Robust controller design for infinite‐dimensional exosystems

Paunonen, Lassi; Pohjolainen, Seppo

doi:10.1002/rnc.2920

Cited by 30 publications

(38 citation statements)

References 34 publications

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“…The time domain theory of robust regulation in [18,27,31] is suitable for more general signal classes than those considered in this paper. The theory there allows the state operator of the exosystem to be a block diagonal operator with an infinite number of nontrivial Jordan blocks on its diagonal.…”

Section: Concluding Discussion and Directions For Future Researchmentioning

confidence: 99%

“…The formulation of the robust regulation problem used ideas that are familiar from time domain theory, i.e., the definition of robustness, the stability types, and the classes of the reference and disturbance signals; see [14,18,31] [14,18,31] is not needed in this article. Further study on the different stability types and realization theory is needed in order to better understand the connection between the two formulations of the seemingly same problem.…”

Section: Concluding Discussion and Directions For Future Researchmentioning

confidence: 99%

“…For the recent development on this subject, see [14,18,31] and the references therein. One can generate very general classes of signals, e.g., general periodic functions, by using infinite-dimensional exosystems.…”

Section: Introductionmentioning

confidence: 99%

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Frequency Domain Robust Regulation of Signals Generated by an Infinite-Dimensional Exosystem

Laakkonen¹,

Pohjolainen²

2015

SIAM J. Control Optim.

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This paper deals with frequency domain robust regulation of signals generated by an infinite-dimensional exosystem. The problem is formulated and the stability types are chosen so that one can generalize the existing finite-dimensional theory to more general classes of infinitedimensional systems and signals. The main results of this article are extensions of the internal model principle, of a necessary and sufficient solvability condition for the robust regulation problem, and of Davison's simple servo compensator for stable plants in the chosen algebraic framework. Introduction.The topic of this article is robust regulation, which is a central problem in mathematical control theory. The robust regulation problem consists of two parts: robust stabilization, and robust asymptotic tracking and disturbance rejection. Asymptotic tracking and disturbance rejection means that the error between the reference and output signals vanishes as time goes to infinity despite some disturbance signals. The reference and disturbance signals are generated by an exogenous system called the exosystem. Robustness of regulation means that the output of the closed loop asymptotically tracks the reference signals even if the plant is perturbed. Robustness is of fundamental importance since small errors in mathematical models of real-world phenomena are unavoidable and stem from various sources such as model simplifications and erroneous parameter estimation.The robust regulation problem of finite-dimensional linear multi-input multioutput (MIMO) systems was solved in the 1970s. Francis, Wonham, and Davison had a central role in this work [5,6,7,9,10]. Many authors have generalized the finitedimensional results to infinite-dimensional and nonlinear systems in the time domain as well as in the frequency domain since then [1,13,15,17,30,32,35,36,37,41].Robust regulation of signals generated by an infinite-dimensional exosystem is a challenging problem [15,39]. For the recent development on this subject, see [14,18,31] and the references therein. One can generate very general classes of signals, e.g., general periodic functions, by using infinite-dimensional exosystems. Signals that are generated by an infinite-dimensional exosystem and their frequency domain counterparts are called infinite-dimensional signals in this paper. The main emphasis in recent research has been on the time domain, while the frequency domain theory has received only a little attention. The purpose of this paper is to develop theory for robust regulation in the frequency domain.There exist multiple different rings of stable transfer functions, each of which has its own special features and is suitable for certain purposes [24]. Thus, it is natural

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Section: Concluding Discussion and Directions For Future Researchmentioning

confidence: 99%

Section: Concluding Discussion and Directions For Future Researchmentioning

confidence: 99%

See 1 more Smart Citation

Frequency Domain Robust Regulation of Signals Generated by an Infinite-Dimensional Exosystem

Laakkonen¹,

Pohjolainen²

2015

SIAM J. Control Optim.

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“…Regulator theory for infinite-dimensional linear systems with bounded control and observation operators has been significantly advanced by a group of researchers at Tampere University of Technology (Finland) who have developed a sophisticated theory of infinite-dimensional exosystems, see for instance [24], [26]- [28], [39], [40]. The state feedback regulator problem for exponentially stabilizable linear plants driven by infinite-dimensional exosystems generating periodic signals was addressed in [27].…”

Section: Introductionmentioning

confidence: 99%

The State Feedback Regulator Problem for Regular Linear Systems

Natarajan

Gilliam

Weiss

2014

IEEE Trans. Automat. Contr.

129

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This paper is about the state feedback regulator problem for infinite-dimensional linear systems. The plant, assumed to be an exponentially stable regular linear system, is driven by a linear (possibly infinite-dimensional) exosystem via a disturbance signal. The exosystem has its spectrum in the closed right half-plane and also generates the reference signal for the plant output. The regulator problem is to design a controller that, while guaranteeing the stability of the closed-loop system without the exosystem, drives the tracking error to zero. A particular version of this problem is the state feedback regulator problem in which the states of the exosystem and the plant are known to the controller. Under suitable assumptions, we show that the latter problem is solvable if and only if a pair of algebraic equations, called the regulator equations, is solvable. We derive conditions, in terms of the transfer function of the plant and eigenvalues of the exosystem, for the solvability of the regulator equations. Three examples illustrating the theory are presented.Index Terms-Infinite-dimensional exosystem, regulator equations, state feedback controller, unbounded control and observation operators.

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“…The finite-dimensional theory has been generalized to infinite-dimensional plants and signals by several authors. See, for instance, [2,6,7,8,9,10,16,17,24] and the references therein. One of the most fundamental results of robust regulation is the internal model principle, which states that any robustly regulating controller contains a suitably reduplicated model of the dynamics to be tracked.…”

Section: Introductionmentioning

confidence: 99%

A fractional representation approach to the robust regulation problem for SISO systems

Laakkonen

Quadrat

2017

Systems & Control Letters

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The purpose of this article is to develop a new approach to the robust regulation problem for plants which do not necessarily admit coprime factorizations. The approach is purely algebraic and allows us dealing with a very general class of systems in a unique simple framework. We formulate the famous internal model principle in a form suitable for plants defined by fractional representations which are not necessarily coprime factorizations. By using the internal model principle, we are able to give necessary and sufficient solvability conditions for the robust regulation problem and to parameterize all robustly regulating controllers.

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Robust controller design for infinite‐dimensional exosystems

Cited by 30 publications

References 34 publications

Frequency Domain Robust Regulation of Signals Generated by an Infinite-Dimensional Exosystem

Frequency Domain Robust Regulation of Signals Generated by an Infinite-Dimensional Exosystem

The State Feedback Regulator Problem for Regular Linear Systems

A fractional representation approach to the robust regulation problem for SISO systems

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