DESIGN AND SIMULATION OF SELF-TUNING PREDICTIVE CONTROL OF TIME-DELAY PROCESSES

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1 DESIGN AND SIMULAION OF SELF-UNING PREDICIVE CONROL OF IME-DELAY PROCESSES Vladimír Bobál,, Marek Kbalčík and Petr Dostál, omas Bata University in Zlín Centre of Polymer Systems, University Institte Department of Process Control, Faclty of Applied Informatics. G. Masaryka Zlín Czech Repblic KEYWORDS ime-delay systems, Model Predictive Control, CARIMA model, Qadratic criterion, Simlation ABSRAC Majority indstrial processes sch as thermal, chemical biological, metallrgical, plastic etc., have time-delays. herefore, the problem of the identification and optimal control of sch systems is of great importance. hese time-delay processes can be effectively handled by the Model-based Predictive Control method. he paper deals with design of an algorithm for self-tning predictive control of sch processes. he self-tning principle is one of possible approaches to control of nonlinear systems or systems with ncertainties. hree types of processes were chosen for simlation verification of the designed self-tning predictive controller. he program system MALAB/SIMULINK was sed for testing and verification of this predictive controller. INRODUCION ime delay is very often encontered in varios technical systems, sch as electric, pnematic and hydralic networks, chemical processes, long transmission lines, robotics, etc. he existence of pre time lag, regardless if it is present in the control or/and the state, may case ndesirable system transient response, or even instability. Conseqently, the problem of controllability, observability, robstness, optimization, adaptive control, pole placement and particlarly stability and robstness stabilization for this class of systems, has been one of the main interests for many scientists and researchers dring the last five decades. For control engineering, sch processes can often be approximated by the FOD (First-Order-ime-Delay model. ime-delay in a process increases the difficlty of controlling it. However the approximation of higherorder process by lower-order model with time-delay provides simplification of the control algorithms. When high performance of the control process is desired or the relative time-delay is very large, the predictive control strategy is one of possible approaches. he predictive control strategy incldes a model of the process in the strctre of the controller. he first time-delay compensation algorithm was proposed by (Smith 957. his control algorithm known as the Smith Predictor (SP contained a dynamic model of the time-delay process and it can be considered as the first model predictive algorithm. First versions of Smith Predictors were designed in the continos-time modifications, see e.g (Normey-Rico and Camacho 7. Becase most of modern controllers are implemented on digital platforms, the discrete versions of the time-delay controllers are more sitable for time-delay compensation in indstrial practice. Most of athors designed the digital timedelay compensators with fixed parameters. However, the time-delay compensators are more sensitive to process parameter variations and therefore reqire an ato-tning or adaptive (self-tning approach in many practical applications. wo adaptive modifications of the digital Smith Predictors are designed in (Hang et al. 989; Bobál et al. and implemented into MALAB/SIMULINK oolbox (Bobál et al. a; Bobál et al. b. Model predictive control (MPC is becoming increasable poplar method in indstrial process control where time-delays are component parts of the system. However, an accrate appropriate model of the process is reqired to ensre the benefits of MPC. Frthermore, pertrbations of a time-delay and parameters of an external linear model may indce complex behaviors (oscillations and instabilities of the closed-loop system. Problems with time-variant model parameters can be solved sing adaptive (selftning approach. MODEL PREDICIVE CONROL Model Predictive Control, also known as Receding Horizon Control (RHC, attracts considerable research attention becase of its nparalleled advantages. hese inclde (L 8: Applicability to a broad class of systems and indstrial applications. Comptational feasibility. Proceedings 7th Eropean Conference on Modelling and Simlation ECMS Webjørn Rekdalsbakken, Robin. Bye, Hoxiang Zhang (Editors ISBN: / ISBN: (CD

2 Systematic approach to obtain a closed-loop control and garanteed stability. Ability to handle hard constraints on the control as well as the system states. Good tracking performance. Robstness with respect to system modeling ncertainty as well as external distrbances. he MPC strategy performs the optimization of a performance index with respect to some ftre control seqence, sing predictions of the otpt signal based on a process model, coping with amplitde constraints on inpts, otpts and states. For a qick comparison of MPC and traditional control scheme, sch as PID control, Fig. shows the difference between the MPC and PID control schemes in which anticipating the ftre is desirable while a PID controller only has capacity of reacting to the past behaviors. he MPC algorithm is very similar to the control strategy sed in driving a car (L 8. follow the desired trajectory. Only the first control action is adopted as the crrent control law, and the procedre is then repeated over the next time horizon, say (k +, k + + N. he term receding horizon is introdced, since the horizon recedes as time proceeds. he basic MPC strategy is shown in Fig., where ( t is the maniplated variable, y ( t is the process otpt and wt is the reference signal, N, N and N are called minimm, maximm and control horizons, respectively. he block diagram is shown in Fig. 3. Crrent (k Process Past y Memory Past Ftre Model Crrent y(k Memory _ Predicted Otpts ŷ + Referenc rajectory w Cost Fnction Optimizer Ftre Errors ê Constraints Figre 3: Block diagram of MPC Calclation of Optimal Control he designed control algorithm is based on the Generalised Predictive Control (GPC method (Clarke et al. 987a,b. he standard cost fnction sed in GPC contains qadratic terms of control error and control increments on a finite horizon into the ftre (Camacho and Bordons 4; Mikleš and Fikar 8 Figre : Difference between the MPC and PID control y(t (t past k- k w (t yt ˆ N k+ N ftre N k+n Figre : Principle of MPC At crrent time k, the driver knows the desired reference trajectory for a finite control horizon, say (k, k + N, and by the taking into accont the car characteristics to decide which control actions (accelerator, brakes, and steering to take in order to time k+n N λ i= N i= N J = yˆ k + i w k+ i + i Δ k+ i where ŷ ( k i + is the process otpt of i steps in the ftre predicted on the base of information available pon the time k, wk ( + i is the seqence of the Δ k + i is the seqence of reference signal and ( the ftre control increments that have to be calclated. Parameters N, N and N are called minimm, maximm and control horizons. he parameter λ ( i can be generally a seqence which affects ftre behavior of the controlled process. he otpt of the model (prediction is compted as the sm of the forced response y n and the free response y yˆ = yn + y ( he free response is that part of the prediction which is determined by past vales of the maniplated variable and past vales of the systems otpt. he forced response is determined by ftre increments of the

3 maniplated variable. he forced response is compted as the mltiplication of the matrix G (Jacobian Matrix of the model and the vector of ftre control increments Δ, which is generally a priori nknown where G g g y GΔ (3 n = g = g3 g g g g g g N N N N N + (4 is matrix containing vales of the step seqence. It follows from ( and (3 that the predictor in a vector form is given by yˆ = GΔ + y (5 By assmption that λ is scalar, the cost fnction ( can be modified to the form J = ( yˆ w ( yˆ w + λδ Δ = (6 = GΔ + y w GΔ + y w + λδ Δ Minimisation of the cost fnction (6 now becomes a direct problem of linear algebra. he soltion in an nconstrained case can be fond by setting partial derivative of J with respect to Δ as zero and yields Δ = ( G G + I G ( w y λ (7 Eqation (7 gives the whole trajectory of the ftre control increments and sch is an open-loop strategy. o close the loop, only the first element is applied to the system and the whole algorithm is recompted at time k+. If we denote the first row of the matrix ( G G + λ I G as K then the actal control increment can be calclated as Δ k = K w y (8 DERIVAION OF PREDICOR An important task is comptation of predictions for arbitrary prediction and control horizons. Dynamics of most of processes reqires horizons of length where it is not possible to compte predictions in a simple straightforward way. Recrsive expressions for comptation of the free response and the matrix G in each sampling period had to be derived. here are several different ways of deriving the prediction eqations for transfer fnction models. Some papers make se of Diophantine eqations to form the prediction eqations (Kwon et al. In (Rossiter 3 matrix methods are sed to compte predictions. We derived a method for recrsive comptation of both the free response and the matrix of the dynamics (Kbalčík and Bobál ; Bobá et al.. Comptation of the predictor for the time-delay system can be obtained by modification of the predictor for the corresponding system withot a time-delay. At first we will consider the second order system withot timedelay and then we will modify the comptation of predictions for the time-delay system. Second Order System withot ime Delay he deterministic model is described by the discrete transfer fnction B( z bz + bz G( z = = (9 A z + az + az Model (9 can be also written in the form A z y k = B z k ( A widely sed model in GPC is the CARIMA model which can be obtained from the nominal model ( by adding a distrbance model C( z A( z y( k = B( z ( k + nc ( k ( Δ where n c ( k is a non-measrable random distrbance that is assmed to have zero mean vale and constant covariance and Δ = z. Inverted Δ is then an integrator. he difference eqation of the second order CARIMA model withot the nknown term n c ( k can be expressed as y( k = ( a y( k + ( a` a y( k ( + ayk 3 + bδk + bδk It was necessary to compte three step-ahead predictions in a straightforward way by establishing of lower predictions to higher predictions. he model order defines that comptation of one step-ahead prediction is based on three past vales of the system otpt. he three step-ahead predictions after modifications can be written in a matrix form ( + yˆ k ( k + = Δ ( k + y k p p p p y k g Δ yˆ k g g yˆ k 3 g3 g + ( ( k 3 4 p p p3 p + 4 y k p3 p3 p33 p 34 Δ where the individal matrix elements in (3 are g = b; g = b a + b ; ( ( g = a a b + a b + a b 3 (3

4 ( ; ; p = a p = a a p3 = a; p4 = b ; p = ( a + ( a a ; p = ( a ( a a + a; p3 = a ( a ; p4 = b ( a ; 3 p3 = ( a + ( a ( a a + a; p3 = a a a + a a + a a p = a a + a a a ; p = b a + a a b. ( ( ; It is possible to divide comptation of the predictions to recrsion of the free response and recrsion of the matrix of the dynamics. Based on the three previos predictions it is repeatedly compted the next row of the free response matrix in the following way: ( ( ( ( p = a p + a a p + a p p = a p + a a p + a p p = a p + a a p + a p p = a p + a a p + a p (4 he first row of the matrix is omitted in the next step and frther prediction is compted based on the three last rows inclding the one compted in the previos step. his procedre is cyclically repeated. It is possible to compte an arbitrary nmber of rows of the matrix. he recrsion of the dynamics matrix is similar. he next element of the first colmn is repeatedly compted in the same way as in the previos case and the remaining colmns are shifted to form a lower trianglar matrix in the way which is obvios from the eqation (3. his procedre is performed repeatedly ntil the prediction horizon is achieved. If the control horizon is lower than the prediction horizon a nmber of colmns in the matrix is redced. Comptation of the new element is performed as follows: ( g = a g + a a g + a g (5 4 3 Second Order System with ime-delay he nominal second order model with d steps of timedelay is considered as B z bz bz G z = z = z (6 + + d + d A z az az where d is a nmber of time-delay steps. he CARIMA model for time-delay system withot the n c k takes the form nknown term d Δ A z y k = z B z Δ k (7 In order to compte the control action it is necessary to determine the predictions from d+ to d+n. he predictor (4 is then modified for an arbitrary nmber of time delay steps to yk ˆ ( + 3 p p p + d + d ( + d 3 yk g k yk ˆ ( 4 p( p ( p ( yk 3 ( g g Δ + = + d + d + d + Δ k ( + yk ˆ ( 5 p yk g3 g ( 3 d p ( 3 d p ( 3+ d 3 g + d g + d p ( + d 4 Δk ( + g + d g3 + d p( + d 4 Δk ( g3 + d g4 + d p k ( 3 ( 3+ d 4 Δ (8 Recrsive comptation of the matrices is analogical to the recrsive comptation described for the second order system withot time-delay. RECURSIVE IDENIFICAION PROCEDURE he regression (ARX model of the following form = + y k Θ k Φ k n k (9 is sed in the identification part of the designed controller predictive algorithm, where ( k = [ a a b b ] Θ ( is the vector of the parameters estimates and ( = ( ( ( ( Φ k y k y k k d k d ( is the regression vector. For calclating of the parameter estimates Θ ˆ ( k is tilized the recrsive least sqares method, its nmerical stability is improved by means of LD decomposition and adaptation is spported by directional forgetting (Klhavý 987; Bobál et al. 5. SIMULAION VERIFICAION OF SELF- UNING MPC A simlation verification of the designed predictive algorithm was performed in MALAB/SIMULINK environment. A typical control scheme, which was sed, is depicted in Fig. 4. his scheme is sed for systems with time-delay of two sample steps. Individal blocks of the Simlink scheme correspond to blocks of the general control scheme presented in Fig. 3. he controller block represents the controlled system. his block consists of the recrsive identification, predictive and optimization parts. his block has two inpts (process otpt y_s and initial condition y_in and three otpts (controller otpt, generating of reference signal w and model parameter estimates aˆ, aˆ ˆ ˆ, b, b. It is possible to inflence the otpt of the process by non-measrable variables the white noise n c and the step v distrbances. he above mentioned predictive controller is not sitable for control of nstable processes. herefore, three types of processes were chosen for simlation verification of digital self-tning predictive controller algorithms. Consider the following continos-time transfer fnctions: c

5 w Initial Condition y_s y_in Controller w Gs b.z -+bz- +a.z -+az - Process Model - Z ime -Delay nc y Control Variables Parmeter Estimates White Noise v Step Distrbance Variable Figre 4: Simlink control scheme Stable non-oscillatory G s = e ( s+ ( 4s+ s G s = e 4s + s+ ( 5s G3 s = e s+ 4s+ 4 Stable oscillatory 3 Non-minimm phase 4s 4s Let s now discretize them with a sampling period = s. he discrete forms of these transfer fnctions. 478z. 76z G ( z = + z. 749z +. 8z. 686z. 4834z G z = + z. 7859z z. 978z. 7783z G ( z = + z. 749z +. 8z 3 were sed in the Simlink control scheme for the verification of the dynamical behavior of individal closed control loops. Simlation control of model G ( z his model was chosen for the complex verification of self-tning MPC properties for time-delay systems. he most important parameters in terms of qality process control are sampling period, penalization factor λ, initial parameter estimates Θ ˆ ( (a priori information and initial diagonal elements of the covariance matrix C ii (. he control performance is dependent also on the variance of the non-measrable noise σ. At first, the model parameter estimates were chosen withot a priori information = [ ] ˆ Θ C ii ( =, σ =., in time t = 5-3 s a step distrbance v(t = 5 affected the systems otpt. Figs. 5 and 6 illstrate an inflence of the penalization factor λ on the control performance. he individal horizons were chosen for all experiments: N = 3, N = 3, N = y - process otpt controller otpt Figre 5: Control of the model G z, λ =. he corses of the control variables oscillate in the initial control interval. When model parameter estimates are converged, the qality of the control process is very good. It is obvios that by increasing λ oscillations of the controller s otpt (k are eliminated which has a positive inflence on the corse of the process otpt y(k. In the sbseqent

6 y - process otpt y - process otpt controller otpt 4 - controller otpt Figre 6: Control of the model G z, λ = Figre 8: Control of the model G z, λ = y - process otpt y - process otpt controller otpt - controller otpt Figre 7: Control of the model G z, λ = Figre 9: Control of the model G z, λ =

7 experiment the model parameter estimates were chosen sing a priori information = [ ] ˆ Θ C ii ( = -3 (an assmption of the parameter estimates dispersion in a narrow interval. he other initial parameters were chosen to be the same as in the previos simlations see Figs he corses of the control variables are satisfactory inclding the initial interval of control. he inflence of the penalization factor λ is also evident on the control corses. Simlation control of model G ( z Simlation control of the model G z (a stable oscillatory model was realized pon similar conditions as in the previos case sing a priori information. he initial vector of parameter estimates has the form ˆ Θ = [ ] he control corses are shown in Fig., the qality of control is very good. Simlation control of model G3 ( z Simlation control of the model 3 G z (the nonminimm phase model was realized pon similar conditions as in the previos cases sing a priori information. he initial vector of parameter estimates has the form ˆ Θ = [ ] he control corses are shown in Fig., the qality of control is very good y - process otpt controller otpt y - process otpt controller otpt Figre : Control of the model G z, λ = Figre : Control of the model 3 CONCLUSION G z, λ = he contribtion presents the self-tning predictive control applied to time-delay processes. he predictive controller is based on the recrsive comptation of predictions by direct se of the CARIMA model. he comptation of predictions was extended for timedelay systems. A linear model with constant coefficients sed in pre model predictive control can not describe the control system in all its modes. herefore, a self-tning approach was applied. It consists of the recrsive identification and the predictive controller. he model parameter estimates obtained from the identification procedre are sed in the self-tning predictive controller. MPC based on minimization of the qadratic criterion was derived and tested. hree models of control processes were sed for simlation verification (the stable non-oscillatory, the

8 stable oscillatory and the non-minimm phase. he designed predictive controller was sccessflly verified not only by simlation bt also in real-time laboratory conditions for control of a heat exchanger. ACKNOWLEDGMEN his article was created with spport of Operational Programme Research and Development for Innovations co-fnded by the Eropean Regional Development Fnd (ERDF, national bdget of Czech Repblic within the framework of the Centre of Polymer Systems project (reg. nmber: CZ..5/../3.. REFERENCES Bobál, V., Böhm, J., Fessl, J. and J. Macháček. 5. Digital Self-tning Controllers: Algorithms, Implementation and Applications. Springer-Verlag, London, 5. Bobál, V., Chalpa, P., Kbalčík, M. and P. Dostál.. Self-tning predictive control of non-linear servomotor. Jornal of Electrical Engineering 6, Bobál, V., Chalpa, P., Dostál, P. and M. Kbalčík.. Design and simlation verification of self-tning Smith predictors. International Jornal of Mathematics and Compters in Simlation 5, Bobál, V., Chalpa, P., Novák, J. and P. Dostál. a. MALAB oolbox for CAD of self-tning of timedelay processes. In Proc. of the International Workshop on Applied Modelling and Simlation, Roma, Bobál, V., Chalpa, P. and J. Novák. b. oolbox for CAD and Verfication of Digital Adaptive Control ime-delay Systems. Available from _Delay_ool.zip. Camacho, E. F. and C. Bordons. 4. Model Predictive Control. Springer-Verlag, London. Clarke, D. W., Mohtadi, C. and P. S. ffs. 987a. Generalized predictive control, part I: the basic algorithm. Atomatica 3, Clarke, D. W., Mohtadi, C. and P. S. ffs. 987b. Generalized predictive control, part II: extensions and interpretations. Atomatica 3, Hang, C. C., Lim, K. W. and B. W. Chong A dalrate digital Smith predictor. Atomatica, -6. Kbalčík, M., and V. Bobál.. echniqes for predictor design in mltivariable predictive control. WSEAS ransactions on Systems and Control 6, Klhavý, R Restricted exponential forgetting in real time identification. Atomatica 3, Kwon, W. H., Choj, H., Byn, D. G. S. Noh. 99. Recrsive soltion of generalized predictive control and its eqivalence to receding horizon tracking control. Atomatica, 8(6, Li, M. Ch. 8. Delay Identification and Model Predictive Control of ime Delayed Systems. Ph.D. hesis, Mc Gill University, Montreal, Canada. Mikleš, J. and M. Fikar. 8. Process Modelling, Optimisation and Control. Springer-Verlag, Berlin. Normey-Rico, J. E. and E. F. Camacho. 7. Control of Dead-time Processes, Springer-Verlag, London. Rossiter, J. A. 3. Model Based Predictive Control: a Practical Approach. CRC Press. Smith, O.J Closed control of loops. Chem. Eng. Progress, 53, 7-9. AUHOR BIOGRAPHIES VLADIMÍR BOBÁL gradated in 966 from the Brno University of echnology, Czech Repblic. He received his Ph.D. degree in echnical Cybernetics at Institte of echnical Cybernetics, Slovak Academy of Sciences, Bratislava, Slovak Repblic. He is now Professor at the Department of Process Control, Faclty of Applied Informatics of the omas Bata University in Zlín, Czech Repblic. His research interests are adaptive and predictive control, system identification and CAD for atomatic control systems. Yo can contact him on address bobal@fai.tb.cz. MAREK KUBALČÍK gradated in 993 from the Brno University of echnology in Atomation and Process Control. He received his Ph.D. degree in echnical Cybernetics at Brno University of echnology in. From 993 to 7 he worked as senior lectrer at the Faclty of echnology, Brno University of echnology. From 7 he has been working as an associate professor at the Department of Process Control, Faclty of Applied Informatics of the omas Bata University in Zlín, Czech Repblic. Crrent work cover following areas: control of mltivariable systems, self-tning controllers, predictive control. His address is: kbalcik@fai.tb.cz. PER DOSÁL stdied at the echnical University of Pardbice, Czech Repblic, where he obtained his master degree in 968 and PhD. degree in echnical Cybernetics in 979. In the year he became professor in Process Control. He is now head of the Department of Process Control, Faclty of Applied Informatics of the omas Bata University in Zlín. His research interests are modelling and simlation of continos-time chemical processes, polynomial methods, optimal and adaptive control. Yo can contact him on address dostalp@fai.tb.cz.