Internal model based tracking and disturbance rejection for stable well-posed systems

Rebarber, Richard; Weiss, George

doi:10.1016/s0005-1098(03)00192-4

Cited by 210 publications

(145 citation statements)

References 32 publications

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“…Output tracking and disturbance rejection of nonsmooth signals with high accuracy have applications in the control of motor and disk drive systems and in power electronics [10]. Output tracking of signals generated by an infinite-dimensional exosystem have been studied using state space methods in [20,17,28,25,30], and using frequency domain techniques in [45,34,46,22]. Robust tracking of nonsmooth periodic functions has also been studied extensively in repetitive control [18,44,41] where the control objective is to achieve precise tracking for a finite number of frequency components of y ref (·).…”

Section: Introductionmentioning

confidence: 99%

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: Introductionmentioning

confidence: 99%

Robust Controllers for Regular Linear Systems with Infinite-Dimensional Exosystems

Paunonen¹

2017

SIAM J. Control Optim.

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“…(In their paper, reference and disturbance signals are more general, containing polynomial parts.) Rebarber and Weiss [13] proved similar results for the more general class of exponentially stable well-posed systems.…”

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confidence: 59%

“…The spectral condition (3.2) (or, alternatively, (3.6)) is the discrete-time analogue of the continuous-time condition in [13]; see [13, To facilitate the proof of Theorem 3.1, we first state and prove the following key lemma, which shows that the transfer function (I + GK ε ) −1 , the so-called sensitivity function, is in…”

Section: Zhenqing Ke Hartmut Logemann and Richard Rebarbermentioning

confidence: 99%

Approximate Tracking and Disturbance Rejection for Stable Infinite-Dimensional Systems Using Sampled-Data Low-Gain Control

Ke¹,

Logemann²,

Rebarber³

2009

SIAM J. Control Optim.

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Abstract. In this paper we solve tracking and disturbance rejection problems for stable infinitedimensional systems using a simple low-gain controller suggested by the internal model principle. For stable discrete-time systems, it is shown that the application of a low-gain controller (depending on only one gain parameter) leads to a stable closed-loop system which asymptotically tracks reference signals r of the form r(k) = N j=1 λ k j r j , where r j ∈ C p and λ j ∈ C with |λ j | = 1 for j = 1, . . . , N. The closed-loop system also rejects disturbance signals which are asymptotically of this form. The discrete-time result is used to derive results on approximate tracking and disturbance rejection for a large class of infinite-dimensional sampled-data feedback systems, with reference signals which are finite sums of sinusoids, and disturbance signals which are asymptotic to finite sums of sinusoids. The results are given for both input-output systems and state-space systems.Key words. discrete-time systems, disturbance rejection, infinite-dimensional systems, internal model principle, low-gain control, sampled-data control, tracking AMS subject classifications. 93C25, 93C55, 93C57, 93C80, 93D15, 93D25 DOI. 10.1137/080716517 1. Introduction. The synthesis of low-gain integral controllers for uncertain stable continuous-time plants has received considerable attention in the last thirty years. Let G be a stable proper rational continuous-time transfer function matrix. The main existence result for robust low-gain integral control states that if all of the eigenvalues of G(0) have positive real parts, then there exists ε * > 0 such that for all ε ∈ (0, ε * ), the controller (ε/s)I stabilizes G. Moreover, the resulting closed-loop system asymptotically tracks arbitrary constant reference signals. This result has been proved by Davison [2] using state-space methods and Morari [11] using frequencydomain methods. This low-gain controller allows stabilization and tracking with very little information about the plant, and it is not based on system identification. The above regulator result has been extended to various classes of (abstract) infinitedimensional continuous-time systems: in [12] for exponentially stable parabolic systems, in [7] for systems in the Callier-Desoer algebra (CD-algebra), and in [9] for exponentially stable regular systems.In the case that the reference and disturbance signals are of the form

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“…Various results in this direction can be found in [33], [42], Vidyasagar [44], and many others. We shall need the following technical result that appears as Lemma 3.3 in [30]. Fig.…”

Section: Input-output Stability Of Coupled Systems With Positive mentioning

confidence: 99%

Stability Properties of Coupled Impedance Passive LTI Systems

Zhao

2017

IEEE Trans. Automat. Contr.

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Abstract-We study the stability of the feedback interconnection of two impedance passive linear time-invariant systems, of which one is finite-dimensional. The closed-loop system is well known to be impedance passive, but no stability properties follow from this alone. We are interested in two main issues: (1) the strong stability of the operator semigroup associated with the closed-loop system, (2) the input-output stability (meaning transfer function in H ∞ ) of the closed-loop system. Our results are illustrated with the system obtained from the non-uniform SCOLE (NASA Spacecraft Control Laboratory Experiment) model representing a vertical beam clamped at the bottom, with a rigid body having a large mass on top, connected with a trolley mounted on top of the rigid body, via a spring and a damper. Such an arrangement called a tuned mass damper (TMD), is used to stabilize tall buildings. We show that the SCOLE-TMD system is strongly stable on the energy state space and that the system is input-output stable from the horizontal force input to the horizontal velocity output.

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Internal model based tracking and disturbance rejection for stable well-posed systems

Cited by 210 publications

References 32 publications

Robust Controllers for Regular Linear Systems with Infinite-Dimensional Exosystems

Robust Controllers for Regular Linear Systems with Infinite-Dimensional Exosystems

Approximate Tracking and Disturbance Rejection for Stable Infinite-Dimensional Systems Using Sampled-Data Low-Gain Control

Stability Properties of Coupled Impedance Passive LTI Systems

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