# H-Shaped Multiple Linear Motor Drive Platform Control System Design Based on an Inverse System Method

^{1}

^{2}

^{3}

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## Abstract

**:**

## 1. Introduction

## 2. An H-Type Platform Driven by Multiple Permanent Magnet Synchronous Linear Motors

#### 2.1. Platform Structure

_{1}and Y

_{2}), which realize the bilateral synchronous drive in Y direction. The stator of the linear motor in X direction is fixed on the crossbeam and the mover of it is connected to the air sliding block, which realizes the motion in the X direction. Both the crossbeam and the slide block are sustained and guided by the air bearing. The planar movement in the XY plane can be realized through the control of the three linear motors. In practical control, there will also be rotation around the axis of Z due to the deviation between the two linear motors in Y direction. In practical control, the goal is to make the deviation approach zero. It can be seen from the above analysis that there is mechanical coupling between the three linear motors.

_{1}and Y

_{2}) moving in the Y direction and their servo systems can be designed to have the same parameters, in practical control process the synchronization of the two motors cannot be guaranteed due to the position change of the motor moving in the Y direction. The position of the linear motor moving in the X direction also changes in the working process, and thus the distance between it and the two linear motors moving in the Y direction also changes accordingly, which influences the force allocated to each motor. Moreover, the machining error is inevitable in the process of manufacturing and installation. In addition, there is always uncertain disturbance during the operation. Therefore, there is high requirement for the control system to make sure the synchronous deviation is within a certain small range.

#### 2.2. Mechanical Analysis

_{1}and motor Y

_{2}, can produce horizontal forces f

_{Y}

_{1}, f

_{Y}

_{2}. Not only can the two forces contribute to the motion in X and Y direction, but also the difference between them can contribute to the rotation movement around Z-axis. With the forces generated by the three linear motors projected on X-axis and Y-axis, we can get the following equations:

_{Y}

_{1}and f

_{Y}

_{2}. According to the control requirement, δ is constrained to be less than 0.1°, thus cosδ approaches 1 and sinδ approaches 0. The above equations can be then further simplified as:

## 3. Pre-Analysis for Permanent Magnet Synchronous Linear Motor Control System

#### 3.1. ABC Coordinate System to d-q Coordinate System

_{q}(proportional to the electromagnetic force) and magnetic field excitation current i

_{d}. The transformation can be divided into two steps.

_{d}and i

_{q}.

#### 3.2. Mathematical Model

_{d}, u

_{q}, i

_{d}, i

_{q}, Ψ

_{d}, Ψ

_{q}, L

_{d}, L

_{q}are dq-axis voltage, current, flux linkage, and inductance respectively. R

_{s}is the primary resistance; Ψ

_{f}is the permanent flux linkage; v is the mover’s velocity; P is the number of pole pairs [19,20,21].

_{T}is the electromagnetic thrust force; and τ is the pole pitch.

_{d}= L

_{q}and thus the equation above can be simplified into equation:

_{q}, current in q-axis, with other parameters fixed.

#### 3.3. PID Controller

_{p}, K

_{i}, K

_{d}are the proportional gain, integral gain and differential gain, respectively. The system input can be seen as the sum of the proportional component, integral component and differential component. The proportional component is essential for decreasing the general control error; the integral component makes the stable state error approach zero and the differential component can improve the dynamic trajectory following performance especially when there is a sudden (high-frequency) change for the system output.

## 4. Inverse System Model and Control System Design

#### 4.1. Inverse System Decoupling

#### 4.1.1. Inverse System Theory

**u**(t) ∈

**R**

^{m}is the input vector,

**y**(t) ∈

**R**

^{r}is the output vector,

**x**(t) ∈

**R**

^{n}is the state variable, and the initial condition is

**x**(t

_{0}) =

**x**

_{0}, r ≤ m.

**y**=

**h**(

**x**,

**u**), we can deduce the α

_{i}-th order derivative of y

_{i}to time variable t, which is shown as:

_{i}(1 ≤ i ≤ r) is defined as:

_{i}< ∞, otherwise it is not derivable.

_{d}(t) is the expected (reference) output of the original system and, y(t) is the real output of the original system. The state variables of the original system are used in the α-th order integration inverse system through feedback. The expected output value can then be the input of the system controller.

#### 4.1.2. Inverse System Condition

_{1}, α

_{2}, ···, α

_{r}} at (

**x**

_{0},

**u**

_{0}). Two basic conditions need to be satisfied as follows.

- At (
**x**_{0},**u**_{0}), the following equation is satisfied:$$\frac{\partial}{\partial {u}_{j}}\left({L}_{f}^{k}{h}_{i}\right)=0,\left(1\le j\le m,1\le i\le r,k\le {\alpha}_{i}-1\right)$$ - The r × m order matrix shown as follows is nonsingular, which indicates the rank of the matrix $\mathrm{rank}\left(A\right)=r$.$$A=\left[\begin{array}{ccc}\frac{\partial {L}_{f}^{{\alpha}_{1}}{h}_{1}}{\partial {u}_{1}}& \cdots & \frac{\partial {L}_{f}^{{\alpha}_{1}}{h}_{1}}{\partial {u}_{m}}\\ \vdots & \cdots & \vdots \\ \frac{\partial {L}_{f}^{{\alpha}_{r}}{h}_{r}}{\partial {u}_{1}}& \cdots & \frac{\partial {L}_{f}^{{\alpha}_{r}}{h}_{r}}{\partial {u}_{m}}\end{array}\right]$$

**x**

_{0},

**u**

_{0}) when the above {α

_{1}, α

_{2}, ···, α

_{r}} exists and they meet the requirements of the following equation where n is the number of system state variables:

#### 4.2. H-Shaped Platform State Equation

#### 4.3. Decoupling Design

**y**= [y

_{1}, y

_{2}, y

_{3}]

^{T}and it contains

**u**= [u

_{1}, u

_{2}, u

_{3}]

^{T}.

**α**= {α

_{1}, α

_{2}, α

_{3}} = {2, 2, 2} = 6, which is equal to the number of state variables. Thus, the system is invertible. Suppose:

#### 4.4. Control System Structure

_{r}, position Y

_{r}and rotation angle around Z-axis δ

_{r}. The feedback actual values are compared with the given values and the difference between them will be used for PID position regulation. The regulation results are used to compare with the state feedback values, and the difference between them is further used as the input of the inverse system model. Vector control with i

_{d}= 0 is chosen as the control strategy because the magnetic field is kept constant and only i

_{q}is used to control the thrust force of the linear motor. The needed force-generating current i

_{q}can then be calculated through the inverse model. After the coordinate transformation, the dq-axis current can be transformed into the three-phase currents, which will be fed to the linear motor after the power amplifier.

## 5. Simulation and Analysis

_{1}and Y

_{2}is analyzed. Second, the proposed control method is compared with the traditional method in terms of its performance for the mover trajectory under disturbance. Lastly, the program has been implemented on NI controller which is a typical industrial controller. The motor in this paper is a maglev permanent magnet linear synchronous motor, whose basic electromagnetic model parameters are show in Table 1.

#### 5.1. Synchronous Deviation Analysis

_{1}or Y

_{2}) causing the synchronous deviation, the controller will control both Y

_{1}and Y

_{2}to decrease the deviation between them. The simulation time is set to be 0.2 s. The given reference position for the two linear motors Y

_{1}and Y

_{2}is 1000 μm. The load force on Y

_{1}and Y

_{2}are 100 N, and at 0.08 s the load force imposed on Y

_{1}is increased to 150 N. The electromagnetic force on Y

_{1}and displacement of Y

_{1}is shown in Figure 6. The result for Y

_{2}is shown in Figure 7.

_{1}as shown in Figure 6c. To keep synchronous with Y

_{1}, the electromagnetic force and displacement of Y

_{2}also changes. The position control of the motor consists of three major phases: acceleration phase, deceleration phase and positioning/stable phase. It can be seen from Figure 6a and Figure 7a that the thrust torque at the start phase approaches 300 N (the maximum motor thrust force), which corresponds to the velocity increase phase in Figure 6b and Figure 7b. When the motor approaches the reference position, it enters the deceleration phase, which occurs at about 0.01 s after the start. Finally, the motor arrives at the given position and its speed also decreases to zero, which indicates the motor is in the positioning or stable phase. In the positioning phase, the position of the motor keeps constant and the thrust force is equal to the load force.

_{1}and Y

_{2}when it is under the disturbance force of 20 N, 35 N and 50 N. It shows the maximal synchronous deviation is 0.025 μm, 0.05 μm, when the load force is 20 N and 35 N respectively. The recovery time under different disturbances is 0.02 s, while the synchronous deviation increases nonlinearly with the increase of disturbance. For the disturbance of 50 N (half the rated load force), the maximal synchronous deviation between them is 0.1 μm with a recovery time of 0.02 s. It proves that the proposed control strategy can meet the high-precision and fast-response dynamic requirements in planar positioning applications.

#### 5.2. H-Platform Trajectory Analysis

_{1}and Y

_{2}) in the Y direction and 50 N (half the rated force for linear motor X) in the X direction, both of them lasting for 0.005 s, are imposed on the platform mover to simulate such random disturbance forces. Figure 9 shows the simulation result for both the proposed new control method and traditional PID method.

#### 5.3. System Real-Time Performance Validation

## 6. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**Figure 6.**Y

_{1}Linear Motor: (

**a**) Electromagnetic force; (

**b**) Velocity; (

**c**) Displacement in Y direction.

**Figure 7.**Y

_{2}Linear Motor: (

**a**)Electromagnetic force; (

**b**) Velocity; (

**c**) Displacement in Y direction.

**Figure 8.**Synchronous deviation between Y

_{1}and Y

_{2}: (

**a**) disturbance force 20 N; (

**b**) disturbance force 35 N; (

**c**) disturbance force 50 N.

**Figure 9.**Comparison of platform trajectory: (

**a**) proposed control model based on inverse system; (

**b**) traditional control model.

**Figure 10.**Circle trajectory tracking analysis: (

**a**) proposed control model; (

**b**) traditional PID control model.

Parameter | Value | Unit |
---|---|---|

Pole pitch τ | 16 | Mm |

Flux of permanent magnet ψ_{f} | 0.211 | Wb |

Winding resistance of each phase R | 2.1 | Ω |

Air gap h | 1 | Mm |

q-axis inductance L_{q} | 0.0163 | H |

d-axis inductance L_{q} | 0.0163 | H |

Mass of mover | 0.6 | Kg |

Rotational inertia J | 0.382 | kg·m^{2} |

force arm in Y direction l | 0.42 | M |

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**MDPI and ACS Style**

Qin, C.; Zhang, C.; Lu, H. H-Shaped Multiple Linear Motor Drive Platform Control System Design Based on an Inverse System Method. *Energies* **2017**, *10*, 1990.
https://doi.org/10.3390/en10121990

**AMA Style**

Qin C, Zhang C, Lu H. H-Shaped Multiple Linear Motor Drive Platform Control System Design Based on an Inverse System Method. *Energies*. 2017; 10(12):1990.
https://doi.org/10.3390/en10121990

**Chicago/Turabian Style**

Qin, Caiyan, Chaoning Zhang, and Haiyan Lu. 2017. "H-Shaped Multiple Linear Motor Drive Platform Control System Design Based on an Inverse System Method" *Energies* 10, no. 12: 1990.
https://doi.org/10.3390/en10121990