Impulse-controllability of system classes of switched differential algebraic equations

Abstract

In this paper, impulse-controllability of system classes containing switched DAEs is studied. We introduce several notions of impulse-controllability of system classes and provide a characterization of strong impulse-controllability of system classes generated by arbitrary switching signals. For a system class generated by switching signals with a fixed-mode sequence, it is shown that either almost all systems are impulse-controllable, or almost all systems are impulse-uncontrollable. Sufficient conditions for all systems in this system class to be impulse-controllable or impulse-uncontrollable are presented. Furthermore, it is observed that even if all systems in a system class are impulse-controllable, knowledge of all the switching times is generally necessary to construct an input that ensures impulse-free solutions. Consequently, it is impossible to design a controller in real time if the switching times in the future are unknown. This phenomenon can be regarded as a causality issue. Therefore, the concept of (quasi-) causal impulse-controllability is introduced, and system classes which are (quasi-) causal are characterized. Finally, necessary and sufficient conditions for a system class to be causal given some dwell-time are stated.