Stability of two-dimensional SISO LTI system with bounded feedback gain that has bounded derivative

TL;DR

This work analyzes the stability of a two-dimensional SISO LTI system with uncertain linear feedback $u = \kappa(t) y$, where $\kappa$ and its derivative are bounded. By augmenting the state with the gain and transforming to polar coordinates, the authors define absolute $r$-stability and formulate a variational problem whose value function $\rho(\kappa,\theta)$ satisfies a Hamilton–Jacobi–Bellman equation in the viscosity sense. They derive sufficient conditions for absolute stability and instability in terms of $\rho(N\pi,\kappa)$ and provide a semi-Lagrangian numerical scheme to approximate $\rho$, enabling a practical stability test. The methodology is illustrated on a power-electronics synchronization example, showing less conservative stability thresholds compared to classical bounds and suggesting pathways to explicit Lyapunov constructions and potential extensions to higher dimensions.

Abstract

We consider a two-dimensional SISO LTI system closed by uncertain linear feedback. The feedback gain is time-varying, bounded, and has a bounded derivative (both bounds are known). We investigate the asymptotic stability of this system under all admissible behaviors of the gain. Note that the situation is similar to the classical absolute stability problem of Lurie--Aizerman with two differences: linearity and derivative constraint. Our method of analysis is therefore inspired by the variational ideas of Pyatnitskii, Barabanov, Margaliot, and others developed for the absolute stability problem. We derive the Hamilton--Jacobi--Bellman equation for a function describing the "most unstable" of the possible portraits of the closed-loop system. A numerical method is proposed for solving the equation. Based on the solution, sufficient conditions are formulated for the asymptotic stability and instability. The method is applied to an equation arising from the analysis of a power electronics synchronization circuit.

Figures (2)

• Figure 1: Possible characteristic directions of the HJB equation \ref{['eq: HJB equation']} depending on the value $\nu^*$ of $\nu$ that attains a minimum in \ref{['eq: HJB equation']}.
• Figure 2: Approximate solution $\rho(\kappa, \theta)$ of the HJB equation \ref{['eq: HJB equation']} in the example, computed for different values of $k$ by the method of Proposition \ref{['pr: numerical method']}. By Theorems \ref{['th: sufficient']} and \ref{['th: necessary']}, absolute stability is lost as $k$ goes below about 0.385.

Theorems & Definitions (17)

• Definition 1
• Remark 1
• Remark 2
• Remark 3
• Definition 2
• Proposition 1
• Remark 4
• Remark 5
• Lemma 1
• proof