# Sampled-Data Control for a Class of Singular Takagi-Sugeno Fuzzy Systems with Application in Truck-Trailer System

^{*}

## Abstract

**:**

## 1. Introduction

- (1)
- How to deal with the obtained results to ensure that the system is not only stable but also regular and impulse-free;
- (2)
- How to construct LKF to obtain less conservative results by adding novel items or introducing new integral inequality;
- (3)
- How to design sampled-data controllers to ensure that the system is asymptotically admissible.

## 2. Problem Formulation

_{i}is the gain matrix. Then, the overall fuzzy controller is described such that:

**Remark 1.**

**Definition**

**1. [45]**

- 1.1
- If there exists a constant$s\in \u2102$($\u2102$represents complex field) satisfying$\mathrm{det}\left(sE-A\right)\ne 0$, then the matrix pairs$\left(E,A\right)$are regular;
- 1.2
- If there exists a scalar function,$V(x)$, which satisfies$V(0)=0,\hspace{1em}V(x)>0$for any non-zero$x(t)$, then the system is stable;
- 1.3
- If$\mathrm{deg}\left(\mathrm{det}\left(sE-A\right)\right)=\mathrm{rank}(E)$, then the matrix pairs$\left(E,A\right)$are impulse free.

**Definition**

**2. [46]**

- 2.1
- If the pair (E, A) is regular and impulse-free, then the system:$$E\dot{x}(t)=Ax(t)+Bu(t)$$
- 2.2
- System (8) is said to be asymptotically admissible if it is regular, impulse-free and asymptotically stable.

- (1)
- That system (7), with $w(t)=0$, is asymptotically admissible.
- (2)
- Under a zero condition, the output $y(t)$ satisfies $\left|\right|y(t)|{|}_{2}\le \gamma ||w(t)|{|}_{2}$ for all nonzero $w(t)\in {L}_{2}\left[0,\infty \right)$, where $\gamma >0$.

**Lemma**

**1.**

## 3. Main Results

**Theorem**

**1.**

**Proof.**

**Theorem**

**2.**

**Proof.**

**Remark 2.**

**Theorem**

**3.**

**Proof.**

## 4. Numerical Examples

**Example 1.**Consider the following fuzzy singular system parameters which have two fuzzy rules.

**Example 3.**The truck-trailer models are expressed as follow [54].

## 5. Conclusions

- (1)
- Through proper transformation, the research results in this paper can be extended to normal sampled-data systems. Hence, the proposed methods have universality;
- (2)
- The input delay approach is proposed to transform the system into a time-delay system so that many novel time-delay methods can be used;
- (3)
- Both the lower and upper bounds of the sampling period were considered, which has a wider application scope;
- (4)
- Reciprocally convex inequality is used to handle integral terms, such as LKF, meaning that the conservatism of the system has been greatly reduced.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**MDPI and ACS Style**

Yang, Y.; Zheng, M.
Sampled-Data Control for a Class of Singular Takagi-Sugeno Fuzzy Systems with Application in Truck-Trailer System. *Symmetry* **2022**, *14*, 1762.
https://doi.org/10.3390/sym14091762

**AMA Style**

Yang Y, Zheng M.
Sampled-Data Control for a Class of Singular Takagi-Sugeno Fuzzy Systems with Application in Truck-Trailer System. *Symmetry*. 2022; 14(9):1762.
https://doi.org/10.3390/sym14091762

**Chicago/Turabian Style**

Yang, Yongcheng, and Minjie Zheng.
2022. "Sampled-Data Control for a Class of Singular Takagi-Sugeno Fuzzy Systems with Application in Truck-Trailer System" *Symmetry* 14, no. 9: 1762.
https://doi.org/10.3390/sym14091762