Abstract
The problem of input-to-state stability (ISS) and its integral version (iISS) are considered for switched nonlinear systems with inputs, resets and possibly unstable subsystems. For the dissipation inequalities associated with the Lyapunov function of each subsystem, it is assumed that the supply functions, which characterize the decay rate and ISS/iISS gains of the subsystems, are nonlinear. The change in the value of Lyapunov functions at switching instants is described by a sum of growth and gain functions, which are also nonlinear. Using the notion of average dwell-time (ADT) to limit the number of switching instants on an interval, and the notion of average activation time (AAT) to limit the activation time for unstable systems, a formula relating ADT and AAT is derived to guarantee ISS/iISS of the switched system. Case studies of switched systems with saturating dynamics and switched bilinear systems are included for illustration of the results.
Highlights
Switched systems—comprising a family of dynamical subsystems orchestrated by a piecewise constant switching signal—provide a mathematical framework for modeling systems where the state trajectories exhibit sudden transitions, either due to instant change in the vector field or due to jumps in some of the state variables [16,17]
We model the switched systems with average dwell-time (ADT) and activation time (AAT) constraints as a hybrid system (16)
We do not work with the strong input-to-state stability (ISS)/integral input-to-state stable (iISS) notions in this paper. This is because it is inferred from [9, Proposition 2.3] that strong ISS/iISS and ISS/iISS are equivalent for switched systems with switching signals satisfying the ADT condition, which is the class of switching signals addressed in our results
Summary
Switched systems—comprising a family of dynamical subsystems orchestrated by a piecewise constant switching signal—provide a mathematical framework for modeling systems where the state trajectories exhibit sudden transitions, either due to instant change in the vector field or due to jumps in some of the state variables [16,17]. Relaxing certain assumptions in these works, along with some developments based on converse results, iISS characterizations via Lyapunov functions for hybrid systems appear in [22] These results do not explicitly work out the stability conditions for switched systems in terms of the data associated with individual subsystems. In our conference paper [18], we only study switched systems whose subsystems are all (i)ISS and the growth of the Lyapunov functions at the switching times is not allowed to depend on the input Such assumptions have been relaxed in this work and a generalization of the results in [21] is obtained with a Lyapunov-function-based approach that allows us to handle additional nonlinearities in the supply functions.
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