2236 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 45, NO. 12, DECEMBER 2000 Output Regulation for Linear Distributed Parameter Systems Christopher I. Byrnes, Fellow, IEEE, István G. Laukó, David S. Gilliam, Member, IEEE, and Victor I. Shubov Abstract—This work extends the geometric theory of output regulation to linear distributed parameter systems with bounded input and output operator, in the case when the reference signal and disturbances are generated by a finite dimensional exogenous system. In particular, it is shown that the full state feedback and error feedback regulator problems are solvable, under the stan- dard assumptions of stabilizability and detectability, if and only if a pair of regulator equations is solvable. For linear distributed parameter systems this represents an extension of the geometric theory of output regulation developed in [10] and [4]. We also pro- vide simple criteria for solvability of the regulator equations based on the eigenvalues of the exosystem and the system transfer func- tion. Examples are given of periodic tracking, set point control, and disturbance attenuation for parabolic systems and periodic tracking for a damped hyperbolic system. Index Terms—Bounded input and output operators, distributed parameter systems, regulator problem. I. INTRODUCTION O NE of the central problems in control theory is to control a fixed plant so that its output tracks a reference signal (and/or rejects a disturbance) produced by an external generator or exogenous system. Generally two versions of this problem are considered. In the first, the state feedback regulator problem, the controller is provided with full information of the state of the plant and exosystem. For the second, and perhaps more re- alistic error feedback regulator problem, only the components of the error are available for measurement. For linear finite-di- mensional systems it has been shown by Francis [10] that the solvability of the regulator problem is equivalent to the solv- ability of a pair of linear matrix Sylvester equations. This in turn can be characterized as a property of the transmission polyno- mials of the composite system formed from the plant and the exosystem, as was shown by Hautus [12]. Francis and Wonham [11] have also shown that any regulator that solves the error feedback problem has to incorporate a model of the exogenous system generating the reference signal which is to be tracked and/or the disturbance that must be rejected. This property is known as the internal model principle. Manuscript received September 18, 1998; revised September 20, 1999. Rec- ommended by Associate Editor, I. Lasiecka. This work was supported in part by grants from AFOSR and Texas ARP. C. I. Byrnes is with the Department of Systems Science and Mathematics, Washington University, St. Louis, MO 63130 USA. I. G. Laukó is with the Center for Research in Scientific Computation, North Carolina State University, Raleigh, NC 27695-8205 USA. D. S. Gilliam and V. I. Shubov are with the Department of Mathematics and Statistics, TexasTech University, Lubbock, TX 79409 USA. Publisher Item Identifier S 0018-9286(00)10665-8. Similar results have been established for finite dimensional nonlinear systems in [4] in case the plant is exponentially stabi- lizable and the exosystem has bounded trajectories that do not trivially converge to zero. In particular, it is shown in [4] that the solvability conditions given by Francis in the linear case can be naturally generalized to the solvability of a pair of non- linear equations, the regulator equations. Geometrically, these nonlinear regulator equations express the existence of a local manifold on which the actual and reference outputs coincide and which can be rendered invariant using feedback. In this paper we develop the geometric methods introduced in [10] and [4] for solving the state and output feedback regulator problems for infinite-dimensional linear control systems, as- suming that the control and observation operators are bounded. We expect to have more to say about the unbounded case in future papers. In particular we derive the regulator equations for a class of distributed parameters systems, obtaining an op- erator Sylvester equation. We also obtain results characterizing the solvability of both state and error feedback regulator prob- lems in terms of solvability of these regulator equations. The main difficulties that arise in developing a geometric theory for the distributed parameter case are obvious: the phase space is infinite dimensional; the state operator is typically unbounded and consequently only densely defined; the error zeroing con- trolled invariant subspace must be contained in the domain of the state operator and the resulting regulator equations may be- come distributed parameter equations. Section II contains a for- mulation of the state and error feedback regulator problems and a discussion of the basic assumptions. In Section III, we present a motivating example of periodic tracking for a controlled heat equation. In Section IV we present our main solvability results in Theorems IV.1 and IV.2. In particular, we give necessary and sufficient conditions for the solvability of both the state and error feedback regulator problems in terms of the solvability of a pair of linear operator equations, the regulator equations. Once a solution of these equations is available, the appropriate state feedback control in the case of the full information problem, or a dynamic controller in the case of error feedback, are obtained which provide a solution to the regulator problem. In Section V, we recall the definitions of transmission and invariant zeros. For certain classes of distributed parameter sys- tems we provide a sequence of results, under various hypotheses on the plant and exosystem, expressing necessary and sufficient conditions for solvability of the requlator equations in terms of nonresonance conditions involving the eigenvalues of the ex- osystem and the plants transmission (or invariant) zeros. Section VI contains several explicit examples of output reg- ulation. In the first example we consider a problem of set point 0018–9286/00$10.00 © 2000 IEEE