Switched Linear Systems: Observability and Observers
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Switched linear systems have long been subject to high interest and intense research efforts, not only because many real world systems happen to exhibit switching behaviors, but also because the control of many complex systems is only possible via the combination of classical continuous control laws with supervisory switching logic. A particularly important problem is that of estimator and observer design, since the state of a system is usually only available through partial, often noise-corrupted, measurements. Even though hybrid estimation has been around for at least thirty years, a veil of mystery has surrounded the concept of ``observability' in switched linear systems. It is not until recently, with the recent renewal of interest toward deterministic hybrid systems, that observer design and observability analysis have fuelled sustained research efforts. It is in this context that this work is grounded. More precisely, the objective of this research is twofold: - To define proper concepts of observability in discrete-time switched linear systems, to characterize them, and to analyze their main properties, among which decidability is of special importance. - To propose and analyze observers - deadbeat and asymptotic - for such systems. The main contributions of this dissertation are as follows. It is shown that pathwise observability, i.e. state observability under arbitrary mode sequences, is decidable. Furthermore, the Kalman-Bertram sampling criterion is carried over to switched linear systems. Under unknown modes, mode and state observability are both characterized through simple linear algebraic tests, and are shown to be decidable in the autonomous case. As for asymptotic observers, a direct algebraic approach is proposed for the class of linear systems subjected to switching in the measurement equation.