# PFC-Based Control of Friction-Induced Instabilities in Drive Systems

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## Abstract

**:**

## 1. Introduction

## 2. Modeling

#### 2.1. Mathematical Model

#### 2.2. Model Discretization

## 3. Friction-Induced Self-Excited Oscillations

#### 3.1. Stability and Zero Dynamics

#### 3.2. Conventional Velocity Control

## 4. PFC-Based Control

#### 4.1. Motivation

- $\tilde{P}\left(s\right)$ is strictly the minimum phase, i.e., zeros ${z}_{i}$ of $\tilde{P}\left(s\right)$ are in the LHP $Re\phantom{\rule{3.33333pt}{0ex}}\left[{z}_{i}\right]<0$;
- the relative degree of the system is 0 or 1.

#### 4.2. PFC Design

#### 4.3. Stability of the Augmented Plant

#### 4.4. PFC-Based Adaptive Feedback Control

## 5. Results

- Scenario 1: adaptation, angular velocity reference tracking and disturbance rejection of the closed-loop system;
- Scenario 2: adaptation and angular velocity reference tracking of the closed-loop system in the presence of measurement noise;
- Scenario 3: adaptation and angular velocity reference tracking of the closed-loop system in the presence of parameters variation.

- $\lambda =0.1$ is chosen as $5\%$ from the reference value $r=2$ rad/s;
- $\xi =1\times {10}^{8}$ is chosen to provide a reasonable adaptation rate;
- ${k}_{0}=1$;
- ${T}_{p}=4$ s for the reference tracking (${T}_{p}=1$ s for the adaptation).

#### 5.1. Scenario 1

#### 5.2. Scenario 2

#### 5.3. Scenario 3

- The static friction torque ${\tau}_{st}$ from (4) is varied in a range of ±50% from its nominal value resulting in a different slope of the friction curve in the operating point of interest;
- The moment of inertia of the second mass ${J}_{2}$ is varied in a range of ±50% from its nominal value.

## 6. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

PFC | Parallel feed-forward compensator |

ASPR | Almost strictly positive real |

PDE | Partial differential equation |

RHP | Right-half plane |

LHP | Left-half plane |

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**Figure 2.**Friction curve (

**left**): region of unstable operating points (red). Low-frequency part of pole-zero map of linearized systems (

**right**).

**Figure 4.**Low-frequency part of root locus of closed-loop system (

**left**) and its zoomed-in view for the 4 low-frequency modes (

**right**). Poles of the open-loop system (blue), poles of the closed-loop system with ${k}_{P}=405$ (green), poles of the closed loop system with ${k}_{P}=2\times {10}^{5}$ (red).

**Figure 5.**Step responses of the closed-loop nonlinear system applying velocity $PI$-controller with gain ${k}_{P}=405$: first mass ${\phi}_{0}$ (solid blue) and second mass ${\phi}_{L}$ (dotted green).

**Figure 6.**Step responses of the closed-loop nonlinear system applying velocity $PI$-controller with gain ${k}_{P}=2\times {10}^{5}$: first mass ${\phi}_{0}$ (solid blue) and second mass ${\phi}_{L}$ (dotted green).

**Figure 7.**Parallel feed-forward interconnection: plant $P\left(s\right)$, PFC $F\left(s\right)$ and extended plant $\tilde{P}\left(s\right)$.

**Figure 8.**The Bode diagram of the linearized system. High-order model (solid blue); reduced order model (dashed green).

**Figure 10.**Overall control scheme: plant P, PFC F, precompensator ${W}_{p}$ and adaptive $\lambda $-tracking controller $k\left(t\right)$.

**Figure 11.**Transient responses of the closed-loop system. Adaptation and angular velocity reference tracking: ${\dot{\phi}}_{0}$ (solid blue), ${\dot{\phi}}_{L}$ (dashed green), $\tilde{y}$ (dash-dot red) and r (dotted gray).

**Figure 12.**Transient responses of the closed-loop system. Angular velocity reference tracking: distributed $\dot{\phi}(z,t)$.

**Figure 13.**Transient responses of the closed-loop system. Disturbance rejection: ${\dot{\phi}}_{0}$ (solid blue), ${\dot{\phi}}_{L}$ (dashed green) and $\tilde{y}$ (dash-dot red). Impulse torques are applied to the second mass at time ${t}_{1}=0$ s (dashed green) and to the first mass ${t}_{1}=30$ s (dash-dot red).

**Figure 14.**Transient responses of the closed-loop system. Adaptation and angular velocity reference tracking in presence of the measurement noise: ${\dot{\phi}}_{0}$ (solid blue), ${\dot{\phi}}_{L}$ (dashed green), $\tilde{y}$ (dash-dot red) and r (dotted gray).

**Figure 16.**Transient responses of the closed-loop system. Adaptation and angular velocity reference tracking in presence of parameter uncertainties: nominal parameter model (solid blue) and uncertain parameter model (dashed green).

Name | Symbol | Value | Unit |
---|---|---|---|

Density of the motor shaft | $\varrho $ | 8000 | [kg/m${}^{3}$] |

Shear modulus of the motor shaft | G | 79.3 $\times {10}^{9}$ | [N/m${}^{2}$] |

Structural damping of the motor shaft | $\gamma $ | $5\times {10}^{-4}$ | [Nms] |

Length of the motor shaft | L | 1500 | [m] |

Moment of inertia of the motor shaft | I | $1\times {10}^{-5}$ | [m${}^{4}$] |

Moments of inertia of the first mass | ${J}_{1}$ | 150 | [kg m${}^{2}$] |

Moments of inertia of the second mass | ${J}_{2}$ | 1500 | [kg m${}^{2}$] |

Viscous friction on the first mass | $\beta $ | 2000 | [Nms] |

Coulomb friction torque | ${\tau}_{co}$ | 165 | [Nm] |

Viscous dissipation torque | ${\tau}_{vi}$ | 20 | [Nms] |

Static friction torque | ${\tau}_{st}$ | 515 | [Nm] |

Stribeck parameter | ${b}_{sb}$ | $0.5$ | [−] |

Slope in the region of zero velocity | ${k}_{fr}$ | $-75$ | [−] |

Cases | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
---|---|---|---|---|---|---|---|---|---|

${t}_{rise}\phantom{\rule{3.33333pt}{0ex}}\left[s\right]$ | 9.27 | 8.22 | 10.46 | 9 | 12.25 | 9.92 | 9.80 | 8.68 | 11.15 |

${t}_{set}\phantom{\rule{3.33333pt}{0ex}}\left[s\right]$ | 16.62 | 21.15 | 20.64 | 16.11 | 13.94 | 20.15 | 17.12 | 28.07 | 21.25 |

$\sigma \phantom{\rule{3.33333pt}{0ex}}[\%]$ | 8.97 | 4.32 | 15.08 | 5.53 | 2.04 | 12.01 | 12.96 | 9.48 | 18.52 |

Cases | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
---|---|---|---|---|---|---|---|---|---|

${t}_{rise}\phantom{\rule{3.33333pt}{0ex}}\left[s\right]$ | 7.44 | 6.40 | 8.62 | 7.26 | 10.46 | 8.12 | 7.91 | 6.71 | 9.28 |

${t}_{set}\phantom{\rule{3.33333pt}{0ex}}\left[s\right]$ | 15 | 19.26 | 23.03 | 14.90 | 14.18 | 18.51 | 15.22 | 26.26 | 34.05 |

$\sigma \phantom{\rule{3.33333pt}{0ex}}[\%]$ | 12.74 | 5.38 | 22.79 | 7.44 | 1.92 | 18 | 19.03 | 11.57 | 28.19 |

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**MDPI and ACS Style**

Golovin, I.; Palis, S.
PFC-Based Control of Friction-Induced Instabilities in Drive Systems. *Machines* **2021**, *9*, 134.
https://doi.org/10.3390/machines9070134

**AMA Style**

Golovin I, Palis S.
PFC-Based Control of Friction-Induced Instabilities in Drive Systems. *Machines*. 2021; 9(7):134.
https://doi.org/10.3390/machines9070134

**Chicago/Turabian Style**

Golovin, Ievgen, and Stefan Palis.
2021. "PFC-Based Control of Friction-Induced Instabilities in Drive Systems" *Machines* 9, no. 7: 134.
https://doi.org/10.3390/machines9070134