Affine parameter-dependent Lyapunov functions and real parametric uncertainty

TLDR

The tests extend quadratic stability and performance by using Lyapunov functions that depend affinely on uncertain parameters. The paper introduces new tests for robust stability and performance of linear systems with real parametric uncertainty. The tests are formulated as linear matrix inequalities, enabling numerical tractability and applicability to both constant and time‑varying parameters, and are less conservative than quadratic stability for slow variations. The tests eliminate the need for frequency sweeps and, according to numerical experiments, often outperform μ‑analysis for time‑invariant parameter uncertainty.

Abstract

This paper presents new tests to analyze the robust stability and/or performance of linear systems with uncertain real parameters. These tests are extensions of the notions of quadratic stability and performance where the fixed quadratic Lyapunov function is replaced by a Lyapunov function with affine dependence on the uncertain parameters. Admittedly with some conservatism, the construction of such parameter-dependent Lyapunov functions can be reduced to a linear matrix inequality (LMI) problem and hence is numerically tractable. These LMI-based tests are applicable to constant or time-varying uncertain parameters and are less conservative than quadratic stability in the case of slow parametric variations. They also avoid the frequency sweep needed in real-/spl mu/ analysis, and numerical experiments indicate that they often compare favorably with /spl mu/ analysis for time-invariant parameter uncertainty.