Sampled-Data Implementation of Derivative-Dependent Control Using Artificial Delays

We study a sampled-data implementation of linear controllers that depend on the output and its derivatives. First, we consider an LTI system of relative degree
$r\geq 2$
that can be stabilized using
$r-1$
output derivatives. Then, we consider PID control of a second-order system. In both cases, the Euler approximation is used for the derivatives giving rise to a delayed sampled-data controller. Given a derivative-dependent controller that stabilizes the system, we show how to choose the parameters of the delayed sampled-data controller that preserves the stability under fast enough sampling. The maximum sampling period is obtained from LMIs that are derived using Taylor's expansion of the delayed terms with the remainders compensated by appropriate Lyapunov–Krasovskii functionals. Finally, we introduce the event-triggering mechanism that may reduce the amount of sampled control signals used for stabilization.