Stability Of Generalized Homogeneous Positive Systems

Nonlinear systems with homogeneous characteristics are not only widely used in the fields of mathematics,chemistry and gene genetics,but also increasingly used in the field of control.Lyapunov theory plays an important role in the stability analysis of nonlinear systems.Similarly,Lyapunov theory also plays an important role in the stability analysis of homogeneous systems.Switched positive system is widely used in many fields,such as congestion control in network communication system,formation flight control in air traffic system,and HIV isolation control in medical system.Switched positive systems have both the unique non-negative characteristics of positive systems and the complex dynamic behavior induced by switched signals.This is a research topic that has both important theoretical significance and broad prospects.However,its research work is not perfect at present,and there are still many research problems to be further solved.On the one hand,this paper studies the asymptotic stability of nonlinear homogeneous systems by introducing a class of generalized homogeneous vector fields.On the other hand,a stability determination method for switched delay systems based on logarithmically compressed average dwell time is proposed for different scenarios of switched systems.This paper is divided into the following four parts:The first part introduces the basic concepts and research background of homogeneous systems and switched positive systems,analyzes the current research status of the two systems,and introduces the definitions and symbols required for proof in this article.In the second part,the stability of the nonlinear system introduced into the generalized homogeneous vector domain is considered at first,and the sufficient conditions for the global asymptotic stability of the system are given by using Lyapunov function.Secondly,for some special cases,the necessary and sufficient conditions for the global asymptotic stability are given.Finally,the decay rate of the system is estimated and quantified.The third part first studies the polynomial stability of different degree homogeneous switched positive systems with time delays.By using the logarithmic compressed average dwell time switching method,sufficient conditions for polynomial stability of different degree homogeneous switched positive systems are given,and the main results are applied to the polynomial stability of Persidskii-type switching systems.Secondly,the polynomial Stability criterion of different degree homogeneous switched systems based on extension mapping are obtained by using similar methods.Finally,the polynomial stability of switched positive systems under extension mapping was studied using the logarithmic compressed average dwell time method,and sufficient conditions for polynomial stability were obtained by combining positive system analysis techniques.The fourth part is a summary of the entire text and prospects for future research.