

119894le 1205910

119894 0 lt 119896

119889le 1205910

119889

(30)


4+

1198981

11199043+ 1198981

21199042+ 1198981

3119904 + 119898

1




lArrrArr 1198981

1gt 0 119898

1

11198981

2minus 1198981

3gt 0

1198981

11198981

21198981

3minus (1198981

3)

2

minus 1198981

4(1198981

1)

2

gt 0 1198981

4gt 0

(31)




120576 ge 0 11987711

(1198981 119898

5) gt 120576 119877

21(1198981 119898

5) gt 120576

11987731

(1198981 119898

5) gt 120576

11987741

(1198981 119898

5) gt 120576 119877

51(1198981 119898

5) gt 120576

lArrrArr 1198981gt 120576 119898

2gt 120576 119898

5gt 120576

11989811198984minus 11989821198983gt 1205761198982

(1198982)2

1198985minus 1198981(1198984)2

minus 119898211989831198984gt 120576 (119898

11198984minus 11989821198983)

(32)



infin

0

119909119879119876119909119889119905 = min119909

119879(0) 119875119909 (0)

119860119879119875 + 119875119860 = minus119876

(33)





119901 119896lowast

119894 119896lowast

119889) = min119866lowast

119888(119904)

119869 (1198981 119898

5)

= min119909(0)119879119875119909 (0)

st

(1) 119860119879119875 + 119875119860 = minus119876

(2)1198981

1gt 0 119898

1

4gt 0 119898

1

11198981

2minus 1198981

3gt 0

1198981

11198981

21198981

3minus (1198981

3)

2

minus 1198981

4(1198981

1)

2

gt 0

(3)1198981gt 120576 119898

2gt 120576 119898

5gt 120576 119898

11198984minus 11989821198983gt 1205761198982

(1198982)2

1198985minus 1198981(1198984)2

minus 119898211989831198984gt 120576 (119898

11198984minus 11989821198983)

(4)1198984= 1198911(1198981 1198982 1198983) 1198985= 1198912(1198981 1198982 1198983)

(5) 0 lt 119896119901le 1205910

119901 0 lt 119896

119894le 1205910

119894 0 lt 119896

119889le 1205910

119889

(34)


1(0) = 1 119909

2(0) = 0 119909

3(0) = 0 and




119894and


119866lowast

119888(119904) = 119896

lowast

119901+

119896lowast

119894

119904

+ 119896lowast

119889119904 119896

lowast

119901=

119898lowast


119894

1198873

119896lowast

119894=

119898lowast

5

1198873

119896lowast

119889=

119898lowast

1minus 1198860

1198871

(35)



1119904119899+sdot sdot sdot+119886

119899minus119908minus2119904119908+3

+

1198981119904119908+2

+ 1198982119904119908+1

+ sdot sdot sdot + 119898119908+2

119904 + 119898119908+3




119889 The

parameters 119896119901 119896119894 and 119896


119901]

(0 1205910

119894] and (0 120591

0



119901 119896119894



119865 (119904) = 119904119899+ 1199011119904119899minus1

+ 1199012119904119899minus2


119904 + 119901119899 (36)

where 1199011 1199012 and 119901





119898

1199011= 1198911(1199021 119902

119898) 119901

119899= 119891119899(1199021 119902

119898)

Ω119902= (1199021 119902

119898)1003816100381610038161003816[119902minus

1 119902+


minus

119898 119902+

119898]

1199021isin [119902minus

1 119902+

1] 119902

119898isin [119902minus

119898 119902+

119898]

(37)


119894and 119902+

119894are the




11987711

(1199021 119902

119898) = 1198911199031

(1199011 119901

119899) gt 120576

119877119899+11

(1199021 119902

119898) = 119891119903119899+1

(1199011 119901

119899) gt 120576

Ω119903120576

= (1199021 119902

119898) | 1198911199031

(1199011 119901

119899) gt 120576

119891119903119899+1

(1199011 119901

119899) gt 120576 (120576 ge 0)

(38)







min(1199021 119902119898)isinΩ119891

1198911199031

(1199011 119901

119899) 119891

119903119899+1(1199011 119901

119899) gt 0

(Ω119891= (Ω119903120576

⋂Ω119902) = 0)

(39)







I Ω1= (119902

1 119902

119898) | 1198911199031

(1199011 119901

119899) gt 1205761

119891119903119899+1

(1199011 119901

119899) gt 1205761

II Ω2= (119902

1 119902

119898) | 1198911199031

(1199011 119901

119899) gt 1205762

119891119903119899+1

(1199011 119901

119899) gt 1205762

(40)




Ω1

1= (1199021 119902

119898) | 1198911199031

(1199011 119901

119899) gt 1205761

Ω1

2= (1199021 119902

119898) | 1198911199031

(1199011 119901

119899) gt 1205762

(41)



1

2will have the

relationship

exist119872lowast

1= (119902lowast

11 119902lowast

21 119902

lowast

1198981) 119891

1199031(119872lowast

1) = 120576lowast

2gt 1205762 (42)


1will have the

relationship

exist119872lowast

2= (119902lowast

12 119902lowast

22 119902

lowast

1198982) 119891

1199031(119872lowast

2) = 120576lowast

1gt 1205761 (43)


1which will



1= (119902lowast

11 119902lowast

21 119902

lowast

1198981) 1205761lt 1198911199031

(119873lowast

1) lt 1205762

lArrrArr 1198911199031

(119873lowast

1) isin (1205761 1205762)


1isin Ω1

1 119873lowast

1notin Ω1

2997904rArr Ω

1

1sup Ω1

2

Case 2 1198911199031

(119873lowast

1) notin (1205761 1205762) 997904rArr Ω

1

1= Ω1

2

(44)


119896


relationship

Case 1 Ω119896

1sup Ω119896

2 (forall119896 = 1 2 119899 119899 + 1)

Case 2 Ω119896

1= Ω119896

2 (forall119896 = 1 2 119899 119899 + 1)

Ω119896

1= (1199021 119902

119898) | 119891119903119896

(1199011 119901

119899) gt 1205761

Ω119896

2= (1199021 119902

119898) | 119891119903119896

(1199011 119901

119899) gt 1205762

(45)


2will have follow-

ing relationships


1sup Ω2) cup (Ω

1= Ω2)

Ω1=

119899+1

⋂

119895=1

Ω119895

1 Ω

2=

119899+1

⋂

119895=1

Ω119895

2

(46)


119905+1gt 120576119905gt 120576119905minus1



Ω119905= (1199021 119902

119898) | 1198911199031

(1199011 119901

119899) gt 120576119905

119891119903119899+1

(1199011 119901

119899) gt 120576119905



119899+1

119905) = (

119899+1

⋂

119895=1

Ω119895

119905)

(119905 = 1 2 3 119905 isin Z+)

Ω119896

119905= (1199021 119902

119898) | 119891119903119896

(1199011 119901

119899) gt 120576119905

(119896 = 1 2 119899 119899 + 1)

(47)


ship

Ω119905+1


2sube Ω1



2Ω2Δ1Ω1)

⋃ (Ω119905+1


2= Ω1)


(48)


119905 then the



(1199021 119902

119898) | 1198911199031

(1199011 119901

119899) gt 120576119905

119891119903119899+1

(1199011 119901

119899) gt 120576119905


Ω119905= lim120576119905rarr+infin

(Ω1

119905cap Ω2


119899+1

119905)


(

119899+1

⋂

119895=1

Ω119895

119905) = Ω

lowast

= (119902lowast

10119902lowast

20 119902

lowast

1198980)



(1199021 119902

119898) | 1198911199031

(1199011 119901

119899) gt 120576119905

119891119903119899+1

(1199011 119901

119899) gt 120576119905


Ω119905= lim120576119905rarr+infin

(Ω1

119905cap Ω2


119899+1

119905)


(

119899+1

⋂

119895=1

Ω119895

119905) = 0

(49)





120576119905997888rarr +infin lim



lowast

119905 120576119905gt 120576lowast

119905 Ω119905= 0

0 lt 120576119905le 120576lowast

119905 Ω119905

= 0

120576lowast

119905gt 120576119905gt 120576119905minus1


Ωlowast

119905sub Ω119905sub Ω119905minus1


(50)


= Ω119905= Ω119905minus1

=



119875 Ω119905+1

= Ω119905= Ω119905minus1


lArrrArr 119875 Ω119905+1

= Ω119905 Ω119905= Ω119905minus1

Ω2= Ω1

(51)


119875 Ω119905+1

= Ω119905 Ω119905= Ω119905minus1

Ω2= Ω1

= 119875 Ω119905+1

= Ω119905 times 119875 Ω

119905= Ω119905minus1


=

119905+1

prod

119895=2

119875 Ω119895= Ω119895minus1

= (119875 Ω2= Ω1)119905

(119905 = 1 2 119905 isin Z+)

(52)



= 119875 (Ω1


119899+1

1) = (Ω

1


119899+1

2)

= 119875

(

119899+1

⋂

119895=1

Ω119895

1) = (

119899+1

⋂

119895=1

Ω119895

2)

lArrrArr 119875(Ω1


119899+1

2) sub Ω

119896

1

(Ω1


119899+1

1) sub Ω

119896

2

(forall119896 = 1 2 119899 119899 + 1)

(53)


2=

Ω1 will yield

119875 (Ω1

2cap Ω2


119899

2) sub Ω

1

1 = 1205821

119875 (Ω1

2cap Ω2


119899

2) sub Ω

119899+1

1 = 1205821

119875 (Ω1

1cap Ω2


119899

1) sub Ω

1

2 = 1205822

119875 (Ω1

1cap Ω2


119899

1) sub Ω

119899+1

2 = 1205822

119875 Ω1= Ω2

= 119875

(

119899+1

⋂

119895=1

Ω119895

2) sub Ω

119896

1 (forall119896 = 1 2 119899 119899 + 1)

times 119875

(

119899+1

⋂

119895=1

Ω119895

1) sub Ω

119896

2 (forall119896 = 1 2 119899 119899 + 1)

=

119899+1

prod

119896=1

119875

(

119899+1

⋂

119895=1

Ω119895

2) sub Ω

119896

1

times

119899+1

prod

119896=1

119875

(

119899+1

⋂

119895=1

Ω119895

1) sub Ω

119896

2

= 120582119899+1

1120582119899+1

2 (0 lt 120582

1lt 1 0 lt 120582

2lt 1)

(54)


119875 Ω119905+1

= Ω119905 Ω119905= Ω119905minus1

Ω2= Ω1 = 120582(119899+1)119905

1120582(119899+1)119905

2

119875 Ω119905+1

Δ119905Ω119905Δ119905minus1

Ω119905minus1


(119899+1)119905

1120582(119899+1)119905

2

(55)


min 119891 (119909)

st ℎ (119909) = 0 119892 (119909) le 0

(56)


where 119891(119909) 119877119899

rarr 119877 ℎ(119909) 119877119899

rarr 119877119898 and 119892(119909)

119877119899



min 119891 (119909) minus 120592

119902

sum

119894=1

ln 120575119894

st ℎ (119909) = 0 119892 (119909) + 120575 = 0

(57)


(1205751 1205752 120575

119902)119879


119871 (119909 119910 119911 120575) = 119891 (119909) minus 120592

119902

sum

119894=1

ln 120575119894+ 119910119879(119892 (119909) + 120575) + 119911

119879ℎ (119909)

(58)


119902)119879

and z = (1199111 1199112 119911




119879

119910 + (nablaℎ (119909))119879119911 = 0

nabla120575119871 (119909 119910 119911 120575) = minus120592119878

minus1

120575119890 + 119910 = 0 lArrrArr minus120592119890 + 119878

120575119884119890 = 0

nabla119910119871 (119909 119910 119911 120575) = 119892 (119909) + 120575 = 0

nabla119911119871 (119909 119910 119911 120575) = ℎ (119909) = 0

(59)

where 119878120575

= diag(1205751 1205752 120575



1 1199102 119910



Δ119909 120575+Δ120575 119911+Δ



(nabla2119891 (119909) + nabla

2119892(119909)119879119910 + nabla

2ℎ(119909)119879119911) Δ119909 + nabla119892(119909)

119879Δ119910


119879119910 + nablaℎ(119909)

119879119911) = 0

minus120592119890 + 119878120575119884119890 + 119878

120575Δ119910 + 119884Δ120575 = 0

119892 (119909) + nabla119892 (119909) Δ119909 + 120575 + Δ120575 = 0

nablaℎ (119909) Δ119909 + ℎ (119909) = 0

(60)


(

119867(119909 119910 119911) 0 nablaℎ(119909)119879

nabla119892(119909)119879

0 119878minus1

120575119884 0 119868

nablaℎ (119909) 0 0 0

nabla119892 (119909) 119868 0 0

)(

Δ119909

Δ120575

Δ119910

Δ119911

)

= (


119879119911

120592119878minus1

120575119890 minus 119910

minusℎ (119909)

minus119892 (119909) minus 120575

)

(61)

119867(119909 119910 119911) = nabla2119891 (119909) + nabla

2119892(119909)119879119910 + nabla

2ℎ(119909)119879119911 (62)


(119909 120575 119911 119910) larr997888 (119909 120575 119911 119910) + 1205771(Δ119909 Δ120575 Δ119911 Δ119910) (63)


1holds the




(0)gt 0 119896 = 0










(119896+1) 120575(119896+1)

119911(119896+1)

119910(119896+1)

) larr (119909(119896)

120575(119896)

119911(119896)

119910(119896)

) +

1205771(Δ119909 Δ120575 Δ119911 Δ119910) 119896 larr 119896 + 1


Step 7 End



1199011(119904) [7]

1198661199012(119904) [15] and 119866

1199013(119904) [6] respectively 119866

1199011(119904) is a third-


1199013(119904) are the SOPTD


1198661199011

(119904) =

15

(1199042+ 09119904 + 5) (119904 + 3)

1198661199012

(119904) =

119890minus05119904

(119904 + 1)2 119866

1199013(119904) =

4119890minus10119904

10119904 + 1

(64)


and 119890minus10119904

asymp (1199042minus 06119904 + 012)(119904

2+ 06119904 + 012) respectively


1199013(119904) are the (minus119904 +

4)(1199043+ 61199042+ 9119904 + 4) and (04119904

2minus 024119904 + 0048)(119904

3+ 07119904

2+



1 1199022 1199023 and 119902

4 The



2 1199023


1(with 119902

2 1199023 and 119902

4


2 1199023 and 119902

4(with 119902




1

















119901119896lowast

119894119896lowast

119889

Plant1

diag (5 5 1 1) 013121 058138 058655

120576 = 0


Plant2

diag (02 1 1 1) 115698 052966 086781

120576 = 05

diag (02 1 1 02) 216962 084500 182140diag (02 2 2 05) 178668 058700 159290

diag (01 08 08 02) 185901 064942 162251diag (04 05 5 1) 122806 062168 142095

Plant3

diag (004 27 158 175) 032447 001537 145256

120576 = 0




1119902211990231199024) 119896

lowast

119901119896lowast

119894119896lowast

119889119905119903(sec) 119905

119889(sec) 119905

119901(sec) 119905

119904(sec) 119872

119901()


0 2 4 6 8 10 12 14

0

02

04

06

08

1

12

Syste

m re

spon

se

minus02

Time (s)

119876 = diag (5 5 1 1)119876 = diag (5 2 1 1)119876 = diag (10 2 1 1)

119876 = diag (10 10 1 1)119876 = diag (5 5 2 1) lowast 2

(a)

0 2 4 6 8 10 12 14

0

02

04

06

08

1

12

Con

trol e

rror

minus02

Time (s)

119876 = diag (5 5 1 1)119876 = diag (5 2 1 1)119876 = diag (10 2 1 1)

119876 = diag (10 10 1 1)119876 = diag (5 5 2 1) lowast 2

(b)



0 1 2 3 4 5 6 7 8 9 100

02

04

06

08

1

12

14

Time (s)

Syste

m re

spon

se

120576 = 05

120576 = 25

120576 = 3

120576 = 4

120576 = 5

(a)

0 1 2 3 4 5 6 7 8 9 10

0

02

04

06

08

1

Time (s)

minus02

minus04

Con

trol e

rror

120576 = 05

120576 = 25

120576 = 3

120576 = 4

120576 = 5

(b)


0 5 10 15 20 25 30

0

02

04

06

08

1

12



Syste

m re

spon

se

Time (s)

minus02

(a)

0 5 10 15 20 25 30

0

02

04

06

08

1

12



Con

trol e

rror

Time (s)

minus02

(b)




119896119894

119896119889

Risetime

119905119903(sec)

Delaytime

119905119889(sec)

Peaktime

119905119901(sec)

Settingtime

119905119904(sec)


Plants



0 2 4 6 8 10 12 14 16 180

05

1

15Sy

stem

resp

onse



Time (s)

(a)

0 2 4 6 8 10 12 14 16 18

0

05

1


minus05


Con

trol e

rror

Time (s)

(b)


0 20 40 60 80 100 120 140 160 180 2000

02

04

06

08

1

12

14



Syste

m re

spon

se

Time (s)

095

105

(a)

0 20 40 60 80 100 120 140 160 180 200

0

02

04

06

08

1



Con

trol e

rror

Time (s)

minus02

minus04

(b)





Time (s)0 5 10 15 20 25 30

0

02

04

06

08

1

12

0 5 10 15 20 25 30

0

02

04

06

08

1

12

Syste

m re

spon

seSy

stem

resp

onse

Con

trol e

rror

Con

trol e

rror

Time (s)

Time (s)Time (s)16 18 20 22 24 26 28

075

08

085

09

095

1

105

11



16 18 20 22 24 26 28

0

005

01

015

02

025

03



minus02 minus02

minus005

119863119906(119905) = 03119906(119905 minus 16)



Methods 119896119901

119896119894

119896119889










16 18 20 22 24 26 28 30086

088

09

092

094

096

098

1

102

104





0 5 10 15 20 25 300

05

1

15

0 5 10 15 20 25 30

0

05

1

18 20 22 24 26 28 30

0

002

004

006

008

01

012

014

Con

trol e

rror

Con

trol e

rror

Syste

m re

spon

seSy

stem

resp

onse

minus05

Time (s) Time (s)

Time (s) Time (s)

minus002

119863119906(119905) = 03119906(119905 minus 18)






200 250 300 350 40007

075

08

085

09

095

1

105

11

200 250 300 350 400

0

005

01

015

02

025

03

0 50 100 150 200 250 300 350 4000

02

04

06

08

1

12

14

0 50 100 150 200 250 300 350 400

0

02

04

06

08

1

095

105

Time (s) Time (s)

Time (s)Time (s)





minus005

Con

trol e

rror

Syste

m re

spon

se

Con

trol e

rror

Syste

m re

spon

se

minus02

minus04

119863119906(119905) = 01119906(119905 minus 180)




1(119904)119876119894119888(119904) = (04465119904 +





119901=

10000 119896119894

= 032549 119896119889

= 0001 120576 = 0 119896119901

=

10000 119896119894

= 010118 119896119889

= 0001 120576 = 0 119896119901

=

10000 119896119894= 022986 119896

119889= 0001 120576 = 0 119896

119901= 6000 119896

119894=



1is set at




0 5 10 15 20 25 300

02

04

06

08

1

12

14

Time (s)

16 18 20 22 24 26 28 30086

088

09

092

094

096

098

1

102

104

0 5 10 15 20 25 30

0

02

04

06

08

1

Time (s)

Time (s) Time (s)

16 18 20 22 24 26 28 30

0

002

004

006

008

01

012

014

minus02

minus04

minus002

120576 = 05

120576 = 25

120576 = 3

120576 = 4

120576 = 5

120576 = 05

120576 = 25

120576 = 3

120576 = 4

120576 = 5



1 the solid







0 200 400 600 800 1000 1200 1400 1600 1800 20000

50

100

150

200

250

300

Time (s)

Liqu

id le

vel o

f CW

TS (m

m)




lowast

119901= 10000 119896lowast

119894= 032549 119896lowast

119889= 0001 and 120576 = 0 in

set point 300mm


119901= 10000 119896lowast

119894= 010118 119896lowast

119889= 0001 and 120576 = 0 in

set point 300mm


119901= 10000 119896lowast

119894= 022986 119896lowast

119889= 0001 and 120576 = 0 in

set point 300mm


lowast

119901= 6000 119896lowast

119894= 013817 119896lowast

119889= 0001 and 120576 = 0 in

set point 300mm


5 Conclusions




119901= 10000 119896lowast

119894= 032549 119896lowast

119889= 0001 and 120576 = 0 in

set point 160mm


119901= 10000 119896lowast

119894= 010118 119896lowast

119889= 0001 and 120576 = 0 in

set point 160mm


1 1199022 1199023 and 119902





2 1199023 and 119902

4(with 119902





4(with 119902

1fixed) Through the


1 1199022 119902

119899+1 The 120576-Routh




119901= 10000 119896lowast

119894= 022986 119896lowast

119889= 0001 and 120576 = 0 in

set point 160mm


lowast

119901= 6000 119896lowast

119894= 013817 119896lowast

119889= 0001 and 120576 = 0 in

set point 160mm



Acknowledgments



References











































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119890(119905)(119899+1)

+ 1198861119890(119905)(119899)

+ 1198862119890(119905)(119899minus1)


119890(119905)(119908+3)

+ 1198981119890(119905)(119908+2)

+ 1198982119890(119905)(119908+1)

+

119908minus1

sum

119895=1

119898119895+2

119890(119905)(119908+1minus119895)

+ 119898119908+2

119890 (119905) + 119898119908+3

119890 (119905) = 0

(11)



1198981= (119886119899minus119908minus1

+ 1198961198891198871) 119898

2= (119886119899minus119908

+ 1198871119896119901+ 1198872119896119889)

119898119895+2

= (119886119899minus119908+119895

+ 119887119895+1

119896119901+ 119887119895119896119894+ 119887119895+2

119896119889)

(119895 = 1 2 119908 minus 1)

119898119908+2

= (119886119899+ 119887119908+1

119896119901+ 119887119908119896119894) 119898

119908+3= 119887119908+1

119896119894

(12)


= 119860119909 (13)






119860 = (

119874119899times1

119868119899times119899

minus119898119908+3

1198621times119899

) 119909 = (119890 (119905) 119890 (119905) 119890 (119905) sdot sdot sdot 119890(119905)(119899)

)

119879

1198621times119899

= (minus119898119908+2



(14)




119901 119896lowast

119894 119896lowast

119889) = min119866lowast

119888(119904)

119869 (1198981 119898

119908+3)

= minint

infin

0

119909119879(119905) 119876119909 (119905) 119889119905

= min

119899

sum

119894=0

int

infin

0

119902119894+1119894+1

(119890(119905)(119894))

2

119889119905

subject to (st)

(1) = 119860119909


(3) 120576 ge 0 11987711

(1198981 119898

119908+3) gt 120576

11987721

(1198981 119898

119908+3) gt 120576

119877119899+11

(1198981 119898

119908+3) gt 120576 119877

119899+21(1198981 119898

119908+3) gt 120576

(4)1198984= 1198911(1198981 1198982 1198983) 119898

119908+3= 119891119908(1198981 1198982 1198983)

(5) 0 lt 119896119901le 1205910

119901 0 lt 119896

119894le 1205910

119894 0 lt 119896

119889le 1205910

119889

(15)


1 119891

119908is the


119889 and

119896lowast

119901 119896lowast119894 and 119896

lowast




119894(119894 = 1 2 119908+3) the


0

119901 1205910119894 and 120591

0


parameters








Command

119903(119905) +

+

minusminus



119910(119905)

Output119863119906(119905)

119866lowast119888 (119904) = 119896

lowast119901 +

119896lowast119894119904+ 119896lowast119889119904



Control valve

ValveWater tank

Tank-1

Tank-2



119876119894119888minus 1198761119888

minus 11987612

= 1198621

119889ℎ1

119889119905

(16)



1 11987612




11987612

minus 1198762119888

= 1198622

119889ℎ2

119889119905

(17)





1198761119888

=

ℎ1

1198771

1198762119888

=

ℎ2

1198772

11987612

=

ℎ1minus ℎ2

11987712

(18)

where1198771and119877




119866ℎ(119904) =

1198671(119904)

119876119894119888(119904)

=

1198871119904 + 1198872

11988601199042+ 1198861119904 + 1198862

=

11988711199041198860+ 11988721198860

1199042+ 11988611199041198860+ 11988621198860

=

1198731(119904)

1198721(119904)

(19)



Tank-1 Tank-2

Control valve

1198771

119876121198671 + ℎ1

11986221198621

1198772

11987621198881198761119888

1198760 + 119876119894119888

11987712

1198672 + ℎ2Valve-3

Valv

e-2

Valv

e-1



1198871= 11987711198771211987721198622 119887

2= 1198771(11987712

+ 1198772)

1198860= 119877111986211198771211987721198622

1198861= 119877111986211198772+ 1198771119862111987712

+ 119877111987721198622+ 1198771211987721198622

(20)

1198862= 1198771+ 11987712

+ 1198772 (21)


119866119901(119904) =

11988711199042+ 1198872119904 + 1198873

11988601199043+ 11988611199042+ 1198862119904 + 1198863

=

119873 (119904)

119872 (119904)

119866119888(119904) = 119896

119901+

119896119894

119904

+ 119896119889119904

(22)


119866 (119904) = 119866119888(119904) 119866119901(119904) =

(119896119901119904 + 119896119894+ 1198961198891199042)119873 (119904)

119904119872 (119904)

(23)


119890 (119904)

119903 (119904)

=

1

(1 + 119866119901(119904) 119866119888(119904))

lArrrArr 119863(119904) 119890 (119904) = 119904119872 (119904) 119903 (119904)

119863 (119904) = (119904119872 (119904) + 119873 (119904) (119896119901119904 + 119896119894+ 1198961198891199042))

deg 119863 (119904) = 4

(24)


(120575119872 (120575) + 119873 (120575) (119896119901120575 + 119896119894+ 1198961198891205752)) 119890 (119905) = 120575119872 (120575) 119903 (119905)

(25)




(1198860+ 1198961198891198871) 119890(119905)(4)

+ (1198861+ 1198961199011198871+ 1198961198891198872)119890 (119905)

+ (1198862+ 1198961198941198871+ 1198961199011198872+ 1198961198891198873) 119890 (119905)

+ (1198863+ 1198961198941198872+ 1198961199011198873) 119890 (119905) + 119896

1198941198873119890 (119905) = 0

(26)


1198981119890(119905)(4)

+ 1198982

119890 (119905) + 119898

3119890 (119905) + 119898

4119890 (119905) + 119898

5119890 (119905) = 0 (27)



1198981= 1198860+ 1198961198891198871 119898

2= 1198861+ 1198961199011198871+ 1198961198891198872 119898

5= 1198961198941198873

1198983= 1198862+ 1198961198941198871+ 1198961199011198872+ 1198961198891198873 119898

4= 1198863+ 1198961198941198872+ 1198961199011198873

(28)


= 119860119909

119909 = (119890 (119905) 119890 (119905) 119890 (119905)

119890 (119905))119879 119860 = (

1198743times1

1198683times3

minus1198981

41198621times3

)

1198621times3

= (minus1198981

3minus 1198981

2minus 1198981

1)

(29)



119894= 119898119894+1

1198981(119894 = 1 2 3 4)



119901 119896lowast

119894 119896lowast

119889) = min119866lowast

119888

119869 (1198981 119898

5)

= minint

infin

0

119909119879(119905) 119876119909 (119905) 119889119905

= min

3

sum

119894=0

int

infin

0

119902119894+1119894+1

(119890(119905)(119894))

2

119889119905

st

(1) = 119860119909


(3) 120576 ge 0 11987711

(1198981 119898

119908+3) gt 120576

11987721

(1198981 119898

119908+3) gt 120576

11987731

(1198981 119898

119908+3) gt 120576 119877

41(1198981 119898

119908+3) gt 120576

11987731

(1198981 119898

119908+3) gt 120576 119877

41(1198981 119898

119908+3) gt 120576

11987751

(1198981 119898

119908+3) gt 120576

(4)1198984= 1198911(1198981 1198982 1198983) 1198985= 1198912(1198981 1198982 1198983)

(5) 0 lt 119896119901le 1205910

119901 0 lt 119896

119894le 1205910

119894 0 lt 119896

119889le 1205910

119889

(30)


4+

1198981

11199043+ 1198981

21199042+ 1198981

3119904 + 119898

1




lArrrArr 1198981

1gt 0 119898

1

11198981

2minus 1198981

3gt 0

1198981

11198981

21198981

3minus (1198981

3)

2

minus 1198981

4(1198981

1)

2

gt 0 1198981

4gt 0

(31)




120576 ge 0 11987711

(1198981 119898

5) gt 120576 119877

21(1198981 119898

5) gt 120576

11987731

(1198981 119898

5) gt 120576

11987741

(1198981 119898

5) gt 120576 119877

51(1198981 119898

5) gt 120576

lArrrArr 1198981gt 120576 119898

2gt 120576 119898

5gt 120576

11989811198984minus 11989821198983gt 1205761198982

(1198982)2

1198985minus 1198981(1198984)2

minus 119898211989831198984gt 120576 (119898

11198984minus 11989821198983)

(32)



infin

0

119909119879119876119909119889119905 = min119909

119879(0) 119875119909 (0)

119860119879119875 + 119875119860 = minus119876

(33)





119901 119896lowast

119894 119896lowast

119889) = min119866lowast

119888(119904)

119869 (1198981 119898

5)

= min119909(0)119879119875119909 (0)

st

(1) 119860119879119875 + 119875119860 = minus119876

(2)1198981

1gt 0 119898

1

4gt 0 119898

1

11198981

2minus 1198981

3gt 0

1198981

11198981

21198981

3minus (1198981

3)

2

minus 1198981

4(1198981

1)

2

gt 0

(3)1198981gt 120576 119898

2gt 120576 119898

5gt 120576 119898

11198984minus 11989821198983gt 1205761198982

(1198982)2

1198985minus 1198981(1198984)2

minus 119898211989831198984gt 120576 (119898

11198984minus 11989821198983)

(4)1198984= 1198911(1198981 1198982 1198983) 1198985= 1198912(1198981 1198982 1198983)

(5) 0 lt 119896119901le 1205910

119901 0 lt 119896

119894le 1205910

119894 0 lt 119896

119889le 1205910

119889

(34)


1(0) = 1 119909

2(0) = 0 119909

3(0) = 0 and




119894and


119866lowast

119888(119904) = 119896

lowast

119901+

119896lowast

119894

119904

+ 119896lowast

119889119904 119896

lowast

119901=

119898lowast


119894

1198873

119896lowast

119894=

119898lowast

5

1198873

119896lowast

119889=

119898lowast

1minus 1198860

1198871

(35)



1119904119899+sdot sdot sdot+119886

119899minus119908minus2119904119908+3

+

1198981119904119908+2

+ 1198982119904119908+1

+ sdot sdot sdot + 119898119908+2

119904 + 119898119908+3




119889 The

parameters 119896119901 119896119894 and 119896


119901]

(0 1205910

119894] and (0 120591

0



119901 119896119894



119865 (119904) = 119904119899+ 1199011119904119899minus1

+ 1199012119904119899minus2


119904 + 119901119899 (36)

where 1199011 1199012 and 119901





119898

1199011= 1198911(1199021 119902

119898) 119901

119899= 119891119899(1199021 119902

119898)

Ω119902= (1199021 119902

119898)1003816100381610038161003816[119902minus

1 119902+


minus

119898 119902+

119898]

1199021isin [119902minus

1 119902+

1] 119902

119898isin [119902minus

119898 119902+

119898]

(37)


119894and 119902+

119894are the




11987711

(1199021 119902

119898) = 1198911199031

(1199011 119901

119899) gt 120576

119877119899+11

(1199021 119902

119898) = 119891119903119899+1

(1199011 119901

119899) gt 120576

Ω119903120576

= (1199021 119902

119898) | 1198911199031

(1199011 119901

119899) gt 120576

119891119903119899+1

(1199011 119901

119899) gt 120576 (120576 ge 0)

(38)







min(1199021 119902119898)isinΩ119891

1198911199031

(1199011 119901

119899) 119891

119903119899+1(1199011 119901

119899) gt 0

(Ω119891= (Ω119903120576

⋂Ω119902) = 0)

(39)







I Ω1= (119902

1 119902

119898) | 1198911199031

(1199011 119901

119899) gt 1205761

119891119903119899+1

(1199011 119901

119899) gt 1205761

II Ω2= (119902

1 119902

119898) | 1198911199031

(1199011 119901

119899) gt 1205762

119891119903119899+1

(1199011 119901

119899) gt 1205762

(40)




Ω1

1= (1199021 119902

119898) | 1198911199031

(1199011 119901

119899) gt 1205761

Ω1

2= (1199021 119902

119898) | 1198911199031

(1199011 119901

119899) gt 1205762

(41)



1

2will have the

relationship

exist119872lowast

1= (119902lowast

11 119902lowast

21 119902

lowast

1198981) 119891

1199031(119872lowast

1) = 120576lowast

2gt 1205762 (42)


1will have the

relationship

exist119872lowast

2= (119902lowast

12 119902lowast

22 119902

lowast

1198982) 119891

1199031(119872lowast

2) = 120576lowast

1gt 1205761 (43)


1which will



1= (119902lowast

11 119902lowast

21 119902

lowast

1198981) 1205761lt 1198911199031

(119873lowast

1) lt 1205762

lArrrArr 1198911199031

(119873lowast

1) isin (1205761 1205762)


1isin Ω1

1 119873lowast

1notin Ω1

2997904rArr Ω

1

1sup Ω1

2

Case 2 1198911199031

(119873lowast

1) notin (1205761 1205762) 997904rArr Ω

1

1= Ω1

2

(44)


119896


relationship

Case 1 Ω119896

1sup Ω119896

2 (forall119896 = 1 2 119899 119899 + 1)

Case 2 Ω119896

1= Ω119896

2 (forall119896 = 1 2 119899 119899 + 1)

Ω119896

1= (1199021 119902

119898) | 119891119903119896

(1199011 119901

119899) gt 1205761

Ω119896

2= (1199021 119902

119898) | 119891119903119896

(1199011 119901

119899) gt 1205762

(45)


2will have follow-

ing relationships


1sup Ω2) cup (Ω

1= Ω2)

Ω1=

119899+1

⋂

119895=1

Ω119895

1 Ω

2=

119899+1

⋂

119895=1

Ω119895

2

(46)


119905+1gt 120576119905gt 120576119905minus1



Ω119905= (1199021 119902

119898) | 1198911199031

(1199011 119901

119899) gt 120576119905

119891119903119899+1

(1199011 119901

119899) gt 120576119905



119899+1

119905) = (

119899+1

⋂

119895=1

Ω119895

119905)

(119905 = 1 2 3 119905 isin Z+)

Ω119896

119905= (1199021 119902

119898) | 119891119903119896

(1199011 119901

119899) gt 120576119905

(119896 = 1 2 119899 119899 + 1)

(47)


ship

Ω119905+1


2sube Ω1



2Ω2Δ1Ω1)

⋃ (Ω119905+1


2= Ω1)


(48)


119905 then the



(1199021 119902

119898) | 1198911199031

(1199011 119901

119899) gt 120576119905

119891119903119899+1

(1199011 119901

119899) gt 120576119905


Ω119905= lim120576119905rarr+infin

(Ω1

119905cap Ω2


119899+1

119905)


(

119899+1

⋂

119895=1

Ω119895

119905) = Ω

lowast

= (119902lowast

10119902lowast

20 119902

lowast

1198980)



(1199021 119902

119898) | 1198911199031

(1199011 119901

119899) gt 120576119905

119891119903119899+1

(1199011 119901

119899) gt 120576119905


Ω119905= lim120576119905rarr+infin

(Ω1

119905cap Ω2


119899+1

119905)


(

119899+1

⋂

119895=1

Ω119895

119905) = 0

(49)





120576119905997888rarr +infin lim



lowast

119905 120576119905gt 120576lowast

119905 Ω119905= 0

0 lt 120576119905le 120576lowast

119905 Ω119905

= 0

120576lowast

119905gt 120576119905gt 120576119905minus1


Ωlowast

119905sub Ω119905sub Ω119905minus1


(50)


= Ω119905= Ω119905minus1

=



119875 Ω119905+1

= Ω119905= Ω119905minus1


lArrrArr 119875 Ω119905+1

= Ω119905 Ω119905= Ω119905minus1

Ω2= Ω1

(51)


119875 Ω119905+1

= Ω119905 Ω119905= Ω119905minus1

Ω2= Ω1

= 119875 Ω119905+1

= Ω119905 times 119875 Ω

119905= Ω119905minus1


=

119905+1

prod

119895=2

119875 Ω119895= Ω119895minus1

= (119875 Ω2= Ω1)119905

(119905 = 1 2 119905 isin Z+)

(52)



= 119875 (Ω1


119899+1

1) = (Ω

1


119899+1

2)

= 119875

(

119899+1

⋂

119895=1

Ω119895

1) = (

119899+1

⋂

119895=1

Ω119895

2)

lArrrArr 119875(Ω1


119899+1

2) sub Ω

119896

1

(Ω1


119899+1

1) sub Ω

119896

2

(forall119896 = 1 2 119899 119899 + 1)

(53)


2=

Ω1 will yield

119875 (Ω1

2cap Ω2


119899

2) sub Ω

1

1 = 1205821

119875 (Ω1

2cap Ω2


119899

2) sub Ω

119899+1

1 = 1205821

119875 (Ω1

1cap Ω2


119899

1) sub Ω

1

2 = 1205822

119875 (Ω1

1cap Ω2


119899

1) sub Ω

119899+1

2 = 1205822

119875 Ω1= Ω2

= 119875

(

119899+1

⋂

119895=1

Ω119895

2) sub Ω

119896

1 (forall119896 = 1 2 119899 119899 + 1)

times 119875

(

119899+1

⋂

119895=1

Ω119895

1) sub Ω

119896

2 (forall119896 = 1 2 119899 119899 + 1)

=

119899+1

prod

119896=1

119875

(

119899+1

⋂

119895=1

Ω119895

2) sub Ω

119896

1

times

119899+1

prod

119896=1

119875

(

119899+1

⋂

119895=1

Ω119895

1) sub Ω

119896

2

= 120582119899+1

1120582119899+1

2 (0 lt 120582

1lt 1 0 lt 120582

2lt 1)

(54)


119875 Ω119905+1

= Ω119905 Ω119905= Ω119905minus1

Ω2= Ω1 = 120582(119899+1)119905

1120582(119899+1)119905

2

119875 Ω119905+1

Δ119905Ω119905Δ119905minus1

Ω119905minus1


(119899+1)119905

1120582(119899+1)119905

2

(55)


min 119891 (119909)

st ℎ (119909) = 0 119892 (119909) le 0

(56)


where 119891(119909) 119877119899

rarr 119877 ℎ(119909) 119877119899

rarr 119877119898 and 119892(119909)

119877119899



min 119891 (119909) minus 120592

119902

sum

119894=1

ln 120575119894

st ℎ (119909) = 0 119892 (119909) + 120575 = 0

(57)


(1205751 1205752 120575

119902)119879


119871 (119909 119910 119911 120575) = 119891 (119909) minus 120592

119902

sum

119894=1

ln 120575119894+ 119910119879(119892 (119909) + 120575) + 119911

119879ℎ (119909)

(58)


119902)119879

and z = (1199111 1199112 119911




119879

119910 + (nablaℎ (119909))119879119911 = 0

nabla120575119871 (119909 119910 119911 120575) = minus120592119878

minus1

120575119890 + 119910 = 0 lArrrArr minus120592119890 + 119878

120575119884119890 = 0

nabla119910119871 (119909 119910 119911 120575) = 119892 (119909) + 120575 = 0

nabla119911119871 (119909 119910 119911 120575) = ℎ (119909) = 0

(59)

where 119878120575

= diag(1205751 1205752 120575



1 1199102 119910



Δ119909 120575+Δ120575 119911+Δ



(nabla2119891 (119909) + nabla

2119892(119909)119879119910 + nabla

2ℎ(119909)119879119911) Δ119909 + nabla119892(119909)

119879Δ119910


119879119910 + nablaℎ(119909)

119879119911) = 0

minus120592119890 + 119878120575119884119890 + 119878

120575Δ119910 + 119884Δ120575 = 0

119892 (119909) + nabla119892 (119909) Δ119909 + 120575 + Δ120575 = 0

nablaℎ (119909) Δ119909 + ℎ (119909) = 0

(60)


(

119867(119909 119910 119911) 0 nablaℎ(119909)119879

nabla119892(119909)119879

0 119878minus1

120575119884 0 119868

nablaℎ (119909) 0 0 0

nabla119892 (119909) 119868 0 0

)(

Δ119909

Δ120575

Δ119910

Δ119911

)

= (


119879119911

120592119878minus1

120575119890 minus 119910

minusℎ (119909)

minus119892 (119909) minus 120575

)

(61)

119867(119909 119910 119911) = nabla2119891 (119909) + nabla

2119892(119909)119879119910 + nabla

2ℎ(119909)119879119911 (62)


(119909 120575 119911 119910) larr997888 (119909 120575 119911 119910) + 1205771(Δ119909 Δ120575 Δ119911 Δ119910) (63)


1holds the




(0)gt 0 119896 = 0










(119896+1) 120575(119896+1)

119911(119896+1)

119910(119896+1)

) larr (119909(119896)

120575(119896)

119911(119896)

119910(119896)

) +

1205771(Δ119909 Δ120575 Δ119911 Δ119910) 119896 larr 119896 + 1


Step 7 End



1199011(119904) [7]

1198661199012(119904) [15] and 119866

1199013(119904) [6] respectively 119866

1199011(119904) is a third-


1199013(119904) are the SOPTD


1198661199011

(119904) =

15

(1199042+ 09119904 + 5) (119904 + 3)

1198661199012

(119904) =

119890minus05119904

(119904 + 1)2 119866

1199013(119904) =

4119890minus10119904

10119904 + 1

(64)


and 119890minus10119904

asymp (1199042minus 06119904 + 012)(119904

2+ 06119904 + 012) respectively


1199013(119904) are the (minus119904 +

4)(1199043+ 61199042+ 9119904 + 4) and (04119904

2minus 024119904 + 0048)(119904

3+ 07119904

2+



1 1199022 1199023 and 119902

4 The



2 1199023


1(with 119902

2 1199023 and 119902

4


2 1199023 and 119902

4(with 119902




1

















119901119896lowast

119894119896lowast

119889

Plant1

diag (5 5 1 1) 013121 058138 058655

120576 = 0


Plant2

diag (02 1 1 1) 115698 052966 086781

120576 = 05

diag (02 1 1 02) 216962 084500 182140diag (02 2 2 05) 178668 058700 159290

diag (01 08 08 02) 185901 064942 162251diag (04 05 5 1) 122806 062168 142095

Plant3

diag (004 27 158 175) 032447 001537 145256

120576 = 0




1119902211990231199024) 119896

lowast

119901119896lowast

119894119896lowast

119889119905119903(sec) 119905

119889(sec) 119905

119901(sec) 119905

119904(sec) 119872

119901()


0 2 4 6 8 10 12 14

0

02

04

06

08

1

12

Syste

m re

spon

se

minus02

Time (s)

119876 = diag (5 5 1 1)119876 = diag (5 2 1 1)119876 = diag (10 2 1 1)

119876 = diag (10 10 1 1)119876 = diag (5 5 2 1) lowast 2

(a)

0 2 4 6 8 10 12 14

0

02

04

06

08

1

12

Con

trol e

rror

minus02

Time (s)

119876 = diag (5 5 1 1)119876 = diag (5 2 1 1)119876 = diag (10 2 1 1)

119876 = diag (10 10 1 1)119876 = diag (5 5 2 1) lowast 2

(b)



0 1 2 3 4 5 6 7 8 9 100

02

04

06

08

1

12

14

Time (s)

Syste

m re

spon

se

120576 = 05

120576 = 25

120576 = 3

120576 = 4

120576 = 5

(a)

0 1 2 3 4 5 6 7 8 9 10

0

02

04

06

08

1

Time (s)

minus02

minus04

Con

trol e

rror

120576 = 05

120576 = 25

120576 = 3

120576 = 4

120576 = 5

(b)


0 5 10 15 20 25 30

0

02

04

06

08

1

12



Syste

m re

spon

se

Time (s)

minus02

(a)

0 5 10 15 20 25 30

0

02

04

06

08

1

12



Con

trol e

rror

Time (s)

minus02

(b)




119896119894

119896119889

Risetime

119905119903(sec)

Delaytime

119905119889(sec)

Peaktime

119905119901(sec)

Settingtime

119905119904(sec)


Plants



0 2 4 6 8 10 12 14 16 180

05

1

15Sy

stem

resp

onse



Time (s)

(a)

0 2 4 6 8 10 12 14 16 18

0

05

1


minus05


Con

trol e

rror

Time (s)

(b)


0 20 40 60 80 100 120 140 160 180 2000

02

04

06

08

1

12

14



Syste

m re

spon

se

Time (s)

095

105

(a)

0 20 40 60 80 100 120 140 160 180 200

0

02

04

06

08

1



Con

trol e

rror

Time (s)

minus02

minus04

(b)





Time (s)0 5 10 15 20 25 30

0

02

04

06

08

1

12

0 5 10 15 20 25 30

0

02

04

06

08

1

12

Syste

m re

spon

seSy

stem

resp

onse

Con

trol e

rror

Con

trol e

rror

Time (s)

Time (s)Time (s)16 18 20 22 24 26 28

075

08

085

09

095

1

105

11



16 18 20 22 24 26 28

0

005

01

015

02

025

03



minus02 minus02

minus005

119863119906(119905) = 03119906(119905 minus 16)



Methods 119896119901

119896119894

119896119889










16 18 20 22 24 26 28 30086

088

09

092

094

096

098

1

102

104





0 5 10 15 20 25 300

05

1

15

0 5 10 15 20 25 30

0

05

1

18 20 22 24 26 28 30

0

002

004

006

008

01

012

014

Con

trol e

rror

Con

trol e

rror

Syste

m re

spon

seSy

stem

resp

onse

minus05

Time (s) Time (s)

Time (s) Time (s)

minus002

119863119906(119905) = 03119906(119905 minus 18)






200 250 300 350 40007

075

08

085

09

095

1

105

11

200 250 300 350 400

0

005

01

015

02

025

03

0 50 100 150 200 250 300 350 4000

02

04

06

08

1

12

14

0 50 100 150 200 250 300 350 400

0

02

04

06

08

1

095

105

Time (s) Time (s)

Time (s)Time (s)





minus005

Con

trol e

rror

Syste

m re

spon

se

Con

trol e

rror

Syste

m re

spon

se

minus02

minus04

119863119906(119905) = 01119906(119905 minus 180)




1(119904)119876119894119888(119904) = (04465119904 +





119901=

10000 119896119894

= 032549 119896119889

= 0001 120576 = 0 119896119901

=

10000 119896119894

= 010118 119896119889

= 0001 120576 = 0 119896119901

=

10000 119896119894= 022986 119896

119889= 0001 120576 = 0 119896

119901= 6000 119896

119894=



1is set at




0 5 10 15 20 25 300

02

04

06

08

1

12

14

Time (s)

16 18 20 22 24 26 28 30086

088

09

092

094

096

098

1

102

104

0 5 10 15 20 25 30

0

02

04

06

08

1

Time (s)

Time (s) Time (s)

16 18 20 22 24 26 28 30

0

002

004

006

008

01

012

014

minus02

minus04

minus002

120576 = 05

120576 = 25

120576 = 3

120576 = 4

120576 = 5

120576 = 05

120576 = 25

120576 = 3

120576 = 4

120576 = 5



1 the solid







0 200 400 600 800 1000 1200 1400 1600 1800 20000

50

100

150

200

250

300

Time (s)

Liqu

id le

vel o

f CW

TS (m

m)




lowast

119901= 10000 119896lowast

119894= 032549 119896lowast

119889= 0001 and 120576 = 0 in

set point 300mm


119901= 10000 119896lowast

119894= 010118 119896lowast

119889= 0001 and 120576 = 0 in

set point 300mm


119901= 10000 119896lowast

119894= 022986 119896lowast

119889= 0001 and 120576 = 0 in

set point 300mm


lowast

119901= 6000 119896lowast

119894= 013817 119896lowast

119889= 0001 and 120576 = 0 in

set point 300mm


5 Conclusions




119901= 10000 119896lowast

119894= 032549 119896lowast

119889= 0001 and 120576 = 0 in

set point 160mm


119901= 10000 119896lowast

119894= 010118 119896lowast

119889= 0001 and 120576 = 0 in

set point 160mm


1 1199022 1199023 and 119902





2 1199023 and 119902

4(with 119902





4(with 119902

1fixed) Through the


1 1199022 119902

119899+1 The 120576-Routh




119901= 10000 119896lowast

119894= 022986 119896lowast

119889= 0001 and 120576 = 0 in

set point 160mm


lowast

119901= 6000 119896lowast

119894= 013817 119896lowast

119889= 0001 and 120576 = 0 in

set point 160mm



Acknowledgments



References











































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Command

119903(119905) +

+

minusminus



119910(119905)

Output119863119906(119905)

119866lowast119888 (119904) = 119896

lowast119901 +

119896lowast119894119904+ 119896lowast119889119904



Control valve

ValveWater tank

Tank-1

Tank-2



119876119894119888minus 1198761119888

minus 11987612

= 1198621

119889ℎ1

119889119905

(16)



1 11987612




11987612

minus 1198762119888

= 1198622

119889ℎ2

119889119905

(17)





1198761119888

=

ℎ1

1198771

1198762119888

=

ℎ2

1198772

11987612

=

ℎ1minus ℎ2

11987712

(18)

where1198771and119877




119866ℎ(119904) =

1198671(119904)

119876119894119888(119904)

=

1198871119904 + 1198872

11988601199042+ 1198861119904 + 1198862

=

11988711199041198860+ 11988721198860

1199042+ 11988611199041198860+ 11988621198860

=

1198731(119904)

1198721(119904)

(19)



Tank-1 Tank-2

Control valve

1198771

119876121198671 + ℎ1

11986221198621

1198772

11987621198881198761119888

1198760 + 119876119894119888

11987712

1198672 + ℎ2Valve-3

Valv

e-2

Valv

e-1



1198871= 11987711198771211987721198622 119887

2= 1198771(11987712

+ 1198772)

1198860= 119877111986211198771211987721198622

1198861= 119877111986211198772+ 1198771119862111987712

+ 119877111987721198622+ 1198771211987721198622

(20)

1198862= 1198771+ 11987712

+ 1198772 (21)


119866119901(119904) =

11988711199042+ 1198872119904 + 1198873

11988601199043+ 11988611199042+ 1198862119904 + 1198863

=

119873 (119904)

119872 (119904)

119866119888(119904) = 119896

119901+

119896119894

119904

+ 119896119889119904

(22)


119866 (119904) = 119866119888(119904) 119866119901(119904) =

(119896119901119904 + 119896119894+ 1198961198891199042)119873 (119904)

119904119872 (119904)

(23)


119890 (119904)

119903 (119904)

=

1

(1 + 119866119901(119904) 119866119888(119904))

lArrrArr 119863(119904) 119890 (119904) = 119904119872 (119904) 119903 (119904)

119863 (119904) = (119904119872 (119904) + 119873 (119904) (119896119901119904 + 119896119894+ 1198961198891199042))

deg 119863 (119904) = 4

(24)


(120575119872 (120575) + 119873 (120575) (119896119901120575 + 119896119894+ 1198961198891205752)) 119890 (119905) = 120575119872 (120575) 119903 (119905)

(25)




(1198860+ 1198961198891198871) 119890(119905)(4)

+ (1198861+ 1198961199011198871+ 1198961198891198872)119890 (119905)

+ (1198862+ 1198961198941198871+ 1198961199011198872+ 1198961198891198873) 119890 (119905)

+ (1198863+ 1198961198941198872+ 1198961199011198873) 119890 (119905) + 119896

1198941198873119890 (119905) = 0

(26)


1198981119890(119905)(4)

+ 1198982

119890 (119905) + 119898

3119890 (119905) + 119898

4119890 (119905) + 119898

5119890 (119905) = 0 (27)



1198981= 1198860+ 1198961198891198871 119898

2= 1198861+ 1198961199011198871+ 1198961198891198872 119898

5= 1198961198941198873

1198983= 1198862+ 1198961198941198871+ 1198961199011198872+ 1198961198891198873 119898

4= 1198863+ 1198961198941198872+ 1198961199011198873

(28)


= 119860119909

119909 = (119890 (119905) 119890 (119905) 119890 (119905)

119890 (119905))119879 119860 = (

1198743times1

1198683times3

minus1198981

41198621times3

)

1198621times3

= (minus1198981

3minus 1198981

2minus 1198981

1)

(29)



119894= 119898119894+1

1198981(119894 = 1 2 3 4)



119901 119896lowast

119894 119896lowast

119889) = min119866lowast

119888

119869 (1198981 119898

5)

= minint

infin

0

119909119879(119905) 119876119909 (119905) 119889119905

= min

3

sum

119894=0

int

infin

0

119902119894+1119894+1

(119890(119905)(119894))

2

119889119905

st

(1) = 119860119909


(3) 120576 ge 0 11987711

(1198981 119898

119908+3) gt 120576

11987721

(1198981 119898

119908+3) gt 120576

11987731

(1198981 119898

119908+3) gt 120576 119877

41(1198981 119898

119908+3) gt 120576

11987731

(1198981 119898

119908+3) gt 120576 119877

41(1198981 119898

119908+3) gt 120576

11987751

(1198981 119898

119908+3) gt 120576

(4)1198984= 1198911(1198981 1198982 1198983) 1198985= 1198912(1198981 1198982 1198983)

(5) 0 lt 119896119901le 1205910

119901 0 lt 119896

119894le 1205910

119894 0 lt 119896

119889le 1205910

119889

(30)


4+

1198981

11199043+ 1198981

21199042+ 1198981

3119904 + 119898

1




lArrrArr 1198981

1gt 0 119898

1

11198981

2minus 1198981

3gt 0

1198981

11198981

21198981

3minus (1198981

3)

2

minus 1198981

4(1198981

1)

2

gt 0 1198981

4gt 0

(31)




120576 ge 0 11987711

(1198981 119898

5) gt 120576 119877

21(1198981 119898

5) gt 120576

11987731

(1198981 119898

5) gt 120576

11987741

(1198981 119898

5) gt 120576 119877

51(1198981 119898

5) gt 120576

lArrrArr 1198981gt 120576 119898

2gt 120576 119898

5gt 120576

11989811198984minus 11989821198983gt 1205761198982

(1198982)2

1198985minus 1198981(1198984)2

minus 119898211989831198984gt 120576 (119898

11198984minus 11989821198983)

(32)



infin

0

119909119879119876119909119889119905 = min119909

119879(0) 119875119909 (0)

119860119879119875 + 119875119860 = minus119876

(33)





119901 119896lowast

119894 119896lowast

119889) = min119866lowast

119888(119904)

119869 (1198981 119898

5)

= min119909(0)119879119875119909 (0)

st

(1) 119860119879119875 + 119875119860 = minus119876

(2)1198981

1gt 0 119898

1

4gt 0 119898

1

11198981

2minus 1198981

3gt 0

1198981

11198981

21198981

3minus (1198981

3)

2

minus 1198981

4(1198981

1)

2

gt 0

(3)1198981gt 120576 119898

2gt 120576 119898

5gt 120576 119898

11198984minus 11989821198983gt 1205761198982

(1198982)2

1198985minus 1198981(1198984)2

minus 119898211989831198984gt 120576 (119898

11198984minus 11989821198983)

(4)1198984= 1198911(1198981 1198982 1198983) 1198985= 1198912(1198981 1198982 1198983)

(5) 0 lt 119896119901le 1205910

119901 0 lt 119896

119894le 1205910

119894 0 lt 119896

119889le 1205910

119889

(34)


1(0) = 1 119909

2(0) = 0 119909

3(0) = 0 and




119894and


119866lowast

119888(119904) = 119896

lowast

119901+

119896lowast

119894

119904

+ 119896lowast

119889119904 119896

lowast

119901=

119898lowast


119894

1198873

119896lowast

119894=

119898lowast

5

1198873

119896lowast

119889=

119898lowast

1minus 1198860

1198871

(35)



1119904119899+sdot sdot sdot+119886

119899minus119908minus2119904119908+3

+

1198981119904119908+2

+ 1198982119904119908+1

+ sdot sdot sdot + 119898119908+2

119904 + 119898119908+3




119889 The

parameters 119896119901 119896119894 and 119896


119901]

(0 1205910

119894] and (0 120591

0



119901 119896119894



119865 (119904) = 119904119899+ 1199011119904119899minus1

+ 1199012119904119899minus2


119904 + 119901119899 (36)

where 1199011 1199012 and 119901





119898

1199011= 1198911(1199021 119902

119898) 119901

119899= 119891119899(1199021 119902

119898)

Ω119902= (1199021 119902

119898)1003816100381610038161003816[119902minus

1 119902+


minus

119898 119902+

119898]

1199021isin [119902minus

1 119902+

1] 119902

119898isin [119902minus

119898 119902+

119898]

(37)


119894and 119902+

119894are the




11987711

(1199021 119902

119898) = 1198911199031

(1199011 119901

119899) gt 120576

119877119899+11

(1199021 119902

119898) = 119891119903119899+1

(1199011 119901

119899) gt 120576

Ω119903120576

= (1199021 119902

119898) | 1198911199031

(1199011 119901

119899) gt 120576

119891119903119899+1

(1199011 119901

119899) gt 120576 (120576 ge 0)

(38)







min(1199021 119902119898)isinΩ119891

1198911199031

(1199011 119901

119899) 119891

119903119899+1(1199011 119901

119899) gt 0

(Ω119891= (Ω119903120576

⋂Ω119902) = 0)

(39)







I Ω1= (119902

1 119902

119898) | 1198911199031

(1199011 119901

119899) gt 1205761

119891119903119899+1

(1199011 119901

119899) gt 1205761

II Ω2= (119902

1 119902

119898) | 1198911199031

(1199011 119901

119899) gt 1205762

119891119903119899+1

(1199011 119901

119899) gt 1205762

(40)




Ω1

1= (1199021 119902

119898) | 1198911199031

(1199011 119901

119899) gt 1205761

Ω1

2= (1199021 119902

119898) | 1198911199031

(1199011 119901

119899) gt 1205762

(41)



1

2will have the

relationship

exist119872lowast

1= (119902lowast

11 119902lowast

21 119902

lowast

1198981) 119891

1199031(119872lowast

1) = 120576lowast

2gt 1205762 (42)


1will have the

relationship

exist119872lowast

2= (119902lowast

12 119902lowast

22 119902

lowast

1198982) 119891

1199031(119872lowast

2) = 120576lowast

1gt 1205761 (43)


1which will



1= (119902lowast

11 119902lowast

21 119902

lowast

1198981) 1205761lt 1198911199031

(119873lowast

1) lt 1205762

lArrrArr 1198911199031

(119873lowast

1) isin (1205761 1205762)


1isin Ω1

1 119873lowast

1notin Ω1

2997904rArr Ω

1

1sup Ω1

2

Case 2 1198911199031

(119873lowast

1) notin (1205761 1205762) 997904rArr Ω

1

1= Ω1

2

(44)


119896


relationship

Case 1 Ω119896

1sup Ω119896

2 (forall119896 = 1 2 119899 119899 + 1)

Case 2 Ω119896

1= Ω119896

2 (forall119896 = 1 2 119899 119899 + 1)

Ω119896

1= (1199021 119902

119898) | 119891119903119896

(1199011 119901

119899) gt 1205761

Ω119896

2= (1199021 119902

119898) | 119891119903119896

(1199011 119901

119899) gt 1205762

(45)


2will have follow-

ing relationships


1sup Ω2) cup (Ω

1= Ω2)

Ω1=

119899+1

⋂

119895=1

Ω119895

1 Ω

2=

119899+1

⋂

119895=1

Ω119895

2

(46)


119905+1gt 120576119905gt 120576119905minus1



Ω119905= (1199021 119902

119898) | 1198911199031

(1199011 119901

119899) gt 120576119905

119891119903119899+1

(1199011 119901

119899) gt 120576119905



119899+1

119905) = (

119899+1

⋂

119895=1

Ω119895

119905)

(119905 = 1 2 3 119905 isin Z+)

Ω119896

119905= (1199021 119902

119898) | 119891119903119896

(1199011 119901

119899) gt 120576119905

(119896 = 1 2 119899 119899 + 1)

(47)


ship

Ω119905+1


2sube Ω1



2Ω2Δ1Ω1)

⋃ (Ω119905+1


2= Ω1)


(48)


119905 then the



(1199021 119902

119898) | 1198911199031

(1199011 119901

119899) gt 120576119905

119891119903119899+1

(1199011 119901

119899) gt 120576119905


Ω119905= lim120576119905rarr+infin

(Ω1

119905cap Ω2


119899+1

119905)


(

119899+1

⋂

119895=1

Ω119895

119905) = Ω

lowast

= (119902lowast

10119902lowast

20 119902

lowast

1198980)



(1199021 119902

119898) | 1198911199031

(1199011 119901

119899) gt 120576119905

119891119903119899+1

(1199011 119901

119899) gt 120576119905


Ω119905= lim120576119905rarr+infin

(Ω1

119905cap Ω2


119899+1

119905)


(

119899+1

⋂

119895=1

Ω119895

119905) = 0

(49)





120576119905997888rarr +infin lim



lowast

119905 120576119905gt 120576lowast

119905 Ω119905= 0

0 lt 120576119905le 120576lowast

119905 Ω119905

= 0

120576lowast

119905gt 120576119905gt 120576119905minus1


Ωlowast

119905sub Ω119905sub Ω119905minus1


(50)


= Ω119905= Ω119905minus1

=



119875 Ω119905+1

= Ω119905= Ω119905minus1


lArrrArr 119875 Ω119905+1

= Ω119905 Ω119905= Ω119905minus1

Ω2= Ω1

(51)


119875 Ω119905+1

= Ω119905 Ω119905= Ω119905minus1

Ω2= Ω1

= 119875 Ω119905+1

= Ω119905 times 119875 Ω

119905= Ω119905minus1


=

119905+1

prod

119895=2

119875 Ω119895= Ω119895minus1

= (119875 Ω2= Ω1)119905

(119905 = 1 2 119905 isin Z+)

(52)



= 119875 (Ω1


119899+1

1) = (Ω

1


119899+1

2)

= 119875

(

119899+1

⋂

119895=1

Ω119895

1) = (

119899+1

⋂

119895=1

Ω119895

2)

lArrrArr 119875(Ω1


119899+1

2) sub Ω

119896

1

(Ω1


119899+1

1) sub Ω

119896

2

(forall119896 = 1 2 119899 119899 + 1)

(53)


2=

Ω1 will yield

119875 (Ω1

2cap Ω2


119899

2) sub Ω

1

1 = 1205821

119875 (Ω1

2cap Ω2


119899

2) sub Ω

119899+1

1 = 1205821

119875 (Ω1

1cap Ω2


119899

1) sub Ω

1

2 = 1205822

119875 (Ω1

1cap Ω2


119899

1) sub Ω

119899+1

2 = 1205822

119875 Ω1= Ω2

= 119875

(

119899+1

⋂

119895=1

Ω119895

2) sub Ω

119896

1 (forall119896 = 1 2 119899 119899 + 1)

times 119875

(

119899+1

⋂

119895=1

Ω119895

1) sub Ω

119896

2 (forall119896 = 1 2 119899 119899 + 1)

=

119899+1

prod

119896=1

119875

(

119899+1

⋂

119895=1

Ω119895

2) sub Ω

119896

1

times

119899+1

prod

119896=1

119875

(

119899+1

⋂

119895=1

Ω119895

1) sub Ω

119896

2

= 120582119899+1

1120582119899+1

2 (0 lt 120582

1lt 1 0 lt 120582

2lt 1)

(54)


119875 Ω119905+1

= Ω119905 Ω119905= Ω119905minus1

Ω2= Ω1 = 120582(119899+1)119905

1120582(119899+1)119905

2

119875 Ω119905+1

Δ119905Ω119905Δ119905minus1

Ω119905minus1


(119899+1)119905

1120582(119899+1)119905

2

(55)


min 119891 (119909)

st ℎ (119909) = 0 119892 (119909) le 0

(56)


where 119891(119909) 119877119899

rarr 119877 ℎ(119909) 119877119899

rarr 119877119898 and 119892(119909)

119877119899



min 119891 (119909) minus 120592

119902

sum

119894=1

ln 120575119894

st ℎ (119909) = 0 119892 (119909) + 120575 = 0

(57)


(1205751 1205752 120575

119902)119879


119871 (119909 119910 119911 120575) = 119891 (119909) minus 120592

119902

sum

119894=1

ln 120575119894+ 119910119879(119892 (119909) + 120575) + 119911

119879ℎ (119909)

(58)


119902)119879

and z = (1199111 1199112 119911




119879

119910 + (nablaℎ (119909))119879119911 = 0

nabla120575119871 (119909 119910 119911 120575) = minus120592119878

minus1

120575119890 + 119910 = 0 lArrrArr minus120592119890 + 119878

120575119884119890 = 0

nabla119910119871 (119909 119910 119911 120575) = 119892 (119909) + 120575 = 0

nabla119911119871 (119909 119910 119911 120575) = ℎ (119909) = 0

(59)

where 119878120575

= diag(1205751 1205752 120575



1 1199102 119910



Δ119909 120575+Δ120575 119911+Δ



(nabla2119891 (119909) + nabla

2119892(119909)119879119910 + nabla

2ℎ(119909)119879119911) Δ119909 + nabla119892(119909)

119879Δ119910


119879119910 + nablaℎ(119909)

119879119911) = 0

minus120592119890 + 119878120575119884119890 + 119878

120575Δ119910 + 119884Δ120575 = 0

119892 (119909) + nabla119892 (119909) Δ119909 + 120575 + Δ120575 = 0

nablaℎ (119909) Δ119909 + ℎ (119909) = 0

(60)


(

119867(119909 119910 119911) 0 nablaℎ(119909)119879

nabla119892(119909)119879

0 119878minus1

120575119884 0 119868

nablaℎ (119909) 0 0 0

nabla119892 (119909) 119868 0 0

)(

Δ119909

Δ120575

Δ119910

Δ119911

)

= (


119879119911

120592119878minus1

120575119890 minus 119910

minusℎ (119909)

minus119892 (119909) minus 120575

)

(61)

119867(119909 119910 119911) = nabla2119891 (119909) + nabla

2119892(119909)119879119910 + nabla

2ℎ(119909)119879119911 (62)


(119909 120575 119911 119910) larr997888 (119909 120575 119911 119910) + 1205771(Δ119909 Δ120575 Δ119911 Δ119910) (63)


1holds the




(0)gt 0 119896 = 0










(119896+1) 120575(119896+1)

119911(119896+1)

119910(119896+1)

) larr (119909(119896)

120575(119896)

119911(119896)

119910(119896)

) +

1205771(Δ119909 Δ120575 Δ119911 Δ119910) 119896 larr 119896 + 1


Step 7 End



1199011(119904) [7]

1198661199012(119904) [15] and 119866

1199013(119904) [6] respectively 119866

1199011(119904) is a third-


1199013(119904) are the SOPTD


1198661199011

(119904) =

15

(1199042+ 09119904 + 5) (119904 + 3)

1198661199012

(119904) =

119890minus05119904

(119904 + 1)2 119866

1199013(119904) =

4119890minus10119904

10119904 + 1

(64)


and 119890minus10119904

asymp (1199042minus 06119904 + 012)(119904

2+ 06119904 + 012) respectively


1199013(119904) are the (minus119904 +

4)(1199043+ 61199042+ 9119904 + 4) and (04119904

2minus 024119904 + 0048)(119904

3+ 07119904

2+



1 1199022 1199023 and 119902

4 The



2 1199023


1(with 119902

2 1199023 and 119902

4


2 1199023 and 119902

4(with 119902




1

















119901119896lowast

119894119896lowast

119889

Plant1

diag (5 5 1 1) 013121 058138 058655

120576 = 0


Plant2

diag (02 1 1 1) 115698 052966 086781

120576 = 05

diag (02 1 1 02) 216962 084500 182140diag (02 2 2 05) 178668 058700 159290

diag (01 08 08 02) 185901 064942 162251diag (04 05 5 1) 122806 062168 142095

Plant3

diag (004 27 158 175) 032447 001537 145256

120576 = 0




1119902211990231199024) 119896

lowast

119901119896lowast

119894119896lowast

119889119905119903(sec) 119905

119889(sec) 119905

119901(sec) 119905

119904(sec) 119872

119901()


0 2 4 6 8 10 12 14

0

02

04

06

08

1

12

Syste

m re

spon

se

minus02

Time (s)

119876 = diag (5 5 1 1)119876 = diag (5 2 1 1)119876 = diag (10 2 1 1)

119876 = diag (10 10 1 1)119876 = diag (5 5 2 1) lowast 2

(a)

0 2 4 6 8 10 12 14

0

02

04

06

08

1

12

Con

trol e

rror

minus02

Time (s)

119876 = diag (5 5 1 1)119876 = diag (5 2 1 1)119876 = diag (10 2 1 1)

119876 = diag (10 10 1 1)119876 = diag (5 5 2 1) lowast 2

(b)



0 1 2 3 4 5 6 7 8 9 100

02

04

06

08

1

12

14

Time (s)

Syste

m re

spon

se

120576 = 05

120576 = 25

120576 = 3

120576 = 4

120576 = 5

(a)

0 1 2 3 4 5 6 7 8 9 10

0

02

04

06

08

1

Time (s)

minus02

minus04

Con

trol e

rror

120576 = 05

120576 = 25

120576 = 3

120576 = 4

120576 = 5

(b)


0 5 10 15 20 25 30

0

02

04

06

08

1

12



Syste

m re

spon

se

Time (s)

minus02

(a)

0 5 10 15 20 25 30

0

02

04

06

08

1

12



Con

trol e

rror

Time (s)

minus02

(b)




119896119894

119896119889

Risetime

119905119903(sec)

Delaytime

119905119889(sec)

Peaktime

119905119901(sec)

Settingtime

119905119904(sec)


Plants



0 2 4 6 8 10 12 14 16 180

05

1

15Sy

stem

resp

onse



Time (s)

(a)

0 2 4 6 8 10 12 14 16 18

0

05

1


minus05


Con

trol e

rror

Time (s)

(b)


0 20 40 60 80 100 120 140 160 180 2000

02

04

06

08

1

12

14



Syste

m re

spon

se

Time (s)

095

105

(a)

0 20 40 60 80 100 120 140 160 180 200

0

02

04

06

08

1



Con

trol e

rror

Time (s)

minus02

minus04

(b)





Time (s)0 5 10 15 20 25 30

0

02

04

06

08

1

12

0 5 10 15 20 25 30

0

02

04

06

08

1

12

Syste

m re

spon

seSy

stem

resp

onse

Con

trol e

rror

Con

trol e

rror

Time (s)

Time (s)Time (s)16 18 20 22 24 26 28

075

08

085

09

095

1

105

11



16 18 20 22 24 26 28

0

005

01

015

02

025

03



minus02 minus02

minus005

119863119906(119905) = 03119906(119905 minus 16)



Methods 119896119901

119896119894

119896119889










16 18 20 22 24 26 28 30086

088

09

092

094

096

098

1

102

104





0 5 10 15 20 25 300

05

1

15

0 5 10 15 20 25 30

0

05

1

18 20 22 24 26 28 30

0

002

004

006

008

01

012

014

Con

trol e

rror

Con

trol e

rror

Syste

m re

spon

seSy

stem

resp

onse

minus05

Time (s) Time (s)

Time (s) Time (s)

minus002

119863119906(119905) = 03119906(119905 minus 18)






200 250 300 350 40007

075

08

085

09

095

1

105

11

200 250 300 350 400

0

005

01

015

02

025

03

0 50 100 150 200 250 300 350 4000

02

04

06

08

1

12

14

0 50 100 150 200 250 300 350 400

0

02

04

06

08

1

095

105

Time (s) Time (s)

Time (s)Time (s)





minus005

Con

trol e

rror

Syste

m re

spon

se

Con

trol e

rror

Syste

m re

spon

se

minus02

minus04

119863119906(119905) = 01119906(119905 minus 180)




1(119904)119876119894119888(119904) = (04465119904 +





119901=

10000 119896119894

= 032549 119896119889

= 0001 120576 = 0 119896119901

=

10000 119896119894

= 010118 119896119889

= 0001 120576 = 0 119896119901

=

10000 119896119894= 022986 119896

119889= 0001 120576 = 0 119896

119901= 6000 119896

119894=



1is set at




0 5 10 15 20 25 300

02

04

06

08

1

12

14

Time (s)

16 18 20 22 24 26 28 30086

088

09

092

094

096

098

1

102

104

0 5 10 15 20 25 30

0

02

04

06

08

1

Time (s)

Time (s) Time (s)

16 18 20 22 24 26 28 30

0

002

004

006

008

01

012

014

minus02

minus04

minus002

120576 = 05

120576 = 25

120576 = 3

120576 = 4

120576 = 5

120576 = 05

120576 = 25

120576 = 3

120576 = 4

120576 = 5



1 the solid







0 200 400 600 800 1000 1200 1400 1600 1800 20000

50

100

150

200

250

300

Time (s)

Liqu

id le

vel o

f CW

TS (m

m)




lowast

119901= 10000 119896lowast

119894= 032549 119896lowast

119889= 0001 and 120576 = 0 in

set point 300mm


119901= 10000 119896lowast

119894= 010118 119896lowast

119889= 0001 and 120576 = 0 in

set point 300mm


119901= 10000 119896lowast

119894= 022986 119896lowast

119889= 0001 and 120576 = 0 in

set point 300mm


lowast

119901= 6000 119896lowast

119894= 013817 119896lowast

119889= 0001 and 120576 = 0 in

set point 300mm


5 Conclusions




119901= 10000 119896lowast

119894= 032549 119896lowast

119889= 0001 and 120576 = 0 in

set point 160mm


119901= 10000 119896lowast

119894= 010118 119896lowast

119889= 0001 and 120576 = 0 in

set point 160mm


1 1199022 1199023 and 119902





2 1199023 and 119902

4(with 119902





4(with 119902

1fixed) Through the


1 1199022 119902

119899+1 The 120576-Routh




119901= 10000 119896lowast

119894= 022986 119896lowast

119889= 0001 and 120576 = 0 in

set point 160mm


lowast

119901= 6000 119896lowast

119894= 013817 119896lowast

119889= 0001 and 120576 = 0 in

set point 160mm



Acknowledgments



References











































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











(1198860+ 1198961198891198871) 119890(119905)(4)

+ (1198861+ 1198961199011198871+ 1198961198891198872)119890 (119905)

+ (1198862+ 1198961198941198871+ 1198961199011198872+ 1198961198891198873) 119890 (119905)

+ (1198863+ 1198961198941198872+ 1198961199011198873) 119890 (119905) + 119896

1198941198873119890 (119905) = 0

(26)


1198981119890(119905)(4)

+ 1198982

119890 (119905) + 119898

3119890 (119905) + 119898

4119890 (119905) + 119898

5119890 (119905) = 0 (27)



1198981= 1198860+ 1198961198891198871 119898

2= 1198861+ 1198961199011198871+ 1198961198891198872 119898

5= 1198961198941198873

1198983= 1198862+ 1198961198941198871+ 1198961199011198872+ 1198961198891198873 119898

4= 1198863+ 1198961198941198872+ 1198961199011198873

(28)


= 119860119909

119909 = (119890 (119905) 119890 (119905) 119890 (119905)

119890 (119905))119879 119860 = (

1198743times1

1198683times3

minus1198981

41198621times3

)

1198621times3

= (minus1198981

3minus 1198981

2minus 1198981

1)

(29)



119894= 119898119894+1

1198981(119894 = 1 2 3 4)



119901 119896lowast

119894 119896lowast

119889) = min119866lowast

119888

119869 (1198981 119898

5)

= minint

infin

0

119909119879(119905) 119876119909 (119905) 119889119905

= min

3

sum

119894=0

int

infin

0

119902119894+1119894+1

(119890(119905)(119894))

2

119889119905

st

(1) = 119860119909


(3) 120576 ge 0 11987711

(1198981 119898

119908+3) gt 120576

11987721

(1198981 119898

119908+3) gt 120576

11987731

(1198981 119898

119908+3) gt 120576 119877

41(1198981 119898

119908+3) gt 120576

11987731

(1198981 119898

119908+3) gt 120576 119877

41(1198981 119898

119908+3) gt 120576

11987751

(1198981 119898

119908+3) gt 120576

(4)1198984= 1198911(1198981 1198982 1198983) 1198985= 1198912(1198981 1198982 1198983)

(5) 0 lt 119896119901le 1205910

119901 0 lt 119896

119894le 1205910

119894 0 lt 119896

119889le 1205910

119889

(30)


4+

1198981

11199043+ 1198981

21199042+ 1198981

3119904 + 119898

1




lArrrArr 1198981

1gt 0 119898

1

11198981

2minus 1198981

3gt 0

1198981

11198981

21198981

3minus (1198981

3)

2

minus 1198981

4(1198981

1)

2

gt 0 1198981

4gt 0

(31)




120576 ge 0 11987711

(1198981 119898

5) gt 120576 119877

21(1198981 119898

5) gt 120576

11987731

(1198981 119898

5) gt 120576

11987741

(1198981 119898

5) gt 120576 119877

51(1198981 119898

5) gt 120576

lArrrArr 1198981gt 120576 119898

2gt 120576 119898

5gt 120576

11989811198984minus 11989821198983gt 1205761198982

(1198982)2

1198985minus 1198981(1198984)2

minus 119898211989831198984gt 120576 (119898

11198984minus 11989821198983)

(32)



infin

0

119909119879119876119909119889119905 = min119909

119879(0) 119875119909 (0)

119860119879119875 + 119875119860 = minus119876

(33)





119901 119896lowast

119894 119896lowast

119889) = min119866lowast

119888(119904)

119869 (1198981 119898

5)

= min119909(0)119879119875119909 (0)

st

(1) 119860119879119875 + 119875119860 = minus119876

(2)1198981

1gt 0 119898

1

4gt 0 119898

1

11198981

2minus 1198981

3gt 0

1198981

11198981

21198981

3minus (1198981

3)

2

minus 1198981

4(1198981

1)

2

gt 0

(3)1198981gt 120576 119898

2gt 120576 119898

5gt 120576 119898

11198984minus 11989821198983gt 1205761198982

(1198982)2

1198985minus 1198981(1198984)2

minus 119898211989831198984gt 120576 (119898

11198984minus 11989821198983)

(4)1198984= 1198911(1198981 1198982 1198983) 1198985= 1198912(1198981 1198982 1198983)

(5) 0 lt 119896119901le 1205910

119901 0 lt 119896

119894le 1205910

119894 0 lt 119896

119889le 1205910

119889

(34)


1(0) = 1 119909

2(0) = 0 119909

3(0) = 0 and




119894and


119866lowast

119888(119904) = 119896

lowast

119901+

119896lowast

119894

119904

+ 119896lowast

119889119904 119896

lowast

119901=

119898lowast


119894

1198873

119896lowast

119894=

119898lowast

5

1198873

119896lowast

119889=

119898lowast

1minus 1198860

1198871

(35)



1119904119899+sdot sdot sdot+119886

119899minus119908minus2119904119908+3

+

1198981119904119908+2

+ 1198982119904119908+1

+ sdot sdot sdot + 119898119908+2

119904 + 119898119908+3




119889 The

parameters 119896119901 119896119894 and 119896


119901]

(0 1205910

119894] and (0 120591

0



119901 119896119894



119865 (119904) = 119904119899+ 1199011119904119899minus1

+ 1199012119904119899minus2


119904 + 119901119899 (36)

where 1199011 1199012 and 119901





119898

1199011= 1198911(1199021 119902

119898) 119901

119899= 119891119899(1199021 119902

119898)

Ω119902= (1199021 119902

119898)1003816100381610038161003816[119902minus

1 119902+


minus

119898 119902+

119898]

1199021isin [119902minus

1 119902+

1] 119902

119898isin [119902minus

119898 119902+

119898]

(37)


119894and 119902+

119894are the




11987711

(1199021 119902

119898) = 1198911199031

(1199011 119901

119899) gt 120576

119877119899+11

(1199021 119902

119898) = 119891119903119899+1

(1199011 119901

119899) gt 120576

Ω119903120576

= (1199021 119902

119898) | 1198911199031

(1199011 119901

119899) gt 120576

119891119903119899+1

(1199011 119901

119899) gt 120576 (120576 ge 0)

(38)







min(1199021 119902119898)isinΩ119891

1198911199031

(1199011 119901

119899) 119891

119903119899+1(1199011 119901

119899) gt 0

(Ω119891= (Ω119903120576

⋂Ω119902) = 0)

(39)







I Ω1= (119902

1 119902

119898) | 1198911199031

(1199011 119901

119899) gt 1205761

119891119903119899+1

(1199011 119901

119899) gt 1205761

II Ω2= (119902

1 119902

119898) | 1198911199031

(1199011 119901

119899) gt 1205762

119891119903119899+1

(1199011 119901

119899) gt 1205762

(40)




Ω1

1= (1199021 119902

119898) | 1198911199031

(1199011 119901

119899) gt 1205761

Ω1

2= (1199021 119902

119898) | 1198911199031

(1199011 119901

119899) gt 1205762

(41)



1

2will have the

relationship

exist119872lowast

1= (119902lowast

11 119902lowast

21 119902

lowast

1198981) 119891

1199031(119872lowast

1) = 120576lowast

2gt 1205762 (42)


1will have the

relationship

exist119872lowast

2= (119902lowast

12 119902lowast

22 119902

lowast

1198982) 119891

1199031(119872lowast

2) = 120576lowast

1gt 1205761 (43)


1which will



1= (119902lowast

11 119902lowast

21 119902

lowast

1198981) 1205761lt 1198911199031

(119873lowast

1) lt 1205762

lArrrArr 1198911199031

(119873lowast

1) isin (1205761 1205762)


1isin Ω1

1 119873lowast

1notin Ω1

2997904rArr Ω

1

1sup Ω1

2

Case 2 1198911199031

(119873lowast

1) notin (1205761 1205762) 997904rArr Ω

1

1= Ω1

2

(44)


119896


relationship

Case 1 Ω119896

1sup Ω119896

2 (forall119896 = 1 2 119899 119899 + 1)

Case 2 Ω119896

1= Ω119896

2 (forall119896 = 1 2 119899 119899 + 1)

Ω119896

1= (1199021 119902

119898) | 119891119903119896

(1199011 119901

119899) gt 1205761

Ω119896

2= (1199021 119902

119898) | 119891119903119896

(1199011 119901

119899) gt 1205762

(45)


2will have follow-

ing relationships


1sup Ω2) cup (Ω

1= Ω2)

Ω1=

119899+1

⋂

119895=1

Ω119895

1 Ω

2=

119899+1

⋂

119895=1

Ω119895

2

(46)


119905+1gt 120576119905gt 120576119905minus1



Ω119905= (1199021 119902

119898) | 1198911199031

(1199011 119901

119899) gt 120576119905

119891119903119899+1

(1199011 119901

119899) gt 120576119905



119899+1

119905) = (

119899+1

⋂

119895=1

Ω119895

119905)

(119905 = 1 2 3 119905 isin Z+)

Ω119896

119905= (1199021 119902

119898) | 119891119903119896

(1199011 119901

119899) gt 120576119905

(119896 = 1 2 119899 119899 + 1)

(47)


ship

Ω119905+1


2sube Ω1



2Ω2Δ1Ω1)

⋃ (Ω119905+1


2= Ω1)


(48)


119905 then the



(1199021 119902

119898) | 1198911199031

(1199011 119901

119899) gt 120576119905

119891119903119899+1

(1199011 119901

119899) gt 120576119905


Ω119905= lim120576119905rarr+infin

(Ω1

119905cap Ω2


119899+1

119905)


(

119899+1

⋂

119895=1

Ω119895

119905) = Ω

lowast

= (119902lowast

10119902lowast

20 119902

lowast

1198980)



(1199021 119902

119898) | 1198911199031

(1199011 119901

119899) gt 120576119905

119891119903119899+1

(1199011 119901

119899) gt 120576119905


Ω119905= lim120576119905rarr+infin

(Ω1

119905cap Ω2


119899+1

119905)


(

119899+1

⋂

119895=1

Ω119895

119905) = 0

(49)





120576119905997888rarr +infin lim



lowast

119905 120576119905gt 120576lowast

119905 Ω119905= 0

0 lt 120576119905le 120576lowast

119905 Ω119905

= 0

120576lowast

119905gt 120576119905gt 120576119905minus1


Ωlowast

119905sub Ω119905sub Ω119905minus1


(50)


= Ω119905= Ω119905minus1

=



119875 Ω119905+1

= Ω119905= Ω119905minus1


lArrrArr 119875 Ω119905+1

= Ω119905 Ω119905= Ω119905minus1

Ω2= Ω1

(51)


119875 Ω119905+1

= Ω119905 Ω119905= Ω119905minus1

Ω2= Ω1

= 119875 Ω119905+1

= Ω119905 times 119875 Ω

119905= Ω119905minus1


=

119905+1

prod

119895=2

119875 Ω119895= Ω119895minus1

= (119875 Ω2= Ω1)119905

(119905 = 1 2 119905 isin Z+)

(52)



= 119875 (Ω1


119899+1

1) = (Ω

1


119899+1

2)

= 119875

(

119899+1

⋂

119895=1

Ω119895

1) = (

119899+1

⋂

119895=1

Ω119895

2)

lArrrArr 119875(Ω1


119899+1

2) sub Ω

119896

1

(Ω1


119899+1

1) sub Ω

119896

2

(forall119896 = 1 2 119899 119899 + 1)

(53)


2=

Ω1 will yield

119875 (Ω1

2cap Ω2


119899

2) sub Ω

1

1 = 1205821

119875 (Ω1

2cap Ω2


119899

2) sub Ω

119899+1

1 = 1205821

119875 (Ω1

1cap Ω2


119899

1) sub Ω

1

2 = 1205822

119875 (Ω1

1cap Ω2


119899

1) sub Ω

119899+1

2 = 1205822

119875 Ω1= Ω2

= 119875

(

119899+1

⋂

119895=1

Ω119895

2) sub Ω

119896

1 (forall119896 = 1 2 119899 119899 + 1)

times 119875

(

119899+1

⋂

119895=1

Ω119895

1) sub Ω

119896

2 (forall119896 = 1 2 119899 119899 + 1)

=

119899+1

prod

119896=1

119875

(

119899+1

⋂

119895=1

Ω119895

2) sub Ω

119896

1

times

119899+1

prod

119896=1

119875

(

119899+1

⋂

119895=1

Ω119895

1) sub Ω

119896

2

= 120582119899+1

1120582119899+1

2 (0 lt 120582

1lt 1 0 lt 120582

2lt 1)

(54)


119875 Ω119905+1

= Ω119905 Ω119905= Ω119905minus1

Ω2= Ω1 = 120582(119899+1)119905

1120582(119899+1)119905

2

119875 Ω119905+1

Δ119905Ω119905Δ119905minus1

Ω119905minus1


(119899+1)119905

1120582(119899+1)119905

2

(55)


min 119891 (119909)

st ℎ (119909) = 0 119892 (119909) le 0

(56)


where 119891(119909) 119877119899

rarr 119877 ℎ(119909) 119877119899

rarr 119877119898 and 119892(119909)

119877119899



min 119891 (119909) minus 120592

119902

sum

119894=1

ln 120575119894

st ℎ (119909) = 0 119892 (119909) + 120575 = 0

(57)


(1205751 1205752 120575

119902)119879


119871 (119909 119910 119911 120575) = 119891 (119909) minus 120592

119902

sum

119894=1

ln 120575119894+ 119910119879(119892 (119909) + 120575) + 119911

119879ℎ (119909)

(58)


119902)119879

and z = (1199111 1199112 119911




119879

119910 + (nablaℎ (119909))119879119911 = 0

nabla120575119871 (119909 119910 119911 120575) = minus120592119878

minus1

120575119890 + 119910 = 0 lArrrArr minus120592119890 + 119878

120575119884119890 = 0

nabla119910119871 (119909 119910 119911 120575) = 119892 (119909) + 120575 = 0

nabla119911119871 (119909 119910 119911 120575) = ℎ (119909) = 0

(59)

where 119878120575

= diag(1205751 1205752 120575



1 1199102 119910



Δ119909 120575+Δ120575 119911+Δ



(nabla2119891 (119909) + nabla

2119892(119909)119879119910 + nabla

2ℎ(119909)119879119911) Δ119909 + nabla119892(119909)

119879Δ119910


119879119910 + nablaℎ(119909)

119879119911) = 0

minus120592119890 + 119878120575119884119890 + 119878

120575Δ119910 + 119884Δ120575 = 0

119892 (119909) + nabla119892 (119909) Δ119909 + 120575 + Δ120575 = 0

nablaℎ (119909) Δ119909 + ℎ (119909) = 0

(60)


(

119867(119909 119910 119911) 0 nablaℎ(119909)119879

nabla119892(119909)119879

0 119878minus1

120575119884 0 119868

nablaℎ (119909) 0 0 0

nabla119892 (119909) 119868 0 0

)(

Δ119909

Δ120575

Δ119910

Δ119911

)

= (


119879119911

120592119878minus1

120575119890 minus 119910

minusℎ (119909)

minus119892 (119909) minus 120575

)

(61)

119867(119909 119910 119911) = nabla2119891 (119909) + nabla

2119892(119909)119879119910 + nabla

2ℎ(119909)119879119911 (62)


(119909 120575 119911 119910) larr997888 (119909 120575 119911 119910) + 1205771(Δ119909 Δ120575 Δ119911 Δ119910) (63)


1holds the




(0)gt 0 119896 = 0










(119896+1) 120575(119896+1)

119911(119896+1)

119910(119896+1)

) larr (119909(119896)

120575(119896)

119911(119896)

119910(119896)

) +

1205771(Δ119909 Δ120575 Δ119911 Δ119910) 119896 larr 119896 + 1


Step 7 End



1199011(119904) [7]

1198661199012(119904) [15] and 119866

1199013(119904) [6] respectively 119866

1199011(119904) is a third-


1199013(119904) are the SOPTD


1198661199011

(119904) =

15

(1199042+ 09119904 + 5) (119904 + 3)

1198661199012

(119904) =

119890minus05119904

(119904 + 1)2 119866

1199013(119904) =

4119890minus10119904

10119904 + 1

(64)


and 119890minus10119904

asymp (1199042minus 06119904 + 012)(119904

2+ 06119904 + 012) respectively


1199013(119904) are the (minus119904 +

4)(1199043+ 61199042+ 9119904 + 4) and (04119904

2minus 024119904 + 0048)(119904

3+ 07119904

2+



1 1199022 1199023 and 119902

4 The



2 1199023


1(with 119902

2 1199023 and 119902

4


2 1199023 and 119902

4(with 119902




1

















119901119896lowast

119894119896lowast

119889

Plant1

diag (5 5 1 1) 013121 058138 058655

120576 = 0


Plant2

diag (02 1 1 1) 115698 052966 086781

120576 = 05

diag (02 1 1 02) 216962 084500 182140diag (02 2 2 05) 178668 058700 159290

diag (01 08 08 02) 185901 064942 162251diag (04 05 5 1) 122806 062168 142095

Plant3

diag (004 27 158 175) 032447 001537 145256

120576 = 0




1119902211990231199024) 119896

lowast

119901119896lowast

119894119896lowast

119889119905119903(sec) 119905

119889(sec) 119905

119901(sec) 119905

119904(sec) 119872

119901()


0 2 4 6 8 10 12 14

0

02

04

06

08

1

12

Syste

m re

spon

se

minus02

Time (s)

119876 = diag (5 5 1 1)119876 = diag (5 2 1 1)119876 = diag (10 2 1 1)

119876 = diag (10 10 1 1)119876 = diag (5 5 2 1) lowast 2

(a)

0 2 4 6 8 10 12 14

0

02

04

06

08

1

12

Con

trol e

rror

minus02

Time (s)

119876 = diag (5 5 1 1)119876 = diag (5 2 1 1)119876 = diag (10 2 1 1)

119876 = diag (10 10 1 1)119876 = diag (5 5 2 1) lowast 2

(b)



0 1 2 3 4 5 6 7 8 9 100

02

04

06

08

1

12

14

Time (s)

Syste

m re

spon

se

120576 = 05

120576 = 25

120576 = 3

120576 = 4

120576 = 5

(a)

0 1 2 3 4 5 6 7 8 9 10

0

02

04

06

08

1

Time (s)

minus02

minus04

Con

trol e

rror

120576 = 05

120576 = 25

120576 = 3

120576 = 4

120576 = 5

(b)


0 5 10 15 20 25 30

0

02

04

06

08

1

12



Syste

m re

spon

se

Time (s)

minus02

(a)

0 5 10 15 20 25 30

0

02

04

06

08

1

12



Con

trol e

rror

Time (s)

minus02

(b)




119896119894

119896119889

Risetime

119905119903(sec)

Delaytime

119905119889(sec)

Peaktime

119905119901(sec)

Settingtime

119905119904(sec)


Plants



0 2 4 6 8 10 12 14 16 180

05

1

15Sy

stem

resp

onse



Time (s)

(a)

0 2 4 6 8 10 12 14 16 18

0

05

1


minus05


Con

trol e

rror

Time (s)

(b)


0 20 40 60 80 100 120 140 160 180 2000

02

04

06

08

1

12

14



Syste

m re

spon

se

Time (s)

095

105

(a)

0 20 40 60 80 100 120 140 160 180 200

0

02

04

06

08

1



Con

trol e

rror

Time (s)

minus02

minus04

(b)





Time (s)0 5 10 15 20 25 30

0

02

04

06

08

1

12

0 5 10 15 20 25 30

0

02

04

06

08

1

12

Syste

m re

spon

seSy

stem

resp

onse

Con

trol e

rror

Con

trol e

rror

Time (s)

Time (s)Time (s)16 18 20 22 24 26 28

075

08

085

09

095

1

105

11



16 18 20 22 24 26 28

0

005

01

015

02

025

03



minus02 minus02

minus005

119863119906(119905) = 03119906(119905 minus 16)



Methods 119896119901

119896119894

119896119889










16 18 20 22 24 26 28 30086

088

09

092

094

096

098

1

102

104





0 5 10 15 20 25 300

05

1

15

0 5 10 15 20 25 30

0

05

1

18 20 22 24 26 28 30

0

002

004

006

008

01

012

014

Con

trol e

rror

Con

trol e

rror

Syste

m re

spon

seSy

stem

resp

onse

minus05

Time (s) Time (s)

Time (s) Time (s)

minus002

119863119906(119905) = 03119906(119905 minus 18)






200 250 300 350 40007

075

08

085

09

095

1

105

11

200 250 300 350 400

0

005

01

015

02

025

03

0 50 100 150 200 250 300 350 4000

02

04

06

08

1

12

14

0 50 100 150 200 250 300 350 400

0

02

04

06

08

1

095

105

Time (s) Time (s)

Time (s)Time (s)





minus005

Con

trol e

rror

Syste

m re

spon

se

Con

trol e

rror

Syste

m re

spon

se

minus02

minus04

119863119906(119905) = 01119906(119905 minus 180)




1(119904)119876119894119888(119904) = (04465119904 +





119901=

10000 119896119894

= 032549 119896119889

= 0001 120576 = 0 119896119901

=

10000 119896119894

= 010118 119896119889

= 0001 120576 = 0 119896119901

=

10000 119896119894= 022986 119896

119889= 0001 120576 = 0 119896

119901= 6000 119896

119894=



1is set at




0 5 10 15 20 25 300

02

04

06

08

1

12

14

Time (s)

16 18 20 22 24 26 28 30086

088

09

092

094

096

098

1

102

104

0 5 10 15 20 25 30

0

02

04

06

08

1

Time (s)

Time (s) Time (s)

16 18 20 22 24 26 28 30

0

002

004

006

008

01

012

014

minus02

minus04

minus002

120576 = 05

120576 = 25

120576 = 3

120576 = 4

120576 = 5

120576 = 05

120576 = 25

120576 = 3

120576 = 4

120576 = 5



1 the solid







0 200 400 600 800 1000 1200 1400 1600 1800 20000

50

100

150

200

250

300

Time (s)

Liqu

id le

vel o

f CW

TS (m

m)




lowast

119901= 10000 119896lowast

119894= 032549 119896lowast

119889= 0001 and 120576 = 0 in

set point 300mm


119901= 10000 119896lowast

119894= 010118 119896lowast

119889= 0001 and 120576 = 0 in

set point 300mm


119901= 10000 119896lowast

119894= 022986 119896lowast

119889= 0001 and 120576 = 0 in

set point 300mm


lowast

119901= 6000 119896lowast

119894= 013817 119896lowast

119889= 0001 and 120576 = 0 in

set point 300mm


5 Conclusions




119901= 10000 119896lowast

119894= 032549 119896lowast

119889= 0001 and 120576 = 0 in

set point 160mm


119901= 10000 119896lowast

119894= 010118 119896lowast

119889= 0001 and 120576 = 0 in

set point 160mm


1 1199022 1199023 and 119902





2 1199023 and 119902

4(with 119902





4(with 119902

1fixed) Through the


1 1199022 119902

119899+1 The 120576-Routh




119901= 10000 119896lowast

119894= 022986 119896lowast

119889= 0001 and 120576 = 0 in

set point 160mm


lowast

119901= 6000 119896lowast

119894= 013817 119896lowast

119889= 0001 and 120576 = 0 in

set point 160mm



Acknowledgments



References











































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












119901 119896lowast

119894 119896lowast

119889) = min119866lowast

119888(119904)

119869 (1198981 119898

5)

= min119909(0)119879119875119909 (0)

st

(1) 119860119879119875 + 119875119860 = minus119876

(2)1198981

1gt 0 119898

1

4gt 0 119898

1

11198981

2minus 1198981

3gt 0

1198981

11198981

21198981

3minus (1198981

3)

2

minus 1198981

4(1198981

1)

2

gt 0

(3)1198981gt 120576 119898

2gt 120576 119898

5gt 120576 119898

11198984minus 11989821198983gt 1205761198982

(1198982)2

1198985minus 1198981(1198984)2

minus 119898211989831198984gt 120576 (119898

11198984minus 11989821198983)

(4)1198984= 1198911(1198981 1198982 1198983) 1198985= 1198912(1198981 1198982 1198983)

(5) 0 lt 119896119901le 1205910

119901 0 lt 119896

119894le 1205910

119894 0 lt 119896

119889le 1205910

119889

(34)


1(0) = 1 119909

2(0) = 0 119909

3(0) = 0 and




119894and


119866lowast

119888(119904) = 119896

lowast

119901+

119896lowast

119894

119904

+ 119896lowast

119889119904 119896

lowast

119901=

119898lowast


119894

1198873

119896lowast

119894=

119898lowast

5

1198873

119896lowast

119889=

119898lowast

1minus 1198860

1198871

(35)



1119904119899+sdot sdot sdot+119886

119899minus119908minus2119904119908+3

+

1198981119904119908+2

+ 1198982119904119908+1

+ sdot sdot sdot + 119898119908+2

119904 + 119898119908+3




119889 The

parameters 119896119901 119896119894 and 119896


119901]

(0 1205910

119894] and (0 120591

0



119901 119896119894



119865 (119904) = 119904119899+ 1199011119904119899minus1

+ 1199012119904119899minus2


119904 + 119901119899 (36)

where 1199011 1199012 and 119901





119898

1199011= 1198911(1199021 119902

119898) 119901

119899= 119891119899(1199021 119902

119898)

Ω119902= (1199021 119902

119898)1003816100381610038161003816[119902minus

1 119902+


minus

119898 119902+

119898]

1199021isin [119902minus

1 119902+

1] 119902

119898isin [119902minus

119898 119902+

119898]

(37)


119894and 119902+

119894are the




11987711

(1199021 119902

119898) = 1198911199031

(1199011 119901

119899) gt 120576

119877119899+11

(1199021 119902

119898) = 119891119903119899+1

(1199011 119901

119899) gt 120576

Ω119903120576

= (1199021 119902

119898) | 1198911199031

(1199011 119901

119899) gt 120576

119891119903119899+1

(1199011 119901

119899) gt 120576 (120576 ge 0)

(38)







min(1199021 119902119898)isinΩ119891

1198911199031

(1199011 119901

119899) 119891

119903119899+1(1199011 119901

119899) gt 0

(Ω119891= (Ω119903120576

⋂Ω119902) = 0)

(39)







I Ω1= (119902

1 119902

119898) | 1198911199031

(1199011 119901

119899) gt 1205761

119891119903119899+1

(1199011 119901

119899) gt 1205761

II Ω2= (119902

1 119902

119898) | 1198911199031

(1199011 119901

119899) gt 1205762

119891119903119899+1

(1199011 119901

119899) gt 1205762

(40)




Ω1

1= (1199021 119902

119898) | 1198911199031

(1199011 119901

119899) gt 1205761

Ω1

2= (1199021 119902

119898) | 1198911199031

(1199011 119901

119899) gt 1205762

(41)



1

2will have the

relationship

exist119872lowast

1= (119902lowast

11 119902lowast

21 119902

lowast

1198981) 119891

1199031(119872lowast

1) = 120576lowast

2gt 1205762 (42)


1will have the

relationship

exist119872lowast

2= (119902lowast

12 119902lowast

22 119902

lowast

1198982) 119891

1199031(119872lowast

2) = 120576lowast

1gt 1205761 (43)


1which will



1= (119902lowast

11 119902lowast

21 119902

lowast

1198981) 1205761lt 1198911199031

(119873lowast

1) lt 1205762

lArrrArr 1198911199031

(119873lowast

1) isin (1205761 1205762)


1isin Ω1

1 119873lowast

1notin Ω1

2997904rArr Ω

1

1sup Ω1

2

Case 2 1198911199031

(119873lowast

1) notin (1205761 1205762) 997904rArr Ω

1

1= Ω1

2

(44)


119896


relationship

Case 1 Ω119896

1sup Ω119896

2 (forall119896 = 1 2 119899 119899 + 1)

Case 2 Ω119896

1= Ω119896

2 (forall119896 = 1 2 119899 119899 + 1)

Ω119896

1= (1199021 119902

119898) | 119891119903119896

(1199011 119901

119899) gt 1205761

Ω119896

2= (1199021 119902

119898) | 119891119903119896

(1199011 119901

119899) gt 1205762

(45)


2will have follow-

ing relationships


1sup Ω2) cup (Ω

1= Ω2)

Ω1=

119899+1

⋂

119895=1

Ω119895

1 Ω

2=

119899+1

⋂

119895=1

Ω119895

2

(46)


119905+1gt 120576119905gt 120576119905minus1



Ω119905= (1199021 119902

119898) | 1198911199031

(1199011 119901

119899) gt 120576119905

119891119903119899+1

(1199011 119901

119899) gt 120576119905



119899+1

119905) = (

119899+1

⋂

119895=1

Ω119895

119905)

(119905 = 1 2 3 119905 isin Z+)

Ω119896

119905= (1199021 119902

119898) | 119891119903119896

(1199011 119901

119899) gt 120576119905

(119896 = 1 2 119899 119899 + 1)

(47)


ship

Ω119905+1


2sube Ω1



2Ω2Δ1Ω1)

⋃ (Ω119905+1


2= Ω1)


(48)


119905 then the



(1199021 119902

119898) | 1198911199031

(1199011 119901

119899) gt 120576119905

119891119903119899+1

(1199011 119901

119899) gt 120576119905


Ω119905= lim120576119905rarr+infin

(Ω1

119905cap Ω2


119899+1

119905)


(

119899+1

⋂

119895=1

Ω119895

119905) = Ω

lowast

= (119902lowast

10119902lowast

20 119902

lowast

1198980)



(1199021 119902

119898) | 1198911199031

(1199011 119901

119899) gt 120576119905

119891119903119899+1

(1199011 119901

119899) gt 120576119905


Ω119905= lim120576119905rarr+infin

(Ω1

119905cap Ω2


119899+1

119905)


(

119899+1

⋂

119895=1

Ω119895

119905) = 0

(49)





120576119905997888rarr +infin lim



lowast

119905 120576119905gt 120576lowast

119905 Ω119905= 0

0 lt 120576119905le 120576lowast

119905 Ω119905

= 0

120576lowast

119905gt 120576119905gt 120576119905minus1


Ωlowast

119905sub Ω119905sub Ω119905minus1


(50)


= Ω119905= Ω119905minus1

=



119875 Ω119905+1

= Ω119905= Ω119905minus1


lArrrArr 119875 Ω119905+1

= Ω119905 Ω119905= Ω119905minus1

Ω2= Ω1

(51)


119875 Ω119905+1

= Ω119905 Ω119905= Ω119905minus1

Ω2= Ω1

= 119875 Ω119905+1

= Ω119905 times 119875 Ω

119905= Ω119905minus1


=

119905+1

prod

119895=2

119875 Ω119895= Ω119895minus1

= (119875 Ω2= Ω1)119905

(119905 = 1 2 119905 isin Z+)

(52)



= 119875 (Ω1


119899+1

1) = (Ω

1


119899+1

2)

= 119875

(

119899+1

⋂

119895=1

Ω119895

1) = (

119899+1

⋂

119895=1

Ω119895

2)

lArrrArr 119875(Ω1


119899+1

2) sub Ω

119896

1

(Ω1


119899+1

1) sub Ω

119896

2

(forall119896 = 1 2 119899 119899 + 1)

(53)


2=

Ω1 will yield

119875 (Ω1

2cap Ω2


119899

2) sub Ω

1

1 = 1205821

119875 (Ω1

2cap Ω2


119899

2) sub Ω

119899+1

1 = 1205821

119875 (Ω1

1cap Ω2


119899

1) sub Ω

1

2 = 1205822

119875 (Ω1

1cap Ω2


119899

1) sub Ω

119899+1

2 = 1205822

119875 Ω1= Ω2

= 119875

(

119899+1

⋂

119895=1

Ω119895

2) sub Ω

119896

1 (forall119896 = 1 2 119899 119899 + 1)

times 119875

(

119899+1

⋂

119895=1

Ω119895

1) sub Ω

119896

2 (forall119896 = 1 2 119899 119899 + 1)

=

119899+1

prod

119896=1

119875

(

119899+1

⋂

119895=1

Ω119895

2) sub Ω

119896

1

times

119899+1

prod

119896=1

119875

(

119899+1

⋂

119895=1

Ω119895

1) sub Ω

119896

2

= 120582119899+1

1120582119899+1

2 (0 lt 120582

1lt 1 0 lt 120582

2lt 1)

(54)


119875 Ω119905+1

= Ω119905 Ω119905= Ω119905minus1

Ω2= Ω1 = 120582(119899+1)119905

1120582(119899+1)119905

2

119875 Ω119905+1

Δ119905Ω119905Δ119905minus1

Ω119905minus1


(119899+1)119905

1120582(119899+1)119905

2

(55)


min 119891 (119909)

st ℎ (119909) = 0 119892 (119909) le 0

(56)


where 119891(119909) 119877119899

rarr 119877 ℎ(119909) 119877119899

rarr 119877119898 and 119892(119909)

119877119899



min 119891 (119909) minus 120592

119902

sum

119894=1

ln 120575119894

st ℎ (119909) = 0 119892 (119909) + 120575 = 0

(57)


(1205751 1205752 120575

119902)119879


119871 (119909 119910 119911 120575) = 119891 (119909) minus 120592

119902

sum

119894=1

ln 120575119894+ 119910119879(119892 (119909) + 120575) + 119911

119879ℎ (119909)

(58)


119902)119879

and z = (1199111 1199112 119911




119879

119910 + (nablaℎ (119909))119879119911 = 0

nabla120575119871 (119909 119910 119911 120575) = minus120592119878

minus1

120575119890 + 119910 = 0 lArrrArr minus120592119890 + 119878

120575119884119890 = 0

nabla119910119871 (119909 119910 119911 120575) = 119892 (119909) + 120575 = 0

nabla119911119871 (119909 119910 119911 120575) = ℎ (119909) = 0

(59)

where 119878120575

= diag(1205751 1205752 120575



1 1199102 119910



Δ119909 120575+Δ120575 119911+Δ



(nabla2119891 (119909) + nabla

2119892(119909)119879119910 + nabla

2ℎ(119909)119879119911) Δ119909 + nabla119892(119909)

119879Δ119910


119879119910 + nablaℎ(119909)

119879119911) = 0

minus120592119890 + 119878120575119884119890 + 119878

120575Δ119910 + 119884Δ120575 = 0

119892 (119909) + nabla119892 (119909) Δ119909 + 120575 + Δ120575 = 0

nablaℎ (119909) Δ119909 + ℎ (119909) = 0

(60)


(

119867(119909 119910 119911) 0 nablaℎ(119909)119879

nabla119892(119909)119879

0 119878minus1

120575119884 0 119868

nablaℎ (119909) 0 0 0

nabla119892 (119909) 119868 0 0

)(

Δ119909

Δ120575

Δ119910

Δ119911

)

= (


119879119911

120592119878minus1

120575119890 minus 119910

minusℎ (119909)

minus119892 (119909) minus 120575

)

(61)

119867(119909 119910 119911) = nabla2119891 (119909) + nabla

2119892(119909)119879119910 + nabla

2ℎ(119909)119879119911 (62)


(119909 120575 119911 119910) larr997888 (119909 120575 119911 119910) + 1205771(Δ119909 Δ120575 Δ119911 Δ119910) (63)


1holds the




(0)gt 0 119896 = 0










(119896+1) 120575(119896+1)

119911(119896+1)

119910(119896+1)

) larr (119909(119896)

120575(119896)

119911(119896)

119910(119896)

) +

1205771(Δ119909 Δ120575 Δ119911 Δ119910) 119896 larr 119896 + 1


Step 7 End



1199011(119904) [7]

1198661199012(119904) [15] and 119866

1199013(119904) [6] respectively 119866

1199011(119904) is a third-


1199013(119904) are the SOPTD


1198661199011

(119904) =

15

(1199042+ 09119904 + 5) (119904 + 3)

1198661199012

(119904) =

119890minus05119904

(119904 + 1)2 119866

1199013(119904) =

4119890minus10119904

10119904 + 1

(64)


and 119890minus10119904

asymp (1199042minus 06119904 + 012)(119904

2+ 06119904 + 012) respectively


1199013(119904) are the (minus119904 +

4)(1199043+ 61199042+ 9119904 + 4) and (04119904

2minus 024119904 + 0048)(119904

3+ 07119904

2+



1 1199022 1199023 and 119902

4 The



2 1199023


1(with 119902

2 1199023 and 119902

4


2 1199023 and 119902

4(with 119902




1

















119901119896lowast

119894119896lowast

119889

Plant1

diag (5 5 1 1) 013121 058138 058655

120576 = 0


Plant2

diag (02 1 1 1) 115698 052966 086781

120576 = 05

diag (02 1 1 02) 216962 084500 182140diag (02 2 2 05) 178668 058700 159290

diag (01 08 08 02) 185901 064942 162251diag (04 05 5 1) 122806 062168 142095

Plant3

diag (004 27 158 175) 032447 001537 145256

120576 = 0




1119902211990231199024) 119896

lowast

119901119896lowast

119894119896lowast

119889119905119903(sec) 119905

119889(sec) 119905

119901(sec) 119905

119904(sec) 119872

119901()


0 2 4 6 8 10 12 14

0

02

04

06

08

1

12

Syste

m re

spon

se

minus02

Time (s)

119876 = diag (5 5 1 1)119876 = diag (5 2 1 1)119876 = diag (10 2 1 1)

119876 = diag (10 10 1 1)119876 = diag (5 5 2 1) lowast 2

(a)

0 2 4 6 8 10 12 14

0

02

04

06

08

1

12

Con

trol e

rror

minus02

Time (s)

119876 = diag (5 5 1 1)119876 = diag (5 2 1 1)119876 = diag (10 2 1 1)

119876 = diag (10 10 1 1)119876 = diag (5 5 2 1) lowast 2

(b)



0 1 2 3 4 5 6 7 8 9 100

02

04

06

08

1

12

14

Time (s)

Syste

m re

spon

se

120576 = 05

120576 = 25

120576 = 3

120576 = 4

120576 = 5

(a)

0 1 2 3 4 5 6 7 8 9 10

0

02

04

06

08

1

Time (s)

minus02

minus04

Con

trol e

rror

120576 = 05

120576 = 25

120576 = 3

120576 = 4

120576 = 5

(b)


0 5 10 15 20 25 30

0

02

04

06

08

1

12



Syste

m re

spon

se

Time (s)

minus02

(a)

0 5 10 15 20 25 30

0

02

04

06

08

1

12



Con

trol e

rror

Time (s)

minus02

(b)




119896119894

119896119889

Risetime

119905119903(sec)

Delaytime

119905119889(sec)

Peaktime

119905119901(sec)

Settingtime

119905119904(sec)


Plants



0 2 4 6 8 10 12 14 16 180

05

1

15Sy

stem

resp

onse



Time (s)

(a)

0 2 4 6 8 10 12 14 16 18

0

05

1


minus05


Con

trol e

rror

Time (s)

(b)


0 20 40 60 80 100 120 140 160 180 2000

02

04

06

08

1

12

14



Syste

m re

spon

se

Time (s)

095

105

(a)

0 20 40 60 80 100 120 140 160 180 200

0

02

04

06

08

1



Con

trol e

rror

Time (s)

minus02

minus04

(b)





Time (s)0 5 10 15 20 25 30

0

02

04

06

08

1

12

0 5 10 15 20 25 30

0

02

04

06

08

1

12

Syste

m re

spon

seSy

stem

resp

onse

Con

trol e

rror

Con

trol e

rror

Time (s)

Time (s)Time (s)16 18 20 22 24 26 28

075

08

085

09

095

1

105

11



16 18 20 22 24 26 28

0

005

01

015

02

025

03



minus02 minus02

minus005

119863119906(119905) = 03119906(119905 minus 16)



Methods 119896119901

119896119894

119896119889










16 18 20 22 24 26 28 30086

088

09

092

094

096

098

1

102

104





0 5 10 15 20 25 300

05

1

15

0 5 10 15 20 25 30

0

05

1

18 20 22 24 26 28 30

0

002

004

006

008

01

012

014

Con

trol e

rror

Con

trol e

rror

Syste

m re

spon

seSy

stem

resp

onse

minus05

Time (s) Time (s)

Time (s) Time (s)

minus002

119863119906(119905) = 03119906(119905 minus 18)






200 250 300 350 40007

075

08

085

09

095

1

105

11

200 250 300 350 400

0

005

01

015

02

025

03

0 50 100 150 200 250 300 350 4000

02

04

06

08

1

12

14

0 50 100 150 200 250 300 350 400

0

02

04

06

08

1

095

105

Time (s) Time (s)

Time (s)Time (s)





minus005

Con

trol e

rror

Syste

m re

spon

se

Con

trol e

rror

Syste

m re

spon

se

minus02

minus04

119863119906(119905) = 01119906(119905 minus 180)




1(119904)119876119894119888(119904) = (04465119904 +





119901=

10000 119896119894

= 032549 119896119889

= 0001 120576 = 0 119896119901

=

10000 119896119894

= 010118 119896119889

= 0001 120576 = 0 119896119901

=

10000 119896119894= 022986 119896

119889= 0001 120576 = 0 119896

119901= 6000 119896

119894=



1is set at




0 5 10 15 20 25 300

02

04

06

08

1

12

14

Time (s)

16 18 20 22 24 26 28 30086

088

09

092

094

096

098

1

102

104

0 5 10 15 20 25 30

0

02

04

06

08

1

Time (s)

Time (s) Time (s)

16 18 20 22 24 26 28 30

0

002

004

006

008

01

012

014

minus02

minus04

minus002

120576 = 05

120576 = 25

120576 = 3

120576 = 4

120576 = 5

120576 = 05

120576 = 25

120576 = 3

120576 = 4

120576 = 5



1 the solid







0 200 400 600 800 1000 1200 1400 1600 1800 20000

50

100

150

200

250

300

Time (s)

Liqu

id le

vel o

f CW

TS (m

m)




lowast

119901= 10000 119896lowast

119894= 032549 119896lowast

119889= 0001 and 120576 = 0 in

set point 300mm


119901= 10000 119896lowast

119894= 010118 119896lowast

119889= 0001 and 120576 = 0 in

set point 300mm


119901= 10000 119896lowast

119894= 022986 119896lowast

119889= 0001 and 120576 = 0 in

set point 300mm


lowast

119901= 6000 119896lowast

119894= 013817 119896lowast

119889= 0001 and 120576 = 0 in

set point 300mm


5 Conclusions




119901= 10000 119896lowast

119894= 032549 119896lowast

119889= 0001 and 120576 = 0 in

set point 160mm


119901= 10000 119896lowast

119894= 010118 119896lowast

119889= 0001 and 120576 = 0 in

set point 160mm


1 1199022 1199023 and 119902





2 1199023 and 119902

4(with 119902





4(with 119902

1fixed) Through the


1 1199022 119902

119899+1 The 120576-Routh




119901= 10000 119896lowast

119894= 022986 119896lowast

119889= 0001 and 120576 = 0 in

set point 160mm


lowast

119901= 6000 119896lowast

119894= 013817 119896lowast

119889= 0001 and 120576 = 0 in

set point 160mm



Acknowledgments



References











































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













I Ω1= (119902

1 119902

119898) | 1198911199031

(1199011 119901

119899) gt 1205761

119891119903119899+1

(1199011 119901

119899) gt 1205761

II Ω2= (119902

1 119902

119898) | 1198911199031

(1199011 119901

119899) gt 1205762

119891119903119899+1

(1199011 119901

119899) gt 1205762

(40)




Ω1

1= (1199021 119902

119898) | 1198911199031

(1199011 119901

119899) gt 1205761

Ω1

2= (1199021 119902

119898) | 1198911199031

(1199011 119901

119899) gt 1205762

(41)



1

2will have the

relationship

exist119872lowast

1= (119902lowast

11 119902lowast

21 119902

lowast

1198981) 119891

1199031(119872lowast

1) = 120576lowast

2gt 1205762 (42)


1will have the

relationship

exist119872lowast

2= (119902lowast

12 119902lowast

22 119902

lowast

1198982) 119891

1199031(119872lowast

2) = 120576lowast

1gt 1205761 (43)


1which will



1= (119902lowast

11 119902lowast

21 119902

lowast

1198981) 1205761lt 1198911199031

(119873lowast

1) lt 1205762

lArrrArr 1198911199031

(119873lowast

1) isin (1205761 1205762)


1isin Ω1

1 119873lowast

1notin Ω1

2997904rArr Ω

1

1sup Ω1

2

Case 2 1198911199031

(119873lowast

1) notin (1205761 1205762) 997904rArr Ω

1

1= Ω1

2

(44)


119896


relationship

Case 1 Ω119896

1sup Ω119896

2 (forall119896 = 1 2 119899 119899 + 1)

Case 2 Ω119896

1= Ω119896

2 (forall119896 = 1 2 119899 119899 + 1)

Ω119896

1= (1199021 119902

119898) | 119891119903119896

(1199011 119901

119899) gt 1205761

Ω119896

2= (1199021 119902

119898) | 119891119903119896

(1199011 119901

119899) gt 1205762

(45)


2will have follow-

ing relationships


1sup Ω2) cup (Ω

1= Ω2)

Ω1=

119899+1

⋂

119895=1

Ω119895

1 Ω

2=

119899+1

⋂

119895=1

Ω119895

2

(46)


119905+1gt 120576119905gt 120576119905minus1



Ω119905= (1199021 119902

119898) | 1198911199031

(1199011 119901

119899) gt 120576119905

119891119903119899+1

(1199011 119901

119899) gt 120576119905



119899+1

119905) = (

119899+1

⋂

119895=1

Ω119895

119905)

(119905 = 1 2 3 119905 isin Z+)

Ω119896

119905= (1199021 119902

119898) | 119891119903119896

(1199011 119901

119899) gt 120576119905

(119896 = 1 2 119899 119899 + 1)

(47)


ship

Ω119905+1


2sube Ω1



2Ω2Δ1Ω1)

⋃ (Ω119905+1


2= Ω1)


(48)


119905 then the



(1199021 119902

119898) | 1198911199031

(1199011 119901

119899) gt 120576119905

119891119903119899+1

(1199011 119901

119899) gt 120576119905


Ω119905= lim120576119905rarr+infin

(Ω1

119905cap Ω2


119899+1

119905)


(

119899+1

⋂

119895=1

Ω119895

119905) = Ω

lowast

= (119902lowast

10119902lowast

20 119902

lowast

1198980)



(1199021 119902

119898) | 1198911199031

(1199011 119901

119899) gt 120576119905

119891119903119899+1

(1199011 119901

119899) gt 120576119905


Ω119905= lim120576119905rarr+infin

(Ω1

119905cap Ω2


119899+1

119905)


(

119899+1

⋂

119895=1

Ω119895

119905) = 0

(49)





120576119905997888rarr +infin lim



lowast

119905 120576119905gt 120576lowast

119905 Ω119905= 0

0 lt 120576119905le 120576lowast

119905 Ω119905

= 0

120576lowast

119905gt 120576119905gt 120576119905minus1


Ωlowast

119905sub Ω119905sub Ω119905minus1


(50)


= Ω119905= Ω119905minus1

=



119875 Ω119905+1

= Ω119905= Ω119905minus1


lArrrArr 119875 Ω119905+1

= Ω119905 Ω119905= Ω119905minus1

Ω2= Ω1

(51)


119875 Ω119905+1

= Ω119905 Ω119905= Ω119905minus1

Ω2= Ω1

= 119875 Ω119905+1

= Ω119905 times 119875 Ω

119905= Ω119905minus1


=

119905+1

prod

119895=2

119875 Ω119895= Ω119895minus1

= (119875 Ω2= Ω1)119905

(119905 = 1 2 119905 isin Z+)

(52)



= 119875 (Ω1


119899+1

1) = (Ω

1


119899+1

2)

= 119875

(

119899+1

⋂

119895=1

Ω119895

1) = (

119899+1

⋂

119895=1

Ω119895

2)

lArrrArr 119875(Ω1


119899+1

2) sub Ω

119896

1

(Ω1


119899+1

1) sub Ω

119896

2

(forall119896 = 1 2 119899 119899 + 1)

(53)


2=

Ω1 will yield

119875 (Ω1

2cap Ω2


119899

2) sub Ω

1

1 = 1205821

119875 (Ω1

2cap Ω2


119899

2) sub Ω

119899+1

1 = 1205821

119875 (Ω1

1cap Ω2


119899

1) sub Ω

1

2 = 1205822

119875 (Ω1

1cap Ω2


119899

1) sub Ω

119899+1

2 = 1205822

119875 Ω1= Ω2

= 119875

(

119899+1

⋂

119895=1

Ω119895

2) sub Ω

119896

1 (forall119896 = 1 2 119899 119899 + 1)

times 119875

(

119899+1

⋂

119895=1

Ω119895

1) sub Ω

119896

2 (forall119896 = 1 2 119899 119899 + 1)

=

119899+1

prod

119896=1

119875

(

119899+1

⋂

119895=1

Ω119895

2) sub Ω

119896

1

times

119899+1

prod

119896=1

119875

(

119899+1

⋂

119895=1

Ω119895

1) sub Ω

119896

2

= 120582119899+1

1120582119899+1

2 (0 lt 120582

1lt 1 0 lt 120582

2lt 1)

(54)


119875 Ω119905+1

= Ω119905 Ω119905= Ω119905minus1

Ω2= Ω1 = 120582(119899+1)119905

1120582(119899+1)119905

2

119875 Ω119905+1

Δ119905Ω119905Δ119905minus1

Ω119905minus1


(119899+1)119905

1120582(119899+1)119905

2

(55)


min 119891 (119909)

st ℎ (119909) = 0 119892 (119909) le 0

(56)


where 119891(119909) 119877119899

rarr 119877 ℎ(119909) 119877119899

rarr 119877119898 and 119892(119909)

119877119899



min 119891 (119909) minus 120592

119902

sum

119894=1

ln 120575119894

st ℎ (119909) = 0 119892 (119909) + 120575 = 0

(57)


(1205751 1205752 120575

119902)119879


119871 (119909 119910 119911 120575) = 119891 (119909) minus 120592

119902

sum

119894=1

ln 120575119894+ 119910119879(119892 (119909) + 120575) + 119911

119879ℎ (119909)

(58)


119902)119879

and z = (1199111 1199112 119911




119879

119910 + (nablaℎ (119909))119879119911 = 0

nabla120575119871 (119909 119910 119911 120575) = minus120592119878

minus1

120575119890 + 119910 = 0 lArrrArr minus120592119890 + 119878

120575119884119890 = 0

nabla119910119871 (119909 119910 119911 120575) = 119892 (119909) + 120575 = 0

nabla119911119871 (119909 119910 119911 120575) = ℎ (119909) = 0

(59)

where 119878120575

= diag(1205751 1205752 120575



1 1199102 119910



Δ119909 120575+Δ120575 119911+Δ



(nabla2119891 (119909) + nabla

2119892(119909)119879119910 + nabla

2ℎ(119909)119879119911) Δ119909 + nabla119892(119909)

119879Δ119910


119879119910 + nablaℎ(119909)

119879119911) = 0

minus120592119890 + 119878120575119884119890 + 119878

120575Δ119910 + 119884Δ120575 = 0

119892 (119909) + nabla119892 (119909) Δ119909 + 120575 + Δ120575 = 0

nablaℎ (119909) Δ119909 + ℎ (119909) = 0

(60)


(

119867(119909 119910 119911) 0 nablaℎ(119909)119879

nabla119892(119909)119879

0 119878minus1

120575119884 0 119868

nablaℎ (119909) 0 0 0

nabla119892 (119909) 119868 0 0

)(

Δ119909

Δ120575

Δ119910

Δ119911

)

= (


119879119911

120592119878minus1

120575119890 minus 119910

minusℎ (119909)

minus119892 (119909) minus 120575

)

(61)

119867(119909 119910 119911) = nabla2119891 (119909) + nabla

2119892(119909)119879119910 + nabla

2ℎ(119909)119879119911 (62)


(119909 120575 119911 119910) larr997888 (119909 120575 119911 119910) + 1205771(Δ119909 Δ120575 Δ119911 Δ119910) (63)


1holds the




(0)gt 0 119896 = 0










(119896+1) 120575(119896+1)

119911(119896+1)

119910(119896+1)

) larr (119909(119896)

120575(119896)

119911(119896)

119910(119896)

) +

1205771(Δ119909 Δ120575 Δ119911 Δ119910) 119896 larr 119896 + 1


Step 7 End



1199011(119904) [7]

1198661199012(119904) [15] and 119866

1199013(119904) [6] respectively 119866

1199011(119904) is a third-


1199013(119904) are the SOPTD


1198661199011

(119904) =

15

(1199042+ 09119904 + 5) (119904 + 3)

1198661199012

(119904) =

119890minus05119904

(119904 + 1)2 119866

1199013(119904) =

4119890minus10119904

10119904 + 1

(64)


and 119890minus10119904

asymp (1199042minus 06119904 + 012)(119904

2+ 06119904 + 012) respectively


1199013(119904) are the (minus119904 +

4)(1199043+ 61199042+ 9119904 + 4) and (04119904

2minus 024119904 + 0048)(119904

3+ 07119904

2+



1 1199022 1199023 and 119902

4 The



2 1199023


1(with 119902

2 1199023 and 119902

4


2 1199023 and 119902

4(with 119902




1

















119901119896lowast

119894119896lowast

119889

Plant1

diag (5 5 1 1) 013121 058138 058655

120576 = 0


Plant2

diag (02 1 1 1) 115698 052966 086781

120576 = 05

diag (02 1 1 02) 216962 084500 182140diag (02 2 2 05) 178668 058700 159290

diag (01 08 08 02) 185901 064942 162251diag (04 05 5 1) 122806 062168 142095

Plant3

diag (004 27 158 175) 032447 001537 145256

120576 = 0




1119902211990231199024) 119896

lowast

119901119896lowast

119894119896lowast

119889119905119903(sec) 119905

119889(sec) 119905

119901(sec) 119905

119904(sec) 119872

119901()


0 2 4 6 8 10 12 14

0

02

04

06

08

1

12

Syste

m re

spon

se

minus02

Time (s)

119876 = diag (5 5 1 1)119876 = diag (5 2 1 1)119876 = diag (10 2 1 1)

119876 = diag (10 10 1 1)119876 = diag (5 5 2 1) lowast 2

(a)

0 2 4 6 8 10 12 14

0

02

04

06

08

1

12

Con

trol e

rror

minus02

Time (s)

119876 = diag (5 5 1 1)119876 = diag (5 2 1 1)119876 = diag (10 2 1 1)

119876 = diag (10 10 1 1)119876 = diag (5 5 2 1) lowast 2

(b)



0 1 2 3 4 5 6 7 8 9 100

02

04

06

08

1

12

14

Time (s)

Syste

m re

spon

se

120576 = 05

120576 = 25

120576 = 3

120576 = 4

120576 = 5

(a)

0 1 2 3 4 5 6 7 8 9 10

0

02

04

06

08

1

Time (s)

minus02

minus04

Con

trol e

rror

120576 = 05

120576 = 25

120576 = 3

120576 = 4

120576 = 5

(b)


0 5 10 15 20 25 30

0

02

04

06

08

1

12



Syste

m re

spon

se

Time (s)

minus02

(a)

0 5 10 15 20 25 30

0

02

04

06

08

1

12



Con

trol e

rror

Time (s)

minus02

(b)




119896119894

119896119889

Risetime

119905119903(sec)

Delaytime

119905119889(sec)

Peaktime

119905119901(sec)

Settingtime

119905119904(sec)


Plants



0 2 4 6 8 10 12 14 16 180

05

1

15Sy

stem

resp

onse



Time (s)

(a)

0 2 4 6 8 10 12 14 16 18

0

05

1


minus05


Con

trol e

rror

Time (s)

(b)


0 20 40 60 80 100 120 140 160 180 2000

02

04

06

08

1

12

14



Syste

m re

spon

se

Time (s)

095

105

(a)

0 20 40 60 80 100 120 140 160 180 200

0

02

04

06

08

1



Con

trol e

rror

Time (s)

minus02

minus04

(b)





Time (s)0 5 10 15 20 25 30

0

02

04

06

08

1

12

0 5 10 15 20 25 30

0

02

04

06

08

1

12

Syste

m re

spon

seSy

stem

resp

onse

Con

trol e

rror

Con

trol e

rror

Time (s)

Time (s)Time (s)16 18 20 22 24 26 28

075

08

085

09

095

1

105

11



16 18 20 22 24 26 28

0

005

01

015

02

025

03



minus02 minus02

minus005

119863119906(119905) = 03119906(119905 minus 16)



Methods 119896119901

119896119894

119896119889










16 18 20 22 24 26 28 30086

088

09

092

094

096

098

1

102

104





0 5 10 15 20 25 300

05

1

15

0 5 10 15 20 25 30

0

05

1

18 20 22 24 26 28 30

0

002

004

006

008

01

012

014

Con

trol e

rror

Con

trol e

rror

Syste

m re

spon

seSy

stem

resp

onse

minus05

Time (s) Time (s)

Time (s) Time (s)

minus002

119863119906(119905) = 03119906(119905 minus 18)






200 250 300 350 40007

075

08

085

09

095

1

105

11

200 250 300 350 400

0

005

01

015

02

025

03

0 50 100 150 200 250 300 350 4000

02

04

06

08

1

12

14

0 50 100 150 200 250 300 350 400

0

02

04

06

08

1

095

105

Time (s) Time (s)

Time (s)Time (s)





minus005

Con

trol e

rror

Syste

m re

spon

se

Con

trol e

rror

Syste

m re

spon

se

minus02

minus04

119863119906(119905) = 01119906(119905 minus 180)




1(119904)119876119894119888(119904) = (04465119904 +





119901=

10000 119896119894

= 032549 119896119889

= 0001 120576 = 0 119896119901

=

10000 119896119894

= 010118 119896119889

= 0001 120576 = 0 119896119901

=

10000 119896119894= 022986 119896

119889= 0001 120576 = 0 119896

119901= 6000 119896

119894=



1is set at




0 5 10 15 20 25 300

02

04

06

08

1

12

14

Time (s)

16 18 20 22 24 26 28 30086

088

09

092

094

096

098

1

102

104

0 5 10 15 20 25 30

0

02

04

06

08

1

Time (s)

Time (s) Time (s)

16 18 20 22 24 26 28 30

0

002

004

006

008

01

012

014

minus02

minus04

minus002

120576 = 05

120576 = 25

120576 = 3

120576 = 4

120576 = 5

120576 = 05

120576 = 25

120576 = 3

120576 = 4

120576 = 5



1 the solid







0 200 400 600 800 1000 1200 1400 1600 1800 20000

50

100

150

200

250

300

Time (s)

Liqu

id le

vel o

f CW

TS (m

m)




lowast

119901= 10000 119896lowast

119894= 032549 119896lowast

119889= 0001 and 120576 = 0 in

set point 300mm


119901= 10000 119896lowast

119894= 010118 119896lowast

119889= 0001 and 120576 = 0 in

set point 300mm


119901= 10000 119896lowast

119894= 022986 119896lowast

119889= 0001 and 120576 = 0 in

set point 300mm


lowast

119901= 6000 119896lowast

119894= 013817 119896lowast

119889= 0001 and 120576 = 0 in

set point 300mm


5 Conclusions




119901= 10000 119896lowast

119894= 032549 119896lowast

119889= 0001 and 120576 = 0 in

set point 160mm


119901= 10000 119896lowast

119894= 010118 119896lowast

119889= 0001 and 120576 = 0 in

set point 160mm


1 1199022 1199023 and 119902





2 1199023 and 119902

4(with 119902





4(with 119902

1fixed) Through the


1 1199022 119902

119899+1 The 120576-Routh




119901= 10000 119896lowast

119894= 022986 119896lowast

119889= 0001 and 120576 = 0 in

set point 160mm


lowast

119901= 6000 119896lowast

119894= 013817 119896lowast

119889= 0001 and 120576 = 0 in

set point 160mm



Acknowledgments



References











































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











(1199021 119902

119898) | 1198911199031

(1199011 119901

119899) gt 120576119905

119891119903119899+1

(1199011 119901

119899) gt 120576119905


Ω119905= lim120576119905rarr+infin

(Ω1

119905cap Ω2


119899+1

119905)


(

119899+1

⋂

119895=1

Ω119895

119905) = 0

(49)





120576119905997888rarr +infin lim



lowast

119905 120576119905gt 120576lowast

119905 Ω119905= 0

0 lt 120576119905le 120576lowast

119905 Ω119905

= 0

120576lowast

119905gt 120576119905gt 120576119905minus1


Ωlowast

119905sub Ω119905sub Ω119905minus1


(50)


= Ω119905= Ω119905minus1

=



119875 Ω119905+1

= Ω119905= Ω119905minus1


lArrrArr 119875 Ω119905+1

= Ω119905 Ω119905= Ω119905minus1

Ω2= Ω1

(51)


119875 Ω119905+1

= Ω119905 Ω119905= Ω119905minus1

Ω2= Ω1

= 119875 Ω119905+1

= Ω119905 times 119875 Ω

119905= Ω119905minus1


=

119905+1

prod

119895=2

119875 Ω119895= Ω119895minus1

= (119875 Ω2= Ω1)119905

(119905 = 1 2 119905 isin Z+)

(52)



= 119875 (Ω1


119899+1

1) = (Ω

1


119899+1

2)

= 119875

(

119899+1

⋂

119895=1

Ω119895

1) = (

119899+1

⋂

119895=1

Ω119895

2)

lArrrArr 119875(Ω1


119899+1

2) sub Ω

119896

1

(Ω1


119899+1

1) sub Ω

119896

2

(forall119896 = 1 2 119899 119899 + 1)

(53)


2=

Ω1 will yield

119875 (Ω1

2cap Ω2


119899

2) sub Ω

1

1 = 1205821

119875 (Ω1

2cap Ω2


119899

2) sub Ω

119899+1

1 = 1205821

119875 (Ω1

1cap Ω2


119899

1) sub Ω

1

2 = 1205822

119875 (Ω1

1cap Ω2


119899

1) sub Ω

119899+1

2 = 1205822

119875 Ω1= Ω2

= 119875

(

119899+1

⋂

119895=1

Ω119895

2) sub Ω

119896

1 (forall119896 = 1 2 119899 119899 + 1)

times 119875

(

119899+1

⋂

119895=1

Ω119895

1) sub Ω

119896

2 (forall119896 = 1 2 119899 119899 + 1)

=

119899+1

prod

119896=1

119875

(

119899+1

⋂

119895=1

Ω119895

2) sub Ω

119896

1

times

119899+1

prod

119896=1

119875

(

119899+1

⋂

119895=1

Ω119895

1) sub Ω

119896

2

= 120582119899+1

1120582119899+1

2 (0 lt 120582

1lt 1 0 lt 120582

2lt 1)

(54)


119875 Ω119905+1

= Ω119905 Ω119905= Ω119905minus1

Ω2= Ω1 = 120582(119899+1)119905

1120582(119899+1)119905

2

119875 Ω119905+1

Δ119905Ω119905Δ119905minus1

Ω119905minus1


(119899+1)119905

1120582(119899+1)119905

2

(55)


min 119891 (119909)

st ℎ (119909) = 0 119892 (119909) le 0

(56)


where 119891(119909) 119877119899

rarr 119877 ℎ(119909) 119877119899

rarr 119877119898 and 119892(119909)

119877119899



min 119891 (119909) minus 120592

119902

sum

119894=1

ln 120575119894

st ℎ (119909) = 0 119892 (119909) + 120575 = 0

(57)


(1205751 1205752 120575

119902)119879


119871 (119909 119910 119911 120575) = 119891 (119909) minus 120592

119902

sum

119894=1

ln 120575119894+ 119910119879(119892 (119909) + 120575) + 119911

119879ℎ (119909)

(58)


119902)119879

and z = (1199111 1199112 119911




119879

119910 + (nablaℎ (119909))119879119911 = 0

nabla120575119871 (119909 119910 119911 120575) = minus120592119878

minus1

120575119890 + 119910 = 0 lArrrArr minus120592119890 + 119878

120575119884119890 = 0

nabla119910119871 (119909 119910 119911 120575) = 119892 (119909) + 120575 = 0

nabla119911119871 (119909 119910 119911 120575) = ℎ (119909) = 0

(59)

where 119878120575

= diag(1205751 1205752 120575



1 1199102 119910



Δ119909 120575+Δ120575 119911+Δ



(nabla2119891 (119909) + nabla

2119892(119909)119879119910 + nabla

2ℎ(119909)119879119911) Δ119909 + nabla119892(119909)

119879Δ119910


119879119910 + nablaℎ(119909)

119879119911) = 0

minus120592119890 + 119878120575119884119890 + 119878

120575Δ119910 + 119884Δ120575 = 0

119892 (119909) + nabla119892 (119909) Δ119909 + 120575 + Δ120575 = 0

nablaℎ (119909) Δ119909 + ℎ (119909) = 0

(60)


(

119867(119909 119910 119911) 0 nablaℎ(119909)119879

nabla119892(119909)119879

0 119878minus1

120575119884 0 119868

nablaℎ (119909) 0 0 0

nabla119892 (119909) 119868 0 0

)(

Δ119909

Δ120575

Δ119910

Δ119911

)

= (


119879119911

120592119878minus1

120575119890 minus 119910

minusℎ (119909)

minus119892 (119909) minus 120575

)

(61)

119867(119909 119910 119911) = nabla2119891 (119909) + nabla

2119892(119909)119879119910 + nabla

2ℎ(119909)119879119911 (62)


(119909 120575 119911 119910) larr997888 (119909 120575 119911 119910) + 1205771(Δ119909 Δ120575 Δ119911 Δ119910) (63)


1holds the




(0)gt 0 119896 = 0










(119896+1) 120575(119896+1)

119911(119896+1)

119910(119896+1)

) larr (119909(119896)

120575(119896)

119911(119896)

119910(119896)

) +

1205771(Δ119909 Δ120575 Δ119911 Δ119910) 119896 larr 119896 + 1


Step 7 End



1199011(119904) [7]

1198661199012(119904) [15] and 119866

1199013(119904) [6] respectively 119866

1199011(119904) is a third-


1199013(119904) are the SOPTD


1198661199011

(119904) =

15

(1199042+ 09119904 + 5) (119904 + 3)

1198661199012

(119904) =

119890minus05119904

(119904 + 1)2 119866

1199013(119904) =

4119890minus10119904

10119904 + 1

(64)


and 119890minus10119904

asymp (1199042minus 06119904 + 012)(119904

2+ 06119904 + 012) respectively


1199013(119904) are the (minus119904 +

4)(1199043+ 61199042+ 9119904 + 4) and (04119904

2minus 024119904 + 0048)(119904

3+ 07119904

2+



1 1199022 1199023 and 119902

4 The



2 1199023


1(with 119902

2 1199023 and 119902

4


2 1199023 and 119902

4(with 119902




1

















119901119896lowast

119894119896lowast

119889

Plant1

diag (5 5 1 1) 013121 058138 058655

120576 = 0


Plant2

diag (02 1 1 1) 115698 052966 086781

120576 = 05

diag (02 1 1 02) 216962 084500 182140diag (02 2 2 05) 178668 058700 159290

diag (01 08 08 02) 185901 064942 162251diag (04 05 5 1) 122806 062168 142095

Plant3

diag (004 27 158 175) 032447 001537 145256

120576 = 0




1119902211990231199024) 119896

lowast

119901119896lowast

119894119896lowast

119889119905119903(sec) 119905

119889(sec) 119905

119901(sec) 119905

119904(sec) 119872

119901()


0 2 4 6 8 10 12 14

0

02

04

06

08

1

12

Syste

m re

spon

se

minus02

Time (s)

119876 = diag (5 5 1 1)119876 = diag (5 2 1 1)119876 = diag (10 2 1 1)

119876 = diag (10 10 1 1)119876 = diag (5 5 2 1) lowast 2

(a)

0 2 4 6 8 10 12 14

0

02

04

06

08

1

12

Con

trol e

rror

minus02

Time (s)

119876 = diag (5 5 1 1)119876 = diag (5 2 1 1)119876 = diag (10 2 1 1)

119876 = diag (10 10 1 1)119876 = diag (5 5 2 1) lowast 2

(b)



0 1 2 3 4 5 6 7 8 9 100

02

04

06

08

1

12

14

Time (s)

Syste

m re

spon

se

120576 = 05

120576 = 25

120576 = 3

120576 = 4

120576 = 5

(a)

0 1 2 3 4 5 6 7 8 9 10

0

02

04

06

08

1

Time (s)

minus02

minus04

Con

trol e

rror

120576 = 05

120576 = 25

120576 = 3

120576 = 4

120576 = 5

(b)


0 5 10 15 20 25 30

0

02

04

06

08

1

12



Syste

m re

spon

se

Time (s)

minus02

(a)

0 5 10 15 20 25 30

0

02

04

06

08

1

12



Con

trol e

rror

Time (s)

minus02

(b)




119896119894

119896119889

Risetime

119905119903(sec)

Delaytime

119905119889(sec)

Peaktime

119905119901(sec)

Settingtime

119905119904(sec)


Plants



0 2 4 6 8 10 12 14 16 180

05

1

15Sy

stem

resp

onse



Time (s)

(a)

0 2 4 6 8 10 12 14 16 18

0

05

1


minus05


Con

trol e

rror

Time (s)

(b)


0 20 40 60 80 100 120 140 160 180 2000

02

04

06

08

1

12

14



Syste

m re

spon

se

Time (s)

095

105

(a)

0 20 40 60 80 100 120 140 160 180 200

0

02

04

06

08

1



Con

trol e

rror

Time (s)

minus02

minus04

(b)





Time (s)0 5 10 15 20 25 30

0

02

04

06

08

1

12

0 5 10 15 20 25 30

0

02

04

06

08

1

12

Syste

m re

spon

seSy

stem

resp

onse

Con

trol e

rror

Con

trol e

rror

Time (s)

Time (s)Time (s)16 18 20 22 24 26 28

075

08

085

09

095

1

105

11



16 18 20 22 24 26 28

0

005

01

015

02

025

03



minus02 minus02

minus005

119863119906(119905) = 03119906(119905 minus 16)



Methods 119896119901

119896119894

119896119889










16 18 20 22 24 26 28 30086

088

09

092

094

096

098

1

102

104





0 5 10 15 20 25 300

05

1

15

0 5 10 15 20 25 30

0

05

1

18 20 22 24 26 28 30

0

002

004

006

008

01

012

014

Con

trol e

rror

Con

trol e

rror

Syste

m re

spon

seSy

stem

resp

onse

minus05

Time (s) Time (s)

Time (s) Time (s)

minus002

119863119906(119905) = 03119906(119905 minus 18)






200 250 300 350 40007

075

08

085

09

095

1

105

11

200 250 300 350 400

0

005

01

015

02

025

03

0 50 100 150 200 250 300 350 4000

02

04

06

08

1

12

14

0 50 100 150 200 250 300 350 400

0

02

04

06

08

1

095

105

Time (s) Time (s)

Time (s)Time (s)





minus005

Con

trol e

rror

Syste

m re

spon

se

Con

trol e

rror

Syste

m re

spon

se

minus02

minus04

119863119906(119905) = 01119906(119905 minus 180)




1(119904)119876119894119888(119904) = (04465119904 +





119901=

10000 119896119894

= 032549 119896119889

= 0001 120576 = 0 119896119901

=

10000 119896119894

= 010118 119896119889

= 0001 120576 = 0 119896119901

=

10000 119896119894= 022986 119896

119889= 0001 120576 = 0 119896

119901= 6000 119896

119894=



1is set at




0 5 10 15 20 25 300

02

04

06

08

1

12

14

Time (s)

16 18 20 22 24 26 28 30086

088

09

092

094

096

098

1

102

104

0 5 10 15 20 25 30

0

02

04

06

08

1

Time (s)

Time (s) Time (s)

16 18 20 22 24 26 28 30

0

002

004

006

008

01

012

014

minus02

minus04

minus002

120576 = 05

120576 = 25

120576 = 3

120576 = 4

120576 = 5

120576 = 05

120576 = 25

120576 = 3

120576 = 4

120576 = 5



1 the solid







0 200 400 600 800 1000 1200 1400 1600 1800 20000

50

100

150

200

250

300

Time (s)

Liqu

id le

vel o

f CW

TS (m

m)




lowast

119901= 10000 119896lowast

119894= 032549 119896lowast

119889= 0001 and 120576 = 0 in

set point 300mm


119901= 10000 119896lowast

119894= 010118 119896lowast

119889= 0001 and 120576 = 0 in

set point 300mm


119901= 10000 119896lowast

119894= 022986 119896lowast

119889= 0001 and 120576 = 0 in

set point 300mm


lowast

119901= 6000 119896lowast

119894= 013817 119896lowast

119889= 0001 and 120576 = 0 in

set point 300mm


5 Conclusions




119901= 10000 119896lowast

119894= 032549 119896lowast

119889= 0001 and 120576 = 0 in

set point 160mm


119901= 10000 119896lowast

119894= 010118 119896lowast

119889= 0001 and 120576 = 0 in

set point 160mm


1 1199022 1199023 and 119902





2 1199023 and 119902

4(with 119902





4(with 119902

1fixed) Through the


1 1199022 119902

119899+1 The 120576-Routh




119901= 10000 119896lowast

119894= 022986 119896lowast

119889= 0001 and 120576 = 0 in

set point 160mm


lowast

119901= 6000 119896lowast

119894= 013817 119896lowast

119889= 0001 and 120576 = 0 in

set point 160mm



Acknowledgments



References











































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










where 119891(119909) 119877119899

rarr 119877 ℎ(119909) 119877119899

rarr 119877119898 and 119892(119909)

119877119899



min 119891 (119909) minus 120592

119902

sum

119894=1

ln 120575119894

st ℎ (119909) = 0 119892 (119909) + 120575 = 0

(57)


(1205751 1205752 120575

119902)119879


119871 (119909 119910 119911 120575) = 119891 (119909) minus 120592

119902

sum

119894=1

ln 120575119894+ 119910119879(119892 (119909) + 120575) + 119911

119879ℎ (119909)

(58)


119902)119879

and z = (1199111 1199112 119911




119879

119910 + (nablaℎ (119909))119879119911 = 0

nabla120575119871 (119909 119910 119911 120575) = minus120592119878

minus1

120575119890 + 119910 = 0 lArrrArr minus120592119890 + 119878

120575119884119890 = 0

nabla119910119871 (119909 119910 119911 120575) = 119892 (119909) + 120575 = 0

nabla119911119871 (119909 119910 119911 120575) = ℎ (119909) = 0

(59)

where 119878120575

= diag(1205751 1205752 120575



1 1199102 119910



Δ119909 120575+Δ120575 119911+Δ



(nabla2119891 (119909) + nabla

2119892(119909)119879119910 + nabla

2ℎ(119909)119879119911) Δ119909 + nabla119892(119909)

119879Δ119910


119879119910 + nablaℎ(119909)

119879119911) = 0

minus120592119890 + 119878120575119884119890 + 119878

120575Δ119910 + 119884Δ120575 = 0

119892 (119909) + nabla119892 (119909) Δ119909 + 120575 + Δ120575 = 0

nablaℎ (119909) Δ119909 + ℎ (119909) = 0

(60)


(

119867(119909 119910 119911) 0 nablaℎ(119909)119879

nabla119892(119909)119879

0 119878minus1

120575119884 0 119868

nablaℎ (119909) 0 0 0

nabla119892 (119909) 119868 0 0

)(

Δ119909

Δ120575

Δ119910

Δ119911

)

= (


119879119911

120592119878minus1

120575119890 minus 119910

minusℎ (119909)

minus119892 (119909) minus 120575

)

(61)

119867(119909 119910 119911) = nabla2119891 (119909) + nabla

2119892(119909)119879119910 + nabla

2ℎ(119909)119879119911 (62)


(119909 120575 119911 119910) larr997888 (119909 120575 119911 119910) + 1205771(Δ119909 Δ120575 Δ119911 Δ119910) (63)


1holds the




(0)gt 0 119896 = 0










(119896+1) 120575(119896+1)

119911(119896+1)

119910(119896+1)

) larr (119909(119896)

120575(119896)

119911(119896)

119910(119896)

) +

1205771(Δ119909 Δ120575 Δ119911 Δ119910) 119896 larr 119896 + 1


Step 7 End



1199011(119904) [7]

1198661199012(119904) [15] and 119866

1199013(119904) [6] respectively 119866

1199011(119904) is a third-


1199013(119904) are the SOPTD


1198661199011

(119904) =

15

(1199042+ 09119904 + 5) (119904 + 3)

1198661199012

(119904) =

119890minus05119904

(119904 + 1)2 119866

1199013(119904) =

4119890minus10119904

10119904 + 1

(64)


and 119890minus10119904

asymp (1199042minus 06119904 + 012)(119904

2+ 06119904 + 012) respectively


1199013(119904) are the (minus119904 +

4)(1199043+ 61199042+ 9119904 + 4) and (04119904

2minus 024119904 + 0048)(119904

3+ 07119904

2+



1 1199022 1199023 and 119902

4 The



2 1199023


1(with 119902

2 1199023 and 119902

4


2 1199023 and 119902

4(with 119902




1

















119901119896lowast

119894119896lowast

119889

Plant1

diag (5 5 1 1) 013121 058138 058655

120576 = 0


Plant2

diag (02 1 1 1) 115698 052966 086781

120576 = 05

diag (02 1 1 02) 216962 084500 182140diag (02 2 2 05) 178668 058700 159290

diag (01 08 08 02) 185901 064942 162251diag (04 05 5 1) 122806 062168 142095

Plant3

diag (004 27 158 175) 032447 001537 145256

120576 = 0




1119902211990231199024) 119896

lowast

119901119896lowast

119894119896lowast

119889119905119903(sec) 119905

119889(sec) 119905

119901(sec) 119905

119904(sec) 119872

119901()


0 2 4 6 8 10 12 14

0

02

04

06

08

1

12

Syste

m re

spon

se

minus02

Time (s)

119876 = diag (5 5 1 1)119876 = diag (5 2 1 1)119876 = diag (10 2 1 1)

119876 = diag (10 10 1 1)119876 = diag (5 5 2 1) lowast 2

(a)

0 2 4 6 8 10 12 14

0

02

04

06

08

1

12

Con

trol e

rror

minus02

Time (s)

119876 = diag (5 5 1 1)119876 = diag (5 2 1 1)119876 = diag (10 2 1 1)

119876 = diag (10 10 1 1)119876 = diag (5 5 2 1) lowast 2

(b)



0 1 2 3 4 5 6 7 8 9 100

02

04

06

08

1

12

14

Time (s)

Syste

m re

spon

se

120576 = 05

120576 = 25

120576 = 3

120576 = 4

120576 = 5

(a)

0 1 2 3 4 5 6 7 8 9 10

0

02

04

06

08

1

Time (s)

minus02

minus04

Con

trol e

rror

120576 = 05

120576 = 25

120576 = 3

120576 = 4

120576 = 5

(b)


0 5 10 15 20 25 30

0

02

04

06

08

1

12



Syste

m re

spon

se

Time (s)

minus02

(a)

0 5 10 15 20 25 30

0

02

04

06

08

1

12



Con

trol e

rror

Time (s)

minus02

(b)




119896119894

119896119889

Risetime

119905119903(sec)

Delaytime

119905119889(sec)

Peaktime

119905119901(sec)

Settingtime

119905119904(sec)


Plants



0 2 4 6 8 10 12 14 16 180

05

1

15Sy

stem

resp

onse



Time (s)

(a)

0 2 4 6 8 10 12 14 16 18

0

05

1


minus05


Con

trol e

rror

Time (s)

(b)


0 20 40 60 80 100 120 140 160 180 2000

02

04

06

08

1

12

14



Syste

m re

spon

se

Time (s)

095

105

(a)

0 20 40 60 80 100 120 140 160 180 200

0

02

04

06

08

1



Con

trol e

rror

Time (s)

minus02

minus04

(b)





Time (s)0 5 10 15 20 25 30

0

02

04

06

08

1

12

0 5 10 15 20 25 30

0

02

04

06

08

1

12

Syste

m re

spon

seSy

stem

resp

onse

Con

trol e

rror

Con

trol e

rror

Time (s)

Time (s)Time (s)16 18 20 22 24 26 28

075

08

085

09

095

1

105

11



16 18 20 22 24 26 28

0

005

01

015

02

025

03



minus02 minus02

minus005

119863119906(119905) = 03119906(119905 minus 16)



Methods 119896119901

119896119894

119896119889










16 18 20 22 24 26 28 30086

088

09

092

094

096

098

1

102

104





0 5 10 15 20 25 300

05

1

15

0 5 10 15 20 25 30

0

05

1

18 20 22 24 26 28 30

0

002

004

006

008

01

012

014

Con

trol e

rror

Con

trol e

rror

Syste

m re

spon

seSy

stem

resp

onse

minus05

Time (s) Time (s)

Time (s) Time (s)

minus002

119863119906(119905) = 03119906(119905 minus 18)






200 250 300 350 40007

075

08

085

09

095

1

105

11

200 250 300 350 400

0

005

01

015

02

025

03

0 50 100 150 200 250 300 350 4000

02

04

06

08

1

12

14

0 50 100 150 200 250 300 350 400

0

02

04

06

08

1

095

105

Time (s) Time (s)

Time (s)Time (s)





minus005

Con

trol e

rror

Syste

m re

spon

se

Con

trol e

rror

Syste

m re

spon

se

minus02

minus04

119863119906(119905) = 01119906(119905 minus 180)




1(119904)119876119894119888(119904) = (04465119904 +





119901=

10000 119896119894

= 032549 119896119889

= 0001 120576 = 0 119896119901

=

10000 119896119894

= 010118 119896119889

= 0001 120576 = 0 119896119901

=

10000 119896119894= 022986 119896

119889= 0001 120576 = 0 119896

119901= 6000 119896

119894=



1is set at




0 5 10 15 20 25 300

02

04

06

08

1

12

14

Time (s)

16 18 20 22 24 26 28 30086

088

09

092

094

096

098

1

102

104

0 5 10 15 20 25 30

0

02

04

06

08

1

Time (s)

Time (s) Time (s)

16 18 20 22 24 26 28 30

0

002

004

006

008

01

012

014

minus02

minus04

minus002

120576 = 05

120576 = 25

120576 = 3

120576 = 4

120576 = 5

120576 = 05

120576 = 25

120576 = 3

120576 = 4

120576 = 5



1 the solid







0 200 400 600 800 1000 1200 1400 1600 1800 20000

50

100

150

200

250

300

Time (s)

Liqu

id le

vel o

f CW

TS (m

m)




lowast

119901= 10000 119896lowast

119894= 032549 119896lowast

119889= 0001 and 120576 = 0 in

set point 300mm


119901= 10000 119896lowast

119894= 010118 119896lowast

119889= 0001 and 120576 = 0 in

set point 300mm


119901= 10000 119896lowast

119894= 022986 119896lowast

119889= 0001 and 120576 = 0 in

set point 300mm


lowast

119901= 6000 119896lowast

119894= 013817 119896lowast

119889= 0001 and 120576 = 0 in

set point 300mm


5 Conclusions




119901= 10000 119896lowast

119894= 032549 119896lowast

119889= 0001 and 120576 = 0 in

set point 160mm


119901= 10000 119896lowast

119894= 010118 119896lowast

119889= 0001 and 120576 = 0 in

set point 160mm


1 1199022 1199023 and 119902





2 1199023 and 119902

4(with 119902





4(with 119902

1fixed) Through the


1 1199022 119902

119899+1 The 120576-Routh




119901= 10000 119896lowast

119894= 022986 119896lowast

119889= 0001 and 120576 = 0 in

set point 160mm


lowast

119901= 6000 119896lowast

119894= 013817 119896lowast

119889= 0001 and 120576 = 0 in

set point 160mm



Acknowledgments



References











































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra














(119896+1) 120575(119896+1)

119911(119896+1)

119910(119896+1)

) larr (119909(119896)

120575(119896)

119911(119896)

119910(119896)

) +

1205771(Δ119909 Δ120575 Δ119911 Δ119910) 119896 larr 119896 + 1


Step 7 End



1199011(119904) [7]

1198661199012(119904) [15] and 119866

1199013(119904) [6] respectively 119866

1199011(119904) is a third-


1199013(119904) are the SOPTD


1198661199011

(119904) =

15

(1199042+ 09119904 + 5) (119904 + 3)

1198661199012

(119904) =

119890minus05119904

(119904 + 1)2 119866

1199013(119904) =

4119890minus10119904

10119904 + 1

(64)


and 119890minus10119904

asymp (1199042minus 06119904 + 012)(119904

2+ 06119904 + 012) respectively


1199013(119904) are the (minus119904 +

4)(1199043+ 61199042+ 9119904 + 4) and (04119904

2minus 024119904 + 0048)(119904

3+ 07119904

2+



1 1199022 1199023 and 119902

4 The



2 1199023


1(with 119902

2 1199023 and 119902

4


2 1199023 and 119902

4(with 119902




1

















119901119896lowast

119894119896lowast

119889

Plant1

diag (5 5 1 1) 013121 058138 058655

120576 = 0


Plant2

diag (02 1 1 1) 115698 052966 086781

120576 = 05

diag (02 1 1 02) 216962 084500 182140diag (02 2 2 05) 178668 058700 159290

diag (01 08 08 02) 185901 064942 162251diag (04 05 5 1) 122806 062168 142095

Plant3

diag (004 27 158 175) 032447 001537 145256

120576 = 0




1119902211990231199024) 119896

lowast

119901119896lowast

119894119896lowast

119889119905119903(sec) 119905

119889(sec) 119905

119901(sec) 119905

119904(sec) 119872

119901()


0 2 4 6 8 10 12 14

0

02

04

06

08

1

12

Syste

m re

spon

se

minus02

Time (s)

119876 = diag (5 5 1 1)119876 = diag (5 2 1 1)119876 = diag (10 2 1 1)

119876 = diag (10 10 1 1)119876 = diag (5 5 2 1) lowast 2

(a)

0 2 4 6 8 10 12 14

0

02

04

06

08

1

12

Con

trol e

rror

minus02

Time (s)

119876 = diag (5 5 1 1)119876 = diag (5 2 1 1)119876 = diag (10 2 1 1)

119876 = diag (10 10 1 1)119876 = diag (5 5 2 1) lowast 2

(b)



0 1 2 3 4 5 6 7 8 9 100

02

04

06

08

1

12

14

Time (s)

Syste

m re

spon

se

120576 = 05

120576 = 25

120576 = 3

120576 = 4

120576 = 5

(a)

0 1 2 3 4 5 6 7 8 9 10

0

02

04

06

08

1

Time (s)

minus02

minus04

Con

trol e

rror

120576 = 05

120576 = 25

120576 = 3

120576 = 4

120576 = 5

(b)


0 5 10 15 20 25 30

0

02

04

06

08

1

12



Syste

m re

spon

se

Time (s)

minus02

(a)

0 5 10 15 20 25 30

0

02

04

06

08

1

12



Con

trol e

rror

Time (s)

minus02

(b)




119896119894

119896119889

Risetime

119905119903(sec)

Delaytime

119905119889(sec)

Peaktime

119905119901(sec)

Settingtime

119905119904(sec)


Plants



0 2 4 6 8 10 12 14 16 180

05

1

15Sy

stem

resp

onse



Time (s)

(a)

0 2 4 6 8 10 12 14 16 18

0

05

1


minus05


Con

trol e

rror

Time (s)

(b)


0 20 40 60 80 100 120 140 160 180 2000

02

04

06

08

1

12

14



Syste

m re

spon

se

Time (s)

095

105

(a)

0 20 40 60 80 100 120 140 160 180 200

0

02

04

06

08

1



Con

trol e

rror

Time (s)

minus02

minus04

(b)





Time (s)0 5 10 15 20 25 30

0

02

04

06

08

1

12

0 5 10 15 20 25 30

0

02

04

06

08

1

12

Syste

m re

spon

seSy

stem

resp

onse

Con

trol e

rror

Con

trol e

rror

Time (s)

Time (s)Time (s)16 18 20 22 24 26 28

075

08

085

09

095

1

105

11



16 18 20 22 24 26 28

0

005

01

015

02

025

03



minus02 minus02

minus005

119863119906(119905) = 03119906(119905 minus 16)



Methods 119896119901

119896119894

119896119889










16 18 20 22 24 26 28 30086

088

09

092

094

096

098

1

102

104





0 5 10 15 20 25 300

05

1

15

0 5 10 15 20 25 30

0

05

1

18 20 22 24 26 28 30

0

002

004

006

008

01

012

014

Con

trol e

rror

Con

trol e

rror

Syste

m re

spon

seSy

stem

resp

onse

minus05

Time (s) Time (s)

Time (s) Time (s)

minus002

119863119906(119905) = 03119906(119905 minus 18)






200 250 300 350 40007

075

08

085

09

095

1

105

11

200 250 300 350 400

0

005

01

015

02

025

03

0 50 100 150 200 250 300 350 4000

02

04

06

08

1

12

14

0 50 100 150 200 250 300 350 400

0

02

04

06

08

1

095

105

Time (s) Time (s)

Time (s)Time (s)





minus005

Con

trol e

rror

Syste

m re

spon

se

Con

trol e

rror

Syste

m re

spon

se

minus02

minus04

119863119906(119905) = 01119906(119905 minus 180)




1(119904)119876119894119888(119904) = (04465119904 +





119901=

10000 119896119894

= 032549 119896119889

= 0001 120576 = 0 119896119901

=

10000 119896119894

= 010118 119896119889

= 0001 120576 = 0 119896119901

=

10000 119896119894= 022986 119896

119889= 0001 120576 = 0 119896

119901= 6000 119896

119894=



1is set at




0 5 10 15 20 25 300

02

04

06

08

1

12

14

Time (s)

16 18 20 22 24 26 28 30086

088

09

092

094

096

098

1

102

104

0 5 10 15 20 25 30

0

02

04

06

08

1

Time (s)

Time (s) Time (s)

16 18 20 22 24 26 28 30

0

002

004

006

008

01

012

014

minus02

minus04

minus002

120576 = 05

120576 = 25

120576 = 3

120576 = 4

120576 = 5

120576 = 05

120576 = 25

120576 = 3

120576 = 4

120576 = 5



1 the solid







0 200 400 600 800 1000 1200 1400 1600 1800 20000

50

100

150

200

250

300

Time (s)

Liqu

id le

vel o

f CW

TS (m

m)




lowast

119901= 10000 119896lowast

119894= 032549 119896lowast

119889= 0001 and 120576 = 0 in

set point 300mm


119901= 10000 119896lowast

119894= 010118 119896lowast

119889= 0001 and 120576 = 0 in

set point 300mm


119901= 10000 119896lowast

119894= 022986 119896lowast

119889= 0001 and 120576 = 0 in

set point 300mm


lowast

119901= 6000 119896lowast

119894= 013817 119896lowast

119889= 0001 and 120576 = 0 in

set point 300mm


5 Conclusions




119901= 10000 119896lowast

119894= 032549 119896lowast

119889= 0001 and 120576 = 0 in

set point 160mm


119901= 10000 119896lowast

119894= 010118 119896lowast

119889= 0001 and 120576 = 0 in

set point 160mm


1 1199022 1199023 and 119902





2 1199023 and 119902

4(with 119902





4(with 119902

1fixed) Through the


1 1199022 119902

119899+1 The 120576-Routh




119901= 10000 119896lowast

119894= 022986 119896lowast

119889= 0001 and 120576 = 0 in

set point 160mm


lowast

119901= 6000 119896lowast

119894= 013817 119896lowast

119889= 0001 and 120576 = 0 in

set point 160mm



Acknowledgments



References











































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













119901119896lowast

119894119896lowast

119889

Plant1

diag (5 5 1 1) 013121 058138 058655

120576 = 0


Plant2

diag (02 1 1 1) 115698 052966 086781

120576 = 05

diag (02 1 1 02) 216962 084500 182140diag (02 2 2 05) 178668 058700 159290

diag (01 08 08 02) 185901 064942 162251diag (04 05 5 1) 122806 062168 142095

Plant3

diag (004 27 158 175) 032447 001537 145256

120576 = 0




1119902211990231199024) 119896

lowast

119901119896lowast

119894119896lowast

119889119905119903(sec) 119905

119889(sec) 119905

119901(sec) 119905

119904(sec) 119872

119901()


0 2 4 6 8 10 12 14

0

02

04

06

08

1

12

Syste

m re

spon

se

minus02

Time (s)

119876 = diag (5 5 1 1)119876 = diag (5 2 1 1)119876 = diag (10 2 1 1)

119876 = diag (10 10 1 1)119876 = diag (5 5 2 1) lowast 2

(a)

0 2 4 6 8 10 12 14

0

02

04

06

08

1

12

Con

trol e

rror

minus02

Time (s)

119876 = diag (5 5 1 1)119876 = diag (5 2 1 1)119876 = diag (10 2 1 1)

119876 = diag (10 10 1 1)119876 = diag (5 5 2 1) lowast 2

(b)



0 1 2 3 4 5 6 7 8 9 100

02

04

06

08

1

12

14

Time (s)

Syste

m re

spon

se

120576 = 05

120576 = 25

120576 = 3

120576 = 4

120576 = 5

(a)

0 1 2 3 4 5 6 7 8 9 10

0

02

04

06

08

1

Time (s)

minus02

minus04

Con

trol e

rror

120576 = 05

120576 = 25

120576 = 3

120576 = 4

120576 = 5

(b)


0 5 10 15 20 25 30

0

02

04

06

08

1

12



Syste

m re

spon

se

Time (s)

minus02

(a)

0 5 10 15 20 25 30

0

02

04

06

08

1

12



Con

trol e

rror

Time (s)

minus02

(b)




119896119894

119896119889

Risetime

119905119903(sec)

Delaytime

119905119889(sec)

Peaktime

119905119901(sec)

Settingtime

119905119904(sec)


Plants



0 2 4 6 8 10 12 14 16 180

05

1

15Sy

stem

resp

onse



Time (s)

(a)

0 2 4 6 8 10 12 14 16 18

0

05

1


minus05


Con

trol e

rror

Time (s)

(b)


0 20 40 60 80 100 120 140 160 180 2000

02

04

06

08

1

12

14



Syste

m re

spon

se

Time (s)

095

105

(a)

0 20 40 60 80 100 120 140 160 180 200

0

02

04

06

08

1



Con

trol e

rror

Time (s)

minus02

minus04

(b)





Time (s)0 5 10 15 20 25 30

0

02

04

06

08

1

12

0 5 10 15 20 25 30

0

02

04

06

08

1

12

Syste

m re

spon

seSy

stem

resp

onse

Con

trol e

rror

Con

trol e

rror

Time (s)

Time (s)Time (s)16 18 20 22 24 26 28

075

08

085

09

095

1

105

11



16 18 20 22 24 26 28

0

005

01

015

02

025

03



minus02 minus02

minus005

119863119906(119905) = 03119906(119905 minus 16)



Methods 119896119901

119896119894

119896119889










16 18 20 22 24 26 28 30086

088

09

092

094

096

098

1

102

104





0 5 10 15 20 25 300

05

1

15

0 5 10 15 20 25 30

0

05

1

18 20 22 24 26 28 30

0

002

004

006

008

01

012

014

Con

trol e

rror

Con

trol e

rror

Syste

m re

spon

seSy

stem

resp

onse

minus05

Time (s) Time (s)

Time (s) Time (s)

minus002

119863119906(119905) = 03119906(119905 minus 18)






200 250 300 350 40007

075

08

085

09

095

1

105

11

200 250 300 350 400

0

005

01

015

02

025

03

0 50 100 150 200 250 300 350 4000

02

04

06

08

1

12

14

0 50 100 150 200 250 300 350 400

0

02

04

06

08

1

095

105

Time (s) Time (s)

Time (s)Time (s)





minus005

Con

trol e

rror

Syste

m re

spon

se

Con

trol e

rror

Syste

m re

spon

se

minus02

minus04

119863119906(119905) = 01119906(119905 minus 180)




1(119904)119876119894119888(119904) = (04465119904 +





119901=

10000 119896119894

= 032549 119896119889

= 0001 120576 = 0 119896119901

=

10000 119896119894

= 010118 119896119889

= 0001 120576 = 0 119896119901

=

10000 119896119894= 022986 119896

119889= 0001 120576 = 0 119896

119901= 6000 119896

119894=



1is set at




0 5 10 15 20 25 300

02

04

06

08

1

12

14

Time (s)

16 18 20 22 24 26 28 30086

088

09

092

094

096

098

1

102

104

0 5 10 15 20 25 30

0

02

04

06

08

1

Time (s)

Time (s) Time (s)

16 18 20 22 24 26 28 30

0

002

004

006

008

01

012

014

minus02

minus04

minus002

120576 = 05

120576 = 25

120576 = 3

120576 = 4

120576 = 5

120576 = 05

120576 = 25

120576 = 3

120576 = 4

120576 = 5



1 the solid







0 200 400 600 800 1000 1200 1400 1600 1800 20000

50

100

150

200

250

300

Time (s)

Liqu

id le

vel o

f CW

TS (m

m)




lowast

119901= 10000 119896lowast

119894= 032549 119896lowast

119889= 0001 and 120576 = 0 in

set point 300mm


119901= 10000 119896lowast

119894= 010118 119896lowast

119889= 0001 and 120576 = 0 in

set point 300mm


119901= 10000 119896lowast

119894= 022986 119896lowast

119889= 0001 and 120576 = 0 in

set point 300mm


lowast

119901= 6000 119896lowast

119894= 013817 119896lowast

119889= 0001 and 120576 = 0 in

set point 300mm


5 Conclusions




119901= 10000 119896lowast

119894= 032549 119896lowast

119889= 0001 and 120576 = 0 in

set point 160mm


119901= 10000 119896lowast

119894= 010118 119896lowast

119889= 0001 and 120576 = 0 in

set point 160mm


1 1199022 1199023 and 119902





2 1199023 and 119902

4(with 119902





4(with 119902

1fixed) Through the


1 1199022 119902

119899+1 The 120576-Routh




119901= 10000 119896lowast

119894= 022986 119896lowast

119889= 0001 and 120576 = 0 in

set point 160mm


lowast

119901= 6000 119896lowast

119894= 013817 119896lowast

119889= 0001 and 120576 = 0 in

set point 160mm



Acknowledgments



References











































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 1 2 3 4 5 6 7 8 9 100

02

04

06

08

1

12

14

Time (s)

Syste

m re

spon

se

120576 = 05

120576 = 25

120576 = 3

120576 = 4

120576 = 5

(a)

0 1 2 3 4 5 6 7 8 9 10

0

02

04

06

08

1

Time (s)

minus02

minus04

Con

trol e

rror

120576 = 05

120576 = 25

120576 = 3

120576 = 4

120576 = 5

(b)


0 5 10 15 20 25 30

0

02

04

06

08

1

12



Syste

m re

spon

se

Time (s)

minus02

(a)

0 5 10 15 20 25 30

0

02

04

06

08

1

12



Con

trol e

rror

Time (s)

minus02

(b)




119896119894

119896119889

Risetime

119905119903(sec)

Delaytime

119905119889(sec)

Peaktime

119905119901(sec)

Settingtime

119905119904(sec)


Plants



0 2 4 6 8 10 12 14 16 180

05

1

15Sy

stem

resp

onse



Time (s)

(a)

0 2 4 6 8 10 12 14 16 18

0

05

1


minus05


Con

trol e

rror

Time (s)

(b)


0 20 40 60 80 100 120 140 160 180 2000

02

04

06

08

1

12

14



Syste

m re

spon

se

Time (s)

095

105

(a)

0 20 40 60 80 100 120 140 160 180 200

0

02

04

06

08

1



Con

trol e

rror

Time (s)

minus02

minus04

(b)





Time (s)0 5 10 15 20 25 30

0

02

04

06

08

1

12

0 5 10 15 20 25 30

0

02

04

06

08

1

12

Syste

m re

spon

seSy

stem

resp

onse

Con

trol e

rror

Con

trol e

rror

Time (s)

Time (s)Time (s)16 18 20 22 24 26 28

075

08

085

09

095

1

105

11



16 18 20 22 24 26 28

0

005

01

015

02

025

03



minus02 minus02

minus005

119863119906(119905) = 03119906(119905 minus 16)



Methods 119896119901

119896119894

119896119889










16 18 20 22 24 26 28 30086

088

09

092

094

096

098

1

102

104





0 5 10 15 20 25 300

05

1

15

0 5 10 15 20 25 30

0

05

1

18 20 22 24 26 28 30

0

002

004

006

008

01

012

014

Con

trol e

rror

Con

trol e

rror

Syste

m re

spon

seSy

stem

resp

onse

minus05

Time (s) Time (s)

Time (s) Time (s)

minus002

119863119906(119905) = 03119906(119905 minus 18)






200 250 300 350 40007

075

08

085

09

095

1

105

11

200 250 300 350 400

0

005

01

015

02

025

03

0 50 100 150 200 250 300 350 4000

02

04

06

08

1

12

14

0 50 100 150 200 250 300 350 400

0

02

04

06

08

1

095

105

Time (s) Time (s)

Time (s)Time (s)





minus005

Con

trol e

rror

Syste

m re

spon

se

Con

trol e

rror

Syste

m re

spon

se

minus02

minus04

119863119906(119905) = 01119906(119905 minus 180)




1(119904)119876119894119888(119904) = (04465119904 +





119901=

10000 119896119894

= 032549 119896119889

= 0001 120576 = 0 119896119901

=

10000 119896119894

= 010118 119896119889

= 0001 120576 = 0 119896119901

=

10000 119896119894= 022986 119896

119889= 0001 120576 = 0 119896

119901= 6000 119896

119894=



1is set at




0 5 10 15 20 25 300

02

04

06

08

1

12

14

Time (s)

16 18 20 22 24 26 28 30086

088

09

092

094

096

098

1

102

104

0 5 10 15 20 25 30

0

02

04

06

08

1

Time (s)

Time (s) Time (s)

16 18 20 22 24 26 28 30

0

002

004

006

008

01

012

014

minus02

minus04

minus002

120576 = 05

120576 = 25

120576 = 3

120576 = 4

120576 = 5

120576 = 05

120576 = 25

120576 = 3

120576 = 4

120576 = 5



1 the solid







0 200 400 600 800 1000 1200 1400 1600 1800 20000

50

100

150

200

250

300

Time (s)

Liqu

id le

vel o

f CW

TS (m

m)




lowast

119901= 10000 119896lowast

119894= 032549 119896lowast

119889= 0001 and 120576 = 0 in

set point 300mm


119901= 10000 119896lowast

119894= 010118 119896lowast

119889= 0001 and 120576 = 0 in

set point 300mm


119901= 10000 119896lowast

119894= 022986 119896lowast

119889= 0001 and 120576 = 0 in

set point 300mm


lowast

119901= 6000 119896lowast

119894= 013817 119896lowast

119889= 0001 and 120576 = 0 in

set point 300mm


5 Conclusions




119901= 10000 119896lowast

119894= 032549 119896lowast

119889= 0001 and 120576 = 0 in

set point 160mm


119901= 10000 119896lowast

119894= 010118 119896lowast

119889= 0001 and 120576 = 0 in

set point 160mm


1 1199022 1199023 and 119902





2 1199023 and 119902

4(with 119902





4(with 119902

1fixed) Through the


1 1199022 119902

119899+1 The 120576-Routh




119901= 10000 119896lowast

119894= 022986 119896lowast

119889= 0001 and 120576 = 0 in

set point 160mm


lowast

119901= 6000 119896lowast

119894= 013817 119896lowast

119889= 0001 and 120576 = 0 in

set point 160mm



Acknowledgments



References











































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 2 4 6 8 10 12 14 16 180

05

1

15Sy

stem

resp

onse



Time (s)

(a)

0 2 4 6 8 10 12 14 16 18

0

05

1


minus05


Con

trol e

rror

Time (s)

(b)


0 20 40 60 80 100 120 140 160 180 2000

02

04

06

08

1

12

14



Syste

m re

spon

se

Time (s)

095

105

(a)

0 20 40 60 80 100 120 140 160 180 200

0

02

04

06

08

1



Con

trol e

rror

Time (s)

minus02

minus04

(b)





Time (s)0 5 10 15 20 25 30

0

02

04

06

08

1

12

0 5 10 15 20 25 30

0

02

04

06

08

1

12

Syste

m re

spon

seSy

stem

resp

onse

Con

trol e

rror

Con

trol e

rror

Time (s)

Time (s)Time (s)16 18 20 22 24 26 28

075

08

085

09

095

1

105

11



16 18 20 22 24 26 28

0

005

01

015

02

025

03



minus02 minus02

minus005

119863119906(119905) = 03119906(119905 minus 16)



Methods 119896119901

119896119894

119896119889










16 18 20 22 24 26 28 30086

088

09

092

094

096

098

1

102

104





0 5 10 15 20 25 300

05

1

15

0 5 10 15 20 25 30

0

05

1

18 20 22 24 26 28 30

0

002

004

006

008

01

012

014

Con

trol e

rror

Con

trol e

rror

Syste

m re

spon

seSy

stem

resp

onse

minus05

Time (s) Time (s)

Time (s) Time (s)

minus002

119863119906(119905) = 03119906(119905 minus 18)






200 250 300 350 40007

075

08

085

09

095

1

105

11

200 250 300 350 400

0

005

01

015

02

025

03

0 50 100 150 200 250 300 350 4000

02

04

06

08

1

12

14

0 50 100 150 200 250 300 350 400

0

02

04

06

08

1

095

105

Time (s) Time (s)

Time (s)Time (s)





minus005

Con

trol e

rror

Syste

m re

spon

se

Con

trol e

rror

Syste

m re

spon

se

minus02

minus04

119863119906(119905) = 01119906(119905 minus 180)




1(119904)119876119894119888(119904) = (04465119904 +





119901=

10000 119896119894

= 032549 119896119889

= 0001 120576 = 0 119896119901

=

10000 119896119894

= 010118 119896119889

= 0001 120576 = 0 119896119901

=

10000 119896119894= 022986 119896

119889= 0001 120576 = 0 119896

119901= 6000 119896

119894=



1is set at




0 5 10 15 20 25 300

02

04

06

08

1

12

14

Time (s)

16 18 20 22 24 26 28 30086

088

09

092

094

096

098

1

102

104

0 5 10 15 20 25 30

0

02

04

06

08

1

Time (s)

Time (s) Time (s)

16 18 20 22 24 26 28 30

0

002

004

006

008

01

012

014

minus02

minus04

minus002

120576 = 05

120576 = 25

120576 = 3

120576 = 4

120576 = 5

120576 = 05

120576 = 25

120576 = 3

120576 = 4

120576 = 5



1 the solid







0 200 400 600 800 1000 1200 1400 1600 1800 20000

50

100

150

200

250

300

Time (s)

Liqu

id le

vel o

f CW

TS (m

m)




lowast

119901= 10000 119896lowast

119894= 032549 119896lowast

119889= 0001 and 120576 = 0 in

set point 300mm


119901= 10000 119896lowast

119894= 010118 119896lowast

119889= 0001 and 120576 = 0 in

set point 300mm


119901= 10000 119896lowast

119894= 022986 119896lowast

119889= 0001 and 120576 = 0 in

set point 300mm


lowast

119901= 6000 119896lowast

119894= 013817 119896lowast

119889= 0001 and 120576 = 0 in

set point 300mm


5 Conclusions




119901= 10000 119896lowast

119894= 032549 119896lowast

119889= 0001 and 120576 = 0 in

set point 160mm


119901= 10000 119896lowast

119894= 010118 119896lowast

119889= 0001 and 120576 = 0 in

set point 160mm


1 1199022 1199023 and 119902





2 1199023 and 119902

4(with 119902





4(with 119902

1fixed) Through the


1 1199022 119902

119899+1 The 120576-Routh




119901= 10000 119896lowast

119894= 022986 119896lowast

119889= 0001 and 120576 = 0 in

set point 160mm


lowast

119901= 6000 119896lowast

119894= 013817 119896lowast

119889= 0001 and 120576 = 0 in

set point 160mm



Acknowledgments



References











































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Time (s)0 5 10 15 20 25 30

0

02

04

06

08

1

12

0 5 10 15 20 25 30

0

02

04

06

08

1

12

Syste

m re

spon

seSy

stem

resp

onse

Con

trol e

rror

Con

trol e

rror

Time (s)

Time (s)Time (s)16 18 20 22 24 26 28

075

08

085

09

095

1

105

11



16 18 20 22 24 26 28

0

005

01

015

02

025

03



minus02 minus02

minus005

119863119906(119905) = 03119906(119905 minus 16)



Methods 119896119901

119896119894

119896119889










16 18 20 22 24 26 28 30086

088

09

092

094

096

098

1

102

104





0 5 10 15 20 25 300

05

1

15

0 5 10 15 20 25 30

0

05

1

18 20 22 24 26 28 30

0

002

004

006

008

01

012

014

Con

trol e

rror

Con

trol e

rror

Syste

m re

spon

seSy

stem

resp

onse

minus05

Time (s) Time (s)

Time (s) Time (s)

minus002

119863119906(119905) = 03119906(119905 minus 18)






200 250 300 350 40007

075

08

085

09

095

1

105

11

200 250 300 350 400

0

005

01

015

02

025

03

0 50 100 150 200 250 300 350 4000

02

04

06

08

1

12

14

0 50 100 150 200 250 300 350 400

0

02

04

06

08

1

095

105

Time (s) Time (s)

Time (s)Time (s)





minus005

Con

trol e

rror

Syste

m re

spon

se

Con

trol e

rror

Syste

m re

spon

se

minus02

minus04

119863119906(119905) = 01119906(119905 minus 180)




1(119904)119876119894119888(119904) = (04465119904 +





119901=

10000 119896119894

= 032549 119896119889

= 0001 120576 = 0 119896119901

=

10000 119896119894

= 010118 119896119889

= 0001 120576 = 0 119896119901

=

10000 119896119894= 022986 119896

119889= 0001 120576 = 0 119896

119901= 6000 119896

119894=



1is set at




0 5 10 15 20 25 300

02

04

06

08

1

12

14

Time (s)

16 18 20 22 24 26 28 30086

088

09

092

094

096

098

1

102

104

0 5 10 15 20 25 30

0

02

04

06

08

1

Time (s)

Time (s) Time (s)

16 18 20 22 24 26 28 30

0

002

004

006

008

01

012

014

minus02

minus04

minus002

120576 = 05

120576 = 25

120576 = 3

120576 = 4

120576 = 5

120576 = 05

120576 = 25

120576 = 3

120576 = 4

120576 = 5



1 the solid







0 200 400 600 800 1000 1200 1400 1600 1800 20000

50

100

150

200

250

300

Time (s)

Liqu

id le

vel o

f CW

TS (m

m)




lowast

119901= 10000 119896lowast

119894= 032549 119896lowast

119889= 0001 and 120576 = 0 in

set point 300mm


119901= 10000 119896lowast

119894= 010118 119896lowast

119889= 0001 and 120576 = 0 in

set point 300mm


119901= 10000 119896lowast

119894= 022986 119896lowast

119889= 0001 and 120576 = 0 in

set point 300mm


lowast

119901= 6000 119896lowast

119894= 013817 119896lowast

119889= 0001 and 120576 = 0 in

set point 300mm


5 Conclusions




119901= 10000 119896lowast

119894= 032549 119896lowast

119889= 0001 and 120576 = 0 in

set point 160mm


119901= 10000 119896lowast

119894= 010118 119896lowast

119889= 0001 and 120576 = 0 in

set point 160mm


1 1199022 1199023 and 119902





2 1199023 and 119902

4(with 119902





4(with 119902

1fixed) Through the


1 1199022 119902

119899+1 The 120576-Routh




119901= 10000 119896lowast

119894= 022986 119896lowast

119889= 0001 and 120576 = 0 in

set point 160mm


lowast

119901= 6000 119896lowast

119894= 013817 119896lowast

119889= 0001 and 120576 = 0 in

set point 160mm



Acknowledgments



References











































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










16 18 20 22 24 26 28 30086

088

09

092

094

096

098

1

102

104





0 5 10 15 20 25 300

05

1

15

0 5 10 15 20 25 30

0

05

1

18 20 22 24 26 28 30

0

002

004

006

008

01

012

014

Con

trol e

rror

Con

trol e

rror

Syste

m re

spon

seSy

stem

resp

onse

minus05

Time (s) Time (s)

Time (s) Time (s)

minus002

119863119906(119905) = 03119906(119905 minus 18)






200 250 300 350 40007

075

08

085

09

095

1

105

11

200 250 300 350 400

0

005

01

015

02

025

03

0 50 100 150 200 250 300 350 4000

02

04

06

08

1

12

14

0 50 100 150 200 250 300 350 400

0

02

04

06

08

1

095

105

Time (s) Time (s)

Time (s)Time (s)





minus005

Con

trol e

rror

Syste

m re

spon

se

Con

trol e

rror

Syste

m re

spon

se

minus02

minus04

119863119906(119905) = 01119906(119905 minus 180)




1(119904)119876119894119888(119904) = (04465119904 +





119901=

10000 119896119894

= 032549 119896119889

= 0001 120576 = 0 119896119901

=

10000 119896119894

= 010118 119896119889

= 0001 120576 = 0 119896119901

=

10000 119896119894= 022986 119896

119889= 0001 120576 = 0 119896

119901= 6000 119896

119894=



1is set at




0 5 10 15 20 25 300

02

04

06

08

1

12

14

Time (s)

16 18 20 22 24 26 28 30086

088

09

092

094

096

098

1

102

104

0 5 10 15 20 25 30

0

02

04

06

08

1

Time (s)

Time (s) Time (s)

16 18 20 22 24 26 28 30

0

002

004

006

008

01

012

014

minus02

minus04

minus002

120576 = 05

120576 = 25

120576 = 3

120576 = 4

120576 = 5

120576 = 05

120576 = 25

120576 = 3

120576 = 4

120576 = 5



1 the solid







0 200 400 600 800 1000 1200 1400 1600 1800 20000

50

100

150

200

250

300

Time (s)

Liqu

id le

vel o

f CW

TS (m

m)




lowast

119901= 10000 119896lowast

119894= 032549 119896lowast

119889= 0001 and 120576 = 0 in

set point 300mm


119901= 10000 119896lowast

119894= 010118 119896lowast

119889= 0001 and 120576 = 0 in

set point 300mm


119901= 10000 119896lowast

119894= 022986 119896lowast

119889= 0001 and 120576 = 0 in

set point 300mm


lowast

119901= 6000 119896lowast

119894= 013817 119896lowast

119889= 0001 and 120576 = 0 in

set point 300mm


5 Conclusions




119901= 10000 119896lowast

119894= 032549 119896lowast

119889= 0001 and 120576 = 0 in

set point 160mm


119901= 10000 119896lowast

119894= 010118 119896lowast

119889= 0001 and 120576 = 0 in

set point 160mm


1 1199022 1199023 and 119902





2 1199023 and 119902

4(with 119902





4(with 119902

1fixed) Through the


1 1199022 119902

119899+1 The 120576-Routh




119901= 10000 119896lowast

119894= 022986 119896lowast

119889= 0001 and 120576 = 0 in

set point 160mm


lowast

119901= 6000 119896lowast

119894= 013817 119896lowast

119889= 0001 and 120576 = 0 in

set point 160mm



Acknowledgments



References











































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










200 250 300 350 40007

075

08

085

09

095

1

105

11

200 250 300 350 400

0

005

01

015

02

025

03

0 50 100 150 200 250 300 350 4000

02

04

06

08

1

12

14

0 50 100 150 200 250 300 350 400

0

02

04

06

08

1

095

105

Time (s) Time (s)

Time (s)Time (s)





minus005

Con

trol e

rror

Syste

m re

spon

se

Con

trol e

rror

Syste

m re

spon

se

minus02

minus04

119863119906(119905) = 01119906(119905 minus 180)




1(119904)119876119894119888(119904) = (04465119904 +





119901=

10000 119896119894

= 032549 119896119889

= 0001 120576 = 0 119896119901

=

10000 119896119894

= 010118 119896119889

= 0001 120576 = 0 119896119901

=

10000 119896119894= 022986 119896

119889= 0001 120576 = 0 119896

119901= 6000 119896

119894=



1is set at




0 5 10 15 20 25 300

02

04

06

08

1

12

14

Time (s)

16 18 20 22 24 26 28 30086

088

09

092

094

096

098

1

102

104

0 5 10 15 20 25 30

0

02

04

06

08

1

Time (s)

Time (s) Time (s)

16 18 20 22 24 26 28 30

0

002

004

006

008

01

012

014

minus02

minus04

minus002

120576 = 05

120576 = 25

120576 = 3

120576 = 4

120576 = 5

120576 = 05

120576 = 25

120576 = 3

120576 = 4

120576 = 5



1 the solid







0 200 400 600 800 1000 1200 1400 1600 1800 20000

50

100

150

200

250

300

Time (s)

Liqu

id le

vel o

f CW

TS (m

m)




lowast

119901= 10000 119896lowast

119894= 032549 119896lowast

119889= 0001 and 120576 = 0 in

set point 300mm


119901= 10000 119896lowast

119894= 010118 119896lowast

119889= 0001 and 120576 = 0 in

set point 300mm


119901= 10000 119896lowast

119894= 022986 119896lowast

119889= 0001 and 120576 = 0 in

set point 300mm


lowast

119901= 6000 119896lowast

119894= 013817 119896lowast

119889= 0001 and 120576 = 0 in

set point 300mm


5 Conclusions




119901= 10000 119896lowast

119894= 032549 119896lowast

119889= 0001 and 120576 = 0 in

set point 160mm


119901= 10000 119896lowast

119894= 010118 119896lowast

119889= 0001 and 120576 = 0 in

set point 160mm


1 1199022 1199023 and 119902





2 1199023 and 119902

4(with 119902





4(with 119902

1fixed) Through the


1 1199022 119902

119899+1 The 120576-Routh




119901= 10000 119896lowast

119894= 022986 119896lowast

119889= 0001 and 120576 = 0 in

set point 160mm


lowast

119901= 6000 119896lowast

119894= 013817 119896lowast

119889= 0001 and 120576 = 0 in

set point 160mm



Acknowledgments



References











































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 5 10 15 20 25 300

02

04

06

08

1

12

14

Time (s)

16 18 20 22 24 26 28 30086

088

09

092

094

096

098

1

102

104

0 5 10 15 20 25 30

0

02

04

06

08

1

Time (s)

Time (s) Time (s)

16 18 20 22 24 26 28 30

0

002

004

006

008

01

012

014

minus02

minus04

minus002

120576 = 05

120576 = 25

120576 = 3

120576 = 4

120576 = 5

120576 = 05

120576 = 25

120576 = 3

120576 = 4

120576 = 5



1 the solid







0 200 400 600 800 1000 1200 1400 1600 1800 20000

50

100

150

200

250

300

Time (s)

Liqu

id le

vel o

f CW

TS (m

m)




lowast

119901= 10000 119896lowast

119894= 032549 119896lowast

119889= 0001 and 120576 = 0 in

set point 300mm


119901= 10000 119896lowast

119894= 010118 119896lowast

119889= 0001 and 120576 = 0 in

set point 300mm


119901= 10000 119896lowast

119894= 022986 119896lowast

119889= 0001 and 120576 = 0 in

set point 300mm


lowast

119901= 6000 119896lowast

119894= 013817 119896lowast

119889= 0001 and 120576 = 0 in

set point 300mm


5 Conclusions




119901= 10000 119896lowast

119894= 032549 119896lowast

119889= 0001 and 120576 = 0 in

set point 160mm


119901= 10000 119896lowast

119894= 010118 119896lowast

119889= 0001 and 120576 = 0 in

set point 160mm


1 1199022 1199023 and 119902





2 1199023 and 119902

4(with 119902





4(with 119902

1fixed) Through the


1 1199022 119902

119899+1 The 120576-Routh




119901= 10000 119896lowast

119894= 022986 119896lowast

119889= 0001 and 120576 = 0 in

set point 160mm


lowast

119901= 6000 119896lowast

119894= 013817 119896lowast

119889= 0001 and 120576 = 0 in

set point 160mm



Acknowledgments



References











































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 200 400 600 800 1000 1200 1400 1600 1800 20000

50

100

150

200

250

300

Time (s)

Liqu

id le

vel o

f CW

TS (m

m)




lowast

119901= 10000 119896lowast

119894= 032549 119896lowast

119889= 0001 and 120576 = 0 in

set point 300mm


119901= 10000 119896lowast

119894= 010118 119896lowast

119889= 0001 and 120576 = 0 in

set point 300mm


119901= 10000 119896lowast

119894= 022986 119896lowast

119889= 0001 and 120576 = 0 in

set point 300mm


lowast

119901= 6000 119896lowast

119894= 013817 119896lowast

119889= 0001 and 120576 = 0 in

set point 300mm


5 Conclusions




119901= 10000 119896lowast

119894= 032549 119896lowast

119889= 0001 and 120576 = 0 in

set point 160mm


119901= 10000 119896lowast

119894= 010118 119896lowast

119889= 0001 and 120576 = 0 in

set point 160mm


1 1199022 1199023 and 119902





2 1199023 and 119902

4(with 119902





4(with 119902

1fixed) Through the


1 1199022 119902

119899+1 The 120576-Routh




119901= 10000 119896lowast

119894= 022986 119896lowast

119889= 0001 and 120576 = 0 in

set point 160mm


lowast

119901= 6000 119896lowast

119894= 013817 119896lowast

119889= 0001 and 120576 = 0 in

set point 160mm



Acknowledgments



References











































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119901= 10000 119896lowast

119894= 032549 119896lowast

119889= 0001 and 120576 = 0 in

set point 160mm


119901= 10000 119896lowast

119894= 010118 119896lowast

119889= 0001 and 120576 = 0 in

set point 160mm


1 1199022 1199023 and 119902





2 1199023 and 119902

4(with 119902





4(with 119902

1fixed) Through the


1 1199022 119902

119899+1 The 120576-Routh




119901= 10000 119896lowast

119894= 022986 119896lowast

119889= 0001 and 120576 = 0 in

set point 160mm


lowast

119901= 6000 119896lowast

119894= 013817 119896lowast

119889= 0001 and 120576 = 0 in

set point 160mm



Acknowledgments



References











































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










References











































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra

















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Documents

Research Article Design of Optimal PID Controller with ...downloads.hindawi.com/journals/mpe/2013/582879.pdf · pole placement method is used to design the PID controller. In [ ],