DIGITAL LINEAR QUADRATIC SMITH PREDICTOR

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1 DIGITAL LINEAR QUADRATIC SMITH PREDICTOR Vladimír Bobál,, Marek Kbalčík, Petr Dostál, and Stanislav Talaš Tomas Bata Universit in Zlín Centre of Polmer Sstems, Universit Institte Department of Process Control, Faclt of Applied Informatics T. G. Masarka Zlín Cech Repblic KEYWORDS Time-dela Sstems, Smith Predictor, LQ Control, Spectral Factoriation, Simlation ABSTRACT Time-delas dead times are fond in man processes in indstr. Time-delas are mainl cased b the time reqired to transport mass, energ or information, bt the can also be cased b processing time or accmlation. The contribtion is focsed on a design of niversal digital algorithm for control of great deal of processes ith time-dela. This reqirement is sccessfll satisfied ith digital Smith Predictor based on Linear Qadratic LQ method. A minimiation of the qadratic criterion is realied sing spectral factoriation. The designed algorithm is sitable for control of stable, nstable and non-minimm phase processes. The algorithms for control of individal processes inflenced b external distrbance ere verified. The program sstem MATLAB/SIMULINK as sed for simlation of designed algorithms. INTRODUCTION Time-delas appear not onl in indstrial processes, sch as thermal, chemical, metallrgical or processes of plastic and rbber materials etc., bt also in other fields, inclding economical and biological sstems. The are cased b some of the folloing phenomena Norm- Rico and Camacho 7: the time needed to transport mass, energ or information, the accmlation of time lags in a great nmbers of lo order sstems connected in series, the reqired processing time for sensors, sch as analers; controllers that need some time to implement a complicated control algorithms or process. The problem of controlling time-dela processes can be solved b several control methods e. g. sing PID controllers, time-dela compensators, model predictive control techniqes. Time-dela in a process increases the difficlt of controlling it. Hoever, the approximation of higherorder process b loer-order model ith time-dela provides simplification of the control algorithms. When high performance of the control process is desired or the relative time-dela is ver large, the predictive control strateg can be sccessfll applied. The predictive control method incldes a model of the process in the strctre of the controller. The first time-dela compensation algorithm as proposed b Smith 957. This control algorithm knon as the Smith Predictor SP contained a dnamic model of the time-dela process and it can be considered as the first model predictive algorithm. First versions of Smith Predictors ere designed in the continos-time modifications, see e.g. Norme-Rico and Camacho 7. Althogh time-dela compensators appeared in the mid 95s, their implementation ith analog techniqe as ver difficlt and these ere not sed in indstr. Becase most of modern controllers are implemented on digital platforms, the discrete versions of the time-dela controllers are more sitable for time-dela compensation in indstrial practice Since 98s digital time-dela compensators can be implemented. The digital time-dela compensators are presented e.g. in Vogel and Edgar 98, Palmor and Halevi 99, Norme-Rico and Camacho 998. Some Self-tning Controller STC modifications of the digital Smith Predictors STCSP are designed in Hang et al. 989; Hang et al. 993; Bobál et al.. To versions of the STCSP ere implemented into MATLAB/SIMULINK Toolbox Bobál et al. a; Bobál et al. b. It is ell knon that classical analog Smith Predictor is not sitable for control of nstable processes. The designed digital LQ Smith Predictor eliminates this draback. The paper is organied in the folloing a. The problem of a control of the time-dela sstems is described in Section. The general principle of the Smith Predictor is described in Section. The discretiation of analoge version, principle of digital Smith Predictor and polnomial to degrees of freedom DOF controller is introdced in Section 3. Primar Linear Qadratic controller of the digital Smith Predictor is proposed in Section. Reslts of simlation experiments are smmed in Section 5. Section 6 concldes the paper. Proceedings 8th Eropean Conference on Modelling and Simlation ECMS Flaminio Sqaoni, Fabio Baronio, Cladia Archetti, Marco Castellani Editors ISBN: / ISBN: CD

2 DIGITAL SMITH PREDICTOR The discrete versions of the SP and their modifications are sitable for time-dela compensation in indstrial practice. PROCESS d e G c - G p - _ ŷ Td ŷ G m - G d - _ ŷ p ê p Figre : Block diagram of a digital Smith Predictor The block diagram of a digital SP see Hang, Lim, and Chong 989; Hang, Tong, and Weng 993 is shon in Fig.. The fnction of the digital version is similar to the classical analog version. The block Gm represents process dnamics ithot the time-dela and is sed to compte an open-loop prediction. The difference beteen the otpt of the process and the model inclding time dela ŷ is the predicted error ê p as shon in Fig., hereas e and v are the error and the measred distrbance, is the reference signal. If there are no modelling errors or distrbances, the error beteen the crrent process otpt and the model otpt ŷ ill be nll. Then the predictor otpt signal ŷ p ill be the time-dela-free otpt of the process. Under these conditions, the controller Gc can be tned, at least in the nominal case, as if the process had no time-dela. The primar main controller Gc can be designed b different approaches for example digital PID control or methods based on algebraic approach. The otard feedback-loop throgh the block Gd in Fig. is sed to compensate for load distrbances and modelling errors. Nmber of higher order indstrial processes can be approximated b a redced order model ith a pre time-dela. In this paper the folloing second-order linear model ith a time-dela is considered B b b G = = d d A a a The term -d represents the pre discrete time-dela. The time-dela is eqal to dt here T is the sampling period. Design of Polnomial DOF Controller Previos simlation experiments demonstrated that polnomial theor is sitable method for design of the digital Smith Predictor. Polnomial control theor is based on the apparats and methods of linear algebra see e.g. Kčera 993. The polnomial Smith Predictor based on the digital pole assignment as designed in Bobál et al.. The design of the controller algorithm is based on the general block scheme of a closed-loop ith to degrees of freedom DOF according to Fig.. Figre : Block diagram of a closed loop DOF control sstem The controlled process is given b the transfer fnction in the form G p Y B = = U A here A and B are the second order polnomials. The controller contains the feedback part G q and the feedforard part G r. Then the digital controllers can be expressed in the form of discrete transfer fnctions G G r q r R = = P p Q q q q = = P p 3 According to the scheme presented in Fig. and eqations it is possible to derive the characteristic polnomial here G r G q A P B Q = D 5 q _ D = d d d d is the forth degree characteristic polnomial. The procedre leading to determination of polnomials Q, R and P in 3 and can be briefl described as follos see Bobál et al. 5. A feedback part of the controller is given b a soltion of the polnomial Diophantine eqation 5. An asmptotic tracking is provided b a feedforard part of the controller given b a soltion of the polnomial Diophantine eqation r S D B R = D 7 For a step-changing reference signal vale, polnomial D - = - - and S is an axiliar polnomial hich does not enter into controller design. G p

3 A feedback controller to control a second-order sstem ithot time-dela ill be derived from eqation 5. A sstem of linear eqations can be obtained sing the ncertain coefficients method b q d a b b a q d a a = 8 b b a a q d3 a b a p d For a step-changing reference signal vale it is possible to derive the polnomial R from eqation 7 b sbstitting = D d d d d R = r = = B b b 3 The DOF controller otpt is then given b k = r k q k q k q k p k p k 9 Minimiation of LQ Criterion The linear qadratic methods tr to minimie the qadratic criterion ith penaliation of the controller otpt k = {[ ] [ ] } J = k k q k here q is the so-called penaliation constant, hich gives the rate of the controller otpt on the vale of the criterion here the constant at the first element of the criterion is considered eqal to one. The standard procedre of the minimiation of the criterion is based on the state description of the sstem and leads to the soltion of the Riccati Eqation. In this paper, criterion minimiation is realied throgh spectral factoriation for the inpt-otpt description of the sstem Bobál et al. 5. If the seqences of the vales of both tracking error and inpt signal are considered as polnomials, it is possible to rerite the criterion sing notation x = x J = E E q U U here E and U are the conjgated polnomials to the polnomials E - and U -, hich means their negative poers are replaced b positive ones. The tracking error polnomial E W Y = B R = W A P B Q and the inpt signal polnomial 3 A R U = W A P B Q are sbstitted into criterion. It can be verified Šebek and Kčera 98 that the criterion is minimal if eqation 5 is valid. The polnomial D - is the reslt of spectral factoriation according to the eqation A qa B B = D D 5 δ here δ is a constant chosen so that d =. The spectral factoriation of a polnomial leaves its stable part nchanged, hile the nstable parts change to reciprocal ones stable. Spectral factoriation of polnomial of the first and the second degree cold be compted simpl; the procedre for the higher degrees mst be performed iterativel. While performing the spectral factoriation of a polnomial of the second degree M = m m m the folloing eqation is solved here δ M M = D D 6 D = d d 7 The prodcts of the polnomials cold be extended as m m m = δ d d 8 δd d δd here the constants of the factoried polnomial on the left side of the eqation 3 are combined into the coefficients m, m and m. Comparing the left and the right side of eqation 3, one obtains m = δ d d ; m = δd d ; m = δd 9 Solving eqations 9, the folloing expressions are derived m m m λ m m m λ λ δ = ; = ; m m d = ; d = δ m δ Solving the spectral factoriation of eqation 5, an identical expression can be sed, bt is necessar to convert the left side of this eqation to the form sed in eqation 6, ths m = q a a b b m = q a aa bb; m = q a

4 PRIMARY LQ CONTROLLER OF DIGITAL SMITH PREDICTOR From the previos paragraph, it is obvios that sing analtical spectral factoriation, onl to parameters of the second degree polnomial D - 7 can be compted. This approach is applicable onl for control of processes ithot time-dela ot of Smith Predictor. The primar controller in the digital Smith Predictor strctre reqires sing the forth degree polnomial D - 6 in eqations 5 and 7. From expression 7 it is obvios that polnomial D = d d have to different real poles α, β or one complex conjgated pole, = α ± jβ in the case of oscillator sstems. These poles mst be inclded into polnomial 3 D = d d d d 3 3 For both to tpes of the processes the sitable pole assignment as designed: st possibilit: Polnomial 8 has to different real poles α, β compted from 7 and ser-defined real poles γ, δ. Then it is possible to rite polnomial 8 as a prodct root of factor α β γ δ D = and it is possible to express its individal parameters as d = α β γ δ d = αβ γδ α β γ δ 5 d3 = α β γδ γ δ αβ d = αβγδ nd possibilit: Polnomial 8 has the complex conjgate pole, = α ± jβ compted from 7 and serdefined real poles γ, δ. Then polnomial 8 has the form α β α β γ δ D = j j 6 and its individal parameters can be expressed as d = α γ δ d = α γ δ α β γδ d3 = αγδ α β γ δ d = α β γδ 7 The control algorithm based on the LQ control method contains the folloing steps:. The parameters of the polnomial M - are compted according to eqations.. The parameters of the polnomial D - are compted according to eqations. 3. If the polnomial has the real poles α, β, its parameters are compted according to eqations 5, otherise, the are compted according to eqations 7.. The controller parameters are compted sing matrix eqation 8 and eqation The controller otpt is given b eqation. Penaliation of the controller otpt is performed b setting q. With increased penaliation constant, the amplitde of the controller otpt decreases and thereb, the flo of the process otpt is smoothened and an possible oscillations or instabilit are damped. SIMULATION VERIFICATION AND RESULTS A simlation verification of the designed predictive algorithm as performed in MATLAB/SIMULINK environment. A tpical control scheme, hich as sed, is depicted in Fig. 3. This scheme is sed for sstems ith time-dela of to sample steps. Individal blocks of the Simlink scheme correspond to blocks of the general control scheme presented in Fig.. It is possible to inflence the otpt of the process ith the non-measrable distrbance d. The above mentioned Smith Predictor has niversal sage for control of great deal of processes ith timedela. Therefore, for tpes of processes ere chosen for simlation verification of controller algorithm. Consider the folloing continos-time transfer fnctions: s Stable non-oscillator G s = e s s s Stable oscillator G s = e s s 5s s 3 Non-minimm phase G3 s = e s s s s Unstable G s = e s s Let s no discretie them ith a sampling period T = s, then the discrete forms are G = G = G = G =

5 ,, Set Point r p - Controller-Feedforard Part Step Fnction s Filter d q.q. -q - p - Controller - Feedback Part - - Integrator Satration bc.sbc ac.s ac.sac Continos-time Process Transport Dela b b - a. -a- b. -3b- b b - Compensator Compensator p ep Figre 3: Simlink control scheme The step responses of individal models are shon in Figs [-], [-].5 [-], [-] [-], [-] Figre : Step response of the model G s Figre 5: Step response of the model G s [-], [-] Figre 6: Step response of the model G s Figre 7: Step response of the model G s The processes hich are described b the above mentioned transfer fnctions ere sed in the Simlink control scheme for the verification of the dnamical 3

6 behavior of individal closed control loops. In time 5 8 s an exponential external distrbance.t =.5 e d t acted on the sstem otpt. The compted poles α, β and ser-defined real poles γ, δ are introdced for individal simlation experiments inclding characteristic polnomial 5. For all experiments, the penaliation factor as chosen q =. Simlation control of model G The poles: α, β =.3 ±.76 i; γ =.; δ =.5 The characteristic polnomial: 3 D = The corses of the control variables are shon in Fig. 8, the qalit of control is ver good. [-], [-], [-] G Figre 8: Control of the model Simlation control of model G The poles: α, β =.5±.38 i; γ =.; δ =.5 The characteristic polnomial: 3 D = The corses of the control variables are shon in Fig. 9, the qalit of control is ver good..5 Simlation control of model G3 The poles: α =.653; β =.35; γ =.; δ =.75 The characteristic polnomial: 3 D = The corses of the control variables are shon in Fig. 9. The process otpt has tpical character for the control of the non-minimm phase sstem ndershoot of in the initial time-interval. The stabilit of a control-loop is ver dependent on pole δ. For small δ the control loop is nstable, for sitable chosen δ, the qalit of control is ver good. [-], [-], [-] G3 Figre : Control of model Simlation control of model G The poles: α, β =.37 ±.7 i; γ =.; δ =.5 The characteristic polnomial: 3 D = The corses of the control variables are shon in Fig., the qalit of control is ver good. [-], [-], [-] [-], [-], [-] Figre : Control of the model G 6 8 G Figre 9: Control of the model CONCLUSION The contribtion presents ne generalied strateg for design of the polnomial digital Smith Predictor for control sstems ith time-dela. The primar

7 controller is based on minimiation of the linear qadratic criterion. Minimiation of criterion is realied throgh spectral factoriation. This controller as derived prposel b analtical a ithot tiliation of nmerical methods to obtain algorithms ith eas implementabilit in indstrial practice. For models of control processes ere sed for simlation verification. Main contribtion of the proposed method is the niversal applicabilit of digital Smith Predictor for nstable processes. The designed predictive controller as sccessfll verified not onl b simlation bt also in real-time laborator conditions for control of a heat exchanger. ACKNOWLEDGEMENTS This article as created ith spport of Operational Programme Research and Development for Innovations co-fnded b the Eropean Regional Development Fnd ERDF, national bdget of Cech Repblic ithin the frameork of the Centre of Polmer Sstems 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. 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. MATLAB Toolbox for CAD of self-tning of timedela processes. In Proc. of the International Workshop on Applied Modelling and Simlation, Roma, 9. Bobál, V., Chalpa, P. and J. Novák. b. Toolbox for CAD and Verfication of Digital Adaptive Control Time-Dela Sstems. Available from _Dela_Tool.ip. Camacho, E. F. and C. Bordons.. Model Predictive Control. Springer-Verlag, London. Hang, C. C., Lim, K. W. and B. W. Chong A dalrate digital Smith predictor. Atomatica, -6. Hang, C. C., Tong, H. L. and K. H. Weng Adaptive Control, North Carolina: Instrment Societ of America. Kčera, V Diophantine eqations in control a srve. Atomatica 9, Norme-Rico, J. E. and E. F. Camacho Dead-time compensators: A nified approach. In Proceedings of IFAC Workshop on Linear Time Dela Sstems LDTS 98, Grenoble, France, -6. Norme-Rico, J. E. and E. F. Camacho. 7. Control of Dead-time Processes, Springer-Verlag, London. Palmor, Z. J. and Y. Halevi. 99. Robstness properties of sampled-data sstems ith dead time compensators. Atomatica 6, Smith, O. J Closed control of loops. Chem. Eng. Progress, 53, 7-9. Šebek, M. and V. Kčera. 98. Polnomial approach to qadratic tracking in discrete linear sstems. IEEE Trans. Atomatic. Control AC-7, 8-5. Vogel, E. F. and T. F. Edgar 98. A ne dead time compensator for digital control. In Proceedings ISA Annal Conference, Hoston, USA, 9-6. AUTHOR BIOGRAPHIES VLADIMÍR BOBÁL gradated in 966 from the Brno Universit of Technolog, Cech Repblic. He received his Ph.D. degree in Technical Cbernetics at Institte of Technical Cbernetics, Slovak Academ of Sciences, Bratislava, Slovak Repblic. He is no Professor at the Department of Process Control, Faclt of Applied Informatics of the Tomas Bata Universit in Zlín, Cech Repblic. His research interests are adaptive and predictive control, sstem identification and CAD for atomatic control sstems. Yo can contact him on address bobal@fai.tb.c. MAREK KUBALČÍK gradated in 993 from the Brno Universit of Technolog in Atomation and Process Control. He received his Ph.D. degree in Technical Cbernetics at Brno Universit of Technolog in. From 993 to 7 he orked as senior lectrer at the Faclt of Technolog, Brno Universit of Technolog. From 7 he has been orking as an associate professor at the Department of Process Control, Faclt of Applied Informatics of the Tomas Bata Universit in Zlín, Cech Repblic. Crrent ork covers folloing areas: control of mltivariable sstems, self-tning controllers, predictive control. His address is: kbalcik@fai.tb.c. PETR DOSTÁL stdied at the Technical Universit of Pardbice, Cech Repblic, here he obtained his master degree in 968 and PhD. degree in Technical Cbernetics in 979. In the ear he became professor in Process Control. He is no head of the Department of Process Control, Faclt of Applied Informatics of the Tomas Bata Universit in Zlín. His research interests are modelling and simlation of continos-time chemical processes, polnomial methods, optimal and adaptive control. Yo can contact him on address dostalp@fai.tb.c. STANISLAV TALAŠ stdied at the Tomas Bata Universit in Zlín, Cech Repblic, here he obtained his master degree in Atomatic Control and Informatics in 3. He no attends PhD. std in the Department of Process Control. His address is talas@fai.tb.c.