Stabilization And Control Design For Several Classes Of Nonlinear Descriptor Systems

The descriptor system is a natural representation of dynamic systems and describes a larger class of systems than the normal system model. In the last three decades, the descriptor system has received a great deal of attention, and many studies have been devoted to the stabilization and H∞control of linear descriptor systems. Generally, it is difficult to design a controller for nonlinear descriptor systems, and accordingly, there are fewer works on nonlinear descriptor systems except several case studies. In recent years, the Hamiltonian function method has received a great deal of attention and obtained many achievements in a series of works. The Hamiltonian function, the sum of potential energy and kinetic energy in physical systems, can play the role of a Lyapunov function for many practical systems. Due to this and its nice structure with clear physical meaning, the Hamiltonian function method has been widely applied to practical control problems, and many effective controllers have been designed.In this paper, we apply the Hamiltonian function method to investigate the stabi-lization and control design problems for several classes of nonlinear descriptor systems. The main contents of this paper are composed of the following five parts:The first part investigates the stabilization and H∞control problems of a class of nonlinear Hamiltonian descriptor systems which the structural matrices are constant ones, and discusses the application to nonlinear descriptor systems. Based on the struc-tural characteristic of the nonlinear Hamiltonian descriptor system, an equivalent non-linear Hamiltonian differential-algebraic system is obtained. The nonlinear Hamilto-nian differential-algebraic system obtained is then transformed into a strictly dissipative Hamiltonian form by a suitable state feedback. With the dissipative form, a stabiliza-tion control law is developed, based on which an H∞controller is then designed for the nonlinear Hamiltonian descriptor systems with external disturbances. Finally, the ob-tained results for the Hamiltonian descriptor system are applied to nonlinear descriptor systems, and a sufficient condition under which the systems can be transformed into a strictly dissipative Hamiltonian form is proposed. Study of examples with simulations shows that the controllers obtained work very well.The second part studies the stabilization and H∞control problems of a class of nonlinear Hamiltonian descriptor systems which the structural matrices are variable ones. The systems describe a larger class of systems than the ones of constant struc-tural matrices. Firstly, the nonlinear Hamiltonian descriptor system is transformed into an dissipative nonlinear Hamiltonian differential-algebraic system by equivalent trans-formation and a suitable state feedback. Based on the dissipative form, a stabilization control law is then proposed, and an H∞controller is designed for the nonlinear Hamil-tonian descriptor systems with external disturbances. Finally, simulation results on nonlinear descriptor circuit systems show the efficiency of the controllers proposed.The third part considers the stabilization,H∞control and robust adaptive control of a class of nonlinear descriptor systems, and a number of new results on the con-trol designs are presents by the dissipative matrix method. Based on the structural characteristic of the nonlinear descriptor system, an equivalent nonlinear dissipative differential-algebraic system is obtained by a suitable state feedback, and two suffi-cient conditions of impulse-free are given for the resulted closed-loop descriptor sys-tem. With the feedback dissipative form, the stabilization problem is then discussed for the nonlinear descriptor system, and a stabilization control design procedure is pre-sented, based on which an admissible H∞controller and an admissible robust adaptive controller are designed for the system with external disturbances and/or parametric un-certainties. Simulation results on nonlinear descriptor circuit systems show that the controllers obtained work very well.The forth part proposes the control designs of a class of nonlinear descriptor sys-tems under state undecomposed status. It is shown that under a suitable state feedback, two sufficient conditions of impulse-free can be established for the resulted closed-loop system. With the established conditions, the stabilization problem is then discussed for the nonlinear descriptor system, and a stabilization control design procedure is pre-sented, based on which an admissible H∞controller and an admissible robust adaptive controller are designed for the system with external disturbances and/or parametric un-certainties. Simulation results on a nonlinear descriptor circuit system show the effi-ciency of the controllers proposed.The fifth part studies the simultaneous stabilization of a class of nonlinear descrip-tor systems via the Hamiltonian function method. Firstly, based on the Hamiltonian realization of the nonlinear descriptor systems and a suitable output feedback, two non-linear descriptor systems are equivalently transformed into two nonlinear Hamiltonian differential-algebraic systems by a nonsingular transformation, and a sufficient con-dition of two closed-loop systems impulsive-free is given. The two systems are then combined to generate an augmented dissipative Hamiltonian differential-algebraic sys-tem by the system-augmentation technique, based on which a simultaneous stabiliza-tion controller, a robust simultaneous stabilization controller, an adaptive simultaneous stabilization controller and a robust adaptive simultaneous stabilization controller are designed for the two systems. Secondly, the case of more than two nonlinear descrip-tor systems is investigated, and four new results are proposed for the simultaneous stabilization, robust simultaneous stabilization, adaptive simultaneous stabilization and robust adaptive simultaneous stabilization, respectively. Finally, two illustrative exam-ples are studied by the results proposed, and simulations show that the simultaneous stabilization controllers obtained work very well.Innovations of the thesis mainly include the following four aspects:●It is first that the stabilization and control design problems are investigated for a class of nonlinear Hamiltonian descriptor systems. Based on the nice structure and dis-sipative properties of nonlinear Hamiltonian descriptor systems, the stabilization and H∞controllers are designed for a class of nonlinear Hamiltonian descriptor systems which the structural matrices are constant ones and variable ones, respectively, and the obtained results for the Hamiltonian descriptor system are applied to nonlinear descrip-tor systems.●It is first that the stabilization and control design are studied for a class of non-linear descriptor systems via the dissipative matrix method. H∞controller and robust adaptive controller are given for a class of nonlinear descriptor systems using the dissi-pative matrix properties.●It is first that the robust controller and robust adaptive controller are designed for a class of nonlinear descriptor systems under state undecomposed.●It is first that simultaneous stabilization is proposed for a class of nonlinear de-scriptor systems. With the Hamiltonian function method, a simultaneous stabilization controller, a robust simultaneous stabilization controller, an adaptive simultaneous sta-bilization controller and a robust adaptive simultaneous stabilization controller are de-signed for two nonlinear descriptor systems and more than two nonlinear descriptor systems, respectively.