Unifying Controller Design for Stabilizing Nonlinear Systems with Norm-Bounded Control Inputs

TL;DR

This paper introduces a unifying controller design method for stabilizing nonlinear systems with norm-bounded inputs, extending classical formulas and optimizing a cost function.

Contribution

It extends Lin-Sontag's universal formula with a generic scaling term, enabling versatile, tailored control designs and explicit optimization solutions.

Findings

Ensures asymptotic stability under input constraints

Provides explicit solutions for optimal scaling terms

Validates effectiveness through simulations

Abstract

This paper revisits a classical challenge in the design of stabilizing controllers for nonlinear systems with a norm-bounded input constraint. By extending Lin-Sontag's universal formula and introducing a generic (state-dependent) scaling term, a unifying controller design method is proposed. The incorporation of this generic scaling term gives a unified controller and enables the derivation of alternative universal formulas with various favorable properties, which makes it suitable for tailored control designs to meet specific requirements and provides versatility across different control scenarios. Additionally, we present a constructive approach to determine the optimal scaling term, leading to an explicit solution to an optimization problem, named optimization-based universal formula. The resulting controller ensures asymptotic stability, satisfies a norm-bounded input constraint, and optimizes a predefined cost function. Finally, the essential properties of the unified controllers are analyzed, including smoothness, continuity at the origin, stability margin, and inverse optimality. Simulations validate the approach, showcasing its effectiveness in addressing a challenging stabilizing control problem of a nonlinear system.