Semiglobal exponential input-to-state stability of sampled-data systems based on approximate discrete-time models
Exact discrete-time models of nonlinear systems are difficult or impossible to obtain, and hence approximate models may be employed for control design. Most existing results provide conditions under which the stability of the approximate model in closed-loop carries over to the stability of the (unknown) exact model but only in a practical sense, i.e. the trajectories of the closed-loop system are ensured to converge to a bounded region whose size can be made as small as desired by limiting the maximum sampling period. In addition, some very stringent conditions exist for the exact model to exhibit exactly the same type of asymptotic stability as the approximate model. In this context, our main contribution consists in providing less stringent conditions by considering semiglobal exponential input-to-state stability (SE-ISS), where the inputs can successfully represent state-measurement and actuation errors. These conditions are based on establishing SE-ISS for an adequate approximate model and are applicable both under uniform and nonuniform sampling. As a second contribution, we show that explicit Runge-Kutta models satisfy our conditions and can hence be employed. An example of control design for stabilization based on approximate discrete-time models is also given.