Convex Structured Controller Design in Finite Horizon

We consider the problem of synthesizing optimal linear feedback policies subject to arbitrary convex constraints on the feedback matrix. This is known to be a hard problem in the usual formulations (H
₂
, H
_∞
, LQR) and previous works have focused on characterizing classes of structural constraints that allow an efficient solution through convex optimization or dynamic programming techniques. In this paper, we propose a new control objective for finite horizon discrete-time problems and show that this formulation makes the problem of computing optimal linear feedback matrices convex under arbitrary convex constraints on the feedback matrix. This allows us to solve problems in decentralized control (sparsity in the feedback matrices), control with delays, and variable impedance control. Although the control objective is nonstandard, we present theoretical and empirical evidence showing that it agrees well with standard notions of control. We show that the theoretical approach carries over to nonlinear systems, although the computational tractability of the extension is not investigated in this paper. We present numerical experiments validating our approach.