Almost-Sure Robust Stabilization of Randomly Switched Linear Systems With Uncontrollable Subsystems

This article investigates state feedback design for achieving almost-sure robust stabilization of randomly switched linear systems whose subsystems are uncontrollable and whose models are subject to nonlinear modeling errors due to, for example, linearization. Complex systems are often uncontrollable under a fixed configuration from a single control input. However, when control actions can be used sequentially and collaboratively through different system configurations, stabilization can be potentially achieved. This design problem encounters some fundamental issues that must be resolved, involving mostly suitable coordinated implementations of state decomposition, feedback pole-placement design, usage of stochastic information on the switching process, coupling of controllable and uncontrollable substates, subsystem interactions, and modeling errors. The common state feedback on controllable substates can lead to unstable closed-loop systems, due to substate coupling. A modified control algorithm is introduced that decouples substates and designs feedback gains simultaneously. Further complications arise when subsystem interaction destabilizes the system. Some structural conditions are shown to be essential for achieving almost-sure stabilization. A design procedure that integrates feedback gain selection and switching information is introduced to achieve almost-sure stability for the closed-loop system. Robustness of the design procedure is established. Examples and simulation case studies are presented to illustrate the main algorithms and stabilization properties.