On information structures, convexity, and linear optimality

Abstract

In 1968, Witsenhausen introduced his celebrated counterexample, which illustrated that when an information pattern is nonclassical, the controllers which optimize an expected quadratic cost may be nonlinear. For the special invited session commemorating the fortieth anniversary of the counterexample, we address one of the four follow-up questions listed in his original paper; namely, whether there is a relation between the convexity of finding the optimal affine controller, and whether that controller is in fact optimal. In particular, we discuss the connections between partially nested structures, for which linear controllers are known to be optimal, and quadratically invariant structures, for which optimal linear control is known to be convex.

Information structures in optimal decentralized control

“…For problems where both are well-defined, this is shown to be equivalent to the partially nested condition [69] discussed in previous sections. When this condition holds, it follows that a controller being admissible is further equivalent to the linear-fractional transformation we encountered earlier lying in the constraint set [71], [73]:…”

“…Choosing P and S, along with the H 2 -norm, to correspond to a given LQG problem with information structure is discussed in [69]. The stabilization constraint is needed in the most typical case where the signals lie in extended spaces and the plant and controller are rational proper systems whose interconnections may thus be unstable.…”

Optimal controller synthesis for a decentralized two-player system with partial output feedback

“…However, this is not always the case. For a broad class of decentralized control problems there exists a linear optimal policy, and finding it amounts to solving a convex optimization problem [2,8,9,17]. The two-player problem is in this class, and so we may without loss of generality restrict our search to linear controllers.…”

Distributed state estimation for linear multi-agent systems with time-varying measurement topology

“…The closely related area of distributed control has also seen a wealth of relevant solutions, such as those in Chen, Wen, Liu, and Wang (2014), Fax (2002), and Tanner and Christodoulakis (2007), and recent work on the subject of quadratic invariance has offered optimal solutions for certain classes of measurement topologies, see e.g. Lessard and Lall (2011) and Rokowitz (2008).…”