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In practical control applications, AC permanent magnet synchronous motors need to work in different response characteristics. In order to meet this demand, a controller which can independently realize the different response characteristics of the motor is designed based on neutrosophic theory and genetic algorithm. According to different response characteristics, neutrosophic membership functions are constructed. Then, combined with the cosine measure theorem and genetic algorithm, the neutrosophic self-tuning PID controller is designed. It can adjust the parameters of the controller according to response requirements. Finally, three kinds of controllers with typical system response characteristics are designed by using Simulink. The effectiveness of the designed controller is verified by simulation results.

Compared with the traditional electric excitation synchronous motor, permanent magnet synchronous motor (PMSM) has the advantages of less loss, high efficiency, and low power consumption. It is excited by a permanent magnet. The structure is simple and the cost is low. The collector ring and brush are omitted, and the reliability is improved. The rotor does not need an excitation current. So, the excitation loss no longer exists. And the efficiency and power factor of the motor are improved. In recent years, the research and application of PMSM have been very popular [

In fact, some parameters of the system are not constants but will change with time, such as manufacturing tolerances, aging of major components, and environmental changes. This will affect the control performance to a certain extent. In order to improve the control performance of the PMSM, Sun et al. [

In order to further solve the problems of uncertainty and inconsistent information, the concept of neutrosophic set was first proposed by Smarandache [

PID algorithm is the most widely used in the field of engineering control. But the tuning of PID parameters is a tedious process. At present, the PID tuning methods studied by scholars include genetic algorithm [

In this paper, the neutrosophic PID tuning control algorithm under different response characteristics is studied. Neutrosophic membership functions are constructed corresponding to different characteristics and the cosine measure theorem and genetic algorithm are used to tune the parameters and get the optimal values.

The paper is organized as follows. In Section

PMSM control system is a high-order, nonlinear, multivariable strong coupling system, so its mathematical model contains time-varying parameters, and the magnetic circuit relationship is also complex.

The section coordinate diagram of PMSM is shown in Figure

Schematic diagram of PMSM section coordinates.

In the coordinate system, the voltage matrix equation of PMSM is as follows:

The voltage equation of AC permanent magnet synchronous motor in the two-phase rotating

The flux linkage equation of PMSM in the

The power of PMSM is equal to the product of phase voltage and phase current of each phase. It can be expressed as follows:

After coordinate transformation, the power is changed in form, but its magnitude is 3/2 of the input power in the

The electromagnetic torque of PMSM not only drives the motor load but also overcomes the friction damping and inertia of the permanent magnet rotor. The torque balance formula is as follows:

By integrating equations (

Since the mathematical model of PMSM is a nonlinear and strong coupling mathematical model, we need to use the vector control method to decouple the mathematical model. And combined with an appropriate control method, the speed control requirements can be achieved. The control method used in this paper is to make

The variable

And each

For convenience,

(see [

Then, the cosine similarity measure between

If

If

The flowchart of the genetic algorithm is shown in Figure

The flowchart of the genetic algorithm.

This paper will adopt the PID control algorithm, which is very effective in engineering. In particular, the incremental PID is only related to the last three errors, which greatly improves the stability of the system. Its specific form is as follows [

The PID parameter self-tuning method is designed based on the neutrosophic theory and genetic algorithm. The self-tuning method needs to consider multiple system response characteristics, that is, a multiobjective programming model problem. In order to comprehensively investigate the advantages and disadvantages of a system, we choose rising time, settling time, peak time, overshoot ratio, undershoot ratio, and steady-state error as the transient characteristics of the control system. According to different response characteristics, neutrosophic membership functions are constructed. Finally, the cosine similarity measure method is used to calculate the measurement value between the transient characteristics and the ideal response characteristics.

The triangular and trapezoidal membership functions are adopted for neutrosophic processing. The six transient characteristics are taken as a whole feature set

Using the neutrosophic theory, the following form is given:

Then, using cosine similarity measure, we can get the similarity of

The rule of PID parameter self-tuning is to minimize

The structure diagram of control system is shown in Figure

Structure diagram of the control system.

The main feature of this control system is that the neutrosophic self-tuning PID control method can realize the control of PMSM with different response characteristics. What users need to do is to give the response characteristics they want by simply adjusting the membership function. Cosine similarity measure method and genetic algorithm in the control system can find the optimal parameters of PID controller automatically, instead of manual adjustment.

In order to facilitate observation in the Simulink module, the main system is first established as shown in Figure

The diagram of the main control system.

The diagram of the control subsystem.

This is a double loop system. The inner loop is the current loop and the outer loop is the velocity loop. The sampling speeds of the angle sensor and speed sensor are set to 1 ms. The speed of the current sensor is faster, set to 0.2 ms. In order to be close to reality, the random white noise is added to the speed sensor with the amplitude of ±0.1 rad/s. The data of the speed sensor are processed by a sliding filter. The average value of five sampling data is taken as the measured value of actual speed. The sampling time of each sensor can be adjusted according to the actual situation.

The two-phase DC to three-phase AC module is shown in Figure

The diagram of two-phase DC to three-phase AC module.

The first module in the two-phase DC to three-phase AC module.

The second module in the two-phase DC to three-phase AC module.

The third module in the two-phase DC to three-phase AC module.

The PMSM used in this simulation is shown in Figure

The PMSM used in simulations.

Simulation parameters of PMSM.

Name | Value |
---|---|

Stator phase resistance | 0.0485 Ω |

Flux linkage | 0.1194 Wb |

0.1 mH | |

0.1 mH | |

Rated speed | 1000 rpm |

Rated power | 100 W |

Moment of inertia | 0.0027 kgm^{2} |

Viscous damping | 0.0624924 F |

Polar logarithm | 1 |

Static friction | 0 |

The simulations are divided into three parts: nonovershooting system, fast response system, and comprehensive response system.

The nonovershooting system expects no overshoot in the response process. The design process of the overshoot ratio neutrosophic membership function should meet the following principles.

The parameters of the true, false, and uncertain membership functions should be selected in the range of small overshoot ratio. The smaller the overshoot requirement is, the larger the true value should be. The larger the overshoot requirement is, the larger the false value should be. The system has relatively low requirements for the other five characteristics. So the neutrosophic membership functions are designed as shown in Figure

Neutrosophic membership function of nonovershooting system.

The desired speed is

Simulation result of the nonovershooting system.

Simulation results of the nonovershooting system.

Name | Value | Name | Value |
---|---|---|---|

Rising time | 0.0007786s | 53.9968 | |

Settling time | 0.0016s | 2402.3231 | |

Overshoot ratio | 0% | 194.1634 | |

Undershoot ratio | 0% | 4275.5304 | |

Peak time | 0.0027s | Optimal value | 0.0951 |

Stead-state error | 0.0010 | Running time | 509.0194s |

The iteration of the group in the simulation is shown in Figure

Population iterative of the nonovershooting system.

The fast response system expects fast response speed. So, the rising time and peak time should be short, and the overshoot and undershoot ratio can be appropriately increased to bring faster speed. The design process of the rising time and peak time neutrosophic membership functions should meet the following principles.

The parameters of the true, false, and uncertain membership functions should be selected in the range of short time. The smaller the response time requirement is, the larger the true value should be. The larger the response time requirement is, the larger the false value should be.

The system has relatively low requirements for the other four characteristics. So, the neutrosophic membership functions are designed as shown in Figure

Neutrosophic membership function of the fast response system.

The desired speed is

Simulation result of the fast response system.

Simulation results of the fast response system.

Name | Value | Name | Value |
---|---|---|---|

Rising time | 0.0006531 s | 205.5004 | |

Settling time | 0.0020 s | 2584.7556 | |

Overshoot ratio | 16.1994% | 51.5292 | |

Undershoot ratio | 8.19% | 8059.1018 | |

Peak time | 0.0013s | Optimal value | 0.1961 |

Stead-state error | 0.0013 | Running time | 707.9218 s |

The iterative result of the population in the simulation is shown in Figure

Population iterative of the fast response system.

The comprehensive response system expects to achieve a balance between response speed and overshoot indexes. The parameters of the overshoot ratio neutrosophic membership function in Figure

Neutrosophic membership function of comprehensive response system.

The desired speed is also

Simulation result of the comprehensive response system.

Simulation results of the comprehensive response system.

Name | Value | Name | Value |
---|---|---|---|

Rising time | 0.0006640 s | 104.1265 | |

Settling time | 0.0015 s | 2693.2360 | |

Overshoot ratio | 8.7895% | 102.5041 | |

Undershoot ratio | 0% | 8310.7028 | |

Peak time | 0.0013s | Optimal value | 0.1741 |

Stead-state error | 0.0017 | Running time | 690.9025 s |

The population iteration diagram is shown in Figure

Population iterative of the comprehensive response system.

In this paper, a neutrosophic self-tuning PID controller is designed for PMSM to match different characteristics. The neutrosophic membership functions of the designed controller can be adjusted according to different response requirements. Then the optimal parameters of the PID controller can be found based on the cosine similarity measure and genetic algorithm. Three kinds of AC permanent magnet motor control systems with different characteristics are designed in simulations. The results show that the designed controller can meet the requirements of different characteristics and has good control accuracy.

It is noted that the determination of the parameters of the six membership functions depends on certain expert experience. The different choices will directly affect the final PMSM control performance. In practice, it is difficult to choose the optimal membership parameters. In future research, the adaptive adjustment method of membership function parameters will be studied.

The data used to support the findings of this study are available from the corresponding author upon request.

The authors declare that there are no conflicts of interest regarding the publication of this paper.