# Parameter Estimation, Robust Controller Design and Performance Analysis for an Electric Power Steering System

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## Abstract

**:**

## 1. Introduction

## 2. Operational Principle and Mathematical Model

#### 2.1. Operational Principle

#### 2.2. Mathematical Model

## 3. Background of the Algorithms

## 4. Identification Analysis of the Motor Parameters

#### 4.1. Simulation Identification

#### 4.2. Identification Experiment

## 5. Identification Analysis and Test Bench Simulation of the EPS System

#### 5.1. Identification Analysis of the EPS System

#### 5.2. Identification Experiment for the EPS System

_{d}) and vehicle speed (v), which is obtained from the vehicle manufacturer. Figure 22 shows the assist torque characteristic curves versus the driver torque with respect to different vehicle speeds. Figure 22 illustrates that the assist torque is proportional to the steering wheel torque, with the assist torque (${T}_{a}$) increasing with decreasing vehicle velocity to ensure maneuverability. In contrast, when the vehicle velocity increases, the assist torque decreases while maintaining the system stability.

## 6. Controller Design

- The sensitivity function: $S=\frac{1}{1\hspace{0.17em}+\hspace{0.17em}G\stackrel{\wedge}{K}}$
- The complementary sensitivity function: $T=\frac{G\stackrel{\wedge}{K}}{1\hspace{0.17em}+\hspace{0.17em}G\stackrel{\wedge}{K}}$. The loop-shaping gain: $L=G\stackrel{\wedge}{K}$

## 7. EPS Test Bench

## 8. Verification of the EPS Control Algorithm

**Step****1.**- Establish the new model based on MATLAB/Simulink and configure the RTI interfaces, including the common I/O, PWM and AD modules.
**Step****2.**- Establish the control algorithm model for the EPS system, filter the collected analogue signals and compile and generate dSPACE executable SDF files.
**Step****3.**- Build the new experimental project in the Control Desk software of dSPACE. The generated SDF files are downloaded into the real-time dSPACE card via the Ethernet network. Finally, the overall EPS frame system is controlled in real time via the control panel of Control Desk.

**Case 1.**PID control algorithm

**Case 2.**Proposed loop-shaping control algorithm

**Step 1:**Select a pre-compensator ${W}_{1}$ and post-compensator ${W}_{2}$. These two shaping functions are added to generate the shaped plant ${G}_{s}$, which is written as follows:

**Step 2:**Given a shaped plant ${G}_{s}$ and A, B, C, D represent the shaped plant in the state-space form. To determine ${\gamma}_{\mathrm{min}}$, there is a unique method, as follows [33].

**Step 3:**Choose $\gamma >{\gamma}_{\mathrm{min}}$; the ${\stackrel{\wedge}{K}}_{\infty}$ controller must satisfy the following equation [13]:

**Step 4:**The controller is synthesized as follows:

_{ris}, maximum overshoot o

_{max}, and settling time t

_{set}enable an objective comparison of the different controllers in Table 11.

## 9. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 3.**Bode plots of the proposed identification algorithms compared with that of the actual model.

**Figure 26.**Overall block diagram of the EPS electric motor parameter identification experiment platform.

**Figure 29.**Comparison of the assist torque response using the PID controller and loop-shaping controller.

**Figure 32.**Actual assist characteristic curve under the PID control algorithm with different velocities.

**Figure 33.**Actual assist characteristic curve using the proposed loop-shaping control algorithm with different velocities.

Motor Parameters | Value | Unit |
---|---|---|

Armature resistance R_{m} | 0.35 | Ω |

Armature inductance L_{m} | 0.001 | H |

Motor back EMF constant K_{t} | 0.0433 | V·s |

Motor rotor inertia J_{m} | 0.00018 | kg·m^{2} |

Hydraulic friction coefficient of Motor rotor B_{m} | 0.0034 | N·m·s |

nb | nf | nk | RT2 | YIC | Cond | AIC |
---|---|---|---|---|---|---|

1 | 1 | 0 | 0.99891 | −10.831 | 27.293 | −5.1153 |

1 | 4 | 0 | 0.96856 | −10.557 | 3.7446 × 10^{9} | −1.7471 |

3 | 3 | 0 | 0.99893 | −9.4845 | 5.0471 × 10^{7} | −5.1219 |

2 | 1 | 0 | 0.99891 | −6.8284 | 1170.7 | −5.1136 |

2 | 2 | 0 | 0.99897 | −11.3369 | 5.1443 × 10^{5} | −5.1377 |

Identification Results | Motor Parameter | |||
---|---|---|---|---|

Sampling Rate (Hz) | Amplitude (%) | Clock Time (ms) | Resistance (Ω) | Inductance (H) |

10k | 20 50 | 200 | −0.374 | 0.0020 |

5k | 20 50 | 200 | −0.759 | 0.0023 |

1k | 20 60 | 200 | 0.096 | 0.0015 |

800 | 20 60 | 200 | 0.101 | 0.0014 |

800 | −60 60 | 500 | 0.281 | 0.0015 |

Identification Results | Evaluation Index | |||
---|---|---|---|---|

Sampling Rate (Hz) | Amplitude (%) | Clock Time (ms) | RT2 | YIC |

700 | 20 60 | 200 | 0.873 | −9.627 |

800 | 20 60 | 200 | 0.903 | −10.562 |

900 | 20 60 | 200 | 0.875 | −10.128 |

800 | −60 60 | 200 | 0.939 | −11.426 |

800 | −60 60 | 400 | 0.954 | −11.882 |

800 | −60 60 | 500 | 0.957 | −11.892 |

800 | −60 60 | 600 | 0.981 | −13.077 |

Identification Algorithm | RT2 | MSE | FIT (%) |
---|---|---|---|

SRIV | 0.957 | 4.643 | 79.01 |

IVSVF | 0.913 | 9.217 | 70.42 |

LSSVF | 0.837 | 17.4 | 59.36 |

nb | nf | nk | RT2 | YIC | Cond | AIC |
---|---|---|---|---|---|---|

3 | 3 | 0 | 0.97269 | −12.621 | 1.6182 × 10^{6} | 1.0582 |

1 | 1 | 0 | 0.91941 | −12.304 | 26.673 | 2.1396 |

2 | 2 | 0 | 0.98023 | −12.598 | 7255.8 | 1.5362 |

3 | 2 | 0 | 0.9628 | −11.754 | 8.3843 × 10^{5} | 1.3774 |

1 | 2 | 0 | 0.93298 | −10.875 | 28,244 | 1.9562 |

EPS Parameters | Value | Unit |
---|---|---|

Steering wheel moment of inertia J_{c} | 0.04 | kg·m^{2} |

Torsional stiffness K_{c} | 115 | N·m/rad |

Steering wheel damping B_{c} | 0.325 | N·m/(rad/s) |

Rack and wheel assembly mass M_{r} | 32 | kg |

Rack damping Br | 653.2 | N/(m/s) |

Tire or rack centring spring rate K_{r} | 91,061 | N/m |

Pinion radius r_{p} | 0.0071 | m |

nb | nf | nk | RT2 | YIC | Cond | AIC |
---|---|---|---|---|---|---|

1 | 2 | 0 | 0.99843 | −16.755 | 3527.9 | −22.109 |

1 | 4 | 0 | 0.99886 | −18.755 | 78,870 | −22.425 |

4 | 3 | 0 | 0.99855 | −10.839 | 9.6734 × 10^{10} | −22.184 |

2 | 2 | 0 | 0.99845 | −10.794 | 47,544 | −22.125 |

3 | 3 | 0 | 0.99859 | −10.478 | 3.8167 × 10^{8} | −22.21 |

Structure of the EPS System | Sampling Rate (Hz) | RT2 | YIC |
---|---|---|---|

nb = 3, nf = 4, nk = 0 | 100 | 0.975 | −9.990 |

500 | 0.965 | −10.014 | |

800 | 0.984 | −12.265 | |

900 | 0.981 | −12.251 | |

1000 | 0.981 | −11.155 |

Identification Algorithm | RT2 | MSE | FIT (%) |
---|---|---|---|

SRIV | 0.981 | 0.204 | 86.28 |

IVSVF | 0.970 | 0.475 | 79.07 |

LSSVF | 0.929 | 0.765 | 73.43 |

Controller | O_{max} (%) | t_{ris} (s) | t_{set} (s) |
---|---|---|---|

PID controller | 23.4 | 0.056 | 0.178 |

Loop-shaping controller | 8.7 | 0.064 | 0.075 |

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**MDPI and ACS Style**

Nguyen, V.G.; Guo, X.; Zhang, C.; Tran, X.K.
Parameter Estimation, Robust Controller Design and Performance Analysis for an Electric Power Steering System. *Algorithms* **2019**, *12*, 57.
https://doi.org/10.3390/a12030057

**AMA Style**

Nguyen VG, Guo X, Zhang C, Tran XK.
Parameter Estimation, Robust Controller Design and Performance Analysis for an Electric Power Steering System. *Algorithms*. 2019; 12(3):57.
https://doi.org/10.3390/a12030057

**Chicago/Turabian Style**

Nguyen, Van Giao, Xuexun Guo, Chengcai Zhang, and Xuan Khoa Tran.
2019. "Parameter Estimation, Robust Controller Design and Performance Analysis for an Electric Power Steering System" *Algorithms* 12, no. 3: 57.
https://doi.org/10.3390/a12030057