Event-Triggered $\mathcal{H}_{\infty}$ Control for Discrete-Time Fuzzy Systems: A Semi-Discrete-Time Looped Functional Approach

This article addresses the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\boldsymbol{\mathcal{H}_{\infty}}$</tex-math></inline-formula> control of discrete-time fuzzy systems with an event-triggered controller. Initially, to promptly update the control input and mitigate the transmission burden, a membership-dependent event-triggered scheme (ETS) is proposed based on accumulative error. An event is triggered when the accumulated error between the current state and the latest control update exceeds a predefined threshold, which works well for systems with small state changes. The proposed ETS utilizes the properties of fuzzy systems, which provides much greater resiliency than the existing works. Then, a semi-discrete-time looped functional (SDTLF) approach is proposed, which does not require the forward difference of the functional to be negative at each triggering instant. Moreover, the constructed SDTLF has the ability to link the ETS and the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\boldsymbol{\mathcal{H}_{\infty}}$</tex-math></inline-formula> performance. A linear matrix inequality-based sufficient control design method is obtained by using the constructed SDTLF. The proposed method co-designs both control gains and event-triggered parameters to ensure that the system satisfies the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\boldsymbol{\mathcal{H}_{\infty}}$</tex-math></inline-formula> performance index. Finally, two examples are utilized to verify the advantages of the proposed method.