# Two-Degrees-of-Freedom PID Control with Kalman Filter for Engraving Machine System

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## Abstract

**:**

## 1. Introduction

## 2. Engraving Machine System

## 3. Design of 2-DOF PID Control

#### 3.1. 2-DOF PID Control Structure

#### 3.2. Design of Disturbance Rejection Controller

- (a)
- Determine $\omega $ according to the expected closed-loop speed.
- (b)
- Gradually increase $K$, and gradually improve the control performance until the speed and disturbance rejection performance satisfy the requirements.
- (c)
- If the system is overshot, the filter $\frac{H(s)}{Q(s)}$ is added to the reference input to suppress the system overshoot.

#### 3.3. Design of the Set Point Tracking Controller

- (a)
- Initialize population parameters:
- (1)
- Population size $M$: Take the number of individuals $M$ in the population according to demand. In general, the larger the population, the more individuals, the better the search ability. However, it may significantly increase the computation burden, and it is generally selected to be 20 to 50.
- (2)
- Individual dimension $n$: Take the individual dimension of the population according to the number of optimized parameters. Since the three parameters ${K}_{P}^{a}$, ${K}_{I}^{a}$ and ${K}_{D}^{a}$ need to be adjusted, the $n$ is taken as $n=3$.
- (3)
- Variation factor $F$: The $F$ determines the population individual differential growth and is used to control population diversity and convergence. The value ranges between 0 and 1. Increasing $F$ can increase the diversity of the population, but it may reduce the convergence speed, and it is easy to jump out of the local extreme value. Reducing $F$ may reduce the difference step size, and it can accelerate the convergence rate such that it will be easier to fall into local optimal values.
- (4)
- Cross factor $CR$: The $CR$ plays a role in balancing global and local search capabilities, and the value is generally between 0 and 1. Increasing $CR$ can improve the diversity of the population and speed up the convergence rate to a certain extent, but too much crossover operation may have too much impact on the population and reduce the convergence rate. However, reducing $CR$ may reduce the diversity of the population, which not only reduces the convergence speed but also may fall into local optimal values. So, in this paper, the $CR$ is taken between 0.3 and 0.6.

- (b)
- Generate the initial population: Randomly generate $M$ individuals satisfying the constraint conditions in a space with dimension $n$. The individual generation mode is

- (c)
- Population variation: Three individuals ${x}_{m1}$, ${x}_{m2}$ and ${x}_{m3}$ are randomly selected from the population, and $i\ne m1\ne m2\ne m3$. The basic variation operation is

- (d)
- Population crossover: The operation of crossover increases the diversity and randomness of the population. The specific operations are

- (e)
- Selection operation: To determine whether ${x}_{i}(j)$ becomes a member of the next generation, compare the fitness function of the crossed vector ${v}_{i}(j)$ and the target vector ${x}_{i}(j)$.

## 4. Kalman Filtering Algorithm

- According to the discrete state space model of the system, the matrix $A$, $B$, $H$ is obtained, and $P(0)$ and $\tilde{x}(0)$ are initialized.
- The covariance matrices $Q$ and $R$ of $w(k)$ and $v(k)$ are set reasonably according to the system characteristics and actual environment.
- Obtain a prior estimate

- 4.
- Update the prior covariance ${P}^{-}(k)$$${P}^{-}(k)=AP(k-1){A}^{T}+BQ{B}^{T}$$
- 5.
- Calculate the Kalman gain $K$$$K={P}^{-}(k){H}^{T}{(H{P}^{-}(k){H}^{T}+R)}^{-1}$$
- 6.
- Compute the best estimate (posterior estimate) $\tilde{x}(k)$$$\tilde{x}(k)={\tilde{x}}^{-}(k)+K(z(k)-H{\tilde{x}}^{-}(k))$$
- 7.
- Update the posterior covariance matrix $P(k)$$$P(k)=(I-KH){P}^{-}(k)$$
- 8.
- Repeat steps 3–7 to achieve the desired target.

## 5. Simulation and Experimental Verification

#### 5.1. Numerical Simulation

#### 5.2. Experimental Example

## 6. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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Parameter Name | Configuration |
---|---|

CPU | Core i5-4210M 2.6 GHz |

RAM | 8 GM |

Operating system | Windows10 64 bit |

Embedded interface board chip | STM32F407ZGT6 168M Hz |

Ethernet communication rate | 100 Mb/s |

CAN communication rate | 1 Mb/s |

Sampling period | 1 ms–5 ms |

AC server | DeltaASDA-A2 |

Permanent magnet synchronous motor | DeltaECMA-C10604RS |

Electronic gear ratio | 1/128 (10,000 pulses/cycles) |

Sensor position accuracy | 5 × 10^{−4} mm |

Range of liabilities for hysteresis | −10 N–10 N |

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## Share and Cite

**MDPI and ACS Style**

Dong, S.; Hao, L.; Shao, Y.; Liu, J.; Han, L.
Two-Degrees-of-Freedom PID Control with Kalman Filter for Engraving Machine System. *Actuators* **2023**, *12*, 399.
https://doi.org/10.3390/act12110399

**AMA Style**

Dong S, Hao L, Shao Y, Liu J, Han L.
Two-Degrees-of-Freedom PID Control with Kalman Filter for Engraving Machine System. *Actuators*. 2023; 12(11):399.
https://doi.org/10.3390/act12110399

**Chicago/Turabian Style**

Dong, Shijian, Leilei Hao, Yiqin Shao, Jun Liu, and Lixin Han.
2023. "Two-Degrees-of-Freedom PID Control with Kalman Filter for Engraving Machine System" *Actuators* 12, no. 11: 399.
https://doi.org/10.3390/act12110399