Research On The Admissibility And Dissipativity Of Descriptor Systems

Descriptor systems theory is an important part of control systems theory and has attracted much attention of scholars at home and abroad.Currently,some achievements have been made in the analysis of admissibility and dissipativity for descriptor systems,but compared to normal systems,the results on this topic are not mature.Many unsolved admissibility and dissipativity problems need to be studied for this kind of systems.In this dissertation,based on the previous work of other scholars,the problem of the analysis of admissibility and dissipativity for descriptor systems is further studied by using the advanced theory of matrix analysis and the linear matrix inequality(LMI)technique.The main results are summarized as follows.(1)The problem of dissipativity for descriptor systems is discussed based on image space analysis.A new image space is proposed to characterize descriptor systems.Based on such characterization,a new necessary and sufficient condition of the dissipativity for descriptor systems is derived.This result can be applied into the descriptor system with impulsive behavior.In addition,the integrability of the supply function of descriptor systems with impulsive behavior is investigated.It should be pointed out that our results include some previous ones as special cases.(2)The problem of controllability and dissipativity for normal systems with deriva-tive inputs is discussed.Firstly,a new controllability criterion of normal systems with input derivatives is given by the rank of system matrices.Secondly,a necessary and suf-ficient condition of the dissipativity for this type of systems is derived.Finally,descriptor systems can be transformed into a special class of normal systems with derivative input by the restricted equivalent transformation and Laplace transformation.Based on this rela-tionship and LMI technique,the dissipativity theorem of descriptor systems is presented.Furthermore,the parametrization form of all solutions to the LMIs in this theorem is given.(3)The problem of admissibility analysis and synthesis for T-S fuzzy descriptor sys-tems is investigated.Firstly,based on nonquadratic fuzzy Lyapunov function and fully using the information of fuzzy membership functions,a relaxed sufficient condition en-suring T-S fuzzy descriptor systems to be admissible is proposed,in which it is not neces-sary to require every fuzzy subsystem to be stable.Based on the admissibility condition,parallel and nonparallel distributed compensation controllers are designed to guarantee the admissibility for closed-loop T-S fuzzy descriptor systems,respectively.Compared with some existing results in this area,our proposed approach has better performance and significantly reduces the conservativeness.(4)The problem of dissipativity analysis and synthesis for a class of T-S fuzzy de-scriptor systems with different derivative matrices is discussed.Different from the results of the previous literature,a new augmented system which is equivalent to the original system is given to deal with the difficulties caused by these different derivative matrices.Based on this new augmented system method and nonquadratic fuzzy Lyapunov function,an admissibility condition is developed in the forms of strict LMIs.Secondly,by the ad-missibility condition and choosing a Lyapunov function as the storage function,the fuzzy controllers are designed to ensure that the resultant closed-loop systems are admissible and dissipative.The advantages of our method over the existing relevant results have also been demonstrated through the simulations.It should be pointed out that other control problems of T-S fuzzy descriptor systems can also follow our proposed method to deal with these different derivative matrices.(5)The problem of admissibility analysis and synthesis for a class of T-S fuzzy descriptor systems with partially unknown membership functions is discussed.In order to fully utilize the information of membership functions,the method that the known part and unknown part are handled respectively is given.Staircase functions are used to approach the known part and the bounds of the unknown part are used to define the unknown part.Based on this method,a relaxed sufficient condition for the existence of PDC controller is obtained and a PDC controller is explicitly constructed by using the augmented system and solving LMIs.In the case that the number of fuzzy rules is relatively large,the proposed method in this dissertation can effectively reduce the amount of computation.