Safe nonlinear control design for input constrained polynomial systems using sum-of-squares programming

Abstract

We study the feedback stabilisation problem for input-affine polynomial systems subject to polytopic input constraints. First, we characterise a subset of the state-space, which we refer to as the stabilisation set, from where the system can be stabilised to the origin by means of a constrained, continuous feedback control law based on a given Control Lyapunov Function (CLF). We show how to express the sufficient conditions for Lyapunov stabilisation starting from this set in terms of semialgebraic set containments that can be associated (and numerically verified) with Sum-of-Squares programs. Second, we propose techniques to reshape and / or enlarge the stabilisation set for a given system and initial CLF by searching over a space of CLFs. Our third contribution relates to the solution of the constrained feedback control problem in real-time. At every time instant, we associate the control input with the solution of a state-dependent Quadratic Program (QP), which remains feasible along the entire trajectory of the closed loop system and thus, asymptotic stabilisation is guaranteed provided that the system had started from the stabilisation set, an estimate of which has been computed off-line a priori.