Stability Analysis And Synthesis Of Impulsive Positive Systems

Positive systems are dynamic systems whose states and outputs are always nonnegative whenever the initial conditions are nonnegative.Applications of positive systems are abundant in various fields,such as economy,biology,physics,chemical industry and so on.In reality,the system dynamic evolution often includes the transient state,i.e.impulse phenomenon.A dynamic system with both positive constraints and impulsive effects is called an impulsive positive system.It contains both positive continuous time dynamics and positive discrete time dynamics,and has complex hybrid characteristics and rich dynamic behavior.Therefore,the research of impulsive positive systems is a challenging subject with great theoretical value and application potential.Therefore,this paper focuses on the state estimation,stability analysis and stabilization of impulsive positive systems.The main contents are as follows:For the state estimation problem of positive systems,the design and synthesis of the impulsive positive observer?IPO?for positive systems are discussed.By constructing a time-varying weighted copositive Lyapunov function,and applying the upper and lower bounds of impulsive intervals method combined with the convex combination technique,sufficient conditions of the existence of the IPO are established.Compared with the continuous positive observer,the IPO can estimate the system state only by utilizing the system output information at some impulsive instants.Then,based on the observation state of IPO,a dynamic output feedback controller is designed to achieve the stabilization of positive system,and an algorithm for solving the IPO gain and controller gain is given.The effectiveness of the proposed IPO and corresponding controller is verified by two numerical simulations based on thyroid hormone metabolism model.When the system state cannot be directly measured,positive observer design for impulsive positive systems with interval uncertainties and time delay is concerned.A copositive Lyapunov-Krasovskii functional with exponential term is constructed.By applying the average impulsive interval method,sufficient conditions for the existence of the positive observer are established.By the linear programming?LP?technique,an algorithm is developed to design the observer gain matrices.The simulation results show that the designed positive observer can realize the state observation of impulsive positive systems.In order to reflect the transient performance of impulsive positive systems in finite time intervals,finite-time practical stability and stabilization problems for positive systems with impulses are investigated.By constructing a time-varying copositive Lyapunov function and utilizing the average impulsive interval approach,the finite-time practical stability criterion is established by exposing different impulsive effects.The relationships between impulsive intervals,impulsive intensity and the range of finite time intervals are revealed.Based on the finite-time practical stability results,the finite-time practical controller design problem is studied to guarantee the positivity and finite-time practical stability of the corresponding closed-loop systems.Three numerical examples show the validity of the theoretical results and illustrate the difference between the traditional Lyapunov stability and the finite-time practical stability.When three different types of impulsive effects?disturbance impulses,"neutral"impulses and stabilizing impulses?are considered,the stability analysis and stabilization problems of impulsive positive systems with time delay are addressed.For each type of impulsive effect,the globally exponential stability criterion is established by utilizing the Lyapunov–Razumikhin stability theory.The effects of different impulses on stability are revealed.On the basis of the obtained stability results,the state-feedback controller design problem is investigated to positively stabilize the impulsive positive systems with time delay under different types of impulsive effects.The simulation results show that the delayed impulsive positive systems can achieve stability and stabilization under different impulses.In order to characterize the anti-jamming performance of impulsive positive systems,L₁-gain analysis and control of impulsive positive systems with interval uncertainty and time delay are investigated.Three types of impulsive effects are considered.By means of the Lyapunov-Razumikhin stability theory,conditions are developed for guaranteeing the robust stability of considered systems with L₁-gain performance.On the basis of the stability result,the L₁-gain controller is designed,the corresponding closed-loop system possesses positivity,robust stability and L₁-gain performance.Two numerical examples are given to verify the effectiveness of the theoretical results.