Input-to-state Stability Of Nonlinear Positive Systems

Positive systems are dynamical systems in which,input,output and state variables are constrained to be nonnegative for all time whenever the initial conditions are nonnegative.This kind of systems have been widely used to model many practical processes such as in economics,biology,ecology and communications.Since the performance of a real control system is affected more or less by uncertainties such as unmodelled dynamics,parameter perturbations,exogenous disturbances,measurement errors etc.,the research on robustness of control systems do always have a vital status in the development of control theory and technology.Aiming at robustness analysis of nonlinear control systems,Sontag’s input-to-state stability(ISS)and its various extensions are developed.And they have been widely applied and studied in various systems.But,as a useful tool of robust analysis for nonlinear systems,ISS has not been used to positive systems for stability analysis and synthesis.In this paper,ISS is applied to continuous-time and discrete-time nonlinear positive systems.For continuous-time and discrete-time positive systems,some new definitions of ISS are introduced.Different to the usual ISS definitions for nonlinear systems,our ISS definitions can fully reflect the positiveness requirements of states and inputs of the positive systems.By introducing the max-separable ISS Lyapunov functions,some ISS criterions are given for general nonlinear positive systems.Based on that,the ISS criterions for lin-ear positive systems and affine nonlinear homogeneous systems are given.Moreover,ISS properties are also analyzed for subhomogeneous cooperative systems.A new definition of finite-time input-to-state stability(FISS)for nonlinear positive systems is given.As useful tools for stability analysis,except the max-separable Lyapunov function,the min-separable Lyapunov-like function is first introduced.Using them,our main results on the solution estimates of subhomogeneous positive systems are provided.From our results,the ISS prop-erties of the systems can be judged only from the differential and algebraic characteristics of the systems.Simulation examples are given to verify the validity of our results.