Growth rates for persistently excited linear systems

Abstract

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 $(A,B)$

Persistently damped transport on a network of circles

“…where a persistently exciting signal α determines the activity of the control u, are usually called persistently excited systems. They have been considered in the finite-dimensional setting in [8,9,11,13,28,29], dealing mostly with problems concerning stabilizability by a linear feedback law (see [12] for a thorough presentation of persistently excited systems). In such systems, the persistently exciting signal α is a convenient tool to model several phenomena, such as failures in links between systems, resource allocation, or other internal or external processes that affect control efficiency.…”

Decay rates for stabilization of linear continuous-time systems with random switching

“…An important motivation for our work comes from the theory of persistently excited control systems, in which one considers systems of the form (1.3)ẋ(t) = Ax(t) + α(t)Bu(t), with x(t) ∈ R d , u(t) ∈ R m , A and B matrices with real entries and appropriate dimensions, and α a (T, µ)-persistently exciting (PE) signal for some positive constants T ≥ µ, i.e., a signal α ∈ L ∞ (R + , [0, 1]) satisfying, for every t ≥ 0, (1.4) t+T t α(s) ds ≥ µ (cf. Chaillet, Chitour, Loría, and Sigalotti [5], Chitour, Colonius, and Sigalotti [9], Chitour, Mazanti, and Sigalotti [10], Chitour and Sigalotti [11], Srikant and Akella [33]). Notice that, when α takes its values in {0, 1}, (1.3) can be seen as a particular case of (1.1) by adding a trivial subsystem (cf.…”

“…t+T t α(s) ds ≥ µ (cf. Chaillet, Chitour, Loría, and Sigalotti [5], Chitour, Colonius, and Sigalotti [9], Chitour, Mazanti, and Sigalotti [10], Chitour and Sigalotti [11], Srikant and Akella [33]). Notice that, when α takes its values in {0, 1}, (1.3) can be seen as a particular case of (1.1) by adding a trivial subsystem (cf.…”

Dwell-time control sets and applications to the stability analysis of linear switched systems

“…The idea of looking at the Lyapunov exponents associated with the restricted class of periodic trajectories has been introduced in [3] in order to relate the asymptotic stability of linear stochastic systems with the behavior of their large deviations, measured in terms of p-th mean Lyapunov exponents. The characterization in terms of periodic trajectories turns out to be useful also to prove the continuity of Lyapunov exponents (see [13] and also [11], where invariant control sets are used to prove continuity for Lyapunov exponents associated with systems subject to persistently exciting inputs).…”