A Convex Information Relaxation for Constrained Decentralized Control Design Problems

Abstract

We describe a convex programming approach to the calculation of lower bounds on the minimum cost of constrained decentralized control problems with nonclassical information structures. The class of problems we consider entail the decentralized output feedback control of a linear time-varying system over a finite horizon, subject to polyhedral constraints on the state and input trajectories, and sparsity constraints on the controller's information structure. As the determination of optimal control policies for such systems is known to be computationally intractable in general, considerable effort has been made in the literature to identify efficiently computable, albeit suboptimal, feasible control policies. The construction of computationally tractable bounds on their suboptimality is the primary motivation for the techniques developed in this note. Specifically, given a decentralized control problem with nonclassical information, we characterize an expansion of the given information structure, which ensures its partial nestedness, while maximizing the optimal value of the resulting decentralized control problem under the expanded information structure. The resulting decentralized control problem is cast as an infinitedimensional convex program, which is further relaxed via a partial dualization and restriction to affine dual control policies. The resulting problem is a finite-dimensional conic program whose optimal value is a provable lower bound on the minimum cost of the original constrained decentralized control problem.The last equality follows from the fact that e T 1 ξ = 1 P-almost surely. To show that W MZ T K 2 0, it suffices to show columnwise inclusion in the second-order cone, i.e.,where s i ∈ L 2 1 is the i th element of the random vector s. By definition, we have that W ξ K 2 0 for all ξ ∈ Ξ. Also, since s i ≥ 0 almost surely, we have that W (s i ξ) K 2 0 almost surely. It follows from the convexity of the second-order cone that W E[s i ξ] K 2 0. APPENDIX D PROOF OF LEMMA 6Define the vector r ∈ R |J| according toJune 5, 2019 DRAFT Define the matrix Ψ := P J M 1/2 , where M 1/2 is the unique square root of the symmetric positive definite matrix M. Note that the matrix M 1/2 is symmetric and positive definite (and hence invertible). It holds that r T P J MP T The second and the last equalities both follow from the fact that η J = Π J P ξ = P J ξ. The fourth equality follows from the definition of the matrix Ψ and the symmetry of the matrix M 1/2 . The fifth equality follows from the fact [33, Prop. 3.2] that Ψ T ΨΨ T † = Ψ † . The sixth equality follows from the fact [33, Prop. 3.1] that Ψ † ΨΨ T = Ψ It holds that E zη T = E E zη T η J = E zη T J L T J = r T P J MP T J L T J June 5, 2019 DRAFT

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