Spatiotemporal systems: gradual variations, identification, adaptation and robustness

Motivated by the increasing size of complex systems by mere interconnection of simple units, this dissertation considers a set of important open research problems related to the stability, identi fication, adaptation and robustness of spatiotemporal systems. First, we consider the l_infinity stability of linear spatiotemporally varying (LSTV) systems when the underlying controllers are designed based on local linear spatiotemporally invariant (LSTI) approximants. We show that the l_infinity to l_infinity performance of global LSTV systems cannot be much worse than the worst frozen spatially and temporally l_infinity to l_infinity performance, given that the rates of variation of the plant and the controller are su ciently small. Next, we consider the problem of system identi fication of LSTI systems where the subsystems cooperatively attempt to identify the dynamics common to every one. We propose a distributed projection algorithm that guarantees to bring the local estimates arbitrarily close to each other for large enough time, hence resulting in a slowly varying spatiotemporal system. Coupled with the results on the stability of LSTV systems, we next propose an indirect adaptive control scheme based on certainty equivalence. Last, we look at the robust l_infinity and l2 stability of LSTI systems and address the necessary and su cient conditions for robust stability in the presence of LSTV perturbations. We also investigate the robust stability of these systems with the underlying perturbations being nonlinear spatiotemporally invariant. We show that the robustness conditions are analogous to the scaled small gain condition (which is equivalent to a spectral radius condition and a linear matrix inequality for the l_infinity and l2 case respectively) derived for standard linear time invariant models subject to linear time varying or nonlinear perturbations. Future research directions are also provided