Stabilizability and optimal control of switched differential algebraic equations

In this thesis control of dynamical systems with switches is considered. Examples of such systems are electronic circuits and mechanical systems. The switches are induced by abrupt structural changes due to component failure or physical switches. In the case of constraints on the dynamics, the state of the system can only take certain values and not only differential equations are involved in modeling the system, but also algebraic equations. An important question in control problems is often how well a certain controller performs. Some controllers require little energy, but induce undesired behavior of the system, whereas others perform well in terms of the systems behavior but require a lot of energy. It turns out that in general an optimal controller does not exist. However, necessary and sufficient conditions for the existence of optimal controller given a quadratic cost functional are presented in this thesis. Besides quantitative properties also some qualitative properties are investigated. The systems considered exhibit discontinuous behavior and Dirac impulses, whereas especially Dirac impulses are practically undesirable. Dirac impulses occur in practice in the form of hydraulic shocks in fluid networks or sparks in electronic circuits. The possibility to avoid Dirac impulses is also studied and necessary and sufficient conditions are given

Determinability and state estimation for switched differential-algebraic equations

International audienceThe problem of state reconstruction and estimation is considered for a class of switched dynamical systems whose subsystems are modeled using linear differential-algebraic equations (DAEs). Since this system class imposes time-varying dynamic and static (in the form of algebraic constraints) relations on the evolution of state trajectories, an appropriate notion of observability is presented which accommodates these phenomena. Based on this notion, we first derive a formula for the reconstruction of the state of the system where we explicitly obtain an injective mapping from the output to the state. In practice, such a mapping may be difficult to realize numerically and hence a class of estimators is proposed which ensures that the state estimate converges asymptotically to the real state of the system