具有时变时滞的离散系统的H∞控制

Vol. 36, No. 4 ACTA AUTOMATICA SINICA April, 2010 HHH∞∞∞ Control for Discrete-time System with Input Time-varying Delay WANG Wei1 ZHANG Huan-Shui2 Abstract This paper is concerned with the finite horizon H∞ control for discrete-time systems with time-varying delay in con- trol and exogenous input channels. By defining a binary variable, the problem is first transformed into the one with multiple input channels, each of which has a constant delay. Then, the problem is solved by using the duality principle between the H∞ control problem and smoothing estimation problem for an associated system without delays. The solution is given in terms of one standard Riccati difference equation (RDE) of the same order as the original system. Key words Discrete-time systems, time-varying delay, H∞ control, Riccati difference equation (RDE) DOI 10.3724/SP.J.1004.2010.00597 During the last decade, the problem of H∞ control for continuous time systems with time delays has been ex- tensively investigated[1−2]. In the discrete-time context, the control problem for systems with input delays has also received some renewed interests due to the applica- tions in network congestion control and networked control systems[3−5]. For discrete-time systems with delays, one might tend to consider augmenting the system and convert a delay problem into a delay-free problem. Although it is certainly possible to do so, the augmentation approach, however, generally results in higher state dimension and thus high computational cost, especially when the system under investigation involves multiple delays and the delays are large[6]. Furthermore, in the state feedback case, the augmentation approach generally leads to a static output feedback control problem, which is non-convex[5]. Most of the results in the aforementioned works are related to the systems with constant time delays. Recently, increasing at- tention has been paid to time-varying delays[7−11]. Most of these works focus on the time-varying delay in state. In this paper, the finite horizon H∞ control for discrete- time systems with time-varying delay in control and exoge- nous input channels is investigated. Under the assumption that time-varying delay is bounded, the problem is first transformed by defining a binary variable into the one with multiple input channels, each of which has a constant delay. Then, the problem is solved by using the duality principle between the H∞ control problem and the smoothing esti- mation problem for an associated system without delays. The causal and strictly causal solutions to the proposed problem are derived. The solvability condition is given in terms of the solution of one Riccati difference equation (RDE) of the same order as the original system. The rest of the paper is organized as follows. In Sec- tion 1, the system under consideration is described and the control problem is formulated. Section 2 presents Manuscript received March 11, 2009; accepted September 11, 2009 Supported by National Basic Research Program of China (973 Pra- gram) (2009CB320600), the Taishan Scholar Construction Engineer- ing by Shandong Government, and National Natural Science Foun- dation for Distinguished Young Scholars of China (60825304) 1. Shenzhen Graduate School, Harbin Institute of Technology, Shenzhen 518055, P.R.China 2. School of Control Science and Engineering, Shandong University, Jinan 250061, P.R.China 更多技术文章， 论文 政研论文下载论文大学下载论文大学下载关于长拳的论文浙大论文封面下载 请登录www.srvee.com 内容版权归作者所有 598 ACTA AUTOMATICA SINICA Vol. 36 some preliminaries. In this section, the proposed H∞ con- trol is converted to an optimization problem associated with a stochastic backward system. Explicit (causal and strictly causal) solutions to the problem are presented in Section 3. A simple simulation is presented in Section 4 to illustrate the proposed results. Some conclusions are drawn in Section 5. The notation used in the paper is fairly standard. AT stands for the transposition of a matrix A, Rn denotes the n-dimensional Euclidean space, Rn×r is the set of all n × r real matrices, and the notation B > 0 means that the matrix B is symmetric and positive definite. diag{· · · } denotes a block-diagonal matrix, whereas col{· · · } stands for a column vector. Ip denotes the p× p identity matrix. 1 Problem statement We consider the following discrete-time linear system for the H∞ control problem x (t+ 1) = Φtx (t) + Γ1,tu(t− ht) + Γ2,te(t− ht) (1) z (t) = Ltx (t) (2) where x (t) ∈ Rn is the measurable state, e(t) ∈ Rp and u(t) ∈ Rq are the exogenous and control inputs, respec- tively; z (t) is a linear combination of the states, which is to be regulated by using control signals. Φt, Γi,t, and Lt are bounded matrices with appropriate dimensions. It is assumed that the exogenous inputs are deterministic sig- nals and are from `2[0, N ], where N is the time-horizon of the control problem under investigation. Assumption 1. ht is the known positive integer time- varying delay term satisfying 0 ≤ ht ≤ m, where m is a constant number. And the initial inputs u(t) = 0 and e(t) = 0 for t < 0. The H∞ full-information control problem under investi- gation can be stated as follows: Given a scalar γ > 0, find a finite-horizon full information control strategy u(t) = F (x (t), {e(τ),u(τ)}tτ=0) such that for all nonzero vectors x 0 and {e(t)}Nt=0 sup (x0,e) 6=0 A B < γ2 (3) where A = xTN+1PN+1xN+1 + N∑ t=0 uT(t− ht)Qut−htu(t− ht) + N∑ t=0 zT(t)Qzt z (t) B = xT0Π −1 0 x 0 + N∑ t=0 eT(t− ht)Qet−hte(t− ht) Π0, PN+1, Q u t , Q e t , and Q z t are positive definite weighting matrices for t ≥ 0. Considering the above performance index, we define J∞N = xT0Π−10 x 0 − γ−2JN (4) where JN = xTN+1PN+1xN+1 + N∑ t=0 vT(t− ht)Qct−htv(t− ht) + N∑ t=0 zT(t)Qzt z (t) (5) with v(t− ht)= [ u(t− ht) e(t− ht) ] , Qct−ht= [ Qut−ht 0 0 −γ2Qet−ht ] It is clear that an H∞ controller u(t) that achieves (3) if and only if it satisfies that J∞N in (4) is positive for all nonzero ( x 0, {e(t)}N−htt=0 ) . Under Assumption 1, the system (1) can be put into the form x (t+ 1) = Φtx (t) + m∑ i=0 βt,iΓtv(t− i) (6) where Γt = [Γ1,t Γ2,t] and the coefficient βt,i, a binary variable, is defined as βt,i = { 1, ht = i 0, otherwise Since ht can only take one value from the finite set {0, · · · ,m} at each time instant t, then we have βt,i × βt,j = 0, i 6= j By using the above notation, JN can be rewritten as JN = xTN+1PN+1xN+1 + m∑ i=0 N∑ t=i vT(t− i)Qct−iv(t− i) + N∑ t=0 zT(t)Qzt z (t) (7) 2 Preliminaries We introduce the following notations for any t ≥ τ ≥ 0: v¯τ (t) =  col{v(t), v(t− 1), · · · , v(τ)}, t− τ < mcol{v(t), v(t− 1), · · · , v(t−m)}, t− τ ≥ m (8) v˜τ (t) =  m∑ j=t−τ+1 βt,jΓtv(t− j), t− τ < m 0, t− τ ≥ m (9) Γ¯τt = { [βt,0Γt, · · · , βt,t−τΓt] , t− τ < m [βt,0Γt, · · · , βt,mΓt] , t− τ ≥ m (10) Q¯τt =  diag{βt,0Qct , βt,1Qct−1, · · · , βt,t−τQcτ}, t− τ < m diag{βt,0Qct , βt,1Qct−1, · · · , βt,mQct−m}, t− τ ≥ m (11) Using above notations, the system (6) can be rewritten as x (t+ 1) = { Φtx (t) + Γ¯ τ t v¯ τ (t) + v˜τ (t), t− τ < m Φtx (t) + Γ¯ τ t v¯ τ (t), t− τ ≥ m (12) and JN in (7) can be rewritten as JN = J τN + m∑ i=0 τ−1∑ t=i vT(t− i)Qct−iv(t− i) + τ−1∑ t=0 zT(t)Qzt z (t) (13) 更多技术文章，论文请登录www.srvee.com 内容版权归作者所有 No. 4 WANG Wei and ZHANG Huan-Shui: H∞ Control for Discrete-time System with · · · 599 where J τN = xTN+1PN+1xN+1 + N∑ t=τ [v¯τ (t)]TQ¯τt v¯ τ (t) + N∑ t=τ zT(t)Qzt z (t) (14) Now, we define the following backward stochastic state- space model associated with (12) and performance index (14): ζζζ(t) = ΦTt ζζζ(t+ 1) + L T t ηηη(t) (15) yyyτ (t) = [Γ¯τt ] Tζζζ(t+ 1) +wwwτ (t), t = τ, · · · , N (16) where ζζζ(N+1), ηηη(t) and www(t) are uncorrelated white noises with zero means and covariances 〈ζζζ(N + 1), ζζζ(N + 1)〉 = PN+1, 〈ηηη(t), ηηη(s)〉 = Qzt δt,s and 〈www(t),www(s)〉 = Q¯τt δt,s, re- spectively. It can be seen that the dimensions of ηηη(t) and yyyτ (t) are respectively dim{ηηη(t)} = n× 1 and dim {yyyτ (t)} = { (t− τ + 1)(q + p)× 1, t− τ < m (m+ 1)(q + p)× 1, t− τ ≥ m and wwwτ (t) has the same dimensions as yyyτ (t). Lemma 1. By making use of the stochastic state-space model (15) and (16), the cost function J τN of (14) can be rewritten in the following quadratic form J τN = [ ξτ v¯τ ]T Πτ [ ξτ v¯τ ] (17) where Π = 〈[ xxxτ0 yyyτ ] , [ xxxτ0 yyyτ ]〉 = [ Rxxxτ0 Rxxxτ0yyyτ Ryyyτxxxτ0 Ryyyτ ] (18) v¯τ = col{v¯τ (τ), · · · , v¯τ (N)} (19) ξτ = col{x (τ), v˜τ (τ), · · · , v˜τ (m+ τ − 1)} (20) xxxτ0 = col{xxx(τ), xxx(τ + 1), · · · , xxx(τ +m)} (21) yyyτ = col{yyyτ (τ), · · · , yyyτ (N)} (22) with v˜τ (·) defined in (9), 〈xxxτ0 ,xxxτ0〉 = Rxxxτ0 , 〈xxxτ0 , yyyτ 〉 = Rxxxτ0yyyτ , and 〈yyyτ , yyyτ 〉 = Ryyyτ . We now decompose the observation yyyτ (t) and the noise vvvτ (t) of the backward stochastic model (15) and (16) as follows: yyyτ (t) = { col{yyy0(t), · · · , yyyt−τ (t)}, t− τ < m col{yyy0(t), · · · , yyym(t)}, t− τ ≥ m (23) vvvτ (t) = { col{vvv0(t), · · · , vvvt−τ (t)}, t− τ < m col{vvv0(t), · · · , vvvm(t)}, t− τ ≥ m (24) where yyyi(t) ∈ Rq+p and vvvi(t) ∈ Rq+p, i = 0, · · · ,m, satisfy yyyi(t) = Γ T t,ixxx(t+ 1) + vvvi(t) (25) with Γt,i = βt,iΓt and 〈vvvi(t), vvvi(s)〉 = βt,iQct−iδt,s. We also define the following re-organized measurement: yˇyyτ (t) =  col{y yy0(t), · · · , yyym(t+m)}, τ ≤ t < N −m col{yyy0(t), · · · , yyyi(t+ i)}, t = N − i (26) and vˇvvτ (t) has the form as vˇvvτ (t) =  col{v vv0(t), · · · , vvvm(t+m)}, τ ≤ t < N −m col{vvv0(t), · · · , vvvi(t+ i)}, t = N − i (27) It is obvious that for τ ≤ t < N −m, yˇyyτ (t) obeys yˇyyτ (t) =  Γ T t,0xxx(t+ 1) ... ΓTt+m,mxxx(t+m+ 1) + vˇvvτ (t) and for t = N − i, yˇyyτ (t) =  Γ T t,0xxx(t+ 1) ... ΓTt+i,ixxx(t+ i+ 1) + vˇvvτ (t) Furthermore, we reorganize the control input v¯τ (t) in the same way as that for the measurement of yyyτ (t) given above, i.e. vˇτ (t) =  col{ m+1︷ ︸︸ ︷ v(t), · · · , v(t)}, τ ≤ t < N −m col{ i+1︷ ︸︸ ︷ v(t), · · · , v(t)}, t = N − i Lemma 2. The cost function J τN of (17) can be rewrit- ten in the following quadratic form J τN = [ ξτ vˇτ ]T Πˇτ [ ξτ vˇτ ] (28) where ξτ is as defined in (20) and vˇτ = col{vˇτ (τ), · · · , vˇτ (N)} yˇyyτ = col{yˇyyτ (τ), · · · , yˇyyτ (N)} Πˇτ = 〈[ xxxτ0 yˇyyτ ] , [ xxxτ0 yˇyyτ ]〉 = [ Rxxxτ0 Rxxxτ0 yˇyyτ Ryˇyyτxxxτ0 Ryˇyyτ ] with xxxτ0 is as in (21). Now, we introduce the following notation y¯yyτ (t) =  m∑ j=0 yyyj(t+ j), τ ≤ t < N −m i∑ j=0 yyyj(t+ j), t = N − i (29) In view of (25), y¯yyτ (t) satisfies y¯yyτ (t) =  m∑ j=0 ΓTt+j,jxxx(t+ j + 1) + v¯vv τ (t), τ ≤ t < N −m i∑ j=0 ΓTt+j,jxxx(t+ j + 1) + v¯vv τ (t), t = N − i where v¯vvτ (t) =  m∑ j=0 vvvj(t+ j), τ ≤ t < N −m i∑ j=0 vvvj(t+ j), t = N − i is a white noise with zero mean and covariance matrix Qτv¯(t) =  m∑ j=0 βt+j,jQ c t , τ ≤ t < N −m i∑ j=0 βt+j,jQ c t , t = N − i (30) 更多技术文章，论文请登录www.srvee.com 内容版权归作者所有 600 ACTA AUTOMATICA SINICA Vol. 36 Lemma 3. The cost function J τN of (28) can be rewrit- ten in the following quadratic form J τN = [ ξτ vτ ]T Π¯τ [ ξτ vτ ] (31) where ξτ is as defined in (20) and vτ = col{v(τ), · · · , v(N)} y¯yyτ = col{y¯yyτ (τ), · · · , y¯yyτ (N)} Π¯τ = 〈[ xxxτ0 y¯yyτ ] , [ xxxτ0 y¯yyτ ]〉 = [ Rxxxτ0 Rxxxτ0 y¯yyτ Ry¯yyτxxxτ0 Ry¯yyτ ] with xxxτ0 given in (21). Lemma 4. The cost function J τN in (31) can be further rewritten as J τN = ξτTPξτ + (vτ − vτ∗)TRy¯yyτ (vτ − vτ∗) (32) where vτ∗ = −R−1y¯yyτ Ry¯yyτxxx(τ)x (τ)− m∑ i=1 R−1y¯yyτ Ry¯yyτxxx(τ+i)v˜ τ (τ + i− 1) (33) P = 〈xxxτ0 − xˆxxτ0 , xxxτ0 − xˆxxτ0〉 (34) and xˆxxτ0 is the projection of state xxx τ 0 onto the linear space L{y¯yyτ (τ), · · · , y¯yyτ (N)}. Given a scalar γ > 0, suppose that for any τ ≤ t ≤ N , the following RDE, with terminal value P τN+1 = PN+1, admits a bounded solution P τt , P τt = Φ T t P τ t+1Φt + L T t Q z tLt −Kτt Mτt (Kτt )T (35) where Mτt = Q¯ τ t + (Γ¯ τ t ) TP τt+1Γ¯ τ t (36) and Kτt is one solution to the following equation Kτt M τ t = Φ T t P τ t+1Γ¯ τ t (37) Denote Φ¯τj = Φ T −Kτj (Γ¯τj )T (38) Φ¯τj,j = I, Φ¯ τ j,s = Φ¯ τ j · · · Φ¯τs−1, s ≥ j (39) Lemma 5. v∗(τ) in (33) can be described by v∗(τ) = − [ S˜ττ (τ) ]T x (τ)− m∑ i=1 [ S˜ττ+i(τ) ]T ×( m∑ j=i βτ+i−1,jΓτ+i−1v ∗(τ + i− j − 1) ) (40) where S˜ττ+i(τ) = min{m,N−τ}∑ j=0 βτ+j,jSττ+i(τ + j)I(τ + j) (41) with I(τ + j) = col{ j+1 blocks︷ ︸︸ ︷ 0, · · · , 0, Iq} (42) Sττ+i(τ + j) = P ττ+i [ (Φ¯ττ+j+1,τ+i) TK˜ττ+j − (Φ¯ττ+j,τ+i)T× Gτ (τ + j)Kττ+j ] , Φ¯ττ,τ+i = 0, 0 ≤ j ≤ i− 1 [In − P ττ+iGτ (τ + i)] Φ¯ττ+i,τ+jKττ+j , i ≤ j ≤ N − τ (43) K˜ττ+j is one solution to the following equation K˜ττ+jM τ τ+j = Γ¯ τ τ+j (44) and Gτ (τ + i) = i∑ l=1 (Φ¯ττ+l,τ+i) TK˜ττ+l−1(Γ¯ τ τ+l−1) TΦ¯ττ+l,τ+i (45) with Φ¯ττ+l,τ+i given by (38). Lemma 6. The linear scalar quadratic form JN in (13) can be rewritten as JN = ξTPξ + N∑ t=0 {v(t)− v∗(t)}T M¯t {v(t)− v∗(t)} (46) where v∗(t) is as shown in (40) and M¯t given as follows M¯t =  m∑ j=0 βt+j,jMt,j , 0 ≤ t < N −m i∑ j=0 βt+j,jMt,j , t = N − i (47) where Mt,j =[ ΓT1,t+jP¯ t+1 j Γ1,t+j +Q u t Γ T 1,t+jP¯ t+1 j Γ2,t+j ΓT2,t+jP¯ t+1 j Γ1,t+j Γ T 2,t+jP¯ t+1 j Γ2,t+j − γ2Qet ] (48) with P¯ t+1j = P 0t+1, j = 0 P 0t+j+1 − ∑j s=1 P 0 t+j+1(Φ¯ 0 t+s+1,t+j+1) TK˜0t+sM 0 t+s× (K˜0t+s) TΦ¯0t+s+1,t+j+1P 0 t+j+1, j = 1, · · · ,m (49) 3 Main results For the convenience of representation of the main results, we first denote M¯t as follows M¯t = [ M¯t(1, 1) M¯t(1, 2) M¯t(2, 1) M¯t(2, 2) ] (50) where M¯t(i, j) is the (i, j)-block of M¯t. Theorem 1 (Causal solution). Consider the system (1) and (2), and the performance index (3). For a given γ > 0, suppose the RDE (35) has a bounded solution. Then, an H∞ controller that solves the full-information control problem exists if and only if 1) Π−10 − γ−2P 00 > 0; 2) M¯t(2, 2)− M¯t(2, 1)M¯−1t (1, 1)M¯t(1, 2) < 0. In this case, the central controller is given by u(t) = { [Iq 0] + M¯ −1 t (1, 1)M¯t(1, 2)[0 Ip] } v∗(t)− M¯−1t (1, 1)M¯t(1, 2)e(t) (51) 更多技术文章，论文请登录www.srvee.com 内容版权归作者所有 No. 4 WANG Wei and ZHANG Huan-Shui: H∞ Control for Discrete-time System with · · · 601 where v∗(t) and M¯t(·, ·) are as shown in (40) and (50), respectively. Proof. Equating the Lower-Diagonal-Upper (LDU) block triangular factorization of M¯t, we obtain[ M¯t(1, 1) M¯t(1, 2) M¯t(2, 1) M¯t(2, 2) ] = [ Iq 0 M¯t(2, 1)M¯ −1 t (1, 1) Ip ] ×[ M¯t(1, 1) 0 0 ∆t ] [ Iq M¯ −1 t (1, 1)M¯t(1, 2) 0 Ip ] where ∆t = M¯t(2, 2)− M¯t(2, 1)M¯−1t (1, 1)M¯t(1, 2). In view of (4) and (46), the above factorization allows us to write J∞N as J∞N =xT0 (Π−10 − γ−2P 00 )x 0 − γ−2 N∑ t=0 [ u(t)− u¯∗(t) e(t)− e∗(t) ]T ×[ M¯t(1, 1) 0 0 ∆t ] [ u(t)− u¯∗(t) e(t)− e∗(t) ] = xT0 (Π −1 0 − γ−2P 00 )x 0 − γ−2 N∑ t=0 {u(t)− u¯∗(t)}T × M¯t(1, 1) {u(t)− u¯∗(t)} − γ−2 N∑ t=0 {e(t)− e∗(t)}T∆t {e(t)− e∗(t)} (52) where we have defined[ u(t)− u¯∗(t) e(t)− e∗(t) ] = [ Iq M¯ −1 t (1, 1)M¯t(1, 2) 0 Ip ] ×[ u(t)− u∗(t) e(t)− e∗(t) ] In other words, we have defined the relation u¯∗(t) = u∗(t) + M¯−1t (1, 1)M¯t(1, 2)e ∗(t)− M¯−1t (1, 1)M¯t(1, 2)e(t) ={ [Iq 0] + M¯ −1 t (1, 1)M¯t(1, 2)[0 Ip] } v∗(t)− M¯−1t (1, 1)M¯t(1, 2)e(t) Note that u¯∗(t) is a function of {x 0, e(0), · · · , e(t)}, where u∗(t) and e∗(t) are functions of {x 0, e(0), · · · , e(t − 1)}. Referring back to (52), we see that since u(t) is only al- lowed to be a function of {x 0, e(0), · · · , e(t)}, it cannot influence the first and third terms in the summation for J∞N . Thus, a necessary condition for J∞N to be positive for all nonzero {x 0, e(0), · · · , e(N)} is that the first and third terms be positive for all such disturbances. In other words, the necessary condition is 1) Π−10 − γ−2P 00 > 0; 2) M¯t(2, 2)− M¯t(2, 1)M¯−1t (1, 1)M¯t(1, 2) < 0. The above conditions are also sufficient, since we can always choose the control signal to be u(t) = u¯∗(t) ¤ Theorem 2 (Strictly causal solution). Consider the system (1) and (2), and the performance (3). For a given γ > 0, suppose the RDE (35) has a bounded solution. Then, an H∞ controller that solves the full-information control problem exists if and only if 1) Π−10 − γ−2P 00 > 0; 2) M¯t(2, 2) < 0. In this case, the central controller is given by u(t) = [Iq 0]v ∗(t) (53) where v∗(t) is as shown in (40). Proof. When the control signal is restricted to be strictly causal, i.e., u(t) is a function of {x 0, e(0), · · · , e(t− 1)}, the only difference in the argument proceeding to The- orem 1 is that we need to replace the LDU block triangular factorization of M¯t with its Upper-Diagonal-Lower (UDL) block triangular factorization[ Iq M¯t(1, 2)M¯ −1 t (2, 2) 0 Ip ] [ Θt 0 0 M¯t(2, 2) ] ×[ Iq 0 M¯−1t (2, 2)M¯t(2, 1) Ip ] where Θt = M¯t(1, 1)− M¯t(1, 2)M¯−1t (2, 2)M¯t(2, 1). In view of (4) and (46), the above factorization allows us to write J∞N as J∞N = xT0 (Π−10 − γ−2P 00 )x 0 − γ−2 N∑ t=0 {uuu(t)− u∗(t)}TΘt {u(t)− u∗(t)} − γ−2 N∑ t=0 {e(t)− e¯∗(t)}T M¯t(2, 2) {e(t)− e¯∗(t)} where we have defined the relation e¯∗(t) = e∗(t) + M¯−1t (2, 2)M¯t(2, 1)u ∗(t)− M¯−1t (2, 2)M¯t(2, 1)u(t) Then proceeding with an argument similar to the one that led to the proof of Theorem 1, we can obtain the results proposed in this theorem. ¤ 4 Illustrative example In this section, we present a simple example to illustrate the previous theoretical results. Consider a scalar dynamic system described in (1) and (2) with the following specifica- tions: Φt ≡ 0.9, Γ1,t ≡ 2, Γ2,t ≡ −1.5, Lt ≡ 1, and the per- formance index in (3) with the following weighting matri- ces: Π0 = 0.2, PN+1 = 0.5, Q u t ≡ 2, Qet ≡ 1.2, and Qzt ≡ 1. In this illustrative example, we consider a special time- varying delay case, where ht ∈ {0,m} with m = 5 is the upper bound of the time-varying delay. And two paths of ht are considered for illustrating the proposed results, as shown in Fig. 1. The time horizon N = 50 and e(t) satis- fies the following equation e(t) = r(s)/10, where r(s) is a normal noise with zero mean and covariance Qr = 0.1. With these specifications of parameters and weighting matrices, the results can be obtained by using the schemes proposed in Sections 2 and 3. It is easy to check that (35) admits bounded solutions in the time horizon under consid- eration and Pt < 0.5. The first condition obtained in Theo- rems 1 and 2 is that Π−10 −γ−2P 00 = 4.9 > 0. ∆t < −0.6375 and M¯t(2, 2) < −0.075 for both cases. So, the conditions for the existence of an H∞ controller in Theorems 1 and 2 are all satisfied. Fig. 2 plots the obtained causal and strictly causal control input signals. The state trajectories associ- ated with the causal and strictly causal control signals are plotted in Fig. 3. The ratio in (3) along the time axis is given in Fig. 4, which shows that the obtained controllers are satisfactory. 更多技术文章，论文请登录www.srvee.com 内容版权归作者所有 602 ACTA AUTOMATICA SINICA Vol. 36 Fig. 1 The time histories of the time-varying delay ht Fig. 2 The time histories of strictly causal and causal controllers Fig. 3 The state trajectories associated with strictly causal and causal controllers Fig. 4 The ratios A/B in (3) along time axis 5 Conclusion This paper has addressed the H∞ control problem for discrete-time systems with time-varying delay. By defining a binary variable, the problem is first transformed into the one with multiple input channels, each of which has a con- stant delay. Then, the proposed problem is solved by using the duality principle between the H∞ control problem and a smoothing estimation problem for an associated system without delays. Necessary and sufficient conditions for the existence of a causal and strictly causal H∞ controller are obtained, and explicit solutions are given for the proposed H∞ problem. References 1 Nagpal K M, Ravi R. H∞ control and estimation prob- lems with delayed measurements: state-