On the Dual of a Mixed H 2 /l 1 Optimisation Problem

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1 International Journal of Automation and Computing 1 (2006) On the Dual of a Mixed H 2 /l 1 Optimisation Problem Jun Wu, Jian Chu National Key Laboratory of Industrial Control Technology Institute of Advanced Process Control, Zhejiang University, Hangzhou , PRC Sheng Chen School of Electronics and Computer Science, University of Southampton Highfield, Southampton SO17 1BJ, UK Abstract: The general discrete-time Single-Input Single-Output (SISO) mixed H 2/l 1 control problem is considered in this paper It is found that the existing results of duality theory cannot be directly applied to this infinite dimension optimisation problem By means of two finite dimension approximate problems, to which duality theory can be applied, the dual of the mixed H 2/l 1 control problem is verified to be the limit of the duals of these two approximate problems Keywords: Optimal control, mixed H 2/l 1 control, duality theory 1 Introduction Most controller synthesis problems can be formulated as follows: Given a plant ˆP, design a controller Ĉ, such that the closed-loop system is stable, and satisfies some given (optimal) performance criteria When the optimal performance criterion is the H -norm (H 2 - norm or l 1 -norm respectively) of the closed-loop transfer function, based on Youla parameterisation [1] a parameterisation of the class of controllers which stabilise plant, the controller synthesis problem can be changed to the H -norm (H 2 -norm or l 1 -norm respectively) model matching problem the problem of finding an optimal stable free parameter, which minimises the H -norm (H 2 -norm or l 1 -norm respectively) of a map of the free parameter Consequently, H control design [2], H 2 control design [3] and l 1 control design [4] have been introduced respectively Mixed performance controls such as mixed H 2 /H control [5], mixed l 1 /H control [6], mixed l 1 /H 2 control [7], and mixed H 2 /l 1 control [8], have been a pole of attraction for many researchers lately Mixed performance control, can directly accommodate realistic situations where a system must satisfy several different performance constraints Based on Youla parameterisation, a mixed performance control problem can be changed into a special optimisation problem, with two kinds of norm The topic of this paper, is mixed H 2 /l 1 control A discrete-time SISO mixed H 2 /l 1 control problem was Manuscript received January 4, 2005; revised October 4, 2005 This work is supported by the National Natural Science Foundation of China (No and No ), the 973 program of China (No2002CB312200), and the program for New Century Excellent Talents in University (NoNCET ) Corresponding author address: sqc@ecssotonacuk addressed by Voulgaris [8,9] by minimising the H 2 -norm of a closed-loop map, while maintaining its l 1 -norm at a prescribed level Based on duality theory, a finite step method was presented to exactly solve this mixed H 2 /l 1 optimisation problem A more general class of discrete-time SISO mixed H 2 /l 1 control problem, within which the Voulgaris problem [8,9] is a special case, was addressed in [10] This class of problem considers minimising the H 2 -norm of the closedloop map, while maintaining the l 1 -norm of another closed-loop map at a prescribed level A methodology using finite-dimension quadratic programming was presented in [11] to obtain converging lower and upper bounds to this class of mixed H 2 /l 1 control problem This methodology was also developed to approximately solve a multi-input multi-output (MIMO) square mixed H 2 /l 1 control problem [2] It was shown in [2] that this methodology, without zero interpolation, avoids some problems present in methods which employ zero interpolation techniques Using zero interpolation techniques, a definition of the MIMO multi-block mixed H 2 /l 1 control problem, was given by Elia & Dahleh [13], and the dual of this MIMO multi-block mixed H 2 /l 1 control problem derived [13] An upper approximation methodology was also presented in [14] for the MIMO multi-block mixed H 2 /l 1 control problem, in which the related approximate problem is to minimize a positive linear combination of the squared H 2 norms, and the l 1 norms, over all stabilizing controllers A mixed H 2 /l 1 control problem, is intrinsically an infinite dimension quadratic programming problem In various minimisation problems, such as linear programming, quadratic programming, convex programming and approximation theory, one comes across a remarkable phenomenon, which is very useful in concrete ap-

2 92 International Journal of Automation and Computing 1 (2006) plications There exists an associated maximisation problem, called dual, involving a different variable, which attains the same optimal value as the original problem called primal Moreover, the value of the variable for which the maximum is attained in the dual problem, can be interpreted as the shadow price [15] Abstractly speaking, there is a duality correspondence, between the primal problem in a vector space, and a dual problem in the normed dual of a constraint space [15,16] A constraint space, is the space where the image of a constraint operator lies An obvious observation of the duality principle, is that the minimum distance from a point to a convex set, is equal to the maximum of the distances from the point, to the hyperplanes separating the point and the convex set This observation is of course completely trivial However, it turns out to be surprisingly useful in concrete applications, where the primal problem has nonlinear constraints or infinite dimensions, and in which the dual problem has linear constraints or finite dimensions This often leads to a simpler indirect method for solving the primal problem, in which the dual problem is solved first The duality relationship between a mixed H 2 /l 1 control problem and its dual uncovers finite-dimension structures in the optimal solution, when such finitedimension structures exist, and provides finitedimension approximations when the problem does not appear to have a finite-dimension structure [9,13] Consequently, duality theory plays an important role in research in H 2 /l 1 problems Elia & Dahleh [13] developed the dual of a MIMO multi-block mixed H 2 /l 1 control problem, based on zero interpolation techniques Nevertheless, for a mixed H 2 /l 1 control problem, defined by zero interpolation techniques, the task of determining an optimal controller still remains, even if an optimal closed-loop map has been determined [12] A closed-loop map needs to satisfy zero interpolation conditions exactly, to guarantee that correct cancellations take place while solving a controller Therefore, an extremely high accuracy closed-loop map is required, in order to determine a correct pole and zero cancellation However, numerical errors are always present which may result in a controller structure different from the optimal All of these difficulties motivate us to develop the dual of a mixed H 2 /l 1 control problem without zero interpolation The general discrete-time SISO mixed H 2 /l 1 control problem is addressed in this paper The H 2 -norm of a closed-loop map is minimised, while the l 1 -norm of another closed-loop map is maintained at a prescribed level It is found, that existing results of duality theory cannot directly be applied to this infinite dimension optimisation problem Two finite dimension approximate problems are constructed, to which duality theory can be applied These two approximate optimisation problems converge to the original infinite dimension optimisation problem, from lower and upper sides, respectively The dual of the general discrete-time SISO mixed H 2 /l 1 control problem, is then shown to be the limit of the duals of the two constructed approximate problems 2 Notation and mathematical preliminaries Let R denote the field of real numbers, R m denote the space of m-dimensional real vectors, and Z + nonnegative integers A causal SISO linear-time-invariant (LTI) transfer function Ĝ can be described as: Ĝ = G(0)+G(1)λ+G(2)λ 2 +, G(k) R, k Z + (1) As Ĝ can be represented uniquely by its impulse response sequence [G(0) G(1) G(2) ] T, Ĝ and its impulse response sequence are not differentiated in notation throughout this paper Define: l e = {Ĝ Ĝ = G(0) + G(1)λ + G(2)λ2 +, G(k) R, k Z + }, (2) l = {Ĝ l e sup G(k) < }, (3) l 2 = {Ĝ l e k (G(k)) 2 < }, (4) l 1 = {Ĝ l e G(k) < } (5) For any Ĝ l, the l -norm of Ĝ is given by: Ĝ = sup G(k) (6) k For any Ĝ l 2, the l 2 -norm of Ĝ is: Ĝ 2 = (G(k)) 2 (7) Ĝ 2 is also the H 2 -norm of Ĝ For any Ĝ l 1, the l 1 -norm of Ĝ is: Ĝ 1 = G(k) (8) It is easy to see that l 1 l 2 l, and that Ĝ1, Ĝ 2 l 1, Ĝ1Ĝ2 l 1 Let X be a normed linear space The space of all bounded linear functionals on X is called the normed

3 Jun Wu et al/ On the Dual of a Mixed H 2 /l 1 Optimisation Problem 93 dual of X, and is denoted X For any x X and x X, < x, x > denotes the value of the bounded linear functional x at the point x The norm of an element x X is: x = sup < x, x > (9) x 1 From standard functional analysis results [16], we have (R m ) = R m, (l 1 ) = l and (l 2 ) = l 2 For any x R N+1 and x R N+1, < x, x >= N (x(k)x (k)) (10) For any x l 1 and x l (or for any x l 2 and x l 2 ), < x, x >= (x(k)x (k)) (11) Given a convex cone P in X, it is possible to define an ordering relation on X as follows: x 1 x 2 if and only if x 1 x 2 P The cone P defining this relation on X is called positive Then it is natural to define a positive cone P inside X in the following way: P = {x X < x, x > 0, x P }, this in turn defines an ordering relation on X For any vector space in this paper, the positive cone which defines an ordering relation is the set consisting of elements with nonnegative point-wise components Let W be a vector space with a positive cone A mapping F : X W is convex if F(tx 1 + (1 t)x 2 ) tf(x 1 ) + (1 t)f(x 2 ) for all x 1, x 2 in X and 0 t 1 The following result of duality theory can be found in [16] Lemma 1 Let f be a real-valued convex function defined on a convex subset Ω of a vector space X, G be a convex mapping of X into a normed space W, and H(x) = Ax b be a map of X into the finite dimension normed space Y Suppose that A is linear, 0 Y is an interior point of {y Y H(x) = y for some x Ω}, there exists an x 1 Ω such that G(x 1 ) < 0 (ie G(x 1 ) is an interior point of the positive cone of W), and H(x 1 ) = 0 Define the minimisation problem: µ = inf G(x) 0 H(x)=0 f(x) (12) Assuming that µ is finite Then the dual problem is: µ = max w W w 0 y Y inf [f(x)+ < G(x), w > + < H(x), y >] 3 Mixed H 2 /l 1 optimisation problem (13) The general discrete-time SISO mixed H 2 /l 1 optimisation problem [11] can be stated as: Given ˆT 1 l 1, ˆT 2 l 2, ˆV1 = [V 1 (0) V 1 (m 1) 1] T R m+1, ˆV 2 = [V 2 (0) V 2 (n 1) 1] T R n+1, and a constant γ, find ˆQ l 1 such that ˆT 2 ˆQˆV 2 2 is minimised and ˆT 1 ˆQˆV 1 1 γ This description of the H 2 /l 1 problem is without zero interpolation conditions Once this H 2 /l 1 problem has been solved, the optimal controller can be determined directly from the optimal ˆQ It should be pointed out that the notation ˆQ does not appear in any H 2 /l 1 problem defined with zero interpolation techniques, and there may exist some problems in determining the optimal controller of a H 2 /l 1 problem defined with zero interpolation techniques, as mentioned in the introduction section For the above mixed H 2 /l 1 optimisation problem, in order to make its feasible region nonempty, it is assumed that γ > inf ˆT 1 ˆQˆV 1 1 (14) ˆQ l 1 In addition, we also assume that all the poles of ˆV 1 are inside the open unit disk in a complex plane, ie for: m (λ λ i ) = λ m +V 1 (m 1)λ m 1 + +V 1 (0), (15) i=1 λ i < 1, i {1,,m} Under these assumptions, we have the following lemma [11] Lemma 2 Suppose that ˆT 1 ˆQˆV 1 1 γ Then ˆQ 1 L, where: L = ˆT γ (16) m (1 λ i ) i=1 Define ξ = { ˆQ l 1 ˆT 1 ˆQˆV 1 1 γ, ˆQ 1 L} From lemma 2, it is easy to see another description of the mixed H 2 /l 1 optimisation problem: µ = inf ˆT 2 ˆQˆV (17) ˆQ ξ Notice the fact that ˆQ 1 L is equal to the linear constraints: ˆQ = ˆQ + ˆQ, ˆQ + 0, ˆQ 0, (18) ˆQ + + ˆQ L Therefore, by additionally setting: ˆT 1 ˆQˆV 1 = ˆΨ = ˆΨ + ˆΨ, ˆΨ + 0, ˆΨ 0, (19) ˆT 2 ˆQˆV 2 = ˆΦ,

4 94 International Journal of Automation and Computing 1 (2006) problem (17) can easily be transformed into: µ = inf ˆΦ 2 2 st ˆΦ = ˆT 2 ˆQ + ˆV2 + ˆQ ˆV2 ˆΨ + ˆΨ = ˆT 1 ˆQ + ˆV1 + ˆQ ˆV1 (20) ˆΨ + + ˆΨ γ, ˆQ + + ˆQ L ˆΦ l 2, 0 ˆΨ + l 1, 0 ˆΨ l 1 0 ˆQ + l 1, 0 ˆQ l 1 To obtain the dual of problem (17), we rewrite (20) in the form of (12) Let X = l 2 l 1 l 1 l 1 l 1, Y = l 1 l 2, W = R 2, ˆΦ ˆΦ l 2 ˆΨ + 0 ˆΨ + l 1 Ω = x = ˆΨ 0 ˆΨ l 1 ˆQ + 0 ˆQ, (21) + l 1 ˆQ 0 ˆQ l 1 I f(x) = x T x, (22) [ ] [ ] 0 E T G(x) = E T 0 0 γ E T E T x, L (23) [ ] [ ] 0 I I H(x) = V 1, V 1, ˆT1 x I 0 0 V 2, V 2, ˆT 2 = Ax b, (24) where: 1 0 I = 1, (25) 0 E = [1 1 ] T, (26) V 1 (0) 0 V 1 (0) V V 1, = 1 (m 1) 1 V 1 (m 1), (27) 1 0 V 2 (0) 0 V 2 (0) V V 2, = 2 (n 1) 1 V 2 (n 1) (28) 1 0 With these definitions, the mixed H 2 /l 1 optimisation problem (17) becomes: µ = inf G(x) 0 H(x)=0 f(x) (29) which has the same form as (12) However, lemma 1 cannot be applied to (17) This is because Y = l 1 l 2 has infinite dimensions, and therefore does not satisfy the conditions of lemma 1 In this paper, this difficulty in setting up the dual of problem (17) is overcome by first considering some approximations of (17) 4 Two approximate problems and their duals N Z +, define: ξ +N = { ˆQ R N+1 ˆQ ξ} (30) The variable N-truncation problem of (17) is constructed as: µ +N = inf ˆQ ξ +N ˆT 2 ˆQˆV (31) The mixed H 2 /l 1 problem (17) can be approximated from the upper side by the variable N-truncation problem (31), as stated in the following lemma [11] Lemma 3 µ +0 µ +1 µ +2 and lim N µ +N = µ N Z +, define the N-th truncation operator Γ N : l e R N+1 as: Γ N Ĝ = G(0) + G(1)λ + + G(N)λ N (32) Similar to the processing for problem (17) in section 3, define X = R 5N+2m+n+5, Y = R N+m+1 R N+n+1, W = R 2, ˆΦ ˆΦ R N+n+1 ˆΨ + 0 ˆΨ + R N+m+1 Ω = x = ˆΨ 0 ˆΨ R N+m+1 ˆQ + 0 ˆQ, (33) + R N+1 ˆQ 0 ˆQ R N+1 I f(x) = x T x + α(n), (34) [ 0 E T G(x) = N+m+1 EN+m+1 T EN+1 T EN+1 T [ ] γ β(n) L ] x (35)

5 Jun Wu et al/ On the Dual of a Mixed H 2 /l 1 Optimisation Problem 95 [ ] 0 I I V1,N+m+1 V H(x) = 1,N+m+1 x I 0 0 V 2,N+n+1 V 2,N+n+1 [ ] ΓN+m ˆT1 = Ax b, (36) Γ N+n ˆT2 where I denotes a finite identity matrix with a proper dimension, α(n) =< ˆT 2 Γ N+n ˆT2, ˆT 2 Γ N+n ˆT2 >, (37) E N+m+1 = [1 1] T R N+m+1, (38) β(n) = ˆT 1 Γ N+m ˆT1 1, (39) V 1 (0) 0 V V 1,N+m+1 = 1 (m 1) V1 (0) 1 V1 (m 1) 0 1 R (N+m+1) (N+1) (40) V 2 (0) 0 V V 2,N+n+1 = 2 (n 1) V2 (0) 1 V2 (n 1) 0 1 R (N+n+1) (N+1) (41) With these definitions, (31) can be expressed in the form of (12) It is easy to see, that f(x) so constructed, is a convex function, Ω is a convex subset, G(x) is a convex map, Y is a finite dimension, A is linear, µ +N is finite, 0 Y is an interior point of {y Y H(x) = y for some x Ω}, there exists x 1 Ω with G(x 1 ) < 0, and H(x 1 ) = 0 Hence, lemma 1 can be directly applied to derive the dual of optimisation problem (31): µ +N = max y 1 RN+m+1 y 2 RN+n+1 0 w 1 R 0 w 2 R inf [< ˆΦ, ˆΦ > +α(n) + < ˆΨ + ˆΨ Γ N+m ˆT1 + V 1,N+m+1 ( ˆQ + ˆQ ), y 1 > + < ˆΦ Γ N+n ˆT2 + V 2,N+n+1 ( ˆQ + ˆQ ), y 2 > + < E T N+m+1 ( ˆΨ + + ˆΨ ) γ + β(n), w 1 > + < E T N+1( ˆQ + + ˆQ ) L, w 2 >] = max y 1 RN+m+1 y 2 RN+n+1 0 w 1 R 0 w 2 R + < ˆΨ +, E N+m+1 w 1 + y 1 > + < ˆΨ, E N+m+1 w 1 y 1 > inf [< ˆΦ, ˆΦ > + < ˆΦ, y2 > + < ˆQ +, E N+1 w 2 + V T 1,N+m+1 y 1 + V T 2,N+n+1 y 2 > + < ˆQ, E N+1 w 2 V T 1,N+m+1y 1 V T 2,N+n+1y 2 > + α(n) < γ β(n), w 1 > < L, w 2 > < Γ N+m ˆT1, y 1 > < Γ N+n ˆT2, y 2 >] In (42), we are sure that: (42) E N+m+1 w 1 + y 1 0 (43) The reason is that, if E N+m+1 w 1 + y 1 < 0, ˆΨ + can be chosen as such a large positive number that µ +N < 0, which contradicts the fact that µ +N 0 Similarly, E N+m+1 w1 y 1 0, (44) E N+1 w2 + V 1,N+m+1 T y 1 + V 2,N+n+1 T y 2 0, (45) E N+1 w 2 V T 1,N+m+1y 1 V T 2,N+n+1y 2 0 (46) Denote: g(x) = < ˆΦ, ˆΦ > + < ˆΦ, y 2 > + < ˆΨ +, E N+m+1 w 1 + y 1 > + < ˆΨ, E N+m+1 w 1 y 1 > + < ˆQ +, E N+1 w 2 + V T 1,N+m+1y 1 + V T 2,N+n+1y 2 > + < ˆQ, E N+1 w 2 V T 1,N+m+1 y 1 V T 2,N+n+1 y 2 > + α(n) < γ β(n), w 1 > < L, w 2 > < Γ N+m ˆT1, y 1 > < Γ N+n ˆT2, y 2 > (47) Obviously, under conditions (43) (46), we have ˆΨ + = 0, ˆΨ = 0, ˆQ+ = 0 and ˆQ = 0 when inf g(x) is achieved Therefore: inf g(x) = inf [< ˆΦ, ˆΦ > + < ˆΦ, y2 > +α(n) ˆΦ l 2 < γ β(n), w 1 > < L, w 2 > < Γ N+m ˆT1, y 1 > < Γ N+n ˆT 2, y 2 >]

6 96 International Journal of Automation and Computing 1 (2006) = inf ˆΦ l 2 [< ˆΦ + y 2 2, ˆΦ + y 2 2 > < y 2 2, y 2 2 > < γ β(n), w 1 > +α(n) < L, w 2 > < Γ N+m ˆT1, y 1 > < Γ N+n ˆT2, y 2 >] = 1 4 < y 2, y 2 > +α(n) < γ β(n), w 1 > < L, w 2 > < Γ N+m ˆT1, y 1 > < Γ N+n ˆT2, y 2 > (48) Consequently, the dual of the variable N-truncation problem (31) is: µ +N =max[ 1 4 < y 2, y 2 > < Γ N+m ˆT 1, y 1 > < γ β(n), w 1 > < L, w 2 > + α(n) < Γ N+n ˆT2, y2 >] (49) st E N+m+1 w 1 y 1 E N+m+1w 1 E N+1 w 2 (V T 1,N+m+1y 1 + V T 2,N+n+1y 2) E N+1 w 2 y 1 R N+m+1, y 2 R N+n+1 N Z +, defines: 0 w 1 R, 0 w 2 R ξ N = { ˆQ l 1 Γ N ( ˆT 1 ˆQˆV 1 ) 1 γ, ˆQ 1 L} (50) Obviously, ξ 0 ξ 1 ξ 2 ξ (51) The constraint N-truncation problem of (17) is constructed as: µ N = inf ˆQ ξ N Γ N ( ˆT 2 ˆQˆV 2 ) 2 2 (52) The mixed H 2 /l 1 optimisation problem (17) can be approximated from the lower side by the constraint N- truncation problem (52), as summarized in the following lemma [11] Lemma 4 µ 0 µ 1 µ 2 and lim N µ N = µ Using the same method for constructing the dual of the variable N-truncation problem, the dual of the constraint N-truncation problem (52) is obtained as: µ N =max[ 1 4 < y 2, y 2 > < γ, w 1 > < L, w 2 > < Γ N ˆT1, y 1 > < Γ N ˆT2, y2 >] (53) st E N+1 w 1 y 1 E N+1 w 1 E N+1 w 2 (UT 1,N+1 y 1 + U T 2,N+1 y 2 ) E N+1w 2 y 1 RN+1, y 2 RN+1 0 w 1 R, 0 w 2 R Here: V 1 (0) 0 U 1,N+1 = R (N+1) (N+1) V 1 (N) V 1 (0) (54) V 2 (0) 0 U 2,N+1 = R (N+1) (N+1) V 2 (N) V 2 (0) (55) 5 The dual of the mixed H 2 /l 1 problem Define: D = ω = y 1 y 2 w 1 w 2 y1 l, y2 l 2 0 w1 R, 0 w2 R E w1 y 1 E w1 E w2 T (V1, y 1 +V2, y T 2) E w2 ϕ(ω) = 1 4 < y 2, y 2 > < γ, w 1 > < L, w 2 > (56) < ˆT 1, y 1 > < ˆT 2, y 2 > (57) Construct an infinite dimension optimisation problem as: υ = sup ϕ(ω) (58) ω D N Z +, define D +N as: y1 l, y 2 l 2 0 w1 D +N = ω R, 0 w 2 R E w1 y 1 E w1 (59) Γ N (E w2) Γ N (V1, y T 1 +V2, T y 2 ) Γ N(E w2 ) The constraint N-truncation problem of (58) can be constructed as: υ +N = sup ϕ(ω) (60) The following proposition is a direct consequence of D +N D Proposition 1 For the optimisation problems (58) and (60), υ +N υ However, sup ϕ(ω) = sup [ 1 4 < Γ N+ny 2, Γ N+ny 2 >

7 Jun Wu et al/ On the Dual of a Mixed H 2 /l 1 Optimisation Problem 97 < γ, w 1 > 1 4 < y 2 Γ N+n y 2, y 2 Γ N+n y 2 > < L, w 2 > < Γ N+m ˆT1, Γ N+m y 1 > < ˆT 1 Γ N+m ˆT1, y 1 Γ N+my 1 > < Γ N+n ˆT2, Γ N+n y 2 > < ˆT 2 Γ N+n ˆT2, y 2 Γ N+ny 2 >] = sup [ 1 4 < Γ N+ny 2, Γ N+ny 2 > < γ, w 1 > < L, w 2 > < (y 2 Γ N+ny 2 )/2 + ( ˆT 2 Γ N+n ˆT2 ), (y 2 Γ N+ny 2 )/2 + ( ˆT 2 Γ N+n ˆT2 ) > + < ˆT 2 Γ N+n ˆT2, ˆT 2 Γ N+n ˆT2 > < Γ N+m ˆT1, Γ N+m y 1 > + w 1 ˆT 1 Γ N+m ˆT1 1 < Γ N+n ˆT2, Γ N+n y 2 >] = sup [ 1 4 < Γ N+ny 2, Γ N+n y 2 > +α(n) < γ β(n), w 1 > < L, w 2 > < Γ N+m ˆT1, Γ N+m y 1 > < Γ N+n ˆT2, Γ N+n y 2 >] (61) and the constraint ω D +N can be changed to: E N+m+1 w 1 Γ N+my 1 E N+m+1w 1, E N+1 w 2 {V T 1,N+m+1(Γ N+m y 1) + V T 2,N+n+1(Γ N+n y 2)} E N+1 w 2, Γ N+m y 1 R N+m+1, Γ N+n y 2 R N+n+1, 0 w1 R, 0 w 2 R (62) This verifies that optimisation problem (60), is exactly optimisation problem (49), which leads to the following proposition Proposition 2 For optimisation problems (49) and (60), µ +N = υ +N N Z +, define: y1 RN+1, y2 RN+1 0 w1 R, 0 w2 R D N = ω E w1 y 1 E w1 (63) E w2 V1, y T 1 + V2, y T 2 E w2 The variable N-truncation problem of (58) can be constructed as: υ N = sup ϕ(ω) (64) ω D N The following proposition is the direct consequence of D N D Proposition 3 For optimisation problems (58) and (64), υ N υ In the same manner, optimisation problem (64) can be transformed into optimisation problem (53), and we have the following proposition Proposition 4 For optimisation problems (53) and (64), µ N = υ N With lemmas 3 and 4, and propositions 1 4, the main result of this paper can be summarized in the following proposition Proposition 5 For optimisation problems (17) and (58), µ = υ 6 Conclusions The dual problem sheds new light on the mixed H 2 /l 1 optimisation problem It can be seen that existing lemma 1 cannot be applied to primal problem (17) directly The idea in verifying the relationship between primal problem (17) and its dual problem (58) is to utilize their corresponding approximation problems (31) and (52), for which lemma 1 can be applied directly It is interesting to notice that the variable truncation in the primal problem becomes the constraint truncation in the dual problem, and the constraint truncation in the primal problem becomes the variable truncation in the dual problem This approach, based on duality theory, is useful in research in the mixed H 2 /l 1 optimisation problem, as it is often the case that the dual problem can be solved far more easily than the primal problem Acknowledgements The authors wishes to thank the support of the United Kingdom Royal Academy of Engineering References [1] B A Francis A Course in H Control Theory, Springer- Verlag, Berlin, 1987 [2] G Zames Feedback and optimal sensitivity: model reference transformations, multiplicative seminorms and approximate inverses IEEE Transactions on Automatic Control, vol 26, no 2, pp , 1981 [3] D C Youla, H A Jabr, J J Bongiorno Modern Wiener- Hopf design of optimal controllers part II: the multivariable case IEEE Transactions on Automatic Control, vol AC-21, no 3, pp , 1976 [4] M Vidyasagar, Optimal rejection of persistent bounded disturbances IEEE Transactions on Automatic Control, vol 31, no 6, pp , 1986 [5] I Kaminer, P P Khargonekar, M A Rotea Mixed H 2 /H control for discrete time systems via convex optimization Automatica, vol 29, no 1, pp 57 70, 1993 [6] M Sznaier, J Bu On the properties of the solutions to mixed l 1 /H control problems, In Preprints 13th IFAC Congress, San Francisco, USA, vol G, pp , 1996 [7] M V Salapaka, M Dahleh, P Voulgaris Mixed objective control synthesis: optimal l 1 /H 2 control SIAM Journal of Control and Optimization, vol 35, no 5, pp , 1997

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