Extended Dissipativity Analysis and Synthesis for Discrete‐Time Lur'e Delayed Systems: Matrix‐Separation‐Based Lyapunov Functional Method

ABSTRACT This paper develops a matrix‐separation‐based Lyapunov functional method to study the extended dissipativity analysis and synthesis issue of discrete‐time Lur'e‐type delayed systems. The advanced idea of matrix‐separation is reflected in Lyapunov functional candidates and estimates summation terms as precisely as feasible. First, we introduce a novel summation inequality based on the matrix‐separation method to provide a bound for the augmented summation that involves common state variables. An improved delay‐product‐type Lyapunov functional is devised by extending non‐positive definite summations as separate subblocks of the state‐augmented summation. Then, by using the matrix injection method to handle the cubic delay term in the functional forward difference, a delay‐variation‐dependent stability criterion and an extended dissipativity criterion are developed under the matrix‐separation‐based method. Subsequently, sufficient conditions for the control design of Lur'e‐type systems are derived. Finally, the effectiveness and superiority of the proposed method are verified through its application to Chua's circuits, neural networks, and a stochastic numerical example.