Stability Researches And Applications For Singular Systems With Time-delays

Singular systems are also known as strong coupled systems,incomplete state systems,descriptor systems.They are established based on the concept of “singular perturbation”,which corresponds to the concept of “regular perturbation” in regular system.The regular perturbation occurs on the right side of the system state equation and is the perturbation of the system parameters,while the singular perturbation on the left side of the state equation is the perturbation of the state.In fact,many systems have rapidly changing variables that exhibit singularly perturbed property,such as constrained robots,non-causal systems,Newman models,and nuclear reactors.For economic and management problems,singular systems also have very important applications.Many practical systems are more convenient and natural to be described by singular systems.Dynamic input-output models,Von-Neumann models and macroeconomic systems are all typical singular dynamic systems.At the same time,the current state of the system is always affected by the past moment.Therefore,singular systems with time-delays have attracted extensive attention from scholars and become a hot research field at present.When singular systems with time-delays are used to solve practical problems,it is necessary to analyze the stability of the systems to ensure the normal operation of the systems.In view of this,it is of great significance to investigate the stability and applications of singular systems with time-delays.The thesis mainly studies the following three aspects:1.Stability of linear uncertain fractional-order singular systems with neutral delaysThe robust asymptotic stability of linear uncertain fractional-order singular systems with neutral delays is investigated.At first,a class of singular fractional-order system with neutral delays is proposed based on the Caputo fractional definition and related theories of singular systems.Then,by using the inequality technique,the sufficient criterion for asymptotic stability of this class of deterministic singular fractional-order system is obtained by constructing an appropriate LyapunovKrasovskii functional.At the same time,based on the assumption that the uncertain terms of the system parameters are bounded,the sufficient criterion for the robust asymptotic stability of this class of uncertain singular fractional-order system is obtained.Finally,three simple examples are given to illustrate the effectiveness and feasibility of the obtained results.2.Stabilization of T-S fuzzy fractional-order rectangular descriptor system with time delaysThe stabilization of a class of T-S fuzzy fractional-order rectangular singular system is investigated without requiring the state delays to be differentiable.Firstly,the T-S fuzzy fractional-order rectangular singular system is transformed into the T-S fuzzy fractional-order square closed-loop system by establishing a kind of appropriate dynamic compensator of proportional and differential type.Secondly,the obtained system is transformed into the equivalent augmented singular system for dynamic behavior analysis.Based on the fractional calculus,Lyapunov direct method and inequality techniques,the sufficient criterion for stabilization of the system is obtained.At the same time,a new inequality equivalent condition is proved.By using this theorem,the above sufficient criterion can be transformed into a set of linear matrix inequalities to facilitate the solution.Finally,a numerical simulation example is given to verify the validity and feasibility of the obtained result.3.Non-cooperative differential games for stochastic singular systems with time delay and its applicationThe non-cooperative differential games for stochastic singular time-delay systems and its application in dynamic input-output model are studied.Firstly,the zero-sum game problems of stochastic singular time-delay systems with finite time and infinite time are studied respectively.By using the method of completing the square,the saddle point equalization strategies for singular stochastic time-delay systems with finite time and infinite time are obtained.Meanwhile,considering the influence of external control on the uncertainty in the system,the zero-sum game problems of stochastic singular time-delay systems with finite time is investigated.Secondly,considering that there is product delay rather than investment delay in dynamic input-output analysis,a new stochastic singular time-delay dynamic input-output model is established.Finally,the game theory is applied to the dynamic input-output model.The rate of investment change of each sector is regarded as the decision control variable of player P1,and the uncertainty of economic system is regarded as the decision control variable of player P2.Thus,the optimal control strategy of the dynamic input-output system can be obtained by solving the saddle-point equilibrium strategy of the corresponding model,and the corresponding conclusion has been obtained.