# Stabilization of Delta Operator Systems with Actuator Saturation via an Anti-Windup Compensator

^{*}

## Abstract

**:**

## 1. Introduction

**Notation**

**1.**

## 2. Problem Statement and Preliminaries

**Remark**

**1.**

**Definition**

**1.**

**Lemma**

**1.**

**Lemma**

**2.**

**Lemma**

**3.**

**Remark**

**2.**

## 3. Main Results

#### New Model Reformulation

**Lemma**

**4.**

**Theorem**

**1.**

**Proof.**

## 4. Anti-Windup Optimization

- Step 1
- For given ${r}_{h1},{r}_{h2}$, and $\mu $, fix a sufficiently large ${d}_{e}$ such that the constrained minimization (27) is feasible. Then, set ${(\tilde{P},\widehat{P},{\tilde{Q}}_{l},{\widehat{Q}}_{l},{\tilde{R}}_{l},{\widehat{R}}_{l},X,\widehat{X},{\sigma}_{P},{\sigma}_{{Q}_{l}},{\sigma}_{{R}_{l}})}_{0}=(\tilde{P},\widehat{P},{\tilde{Q}}_{l},{\widehat{Q}}_{l},{\tilde{R}}_{l},{\widehat{R}}_{l},X,\widehat{X},{\sigma}_{P},{\sigma}_{{Q}_{l}},{\sigma}_{{R}_{l}})$. Fix a sufficiently small ${D}_{e}$, and set ${d}_{e}={d}_{e}+{D}_{e}$.
- Step 2
- Solve the following LMI minimization problem:$\mathrm{min}\mathrm{tr}(\tilde{P}{\widehat{P}}_{0}+{\tilde{Q}}_{l}{\widehat{Q}}_{{l}_{0}}+{\tilde{R}}_{l}{\widehat{R}}_{{l}_{0}}+(X+{X}^{T})({\widehat{X}}_{0}+{\widehat{X}}_{0}^{T})+{\tilde{P}}_{0}\widehat{P}+{\tilde{Q}}_{{l}_{0}}{\widehat{Q}}_{l}+{\tilde{R}}_{{l}_{0}}{\widehat{R}}_{l}+({X}_{0}+{X}_{0}^{T})(\widehat{X}+{\widehat{X}}^{T}))$subject to LMIs in (27).
- Step 3
- Substitute the new matrix variables into (27). If the result is feasible, then set ${d}_{e}={d}_{e}+{D}_{e}$ and the new solution as ${(\tilde{P},\widehat{P},{\tilde{Q}}_{l},{\widehat{Q}}_{l},{\tilde{R}}_{l},{\widehat{R}}_{l},X,\widehat{X},{\sigma}_{P},{\sigma}_{{Q}_{l}},{\sigma}_{{R}_{l}})}_{0}$, and repeat Step 2; otherwise, ${d}_{e}={d}_{e}-{D}_{e}$ is the required estimate: Stop.

**Remark**

**3.**

**Remark**

**4.**

**Remark**

**5.**

## 5. Numerical Examples

**Example**

**1.**

**Example**

**2.**

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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${\mathit{r}}_{\mathit{h}1}$ | ${\mathit{r}}_{\mathit{h}2}$ | ${\mathit{E}}_{\mathit{c}}$ |
---|---|---|

1 | 21 | ${\left[\begin{array}{cc}15.0338& -2.4257\end{array}\right]}^{T}$ |

3 | 23 | ${\left[\begin{array}{cc}14.7327& -2.2891\end{array}\right]}^{T}$ |

5 | 25 | ${\left[\begin{array}{cc}15.2971& -2.6786\end{array}\right]}^{T}$ |

7 | 27 | ${\left[\begin{array}{cc}14.3278& 2.3199\end{array}\right]}^{T}$ |

9 | 29 | ${\left[\begin{array}{cc}14.9153& -0.2678\end{array}\right]}^{T}$ |

11 | 31 | ${\left[\begin{array}{cc}15.1340& -0.3745\end{array}\right]}^{T}$ |

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**MDPI and ACS Style**

Rachid, H.; Lamrabet, O.; Tissir, E.H.
Stabilization of Delta Operator Systems with Actuator Saturation via an Anti-Windup Compensator. *Symmetry* **2019**, *11*, 1084.
https://doi.org/10.3390/sym11091084

**AMA Style**

Rachid H, Lamrabet O, Tissir EH.
Stabilization of Delta Operator Systems with Actuator Saturation via an Anti-Windup Compensator. *Symmetry*. 2019; 11(9):1084.
https://doi.org/10.3390/sym11091084

**Chicago/Turabian Style**

Rachid, Hafsaa, Ouarda Lamrabet, and El Houssaine Tissir.
2019. "Stabilization of Delta Operator Systems with Actuator Saturation via an Anti-Windup Compensator" *Symmetry* 11, no. 9: 1084.
https://doi.org/10.3390/sym11091084