Normalized P-D and Intermittent Hybrid <i>H</i> <sub>∞</sub> Control for Delayed Descriptor Systems via Impulsive-Inputs-Dependent Conditions
Guangming Zhuang, Yiqun Liu, Jianwei Xia, Xiangpeng Xie
Abstract
The problem of normalized proportional-derivative (P-D) and intermittent hybrid <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$H_{\infty}$</tex-math> </inline-formula> control for delayed descriptor impulsive systems (DDISs) is studied in this work. The P-D and intermittent hybrid state feedback controller is designed, such that normalisation and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$H_{\infty}$</tex-math> </inline-formula> stabilisation of DDISs can be realised. A novel impulsive-inputs-dependent Lyapunov-Krasovskii functional based on a couple of auxiliary functions is constructed, which is continuous along the system state trajectories without imposing additional conditions on impulsive dynamics. Improved <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$H_{\infty}$</tex-math> </inline-formula> stability criterion is proposed in the form of linear matrix inequalities by utilizing free weighting matrix technique, 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">$H_{\infty}$</tex-math> </inline-formula> stability criterion is described as a single condition based on a combination of discrete and continuous dynamics. Finally, the usefulness of the method set forth in this work is demonstrated by a two-loop circuit network. <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Note to Practitioners</i> —This work is motivated by realizing normalized P-D and intermittent hybrid <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$H_{\infty}$</tex-math> </inline-formula> control for delayed descriptor impulsive systems. In practical engineering, many dynamic systems have to be formulated as descriptor systems when modeling, such as circuit systems, vehicle suspension systems, and so on. It is worth pointing out that infinite dynamic modes will occur due to the algebraic constraints in descriptor systems, and can lead to internal impulsive behavior. At the same time, these systems are often accompanied by external impulsive perturbations, such as the sudden failure of a component in circuit systems, etc. The internal and external impulses can cause serious damage to the system performances and stability. Most of the existing control schemes are not well suited for such practical systems. In addition, due to the presence of time-varying delays, the delayed dynamic systems can easily collapse and cause significant damages. In this work, the P-D and intermittent hybrid <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$H_{\infty}$</tex-math> </inline-formula> controller is designed so as to deal with the above challenges.