# Discrete-Time System Conditional Optimization Based on Takagi–Sugeno Fuzzy Model Using the Full Transfer Function

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## Abstract

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## 1. Introduction

## 2. Takagi–Sugeno Fuzzy Model

## 3. Parallel Distributed Compensation

## 4. Results

#### 4.1. Takagi–Sugeno Model of the DC Motor Based on Linearized Models

#### 4.2. Conditional Optimization under the Influence of Nonzero Initial Conditions

#### 4.2.1. System Description

**Plant**

**Controller**

**The system**

**Full transfer function matrix**

#### 4.2.2. Relative Stability

#### 4.2.3. Performance Index

#### 4.3. Control Systems Design

#### 4.4. Simulation and Experimental Results

## 5. The Major Contributions of the Work

- Three linear discrete-time mathematical models of the DC motor have been identified. A Takagi–Sugeno fuzzy model was constructed using these linear models, which represent the plant behavior around its nominal values. The membership functions are uniformly distributed, with their centers located at these nominal points;
- The characteristic polynomial of the full transfer function, rather than the traditional one, is used in this paper to carry out the conditional optimization synthesis technique. More specifically, the characteristic polynomial of the row nondegenerate full transfer function is the only one suitable and acceptable to be utilized for objective testing of system stability and optimization. The most general and realistic case of optimization was considered, thanks to the full transfer function, in which the error is the result of the simultaneous action of nonzero initial conditions and external input. Optimal parameters for three first-order PS controllers at zero and nonzero initial conditions are determined, considering that the individual closed-loop systems have a damping coefficient $\zeta =0.7$. The synthesis of the PDC controller, which uses the same membership functions as the TS fuzzy model, was performed in two cases. In the first case, the PDC controller is built by local linear first-order PS controllers, whose parameters are determined at zero initial conditions. In the second case, the PDC controller is composed of local linear controllers whose parameters are determined under nonzero initial conditions;
- Simulation and experimental results of comparing these two PDC controllers are presented to show that this paper’s technique is more comprehensive than the classical one because it considers nonzero initial conditions that the plant starts working from. The classical method offers parameters that should be universally optimal for any initial conditions, which is obviously not the case.

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 2.**The loci in the $\alpha \beta $ parameter plane of the constant damping coefficient $\zeta =0.7$.

**Figure 5.**The constant damping coefficient curve with optimal values applied to it for nonzero and zero initial conditions.

**Figure 6.**Simulation and experimental results of designed PDC controllers that take into account zero and nonzero initial conditions.

**Figure 7.**Simulation and experimental control signals of designed PDC controllers that take into account zero and nonzero initial conditions.

i | ${\mathbf{\Omega}}_{\mathit{Ni}}$ [rad/s] | ${\mathit{U}}_{\mathit{Ni}}$ [V] | ${\mathit{G}}_{\mathit{i}}\mathbf{\left(}\mathit{z}\mathbf{\right)}$ | ${\mathit{a}}_{\mathit{i}}$ | ${\mathit{b}}_{\mathit{i}}$ | ${\mathit{\alpha}}_{\mathit{i}}$ | ${\mathit{\beta}}_{\mathit{i}}$ |
---|---|---|---|---|---|---|---|

1 | 0.62 | 0.5 | $\frac{0.1023}{z-0.9398}$ | 0.9398 | 0.1023 | −0.0138 | 0 |

2 | 4.02 | 2.5 | $\frac{0.0953}{z-0.9444}$ | 0.9444 | 0.0953 | −0.0149 | 0 |

3 | 7.56 | 4.5 | $\frac{0.1042}{z-0.9443}$ | 0.9443 | 0.1042 | −0.0477 | 0 |

i | ${\mathit{u}}_{\mathit{i}}\left(0\right)$ | ${\mathit{u}}_{\mathit{i}}\left(1\right)$ | ${\mathit{y}}_{\mathit{i}}\left(0\right)$ | ${\mathit{r}}_{\mathit{i}}\left(0\right)$ | ${\mathit{I}}_{\mathit{i}\phantom{\rule{4pt}{0ex}}\mathit{min}\phantom{\rule{4pt}{0ex}}\mathit{nonzero}}$ | ${\mathit{K}}_{\mathit{i}}$ | ${\mathit{K}}_{\mathit{Si}}$ |
---|---|---|---|---|---|---|---|

1 | 1 | 1 | 1.68 | 4.02 | 469.88 | 7.5919 | 43.2051 |

2 | −1 | −1 | −1.72 | 0.62 | 1502.26 | 6.9531 | 36.8605 |

3 | −3 | −3 | −5.26 | −2.92 | 3487.25 | 6.4655 | 34.1012 |

i | ${\mathit{u}}_{\mathit{i}}\left(0\right)$ | ${\mathit{u}}_{\mathit{i}}\left(1\right)$ | ${\mathit{y}}_{\mathit{i}}\left(0\right)$ | ${\mathit{r}}_{\mathit{i}}\left(0\right)$ | ${\mathit{I}}_{\mathit{i}\phantom{\rule{4pt}{0ex}}\mathit{min}\phantom{\rule{4pt}{0ex}}\mathit{zero}}$ | ${\mathit{K}}_{\mathit{i}}$ | ${\mathit{K}}_{\mathit{Si}}$ |
---|---|---|---|---|---|---|---|

1 | 0 | 0 | 0 | 0 | 1491.02 | 8.7564 | 9.8506 |

2 | 0 | 0 | 0 | 0 | 2500.6 | 8.0313 | 9.0201 |

3 | 0 | 0 | 0 | 0 | 4351.55 | 7.3831 | 8.3161 |

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**MDPI and ACS Style**

Jovanović, R.; Zarić, V.; Bučevac, Z.; Bugarić, U. Discrete-Time System Conditional Optimization Based on Takagi–Sugeno Fuzzy Model Using the Full Transfer Function. *Appl. Sci.* **2022**, *12*, 7705.
https://doi.org/10.3390/app12157705

**AMA Style**

Jovanović R, Zarić V, Bučevac Z, Bugarić U. Discrete-Time System Conditional Optimization Based on Takagi–Sugeno Fuzzy Model Using the Full Transfer Function. *Applied Sciences*. 2022; 12(15):7705.
https://doi.org/10.3390/app12157705

**Chicago/Turabian Style**

Jovanović, Radiša, Vladimir Zarić, Zoran Bučevac, and Uglješa Bugarić. 2022. "Discrete-Time System Conditional Optimization Based on Takagi–Sugeno Fuzzy Model Using the Full Transfer Function" *Applied Sciences* 12, no. 15: 7705.
https://doi.org/10.3390/app12157705