Control systems design with enlarged stability domains for nonlinear constrained systems via sum‐of‐squares optimization

Abstract

Designing safe control laws for nonlinear systems is challenging, especially when ensuring stability under actuator saturation and state constraints. A key aspect is embedding controllers with a Region of Attraction (ROA), which defines initial conditions guaranteeing convergence to a stable equilibrium point (EP). Controllers with enlarged ROAs are desirable for improved stability, performance, and robustness, but their design is a complex and computationally intensive task. This paper proposes a new methodology for designing controllers with expanded ROAs for nonlinear polynomial systems facing actuator saturation and state constraints, utilizing the Sum‐of‐Squares (SOS) optimization technique. Our approach proposes the use of multiple variable‐sized regions defined by multiple positive definite functions known as Shape Functions and the S‐procedure within the SOS optimization framework. This approach synthesizes a control law with enlarged ROA, even in the presence of constraints in the system. We developed a numerically efficient algorithm to solve this problem, yielding superior results. Extensive numerical simulations demonstrate the effectiveness of our approach. Additionally, we verified the estimated ROA under the designed control law using Monte Carlo simulations, confirming the robustness and practical applicability of our proposed method.