# Stability Analysis of Linear Feedback Systems in Control

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## Abstract

**:**

## 1. Introduction

#### Overview of the Article

## 2. Preliminaries

**Definition**

**1.**

**Definition**

**2.**

**Definition**

**3.**

**Definition**

**4.**

**Definition**

**5.**

**Theorem**

**1**

**.**An input–output system is well-posed and stable for perturbation Δ having 2-norm bounded above by 1 if and only if

**Theorem**

**2**

**.**For two structured uncertainties ${\mathbb{B}}_{1}\subset {\mathbb{B}}_{2}$, then

## 3. Reformulation of $\mathsf{\mu}$-Values

**Definition**

**6.**

**Definition**

**7.**

**Definition**

**8.**

## 4. Proposed Methodology

#### 4.1. Inner Algorithm

#### 4.1.1. The Basic Theory

#### 4.1.2. Approximation of an Extremizers

**Theorem**

**3.**

#### 4.1.3. Gradient System of ODE’s

#### 4.2. Outer Algorithm

**Theorem**

**4.**

#### Choice of Suitable Initial Value Matrix and Initial Perturbation Level

## 5. Linear Feedback Systems in the Form of Toeplitz Matrices

#### 5.1. Toeplitz Matrix for an Inverse System

#### 5.2. Toeplitz Matrix for White Additive Noise

#### 5.3. Toeplitz Matrix for Additive Colored and White Noise

## 6. Numerical Experimentation

**Example**

**1.**

**mussv**, we compute an admissible perturbation $VDelta$ as

**Example**

**2.**

**mussv**, we compute an admissible perturbation $VDelta$ as

**Example**

**3.**

**mussv**computes an admissible perturbation $VDelta$ as

**Example**

**4.**

**mussv**, we compute an admissible perturbation $VDelta$

**Example**

**5.**

**mussv**, we compute an admissible perturbation $VDelta$ as

**Example**

**6.**

**mussv**, we compute an admissible perturbation $VDelta$ with

## 7. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**MATLAB interface for computing pseudo-spectrum. The graphical representation show the pseudo-spectrum of the 5-dimensional real valued matrix (Example 1, Section 6).

**Figure 2.**MATLAB interface for computing pseudo-spectrum. The graphical representation show the pseudo-spectrum of the 6-dimensional real valued matrix (Example 2, Section 6).

**Figure 3.**MATLAB interface for computing pseudo-spectrum. The graphical representation show the pseudo-spectrum of the 6-dimensional real valued matrix (Example 3, Section 6).

**Figure 4.**MATLAB interface for computing pseudo-spectrum. The graphical representation show the pseudo-spectrum of the 4-dimensional real valued matrix (Example 4, Section 6).

**Figure 5.**MATLAB interface for computing pseudo-spectrum. The graphical representation show the pseudo-spectrum of the 6-dimensional real valued matrix (Example 5, Section 6).

**Figure 6.**MATLAB interface for computing pseudo-spectrum. The graphical representation show the pseudo-spectrum of the 4-dimensional real valued matrix (Example 6, Section 6).

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**MDPI and ACS Style**

Rehman, M.-U.; Alzabut, J.; Anwar, M.F.
Stability Analysis of Linear Feedback Systems in Control. *Symmetry* **2020**, *12*, 1518.
https://doi.org/10.3390/sym12091518

**AMA Style**

Rehman M-U, Alzabut J, Anwar MF.
Stability Analysis of Linear Feedback Systems in Control. *Symmetry*. 2020; 12(9):1518.
https://doi.org/10.3390/sym12091518

**Chicago/Turabian Style**

Rehman, Mutti-Ur, Jehad Alzabut, and Muhammad Fazeel Anwar.
2020. "Stability Analysis of Linear Feedback Systems in Control" *Symmetry* 12, no. 9: 1518.
https://doi.org/10.3390/sym12091518