Section: New Results

New results: switched systems

• In [2] we study the control system $\dot{x}=Ax+\alpha \left(t\right)bu$ where the pair $(A,b)$ is controllable, $x\in {ℝ}^2$, $u\in ℝ$ is a scalar control and the unknown signal $\alpha :{ℝ}_{+}\to [0,1]$ is $(T,\mu )$-persistently exciting (PE), i.e., there exists $T\ge \mu >0$ such that, for all $t\in {ℝ}_{+}$, ${\int }_t^{t+T}\alpha \left(s\right)ds\ge \mu$. We are interested in the stabilization problem of this system by a linear state feedback $u=-Kx$. In [2] , we positively answer a question asked in [52] and prove the following: Assume that the class of $(T,\mu )$-PE 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 $u=-Kx$ where $K$ only depends on $(A,b)$ and $T,\mu ,M$ so that, for every $M$-Lipschitzian $(T,\mu )$-PE signal, the rate of exponential decay of the time-varying system $\dot{x}=(A-\alpha \left(t\right)bK)x$ is greater than $C$.

  In [20] we consider a family of linear control systems $\dot{x}=Ax+\alpha Bu$ where $\alpha$ belongs to a given class of persistently exciting signals. We seek maximal $\alpha$-uniform stabilisation and destabilisation by means of linear feedbacks $u=Kx$. We extend previous results obtained for bidimensional single-input linear control systems to the general case as follows: if the pair $(A,B)$ verifies a certain Lie bracket generating condition, then the maximal rate of convergence of $(A,B)$ is equal to the maximal rate of divergence of $(-A,-B)$. We also provide more precise results in the general single-input case, where the above result is obtained under the sole assumption of controllability of the pair $(A,B)$.

  The paper [24] considers the stabilization to the origin of a persistently excited linear system by means of a linear state feedback, where we suppose that the feedback law is not applied instantaneously, but after a certain positive delay (not necessarily constant). The main result is that, under certain spectral hypotheses on the linear system, stabilization by means of a linear delayed feedback is indeed possible, generalizing a previous result already known for non-delayed feedback laws.

  Several problems and results related with persistent excitation and stabilization are discussed in the survey [11] . These problems and results deal with both finite- and infinite-dimensional systems.

• In [7] we consider several time-discretization algorithms for singularly perturbed switched systems. The algorithms correspond to different sampling times and the discretization procedure respects the splitting of each mode in fast and slow dynamics. We study whether such algorithms preserve the asymptotic or quadratic stability of the original continuous-time singularly perturbed switched system.

• In [10] we consider affine switched systems as perturbations of linear ones, the equilibria playing the role of perturbation parameters. We study the stability properties of an affine switched system under arbitrary switching, assuming that the corresponding linear system is uniformly exponentially stable. It turns out that the affine system admits a minimal invariant set $\Omega$, whose properties we investigate. In the two-dimensional bi-switched case when both subsystems have non-real eigenvalues we are able to characterize $\Omega$ completely and to prove that all trajectories of the system converge to $\Omega$. We also explore the behavior of minimal-time trajectories in $\Omega$ by constructing optimal syntheses.

• In [21] we give a collection of converse Lyapunov–Krasovskii theorems for uncertain retarded differential equations. We show that the existence of a weakly degenerate Lyapunov–Krasovskii functional is a necessary and sufficient condition for the global exponential stability of the linear retarded functional differential equations. This is carried out using the switched system transformation approach.