Entropy of controlled invariant subspaces

Abstract

Abstract. For continuous-time linear control systems, a concept of entropy for controlled and almost controlled invariant subspaces is introduced. Upper bounds for the entropy in terms of the eigenvalues of the autonomous subsystem are derived. September 15, 2009Key words. invariance entropy, topological entropy, geometric control, almost (A,B)-invariant subspaces, eigenvalue bounds.AMS subject classifications. 94A17, 37B40, 93C151. Introduction. Controlled and conditioned invariant subspaces of linear dynamical systems play a crucial role in understanding controller design problems such as disturbance decoupling, filtering, robust observer design, and high gain state feedback. In fact, starting form the early work of Basile-Morro [1] and Wonham [14], controlled invariant subspaces became a cornerstone of geometric control theory. In this paper, we begin an investigation of how geometric control design via controlled invariant subspaces is affected by entropy estimates and associated data rate constraints. The main motivation for this circle of ideas comes from the increasing needs of controlling systems with communication constraints, i.e. for systems where the state passes through a communication channel and may thus not be fully available to the controller.As a starting point for such an investigation, we associate to any almost (A, B)-invariant subspace V of a linear control system a number, called the invariance entropy of V, that measures how difficult it is, using open loop controls, to keep the system in V . It is defined by the exponential growth rate of the number of controls necessary to keep the system in an arbitrarily small ε-neighborhood of V . More generally, by extending the familiar notion of topological entropy for the flow defined by A, we define the entropy of an arbitary linear subspace V of the state space. We show that the invariance entropy is finite for any almost (A, B)-invariant subspace and derive upper bounds in terms of the sum of the eigenvalues of A with positive real part. Sharper upper bounds are derived for specific classes of linear systems.Our approach partially extends and follows that by Colonius and Kawan [4], where an entropy-like notion was proposed for controlled invariance of compact subsets of the state space of general control systems. Their approach in turn has been motivated by the work of Nair et al. [8] on feedback entropies for nonlinear discrete-time systems. The entropy notion considered here may be regarded as a lower bound for the minimum data rate (take the logarithm with base 2 instead of the natural logarithm used, for convenience, in the present paper.) More explicit relations to data rates are given in Kawan's PhD thesis [9].

A note on topological feedback entropy and invariance entropy

“…Later, Colonius and Kawan [4] introduced invariance entropy for continuous-time control systems. In the doctoral thesis [11] by Kawan and in his subsequent papers [12,13,14], as well as in Colonius and Kawan [5] and Colonius and Helmke [3] this notion has been further elaborated and used to analyze properties of control systems. The forthcoming monograph [15] will give a comprehensive presentation of these results.…”

Entropy properties of deterministic control systems

“…Further work: The encyclopedia entry [24] gives condensed information on topological invariance entropy. The growth rate of the number of open-loop control functions as a measure for the information needed to achieve invariance can also be used for other control tasks: In [3] this is done for exponential stabilization; in [7] and in [4] for controlled invariant subspaces. The contribution [6] discusses generalizations of topological invariance entropy to semigroup actions.…”

“…and for k ∈ N the measure µ is also conditionally invariant for T k Q with constant ρ k . Again we suppose that the compact set Q satisfies the invariance condition (7). Since µ lives in U ×Q we construct certain partitions for U × Q whose entropy with respect to µ will be used to define the metric invariance entropy.…”

“…Theorem 7: Let Q be a compact subset of the state space of control system (5) satisfying condition (7). Consider for the associated skew product map T from (6) the topological invariance entropy as well as the feedback invariance µentropy and the controlled invariance µ-entropy with respect to a conditionally invariant measure µ.…”

On a connection between the reliability of multi-channel systems and the notion of controlled-invariance entropy

“…Notably, in the paper [17], Nair et al have introduced the notion of topological feedback entropy, which is based on the ideas of [1], to quantify the minimum rate at which deterministic discrete-time dynamical systems generate information relevant to the control objective of set-invariance. More recently, the notion of controlled-invariance entropy (as well as the notion of almost invariance entropy) has been studied for continuous-time control systems in [6], [8], [14] and [9] based on the metric-space technique of [5]. It is noted that such an invariant entropy provides a measure of the smallest growth rate for the number of open-loop control functions that are needed to confine the states within an arbitrarily small distance (in the sense of gap metric) from a given subspace.…”