Research On Analysis For High-order Descriptor Systems And Cooperative Optimal Control

Second-order and high-order descriptor systems are suitable for the modeling of most physical systems,including vibration analysis of mechanical systems,fluid mechanics and other engineering applications.While the traditional linearization method may bring many disadvantages,this thesis is to explore a number of basic problems of second-order and high-order descriptor systems for control under the framework of original system parameters.Then,combing the descriptor system theory and the graph theory,the thesis also studies the consensus control problem of descriptor multi-agent systems,aiming to expand the new filed of descriptor multi-agent systems and design more practical controllers.The main contributions of this thesis are as follows:The complete solution of nonhomogeneous high-order descriptor systems under the nonzero initial state and initial input conditions is investigated.Based on the SmithMac Millan form of a polynomial matrix at infinity,it is shown that the state response of the nonhomogeneous high-order descriptor systems can be completely decomposed into the slow state response and the fast state response.Based on the structure of the proposed complete solution,the reachability and controllability of high-order descriptor systems are analyzed and the relevant algebraic criterions are also provided.The optimal control problem of second-order descriptor systems is studied.By virtue of the constructed quadratic performance index,a novel second-order generalized Riccati equation is derived directly by the original coefficient matrices,and a sufficient condition for the existence of the optimal controller of the second-order descriptor system is given.By using matrix transformation and singular value decomposition theory,the nonlinear matrix equation is transformed into the linear matrix equation,and the analytic expression of the optimal solution for the second-order descriptor system is obtained.The corresponding results are extended to the high-order descriptor systems.The positive realness of second-order and high-order descriptor systems is studied.Without any linearization,necessary and sufficient conditions for strictly positive realness and extended strictly positive realness of second-order descriptor systems are given,respectively,and the solutions for the positive real constraint equations are constructed,which are represented by the solutions of the above second-order generalized Riccati equation.Applying the polynomial matrix decomposition technology,a necessary and sufficient condition for the extended strictly positive realness of the high-order descriptor system is derived.The optimal consensus problems of homogenous descriptor multi-agent systems with different communication topologies are studied.Under the nonnegative topology,a state feedback consensus protocol is designed,and the sufficient conditions for the existence of the local optimal and the global optimal consensus protocols for descriptor multi-agent systems are obtained,respectively.Under the signed graph,the existence condition for the descriptor multi-agent systems to be admissible bipartite consensus is established,and the optimal design algorithm of the bipartite consensus protocol is given based on the proposed consensus conditions.The cooperative optimal output regulation problem of heterogeneous descriptor multi-agent systems over the directed topology is studied.Firstly,a distributed observer is proposed to estimate the state information of the exogenous system.Then,the distributed state feedback controller and the distributed dynamic output feedback controller based on the normal reduced-order observer are designed,the sufficient conditions for the solvability of the optimal output regulation problem under the two control protocols are determined,and the design method of the optimal control gain is given,respectively.