Derivative Feedback Control For Singular Systems With Uncertainties In The Derivative Matrix

Descriptor system form contains the standard state-space form as a special case and can represent a much wider class of systems than the standard state-space coun-terpart. It is often convenient and natural to model many practical systems by using descriptor systems. Recently, the theories and practical applications of descriptor systems have attracted much attention of many scholars at home and abroad, and many results and various research methods for the standard state-space systems have been perfectly extended to descriptor systems. Compared with the standard state-space systems, the singularity of the derivative matrix of descriptor systems is a source of its distinguishing structural features and special characteristics. The arbitrarily small unstructured perturbations in the derivative matrix may destroy regularity, cause impulse and even instability. However, the derivative matrix of a descriptor system, which, just as the other matrices in state and input, is subject to possible perturbations due to the fact that all mathematical models of a physical system suffer from inaccuracies that result from non-exact measurements or from the general inability to capture all phenomena that are involved in the dynamics of the considered system. Up to now, the results of descriptor systems without param-eter uncertainties in the derivative matrix have been approaching towards maturity, but in the case of descriptor systems with parameter uncertainties appearing in the derivative matrix, how to control the systems under consideration to have the satisfied performances remains further study.In view of these facts that the admissibility of a descriptor system may be fragile to arbitrarily small perturbations in the derivative matrix and the derivative feedback has a well engineering motivation, this dissertation mainly investigates derivative feedback control problems for descriptor systems with parameter uncer-tainties in the derivative matrix by using linear matrix inequality (LMI) technique. The main contents include:guaranteed cost control, H∞control for both descriptor systems and time-delayed descriptor systems under derivative feedbacks; positive real control for descriptor systems under derivative feedbacks; proportional-plus-derivative observer design, respectively, for both continuous descriptor systems and discrete counterpart and stability analysis of a class of discrete linear systems. The main results are summarized as follows:(1) The design of guaranteed cost controllers for descriptor systems with pa-rameter uncertainties in the derivative matrix is studied by applying a derivative feedback. In this dissertation, a nonstandard quadratic performance index, which is different from ones in the existing literature, contains not only quadratic state and input constrains but only quadratic derivative state constrain. By considering the forms of derivative feedback controllers and quadratic performance index, and ap-plying LMIs and free-weighting matrix, a necessary and sufficient condition for the existence of derivative feedback guaranteed cost controller is obtained and a deriva-tive feedback guaranteed cost controller is explicitly constructed. Furthermore, the existing condition and design of the optimal derivative feedback guaranteed cost controller is given.(2) The design of H∞controller for descriptor systems with parameter uncer-tainties in the derivative matrix is studied by applying a derivative feedback. Firstly, a new LMI characterization of bounded real lemma for descriptor systems, which guarantees a system to be stable with γ-disturbance rejection, is presented. The new version of bounded real lemma can be regarded as an extension of the existing one for the standard state-space systems to normal descriptor form representations. Based on this lemma, a necessary and sufficient condition for the existence of de-sired derivative feedback H∞controllers for uncertain descriptor systems is derived, and further the explicit expression of a derivative feedback H∞controller is con-structed by solving LMIs. Secondly, taking into account the practical importance of guaranteed cost control, we investigate the existence conditions and design of deriva-tive feedback H∞guaranteed cost controller for descriptor systems with parameter uncertainties in the derivative matrix by using LMIs. Furthermore, the optimal derivative feedback H∞guaranteed cost controller design problem is also studied. Finally, the existing condition and design of memory derivative feedback controller for time-delayed descriptor systems are given by using LMI technique.(3) The design of positive real controller for descriptor systems with parameter uncertainties in the derivative matrix is studied by applying a derivative feedback. A new version of positive real lemma, which can be regarded as an extension of the existing one for the standard state-space systems, is derived. Based on the lemma, a necessary and sufficient condition, which guarantees the resulting closed-loop system to be quadratically normal and quadratically stable with extended strictly positive real, is proposed, and a desired derivative feedback positive real controller is con- structed by solving LMIs.(4) The proportional-plus-derivative observer design, respectively, for both continuous descriptor systems and discrete counterpart, is investigated. Several equivalent necessary and sufficient conditions for the existence of a proportional-plus-derivative state observer are proposed, and the design of a proportional-plus-derivative state observer is also given by LMIs. The proposed proportional-plus-derivative state observer, whose derivative matrix is nonsingular, is more practical in physics and more robust of stability compared to the usual proportional observer of descriptor systems.(5) The quadratic stability analysis of a class of discrete linear systems with un-certainties in the derivative matrix is studied. In context of continuous counterpart, the usual methods are to convert them into the corresponding standard state-space systems when studying their stability. However, these transformations inevitably introduce conservatism and may result in some numerical problems. Without re-sorting to any conversion and by introducing slack matrix variables, a necessary and sufficient condition for quadratic stability for the considered systems is proposed by using LMIs.Finally, the main work of the dissertation is summarized, and the research direction of further work is proposed.