Design of a Discrete-time Output-feedback Based Repetitive-control System

International Journal of Automation and Computing, Aug 2013

This paper deals with the problem of designing a robust discrete output-feedback based repetitive-control system for a class of linear plants with periodic uncertainties. The periodicity of the repetitive-control system is exploited to establish a two-dimensional (2D) model that converts the design problem into a robust stabilization problem for a discrete 2D system. By employing Lyapunov stability theory and the singular-value decomposition of the output matrix, a linear-matrix-inequality (LMI) based stability condition is derived. The condition can be used directly to design the gains of the repetitive controller. Two tuning parameters in the LMI enable the preferential adjustment of control and learning. A numerical example illustrates the design procedure and demonstrates the validity of the method.

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Design of a Discrete-time Output-feedback Based Repetitive-control System

Lan Zhou 2 3 4 Jin-Hua She 1 3 4 Min Wu 0 3 4 0 School of Information Science and Engineering, Central South University , Changsha 410083, China 1 School of Computer Science, Tokyo University of Technology , Tokyo 192 2 School of Information and Electrical Engineering, Hunan University of Science and Technology , Xiangtan 411201, China 3 Min Wu received the B. Sc . and M. Sc . degrees in engineering from Central South University , China in 1983 and 1986, respec- tively, and the Ph. D. degree in engineering from Tokyo Institute of Technology , Japan in 1999. Since 1986, he has been a fac- ulty member with Central South Univer- sity, where he is currently a professor of au- tomatic control engineering with the School of Information Science and Engineering. He was a visiting scholar at the Department of Electrical Engineer- ing, Tohoku University , Japan from 1989 to 1990, a visiting re- search scholar at the Department of Control and Systems En- gineering, Tokyo Institute of Technology , Japan from 1996 to 1999, and a visiting professor at the School of Mechanical , Ma- terials, Manufacturing Engineering and Management, University of Nottingham, UK from 2001 to 2002. He received the Control Engineering Practice Paper Prize of the International Federation of Automatic Control (IFAC) in 1999 (jointly with M. Nakano and J. She). He is a member of the Nonferrous Metals Society of China and the China Association of Automation. His research interests include robust control and its applica- tion , process control, and intelligent control 4 Jin-Hua She received the B. Sc . degree in engineering from Central South Uni- versity , China in 1983, the M. Sc . and the Ph. D. degrees in engineering from the Tokyo Institute of Technology , Japan in 1990 and 1993. In 1993 , he joined the De- partment of Mechatronics, School of En- gineering, Tokyo University of Technology , Japan. In 2008 , he transferred to the School of Computer Science, Tokyo University of Technology, where he is currently a professor. He received the Control Engineering Practice Paper Prize of the International Federation of Automatic Control (IFAC) in 1999 (jointly with M. Wu and M. Nakano). His research interests include the application of control theory , repetitive control, process control, internet-based engineering ed- ucation , and robotics This paper deals with the problem of designing a robust discrete output-feedback based repetitive-control system for a class of linear plants with periodic uncertainties. The periodicity of the repetitive-control system is exploited to establish a two-dimensional (2D) model that converts the design problem into a robust stabilization problem for a discrete 2D system. By employing Lyapunov stability theory and the singular-value decomposition of the output matrix, a linear-matrix-inequality (LMI) based stability condition is derived. The condition can be used directly to design the gains of the repetitive controller. Two tuning parameters in the LMI enable the preferential adjustment of control and learning. A numerical example illustrates the design procedure and demonstrates the validity of the method. Introduction By repeating the same operation, a person gradually becomes skilled, and flnally can operate with great eciency and precision. In fact, the repetition of operation is a process of learning and gradual progress. Based on the internal model principle[1], Inoue et al.[2] added a human-like learning capability to a control system and devised a new control strategy called repetitive control (RC). Fig. 1 shows the conflguration of a discrete-time repetitive-control system (RCS). Conflguration of a basic discrete-time repetitive-control In Fig. 1, G(z) is a compensated plant, and z is a shift operator. The part enclosed by the dotted line is a repetitive controller that contains a pure delay with a positivefeedback loop. For a given periodic reference input r(k) with period of N , a repetitive controller gradually improves the tracking precision through repeated learning actions, which involve adding the control input of the previous period v(k N ) to that of the present period to regulate the control input. As a result, the tracking error is reduced step by step. Finally, the output tracks the reference input without any steady-state error. RC is similar to iterative learning control (ILC)[3], which is another well-known method that makes use of the previous control trials. However, the difierence between RC and ILC regarding the initial state of every period makes the issues of interest (stabilization, control system design, etc.) for these two methods very difierent. An RCS has its inherent two-dimensional (2D) structural characteristics. It actually involves two difierent actions: Control within each repetition period and learning between periods. However, most analysis and design methods for an RCS developed for one-dimensional (1D) space ignore the difierence between these two actions and only consider their overall efiect. Consequently, they are incapable of making fundamental improvements in transient performance[46]. For the case of continuous RC, Zhou et al.[7; 8] employed 2D system theory[9; 10] to design a robust RCS. Mapping the RCS to 2D space yields a continuous-discrete 2D model. Unlike the 1D methods, the method in [7, 8] enables preferential adjustment of control and learning. The resulting control system exhibits both satisfactory robustness and good tracking performance. RCS has been studied mainly in the continuous time domain. However, in practical control systems, digital implementation of repetitive controller is simpler than analog counterpart. So, developing a design method for a discrete-time robust RCS has practical signiflcance. References [11,12] proposed a design method for discrete-time robust RCS, which is based on a discrete 2D model. It converted the system design to the robust stabilization problem for a discrete 2D system. However, the whole state of the plant is needed for the design of the controller, which is unfortunately unavailable in many practical applications. A design method that employs only the output of a plant is more practical. This paper presents a method of designing a discretetime robust RCS based on static output-feedback for a class of linear plants with periodic uncertainties. First, applying the lifting technique, a new discrete 2D model is established that accurately describes the characteristics of RCS and allows us to adjust control and learning preferentially. Then, by employing a 2D Lyapunov functional and the singularvalue decomposition (SVD) of the output matrix, a linearmatrix-inequality (LMI) based sucient stability condition is derived. Two tuning parameters in the LMI are used to adjust control and learning. Finally, a numerical example demonstrates the efiectiveness of the design method. Throughout this paper, Z+ is the set of non-negative integers, Cp is the p-dimensional vector space over complex numbers, @ is the linear space of all the functions from f0; 1; 2; 3; ; N 1g to Cp, l2(Z+; Cp) is the linear space of square integrable functions from Z+ to Cp, and l2(Z+; @) is the linear space of all the functions from Z+ to @. Problem description This paper considers the discrete-time RCS in Fig. 2. It is assumed that the process dynamics have been sampled by the zeroth-order hold method at a uniform rate Ts. The dynamics model of the repetitive controller CR(z) is v(k) = e(k) + v(k N ); where e(k) = r(k) y(k) is the tracking error. In Fig. 2, the single-input single-output (SISO) compensated plant with a relative degree of zero and time-varying structured uncertainties is ( x(k + 1) = (A + A(k))x(k) + (B + B(k))u(k) y(k) = Cx(k) + Du(k) where x(k) 2 Rn is the state of the plant, and u(k) 2 R and y(k) 2 R are the control input and output, respectively. A, B, C and D are real constant matrices. Assume that the structured uncertainties of the plant are h A(k) B(k) i = M F (k) h N0 where M; N0 and N1 are known constant matrices, and F (k) is a real, unknown, and time-varying matrix with Lebesgue measurable elements satisfying F T(k)F (k) 6 I; Fig. 2 Conflguration of the discrete output-feedback-based repetitive-control system We make the following assumption. Assumption 1. The uncertainties A(k) and B(k) vary periodically with the same period as that of the reference input r(k), i.e., ( A(k + N ) = A(k) B(k + N ) = B(k) From (3) and (5), we have F (k + N ) = F (k): The linear control law based on output feedback is u(k) = Kev(k) + Kyy(k); Ke; Ky 2 R where Ke is the feedback gain of the repetitive controller and Ky is the output feedback gain. This paper considers the following design problem: Design suitable control gains Ke and Ky in (7) such that the RCS in Fig. 2 is robustly stable for the admissible uncertainties (3). Equations (1), (2) and (7) describe the dynamics of the RCS in Fig. 2 in the time domain. Note that (7) can be rewritten as u(k) = Ke[e(k) + v(k N )] + Kyy(k) = Kev(k N ) + fKer(k) + (Ky Ke)y(k)g: (8) Equation (8) shows that the control law u(k) contains two kinds of information: Information on the previous period (flrst term on the right-hand side) and information on the present period (second term). Since we can view the past state as a kind of experience, RC actually involves two difierent types of actions: Control and learning. In other words, we use the information on the present and previous periods respectively to produce the present control input. A design method that enables independent adjustment of control and learning can potentially provide better transient performance. But from (8), we cannot preferentially adjust control and learning actions by directly changing the control gains Ke and Ky in (7). To solve this problem, we present a 2D description for the RCS in Fig. 2 in the following. First, focusing on the periodicity of RC, we employ the lifting technique[13] (Fig. 3) to slice the time axis into intervals of length N . That converts the vector-valued discretetime signal sequence f(k)g into the function-valued signal sequence fi(j)g. Its element is denoted as (i; j) in this paper. That is, Fig. 3 Mapping LD LD : l2(Z+; Cp) ! l2(Z+; @) (i; j) = i(j) = LD((k)) lL2D(Z +is; aCnp)isaonmdelt2r(icZ,+;iso@m).orphic transformation between Since the stability of the system does not depend on an exogenous signal, we set r(k) = 0. Then, from (1), (2) and (8), we get the following 2D representation for the RCS in Fig. 2. (x(i;j +1) = (A+A(i; j))x(i; j)+(B +B(i; j))u(i; j) v(i; j) = Cx(i; j)Du(i; j)+v(i1; j) with the 2D feedback control law of the form u(i; j) = Fev(i 1; j) + FpCx(i; j) Fe = Fp = Its dynamic boundary conditions are x(i + 1; 0) = x(i; N ); i 2 Z+: In (9), the flrst equation describes the control action within one repetition period, and the second equation describes the learning behavior between two adjacent periods. We can adjust them preferentially by changing Fp and Fe in (10) and thereby accelerate the convergence of the tracking error. This is a big advantage over the 1D methods. Note that there is a coupling relationship between control and learning, and we cannot adjust them completely and independently. This can be observed from the second equation of (9). Meanwhile, from (11), the control gains in Fig. 2 can be rewritten as Ke = Ky = Fe + Based on the 2D model (9), the design of the robust RCS in Fig. 2 is reformulated as the following problem: Find an admissible control law (10) for discrete 2D system (9) under boundary conditions of the form (12) such that the system is robustly stable and achieves fast convergence. Design of discrete output-feedback based RCS Substituting the control input (10) into (9) yields a representation of the 2D closed-loop RCS (i; j) = Al(i; j) + Bl(i; j) A + BFpC C DFpC N0 + N1FpC F (i; j)(i; j): T(i; j)(i; j) 6 T(i; j)T(i; j): Below, we derive a sucient robust stability condition for the closed-loop 2D system (14) by constructing a 2D Lyapunov functional and combining the SVD of the output matrix. Deflnition 1[14]. Assume that the output matrix C in (2) has full row rank and rank(C) = m. The SVD of C is C = U h S where S 2 Rmm is a diagonal matrix with positive, diagonal elements in decreasing order, 0 2 Rm(nm) is a zero matrix, U 2 Rmm and V 2 Rnn are unitary matrices. For the SVD (17), the following lemma presents an equivalent condition for matrix equation CX = X C. Lemma 1[15]. For a given C 2 Rmn with rank(C) = m, if X 2 Rnn is a symmetric matrix, then there exists a matrix X 2 Rmm such that CX = X C holds if and only if X = V diagfX11; X22g V T 1) = 2) S11 < 0 and S22 S1T2S111S12 < 0 0(x) "1(x) < 0; 8x 2 Rn f0g : The above lemmas lead to Theorem 1. Theorem 1. For given positive scalars fi and fl, if there exist symmetric, positive-deflnite matrices X11, X22, X2, and arbitrary matrices W1 and W2 such that the LMI 0 0 I X1 = V diagfX11; X22g V T 14 = fiX1AT + fiCTW1TBT 15 = fiX1CT fiCTW1TDT 16 = fiX1N0T + fiCTW1TN1T 24 = flW2TBT 25 = flX2 flW2TDT 26 = flW2TN1T then the discrete RCS (14) in Fig. 2 is asymptotically stable. And the 2D feedback gains in (10) are where U and S are deflned in SVD (17). Proof. Let Choose a Lyapunov functional candidate to be Fe = W2X21 Fp = W1U SX111S1U T P1 = X11; P2 = X21: V (i; j) = V1(i; j) + V2(i; j) Consider the associated increment V (i; j) as Along the time trajectory of (14), we have V (i; j) = ~T(i; j)(P + A^lTP A^l)~(i; j) 2 AT + CTFpTBT Furthermore, (15) and (25) yield V (i; j)[T(i; j)(i; j)T(i; j)T(i; j)] = ~T(i; j)~(i; j) 2 A~lT11 4 M T A~lT11 = AT + CTFpTBT A~lT12 = CT CTFpTDT A~lT13 = N0T + CTFpTN1T: Combining (16) and (26), and applying S-procedure (Lemma 3) to (26), we flnd that if < 0, then for any ~(i; j) 6= 0, V (i; j) < 0; thus the closed-loop system (14) is asymptotically stable. Also, from Schur complement Lemma 2, < 0 is equivalent to the following matrix inequality 16 = N0T + CTFpTN1T 24 = fi1 FeTBT 25 = Since X1 = V diagfX11; X22g V T, from Lemma 1, there exists X1 such that CX1 = X1C X11 = U SX111S1U T: W1 = FpX1; W2 = FeX2: Pre-multiplying and post-multiplying the matrix on the left-hand side of (27) by diagffiX1; flX2; I; fiX1; flX2; Ig and substituting (28) and (30) into the corresponding matrix, we have LMI (18). So, if LMI (18) holds, then the discrete closed-loop RCS (14) is robustly stable for the admissible uncertainties (3). Finally, we obtain (20) from (29) and (30). Remark 1. Theorem 1 provides an LMI-based sucient robust stability condition for discrete 2D system (9) under control law (10). The condition can be easily used to directly design the controller parameters of the RCS in Fig. 2 using the feasp solver in robust control toolbox[18]. The two tuning parameters fi and fl in LMI (18) manipulate the preferential adjustment of control and learning. More speciflcally, fi adjusts the weighting matrix P1 in V1(i; j), and fl adjusts the weighting matrix P2 in V2(i; j). Note that V1(i; j) and V2(i; j) are quadratic terms related directly to control and learning. Accordingly, fi and fl regulate the feasible solutions Fp and Fe in (20), respectively. In addition, from Theorem 1, we can derive a sucient stability condition for the nominal discrete 2D closed-loop system (i; j) = Al(i; j) where (i; j), (i; j) and Al are deflned in (14). Corollary 1. For given positive scalars fi and fl, if there exist symmetric, positive-deflnite matrices X11, X22, X2, and arbitrary matrices W1 and W2 such that the following LMI holds where X1 is deflned in (19), and then the 2D system (31) is asymptotically stable. And the 2D feedback gains in (10) are ~ 13 = fiX1AT + fiCTW1TBT ~ 14 = fiX1CT fiCTW1TDT ~ 24 = flX2 flW2TDT Fe = W2X21 Fp = W1U SX111S1U T: Numerical example Consider the problem of designing a controller for a direct current motor driven manipulator with a proportional integral regulator. The control input is the voltage applied to the armature, and the output is the rotational torque of the manipulator. The state-space model of the motor can be described using (2). Assume that the parameters of the uncertain plant (2) are ; N0 = ; F (k) = ; B = and the sampling frequency is fs = T1s = 10 Hz. We consider the problem of tracking the reference input r(k) = sin k + 0:5 sin 2k + 0:5 sin 3k and then the repetition period of the RCS in Fig. 2 is Choose the performance index N = = 20: J5 = 1 X4 X20 e2(iN + j) as a criterion for the selection of the tuning parameters fi and fl. Fig. 4 shows the efiect of adjusting the tuning parameters through simulation on three parameter sets: The performance indices are J5a = 3:3208; J5b = 0:8068; J5c = 0:1841: Note that, in Fig. 4, the tracking error in the flrst period shows only the control performance because there is no learning behavior within this period, and the convergence speeds of the tracking error in difierent periods show difierent learning eciencies. Comparing the simulation results of sets (a) and (b), it is clear that the tracking speed is accelerated by tuning fl from 0.05 to 0.13. And comparing the simulation results of sets (b) and (c), we know that the control performance is greatly improved by tuning fi from 0.6 to 0.2. These simulation results show that adjusting fi can mainly afiect control, and adjusting fl can mainly affect learning. Note that, due to the coupling relationship between control and learning, tuning fi and fl also strongly inuences learning and control, respectively. This can be observed from Fig. 4. Among the three parameter sets, (c) provides the best overall control and learning performance. In this way, we use the optimization algorithm min J5 s.t. LMI (18) to calculate the best tuning parameters. In addition, using the guaranteed cost algorithm in [11], we carried out simulation (in Fig. 6) for the uncertain plant (34). We found that J [11] = 0:4248. The comparison of the 5 simulation results (Figs. 5 and 6) shows that the preferential adjustment of control and learning greatly improves the transient performance. Fig. 4 Tracking errors for parameter sets (38) A combination of (40) and the flxed-step method in the ranges yields the best parameters set fi = 0:2; fl = 0:99: From Theorem 1, we obtain the corresponding control gains Ke = 68:9590; Ky = 2:9357: Simulation results in Fig. 5 show that the system is asymptotically stable for the admissible uncertainties and it enters into the steady state in the third period. Moreover, J5 = 0:0034. Based on the above analysis, we present a design algorithm for the output-feedback-based RCS in Fig. 2. Algorithm 1. Step 1. Use (40) to flnd fi and fl that minimize J5. Step 2. Use Theorem 1 to calculate Fe and Fp. Step 3. Use (13) to calculate Ke and Ky. Fig. 5 Simulation results for output-feedback-based RCS for fi = 0:2 and fl = 0:99 Fig. 6 Tracking error using the method in [11] Conclusions This paper described an LMI-based design method for a discrete-time robust RCS based on static output-feedback for a class of linear plants with periodic uncertainties. Exploiting the inherent 2D structural characteristics of RC, we established a discrete 2D model, making it possible to preferentially adjust control and learning by means of the gains in the 2D control law. The stability theory of 2D systems and the SVD of the output matrix were applied to derive an LMI-based sucient stability condition for the closed-loop system. The two tuning parameters in the condition manipulated the preferential adjustment of control and learning. Finally, a numerical example illustrated the design and tuning procedures, and simulation results demonstrated the efiectiveness of the method. Lan Zhou received the B. Sc. degree from Hunan Normal University, China in 1998, and the M. Sc. degree from Central South University, China in 2006. From 2008 to 2010, she was a joint cultivation doctoral candidate of Japan and China. She received her Ph. D. degree in control science and engineering from Central South University, China in 2011. She is an associate professor of control theory and control engineering with the School of Information and Electrical Engineering, Hunan University of Science and Technology, China. Her research interests include robust control, repetitive control, and control application. E-mail: (Corresponding author)


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Lan Zhou, Jin-Hua She, Min Wu. Design of a Discrete-time Output-feedback Based Repetitive-control System, International Journal of Automation and Computing, 2013, 343-349, DOI: 10.1007/s11633-013-0730-0