Practical Stability Of Nonlinear Positive Systems

The stability analysis is one of the key problems of the control theory.Practical stability,as one of the research directions of modern motion stability theory,mainly studies the motion of the given initial estimation region and the subsequent deviation estimation region,which is not weaker than the Lyapunov stability.In addition,nonlinearity is a very common phenomenon in the real world,and the variables involved in many nonlinear systems are nonnegative,for instance,density,absolute temperature,concentration,and so on.Such systems are called nonlinear positive systems.In this paper,we primarily study the practical stability of several kinds of nonlinear systems,which can be researched by the method and theory of positive system.The main contributions are as follows:Firstly,in view of the fact that the past researches on practical stability are based on the definition of norm,a novel concept on practical stability which is more suitable for the positive system is proposed.Then applying to the comparison principle,the sufficient conditions on practical stability(PS)and uniformly practical stability(UPS)are given respectively.For the linear time-varying positive systems with disturbances,the disturbances are divided into four different cases.And the sufficient criterion on PS of the linear time-varying positive systems with disturbances is obtained by Bellman inequality,Bihari inequality and Bellman-Bihari inequality respectively.At the same time,the effectiveness of the results are verified by numerical simulations.Secondly,inspired by the first part,nonlinear switched positive systems are studied by multiple max-separable Lyapunov functions.Sufficient conditions on PS and UPS for the nonlinear switched positive systems are presented.Linear time-varying switched positive systems with disturbances are considered as well,and sufficient conditions of PS and UPS are given by the relationship of subsystems.Corresponding examples are provided to illustrate the effectiveness of our results.Thirdly,for nonlinear positive systems with time delay,max-separable LyapunovKrasovskii function is introduced.By max-separable Lyapunov-Krasovskii function,sufficient condition on PS for nonlinear positive system with time delay is given.Then,sufficient condition on PS of nonlinear positive system with multi-delays are presented directly.Similarly,for linear time-varying positive systems with single delay and multi-delays,sufficient conditions on PS are given by max-separable Lyapunov-Krasovskii function,respectively.In addition,we also prove that linear time-varying positive system with state delay can be practically stabilizable by using state feedback.Numerical examples are given to verify the correctness of the conclusions.