Controllers design for two interconnected systems via unbiased observers
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1 Preprints of the 19th World Congress The nternational Federation of Automatic Control Cape Town, South Africa. August 24-29, 214 Controllers design for two interconnected systems via unbiased observers Harouna Souley Ali Marouane Alma Mohamed Darouach Université de Lorraine (UT de Longwy), CRAN UMR 739 CNRS 186 Rue de Lorraine, 544 Cosnes et Romain, France ( Abstract: This paper proposes functional observer based controller for each subsystem of two interconnected system. The new approach that we use to express the control error permits to use some results on large scale delay systems to solve the problem. The proposed controllers are free of the interconnection terms, and the observation errors are unbiased. The observers based controllers are designed via some matrices gains which are obtained via LMs (i.e. via a tractable approach) from the Lyapunov stability results. An algorithm is given for the computation of the solutions. The result can be extended for the H control of linear and nonlinear large scale systems. Keywords: nterconnected systems, Filtering techniques, LM techniques, Observer based control 1. NTRODUCTON The study of large-scale dynamical systems is of great importance during the last years. This is due to the fact that physical systems are generally interconnected in practice giving a large-scale systems. n fact many practical control applications can often include electrical power systems, nuclears reactors, chemical process control systems, transportation systems, computer communication, economic systems and so on (Mahmoud 1997, Nguyen and Bagajewicz 28). Several dynamics subsystems can be distinguished (Chen and Lee 29) and delays generally arise in the processing of information transmission. This can cause instabilities and oscillations in these systems. For these reasons, many studies have been devoted to the analysis of stability of these systems (see (Siljak 1978, Lyou and al. 1984, Hmamed. 1986, Huang and al. 1995) and the references therein). The stability of such systems is generally based on checking the sign of the eigenvalues of the stability matrix (Chen and Lee 29). This is not a very tractable method, especially for the synthesis step. n (Chen. 26), a more tractable LM approach has been proposed for the stability analysis. Notice that in most practical situations, the complete state measurements are not available for each subsystem. But we know that for the control purpose, the availability of the states of a functional of the states of each subsystem is required (Dhbaibi and al. 28). Many works treat this problem by two ways generally. The first one is by designing an observer containing the interconnection term (see Sundareshan and Huang. 1984, Siljak 1991 and the references therein). the second method proposes an observer without interconnection term (see Pagilla and Zhu 25, Dhbaibi and al. 26and the references therein). n (Trinh and Aldeen 1997) a reduced order observer is proposed which dynamics are given from the set of unstable and/or poorly damped eigenvalues of the considered Linear Time nvariant system. n this paper, we propose a functional observer based controller to estimate directly a stabilizing control for a wide class of large scale systems, constituted by subsystems with interconnection terms between each other. Our proposed functional filter is without interconnection terms which makes its implementation more easy. 2. PROBLEM FORMULATON Let us consider a large-scale time-delay system S which is described as an interconnection of n subsystems S 1, S 2... S N, that are represented by (see Chen and Lee 29) S i : ẋ i (t) = A i x i (t) + A ii x i (t d ii ) + A ij x j (t d ij ) + B i u i (t) (1a) j i y i (t) = C i x i (t) (1b) for i, j = 1...N; i j and x i (t) R ni is the state vector for the i th system, y i (t) R pi is its measured output and u i (t) R ni is the control input. A i, A ij, B i and C i are constant matrices with appropriate dimensions and d ij denotes the physical communication delays in the interconnections of the large-scale system S. n our Purpose, we suppose that all the B i matrices are equal to the identity matrix: B i = n. Our objective is to design an observer-based controller with the following structure η i (t) = N i η i (t) + N ri η i (t d ii ) + J i y i (t) + J ri y i (t d ii ) (2a) û i (t) = η i (t) + E i y i (t) (2b) η i (t) R ni is the state vector for the i th system. t must be noticed that both the filter and the controller Copyright 214 FAC 188
2 19th FAC World Congress Cape Town, South Africa. August 24-29, 214 do not contain an interconnection term, which makes it more easy to be designed for each subsystem. Problem 1. The objective is to establish a functional observer (6a) and a control law (6b)such that: (i) lim t u i (t) K i x i (t) = (ii) the resulting closed-loop system (1)-(6) is asymptotically stable the observer (6) is unbiased i.e. its dynamics does not depend explicitly on the subsystem state x i (t). The approach used in this paper is to design the controller into two steps. The first one consists to use the item (i) of problem 1 i.e. to search for a state feedback gain K i verifying for subsystem (1). Once this gain is found we then search in a second step, an unbiased controller via observer (6) permitting to construct this gain K i. 3. DESGN OF THE STATE FEEDBACK GAN Here,we suppose that we have access to all the state vector, y i (t) = x i (t), such that u i (t) = K i x i (t). By replacing the control input u i (t) by K i x i (t) in each subsystem (1), then we obtain the following closed loop subsystems: S i : ẋ i (t) = A i x i (t) + A ii x i (t d ii ) + A ij x j (t d ij ) + K i x i (t) j i = (A i + K i )x i (t) + A ii x i (t d ii ) + A ij x j (t d ij ) (3) j i K i (t) R ni ni Using the results of Chen. 26, the first condition to stabilize x(t) is to get the matrix (A i + K i ) Hurwitz. The gain matrix K i can be obtained by resolving the following simple LM: (A i + K i ) T P i + P i (A i + K i ) <, P i > (4) By choosing: Y i = P i K i, we solve: A T i P i + Y i + P i A i + Y i T <, P i > (5) and we can obtain : K i = X i 1 Y i So, after this first step, we have K i such that 3 must be stable with A i + K i Hurwitz. 4. OBSERVER BASED CONTROLLER DESGN Now, we design an observer: η i (t) = N i η i (t) + N ri η i (t d ii ) + J i y i (t) + J ri y i (t d ii ) (6a) û i (t) = η i (t) + E i y i (t) (6b) which estimates the control input. Remark 1. Notice that for simplicity purpose, we consider in this paper only two interconnected systems. Nevertheless, our results can be obviously extended to a large-scale system composed with N interconnected systems. The estimation error is defined as: e i (t) = u i (t) û i (t) ψ i = K i E i C i. then = K i x i (t) η i (t) E i y i (t) = ψ i x i (t) η i (t) (7) ė i (t) = ψ i ẋ i (t) η i (t) (8) By replacing ẋ i (t) and η i (t) from (3) and (6) respectively, we obtain: ė i (t) = ψ i (A i + K i )x i (t) + A ii x i (t d ii ) + A ij x j (t d ij ) N i η i (t) N ri η i (t d ii ) J i C i x i (t) J ri C i x i (t d ii ) Using the fact that : η i (t) = ψ i x i (t) e i (t), the error dynamic can be written as: ė i (t) = ψ i (A i + K i )x i (t) + A ii x i (t d ij ) + A ij x j (t d ij ) J i C i x i (t) J ri C i x i (t d ii ) N i ψ i x i (t) e i (t) N ri ψ i x i (t d ii ) e i (t d ii ) = ψ i (A i + K i ) N i ψ i J i C i x i (t) + ψ i A ij x j (t d ij ) +ψ i A ii J ri C i N ri ψ i x i (t d ii ) +N i e i (t) + N ri e i (t d ii ) (9) f we want to obtain the estimation error dynamics independant from x i, we consider the following unbiasedness conditions: ψ i (A i + K i ) N i ψ i J i C i = (1a) ψ i A ii J ri C i N ri ψ i = (1b) then ė i (t) = N i e i (t) + N ri e i (t d ii ) + ψ i A ij x j (t d ij )(11) By taking B i = A i + K i, the unbiasedness condition for the free of delay part is given by: which can be written as: ψ i B i N i ψ i J i C i = (12) Ki N i L i E i C i = K i B i (13) C i B i L i = J i N i E i. n the same way, the unbiasedness condition for the delayed part can be written in the following compact form: Ki N ri L ri E i C i C i A ii = K i A ii (14) by taking L ri = J ri N ri E i. These two conditions can be rewritten in the following form: 189
3 19th FAC World Congress Cape Town, South Africa. August 24-29, 214 K i K i N i N ri L i L ri E i C i C = K ib i K i A ii (15) i C i B i C i A ii or in the form: with: χ i Γ i = Φ i (16) χ i = N i N ri L i L ri E i (17) K i K i Γ i = C i (18) C i C i B i C i A ii Φ i = K i B i K i A ii (19) The matrix equation (16) has a solution χ i if and only if: Γi rank = rankγ Φ i (2) i then the solution is given by: χ i = Φ i Γ + i Z i ( Γ i Γ + i ) (21) Γ + i is a generalized inverse and Z i is an arbitrary matrix. Now we can obtain the different above matrices such that: N i = χ i = N i 1 Z i Ni 2 (22) Ni 1 = Φ i Γ + i, and : N i 2 = ( Γ i Γ + i ) (23) N 1 ri = Φ i Γ + i L 1 i = Φ i Γ + i N ri = χ i = N ri 1 Z i Nri 2 (24), and : N ri 2 = ( Γ i Γ + i ) (25) L i = χ i = L1 i Z i L 2 i (26), and : L2 i = ( Γ i Γ + i ) (27) L 1 ri = Φ i Γ + i E 1 i = Φ i Γ + i L ri = χ i = L1 ri Z i L 2 ri (28), and : L2 ri = ( Γ i Γ + i ) (29) E i = χ i = E1 i Z i Ei 2 (3), and : E2 i = ( Γ i Γ + i ) (31) Once L i, N i and E i determinated, we compute J i using: J i = L i + N i E i (32) Similarely we get J ri using L ri, N ri and E i as: J ri = L ri + N ri E i (33) then, with some algebraic manipulations, the error dynamic can be rewritten as: ė i (t) = (N 1 i Z i N 2 i )e i (t) + (N 1 ri Z i N 2 ri)e i (t d ii ) : with: +(Q 1 i + Z i Q 2 i )x j (t d ij ) (34) Q 1 i = µ 1 i A ij (35) Q 2 i = µ 2 i A ij (36) µ 1 i = K i E 1 i C i (37) µ 2 i = E 2 i C i (38) n order to study the stability of the error, we introduce the following new augmented state : xi ē i = (39) e i n fact that we don t have access to all state vector, the applied control u i (t) of (1a) is replaced by û i (t) such that S i : ẋ i (t) = A i x i (t) + A ii x i (t d ii ) + A ij x j (t d ij ) + û i (t) (4) Since e i (t) = u i (t) û i (t), so we can replace û i (t) in (4) by: and we obtain: û i (t) = K i x i (t) e i (t) (41) 181
4 19th FAC World Congress Cape Town, South Africa. August 24-29, 214 S i : ẋ i (t) = (A i + K i )x i (t) + A ii x i (t d ii ) + A ij x j (t d ij ) e i (t) (42) Now, let us consider dynamics of the augmented state ē i, then we have: ẋi Ai + K ē i = = i xi (t) ė i Ni 1 Z i Ni 2 e i (t) Aii xi (t d + ii ) Nri 1 Z i Nri 2 e i (t d ii ) A + ij xj (t d ij ) Q 1 i + Z i Q 2 (43) i e j (t d ij ) Then the error dynamics can be rewritten in a compact form as ( for i, j = 1, 2; i j) ē i = A i ē i + A ii ē i (t d ii ) + A ij ē j (t d ij ) (44) The expressions of matrices A i, A ii and A ij are directly deduced from equation (43). For this purpose and similarly to (Chen. 26), let us consider the following subsystem (for i, j N = {1, 2,..., N}) ẋ i (t) = F i x i (t) + F ij x j (t h ij ), t (45) i=1 The stability condition of system (45) is given through the following lemma. Lemma 1.(Chen. 26) The system (45) is asymptotically stable with h ij R + provided that F i is Hurwitz, and there exist matrices P i and V ij such that the following LM condition holds. Ξ11 Ξ 12 Ξ T < (46) 12 Ξ 22 Ξ 11 = bdiag Φ 1, Φ 2,, Φ N (47) with Φ i = Fi T P i + P i F i + V ij (48) Ξ 12 = bdiag 1, 2,, N (49) Ξ 22 = bdiag 1, 2,, N (5) i = F1iP T 1 FNiP T N (51) i = bdiag V 1i, V 2i,, V Ni (52) Proof. The proof is given in (Chen. 26), Theorem 1, by using Lyapunov method. The idea is to apply Lemma1 to system (44) in order to obtain the stability condition, in closed loop, of the two interconnected systems (1). The result is given in the following lemma Lemma 2. The new error state is asymptotically stable with d ij R + provided that A 1, A 2 are Hurwitz, and there exist matrices X 1, X 2, W 11, W 12, W 21 and W 22 such that the following LM condition holds. (1, 1) A T 11X 1 A T 21X 2 (2, 2) A T 12X 1 A T 22X 2 X 1 A 11 W 11 < (53) X 2 A 21 W 21 X 1 A 12 W 12 X 2 A 22 W 22 with (1, 1) = A T 1 X 1 + X 1 A 1 + W 11 + W 12 (54) (2, 2) = A T 2 X 2 + X 2 A 2 + W 21 + W 22 (55) Now, we can state the following theorem witch gives in terms of LM the stability condition of the error system of the two interconnected systems. Theorem 1. The new system error is asymptotically stable if there exist matrices X 11 = X11 T >, X 12 = X12 T >, X 21 = X21 T >, X 22 = X22 T >, Y 12, Y 22, X 22, W 111 = W111 T >, W 112, W 113 = W113 T >, W 121 = W121 T >, W 122, W 123 = W123 T >, W 211 = W211 T >, W 212, W 213 = W213 T >, W 221 = W221 T >, W 222, W 223 = W223 T > such that B T 1 X 11 + X 11 B 1 X 12 K 1 B 1 + N1 1T X 12 K 1 + N1 2T Y12K T 1 B T 1 K1 T X 12 + K1 T X 12 N1 1 + K1 T Y 12 N1 2 N1 1T X 12 + X 12 N1 1 + N1 2T Y12 T + Y 12 N1 2 < (56) B T 2 X 21 + X 21 B 2 X 22 K 2 B 2 + N2 1T X 22 L 2 +N2 2T Y22K T 2 B T 2 K2 T X2 2 + K2 T X 22 N2 1 + K2 T Y 22 N2 2 N2 2T Y22 T + X 22 N1 2 + N2 1T X 22 + Y 22 N2 2 < (57) (k, l) < (58) for 1 k, l 12 such that (1, 1) = B T 1 X 11 + X 11 B 1 + W W 121 (1, 2) = (2, 1) T = B T 1 K T 1 X 12 + K T 1 X 12 N K T 1 Y 12 N W W 122 (1, 5) = (5, 1) T = A T 11X 11 + X 11 A 11 + W W 121 (1, 6) = (6, 1) T = A T 11K T 1 X 12 + K T 1 X 12 N 1 r1 + K T 1 Y 12 N 2 r1 + W W 122 (1, 7) = (7, 1) T = A T 21X 21 + Q 1T 2 X 22 K 2 + Q 2T 2 Y T 22K 2 (1, 8) = (8, 1) T = A T 21K T 2 X 22 + Q 1T 2 X 22 + Q 2T 2 Y T 22 (2, 2) = N1 1T X 12 + X 12 N1 1 + N1 2T Y12 T + Y 12 N1 2 + W W 123 (3, 3) = B T 2 X 21 + X 21 B 2 + W W 221 (3, 4) = (4, 3) T = B T 2 K T 2 X 22 + K T 2 X 22 N K T 2 Y 22 N W W 222 (3, 9) = (9, 3) T = A T 12X 11 + Q 1T 1 X 12 K 1 + Q 2T 1 Y T 12K 1 (3, 1) = (1, 3) T = A T 12K T 1 X 12 + Q 1T 1 X 12 + Q 2T 1 Y T 12 (3, 11) = (11, 3) T = A T 22X 21 + X 21 A 22 + W W 221 (3, 12) = (12, 3) T = A T 22K T 2 X 22 + K T 2 X 22 N 1 r2 + K T 2 Y 22 N 2 r2 + W W 222 (4, 4) = N2 1T X 22 + X 22 N2 1 + N2 2T Y22 T + Y 22 N2 2 + W W
5 19th FAC World Congress Cape Town, South Africa. August 24-29, 214 (5, 5) = W 111 (5, 6) = (6, 5) T = W 112 (6, 6) = W 113 (7, 7) = W 211 (7, 8) = (8, 7) T = W 212 (8, 8) = W 213 (9, 9) = W 121 (9, 1) = (1, 9) T = W 122 (1, 1) = W 123 (11, 11) = W 221 (11, 12) = (12, 11) T = W 222 (12, 12) = W 223 Any other block of the above matrix is equal to. Once the LMs are verified, the matrix gain Z 1 and Z 2 are given by Z 1 = X12 1 Y 12 (59) and Z 2 = X 1 22 Y 22 (6) Proof. We just give a sketch of proof due to lack of place; in fact the whole proof is derived from that of (Chen. 26) and is too long. Notice that we express the Hurwitzness of matrices A 1 and A 2 as required by Lemma 2 by using Lyapunov approach which give the LM (56) and (57). The LM (58) is obtained from inequality (53) of Lemma 2 matrices A 1, A 2, A 12 and A 21 are given by (44). Then matrices X 1, X 2, W 11, W 12, W 21 and W 22 of Lemma 2 are chosen as follows X11 K X 1 = 1 T X 12 X21 K X X 12 K 1 X 2 = 2 T X X 22 K 2 X 22 (61) W111 W W 11 = 112 W121 W W112 T W W 12 = W122 T (62) W 123 W211 W W 21 = 212 W221 W W212 T W W 22 = W222 T (63) W 223 Notice that instead of taking diagonal lyapunov matrices X 1 and X 2 to make the inequality be LM, we consider a more general class as shown by (61). So it suffices to take Y 12 = X 12 Z 1 and Y 22 = X 22 Z 2 to obtain LM (58). The observers (6) are easily reconstructed using the gain Z i for i = 1, 2 using equations (22) to (33). The following algorithm permits to compute the observers easily. The observer design algorithm The different steps of the functional filter design are summarized as follows : (1) Step 1 : Compute Γ i and verify that rank condition is satisfied ; (2) Step 2 : Compute matrices Ni 1 and Ni 2, E1 i and Ei 2, Q 1 i and Q2 i for i = 1, 2 (3) Step 3 : Obtain matrices X 12, X 22, Y 12, Y 22 through resolution of LMs. (4) Step 4 : Determine matrix gains Z 1 and Z 2. (5) Step 5 : Compute the filter matrices N i, E i, J i. 5. CONCLUSON This paper presented a computationally tractable solution via LM to the observation based control problem of interconnected systems. A new method is proposed which permits to first write the functional control error, and after that, using results on the stability of large scale time delayed systems, a functional filter is easily designed for each interconnected subsystem. Notice that the proposed filters are free of interconnection terms which make the design easy. The future works will concern the design of functional H filters based controllers of more general classes of interconnected systems. REFERENCES Chen, J. (26). LM based stability criteria for largescale time-delays systems. Japan Society Mechanical Engineering, nt Journal Series C, vol. 49, pp P. Gahinet and P. Apkarian (1994). A Linear Matrix nequality approach to Hinfinity control. nternational Journal of Robust and Nonlinear Control, vol. 4, pp C. Scherer and P. Gahinet and M. Chilali (1997). Multiobjective output-feedback control via LM optimization. EEE Transactions on Automatic Control, vol. 42, pp Chen, C. and Lee, C. (29). Robust stability of homogeneous large-scale bilinear systems with time-delays and uncertainties. J. of Process Control, vol. 19, pp Darouach, M. (2). Existence and design of functional observers for linear systems EEE Trans. Aut. Contr., vol. 45, pp Dhbaibi, S., Tlili, A. and Braiek, N. B. (26). Commande avec observateur d tat lpv dcentralis des systmes interconnects : application un systme de production et transport d nergie, Conference nternationale Francophone d Automatique, Bordeaux, France. Dhbaibi, S., Tlili, A. and Braiek, N. B. (28). Decentralized observer based H decentralized control for interconnected systems nternational Congress for global science and technology nternational Journal on Automatic Control and System Engineering, vol. 8, pp Hmamed, A. (1986). Note on the stability of large scale systems with delays. nt. J. Syst. Sci., vol. 17, pp Huang, S., Shao, H. and Zhang, Z. (1995). analysis of large scale systems with delays. Contr. Letters, vol. 25, pp Stability Syst. & Lyou, J., Kim, Y. and Bien, Z. (1984). A note on the stability of a class of interconnected dynamic systems nt. J. Contr., vol. 39, pp
6 19th FAC World Congress Cape Town, South Africa. August 24-29, 214 Mahmoud, M. (1997). Adaptive stabilization of a class of interconnected systems Computer and Electrical Engineering, vol. 23, pp Nguyen, D. and Bagajewicz, M. (28). Design of nonlinear sensor networks for process plants. ndustrial and Engineering Chemistry Research, vol. 47, pp Pagilla, P. and Zhu, Y. (25). A decentralized output feedback controller for a class of large-scale interconnected nonlinear systems, Trans. of the ASME, J. Dyn. Syst., Meas. & Cont., vol. 127, pp Siljak, D. (1978). Large-scale dynamic systems : stability and structure. North-Holland, New York: Elsevier. Siljak, D. (1991). Decentralized Control of Complex Systems. New York: Academic Press. Souley Ali, H., Darouach, M. and Zasadzinski, M. (26). Approche LM pour la synthse des filtres H non biaiss. Conference nternationale Francophone d Automatique, Bordeaux, France. Sundareshan, K. and Huang, P. (1984). On the design of a decentralized observation scheme for large scale systems EEE Trans. Aut. Contr., vol. 29, pp Trinh, H. and Aldeen, M. (1997). Reduced-order observer for large-scale systems EE Proc. - Part D, Contr. Theory & Applications, vol. 144, pp
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