Piecewise Polynomial Policy Iterations for Synthesis of Optimal Control Laws in Input-Saturated Systems

This work proposes a policy iteration procedure for the synthesis of optimal and globally stabilizing control policies for Linear Time Invariant (LTI) Asymptotically Null-controllable with Bounded Inputs (ANCBI) systems. This class includes systems with eigenvalues on the imaginary axis (possibly repeated) but no pole with positive real part. The proposed policy iteration relies on a class of piecewise quadratic Lyapunov functions which is non-differentiable, but continuous, and polynomial in both the state and the deadzone functions of the input signals. The second step of the policy iteration is based on a piecewise control policy improvement. An important aspect of the proposed piecewise policy is that at each step of the iteration the computed policy is globally stabilizing and the existence of an improving value function is guaranteed as well. The solution to the inequalities which is required to hold at each step of the policy iteration, is obtained by solving Sum-of-Squares Programs (SOSP) that can be efficiently implemented with semidefinite programming (SDP) solvers.