Robustness in the face of polytopic initial conditions uncertainty and polytopic system matrices uncertainty in finite-horizon linear H∞-analysis

This paper addresses a linear finite-horizon robust optimal H∞ analysis problem, where the system matrices and the system initial conditions (ICs, x0) are concurrently uncertain, both in a polytopic manner. Current finite-horizon H∞ analysis practice assumes x0 ∈R ; that is, allows infinite ICs uncertainty. This assumption is unrealistically conservative, and incompatible with the prevalent robust H∞ analysis practice of attributing finite uncertainty to the systems’s parameters/matrices. Here, the ICs uncertainty model is analogous to the (convex) uncertainty model of the system matrices. The development applies H∞ ‘first principles’, and exploits convexity over the matrices uncertainty polytope, over the ICs uncertainty polytope, and over time (‘time-convexity’). A conjecture regarding polytopic final-state convexity in this setup is given, and applied, to overcome non-convexity of the statetransition matrix with respect to the system matrices. A detailed numerical example shows a dramatic advantage of the methods which do not constrain the final Lyapunov function.