An improved time-delay implementation of derivative-dependent feedback

We consider an LTI system of relative degree that can be stabilized using output derivatives. The derivatives are approximated by finite differences leading to a time-delayed feedback. We present a new method of designing and analyzing such feedback under continuous-time and sampled measurements. This method admits essentially larger time-delay/sampling period compared to the existing results and, for the first time, allows to use consecutively sampled measurements in the sampled-data case. The main idea is to present the difference between the derivative and its approximation in a convenient integral form. The kernel of this integral is hard to express explicitly but we show that it satisfies certain properties. These properties are employed to construct the Lyapunov–Krasovskii functional that leads to LMI-based stability conditions. If the derivative-dependent control exponentially stabilizes the system, then its time-delayed approximation stabilizes the system with the same decay rate provided the time-delay (for continuous-time measurements) or the sampling period (for sampled measurements) are small enough.