Generalization of an input-to-state stability preserving Runge-Kutta method for nonlinear control systems

Given a nonlinear control system that is input-to-state stable (ISS) it is assured that the states of the system remain bounded if the input is bounded and the system is global asymptotic stable (GAS) in the sense of Lyapunov in the absence of inputs. Recently, Hu and Liu (2007) [7] studied under which condition ISS of a continuous time nonlinear control system implies ISS of the discrete time system obtained by an implicit Runge-Kutta (RK) method. In this contribution, we extend those results to explicit RK methods. This represents an important extension with respect to applications like system identification where explicit RK methods are presupposed or real-time applications where the computational burden of implicit RK methods is prohibitive.