On the Design of Rational Polynomial State Feedback Controllers

TL;DR

The paper addresses stabilizing nonlinear systems with rational state-feedback controllers using Sum of Squares programming to convexify the design problem. It introduces a convex iterative synthesis method that decouples the controller from the plant by embedding the controller as a constraint and co-designing a Lyapunov function via a polynomial multiplier $\lambda(x,u)$, enabling joint optimization at each step. The framework generalizes and unifies several existing rational-controller approaches, including Sontag-style analytical cancellation and completing-the-square techniques, as special cases. Numerical results on inverted-pendulum, rational-function, and polynomial systems show larger regions of attraction and improved performance for rational controllers compared with polynomial controllers and traditional iterative methods, highlighting practical gains in stability robustness and control effort under constraints.

Abstract

One of the desirable objectives in feedback control design is to formulate and solve the design problem as an optimisation problem that is convex, so that an optimal solution can be found efficiently. Unfortunately many control design problems are non-convex: approximations, relaxations, or iterative schemes are usually employed to solve them. Several such approaches have been developed in the literature, for example Sum-of-Squares (SOSs) methods have been used for systems described by polynomial dynamics. Alternatively, and relevant to this paper, one can choose a (non-unique) linear-like representation of the system and solve the resulting state-dependent Linear Matrix Inequalities (LMIs) or use SOSs optimisation techniques to derive a control law. This SOS method has been shown to effectively design polynomial and rational controllers for nonlinear polynomial systems, offering a broader class of controllers and the potential for improved performance and robustness guarantees. In this paper, we start off by considering rational functions as controllers for nonlinear systems and propose a procedure for designing such controllers by iteratively solving convex SOS optimisation problems. Our approach decouples the controller structure from the system dynamics and incorporates it as a constraint within the optimisation problem, which results in an optimisation that co-designs aspects of the controller and the Lyapunov function at the same iterative step. We theoretically establish the properties of this procedure by showing that several existing rational controller design methods can be recovered as special cases of this procedure. The proposed method is evaluated on various nonlinear benchmark system examples, demonstrating improved performance and robustness over both polynomial controllers and rational controllers obtained by existing approaches.

Figures (9)

• Figure 1: Comparison between $\mathrm{Rtanh}_1(x)$, $\mathrm{Rtanh}_2(x)$ and $\mathrm{tanh}(x)$
• Figure 2: Function Classes: Approximation Power vs. Model Complexity.
• Figure 3: A comparison between traditional iterative approaches and the proposed approach.
• Figure 4: The trajectories under rational controller from the proposed method in the constrained inverted pendulum case.
• Figure 5: Size of the ROAs at different iteration counts.

Theorems & Definitions (3)

• Definition 1
• Proposition 1: mnew5
• Proposition 2