# Robust Stability of Switched-Interval Positive Linear Systems with All Modes Unstable Using the Φ-Dependent Dwell Time Technique

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## Abstract

**:**

## 1. Introduction

## 2. Problem Formulation and Preliminaries

**Definition 1**

**Definition 2**

**.**A matrix $\mathfrak{G}$ is said to be a Metzler matrix if its non-diagonal elements are positive or zero.

**Lemma 1**

**Lemma 2**

## 3. Main Results

#### 3.1. Continuous-Time Case

**Theorem 1.**

**Proof.**

**Remark 1.**

**Remark 2.**

**Remark 3.**

**Remark 4.**

**Theorem 2.**

**Proof.**

#### 3.2. Discrete-Time Case

**Theorem 3.**

**Proof.**

**Theorem 4.**

## 4. Illustrative Example

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 2.**The state response of the system with group one of the $\Phi $-dependent dwell time technique.

**Figure 3.**The state response of the system with group two of the $\Phi $-dependent dwell time technique.

**Figure 4.**The state response of the system with group three of the $\Phi $-dependent dwell time technique.

$\mathbb{R}$ | The set of real numbers |

${\mathbb{R}}^{n}$ | The set of n-dimensional real vectors |

${\mathbb{R}}^{n\times n}$ | The space of $n\times n$ real matrices |

$\mathbb{N}$ (${\mathbb{N}}_{+}$) | The set of nonnegative (positive) integers |

${\mathfrak{G}}^{T}$ | The transpose of a matrix $\mathfrak{G}$ |

$\iota \u2ab00(\iota \succ 0)$ | Each component of vector $\iota $ is nonnegative (positive) |

$\underline{\delta}\left(\iota \right)\left(\overline{\delta}\left(\iota \right)\right)$ | The minimal (maximal) components of vector $\iota $ |

$\mathfrak{G}\u2ab00(\mathfrak{G}\succ 0)$ | Each component of matrix $\mathfrak{G}$ is nonnegative (positive) |

${\parallel x\parallel}_{1}$ | 1-norm of $x\left(t\right)$; i.e., ${\parallel x\parallel}_{1}={\sum}_{i=1}^{n}\parallel {x}_{i}\parallel $ |

${\parallel x\parallel}_{2}$ | Euclidean vector norm of $x\left(t\right)$; i.e., ${\parallel x\parallel}_{2}={\left({\sum}_{i=1}^{n}{x}_{i}^{2}\right)}^{\frac{1}{2}}$ |

⟺ | If and only if |

$\mathbf{\Phi}$-Dependent Dwell Time | |||||
---|---|---|---|---|---|

Technique | Dwell Time $\mathfrak{K}=\left\{\mathbf{1}\right\}$ | $\mathfrak{K}=\{\mathbf{1},\mathbf{2}\}$ | Mode-Dependent Dwell Time $\mathfrak{K}=\{\mathbf{1},\mathbf{2},\mathbf{3}\}$ | ||

$\Phi $ | ${\Phi}_{1}=\{1,2,3\}$ | ${\Phi}_{1}=\{1,2\}$ | ${\Phi}_{1}=\{1,3\}$ | ${\Phi}_{1}=\left\{1\right\}$ | ${\Phi}_{1}=\left\{1\right\}$ |

${\Phi}_{2}=\left\{3\right\}$ | ${\Phi}_{2}=\left\{2\right\}$ | ${\Phi}_{2}=\{2,3\}$ | ${\Phi}_{2}=\left\{2\right\}$ | ||

${\Phi}_{3}=\left\{3\right\}$ | |||||

$\lambda $ | ${\lambda}_{1}=0.5$ | ${\lambda}_{1}=0.5$ | ${\lambda}_{1}=0.5$ | ${\lambda}_{1}=0.5$ | ${\lambda}_{1}=0.5$ |

${\lambda}_{2}=0.55$ | ${\lambda}_{2}=0.57$ | ${\lambda}_{2}=0.6$ | ${\lambda}_{2}=0.6$ | ||

${\lambda}_{3}=0.55$ | |||||

${\mu}_{1}=0.75$ | ${\mu}_{1}=0.7$ | ${\mu}_{1}=0.7$ | ${\mu}_{1}=0.7$ | ${\mu}_{1}=0.75$ | |

$\mu $ | ${\mu}_{2}=0.65$ | ${\mu}_{2}=0.6$ | ${\mu}_{2}=0.68$ | ${\mu}_{2}=0.7$ | |

${\mu}_{3}=0.65$ | |||||

${\iota}_{1,0}$ | $\left[\begin{array}{c}0.0100\\ 0.0116\end{array}\right]$ | $\left[\begin{array}{c}0.0100\\ 0.0124\end{array}\right]$ | $\left[\begin{array}{c}0.0100\\ 0.0124\end{array}\right]$ | $\left[\begin{array}{c}0.0100\\ 0.0127\end{array}\right]$ | $\left[\begin{array}{c}0.0100\\ 0.0119\end{array}\right]$ |

${\iota}_{1,1}$ | $\left[\begin{array}{c}0.0177\\ 0.0133\end{array}\right]$ | $\left[\begin{array}{c}0.0234\\ 0.0143\end{array}\right]$ | $\left[\begin{array}{c}0.0255\\ 0.0143\end{array}\right]$ | $\left[\begin{array}{c}0.0226\\ 0.0143\end{array}\right]$ | $\left[\begin{array}{c}0.0214\\ 0.0133\end{array}\right]$ |

${\iota}_{2,0}$ | $\left[\begin{array}{c}0.0132\\ 0.0100\end{array}\right]$ | $\left[\begin{array}{c}0.0164\\ 0.0100\end{array}\right]$ | $\left[\begin{array}{c}0.0178\\ 0.0100\end{array}\right]$ | $\left[\begin{array}{c}0.0158\\ 0.0100\end{array}\right]$ | $\left[\begin{array}{c}0.0161\\ 0.0100\end{array}\right]$ |

${\iota}_{2,1}$ | $\left[\begin{array}{c}0.0157\\ 0.0133\end{array}\right]$ | $\left[\begin{array}{c}0.0195\\ 0.0143\end{array}\right]$ | $\left[\begin{array}{c}0.0211\\ 0.0167\end{array}\right]$ | $\left[\begin{array}{c}0.0189\\ 0.0147\end{array}\right]$ | $\left[\begin{array}{c}0.0195\\ 0.0143\end{array}\right]$ |

${\iota}_{3,0}$ | $\left[\begin{array}{c}0.0118\\ 0.0100\end{array}\right]$ | $\left[\begin{array}{c}0.0137\\ 0.0100\end{array}\right]$ | $\left[\begin{array}{c}0.0126\\ 0.0100\end{array}\right]$ | $\left[\begin{array}{c}0.0128\\ 0.0100\end{array}\right]$ | $\left[\begin{array}{c}0.0137\\ 0.0100\end{array}\right]$ |

${\iota}_{3,1}$ | $\left[\begin{array}{c}0.0133\\ 0.0154\end{array}\right]$ | $\left[\begin{array}{c}0.0154\\ 0.0190\end{array}\right]$ | $\left[\begin{array}{c}0.0143\\ 0.0177\end{array}\right]$ | $\left[\begin{array}{c}0.0147\\ 0.0187\end{array}\right]$ | $\left[\begin{array}{c}0.0154\\ 0.0183\end{array}\right]$ |

${\mathfrak{D}}_{1}$ | $[0.5,0.55]$ | $[0.5,0.7]$ | $[0.5,0.7]$ | $[0.4,0.7]$ | $[0.4,0.55]$ |

${\mathfrak{D}}_{2}$ | $[0.4,0.75]$ | $[0.4,0.85]$ | $[0.45,0.6]$ | $[0.45,0.5]$ | |

${\mathfrak{D}}_{3}$ | $[0.4,0.53]$ | ||||

${\tau}_{1}$ | $0.5$ | $0.6$ | $0.6$ | $0.55$ | $0.4$ |

${\tau}_{2}$ | $0.5$ | $0.6$ | $0.45$ | $0.5$ | $0.45$ |

${\tau}_{3}$ | $0.5$ | $0.5$ | $0.6$ | $0.5$ | $0.5$ |

State response | Figure 1 | Figure 2 | Figure 3 | Figure 4 | Figure 5 |

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**MDPI and ACS Style**

Yu, Q.; Jiang, X.
Robust Stability of Switched-Interval Positive Linear Systems with All Modes Unstable Using the Φ-Dependent Dwell Time Technique. *Axioms* **2023**, *12*, 686.
https://doi.org/10.3390/axioms12070686

**AMA Style**

Yu Q, Jiang X.
Robust Stability of Switched-Interval Positive Linear Systems with All Modes Unstable Using the Φ-Dependent Dwell Time Technique. *Axioms*. 2023; 12(7):686.
https://doi.org/10.3390/axioms12070686

**Chicago/Turabian Style**

Yu, Qiang, and Xiujuan Jiang.
2023. "Robust Stability of Switched-Interval Positive Linear Systems with All Modes Unstable Using the Φ-Dependent Dwell Time Technique" *Axioms* 12, no. 7: 686.
https://doi.org/10.3390/axioms12070686