Output Regulation Theory for Linear Systems with InfiniteDimensional and Periodic Exosystems
Abstract:
For generating our reference and disturbance signals we consider two separate methods, namely, a timeinvariant infinitedimensional exosystem and a periodically timedependent finitedimensional exosystem. We will see that the chosen method has a considerable effect on the properties of the resulting control law as well as on the behavior of the controlled closedloop system. One of the main differences in these respective theories of output regulation is that the control law designed based on the infinitedimensional exosystem is guaranteed to be robust with respect to a class of perturbations preserving the stability of the closedloop system.
The first main result of this thesis is the generalization of the wellknown internal model principle of finitedimensional control theory for distributed parameter systems with infinitedimensional exosystems. On a general level this result states that in order for a controller to solve the robust output regulation problem related to a given signal generator, the controller must be able to reproduce the dynamics of this exosystem. In addition to its theoretical significance the internal model principle can also be applied in the construction of controllers solving the robust output regulation problem. Our proof of this result is based on a close connection between the behavior of the state of the closedloop system and an associated Sylvester operator equation. In particular, the controllers achieving asymptotic tracking of the reference signals can be characterized using the solvability of certain constrained Sylvester equations, and the robustness of this property can be expressed as a condition involving equations of this type.
The second main contribution of this thesis consists of the development of the theory of output regulation for infinitedimensional systems with periodically timedependent exosystems. In particular this also includes designing nonautonomous controllers achieving asymptotic output tracking and disturbance rejection. Our treatment shows that it is possible to study the output regulation problem for a distributed parameter system together with a nonautonomous exosystem using methods similar to the ones familiar from case of a timeinvariant signal generator. In particular, the solvability of the problem related to a given periodic exosystem can be characterized using a periodically timedependent version of the wellknown regulator equations if the associated Sylvester operator equation is replaced with an infinitedimensional Sylvester differential equation.
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