Observer-Based Fuzzy <i>l</i> <sub>2</sub>–l<sub>∞</sub> Control for Discrete-Time Nonlinear Systems
Xiao‐Heng Chang, Xu Han
Abstract
In this article, an observer-based <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$l_{2}$</tex-math></inline-formula> – <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$l_\infty$</tex-math></inline-formula> performance analysis and controller design problem is proposed for a category of discrete-time nonlinear control systems. First of all, the closed-loop system with external disturbance in both state and output variables under consideration is represented as a discrete-time Takagi–Sugeno (T–S) fuzzy model and is described by the descriptor representation method. Subsequently, a new Lyapunov function is suggested that satisfies the prescribed <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$l_{2}$</tex-math></inline-formula> – <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$l_\infty$</tex-math></inline-formula> performance while ensuring stability. The application of the linear matrix inequality technique is employed to establish an analysis criterion for the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$l_{2}$</tex-math></inline-formula> – <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$l_\infty$</tex-math></inline-formula> performance. By defining the structure of matrix variables, new design conditions of observer-based <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$l_{2}$</tex-math></inline-formula> – <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$l_\infty$</tex-math></inline-formula> controller are given. Ultimately, an instance demonstrates the effectiveness of the proposed controller design criterion.