#
A Unified Framework for the H_{∞} Mixed-Sensitivity Design of Fixed Structure Controllers through Putinar Positivstellensatz ^{ †}

^{*}

^{†}

*H*

_{∞}mixed-sensitivity design with fixed structure controller through Putinar positivstellensatz. In Proceedings of the American Control Conference (ACC), Philadelphia, PA, USA, 10–12 July 2019; pp. 1806–1811.

## Abstract

**:**

## 1. Introduction

## 2. Notations and Background

**Definition**

**1.**

**Result**

**1.**

**Definition**

**2.**

**Result**

**2.**

**Definition**

**3.**

**Result**

**3.**

## 3. Problem Formulation

**Remark**

**1.**

**Definition**

**4.**

**Definition**

**5.**

**Result**

**4.**

**Result**

**5.**

**Definition**

**6.**

**Remark**

**2.**

## 4. An SOS Approach to Mixed Sensitivity Design with Fixed Structure Controller

#### 4.1. Mathematical Description of the Set $\mathcal{S}$

#### 4.1.1. Routh’s Stability Criterion

**Result**

**6.**

#### 4.1.2. Jury’s Stability Criterion

**Result**

**7.**

**Remark**

**3.**

**Remark**

**4.**

#### 4.2. Polynomial Description of the Set $\mathcal{P}$

**Result**

**8.**

**Proof.**

#### 4.3. SOS Relaxation of the Set $\mathcal{D}$

**Result**

**9.**

**Result 10.**(Putinar’s Positivstellensatz [39])

**Result**

**11.**

**Proof.**

**Remark**

**5.**

## 5. Numeric Examples

#### 5.1. Design of CT Controller

#### 5.2. DT Controller Design

#### 5.3. Comparison with Hinfstruct

## 6. Experimental Example

## 7. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**Comparison between $|{W}_{u}\left(j\omega \right)|$ (dotted) and $|{W}_{2}\left(j\omega \right)|$ (solid).

**Figure 3.**Comparison between $|{{W}_{1}}^{-1}\left(j\omega \right)|$(solid) and $|{S}_{n}\left(j\omega \right)|$ (dashed).

**Figure 4.**Comparison between $|{{W}_{2}}^{-1}\left(j\omega \right)|$ (solid) and $|{T}_{n}\left(j\omega \right)|$ (dashed).

**Figure 5.**Comparison between $|{W}_{u}\left({e}^{j\omega}\right)|$ (dotted) and $|{W}_{2}\left({e}^{j\omega}\right)|$ (solid).

**Figure 6.**Comparison between $|{W}_{1}^{-1}\left({e}^{j\omega}\right)|$ (solid) and $|{S}_{n}\left({e}^{j\omega}\right)|$ (dotted).

**Figure 7.**Comparison between $|{\widehat{W}}_{2}^{-1}\left({e}^{j\omega}\right)|$ (solid) and $|{T}_{n}\left({e}^{j\omega}\right)|$ (dotted).

**Figure 8.**Comparison between $|{W}_{1}^{-1}\left(j\omega \right)|$ (solid) and $|{S}_{n}\left(j\omega \right)|$ (dotted).

**Figure 9.**Comparison between $|{W}_{2}^{-1}\left(j\omega \right)|$ (solid) and $|{T}_{n}\left(j\omega \right)|$ (dotted).

**Figure 11.**Comparison between $|{W}_{u}\left(j\omega \right)|$(dotted) and $|{W}_{d2}\left(j\omega \right)|$ (solid).

**Figure 12.**Comparison between $|{W}_{1}^{-1}\left(j\omega \right)|$ (solid) and $|{S}_{n}\left(j\omega \right)|$ (dotted).

**Figure 13.**Comparison between $|{W}_{2}^{-1}\left(j\omega \right)|$ (solid) and $|{T}_{n}\left(j\omega \right)|$ (dotted).

**Figure 14.**Magnetic levitation system response to square wave reference signal: reference $w\left(t\right)$ (solid square-wave), magnetic levitation system output $y\left(t\right)$ (solid) and linearized ${G}_{n}\left(s\right)$ system output (dashed) responses.

${\mathit{a}}_{\mathit{n}}$ | ${\mathit{a}}_{\mathit{n}-2}$ | ${\mathit{a}}_{\mathit{n}-4}$ | ⋯ |
---|---|---|---|

${a}_{n-1}$ | ${a}_{n-3}$ | ${a}_{n-5}$ | ⋯ |

${b}_{1}$ | ${b}_{2}$ | ${b}_{3}$ | ⋯ |

${c}_{1}$ | ${c}_{2}$ | ${c}_{3}$ | ⋯ |

${d}_{1}$ | ${d}_{2}$ | ${d}_{3}$ | ⋯ |

⋮ | ⋮ | ⋮ | ⋱ |

Row Number | ${\mathit{z}}^{0}$ | ${\mathit{z}}^{1}$ | ${\mathit{z}}^{2}$ | … | ${\mathit{z}}^{\mathit{n}-\mathit{k}}$ | … | ${\mathit{z}}^{\mathit{n}-1}$ | ${\mathit{z}}^{\mathit{n}}$ |
---|---|---|---|---|---|---|---|---|

1 | ${a}_{0}$ | ${a}_{1}$ | ${a}_{2}$ | … | ${a}_{n-k}$ | … | ${a}_{n-1}$ | ${a}_{n}$ |

2 | ${a}_{n}$ | ${a}_{n-1}$ | ${a}_{n-2}$ | … | ${a}_{k}$ | … | ${a}_{1}$ | ${a}_{0}$ |

3 | ${b}_{0}$ | ${b}_{1}$ | ${b}_{2}$ | … | ${b}_{n-k}$ | … | ${b}_{n-1}$ | |

4 | ${b}_{n-1}$ | ${b}_{n-2}$ | ${b}_{n-3}$ | … | ${b}_{k-1}$ | … | ${b}_{0}$ | |

5 | ${c}_{0}$ | ${c}_{1}$ | ${c}_{2}$ | … | ${c}_{n-k}$ | … | ||

6 | ${c}_{n-2}$ | ${c}_{n-3}$ | ${c}_{n-4}$ | … | ${c}_{k-2}$ | … | ||

⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ | … | ||

$2n-2$ | ${p}_{4}$ | ${p}_{3}$ | ${p}_{2}$ | ${p}_{1}$ | ||||

$2n-3$ | ${q}_{0}$ | ${q}_{1}$ | ${q}_{2}$ |

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**MDPI and ACS Style**

Razza, V.; Salam, A. A Unified Framework for the *H*_{∞} Mixed-Sensitivity Design of Fixed Structure Controllers through Putinar Positivstellensatz . *Machines* **2021**, *9*, 176.
https://doi.org/10.3390/machines9080176

**AMA Style**

Razza V, Salam A. A Unified Framework for the *H*_{∞} Mixed-Sensitivity Design of Fixed Structure Controllers through Putinar Positivstellensatz . *Machines*. 2021; 9(8):176.
https://doi.org/10.3390/machines9080176

**Chicago/Turabian Style**

Razza, Valentino, and Abdul Salam. 2021. "A Unified Framework for the *H*_{∞} Mixed-Sensitivity Design of Fixed Structure Controllers through Putinar Positivstellensatz " *Machines* 9, no. 8: 176.
https://doi.org/10.3390/machines9080176