# Stability Analysis of an LTI System with Diagonal Norm Bounded Linear Differential Inclusions

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## Abstract

**:**

## 1. Introduction

#### Overview of the Article

## 2. Preliminaries

**Definition**

**1.**

**Definition**

**2.**

**Definition**

**3.**

**Definition**

**4.**

**Definition**

**5**

**Definition**

**6**

## 3. Diagonal Norm Bounded Linear Differential Inclusion

#### An Equivalent System

- ${x}^{t}\left(A+{B}_{p}{(I-{D}_{qp,i})}^{-1}{\delta}_{i}\left(t\right){C}_{q,i}\right)x=0,$
- ${x}^{t}\left(A+{B}_{p}{(I-{D}_{qp,i})}^{-1}{\delta}_{i}\left(t\right){C}_{q,i}\right)x>0.$

- ${x}^{t}\left(A+{B}_{p}{(I-{D}_{qp,i})}^{-1}{\delta}_{i}\left(t\right){C}_{q,i}\right)x<0.$

## 4. Systems of ODE’s to Shift Negative Spectrum

#### 4.1. Optimization Problem

#### 4.2. Euler’s Method

## 5. Alternative Way to Compute Strictly Positive Spectrum

- ${B}^{t}=B,$ Symmetric matrix
- ${b}_{ij}={b}_{ji}\in \left[-1,1\right]\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\forall i,j,$
- $Diag\left(B\right)=1,$
- ${\lambda}_{i}\left(B\right)>0\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\forall i.$

#### Construction of $\widehat{B}$

## 6. Conclusions

- •
- Comparison of different numerical techniques to determine the suitable choice of perturbation level,
- •
- Stability analysis of some practical examples from linear time-invariant systems with norm bounded differential inclusion and diagonal norm bounded linear differential inclusions,
- •
- Stability analysis of linear time-variant systems in control.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**MDPI and ACS Style**

Rehman, M.-U.; Iqbal, S.; Alzabut, J.; El-Nabulsi, R.A.
Stability Analysis of an LTI System with Diagonal Norm Bounded Linear Differential Inclusions. *Symmetry* **2021**, *13*, 152.
https://doi.org/10.3390/sym13010152

**AMA Style**

Rehman M-U, Iqbal S, Alzabut J, El-Nabulsi RA.
Stability Analysis of an LTI System with Diagonal Norm Bounded Linear Differential Inclusions. *Symmetry*. 2021; 13(1):152.
https://doi.org/10.3390/sym13010152

**Chicago/Turabian Style**

Rehman, Mutti-Ur, Sohail Iqbal, Jehad Alzabut, and Rami Ahmad El-Nabulsi.
2021. "Stability Analysis of an LTI System with Diagonal Norm Bounded Linear Differential Inclusions" *Symmetry* 13, no. 1: 152.
https://doi.org/10.3390/sym13010152