Observer and Compensator Design via Quotient Signal Modules

The project is devoted to the design and parametrization of observers and of stabilizing compensators, both for one- and multidimensional and for continuous and discrete systems, and is planned for a total time of two years. Intended participants are Dipl.-Ing. I. Blumthaler, Dipl.-Ing. Dr. M. Scheicher and Prof.em. Dr. U. Oberst. The research on observers completes Blumthaler`s recent work on one-dimensional functional observers with respect to spectral assignability, robustness and minimality and primarily extends it to the multidimensional situation. In our new framework of compensator design the plant and the compensator are {\em generalized} input/output systems which are partially interconnected to form the well-posed or regular full interconnected control system which is a stable input/out-put system whose transfer matrix or submatrices realize additional tasks like tracking, disturbance rejection, model matching, decoupling and others. The existence and parametrization of such stabilizing compensators will be discussed. The framework generalizes the standard output feedback configurations with two-parameter (two-degree-of-freedom) compensators and uses Willems` idea of control by interconnection, but differs decisively from the partial interconnection model studied by Willems, Trentelman et al. Our approach is also behavioral, uses the categorical duality between behaviors and modules and employs, as an important technical ingredient, a new mathematical technique for stability and stabilization questions, viz. that of injective cogenerator quotient signal modules in the one-dimensional case and Gabriel localization as its counterpart in the multidimensional situation. The new technique especially permits the treatment of different types of stability simultaneously. In the planned behavioral approach to compensator design the full interconnected control behavior and especially its autonomous part and not only its transfer matrix or an induced manifest controlled behavior are studied. It is expected that the new technique furnishes substantially simpler proofs and new and sharper results also for the standard stabilizing compensators by output feedback. This expectation is justified by Blumthaler`s successful use of this technique for the construction and parametrization of one-dimensional functional observers which generalized and sharpened earlier work of Wolovich, Valcher, Willems, Fuhrmann and others on behavioral observer theory and especially investigated exact, dead-beat, asymptotic and tracking observers simultaneously. The significance of the new results in the standard one-dimensional cases will be carefully elaborated. Whereas the design of one-dimensional observers and of stabilizing compensators belong to the most important subjects of control engineering and have therefore been investigated by many prominent researchers the multidimensional counterparts have not yet been studied extensively. The latter use the same engineering ideas, require, however, more advanced mathematical tools from algebra and analysis. Two-dimensional observers, mainly for discrete systems described by state space equations according to Fornasini/Marchesini or Roesser and of the dead-beat type, have been investigated by Bisiacco and Valcher in a recent series of papers. Stabilizing compensators for multidimensional discrete systems described by convolution with proper rational transfer matrices and for stability defined via structurally stable polynomials and rational functions have been studied by various authors, the first results from thirty years ago being due to Bose. The problem of tracking and disturbance rejection in this context is the subject of a recent paper by Xu. Regular interconnection of multidimensional behaviors according to Rocha et al. will be used.

The main goals of the project were the construction and parametrization of multidimensional observers and compensators of a primary system, usually called plant, and were reached to a large extent. An observer is a second system which uses some measured components (dependent variables) of the plant to approximate (estimate) other components that are needed for controlling the plant. A stabilizing compensator is a second system that is connected to the plant such that the interconnected system is stable and performs desired tasks, for instance the approximation or tracking of a reference signal. The considered systems are one-dimensional, i.e., the signals are functions of only one independent variable (usually the time), or multidimensional if more independent variables (for instance, also space coordinates) are used. The systems are continuous (analog) resp. discrete (digital) if the independent variables are real numbers resp. integers. A computer is a typical one-dimensional discrete system whereas image processing uses at least two-dimensional, analog or digital signals. The emphasis of our work was on discrete multidimensional systems. Approximation of a signal by another one signi?es that the difference signal is stable (small, negligible) in a suitable sense. Therefore an essential part of the project was devoted to the development of multidimensional stability.The design of stabilizing compensators for one-dimensional systems belongs to the most important and dif?cult tasks of control engineering and has therefore been treated by many prominent researchers and exposed in many well-known textbooks. In spite of our predecessors outstanding results our new localization technique enables new results and proofs even in the one-dimensional situation. For higher dimensional systems both the technical (applied, engineering) side and the mathematical description and theory of observers and compensators are less developed and require new and more advanced mathematical tools. Injective cogenerator signal modules, Serre categories of stable modules and systems and their associated Gabriel localization are among such tools.