Input-to-State Stabilization of Second-Order Systems by Filtered Derivative-Dependent Feedback via Time-Delay Implementation

We study input-to-state stabilization of the second-order systems subject to an unknown disturbance. To reduce the sensitivity to the disturbances, we consider a filtered derivative-dependent feedback by employing a low-pass filter. We use the finite difference method with a constant parameter to mimic the derivative leading to filtered time-delayed feedback. An appropriate Lyapunov functional is presented for finding sufficient input-to-state stability (ISS) conditions in terms of linear matrix inequalities (LMIs). Then we find the upper bound of the constant parameter that preserves ISS of the resulting closed-loop system from the LMIs. We also propose to implement its sampled-data by sampling measurements only at discrete-time instants. Finally, a numerical example is given to illustrate the efficiency of the presented method.