Robust Dissipative Control For Uncertain Systems

Originated from electrical network, dissipative theory owns extensive engineering backgrounds, and it has many important applications in adaptive control, nonlinear control and robust control, etc. Moreover, dissipative control can unify H∞control and passive control, and provide a more flexible and less conservative control design. In addation, dissipative control can provided an important method for robust control design. On another hand, since it is difficult to describe practical systems exactly and many uncertainties may appear when the systems is running, dissipative theory can't be used effectively without robust dissipative control problems settled. Therefore, it has both theoretical significance and applied value to study robust dissipative control problems.This dissertation investigates thoroughly the robust dissipative control problem for uncertain linear systems using linear matrix inequality (LMI) as a main tool. A state feedback or output feedback controller is designed such that the resultant closed-loop system is robustly asymptotically stable and strict quadratic dissipative. The systems considered include continuous-time system, discrete-time system, delay system and descriptor system. The uncertainties concerned contain norm-bounded uncertainty, positive real uncertainty, generalized positive real uncertainty and the uncertainty characterized by a dissipative system. By establishing equivalence between dissipativeness and positive realness, the characterization of strict quadratic dissipative systems is derived and sufficient and necessary conditions for the uncertain system to be strict quadratic dissipative are obtained. Then the solvability of robust dissipative control problems is studied and steps for synthesizing the controllers are given. It is shown that the robustly dissipative analysis and synthesis problems can be reduced to problems of solving some LMIs. In addition, by augmenting the nominal systems robust dissipative analysis and synthesis problems can be converted to those of related systems without uncertainties. Some existing results of the robust passive control problem, the robust H∞control problem and the robust dissipative control problem have been extended in this paper. The main contributions of this dissertation are summed as follows:In the 1st chapter of this dissertation, the born and development of dissipative theory are reviewed and main results of robust dissipative control for linear systems are revisited. Some approaches and problems are given in the study of the dissipative control problems, and the main work of this dissertation is introduced.The 2nd chapter addresses the robust positive real problem for continuous-time systems with perturbation-independent parameters. The commonly used norm-bounded uncertainty is extended here and some existing results of the robust positive control problem are generalized.In contrast with the norm-bounded uncertainty, positive real uncertainty is considered in Chapter 3. The uncertainty has roots in engineering and can be viewed as a generalization of the feedback gain. Conditions are characterized under which the uncertain system is robustly stable and strictly positive real. The robust positive real control of discrete-time systems is discussed here. Both state feedback and output feedback are considered.Chapter 4 presents robust positive real analysis and synthesis for continuous-time systems, where the generalized positive real uncertainties are considered. The extension of the uncertainty covers that of the norm-bounded and positive real uncertainty. Conditions for the uncertain system to be robust strictly positive real are derived and solutions to the robust strict positive real synthesis problem are obtained.The 5th chapter is about the robust dissipative control problem for linear continuous-time systems with norm-bounded parametric uncertainty. For the system without uncertainty, the equivalence between quadratic dissipative and positive realness is established by augmenting the system, and the characterization of strictly quadratic dissipative linear system is thus obtained using the Positive Real Lemma. For the uncertainty system, conditions are derived for it to be strictly quadratic dissipative, and solutions to the robust dissipative control problem is obtained via state feedback and output feedback. The results of this chapter unify existing results on H∞and positive real control and provide a more flexible and less conservative robust control design.Being parallel to chapter 5, the 6th chapter investigates the robust dissipative control problem for linear discrete-time systems. The similar results are obtained here.Chapter 7 centers on the robust dissipative control problem for continoustime delay systems with dissipative uncertainties, namely designing of an output feedback law achieving robust asymptotical stable and strict quadratic dissipative. For the system without uncertainties, conditions are derived for the system to be asymptotically stable and strictly quadratic dissipative, and an output feedback controller is obtained such that the closed-loop system is asymptotically stable and strictly quadratic dissipative. As for the uncertain system, the structured uncertainty characterized by a dissipative system is considered. The dissipative uncertainty is quite general and includes the commonly used types of uncertainties such as norm-bounded uncertainty, positive real uncertainty and generalized positive real uncertainty as special cases. Conditions are invesgated for the uncertain system to be robustly asymptotically stable and strict quadratic dissipative. By augmenting the nominal system, it is show that the robust dissipative control problem is reduced to that without uncertainties. The results obtained in this chapter unify existing results on H∞and positive real control for delay systems, and generalize results of dissipative control problems from normal systems to delay systems.The results of last Chapter are generalized to the discrete-time case in the 8th chapter. Conditions are characterized for the uncertain discrete-time systems to be robustly asymptotically stable and strict quadratic dissipative. Output feedback dissipative control problems are also addressed here.Conditions for positive real descriptor systems are improved and re-expressed by strict LMIs in Chapter 9. The original non-strict LMI constraints, which may cause numerical problems, are removed. The robust positive real control problem for descriptor systems with norm-bounded uncertainty is explored based on the new positive real conditions. Conditions for state feedback controllers to exist are derived and the controllers are given using an LMI approach.Some concluding remarks are given in Chapter 10. The main contents and some problems related to this paper are given here.Besides, simulations are made for main design schemes and the results demonstrate the effectiveness of the proposed approaches.