Nominal parameters of the two-mass resonance model.
Motor drive systems are indispensable for applications in the industrial field. High-speed and high-accuracy control is required for motor drive systems. However, solutions to meet these requirements can cause mechanical resonance vibrations to occur in the system as a result of miniaturization and system weight reduction. It is therefore necessary to model these systems as multi-mass resonance systems with multiple masses and finite rigid shafts, gears, and loads. In addition, vibration suppression control should be applied to these systems. This chapter provides two off-line tuning methods for a digital proportional-integral-derivative (PID)-type controller for a two-mass resonance system to suppress its mechanical resonance vibrations. These methods include a coefficient diagram method and a fictitious reference iterative tuning method. The former method uses a nominal mathematical model of the object while the latter method uses only the initial experimental data without use of the mathematical model. In this chapter, the two methods are compared. A controller is proposed that consists of a modified integral-proportional derivative (I-PD) speed controller and a proportional-integral (PI) current controller, and requires no information about the load side state variables. Finally, the effectiveness of the proposed method is confirmed through computer simulations and experimental results.
- two-mass resonance system
- vibration suppression control
- modified I-PD controller
- coefficient diagram method
- fictitious reference iterative tuning
Motor drive systems are used in a wide range of applications, including industrial robots, home electrical appliances, automobiles, steel rolling mills, computers, and space work surfaces. In general, motor drive systems consist of electric motors, gears, belts, flexible shafts, and mechanical load equipment. Recently, the overall stiffness of these systems has been decreasing because of demands for high-speed and high-accuracy system responses, miniaturization, system weight reduction, and low system costs. Additionally, the system constructions have become more complex and the central processing unit (CPU) processing speeds that are required to perform the system calculations have increased exponentially. Consequently, torsional resonance vibrations occur between the motor and the load side. It is therefore necessary to model the system as a multi-mass resonance system, which is composed of several masses with finite rigid shafts, gears, and loads. In addition, a vibration suppression control method should be applied to the system.
The first-order approximation model of the multi-mass resonance system has the form of a two-mass resonance model. Several control methods are effective for control of a two-mass resonance system [1–5]. PID-type controllers are the most commonly used controllers for industrial applications because of their simplicity and their practicality for use with multi-mass systems. Various PID controller design methods have been proposed; examples of these methods include the limit sensitivity method, the Ziegler and Nichols tuning method, and methods that use the system polynomial. These controller design methods, which are called model-based design methods, may be able to produce the required results in cases where both the system equation and the real system’s parameters are unknown.
In the industrial fields in which many typical motor drive systems are used, experienced technicians often adjust the control system on site to suit the needs of the manufacturing equipment. However, engineer shortages in these fields are becoming a serious problem. It is therefore essential to develop a simple controller design method for industrial applications.
Under these circumstances, and to save both the time required and the cost of tuning the controllers for the motor drive systems, some direct controller tuning methods have been proposed based on the transient response data from closed-loop systems, without modeling of the plant. The fictitious reference iterative tuning (FRIT) method is one of the most promising candidate methods for practical direct parameter tuning [6, 7]. Using the FRIT method, the controller gains can be designed using only single-shot experimental input-output data without knowledge of the model parameters of the object to be controlled.
This chapter introduces two types of approaches to PID-type controller design for suppression of the two-mass resonance system. The first method is based on the assumption that the mathematical model of the object to be controlled is known. This design method is called the coefficient diagram method (CDM) [8–10]. The CDM is an algebraic approach designed to produce the characteristic polynomial directly in the parameter space. The design of the coefficient diagram of the control system is performed using a differential evolution (DE) procedure to obtain the optimal controller gains in a short time [11–13]. The second design method is a FRIT method, which is a PID controller design approach based on one-shot experimental data only and does not use the mathematical model of the object to be controlled [14–16]. The effectiveness of the two proposed design methods is confirmed using a combination of computer simulations and experiments.
2. Two-mass speed control system applying PID-type control
2.1. Description of two-mass resonance model
The model consists of two rigid masses and a torsional shaft. The multiple masses on the load side of the actual system are approximated as one inertial element. Similarly, several shafts and gears are approximated as a single torsional shaft. The typical two-mass model is depicted schematically in Figure 1 below [14–17].
The research in this case deals with a normalized model to provide generality for the design of the proposed control system. Eqs. (5)–(8) show the normalized state equations. The state equation parameters are normalized as shown in Eq. (9), where the suffix
Therefore, the unit for all state variables is [pu]. Figure 2 shows a block diagram of the normalized two-mass resonance model.
Eq. (10) gives the resonance angular frequency, the anti-resonance angular frequency, and the inertia ratio, respectively.
The next equation gives the simplified transfer function for the two-mass mechanical element, in which the input and the output are
The nominal parameters for the two-mass resonance model in this chapter are given in Table 1 below. In this chapter, the proposed CDM method is evaluated through computer simulations, while the proposed FRIT method is evaluated experimentally using the experimental setup shown below.
|2.744 × 10−4 (kgm2)||2.940 × 10−4 (kgm2)|
|18.5 (Nm/rad)||2.884 (Ω)|
|6.676 (mH)||25.0 (V)|
|0.2778 (Nm/A)||0.2778 (V/(rad sec))|
Figure 3 shows a photograph of the experimental system that was constructed in this research. The two-mass resonance system is simulated using the dc servo motor and a dc generator with a finite rigid coupling. The controller is realized using a digital signal processor that calculates the pulse-width modulation (PWM) signal to send to a four-quadrant dc chopper .
The digital signal processor (DSP) board (PE-PRO/F28335 Starter Kit, Myway Plus Corp.), consists of the DSP (TMS320F28335PGFA), a digital input/output (I/O), ABZ counters for the encoder signals, analogue-to-digital (A/D) converters and digital-to-analogue (D/A) converters . The motor and load angles and the angular speeds are detected using 5000 pulses-per-revolution encoders. The dc servo motor current is measured using a current sensor and the A/D converter.
The control period (
2.2. Modified-IPD speed controller and PI current controller
In this chapter, classical PID speed and current controllers are used to suppress the resonance vibrations for the two-mass resonance model. In general, the PI controller, which consists of a proportional controller and an integral controller that are placed in parallel to determine the speed error, is used as the angular speed controller. However, because the resonance system has a complex structure, it is difficult to suppress the vibrations using the classic PI controller alone. Therefore, the I-PD controller is used in this chapter and a first-order lag element is also used to increase the degrees of freedom for the controller design. Additionally, a simple PI controller is used to realize the high-speed torque response for the current minor loop. The continuous control system proposed here is shown in Figure 6. In the figure,
3. Controller design using coefficient diagram method
3.1. Coefficient diagram method
First, this section explains the coefficient diagram method (CDM) for design of the proposed controller, which is required to suppress the resonance vibrations for the two-mass resonance model. The CDM is an algebraic approach that is used to design the characteristic polynomial directly in the parameter space. In the CDM, a coefficient diagram (CD) is used to perform the controller design. The CD provides the ability to analyze the time response, stability and robustness qualities of the controller using a diagram. In the CD, the vertical axis shows the coefficient of the characteristic polynomial (
In the CDM, the stability indices (
The equivalent time constant
In the CDM, use of the standard values of the stability indices is recommended, and these values are listed as follows:
This form is called “the standard form of the stability indices.”
3.2. Design method for the controller gains using the CDM
The CDM is a very simple and effective method for controller design. However, in higher order systems, it is difficult to complete the design by trial and error alone. In this work, the design of the CD for the control system is performed using the differential evolution (DE) method. The DE method is an optimization search method [12, 13].
In the proposed design method, the reference value of the equivalent time constant
The stability indices are then computed using these calculated coefficients and the evaluation function
3.3. Simulation results
Figure 8 shows an example of the simulation results in the form of the angular speed step responses for the control input. The resonance vibrations can be seen in this figure.
Figure 9 shows the frequency response characteristics of the two-mass resonance model from the control input
Table 2 shows the controller gains that were designed using the proposed method, where the reference time constant
|Gain name||Value||Gain name||Value||Gain name||Value|
|2.792 × 102||9.007 × 103||3.522|
Figure 12 shows the simulation results, where the speed reference command input
4. Controller design using fictitious reference iterative tuning (FRIT)
In this section, an off-line tuning method for the vibration suppression-type speed and current controller gains for the two-mass system is proposed based on the FRIT method; this method uses only single-shot experimental input-output data and does not use either the model parameters or the state equation of the two-mass resonance model. While most FRIT designs only use one state variable, this method uses specific multiple state variables to design the controller gains when using the FRIT method [6, 7, 14–16].
Figure 14 shows a typical control system, in which
Initially, as shown in Figure 15, a single-shot experiment is performed using the initial controller gains
The optimal controller gains that are required to achieve
4.2. Vibration suppression controller design method by FRIT
Figure 17 shows a simplified form of the proposed vibration suppression control system, where
Therefore, the fictitious reference signal can be calculated using the following equation without any need for the two-mass resonance model.
The differential evolution method is then used to search for the optimal gains. The performance index function
4.3. Experimental results
Figure 18 shows an example of the experimental results in the form of the angular speed step response of the control input. The resonance vibrations can again be observed in a similar manner to the case of the simulation results shown in Figure 8 above.
Figure 19 shows the gain characteristics of the frequency responses from the experimental results shown in Figure 18, which relate the input voltage to the motor angular speed
Figure 20 shows the experimental results obtained when using the general PI speed and current controller for comparison with the effects of the proposed control system. Figure 21 shows the gain characteristics for the frequency responses shown in Figure 20, which relate
Both the initial rise and the oscillation can be observed in these figures.
Table 3 shows the results for the controller gains determined using the proposed FRIT design method with searching by the DE method, where
|Gain name||Value||Gain name||Value||Gain name||Value|
|1.24||17.0||6.53 × 10−3|
|8.59 × 10−4||2.37||135|
Figure 28 shows the gain characteristics of the frequency responses of the proposed control system, where these characteristics are shown from the perspectives of
This chapter has proposed two design methods for the controller gains required for vibration suppression control in a two-mass resonance system. The proposed controller consists of a modified-IPD speed controller and a PI current controller. The proposed controller design methods are based on application of the coefficient diagram method (CDM) and application of the fictitious reference iterative tuning (FRIT) method. Both methods use the motor side variables only, including the motor’s angular speed and the armature current. The CDM method uses the coefficient of the characteristic polynomial of the control system and can determine the control performance based on the shape of the coefficient diagram and the stability indices. In this chapter, the fitting performances for the standard form of the stability indices and the coefficient values were used to determine the controller gains, which were designed using the differential evolution method. The effectiveness of the proposed CDM was confirmed by the simulation results. The FRIT method can be used to design the controller without knowledge of the model state equations and their parameters. Furthermore, a fictitious signal that was calculated using the initial experimental data for multi-state variables was also proposed in this chapter. The effectiveness of the proposed FRIT method was confirmed using the experimental results. Consequently, the CDM and the FRIT method were shown to produce the same design performance. The CDM is useful for controller design when the mathematical model and the object parameters are known. The FRIT method is effective when the mathematical model is unknown but the initial experimental data can be observed.