Systematic, Lyapunov-Based, Safe and Stabilizing Controller Synthesis for Constrained Nonlinear Systems

A controller synthesis method for state- and input-constrained nonlinear systems is presented that seeks continuous piecewise affine (CPA) Lyapunov-like functions and controllers simultaneously. Nonconvex optimization problems are formulated on triangulated subsets of the admissible states that can be refined to meet primary control objectives, such as stability and safety, alongside secondary performance objectives. A multistage design is also given that enlarges the region of attraction (ROA) sequentially while allowing different performance metrics for each stage. A boundary for the closed-loop system's ROA is obtained from the resulting Lipschitz Lyapunov function. For control-affine nonlinear systems, the nonconvex problem is formulated as a series of conservative, but convex, well-posed optimization problems. These iteratively decrease the cost function until the design objectives are met. Since the resulting CPA Lyapunov-like functions are also Lipschitz control (or barrier) Lyapunov functions, they can be used in online quadratic programming to find minimum-norm control inputs. Numerical examples are provided to demonstrate the effectiveness of the method.