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 ε Rⁿ; 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.