# T-S fuzzy model

## Model description:

Consider a T-S fuzzy model

Plant Rule $i$: If $x_1(t)$ is $F_1^1(x_1(t))$

Then $x(t+1) = A_ix(t)+B_iu(t),$

where

\begin{align*} A_1 &=\left[\matrix{-a & 2\cr -0.1 & b}\right], A_2=\left[\matrix{-a & 2\cr-0.1 & b }\right], A_3=\left[\matrix{-0.9 & 0.5\cr -0.1 & -1.7}\right] \\ B_1 &=\left[\matrix{b\cr 4}\right], B_2=\left[\matrix{b\cr 4.8}\right], B_3=\left[\matrix{3\cr 0.1}\right]. \end{align*}

The parameters $a$ and $b$ are adjusted to compare the relaxation of stabilization conditions.

## Publication details:

 Title Stabilization of discrete-time nonlinear control systems - Multiple fuzzy Lyapunov function approach Publication Type Conference Paper Year of Publication 2009 Authors Kau, Shih-Wei, Huang Xin-Yuan, Shiu Sheng-Yu, and Fang Chun-Hsiung Conference Name International Conference on Information and Automation, 2009. ICIA '09. Date Published 06/2009 Publisher IEEE Conference Location Zhuhai, Macau ISBN Number 978-1-4244-3607-1 Accession Number 10837484 Keywords discrete time systems, fuzzy control, linear matrix inequalities, Lyapunov methods, nonlinear control systems, stability Abstract This paper deals with the stabilization problem for discrete-time nonlinear systems that are represented by the Takagi - Sugeno fuzzy model. By the multiple fuzzy Lyapunov function and the three-index algebraic combination technique, a new stabilization condition is developed. The condition is expressed in the form of linear matrix inequalities (LMIs) and proved to be less conservative than existing results in the literature. Finally, a truck-trailer system is given to illustrate the novelty of the proposed approach. DOI 10.1109/ICINFA.2009.5204890