# State space output regulation theory for infinite-dimensional linear systems and bounded uniformly continuous exogenous signals

##### Immonen, E. (2006)

Immonen, E.

Tampere University of Technology

2006

Sähkötekniikan osasto - Department of Elecrical Engineering

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**Julkaisun pysyvä osoite on**

https://urn.fi/URN:NBN:fi:tty-200810021035

##### Tiivistelmä

In this thesis we develop a state space output regulation theory for linear infinite-dimensional systems and bounded uniformly continuous exogenous reference/disturbance signals. The output regulation problems that we study involve the construction of such controllers which (i) stabilize the closed loop system consisting of the plant and the controller appropriately, (ii) achieve asymptotic tracking of the reference signals and rejection of the disturbance signals, and (iii) preferrably do this robustly with respect to small parameter variations in the control system.

We show how bounded uniformly continuous reference/disturbance signals are best generated using a (possibly infinite-dimensional) exogenous system. This exosystem utilizes a strongly continuous group of isometries on some Banach space and two bounded observation operators. The regulation of all signals in certain Banach subspaces of bounded uniformly continuous functions is shown to be equivalent to the regulation of all signals generated by such exosystems, with a suitable choice of the free parameters.

We conduct an extensive study of three controller configurations feedforward controllers, error feedback controllers and hybrid feedforward-feedback controllers for output regulation purposes. In particular, complete characterizations for the solvability of the three output regulation problems are obtained in terms of solutions of certain constrained operator Sylvester equations (regulator equations). We illustrate the abstract results with various examples and case studies, particularly from repetitive control applications.

We study robustness of the devised error feedback controllers using perturbation techniques. We also prove such a state space generalization of the Internal Model Principle which does not utilize any purely finite-dimensional concepts. This result describes the necessary and sufficient structure of all robustly regulating error feedback controllers, under appropriate closed loop stability assumptions.

We introduce the practical output regulation problem in which asymptotic tracking and disturbance rejection with a given accuracy only is required. Using perturbation techniques we present upper bounds for the norms of perturbations to the closed loop control systems parameters such that practical output regulation with a desired accuracy occurs. Our results treat the above three controller configurations in a unified way.

Finally, we present a general methodology for the solution of the regulator equations in two (separate) cases. In the first case we assume that the plant is a single-input single-output (SISO) system, whereas in the second case we assume that the spectrum of the exosystem s generator is a discrete set. Both of these cases are important in practice, and they cover most of the applications that we have in mind in particular the repetitive control problems for infinite-dimensional linear systems.

We show how bounded uniformly continuous reference/disturbance signals are best generated using a (possibly infinite-dimensional) exogenous system. This exosystem utilizes a strongly continuous group of isometries on some Banach space and two bounded observation operators. The regulation of all signals in certain Banach subspaces of bounded uniformly continuous functions is shown to be equivalent to the regulation of all signals generated by such exosystems, with a suitable choice of the free parameters.

We conduct an extensive study of three controller configurations feedforward controllers, error feedback controllers and hybrid feedforward-feedback controllers for output regulation purposes. In particular, complete characterizations for the solvability of the three output regulation problems are obtained in terms of solutions of certain constrained operator Sylvester equations (regulator equations). We illustrate the abstract results with various examples and case studies, particularly from repetitive control applications.

We study robustness of the devised error feedback controllers using perturbation techniques. We also prove such a state space generalization of the Internal Model Principle which does not utilize any purely finite-dimensional concepts. This result describes the necessary and sufficient structure of all robustly regulating error feedback controllers, under appropriate closed loop stability assumptions.

We introduce the practical output regulation problem in which asymptotic tracking and disturbance rejection with a given accuracy only is required. Using perturbation techniques we present upper bounds for the norms of perturbations to the closed loop control systems parameters such that practical output regulation with a desired accuracy occurs. Our results treat the above three controller configurations in a unified way.

Finally, we present a general methodology for the solution of the regulator equations in two (separate) cases. In the first case we assume that the plant is a single-input single-output (SISO) system, whereas in the second case we assume that the spectrum of the exosystem s generator is a discrete set. Both of these cases are important in practice, and they cover most of the applications that we have in mind in particular the repetitive control problems for infinite-dimensional linear systems.

##### Kokoelmat

- Väitöskirjat [4551]