Section: New Results

New results: switched systems

New results on switched systems have been obtained in three directions:

• Discrete-time systems. In [14] we dealt with the stability properties of linear discrete-time switched systems with polytopic sets of modes. The most classical and viable way of studying the uniform asymptotic stability of such a system is to check for the existence of a quadratic Lyapunov function. It is known from the literature that letting the Lyapunov function depend on the time-varying switching parameter improves the chance that a quadratic Lyapunov function exists. The contribution of [14] is twofold. We first proved that under a non-degeneracy assumption the dependence on the switching function can be actually assumed to be linear with no prejudice on the effectiveness of the method. Moreover, we showed that no gain is obtained even if we allow the Lyapunov function to depend on the time. Second, we introduced the notion of eventual accessible sets and we showed that, in the degenerate case, it leads to a relaxation of the LMI conditions to check stability of switched linear systems. As a consequence, equivalence between different notions of quadratic stability can still be established under an additional assumption but, in general, allowing the Lyapunov function to depend on time leads to less conservative LMI conditions, as we explicitly showed through an example. We also discussed the case where the variation of the switching parameter is bounded by a prescribed constant between two subsequent times.

• Continuous-time systems subject to persistent-excitation. In [11] we studied linear control systems for which the controlled part can be switched off by a signal subject to a persistent excitation condition. We were interested in the stabilization problem of this system by a linear state feedback and we positively answered a question asked in [41] , proving the following: Assume that the class of persistently exciting signals is restricted to those which are $M$-Lipschitzian, where $M>0$ is a positive constant. Then, given any $C>0$, there exists a linear state feedback depending on the class of signals under consideration (but not an individual signal) so that the rate of exponential decay of the time-varying system associated with any signal is greater than $C$.

• Infinite-dimensional continuous-time systems. In [13] we partially extended the analysis of finite-dimensional systems subject to persistently exciting signals to the case of systems driven by PDEs. More precisely, we studied the asymptotic stability of a dissipative evolution in a Hilbert space subject to intermittent damping. We observed that, even if the intermittence satisfies a persistent excitation condition, if the Hilbert space is infinite-dimensional then the system needs not being asymptotically stable (not even in the weak sense). Exponential stability is recovered under a generalized observability inequality, allowing for time-domains that are not intervals. Weak asymptotic stability is obtained under a similarly generalized unique continuation principle. Strong asymptotic stability is proved for intermittences that do not necessarily satisfy some persistent excitation condition, evaluating their total contribution to the decay of the trajectories of the damped system. Our results are discussed using the example of the wave equation and the linear Schrödinger equation.