On the parametrization of conditioned invariant subspaces and observer theory

“…via a unique monic polynomial t of degree d; see Fuhrmann and Helmke [7]. Let V * (V * ) denote the largest (smallest) shift invariant subspace of X d contained in V (containing V ).…”

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confidence: 99%

Entropy of controlled invariant subspaces

Colonius

Helmke

2014

Z Angew Math Mech

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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].

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“…Note that the block-Toeplitz matrices X ∈ Ker(ad A * ) can be interpreted as partial reachability matrices X = (g 1 , F g 1 , F 2 g 1 , g 2 ), where F := A and g 1 := (a, b, c, d) , g 2 := (0, 0, f, e) . Indeed, this is no coincidence, as can be seen from [10].…”

Section: A Resolution Of Singularities For Nilpotent Matricesmentioning

confidence: 91%

“…spectral factorization, linear quadratic control, H ∞ and game theory, as well as observer theory, filtering and estimation. However, it is only until recently, that first attempts have been made towards a better understanding of the geometry of the set of conditioned invariant subspaces Inv k (C, A); see [9], [10], [19], [6], [17]. The recent Ph.D. thesis [29] contains a comprehensive summary.…”

Section: Introductionmentioning

confidence: 99%

Conditioned Invariant Subspaces and the Geometry of Nilpotent Matrices

Helmke

Trumpf

New Directions and Applications in Control Theory

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On the parametrization of conditioned invariant subspaces and observer theory

Cited by 33 publications

References 20 publications

An Interpretation of Rosenbrock's Theorem via Local Rings

An Interpretation of Rosenbrock's Theorem via Local Rings

Entropy of controlled invariant subspaces

Conditioned Invariant Subspaces and the Geometry of Nilpotent Matrices

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