# Unstructured Uncertainty Based Modeling and Robust Stability Analysis of Textile-Reinforced Composites with Embedded Shape Memory Alloys

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## Abstract

**:**

## 1. Introduction

## 2. System Description

## 3. Model Identification

^{®}is used [28]. This toolbox offers an easy and efficient way to generate various mathematical models of systems such as continuous-time and discrete-time transfer functions as well as state space models. To describe the characteristics of the SMA actuator over a wide range of operation points, open-loop tests of the actuator using step inputs are conducted. The input and output values of these experiments are used to identify the mathematical model (Equation (1)), of the system. Each step response yields different values of the parameters K and T. Over all experiments we obtained the following upper and lower bounds of these parameters [21]:

## 4. Classical Controller Design

## 5. Robust Control

#### 5.1. Uncertainty Models

- Additive Uncertainty$$G\left(s\right)=\tilde{G}\left(s\right)+W\left(s\right)\Delta \left(s\right).$$
- Multiplicative Uncertainty$$G\left(s\right)=\tilde{G}\left(s\right)\left(\right)open="("\; close=")">1+W\left(s\right)\Delta \left(s\right)$$
- Feedback Uncertainty$$G\left(s\right)=\frac{\tilde{G}\left(s\right)}{1+W\left(s\right)\Delta \left(s\right)\tilde{G}\left(s\right)}.$$
- Multiplicative Feedback Uncertainty$$G\left(s\right)=\frac{\tilde{G}\left(s\right)}{1+W\left(s\right)\Delta \left(s\right)}.$$

- Multiplicative Uncertainty$$W\left(s\right)=\frac{s(\overline{K}\tilde{T}-\tilde{K}\underline{T})+\overline{K}-\tilde{K}}{\tilde{K}(s\underline{T}+1)},\phantom{\rule{1.em}{0ex}}\Delta \left(s\right)=\left(\right)open="("\; close=")">\frac{s\underline{T}+1}{sT+1}.$$
- Feedback Uncertainty$$W\left(s\right)=\frac{s(\overline{K}\tilde{T}-\tilde{K}\underline{T})+\overline{K}-\tilde{K}}{\underline{K}\tilde{K}},\phantom{\rule{1.em}{0ex}}\Delta \left(s\right)=\left(\right)open="("\; close=")">\frac{\underline{K}}{K}.$$
- Multiplicative Feedback Uncertainty$$W\left(s\right)=\frac{s\left(\tilde{K}\overline{T}-\underline{K}\tilde{T}\right)+\tilde{K}-\underline{K}}{\underline{K}(1+s\tilde{T})},\phantom{\rule{1.em}{0ex}}\Delta \left(s\right)=\left(\right)open="("\; close=")">\frac{\underline{K}}{K}.$$

#### 5.2. Robust Stability Analysis

#### 5.3. Robust PI Control

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Test bench of textile-reinforced composite specimen with integrated shape memory alloys (SMAs).

**Figure 3.**Four consecutive heating up and cooling down cycles of shape memory alloy with hysteresis.

**Figure 5.**Deflection of the controlled system for 22 mm reference with classically designed proportional integral (PI) controller.

**Figure 6.**Uncertainty models of the plant’s transfer function G(s). (

**a**) Additive uncertainty; (

**b**) Multiplicative uncertainty; (

**c**) Feedback uncertainty; (

**d**) Multiplicative feedback uncertainty.

**Figure 7.**Level sets of $\parallel WC\tilde{S}{\parallel}_{\infty}$ and $\parallel W\tilde{T}{\parallel}_{\infty}$ for different values of ${K}_{p}$ and ${K}_{i}$ for the system with additive and multiplicative uncertainty.

**Figure 8.**Level sets of $\parallel W\tilde{S}{\parallel}_{\infty}$ for different values of ${K}_{p}$ and ${K}_{i}$ for the system with multiplicative feedback uncertainty.

**Figure 9.**Bode diagram of $WC\tilde{S}$ for ${K}_{p}=2$ and ${K}_{i}=0.004$ considering multiplicative uncertainty.

**Figure 10.**Deflection of the controlled system with multiplicative uncertainty for 22 mm reference using a robust PI controller.

**Figure 11.**Bode diagram of $W\tilde{S}$ for ${K}_{p}=3.2$ and ${K}_{i}=0.16$ considering multiplicative feedback uncertainty.

**Figure 12.**Deflection of the controlled system with multiplicative feedback uncertainty for 22 mm reference using a robust PI controller.

Uncertainty Type | Robust Stability Condition |
---|---|

$\tilde{G}+W\Delta $ | $\parallel WC\tilde{S}{\parallel}_{\infty}<1$ |

$\tilde{G}\phantom{\rule{0.166667em}{0ex}}(1+W\Delta )$ | $\parallel W\tilde{T}{\parallel}_{\infty}<1$ |

$\frac{\tilde{G}}{1+W\Delta \tilde{G}}$ | $\parallel W\tilde{G}\tilde{S}{\parallel}_{\infty}<1$ |

$\frac{\tilde{G}}{1+W\Delta}$ | $\parallel W\tilde{S}{\parallel}_{\infty}<1$ |

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**MDPI and ACS Style**

Keshtkar, N.; Röbenack, K.
Unstructured Uncertainty Based Modeling and Robust Stability Analysis of Textile-Reinforced Composites with Embedded Shape Memory Alloys. *Algorithms* **2020**, *13*, 24.
https://doi.org/10.3390/a13010024

**AMA Style**

Keshtkar N, Röbenack K.
Unstructured Uncertainty Based Modeling and Robust Stability Analysis of Textile-Reinforced Composites with Embedded Shape Memory Alloys. *Algorithms*. 2020; 13(1):24.
https://doi.org/10.3390/a13010024

**Chicago/Turabian Style**

Keshtkar, Najmeh, and Klaus Röbenack.
2020. "Unstructured Uncertainty Based Modeling and Robust Stability Analysis of Textile-Reinforced Composites with Embedded Shape Memory Alloys" *Algorithms* 13, no. 1: 24.
https://doi.org/10.3390/a13010024