State estimation-based robust control design for periodic piecewise systems with time-varying delays

In this article, the issue of finite-time boundedness is investigated for a class of uncertain periodic piecewise systems subject to time-varying input/state delays and external disturbances. The main aim of this article is to design an $ \mathcal {H}_{\infty } $ H∞ observer-based control protocol for guaranteeing the resulting closed-loop system to be finite-time bounded. The state dynamics of the system are not always directly measurable, so that a system states are reconstructed by constructing a periodic piecewise observer system to get the required result. Subsequently, the Lyapunov–Krasovskii functional by the decomposition of periodic positive-definite matrices is built and in conjunction with convex optimisation technique and extended Wirtinger's integral inequality, the sufficient conditions are derived to assure the finite-time boundedness of the closed-loop system. In particular, the corresponding periodic piecewise time-varying controller and observer gain matrices can be obtained as solutions to a set of derived sufficient linear matrix inequality-based conditions. Finally, the efficacy of the proposed control design is examined by utilizing two numerical examples with simulation results including a vibration system model.