Duality of the Optimal Distributed Control for Spatially Invariant Systems

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1 2014 American Control Conference (ACC) June 4-6, Portland, Oregon, USA Duality of the Optimal Distributed Control for Spatially Invariant Systems Seddik M. Djouadi and Jin Dong Abstract We consider the problem of optimal distributed control of spatially invariant systems. We develop an inputoutput framework for problems of this class. Spatially invariant systems are viewed as multiplication operators from a particular Hilbert function space into itself in the Fourier domain. Optimal distributed performance is then posed as a distance minimization in a general L-infinity space from a vector function to a subspace with a mixed L and H space structure. In this framework, a generalized version of the Youla parametrization plays a central role. The duality structure of the problem is characterized by computing various dual and pre-dual spaces. The annihilator and preannihilator subspaces are also calculated for the dual and predual problems. Furthermore, the latter is used to show the existence of optimal distributed controllers and dual extremal functions under certain conditions. The dual and pre-dual formulations lead to finite dimensional convex optimizations which approximate the optimal solution within desired accuracy. These optimizations can be solved using convex programming methods. Our approach is purely input-output and does not use any state space realization. I. INTRODUCTION There has been resonant interest in analysis and synthesis of distributed coordination and control algorithms for spatially interconnected systems. For recent work on this class and some of the background for the present work, we refer the reader to [1] [5], and the references therein. The basic idea for this spatially distributed problem is to perform distributed computations over a network to implicitly solve a global optimization problem. A networked system is a collection of dynamic units that interact over an information exchange network. Such systems are ubiquitous in diverse areas of science and engineering [6]. There are many important problems that have been cast in the form of a large-scale finite-dimensional or an infinitedimensional constraint optimization problem [7]. Such problems can range from physical, biological to mechanical and social systems [8] [11]. Distributed control has become a successful strategy to handle such design issues as coordinated control, formation control and synchronization of multi-agent systems [12] [17]. Even if the subsystems interact locally, the optimal controller will need global information to produce the feedback signal. Standard control design techniques are inadequate S. Djouadi is an Associate Professor with the Department of Electrical Engineering and Computer Science, University of Tennessee, Knoxville, TN 37996, djouadi@eecs.utk.edu J. Dong is a Ph.D candidate with the Department of Electrical Engineering and Computer Science, University of Tennessee, Knoxville, TN 37996, jdong@utk.edu since most optimal control techniques cannot handle systems of very high dimensions and with a large number of inputs and outputs. A preferred alternative is to have control signals computed using only local communication among neighboring subsystems as motivated in [1] and [2]. Using an approach based on spatial Fourier transforms and operator theory, Bamieh et al. discussed the optimal control of linear spatially invariant systems with standard linear quadratic(lq) criterion in [1]. As pointed out in [18], the above results are valid when system operator could generate a semigroup on L 2 (R n ). After the recent advances in communication technologies, the design of distributed controllers for physically interconnected systems has become an attractive and fruitful research direction [4], [19]. A body of literature has been worked out for the spatially distributed systems, where all signals are functions of both spatial and temporal variables. The linear matrix inequality (LMI) conditions for spatially interconnected systems consisting of homogeneous units are introduced in [4], [20]. Control synthesis results have employed consensus-based observer to guarantee leaderless synchronization of multiple identical linear dynamic systems under switching communication topologies [21]; neighborbased observer to solve the synchronization problem for general linear time-invariant systems [22]; and individualbased observer with low-gain technique to synchronize a group of linear systems [23]. Synchronization of multiple heterogeneous linear systems has been studied in [3], [5], where the interconnection topology is represented by an arbitrary graph, using a conservative analysis-lmi. A similar problem is investigated under both fixed and switching communication topologies heterogeneous spatially distributed systems [24], [25]. In this paper, we focus our investigation on spatially invariant systems. We show that the spatially invariant systems can be viewed as multiplication operators from a particular Hilbert function space into itself in the Fourier domain. A key distinctive feature of this paper with respect to the existing literature, is that we propose a new technique to pose the optimal distributed performance as a distance minimization in a general L space, from a vector function to a subspace with a mixed L and H space structure via a spatialtemporal Youla parametrization [26]. In this framework, tools from functional analysis are borrowed to characterize the pre-dual and dual optimizations along the lines of [27]. The duality structure of the problem is characterized by computing various dual and pre-dual spaces. The annihilator and pre-annihilator subspaces are also calcu /$ AACC 2214

2 lated for the dual and pre-dual problems. The latter is used to show the existence of optimal distributed controllers and dual extremal functions under certain conditions. We discuss how the dual and pre-dual formulations lead to finite dimensional convex optimizations which approximate the optimal solution within desired tolerance. These optimizations can be solved by convex programming methods. Our paper is organized as follows. In Section II, we introduce the formalism of discrete spatio-temporal invariant systems. As discussed in Section III, optimal distributed performance for these systems can be viewed as a distance minimization problem. The pre-dual characterization is illustrated in Section IV which is crucial in computing the optimal solution that we tackle via proving the existence of a pre-dual space and a pre-annihilator. It is followed by the dual characterization discussed in Section V. A discussion of a numerical solution is given in Section VI to demonstrate that the reduced finite dimensional optimization problems can estimate the optimal solutions within desired accuracy. Finally, we close the paper with some concluding remarks in Section VII. II. DISCRETE SPATIO-TEMPORAL INVARIANT SYSTEMS Following [26], [28], we consider signals that are both functions of discrete time t and discrete space i, denoted u(t, i). Spatially invariant spatio-temporal systems act on signals by convolution. If y(t, i) denotes the output of a spatially invariant systems G then y = Gu, where y(t, i) = ĝ(t τ, i j)u(τ, j) (1) j= τ= where ĝ(t, i) denotes the spatio-temporal impulse response of G. We assume temporal causality [26], [28], that is, ĝ(t, i) = 0, for t < 0 (2) The λ-transform of ĝ(t, i) is defined as g(λ, i) = ĝ(t, i)λ t (3) t=0 The spatial-temporal transfer functions is given by: G(z, λ) := g(λ, i)z i (4) i= The input-output relationship is given by the expression: Y (z, λ) = G(z, λ)u(z, λ) (5) where U(z, λ) is the transform of u(t, i), and Y (z, λ) is the transform of y(t, i). The system G(z, λ) can be viewed as a multiplication operator on L 2 (T, ) where T is the unit circle and (D) is the closed (open) unit disc of the complex domain C. If we assume that G(z, λ) is stable, then [26], [28]: G(z, λ) : L 2 (T, ) L 2 (T, ), u Gu = G(e iθ, λ)u(e iθ, λ), where θ [0, 2π), λ 1. Then we define the operator induced norm as [26], [28]: where G = u 2 2 = sup Gu 2 = G u 2 1 = esssup G(e iθ, λ) < 0 θ<2π λ 1 G(e iθ, λ)u(e iθ, λ) 2 dθdλ III. PROBLEM FORMULATION Consider the standard feedback configuration of Figure 1, where w is the extertal disturbance, z is the controlled output, y is the measurement signal, and u is the control for all spatio-temporal sequences. The plant G and controller K are spatially and temporally invariant systems. w Fig. 1. Standard Feedback Configuration We asume that G is stable, and the transmission from w to z is denoted by T zw. We have: ( ) ( )( ) z G11 G 12 w = (6) y G 21 G 22 u All stabilizing spatio-temporal invariant controllers [26] with Q stable, denote K = Q(I G 22 Q) 1 (7) Q H (L ( )) (8) The subspace H (L ( )) is defined as follows: For f H (L ( )), for each θ [0, 2π), f(e iθ, ) L (T), and for each λ, f(, λ) H. In other words, all functions f H (L ( )) can be viewed as H functions which take values in L ( ), i.e., L ( ) valued H functions. This interpretation is carried out to other subspaces in a similar vein. With the parametrization (7), optimal disturbance rejection can be formulated as: p : = inf Kstabilizing sup w 2 1 T zw w 2 (9) = inf Q H (L ( )) T 1 T 2 Q Using a spatio-temporal inner-outer factorization [26], we have: T 2 (e jθ, λ) = T 2in (e jθ, λ)t 2out (e jθ, λ) (10) 2215

3 where T 2in (e jθ, λ) is an isometry and T 2out (e jθ, λ) is (temporally) causally invertible. Therefore, the optimal performance index (9) can be written as: where ψ := inf T2inT 1 T 2out Q (11) Q H (L ( )) T 2in := T 2in (z 1, λ 1 ) (12) after absorbing T 2out in Q and denote the product by Q by abuse of notation. Then ψ = inf T2inT 1 Q (13) Q H (L ( )) Note that T 2in T 1 L (T ) such that (13) can be viewed as a distance minimization from the function T 2in T 1 to the subspace S of L (T ) defined by: S := H (L ( )) (14) In the next section, we will show the existence of optimal controllers using duality theory. IV. PRE-DUAL CHARACTERIZATION Let B be a Banach space with norm. Its dual space denoted B is the space of bounded linear functionals defined on B. Isometric isomorphism between Banach space is denoted by. B is said to be the predual space of B if B B. For a subset M of B, the annihilator of M in B is denoted M and is defined by [29]: M := {Φ B : Φ(m) = 0, m M} (15) In other words, M is the set of bounded linear functionals on B which vanish on M. It is a sort of generation of orthogonal subspace in the Banach space setting. Similarly, if M is a subset of B, then the pre-annihilator of M in B is denoted M, which is defined by [29]: M := {b B : ψ(b) = 0, ψ M} (16) Obviously, the preannihilator satisfies ( M) M. Following standard result of Banach space duality theory [29], the existence of a pre-annihilator implies that the following identity holds: min m M b m = sup < b, b > (17) b M b 1 where <, > denotes the duality product. It is readily seen that for problem (13): B = L (T ), b = T 2in L (T ) M = S = H (L (T)) (18) To apply the pre-duality result (17), we need to compute the pre-dual space of L (T ) and the pre-annihilator of S, S. Let us first characterize the pre-dual space of L (T ). In order to do so, define the Banach space L 1 (T ) of measurable and absolutely integrable functions on T under the L 1 -norm for f L 1 (T ) f 1 := f(e iθ, λ) dθdλ (19) T To show that L (T ) is isometrically isomorphic to the dual space of L 1 (T ), let us introduce the concept of tensor product of spaces. Let X, Y and Z be linear spaces over the same scalar field K(R or C). A function ϕ : X Y Z is bilinear if ϕ(x, ) : Y Z is linear for each x X, and ϕ(, y) : X Z for each y Y. The set of all bilinear functions from X Y into Z is denoted by B(X, Y ; Z). If Z = K, it is denoted simply by B(X, Y ). For a linear space X, the space of all linear functionals on X is denoted by X. For x X, y Y, the elementary tensor denoted x y is the element of B(X, Y ) defined by [29]: (x y)(ϕ) = ϕ(x, y), ϕ B(X, Y ) (20) The tensor product X Y is the linear span of all elementary tensors {x y : x X, y Y }. Therefore, if z X Y, we then have: z = λ i x i y i (21) for some certain {λ 1,, λ n } which are scalars, {x 1,, x n } X, {y 1,, y n } Y and n N is arbitrary. For any z X Y, define the tensor norm [28]: γ(z) = inf{ x i y i : x i X, y i Y, z = x i y i } In general, the space X Y under the norm γ( ) is not complete. We will denote its completion by X γ Y. A result in [30] asserts that if Γ and Γ are σ- finite measure spaces, then L 1 (Γ Γ) = L 1 (Γ) γ L 1 ( Γ) (22) In our case, T and are finite measure spaces and therefore σ-finite measure spaces. Hence it follows that: L 1 (T ) = L 1 (T) γ L 1 ( ) And for z L 1 (T) L 1 ( ), γ(z) = inf{ x i (e iθ ) 1 y i (λ) 1 : x i ( ) L 1 (T), where and y i ( ) L 1 ( ), z = x i (e iθ ) 1 = x i y i } (23) x i (e iθ ) dθ (24) y i (λ) 1 = y i (λ) dλ (25) 2216

4 The dual space of L 1 (T ), denoted L 1 (T ), can then be identified with L (T ): L 1 (T ) ( L 1 (T) γ L 1 ( ) ) L (T ) (26) by the Steinhaus-Nikodym theorem. That is for each F L (T ), define ϕ F L 1 (T ) such that: ϕ F (f, g) = F (e iθ, λ)f(e iθ )g(λ)dθdλ (27) Expression (27) characterizes every bounded linear functionals on L 1 (T ). Next, the pre-annihilator of S = H (L ( )) in L (T ) is computed. First note that the pre-annihilator of H ( ) in L (T) is: H 1 0 (T) := {f L 1 (T) : ˆf(n) = 0, n 0} where ˆf(n) denotes the n-th Fourier coefficients of f. That is g H ( ), f H0 1 (D) we have: g(e iθ )f(e iθ ) = 0 (28) To compute the pre-annihilator of S = H (L ( )), it suffices to notice that for each f H0 1 (L 1 (T)), i.e. f(z, λ) with z D, λ and for each fixed λ, f(, λ) H0 1 and for each fixed z D, f(z, ) L 1 ( ). Hence, for F H (L ( )) and f H0 1 (L 1 ( )), F (e iθ, λ)f(e iθ, λ)dθ dλ = 0 (29) } {{ } =0 Then the pre-annihilator of S is: S = H 1 0 (L 1 ( )) (30) The existence of a pre-dual space L 1 (T) γ L 1 ( ) and a pre-annihilator S implies the following theorem which is a standard result in Banach space duality theory relating the distance from a vector to a subspace and an extremal functional in the predual (Theorem 2 in [29]). Theorem 1: There exists at least one optimal Q 0 H (L (T)) achieving optimal performance µ in (9). Moreover the follwing identities hold: µ = inf T2inT 1 Q = T2inT 1 Q 0 Q H (L ( )) = sup T2inT 1(e iθ, λ)f (e iθ, λ)dθdλ (31) F H 0 1(L1 (T)) F 1 1 The optimal controller can then be computed by letting and Q = T 1 2out Q 0 K = Q 0 (I G 22 Q 0 ) 1 Note that the supremum in the pre-dual characterization is in general not attained. However, if we assume that T 2in T 1 is continuous on T, then it is possible to show that in fact the supremum is achieved. This is carried out in the next section. V. DUAL CHARACTERIZATION Let s first introduce the space of continuous functions on T, which is denoted C(T ) under the sup-norm: sup G(e iθ, λ) <, for G C(T ) (32) θ λ As mentioned before, in this section we make the following assumption: (A 1 ) T 2inT 1 C(T ) (33) that is T 2in T 1 is continuous on T. Since T and are compact, the dual space of C(T ), henceforth denoted C(T ), is isometrically isomorphic to the space of Borel measure M(T ) on T under the total variation norm for µ M(T ): µ = µ (T ) (34) where µ is the total variation of µ. The isometric isomorphism is given by: For µ M(T ) and f C(T ) letting I µ (f) = f(e iθ, λ)dµ(θ, λ) (35) The map µ I µ is the isometric isomorphism from M(T ) to C(T ) [31]. Banach duality states that for a Banach space B, and a subspace M of B, we have inf m M b m = max m(b) (36) m M m 1 where M is the annihilator of M in B, the dual space of B, defined in (15). Define a subspace of C(T ) as follows: S c = S C(T ) = H (L (T)) C(T ) (37) In the following lemma we establish that the distance from T 2in T 1 C(T ) to S c is the same as to S. Observe that with assumption (A 1 ), functions in S are continuous in the second variable λ. Lemma 1 : µ = T 2inT 1 Q 0 = inf T 2in Q (38) Proof: First notice that since S c S, then µ inf T 2in Q since the infimum in the definition of ψ is taken over a larger subspace. For the reverse inequality, let F := T 2in T 1 and 0 < r < 1, call: F r (e iθ, λ) := F (re iθ, λ) Q or (e iθ, λ) := Q o (re iθ, λ) (39) 2217

5 Then, F Q or = F F r + F r Q or F r Q or + F F r (40) Now, note that (F r Q or ) is bounded above by F Q o, since in the latter norm the supremum is taken over a larger set, that is, for g(e iθ,λ ) H (L T), and g r is a non-decreasing function of r for r [0, 1]. Therefore, F Q or F Q o + F F r (41) Considering T 2in T 1 is continuous on T, and the definition that F := T2in T 1, then F is also continuous. Therefore, ɛ > 0, there exists 0 < r < 1 such that and Q or being in S c implies: F F r < ɛ (42) F Q or D µ + ɛ (43) inf F Q µ + ɛ (44) Since ɛ is arbitrary, therefore: inf F Q µ (45) so the lemma holds. Define the space A known as the disc algebra A 0 = C(T) H. So S c can be written in the following form: S c = A 0 (C( )) (46) To compute the annihilator of S c, Sc in M(T ) it suffices to notice that the annihilator of A 0 in M(T) is H0 1 (T), and use a similar argument as (29) S c = H 1 0 (M( )) (47) Using the duality theory result (36), we deduce the following theorem: Theorem 2: µ = max φ(t2int 1 ) φ H 0 1 (M( )) φ 1 = max φ H 0 1 (M( )) φ 1 = T2inT 1 (e iθ, λ)dφ(e iθ, λ) T 2inT 1 (e iθ, λ)dφ 0 (e iθ, λ) (48) where φ 0 (, ) is the dual extremal functional, and φ H 1 0 (M( )) means for each fixed θ [0, 2π), Φ(e iθ, ) is a bounded Borel measure on and for each fixed λ, dφ(, λ) = G(e iθ )dθ for some function G( ) H 1 0. Moreover, by Lemma 1 under assumption (A 1 ) the search of Q can be restricted to the subspace S c. This will play an important role in finding a numerical solution as discussed in the next section. VI. DISCUSSION OF A NUMERICAL SOLUTION The infimum in (38) is termed the primal optimization and corresponds to the following representation: µ = inf Q S c T 2inT 1 Q (49) if Q is restricted to the subspace P mn consisting of polynomial in two variables of the form: m P mn (z, λ) := α ij z i λ j ; α ij R, j= m i=0 for z = 1, λ 1 (Note that these polynomials are analytic in the first variable z for z < 1 since Q is analytic in z for z < 1.) then we get an upper bound for (49) that is for µ mn := inf Q P mn T 2inT 1 Q (50) That is, µ mn µ since P mn S c, the infimum being taken over a smaller subspace. Since the polynomials P mn are dense in H (L ( )), therefore we have: µ mn µ as m, n, (51) i.e., µ mn converges to the optimal µ from above. The optimization problem (50) is finite dimensional, since reduces to searching for the coefficients {α ij } n,m i=0,j= m that minimize T2in T 1 Q. Now, we turn our attention to the dual problem (31) which is: µ = sup T2inT 1(e iθ, λ)f (e iθ, λ)dθdλ (52) F H 0 1(L1 ( )) F 1 1 By restricting the search to polynomials Pkl of two variables of the form: P kl (z, λ) := k j= k l β ij z i λ j ; β ij R, for z = 1, λ 1 (53) with norm P kl 1 1. Note in the sum over i we start from 1 since P kl (z, λ) H0 1 (L 1 ( )). We get the finite dimensional optimization: µ kl := sup T2inT 1(e iθ, λ)p kl (e iθ, λ)dθdλ (54) P kl P kl P kl 1 1 since the search in (54) is over the coefficients {β ij } l,k,j= k. Moreover, P kl is a subspace of H0 1 (L 1 ( )), then µ kl µ, since the supremum is taken over a smaller set, i.e., we get a lower bound for µ. Polynomials of the form (54) are dense in H0 1 (L 1 ( )), therefore we have: µ kl µ as k, l (55) 2218

6 In other words, µ kl converges to the optimal µ from below. Combining (51) and (55), we then have that: µ kl µ µ mn as k, l, m, n (56) squeezing the optimum within desired accuracy by taking large enough k, l, m and n. Therefore the finite dimensional optimization (50) and (53) estimate µ within desired tolerance and compute the corresponding Q in P mn, which in turn leads to the computation of distributed spatially invariant controllers K as close as desired to the optimal ones through the parametrization (7). Solving such problems are then applications of finite variable convex programming. VII. CONCLUSION AND FUTURE WORK In this paper, the duality structure of optimal distributed control of spatially invariant systems was characterized by computing the pre-dual and dual spaces after formulating the problem as a distance minimization. The pre-annihilator and annihilator subspaces were computed explicitly showing that an optimal distributed control exists. 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