Design of Discrete-time Repetitive Control System Based on Two-dimensional Model

International Journal of Automation and Computing 9(2), April 212, 165-17 DOI: 1.17/s11633-12-629-1 Design of Discrete-time Repetitive Control System Based on Two-dimensional Model Song-Gui Yuan 1,2 Min Wu 1 Bao-Gang Xu 1 Rui-Juan Liu 1 1 School of Information Science and Engineering, Central South University, Changsha 4183, PRC 2 School of Electrical and Information Engineering, Changsha University of Science and Technology, Changsha 4176, PRC Abstract: This paper presents a novel design method for discrete-time repetitive control systems (RCS) based on two-dimensional (2D) discrete-time model. Firstly, the 2D model of an RCS is established by considering both the control action and the learning action in RCS. Then, through constructing a 2D state feedback controller, the design problem of the RCS is converted to the design problem of a 2D system. Then, using 2D system theory and linear matrix inequality (LMI) method, stability criterion is derived for the system without and with uncertainties, respectively. Parameters of the system can be determined by solving the LMI of the stability criterion. Finally, numerical simulations validate the effectiveness of the proposed method. Keywords: Linear systems, learning control, discrete-time repetitive control, two-dimensional (2D) systems, linear matrix inequality. 1 Introduction Periodic control tasks are common in industrial practice, such as robot handling, disk driver, rotating machinery and so on. To achieve the high control performance in such periodic control, Inoue et al. 1 proposed a specialized control strategy named repetitive control in 1981. Based on the internal model principle, repetitive control can track (or reject) periodic signals completely by constructing a periodic signal generator in the stabilized feedback system. Therefore, repetitive control has been widely concerned, and much effort has been devoted to theories 2 7 and industrial applications, including power electronics systems 8, disk driver systems 9, rotating machinery 1, and so on. Similar to iterative learning control (ILC), repetitive control is a branch of learning control, which improves system control performance periodically by introducing the learning ability 11,12. There are actually two dynamic behaviors in a repetitive control system (RCS), one is continuous control behavior within a repetitive period and the other is discrete learning behavior between periods. In order to explore and improve the performance of an RCS, it is important to consider such two behaviors fully in the design of an RCS. For continuous time RCSs, considering these two behaviors, Wu et al. 13 15 proposed a series of design methods based on the continuous-discrete two-dimensional (2D) model by using linear matrix inequality (LMI) method, which provide a simple and effective way for the RCS design. However, in practice, an RCS is mostly implemented digitally. By applying these methods in 13-15, the repetitive controller designed has to be discretized and discretization error will inevitably be introduced in the discretization process. To eliminate the discretization error and ensure system performance, design of the RCS directly in discrete-time domain has been concerned by many scholars 3, 4, 8 1. But these studies have not considered the control behavior and Manuscript received May 7, 211; revised July 21, 211 This work was supported by National Natural Science Foundation of China (Nos. 697445 and 667416) and the Research Foundation of Education Bureau of Hunan Province, China (No. 8C9). the learning behavior separately and are unable to regulate these two behaviors separately; moreover, the design procedure is tedious. To overcome the above disadvantages, with consideration of the above mentioned two behaviors, this paper presents a new design method for the discrete-time RCS based on 2D discrete-time model by employing two-dimensional system theory. Moreover, by applying LMI method, the design procedure becomes more simple. In the paper, the design problem of an RCS is formulated first; then, the 2D discrete model of the RCS is obtained. By changing the design problem of the RCS into the state feedback control design problem of a 2D system, separation of the control behavior and the learning behavior is realized. Then, robust stability condition and the controller design method are obtained by using the 2D system theory and LMI method. Finally, a numerical simulation is given to demonstrate the validity of the proposed method. Notations: R n denotes the n-dimensional real space; R n m and I denote the set of n m real matrices and the unit matrix with appropriate dimensions, respectively. For the symmetric matrices, denotes generically each of its symmetric blocks. The symbol X > (X < ) and diagw 1, W 2} stand for positive (negative) definite matrix and diagonal matrix with W i as its i-th diagonal element, respectively. 2 Problem description Consider the single-input single-output (SISO) discrete RCS shown in Fig. 1. r(k), e(k), and v(k) are periodic reference input, output error and repetitive controller output, respectively. L (= T/T s) in the repetitive controller is the sampling times per period of r(k), where T is the fundamental period of r(k) and T s is the sampling period. F e is the feed-forward gain and F p is the state feedback gain. Consider the discrete-time control plant with time-varying structured uncertainties as follows: x(k + 1)=(A + A(k))x(k) + (B + B(k))u(k) y(k) = Cx(k) + Du(k) (1)

166 International Journal of Automation and Computing 9(2), April 212 Fig. 1 Configuration of the repetitive control system where x(k) R n is the state of the plant, u(k) R is the control input, y(k) R is the output of the system, A R n n, B R n 1, C R n and D R are system matrices. And uncertainties of the plant are A(k) B(k) = H Γ(k) E 1 E 2 (2) where H, E 1 and E 2 are known constant matrices, and Γ(k) R n n is an uncertain time-varying matrix with Lebesgue measurable element which satisfies Γ T (k)γ(k) I. With an abuse of notation, we will write A, B, and Γ instead of A(k), B(k), and Γ(k) in the following. When A = and B =, (1) is the nominal control plant. The design problem of the RCS in Fig. 1 can be described as following: find a suitable feed-forward gain F e and a state feedback gain F p to ensure the robust stability of the system and make the system achieve required control performance, including convergence speed and steady-state error, etc. Through analyzing the RCS in Fig. 1, it can be seen that, by introducing the repetitive controller, the output error e(k L) in the former period is recorded and then it is introduced in the current period. In this way, the output v(k) of the repetitive controller and the control input u(k) are modified in sequence, and thus the output error e(k) is further reduced periodically. It is obvious that the control input u(k) is decided by the current state x(k), the current periodical output error e(k), and the former period output error e(k L). Therefore, the performance of the RCS is dependent on the learning behavior and the control behavior together. In order to make better use of these two kinds of behavior to realize satisfactory performance, it is necessary to describe and regulate them independently. To this end, we build a 2D discrete model of the RCS in Fig. 1 and transform the design problem of the RCS into the design problem of a 2D discrete system. The transform process is stated as follows: The output of the repetitive controller is v(k) = v(k L) + e(k) (3) where the output error is e(k) = r(k) y(k). The control law of the system is u(k) = F ev(k) + F px(k). (4) Define x(k + 1) = x(k + 1) x(k + 1 L) = (A + A) x(k) + (B + B) u(k) (5) e(k) = e(k L) C x(k) D u(k) (6) u(k) = u(k) u(k L) = F e v(k) + F p x(k) = F ee(k) + F p x(k). Introduce variables p ( ) and q ( L) to demonstrate the learning process and control process, respectively, where p is the iteration times of the learning behavior and q is the discrete time variable. For a variable ξ, define ξ(p, q) = ξ(pl + q) = ξ(k) ξ(p 1, q) = ξ((p 1)L + q) = ξ(k L). From (5), (6), and (7), the 2D model of the RCS is x(p, q + 1) = e(p, q) A + A x(p, q) B + B + u(p, q). C 1 e(p 1, q) D Combining (6) and (7) yields where u(p, q) = K p K e x(p, q) e(p 1, q) F e K e = 1 + DF e Fp FeC K p =. 1 + DF e (7) (8) (9) (1) Substitute (9) into (8), we obtain a closed loop 2D system x(p, q + 1) = e(p, q) (A + A) + (B + B)K p (B + B)K e x(p, q) C DK p 1 DK e e(p 1, q) (11)

S. G. Yuan et al. / Design of Discrete-time Repetitive Control System Based on Two-dimensional Model 167 where x(p, q + 1) and e(p, q) are the states of the 2D discrete system (11). Thus, the RCS design problem is converted to the 2D state feedback controller (9) design problem of the 2D discrete system (11). For most of the existing RCS design methods, K p and K e are mixed in F p and F e, which makes it difficult to adjust these two behaviors. In this paper, the control process x(p, q+1) and the learning process e(p, q) can be described separately, so the control behavior and the learning behavior can be regulated separately by adjusting the parameters of the feedback controller (9). In the following, by applying the 2D system theory and the LMI method, the stability criteria and the design methods are derived for the nominal system and uncertain system, respectively. Three necessary lemmas cited from 16 18 respectively are given first. Lemma 1 16. Consider the 2D discrete-time system x(i, j + 1) = Ax(i, j) + B y(i 1, j) + Bu(i, j) (12) y(i, j) = Cx(i, j) + D y(i 1, j) + Du(i, j) if there exist positive definite matrices P > and Q > such that the following LMI ÂT 1 P Â1 + Q P ÂT 1 P Â2 Â T 2 P Â2 Q < holds, then system (12) is stable when u(i, j) =, where A B Â 1 =, Â 2 =. C D Lemma 2 (Schur complement) 17. For any given constant symmetric matrix Σ = Σ T, the following assertions are equivalent: Σ 11 Σ 12 1) Σ = < ; Σ 22 2) Σ 11 <, Σ 22 Σ T 12Σ 1 11 Σ12 < ; 3) Σ 22 <, Σ 11 Σ 12Σ 1 22 ΣT 12 <. Lemma 3 18. For matrices with appropriate dimensions Q = Q T, U, V, W, and R = R T >, if and only if there exists a constant ɛ (> ) such that Q+ɛUU T +ɛ 1 W T RW < holds, then Q+UV W +(UV W ) T < holds for all matrices V satisfying V T V R. 3 RCS design for nominal system The 2D closed-loop system with nominal plant (1) is x(p, q + 1) = e(p, q) (13) A + BK p BK e x(p, q). C DK p 1 DK e e(p 1, q) According to Lemma 1, we have the following theorem. Theorem 1. If there exist positive definite matrices P > and Q > such that the following LMI ÃT 1 P Ã1 + Q P ÃT 1 P Ã2 Ã T 2 P Ã2 Q < (14) holds, then the closed-loop system (13) is asymptotically stable, that is, the control law u(k) = K e Kp + KeC v(k) + x(k) (15) 1 DK e 1 DK e ensures the stability of the RCS in Fig. 1, where Ā 1 = Ã 1 = Ā1 + B 1 K, Ã2 = Ā2 + B 2 K, A, Ā 2 =, B1 = C 1 B 2 =, K = K p K e. D Proof. Reformulate the closed-loop system (13) in the form of (12), we have x(p, q + 1) = (A + BK p) x(p, q) + BK ee(p 1, q) e(p, q) = ( C DK e) x(p, q) + (1 DK e)e(p 1, q). Based on Lemma 1, if LMI (14) holds, then system (13) is stable. Simultaneously, according to (4) and (1), control law (15) is obtained. Since Theorem 1 requires that the parameters of state feedback controller (9) be known and repeatedly tuning is needed to achieve satisfactory performance, it is tedious to determine the parameters by using Theorem 1. Therefore, to simplify the design process, the following theorem is given. Theorem 2. For the nominal control plant (1), if there exist positive definite matrices P > and Q >, and a matrix N with appropriate dimensions such that the following LMI Z Y Ω 13 Z Ω 23 < (16) holds, where Ω13 = Y ĀT 1 + N T BT 1 Ω 23 = Y ĀT 2 + N T BT 2 then the state feedback controller (9) guarantees that the closed-loop system (13) is asymptotically stable, and the gain of controller (9) is K = K p K e = NY 1. (17) Proof. By applying Lemma 2, matrix inequality (14) is equivalent to the following matrix inequality Q P Ã T 1 P Q ÃT 2 P <. (18) P Define Y = P 1, Z = Y Q Y, N = KY and multiply the left and right hand sides of (18) by diagy, Y, Y }, then (18) is equivalent to LMI (16). Therefore, according to Theorem 1, system (13) is stable when LMI (16) holds. B,

168 International Journal of Automation and Computing 9(2), April 212 Remark 1. Parameters of the RCS can be determined easily by using Theorem 2. Firstly, solve LMI (16) for matrices Y, Z, and N; then, substitute Y and N into (17) to obtain the feedback gain, K = Kp K e ; finally, calculate F = F p F e from (1). Remark 2. Theorem 2 provides a sufficient condition for the convergence of the steady-state error in the RCS. Compared with the complex stability conditions in 19 21, the sufficient condition in Theorem 2 is simple, and it is easy to use the sufficient condition to validate the stability of the system using Matlab LMI toolbox. A state feedback controller has also been used in 22, but repeatedly tuning of the parameters is required as the linear quadratic regulator method was adopted to design the controller. 4 RCS design for uncertain system For the control plant with time-varying uncertainties (1), the following theorem gives the design method of the control law (9) to ensure the robust stability of the closed-loop system (11). Theorem 3. For the plant with time-varying uncertainties (1), if there exist positive definite matrices Y > and Z >, a matrix N with appropriate dimensions, and positive constants ε 1 and ε 2 such that the following LMI Z Y Y ĀT + N 1 T B 1 T Y ĒT 1 N TĒT 2 Z Y ĀT + N 2 T B 2 T + ɛ 1 H HT + ɛ 2 H HT ɛ 1I ɛ 2I < (19) holds, then state feedback controller (9) guarantees the robust stability of the closed-loop system (11), that is, the RCS in Fig. 1 is robustly stable, and correspondingly the controller gain is K = K p K e = NY 1. (2) Proof. According to Theorem 2, if there exist positive definite matrices Y > and Z >, and a matrix N with appropriate dimensions such that the following matrix inequality holds, where Z Y Ω13 Z Ω 23 < (21) Ω 13 = Y (Ā1 + Ā1)T + N T ( B 1 + B 1) T, Ā1 = A, B 1 = B then the 2D state feedback controller (9) guarantees the robust stability of system (11). Inequality (21) can be decomposed into Z Y Ω13 Z Ω 23 = Z Y Ω 13 Z Ω 23 + Y ĀT 1 + N T B 1 T and the second matrix on the right of the above equation can be written as Y ĀT 1 + N T B 1 T = where with Ẽ 1 = H ΓẼ1 + ( H ΓẼ1)T + H ΓẼ2 + ( H ΓẼ2)T H = H Ē 1Y H = Ē 1 = H, Γ = E 1, Ẽ 2 = Γ Γ Γ, Ē 2N Γ, Γ =, Γ E 2, Ē 2 =., According to Lemma 3, if there exist positive constants ε 1 and ɛ 2 such that the following matrix inequality Z Y Ω 13 Z Ω 23 + (22) ɛ 1 H HT + ɛ 1 1 Ẽ1 T Ẽ1 + ɛ2 H H T + ɛ 1 2 Ẽ2 T Ẽ2 < holds, then matrix inequality (21) holds. And by using Lemma 2, matrix inequality (22) is equivalent to LMI (19). Therefore, if LMI (19) is feasible, then the 2D state feedback controller (9) guarantees the robust stability of system (11). Remark 3. Based on the solutions of LMI (19), according to (2) and (1) we can obtain the parameters of the RCS, F = F p F e, which guarantees the robust stability of the RCS. Remark 4. For the system with time-varying uncertainties (2), Theorem 3 provides an RCS design method while the methods in 19 22 are no longer applicable in this case. 5 Numerical simulations Assume the control plant (1) has the following parameters.93 A =, B =, C = 1,.3.82.6

S. G. Yuan et al. / Design of Discrete-time Repetitive Control System Based on Two-dimensional Model 169 1 2 H =, E 1 =, E 2 =,.2.6 1 sin 2πk Γ = 2 sin 2πk 2 and the sampling period is T s =.5 s, consider the tracking of the periodic reference input with fundamental period T = 1 s r(k) = sin 2πk 4πk 6πk +.5sin +.5sin 2 2 2. (23) According to (23), L = T/T s = 2. For the nominal plant (1), by using Theorem 2, the gains of 2D controller (9) are K p =.9837.245 and K e =.791. Consequently, parameters of the RCS are F p =.9223 1.1673 and F e = 3.7651. Simulation results are shown in Fig. 2. It can be seen that the RCS converges to steady-state after 3 periods and the steadystate error converges to zero. Thus, high accurate tracking of the reference input is achieved. Fig. 2 Simulation results for the nominal system For the uncertain plant (1), by using Theorem 3, the gains of controller (9) are K p =.453.2851 and K e =.3584. Correspondingly, parameters of the RCS are F p =.1474.4443 and F e =.5586. Fig. 3 shows the simulation results. It is clear that for admissible uncertainties (2), the designed RCS achieves robust stability and converges to steady-state after 6 periods while achieving tracking of the reference input accurately. Therefore, the proposed design method for the RCS based on 2D model can deal with the system without and with uncertainties effectively. And the resulted RCS is able to achieve high control performance. Compared with the methods in 19, 21, feed-forward controller is not necessary here to achieve high control performance. Fig. 3 6 Conclusion Simulation results for the uncertain system This paper has proposed a novel design method for a class of discrete-time RCSs with time-varying uncertainties based on 2D model. The basic idea of the method is: establish the 2D model of the RCS and then convert the design of the RCS into the design of a 2D system with a 2D state feedback controller. This method has achieved the separation of the control action and the learning action in the RCS. Stability criterion has been derived for the system without and with uncertainties, respectively. By solving the LMI of the stability criterion, parameters of the system can be determined directly. The structure of the RCS in this paper is simple, and by applying the proposed method, parameters of the RCS which ensures satisfactory performance can be determined easily and system uncertainties can be effectively dealt with. A numerical example has been given to validate the effectiveness and advantages of the proposed method. Future study will be focused on how the learning behavior and the control behavior affect the error convergence speed and steady-state error, respectively. Then, we will extend these results to the control plant without the direct path, i.e., the feed-forward matrix D =, to widen the application of this method. References 1 T. Inoue, M. Nakano, S. Iwai. High accuracy control of servomechanism for repeated contouring. In Proceedings of the 1th Annual Symposium on Incremental Motion Control System and Devices, Chicago, USA, pp. 258 292, 1981. 2 S. Hara, Y. Yamamoto, T. Omata, M. Nakano. Repetitive control system: A new type servo system for periodic exogenous signals. IEEE Transactions on Automatic Control, vol. 33, no. 7, pp. 659 668, 1988. 3 M. Tomizuka, T. C. Tsao, K. K. Chew. Analysis and synthesis of discrete-time repetitive controllers. ASME Transactions of Dynamic Systems, Measurement, and Control, vol. 111, no. 3, pp. 353 358, 1989.

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Srinivasan. Discrete-time repetitive control systems design using the regeneration spectrum. ASME Transactions of Dynamic Systems, Measurement, and Control, vol. 115, no. 2A, pp. 228 237, 1993. 2 C. Smith, M. A. Tomizuka. A cost effective repetitive control and its design. In Proceedings of American Control Conference, IEEE, Chicago, USA, vol. 2, pp. 1169 1174, 2. 21 C. M. Chang, T. S. Liu. Application of discrete wavelet transform to repetitive control. In Proceedings of American Control Conference, IEEE, Anchorage, USA, vol. 6, pp. 456 4565, 22. 22 J. H. She, Y. Pan, M. Nakano. Repetitive control system with variable structure controller. In Proceedings of the 6th International Workshop on Variable Structure Systems, Gold Coast, Australia, pp. 273 282, 2. Song-Gui Yuan received his B. Sc. degree in automation from Xiangtan University in 1993, and M. Sc. degree in Technology of Computer Application from Changsha University of Science and Technology, PRC in 23. In 1998, he was a lecturer at Changsha Communications Institute. Currently, he is an assistant professor at the School of Electrical and Information Engineering, Changsha University of Science and Technology, PRC. His research interests include robust control and repetitive control theory. E-mail: cscuysg@yahoo.com.cn Min Wu received his B. Sc. and M. Sc. degrees in engineering from Central South University, Changsha, PRC in 1983 and 1986, respectively. He received the Ph. D. degree in engineering from Tokyo Institute of Technology, Japan in 1999. Since July 1986, he has been with Central South University, where he is currently a professor of automatic control engineering at the School of Information Science and Engineering. He was a visiting scholar in the Department of Electrical Engineering, Tohoku University, Japan from 1989 to 199, a visiting research scholar in the Department of Control and Systems Engineering, Tokyo Institute of Technology, Tokyo from 1996 to 1999, and a visiting professor at the School of Mechanical, Materials, Manufacturing and Management, University of Nottingham, England from 21 to 22. He is an IEEE senior member and a member of Nonferrous Metals Society of China and China Automation Association. He received the Best Paper Award at the International Federation of Automatic Control in 1999 (jointly with M. Nakano and J. H. She). His research interests include robust control and its application, process control and intelligent control. E-mail: min@csu.edu.cn (Corresponding author) Bao-Gang Xu received his B. Sc. degree in engineering from the Central South University, PRC in 29. Currently, he is a master student in the Department of Control Engineering at the School of Information Science and Engineering, Central South University, PRC. His research interests include mechatronics and application of control theory. E-mail: xubaogangcsu@csu.edu.cn Rui-Juan Liu received her B. Sc. and M. Sc. degrees in mathematics from Changsha University of Science and Technology, PRC in 24 and 28, respectively. Currently, she is a Ph. D. candidate at the School of Information Science and Engineering, Central South University, PRC. Her research interests include robust control and its application. E-mail: liuruijuan@csu.edu.cn