Equivalence of rational representations of behaviors

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

doi:10.1016/j.sysconle.2010.11.003

Cited by 14 publications

(9 citation statements)

References 17 publications

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“…Remark 4.6. Theorem 4.3 substantially differs from the equivalence results in Gottimukkala, Fiaz, and Trentelman (2011), Trentelman (2010) and Yamamoto (2007, 2008) where C ∞ behaviors are defined as kernels of rational differential operators P. In Gottimukkala et al (2011), it is shown that the controllable parts of the C ∞ kernels of rational operators P and Q coincide if and only if there exists a unitary matrix U ∈ UL ∞, * such that P = UQ .…”

Section: Theorem 43 (Inclusion and Equivalence) Let Two Systems In Thementioning

confidence: 78%

Rational representations and controller synthesis of L2 behaviors

Mutsaers¹,

Weiland²

2012

Automatica

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Section: Theorem 43 (Inclusion and Equivalence) Let Two Systems In Thementioning

confidence: 78%

Rational representations and controller synthesis of L2 behaviors

Mutsaers¹,

Weiland²

2012

Automatica

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“…Proof of theorem Let

G = M_{1} N_{1}^{- 1}, \overset{G}{true} = P_{1}^{- 1} Q_{1}

and

\overset{K}{true} = P_{2}^{- 1} Q_{2}

be coprime factorizations over

double-struckR [ξ]

. Then, we have

P = \ker Q_{1} ((), \frac{normald}{normald t}) = im M_{1} ((), \frac{normald}{normald t})

(using Lemma 7.2 in ) and

C = \ker Q_{2} ((), \frac{normald}{normald t})

. Immediately, we then have

P \cap C = \ker ([], \begin{matrix} Q_{1} (\frac{d}{d t}) \\ Q_{2} (\frac{d}{d t}) \end{matrix}) = M_{1} ((), \frac{normald}{normald t}) \ker (Q_{2} M_{1}) ((), \frac{normald}{normald t})

.…”

Section: Robust Stabilizationmentioning

confidence: 96%

“…Representations of

frakturB

as in are called rational image representations . For more details on rational image representations, we refer the reader to .…”

Section: Linear Differential Systems and Their Representationsmentioning

confidence: 99%

Rational representations and robust stabilization in the behavioral framework

Gottimukkala

Trentelman

2014

Intl J Robust & Nonlinear

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In this paper, we study the problem of robust stabilization for linear differential systems in the behavioral framework. We study the existence of controllers that regularly stabilize all plants in a given neighborhood around a certain nominal plant. We call such controllers robustly stabilizing controllers. This problem was studied in the behavioral framework by Trentelman, H.L et al. In contrast to their work, however, in the present paper, we study the problem considering neighborhoods that are defined entirely representation free. These neighborhoods are induced by different kinds of concepts of distance between behaviors. As one of our main results we obtain that a given controller regularly stabilizes all plants in one of these neighborhoods if and only if it regularly stabilizes all plants in all of the other neighborhoods.if G D P 1 Q is a left coprime factorization of G over ROE . Representations of B as in (4) are called rational image representations. For more details on rational image representations, we refer the reader to [15,17,18]. The number of columns of G in any rational image representation of B, with G full column rank, is equal to m.B/. The number of rows of Q G in rational kernel representation of B, with Q G full row rank, is equal to p.B/. DISTANCE BETWEEN BEHAVIORSIn this section, we review the relevant material on the notion of distance between behaviors. We refer to [15] for a more detailed treatment of this subject.

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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%

“…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. This is equivalent to saying that the kernel representation is minimal, see [5] and [6].…”

Section: Polynomial and Rational Kernel Representationsmentioning

confidence: 99%

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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Equivalence of rational representations of behaviors

Cited by 14 publications

References 17 publications

Rational representations and controller synthesis of L2 behaviors

Rational representations and controller synthesis of L2 behaviors

Rational representations and robust stabilization in the behavioral framework

A converse to the deterministic separation principle

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