A first-order Lyapunov robustness method for linear systems with uncertain parameters

A method for stability-robustness analysis based on a quadratic Lyapunov function that varies linearly with uncertainty parameters is derived. Linear time-invariant systems with structured uncertainties are discussed. The Lyapunov function is optimized numerically to maximize the robustness region in parameter space. Numerical results are given for four examples in which the first-order method is compared to previous Lyapunov methods for robustness analysis. While the zero-order method is slightly better than the first-order method for one example, the first-order method is clearly superior in the other three, more realistic, examples. The first-order method is especially superior for applications to active control of flexible structures, where robustness with respect to unmodeled coupling between modeled modes are important issues. For such applications, the first-order method is much better at detecting the increased robustness associated with increased separation between frequencies