and the nite horizon cost index with the nite terminal weighting matrix F > : N?1 X J(z r ; u; w) = [z(n)? z r (N)] T F [z(n)? z r (N)] + t= [kz? z r

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1 Intervalwise Receding Horizon H 1 -Tracking Control for Discrete Linear Periodic Systems Ki Baek Kim, Jae-Won Lee, Young Il. Lee, and Wook Hyun Kwon School of Electrical Engineering Seoul National University, Seoul , KOREA FAX: , Tel: whkwon@cisl.snu.ac.kr Abstract In this paper, a xed-horizon H 1 tracking control (HTC) for discrete time-varying systems is obtained via the dynamic game theory in state-feedback case. From HTC, an intervalwise receding horizon H 1 tracking control (IHTC) for discrete periodic systems is suggested using the intervalwise strategy. The conditions under which IHTC stabilizes the closed-loop system are proposed. Under proposed stability conditions, it is shown that IHTC guarantees the H 1 -norm bound and that IHTC with integral action provides zero oset for a constant command signal. The performance of IHTC is compared with that of RHTC via simulation studies for a discrete periodic system. 1 Introduction The receding horizon control strategy has been developed as a proper control strategy for tracking performance and time-varying systems. It is well known that this strategy presents more practical aspects in the applications to real systems than the innite horizon control strategy, because it needs only information for only a nite future time. As an approach to overcome this tracking problem, the receding horizon strategy has been developed. The receding horizon strategy is obtaining a solution to optimize a nite future cost horizon. There are two receding horizon strategies, the pointwise and intervalwise one. As well shown in [3], in the pointwise strategy, the terminal point of a xedlength nite cost horizon continuously recedes at each time instant. In the intervalwise strategy the terminal point is kept xed for a nite cost horizon and, after one period, the terminal point moves by one period and xed by the next period. The intervalwise strategy presents some advantages over the pointwise one in some respects. It has much less computation burdens than the pointwise one, since the intervalwise one requires calculation of control gain per a period of every cost horizon while the other one requires it per every time instant. During the horizon in which the optimal solutions are implemented, the intervalwise strategy is optimal, while the pointwise one is suboptimal. Hence the tracking performance of the intervalwise strategy is superior to the other one. The pointwise strategy has been developed for general timevarying systems [8], [1], [11], while the intervalwise strategy only for periodic and time-invariant systems [2], [3], [9]. There has been a few studies on the receding horizon tracking problems and its stability property in the H 1 problem [4], [5], [11]. But the intervalwise receding horizon strategy has not been investigated in the tracking problems and the H 1 problem to authors's knowledge. In this paper, an intervalwise receding horizon H 1 tracking control (IHTC) for discrete periodic systems is proposed. Our xed nite horizon H 1 tracking control (HTC) which is rst obtained to derive IHTC is dierent from that of [5]. The solution (HTC) is obtained via the dynamic game theory as shown in [6], [7]. The condidions under which closed loop stability, zero-oset tracking error, and innite horizon H 1 norm bound are guaranteed with IHTC are proposed, respectively. 2 H 1 tracking control for discrete time-varying systems We derive a nite horizon H 1 -tracking control (HTC) using the previous result [6], [7] in which only the regulation problem is dealt with. Consider the following discrete time-varying system: x(t + 1) = Ax + B 1 w + B 2 u (1) Cx yr z = ; z u r = x 2 R n ; u 2 R m ; w 2 R l ; z 2 R p+m

2 and the nite horizon cost index with the nite terminal weighting matrix F > : N?1 X J(z r ; u; w) = [z(n)? z r (N)] T F [z(n)? z r (N)] + t= [kz? z r k 2 2? 2 kwk 2 2] (2) is the disturbance attenuation level and y r (1), y r (2),, y r (N) are tracking commands which are assumed to be available over the future horizon N. In the following theorem, we introduce the existing result on the nite horizon H 1 -regulation problem the tracking commands y r = for 8 t. From now on, we substitute B 1 with B =?1 B 1 without loss of generality. Theorem 1 [6] When y r =, the dynamic game theory described by (1), (2) admits a unique feedback saddle-point solution, if and only if [I? B T M(t + 1)B ] > over t 2 [; N? 1] the sequence of M over t 2 [1; N] are generated by the following equation with M(N) = Q f : M = Q + A T M(t + 1)?1 A (3) = I + [B 2 B T 2??2 B 1 B T 1 ]M(t + 1) Then the unique saddle-point solution is given for t 2 [; N? 1] by u =?B T 2 M(t + 1)?1 Ax (4) w =?2 B T 1 M(t + 1)?1 Ax (5) Now in order to derive a nite horizon H 1 tracking control (HTC), we modify the notation shown in [1] as follows : Using, P = M[I? B (t? 1)B T (t? 1)M]?1 (6) (A? A 12A 22 A 21 )?1 = A 11? A 11 A 12 (A 21 A 11 A 12 +A?1 22 )?1 A 21 A 11 (7) and (I + )?1 = (I + )?1 (8) we can change (3)-(5) as follows: u =?[I + B T 2 P (t + 1)B 2 ]?1 B T 2 P (t + 1)Ax (9) w =?2 B T 1 [I + P (t + 1)B 2 B T 2 ]?1 P (t + 1)Ax (1) P = A T P (t + 1)A + P B (t? 1)[I + B T (t? 1)P B (t? 1)]?1 B T (t? 1)P?A T P (t + 1)B 2 [I + B T 2 P (t + 1) B 2 ]?1 B T 2 P (t + 1)A + Q (11) From the above modied equations, we derive the following result. Theorem 2 If and only if [I? B T r M(t + 1)B r ] > over t 2 [; N? 1], the unique saddle-point solution of HTC for discrete time-varying systems is given by u =?[I + B T 2 P (t + 1)B 2 ]?1 B T 2 [P (t + 1)Ax + g(t + 1)] (12) w =?2 B T 1 [I + P (t + 1)B 2 B T 2 ]?1 [P (t + 1)Ax + g(t + 1)] (13) g = [I + P B (t? 1)B T (t? 1)]fA T [I + P (t + 1)B 2 B T 2 ]?1 g(t + 1)? C T y r g g(n) =?fi + Q f B (N? 1)[I? B T (N? 1)Q f B (N? 1)]?1 B T (N? 1)gC T (N)F y r (N) P (N) = Q f [I? B (N? 1)B T (N? 1)Q f ]?1 Proof: It is well known that for a given p n (p n) full rank matrix C, there always exist some n p matrices L such that CL = I pp. Let ~x = Ly r. (2) is then rewritten with Q f = C T (N)F C(N) and Q = C T C as: N?1 X J = [x(n)? ~x(n)] T Q f [x(n)? ~x(n)] + t= We dene: x ^x = 1 A ^A = 1 ^Q = [(x? ~x) T Q(x? ~x) +kuk 2 2? 2 kwk 2 2] (14) B1 ; ^B1 = Q?C T y r?y T r C y T r y r Hence (1) and (14) are written as: B2 ; ^B2 = ; ^C = C?yr ^x(t + 1) = ^A^x + ^B1 w + ^B2 u(15) ^C^x ^z = ; ^Qf = u ^CT (N)F ^C(N) X N?1 J = ^x T (N) ^Qf ^x(n) + [^x T ^Q^x + kuk 2 2 t=? 2 kwk 2 2] (16) The dynamic game theory described by (15)-(16) admits a unique feedback saddle-point solution, if and

3 only if I? ^BT ^M(t + 1) ^B > over t 2 [; N? 1] [6], [7]. Let P P12 ^P = P12 T ; g = P P We know that [I? ^BT ^M(t + 1) ^B ] = [I? B T M(t + 1)B ]. Using (7)-(8), (12) and (13) are obtained from (9)-(11) with A, B 1, B 2, and P replaced by ^A, ^B1, ^B2, and ^P. 3 Stability of IHTC for discrete periodic systems Consider a generally time-varying matrix function L(). The symbol L () will denote a T -periodic matrix function such that L =L for t + T? 1 and L (t + T )=L for 8t. From the result of the previous section, we propose an intervalwise receding horizon H 1 -tracking control (IHTC) which stabilizes discrete T -periodic systems. Assume that N T + 1. Here N is both the cost horizon and the horizon that the tracking signal is given. Let the initial point be and Q f be the xed value. Among the solutions obtained over [; +N], we use the solutions over [; +T ]. Next the initial point moves to + T and the terminal point of the cost horizon moves to + T + N. This procedure repeats. Therefore P () is T-periodical. Let us make a T-periodic Riccati equation (T -PRE). We dene the following notations with k. E = A T P (t + 1)A? A T P (t + 1)B 2 [I +B T 2 P (t + 1)B 2 ]?1 B T 2 P (t + 1)A Q = Q + E( + T )? E() if t = + (k + 1)T Q otherwise Lemma 1 If A is nonsingular for 8t, Q () makes the solutions of the following Riccati equation T- periodic, i.e., it makes the following T -PRE. P = A T P (t + 1)A + Q? A T P (t + 1) B 2 [I + B T 2 P (t + 1)B 2 ]?1 B T 2 P (t + 1)A + P B (t? 1)[I + B T (t? 1) P B (t? 1)]?1 B T (t? 1)P (17) Proof: First, we will obtain the solutions over [ + 1; + N] from (11). Let the initial value be. Then from (17), we obtain at the terminal time t = T : P (T )? P (T )B (T? 1)[I + B T (T? 1)P (T ) B (T? 1)]?1 B T (T? 1)P (T )? Q (T ) = A T (T )P (T + 1)A(T )? A T (T )P (T + 1)B 2 (T )[I + B T 2 (T )P (T + 1)B 2 (T )]?1 B T 2 (T )P (T + 1)A(T ) (18) From the denition of Q (T ), the left side of (18) equals to E(). From (18), we observe that P (T +1) = P (1). Similarly P ((k + 1)T + i + 1) = P (i + 1) for k and i 2 [; T? 1]. Now the stabilizing property of the solution is derived in forward time, i.e., P +kt (). Consider a periodic matrix function X() with X(t+T )=X, and let the state feedback gain be?? =?[I + B T 2 X(t + 1)B 2 ]?1 B T 2 X(t+1)A Then, X() will be said to be stabilizing if the matrix A() + B 2 ()?() is asymptotically stable. If we replace P and Q with P +kt and Q +kt respectively, we obtain P +kt = F T k P +kt (t + 1)F k + Q +kt +K T k K k + P +kt B (t? 1) [I + B T (t? 1)P +kt B (t? 1)]?1 B T (t? 1)P +kt (19) K k =?[I + B T 2 P +kt (t + 1)B 2 ]?1 B T 2 P +kt (t + 1)A F k = A + B 2 K k Before stating the following theorem, we dene : = Q +kt + K T k K k + P +kt B (t? 1)[I + B T (t? 1)P +kt B (t? 1)]?1 B T (t? 1)P +kt (2) Theorem 3 y r over next N-horizon at + kt, i.e. [ + kt + 1; + kt + N] is given for k. Let P () be the solution of (11) and assume that 1) > or M?1 >?2 B 1 (t? 1)B T 1 (t? 1) on t 2 P [ + 1; + T ] +T 2) t= +1 > 3) (C(); A()) is completely observable 4) A is nonsingular for 8t Then, the periodic matrix functions P +kt () are stabilizing for each k. [I? B T M(t + 1)B > ] is satised from 1) since [I? B T M(t + 1)B ] = [I + B T P (t + 1)B ]?1. Proof: By (19) and (2), P +kt ( +1) can be denoted as follows: P +kt ( + 1) = T F k ( + 1)P +kt ( + T + 1) Fk ( + 1) + X+T t= +1 T F k (t; + 1) F k (t; + 1) F k ( + T; ) = F k ( + T? 1) F k ( + 1)F k (); Fk () = F k ( + T; ):

4 Let v be an eigenvector of Fk ( + 1) associated with the eigenvalue. Then we obtain: (1? jj 2 )v P +kt ( + 1)v = X+T t= +1 v T F k (t; + 1) F k (t; + 1)v From the assumption, we observe that all characteristic multipliers of Fk ( + 1) belong to the open-unit disk. Therefore P +kt () is stabilizing [2]. Let us consider the assumptions 1) and 2) of Theorem 3. The assumption 2) seems not to be a strong condition. In LQ problems, it is well known that P () > under some basic conditions such as controllability and observability conditions. But in H 1 - problems, no proper condition satisfying P () > has been found. In Section 6, it will be shown that quite a small satises 1). Consider discrete time-invariant systems. A discrete time-invariant system as shown in [2] can be viewed as a periodic system of an arbitrary period. Then, we can derive a time-invariant version of Theorem 3. Corollary 1 Let P () be the solution of (11) for discrete time-invariant systems with 3) and 4) of Theorem 3. If there exists an integer T such that: 1) P > or M?1 >?2 B 1 B T 1 on [ + 1; + T ] +T 2) t= +1 > then, the T -periodic matrix functions P kt () are stabilizing for each k 4 The stabilizing IHTC with integral action In this section, we investigate the zero oset property of the proposed stabilizing IHTC when the tracking command is constant and the system is time-invariant. It is well known that such a property can be obtained by introducing the following incremental state-space model: x e (t + 1) = A e x e + B 1e w + B 2e u(21) y = C e x e Ce x z = e yr I CA ; z u r = ; A e = A y x e = ; B x 1e = CB1 ; B B 2e 1 = CB2 B 2 C e = I The dynamic game theory based on the (21) gives the following control: u =?[I + B T 2e P e(t + 1)B 2e ]?1 B T 2e[P e (t + 1) A e x e + g e (t + 1)] (22) P e () and g e () are obtained from (11) and g of Section 2 with A, B, and C replaced by A e, B e, and C e. It is noted that the IHTC based on the incremental control u in (22) is stabilizing from Theorem 3 in the previous section. Now we will show that the stabilizing IHTC with integral action provides the zero oset. Corollary 2 The stabilizing IHTC with integral action provides the zero oset. proof: g e is derived similarly to [1]: g e =? T e (T + 1; t + 1)C T e F y r? TX j=t+1 [ T e (j; t + 1)B ce (j? 1)C T e y r]; t 2 [1; T? 1] e (t; t ) = A ce (t? 1)A ce (t? 1) A ce (t ) A ce = [I + B 2e B2e T P e]?1 A e [I + B e Be T P e(t? 1)] B ce = [I + P e B e B e ]?1 ; B e =?1 B 1e Now, we demonstrate the following fact: P e I e = T e (T + 1; t + 1)C T e F + TX j=t+1 [ T e (j; t + 1) B ce (j? 1)C T e ]; t 2 [1; T ] (23) Dene with H T (i; t + 1) = e (i; t + 1): G = H T T (T + 1; t + 1)C T e F + TX B ce (j? 1)C T e ; j=t+1 [H T T (j; t + 1) If t = T, it is clear that P e (T )I e = G(T ). Assuming that (23) is true when t = n + 1, we obtain: P e (n + 1)I e = G(n + 1) Let t = n then TX = H T T (T + 1; n + 2)C T e F + j=n+2 [H T T (j; n + 2)B ce (j? 1)C T e ] P e (n)i e = A T e fi? P e(n + 1)B 2e [I + B T 2e P e(n + 1) B 2e ]?1 B T 2egP e (n + 1)I e + C T e + P e (n) B e [I + B T e P e(n)b e ]?1 B T e P e(n)i e TX = H T T (T + 1; n + 1)C T e F + j=n+1 = G(n) [H T T (j; n + 1)B ce (j? 1)C T e ]

5 This means that (23) is true. Using this fact, the control u can be written with e = y? y r as u =?[I + B T 2e P e(t + 1)B 2e ]?1 B T 2e P e(t + 1)A e [e x] T (24) If we dene x E = [e x] T, we get x E = A e x E + B 1e w + B 2e u e = C e x E Since the above system is stable with the control (24), e! as t! 1, which means y! y r. 5 The H 1 -norm bound of the stabilizing IHTC In this section, we show that the stabilizing IHTC guarantees the H 1 -norm bound when y r () =. Theorem 4 Assume that E( + T )-E(). With the stabilizing IHTC u, the H 1 -norm bound of the closed-loop system is guaranteed, i.e., kt zw k 1 < (25) proof: Here for convenience, we denote P +kt ; M +kt ; F k ; and Q +kt as P ; M; F, and Q each other. By assumption, M >. Therefore?x T ()M()x() < 1X t= [x T (t + 1)M(t + 1)x(t + 1)?x T Mx] (26) Let = x T (t + 1)M(t + 1)x(t + 1)? x T Mx. Then, = [F x + B 1 w] T M(t + 1)[F x +B 1 w]? x T Mx = x T [X + Y ]x? W T [I? B T M(t + 1)B ]W + 2 w T w W = w? [I? B T M(t + 1)B ]?1 B T M(t + 1)F x X = F T M(t + 1)F? M Y = F T M(t + 1)B [I? B T By (6) and (7), M(t + 1)B ]?1 B T M(t + 1)F [I? B T M(t + 1)B ]?1 = I + B T P (t + 1)B (27) By (6) and (27), X and Y can be written as follows: X = F T P (t + 1)F? M? F T P (t + 1)B [I + B T P (t + 1)B ]?1 B T P (t + 1)F Y = F T fi? P (t + 1)B [I + B T P (t + 1)B ]?1 Denoting B T gp (t + 1)B [I + B T P (t + 1)B ] B T P (t + 1)fI? B [I + B T P (t + 1) B ]?1 B T P (t + 1)gF = B T P (t + 1)B = B T P (t + 1)F?F T P (t + 1)B [I + B T P (t + 1)B ]?1 B T P (t + 1)F + Y = T [?(I + )?1 + (I? (I + )?1 )(I + )(I? (I + )?1 )] = using = ( + I? I) (28) Using (19) and (28), X+Y =? Q?K T k K k Therefore =?x T [ Q + kkk k 2 2]x? W T [I? B T M(t + 1)B ]W + 2 kwk 2 2 <?x T [ Q + kkk k 2 2]x + 2 kwk 2 2?z T z + 2 kwk 2 2 (29) When x() =, we know from the relations of (26) and (29) that (25) is satised. 6 Simulation studies We demonstrate the properties of the proposed IHTC through simulation studies. the tracking performance of IHTC is compared with that of RHTC which is known to show good performances. [1]. We consider the following T -periodic system matrices: 1 + :1 B 1 = A = :9 + :9 C = q + :1q :1 + :1 :7 + :7 ; B 2 = :2 + :2 :8 + :8 :1q = cos( 2t T ); q is tuning parameter; F = 1; R = I(RHT C):

6 the external disturbances. Another advantage is that computation burdens are lessened and the IHTC can easily applied to real-time tracking systems In this simulation we assume that T = 2 and = :7. Using these values, we obtain a stabilizing IHTC. We make the values of disturbance as multiplying 2% of the tracking command by random signal which has a normal distribution between?:5 and :5. We select the cost horizon as T + 1 for the both cases of IHTC and RHTC. Fig.1 shows outputs of IHTC and RHTC for the given command signal. Fig.2 shows the dierence between the output and the command signal. In Fig.1, solid curve represents the tracking command. In Fig.1-Fig.2, '.' represents the result of IHTC and '{' represents the result of RHTC. Fig.1 and Fig.2 show that the performance of IHTC is better than that of RHTC. This result is also the same as that when there is no disturbance. If we increase the cost horizon, the performance of RHTC becomes better. Also in that case, the performance of IHTC is also a little better than that of RHTC. The results for time-invariant systems are similar to those for T- periodic systems. 7 Conclusion In this paper, a xed nite horizon H 1 -tracking control (HTC) for discrete time-varying systems is rst derived. And then, an intervalwise receding horizon H 1 -tracking control (IHTC) is proposed for discrete periodic systems. It is shown that the proposed IHTC guarantees closed loop stability, innite horizon H 1 - norm bound, and zero oset tracking error under the proposed conditions. Through the example, it is shown that the performance of the proposed IHTC presents better tracking performance than the existing pointwise receding horizon control which is proposed in [1]. Specially when the cost horizon is near the system order, the performance of IHTC is shown to be very good compared with that of the pointwise one for some systems. References [1] I. Yaesh and U. Shaked, "Minimum H1 Norm Regulation of Linear Discrete-Time Systems and Its Relation to Linear Quadratic Discrete Games," IEEE Trans. Automat. Contr., vol. AC-35, pp , 199. [2] G. D. Nicolao, "Cyclomonotonicity and Stabilizability Properties of Solutions of the Dierence Periodic Riccati Equation," IEEE Trans. Automat. Contr., vol. AC-37, pp , [3] G. D. Nicolao and S. Strada, "What is the easiest way to stabilize a Linear Periodic System," ECC95., [4] U. Shaked and C. E. DeSouza, "Continuous-Time Tracking Problems in an H 1 Setting:A Game Theory Approach," IEEE Trans. Automat. Contr., vol. AC-4, pp , [5] A. Cohen and U. Shaked, "Linear Discrete-Time H 1 -Optimal Tracking with Preview," Pro. of the 34th CDC, New Orleans, LA, pp , December [6] T. Basar and P. Bernhard, "H 1 -Optimal Control and Related Minimax Design Problems : A Dynamic Game Approach," Birkhauser Boston Basel Berlin, [7] T. Basar, "A Dynamic Games Approach to Controller Design: Disturbance Rejection in Discrete- Time," IEEE Trans. Automat. Contr., vol. AC-36, pp , [8] W. H. Kwon and A. E. Pearson, "On Feedback Stabilization of Time-Varying Discrete Linear Systems," IEEE Trans. Automat. Contr., vol. AC-23, no. 3, pp , [9] W. H. Kwon and A. E. Pearson, "Linear Systems with Two-Point Boundary Lyapunov and Riccati Equations," IEEE Trans. Automat. Contr., vol. AC-25, no. 2, [1] W. H. Kwon and D. G. Byun, "Receding horizon tracking control as a predictive control and its stability properties," Int.J.Control., vol. 5, no. 5, pp , [11] Sanjay Lall and Keith Glover, "A game theoretic approach to moving horizon control," Oxford University Press., Edited by David Clarke, One of the advantages of the proposed IHTC is that it can show very good tracking performance in spite of

H 1 optimisation. Is hoped that the practical advantages of receding horizon control might be combined with the robustness advantages of H 1 control.

H 1 optimisation. Is hoped that the practical advantages of receding horizon control might be combined with the robustness advantages of H 1 control. A game theoretic approach to moving horizon control Sanjay Lall and Keith Glover Abstract A control law is constructed for a linear time varying system by solving a two player zero sum dierential game