Stability Analysis And Design For Some Classes Of Nonlinear Descriptor Systems

Stability is a fundamental problem in control theory. Now, there have been many significant results on stability problem of linear descriptor systems. However, investigation on stability problem of nonlinear descriptor systems is premature. This thesis investigates the problems of practical stability, strongly absolute stability, input-state stability and observer design for nonlinear descriptor systems and the problems of absolute stability and multi-objective control for nonlinear singularly perturbed systems by using Lyapunov stability theory, comparison principle, S-procedure and linear matrix inequality (LMI) etc.. The main results are summarized as follows:(1) Practical stability of nonlinear descriptor systems is studied. The basic con-cepts and results on practical stability for normal systems are generalized to descrip-tor systems. Firstly, some sufficient conditions for nonlinear descriptor systems to be practically stable are derived by Lyapunov stability theory and comparison princi-ple. The obtained results are used to analyze practical stability of linear descriptor systems and nonlinear circuits systems. Secondly, practical stability of nonlinear descriptor systems with disturbance inputs is studied by comparison principle. Two problems are solved:one is to describe an admissible input set such that the system is practically stable; the other is to determine if the system is practically stable for a given admissible input set. Thirdly, practical stability in terms of two measurements is investigated for nonlinear descriptor systems with time delays and some sufficient conditions are derived by using Lyapunov stability theory and comparison principle. Finally, the problem of practical stabilization of nonlinear descriptor systems is in-vestigated. A controller design method is derived by introducing a new comparison principle.(2) Strongly absolute stability of Lur'e descriptor systems (LDS) is studied. Circle criterion and Popov criterion are derived. Firstly, the concept of strongly absolute stability of LDS is defined and the positive realness of descriptor systems is discussed. Secondly, single input and single output LDS is considered and the graphical representation of circle criterion is given. If the feed forward is impulsive-free, the existing circle criterion is directly generalized to descriptor systems by the classical Nyquist stability criterion. If the feed forward is not impulsive-free, the classical Nyquist stability criterion is useless. We propose a Nyquist like stability criterion by which a more general circle criterion is established. Thirdly, multiple input and multiple output LDS is considered and an LMI-based circle criterion is derived by a generalized Lyaponov function and S-procedure. Then, the Popov cri-terion for normal systems is generalized to descriptor systems. Finally, a generalized Lur'e Lyaponov function (GLLF) is constructed. It is shown that LDS is strongly absolutely stable if there exists a GLLF whose derivative along the trajectories of LDS is negatively definite. It is further proved that the presented Popov criterion is only a sufficient condition for the existence of the GLLF. To get a less conservative criterion, a Popov like criterion which is a necessary and sufficient condition for the existence of the GLLF is derived.(3) Input-state stability of LDS with disturbances is investigated. Firstly, the notion of input-state stability (ISS) for nonlinear descriptor systems is defined based on the concept of ISS for normal systems and the characteristics of descriptor sys-tems. Then, an LMI-based sufficient condition for ISS of LDS is derived by the classical ISS theory. Furthermore, a state feedback controller design method is pro-posed, such that the closed-loop system is ISS.(4) Observer design for a class of nonlinear descriptor systems is studied. The involved nonlinear term satisfies a given quadratic inequality. Under this condition, the error system is expressed by an LDS. As a result, the convergency problem of the estimate error is reduced to the stability of the LDS. By virtue of the basic idea of absolute stability, a unified design method for full-order and reduced-order observer is derived. A class of nonlinear descriptor systems with disturbances is considered. Both of the state equation and the output equation of the systems contain slope-restricted nonlinear terms. An H∞observer is designed such that the error system is exponentially stable and the decay rate is bigger than or equal to a given constant and the H∞performance of the error system is less than or equal to a prescribed level. Furthermore, two convex optimization algorithms are given to optimize the decay rate and the H∞performance, respectively.(5) Absolute stability and multi-objective control of nonlinear singularly per-turbed systems are studied. Firstly, we propose two lemmas which are the theory basis for constructingε-dependent Lyapunov functions. Then, circle criterion and Popov criterion for absolute stability of Lur'e singularly perturbed systems are de-rived by usingε-dependent quadratic Lyapunov function and Lur'e Lyapunov func-tion, respectively. Compared with the existing results, the obtained methods do not depend on the decomposition of the original system and can produce an determinate upper bound for the perturbation parameter. Finally, the problem of multi-objective control of T-S fuzzy singularly perturbed systems is considered. In this problem, a given upper bound for the perturbation parameter is one of the design objectives. Using anε-dependent Lyapunov function, an LMI-based controller design method is given.