Realization and elimination in rational representations of behaviors

Gottimukkala, Sasanka V.; Trentelman, Harry L.; Fiaz, Shaik

doi:10.1016/j.sysconle.2013.04.007

Cited by 3 publications

(3 citation statements)

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“…For a given rational matrix R, we call a representation of B as R( d dt )w = 0 a rational kernel representation of B and write B = kerR( d dt ). For additional material on rational representations we refer to [6] and [7]. In this paper we will often assume that the rational matrices R(s) used in kernel representations have full row rank over the field of real rational functions.…”

Section: Polynomial and Rational Kernel Representationsmentioning

confidence: 99%

“…We now study the related question whether the triangular structure in (7) admits a similar triangular structure in a representation of the controllable part. More specifically, assume that the behavior B is represented by the rational kernel representation associated with (7), does there exist a representation of its controllable part with compatible triangular structure. This indeed turns out to be the case as is shown in the next lemma.…”

Section: Elimination Of Variablesmentioning

confidence: 99%

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A converse to the deterministic separation principle

Trumpf

Trentelman

2017

Systems & Control Letters

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In the classical theory of finite-dimensional linear time-invariant systems in state space form the term deterministic separation principle refers to the observation that a stabilizing output feedback controller can be constructed by first constructing an asymptotic state observer that is then coupled to a stabilizing state feedback controller. In this paper we discuss the following converse problem: Can every stabilizing output feedback controller be realized as interconnection of an asymptotic state observer and a stabilizing state feedback controller? We will provide an affirmative answer to this question (modulo a number of technicalities) in a behavioral setting and with the help of rational representations.

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Section: Polynomial and Rational Kernel Representationsmentioning

confidence: 99%

Section: Elimination Of Variablesmentioning

confidence: 99%

A converse to the deterministic separation principle

Trumpf

Trentelman

2017

Systems & Control Letters

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“…Thus, B ON is a latent variable representation of its external behavior given by The following result from [16,Theorem 5.8] will be used in the rest of the paper. …”

Section: Driving Variable and Output Nulling Representations Of Behavmentioning

confidence: 99%

Distance between Behaviors and Rational Representations

Trentelman

Gottimukkala

2013

SIAM J. Control Optim.

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Abstract. In this paper we study notions of distance between behaviors of linear differential systems. We introduce four metrics on the space of all controllable behaviors which generalize existing metrics on the space of input-output systems represented by transfer matrices. Three of these are defined in terms of gaps between closed subspaces of the Hilbert space L 2 (R). In particular we generalize the "classical" gap metric. We express these metrics in terms of rational representations of behaviors. In order to do so, we establish a precise relation between rational representations of behaviors and multiplication operators on L 2 (R). We introduce a fourth behavioral metric as a generalization of the well-known ν-metric. As in the input-output framework, this definition is given in terms of rational representations. For this metric, however, we establish a representation-free, behavioral characterization as well. We make a comparison between the four metrics and compare the values they take and the topologies they induce. Finally, for all metrics we make a detailed study of necessary and sufficient conditions under which the distance between two behaviors is less than one. For this, both behavioral as well as state space conditions are derived in terms of driving variable representations of the behaviors. 1. Introduction. This paper deals with notions of distance between systems. In the context of linear systems with inputs and outputs, several concepts of distance have been studied in the past. Perhaps the most well-known distance concept is that of gap metric introduced by Zames and El-Sakkary in [28] and extensively used by Georgiou and Smith in the context of robust stability in [7]. The distance between two systems in the gap metric can be calculated, but the calculation is by no means easy and requires the solution of an H ∞ optimization problem; see [6]. A distance concept which is equally relevant in the context of robust stability is the so-called ν-gap, introduced by Vinnicombe in [22], [21]. Computation of the ν-gap between two systems is much easier than that of the ordinary gap and basically requires computation of the winding number of a certain proper rational function, followed by computation of the L ∞ -norm of a given proper rational matrix. A third distance concept is that of L 2 -gap, which is the most easy to compute but which is not at all useful in the context of robust stability, as shown in [21]. More recently an alternative notion of gap for linear input-output systems was introduced by Ball and Sasane in [13], allowing also nonzero initial conditions of the system. In this paper we will put the above four distance concepts into a more general, behavioral context, extending them to a framework in which the systems are not necessarily identified with their representations (e.g., transfer matrices), but in which, instead, their behaviors, i.e., the spaces of all possible trajectories of the systems, form

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A Complement on Elimination and Realization in Rational Representations

Trentelman

Gottimukkala²

2015

Mathematical Control Theory I

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Realization and elimination in rational representations of behaviors

Cited by 3 publications

References 11 publications

A converse to the deterministic separation principle

A converse to the deterministic separation principle

Distance between Behaviors and Rational Representations

A Complement on Elimination and Realization in Rational Representations

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