What Periodic Signals Can an Exponentially Stabilizable Linear Feedforward Control System Asymptotically Track?

Immonen, Eero; Pohjolainen, Seppo

doi:10.1137/040613093

Cited by 22 publications

(39 citation statements)

References 15 publications

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“…However, Assumption 3 can also be used if {iω k } k ∩ σ(A) = ∅, which is the case, e.g., for the heat system in Section 7. For examples on verifying Assumption 3 and locating the zeros of infinite-dimensional systems, see [21,Sec. 5], [17,Sec.…”

Section: The System the Controller And The Closed-loop Systemmentioning

confidence: 99%

“…In the case of an infinite-dimensional exosystem the reference and disturbance signals are required to have a certain minimum level of smoothness in order for the robust output regulation problem to be solvable, and the exact level depends on the behaviour of the transfer function P (λ) = C Λ R(λ, A)B + D of the plant at the frequencies λ = iω k of the exosystem [21,17]. More precisely, faster growth of the norms P (iω k ) † of the Moore-Penrose pseudoinverses of P (iω k ) as |k| → ∞ leads to a higher minimal level of smoothness for y ref (t) and w(t).…”

Section: Introductionmentioning

confidence: 99%

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Robust Controllers for Regular Linear Systems with Infinite-Dimensional Exosystems

Paunonen¹

2017

SIAM J. Control Optim.

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Abstract. We construct two error feedback controllers for robust output tracking and disturbance rejection of a regular linear system with nonsmooth reference and disturbance signals. We show that for sufficiently smooth signals the output converges to the reference at a rate that depends on the behaviour of the transfer function of the plant on the imaginary axis. In addition, we construct a controller that can be designed to achieve robustness with respect to a given class of uncertainties in the system, and present a novel controller structure for output tracking and disturbance rejection without the robustness requirement. We also generalize the internal model principle for regular linear systems with boundary disturbance and for controllers with unbounded input and output operators. The construction of controllers is illustrated with an example where we consider output tracking of a nonsmooth periodic reference signal for a two-dimensional heat equation with boundary control and observation, and with periodic disturbances on the boundary. IntroductionThe purpose of this paper is to construct controllers for robust output regulation of a regular linear system 1 [39, 40, 36]on an infinite-dimensional Banach space X. The main goal in the control problem is to achieve asymptotic convergence of the output y(t) to a given reference signal y ref (t) despite external disturbance signals w(t). In addition, it is required that the controller is robust in the sense that output tracking is achieved even under perturbations and uncertainties in the operators (A, B, B d , C, D) of the plant. The class of regular linear systems facilitates the study of robust output tracking and disturbance rejection for many important classes partial differential equations with boundary control and observation with corresponding unbounded operators B, B d and C [9,16,47,25]. In this paper we continue the work on designing robust controllers for regular linear systems begun recently in [26].The reference signal y ref (t) and the disturbance signals w(t) considered in the robust output regulation problem are assumed to be generated by an exosystem of the forṁ

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Section: The System the Controller And The Closed-loop Systemmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Robust Controllers for Regular Linear Systems with Infinite-Dimensional Exosystems

Paunonen¹

2017

SIAM J. Control Optim.

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“…We next show that the same regulator equations as in References [5,11,12] provide a sufficient condition for the solvability of the feedforward-feedback output regulation problem under the assumption of closed-loop stability.…”

Section: A Solution Of the Feedforward-feedback Regulation Problemmentioning

confidence: 96%

“…In the closed-loop system ' zðtÞ ¼ AzðtÞ þ BJxðtÞ þ ðBG þ PÞwðtÞ; t50 ð4aÞ We hasten to emphasize that the control law uðtÞ ¼ JxðtÞ þ GwðtÞ above does not involve a direct feedback from the state of the plant, as in References [11,12,18]. Furthermore, the feedforward part of the control is given by vðtÞ ¼ GwðtÞ ¼ Gy ref ðÁ þ tÞ; t50; where wð0Þ ¼ y ref ðÁÞ is the reference signal we want to track.…”

Section: The Feedforward-feedback Regulation Problemmentioning

confidence: 99%

“…In fact, there exist pairs ðA; BÞ which can only be stabilized strongly by a bounded feedback K; but sðA þ BKÞ \ iR ¼ |; and only AEi1 are points of accumulation for sðA þ BKÞ: This phenomenon is quite common in infinite-dimensional pole-placement problems; the reader is referred to Reference [32] for more details. We also assume throughout this section that plant (1) is a SISO system, but we do not restrict our attention to the case of p-periodic reference/disturbance signals as in References [12,18]. Our approach for solving the regulator equations (5) We can in this case choose the spaces H n ¼ spanff k j À n4k4ng for each n 2 N: This observation was the foundation of the method for solving the regulator equations (5) in References [12,18]; the same method has been shown to work for finite-dimensional exogenous systems in Reference [5].…”

Section: Solution Of the Regulator Equations For Siso Systemsmentioning

confidence: 99%

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A feedforward–feedback controller for infinite-dimensional systems and regulation of bounded uniformly continuous signals

Immonen

2006

Int. J. Robust Nonlinear Control

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SUMMARYWe design a controller for infinite-dimensional linear systems (with bounded control, observation and feedthrough operators) which, under certain assumptions, achieves asymptotic tracking of arbitrary bounded uniformly continuous reference signals in the presence of disturbances. The proposed controller is of feedforward-feedback type: The dynamic feedback part is used to stabilize the closed-loop system consisting of the plant and the controller, whereas the feedforward part is tuned using the regulator equations to achieve the regulation of desired signals. We also completely solve the regulator equations for SISO systems, and we discuss robustness properties of the proposed controller. A useful feature in our design is that the feedforward part of the controller can be designed independently of the feedback part. This automatically leads to a degree of robustness in the stabilizing part of the controller, which is not present in the existing state feedback controllers solving the same output regulation problem.

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Robust controller design for infinite‐dimensional exosystems

Paunonen

Pohjolainen

2012

Intl J Robust & Nonlinear

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SUMMARYIn this paper, we consider robust output regulation of distributed parameter systems with infinite‐dimensional exosystems capable of generating polynomially growing signals. We design an observer‐based error feedback controller solving the control problem. The controller is chosen in such a way that it incorporates an internal model of the infinite‐dimensional exosystem. The remaining parameters of the controller are chosen to stabilize the closed‐loop system strongly. We also analyze the classes of signals generated by the exosystem. In particular, we explore the connection between the smoothness properties of the reference and disturbance signals and the strictness of the conditions required for the existence of a controller solving the robust output regulation problem. Copyright © 2012 John Wiley & Sons, Ltd.

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What Periodic Signals Can an Exponentially Stabilizable Linear Feedforward Control System Asymptotically Track?

Cited by 22 publications

References 15 publications

Robust Controllers for Regular Linear Systems with Infinite-Dimensional Exosystems

Robust Controllers for Regular Linear Systems with Infinite-Dimensional Exosystems

A feedforward–feedback controller for infinite-dimensional systems and regulation of bounded uniformly continuous signals

Robust controller design for infinite‐dimensional exosystems

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