Reduced conservatism in stability robustness bounds by state transformation

This note addresses the issue of "conservatism" in the time domain stability robustness bounds obtained by the Lyapunov approach. A state transformation is employed to improve the upper bounds on the linear time-varying perturbation of an asymptotically stable linear time-invariant system for robust stability. This improvement is due to the variance of the conservatism of the Lyapunov approach with respect to the basis of the vector space in which the Lyapunov function is constructed. Improved bounds are obtained, using a transformation, on elemental and vector norms of perturbations (i.e., structured perturbations) as well as on a matrix norm of perturbations (i.e., unstructured perturbations). For the case of a diagonal transformation, an algorithm is proposed to find the "optimal" transformation. Several examples are presented to illustrate the proposed analysis.