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Model of ultrasonic motor is the foundation of the design of ultrasonic motor's speed and position controller. A two-input and one-output dynamic Takagi-Sugeno model of ultrasonic motor driving system is worked out using fuzzy reasoning modeling method in this paper. Many fuzzy reasoning modeling methods are sensitive to the initial values and easy to fall into local minimum, and have a large amount of calculation. In order to overcome these defects, equalized universe method is used in this paper to get clusters centers and obtain fuzzy clustering membership functions, and then, the unknown parameters of the conclusions of fuzzy rules are identified using least-square method. Different experimental data that are tested with different operational conditions are used to examine the validity of the fuzzy model. Comparison between experimental data and calculated data of the model indicates that the model can well describe the nonlinear characteristics among the frequency, amplitude of driving voltage and rotating speed. The proposed fuzzy model can be used to analyze the performance of ultrasonic motor driving system, and also can be used to design the speed and position controller of ultrasonic motor.

The model of ultrasonic motor system is the foundation of analyzing ultrasonic motor’s performance and is also an important premise of designing motor controller and trying to improve the control performance. The energy conversion processes of ultrasonic motor have several stages [

Modeling of ultrasonic motor can adopt different methods. The equivalent circuit of ultrasonic motor is a kind of model that can be used to analyze the basic characteristics of ultrasonic motors [

In recent years, fuzzy modeling method based on fuzzy reasoning is gradually arisen. The same as in the neural network model, fuzzy model is also based on experimental data, easy to show the nonlinear information. People’s fuzzy thinking process is simulated by fuzzy reasoning. So, fuzzy model is more easily mixed and makes use of people’s relevant experience knowledge. This is different from other methods. The complexity of fuzzy model is relatively low. So, another effective way is provided for the modeling of nonlinear complex system. The fuzzy modeling method is rarely used in the field of motor. The fuzzy method is mostly used to realize rotating speed and position control [

This paper works out the appropriate fuzzy modeling method to obtain the dynamic model of ultrasonic motor system. Equalized universe method is used to get clusters centers and obtain fuzzy clustering membership functions. And then least-square-method is used to identify the unknown parameters of the conclusions of fuzzy rules. The two-input and one-output Takagi-Sugeno model of ultrasonic motor system is established, and the model can well show the nonlinear dynamic relationship among the amplitude of driving voltage, frequency, and rotating speed.

The block diagram of experimental system is shown in Figure

Structure of the experimental system for speed control.

Supply voltage of the circuit is DC12V. Phase-shift PWM method is used in the circuit to control the H-bridge driver as shown in Figure _{d} is the capacitance due to the piezoelectric element's dielectric properties called the “tank capacitance.”

Structure of the driving and control circuit.

Equivalent circuit of ultrasonic motor’s stator.

PWM1~PWM8 are the PWM control signals for

Specification of USR60 motor.

Item | Value |
---|---|

Driving voltage | 130 Vrms |

Rated torque | 0.5 Nm |

Rated output power | 5.0 W |

Rated rotating speed | 100 r/min |

Max. torque | 1.0 Nm |

Temperature range | −10 ~ + 55°C |

Weight | 260 g |

The controllable variables that can be used for rotating speed control are the amplitude of driving voltage, frequency, and phase difference of the two-phase voltages. The theoretical analyses and related experimental studies show that if the phase difference of two-phase voltages is set to ±90°, better operation performance can be maintained and the energy conversion efficiency of ultrasonic motor is higher. In this paper, the phase difference of two-phase voltages is set to ±90°. The remainder controllable variables, amplitude and frequency of driving voltage can be adjusted to realize the aim of speed control. Therefore, the dynamic data, which contain the amplitude of driving voltage, frequency, and speed, should be measured in the experiments. A dynamic fuzzy model, which uses the amplitude of driving voltage and frequency as input and rotating speed as output, is established.

Experiment process is designed as follows. The step response of rotating speed is measured by setting

Tested data of speed control.

The basic structure of fuzzy model is shown in Figure _{i}

Basic structure of fuzzy model.

According to the tested input and output data, structure identification and parameter identification are required in establishing the T-S model of ultrasonic motor system. Determining the appropriate model structure is the basis of parameter identification. The first step of structure identification is to confirm the input variables and the input space. This paper adopts equalized universe method to divide the fuzzy input space. First, set a clustering number which is the number of membership functions and rules. Secondly, divide the input space. Equalized universe method is used to determine the characters of Gaussian membership function, such as value of the center and width. The structure identification of T-S model is completed. The unknown parameters of conclusion section of T-S model can be determined using the least square method. After the model is established, the validation calculation should be carried on. If the error of model output is too large, the number of rules can be increased to improve the precision.

The steps of this fuzzy modeling method can be summarized as follows.

Determine the input variables, the number

According to the analysis of tested data, determine the corresponding domain space

Determine the centers

Calculate the distance

Calculate the membership degree of modeling data

The unknown parameters of conclusion section of T-S model are obtained by the least-square method.

If the precision of model is not satisfactory after validation calculation, change the number

Dynamic fuzzy model reflects the dynamic characteristics. The characteristics of ultrasonic motor system for speed control are related with time. Output variable of the model is the rotating speed value

The 9 groups of data which are obtained through the above experiments are used for the fuzzy modeling. Among them, the 7 groups are used to model, and the other 2 groups are used as validation data. By studying the tested data, the domain space of

In accordance with the above modeling steps, the process of T-S dynamic fuzzy model is an optimization process. According to the experience and tested data analysis, the value of rotating speed

Membership functions of

After parameter identification, the five fuzzy rules are shown in Table _{i}_{i}_{i}_{i}

Fuzzy rules.

_{i} | _{1} | _{2} | _{3} | _{4} | _{5} | |

Premise part | _{1} | _{2} | _{3} | _{4} | _{5} | |

_{1} | _{2} | _{3} | _{4} | _{5} | ||

_{1} | _{2} | _{3} | _{4} | _{5} | ||

_{1} | _{2} | _{3} | _{4} | _{5} | ||

_{1} | _{2} | _{3} | _{4} | _{5} | ||

Conclusion part | _{0} | 654.5 | 386.7 | 187.0 | 7.896 | |

_{1} | 0.0062 | |||||

_{2} | 0.0535 | 0.1561 | 0.0522 | 0.1099 | ||

_{3} | 38.94 | 29.29 | 5.856 | 9.296 | ||

_{4} | 0.5271 | |||||

_{5} | 0.7883 | 0.9124 | 0.9308 | 1.011 | 0.9904 |

The comparisons of model output data and tested data are shown in Figures

Comparison between model output and tested data (

Comparison between model output and tested data (

The precision of the model is not high enough as shown in Figures

Final fuzzy rules.

_{i} | _{1} | _{2} | _{3} | _{4} | _{5} | _{6} | _{7} | _{8} | _{9} | _{10} | |
---|---|---|---|---|---|---|---|---|---|---|---|

Premise part | _{1} | _{2} | _{3} | _{4} | _{5} | _{6} | _{7} | _{8} | _{9} | _{10} | |

_{1} | _{2} | _{3} | _{4} | _{5} | _{6} | _{7} | _{8} | _{9} | _{10} | ||

_{1} | _{2} | _{3} | _{4} | _{5} | _{6} | _{7} | _{8} | _{9} | _{10} | ||

_{1} | _{2} | _{3} | _{4} | _{5} | _{6} | _{7} | _{8} | _{9} | _{10} | ||

_{1} | _{2} | _{3} | _{4} | _{5} | _{6} | _{7} | _{8} | _{9} | _{10} | ||

_{1} | _{2} | _{3} | _{4} | _{5} | _{6} | _{7} | _{8} | _{9} | _{10} | ||

_{1} | _{2} | _{3} | _{4} | _{5} | _{6} | _{7} | _{8} | _{9} | _{10} | ||

_{1} | _{2} | _{3} | _{4} | _{5} | _{6} | _{7} | _{8} | _{9} | _{10} | ||

_{1} | _{2} | _{3} | _{4} | _{5} | _{6} | _{7} | _{8} | _{9} | _{10} | ||

_{1} | _{2} | _{3} | _{4} | _{5} | _{6} | _{7} | _{8} | _{9} | _{10} | ||

Conclusion part | _{0} | 180.9 | 1249 | 145.2 | 255.8 | 169.2 | 130.1 | 13.5 | 30.78 | 59.04 | 38.81 |

_{1} | 0.0937 | 0.2 | 0.1259 | 0.4378 | 0.3607 | 0.1464 | 0.2738 | 0.8669 | 1.252 | ||

_{2} | |||||||||||

_{3} | 0.6663 | 0.007595 | 0.7137 | 0.1353 | 1.266 | 1.416 | 0.6314 | 1.586 | 1.617 | 7.234 | |

_{4} | 0.08016 | 0.0179 | |||||||||

_{5} | 32.89 | 37.2 | 155.9 | 49.56 | 88.89 | 18.67 | 0.8444 | 22.93 | 134.8 | 22.62 | |

_{6} | 44.22 | ||||||||||

_{7} | 173.1 | 275.5 | 53.49 | 51.14 | |||||||

_{8} | 65.63 | 63.41 | 26.74 | 91.09 | 28.54 | ||||||

_{9} | 0.153 | 0.428 | 0.4712 | 0.0339 | 0.8306 | 0.3826 | 0.1745 | ||||

_{10} | 0.7901 | 0.1586 | 0.486 | 1.49 | 1.079 | 0.9205 | 0.1608 | 0.6079 | 0.807 | 1.076 |

Membership functions of

The model output data are given in Figures

Step response comparison between model output and tested data (

The curve of drive voltage and frequency

Comparison between model output and tested data

Step response comparison between model output and tested data (

The curve of drive voltage and frequency

Comparison between model output and tested data

Step response comparison between model output and tested data (

The curve of drive voltage and frequency

Comparison between model output and tested data

Step response comparison between model output and tested data (

The curve of drive voltage and frequency

Comparison between model output and tested data

Step response comparison between model output and tested data (

The curve of drive voltage and frequency

Comparison between model output and tested data

Slope response comparison between model output and tested data (

The curve of drive voltage and frequency

Comparison between model output and tested data

Slope response comparison between model output and tested data (

The curve of drive voltage and frequency

Comparison between model output and tested data

Based on the experimental data, the two-input and one-output dynamic model for speed control of ultrasonic motor is worked out using dynamic fuzzy modeling method. The model structure and parameters are identified using equalized universe method and the least-square method, respectively. Using the proposed modeling methods, a few fuzzy rules are needed. The structure of model is simpler. The amount of online calculation is smaller. Comparison between model output and tested data shows that the model can well simulate the nonlinear relationship among the amplitude of driving voltage, frequency, and rotating speed.

This work shows that the fuzzy modeling method is simple. Besides the tested data, membership functions and fuzzy rules can be obtained without other prior knowledge. Fuzzy model has a strong ability of approximating to the nonlinear characteristics and is suitable for ultrasonic motor which is a strong nonlinear system.