Characterization of Upper Bound on Cost for Linear Quadratic Problems With a Norm-Bounded Gain Matrix

This work proposes a new technique to characterize the suboptimality of stabilizing control laws for the infinite-horizon linear quadratic regulation problem, wherein the norm of the state feedback gain matrix is conditioned to be lesser than a given bound. In contrast to existing techniques where the upper bound on the cost is pre-specified, this work proposes to compute an upper bound associated with the suboptimal control law(s). To derive the norm-bounded state feedback gain matrix, sufficient conditions of Krotov global optimal control framework are analyzed for the constrained linear quadratic problem, which typically renders the optimal control problem non-convex. To address this issue, suitable convex optimization problems are formulated to compute the upper bound independent of initial conditions. The effectiveness of the proposed technique and its comparison with state-of-the-art methods are demonstrated through numerical examples.