Research Article Adaptive Robust Backstepping Control of ...downloads.hindawi.com/journals/mpe/2016/3690240.pdf · Backstepping control (BC) is one of the most popu-lar nonlinear

Research ArticleAdaptive Robust Backstepping Control ofPermanent Magnet Synchronous Motor Chaotic System withFully Unknown Parameters and External Disturbances

Yang Yu Xudong Guo and Zengqiang Mi

State Key Laboratory of Alternate Electrical Power System with Renewable Energy Sources North China Electric Power UniversityBaoding 071003 China

Correspondence should be addressed to Yang Yu ncepu yy163com

Received 15 December 2015 Revised 16 March 2016 Accepted 30 March 2016

Academic Editor Shengjun Wen

Copyright copy 2016 Yang Yu et alThis is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

The chaotic behavior of permanent magnet synchronous motor is directly related to the parameters of chaotic system Theparameters of permanentmagnet synchronousmotor chaotic system are frequently unknownHence chaotic control of permanentmagnet synchronous motor with unknown parameters is of great significance In order to make the subject more general andfeasible an adaptive robust backstepping control algorithm is proposed to address the issues of fully unknown parametersestimation and external disturbances inhibition on the basis of associating backstepping control with adaptive control Firstlythe mathematical model of permanent magnet synchronous motor chaotic system with fully unknown parameters is constructedand the external disturbances are introduced into the model Secondly an adaptive robust backstepping control technology isemployed to design controller In contrastwith traditional backstepping control the proposed controller ismore concise in structureand avoids many restricted problems The stability of the control approach is proved by Lyapunov stability theory Finally theeffectiveness and correctness of the presented algorithm are verified through multiple simulation experiments and the resultsshow that the proposed scheme enablesmaking permanentmagnet synchronousmotor operate away from chaotic state rapidly andensures the tracking errors to converge to a small neighborhood within the origin rapidly under the full parameters uncertaintiesand external disturbances

1 Introduction

In recent years the permanent magnet synchronous motor(PMSM) is utilized widely in various industrial fields dueto its constantly dropping production cost simple structurehigh torque and high efficiency However Hemati foundthat PMSM would generate chaotic behavior with systemparameters entering into a certain region [1] Previous studieshave shown that the chaotic movement of PMSM willproduce irregular oscillations of torque and speed exacerbatecurrent noise and worsen operation performance and mayeven damage the entire drive system Therefore research onPMSM chaos phenomenon has attracted extensive attentionworldwide [2ndash5] and further studying on the controlmethodof PMSM chaos is of extreme significance [6ndash8]

The nonlinear characteristics of PMSM such as multi-variability strong coupling and high dimension make it dif-ficult to control for traditional linear control theory Hence

a variety of modern and nonlinear control algorithms areintroduced to suppress PMSM chaotic behavior In terms ofthese control algorithms whether or not relying on the modelparameters the previous control methods can primarily bedivided into two categories The first type is on the basisof accurate model parameters such as entrainment andmigration control [9] exact feedback linearization control[10] and decoupling control [11] However the accuracy ofthese control methods directly depends on PMSM modelparameters if the system parameters deviate from the ratedvalues the control performance will go bad The second typeis based on unknown parameters which have become theresearch focus of PMSM chaos suppression recently mainlyincluding sliding mode variable structure control [12 13]fuzzy control [14] and 119867

infincontrol [15] However sliding

mode variable structure control requires uncertain parame-ters to satisfy certain matching conditions fuzzy control is

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2016 Article ID 3690240 14 pageshttpdxdoiorg10115520163690240

2 Mathematical Problems in Engineering

dependent on the fuzzification of Takagi-Sugeno and 119867infin

control is inclined to ignore the operating states under specialconditions [16] In essence PMSM chaotic system is highlysensitive to initial states and parameters and PMSM modelparameters are susceptible to the temperature and humidityof the surrounding environment Therefore PMSM chaoticrepression with unknownmodel parameters has applicabilityto a broader field and ismore in linewith reality [17] Actuallythe adaptive control (AC) provides a natural routine forPMSM chaotic control with unknown parameters which hasbeen presented in literatures [12 13 18]

Backstepping control (BC) is one of the most popu-lar nonlinear control methods newly proposed to addressparameter uncertainty specifically the uncertainty not sat-isfying matching condition which has been successfullyapplied to many engineering fields such as motor drivetemperature control of boiler main steam and rocket loca-tion tracking The core idea of BC is that complex high-dimensional nonlinear systems are decomposed into manysimple low-dimensional subsystems and virtual control vari-ables are introduced to backstepping process to design con-crete controllers In addition BC has been successfullyapplied to suppress Liu chaotic system [19] and Chen chaoticsystem [20] Therefore the idea of combining BC with ACprovides a useful and feasible train of thought to controlPMSM chaotic systemwith unknown parameters Literatures[21 22] have exactly practiced this idea

However the conventional backstepping approach is con-fronted with two major problems of solving complicatedldquoregression matrixrdquo [23] and encountering ldquoexplosion oftermsrdquo [24] In [25] the complexity of regression matrix issufficiently manifested which almost occupies one full pageNevertheless explosion of terms is an inherent shortcomingand is induced by repeated differentiations of virtual vari-ables particularly in design of adaptive backstepping con-troller [26] Additionally integration of BC with AC is fre-quently facedwith the singularity arising from any estimationterm emerging as a denominator of any control input Theoverparameterization caused by the number of estimationslarger than actual system parameters hinders the conven-tional adaptive backstepping control

In addition to the above problems to the extent of ourknowledgemostly existing literatures on PMSMchaotic con-trol only concentrate on the cases of single unknown param-eter and partial unknown parameters [21 22] and there isno way to address the issue of fully unknown parametersFurthermore the existing researches mainly aim at the sit-uation of sudden power failure during PMSM operation [16]the existing conclusions lack the generality Hence throughcombination of BC and AC not only does this paper studythe control issue of PMSM chaos suppression with fullyunknown parameters but also the external disturbances aretaken into account in PMSM chaos model Newly adaptiveupdating laws of unknown parameters are designed to totallyestimate unknown parameters of PMSM chaotic model andadaptive robust backstepping controllers on the basis ofadaptive estimations and external disturbances are developedto drive PMSM to escape out of chaotic state quickly inhibitthe external disturbances and accomplish the given signals

tracking rapidlyThemethod proposed in this paper expandsthe applied range of backstepping control theory in PMSMchaotic systemMoreover the study of chaos control problemwith totally unknown parameters and external disturbancesis more general and practical and the results and conclusionsobtained are more applicable

2 PMSM Chaotic Model with FullyUnknown Parameters

For a PMSM its mathematical model in 119889119902 axis coordinatesystem can be described as follows [16]

=

3119901120601119898

2119869119902minus119861

119869 minus

120591119897

119869

119902= minus

119877

119871119902

119902minus 119901 sdot

119889minus119901120601119898

119871119902

sdot +1

119871119902

119902

119889= minus

119877

119871119889

119889+ 119901 sdot

119902+

1

119871119889

119889

(1)

where is the mechanical angular velocity of the rotatingrotor

119889and 119902are 119889 axis and 119902 axis currents of stator winding

respectively 119889and

119902are 119889 axis and 119902 axis voltages of stator

winding 119901 is the number of rotor pole pairs 120601119898is the flux

generated by permanent magnets 119869 is the moment of inertia119861 is the viscous damping coefficient 120591

119897is the load torque 119877 is

the phase resistance of the stator windings and 119871119889and 119871

119902are

119889 axis and 119902 axis inductances of stator winding respectivelyFor a PMSM with uniform air gap 119871

119889= 119871119902 Hence we use 119871

to substitute 119871119889and 119871

119902in the following paper

Selecting the affine transformation [

119902

119889

] =

[1119901120591 0 0

0 119896 0

0 0 119896

] [

120596

119894119902

119894119889

] and time scale transformation = 120591119905 thePMSMmathematicalmodel described in (1) can be convertedinto dimensionless form as follows

=1

120575(119894119902minus 120596) minus 120591

119897

119894119902= minus119894119902minus 120596 sdot 119894

119889+ 120574 sdot 120596 + 119906

119902

119894119889= minus119894119889+ 120596 sdot 119894

119902+ 119906119889

(2)

where 120591 = 119871119877 119896 = 211986131199012

120591120601119898 119906119889= (1119896119877)

119889 120574 = minus120601

119898119896119871

119906119902= (1119896119877)

119902 120575 = 119869119861120591 and 120591

119897= 1199011205912

120591119897119869

As presented in (2) the dynamic performance of PMSMdepends on three parameters 120575 120574 and 120591

119897 Considering the

most general case let 120575 = 02 120574 = 50 120591119897= 32 119906

119889= minus06

and 119906119902

= 08 If the initial state is selected as (120596 119894119902 119894119889) =

(0 0 0) PMSM system will run on a chaotic state and displaythe chaotic behavior A typical chaotic attractor of PMSM ismanifested in Figure 1

In reality the three parameters 120575 120574 and 120591119897in (2) tend to

be unknownor to have uncertainties resulting fromoperatingconditions In other words when all the parameters 120575 120574and 120591119897cannot be determined (2) actually represents PMSM

chaotic system model with fully unknown parameters

Mathematical Problems in Engineering 3

minus20 minus15 minus10 minus5 0 5 10 15

minus50minus40minus30minus20minus100102030

minus100

102030405060708090

i d

iq120596

Figure 1 Chaotic attractor of PMSM system

3 Design of Adaptive Robust Controller withBackstepping Approach

Taking a more general situation into account the PMSMchaotic model described in (2) is immersed by external dis-turbances The model can be rewritten as follows

=1

120575(119894119902minus 120596) minus 120591

119897

119894119902= minus119894119902minus 120596 sdot 119894

119889+ 120574 sdot 120596 + 119906

119902+ Δ1(x 119905)

119894119889= minus119894119889+ 120596 sdot 119894

119902+ 119906119889+ Δ2(x 119905)

(3)

where Δ1(x 119905) and Δ

2(x 119905) represent the external distur-

bances x indicates the system states and x = (1199091 1199092 1199093) =

(120596 119894119902 119894119889)

31 Control Objective and Assumptions Control problem inthe paper can be described as follows for PMSM chaoticsystem (3) with fully unknown parameters 120575 120574 and 120591

119897and

external disturbances Δ1and Δ

2 adaptive laws of unknown

parameters 120575 120574 and 120591119897are designed and adaptive robust

controllers 119906119889and 119906119902are constructed to ensure PMSMbreaks

away from chaos rapidly and runs into an expected orbitSimultaneously the fully unknownparameters 120575 120574 and 120591

119897can

be estimated accurately and the external disturbances can beinhibited effectively

For convenience of controller design the control systemis supposed to hold some reasonable assumptions as follows

Assumption 1 The state variables for PMSM chaotic system(120596 119894119902 119894119889) are observable

Assumption 2 The external disturbances Δ119894(x 119905) satisfy the

condition |Δ119894(x 119905)| le 119889

119894(x)119891119894(119905) 119894 = 1 2 where 119889

119894(x) is

a known function 119891119894(119905) is an unknown but bounded time-

varying function and |119891119894(119905)| le 119891

119894max where 119891119894max is a

constant

Assumption 3 Thedesired speed and 119889 axis current referencesignals 120596

lowast and 119894lowast

119889and their derivatives are known and

bounded

The estimated values of unknown system parameters aredescribed as and

119897 then the estimation errors and

119897can be expressed as follows

= minus 120575

= minus 120574

119897= 119897minus 120591119897

(4)

32 Controller Design The essence of adaptive robust back-stepping controller is to design controller through combina-tion of backstepping method and adaptive approach thena reasonably stable function is built in accordance withLyapunov stability theory to guarantee error variables to beeffectively stabilized and meanwhile ensure the output ofclosed loop system tracks reference signals quickly On thebasis of this the adaptive robust backstepping controller isdesigned as follows

Step 1 For the speed reference signal 120596lowast define the trackingerror 119890

120596as follows

119890120596= 120596 minus 120596

lowast

(5)

Taking PMSM chaotic system model (3) into account thederivative of (5) can be written as

119890120596= minus

lowast

=1

120575(119894119902minus 120596) minus 120591

119897minus lowast

(6)

Define the tracking error 119890119902of 119902 axis stator current 119894

119902as

follows119890119902= 119894119902minus 119894lowast

119902 (7)

where 119894lowast119902is the expected output value of 119894

119902

For the119889 axis current reference signal 119894lowast119889 its tracking error

119890119889is defined as follows

119890119889= 119894119889minus 119894lowast

119889 (8)

By substitution of (7) into (6) we can obtain

119890120596=

1

120575(119890119902+ 119894lowast

119902minus 120596) minus 120591

119897minus lowast

=1

120575(119894lowast

119902minus 120596) minus

119897+ 119897minus lowast

(9)

Let

119894lowast

119902= 120596 + (

119897+ lowast

minus 1198961119890120596) (10)

119894lowast

119889= 0 (11)

where 1198961represents the positive control gain

Through substitution of (10) into (9) (12) can be obtained

119890120596=

1

120575119890119902+

120575(119897+ lowast

minus 1198961119890120596) minus 119897+ 119897minus lowast

=1

120575119890119902+120575 +

120575(119897+ lowast


=1

120575119890119902+

120575(119897+ lowast

minus 1198961119890120596) + 119897minus 1198961119890120596

(12)


Lyapunov function 1198811is selected as follows

1198811=

1

21198902

120596 (13)

Then the derivative of 1198811can be described as

1= 119890120596119890120596

= 119890120596(1

120575119890119902+

120575(119897+ lowast

minus 1198961119890120596) + 119897minus 1198961119890120596)

(14)

Step 2 To stabilize the output 119902 axis current of PMSM thederivative of 119890

119902is conducted as follows

119890119902= 119894119902minus 119894lowast

119902= minus119894119902minus 120596 sdot 119894

119889+ 120574 sdot 120596 + 119906

119902+ Δ1minus [

+

120575 (119897+ lowast

minus 1198961sdot 119890120596) + ( 120591

119897+ lowast

minus 1198961sdot 119890120596)]

= minus119894119902minus 120596 sdot 119894

119889+ 120574 sdot 120596 + 119906

119902+ Δ1minus [

+

120575 (119897+ lowast

minus 1198961sdot 119890120596) + ( 120591

119897+ lowast

minus 1198961sdot 119890120596)]

= minus119894119902minus 120596 sdot 119894

119889+ sdot 120596 minus sdot 120596 + 119906

119902+ Δ1minus

120575 (119897

+ lowast

minus 1198961sdot 119890120596) minus minus ( 120591

119897+ lowast

minus 1198961sdot 119890120596) = minus119894

119902

minus 120596 sdot 119894119889+ sdot 120596 minus sdot 120596 + 119906

119902+ Δ1minus

120575 (119897+ lowast

minus 1198961

sdot 119890120596) minus 119890120596minus lowast

minus ( 120591119897+ lowast

) + sdot 1198961(120596 minus

lowast

)

(15)

By substitution of (12) into (15) we can get119890119902= minus119894119902minus 120596 sdot 119894

119889+ sdot 120596 minus sdot 120596 + 119906

119902+ Δ1

minus

120575 (119897+ lowast

minus 1198961sdot 119890120596) minus

1

120575119890119902

minus

120575(119897+ lowast

minus 1198961sdot 119890120596) minus 119897+ 1198961sdot 119890120596minus lowast

minus ( 120591119897+ lowast

) + sdot 1198961(120596 minus

lowast

)

(16)

Combined with the mathematical model of PMSM chaoticsystem (16) can be further calculated as follows

119890119902= minus119894119902minus 120596 sdot 119894

119889+ sdot 120596 minus sdot 120596 + 119906

119902+ Δ1

minus

120575 (119897+ lowast

minus 1198961sdot 119890120596) minus

1

120575119890119902

minus

120575(119897+ lowast


minus ( 120591119897+ lowast

) + sdot 1198961sdot

119894119902minus 120596 minus 120591

119897

120575minus sdot 119896

1sdot lowast

= minus119894119902minus 120596 sdot 119894

119889+ sdot 120596 minus sdot 120596 + 119906

119902+ Δ1

minus

120575 (119897+ lowast

minus 1198961sdot 119890120596) minus

1

120575119890119902

minus

120575(119897+ lowast


minus ( 120591119897+ lowast

) +

120575sdot 1198961sdot (119894119902minus 120596) minus sdot 119896

1sdot 120591119897minus

sdot 1198961sdot lowast

= minus119894119902minus 120596 sdot 119894

119889+ sdot 120596 minus sdot 120596 + 119906

119902+ Δ1

minus120575 (119897+ lowast

minus 1198961sdot 119890120596) minus

1

120575119890119902

minus

120575(119897+ lowast


minus ( 120591119897+ lowast

) +


1

sdot (119897minus 119897) minus sdot 119896

1sdot lowast

= minus119894119902minus 120596 sdot 119894

119889+ sdot 120596 minus sdot 120596 + 119906

119902+ Δ1


minus 1198961sdot 119890120596) minus

1

120575119890119902

minus

120575(119897+ lowast


minus ( 120591119897+ lowast

) +


1sdot 119897+

sdot 1198961sdot 119897minus sdot 119896

1sdot lowast

(17)

By substitution of = + 120575 into (17) the following equationcan be obtained

119890119902= minus119894119902minus 120596 sdot 119894

119889+ sdot 120596 minus sdot 120596 + 119906

119902+ Δ1

minus

120575 (119897+ lowast

minus 1198961119890120596) minus

1

120575119890119902

minus

120575(119897+ lowast


minus ( 120591119897+ lowast

) +120575 +


1sdot 119897

+ sdot 1198961sdot 119897minus sdot 119896

1sdot lowast

= minus119894119902minus 120596 sdot 119894

119889+ sdot 120596 minus sdot 120596 + 119906

119902+ Δ1

minus

120575 (119897+ lowast

minus 1198961119890120596) minus

1

120575119890119902

minus

120575(119897+ lowast


minus ( 120591119897+ lowast

) + 1198961sdot (119894119902minus 120596) +

120575sdot 1198961sdot (119894119902minus 120596)

minus sdot 1198961sdot 119897+ sdot 119896

1sdot 119897minus sdot 119896

1sdot lowast

(18)

Lyapunov function 1198812is chosen as follows

1198812= 1198811+1

21198902

119902 (19)



2= 1+ 119890119902119890119902= 119890120596(1

120575119890119902+

120575(119897+ lowast

minus 1198961119890120596) + 119897

minus 1198961119890120596) + 119890119902(minus119894119902minus 120596 sdot 119894

119889+ sdot 120596 minus sdot 120596 + 119906

119902

+ Δ1minus

120575 (119897+ lowast

minus 1198961119890120596) minus

1

120575119890119902

minus

120575(119897+ lowast


minus ( 120591119897+ lowast

) + 1198961sdot (119894119902minus 120596) +

120575sdot 1198961sdot (119894119902minus 120596)



1sdot lowast

)

(20)

The first control variable is selected as

119906119902= 119906119902119904+ 119906119902119903 (21)

where 119906119902119904

and 119906119902119903

are the model compensation and robustcontrol inputs respectively

Then 119906119902119904and 119906

119902119903can be respectively chosen as

119906119902119904

= minus1198962sdot 119890119902+ 119894119902+ 120596 sdot 119894

119889minus sdot 120596

+

120575 (119897+ lowast

minus 1198961119890120596) + lowast

+ ( 120591119897+ lowast

) +

sdot 1198961sdot 119897+ sdot 119896

119903sdot lowast

minus 1198961(119894119902minus 120596)

(22)

119906119902119903

= minus1198892

1(x)

41205761

119890119902 (23)

where 1198962is another positive control gain and 120576

1is a positive

number chosen arbitrarilyBy substitution of (21) and (22) into (20) we can acquire

2= 1+ 119890119902119890119902= 119890120596(1

120575119890119902+

120575(119897+ lowast

minus 1198961119890120596) + 119897

minus 1198961119890120596) + 119890119902(minus1198962119890119902minus

1

120575sdot 119890119902+ 1198961119890120596minus sdot 120596 + 119906

119902119903

+ Δ1minus

120575((119897+ lowast

minus 1198961119890120596) minus 1198961sdot (119894119902minus 120596))

+ 119897( sdot 1198961minus 1)) = 119890

120596(1

120575119890119902

+

120575(119897+ lowast

minus 1198961119890120596) + 119897minus 1198961119890120596) + 119890119902(minus1198962119890119902

minus1

120575sdot 119890119902+ 1198961119890120596minus sdot 120596 minus

1198892

1(x)

41205761

119890119902+ Δ1

minus

120575((119897+ lowast


+ 119897( sdot 1198961minus 1))

(24)

Additionally

119890119902(minus

1198892

1(x)

41205761

119890119902+ Δ1) = minus

1198892

1(x)

41205761

1198902

119902+ Δ1119890119902

le minus1198892

1(x)

41205761

1198902

119902+ 1198891(x) 1198911max

10038161003816100381610038161003816119890119902

10038161003816100381610038161003816

= minus(

1198891(x) 10038161003816100381610038161003816119890119902

10038161003816100381610038161003816

2radic1205761

minus radic12057611198911max)

2

+ 12057611198912

1max

(25)

Step 3 Differentiating the tracking error 119890119889of 119889 axis current

119894119889 we can get

119890119889= 119894119889minus 119894lowast

119889= minus119894119889+ 120596 sdot 119894

119902+ 119906119889+ Δ2minus 119894lowast

119889 (26)

Lyapunov function 1198813is chosen as

1198813= 1198812+1

21198902

119889 (27)

Then the derivative of 1198813can be represented as

3= 2+ 119890119889119890119889

= 2+ 119890119889(minus119894119889+ 120596 sdot 119894

119902+ 119906119889+ Δ2minus 119894lowast

119889)

(28)

In terms of (28) 119889 axis output stator voltage 119906119889can be cal-

culated

119906119889= 119906119889119904+ 119906119889119903 (29)

where

119906119889119904

= minus1198963sdot 119890119889+ 119894119889minus 120596 sdot 119894

119902+ 119894lowast

119889 (30)

119906119889119903

= minus1198892

2(x)

41205762

119890119889 (31)

where 1198963is the positive control gain and 120576

2is a positive num-

ber chosen arbitrarilyBy substitution of (30) and (31) into (26) and (28) respec-

tively the following equations can be acquired

119890119889= minus1198963sdot 119890119889+ 119906119889119903

+ Δ2= minus1198963sdot 119890119889minus1198892

2(x)

41205762

119890119889+ Δ2 (32)

3= 2+ 119890119889119890119889

= 2+ 119890119889(minus1198963sdot 119890119889minus1198892

2(x)

41205762

119890119889+ Δ2)

(33)


119890119889(minus

1198892

2(x)

41205762

119890119889+ Δ2) = minus

1198892

2(x)

41205762

1198902

119889+ Δ2119890119889

le minus1198892

2(x)

41205762

1198902

119889+ 1198892(x) 1198912max

10038161003816100381610038161198901198891003816100381610038161003816

= minus(1198892(x) 1003816100381610038161003816119890119889

1003816100381610038161003816

2radic1205762


2

+ 12057621198912

2max

(34)

Step 4 Lyapunov function 119881 of PMSM chaotic systemwith fully unknown parameters and external disturbances isselected as follows

119881 = 1198813+

1

21205791

2

119897+

1

21205792

2

+1

2120575 sdot 1205793

2

(35)

where 1205791 1205792 and 120579

3represent positive adaptive gains

Combined with equations 120575 =

120575 120574 = 120574 and 120591

119897= 120591119897

derivative of selected Lyapunov function 119881 can be calculatedas follows

= 3+

119897

1205791

120591119897+

1205792

120574 +

120575 sdot 1205793

120575 =

2+ 119890119889(minus1198963sdot 119890119889

+ 119906119889119903

+ Δ2) = 1+ 119890119902(minus1198962119890119902minus

1

120575sdot 119890119902+ 1198961119890120596minus

sdot 120596 + 119906119902119903+ Δ1

minus

120575((119897+ lowast


+ 119897( sdot 1198961minus 1)) + 119890

119889(minus1198963sdot 119890119889+ 119906119889119903

+ Δ2)

= 119890120596(1

120575119890119902+

120575(119897+ lowast

minus 1198961119890120596) + 119897minus 1198961119890120596)

+ 119890119902(minus1198962119890119902minus

1

120575sdot 119890119902+ 1198961119890120596minus sdot 120596 + 119906

119902119903+ Δ1

minus

120575((119897+ lowast


+ 119897( sdot 1198961minus 1)) + 119890

119889(minus1198963sdot 119890119889+ 119906119889119903

+ Δ2) =

1

120575

sdot 119890119902119890120596+

120575(119897+ lowast

minus 1198961119890120596) sdot 119890120596+ 119897119890120596minus 11989611198902

120596

minus 11989631198902

119889minus 11989621198902

119902minus 120596119890

119902minus

1

1205751198902

119902+ 119890119902(119906119902119903+ Δ1)

+ 119890119889(119906119889119903

+ Δ2) minus

120575((119897+ lowast

minus 1198961119890120596)

minus 1198961(119894119902minus 120596)) sdot 119890

119902+ 119897( sdot 1198961minus 1) 119890

119902+ 1198961119890120596119890119902

+119897

1205791

120591119897+

1205792

120574 +

1205751205793

120575 =

1

120575119890119902119890120596+ 1198961119890120596119890119902

+

120575[(119897+ lowast

minus 1198961119890120596) 119890120596minus (119897+ lowast

minus 1198961119890120596) 119890119902

+ 1198961(119894119902minus 120596) 119890

119902+

120575

1205793

] minus 11989611198902

120596minus 11989621198902

119889minus 11989631198902

119902

+ 119897[119890120596minus 119890119902+ 1198961119890119902+

120591119897

1205791

] + [minus120596119890119902+

120574

1205792

]

minus1

1205751198902

119902+ 119890119902(minus

1198892

1(x)

41205761

119890119902+ Δ1) + 119890119889(minus

1198892

2(x)

41205762

119890119889

+ Δ2)

(36)

In terms of (36) the adaptive laws of unknown parameters 120575120574 and 120591

119897can be selected respectively as follows

120575 = minus120579

3[(119897+ lowast

minus 1198961119890120596) sdot (119890120596minus 119890119902)

+ 1198961(119894119902minus 120596) 119890

119902]

(37)

120574 = 1205792120596119890119902 (38)

120591119897= minus1205791[119890120596minus 119890119902+ 1198961119890119902] (39)

By substitution of (37) (38) and (39) into (36) (36) can besimplified as follows

=1

120575119890119902119890120596+ 1198961119890119902119890120596minus 11989611198902

120596minus 11989631198902

119889minus 11989621198902

119902minus

1

1205751198902

119902

+ 119890119902(minus

1198892

1(x)

41205761

119890119902+ Δ1)

+ 119890119889(minus

1198892

2(x)

41205762

119890119889+ Δ2)

(40)

33 Stability Analysis

Theorem 4 For PMSM chaotic system model (3) with fullyunknown parameters and external disturbances design ofadaptive control laws (37) (38) and (40) and selection ofsuitable controller gains 119896

1 1198962 1198963 1205761 and 120576

2and adaptive

gains 1205791 1205792 and 120579

3 the proposed adaptive robust backstepping

controllers (21) and (29) can ensure the tracking error signals(5) (7) and (8) of PMSM chaotic systems are asymptoticallystableThat is to say PMSMchaotic system can run out of chaosquickly through the proposed controllers (21) and (29) and trackthe given reference signals

Through stability analysis we want to verify the correct-ness of the theorem


According to (40) new expression can be obtained as fol-lows through some mathematical computations

=1

120575119890119902119890120596+ 1198961119890119902119890120596minus 11989611198902

120596minus 11989631198902

119889minus 11989621198902

119902minus

1

1205751198902

119902

+ 119890119902(minus

1198892

1(x)

41205761

119890119902+ Δ1) + 119890119889(minus

1198892

2(x)

41205762

119890119889+ Δ2)

= minus1

2120575(119890120596minus 119890119902)2

+1

21205751198902

120596+

1

21205751198902

119902minus1198961

2(119890120596minus 119890119902)2

+1198961

21198902

120596+1198961

21198902

119902minus 11989611198902

120596minus 11989631198902

119889minus 11989621198902

119902minus

1

1205751198902

119902

+ 119890119902(minus

1198892

1(x)

41205761

119890119902+ Δ1) + 119890119889(minus

1198892

2(x)

41205762

119890119889+ Δ2)

= minus1

2120575(119890120596minus 119890119902)2

minus1198961

2(119890120596minus 119890119902)2

minus 11989631198902

119889

+ (1

2120575+1198961

2minus 1198961) 1198902

120596+ (

1

2120575+1198961

2minus 1198962minus

1

120575) 1198902

119902

+ 119890119902(minus

1198892

1(x)

41205761

119890119902+ Δ1)

+ 119890119889(minus

1198892

2(x)

41205762

119890119889+ Δ2)

(41)

Appropriate controller gains 1198961and 1198962are selected as follows

(1

2120575+1198961

2minus 1198961) lt 0

(1

2120575+1198961

2minus 1198962minus

1

120575) lt 0

(42)

Equation (42) can be replaced by the following

1198961gt

1

120575

1198961minus 21198962lt

1

120575

(43)

Then by substitution of (43) into (41) we can obtain

= minus1

2120575(119890120596minus 119890119902)2

minus1198961

2(119890120596minus 119890119902)2

minus 11989631198902

119889

+ (1

2120575+1198961

2minus 1198961) 1198902

120596+ (

1

2120575+1198961

2minus 1198962minus

1

120575) 1198902

119902

+ 119890119902(minus

1198892

1(x)

41205761

119890119902+ Δ1)

+ 119890119889(minus

1198892

2(x)

41205762

119890119889+ Δ2)

le minus1

2120575(119890120596minus 119890119902)2

minus1198961

2(119890120596minus 119890119902)2

minus 11989631198902

119889minus 11989641198902

120596

minus 11989651198902

119902minus (

1198891(x) 10038161003816100381610038161003816119890119902

10038161003816100381610038161003816

2radic1205761


2

+ 12057611198912

1max

minus (1198892(x) 1003816100381610038161003816119890119889

1003816100381610038161003816

2radic1205762


2

+ 12057621198912

2max

le minus1

2120575(119890120596minus 119890119902)2

minus1198961

2(119890120596minus 119890119902)2

minus 11989631198902

119889minus 11989641198902

120596

minus 11989651198902

119902minus (

1198891(x) 10038161003816100381610038161003816119890119902

10038161003816100381610038161003816

2radic1205761


2

minus (1198892(x) 1003816100381610038161003816119890119889

1003816100381610038161003816

2radic1205762


2

+ 1205760

(44)

where 1198964= minus(12120575 + 119896

12 minus 119896

1) ge 0 119896

5= minus(12120575 + 119896

12 minus

1198962minus 1120575) ge 0 and 120576

0= 12057611198912

1max + 12057621198912

2maxLet

119882(119894 (119905)) = minus1

2120575(119890120596minus 119890119902)2

minus1198961

2(119890120596minus 119890119902)2

minus 11989631198902

119889

minus 11989641198902

120596minus 11989651198902

119902

minus (

1198891(x) 10038161003816100381610038161003816119890119902

10038161003816100381610038161003816

2radic1205761


2

minus (1198892(x) 1003816100381610038161003816119890119889

1003816100381610038161003816

2radic1205762


2

+ 1205760

(45)

where 119894(119905) = (119890120596 119890119902 119890119889)

By integration of (45) we can get

int

119905

1199050

119882(119894 (119905)) 119889119905 = minusint

119905

1199050

(119894 (119905)) 119889119905 997904rArr

int

119905

1199050

119882(119894 (119905)) 119889119905 = 119881 (1199050) minus 119881 (119905)

(46)

Since119881(1199050) is bounded and119881(119905) is bounded and nonincreas-

ing hence

lim119905rarrinfin

int

119905

1199050

119882(119894 (119905)) 119889119905 lt infin (47)

Moreover 119882(119894(119905)) is uniform continuous and (119894(119905)) isbounded In accordance with Barbalatrsquos Lemma the follow-ing equation can be obtained

lim119905rarrinfin

119882(119894 (119905)) = 0 (48)

Apparently through selection of suitable controller gains 1198961

1198962 1198963 1205761 and 120576

2 can be ensured to be negative definite

The above derivation has proved that the selected suitablecontroller gains 119896

1 1198962 1198963 1205761 and 120576

2and adaptive gains 120579

1 1205792


minus25

minus20

minus15

minus10

120596minus5

0

5

10

15

20

10 20 30 40 50 60 70 80 90 1000t (s)

Figure 2 The 120596 curve of PMSM chaotic system with no controlinputs 119906

119889and 119906

119902

and 1205793can make the inequalities of 119881 ge 0 and lt 0 hold

In addition equation of 119881 = 0 is not satisfied until 119890120596= 119890119902=

119890119889= 119897= = = 0 In summary PMSM chaotic system

is globally asymptotically stable at the equilibrium point of(119890120596 119890119902 119890119889) = (0 0 0)

4 Numerical Simulation and Discussions

In order to illustrate the superiority of the proposed approachadequately the simulation is carried out in MATLAB envi-ronment for three cases under the initial condition of(120596 119894119902 119894119889) = (0 0 0) Let (120596 119894

119902 119894119889) = (119909

1 1199092 1199093) and the con-

trol parameters are selected as 1198961= 10 119896

2= 30000 119896

3= 5

and 1205761= 1205762= 001 the adaptive gains are chosen as 120579

1= 62

1205792= 100 and 120579

3= 006 The simulation time is chosen as

100 s and the designed controller is put into effect at the timeof 20 s

41 Test-I The PMSM chaotic system is tested with theparameters 120575 = 02 120574 = 50 and 120591

119897= 32 In order to be

consistent with the reality better we assume that the threeparameters of PMSM chaotic system are all unknown withthe initial condition of (120575 120574 120591

119897) = (0 0 0) and the expected

reference signals are set as 120596lowast = 10 and 119894lowast

119889= 1 Further-

more the external disturbances Δ1(x 119905) = 20119909

3sin(5119905) and

Δ2(x 119905) = 10 sin(5119905) are injected into the PMSM chaotic sys-

tem The simulation results given in Figures 2ndash4 apparentlyshow PMSM runs in a chaotic state with no control inputsTherefore introduction of the presented control approach tosuppress chaos in PMSM system will be of great importanceand necessity Figures 5ndash12 show that the proposed controlleris utilized to control the PMSMchaotic systemwhere Figures5ndash7 display the curves of state variables changing over timefor PMSM chaotic system which demonstrate the PMSMsystem stays away from the previous chaotic state when thedesigned controller is added to PMSM chaotic system andtrack the desired signals accurately and rapidly Furthermore

10 20 30 40 50 60 70 80 90 1000t (s)

minus50

minus40

minus30

minus20

minus10

0

10

20

30

40

i q

Figure 3 The 119894119902curve of PMSM chaotic system with no control

inputs 119906119889and 119906

119902

10 20 30 40 50 60 70 80 90 1000t (s)

0

10

20

30

40

50

60

70

80

90

100i d


inputs 119906119889and 119906

119902

minus25

minus20

minus15

minus10

120596 minus5

0

5

10

15

10 20 30 40 50 60 70 80 90 1000t (s)

Figure 5The120596 curve of PMSMchaotic system added the controllerinputs 119906

119889and 119906

119902


minus50

minus40

minus30

minus20

minus10

0

10

20

30

40

i q

10 20 30 40 50 60 70 80 90 1000t (s)

Figure 6The 119894119902curve of PMSMchaotic system added the controller

inputs 119906119889and 119906

119902

0

10

20

30

40

50

60

70

80

90

100

i d

10 20 30 40 50 60 70 80 90 1000t (s)


inputs 119906119889and 119906

119902

minus273

minus272

minus271

minus27

minus269

minus268

minus267

60 61 62 63minus27288

minus27282

30 40 50 60 70 80 90 10020t (s)

times106

times106

ud

Figure 8 The controller input 119906119889

95

952

954

956

958

96

962

964

966

60 6005 601957958959

96961962

times104

times104

uq

30 40 50 60 70 80 90 10020t (s)


minus02

minus015

minus01

minus005

0

005Es

timat

ion

erro

r of120575

30 40 50 60 70 80 90 10020t (s)

Figure 10 The curve of estimation error of 120575

Estim

atio

n er

ror o

f120574

minus50

minus40

minus30

minus20

minus10

0

10

30 40 50 60 70 80 90 10020t (s)



minus35

minus3

minus25

minus2

minus15

minus1

minus05

0

05

30 40 50 60 70 80 90 10020t (s)

Estim

atio

n er

ror o

f120591l

Figure 12 The 119897curve of estimation error of 120591

119897

10 20 30 40 50 60 70 80 90 1000t (s)

minus20

minus15

minus10

minus5

0120596

5

10

15

20

25

Figure 13 The 120596 curve of PMSM chaotic system added the con-troller inputs 119906

119889and 119906

119902

Figures 5ndash7 indicate the proposed controller shown in Figures8-9 can inhibit the external disturbances

Figures 10ndash12 show the estimated errors and 119897

of unknown parameters 120575 120574 and 120591119897for PMSM chaotic

systemwhich testify the effectiveness of constructed adaptivelaws and demonstrate the proposed approach has a goodrobustness against the uncertainties in system parameters

42 Test-II In reality the motor parameters are frequentlyvarying with the design values As a result the parameters120575 120574 and 120591

119897in Test-I are changed into 120575 = 01 120574 = 25 and

120591119897= 16 in Test-II respectively Simultaneously the expected

reference signals are also changed and set as 120596lowast = 20 and119894lowast

119889= 0 In a word the unknown motor parameters and

expected reference signals all differ from Test-I which isable to validate the proposed control algorithm better Thesimulation results are shown in Figures 13ndash20 Figures 13ndash15indicate the motorrsquos output states 120596 119894

119902 and 119894

119889added the

10 20 30 40 50 60 70 80 90 1000t (s)

minus30

minus20

minus10

0

10

20

30

40

i q

Figure 14 The 119894119902curve of PMSM chaotic system added the con-

troller inputs 119906119889and 119906

119902

0

10

20

30

40

50

10 20 30 40 50 60 70 80 90 1000t (s)

i d

minus10



119902

minus51

minus5095

minus509

minus5085

minus508

minus5075

469

469

246

94

469

646

98 47

minus5095minus509minus5085minus508minus5075minus507

times104

times104

ud

30 40 50 60 70 80 90 10020t (s)



minus3000

minus2500

minus2000

minus1500

minus1000

minus500

0

500

1000

469

469

246

94

469

646

98 47

minus2000minus1000

01000

uq

30 40 50 60 70 80 90 10020t (s)


Estim

atio

n er

ror o

f120575

30 40 50 60 70 80 90 10020t (s)

minus01

minus008

minus006

minus004

minus002

0

002

004

006

008

01


Estim

atio

n er

ror o

f120574

minus25

minus20

minus15

minus10

minus5

0

5

30 40 50 60 70 80 90 10020t (s)


minus2

0

2

4

6

8

10

12

14

16

18

20

30 40 50 60 70 80 90 10020t (s)

Estim

atio

n er

ror o

f120591l


119897

0

5

10

15

20

25

minus20

minus15

minus10

120596

minus5

10 20 30 40 50 60 70 80 90 1000t (s)


119889and 119906

119902

controller inputs 119906119889and 119906

119902shown in Figures 16-17 which

demonstrate that the designed controller can guarantee theoutputs track references well and suppress the external dis-turbances effectively Figures 18ndash20 indicate that the designedadaptive law can estimate the fully unknown parametersprecisely even if the fully unknown parameters are changed

43 Test-III For Test-III the external disturbances are en-larged in addition to changing the unknown motor param-eters and expected reference signals on the basis of Test-IIwhich are described as Δ

1(x 119905) = 40119909

3sin(5119905) and Δ

2(x 119905) =

20 sin(5119905) The control difficulty in Test-III is larger thanthe previous two experiments and Test-III is a more generalinstance to verify the controllerrsquos performance The sim-ulation results are shown in Figures 21ndash28 Figures 21ndash23give the curves of the state variables 120596 119894

119902 and 119894

119889 which

manifest these variables are controlled to their references andchaos is eliminated when adding the proposed controllers


10 20 30 40 50 60 70 80 90 1000t (s)

minus20

minus10

0

10

20

30

40

i q



119902

minus10

0

10

20

30

40

50

i d

10 20 30 40 50 60 70 80 90 1000t (s)



119902

minus5085

minus508

minus5075

minus507

minus5065

469

469

5 4747

05

471

minus5085minus508minus5075

times104

times104

ud

30 40 50 60 70 80 90 10020t (s)


minus2500

minus2000

minus1500

minus1000

minus500

0

500

1000

469

469

5 4747

05

471

minus1000minus500

0500

1000

uq

30 40 50 60 70 80 90 10020t (s)


Estim

atio

n er

ror o

f120575

minus02

minus015

minus01

minus005

0

005

01

015

02

30 40 50 60 70 80 90 10020t (s)


Estim

atio

n er

ror o

f120574

30 40 50 60 70 80 90 10020t (s)

minus25

minus20

minus15

minus10

minus5

0

5



30 40 50 60 70 80 90 10020t (s)

minus01

minus008

minus006

minus004

minus002

0

002

004

Estim

atio

n er

ror o

f120591l


119897

119906119889and 119906

119902shown in Figures 24-25 Figures 21ndash23 also illus-

trate the enlarged external disturbances are restrained bythe controllers The estimation errors of the fully unknownparameters are provided in Figures 26ndash28 which proves theeffectiveness of the adaptive laws again

Remark 5 Previous researches on parameter estimation ofPMSM chaotic system mostly assumed that only partialparameters of the system are unknownThe paper takes fullynondeterministic parameters 120575 120574 and 120591

119897into account it

undoubtedly extends the theory of parameter estimation forPMSM chaotic system

Remark 6 The action time of control inputs is 20 s in thesimulation The aim of doing this is to observe the effect ofthe control approach better In reality as long as the chaosoccurs the controller will be put into effect

Remark 7 On the basis of considering fully unknown param-eters the external disturbances are introduced into thePMSM chaotic model Hence the designed control consistsof two parts One is to guarantee the state variables to trackthe reference signals another is to suppress the external dis-turbances In general the simultaneous consideration of fullyunknown parameters and external disturbances makes theresearch results more general and practical

5 Conclusions

In this paper a control approach is proposed to address thecontrol issue of chaos in PMSM system with fully unknownparameters and external disturbances Main conclusions areacquired as the following

(1) Through combination of adaptive control with back-stepping control the presented adaptive robust back-stepping control scheme resolves the main problemsof the conventional backstepping algorithm encoun-tered And the stability of the designed controller isproved by Lyapunov theory

(2) The simulation results show that the designed con-troller is able to make the PMSM operate out of cha-otic state quickly and the adaptive laws are establishedto estimate the unknown parameters accurately Fur-thermore the proposed algorithm can ensure theunknown parameters converge to the actual valuesfast and restrain the external disturbance effectively

(3) The design method in this paper is simple and effect-ive For PMSM chaotic system with fully unknownparameters the control variables in proposed ap-proach can be self-adjusted with the changing of sys-tem parameters Therefore our findings are morepractical and more convenient for engineering appli-cations Future research will discuss the application ofthe proposed control approach into practical imple-mentation

Competing Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work is partially supported by the National NaturalScience Foundation of China (Grant no 51407077) and theFundamental Research Funds for the Central Universities ofMinistry of Education of China (Grant no 2014MS93)

References

[1] N Hemati ldquoStrange attractors in brushless DC motorsrdquo IEEETransactions on Circuits and Systems I FundamentalTheory andApplications vol 41 no 1 pp 40ndash45 1994

[2] Z Li J B Park Y H Joo B Zhang and G Chen ldquoBifurcationsand chaos in a permanent-magnet synchronous motorrdquo IEEETransactions on Circuits and Systems I FundamentalTheory andApplications vol 49 no 3 pp 383ndash387 2002

[3] Z Jing C Yu and G Chen ldquoComplex dynamics in a perma-nent-magnet synchronous motor modelrdquo Chaos Solitons andFractals vol 22 no 4 pp 831ndash848 2004

[4] C-L Li S-M Yu and X-S Luo ldquoFractional-order permanentmagnet synchronous motor and its adaptive chaotic controlrdquoChinese Physics B vol 21 no 10 Article ID 100506 2012

[5] Y-YHou ldquoFinite-time chaos suppression of permanentmagnetsynchronous motor systemsrdquo Entropy vol 16 no 4 pp 2234ndash2243 2014

[6] H P Ren and D Liu ldquoNonlinear feedback control of chaos inpermanent magnet synchronous motorrdquo IEEE Transactions onCircuits and Systems II Express Briefs vol 53 no 1 pp 45ndash502006

[7] A Lorıa ldquoRobust linear control of (chaotic) permanent-magnetsynchronous motors with uncertaintiesrdquo IEEE Transactions onCircuits and Systems I Regular Papers vol 56 no 9 pp 2109ndash2122 2009

[8] M Messadi A Mellit K Kemih and M Ghanes ldquoPredictivecontrol of a chaotic permanent magnet synchronous generatorin a wind turbine systemrdquo Chinese Physics B vol 24 no 1Article ID 010502 2015


[9] Z Li B Zhang and Z Y Mao ldquoEntrainment and migrationcontrol of permanent-magnet synchronous motor systemrdquoControl Theory and Applications vol 19 no 1 pp 53ndash56 2002

[10] H C Cho K S Lee and M S Fadali ldquoAdaptive control ofpmsm systems with chaotic nature using lyapunov stabilitybased feedback linearizationrdquo International Journal of Innova-tive Computing Information and Control vol 5 no 2 pp 479ndash488 2009

[11] D Q Wei B Zhang X S Luo S Y Zeng and D Y QiuldquoEffects of couplings on the collective dynamics of permanent-magnet synchronous motorsrdquo IEEE Transactions on Circuitsand Systems II Express Briefs vol 60 no 10 pp 692ndash696 2013

[12] G Maeng and H H Choi ldquoAdaptive sliding mode control ofa chaotic nonsmooth-air-gap permanent magnet synchronousmotor with uncertaintiesrdquo Nonlinear Dynamics vol 74 no 3pp 571ndash580 2013

[13] T-B-TNguyen T-L Liao and J-J Yan ldquoAdaptive slidingmodecontrol of chaos in permanent magnet synchronous motor viafuzzy neural networksrdquoMathematical Problems in Engineeringvol 2014 Article ID 868415 11 pages 2014

[14] J P Yu B Chen and H S Yu ldquoFuzzy-approximation-basedadaptive control of the chaotic permanent magnet synchronousmotorrdquo Nonlinear Dynamics vol 69 no 3 pp 1479ndash1488 2012

[15] Y-F Yang M-Z Luo S-B Xing X-X Han and H-Q ZhuldquoAnalysis of chaos in permanentmagnet synchronous generatorand optimal output feedback 119867

infincontrolrdquo Acta Physica Sinica

vol 64 no 4 Article ID 040504 2015[16] J-H Hao X-W Wang and H Zhang ldquoChaotic robust control

of permanent magnet synchronous motor system under uncer-tain factorsrdquo Acta Physica Sinica vol 63 no 22 Article ID220203 2014

[17] S H Luo ldquoAdaptive fuzzy dynamic surface control for thechaotic permanent magnet synchronous motor using Nuss-baum gainrdquo Chaos vol 24 no 3 Article ID 033135 2014

[18] D Q Wei X S Luo B H Wang and J Q Fang ldquoRobust adap-tive dynamic surface control of chaos in permanent magnetsynchronous motorrdquo Physics Letters A vol 363 no 1-2 pp 71ndash77 2007

[19] A N Njah ldquoTracking control and synchronization of the newhyperchaotic Liu system via backstepping techniquesrdquo Nonlin-ear Dynamics vol 61 no 1-2 pp 1ndash9 2010

[20] X Wang and J Zhang ldquoTracking control and the Backsteppingdesign of synchronization controller for Chen systemrdquo Interna-tional Journal ofModern Physics B vol 25 no 28 pp 3815ndash38242011

[21] N Chen S Q Xiong B Liu and W H Gui ldquoAdaptive back-stepping control of permanent magnet synchronous motorchaotic systemrdquo Journal of Central South University (Science andTechnology) vol 45 no 1 pp 99ndash104 2014

[22] X Ge and J Huang ldquoChaos control of permanent magnet syn-chronous motorrdquo in Proceedings of the 8th International Confer-ence on Electrical Machines and Systems pp 484ndash488 NanjingChina September 2006

[23] C M Kwan and F L Lewis ldquoRobust backstepping control ofinductionmotors using neural networksrdquo IEEE Transactions onNeural Networks vol 11 no 5 pp 1178ndash1187 2000

[24] A Stotsky I Kanellakopoulus and P Kokotovic Nonlinear andAdaptive Control Design John Wiley amp Sons New York NYUSA 1995

[25] A Harb and W Ahmad ldquoControl of chaotic oscillators using anonlinear recursive backstepping controllerrdquo in Proceedings of

the IASTED Conference on Applied Simulations and Modelingpp 451ndash453 Crete Greece June 2002

[26] J-H Hu and J-B Zou ldquoAdaptive backstepping control of per-manent magnet synchronous motors with parameter uncer-taintiesrdquo Control and Decision vol 21 no 11 pp 1264ndash12692006

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

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Applied MathematicsJournal of


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Stochastic AnalysisInternational Journal of


dependent on the fuzzification of Takagi-Sugeno and 119867infin

control is inclined to ignore the operating states under specialconditions [16] In essence PMSM chaotic system is highlysensitive to initial states and parameters and PMSM modelparameters are susceptible to the temperature and humidityof the surrounding environment Therefore PMSM chaoticrepression with unknownmodel parameters has applicabilityto a broader field and ismore in linewith reality [17] Actuallythe adaptive control (AC) provides a natural routine forPMSM chaotic control with unknown parameters which hasbeen presented in literatures [12 13 18]

Backstepping control (BC) is one of the most popu-lar nonlinear control methods newly proposed to addressparameter uncertainty specifically the uncertainty not sat-isfying matching condition which has been successfullyapplied to many engineering fields such as motor drivetemperature control of boiler main steam and rocket loca-tion tracking The core idea of BC is that complex high-dimensional nonlinear systems are decomposed into manysimple low-dimensional subsystems and virtual control vari-ables are introduced to backstepping process to design con-crete controllers In addition BC has been successfullyapplied to suppress Liu chaotic system [19] and Chen chaoticsystem [20] Therefore the idea of combining BC with ACprovides a useful and feasible train of thought to controlPMSM chaotic systemwith unknown parameters Literatures[21 22] have exactly practiced this idea

However the conventional backstepping approach is con-fronted with two major problems of solving complicatedldquoregression matrixrdquo [23] and encountering ldquoexplosion oftermsrdquo [24] In [25] the complexity of regression matrix issufficiently manifested which almost occupies one full pageNevertheless explosion of terms is an inherent shortcomingand is induced by repeated differentiations of virtual vari-ables particularly in design of adaptive backstepping con-troller [26] Additionally integration of BC with AC is fre-quently facedwith the singularity arising from any estimationterm emerging as a denominator of any control input Theoverparameterization caused by the number of estimationslarger than actual system parameters hinders the conven-tional adaptive backstepping control

In addition to the above problems to the extent of ourknowledgemostly existing literatures on PMSMchaotic con-trol only concentrate on the cases of single unknown param-eter and partial unknown parameters [21 22] and there isno way to address the issue of fully unknown parametersFurthermore the existing researches mainly aim at the sit-uation of sudden power failure during PMSM operation [16]the existing conclusions lack the generality Hence throughcombination of BC and AC not only does this paper studythe control issue of PMSM chaos suppression with fullyunknown parameters but also the external disturbances aretaken into account in PMSM chaos model Newly adaptiveupdating laws of unknown parameters are designed to totallyestimate unknown parameters of PMSM chaotic model andadaptive robust backstepping controllers on the basis ofadaptive estimations and external disturbances are developedto drive PMSM to escape out of chaotic state quickly inhibitthe external disturbances and accomplish the given signals

tracking rapidlyThemethod proposed in this paper expandsthe applied range of backstepping control theory in PMSMchaotic systemMoreover the study of chaos control problemwith totally unknown parameters and external disturbancesis more general and practical and the results and conclusionsobtained are more applicable

2 PMSM Chaotic Model with FullyUnknown Parameters

For a PMSM its mathematical model in 119889119902 axis coordinatesystem can be described as follows [16]

=

3119901120601119898

2119869119902minus119861

119869 minus

120591119897

119869

119902= minus

119877

119871119902

119902minus 119901 sdot

119889minus119901120601119898

119871119902

sdot +1

119871119902

119902

119889= minus

119877

119871119889

119889+ 119901 sdot

119902+

1

119871119889

119889

(1)

where is the mechanical angular velocity of the rotatingrotor

119889and 119902are 119889 axis and 119902 axis currents of stator winding

respectively 119889and

119902are 119889 axis and 119902 axis voltages of stator

winding 119901 is the number of rotor pole pairs 120601119898is the flux

generated by permanent magnets 119869 is the moment of inertia119861 is the viscous damping coefficient 120591

119897is the load torque 119877 is

the phase resistance of the stator windings and 119871119889and 119871

119902are

119889 axis and 119902 axis inductances of stator winding respectivelyFor a PMSM with uniform air gap 119871

119889= 119871119902 Hence we use 119871

to substitute 119871119889and 119871

119902in the following paper

Selecting the affine transformation [

119902

119889

] =

[1119901120591 0 0

0 119896 0

0 0 119896

] [

120596

119894119902

119894119889

] and time scale transformation = 120591119905 thePMSMmathematicalmodel described in (1) can be convertedinto dimensionless form as follows

=1

120575(119894119902minus 120596) minus 120591

119897

119894119902= minus119894119902minus 120596 sdot 119894

119889+ 120574 sdot 120596 + 119906

119902

119894119889= minus119894119889+ 120596 sdot 119894

119902+ 119906119889

(2)

where 120591 = 119871119877 119896 = 211986131199012

120591120601119898 119906119889= (1119896119877)

119889 120574 = minus120601

119898119896119871

119906119902= (1119896119877)

119902 120575 = 119869119861120591 and 120591

119897= 1199011205912

120591119897119869

As presented in (2) the dynamic performance of PMSMdepends on three parameters 120575 120574 and 120591

119897 Considering the

most general case let 120575 = 02 120574 = 50 120591119897= 32 119906

119889= minus06

and 119906119902

= 08 If the initial state is selected as (120596 119894119902 119894119889) =

(0 0 0) PMSM system will run on a chaotic state and displaythe chaotic behavior A typical chaotic attractor of PMSM ismanifested in Figure 1

In reality the three parameters 120575 120574 and 120591119897in (2) tend to

be unknownor to have uncertainties resulting fromoperatingconditions In other words when all the parameters 120575 120574and 120591119897cannot be determined (2) actually represents PMSM

chaotic system model with fully unknown parameters




minus100

102030405060708090

i d

iq120596




=1

120575(119894119902minus 120596) minus 120591

119897

119894119902= minus119894119902minus 120596 sdot 119894

119889+ 120574 sdot 120596 + 119906

119902+ Δ1(x 119905)

119894119889= minus119894119889+ 120596 sdot 119894

119902+ 119906119889+ Δ2(x 119905)

(3)




(120596 119894119902 119894119889)


119897and






119897can





condition |Δ119894(x 119905)| le 119889

119894(x)119891119894(119905) 119894 = 1 2 where 119889

119894(x) is



119894max where 119891119894max is a

constant




bounded




= minus 120575

= minus 120574

119897= 119897minus 120591119897

(4)



120596as follows

119890120596= 120596 minus 120596

lowast

(5)


119890120596= minus

lowast

=1

120575(119894119902minus 120596) minus 120591

119897minus lowast

(6)


119902as


119902 (7)


119902



119890119889= 119894119889minus 119894lowast

119889 (8)


119890120596=

1

120575(119890119902+ 119894lowast

119902minus 120596) minus 120591

119897minus lowast

=1

120575(119894lowast



(9)

Let

119894lowast

119902= 120596 + (

119897+ lowast

minus 1198961119890120596) (10)

119894lowast

119889= 0 (11)



119890120596=

1

120575119890119902+

120575(119897+ lowast


=1

120575119890119902+120575 +

120575(119897+ lowast


=1

120575119890119902+

120575(119897+ lowast

minus 1198961119890120596) + 119897minus 1198961119890120596

(12)



1198811=

1

21198902

120596 (13)


1= 119890120596119890120596

= 119890120596(1

120575119890119902+

120575(119897+ lowast

minus 1198961119890120596) + 119897minus 1198961119890120596)

(14)



119890119902= 119894119902minus 119894lowast

119902= minus119894119902minus 120596 sdot 119894

119889+ 120574 sdot 120596 + 119906

119902+ Δ1minus [

+

120575 (119897+ lowast

minus 1198961sdot 119890120596) + ( 120591

119897+ lowast

minus 1198961sdot 119890120596)]

= minus119894119902minus 120596 sdot 119894

119889+ 120574 sdot 120596 + 119906

119902+ Δ1minus [

+

120575 (119897+ lowast

minus 1198961sdot 119890120596) + ( 120591

119897+ lowast

minus 1198961sdot 119890120596)]

= minus119894119902minus 120596 sdot 119894

119889+ sdot 120596 minus sdot 120596 + 119906

119902+ Δ1minus

120575 (119897

+ lowast


119897+ lowast

minus 1198961sdot 119890120596) = minus119894

119902


119902+ Δ1minus

120575 (119897+ lowast

minus 1198961


minus ( 120591119897+ lowast

) + sdot 1198961(120596 minus

lowast

)

(15)


119889+ sdot 120596 minus sdot 120596 + 119906

119902+ Δ1

minus

120575 (119897+ lowast

minus 1198961sdot 119890120596) minus

1

120575119890119902

minus

120575(119897+ lowast


minus ( 120591119897+ lowast

) + sdot 1198961(120596 minus

lowast

)

(16)


119890119902= minus119894119902minus 120596 sdot 119894

119889+ sdot 120596 minus sdot 120596 + 119906

119902+ Δ1

minus

120575 (119897+ lowast

minus 1198961sdot 119890120596) minus

1

120575119890119902

minus

120575(119897+ lowast


minus ( 120591119897+ lowast


119894119902minus 120596 minus 120591

119897

120575minus sdot 119896

1sdot lowast

= minus119894119902minus 120596 sdot 119894

119889+ sdot 120596 minus sdot 120596 + 119906

119902+ Δ1

minus

120575 (119897+ lowast

minus 1198961sdot 119890120596) minus

1

120575119890119902

minus

120575(119897+ lowast


minus ( 120591119897+ lowast

) +


1sdot 120591119897minus


= minus119894119902minus 120596 sdot 119894

119889+ sdot 120596 minus sdot 120596 + 119906

119902+ Δ1


minus 1198961sdot 119890120596) minus

1

120575119890119902

minus

120575(119897+ lowast


minus ( 120591119897+ lowast

) +


1


1sdot lowast

= minus119894119902minus 120596 sdot 119894

119889+ sdot 120596 minus sdot 120596 + 119906

119902+ Δ1


minus 1198961sdot 119890120596) minus

1

120575119890119902

minus

120575(119897+ lowast


minus ( 120591119897+ lowast

) +


1sdot 119897+


1sdot lowast

(17)


119890119902= minus119894119902minus 120596 sdot 119894

119889+ sdot 120596 minus sdot 120596 + 119906

119902+ Δ1

minus

120575 (119897+ lowast

minus 1198961119890120596) minus

1

120575119890119902

minus

120575(119897+ lowast


minus ( 120591119897+ lowast

) +120575 +


1sdot 119897


1sdot lowast

= minus119894119902minus 120596 sdot 119894

119889+ sdot 120596 minus sdot 120596 + 119906

119902+ Δ1

minus

120575 (119897+ lowast

minus 1198961119890120596) minus

1

120575119890119902

minus

120575(119897+ lowast


minus ( 120591119897+ lowast

) + 1198961sdot (119894119902minus 120596) +

120575sdot 1198961sdot (119894119902minus 120596)



1sdot lowast

(18)


1198812= 1198811+1

21198902

119902 (19)



2= 1+ 119890119902119890119902= 119890120596(1

120575119890119902+

120575(119897+ lowast

minus 1198961119890120596) + 119897


119889+ sdot 120596 minus sdot 120596 + 119906

119902

+ Δ1minus

120575 (119897+ lowast

minus 1198961119890120596) minus

1

120575119890119902

minus

120575(119897+ lowast


minus ( 120591119897+ lowast

) + 1198961sdot (119894119902minus 120596) +

120575sdot 1198961sdot (119894119902minus 120596)



1sdot lowast

)

(20)


119906119902= 119906119902119904+ 119906119902119903 (21)

where 119906119902119904

and 119906119902119903


Then 119906119902119904and 119906


119906119902119904

= minus1198962sdot 119890119902+ 119894119902+ 120596 sdot 119894

119889minus sdot 120596

+

120575 (119897+ lowast

minus 1198961119890120596) + lowast

+ ( 120591119897+ lowast

) +

sdot 1198961sdot 119897+ sdot 119896

119903sdot lowast

minus 1198961(119894119902minus 120596)

(22)

119906119902119903

= minus1198892

1(x)

41205761

119890119902 (23)


1is a positive


2= 1+ 119890119902119890119902= 119890120596(1

120575119890119902+

120575(119897+ lowast

minus 1198961119890120596) + 119897


1

120575sdot 119890119902+ 1198961119890120596minus sdot 120596 + 119906

119902119903

+ Δ1minus

120575((119897+ lowast


+ 119897( sdot 1198961minus 1)) = 119890

120596(1

120575119890119902

+

120575(119897+ lowast


minus1


1198892

1(x)

41205761

119890119902+ Δ1

minus

120575((119897+ lowast


+ 119897( sdot 1198961minus 1))

(24)

Additionally

119890119902(minus

1198892

1(x)

41205761

119890119902+ Δ1) = minus

1198892

1(x)

41205761

1198902

119902+ Δ1119890119902

le minus1198892

1(x)

41205761

1198902

119902+ 1198891(x) 1198911max

10038161003816100381610038161003816119890119902

10038161003816100381610038161003816

= minus(

1198891(x) 10038161003816100381610038161003816119890119902

10038161003816100381610038161003816

2radic1205761


2

+ 12057611198912

1max

(25)


119894119889 we can get

119890119889= 119894119889minus 119894lowast

119889= minus119894119889+ 120596 sdot 119894

119902+ 119906119889+ Δ2minus 119894lowast

119889 (26)


1198813= 1198812+1

21198902

119889 (27)


3= 2+ 119890119889119890119889

= 2+ 119890119889(minus119894119889+ 120596 sdot 119894

119902+ 119906119889+ Δ2minus 119894lowast

119889)

(28)


culated

119906119889= 119906119889119904+ 119906119889119903 (29)

where

119906119889119904


119902+ 119894lowast

119889 (30)

119906119889119903

= minus1198892

2(x)

41205762

119890119889 (31)


2is a positive num-



119890119889= minus1198963sdot 119890119889+ 119906119889119903


2(x)

41205762

119890119889+ Δ2 (32)

3= 2+ 119890119889119890119889

= 2+ 119890119889(minus1198963sdot 119890119889minus1198892

2(x)

41205762

119890119889+ Δ2)

(33)


119890119889(minus

1198892

2(x)

41205762

119890119889+ Δ2) = minus

1198892

2(x)

41205762

1198902

119889+ Δ2119890119889

le minus1198892

2(x)

41205762

1198902

119889+ 1198892(x) 1198912max

10038161003816100381610038161198901198891003816100381610038161003816

= minus(1198892(x) 1003816100381610038161003816119890119889

1003816100381610038161003816

2radic1205762


2

+ 12057621198912

2max

(34)


119881 = 1198813+

1

21205791

2

119897+

1

21205792

2

+1

2120575 sdot 1205793

2

(35)

where 1205791 1205792 and 120579



120575 120574 = 120574 and 120591

119897= 120591119897


= 3+

119897

1205791

120591119897+

1205792

120574 +

120575 sdot 1205793

120575 =

2+ 119890119889(minus1198963sdot 119890119889

+ 119906119889119903

+ Δ2) = 1+ 119890119902(minus1198962119890119902minus

1

120575sdot 119890119902+ 1198961119890120596minus

sdot 120596 + 119906119902119903+ Δ1

minus

120575((119897+ lowast


+ 119897( sdot 1198961minus 1)) + 119890

119889(minus1198963sdot 119890119889+ 119906119889119903

+ Δ2)

= 119890120596(1

120575119890119902+

120575(119897+ lowast

minus 1198961119890120596) + 119897minus 1198961119890120596)

+ 119890119902(minus1198962119890119902minus

1

120575sdot 119890119902+ 1198961119890120596minus sdot 120596 + 119906

119902119903+ Δ1

minus

120575((119897+ lowast


+ 119897( sdot 1198961minus 1)) + 119890

119889(minus1198963sdot 119890119889+ 119906119889119903

+ Δ2) =

1

120575

sdot 119890119902119890120596+

120575(119897+ lowast

minus 1198961119890120596) sdot 119890120596+ 119897119890120596minus 11989611198902

120596

minus 11989631198902

119889minus 11989621198902

119902minus 120596119890

119902minus

1

1205751198902

119902+ 119890119902(119906119902119903+ Δ1)

+ 119890119889(119906119889119903

+ Δ2) minus

120575((119897+ lowast

minus 1198961119890120596)

minus 1198961(119894119902minus 120596)) sdot 119890

119902+ 119897( sdot 1198961minus 1) 119890

119902+ 1198961119890120596119890119902

+119897

1205791

120591119897+

1205792

120574 +

1205751205793

120575 =

1

120575119890119902119890120596+ 1198961119890120596119890119902

+

120575[(119897+ lowast


minus 1198961119890120596) 119890119902

+ 1198961(119894119902minus 120596) 119890

119902+

120575

1205793

] minus 11989611198902

120596minus 11989621198902

119889minus 11989631198902

119902

+ 119897[119890120596minus 119890119902+ 1198961119890119902+

120591119897

1205791

] + [minus120596119890119902+

120574

1205792

]

minus1

1205751198902

119902+ 119890119902(minus

1198892

1(x)

41205761

119890119902+ Δ1) + 119890119889(minus

1198892

2(x)

41205762

119890119889

+ Δ2)

(36)



120575 = minus120579

3[(119897+ lowast

minus 1198961119890120596) sdot (119890120596minus 119890119902)

+ 1198961(119894119902minus 120596) 119890

119902]

(37)

120574 = 1205792120596119890119902 (38)

120591119897= minus1205791[119890120596minus 119890119902+ 1198961119890119902] (39)


=1

120575119890119902119890120596+ 1198961119890119902119890120596minus 11989611198902

120596minus 11989631198902

119889minus 11989621198902

119902minus

1

1205751198902

119902

+ 119890119902(minus

1198892

1(x)

41205761

119890119902+ Δ1)

+ 119890119889(minus

1198892

2(x)

41205762

119890119889+ Δ2)

(40)



1 1198962 1198963 1205761 and 120576

2and adaptive

gains 1205791 1205792 and 120579






=1

120575119890119902119890120596+ 1198961119890119902119890120596minus 11989611198902

120596minus 11989631198902

119889minus 11989621198902

119902minus

1

1205751198902

119902

+ 119890119902(minus

1198892

1(x)

41205761

119890119902+ Δ1) + 119890119889(minus

1198892

2(x)

41205762

119890119889+ Δ2)

= minus1

2120575(119890120596minus 119890119902)2

+1

21205751198902

120596+

1

21205751198902

119902minus1198961

2(119890120596minus 119890119902)2

+1198961

21198902

120596+1198961

21198902

119902minus 11989611198902

120596minus 11989631198902

119889minus 11989621198902

119902minus

1

1205751198902

119902

+ 119890119902(minus

1198892

1(x)

41205761

119890119902+ Δ1) + 119890119889(minus

1198892

2(x)

41205762

119890119889+ Δ2)

= minus1

2120575(119890120596minus 119890119902)2

minus1198961

2(119890120596minus 119890119902)2

minus 11989631198902

119889

+ (1

2120575+1198961

2minus 1198961) 1198902

120596+ (

1

2120575+1198961

2minus 1198962minus

1

120575) 1198902

119902

+ 119890119902(minus

1198892

1(x)

41205761

119890119902+ Δ1)

+ 119890119889(minus

1198892

2(x)

41205762

119890119889+ Δ2)

(41)


(1

2120575+1198961

2minus 1198961) lt 0

(1

2120575+1198961

2minus 1198962minus

1

120575) lt 0

(42)


1198961gt

1

120575

1198961minus 21198962lt

1

120575

(43)


= minus1

2120575(119890120596minus 119890119902)2

minus1198961

2(119890120596minus 119890119902)2

minus 11989631198902

119889

+ (1

2120575+1198961

2minus 1198961) 1198902

120596+ (

1

2120575+1198961

2minus 1198962minus

1

120575) 1198902

119902

+ 119890119902(minus

1198892

1(x)

41205761

119890119902+ Δ1)

+ 119890119889(minus

1198892

2(x)

41205762

119890119889+ Δ2)

le minus1

2120575(119890120596minus 119890119902)2

minus1198961

2(119890120596minus 119890119902)2

minus 11989631198902

119889minus 11989641198902

120596

minus 11989651198902

119902minus (

1198891(x) 10038161003816100381610038161003816119890119902

10038161003816100381610038161003816

2radic1205761


2

+ 12057611198912

1max

minus (1198892(x) 1003816100381610038161003816119890119889

1003816100381610038161003816

2radic1205762


2

+ 12057621198912

2max

le minus1

2120575(119890120596minus 119890119902)2

minus1198961

2(119890120596minus 119890119902)2

minus 11989631198902

119889minus 11989641198902

120596

minus 11989651198902

119902minus (

1198891(x) 10038161003816100381610038161003816119890119902

10038161003816100381610038161003816

2radic1205761


2

minus (1198892(x) 1003816100381610038161003816119890119889

1003816100381610038161003816

2radic1205762


2

+ 1205760

(44)

where 1198964= minus(12120575 + 119896

12 minus 119896

1) ge 0 119896

5= minus(12120575 + 119896

12 minus

1198962minus 1120575) ge 0 and 120576

0= 12057611198912

1max + 12057621198912

2maxLet

119882(119894 (119905)) = minus1

2120575(119890120596minus 119890119902)2

minus1198961

2(119890120596minus 119890119902)2

minus 11989631198902

119889

minus 11989641198902

120596minus 11989651198902

119902

minus (

1198891(x) 10038161003816100381610038161003816119890119902

10038161003816100381610038161003816

2radic1205761


2

minus (1198892(x) 1003816100381610038161003816119890119889

1003816100381610038161003816

2radic1205762


2

+ 1205760

(45)

where 119894(119905) = (119890120596 119890119902 119890119889)


int

119905

1199050

119882(119894 (119905)) 119889119905 = minusint

119905

1199050

(119894 (119905)) 119889119905 997904rArr

int

119905

1199050

119882(119894 (119905)) 119889119905 = 119881 (1199050) minus 119881 (119905)

(46)


ing hence

lim119905rarrinfin

int

119905

1199050

119882(119894 (119905)) 119889119905 lt infin (47)


lim119905rarrinfin

119882(119894 (119905)) = 0 (48)


1198962 1198963 1205761 and 120576



1 1198962 1198963 1205761 and 120576


1 1205792


minus25

minus20

minus15

minus10

120596minus5

0

5

10

15

20

10 20 30 40 50 60 70 80 90 1000t (s)


119889and 119906

119902







119902 119894119889) = (119909

1 1199092 1199093) and the con-


2= 30000 119896

3= 5


1= 62

1205792= 100 and 120579








119889= 1 Further-


3sin(5119905) and



10 20 30 40 50 60 70 80 90 1000t (s)

minus50

minus40

minus30

minus20

minus10

0

10

20

30

40

i q


inputs 119906119889and 119906

119902

10 20 30 40 50 60 70 80 90 1000t (s)

0

10

20

30

40

50

60

70

80

90

100i d


inputs 119906119889and 119906

119902

minus25

minus20

minus15

minus10

120596 minus5

0

5

10

15

10 20 30 40 50 60 70 80 90 1000t (s)


119889and 119906

119902


minus50

minus40

minus30

minus20

minus10

0

10

20

30

40

i q

10 20 30 40 50 60 70 80 90 1000t (s)


inputs 119906119889and 119906

119902

0

10

20

30

40

50

60

70

80

90

100

i d

10 20 30 40 50 60 70 80 90 1000t (s)


inputs 119906119889and 119906

119902

minus273

minus272

minus271

minus27

minus269

minus268

minus267

60 61 62 63minus27288

minus27282

30 40 50 60 70 80 90 10020t (s)

times106

times106

ud


95

952

954

956

958

96

962

964

966

60 6005 601957958959

96961962

times104

times104

uq

30 40 50 60 70 80 90 10020t (s)


minus02

minus015

minus01

minus005

0

005Es

timat

ion

erro

r of120575

30 40 50 60 70 80 90 10020t (s)


Estim

atio

n er

ror o

f120574

minus50

minus40

minus30

minus20

minus10

0

10

30 40 50 60 70 80 90 10020t (s)



minus35

minus3

minus25

minus2

minus15

minus1

minus05

0

05

30 40 50 60 70 80 90 10020t (s)

Estim

atio

n er

ror o

f120591l


119897

10 20 30 40 50 60 70 80 90 1000t (s)

minus20

minus15

minus10

minus5

0120596

5

10

15

20

25


119889and 119906

119902











119902 and 119894

119889added the

10 20 30 40 50 60 70 80 90 1000t (s)

minus30

minus20

minus10

0

10

20

30

40

i q



119902

0

10

20

30

40

50

10 20 30 40 50 60 70 80 90 1000t (s)

i d

minus10



119902

minus51

minus5095

minus509

minus5085

minus508

minus5075

469

469

246

94

469

646

98 47


times104

times104

ud

30 40 50 60 70 80 90 10020t (s)



minus3000

minus2500

minus2000

minus1500

minus1000

minus500

0

500

1000

469

469

246

94

469

646

98 47

minus2000minus1000

01000

uq

30 40 50 60 70 80 90 10020t (s)


Estim

atio

n er

ror o

f120575

30 40 50 60 70 80 90 10020t (s)

minus01

minus008

minus006

minus004

minus002

0

002

004

006

008

01


Estim

atio

n er

ror o

f120574

minus25

minus20

minus15

minus10

minus5

0

5

30 40 50 60 70 80 90 10020t (s)


minus2

0

2

4

6

8

10

12

14

16

18

20

30 40 50 60 70 80 90 10020t (s)

Estim

atio

n er

ror o

f120591l


119897

0

5

10

15

20

25

minus20

minus15

minus10

120596

minus5

10 20 30 40 50 60 70 80 90 1000t (s)


119889and 119906

119902





1(x 119905) = 40119909

3sin(5119905) and Δ

2(x 119905) =


119902 and 119894

119889 which



10 20 30 40 50 60 70 80 90 1000t (s)

minus20

minus10

0

10

20

30

40

i q



119902

minus10

0

10

20

30

40

50

i d

10 20 30 40 50 60 70 80 90 1000t (s)



119902

minus5085

minus508

minus5075

minus507

minus5065

469

469

5 4747

05

471


times104

times104

ud

30 40 50 60 70 80 90 10020t (s)


minus2500

minus2000

minus1500

minus1000

minus500

0

500

1000

469

469

5 4747

05

471

minus1000minus500

0500

1000

uq

30 40 50 60 70 80 90 10020t (s)


Estim

atio

n er

ror o

f120575

minus02

minus015

minus01

minus005

0

005

01

015

02

30 40 50 60 70 80 90 10020t (s)


Estim

atio

n er

ror o

f120574

30 40 50 60 70 80 90 10020t (s)

minus25

minus20

minus15

minus10

minus5

0

5



30 40 50 60 70 80 90 10020t (s)

minus01

minus008

minus006

minus004

minus002

0

002

004

Estim

atio

n er

ror o

f120591l


119897

119906119889and 119906








5 Conclusions





Competing Interests


Acknowledgments


References






































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












minus100

102030405060708090

i d

iq120596




=1

120575(119894119902minus 120596) minus 120591

119897

119894119902= minus119894119902minus 120596 sdot 119894

119889+ 120574 sdot 120596 + 119906

119902+ Δ1(x 119905)

119894119889= minus119894119889+ 120596 sdot 119894

119902+ 119906119889+ Δ2(x 119905)

(3)




(120596 119894119902 119894119889)


119897and






119897can





condition |Δ119894(x 119905)| le 119889

119894(x)119891119894(119905) 119894 = 1 2 where 119889

119894(x) is



119894max where 119891119894max is a

constant




bounded




= minus 120575

= minus 120574

119897= 119897minus 120591119897

(4)



120596as follows

119890120596= 120596 minus 120596

lowast

(5)


119890120596= minus

lowast

=1

120575(119894119902minus 120596) minus 120591

119897minus lowast

(6)


119902as


119902 (7)


119902



119890119889= 119894119889minus 119894lowast

119889 (8)


119890120596=

1

120575(119890119902+ 119894lowast

119902minus 120596) minus 120591

119897minus lowast

=1

120575(119894lowast



(9)

Let

119894lowast

119902= 120596 + (

119897+ lowast

minus 1198961119890120596) (10)

119894lowast

119889= 0 (11)



119890120596=

1

120575119890119902+

120575(119897+ lowast


=1

120575119890119902+120575 +

120575(119897+ lowast


=1

120575119890119902+

120575(119897+ lowast

minus 1198961119890120596) + 119897minus 1198961119890120596

(12)



1198811=

1

21198902

120596 (13)


1= 119890120596119890120596

= 119890120596(1

120575119890119902+

120575(119897+ lowast

minus 1198961119890120596) + 119897minus 1198961119890120596)

(14)



119890119902= 119894119902minus 119894lowast

119902= minus119894119902minus 120596 sdot 119894

119889+ 120574 sdot 120596 + 119906

119902+ Δ1minus [

+

120575 (119897+ lowast

minus 1198961sdot 119890120596) + ( 120591

119897+ lowast

minus 1198961sdot 119890120596)]

= minus119894119902minus 120596 sdot 119894

119889+ 120574 sdot 120596 + 119906

119902+ Δ1minus [

+

120575 (119897+ lowast

minus 1198961sdot 119890120596) + ( 120591

119897+ lowast

minus 1198961sdot 119890120596)]

= minus119894119902minus 120596 sdot 119894

119889+ sdot 120596 minus sdot 120596 + 119906

119902+ Δ1minus

120575 (119897

+ lowast


119897+ lowast

minus 1198961sdot 119890120596) = minus119894

119902


119902+ Δ1minus

120575 (119897+ lowast

minus 1198961


minus ( 120591119897+ lowast

) + sdot 1198961(120596 minus

lowast

)

(15)


119889+ sdot 120596 minus sdot 120596 + 119906

119902+ Δ1

minus

120575 (119897+ lowast

minus 1198961sdot 119890120596) minus

1

120575119890119902

minus

120575(119897+ lowast


minus ( 120591119897+ lowast

) + sdot 1198961(120596 minus

lowast

)

(16)


119890119902= minus119894119902minus 120596 sdot 119894

119889+ sdot 120596 minus sdot 120596 + 119906

119902+ Δ1

minus

120575 (119897+ lowast

minus 1198961sdot 119890120596) minus

1

120575119890119902

minus

120575(119897+ lowast


minus ( 120591119897+ lowast


119894119902minus 120596 minus 120591

119897

120575minus sdot 119896

1sdot lowast

= minus119894119902minus 120596 sdot 119894

119889+ sdot 120596 minus sdot 120596 + 119906

119902+ Δ1

minus

120575 (119897+ lowast

minus 1198961sdot 119890120596) minus

1

120575119890119902

minus

120575(119897+ lowast


minus ( 120591119897+ lowast

) +


1sdot 120591119897minus


= minus119894119902minus 120596 sdot 119894

119889+ sdot 120596 minus sdot 120596 + 119906

119902+ Δ1


minus 1198961sdot 119890120596) minus

1

120575119890119902

minus

120575(119897+ lowast


minus ( 120591119897+ lowast

) +


1


1sdot lowast

= minus119894119902minus 120596 sdot 119894

119889+ sdot 120596 minus sdot 120596 + 119906

119902+ Δ1


minus 1198961sdot 119890120596) minus

1

120575119890119902

minus

120575(119897+ lowast


minus ( 120591119897+ lowast

) +


1sdot 119897+


1sdot lowast

(17)


119890119902= minus119894119902minus 120596 sdot 119894

119889+ sdot 120596 minus sdot 120596 + 119906

119902+ Δ1

minus

120575 (119897+ lowast

minus 1198961119890120596) minus

1

120575119890119902

minus

120575(119897+ lowast


minus ( 120591119897+ lowast

) +120575 +


1sdot 119897


1sdot lowast

= minus119894119902minus 120596 sdot 119894

119889+ sdot 120596 minus sdot 120596 + 119906

119902+ Δ1

minus

120575 (119897+ lowast

minus 1198961119890120596) minus

1

120575119890119902

minus

120575(119897+ lowast


minus ( 120591119897+ lowast

) + 1198961sdot (119894119902minus 120596) +

120575sdot 1198961sdot (119894119902minus 120596)



1sdot lowast

(18)


1198812= 1198811+1

21198902

119902 (19)



2= 1+ 119890119902119890119902= 119890120596(1

120575119890119902+

120575(119897+ lowast

minus 1198961119890120596) + 119897


119889+ sdot 120596 minus sdot 120596 + 119906

119902

+ Δ1minus

120575 (119897+ lowast

minus 1198961119890120596) minus

1

120575119890119902

minus

120575(119897+ lowast


minus ( 120591119897+ lowast

) + 1198961sdot (119894119902minus 120596) +

120575sdot 1198961sdot (119894119902minus 120596)



1sdot lowast

)

(20)


119906119902= 119906119902119904+ 119906119902119903 (21)

where 119906119902119904

and 119906119902119903


Then 119906119902119904and 119906


119906119902119904

= minus1198962sdot 119890119902+ 119894119902+ 120596 sdot 119894

119889minus sdot 120596

+

120575 (119897+ lowast

minus 1198961119890120596) + lowast

+ ( 120591119897+ lowast

) +

sdot 1198961sdot 119897+ sdot 119896

119903sdot lowast

minus 1198961(119894119902minus 120596)

(22)

119906119902119903

= minus1198892

1(x)

41205761

119890119902 (23)


1is a positive


2= 1+ 119890119902119890119902= 119890120596(1

120575119890119902+

120575(119897+ lowast

minus 1198961119890120596) + 119897


1

120575sdot 119890119902+ 1198961119890120596minus sdot 120596 + 119906

119902119903

+ Δ1minus

120575((119897+ lowast


+ 119897( sdot 1198961minus 1)) = 119890

120596(1

120575119890119902

+

120575(119897+ lowast


minus1


1198892

1(x)

41205761

119890119902+ Δ1

minus

120575((119897+ lowast


+ 119897( sdot 1198961minus 1))

(24)

Additionally

119890119902(minus

1198892

1(x)

41205761

119890119902+ Δ1) = minus

1198892

1(x)

41205761

1198902

119902+ Δ1119890119902

le minus1198892

1(x)

41205761

1198902

119902+ 1198891(x) 1198911max

10038161003816100381610038161003816119890119902

10038161003816100381610038161003816

= minus(

1198891(x) 10038161003816100381610038161003816119890119902

10038161003816100381610038161003816

2radic1205761


2

+ 12057611198912

1max

(25)


119894119889 we can get

119890119889= 119894119889minus 119894lowast

119889= minus119894119889+ 120596 sdot 119894

119902+ 119906119889+ Δ2minus 119894lowast

119889 (26)


1198813= 1198812+1

21198902

119889 (27)


3= 2+ 119890119889119890119889

= 2+ 119890119889(minus119894119889+ 120596 sdot 119894

119902+ 119906119889+ Δ2minus 119894lowast

119889)

(28)


culated

119906119889= 119906119889119904+ 119906119889119903 (29)

where

119906119889119904


119902+ 119894lowast

119889 (30)

119906119889119903

= minus1198892

2(x)

41205762

119890119889 (31)


2is a positive num-



119890119889= minus1198963sdot 119890119889+ 119906119889119903


2(x)

41205762

119890119889+ Δ2 (32)

3= 2+ 119890119889119890119889

= 2+ 119890119889(minus1198963sdot 119890119889minus1198892

2(x)

41205762

119890119889+ Δ2)

(33)


119890119889(minus

1198892

2(x)

41205762

119890119889+ Δ2) = minus

1198892

2(x)

41205762

1198902

119889+ Δ2119890119889

le minus1198892

2(x)

41205762

1198902

119889+ 1198892(x) 1198912max

10038161003816100381610038161198901198891003816100381610038161003816

= minus(1198892(x) 1003816100381610038161003816119890119889

1003816100381610038161003816

2radic1205762


2

+ 12057621198912

2max

(34)


119881 = 1198813+

1

21205791

2

119897+

1

21205792

2

+1

2120575 sdot 1205793

2

(35)

where 1205791 1205792 and 120579



120575 120574 = 120574 and 120591

119897= 120591119897


= 3+

119897

1205791

120591119897+

1205792

120574 +

120575 sdot 1205793

120575 =

2+ 119890119889(minus1198963sdot 119890119889

+ 119906119889119903

+ Δ2) = 1+ 119890119902(minus1198962119890119902minus

1

120575sdot 119890119902+ 1198961119890120596minus

sdot 120596 + 119906119902119903+ Δ1

minus

120575((119897+ lowast


+ 119897( sdot 1198961minus 1)) + 119890

119889(minus1198963sdot 119890119889+ 119906119889119903

+ Δ2)

= 119890120596(1

120575119890119902+

120575(119897+ lowast

minus 1198961119890120596) + 119897minus 1198961119890120596)

+ 119890119902(minus1198962119890119902minus

1

120575sdot 119890119902+ 1198961119890120596minus sdot 120596 + 119906

119902119903+ Δ1

minus

120575((119897+ lowast


+ 119897( sdot 1198961minus 1)) + 119890

119889(minus1198963sdot 119890119889+ 119906119889119903

+ Δ2) =

1

120575

sdot 119890119902119890120596+

120575(119897+ lowast

minus 1198961119890120596) sdot 119890120596+ 119897119890120596minus 11989611198902

120596

minus 11989631198902

119889minus 11989621198902

119902minus 120596119890

119902minus

1

1205751198902

119902+ 119890119902(119906119902119903+ Δ1)

+ 119890119889(119906119889119903

+ Δ2) minus

120575((119897+ lowast

minus 1198961119890120596)

minus 1198961(119894119902minus 120596)) sdot 119890

119902+ 119897( sdot 1198961minus 1) 119890

119902+ 1198961119890120596119890119902

+119897

1205791

120591119897+

1205792

120574 +

1205751205793

120575 =

1

120575119890119902119890120596+ 1198961119890120596119890119902

+

120575[(119897+ lowast


minus 1198961119890120596) 119890119902

+ 1198961(119894119902minus 120596) 119890

119902+

120575

1205793

] minus 11989611198902

120596minus 11989621198902

119889minus 11989631198902

119902

+ 119897[119890120596minus 119890119902+ 1198961119890119902+

120591119897

1205791

] + [minus120596119890119902+

120574

1205792

]

minus1

1205751198902

119902+ 119890119902(minus

1198892

1(x)

41205761

119890119902+ Δ1) + 119890119889(minus

1198892

2(x)

41205762

119890119889

+ Δ2)

(36)



120575 = minus120579

3[(119897+ lowast

minus 1198961119890120596) sdot (119890120596minus 119890119902)

+ 1198961(119894119902minus 120596) 119890

119902]

(37)

120574 = 1205792120596119890119902 (38)

120591119897= minus1205791[119890120596minus 119890119902+ 1198961119890119902] (39)


=1

120575119890119902119890120596+ 1198961119890119902119890120596minus 11989611198902

120596minus 11989631198902

119889minus 11989621198902

119902minus

1

1205751198902

119902

+ 119890119902(minus

1198892

1(x)

41205761

119890119902+ Δ1)

+ 119890119889(minus

1198892

2(x)

41205762

119890119889+ Δ2)

(40)



1 1198962 1198963 1205761 and 120576

2and adaptive

gains 1205791 1205792 and 120579






=1

120575119890119902119890120596+ 1198961119890119902119890120596minus 11989611198902

120596minus 11989631198902

119889minus 11989621198902

119902minus

1

1205751198902

119902

+ 119890119902(minus

1198892

1(x)

41205761

119890119902+ Δ1) + 119890119889(minus

1198892

2(x)

41205762

119890119889+ Δ2)

= minus1

2120575(119890120596minus 119890119902)2

+1

21205751198902

120596+

1

21205751198902

119902minus1198961

2(119890120596minus 119890119902)2

+1198961

21198902

120596+1198961

21198902

119902minus 11989611198902

120596minus 11989631198902

119889minus 11989621198902

119902minus

1

1205751198902

119902

+ 119890119902(minus

1198892

1(x)

41205761

119890119902+ Δ1) + 119890119889(minus

1198892

2(x)

41205762

119890119889+ Δ2)

= minus1

2120575(119890120596minus 119890119902)2

minus1198961

2(119890120596minus 119890119902)2

minus 11989631198902

119889

+ (1

2120575+1198961

2minus 1198961) 1198902

120596+ (

1

2120575+1198961

2minus 1198962minus

1

120575) 1198902

119902

+ 119890119902(minus

1198892

1(x)

41205761

119890119902+ Δ1)

+ 119890119889(minus

1198892

2(x)

41205762

119890119889+ Δ2)

(41)


(1

2120575+1198961

2minus 1198961) lt 0

(1

2120575+1198961

2minus 1198962minus

1

120575) lt 0

(42)


1198961gt

1

120575

1198961minus 21198962lt

1

120575

(43)


= minus1

2120575(119890120596minus 119890119902)2

minus1198961

2(119890120596minus 119890119902)2

minus 11989631198902

119889

+ (1

2120575+1198961

2minus 1198961) 1198902

120596+ (

1

2120575+1198961

2minus 1198962minus

1

120575) 1198902

119902

+ 119890119902(minus

1198892

1(x)

41205761

119890119902+ Δ1)

+ 119890119889(minus

1198892

2(x)

41205762

119890119889+ Δ2)

le minus1

2120575(119890120596minus 119890119902)2

minus1198961

2(119890120596minus 119890119902)2

minus 11989631198902

119889minus 11989641198902

120596

minus 11989651198902

119902minus (

1198891(x) 10038161003816100381610038161003816119890119902

10038161003816100381610038161003816

2radic1205761


2

+ 12057611198912

1max

minus (1198892(x) 1003816100381610038161003816119890119889

1003816100381610038161003816

2radic1205762


2

+ 12057621198912

2max

le minus1

2120575(119890120596minus 119890119902)2

minus1198961

2(119890120596minus 119890119902)2

minus 11989631198902

119889minus 11989641198902

120596

minus 11989651198902

119902minus (

1198891(x) 10038161003816100381610038161003816119890119902

10038161003816100381610038161003816

2radic1205761


2

minus (1198892(x) 1003816100381610038161003816119890119889

1003816100381610038161003816

2radic1205762


2

+ 1205760

(44)

where 1198964= minus(12120575 + 119896

12 minus 119896

1) ge 0 119896

5= minus(12120575 + 119896

12 minus

1198962minus 1120575) ge 0 and 120576

0= 12057611198912

1max + 12057621198912

2maxLet

119882(119894 (119905)) = minus1

2120575(119890120596minus 119890119902)2

minus1198961

2(119890120596minus 119890119902)2

minus 11989631198902

119889

minus 11989641198902

120596minus 11989651198902

119902

minus (

1198891(x) 10038161003816100381610038161003816119890119902

10038161003816100381610038161003816

2radic1205761


2

minus (1198892(x) 1003816100381610038161003816119890119889

1003816100381610038161003816

2radic1205762


2

+ 1205760

(45)

where 119894(119905) = (119890120596 119890119902 119890119889)


int

119905

1199050

119882(119894 (119905)) 119889119905 = minusint

119905

1199050

(119894 (119905)) 119889119905 997904rArr

int

119905

1199050

119882(119894 (119905)) 119889119905 = 119881 (1199050) minus 119881 (119905)

(46)


ing hence

lim119905rarrinfin

int

119905

1199050

119882(119894 (119905)) 119889119905 lt infin (47)


lim119905rarrinfin

119882(119894 (119905)) = 0 (48)


1198962 1198963 1205761 and 120576



1 1198962 1198963 1205761 and 120576


1 1205792


minus25

minus20

minus15

minus10

120596minus5

0

5

10

15

20

10 20 30 40 50 60 70 80 90 1000t (s)


119889and 119906

119902







119902 119894119889) = (119909

1 1199092 1199093) and the con-


2= 30000 119896

3= 5


1= 62

1205792= 100 and 120579








119889= 1 Further-


3sin(5119905) and



10 20 30 40 50 60 70 80 90 1000t (s)

minus50

minus40

minus30

minus20

minus10

0

10

20

30

40

i q


inputs 119906119889and 119906

119902

10 20 30 40 50 60 70 80 90 1000t (s)

0

10

20

30

40

50

60

70

80

90

100i d


inputs 119906119889and 119906

119902

minus25

minus20

minus15

minus10

120596 minus5

0

5

10

15

10 20 30 40 50 60 70 80 90 1000t (s)


119889and 119906

119902


minus50

minus40

minus30

minus20

minus10

0

10

20

30

40

i q

10 20 30 40 50 60 70 80 90 1000t (s)


inputs 119906119889and 119906

119902

0

10

20

30

40

50

60

70

80

90

100

i d

10 20 30 40 50 60 70 80 90 1000t (s)


inputs 119906119889and 119906

119902

minus273

minus272

minus271

minus27

minus269

minus268

minus267

60 61 62 63minus27288

minus27282

30 40 50 60 70 80 90 10020t (s)

times106

times106

ud


95

952

954

956

958

96

962

964

966

60 6005 601957958959

96961962

times104

times104

uq

30 40 50 60 70 80 90 10020t (s)


minus02

minus015

minus01

minus005

0

005Es

timat

ion

erro

r of120575

30 40 50 60 70 80 90 10020t (s)


Estim

atio

n er

ror o

f120574

minus50

minus40

minus30

minus20

minus10

0

10

30 40 50 60 70 80 90 10020t (s)



minus35

minus3

minus25

minus2

minus15

minus1

minus05

0

05

30 40 50 60 70 80 90 10020t (s)

Estim

atio

n er

ror o

f120591l


119897

10 20 30 40 50 60 70 80 90 1000t (s)

minus20

minus15

minus10

minus5

0120596

5

10

15

20

25


119889and 119906

119902











119902 and 119894

119889added the

10 20 30 40 50 60 70 80 90 1000t (s)

minus30

minus20

minus10

0

10

20

30

40

i q



119902

0

10

20

30

40

50

10 20 30 40 50 60 70 80 90 1000t (s)

i d

minus10



119902

minus51

minus5095

minus509

minus5085

minus508

minus5075

469

469

246

94

469

646

98 47


times104

times104

ud

30 40 50 60 70 80 90 10020t (s)



minus3000

minus2500

minus2000

minus1500

minus1000

minus500

0

500

1000

469

469

246

94

469

646

98 47

minus2000minus1000

01000

uq

30 40 50 60 70 80 90 10020t (s)


Estim

atio

n er

ror o

f120575

30 40 50 60 70 80 90 10020t (s)

minus01

minus008

minus006

minus004

minus002

0

002

004

006

008

01


Estim

atio

n er

ror o

f120574

minus25

minus20

minus15

minus10

minus5

0

5

30 40 50 60 70 80 90 10020t (s)


minus2

0

2

4

6

8

10

12

14

16

18

20

30 40 50 60 70 80 90 10020t (s)

Estim

atio

n er

ror o

f120591l


119897

0

5

10

15

20

25

minus20

minus15

minus10

120596

minus5

10 20 30 40 50 60 70 80 90 1000t (s)


119889and 119906

119902





1(x 119905) = 40119909

3sin(5119905) and Δ

2(x 119905) =


119902 and 119894

119889 which



10 20 30 40 50 60 70 80 90 1000t (s)

minus20

minus10

0

10

20

30

40

i q



119902

minus10

0

10

20

30

40

50

i d

10 20 30 40 50 60 70 80 90 1000t (s)



119902

minus5085

minus508

minus5075

minus507

minus5065

469

469

5 4747

05

471


times104

times104

ud

30 40 50 60 70 80 90 10020t (s)


minus2500

minus2000

minus1500

minus1000

minus500

0

500

1000

469

469

5 4747

05

471

minus1000minus500

0500

1000

uq

30 40 50 60 70 80 90 10020t (s)


Estim

atio

n er

ror o

f120575

minus02

minus015

minus01

minus005

0

005

01

015

02

30 40 50 60 70 80 90 10020t (s)


Estim

atio

n er

ror o

f120574

30 40 50 60 70 80 90 10020t (s)

minus25

minus20

minus15

minus10

minus5

0

5



30 40 50 60 70 80 90 10020t (s)

minus01

minus008

minus006

minus004

minus002

0

002

004

Estim

atio

n er

ror o

f120591l


119897

119906119889and 119906








5 Conclusions





Competing Interests


Acknowledgments


References






































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











1198811=

1

21198902

120596 (13)


1= 119890120596119890120596

= 119890120596(1

120575119890119902+

120575(119897+ lowast

minus 1198961119890120596) + 119897minus 1198961119890120596)

(14)



119890119902= 119894119902minus 119894lowast

119902= minus119894119902minus 120596 sdot 119894

119889+ 120574 sdot 120596 + 119906

119902+ Δ1minus [

+

120575 (119897+ lowast

minus 1198961sdot 119890120596) + ( 120591

119897+ lowast

minus 1198961sdot 119890120596)]

= minus119894119902minus 120596 sdot 119894

119889+ 120574 sdot 120596 + 119906

119902+ Δ1minus [

+

120575 (119897+ lowast

minus 1198961sdot 119890120596) + ( 120591

119897+ lowast

minus 1198961sdot 119890120596)]

= minus119894119902minus 120596 sdot 119894

119889+ sdot 120596 minus sdot 120596 + 119906

119902+ Δ1minus

120575 (119897

+ lowast


119897+ lowast

minus 1198961sdot 119890120596) = minus119894

119902


119902+ Δ1minus

120575 (119897+ lowast

minus 1198961


minus ( 120591119897+ lowast

) + sdot 1198961(120596 minus

lowast

)

(15)


119889+ sdot 120596 minus sdot 120596 + 119906

119902+ Δ1

minus

120575 (119897+ lowast

minus 1198961sdot 119890120596) minus

1

120575119890119902

minus

120575(119897+ lowast


minus ( 120591119897+ lowast

) + sdot 1198961(120596 minus

lowast

)

(16)


119890119902= minus119894119902minus 120596 sdot 119894

119889+ sdot 120596 minus sdot 120596 + 119906

119902+ Δ1

minus

120575 (119897+ lowast

minus 1198961sdot 119890120596) minus

1

120575119890119902

minus

120575(119897+ lowast


minus ( 120591119897+ lowast


119894119902minus 120596 minus 120591

119897

120575minus sdot 119896

1sdot lowast

= minus119894119902minus 120596 sdot 119894

119889+ sdot 120596 minus sdot 120596 + 119906

119902+ Δ1

minus

120575 (119897+ lowast

minus 1198961sdot 119890120596) minus

1

120575119890119902

minus

120575(119897+ lowast


minus ( 120591119897+ lowast

) +


1sdot 120591119897minus


= minus119894119902minus 120596 sdot 119894

119889+ sdot 120596 minus sdot 120596 + 119906

119902+ Δ1


minus 1198961sdot 119890120596) minus

1

120575119890119902

minus

120575(119897+ lowast


minus ( 120591119897+ lowast

) +


1


1sdot lowast

= minus119894119902minus 120596 sdot 119894

119889+ sdot 120596 minus sdot 120596 + 119906

119902+ Δ1


minus 1198961sdot 119890120596) minus

1

120575119890119902

minus

120575(119897+ lowast


minus ( 120591119897+ lowast

) +


1sdot 119897+


1sdot lowast

(17)


119890119902= minus119894119902minus 120596 sdot 119894

119889+ sdot 120596 minus sdot 120596 + 119906

119902+ Δ1

minus

120575 (119897+ lowast

minus 1198961119890120596) minus

1

120575119890119902

minus

120575(119897+ lowast


minus ( 120591119897+ lowast

) +120575 +


1sdot 119897


1sdot lowast

= minus119894119902minus 120596 sdot 119894

119889+ sdot 120596 minus sdot 120596 + 119906

119902+ Δ1

minus

120575 (119897+ lowast

minus 1198961119890120596) minus

1

120575119890119902

minus

120575(119897+ lowast


minus ( 120591119897+ lowast

) + 1198961sdot (119894119902minus 120596) +

120575sdot 1198961sdot (119894119902minus 120596)



1sdot lowast

(18)


1198812= 1198811+1

21198902

119902 (19)



2= 1+ 119890119902119890119902= 119890120596(1

120575119890119902+

120575(119897+ lowast

minus 1198961119890120596) + 119897


119889+ sdot 120596 minus sdot 120596 + 119906

119902

+ Δ1minus

120575 (119897+ lowast

minus 1198961119890120596) minus

1

120575119890119902

minus

120575(119897+ lowast


minus ( 120591119897+ lowast

) + 1198961sdot (119894119902minus 120596) +

120575sdot 1198961sdot (119894119902minus 120596)



1sdot lowast

)

(20)


119906119902= 119906119902119904+ 119906119902119903 (21)

where 119906119902119904

and 119906119902119903


Then 119906119902119904and 119906


119906119902119904

= minus1198962sdot 119890119902+ 119894119902+ 120596 sdot 119894

119889minus sdot 120596

+

120575 (119897+ lowast

minus 1198961119890120596) + lowast

+ ( 120591119897+ lowast

) +

sdot 1198961sdot 119897+ sdot 119896

119903sdot lowast

minus 1198961(119894119902minus 120596)

(22)

119906119902119903

= minus1198892

1(x)

41205761

119890119902 (23)


1is a positive


2= 1+ 119890119902119890119902= 119890120596(1

120575119890119902+

120575(119897+ lowast

minus 1198961119890120596) + 119897


1

120575sdot 119890119902+ 1198961119890120596minus sdot 120596 + 119906

119902119903

+ Δ1minus

120575((119897+ lowast


+ 119897( sdot 1198961minus 1)) = 119890

120596(1

120575119890119902

+

120575(119897+ lowast


minus1


1198892

1(x)

41205761

119890119902+ Δ1

minus

120575((119897+ lowast


+ 119897( sdot 1198961minus 1))

(24)

Additionally

119890119902(minus

1198892

1(x)

41205761

119890119902+ Δ1) = minus

1198892

1(x)

41205761

1198902

119902+ Δ1119890119902

le minus1198892

1(x)

41205761

1198902

119902+ 1198891(x) 1198911max

10038161003816100381610038161003816119890119902

10038161003816100381610038161003816

= minus(

1198891(x) 10038161003816100381610038161003816119890119902

10038161003816100381610038161003816

2radic1205761


2

+ 12057611198912

1max

(25)


119894119889 we can get

119890119889= 119894119889minus 119894lowast

119889= minus119894119889+ 120596 sdot 119894

119902+ 119906119889+ Δ2minus 119894lowast

119889 (26)


1198813= 1198812+1

21198902

119889 (27)


3= 2+ 119890119889119890119889

= 2+ 119890119889(minus119894119889+ 120596 sdot 119894

119902+ 119906119889+ Δ2minus 119894lowast

119889)

(28)


culated

119906119889= 119906119889119904+ 119906119889119903 (29)

where

119906119889119904


119902+ 119894lowast

119889 (30)

119906119889119903

= minus1198892

2(x)

41205762

119890119889 (31)


2is a positive num-



119890119889= minus1198963sdot 119890119889+ 119906119889119903


2(x)

41205762

119890119889+ Δ2 (32)

3= 2+ 119890119889119890119889

= 2+ 119890119889(minus1198963sdot 119890119889minus1198892

2(x)

41205762

119890119889+ Δ2)

(33)


119890119889(minus

1198892

2(x)

41205762

119890119889+ Δ2) = minus

1198892

2(x)

41205762

1198902

119889+ Δ2119890119889

le minus1198892

2(x)

41205762

1198902

119889+ 1198892(x) 1198912max

10038161003816100381610038161198901198891003816100381610038161003816

= minus(1198892(x) 1003816100381610038161003816119890119889

1003816100381610038161003816

2radic1205762


2

+ 12057621198912

2max

(34)


119881 = 1198813+

1

21205791

2

119897+

1

21205792

2

+1

2120575 sdot 1205793

2

(35)

where 1205791 1205792 and 120579



120575 120574 = 120574 and 120591

119897= 120591119897


= 3+

119897

1205791

120591119897+

1205792

120574 +

120575 sdot 1205793

120575 =

2+ 119890119889(minus1198963sdot 119890119889

+ 119906119889119903

+ Δ2) = 1+ 119890119902(minus1198962119890119902minus

1

120575sdot 119890119902+ 1198961119890120596minus

sdot 120596 + 119906119902119903+ Δ1

minus

120575((119897+ lowast


+ 119897( sdot 1198961minus 1)) + 119890

119889(minus1198963sdot 119890119889+ 119906119889119903

+ Δ2)

= 119890120596(1

120575119890119902+

120575(119897+ lowast

minus 1198961119890120596) + 119897minus 1198961119890120596)

+ 119890119902(minus1198962119890119902minus

1

120575sdot 119890119902+ 1198961119890120596minus sdot 120596 + 119906

119902119903+ Δ1

minus

120575((119897+ lowast


+ 119897( sdot 1198961minus 1)) + 119890

119889(minus1198963sdot 119890119889+ 119906119889119903

+ Δ2) =

1

120575

sdot 119890119902119890120596+

120575(119897+ lowast

minus 1198961119890120596) sdot 119890120596+ 119897119890120596minus 11989611198902

120596

minus 11989631198902

119889minus 11989621198902

119902minus 120596119890

119902minus

1

1205751198902

119902+ 119890119902(119906119902119903+ Δ1)

+ 119890119889(119906119889119903

+ Δ2) minus

120575((119897+ lowast

minus 1198961119890120596)

minus 1198961(119894119902minus 120596)) sdot 119890

119902+ 119897( sdot 1198961minus 1) 119890

119902+ 1198961119890120596119890119902

+119897

1205791

120591119897+

1205792

120574 +

1205751205793

120575 =

1

120575119890119902119890120596+ 1198961119890120596119890119902

+

120575[(119897+ lowast


minus 1198961119890120596) 119890119902

+ 1198961(119894119902minus 120596) 119890

119902+

120575

1205793

] minus 11989611198902

120596minus 11989621198902

119889minus 11989631198902

119902

+ 119897[119890120596minus 119890119902+ 1198961119890119902+

120591119897

1205791

] + [minus120596119890119902+

120574

1205792

]

minus1

1205751198902

119902+ 119890119902(minus

1198892

1(x)

41205761

119890119902+ Δ1) + 119890119889(minus

1198892

2(x)

41205762

119890119889

+ Δ2)

(36)



120575 = minus120579

3[(119897+ lowast

minus 1198961119890120596) sdot (119890120596minus 119890119902)

+ 1198961(119894119902minus 120596) 119890

119902]

(37)

120574 = 1205792120596119890119902 (38)

120591119897= minus1205791[119890120596minus 119890119902+ 1198961119890119902] (39)


=1

120575119890119902119890120596+ 1198961119890119902119890120596minus 11989611198902

120596minus 11989631198902

119889minus 11989621198902

119902minus

1

1205751198902

119902

+ 119890119902(minus

1198892

1(x)

41205761

119890119902+ Δ1)

+ 119890119889(minus

1198892

2(x)

41205762

119890119889+ Δ2)

(40)



1 1198962 1198963 1205761 and 120576

2and adaptive

gains 1205791 1205792 and 120579






=1

120575119890119902119890120596+ 1198961119890119902119890120596minus 11989611198902

120596minus 11989631198902

119889minus 11989621198902

119902minus

1

1205751198902

119902

+ 119890119902(minus

1198892

1(x)

41205761

119890119902+ Δ1) + 119890119889(minus

1198892

2(x)

41205762

119890119889+ Δ2)

= minus1

2120575(119890120596minus 119890119902)2

+1

21205751198902

120596+

1

21205751198902

119902minus1198961

2(119890120596minus 119890119902)2

+1198961

21198902

120596+1198961

21198902

119902minus 11989611198902

120596minus 11989631198902

119889minus 11989621198902

119902minus

1

1205751198902

119902

+ 119890119902(minus

1198892

1(x)

41205761

119890119902+ Δ1) + 119890119889(minus

1198892

2(x)

41205762

119890119889+ Δ2)

= minus1

2120575(119890120596minus 119890119902)2

minus1198961

2(119890120596minus 119890119902)2

minus 11989631198902

119889

+ (1

2120575+1198961

2minus 1198961) 1198902

120596+ (

1

2120575+1198961

2minus 1198962minus

1

120575) 1198902

119902

+ 119890119902(minus

1198892

1(x)

41205761

119890119902+ Δ1)

+ 119890119889(minus

1198892

2(x)

41205762

119890119889+ Δ2)

(41)


(1

2120575+1198961

2minus 1198961) lt 0

(1

2120575+1198961

2minus 1198962minus

1

120575) lt 0

(42)


1198961gt

1

120575

1198961minus 21198962lt

1

120575

(43)


= minus1

2120575(119890120596minus 119890119902)2

minus1198961

2(119890120596minus 119890119902)2

minus 11989631198902

119889

+ (1

2120575+1198961

2minus 1198961) 1198902

120596+ (

1

2120575+1198961

2minus 1198962minus

1

120575) 1198902

119902

+ 119890119902(minus

1198892

1(x)

41205761

119890119902+ Δ1)

+ 119890119889(minus

1198892

2(x)

41205762

119890119889+ Δ2)

le minus1

2120575(119890120596minus 119890119902)2

minus1198961

2(119890120596minus 119890119902)2

minus 11989631198902

119889minus 11989641198902

120596

minus 11989651198902

119902minus (

1198891(x) 10038161003816100381610038161003816119890119902

10038161003816100381610038161003816

2radic1205761


2

+ 12057611198912

1max

minus (1198892(x) 1003816100381610038161003816119890119889

1003816100381610038161003816

2radic1205762


2

+ 12057621198912

2max

le minus1

2120575(119890120596minus 119890119902)2

minus1198961

2(119890120596minus 119890119902)2

minus 11989631198902

119889minus 11989641198902

120596

minus 11989651198902

119902minus (

1198891(x) 10038161003816100381610038161003816119890119902

10038161003816100381610038161003816

2radic1205761


2

minus (1198892(x) 1003816100381610038161003816119890119889

1003816100381610038161003816

2radic1205762


2

+ 1205760

(44)

where 1198964= minus(12120575 + 119896

12 minus 119896

1) ge 0 119896

5= minus(12120575 + 119896

12 minus

1198962minus 1120575) ge 0 and 120576

0= 12057611198912

1max + 12057621198912

2maxLet

119882(119894 (119905)) = minus1

2120575(119890120596minus 119890119902)2

minus1198961

2(119890120596minus 119890119902)2

minus 11989631198902

119889

minus 11989641198902

120596minus 11989651198902

119902

minus (

1198891(x) 10038161003816100381610038161003816119890119902

10038161003816100381610038161003816

2radic1205761


2

minus (1198892(x) 1003816100381610038161003816119890119889

1003816100381610038161003816

2radic1205762


2

+ 1205760

(45)

where 119894(119905) = (119890120596 119890119902 119890119889)


int

119905

1199050

119882(119894 (119905)) 119889119905 = minusint

119905

1199050

(119894 (119905)) 119889119905 997904rArr

int

119905

1199050

119882(119894 (119905)) 119889119905 = 119881 (1199050) minus 119881 (119905)

(46)


ing hence

lim119905rarrinfin

int

119905

1199050

119882(119894 (119905)) 119889119905 lt infin (47)


lim119905rarrinfin

119882(119894 (119905)) = 0 (48)


1198962 1198963 1205761 and 120576



1 1198962 1198963 1205761 and 120576


1 1205792


minus25

minus20

minus15

minus10

120596minus5

0

5

10

15

20

10 20 30 40 50 60 70 80 90 1000t (s)


119889and 119906

119902







119902 119894119889) = (119909

1 1199092 1199093) and the con-


2= 30000 119896

3= 5


1= 62

1205792= 100 and 120579








119889= 1 Further-


3sin(5119905) and



10 20 30 40 50 60 70 80 90 1000t (s)

minus50

minus40

minus30

minus20

minus10

0

10

20

30

40

i q


inputs 119906119889and 119906

119902

10 20 30 40 50 60 70 80 90 1000t (s)

0

10

20

30

40

50

60

70

80

90

100i d


inputs 119906119889and 119906

119902

minus25

minus20

minus15

minus10

120596 minus5

0

5

10

15

10 20 30 40 50 60 70 80 90 1000t (s)


119889and 119906

119902


minus50

minus40

minus30

minus20

minus10

0

10

20

30

40

i q

10 20 30 40 50 60 70 80 90 1000t (s)


inputs 119906119889and 119906

119902

0

10

20

30

40

50

60

70

80

90

100

i d

10 20 30 40 50 60 70 80 90 1000t (s)


inputs 119906119889and 119906

119902

minus273

minus272

minus271

minus27

minus269

minus268

minus267

60 61 62 63minus27288

minus27282

30 40 50 60 70 80 90 10020t (s)

times106

times106

ud


95

952

954

956

958

96

962

964

966

60 6005 601957958959

96961962

times104

times104

uq

30 40 50 60 70 80 90 10020t (s)


minus02

minus015

minus01

minus005

0

005Es

timat

ion

erro

r of120575

30 40 50 60 70 80 90 10020t (s)


Estim

atio

n er

ror o

f120574

minus50

minus40

minus30

minus20

minus10

0

10

30 40 50 60 70 80 90 10020t (s)



minus35

minus3

minus25

minus2

minus15

minus1

minus05

0

05

30 40 50 60 70 80 90 10020t (s)

Estim

atio

n er

ror o

f120591l


119897

10 20 30 40 50 60 70 80 90 1000t (s)

minus20

minus15

minus10

minus5

0120596

5

10

15

20

25


119889and 119906

119902











119902 and 119894

119889added the

10 20 30 40 50 60 70 80 90 1000t (s)

minus30

minus20

minus10

0

10

20

30

40

i q



119902

0

10

20

30

40

50

10 20 30 40 50 60 70 80 90 1000t (s)

i d

minus10



119902

minus51

minus5095

minus509

minus5085

minus508

minus5075

469

469

246

94

469

646

98 47


times104

times104

ud

30 40 50 60 70 80 90 10020t (s)



minus3000

minus2500

minus2000

minus1500

minus1000

minus500

0

500

1000

469

469

246

94

469

646

98 47

minus2000minus1000

01000

uq

30 40 50 60 70 80 90 10020t (s)


Estim

atio

n er

ror o

f120575

30 40 50 60 70 80 90 10020t (s)

minus01

minus008

minus006

minus004

minus002

0

002

004

006

008

01


Estim

atio

n er

ror o

f120574

minus25

minus20

minus15

minus10

minus5

0

5

30 40 50 60 70 80 90 10020t (s)


minus2

0

2

4

6

8

10

12

14

16

18

20

30 40 50 60 70 80 90 10020t (s)

Estim

atio

n er

ror o

f120591l


119897

0

5

10

15

20

25

minus20

minus15

minus10

120596

minus5

10 20 30 40 50 60 70 80 90 1000t (s)


119889and 119906

119902





1(x 119905) = 40119909

3sin(5119905) and Δ

2(x 119905) =


119902 and 119894

119889 which



10 20 30 40 50 60 70 80 90 1000t (s)

minus20

minus10

0

10

20

30

40

i q



119902

minus10

0

10

20

30

40

50

i d

10 20 30 40 50 60 70 80 90 1000t (s)



119902

minus5085

minus508

minus5075

minus507

minus5065

469

469

5 4747

05

471


times104

times104

ud

30 40 50 60 70 80 90 10020t (s)


minus2500

minus2000

minus1500

minus1000

minus500

0

500

1000

469

469

5 4747

05

471

minus1000minus500

0500

1000

uq

30 40 50 60 70 80 90 10020t (s)


Estim

atio

n er

ror o

f120575

minus02

minus015

minus01

minus005

0

005

01

015

02

30 40 50 60 70 80 90 10020t (s)


Estim

atio

n er

ror o

f120574

30 40 50 60 70 80 90 10020t (s)

minus25

minus20

minus15

minus10

minus5

0

5



30 40 50 60 70 80 90 10020t (s)

minus01

minus008

minus006

minus004

minus002

0

002

004

Estim

atio

n er

ror o

f120591l


119897

119906119889and 119906








5 Conclusions





Competing Interests


Acknowledgments


References






































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











2= 1+ 119890119902119890119902= 119890120596(1

120575119890119902+

120575(119897+ lowast

minus 1198961119890120596) + 119897


119889+ sdot 120596 minus sdot 120596 + 119906

119902

+ Δ1minus

120575 (119897+ lowast

minus 1198961119890120596) minus

1

120575119890119902

minus

120575(119897+ lowast


minus ( 120591119897+ lowast

) + 1198961sdot (119894119902minus 120596) +

120575sdot 1198961sdot (119894119902minus 120596)



1sdot lowast

)

(20)


119906119902= 119906119902119904+ 119906119902119903 (21)

where 119906119902119904

and 119906119902119903


Then 119906119902119904and 119906


119906119902119904

= minus1198962sdot 119890119902+ 119894119902+ 120596 sdot 119894

119889minus sdot 120596

+

120575 (119897+ lowast

minus 1198961119890120596) + lowast

+ ( 120591119897+ lowast

) +

sdot 1198961sdot 119897+ sdot 119896

119903sdot lowast

minus 1198961(119894119902minus 120596)

(22)

119906119902119903

= minus1198892

1(x)

41205761

119890119902 (23)


1is a positive


2= 1+ 119890119902119890119902= 119890120596(1

120575119890119902+

120575(119897+ lowast

minus 1198961119890120596) + 119897


1

120575sdot 119890119902+ 1198961119890120596minus sdot 120596 + 119906

119902119903

+ Δ1minus

120575((119897+ lowast


+ 119897( sdot 1198961minus 1)) = 119890

120596(1

120575119890119902

+

120575(119897+ lowast


minus1


1198892

1(x)

41205761

119890119902+ Δ1

minus

120575((119897+ lowast


+ 119897( sdot 1198961minus 1))

(24)

Additionally

119890119902(minus

1198892

1(x)

41205761

119890119902+ Δ1) = minus

1198892

1(x)

41205761

1198902

119902+ Δ1119890119902

le minus1198892

1(x)

41205761

1198902

119902+ 1198891(x) 1198911max

10038161003816100381610038161003816119890119902

10038161003816100381610038161003816

= minus(

1198891(x) 10038161003816100381610038161003816119890119902

10038161003816100381610038161003816

2radic1205761


2

+ 12057611198912

1max

(25)


119894119889 we can get

119890119889= 119894119889minus 119894lowast

119889= minus119894119889+ 120596 sdot 119894

119902+ 119906119889+ Δ2minus 119894lowast

119889 (26)


1198813= 1198812+1

21198902

119889 (27)


3= 2+ 119890119889119890119889

= 2+ 119890119889(minus119894119889+ 120596 sdot 119894

119902+ 119906119889+ Δ2minus 119894lowast

119889)

(28)


culated

119906119889= 119906119889119904+ 119906119889119903 (29)

where

119906119889119904


119902+ 119894lowast

119889 (30)

119906119889119903

= minus1198892

2(x)

41205762

119890119889 (31)


2is a positive num-



119890119889= minus1198963sdot 119890119889+ 119906119889119903


2(x)

41205762

119890119889+ Δ2 (32)

3= 2+ 119890119889119890119889

= 2+ 119890119889(minus1198963sdot 119890119889minus1198892

2(x)

41205762

119890119889+ Δ2)

(33)


119890119889(minus

1198892

2(x)

41205762

119890119889+ Δ2) = minus

1198892

2(x)

41205762

1198902

119889+ Δ2119890119889

le minus1198892

2(x)

41205762

1198902

119889+ 1198892(x) 1198912max

10038161003816100381610038161198901198891003816100381610038161003816

= minus(1198892(x) 1003816100381610038161003816119890119889

1003816100381610038161003816

2radic1205762


2

+ 12057621198912

2max

(34)


119881 = 1198813+

1

21205791

2

119897+

1

21205792

2

+1

2120575 sdot 1205793

2

(35)

where 1205791 1205792 and 120579



120575 120574 = 120574 and 120591

119897= 120591119897


= 3+

119897

1205791

120591119897+

1205792

120574 +

120575 sdot 1205793

120575 =

2+ 119890119889(minus1198963sdot 119890119889

+ 119906119889119903

+ Δ2) = 1+ 119890119902(minus1198962119890119902minus

1

120575sdot 119890119902+ 1198961119890120596minus

sdot 120596 + 119906119902119903+ Δ1

minus

120575((119897+ lowast


+ 119897( sdot 1198961minus 1)) + 119890

119889(minus1198963sdot 119890119889+ 119906119889119903

+ Δ2)

= 119890120596(1

120575119890119902+

120575(119897+ lowast

minus 1198961119890120596) + 119897minus 1198961119890120596)

+ 119890119902(minus1198962119890119902minus

1

120575sdot 119890119902+ 1198961119890120596minus sdot 120596 + 119906

119902119903+ Δ1

minus

120575((119897+ lowast


+ 119897( sdot 1198961minus 1)) + 119890

119889(minus1198963sdot 119890119889+ 119906119889119903

+ Δ2) =

1

120575

sdot 119890119902119890120596+

120575(119897+ lowast

minus 1198961119890120596) sdot 119890120596+ 119897119890120596minus 11989611198902

120596

minus 11989631198902

119889minus 11989621198902

119902minus 120596119890

119902minus

1

1205751198902

119902+ 119890119902(119906119902119903+ Δ1)

+ 119890119889(119906119889119903

+ Δ2) minus

120575((119897+ lowast

minus 1198961119890120596)

minus 1198961(119894119902minus 120596)) sdot 119890

119902+ 119897( sdot 1198961minus 1) 119890

119902+ 1198961119890120596119890119902

+119897

1205791

120591119897+

1205792

120574 +

1205751205793

120575 =

1

120575119890119902119890120596+ 1198961119890120596119890119902

+

120575[(119897+ lowast


minus 1198961119890120596) 119890119902

+ 1198961(119894119902minus 120596) 119890

119902+

120575

1205793

] minus 11989611198902

120596minus 11989621198902

119889minus 11989631198902

119902

+ 119897[119890120596minus 119890119902+ 1198961119890119902+

120591119897

1205791

] + [minus120596119890119902+

120574

1205792

]

minus1

1205751198902

119902+ 119890119902(minus

1198892

1(x)

41205761

119890119902+ Δ1) + 119890119889(minus

1198892

2(x)

41205762

119890119889

+ Δ2)

(36)



120575 = minus120579

3[(119897+ lowast

minus 1198961119890120596) sdot (119890120596minus 119890119902)

+ 1198961(119894119902minus 120596) 119890

119902]

(37)

120574 = 1205792120596119890119902 (38)

120591119897= minus1205791[119890120596minus 119890119902+ 1198961119890119902] (39)


=1

120575119890119902119890120596+ 1198961119890119902119890120596minus 11989611198902

120596minus 11989631198902

119889minus 11989621198902

119902minus

1

1205751198902

119902

+ 119890119902(minus

1198892

1(x)

41205761

119890119902+ Δ1)

+ 119890119889(minus

1198892

2(x)

41205762

119890119889+ Δ2)

(40)



1 1198962 1198963 1205761 and 120576

2and adaptive

gains 1205791 1205792 and 120579






=1

120575119890119902119890120596+ 1198961119890119902119890120596minus 11989611198902

120596minus 11989631198902

119889minus 11989621198902

119902minus

1

1205751198902

119902

+ 119890119902(minus

1198892

1(x)

41205761

119890119902+ Δ1) + 119890119889(minus

1198892

2(x)

41205762

119890119889+ Δ2)

= minus1

2120575(119890120596minus 119890119902)2

+1

21205751198902

120596+

1

21205751198902

119902minus1198961

2(119890120596minus 119890119902)2

+1198961

21198902

120596+1198961

21198902

119902minus 11989611198902

120596minus 11989631198902

119889minus 11989621198902

119902minus

1

1205751198902

119902

+ 119890119902(minus

1198892

1(x)

41205761

119890119902+ Δ1) + 119890119889(minus

1198892

2(x)

41205762

119890119889+ Δ2)

= minus1

2120575(119890120596minus 119890119902)2

minus1198961

2(119890120596minus 119890119902)2

minus 11989631198902

119889

+ (1

2120575+1198961

2minus 1198961) 1198902

120596+ (

1

2120575+1198961

2minus 1198962minus

1

120575) 1198902

119902

+ 119890119902(minus

1198892

1(x)

41205761

119890119902+ Δ1)

+ 119890119889(minus

1198892

2(x)

41205762

119890119889+ Δ2)

(41)


(1

2120575+1198961

2minus 1198961) lt 0

(1

2120575+1198961

2minus 1198962minus

1

120575) lt 0

(42)


1198961gt

1

120575

1198961minus 21198962lt

1

120575

(43)


= minus1

2120575(119890120596minus 119890119902)2

minus1198961

2(119890120596minus 119890119902)2

minus 11989631198902

119889

+ (1

2120575+1198961

2minus 1198961) 1198902

120596+ (

1

2120575+1198961

2minus 1198962minus

1

120575) 1198902

119902

+ 119890119902(minus

1198892

1(x)

41205761

119890119902+ Δ1)

+ 119890119889(minus

1198892

2(x)

41205762

119890119889+ Δ2)

le minus1

2120575(119890120596minus 119890119902)2

minus1198961

2(119890120596minus 119890119902)2

minus 11989631198902

119889minus 11989641198902

120596

minus 11989651198902

119902minus (

1198891(x) 10038161003816100381610038161003816119890119902

10038161003816100381610038161003816

2radic1205761


2

+ 12057611198912

1max

minus (1198892(x) 1003816100381610038161003816119890119889

1003816100381610038161003816

2radic1205762


2

+ 12057621198912

2max

le minus1

2120575(119890120596minus 119890119902)2

minus1198961

2(119890120596minus 119890119902)2

minus 11989631198902

119889minus 11989641198902

120596

minus 11989651198902

119902minus (

1198891(x) 10038161003816100381610038161003816119890119902

10038161003816100381610038161003816

2radic1205761


2

minus (1198892(x) 1003816100381610038161003816119890119889

1003816100381610038161003816

2radic1205762


2

+ 1205760

(44)

where 1198964= minus(12120575 + 119896

12 minus 119896

1) ge 0 119896

5= minus(12120575 + 119896

12 minus

1198962minus 1120575) ge 0 and 120576

0= 12057611198912

1max + 12057621198912

2maxLet

119882(119894 (119905)) = minus1

2120575(119890120596minus 119890119902)2

minus1198961

2(119890120596minus 119890119902)2

minus 11989631198902

119889

minus 11989641198902

120596minus 11989651198902

119902

minus (

1198891(x) 10038161003816100381610038161003816119890119902

10038161003816100381610038161003816

2radic1205761


2

minus (1198892(x) 1003816100381610038161003816119890119889

1003816100381610038161003816

2radic1205762


2

+ 1205760

(45)

where 119894(119905) = (119890120596 119890119902 119890119889)


int

119905

1199050

119882(119894 (119905)) 119889119905 = minusint

119905

1199050

(119894 (119905)) 119889119905 997904rArr

int

119905

1199050

119882(119894 (119905)) 119889119905 = 119881 (1199050) minus 119881 (119905)

(46)


ing hence

lim119905rarrinfin

int

119905

1199050

119882(119894 (119905)) 119889119905 lt infin (47)


lim119905rarrinfin

119882(119894 (119905)) = 0 (48)


1198962 1198963 1205761 and 120576



1 1198962 1198963 1205761 and 120576


1 1205792


minus25

minus20

minus15

minus10

120596minus5

0

5

10

15

20

10 20 30 40 50 60 70 80 90 1000t (s)


119889and 119906

119902







119902 119894119889) = (119909

1 1199092 1199093) and the con-


2= 30000 119896

3= 5


1= 62

1205792= 100 and 120579








119889= 1 Further-


3sin(5119905) and



10 20 30 40 50 60 70 80 90 1000t (s)

minus50

minus40

minus30

minus20

minus10

0

10

20

30

40

i q


inputs 119906119889and 119906

119902

10 20 30 40 50 60 70 80 90 1000t (s)

0

10

20

30

40

50

60

70

80

90

100i d


inputs 119906119889and 119906

119902

minus25

minus20

minus15

minus10

120596 minus5

0

5

10

15

10 20 30 40 50 60 70 80 90 1000t (s)


119889and 119906

119902


minus50

minus40

minus30

minus20

minus10

0

10

20

30

40

i q

10 20 30 40 50 60 70 80 90 1000t (s)


inputs 119906119889and 119906

119902

0

10

20

30

40

50

60

70

80

90

100

i d

10 20 30 40 50 60 70 80 90 1000t (s)


inputs 119906119889and 119906

119902

minus273

minus272

minus271

minus27

minus269

minus268

minus267

60 61 62 63minus27288

minus27282

30 40 50 60 70 80 90 10020t (s)

times106

times106

ud


95

952

954

956

958

96

962

964

966

60 6005 601957958959

96961962

times104

times104

uq

30 40 50 60 70 80 90 10020t (s)


minus02

minus015

minus01

minus005

0

005Es

timat

ion

erro

r of120575

30 40 50 60 70 80 90 10020t (s)


Estim

atio

n er

ror o

f120574

minus50

minus40

minus30

minus20

minus10

0

10

30 40 50 60 70 80 90 10020t (s)



minus35

minus3

minus25

minus2

minus15

minus1

minus05

0

05

30 40 50 60 70 80 90 10020t (s)

Estim

atio

n er

ror o

f120591l


119897

10 20 30 40 50 60 70 80 90 1000t (s)

minus20

minus15

minus10

minus5

0120596

5

10

15

20

25


119889and 119906

119902











119902 and 119894

119889added the

10 20 30 40 50 60 70 80 90 1000t (s)

minus30

minus20

minus10

0

10

20

30

40

i q



119902

0

10

20

30

40

50

10 20 30 40 50 60 70 80 90 1000t (s)

i d

minus10



119902

minus51

minus5095

minus509

minus5085

minus508

minus5075

469

469

246

94

469

646

98 47


times104

times104

ud

30 40 50 60 70 80 90 10020t (s)



minus3000

minus2500

minus2000

minus1500

minus1000

minus500

0

500

1000

469

469

246

94

469

646

98 47

minus2000minus1000

01000

uq

30 40 50 60 70 80 90 10020t (s)


Estim

atio

n er

ror o

f120575

30 40 50 60 70 80 90 10020t (s)

minus01

minus008

minus006

minus004

minus002

0

002

004

006

008

01


Estim

atio

n er

ror o

f120574

minus25

minus20

minus15

minus10

minus5

0

5

30 40 50 60 70 80 90 10020t (s)


minus2

0

2

4

6

8

10

12

14

16

18

20

30 40 50 60 70 80 90 10020t (s)

Estim

atio

n er

ror o

f120591l


119897

0

5

10

15

20

25

minus20

minus15

minus10

120596

minus5

10 20 30 40 50 60 70 80 90 1000t (s)


119889and 119906

119902





1(x 119905) = 40119909

3sin(5119905) and Δ

2(x 119905) =


119902 and 119894

119889 which



10 20 30 40 50 60 70 80 90 1000t (s)

minus20

minus10

0

10

20

30

40

i q



119902

minus10

0

10

20

30

40

50

i d

10 20 30 40 50 60 70 80 90 1000t (s)



119902

minus5085

minus508

minus5075

minus507

minus5065

469

469

5 4747

05

471


times104

times104

ud

30 40 50 60 70 80 90 10020t (s)


minus2500

minus2000

minus1500

minus1000

minus500

0

500

1000

469

469

5 4747

05

471

minus1000minus500

0500

1000

uq

30 40 50 60 70 80 90 10020t (s)


Estim

atio

n er

ror o

f120575

minus02

minus015

minus01

minus005

0

005

01

015

02

30 40 50 60 70 80 90 10020t (s)


Estim

atio

n er

ror o

f120574

30 40 50 60 70 80 90 10020t (s)

minus25

minus20

minus15

minus10

minus5

0

5



30 40 50 60 70 80 90 10020t (s)

minus01

minus008

minus006

minus004

minus002

0

002

004

Estim

atio

n er

ror o

f120591l


119897

119906119889and 119906








5 Conclusions





Competing Interests


Acknowledgments


References






































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










119890119889(minus

1198892

2(x)

41205762

119890119889+ Δ2) = minus

1198892

2(x)

41205762

1198902

119889+ Δ2119890119889

le minus1198892

2(x)

41205762

1198902

119889+ 1198892(x) 1198912max

10038161003816100381610038161198901198891003816100381610038161003816

= minus(1198892(x) 1003816100381610038161003816119890119889

1003816100381610038161003816

2radic1205762


2

+ 12057621198912

2max

(34)


119881 = 1198813+

1

21205791

2

119897+

1

21205792

2

+1

2120575 sdot 1205793

2

(35)

where 1205791 1205792 and 120579



120575 120574 = 120574 and 120591

119897= 120591119897


= 3+

119897

1205791

120591119897+

1205792

120574 +

120575 sdot 1205793

120575 =

2+ 119890119889(minus1198963sdot 119890119889

+ 119906119889119903

+ Δ2) = 1+ 119890119902(minus1198962119890119902minus

1

120575sdot 119890119902+ 1198961119890120596minus

sdot 120596 + 119906119902119903+ Δ1

minus

120575((119897+ lowast


+ 119897( sdot 1198961minus 1)) + 119890

119889(minus1198963sdot 119890119889+ 119906119889119903

+ Δ2)

= 119890120596(1

120575119890119902+

120575(119897+ lowast

minus 1198961119890120596) + 119897minus 1198961119890120596)

+ 119890119902(minus1198962119890119902minus

1

120575sdot 119890119902+ 1198961119890120596minus sdot 120596 + 119906

119902119903+ Δ1

minus

120575((119897+ lowast


+ 119897( sdot 1198961minus 1)) + 119890

119889(minus1198963sdot 119890119889+ 119906119889119903

+ Δ2) =

1

120575

sdot 119890119902119890120596+

120575(119897+ lowast

minus 1198961119890120596) sdot 119890120596+ 119897119890120596minus 11989611198902

120596

minus 11989631198902

119889minus 11989621198902

119902minus 120596119890

119902minus

1

1205751198902

119902+ 119890119902(119906119902119903+ Δ1)

+ 119890119889(119906119889119903

+ Δ2) minus

120575((119897+ lowast

minus 1198961119890120596)

minus 1198961(119894119902minus 120596)) sdot 119890

119902+ 119897( sdot 1198961minus 1) 119890

119902+ 1198961119890120596119890119902

+119897

1205791

120591119897+

1205792

120574 +

1205751205793

120575 =

1

120575119890119902119890120596+ 1198961119890120596119890119902

+

120575[(119897+ lowast


minus 1198961119890120596) 119890119902

+ 1198961(119894119902minus 120596) 119890

119902+

120575

1205793

] minus 11989611198902

120596minus 11989621198902

119889minus 11989631198902

119902

+ 119897[119890120596minus 119890119902+ 1198961119890119902+

120591119897

1205791

] + [minus120596119890119902+

120574

1205792

]

minus1

1205751198902

119902+ 119890119902(minus

1198892

1(x)

41205761

119890119902+ Δ1) + 119890119889(minus

1198892

2(x)

41205762

119890119889

+ Δ2)

(36)



120575 = minus120579

3[(119897+ lowast

minus 1198961119890120596) sdot (119890120596minus 119890119902)

+ 1198961(119894119902minus 120596) 119890

119902]

(37)

120574 = 1205792120596119890119902 (38)

120591119897= minus1205791[119890120596minus 119890119902+ 1198961119890119902] (39)


=1

120575119890119902119890120596+ 1198961119890119902119890120596minus 11989611198902

120596minus 11989631198902

119889minus 11989621198902

119902minus

1

1205751198902

119902

+ 119890119902(minus

1198892

1(x)

41205761

119890119902+ Δ1)

+ 119890119889(minus

1198892

2(x)

41205762

119890119889+ Δ2)

(40)



1 1198962 1198963 1205761 and 120576

2and adaptive

gains 1205791 1205792 and 120579






=1

120575119890119902119890120596+ 1198961119890119902119890120596minus 11989611198902

120596minus 11989631198902

119889minus 11989621198902

119902minus

1

1205751198902

119902

+ 119890119902(minus

1198892

1(x)

41205761

119890119902+ Δ1) + 119890119889(minus

1198892

2(x)

41205762

119890119889+ Δ2)

= minus1

2120575(119890120596minus 119890119902)2

+1

21205751198902

120596+

1

21205751198902

119902minus1198961

2(119890120596minus 119890119902)2

+1198961

21198902

120596+1198961

21198902

119902minus 11989611198902

120596minus 11989631198902

119889minus 11989621198902

119902minus

1

1205751198902

119902

+ 119890119902(minus

1198892

1(x)

41205761

119890119902+ Δ1) + 119890119889(minus

1198892

2(x)

41205762

119890119889+ Δ2)

= minus1

2120575(119890120596minus 119890119902)2

minus1198961

2(119890120596minus 119890119902)2

minus 11989631198902

119889

+ (1

2120575+1198961

2minus 1198961) 1198902

120596+ (

1

2120575+1198961

2minus 1198962minus

1

120575) 1198902

119902

+ 119890119902(minus

1198892

1(x)

41205761

119890119902+ Δ1)

+ 119890119889(minus

1198892

2(x)

41205762

119890119889+ Δ2)

(41)


(1

2120575+1198961

2minus 1198961) lt 0

(1

2120575+1198961

2minus 1198962minus

1

120575) lt 0

(42)


1198961gt

1

120575

1198961minus 21198962lt

1

120575

(43)


= minus1

2120575(119890120596minus 119890119902)2

minus1198961

2(119890120596minus 119890119902)2

minus 11989631198902

119889

+ (1

2120575+1198961

2minus 1198961) 1198902

120596+ (

1

2120575+1198961

2minus 1198962minus

1

120575) 1198902

119902

+ 119890119902(minus

1198892

1(x)

41205761

119890119902+ Δ1)

+ 119890119889(minus

1198892

2(x)

41205762

119890119889+ Δ2)

le minus1

2120575(119890120596minus 119890119902)2

minus1198961

2(119890120596minus 119890119902)2

minus 11989631198902

119889minus 11989641198902

120596

minus 11989651198902

119902minus (

1198891(x) 10038161003816100381610038161003816119890119902

10038161003816100381610038161003816

2radic1205761


2

+ 12057611198912

1max

minus (1198892(x) 1003816100381610038161003816119890119889

1003816100381610038161003816

2radic1205762


2

+ 12057621198912

2max

le minus1

2120575(119890120596minus 119890119902)2

minus1198961

2(119890120596minus 119890119902)2

minus 11989631198902

119889minus 11989641198902

120596

minus 11989651198902

119902minus (

1198891(x) 10038161003816100381610038161003816119890119902

10038161003816100381610038161003816

2radic1205761


2

minus (1198892(x) 1003816100381610038161003816119890119889

1003816100381610038161003816

2radic1205762


2

+ 1205760

(44)

where 1198964= minus(12120575 + 119896

12 minus 119896

1) ge 0 119896

5= minus(12120575 + 119896

12 minus

1198962minus 1120575) ge 0 and 120576

0= 12057611198912

1max + 12057621198912

2maxLet

119882(119894 (119905)) = minus1

2120575(119890120596minus 119890119902)2

minus1198961

2(119890120596minus 119890119902)2

minus 11989631198902

119889

minus 11989641198902

120596minus 11989651198902

119902

minus (

1198891(x) 10038161003816100381610038161003816119890119902

10038161003816100381610038161003816

2radic1205761


2

minus (1198892(x) 1003816100381610038161003816119890119889

1003816100381610038161003816

2radic1205762


2

+ 1205760

(45)

where 119894(119905) = (119890120596 119890119902 119890119889)


int

119905

1199050

119882(119894 (119905)) 119889119905 = minusint

119905

1199050

(119894 (119905)) 119889119905 997904rArr

int

119905

1199050

119882(119894 (119905)) 119889119905 = 119881 (1199050) minus 119881 (119905)

(46)


ing hence

lim119905rarrinfin

int

119905

1199050

119882(119894 (119905)) 119889119905 lt infin (47)


lim119905rarrinfin

119882(119894 (119905)) = 0 (48)


1198962 1198963 1205761 and 120576



1 1198962 1198963 1205761 and 120576


1 1205792


minus25

minus20

minus15

minus10

120596minus5

0

5

10

15

20

10 20 30 40 50 60 70 80 90 1000t (s)


119889and 119906

119902







119902 119894119889) = (119909

1 1199092 1199093) and the con-


2= 30000 119896

3= 5


1= 62

1205792= 100 and 120579








119889= 1 Further-


3sin(5119905) and



10 20 30 40 50 60 70 80 90 1000t (s)

minus50

minus40

minus30

minus20

minus10

0

10

20

30

40

i q


inputs 119906119889and 119906

119902

10 20 30 40 50 60 70 80 90 1000t (s)

0

10

20

30

40

50

60

70

80

90

100i d


inputs 119906119889and 119906

119902

minus25

minus20

minus15

minus10

120596 minus5

0

5

10

15

10 20 30 40 50 60 70 80 90 1000t (s)


119889and 119906

119902


minus50

minus40

minus30

minus20

minus10

0

10

20

30

40

i q

10 20 30 40 50 60 70 80 90 1000t (s)


inputs 119906119889and 119906

119902

0

10

20

30

40

50

60

70

80

90

100

i d

10 20 30 40 50 60 70 80 90 1000t (s)


inputs 119906119889and 119906

119902

minus273

minus272

minus271

minus27

minus269

minus268

minus267

60 61 62 63minus27288

minus27282

30 40 50 60 70 80 90 10020t (s)

times106

times106

ud


95

952

954

956

958

96

962

964

966

60 6005 601957958959

96961962

times104

times104

uq

30 40 50 60 70 80 90 10020t (s)


minus02

minus015

minus01

minus005

0

005Es

timat

ion

erro

r of120575

30 40 50 60 70 80 90 10020t (s)


Estim

atio

n er

ror o

f120574

minus50

minus40

minus30

minus20

minus10

0

10

30 40 50 60 70 80 90 10020t (s)



minus35

minus3

minus25

minus2

minus15

minus1

minus05

0

05

30 40 50 60 70 80 90 10020t (s)

Estim

atio

n er

ror o

f120591l


119897

10 20 30 40 50 60 70 80 90 1000t (s)

minus20

minus15

minus10

minus5

0120596

5

10

15

20

25


119889and 119906

119902











119902 and 119894

119889added the

10 20 30 40 50 60 70 80 90 1000t (s)

minus30

minus20

minus10

0

10

20

30

40

i q



119902

0

10

20

30

40

50

10 20 30 40 50 60 70 80 90 1000t (s)

i d

minus10



119902

minus51

minus5095

minus509

minus5085

minus508

minus5075

469

469

246

94

469

646

98 47


times104

times104

ud

30 40 50 60 70 80 90 10020t (s)



minus3000

minus2500

minus2000

minus1500

minus1000

minus500

0

500

1000

469

469

246

94

469

646

98 47

minus2000minus1000

01000

uq

30 40 50 60 70 80 90 10020t (s)


Estim

atio

n er

ror o

f120575

30 40 50 60 70 80 90 10020t (s)

minus01

minus008

minus006

minus004

minus002

0

002

004

006

008

01


Estim

atio

n er

ror o

f120574

minus25

minus20

minus15

minus10

minus5

0

5

30 40 50 60 70 80 90 10020t (s)


minus2

0

2

4

6

8

10

12

14

16

18

20

30 40 50 60 70 80 90 10020t (s)

Estim

atio

n er

ror o

f120591l


119897

0

5

10

15

20

25

minus20

minus15

minus10

120596

minus5

10 20 30 40 50 60 70 80 90 1000t (s)


119889and 119906

119902





1(x 119905) = 40119909

3sin(5119905) and Δ

2(x 119905) =


119902 and 119894

119889 which



10 20 30 40 50 60 70 80 90 1000t (s)

minus20

minus10

0

10

20

30

40

i q



119902

minus10

0

10

20

30

40

50

i d

10 20 30 40 50 60 70 80 90 1000t (s)



119902

minus5085

minus508

minus5075

minus507

minus5065

469

469

5 4747

05

471


times104

times104

ud

30 40 50 60 70 80 90 10020t (s)


minus2500

minus2000

minus1500

minus1000

minus500

0

500

1000

469

469

5 4747

05

471

minus1000minus500

0500

1000

uq

30 40 50 60 70 80 90 10020t (s)


Estim

atio

n er

ror o

f120575

minus02

minus015

minus01

minus005

0

005

01

015

02

30 40 50 60 70 80 90 10020t (s)


Estim

atio

n er

ror o

f120574

30 40 50 60 70 80 90 10020t (s)

minus25

minus20

minus15

minus10

minus5

0

5



30 40 50 60 70 80 90 10020t (s)

minus01

minus008

minus006

minus004

minus002

0

002

004

Estim

atio

n er

ror o

f120591l


119897

119906119889and 119906








5 Conclusions





Competing Interests


Acknowledgments


References






































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











=1

120575119890119902119890120596+ 1198961119890119902119890120596minus 11989611198902

120596minus 11989631198902

119889minus 11989621198902

119902minus

1

1205751198902

119902

+ 119890119902(minus

1198892

1(x)

41205761

119890119902+ Δ1) + 119890119889(minus

1198892

2(x)

41205762

119890119889+ Δ2)

= minus1

2120575(119890120596minus 119890119902)2

+1

21205751198902

120596+

1

21205751198902

119902minus1198961

2(119890120596minus 119890119902)2

+1198961

21198902

120596+1198961

21198902

119902minus 11989611198902

120596minus 11989631198902

119889minus 11989621198902

119902minus

1

1205751198902

119902

+ 119890119902(minus

1198892

1(x)

41205761

119890119902+ Δ1) + 119890119889(minus

1198892

2(x)

41205762

119890119889+ Δ2)

= minus1

2120575(119890120596minus 119890119902)2

minus1198961

2(119890120596minus 119890119902)2

minus 11989631198902

119889

+ (1

2120575+1198961

2minus 1198961) 1198902

120596+ (

1

2120575+1198961

2minus 1198962minus

1

120575) 1198902

119902

+ 119890119902(minus

1198892

1(x)

41205761

119890119902+ Δ1)

+ 119890119889(minus

1198892

2(x)

41205762

119890119889+ Δ2)

(41)


(1

2120575+1198961

2minus 1198961) lt 0

(1

2120575+1198961

2minus 1198962minus

1

120575) lt 0

(42)


1198961gt

1

120575

1198961minus 21198962lt

1

120575

(43)


= minus1

2120575(119890120596minus 119890119902)2

minus1198961

2(119890120596minus 119890119902)2

minus 11989631198902

119889

+ (1

2120575+1198961

2minus 1198961) 1198902

120596+ (

1

2120575+1198961

2minus 1198962minus

1

120575) 1198902

119902

+ 119890119902(minus

1198892

1(x)

41205761

119890119902+ Δ1)

+ 119890119889(minus

1198892

2(x)

41205762

119890119889+ Δ2)

le minus1

2120575(119890120596minus 119890119902)2

minus1198961

2(119890120596minus 119890119902)2

minus 11989631198902

119889minus 11989641198902

120596

minus 11989651198902

119902minus (

1198891(x) 10038161003816100381610038161003816119890119902

10038161003816100381610038161003816

2radic1205761


2

+ 12057611198912

1max

minus (1198892(x) 1003816100381610038161003816119890119889

1003816100381610038161003816

2radic1205762


2

+ 12057621198912

2max

le minus1

2120575(119890120596minus 119890119902)2

minus1198961

2(119890120596minus 119890119902)2

minus 11989631198902

119889minus 11989641198902

120596

minus 11989651198902

119902minus (

1198891(x) 10038161003816100381610038161003816119890119902

10038161003816100381610038161003816

2radic1205761


2

minus (1198892(x) 1003816100381610038161003816119890119889

1003816100381610038161003816

2radic1205762


2

+ 1205760

(44)

where 1198964= minus(12120575 + 119896

12 minus 119896

1) ge 0 119896

5= minus(12120575 + 119896

12 minus

1198962minus 1120575) ge 0 and 120576

0= 12057611198912

1max + 12057621198912

2maxLet

119882(119894 (119905)) = minus1

2120575(119890120596minus 119890119902)2

minus1198961

2(119890120596minus 119890119902)2

minus 11989631198902

119889

minus 11989641198902

120596minus 11989651198902

119902

minus (

1198891(x) 10038161003816100381610038161003816119890119902

10038161003816100381610038161003816

2radic1205761


2

minus (1198892(x) 1003816100381610038161003816119890119889

1003816100381610038161003816

2radic1205762


2

+ 1205760

(45)

where 119894(119905) = (119890120596 119890119902 119890119889)


int

119905

1199050

119882(119894 (119905)) 119889119905 = minusint

119905

1199050

(119894 (119905)) 119889119905 997904rArr

int

119905

1199050

119882(119894 (119905)) 119889119905 = 119881 (1199050) minus 119881 (119905)

(46)


ing hence

lim119905rarrinfin

int

119905

1199050

119882(119894 (119905)) 119889119905 lt infin (47)


lim119905rarrinfin

119882(119894 (119905)) = 0 (48)


1198962 1198963 1205761 and 120576



1 1198962 1198963 1205761 and 120576


1 1205792


minus25

minus20

minus15

minus10

120596minus5

0

5

10

15

20

10 20 30 40 50 60 70 80 90 1000t (s)


119889and 119906

119902







119902 119894119889) = (119909

1 1199092 1199093) and the con-


2= 30000 119896

3= 5


1= 62

1205792= 100 and 120579








119889= 1 Further-


3sin(5119905) and



10 20 30 40 50 60 70 80 90 1000t (s)

minus50

minus40

minus30

minus20

minus10

0

10

20

30

40

i q


inputs 119906119889and 119906

119902

10 20 30 40 50 60 70 80 90 1000t (s)

0

10

20

30

40

50

60

70

80

90

100i d


inputs 119906119889and 119906

119902

minus25

minus20

minus15

minus10

120596 minus5

0

5

10

15

10 20 30 40 50 60 70 80 90 1000t (s)


119889and 119906

119902


minus50

minus40

minus30

minus20

minus10

0

10

20

30

40

i q

10 20 30 40 50 60 70 80 90 1000t (s)


inputs 119906119889and 119906

119902

0

10

20

30

40

50

60

70

80

90

100

i d

10 20 30 40 50 60 70 80 90 1000t (s)


inputs 119906119889and 119906

119902

minus273

minus272

minus271

minus27

minus269

minus268

minus267

60 61 62 63minus27288

minus27282

30 40 50 60 70 80 90 10020t (s)

times106

times106

ud


95

952

954

956

958

96

962

964

966

60 6005 601957958959

96961962

times104

times104

uq

30 40 50 60 70 80 90 10020t (s)


minus02

minus015

minus01

minus005

0

005Es

timat

ion

erro

r of120575

30 40 50 60 70 80 90 10020t (s)


Estim

atio

n er

ror o

f120574

minus50

minus40

minus30

minus20

minus10

0

10

30 40 50 60 70 80 90 10020t (s)



minus35

minus3

minus25

minus2

minus15

minus1

minus05

0

05

30 40 50 60 70 80 90 10020t (s)

Estim

atio

n er

ror o

f120591l


119897

10 20 30 40 50 60 70 80 90 1000t (s)

minus20

minus15

minus10

minus5

0120596

5

10

15

20

25


119889and 119906

119902











119902 and 119894

119889added the

10 20 30 40 50 60 70 80 90 1000t (s)

minus30

minus20

minus10

0

10

20

30

40

i q



119902

0

10

20

30

40

50

10 20 30 40 50 60 70 80 90 1000t (s)

i d

minus10



119902

minus51

minus5095

minus509

minus5085

minus508

minus5075

469

469

246

94

469

646

98 47


times104

times104

ud

30 40 50 60 70 80 90 10020t (s)



minus3000

minus2500

minus2000

minus1500

minus1000

minus500

0

500

1000

469

469

246

94

469

646

98 47

minus2000minus1000

01000

uq

30 40 50 60 70 80 90 10020t (s)


Estim

atio

n er

ror o

f120575

30 40 50 60 70 80 90 10020t (s)

minus01

minus008

minus006

minus004

minus002

0

002

004

006

008

01


Estim

atio

n er

ror o

f120574

minus25

minus20

minus15

minus10

minus5

0

5

30 40 50 60 70 80 90 10020t (s)


minus2

0

2

4

6

8

10

12

14

16

18

20

30 40 50 60 70 80 90 10020t (s)

Estim

atio

n er

ror o

f120591l


119897

0

5

10

15

20

25

minus20

minus15

minus10

120596

minus5

10 20 30 40 50 60 70 80 90 1000t (s)


119889and 119906

119902





1(x 119905) = 40119909

3sin(5119905) and Δ

2(x 119905) =


119902 and 119894

119889 which



10 20 30 40 50 60 70 80 90 1000t (s)

minus20

minus10

0

10

20

30

40

i q



119902

minus10

0

10

20

30

40

50

i d

10 20 30 40 50 60 70 80 90 1000t (s)



119902

minus5085

minus508

minus5075

minus507

minus5065

469

469

5 4747

05

471


times104

times104

ud

30 40 50 60 70 80 90 10020t (s)


minus2500

minus2000

minus1500

minus1000

minus500

0

500

1000

469

469

5 4747

05

471

minus1000minus500

0500

1000

uq

30 40 50 60 70 80 90 10020t (s)


Estim

atio

n er

ror o

f120575

minus02

minus015

minus01

minus005

0

005

01

015

02

30 40 50 60 70 80 90 10020t (s)


Estim

atio

n er

ror o

f120574

30 40 50 60 70 80 90 10020t (s)

minus25

minus20

minus15

minus10

minus5

0

5



30 40 50 60 70 80 90 10020t (s)

minus01

minus008

minus006

minus004

minus002

0

002

004

Estim

atio

n er

ror o

f120591l


119897

119906119889and 119906








5 Conclusions





Competing Interests


Acknowledgments


References






































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










minus25

minus20

minus15

minus10

120596minus5

0

5

10

15

20

10 20 30 40 50 60 70 80 90 1000t (s)


119889and 119906

119902







119902 119894119889) = (119909

1 1199092 1199093) and the con-


2= 30000 119896

3= 5


1= 62

1205792= 100 and 120579








119889= 1 Further-


3sin(5119905) and



10 20 30 40 50 60 70 80 90 1000t (s)

minus50

minus40

minus30

minus20

minus10

0

10

20

30

40

i q


inputs 119906119889and 119906

119902

10 20 30 40 50 60 70 80 90 1000t (s)

0

10

20

30

40

50

60

70

80

90

100i d


inputs 119906119889and 119906

119902

minus25

minus20

minus15

minus10

120596 minus5

0

5

10

15

10 20 30 40 50 60 70 80 90 1000t (s)


119889and 119906

119902


minus50

minus40

minus30

minus20

minus10

0

10

20

30

40

i q

10 20 30 40 50 60 70 80 90 1000t (s)


inputs 119906119889and 119906

119902

0

10

20

30

40

50

60

70

80

90

100

i d

10 20 30 40 50 60 70 80 90 1000t (s)


inputs 119906119889and 119906

119902

minus273

minus272

minus271

minus27

minus269

minus268

minus267

60 61 62 63minus27288

minus27282

30 40 50 60 70 80 90 10020t (s)

times106

times106

ud


95

952

954

956

958

96

962

964

966

60 6005 601957958959

96961962

times104

times104

uq

30 40 50 60 70 80 90 10020t (s)


minus02

minus015

minus01

minus005

0

005Es

timat

ion

erro

r of120575

30 40 50 60 70 80 90 10020t (s)


Estim

atio

n er

ror o

f120574

minus50

minus40

minus30

minus20

minus10

0

10

30 40 50 60 70 80 90 10020t (s)



minus35

minus3

minus25

minus2

minus15

minus1

minus05

0

05

30 40 50 60 70 80 90 10020t (s)

Estim

atio

n er

ror o

f120591l


119897

10 20 30 40 50 60 70 80 90 1000t (s)

minus20

minus15

minus10

minus5

0120596

5

10

15

20

25


119889and 119906

119902











119902 and 119894

119889added the

10 20 30 40 50 60 70 80 90 1000t (s)

minus30

minus20

minus10

0

10

20

30

40

i q



119902

0

10

20

30

40

50

10 20 30 40 50 60 70 80 90 1000t (s)

i d

minus10



119902

minus51

minus5095

minus509

minus5085

minus508

minus5075

469

469

246

94

469

646

98 47


times104

times104

ud

30 40 50 60 70 80 90 10020t (s)



minus3000

minus2500

minus2000

minus1500

minus1000

minus500

0

500

1000

469

469

246

94

469

646

98 47

minus2000minus1000

01000

uq

30 40 50 60 70 80 90 10020t (s)


Estim

atio

n er

ror o

f120575

30 40 50 60 70 80 90 10020t (s)

minus01

minus008

minus006

minus004

minus002

0

002

004

006

008

01


Estim

atio

n er

ror o

f120574

minus25

minus20

minus15

minus10

minus5

0

5

30 40 50 60 70 80 90 10020t (s)


minus2

0

2

4

6

8

10

12

14

16

18

20

30 40 50 60 70 80 90 10020t (s)

Estim

atio

n er

ror o

f120591l


119897

0

5

10

15

20

25

minus20

minus15

minus10

120596

minus5

10 20 30 40 50 60 70 80 90 1000t (s)


119889and 119906

119902





1(x 119905) = 40119909

3sin(5119905) and Δ

2(x 119905) =


119902 and 119894

119889 which



10 20 30 40 50 60 70 80 90 1000t (s)

minus20

minus10

0

10

20

30

40

i q



119902

minus10

0

10

20

30

40

50

i d

10 20 30 40 50 60 70 80 90 1000t (s)



119902

minus5085

minus508

minus5075

minus507

minus5065

469

469

5 4747

05

471


times104

times104

ud

30 40 50 60 70 80 90 10020t (s)


minus2500

minus2000

minus1500

minus1000

minus500

0

500

1000

469

469

5 4747

05

471

minus1000minus500

0500

1000

uq

30 40 50 60 70 80 90 10020t (s)


Estim

atio

n er

ror o

f120575

minus02

minus015

minus01

minus005

0

005

01

015

02

30 40 50 60 70 80 90 10020t (s)


Estim

atio

n er

ror o

f120574

30 40 50 60 70 80 90 10020t (s)

minus25

minus20

minus15

minus10

minus5

0

5



30 40 50 60 70 80 90 10020t (s)

minus01

minus008

minus006

minus004

minus002

0

002

004

Estim

atio

n er

ror o

f120591l


119897

119906119889and 119906








5 Conclusions





Competing Interests


Acknowledgments


References






































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










minus50

minus40

minus30

minus20

minus10

0

10

20

30

40

i q

10 20 30 40 50 60 70 80 90 1000t (s)


inputs 119906119889and 119906

119902

0

10

20

30

40

50

60

70

80

90

100

i d

10 20 30 40 50 60 70 80 90 1000t (s)


inputs 119906119889and 119906

119902

minus273

minus272

minus271

minus27

minus269

minus268

minus267

60 61 62 63minus27288

minus27282

30 40 50 60 70 80 90 10020t (s)

times106

times106

ud


95

952

954

956

958

96

962

964

966

60 6005 601957958959

96961962

times104

times104

uq

30 40 50 60 70 80 90 10020t (s)


minus02

minus015

minus01

minus005

0

005Es

timat

ion

erro

r of120575

30 40 50 60 70 80 90 10020t (s)


Estim

atio

n er

ror o

f120574

minus50

minus40

minus30

minus20

minus10

0

10

30 40 50 60 70 80 90 10020t (s)



minus35

minus3

minus25

minus2

minus15

minus1

minus05

0

05

30 40 50 60 70 80 90 10020t (s)

Estim

atio

n er

ror o

f120591l


119897

10 20 30 40 50 60 70 80 90 1000t (s)

minus20

minus15

minus10

minus5

0120596

5

10

15

20

25


119889and 119906

119902











119902 and 119894

119889added the

10 20 30 40 50 60 70 80 90 1000t (s)

minus30

minus20

minus10

0

10

20

30

40

i q



119902

0

10

20

30

40

50

10 20 30 40 50 60 70 80 90 1000t (s)

i d

minus10



119902

minus51

minus5095

minus509

minus5085

minus508

minus5075

469

469

246

94

469

646

98 47


times104

times104

ud

30 40 50 60 70 80 90 10020t (s)



minus3000

minus2500

minus2000

minus1500

minus1000

minus500

0

500

1000

469

469

246

94

469

646

98 47

minus2000minus1000

01000

uq

30 40 50 60 70 80 90 10020t (s)


Estim

atio

n er

ror o

f120575

30 40 50 60 70 80 90 10020t (s)

minus01

minus008

minus006

minus004

minus002

0

002

004

006

008

01


Estim

atio

n er

ror o

f120574

minus25

minus20

minus15

minus10

minus5

0

5

30 40 50 60 70 80 90 10020t (s)


minus2

0

2

4

6

8

10

12

14

16

18

20

30 40 50 60 70 80 90 10020t (s)

Estim

atio

n er

ror o

f120591l


119897

0

5

10

15

20

25

minus20

minus15

minus10

120596

minus5

10 20 30 40 50 60 70 80 90 1000t (s)


119889and 119906

119902





1(x 119905) = 40119909

3sin(5119905) and Δ

2(x 119905) =


119902 and 119894

119889 which



10 20 30 40 50 60 70 80 90 1000t (s)

minus20

minus10

0

10

20

30

40

i q



119902

minus10

0

10

20

30

40

50

i d

10 20 30 40 50 60 70 80 90 1000t (s)



119902

minus5085

minus508

minus5075

minus507

minus5065

469

469

5 4747

05

471


times104

times104

ud

30 40 50 60 70 80 90 10020t (s)


minus2500

minus2000

minus1500

minus1000

minus500

0

500

1000

469

469

5 4747

05

471

minus1000minus500

0500

1000

uq

30 40 50 60 70 80 90 10020t (s)


Estim

atio

n er

ror o

f120575

minus02

minus015

minus01

minus005

0

005

01

015

02

30 40 50 60 70 80 90 10020t (s)


Estim

atio

n er

ror o

f120574

30 40 50 60 70 80 90 10020t (s)

minus25

minus20

minus15

minus10

minus5

0

5



30 40 50 60 70 80 90 10020t (s)

minus01

minus008

minus006

minus004

minus002

0

002

004

Estim

atio

n er

ror o

f120591l


119897

119906119889and 119906








5 Conclusions





Competing Interests


Acknowledgments


References






































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










minus35

minus3

minus25

minus2

minus15

minus1

minus05

0

05

30 40 50 60 70 80 90 10020t (s)

Estim

atio

n er

ror o

f120591l


119897

10 20 30 40 50 60 70 80 90 1000t (s)

minus20

minus15

minus10

minus5

0120596

5

10

15

20

25


119889and 119906

119902











119902 and 119894

119889added the

10 20 30 40 50 60 70 80 90 1000t (s)

minus30

minus20

minus10

0

10

20

30

40

i q



119902

0

10

20

30

40

50

10 20 30 40 50 60 70 80 90 1000t (s)

i d

minus10



119902

minus51

minus5095

minus509

minus5085

minus508

minus5075

469

469

246

94

469

646

98 47


times104

times104

ud

30 40 50 60 70 80 90 10020t (s)



minus3000

minus2500

minus2000

minus1500

minus1000

minus500

0

500

1000

469

469

246

94

469

646

98 47

minus2000minus1000

01000

uq

30 40 50 60 70 80 90 10020t (s)


Estim

atio

n er

ror o

f120575

30 40 50 60 70 80 90 10020t (s)

minus01

minus008

minus006

minus004

minus002

0

002

004

006

008

01


Estim

atio

n er

ror o

f120574

minus25

minus20

minus15

minus10

minus5

0

5

30 40 50 60 70 80 90 10020t (s)


minus2

0

2

4

6

8

10

12

14

16

18

20

30 40 50 60 70 80 90 10020t (s)

Estim

atio

n er

ror o

f120591l


119897

0

5

10

15

20

25

minus20

minus15

minus10

120596

minus5

10 20 30 40 50 60 70 80 90 1000t (s)


119889and 119906

119902





1(x 119905) = 40119909

3sin(5119905) and Δ

2(x 119905) =


119902 and 119894

119889 which



10 20 30 40 50 60 70 80 90 1000t (s)

minus20

minus10

0

10

20

30

40

i q



119902

minus10

0

10

20

30

40

50

i d

10 20 30 40 50 60 70 80 90 1000t (s)



119902

minus5085

minus508

minus5075

minus507

minus5065

469

469

5 4747

05

471


times104

times104

ud

30 40 50 60 70 80 90 10020t (s)


minus2500

minus2000

minus1500

minus1000

minus500

0

500

1000

469

469

5 4747

05

471

minus1000minus500

0500

1000

uq

30 40 50 60 70 80 90 10020t (s)


Estim

atio

n er

ror o

f120575

minus02

minus015

minus01

minus005

0

005

01

015

02

30 40 50 60 70 80 90 10020t (s)


Estim

atio

n er

ror o

f120574

30 40 50 60 70 80 90 10020t (s)

minus25

minus20

minus15

minus10

minus5

0

5



30 40 50 60 70 80 90 10020t (s)

minus01

minus008

minus006

minus004

minus002

0

002

004

Estim

atio

n er

ror o

f120591l


119897

119906119889and 119906








5 Conclusions





Competing Interests


Acknowledgments


References






































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










minus3000

minus2500

minus2000

minus1500

minus1000

minus500

0

500

1000

469

469

246

94

469

646

98 47

minus2000minus1000

01000

uq

30 40 50 60 70 80 90 10020t (s)


Estim

atio

n er

ror o

f120575

30 40 50 60 70 80 90 10020t (s)

minus01

minus008

minus006

minus004

minus002

0

002

004

006

008

01


Estim

atio

n er

ror o

f120574

minus25

minus20

minus15

minus10

minus5

0

5

30 40 50 60 70 80 90 10020t (s)


minus2

0

2

4

6

8

10

12

14

16

18

20

30 40 50 60 70 80 90 10020t (s)

Estim

atio

n er

ror o

f120591l


119897

0

5

10

15

20

25

minus20

minus15

minus10

120596

minus5

10 20 30 40 50 60 70 80 90 1000t (s)


119889and 119906

119902





1(x 119905) = 40119909

3sin(5119905) and Δ

2(x 119905) =


119902 and 119894

119889 which



10 20 30 40 50 60 70 80 90 1000t (s)

minus20

minus10

0

10

20

30

40

i q



119902

minus10

0

10

20

30

40

50

i d

10 20 30 40 50 60 70 80 90 1000t (s)



119902

minus5085

minus508

minus5075

minus507

minus5065

469

469

5 4747

05

471


times104

times104

ud

30 40 50 60 70 80 90 10020t (s)


minus2500

minus2000

minus1500

minus1000

minus500

0

500

1000

469

469

5 4747

05

471

minus1000minus500

0500

1000

uq

30 40 50 60 70 80 90 10020t (s)


Estim

atio

n er

ror o

f120575

minus02

minus015

minus01

minus005

0

005

01

015

02

30 40 50 60 70 80 90 10020t (s)


Estim

atio

n er

ror o

f120574

30 40 50 60 70 80 90 10020t (s)

minus25

minus20

minus15

minus10

minus5

0

5



30 40 50 60 70 80 90 10020t (s)

minus01

minus008

minus006

minus004

minus002

0

002

004

Estim

atio

n er

ror o

f120591l


119897

119906119889and 119906








5 Conclusions





Competing Interests


Acknowledgments


References






































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










10 20 30 40 50 60 70 80 90 1000t (s)

minus20

minus10

0

10

20

30

40

i q



119902

minus10

0

10

20

30

40

50

i d

10 20 30 40 50 60 70 80 90 1000t (s)



119902

minus5085

minus508

minus5075

minus507

minus5065

469

469

5 4747

05

471


times104

times104

ud

30 40 50 60 70 80 90 10020t (s)


minus2500

minus2000

minus1500

minus1000

minus500

0

500

1000

469

469

5 4747

05

471

minus1000minus500

0500

1000

uq

30 40 50 60 70 80 90 10020t (s)


Estim

atio

n er

ror o

f120575

minus02

minus015

minus01

minus005

0

005

01

015

02

30 40 50 60 70 80 90 10020t (s)


Estim

atio

n er

ror o

f120574

30 40 50 60 70 80 90 10020t (s)

minus25

minus20

minus15

minus10

minus5

0

5



30 40 50 60 70 80 90 10020t (s)

minus01

minus008

minus006

minus004

minus002

0

002

004

Estim

atio

n er

ror o

f120591l


119897

119906119889and 119906








5 Conclusions





Competing Interests


Acknowledgments


References






































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










30 40 50 60 70 80 90 10020t (s)

minus01

minus008

minus006

minus004

minus002

0

002

004

Estim

atio

n er

ror o

f120591l


119897

119906119889and 119906








5 Conclusions





Competing Interests


Acknowledgments


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









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