# Observer-Based PID Control Strategy for the Stabilization of Delayed High Order Systems with up to Three Unstable Poles

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

## 2. Problem Statement

#### Preliminary Results

**Lemma**

**1**

**Lemma**

**2**

**Remark**

**1.**

**Lemma**

**3**

## 3. Main Results

#### 3.1. System with Three Unstable Poles

#### 3.1.1. State Feedback Controller

**Lemma**

**4.**

**Proof.**

#### 3.1.2. Auxiliary Output Injection Structure

**Lemma**

**5.**

**Proof.**

#### 3.1.3. Observer-Based PID Control Scheme

**Theorem**

**1.**

**Proof.**

#### 3.1.4. Controller Parameters Selection

- (a)
- Consider Figure 1. For ${G}_{s}\left(s\right){G}_{{\alpha}_{1}}\left(s\right)$ we obtain a state space realization (matrices A and B), then chose $\mathbb{F}$ such that the eigenvalues of $(A-B\mathbb{F})$ become $[{d}_{1},{d}_{2},\cdots ,{d}_{n-2}]$ and relation (18) is satisfied, i.e., placing the new poles as far from the origin as required;
- (b)
- Consider Figure 2. Select ${g}_{2}$ and ${g}_{3}$ to move poles ${\alpha}_{2}$ and ${\alpha}_{3}$ to positions ${c}_{2}$ and ${c}_{3}$, satisfying relation (24). Again the new poles should be placed far from the origin. Then, we use a Nyquist diagram to select ${g}_{1}$, stabilizing the new subsystem: $\frac{{e}^{\tau s}}{(s-{\alpha}_{1})(s-{c}_{2})(s-{c}_{3})}$. Parameter ${g}_{1}$ must be such that the Nyquist diagram encircle once the point $(-1,0j)$ in counter-clockwise direction;
- (c)
- Consider Figure 1. The PID controller must stabilize the closed loop subsystem (17); a system with two unstable poles and $n-2$ stable (relocated by $\mathbb{F}$) poles. The existence of a PID controller is guarantee by relation (18) and the parameters can be selected by the methodology proposed in [25] or, in an alternative way, trough trail and error, by using again a Nyquist diagram, noting that a PID controller is equivalent to a pole at the origin, two free zeros and a free gain. The location of the two free zeros must be such that there exists a free gain value making the Nyquist diagram to encircle twice the point $(-1,0j)$ in counter-clockwise direction.

#### 3.2. System with Two Unstable Poles

#### 3.2.1. State Feedback Controller

**Corollary**

**1.**

**Proof.**

#### 3.2.2. Output Injection Structure

**Corollary**

**2.**

**Proof.**

#### 3.2.3. Observer-Based Control Scheme for the Case of Two Unstable Poles

**Theorem**

**2.**

**Proof.**

**Remark**

**3.**

#### 3.3. System with One Unstable Pole

**Theorem**

**3.**

**Remark**

**4.**

**Remark**

**5.**

## 4. Numerical Experiments

#### 4.1. Example 1: Delayed System with One Unstable Pole

#### 4.2. Example 2: Fourth Order Linear System with Delay and Two Unstable Poles

#### 4.3. Example 3: Three Unstable Poles

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**MDPI and ACS Style**

Cruz-Díaz, C.; del Muro-Cuéllar, B.; Duchén-Sánchez, G.; Márquez-Rubio, J.F.; Velasco-Villa, M.
Observer-Based PID Control Strategy for the Stabilization of Delayed High Order Systems with up to Three Unstable Poles. *Mathematics* **2022**, *10*, 1399.
https://doi.org/10.3390/math10091399

**AMA Style**

Cruz-Díaz C, del Muro-Cuéllar B, Duchén-Sánchez G, Márquez-Rubio JF, Velasco-Villa M.
Observer-Based PID Control Strategy for the Stabilization of Delayed High Order Systems with up to Three Unstable Poles. *Mathematics*. 2022; 10(9):1399.
https://doi.org/10.3390/math10091399

**Chicago/Turabian Style**

Cruz-Díaz, César, Basilio del Muro-Cuéllar, Gonzalo Duchén-Sánchez, Juan Francisco Márquez-Rubio, and Martín Velasco-Villa.
2022. "Observer-Based PID Control Strategy for the Stabilization of Delayed High Order Systems with up to Three Unstable Poles" *Mathematics* 10, no. 9: 1399.
https://doi.org/10.3390/math10091399