Robust root clustering for linear uncertain systems using generalized lyapunov theory

In this paper, the problem of matrix root clustering in sub-regions of complex plane for linear state space models with real parameter uncertainty is considered. The nominal matrix root clustering theory of Gutman and Jury (1981, IEEE Trans. Aut. Control, AC-26, 403) using Generalized Lyapunov Equation is extended to the perturbed matrix case and bounds are derived on the perturbation to maintain root clustering inside a given region. The theory allows us to get an explicit relationship between the parameters of the root clustering region and the uncertainty range of the parameter space. The current literature available on perturbation bounds for robust stability becomes a special case of this unified theory.