Control of Switched Linear Systems: Adaptation and Robustness

As a special class of hybrid systems, switched systems have attracted a lot of attention in the last decade due to theoretical and practical interests. When controlling switched systems, a ubiquitous problem is the presence of large parametric uncertainties and external disturbances. However, the state of the art on adaptive and robust control of switched linear systems is not satisfactory and due to the existence of theoretical gaps between adaptive and robust control for switched linear systems and non-switched linear systems. To this end, this thesis has been successfully closed some theoretical gaps, which is divided into two parts.
In the first part of this thesis, to start with, we have extended the state-of-the-art results using extended notions of dwell time and of average dwell time: mode-dependent dwell time and mode-dependent average dwell time, respectively. This gives rise to less conservative switching signals. To address the cases in which the next subsystem to be switched on is known, we propose a new time-dependent switching scheme: mode-mode-dependent dwell time, which not only exploits the information of the current subsystem, but also of the next subsystem. Subsequently, an adaptive law for uncertain switched linear systems has been introduced, which fills the theoretical gaps between adaptive control of non-switched linear systems and of switched linear systems. The proposed adaptive law and switching law based on dwell time guarantee asymptotic convergence of the tracking error to zero and, with a persistent exciting reference input, convergence of parameter estimates to nominal parameters asymptotically. To conclude the first part of this thesis, the adaptive law for switched linear systems has been modified using the ideas of parameter projection and leakage method, depending on the available a priori information: when the bounds of uncertain parameters are known, parameter projection is adopted; otherwise, the leakage method is used. The resulting adaptive closed loop system is shown to be global uniform ultimate bounded in the presence of external disturbances.
In the second part of this thesis, adaptive and robust stabilization of switched linear systems have been investigated. Based on the stability conditions, adaptive stabilization of uncertain asynchronously switched systems is studied. Furthermore, in the presence of discontinuous time-varying delays, neither Krasovskii nor Razumikhin techniques can be successfully applied to adaptive stabilization of uncertain switched time-delay systems. A new adaptive control scheme for switched time-delay systems is developed that can handle impulsive behavior in states and time-varying delays with discontinuities. At the core of the proposed scheme is a Lyapunov function with a dynamically time-varying coefficient, which allows the Lyapunov function to be non-increasing at the switching instants. The control scheme substantially enlarges the class of uncertain switched systems for which the adaptive stabilization problem can be solved. Furthermore, in the presence of switching delays between a mode change and activation of its corresponding controller, enhanced stability criteria are investigated, whose novelty consists in continuity of the Lyapunov function at the switching instants and discontinuity when the system modes and controller modes are matched. The proposed Lyapunov function can be used to guarantee a finite non-weighted L2 gain for asynchronously switched systems, for which methods proposed in literature are inconclusive.