Sampled-data implementation of extended PID control using delays

We study the sampled-data implementation of extended PID control using delays for the nth-order stochastic nonlinear systems. The derivatives are approximated by finite differences giving rise to a delayed sampled-data controller. An appropriate Lyapunov–Krasovskii (L-K) method is presented to derive linear matrix inequalities (LMIs) for the exponential stability of the resulting closed-loop system. We show that with appropriately chosen gains, the LMIs are always feasible for small enough sampling period and stochastic perturbation. We further employ an event-triggering condition that allows to reduce the number of sampled control signals used for stabilization and provide (Formula presented.) -gain analysis. Finally, three numerical examples illustrate the efficiency of the presented approach.