Discrete-Time Solution to the Performance Guaranteeing $H_{\infty}$ Event-Triggered Control Problem

We study event-triggered control for discrete systems with the
$H_{\infty}$
(induced
$\ell_{2}$
) performance measure. We construct event-triggered controllers generating sampling intervals no smaller than those of the optimal time-triggered controller under the same
$H_{\infty}$
performance bound
$\gamma$
. The design philosophy is based on a parametrization of discrete, possibly nonlinear and time varying,
$\gamma$
-suboptimal controllers and triggering events via the
$Q$
parameter that renders the parametrization sampled-data. Although this is similar to our previous event-triggered design for continuous-time systems, the lack of continuity of discrete behaviors constitutes non-trivial differences that require special treatment. In particular, dynamic event-triggering is proposed to compensate for premature triggering, which can only be detected a posteriori. As a result of the discrete-time nature of the problem, there appears to be a wider class of signals that causes our event-triggered controllers to generate the optimal time-triggered sampling pattern. We characterize a narrow subclass, the continuous-time counterpart, through the associated difference Riccati equation.