Robust Controllers for Regular Linear Systems with Infinite-Dimensional Exosystems

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.Comment: 30 pages, 3 figures, to appear in SIAM Journal on Control & Optimizatio

Controller Design for Robust Output Regulation of Regular Linear Systems

We present three dynamic error feedback controllers for robust output regulation of regular linear systems. These controllers are (i) a minimal order robust controller for exponentially stable systems (ii) an observer-based robust controller and (iii) a new internal model based robust controller structure. In addition, we present two controllers that are by construction robust with respect to predefined classes of perturbations. The results are illustrated with an example where we study robust output tracking of a sinusoidal reference signal for a two-dimensional heat equation with boundary control and observation.Comment: 26 pages, 2 figures, to appear in IEEE Transactions on Automatic Contro

The Internal Model Principle for Systems with Unbounded Control and Observation

In this paper the theory of robust output regulation of distributed parameter systems with infinite-dimensional exosystems is extended for plants with unbounded control and observation. As the main result, we present the internal model principle for linear infinite-dimensional systems with unbounded input and output operators. We do this for two different definitions of an internal model found in the literature, namely, the p-copy internal model and the $\mathcal{G}$-conditions. We also introduce a new way of defining an internal model for infinite-dimensional systems. The theoretic results are illustrated with an example where we consider robust output tracking for a one-dimensional heat equation with boundary control and pointwise measurements.Comment: 38 pages, 2 figures, in revie

Approximate robust output regulation of boundary control systems

We extend the internal model principle for systems with boundary control and boundary observation, and construct a robust controller for this class of systems. However, as a consequence of the internal model principle, any robust controller for a plant with infinite-dimensional output space necessarily has infinite-dimensional state space. We proceed to formulate the approximate robust output regulation problem and present a finite-dimensional controller structure to solve it. Our main motivating example is a wave equation on a bounded multidimensional spatial domain with force control and velocity observation at the boundary. In order to illustrate the theoretical results, we construct an approximate robust controller for the wave equation on an annular domain and demonstrate its performance with numerical simulations.Comment: 29 pages, 4 figure

Reduced Order Controller Design for Robust Output Regulation

We study robust output regulation for parabolic partial differential equations and other infinite-dimensional linear systems with analytic semigroups. As our main results we show that robust output tracking and disturbance rejection for our class of systems can be achieved using a finite-dimensional controller and present algorithms for construction of two different internal model based robust controllers. The controller parameters are chosen based on a Galerkin approximation of the original PDE system and employ balanced truncation to reduce the orders of the controllers. In the second part of the paper we design controllers for robust output tracking and disturbance rejection for a 1D reaction-diffusion equation with boundary disturbances, a 2D diffusion-convection equation, and a 1D beam equation with Kelvin-Voigt damping.Comment: Revised version with minor improvements and corrections. 28 pages, 9 figures. Accepted for publication in the IEEE Transactions on Automatic Contro

Robust Output Tracking for a Room Temperature Model with Distributed Control and Observation

We consider robust output regulation of a partial differential equation model describing temperature evolution in a room. More precisely, we examine a two-dimensional room model with the velocity field and temperature evolution governed by the incompressible steady state Navier-Stokes and advection-diffusion equations, respectively, which coupled together form a simplification of the Boussinesq equations. We assume that the control and observation operators of our system are distributed, whereas the disturbance acts on a part of the boundary of the system. We solve the robust output regulation problem using a finite-dimensional low-order controller, which is constructed using model reduction on a finite element approximation of the model. Through numerical simulations, we compare performance of the reduced-order controller to that of the controller without model reduction as well as to performance of a low-gain robust controller.Comment: 12 pages, 5 figures. Accepted for publication in the Proceedings of the 24th International Symposium on Mathematical Theory of Networks and Systems, 23-27 August, 202

Robust Regulation for Infinite-Dimensional Systems and Signals in the Frequency Domain

In this thesis, the robust output regulation problem is studied both in the time domain and in the frequency domain. The problem to be addressed is to find a stabilizing controller for a given plant so that every signal generated by an exogenous system, or shortly exosystem, is asymptotically tracked despite perturbations in the plant or some external disturbances. The exosystem generating the reference and disturbance signals is assumed to be infinite-dimensional. The main contribution of this thesis is to develop the robust regulation theory for an infinite-dimensional exosystem in the frequency domain framework. In order to do that, the time domain theory is studied in some detail and new results that emphasize the smoothness requirement on the reference and disturbance signals due to infinite-dimensionality of the exosystem are presented. Two types of controllers are studied, the feedforward controllers and the error feedback controllers, the latter of which facilitate robust regulation. These results exploit the structure at infinity of tha plant transfer function. In this thesis, a new definition of the structure at infinity suitable for infinite-dimensional systems is developed and its properties are studied. The frequency domain theory developed is based on the insights into the corresponding time domain theory. By following some recent time domain ideas the type of robustness and stability types are chosen so that they facilitate the use of an infinite-dimensional exosystem. The robustness is understood in the sense that stability should imply regulation. The chosen stability types resemble the time domain polynomial and strong stabilities and allow robust regulation of signals that have an infinite number of unstable dynamics along with transfer functions vanishing at infinity. The main contribution of this thesis is the formulation of the celebrated internal model principle in the frequency domain terms in a rather abstract algebraic setting. Unlike in the existing literature, no topological aspect of the problem is needed because of the adopted definition of robustness. The plant transfer function is only assumed to have a right or a left coprime factorization but not necessarily both. The internal model principle leads to a necessary and sufficient condition for the solvability of the robust regulation problem. The second main contribution of the thesis is to design frequency domain controllers for infinite-dimensional systems and exosystems. In this thesis, the Davison’s simple controller design for stable plants is extended to infinite-dimensional systems and exosystems. Then a controller design procedure for unstable plants containing two phases is proposed. In the first phase, a stabilizing controller is constructed for a given plant. The second phase is to design a robustly regulating controller for a stable part of the plant. This design procedure nicely combines with the Davison’s type controllers and is especially suitable for infinite-dimensional plants with transfer functions in the Callier-Desoer class of transfer functions