Analysis Of The Dynamical Behaviors Of Two-dimensional Positive Systems

Positive systems are proposed to characterize a class of special systems whose variables are nonnegative.Compared with general systems,one knows that the characteristic of this special type of systems lies in that the state variables and system output can only be obtained in the cone of the first quadrant instead of the whole space.Consequently,many conclusions concerning general systems are no longer applicable to positive systems,which is also the main reason for slow development of positive systems.In recent years,positive systems have drawn much research attention due to their universal existence and extensive application.Most of the existing results are mainly focused on the case of one-dimensional?1-D?systems,whereas the corresponding parts on the two-dimensional?2-D?positive systems are relatively few.Unlike the traditional 1-D systems,2-D systems possess mutually independent evolutionary processes along two independent directions,which will bring great difficulties in calculation and analysis.With the development of science and technology and the demand of engineering applica-tion,there has been some progress on 2-D systems.However,the corresponding research for2-D positive systems is still on the initial stage,and there are many difficult problems to be solved urgently.The purpose of this dissertation is to investigate the dynamic characteristics of 2-D positive systems which suffer from the influence of different external factors,such as stability,stabilization,filtering,performance analysis,asynchronous control,and so on.The organization of this dissertation is as follows:In the first chapter,some related background and research status are reviewed,and the main contents,contributions and innovations of this dissertation are also provided.In the second chapter,a class of 2-D delayed switched Takagi-Sugeno?T-S?fuzzy positive systems described by the Fornasini-Marchesini?F-M?state space model are studied.It can be described by the form of weighted sum with a family of linear systems whose weight coefficients are corresponding membership functions.Furthermore,based on the Lyapunov stability theory and the idea of average dwell time?ADT?,the criteria are derived under which the proposed system is exponentially stable and has l₁-gain performance.In the third chapter,a class of 2-D switched T-S fuzzy positive systems described by the Roesser state space model are analyzed.Similar to chapter 2,the considered system can also be described by the form of weighted sum with a family of linear systems and corresponding membership functions.When the system is unstable,a dynamic output feedback controller is designed to achieve the stabilization effect.It is worth noting that the special property of the Moore-Penrose generalized inverse is skillfully utilized in designing the desired controller.In the fourth chapter,a class of special 2-D systems are considered.The two independent variables concerning with the system states are of different types.More specifically,one of them is continuous and the other is discrete.It is a differential equation with respect to the continuous variable but a difference equation with respect to the discrete variable.In addition,the state variable and output variable of the original system can be estimated via both the lower-bounding filter and the upper-bounding one,which make the estimation more reliable and accurate.Furthermore,two different types of filters,i.e.,the observer-based style and the general style,are analyzed respectively,and the advantages/disadvantages for both of them are also illustrated.In the fifth chapter,actuator saturation,parameter uncertainty and switching phenomenon are introduced into the 2-D positive systems,and the robust finite-horizon stability and sta-bilization problems are investigated.Finite-horizon stability is different from the exponen-tial/asymptotic stability that studied in previous chapters.It focuses on the behavior of the system in a particular region.In addition,when it comes to the stabilization problem,observer-based state feedback controller needs to be designed to solve the unmeasurability of the system state.What is more,the results mentioned in this chapter can be transformed into the form of linear programming?LP?,which can be solved by the standard software and thus more convenient for application.In the sixth chapter,impulse,parameter uncertainty and asynchronous switching phe-nomenon are introduced into the 2-D positive systems,and the issue of asynchronous control is investigated.Asynchronous control is more general due to the fact that the asynchronism between system modes and corresponding mode-dependent controller candidates is considered.In addition,by choosing a suitable co-positive Lyapunov function and utilizing the mode-dependent average dwell time?MDADT?technique,the issue of exponential stability and weighted l₁control is dealt with.It is worth noting that the value of the MDADT given in this chapter is closely related to the maximal asynchronous time.In the seventh chapter,the research work of this dissertation is summarized comprehen-sively,and the future directions are also prospected.