Construction of Continuous and Piecewise Affine Feedback Stabilizers for Nonlinear Systems

In this article, two methods for constructing continuous and piecewise affine (CPA) feedback stabilizers for nonlinear systems are presented. First, a construction based on a piecewise affine interpolation of Sontag's “universal” formula is developed. Stability of the corresponding closed-loop system is verified a posteriori by means of a CPA control Lyapunov function and subsequently solving a feasibility problem. Second, we develop a procedure for computing CPA feedback stabilizers via linear programming, which allows for the optimization of a control-oriented criterion in the synthesis procedure. Stability conditions are a priori specified in the linear program, which removes the necessity for a posteriori verification of closed-loop stability. We illustrate the developed methods via two application-inspired examples considering the stabilization of an inverted pendulum and the stabilization of a healthy equilibrium of the hypothalamic-pituitary-adrenal axis.