Smith predictor based-sliding mode controller for integrating processes with elevated deadtime

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ISATRANSACTIONS®

ISA Transactions 43~2004! 257–270

Smith predictor based–sliding mode controller for integratinprocesses with elevated deadtime

Oscar Camacho,a,* Francisco De la Cruzba Postgrado en Automatizacioń e Instrumentacioń, Grupo en Nuevas Estrategias de Control Aplicado, Universidad de los Ande

Merida 5101, Venezuelab Departamento de Ingenierıá Electronica, Vice Rectorado Barquisimeto Universidad Nac. Exp. Politećnica

‘‘Antonio Josede Sucre,’’ Barquisimeto 3001, Venezuela

~Received 23 October, 2002; accepted 25 August 2003!

Abstract

An approach to control integrating processes with elevated deadtime using a Smith predictor sliding mode cois presented. A PID sliding surface and an integrating first-order plus deadtime model have been used to synthcontroller. Since the performance of existing controllers with a Smith predictor decrease in the presence of merrors, this paper presents a simple approach to combining the Smith predictor with the sliding mode concept,a proven, simple, and robust procedure. The proposed scheme has a set of tuning equations as a functicharacteristic parameters of the model. For implementation of our proposed approach, computer based icontrollers that execute PID algorithms can be used. The performance and robustness of the proposed contcompared with the Matausˇek-Micic scheme for linear systems using simulations. © 2004 ISA—The InstrumentaSystems, and Automation Society.

Keywords: Sliding mode control; Smith predictor; Integrating process; Deadtime

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1. Introduction

The presence of time delays in many industrprocesses is a well-recognized problem. Tachievable performance of typical feedback cotrol systems can decline if a process has a retively large time delay compared to the dominatime constant@1#. In the case of integral processedeadtime compensation is necessary in ordeenhance the performance of the control system

O’Dwyer @2,3# considered a wide variety omethods for the compensation of processes wtime delay, in both the continuous time and d

*Corresponding author. Tel:158-274-2402891; fax:158-274-2402890.E-mail address: [email protected]

0019-0578/2004/$ - see front matter © 2004 ISA—The Instru

crete time domains. The compensators discusare:

1. PID controllers and its variations.2. Lead, lag or lead-lag controllers.3. The Smith predictor@4# and its variations.4. Direct synthesis methods, which are typica

based on designing the controller to meet aquired output specification; pole placement cotrollers are an example.

5. Optimal controller design methods, whicmay be based on a minimum variance or linequadratic control strategy.

6. Predictive controllers.7. Other compensation strategies for proces

with time delays, including fuzzy implementationneural networks, and expert systems.

The wide spectrum of methods covered anddependence of the choice of compensator met

mentation, Systems, and Automation Society.

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258 Camacho, De La Cruz / ISA Transactions 43 (2004) 257–270

on the application mean that an overall conclusas to the best method to use is not appropriaViewing the compensation problem from a varieof perspectives, it appears that the Smith predicis the optimal ~or a component of the optimal!controller for dominant delay processes@2#. Inthose works, the compensation of integrating pcess with dead time is only achieved using Pcontrollers, Smith predictors, or its variationOther approaches have not been reported forkind of system.

The Smith predictor~SP!, or deadtime compensator~DTC!, as it is also known, has many weapoints, including possible instability and poor peformance under modeling errors, and poorsponse to disturbances@1#. In addition, the originalstructure of the SP cannot reject constant load dturbance for processes with integration@5#. Toovercome this obstacle many variations of SP habeen proposed.

Sliding mode control~SMC! is a robust andsimple procedure to develop controllers for lineand nonlinear processes@6#. The design of a slid-ing mode controller~SMCr! depends on the process model, and the number of tuning parameis proportional to the model order. Camacho aSmith @7# developed a simple and practicmethod for the design of a SMCr based on a siplified model of the actual process.

The SP performs well for eliminating deadtimand the SMCr is a proven robust controller. Itdesired to combine them into a single contrstructure that preserves the good qualities of btechniques and improves the bad qualities ofSP. A robust controller for integral processes wdeadtime, the Smith predictor sliding mode cotroller ~SPSMCr!, will be developed.

This paper is organized as follows: Sectionintroduces the basic concepts of the sliding mocontroller, the SP architecture and a review of tmodified SP for integrating processes that hapreviously been proposed. Section 3 covers thevelopment of the SPSMCr based on an integratfirst-order process model with deadtime. Sectioshows the SPSMCr implementation using a Palgorithm of a computer based industrial controler. Section 5 provides simulation results to illutrate the approach and to compare it with tperformance and robustness of a previous scheFinally, some conclusions are presentedSection 6.

.

-

.

2. Basic concepts

2.1. Sliding mode control (SMC)

Sliding mode control is a technique derivefrom variable structure~VSC! which was origi-nally studied by Utkin@8#. A controller designedusing the SMC method is particularly appealindue to its ability to deal with nonlinear systemand time-varying systems@9#. The robustness tothe uncertainties becomes an important aspecdesigning any control system.

The idea behind SMC is to define a surfaalong which the process can slide to its desirfinal value. Fig. 1 depicts the SMC objective. Thstructure of the controller is intentionally altereas its state crosses the surface in accordance wprescribed control law. Thus the first step in SMis to define the sliding surfaceS(t) which repre-sents a desired global behavior, like stability atracking performance.

The S(t) selected in this work, presented bSlotine and Li@9#, is an integral-differential equation acting on the tracking-error expression:

S~ t !5 f S e~ t !,E e~ t !dt,de~ t !

dt,l,nD , ~1!

wheree(t) is the tracking error, that is, the difference between the reference value or set pointr (t)and the output measurementx(t), namely, e(t)5r (t)2x(t). l is a tuning parameter, which helpto defineS(t). This term is selected by the designer, and determines the performance of the stem on the sliding surface.n is the system order.

The control objective is to ensure that the cotrolled variable be equal to its reference valueall times, meaning thate(t) and its derivatives

Fig. 1. Graphical interpretation of SMC.

259Camacho, De La Cruz / ISA Transactions 43 (2004) 257–270

Fig. 2. Chattering reduction using a saturation function~a: d50; b: d50.01; c:d50.1; d:d51.0!.

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must be zero. Once the reference value is reachEq. ~1! indicates thatS(t) has reached a constavalue, meaning thate(t) is zero at all times; it isdesired to set

dS~ t !

dt50. ~2!

Once the sliding surface has been selected, attion must be turned to design the control law thdrives the controlled variable to its reference valand satisfies Eq.~2!. The SMC control lawM (t)consists of two additive parts: a continuous pUC(t) and a discontinuous partUD(t). That is,

M ~ t !5UC~ t !1UD~ t !. ~3!

The continuous part is given by

UC~ t !5 f „x~ t !,r ~ t !…, ~4!

where f „x(t),r (t)… is a function of the controlledvariable and the reference value.

The discontinuous part is nonlinear and repsents the switching element of the control laThis part of the controller is discontinuous acrothe sliding surface. Mainly,UD(t) is designedbased on a relaylike function„i.e., UD(t)5a sgn(@S(t)#)…, because it allows for changes btween the structures with a hypothetical infinitefast speed. In practice, however, it is impossible

,

-

achieve the high switching control because of tpresence of finite time delays for control comptations or limitations of the physical actuatorthus causing chattering around of the sliding sface @8,9#. Chattering is a high-frequency oscillation around the desired equilibrium point. It is undesirable in practice, because it involves hicontrol activity and can excite high-frequency dnamics ignored in the modeling of the syste@9,10#. The aggressiveness to reach the sliding sface depends on the control gain~i.e.,a!, but if thecontroller is too aggressive it can collaborate wthe chattering. To reduce the chattering, oneproach is to replace the relaylike function bysaturation or sigma function, which can be writteas follows:

UD~ t !5KD

S~ t !

uS~ t !u1d, ~5!

whereKD is the tuning parameter responsible fthe reaching mode.d is a tuning parameter used treduce the chattering problem. Fig. 2 shows teffect of d variations in the shape of saturatiofunction. Fig. 3 shows the effect ofKD variationson the system trajectory on the phase plane fran initial state to a final state.


Fig. 3. Graphical interpretation forKD variations.K1,K2,K3.

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In summary, the control law usually results infast motion to bring the state onto the sliding suface, and a slower motion to proceed until a dsired state is reached.

2.2. Smith predictor: Basic concepts andprevious schemes for integrating processes

As stated before, SP is a popular schemedeadtime compensation. Fig. 4 shows the architture of the SP. The process transfer functionGp(s)5G(s)e2t0s which is assumed to consist oa rational stable transfer functionG(s) and adeadtimet0 . A model of the process without deadtime, Gm(s), is used to predict the effect of th

-

control action on the process output and tocrease the performance of the system. The diffence between the output of the process andmodel is fed back in order to correct modelinerrors and load disturbances.

If there is no process/model mismatch,Gm(s)5G(s) and t0m5t0 , then the modeling errorem(t)5y(t)2ym(t)50. Since the deadtime isseparated from the model, andem50, the feed-back only consists of the model without delaTherefore the deadtime is isolated and compsated@1#, and thus, for controller design purposeit can be ignored. However, the SP cannot be uwith its original structure to control processes wiintegration since a constant load disturbance

Fig. 4. Smith predictor scheme.


Fig. 5. The modified SP proposed by Watanabe and Ito.

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sults in a steady-state error@5#. To overcome thisobstacle many variations of SP have been pposed.

Watanabe and Ito@5# proposed a modification othe SP as shown in Fig. 5. The system can rejeload disturbance if the time delay of the processexactly known. Otherwise, there will be a smasteady-state error@12#. Simulations studies havshown that the setpoint and load disturbanceseither very oscillatory or highly damped when thprocess has a large dead time@13#.

Astrom et al. @13# proposed a new SP structuras shown in Fig. 6, where the disturbance respois decoupled from the setpoint response. The ctroller has four adjustable parameters but a stematic tuning method was not given. The authconsidered only the restricted integrating proceG(s)5e2t0s/s.

Zhang and Sun@12# improved the results of Astrom et al. retaining the separation nature of loaresponse from the setpoint response. The strucof the modified SP is shown in Fig. 7.M (s) wasrecommended as

e

M ~s!5sMo~s!

12sT21Mo~s!e2t0s,

Mo~s!5~uT12lT!s1T

~ls11!2Ts. ~6!

Eq. ~6! contains a positive feedback loop that ispotential instability source, resulting in limited robustness@15#. Zhang and Sun rely on a generguideline to tuneKr andl rather than a systematiapproach@16#. The proposed scheme does not pvide much better performance than Astro¨m’s @12#.When the process is described by a high-ordmodel, the controller is a derivative or a derivativwith lag and a more complexM (s) will result.

Matausek and Micic @17,18# proposed a modi-fied SP and gave a simple controller tuninmethod. Their scheme, given in Fig. 8, has a cotroller to remove the load disturbance although tsetpoint and load responses cannot be decoupIn a first paper@17#, F(s) is a proportional con-troller Ko , obtaining a simple structure with threadjustable parameters. However, for high valu

Fig. 6. The SP structure proposed by Astromet al.


Fig. 7. The structure of Zhan and Sun’s SP.

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of the deadtime, the scheme has poor disturbarejection @19# and becomes significantly oscillatory with time delay deviation@15#.

A second Matausˇek and Micic’s modified SP@18#, whereF(s) is a lead-lag compensator, provides considerably faster load disturbance rejtion but requires a trial-and-error procedure of tmain controller gain in a tradeoff between closeloop system performance and stability/robustneThis scheme will be used to compare the propocontroller.

The Tian and Gao’s control scheme@15# has thesame structure as Astrom’s SP and a parametuning as in Ref.@18#. The structure of this SPincludes four controllers, as shown in Fig. 9.

A local proportional feedbackKo is introducedto prestabilize the integrator process. The intduction of Go(s) eliminates the effect ofKo onsetpoint tracking. To compensate the phasecaused by integrator and time delayproportional-derivative controllerGc(s) is sug-gested, where

.

’

Go~s!511KoKp

se2t0s; Gc~s!5Kc~11Tds!.

~7!

This structure behaves slightly better than the lest Matausˇek and Micic’s scheme~setpoint ITAEindex reduction,0.5%! @15#, despite the addedstructural complexity, and their tuning strategy cbe applied to only pure integrator plus time delmodels@16#.

3. Smith predictor based sliding modecontroller for integrating processes

The Smith predictor based sliding mode controler ~SPSMCr! proposed in this paper uses the stadard SP architecture while the controller is a sliing mode controller~SMC!. The block diagram ofthe proposed scheme is shown in Fig. 10. As mtioned, the original structure of SP is ineffectivfor integrator processes because it cannot reje

Fig. 8. Modified SP proposed by Matausˇek and Micic.


Fig. 9. Tian and Gao’s control scheme.

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constant load disturbance@5# and proportional-derivative controllerGd(s) is used for load distur-bance rejection, where

Gd~s!5Ko~Tds11!. ~8!

This controller will be discussed later.To develop SPSMCr, an integrating first-ord

plus deadtime~IFOPDT! process model given inEq. ~9! is considered,

Gp~s!5G~s!e2t0s5K

s~ts11!e2t0s, ~9!

whereK is the process gain,t0 is the process deadtime, andt is the process time constant. If thereno process/model mismatch, the model transfunction without deadtime is

Gm~s!5X1~s!

M ~s!5

Km

s~tms11!, ~10!

whereKm is the model gain andtm is the modeltime constant.

Since the deadtime term has been isolated usa SP structure, we can ignore it in the SMC dsign. Then, transforming Eq.~10! into differentialequation form,

tm

d2X1~ t !

dt21

dX1~ t !

dt5KmM ~ t ! ~11!

and

d2X1~ t !

dt25

1

tmFKmM ~ t !2

dX1~ t !

dt G . ~12!

In this case, we use an IFOPDT process andn52. Then, from Eq.~1!, we selected asS(t) anintegral-differential equation acting on the tracing error expression represented by

S~ t !5de~ t !

dt1l1e~ t !1l0E

0

te~ t !dt. ~13!

Eq. ~13! represents a PID surface. The parametl1 andl0 can be chosen independently and, in ocase, they were selected to obtain an overdamresponse.

From Eq.~2!,

dS~ t !

dt5

d2e~ t !

dt21l1

de~ t !

dt1l0e~ t !50

~14!

Fig. 10. Smith predictor based sliding mode controller.

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but e(t)5R(t)2X1(t) and substituting into theabove equation gives

d2R~ t !

dt22

d2X1~ t !

dt21l1

dR~ t !

dt2l1

dX1~ t !

dt

1l0e~ t !50. ~15!

Camacho@11# has shown that the derivates of threference value can be discarded without anyfect on the control performance. Thus

d2X1~ t !

dt252l1

dX1~ t !

dt1l0e~ t !. ~16!

Substituting Eq.~12! into Eq. ~16!,

2l1

dX1~ t !

dt1l0e~ t !5

1

tmFKmM ~ t !2

dX1~ t !

dt G .~17!

Thus the continuous part of the controller is

UC~ t !51

KmF ~12tml1!

dX1~ t !

dt1tml0e~ t !G .

~18!

Then, the complete SPSMCr can be represente

M ~ t !51

KmF ~12tml1!

dX1~ t !

dt1tml0e~ t !G

1KD

S~ t !

uS~ t !u1d~19!

with

S~ t !5sgn~K !F2dX1~ t !

dt1l1e~ t !

1l0E0

te~ t !dtG . ~20!

The functionsgn(K) in Eq. ~20! is included in thesliding surface equation to guarantee the approate action of the controller for the given systeNote thatsgn(K) only depends on the static gaof the plant; therefore it never switches@6#.

Eqs. ~19! and ~20! define the controller equations to be used in the SPSMCr, which cansimplified by setting

l151

tm@5#@ time#21. ~21!

s

Furthermore, to assure that the sliding surfacehaves as a critical or overdamped system,l0should be

l0<l1

2

4@5#@ time#22. ~22!

Numerous simulations showed that the valuesl1 andl0 are a function of the controllability relationship,CR5t0 /tm , and the following valuesprovide satisfactory system performance andbustness with time delay deviation:

l15H 4

tm@5#@ time#21 if CR<4

1.5

tm@5#@ time#21 if CR>4,

~23!

l05l1

2

8@5#@ time#22. ~24!

The parametersd and KD have a relationshipwith system speed, overshot, and chatteriBased on previous approaches@6,11,14# where theNelder-Mead searching algorithm was used,tuning parameters of the controller discontinuopart are

KD50.75

uKmu S t0

tmD 20.76

@5#@ fraction CO#,

~25!

[email protected]~ uKmuKDl1!#

@5#@ fraction TO/time#. ~26!

3.1. Disturbance rejection

As mentioned, many modified SP for integratprocesses with different structures have been pposed in the literature to remove the steady-sterror produced by a constant load disturban

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Tian and Gao@15# and Matausˇek and Micic@18#added derivative action to their proposed DTCovercome this problem. A proportional-derivativcontroller Gd(s), given in Eq.~8!, is used in theproposed SPSMCr to enable the load disturbarejection. The parameters of this controller, usia50.4, Fpm564°51.117 rad, as recommendedin Ref. @18#, are

Ko5

p

22Fpm

Km~tm1t0!A~12a!1S p

22FpmD 2

a2

50.7239

Km~tm1t0!, ~27!

Td5a~tm1t0!50.4~tm1t0!. ~28!

4. SPSMCr implementation using a PIDalgorithm

Computer based industrial controllers that eecute PID algorithms~programmable logic controllers and remote terminal units! can be used forSPSMCr implementation. The more common Palgorithm is based on the following equation:

MV~ t !5KcF2tD

dCV~ t !

dt1e~ t !

11

t i E0

te~ t !dtG , ~29!

whereMV(t) is the manipulated variable,t i is theintegral time,tD is the derivative term, andCV(t)

is the controlled variable. Note that the derivatiterm is calculated using the measured controlvariable, not the error. Fig. 11 shows the blodiagram that implements an algorithm for a PID

As can be observed, Eqs.~20! and~29! are simi-lar. The termX1(t) in Eq. ~20! is the sameCV(t)in Eq. ~29!. Therefore, to represent the sliding sufaceS(t), the next step is to tune the PID basedthe SPSMCr tuning equations as follows:

Kc5l1 ,

t i5l1 /l0 , ~30!

tD51/l1 .

With these equations the PID algorithm canchanged, representingS(t).

Then, to achieve the implementation, the slidisurface valueS(t) is calculated from the PID output. Because the continuous and discontinuoparts of the controller are algebraic equations, thare easily programmable. The block diagram thimplements an algorithm for the SPSMCrshown in Fig. 12.

Fig. 11. Implementation of PID algorithm in an industricontroller.

Fig. 12. Implementation of SPSMCr using an industrial controller.


Fig. 13. System responses to a setpoint change for differentl1 values. : l151.33; • • • : l152.67; : l1

50.33.

Fig. 14. System responses to a setpoint change for differentl0 values. : l050.22; • • • : l050.44; : l0

50.056.

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5. Simulation results

To illustrate the SPSMCr performance, two eamples are given. In example 1, the systemsponse to different parameter values is presento show the relationship with the system perfomance. In example 2, the response of an integing system with a long deadtime is shown and tresults are compared to those obtained usingproposed controller in Ref.@18#.

Example 1Let us consider a process with IFOPDT trans

function G1(s),

G1~s!51

s~3s11!e26s, ~31!

with CR5t0 /t52. Using Eqs.~23!–~26!, the pa-rameters of SPSMCr are

l154

tm5

4

351.333, l05

l12

850.222,

KD50.75

K S t0

tmD 20.76

50.75

1 S 6

3D 20.76

50.443,

d52* @0.6810.12~KmKDl1!#

52* @0.6810.12~1!~0.443!~1.333!#51.5,

Ko50.7239

Km~tm1t0!5

0.7239

1~316!50.0809,

Td50.4~tm1t0!50.4~9!53.6.

Figs. 13 and 14 show the process output to apoint change whenl1 andl0 variations were in-troduced. In these figures, the dotted lines corspond to responses using the values obtained wEqs. ~23! and ~24!. The smaller values ofl1 in-crease speed and overshoot. The smaller valuel0 increase speed response and decrease setime. Numerous simulations showed that as spincreases, the system robustness decreases.

Fig. 15 shows the system responses to a setp

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-

fg

t

change when reductions ofd value were intro-duced. Fig. 15~a! corresponds to thed value ob-tained with Eq.~25!. Fig. 15~b! corresponds tod50. The smallerd increases speed and chatteing.

Fig. 16 shows the system responses to a setpchange when reductions ofKD value were intro-duced. In this figure, the solid lines correspondresponses toKD value using Eq.~26!. A largerKD

increase speed and overshot system andsmallerKD values increase settling time.

Example 2Let us consider a process with a transfer fun

tion G2(s), which has been investigated in Ref@15,17–19#,

Fig. 15. System responses using SPSMCr forG1(s): ~a!d51.5 and~b! d50.


Fig. 16. System responses to a setpoint change for differentKD values. : KD50.44; : KD50.25; • • • :KD53.54.

he

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ndint

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e-re-

st-ithme.re

rfor-%

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G2~s!

51

s~s11!~0.5s11!~0.2s11!~0.1s11!e220s.

~32!

The corresponding IFOPDT model is

G2m~s!51

s~1.28s11!e220.64s ~33!

with CR516.125.In this example, the SPSMCr is compared to t

controller proposed in Ref.@18# and denoted byMM99. For MM99, the equivalent time constanTe is set to 2.4 as in Ref.@18# to improve robust-ness. The rest of the parameters of MM99 aSPSMCr are given in Table 1. A unit step setpo

Table 1Controller tuning parameters for processG2(s).

MM99 SPSMCr

Tr Kr Ko Td l1 l0 KD d Ko Td

2.4 0.417 0.027 10.56 1.172 0.172 0.09 0.139 0.033 8

s introduced at timet50 and a load disturbanced520.1 is introduced at timet570. When themodel is exact, Fig. 17~a! shows the response oboth control schemes. Fig. 17~b! shows the effectof a 20% deadtime modeling error on system pformance. The MM99 scheme becomes unstab

6. Conclusions

In this paper, a robust control scheme for intgrating systems with deadtime using a Smith pdictor sliding mode controller~SPSMCr! has beenpresented. Sliding mode control improves robuness and stability to the scheme while the Smpredictor isolates and compensates the deadtiThis new method allows for a robust controllewhile compensating for the deadtime within thprocess. The robustness of the steady-state pemance is improved by the sliding mode with 20modeling errors. This controller combines the simplicity of the SP architecture implementation anthe robustness of a sliding mode controller. Bsides, the disturbance rejection for integrator pcesses of the original structure of SP has beenproved, adding a proportional-derivative controll


Fig. 17. Responses of the proposed SPSMCr and MM99 forG2(s). ~a! Nominal case,~b! 10% error int0 .

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originally proposed by Matausˇek and Micic@18#.The new proposed controller has a simple tunprocedure if an IFOPDT model is available.

References

@1# Tan, K. K., Lee, T. H., and Leu, F. M., Predictive Pversus Smith control for dead-time compensation. ITrans.40, 17–29~2001!.

@2# O’Dwyer, A., The estimation and compensation

processes with time delays. Ph.D. thesis, Dublin CUniversity, Ireland, 1996.

@3# O’Dwyer, A., A survey of techniques for the estimation and compensation of processes with time delTechnical Report Number: AOD. 00.03, Dublin Insttute of Technology, Ireland, 2000.

@4# Smith, O. J. M., Close control loops with dead timChem. Eng. Prog.53, 217–219~1957!.

@5# Watanabe, K. and Ito, M., A process-model control flinear systems with delay. IEEE Trans. Autom. Contr26, 1261–1269~1981!.

ith

ol:ss.

g

l.

dens.

neD.

-d.

e-n-

t-nt.

s,

es

thand

or

e-

n-

ingte-n-

,

s-

nsa-ons

-

r-

-

t

in

ode


@6# Camacho, O., Rojas, R., and Garcıá, W. M., Variablestructure control applied to chemical processes winverse response. ISA Trans.38, 55–72~1999!.

@7# Camacho, O. and Smith, C. A., Sliding mode contrAn approach to regulate nonlinear chemical proceISA Trans.39, 205–218~2000!.

@8# Utkin, V. I., Variable structure systems with slidinmodes. IEEE Trans. Autom. ControlAC-„22…, 212–222 ~1997!.

@9# Slotine, J. J. and Li, W., Applied Nonlinear ControPrentice-Hall, Englewood Cliffs, NJ, 1991.

@10# Camacho, O. and Rojas, R., A general sliding mocontroller for nonlinear chemical processes. TraASME 122, 650–655~2000!.

@11# Camacho, O., A New Approach to Design and TuSliding Mode Controller for Chemical Process. Ph.Dissertation, University of South Florida, 1996.

@12# Zhang, W. D. and Sun, Y. X., Modified Smith predictor for controlling integrator/time delay processes. InEng. Chem. Res.35, 2769–2772~1996!.

@13# Astrom, K. J., Hang, C. C., and Lim, B. C., A newsmith predictor for controlling a process with an intgrator and long dead-time. IEEE Trans. Autom. Cotrol 39, 343–345~1994!.

@14# Camacho, O., Rojas, R., Garcıá, W., and Alvarez, A.,Sliding Mode Control: A Robust Approach to Integraing Systems with Dead Time. Proc. Second IEEE ICaracas Conf on Devices, Circuits and SystemICCDSC-98, Margarita, Venezuela, 1998.

@15# Tian, Y. and Gao, F., Control of integrator processwith dominant time delay. Ind. Eng. Chem. Res.38,2979–2983~1999!.

@16# Kwak, H. J., Sung, S. W., and Lee, I., Modified Smipredictors for integrating processes: Comparisonsproposition. Ind. Eng. Chem. Res.40, 1500–1506~2001!.

@17# Matausek, M. R. and Micic, A. D., A modified Smithpredictor for controlling a process with an integratand long dead-time. IEEE Trans. Autom. Control41,1199–1203~1996!.

@18# Matausek, M. R. and Micic, A. D., On the modifiedSmith predictor for controlling a process with an int

grator and long dead-time. IEEE Trans. Autom. Cotrol 44, 1603–1606~1999!.

@19# Normay-Rico, J. E. and Camacho, E. F., Robust tunof dead-time compensators for processes with an ingrator and long dead-time. IEEE Trans. Autom. Cotrol 44, 1597–1603~1996!.

Oscar Camacho received theElectrical Engineering, andM.S. in Control Engineeringdegrees from Universidad deLos Andes ~ULA !, Merida,Venezuela in 1984 and 1992respectively, and the M.E. andPh.D. in Chemical Engineeringat University of South Florida~USF!, Tampa-Florida, in 1994and 1996, respectively. He haheld teaching and research positions at ULA, CIED-PDVSA, and USF. His current

research interests include sliding mode control, deadtime competion, and hybrid systems. He is the author of more than 50 publicatiin journals and conference proceedings.

Francisco De la Cruz re-ceived the Electronic Engi-neering degree from the Instituto Universitario Politećnico~IUP! de Barquisimeto, Ven-ezuela in 1980 and the Mastedegree in Electronic Engineering from the Universidad Si-mon Bolivar, Caracas, Venezuela in 1990. Since 1980, hehas been at the Departmenof Electronic Engineering,Universidad Politećnica~UNEXPO!, Barquisimeto,

Venezuela. He is currently working towards the Doctoral DegreeApplied Sciences, Universidad de Los Andes, Me´rida, Venezuela. Hiscurrent research interests include systems with delays, sliding mcontrol, and predictive control.

Documents

Smith predictor based-sliding mode controller for integrating processes with elevated deadtime