Balancing dwell times for switching linear systems

Switching Linear Systems (SLSs) are a subclass of hybrid systems characterized by a Finite State Machine (FSM) and a set of linear dynamical systems, each corresponding to a state of the FSM. The transition between two di?erent states of the FSM is caused by external uncontrollable events that act as discrete disturbances. In the past few years structural properties of SLSs have been the topic of intensive study and in particular much work has been devoted to the attempt of characterizing their stability and/or stabilizability properties. We focus on the class of uncontrolled SLSs with a dwell time associated to each transition. Loosely, a dwell time function assigns to each transition a dwell time that serves as a minimal delay for the transitions. Notice that in our setting the dwell time is associated with transitions rather than with locations. The motivation to use the notion of dwell time function lies in the possibility to quantify the balance between long delays for some transitions and short delays for others. For instance, in a cycle of transitions, instantaneous transitions could be compensated by long delays elsewhere in the cycle. A recent result (see ”Can linear stabilizability analysis be generalized to switching systems?” by E. De Santis, M.D. Di Benedetto, G. Pola), which extends Kalman decomposition to the class of controlled SLSs, shows that a controlled SLS is asymptotically stabilizable if and only if an uncontrolled SLS, appropriately associated to the controlled SLS, is asymptotically stable. Then, the stabilizability problem for the class of controlled SLSs directly translates to the stability analysis of uncontrolled SLSs. Therefore, we focus on stability problems for uncontrolled SLSs. It is well-known that if transitions are sufficiently delayed and if the dynamics in each location is asymptotically stable, then the uncontrolled SLS is asymptotically stable. The case of stability under quadratic Lyapunov function analysis is investigated. In this special case, an explicit condition on the dwell time function that ensures asymptotic stability is provided. Although this condition is based on estimates and is therefore conservative, it yields the possibility of instantaneous transitions when applied to the case where a common quadratic Lyapunov function exists. With respect to previous work in this research area, our approach gives less conservative conditions.