Global stabilization of nonlinear systems via hybrid implementation of dynamic continuous-time local controllers

doi:10.1016/j.automatica.2019.04.002

Automatica

Volume 106, August 2019, Pages 401-405

https://doi.org/10.1016/j.automatica.2019.04.002 Get rights and content

Abstract

Given a continuous-time system and a dynamic control law such that the closed-loop system satisfies standard Lyapunov conditions for local asymptotic stability, we propose a hybrid implementation of the continuous-time control law. We demonstrate that subject to certain “relaxed” conditions, the hybrid implementation yields global asymptotic stability properties. These conditions can be further specialized to yield local/regional asymptotic stability with an enlarged basin of attraction with respect to the original control law. Two illustrative numerical examples are provided to demonstrate the main results.

Section snippets

Introduction and preliminaries

Consider a nonlinear system described by the equation $\dot{x} = f (x, u),$ where $x (t) \in R^{n}$ denotes the state, $u (t) \in R^{m}$ is the control input and the mapping $f : R^{n} \times R^{m} \to R^{n}$ is assumed to be $C^{k}$ for some sufficiently large $k \in N$ . Suppose, in addition, that $f (0, 0) = 0$ , namely the origin is an equilibrium point of the unforced system.

Definition 1

Consider the nonlinear system $\dot{ξ} = α (x, ξ), u = β (x, ξ),$ with state $ξ (t) \in R^{s}$ , $s \in N$ , where the mappings $α : R^{n} \times R^{s} \to R^{s}$ , $α (0, 0) = 0$ , and $β : R^{n} \times R^{s} \to R^{m}$ , $β (0, 0) = 0$ , are in $C^{k}$ . Then, system (2) is a globally

Hybrid implementation of dynamic continuous-time controllers

In this section we discuss a control design technique – based on the knowledge of a locally stabilizing controller for system (1) – that guarantees global convergence. Moreover, we formalize the notion of hybrid implementation of the dynamic control law (2).

Assumption 1

A locally stabilizing controller of the form of (2), together with the underlying functions $V$ and $ρ$ , is given for (1).

To provide a concise statement of the results, let $L (x, ξ) ≜ \frac{\partial V (x, ξ)}{\partial x} f (x, β (x, ξ)) + \frac{\partial V (x, ξ)}{\partial ξ} α (x, ξ),$ for any $(x, ξ) \in R^{n} \times R^{s}$ , and

Numerical simulations

To corroborate the above theoretical analysis, in this section, we present two examples of application of the hybrid implementation of local controllers.

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^☆: The material in this paper was not presented at any conference. This paper was recommended for publication in revised form by Associate Editor Tingshu Hu under the direction of Editor André L. Tits.

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Technical communiqueGlobal stabilization of nonlinear systems via hybrid implementation of dynamic continuous-time local controllers☆

Abstract

Section snippets

Introduction and preliminaries

Hybrid implementation of dynamic continuous-time controllers

Numerical simulations

Systems & Control Letters

A variation on a random coordinate minimization method for constrained polynomial optimization

IEEE Control Systems Letter

Analysis and design of event-triggered control algorithms using hybrid systems tools

Quantifier elimination for the elementary theory of real closed fields by cylindrical algebraic decomposition

Lecture Notes in Computer Science

Hybrid dynamical systems

IEEE Control Systems

Hybrid dynamical systems: modeling, stability, and robustness

Technical communique
Global stabilization of nonlinear systems via hybrid implementation of dynamic continuous-time local controllers☆