Simultaneous Controller and Lyapunov Function Design for Constrained Nonlinear Systems

TL;DR

This paper introduces an offline semi-definite programming method to design controllers and Lyapunov functions for constrained nonlinear systems, avoiding complex invariant set calculations and explicitly finding a CPA Lyapunov function.

Contribution

It presents a novel approach that simultaneously designs controllers and Lyapunov functions for constrained nonlinear systems without relying on invariant sets or prior CLFs.

Findings

Efficient SDP-based method for controller and Lyapunov function design.

Explicit CPA Lyapunov functions for constrained systems.

Systematic approach to determine the region of attraction.

Abstract

This paper presents a method to stabilize state and input constrained nonlinear systems using an offline optimization on variable triangulations of the set of admissible states. For control-affine systems, by choosing a continuous piecewise affine (CPA) controller structure, the non-convex optimization is formulated as iterative semi-definite programming (SDP), which can be solved efficiently using available software. The method has very general assumptions on the system's dynamics and constraints. Unlike similar existing methods, it avoids finding terminal invariant sets, solving non-convex optimizations, and does not rely on knowing a control Lyapunov function (CLF), as it finds a CPA Lyapunov function explicitly. The method enforces a desired upper-bound on the decay rate of the state norm and finds the exact region of attraction. Thus, it can be also viewed as a systematic approach for finding Lipschitz CLFs in state and input constrained control-affine systems. Using the CLF, a minimum norm controller is also formulated by quadratic programming for online application.