Structure and Output Feedback in Singular Systems

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Generalized state-space systems of the form Ex= Ax + Bu, with E generally singular, arise naturally in the study of robots, networks, large-scale interconnected systems, power systems, economic systems, delay-differential systems, neural networks, and elsewhere. Our main contentions are that there are some serious limitations in state-variable theory and that singular systems provide a more general context that yields important insight into some deep questions about state-variable systems. Two important cases will be studied in the proposed research. First, we point out that the inverse of a state-space system is not a state-variable system. Indeed, the inverse generally contains derivatives of the output, so that it is a singular system. Therefore, singular systems, not state-variable systems, provide the natural dynamical formulation for the study of issues related to inversion. This includes high-grain feedback, singular optimal control (in the traditional sense of incomplete control weighting), and other important problems as well as the usual system inversion problems. Our first goal is, therefore, to study the extensions and properties of the Singular System Structure Algorithm which has been developed in connection with inversion and optimal control of singular systems. Second, if proportional-plus-derivative output feedback is applied to a state-space system, the result is a singular and not a state- variable system. These systems therefore provide a nature formulation for carrying out our second goal, the study of proportional-plus derivative output feedback in state-variable and singular systems. Recent results make it clear that using this technique, far more can be achieved than the assignment of the traditional min(n, m+p-1) poles.