Control Lyapunov Functions for Optimality in Sontag-Type Control

Given a Control Lyapunov Function (CLF), Son-tag’s famous Formula provides a nonlinear state-feedback guaranteeing asymptotic stability of the setpoint. At the same time, a cost function that depends on the CLF is minimized. While there exist methods to construct CLFs for certain classes of systems, the impact on the resulting performance is unclear. This article aims to make two contributions to this problem: (1) We show that using the value function of an LQR design as CLF, the resulting Sontag-type controller minimizes a classical quadratic cost around the setpoint and a CLF-dependent cost within the domain where the CLF condition holds. We also show that the closed-loop system is stable within a local region at least as large as that generated by the LQR. (2) We show a related CLF design for feedback-linearizable systems resulting in a global CLF in a straight-forward manner; The Sontag design then guarantees global asymptotic stability while minimizing a quadratic cost at the setpoint and a CLF-dependent cost in the whole state-space. Both designs are constructive and easily applicable to nonlinear multi-input systems under mild assumptions.