Towards Networked Control Systems with guaranteed stability: Using weakly hard real-time constraints to model the loss process

Within this paper, we consider the feedback control of an unstable system, which runs in open loop from time to time due to failures like packet losses in a Networked Control System. Up to now, this problem is most often analyzed with a probabilistic description of the failure process. This approach allows to guarantee stability in the mean square sense, i.e., the convergence of the first and second moment of the state, but not in the classical sense of Lyapunov. Moreover, even when the closed loop is stable in the mean square sense, there is no guarantee that the state does not exceed an arbitrarily large bound. In contrast to these works, we use the ideas of weakly hard real-time constraints to describe the failure process in a non-probabilistic and more precise way. In doing so, we derive conditions to guarantee stability of the entire system in the classical sense of Lyapunov despite failures in updating the control input.