[Paper Review] An Extension of Barbashin-Krasovski-LaSalle Theorem to a Class of Nonautonomous Systems

Radu Bălan|ArXiv.org|Jun 22, 2005

Stability and Control of Uncertain Systems4 references11 citations

TL;DR

This paper extends the Barbashin-Krasovski-LaSalle invariance principle to a class of nonautonomous systems where the vector field is time-invariant on the zero-set of the derivative of a Lyapunov function. By requiring the restricted dynamics on this set to be time-invariant and asymptotically stable, the full system achieves uniform asymptotic stability. The key contribution is a nonlinear extension of the linear detectability result and a sufficient condition for robust stabilizability of affine control systems via Hamilton-Jacobi equations.

ABSTRACT

In this paper we give an extension of the Barbashin-Krasovski-LaSalle Theorem to a class of time-varying dynamical systems, namely the class of systems for which the restricted vector field to the zero-set of the time derivative of the Liapunov function is time invariant and this set includes some trajectories. Our goal is to improve the sufficient conditions for the case of uniform asymptotic stability of the equilibrium. We obtain an extension of an well-known linear result to the case of zero-state detectability (given (C,A) a detectable pair, if there exists a positive semidefinite matrix P>=0 such that: A^TP+PA+C^TC=0 then A is Hurwitz - i.e. it has all eigenvalues with negative real part) as well as a result about robust stabilizability of nonlinear affine control systems.

Motivation & Objective

• To extend the Barbashin-Krasovski-LaSalle invariance principle to time-varying systems where the vector field is time-invariant on the zero-set of the Lyapunov function derivative.
• To establish sufficient conditions for uniform asymptotic stability of equilibrium points in nonautonomous systems under weaker assumptions than classical LaSalle theorems.
• To generalize the linear detectability result (if (C,A) is detectable and A^TP + PA + C^TC = 0 has P ≥ 0, then A is Hurwitz) to nonlinear systems using strong zero-state detectability.
• To provide a sufficient condition for robust stabilizability of nonlinear affine control systems under time-invariant and time-varying perturbations using a Hamilton-Jacobi inequality.

Proposed method

• Introduce a C¹ Lyapunov function V with non-increasing derivative W(x) ≤ 0 along trajectories.
• Define the zero-set E = {x ∈ D | W(x) = 0}, and assume f(t,x) is time-invariant on E.
• Restrict the dynamics to the maximal positive invariant set N ⊂ E and analyze its stability.
• Apply Theorem 1: if the restricted dynamics on N is asymptotically stable, then the full system is uniformly asymptotically stable.
• Use the Hamilton-Jacobi equation ∇V·f + ∇V·g(φ(h) + p(h,t)) ≤ 0 to derive robust stabilizability conditions.
• Define two classes of perturbations P₁ (time-invariant) and P₂ (time-varying) and show that W(x) ≤ 0 holds in both cases, leading to stability via Theorem 1.

Experimental results

Research questions

• RQ1 Under what conditions can the classical LaSalle invariance principle be extended to nonautonomous systems with time-varying vector fields?
• RQ2 How can the stability of the full system be determined when the vector field is time-invariant only on the zero-set of the Lyapunov derivative?
• RQ3 Can the linear detectability result be generalized to nonlinear systems using a Lyapunov-based approach?
• RQ4 What conditions ensure robust stabilizability of affine nonlinear control systems under sector-bounded perturbations?
• RQ5 How does strong zero-state detectability relate to the existence of a positive semidefinite solution to a Hamilton-Jacobi inequality?

Key findings

• The full system is uniformly asymptotically stable if the dynamics restricted to the maximal positive invariant set N ⊂ E is asymptotically stable.
• A nonlinear extension of the linear detectability result is established: if (h,f) is strongly zero-state detectable and a solution P ≥ 0 exists to AᵀP + PA + CᵀC = 0, then A is Hurwitz.
• For the planar system ẋ₁ = -x₁³ + u, ẋ₂ = -x₂³, y = x₂³, the feedback φ(y) = y robustly stabilizes the system for any a > 0, with V(x₁,x₂) = (a²/4)x₂⁴ as a valid Lyapunov function.
• The robust stabilizability condition holds for perturbations in P₁ ∪ P₂, with the time derivative of V bounded by W(x) ≤ 0, where W(x) = -(2aε - ε²)(φ(h(x)))² for P₂ and W(x) = (p(h(x)))² - a²(φ(h(x)))² for P₁.
• The existence of a positive semidefinite solution to the Hamilton-Jacobi equation ensures robust stabilizability under both time-invariant and time-varying perturbations.
• The result implies that feedbacks satisfying |φ(y)| ≤ a|y| can stabilize the system robustly for arbitrarily large a > 0, as demonstrated in Example 3.