# Cooperative Control for Dual Permanent Magnet Motor System with Unified Nonlinear Predictive Control

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Contour Errors of Traditional Dual Motor System

**p***is the point of reference position trajectory,

**p**is the point of actual position trajectory, β is the angle between the position trajectory tangential speed and x-axis positive direction, |

**pp***| is the tracking error

**e**, and R is radius of the curve position trajectory.

_{x}, e

_{y}, and the contour error, ε, of the linear position trajectory can be expressed as:

_{x}= −sin β, C

_{y}= cos β.

_{x}, e

_{y}, and the contour error, ε, of the curve position trajectory can be expressed as:

_{x}= −sin β + e

_{x}/2R, C

_{y}= cos β + e

_{y}/2R.

_{x}* and p

_{y}* are the reference position of each motor. p

_{x}and p

_{y}are the actual positions of each motor. p

_{cx}and p

_{cx}are the position compensation values of each motor after adding the cross-coupling structure. K

_{px}and K

_{py}are the proportional coefficients of the position loop. W

_{ox}(s) and W

_{oy}(s) are the position control system of motors. K

_{c}is the gain coefficient of the cross-coupling controller.

_{c}gradually increases, the position tracking errors, and the contour error are smaller under the premise that the system stability is guaranteed. Nevertheless, when K

_{c}is too large, the position tracking precision of servo system decreases. In addition, when the reference position signal is, the high-order signa and the tracking speed of position will be limited by the proportional coefficients, K

_{px}and K

_{py}, which makes it difficult to achieve fast and precise tracking control of the desired position trajectory.

## 3. Design of UNPC System

#### 3.1. Mathematical Model of PMSM Systems

**x**= (i

_{dx}, i

_{qx}, ω

_{x}, θ

_{x,}i

_{dy}, i

_{qy}, ω

_{y},θ

_{y})

^{T}are the vectors of the states (the d-axis and the q-axis components of the armature current and rotor speed and rotational angel of the motors),

**u**= (u

_{dx}, u

_{qx}, u

_{dy}, u

_{qy})

^{T}are the control input vectors (the d-axis and q-axis components of the motor’s stator voltage),

**y**= (i

_{dx}, θ

_{x}, i

_{dy}, θ

_{y})

^{T}are the output vectors (the d-axis components of the armature current and the rotational angel of motors),

**b**represents the disturbances caused by the parameter mismatch of motors and load torque, and

**b**is given by

**f**(

**x**),

**g**

_{1}(

**x**), and

**g**

_{2}(

**x**) are defined as:

**I**

_{4×4}is the identity matrix,

**0**

_{4×2}and

**0**

_{4×4}are the zero matrix,

**f**

_{i}(i = x, y), and

**g**

_{11}are given by:

_{s}, L

_{s}, ψ

_{f}, p, J

_{m}, and B are the stator resistance, inductance, rotor permanent magnetic flux, pole pairs, inertia, and friction coefficient of the motor, respectively.

_{i}as the displacement of the end device of individual motor,

**Θ**

_{i}is the movement of the end device of every motor when the motor rotates one turn; then, there is

#### 3.2. Design of Nonlinear Disturbance Observer

**b**(t) should be obtained through real-time computing, which is used for the feed-forward compensation control.

**q**(

**x**) is the nonlinear function which needs to be designed,

**z**is the internal state variable of the NL-DOB, and the gain

**L**of the NL-DOB is given by:

**q**(

**x**) can be designed as:

**L**= diag(l

_{d}, l

_{q}, l

_{ω}, l

_{θ}, l

_{d}, l

_{q}, l

_{ω}, and l

_{θ}) and l

_{d}, l

_{q}, l

_{ω}, and l

_{θ}are the constants.

**b**(t) is assumed to change slowly and is bounded, so that:

**P**

_{b}is a real positive definite symmetric matrix, and

**P**

_{b}= diag(p

_{1}, p

_{2}, p

_{3}, p

_{4}, p

_{5}, p

_{6}, p

_{7}, and p

_{8}). By taking derivative of (19) with respect to time, we can get

**L**(l

_{d}, l

_{q}, l

_{ω}, and l

_{θ}> 0), the disturbance observer is asymptotically stable. That is, when t → ∞,

**e**

_{b}= 0.

**b**(t) as soon as possible, a suitable

**L**needs to be designed. For this reason, the time constants of the observer shown in Equation (13) is deduced, and the parameters design principle of NL-DOB are given by

_{d}, 1/l

_{q}, 1/l

_{ω}, and 1/l

_{θ}, respectively. And the poles of observer are −l

_{d}, −l

_{q}, −l

_{ω}and −l

_{θ}, respectively Therefore, when the absolute values of l

_{d}, l

_{q}, l

_{ω}, and l

_{θ}are larger, the response speed of the observer is faster. Nevertheless, the absolute values of l

_{d}, l

_{q}, l

_{ω}, and l

_{θ}are too large, which will cause measurement noise in a certain extent.

#### 3.3. Design of Unified Nonlinear Predictive Controller

_{1}and T

_{2}represent the predictive horizon of the current-loop and the position-loop, respectively.

**E**

_{1}(t + τ) is the tracking error of d-axis current,

**E**

_{2}(t + τ) is the tracking error of the rotational angel of motors, and ε(t + τ) is the contour error, and

**E**

_{1},

**E**

_{2}, and ε are given by:

**C**= (C

_{x}C

_{y}) is the gain matrix of the contour error. ${i}_{\mathrm{dx}}^{*}$ and ${i}_{\mathrm{dy}}^{*}$ are the reference values of the d-axis current of the x-, y-axis motor, ${\theta}_{x}^{*}$ and ${\theta}_{y}^{*}$ are the reference values of the rotational angel of the x-, y-axis motor.

**I**is the identity matrix.

_{di}(t + τ) and θ

_{i}(t + τ) (i = x, y) are expressed as:

**u**= 0, the control law of the dual-PMSM system were obtained as:

## 4. Stability Analysis and Parameters Tuning

_{1}and T

_{2}are positive, the real parts of the closed-loop eigenvalues of the UNPC system are negative. Thus, the system is asymptotically stable.

_{1}and T

_{2}can be presented in Figure 4. It should be noted that s

_{1}is the current pole, s

_{2}and s

_{3,4}are the position poles, and the direction of the arrow represents the increasing direction of T

_{1}and T

_{2}. Then, we can know that:

- (1)
- With the increasing in the T
_{1}and T_{2}, the poles s_{1}, s_{2}, and s_{3,4}will continue to approach the imaginary axis, the dynamic response speed of the system will slow down, and the stability of the system will deteriorate; - (2)
- The position poles s
_{3,4}are closer to the imaginary axis than the pole s_{2}. That is, when the T_{2}changes, the poles s_{3,4}will play a dominant role in the position tracking performance. - (3)
- With the increasing in T
_{1}and T_{2}, the position poles s_{3,4}are closer to the imaginary axis than the current pole s_{1}, that is, the position response speed is slower than the current response speed of the PMSM control system. Thus, the corresponding requirement is T_{2}> T_{1}.

_{1}and T

_{2}should not be selected too large, so that it ensures the system has a better dynamic performance and certain stability.

## 5. Experimental Verification

_{s}as 100 μs, and the prediction horizon T

_{1}and T

_{2}were 20 and Ts, respectively. The tracking performance of the position trajectory was evaluated by means of the maximum value of the contour error, ε

_{max}, and the mean value of the steady-state error, δ

_{rms}, and the calculating formulas were as follows:

_{x}(k), y-axis tracking error e

_{y}(k), contour error ε(k) at kT

_{s}.

_{q}, was used to evaluate its performance, and the calculating formula is given by:

#### 5.1. Linear Trajectory Comparative Experiments

_{x}, p

_{y}, the speeds of motors n

_{x}, n

_{y}, q-axis currents, i

_{qx}, i

_{qy}, position tracking errors, e

_{x}, e

_{y}, and contour error, ε, are given in Figure 5b.

_{max}) were 2.77, 2.29, and 36.28 mm, respectively. When the PI-CCC control strategy was adopted, the mean values of the steady-state tracking errors of the x-, y-axis position and the maximum contour error (ε

_{max}) are 2.19, 1.67, and 24.8 mm, respectively. When Case III was adopted, the mean values of the steady-state tracking errors of the x-, y-axis position and the maximum contour error (ε

_{max}) were 1.87, 1.58, and 14.7 mm, respectively. Nevertheless, adopting the UNPC strategy, the mean values of the steady-state tracking errors of the x-, y-axis position and the maximum contour error (ε

_{max}) were 1.62, 0.79, and 10.1, respectively. Through the above experimental results, it can be known that compared with the traditional PI single-axis decoupling control strategy, the UNPC control strategy can reduce the mean values of the steady-state tracking errors of the x-, y-axis position and the maximum contour error (ε

_{max}) by 41.5%, 65.5%, 72.1%, and compared with the PI-CCC control strategy, the UNPC control strategy can reduce the mean values of the steady-state tracking errors of the x-, y-axis position and the maximum contour error (ε

_{max}) by 26.0%, 52.1%, and 59.25%, and compared with Case III, the UNPC control strategy can reduce the mean values of the steady-state tracking errors of the x-, y-axis position and the maximum contour error, ε

_{max}, by 13.36%, 50.0%, and 31.29%. Hence, when utilizing the UNPC strategy, the output trajectory can track the desired trajectory accurately and the position tracking effect is better.

#### 5.2. Circular Trajectory Experiments

_{max}) and the mean values of the contour error (ε

_{rms}) were 16.5 mm and 3.45 mm, respectively. Nevertheless, when the proposed UNPC was applied, the maximum contour error (ε

_{max}) and the mean values of the contour error (ε

_{rms}) were 12.3 mm and 2.21 mm, respectively. Thus, compared with the PI-CCC control strategy, the proposed UNPC can reduce the maximum contour error (ε

_{max}) and the mean values of the contour error (ε

_{rms}) by 25.5% and 35.9%. The above phenomenon shows that the proposed UNPC can more accurately track the s trajectory than the PI-CCC method.

#### 5.3. Motor Parameters Mismatch Experiment

_{m}, stator inductance, L

_{s}, and stator resistance, R

_{s}, in the controller are equal to 1.0 J

_{m}, 1.0 L

_{S}, 1.0 R

_{s}, 0.8 J

_{m}, 0.8 L

_{S}, 0.8 R

_{s}, and 1.2 J

_{m}, 1.2 L

_{S}, and 1.2R

_{s}, respectively. Let the dual motor system follow the 45° slope position signal, the experimental results are shown in Figure 7.

_{q}) of the q-axis current and the harmonic content of stator current change slightly. The experimental results indicate that the designed controller in this paper has a good steady-state control performance in case of that the motor’s parameters are mismatch.

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Conflicts of Interest

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**Figure 1.**Schematic diagram of the contour errors of the trajectory: (

**a**) linear position trajectory; (

**b**) curve position trajectory.

**Figure 5.**Experimental comparison of traditional control strategy and UNPC: (

**a**) output trajectory of diamond; (

**b**) waveforms of steady-state experimental.

**Figure 6.**Circular tracking experiment of PI-CCC and UNPC: (

**a**) Output trajectory of circular; (

**b**) waveforms of steady-state experiment.

**Figure 7.**Experimental results of UNPC under the condition of motor parameters mismatch: (

**a**) 1.0 J

_{m}, 1.0 L

_{S}, and 1.0 R

_{s}; (

**b**) 0.8 J

_{m}, 0.8 L

_{S}; (

**c**) 1.2 J

_{m}, 1.2 L

_{S}; (

**d**) 0.8 R

_{S}; (

**e**) 1.2 R

_{S}.

Parameters | Symbol | Value | Units |
---|---|---|---|

Rated power | P_{N} | 2.3 | kW |

Rated torque | T_{N} | 15 | Nm |

Number of pole–pairs | p | 2 | - |

Stator resistance | R_{s} | 0.635 | Ω |

Stator inductance | L_{s} | 4.025 | mH |

Permanent magnet flux | ψ_{f} | 0.5 | Wb |

Inertia | J_{m} | 0.00272 | kgm^{2} |

Friction coefficient | B | 0.002 | - |

Movement of the end | Θ_{i} | 0.145 | m |

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## Share and Cite

**MDPI and ACS Style**

Zhou, Z.; Xu, Z.; Zhang, G.; Geng, Q.
Cooperative Control for Dual Permanent Magnet Motor System with Unified Nonlinear Predictive Control. *World Electr. Veh. J.* **2021**, *12*, 266.
https://doi.org/10.3390/wevj12040266

**AMA Style**

Zhou Z, Xu Z, Zhang G, Geng Q.
Cooperative Control for Dual Permanent Magnet Motor System with Unified Nonlinear Predictive Control. *World Electric Vehicle Journal*. 2021; 12(4):266.
https://doi.org/10.3390/wevj12040266

**Chicago/Turabian Style**

Zhou, Zhanqing, Zhengchao Xu, Guozheng Zhang, and Qiang Geng.
2021. "Cooperative Control for Dual Permanent Magnet Motor System with Unified Nonlinear Predictive Control" *World Electric Vehicle Journal* 12, no. 4: 266.
https://doi.org/10.3390/wevj12040266