Optimization of highly uncertain feedback systems in H-infinity

This dissertation presents a new mathematical framework to optimize performance of multi-input multi-output feedback systems subject to large dynamical uncertainty. Optimal performance is represented by two-disc type optimization problems defined in appropriate function spaces. These optimizations are interpreted as shortest distance minimizations in special vector valued Hinfinity spaces. Characterization of various predual and dual maximizations shows existence of optimal solutions. Alignment conditions are exploited to show that the optimal solution is flat or "allpass", therefore generalizing a result obtained previously for single-input single-output systems. Detailed analysis gave further quantitative results, in particular absolute continuity of extremal measures is proven. This lead to extremal identities which provide a test of optimality.;A novel operator theoretic framework is next developed. Key multiplication operators acting on particular vector-valued Hardy spaces are introduced. Subsequently, the optimizations are shown to be equal to the induced norms of specific operators. The latter are Banach space projections of multiplication operators, therefore analogous to the Sarason operator well known in the standard Hinfinity theory. Further computations show that these operators are in fact combinations of multiplication and Toepltiz operators. Explicit formulas for the optimal controllers are provided through existence of maximal vectors. Then "infinite matrix" representation with respect to a canonical basis is given, and the norms of the relevant operators are approximated by special matrix norms.;These results are further generalized to unstable systems using coprime factorization techniques with similar conclusions. Relation to the standard two-block Hinfinity problem is investigated in the context of duality and operator theory. The optimal solution is then shown to be flat, implying that a well known Hankel-Toeplitz operator achieves its norm on its discrete spectrum for (possibly) infinite dimensional systems.;Finally, the optimal robust disturbance attenuation problem for continuous time-varying plants subject to continuous time-varying uncertainty, is shown to reduce to finding the smallest fixed point of a two-disc type optimization problem under continuous time-varying control laws. Duality is then applied in the context of nest algebra of causal stable systems, and shows existence of optimal continuous time-varying controllers. It is also shown that for time-invariant nominal plants under time-varying uncertainty, continuous time-varying control laws offer no advantage over time-invariant ones.