Invertibility and input-to-state stability of switched systems and applications in adaptive control

This dissertation aims to study basic properties---invertibility and stability---of switched systems and their applications in control.; We formulate the invertibility problem for switched linear systems, which concerns finding conditions on a switched system so that one can uniquely recover the switching signal and the input from an output and an initial state. In solving the invertibility problem, we introduce the concept of singular pairs and we provide a necessary and sufficient solution for invertibility of switched linear systems, which says that every subsystem should be invertible and there is no singular pair. We propose a switching inversion algorithm for switched systems to find inputs and switching signals that generates a given output starting at a given initial state.; Another result of the dissertation addresses stability of switched nonlinear systems. Unlike switched linear systems where there always exists a constant switching gain among the Lyapunov functions of the subsystems, the existence of such constant gains is not guaranteed for switched nonlinear systems in general. We provide conditions on how slow the switching signals should be in order to guarantee input-to-state stability (or asymptotic stability) of a switched system if all the subsystems are input-to-state stable (or asymptotically stable, respectively), both when a constant switching gain exists and when it does not. The slowly switching conditions are characterized via switching profiles, dwell-time switching, and average dwell-time switching.; As control applications of switched systems, we apply our stability, results for switched systems to the problem of adaptively controlling uncertain nonlinear plants and linear time-varying plants. For uncertain nonlinear plants with hounded noise and disturbances, we show that using supervisory control, all the closed-loop signals can be kept bounded for arbitrary initial conditions when the controllers provide the ISS property with respect to the estimation errors. We also show that supervisory control is capable of stabilizing uncertain linear plants with large parameter variation in the presence of unmodeled dynamics and hounded noise and disturbances, provided that the unmodeled dynamics are small enough and the parameters vary slowly enough as described by switching profiles.