Optimal Control Theory for Infinite Dimensional Systems

Abstract

In each time domain, LTR is described in three chapters, organized as: preliminary analysis, detailed analysis, and design. The two chapters on preliminary analysis deal mainly with controller structures, i.e., especially with observer architectures. It is demonstrated how different observer architectures have different advantages and drawbacks. It turns out that the recovery properties of the various controller types all can be characterized in terms of a certain matrix valued function, the recovery matrix. Different variations of the recovery problem are then restated in terms of the recovery matrix. In the two chapters containing a detailed analysis, LTR is investigated by means of the special coordinate basis. The analysis is quite insightful and sheds light on a variety of related problems also. In this context the authors introduce a new concept, recovery in subspaces, where the idea basically is to extract for a given system the properties that actually can be recovered and then concentrate upon these. The chapters dealing with synthesis characterize the degrees of freedom involved with recovery design. Explicit design methods include: eigenstructure assignment, and optimization, and subspace recovery design. Also, both for the continuous and for the discrete-time case, closed-loop recovery is discussed. Historically, LTR was formulated as open-loop problems, but obviously, if the design objective is a concrete, closed-loop property, it is more natural to pose the whole problem in a closed-loop formulation. The treatment and the applied methods are in form and spirit similar to the open-loop results. The final chapter of the book discusses controller architecture in the light of the design problems that have been given treatment in the preceding chapters. As a main point here is the introduction of the so-called CSS architecture (the acronym is formed by the initials of the authors). It is demonstrated that in most cases this controller architecture does significantly better than the classical architectures. One problem, though, is that a CSS controller by construction is internally stable. Since there exist a class of unstable systems that can be stabilized only by unstable controllers, the CSS controllers do not apply always. The CSS architecture is formulated both in terms of full-order and reducedorder controllers. Since Loop Transfer Recovery: Analysis and Designis the first book dedicated to this subject, it is not directly comparable with other monographs. It does, however, in an elegant way combine the tradition from classical monographs on observer theory [6], on loopshaping [7], and on geometric methods [8]. In comparison to these references Loop Transfer Recovery: Analysis and Design offers a broader and more unified approach to multivariable control synthesis.