31 5 2014 5 DOI: 10.7641/CA.2014.30666 Control heory & Applications Vol. 31 No. 5 May 2014 H,, (, 250100), (LMI),, H,,,,, H,, ; H ; ; ; P273 A Resilient static output feedback robust H control for controlled positive systems SONG Shi-jun, FENG Jun-e, MENG Min (School of Mathematics, Shandong University, Jinan Shandong 250100, China) Abstract: Firstly, by means of a small scalar, a sufficient condition of resilient static output feedback robust H control for controlled positive systems is obtained via linear matrix inequality (LMI) and bounded real lemma, which is useful for the design of resilient static output feedback controller because the controller parameter is separated from the unknown parameter. hen, based on the above condition, an inference for the special case of the control matrix is given and a sufficient condition of resilient static output feedback robust H control for more general controlled positive systems is established. Furthermore, the desired controller gain matrix can be derived via the cone complementarity linearization techniques directly. Finally, a numerical simulation to show the validity of the results is presented. Key words: positive systems; robust H control; resilient static output feedback; cone complementarity linearization; linear matrix inequality 1 (Introduction), [1 3]., [4 8].,,, H,.,. [9] [10], H [11 13]., H [14 15].,, [16 17].,, [18 19], H,, H,, H,, 2013 07 04; 2014 01 15. E-mail: fengjune@sdu.edu.cn. (61174141, 61374025); (JQ201219); (BS2011SF009, BS2011DX019).
672 31 R n n x 1 c 11 x 1 c 12 x 1 c 1n, R m n m n, I x XC = 2 c 21 x 2 c 22 x 2 c 2n, L 2 [0, ) [0, )..., ; [A] ij A i j x n c n1 x n c n2 x n c nn, A Metzler, A, XC, XC 0., A 0(A 0) A XA Metzler ( ), A B A B 0, A [A m, A M ] A m A A M, A > 0(A < ẋ(t) = Ax(t) + Bw(t) + B 1 u(t), 0) A ( ) ; µ(a) z(t) = Cx(t) + D 1 u(t), A, µ(a) = max{re λ : λ σ(a)}, (3) y(t) = C 1 x(t), σ(a) A x(0) = x 0, 2 (Problem formulation and preliminaries) ẋ(t) = Ax(t) + Bw(t), z(t) = Cx(t) + Dw(t), x(0) = x 0, (1) x(t) R n, w(t) R p, w(t) L 2 [0, ), z(t) R s, A, B, C, D, 1 [2] x 0 0, w(t) 0, x(t) 0, z(t) 0, t > 0, (1) 1 [2] (1)A Metzler, B 0, C 0, D 0. 2 [3] Metzler M, N R n n, M N, µ(m) µ(n). 3 [10] γ > 0, (1) (s) = C(sI A) 1 B + D (s) < γ, P > 0, P A + A P P B C B P γi D < 0. (2) C D γi 4 X, C 0, A Metzler, XC 0, XA Metzler x 1 c 11 c 12 c 1n x X = 2..., C = c 21 c 22 c 2n..., xn c n1 c n2 c nn x 1, x 2,, x n > 0, c ij 0, 1 i, j n, x(t) R n, u(t) R m, w(t) R p, w(t) L 2 [0, ), z(t) R s, y(t) R q, A [A m, A M ], A m, A M, B 1, C 1, D 1, B, C (3) u(t) = (K + K)y(t), (4) K, K [K m, K M ], K m K M (4), { ẋ(t) = A c x(t) + Bw(t), z(t) = C c x(t), A c = A + B 1 KC 1 + B 1 KC 1, C c = C + D 1 KC 1 + D 1 KC 1. (5) (4), (5) zw (s) = C c (si A c ) 1 B zw (s) < γ, (6) γ > 0, 1 K m = K M, K M 0. 1, 1 K m, K M, K m K M, K = K M + K m, k = K M K m, 2 2 (4) u(t) = (K + K + K K )y(t) = (F + F )y(t), (7) F = K + K, F = K K, F [ k, k ],, (4) (7) 3 (Main results), B 1, D 1
5 H 673 B 1, D 1, B 2 0, B 3 0, D 2 0, P A d + A d P P B Cd D 3 0 B 1 = B 2 + B 3, D 1 = D 2 + D 3, γi 0 < 0, (18) 1 γ > 0, ε > 0 (3), A [A m, A M ], B 0, C 0, C 1 0, K P > 0, Q > 0, ε 1 P I + εa d P B Cd εq 0 0 < 0, (8) 0 P Q = I, (9) [A f ] ij 0, 1 i j n, (10) [C f ] ij 0, 1 i, j n, (11) A d = A M + B 1 KC 1 + B 2 K M C 1 + B 3 K m C 1, C d = C + D 1 KC 1 + D 2 K M C 1 + D 3 K m C 1, A f = A m + B 1 KC 1 + B 2 K m C 1 + B 3 K M C 1, C f = C + D 1 KC 1 + D 2 K m C 1 + D 3 K M C 1, (4), (5) zw (s) (6). A [A m, A M ], K [K m, K M ], B 2 0, B 3 0, D 2 0, D 3 0, C 1 0 A d A c A f, (12) C d C c C f. (13) (10)--(11), A f Metzler, C f 0, A c, A d Metzler, C c, C d 0, 1, (5) (8) (9), Schur, ε 1 P + (I + εa d )ε 1 P (I + εa d ) + γ 1 P BB P + γ 1 C d C d < 0, (14) P A d + A d P + εa d P A d + γ 1 P BB P + γ 1 C d C d < 0. (15) P, ε > 0, (15) (16), A d P A d 0. (16) P A d + A d P + γ 1 P BB P + γ 1 C d C d < 0. (17) Schur,, P A d + A d P P B Cd µ γi 0 < 0. (19), A c, A d Metzler, B 0, P, 4, P A c P A d Metzler, P B 0. A d A c, P A d P A c. (11)(13), P A c + A c P P B Cc γi 0 P A d + A d P P B Cd γi 0 (20) Metzler 2 P A c + A c P P B Cc µ γi 0 P A d + A d P P B Cd µ γi 0, (21) P A c + A c P P B Cc µ γi 0 < 0, (22) P A c + A c P P B Cc γi 0 < 0, (23) 3, (5) zw (s)(6). 2 1 (3) H, 1, LMI, 1, K P, 1, (3), B 1, D 1, B 1 0, D 1 0, 1 γ > 0, ε > 0 (3), A [A m, A M ], B 0, B 1 0, C 0, D 1 0, C 1 0, K
674 31 P > 0, Q > 0, ε 1 P I + εa d P B Cd εq 0 0 < 0, (24) 0 P Q = I, (25) [A f ] ij 0, 1 i j n, (26) [C f ] ij 0, 1 i, j n, (27) A d = A M + B 1 KC 1 + B 1 K M C 1, C d = C + D 1 KC 1 + D 1 K M C 1, A f = A m + B 1 KC 1 + B 1 K m C 1, C f = C + D 1 KC 1 + D 1 K m C 1, (4), (5) zw (s) (6). 1, B 2 = B 1, B 3 = 0, D 2 = D 1, D 3 = 0, 3 B 1 0, D 1 0, B 1 0, D 1 0, B 1 0, D 1 0,,, (3), B 1, C 1, D 1, B 1 [B m, B M ], C 1 [C m, C M ], D 1 [D m, D M ], B m 0, C m 0, D m 0, (3) H K,, K = K 1 + K 2, K 1 0, K 2 0, (4), u(t) = K 1 y(t) + K 2 y(t) + Ky(t), (28) 2 γ > 0, ε > 0 (3), A [A m, A M ], B 1 [B m, B M ], C 1 [C m, C M ], D 1 [D m, D M ], B m 0, C m 0, D m 0, B 0, C 0, K 1 0, K 2 0, P > 0, Q > 0, ε 1 P I + εa d P B Cd εq 0 0 < 0, (29) 0 P Q = I, (30) [A f ] ij 0, 1 i j n, (31) [C f ] ij 0, 1 i, j n, (32) A d = A M +B M K 1 C M +B m K 2 C m +B M K M C M, C d = C +D M K 1 C M +D m K 2 C m +D M K M C M, A f = A m +B m K 1 C m +B M K 2 C M +B M K m C M, C f = C +D m K 1 C m +D M K 2 C M +D M K m C M, (28), (5) zw (s) (6). A [A m, A M ], K [K m, K M ], B 1 [B m, B M ], C 1 [C m, C M ], D 1 [D m, D M ], K = K 1 + K 2, K 1 0, K 2 0, B m 0, C m 0, D m 0, A d A c A f, (33) 1, 4 C d C c C f, (34) 2(3) H, [20], ( 2)K. 1 1 (29)(31) (32), ( ˆP 0, ˆQ 0, ˆK 10, ˆK 20 ). i = 0, ˆP 0, ˆQ 0 2 ( ˆP i, ˆQ i ), min {tr( ˆP ˆQ i + ˆP i ˆQ)}, ˆP, ˆQ, ˆK 1, ˆK 2 (29)(31) (32). ( ˆP, ˆQ, ˆK 1, ˆK 2 ). 3 (31) (32) ˆP A s + A ˆP ˆP s B Cs γi 0 < 0, A s =A M +B M ˆK1 C M +B m ˆK2 C m +B M K M C M, C s =C +D M ˆK1 C M +D m ˆK2 C m +D M K M C M, u(t) = ˆK 1 y(t) + ˆK 2 y(t) + Ky(t), 4 i > N 0, N 0,., i = i + 1, ( ˆP i, ˆQ i ) = ( ˆP, ˆQ), 2. 5 1, ε > 0 (29) (32),,, ε, (29) (32),
5 H 675 (28) K 3.7406 63.8913, K, K m K 1 = 0.7670, K 2 = 0.0001, K M, 0.1418 0.0689 2 1 1K m, K M. 2 X R m q,, X, K m = K m X, K M = K M + X, K m, K M 2 (29) (32) 3, K m + X, K M X K, K m + X K, K M X K,, 2. 6 2 K, K, 4 (Numerical simulation) (3), 0.342 1.94 1.45 A m = 0.1 3.87 0, 0.1 0 2.91 0.158 2.06 1.55 A M = 0.142 3.73 0, 0.2 0 2.55 0.22 0 0 0.19 0 0 B = 0 0.32 0, B m = 0 0.18 0, 0 0 0.42 0 0 0.17 0.21 0 0 B M = 0 0.22 0, C = [1 0 0], 0 0 0.23 C m =[0.9 0 0], C M =[1.1 0 0], D m =[0 0.5 0], 0.1 0.1 D M = [0 1 0], K M = 0.1, K m = 0.1. 0.1 0.1, H γ = 0.2, ε = 0.1., A M Metzler, A m Metzler, (3) A M 0.0393, 2.6616, 3.8157, (3) 2, MALAB LMI 2.0155 0 0 P = 0 2.0155 0, 0 0 2.0155 0.4962 0 0 Q = 0 0.4962 0, 0 0 0.4962, K = 60.1507 0.7669 0.0729. x(0) = [1 2 3], 1, 2,,, 1 Fig. 1 State variables of open-loop system 2 Fig. 2 State variables of close-loop system, 0.001 X = 0.001, 2, 0.001 315, 2,, 314 K m, K M K, 0.414 0.414 K m = 0.414, K M = 0.414. 0.414 0.414 5 (Conclusions) H, H,
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