Control of Delayed Integrating Processes Using Two Feedback Controllers R MS Approach

Transcription

1 Proceeding of the 7th WSEAS International Conference on SYSTEM SCIENCE and SIMULATION in ENGINEERING (ICOSSSE '8) Control of Delayed Integrating Procee Uing Two Feedback Controller R MS Approach LIBOR PEAŘ, ROMAN PROOP Department of Automation and Control Toma Bata Univerity in Zlín Nad Stráněmi 45, 765, Zlín CZECH REPUBLIC Abtract: - The objective of thi paper i to demontrate the utilization of algebraic controller deign in an unconventional ring while control integrating procee with time delay. In contrat to many other method, the propoed method i not baed on the time delay approximation. A control tructure combining a imple feedback loop and a two-degree-of-freedom control tructure i conidered. Thi tructure can be conceived a a imple feedback loop with inner tabilizing loop. The control deign i performed in the ring of retarded quaipolynomial (RQ) meromorphic function (R MS ) - an algebraic method baed on the olution of the Bézout equation with Youla-učera parameterization i preented. Final controller may be of o-called aniochronic type and enure feedback loop tability, tracking of the tep reference and load diturbance attenuation. Among many poible tuning method, the dominant pole aignment method i adopted. ey-word: - Time delay ytem, algebraic control, Bézout identity, Youla-učera parameterization Introduction Integrating model appear while modeling ma or energy accumulation, rotation of machinerie, etc. Thee procee contain undeirable pole which need to be hifted by the feedback loop. A well, the great deal of technological and other procee own an input-output time delay. The preence of a delay entail problem with controller deign due to the fact that the delay ignificantly influence the feedback propertie of a control ytem. The combination of integrating behavior of the ytem and delay make a controller deign more difficult. There have been invetigated variou principle for control of integrating procee with a time delay. Some of thee method utilize tandard PI or PID controller, e.g. [-[3. Some idea are baed on generalized Smith predictor, e.g. [4-[6 or predictive approache, mainly uing tate-pace decription [7. One of ignificant approache in modern control theory i a family of algebraic method. Unlike ome traditional tate-pace model, algebraic tool are baed on fractional decription of ytem. Any tranfer function can be expreed a a ratio of two element in the appropriate ring. Traditional tranfer function i repreented by a polynomial fraction. Thi decription i employed for algebraic control trategy for integrating delayed ytem in [8 where the control tructure with two feedback controller (Fig.) i conidered. Another frequently ued ring, beide polynomial, i deigned a R PS (ring of Hurwitz table and proper rational function [9, [. Algebraic control philoophy in thi ring then exploit Bézout identity (Diophantine equation) along with Youla-učera parameterization to obtain table and proper controller. Algebraic controller deign method mentioned above require rational approximation of exponential expreing delay, uually via firt order Padé approximation. Thi contribution preent an algebraic method avoiding any time delay approximation. A ring of table and proper retarded quaipolynomial meromorphic function (R MS ) for thi purpoe i utilized. A term of thi ring i a ratio of two quaipolynomial where the denominator quaipolynomial i table and the whole ratio i proper with repect to highet -power. The only effort to deign controller in thi ring for integration delayed ytem i in [ where a imple DOF control tructure wa utilized. In thi paper, an algebraic approach baed on Bézout identity and the Youla-učera parameterization uing control ytem with two controller i conidered. The preented feedback ytem can be comprehend and olved in double meaning. Firt, it i poible to take the ytem a a whole. It mean that overall input-output tranfer function are utilized for controller deign. Thi approach wa preented in [2 where it i compared with linear quadratic (LQ) polynomial approach [8. Second, the control ytem can be viewed a a imple control feedback with inner (tabilizing) feedback loop. In thi cae, inner loop i olved firt and the main loop ubequently. The final controller enure (in both cae feedback loop tability, tep reference tracking and load diturbance attenuation, and they are tuned by pole ISSN: ISBN:

2 Proceeding of the 7th WSEAS International Conference on SYSTEM SCIENCE and SIMULATION in ENGINEERING (ICOSSSE '8) aignment method decribed in [3. The paper dicue and compare reult of both cae. Illutrative example alo demontrate and verifie the uefulne and applicability of the propoed method. where m ( i an appropriate quaipolynomial of degree one. The uitable form of m ( i dicued in the Section 4 where controller deign for inner feedback loop i decribed. 2 Sytem Decription in R MS Ring 2. R MS Ring Algebraic control method are baed on input-output ytem formulation in the form of tranfer function. Conventional tranfer function a a ratio of two polynomial are not directly applicable for model containing delay due to exponential reulting from the Laplace tranform of delay. In order to expre the numerator and denominator in polynomial, the firt order Padé approximation i then uually utilized. However, there i alo poible to ue another way. Rational approximation can be avoided o that the tranfer function can be performed in the ring of table and proper RQ-meromorphic function, R MS. Any function over thi ring i a ratio of two retarded quaipolynomial y(/x( in general. A denominator quaipolynomial x( of degree n mean n n h i x( + x exp( ϑ () i j where retarded refer to the fact that the highet - power i not affected by exponential. Quaipolynomial () i table iff it own no finite zero uch that Re { }. In other word, a term in R MS ring i analytic in the right half complex plane. Stability can be verified by the Michailov tability criterion, ee e.g. in [4. The numerator y( of an element in R MS can be factorized in the form y( ~ y( exp( τ ), where τ > and ~ y ( ) i a retarded quaipolynomial of degree l l h ~ ( l ) + ~ i y y exp( τ (2) i j The quaipolynomial fraction i called proper iff l n. 2. Integrating Delayed Plant in R MS R MS ring can be naturally utilized for decription of ytem with delay in both left and right ide of appropriate differential equation. The tranfer function of the plant or the controller i then expreed a a ratio of two term in R MS ring. Thi contribution deal with the integrating time delay ytem incribed with the tranfer function exp( τ ) m () () G exp( τ ; A(, R MS A( m () (3) 3 Control Sytem with Two Controller Controller deign decribed in Section 4 have been implemented for a imple feedback loop or internal model tructure (IMC) up to thi day. In thi paper, the control ytem tructure with two controller i utilized, ee Fig.. Fig. - Propoed control tructure with two controller. In the cheme, W( i the reference ignal, D( i the load diturbance, E( i the control error, U ( i the controller output, U( i the plant input, and Y( i the plant output (controlled value) in the Laplace tranform. The plant tranfer function i depicted a G (, the inner feedback controller i G Q (, and G R ( repreent outer controller. The following baic tranfer function can be derived in the control ytem in general: Y ( Y ( GWY ( ; GDY ( W ( M ( D( M ( E( A( + GWE ( W ( M ( E( GDE ( D( M ( (4) where GR ( ; GQ ( (5) M( A( + [ + (6) and, and are from R MS and the numerator of M( correpond to the characteritic quaipolynomial of the cloed loop. ISSN: ISBN:

3 Proceeding of the 7th WSEAS International Conference on SYSTEM SCIENCE and SIMULATION in ENGINEERING (ICOSSSE '8) 4 Algebraic controller deign in R MS ring The algebraic controller deign preented in thi paper uppoe that all tranfer function and ignal in the control ytem are in the form of a ratio of term in R MS. The control ytem cheme in Fig. can be graped either a the whole ytem correponding to tranfer function (4) or a an inner feedback loop with controller G Q and outer loop with controller G R. Algebraic control deign for the former wa hown in [2. Thi contribution concern in the latter idea. Baic requirement on the control ytem are cloedloop tability, aymptotical reference tracking and load diturbance attenuation. To avoid the preence of input diturbance in the inner feedback for controller deign, let the control ytem cheme be rearranged a in Fig.2. All tranfer function in (4) are till valid; however, controller deign for the inner loop exclude the diturbance. The idea i that inner feedback pretabilize the controlled proce, i.e. zero pole i moved to the left, and the outer feedback controller enure mentioned requirement for pre-tabilized ytem G. Fig.2 - Reconfigured control cheme 4. Inner loop pre-tabilization Let an integrating delayed plant be pre-tabilized uing a proportional controller G Q q. The condition for cloed-loop tability i given by Diophantine equation exp( τ q + (7) m () m () The natural tak i to find a uitable table quaipolynomial m (). The olution of (6) give m () q (8) exp( τ The requirement i q to be real; therefore the implet m ( ha to be of the form m ( q exp( τ + (9) Stability of m ( can be tudied e.g. uing Michajlov criterion, which reult in π q Am 2 τ () where A m > i the gain margin ( Am correpondence to tability border). Thu, tranfer function of the inner feedback loop i exp( τ G ( () + q exp τ 4.2 Sytem tabilization Now the tak i to control G uing a imple feedback with controller G R. However, the numerator and denominator in () are not from R MS and thu the tranfer function mut be factorized a exp( τ m ( G ( (2) exp( τ m ( where m ( i a table (quai)polynomial. In order to have (2) a imple a poible, let m ( + λ (3) where λ > i a electable real parameter which bring an additional degree of freedom. Naturally one can take another m (; an example i preented in Section 4.4. Cloed loop tability i enured by a olution of Diophantine equation + q exp exp τ τ + (4) + λ + λ A particular olution + λ exp( τ R (, P ( (5) exp( τ it i poible to parameterize uing the Youla-učera parameterization exp( τ R ( + A( + + λ P ( + λ exp( τ exp( τ ; R MS exp τ + λ (6) to fulfill other control requirement via appropriate choice of. 4.3 Reference Tracking and Diturbance Rejection Parameterization (5) enable to find the olution of (3), o that requirement of reference tracking and diturbance rejection are accomplihed. Both input are conidered a tep function ISSN: ISBN:

4 Proceeding of the 7th WSEAS International Conference on SYSTEM SCIENCE and SIMULATION in ENGINEERING (ICOSSSE '8) w HW ( mw ( H D ( md ( W ( ; D( F ( F ( W D mw ( md ( (7) where m w ( and m d ( are arbitrary table (quai)polynomial of degree one and H W (, H D (, F W (, F D ( RMS. The requirement i both F W ( and F D ( divide ; in other word control error, E(, mut be from R MS. It mean that the numerator of contain at leat one zero root. Let λ ( + λ) (8) exp( τ Then the feedback controller read λ [ exp( τ exp( τ G R ( + λ[ exp( τ (9) + q exp τ λ q [ + exp( τ [ + λ[ exp( τ Note that thi controller i of o-called aniochronic tructure [4, i.e. it own delay in the feedback. Recall that the inner-feedback controller i proportional, G Q q ; however, in term of algebraic philoophy and Fig. it can be written alo a q[ + λ[ exp( τ exp( τ G Q ( (2) + λ[ exp( τ + q exp τ 4.4 An Alternative Solution A wa mentioned in Section 4., table (quai)polynomial m ( can be choen other than in (3). Another natural choice i m ( m ( exp( τ (2) which give the inner-feedback tranfer function exp( τ exp( τ exp( τ exp( τ G ( (22) exp( τ exp( τ In thi cae, tabilizing Diophantine equation exp( τ + (23) exp( τ ha one of particular olution d ( τ exp( τ exp( τ exp R (, P ( (24) + q and chooing Z ( q in the Youla-učera parameterization, the final outer-feedback controller tructure enuring reference tracking and diturbance rejection i then [ + q exp( τ q G R ( (25) which i generalized (delayed) PI controller. The innerfeedback controller i G Q q again. 4.5 Comparion to direct controller deign Control ytem in Fig. can be conceived alo a one ytem without eparation of inner feedback loop. Algebraic controller deign in R MS for thi philoophy wa propoed in [2 where the following two et of (alternative) controller were derived λ γ R α + + αλ GR ( + λ [ exp( τ (26) λ ( γ ) α Q + GQ ( + λ exp τ [ ( τ γ 2α + exp GR ( (27) GQ ( ( γ ) 2α where α i a electable real parameter contrained in the ame way a q in (), another electable real parameter γ, repreent the ditribution of the olution, ee detail in [2, and λ ha analogou function to λ. 5 Controller Tuning The final et of controller, (9), (2), (25) (27), till own unknown parameter that have to be et properly. There are naturally plenty of approache olving the problem of controller tuning. The well applicable and relatively imple tuning method wa decribed e.g. in [3. Thi method enable to et the deired dominant pole of the cloed loop, the maximum number of which i given by the number, k, of unknown parameter in the characteritic quaipolynomial. If the dominant pole are denoted aσ i, i... k, the characteritic equation a m(, and a vector of unknown parameter a v, then the following ytem of k linear equation i obtained m( σ i, v) ; i... k (28) ISSN: ISBN:

5 Proceeding of the 7th WSEAS International Conference on SYSTEM SCIENCE and SIMULATION in ENGINEERING (ICOSSSE '8) For complex pole, one root from each complex pair i taken and (28) i divided into two equation of the form Re{ m( σ i, v) } (29) Im{ m( σ i, v) } The ignificant feature i that et (28) or (29) i linear with repect to unknown parameter, which make the olution eay to find. In the cae of delayed integrator, the ytem given by outer controller (25) ha the characteritic quaipolynomial m( ( + q ) 2 exp( τ ) (3) with unknown parameter q. Since there i a ingle parameter to be found, q, the only double real dominant root or a conjugate pair of complex root can be precribed. Moreover, tability condition () mut be conidered. In cae of real pole, the optimal choice i the precription of leftmot dominant root. It can be proved that thi optimal double real pole i enured by the option q τe (3) which correpond with dominant pole σ,2 τ In cae of conjugate pole, the olution of (29) give the ame real q if a complex root atifie Re{ σ i} Im{ σ i} / tan( Im{ σ i} τ ) (32) Thee root (real or complex) mut be choen carefully becaue ome attempt to place them exceively to the left in the complex plane can lead into the following ituation. Due to the fact that the quaipolynomial (3) ha the infinity number of root, the chain of complex root can move to the right near to the tability border (imaginary axi and thu thee root can take over the role of dominant pole of the ytem. The characteritic of the quaipolynomial feedback ytem with outer controller (9) i m ( [ + q exp( τ ( + λ) (33) In thi cae, two parameter are to be et. Again, condition () mut be fulfilled and λ >. The tak of pole placement can be olved eparately for each of factor in (33). In [8, the uggetion for the choice of λ i 2 λ (34) τ which, in comparion with (3), doe not allow for the dominant pole. Coefficient γ in (26) and (27) influence the feedback ytem behavior a well becaue it appear in the numerator of cloed loop tranfer function. However, it doe not impact the pectrum of the ytem. In thi paper, variou value of γ are et randomly. 6 Illutrative example Thi imulation example compoed in Matlab- Simulink environment demontrate the uability of the propoed controller deign method in R MS. For the ake of limit pace, imulation how output only and the control action value are commented if neceary. Hence, let and τ 5. Aume controller (9) and (2) firt. The highet value of q according to (3) when a double real root i choen i q.736 (for σ,2 -.2). Thi choice correponding to gain margin A m 4.27 i confronted with the dominant pole σ -.8 ±.j, i.e. q.835, A m Parameter λ i choen according to (34) a λ. 4. The control performance of thi controller i compared with controller (9) and (25), with the ame etting, in Fig. 3. Fig.3 - Step etpoint and load diturbance repone Controller: (9), (2); Controller2: (9), (25);, τ 5, λ.4, d -.. A can be een in Fig.3, a change of q for controller (9) and (2) doe not affect etpoint repone. Diturbance repone for thee controller i much better then for (9) and (25); however, thi performance i afflicted by higher control action peak. Finally, the repone uing (9) and (2) i compared with repone of the feedback ytem containing controller (26) and (27) obtained via direct controller deign in Fig.4. Thee controller have the conitent etting with controller (9) and (2), i.e. λ λ. 4, q α.736, and equable ditribution of the olution, i.e. γ. 5. Fig.4 clearly indicate that controller (9)- (2) and (26) caue identical diturbance repone. The advantage of the latter controller init in the poibility of etting the ditribution of the olution uing γ. However, propoed alternative approach via conideration of the inner feedback loop give comparable imulation reult. ISSN: ISBN:

6 Proceeding of the 7th WSEAS International Conference on SYSTEM SCIENCE and SIMULATION in ENGINEERING (ICOSSSE '8) Fig.4 - Step etpoint and load diturbance repone Controller: (9), (2); Controller3: (26); Controller4: (27);, τ 5, λ λ. 4, q α.736, γ.5, d Concluion In thi contribution, the problem of algebraic control deign in the ring of table and proper RQ meromorphic function for integrating time delay procee ha been invetigated. The propoed method doe not involve the delay approximation. The controller tructure i derived through the olution of the Bézout equation with Youla- učera parameterization. The methodology enable to find variou controller that atify requirement on cloed loop tability, tep reference tracking and tep load diturbance attenuation. The control ytem combine conventional DOF and 2DOF cheme and it i conceived a a inner (pre-tabilizing) feedback loop plu outer loop. The final controller are tuned uing dominant pole aignment method. The efficiency and uability of the propoed method i verified on a imulation example. 4 th World Congre of IFAC, Beiing, P. R. China, 999, pp [6 W. D. Zhang, H. Wang, W. Wang, X. Xu, Novel controller for integrating and table procee with long time delay, In: Proc. of 5 th World Congre of IFAC, Barcelona, Spain, 22, pp [7 M. Flie, R. Marque, H. Mounier, An extenion of predictive control, PID regulator and Smith predictor to ome linear delay ytem, Int. J. Control, Vol. 75, 22, pp [8 P. Dotál, F. Gazdoš, V. Bobál, Deign of controller for procee with time delay by polynomial method, In: Proc. of the European Control Conf., o, Greece, 27. [9 R. Prokop, J. P. Corriou, Deign and analyi of imple robut controller, Int. J. Control, Vol. 66, 997, pp [ V. učera, Diophantine equation in control - a urvey, Automatica, Vol. 29, 993, pp [ P. Zítek, V. učera, Algebraic deign of aniochronic controller for time delay ytem, Int. J. Control, Vol. 76, 23, pp [2 L. Pekař, R. Prokop, Deign of Controller for Delayed Integration Procee Uing RMS Ring. In: 6 th Mediterranean Conference on Control and Automation, Ajaccio-Corica, France, 28, pp [3 P. Zítek, V. učera, T. Vyhlídal, Meromorphic Stabilization and Control of Time Delay Sytem, Preprint of 5 th World Congre of IFAC [CD- ROM, Prague, Czech Republic, 25. [4 P. Zítek, A. Víteček, The deign of control of ubytem with delay and nonlinearitie (in Czech). Prague, Czech Republic, ČVUT publihing, 999. Reference: [ Q-C. Zhong, L. Mirkin, Control of integral procee with dead-time. 2. Quantitativeanalyi, IEE Proc. Control Theory Appl., Vol.49, No.4, 22, pp [2 W. D. Zhang, X. M. Xu, Quantitative performance deign for integrating procee with time delay, Automatica, Vol. 35, 999, pp [3 L. Wang, W. R. Cluett, Tuning of PID controller for integrating procee, IEE Proc. Control Theory Appl., Vol. 44, 997, pp [4 S. Majhi, D. P. Atherton, Modified Smith predictor and controller for procee with time delay, IEE Proc. Control Theory Appl., Vol. 46, 999, pp [5 Y. Tian, F. Gao, M. O. Tadé, J. Tang, Control of integrator procee with long dead-time, In: Proc. of ISSN: ISBN: