Elsevier

Automatica

Volume 47, Issue 11, November 2011, Pages 2468-2473
Automatica

Brief paper
Output regulation for linear distributed-parameter systems using finite-dimensional dual observers

https://doi.org/10.1016/j.automatica.2011.08.033Get rights and content

Abstract

In this article, the solution of the output regulation problem is considered for linear infinite-dimensional systems where the outputs to be controlled cannot be measured. It is shown that this problem can be solved by a finite-dimensional dual observer that is directly implementable so that the separation principle can be applied for the stabilization as in finite dimensions. A parametric design of these dual observers is proposed for Riesz-spectral systems that allows to achieve a low controller order and a desired control performance for the closed-loop system. The presented results are illustrated by determining a finite-dimensional regulator for an Euler–Bernoulli beam with Kelvin–Voigt damping that achieves tracking for steplike reference inputs and that asymptotically rejects sinusoidal disturbances.

Introduction

A classical problem in control theory is output regulation, i.e. to design a stabilizing controller such that the closed-loop system tracks a given reference input and rejects disturbances. If these exogenous signals can be modeled by a signal process, a systematic solution can be derived. The output regulation theory was first developed for lumped-parameter systems and is well documented in, for example, Knobloch et al. (1993) for linear systems and in Byrnes et al. (1997) for nonlinear systems. Generally, there exist two different approaches for solving the output regulation problem. The first one originates from the work of Davison (see Davison, 1976), where the signal process is included into the controller and is driven by the output tracking error. Thus, this approach assumes that the outputs to be controlled can be measured. The main advantage of this method is its inherent robustness property. Consequently, this approach has been intensively studied also for infinite-dimensional systems in the past three decades. In Pohjolainen (1982), an extension of the classical PI-controller was proposed (see also Chentouf and Wang, 2008), which was later extended to general classes of reference and disturbance signals in Hämäläinen and Pohjolainen (2000), Immonen (2007), Paunonen and Pohjolainen (2010) and to well-posed systems in Rebarber and Weiss (2003). The second method for achieving output regulation traces back to the work of Johnson in Johnson (1971), where an observer containing the signal process is designed that estimates the state of the system and of the signal process. These estimates are used to implement a state feedback that achieves output regulation. Different from Davison’s approach, the outputs to be controlled need not be measured. Especially for distributed-parameter systems this property is of interest if the measurements and the output to be controlled do not coincide because they describe the system behavior at different spatial positions. However, Johnson’s approach does not share the robustness properties of the approach due to Davison. Compared to this method, Johnson’s approach for output regulation has received lesser attention. The geometric theory of output regulation introduced in Francis (1977) for lumped-parameter systems was extended to infinite dimensions in Byrnes et al. (2000). Furthermore, on the basis of the results in Francis and Wonham (1975) an infinite-dimensional robust compensator was designed which is not directly implementable. For systems with discrete spectrum and bounded input and output operators, the direct design of a finite-dimensional compensator achieving output regulation was considered in Schumacher (1983). The aim of that work was to establish an existence result for the finite-dimensional regulator. The proposed design procedure, however, does not allow to systematically determine a controller of low order for systems with unobservable outputs that also ensures an acceptable control performance.

In this article, the design of finite-dimensional compensators for output regulation on the basis of Johnson’s approach is considered for infinite-dimensional systems. In order solve this problem by applying the separation principle for the compensator design dual observers are used instead of classical state observers. The concept of a dual observer was introduced in the seminal paper (Luenberger, 1971) by Luenberger for finite-dimensional systems. He demonstrated that the problem of implementing a state feedback can also be solved using dual observers on the basis of the separation principle. Different from the classical state observers, dual observers are finite-dimensional when solving the output regulation problem for infinite-dimensional systems provided that the signal process is finite-dimensional. Thus, no approximation of the resulting compensator is needed. This approach is formulated for linear infinite-dimensional systems where the linear system operator is the infinitesimal generator of a C0-semigroup and where the input and output operators are bounded linear operators. The design of the dual observer amounts to determining a static output feedback controller for an infinite-dimensional system. This problem can be solved using the parametric approach presented in Deutscher and Harkort (2010) if the system operator is a Riesz-spectral operator and generates an analytic C0-semigroup, i.e. for a class of distributed-parameter systems. Different from the considered system class in Schumacher (1983), these systems may have accumulation points in their spectra. The parametric approach has the advantage that the static output feedback controller is designed systematically using the eigenvalue assignment on the basis of a parameter optimization. Thus, the available degrees of freedom in the compensator can be determined such that a low controller order and a desired control performance of the resulting closed-loop system are achieved.

The next section states the problems considered in this article. Then, a dynamic feedforward controller is designed in Section 3 that achieves output regulation for appropriate initial conditions. In order to take initial errors into account, the feedforward controller is extended to a dual observer in Section 4 which results in a finite-dimensional compensator achieving output regulation. Section 5 presents the parametric design of compensators for Riesz-spectral systems. The proposed design procedure is demonstrated by means of an Euler–Bernoulli beam with an accumulation point in its spectrum for steplike reference inputs and a sinusoidal disturbance.

Section snippets

Problem formulation

Consider the linear infinite-dimensional system ẋ(t)=Ax(t)+Bu(t)+Gd(t),t>0,x(0)Hy(t)=Cx(t),t0ym(t)=Cmx(t),t0 with the input u(t)Rp and the unmeasurable disturbance d(t)Rq. The output y(t)Rp is the output to be controlled and ym(t)Rm is the measurement. In the following, it is not required that the output y can be measured. The state x(t)H is an element of the infinite-dimensional and complex Hilbert space H with inner product , which is used as state space. The input operator B and

Output regulation by feedforward control

Consider the finite-dimensional feedforward controller[ṙv(t)ṙx(t)]=[S00F][rv(t)rx(t)],t>0u(t)=[PvPx][rv(t)rx(t)],t0 with initial conditions rv(0)=rv0Cnv and rx(0)=rx0Cnx. This controller contains a model of the signal process (8), so that an input can be generated which on the one hand compensates the influence of the disturbance d and on the other hand assures tracking of w in the steady state. The second subsystem in (17) with the system matrix F is introduced to obtain additional

Output regulation using a dual observer

Since the state x of the system (1) and the signal process state v in (8) cannot be measured the initial values rv(0) and rx(0) of the feedforward controller (17)–(18) cannot be chosen such that (20) holds if this is possible at all. Consequently, an initial error xe(0)Πr(0)0 results so that, in general, output regulation is not achieved. However, by stabilizing the dynamics of xeΠr an asymptotically vanishing tracking error xeΠr can be obtained ensuring (16). To this end, the feedforward

Parametric design of the dual observer

In order to stabilize the error dynamics (29), consider the fictitious plantẋe(t)Πṙ(t)=Ae(xe(t)Πr(t))+B̃eū(t) with the input ū(t) and the output ȳ(t)=C̃m(xe(t)Πr(t)). Then, the error dynamics (29) result if the static output feedbackū(t)=Rȳ(t) is applied to (34)–(35). Consequently, determining a stabilizing feedback gain R for the error dynamics (29) amounts to computing a stabilizing static output feedback (36) for (34). The next lemma characterizes a class of distributed-parameter

Example

Consider the simply-supported Euler–Bernoulli beam with Kelvin–Voigt damping t2f(z,t)+z4f(z,t)+2δtz4f(z,t)=b(z)u(t)+g(z)d(t),t>0,z(0,1)f(0,t)=z2f(0,t)=f(1,t)=z2f(1,t)=0,t>0f(z,0)=f0(z),tf(z,0)=f1(z),z[0,1]y(t)=01c(z)f(z,t)dz,t0ym(t)=01cm(z)f(z,t)dz,t0 where f(z,t), z[0,1], is the transverse displacement of the beam along the spatial coordinate z and δ=0.01 is the damping constant. The input of the system is a nearly pointlike actor force u that is modeled by the spatial

Concluding remarks

The design of regulators for distributed-parameter systems with boundary control using the presented results is also possible since many boundary control problems can be converted to equivalent distributed control problems (see Curtain and Zwart, 1995). Since the existence result for the dual observer is also valid for linear time-delay systems future work considers the extension of the parametric dual observer design to these systems.

Joachim Deutscher was born in Schweinfurt, Germany, in 1970. He received the Dipl.-Ing. (FH) degree in Electrical Engineering from Fachhochschule Würzburg-Schweinfurt-Aschaffenburg, Germany, in 1996, the Dipl.-Ing. Univ. degree in Electrical Engineering and the Dr.-Ing. degree from Friedrich-Alexander Universität Erlangen-Nürnberg (FAU), Germany, in 1999 and 2003, respectively. After his habilitation in 2010 at the Faculty of Engineering (FAU), he is Privatdozent (Associate Professor) for

References (22)

  • R. Rebarber et al.

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

    Automatica

    (2003)
  • J.M. Schumacher

    Finite-dimensional regulators for a class of infinite-dimensional systems

    Systems & Control Letters

    (1983)
  • M.J. Balas

    Active control of flexible systems

    Journal of Optimization Theory and Applications

    (1978)
  • Ch.I. Byrnes et al.

    Geometric theory of output regulation for linear distributed parameter systems

    Frontiers in Applied Mathematics

    (2003)
  • Ch.I. Byrnes et al.

    Output regulation for linear distributed parameter systems

    IEEE Transactions on Automatic Control

    (2000)
  • Ch.I. Byrnes et al.

    Output regulation for uncertain nonlinear systems

    (1997)
  • B. Chentouf et al.

    A Riesz basis methodology for proportional and integral output regulation of a one-dimensional diffusive-wave equation

    SIAM Journal on Control and Optimization

    (2008)
  • R.F. Curtain et al.

    An introduction to infinite-dimensional linear systems theory

    (1995)
  • E.J. Davison

    The robust control of a servomechanism problem for linear time-invariant multivariable systems

    IEEE Transactions on Automatic Control

    (1976)
  • J. Deutscher et al.

    A parametric approach to finite-dimensional control of linear distributed-parameter systems

    International Journal of Control

    (2010)
  • B.A. Francis

    The linear multivariable regulator problem

    SIAM Journal on Control and Optimization

    (1977)
  • Cited by (0)

    Joachim Deutscher was born in Schweinfurt, Germany, in 1970. He received the Dipl.-Ing. (FH) degree in Electrical Engineering from Fachhochschule Würzburg-Schweinfurt-Aschaffenburg, Germany, in 1996, the Dipl.-Ing. Univ. degree in Electrical Engineering and the Dr.-Ing. degree from Friedrich-Alexander Universität Erlangen-Nürnberg (FAU), Germany, in 1999 and 2003, respectively. After his habilitation in 2010 at the Faculty of Engineering (FAU), he is Privatdozent (Associate Professor) for Automatic Control at the Lehrstuhl für Regelungstechnik (FAU), where he leads the infinite-dimensional systems group. His research interests include the control of infinite-dimensional systems, nonlinear control theory and the application of polynomial matrix methods in control. He has co-authored a book on state control: Design of observer-based compensators (Springer, 2009).

    This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor George Weiss under the direction of Editor Miroslav Krstic.

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