A Design Method for Smith Predictors for Minimum-Phase Time-Delay Plants

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1 00 ECTI TRANSACTIONS ON COMPUTER AND INFORMATION TECHNOLOGY VOL., NO.2 NOVEMBER 2005 A Design Method for Smith Predictors for Minimum-Phase Time-Delay Plants Kou Yamada Nobuaki Matsushima, Non-members ABSTRACT In this paper, we examine a design method for a modified Smith predictor for minimum-phase timedelay plants. The modified Smith predictor is well known as an effective time-delay compensator for a plant with large time delays, several papers on the modified Smith predictor have been published. However, the parameterization of all stabilizing modified Smith predictors has not been obtained. If this can be obtained, we can express existing proposals for modified Smith predictors in a uniform manner, the modified Smith predictor can be designed systematically. The purpose of this paper is to propose the parameterization of all stabilizing modified Smith predictors for minimum-phase time-delay plants. The control characteristics of the control system using the parameterization of all stabilizing modified Smith predictors are also given. Finally, numerical examples for stable plants unstable plants are illustrated to show the effectiveness of the proposed parameterization. Keywords: Minimum-Phase System, Time-Delay System, Smith Predictor, Parameterization. INTRODUCTION In this paper, we examine a design method for Smith predictors for minimum-phase time-delay plants. Proposed by Smith to overcome time delays [], it is well known as an effective time-delay compensator for a stable plant with large time delays [ 2]. The Smith predictor in [] cannot be used for plants having an integral mode, because a step disturbance will result in a steady state error [2 4]. To overcome this problem, Watanabe Ito [4], Astrom, Hang Lim [9], Matusek Micic [0] proposed a design method for a modified Smith predictor for time-delay plants with an integrator. Watanabe Sato exped the result in [4] proposed a design method for modified Smith predictors for multivariable systems with multiple delays in inputs outputs [5]. Because the modified Smith predictor cannot be Manuscript received on July 30, 2005 ; revised on August 22, The authors are with Department of Mechanical System Engineering, Gunma University -5- Tenjincho, Kiryu Japan. yamada@me.gunma-u.ac.jp used for unstable plants [2 ], De Paor [6], De Paor Egan [8] Kwak, Sung, Lee Park [2] proposed a design method for modified Smith predictors for unstable plants. Thus, several design methods of modified Smith predictors have been published. On the other h, another important control problem is the parameterization problem, the problem of finding all stabilizing controllers for a plant [3 2]. This problem was considered in [20, 2]. However, the parameterization of all stabilizing modified Smith predictors has not been obtained. If this result could be obtained, we could express previous studies of modified Smith predictors in a uniform manner. In addition, modified Smith predictors could be designed systematically. The purpose of this paper is to propose the parameterization for minimum-phase time-delay plants. First, the structure necessary characteristics of modified Smith predictors described in past studies in [ 2] are defined. Next, the parameterization of all stabilizing modified Smith predictors for minimumphase time-delay plants is proposed, for both stable unstable plants. The control characteristics of the control systems using this parameterization are also given. Finally, a numerical example is presented to show the effectiveness of the proposed parameterization. This paper is organized as follows: In Section 2. the Smith predictor is introduced briefly the problem considered in this paper is explained. In Section 3. Section 4., the parameterizations of all stabilizing modified Smith predictors for stable unstable plants are given, respectively. In Section 3. Section 4., we clarify the control characteristics using the parameterization of all stabilizing modified Smith predictors. Simple numerical examples are illustrated in Section 5. Notation R The set of real numbers. Rs The set of real rational functions with s. RH The set of stable proper real rational functions. H The set of stable causal functions. U The set of unimodular functions on RH. That is, Us U implies both Us RH U s RH. R{ } The real part of { }.

2 A design method for Smith predictors for minimum-phase time-delay plants 0 2. MODIFIED SMITH PREDICTOR Consider the control system: { y = Gse u + d u = Cs r y, where Gse is the single-input/single-output time-delay plant with time-delay T > 0, Cs is the controller, y R is the output, u R is the input, d R is the disturbance r R is the reference input. Gs is assumed to be coprime of minimum phase, that is, Gs has no zeros in the closed right half plane. According to [ 2], the modified Smith predictor Cs is decided by the form: C s, 2 + C 2 se where C s Rs C 2 s Rs. In addition, using the modified Smith predictor in [ 2], the transfer function from r to y of the control system in, written as y = CsGse r 3 + CsGse has a finite number of poles. That is, the transfer function from r to y of the control system in is written as y = Ḡse r, 4 where Ḡs RH. Therefore, we call Cs the modified Smith predictor if Cs takes the form of 2 the transfer function from r to y of the control system in has a finite number of poles. The problem considered in this paper is to obtain the parameterization of all modified Smith predictors Cs that make the control system in stable. In Section 3., we propose the parameterization of all stabilizing modified Smith predictors Cs for stable plants. In Section 4., we exp the result in Section 3. propose the parameterization of all stabilizing modified Smith predictors Cs for unstable plants. 3. THE PARAMETERIZATION OF ALL STABILIZING MODIFIED SMITH PRE- DICTORS FOR STABLE PLANTS The parameterization of all stabilizing modified Smith predictors for the stable plant Gse is summarized in the following theorem. Theorem : Gse is assumed to be stable. The parameterization of all stabilizing modified Smith predictors Cs takes the form Qs, 5 QsGse where Qs RH is any function. Proof: First, the necessity is shown. If the controller Cs in 2 makes the control system in stable makes the transfer function from r to y of the control system in have a finite number of poles, then Cs takes the form of 5. From the assumption that the controller Cs in 2 makes the transfer function from r to y of the control system in have a finite number of poles, CsGse + CsGse = C sgse + C 2 s + C sgs e 6 has a finite number of poles. This implies that is necessary, that is: C 2 s = C sgs 7 C s. 8 C sgse From the assumption that Cs in 2 makes the control system in stable, CsGse / + CsGse, Cs/+CsGse, Gse /+ CsGse / + CsGse are stable. From simple manipulation 7, we have CsGse + CsGse = C sgse, 9 Cs + CsGse = C s, 0 Gse + CsGse = C sgse Gse + CsGse = C sgse. 2 It is obvious that the necessary condition for all the transfer functions in 9, 0, 2 to be stable is C s RH. Using Qs RH, let C s be C s = Qs, 3 we find that Cs takes the form of 5. Thus, the necessity has been shown. Next, the sufficiency is shown. If Cs takes the form of 5, then the controller Cs makes the control system in stable makes the transfer function from r to y of the control system in have a finite number of poles. From simple manipulation, we have CsGse + CsGse = QsGse, 4

3 02 ECTI TRANSACTIONS ON COMPUTER AND INFORMATION TECHNOLOGY VOL., NO.2 NOVEMBER 2005 Cs = Qs, 5 + CsGse Gse + CsGse = QsGse Gse 6 + CsGse = QsGse. 7 From the assumption that Gse is stable Qs RH, 4, 5, 6 7 are all stable. In addition, because the transfer function from r to y of the control system in takes the form 4 Q RH, the transfer function from r to y of the control system in has a finite number of poles. We have thus proved Theorem. Note : Note that because the proof of Theorem does not require the assumption that Gs is of minimum phase, even if Gse is of non-minimum phase, the parameterization of all stabilizing modified Smith predictors is given by Theorem. Next, we explain the control characteristics of the control system using the parameterization of all stabilizing modified Smith predictors in 5. The transfer function from the reference input r to the output y of the control system in takes the form y = QsGse r. 8 Therefore, for the output y to follow the step reference input r = /s without steady state error, Q0G0 = 9 must be satisfied. Qs is chosen by where Qs = qs Gs, 20 qs = + sτ α, 2 where τ > 0 α is a positive integer that makes Qs in 20 proper. The disturbance attenuation characteristics are as follows. The transfer function from the disturbance d to the output y of the control system in is given by y = QsGse d. 22 Therefore, to attenuate the step disturbance d = /s effectively, Qs must satisfy Q0G0 =. 23 That is, when Qs is chosen according to 20, the control system in 20 can attenuate not only the step disturbance d effectively but also a disturbance with frequency component ω satisfying qjωe jωt THE PARAMETERIZATION OF ALL STABILIZING MODIFIED SMITH PRE- DICTORS FOR UNSTABLE PLANTS In this section, we exp the result in Section 3. propose the parameterization of all stabilizing modified Smith predictors Cs for unstable minimum phase plants. This parameterization is summarized in the following theorem. Theorem 2: Gse is assumed to be unstable to be of minimum phase. For simplicity, the unstable poles of Gse are assumed to be distinct. That is, when s i i =,, n denote unstable poles of Gs, s i s j i j; i =,, n; j =,, n. Under these assumptions, there exists Ḡus U satisfying Ḡ u s i =, 25 sit G s s i e where G s s is a stable minimum-phase function of Gs, that is, when Gs is factorized as Gs = G u sg s s, 26 G u s is the unstable biproper minimum phase function G s s is the stable minimum phase function. Using these functions, the parameterization of all stabilizing modified Smith predictors Cs is written as where C f s is given by C f s, 27 C f sgse C f s = Ḡus + Qs G u s G u s 28 Qs RH is any function. The proof of Theorem 2 requires the following Lemma. Lemma : Gs is assumed to be unstable to be of minimum phase. For simplicity, the unstable poles s i i =,, n of Gse are assumed to be distinct. Under these assumptions, there exists Ḡ u s U satisfying 25, where G s s is the stable function of Gs. Proof: From the assumption that Gs is of minimum phase, G s s is also of minimum phase. Therefore, for all s i on the real axis, / G s s i e sit are all the same sign. From Theorem in [6], there exists Ḡus U satisfying 25. We have thus proved Lemma. Using this Lemma, we shall show the proof of Theorem 2. Proof: First, the necessity is shown. If the controller Cs in 2 makes the control system in stable makes the transfer function from r to y of the control system in have a finite number of

4 A design method for Smith predictors for minimum-phase time-delay plants 03 poles, then Cs takes the form 27. From the same assumption, CsGse + CsGse = C sgse + C 2 s + C sgs e 29 has a finite number of poles. This implies that C 2 s = C sgs 30 is satisfied, that is, Cs is necessarily C s. 3 C sgse From the assumption that Cs in 2 makes the control system in stable, CsGse / + CsGse, Cs/+CsGse, Gse /+ CsGse / + CsGse are stable. From simple manipulation 30, we have CsGse + CsGse = C sgse, 32 Cs + CsGse = C s, 33 Gse + CsGse = C sgse Gse 34 + CsGse = C sgse. 35 It is obvious that the necessary condition for all the transfer functions in 32, to be stable is that C sgs is stable. This implies that C s must take the form C s = C s G u s, 36 where C s RH. From the assumption that the transfer function in 34 is stable from 36, for s i i =,, n, which are the unstable poles of Gs, C s i Gs i e sit = C s i G s s i e sit = 0 i =,, n 37 must be satisfied. From Lemma, there exists Ḡ u s U satisfying Ḡus i G s s i e sit = 0 i =,, n.38 Note that the condition in 38 is equivalent to 25, because G s s the stable minimum phase function, that is G s s i 0. Using Ḡus U satisfying 38, C s is rewritten as C s = Ḡus + C s Ḡus Ḡ u s. 39 Because Ḡus U C s RH, C s Ḡ u s/ḡus is stable. In addition, because C s Ḡ u s/ḡus takes the form C s Ḡus Ḡ u s = C s Ḡ u s 40 both C s /Ḡus are proper, C s Ḡ u s/ḡus is proper. Therefore, C s Ḡ u s/ḡus RH. From 37 38, C s i Ḡus i = 0 i =,, n 4 holds. This implies that s i i =,, n, which are zeros of G u s, are zeros of C s Ḡus, because Ḡu U C s RH. When we rewrite C s Ḡus/Ḡus as C s Ḡus Ḡ u s = Qs G u s, 42 then Qs RH, because /G u s RH. In this way, it is shown that if the controller Cs in 2 makes the control system in stable makes the transfer function from r to y of the control system in have a finite number of poles, then Cs is written as 27. Next, the sufficiency is shown. If Cs takes the form 27, then the controller Cs makes the control system in stable makes the transfer function from r to y of the control system in have a finite number of poles. After simple manipulation, we have CsGse + CsGse = Ḡus + Qs G s se, 43 G u s Cs Ḡus = + CsGse G u s + Qs, 44 G u s Gse + CsGse { = Ḡus + Qs } G s se Gse G u s 45 + CsGse = Ḡus + Qs G u s G s se. 46

5 04 ECTI TRANSACTIONS ON COMPUTER AND INFORMATION TECHNOLOGY VOL., NO.2 NOVEMBER 2005 Because Ḡus U, Qs RH, G s s RH, the transfer functions in 43, are stable. If the transfer function in 45 is unstable, the unstable poles of the transfer function in 45 are unstable poles of Gs. From the assumption that Ḡus satisfies 25, the unstable poles of Gs are not the poles of { Ḡus + Qs G u s G s se } Gse. Therefore, the transfer function in 45 is stable. In addition, because the transfer function from r to y of the control system in is given by 43 Ḡus, Qs, /G u s RH, the transfer function from r to y of the control system in has a finite number of poles. We have thus proved Theorem 2. Note 2: Ḡ u s satisfying 25 is obtained using the method given in the proof of Theorem in [6]. The modified Smith predictor in 27 is explained based on the frequency domain. In the time domain, using the modified Smith predictor in 27, the control input ut is given as follows. From the assumption that the unstable poles of Gse are distinct, from 25, G u s Ḡus+Qs/G u sgs can be rewritten as G u s = i= Ḡ u s + Qs Gs = G u s respectively, where c i s s i + δ 47 i= c i e sit s s i + Qs, 48 c i = s s i G u s s=si i =,..., n. 49 Qs RH holds true because the unstable poles of Ḡ u s + Qs/G u sgs are equal to s i i =,..., n. δ 0 R is satisfied, because G u s is biproper. Using δ, Qs ci i =,..., n in 47, 48 49, the control input ut is given by ut 0 = δ Qput T δ + δ Ḡup + Qp G u p T i= c i e siτ ut + τdτ rt yt, 50 where p is the differential operator, i.e. pyt = dyt dt. The control input ut in 50 is obviously realizable, because Qs RH, R{s i } 0i =,..., n, Ḡ u s + Qs/G u s RH ut + τ, T τ 0, is the past history of the control input over the finite interval T. The fact that the control input ut in 50 is written in the frequency domain as Cs in 27 is confirmed as follows: Taking the Laplace transformation of 50 yields δus = Qse us 0 T i= c i e siτ use τs dτ + Ḡus + Qs rs ys G u s = Qse n c i us s s i= i us + Ḡus i= c i e sit s s i e + Qs G u s rs ys. 5 From 47, 48, 5 simple manipulation, we have Ḡ u s + Qs rs ys G u s us = G u s Ḡus + Qs Gse G u s C f s = rs ys. 52 C f sgse From the above equation, we find that the control input ut in 50 is written in the frequency domain as Cs in 27. Next, we explain the control characteristics of the control system using the parameterization of all stabilizing modified Smith predictors in 27. The transfer function from the reference input r to the output y of the control system in is written as y = Ḡus + Qs G s se r. 53 G u s Therefore, when Gs has a pole at the origin, for the output y to follow the step reference input r = /s without steady state error, Ḡ u 0G s 0 = 54 must be satisfied. Because Ḡus U satisfies 38, 54 holds true. This implies that when Gs has a pole at the origin, the output y follows the step reference input r without steady state error, independent of Qs RH in 27. On the other h, when Gs has no pole at the origin, for the output y to follow the step reference input r = /s without steady state error, Ḡ u 0 + Q0 G s 0 = 55 G u 0 must hold. From simple manipulation, if Qs is chosen satisfying Q0 = G u 0 Ḡ u 0G s 0, 56

6 A design method for Smith predictors for minimum-phase time-delay plants 05 then the output y follows the step reference input without steady state error. The disturbance attenuation characteristics are as follows. The transfer function from the disturbance d to the output y of the control system in is given by { y = Ḡus + Qs } G s se d. 57 G u s Therefore, to attenuate the step disturbance d = /s effectively, Qs must satisfy Ḡ u 0 + Q0 G s 0 =. 58 G u 0 That is, when Qs is chosen satisfying 56, the step disturbance d is attenuated effectively. 5. NUMERICAL EXAMPLE In this section, numerical examples for stable unstable plants are presented to show the effectiveness of the proposed parameterization of all stabilizing modified Smith predictors. Let us consider the problem of finding the parameterization for the stable plant Gse written as where Gse = Gs = s 2 + 3s + 4 e 4s, 59 s + s 2 + 3s T = 4[sec]. From Theorem, the parameterization for Gse in 59 is given by 5. For the output y to follow the step reference input without steady state error, Qs RH is chosen using 20, where qs is qs = + 0.s 2. 6 The response of the output y for the step reference input r is shown in Fig.. Figure shows that the control system is stable the output y follows the step reference input r without steady state error. Next we show a numerical example for an unstable plant. Let us consider the problem of finding the parameterization for the unstable plant Gse given by where Gse = s + s 2 + 3s e s, 62 Gs = s + s 2 + 3s 63 yt t[sec] Fig.: Step response T = [sec]. Gs is factorized by 26 as G s s = G u s = s + 2 s 64 s + s + 2s One Ḡus in 28 satisfying 25 is given by Ḡ u s = 62s + 3s From Theorem 2, the parameterization of all stabilizing modified Smith predictors for Gse in 62 is given by 27, where C f s = 6s2s + 3s + s + 2 Qs RH. choosing Qs RH as we have C f s = + s s + 2 Qs 67 Qs = s + 2 s + 3, 68 6s2s + 2s + 3 3s + s + 2s Because Gs in 63 has a pole at the origin, from the discussion in the preceding section, the output y follows the step reference input r without steady state error. The step response of the control system in using C f s in 69 is shown in Fig. 2. Figure 2 shows that the control system in is stable the output y follows the step reference input r without steady state error. In this way, we find that by using the results in this paper, we can easily obtain the parameterization for timedelay plants.

7 06 ECTI TRANSACTIONS ON COMPUTER AND INFORMATION TECHNOLOGY VOL., NO.2 NOVEMBER 2005 yt t[sec] Fig.2: 6. CONCLUSION Step response In this paper, we proposed the parameterization for minimum-phase time-delay plants. First, the parameterization for stable plants, which are not necessarily of minimum phase, was proposed. Next, we exped the result of the parameterization for stable plants proposed the parameterization of all stabilizing modified Smith predictors for unstable plants. The control characteristics of the control system using the parameterization design method of Qs in 5 27 were also given. Finally, numerical examples were presented to show the effectiveness of the proposed parameterization. References [] O.J.M. Smith, A controller to overcome deadtime, ISA Journal, Vol. 6, pp , 959. [2] S. Sawano, Analog study of process-model control systems, Journal of the Society of Instrument Control Engineers, Vol., pp , 962. [3] C.C. Hang, F.S. Wong, Modified Smith predictors for the control of processes with dead time, Proc. ISA Annual Conf., 979, pp ,. [4] K. Watanabe, M. Ito, A process-model control for linear systems with delay, IEEE Transactions on Automatic Control, Vol. 26, pp , 98. [5] K. Watanabe, M. Sato, A process-model control for multivariable systems with multiple delays in inputs outputs subject to unmeasurable disturbances, International Journal of Control, Vol. 39, pp. -7, 984. [6] A.M. De Paor, A modified Smith predictor controller for unstable processes with time delay, International Journal of Control, Vol. 4, pp. 025, 985. [7] P.B. Despe, R.H. Ash, Computer process control, ISA Pub., 988. [8] A.M. De Paor, R.P.K. Egan, Extension partial optimization of a modified Smith predictor controller for unstable processes with time delay, International Journal of Control, Vol. 50, pp. 35, 989. [9] K.J. Astrom, C.C. Hang, B.C. Lim, A new Smith predictor for controlling a process with an integrator long dead-time, IEEE Transactions on Automatic Control, Vol. 39, pp , 994. [0] M.R. Matusek, A.D. Micic, A modified Smith predictor for controlling a process with an integrator long dead-time, IEEE Transactions on Automatic Control, Vol. 4, pp , 996. [] K. Watanabe, A new modified Smith predictor control for time-delay systems with an integrator, Proceedings of the 2nd Asian Control Conference, 3 997, pp [2] H.J. Kwak, S.W. Sung, I.B. Lee, J.Y. Park, A modified Smith predictor with a new structure for unstable processes, Ind. Eng. Chem. Res., Vol. 38, pp , 999. [3] G. Zames, Feedback optimal sensitivity: model reference transformations, multiplicative seminorms approximate inverse, IEEE Transactions on Automatic Control, Vol. 26, pp , 98. [4] D.C. Youla, H. Jabr, J.J. Bongiorno, Modern Wiener-Hopf design of optimal controllers. Part I, IEEE Transactions on Automatic Control, Vol. 2, pp. 3-3, 976. [5] C.A. Desoer, R.W. Liu, J. Murray, R. Saeks, Feedback system design: The fractional representation approach to analysis synthesis, IEEE Transactions on Automatic Control, Vol. 25, pp , 980. [6] M. Vidyasagar, Control System Synthesis A factorization approach, MIT Press, 985. [7] M. Morari, E. Zafiriou, Robust Process Control, Prentice-Hall, 989. [8] J.J. Glaria, G.C. Goodwin, A parameterization for the class of all stabilizing controllers for linear minimum phase systems, IEEE Transactions on Automatic Control, Vol. 39, pp , 994. [9] K. Yamada, A parameterization for the class of all proper stabilizing controllers for linear minimum phase systems, Preprints of the 9th IFAC/IFORS/IMACS/IFIP/ Symposium on Large Scale Systems: Theory Applications, 200, pp [20] E. Nobuyama, T. Kitamori, Spectrum assignment parameterization of all stabilizing com-

8 A design method for Smith predictors for minimum-phase time-delay plants 07 pensators for time-delay systems, Proceedings of the 29th Conference on Decision Control, Honolulu, Hawaii, 990, pp [2] E. Nobuyama, T. Kitamori, Parameterization of all stabilizing compensators in time-delay systems, Transactions of the Society of Instrument Control Engineers, Vol. 27, pp. 5-22, 99. Kou Yamada was born in Akita, Japan, in 964. He received the B.S. M.S. degrees from Yamagata University, Yamagata, Japan, in , respectively, the Dr. Eng. degree from Osaka University, Osaka, Japan in 997. From 99 to 2000, he was with the Department of Electrical Information Engineering, Yamagata University, Yamagata, Japan, as a research associate. Since 2000, he has been an associate professor in the Department of Mechanical System Engineering, Gunma University, Gunma, Japan. His research interests include robust control, repetitive control, process control control theory for inverse systems infinite-dimensional systems. Dr. Yamada received 2005 Yokoyama Award in Science Technology. Nobuaki Matsushima was born in Gunma, Japan, in 98. He received the B.S. degree in Mechanical System Engineering from Gunma University, Gunma Japan, in He is currently M.S. cidate in Mechanical System Engineering at Gunma University. His research interests include process control control theory for infinite-dimensional systems.

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