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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.

This article investigates the problem of joint continuous and discrete state estimation of randomly switched linear systems in which subsystems may not be observable. Estimation of both continuous state and discrete sequence simultaneously based on the same output observations is a challenging task that is inherently nonlinear and often infinite dimensional. This article presents necessary and sufficient conditions when joint estimation is possible without using a probing input. When such conditions are not satisfied, a suitably designed input must be used to achieve the goal of jointly detecting the subsystem and estimating the internal state. This article employs certain structures of randomly switched linear systems to develop algorithms that use finite-dimensional estimators for continuous states and sampled data for detecting the discrete states. The convergence analysis shows that this framework can achieve convergence. Examples and simulation case studies are presented to illustrate the main results of this article. The findings of this article can be used to form a supporting foundation for robust control.

In this article, we model a hybrid system by using a hybrid machine consisting of an automaton and a set of continuous-variable subsystems. We investigate the problem of determining the (current) discrete state of the hybrid system. Extending the terminologies in discrete-event systems, a hybrid system is detectable if its discrete state can be uniquely determined. We derive conditions for a hybrid system to be detectable. If a hybrid system is detectable, we determine its current discrete state using a two-level approach. At the discrete-event level, we construct a discrete-event observer to obtain a discrete state estimate that consists of all possible discrete states that the hybrid system may be in. At the continuous-variable level, a set of continuous-variable observers are constructed, if necessary, to uniquely determine the discrete state. An example of a dc microgrid is used to illustrate the theoretical results.