Reachable set estimation and controller design for distributed delay systems with bounded disturbances

This paper is concerned with the problems of reachable set estimation and state-feedback controller design for linear systems with distributed delays and bounded disturbance inputs. The disturbance inputs are assumed to be either unit-energy bounded or unit-peak bounded. First, based on the Lyapunov-Krasovskii functional approach and the delay-partitioning technique, delay-dependent conditions for estimating the reachable set of the considered system are derived. These conditions guarantee the existence of an ellipsoid that contains the system state under zero initial conditions. Second, the reachable set estimation is taken into account in the controller design. Here, the purpose is to determine an ellipsoid and find a state-feedback controller such that the determined ellipsoid contains the reachable set of the resulting closed-loop system. Sufficient conditions for the solvability of the control synthesis problem are obtained. Based on these results, the problem of how to design a controller such that the state of the resulting closed-loop system is contained in a prescribed ellipsoid is studied. Finally, numerical examples and simulation results are provided to show the effectiveness of the proposed analysis and design methods.