Low-frequency Vibration Suppression Control in a Two-mass System by Using a Torque Feed-forward and Disturbance Torque Observer

Abstract

Given that elastic connection is often used between motor drives and load devices in industrial applications, vibration often occurs at the load side. Vibration suppression is a crucial problem that needs to be addressed to achieve a high-performance servo-control system. Scholars have presented many strategies to suppress vibration. In this study, we propose a method to diminish vibration by using a torque feed-forward and disturbance torque observer. We analyze the system performance and explain the principle of the proposed vibration suppression method based on the transfer functions of the system. The design of controller parameters is another important issue in practical applications. We accordingly provide a succinct outline of the design specifications based on the coefficient diagram method. Furthermore, we build a model under the Simulink environment and conduct experiments to validate the proposed method. Results show that speed and position vibrations are successfully suppressed by the proposed method.

I. INTRODUCTION

In practical industrial applications, shaft couplings are often used in mechanical transmission devices. However, these elastic connections may cause mechanical resonance that severely deteriorates the performance of controllers and induces speed and position vibrations in the control system. Therefore, mechanical resonance in servo drive systems is a crucial problem that needs to be addressed to achieve high-dynamic and high-precision responses for modern servo-control systems.

Many methods are proposed to suppress the vibration in two-inertia systems. A μ-synthesis with a descriptor form representation is described; its use can achieve strong robustness and vibration suppression [1]. A modified speed-loop controller is presented to improve the performance of present speed controllers [2]. Various controller structures are elucidated to reduce torsional vibration, and relative controller parameters are designed according to the inertia ratio and the phase lag in speed loop [3]. A common approximator using a radical basis function network is applied to the speed loop to weaken the vibration [4]. Unlike the traditional disturbance observer, this method features self-tuning capability. Self-adapting fuzzy control can also be utilized to reduce torsional vibration. A neural network is applied to reconcile load speed and torsional torque to decrease the vibration in a two-inertia system [5]. A method based on a sliding-mode fuzzy controller is also applied to suppress the vibration in a two-inertia system [6]. Saarakkala proposed a model-based two-degrees-of-freedom state-space speed-controller design for a two-mass mechanical system [18] and presented a discrete-time polynomial method for parameter identification of two-mass mechanical loads [19]. An immersion and invariance approach is applied to suppress the vibration in two-mass systems [20]. A fuzzy Luenberger observer to ensure that fitting observer closed-loop poles is used for a system with flexible joint [21]. Jong-Sun Ko presented a neural-load torque compensation method to obtain good precision position control [22].

The design of controller parameters is also an important topic. Complex controllers are generally of high order [7], [8]. However, low-order controllers are more practical in industrial applications. The parameters of high-order controllers are difficult to tune. Therefore, a simple and reliable way to design controller parameters will be useful for practical applications.

Traditional and modern control theories for the design of controller parameters are well known. The polynomial method is another approach to finding approximate parameters for controllers. This method is actually an algebraic design method that uses polynomial expressions. According to the order of the transfer functions of a closed-loop control system and its characteristic polynomial, the coefficients of the controller and polynomial can be calculated by some specifications. Naslin studied the relationship between characteristic ratio and system response in the 1940s [9]. Manabe made a further contribution by proposing the coefficient diagram method (CDM) based on Naslin’s theory. CDM successfully solves many controller design problems in practical applications. In this study, we select the CDM to design controller parameters.

This study aims to develop a strategy to suppress the vibration in a two-inertia system and to find a simple and reliable way to design controller parameters. The entire system is expected to have fast response and strong robustness by using the proposed method. Theoretical analysis, simulation, and experimental results are presented to verify the proposed method.

 

II. PRELIMINARIES

Fig. 1 shows the standard single-input single-output block diagram of CDM, where Ac(s), Br(s), and Bf (s) are the controller polynomials; yr, u, d, and n are the input reference, the control, the disturbance, and the measurement noise variables, respectively. The plant to be controlled is represented by G(s) transfer function given by

Fig. 1.CDM block diagram.

The command and feedback transfer functions of the controller Gc(s) and Gf(s) are expressed in the following forms:

In this study, P(s) is the characteristic polynomial of the closed-loop system given by

Given that only three components Ac(s), Br(s), Bf(s) are to be designed, only three transfer functions are needed for design. Based on the CDM, the following equations are selected as standard [10]:

Equ. (4a) determines the command-response characteristics. Equ. (4b) is for the disturbance rejection characteristics. Equ. (4c) is used to check the robustness. Thus, the three basic equations are taken as system performance specifications in the CDM design. The design of P(s) can satisfy the requirements on Equs. (4b) and (4c), and Br(s) is tuned to satisfy the specification on Equ. (4a).

We explain the CDM design procedure as follows. Some mathematical relations are first described. The general form of P(s) is given by

The CDM has two design parameters with respect to characteristic polynomial coefficients; these parameters are the equivalent time constant τ and the stability index γi. The relations among τ, γi, and ai are given by

[11] stated that “the system of any order can stay stable if all the 4th order polynomials are stable with the margin of 1.12.” Hence, the design guides for stability are obtained as follows:

In the CDM, the stability indices are recommended as follows [12], [13]:

We then design Ac(s) and Bf(s) based on Equ. (8). The denominator Ap(s) and the numerator Bp(s) polynomials can be denoted as

Controller polynomials are given by

Equ. (3) can be rewritten as

where ai, bi, and ci parameters are known. Thus, li and ki can be calculated. Given that Br(s) is adjusted to satisfy command-response characteristics, we can design it based on practical conditions, as in section IV. B.

 

III. MATHEMATICAL MODEL OF A TWO-INERTIA SYSTEM

Fig. 2(a) shows the diagram of a two-inertia system. Fig. 2(b) describes the classical speed-loop-control diagram of a two-inertia system. The definitions of the corresponding parameters are as follows:

Fig. 2.(a) Mechanical model diagram of a two-inertia system. (b) Classical speed-control diagram of a two-mass system.

The state equations of the two-inertia system can be written as follows:

The damping losses assumed to be small can be neglected without affecting the precision of the following analysis significantly. The transfer functions HLM(s), HMM(s), and HRM(s) are as follows:

where Δ(s) is the characteristic polynomial given by

With further simplification of the above equations, HLM(s) and HMM(s) can be rewritten as

where ωres and ωant are the resonance frequency and the anti-resonance frequency, respectively, defined as

Based on Equs. (16) and (17), the increment in the motor inertia decreases the resonance frequency, whereas it has no influence on the anti-resonance frequency. The system performance is seriously damped at the anti-resonance frequency and has a resonance peak at the resonance frequency. The details from the frequency response of a two-inertia system are presented in Fig. 3. The high gain at the resonance point essentially causes position and speed vibrations. Hence, a method to offset the resonance peak can suppress vibrations.

Fig. 3.Response frequency of a two-inertia system.

 

IV. VIBRATION SUPPRESSION DESIGN

In this section, a method based on a torque feed-forward and disturbance torque observer to suppress the low-frequency vibrations in a two-mass system is proposed. The schematic of the proposed method is shown in Fig. 4. The torque feed-forward compensation is derived from the speed reference, and the disturbance torque observer adopts a low-order observer to estimate the disturbance torque.

Fig. 4.Diagram of the proposed control strategy for a two-inertia system.

The following sections describe the proposed method and the design of controller parameters in detail.

A. Disturbance Observer

Disturbance observer is applied to obtain a robust control system. In this study, we apply a low-order Luenberger observer to estimate the disturbance torque. For simplicity, Fig. 5 shows the principle of the proposed disturbance compensation method. According to Fig. 5, we can obtain the transfer function between TL and ωm as Equ. (18). Compared with the transfer function under the traditional proportional–integral (PI) control, which is denoted by Equ. (19), Equ. (18) equals Equ. (19) multiplied by s2/(2ks + k2 − s2). s2/(2ks + k2 − s2) has high-pass filter features. Thus, disturbance torque is seriously damped under the frequency k(rad/s). Similarly, the influence of measurement noise on the disturbance observer is damped by a low-pass filter factor 1 - Jm (2ks + k2)s/[KT(s + k)2(kps + ki)] compared with PI control, that is, Equs. (20) and (21). Therefore, the effect of high-frequency noise can be diminished.

Fig. 5.Disturbance observer.

According to Equs. (18)-(20), a large k is helpful for disturbance rejection, whereas a small k can improve the capability of measurement noise rejection. Accordingly, k is determined by considering both Equs. (18) and (20). When the system cut-off frequency is near we recommend that k be not larger than

B. Parameter Design of Feed-forward Compensator

Calculating torque feed-forward compensation by a classical derivative is usually used. However, classical differentiation is sensitive to noise, thereby limiting its practical application. In this study, we apply Equ. (22) to obtain the torque feed-forward value. We assume that the cut-off frequency is ωcf and the gain is KG .

Fig. 6 shows the frequency response of Equ. (22). From Fig. 6, the response characteristics of Equ. (22) are compatible to the process signals that contain a certain frequency.

Fig. 6.Bode of Equ. (22) when KG = 2ωcff,ωcf = 62.8rad/s.

The feed-forward compensator procedure is explained hereafter. For simplicity, the transfer function of current loop equals 1 because of the high bandwidth of the current loop, and we assume that DM = DL = 0. The block diagram of transfer functions is shown in Fig. 7. Given the design of the disturbance observer in section A, we neglect this part to reduce computational effort.

Fig. 7.Control structure of a two-mass system.

From Fig. 7, the motor speed, the torsional torque, and the electrical torque are represented as

The transfer functions from the reference speed to the motor speed, from the motor speed to the load torque, the electrical torque to the reference speed, and the torsional torque to the reference speed are described as

Thus, the characteristic polynomial can be derived from the above equations.

In this section, the CDM that is previously discussed is applied to design control parameters.

According to the discussion in [17], γi ≥ 2 is defined for all i = 1,2,...n-1, the step response overshoot is ensured to be small, and a large γi leads to great damping. The conclusion can be applied to realize the damping at the resonant frequency by setting an appropriate γi. Compared with all-pole systems, two-inertia systems require a particularly large γ1 because the characteristic ratios with a low index i have an advantageous effect on frequency-domain specifications. Thus, according to the CDM report, γ1 = 2.5, γn-1 = γn-2 = ⋯ γ2 = 2 is recommended considering rapid response and robustness. The elimination of Equ. (24) by Equ. (6b) is thus expressed as

The basic idea of the design of Gcf (s) is to offset the resonance peak of the command response at the resonance point. According to Fig. 7, we can derive that the transfer function between ωc and ωL relates to Equ. (26a).

To decrease the gain at the resonance point, the magnitude of the numerator of Equ. (26a) should equal the magnitude of the denominator as much as possible, that is, Equ. (26b). Therefore, Equ. (27) should be satisfied.

When Kp and KI are known, ωcf and KG can be calculated based on Equ. (27). In this study, we use the simulation parameters in Table I to verify the availability of the design specifications, that is, Equs. (25) and (27). According to Table I, ωant = 128rad / s, ωres = 141rad / s. The controller parameters calculated by Equs. (25)–(27) are shown in Table II. The elimination of Equ. (26) by the parameters in Tables I and II derives the bode diagrams of the command response and the disturbance torque response, as shown in Fig. 8. Fig. 8(a) shows the speed frequency response difference between the proposed method and the classical PI control. On the one hand, the command response gain of the classical PI controller (+15 db) is higher than the gain of the proposed method (−9 db) at the resonance frequency. On the other hand, the magnitude of the disturbance frequency response with the proposed method (−12 db) is smaller than that (−3 db) with the classical PI control at the resonance point.

TABLE ISIMULATION PARAMETERS

TABLE IIDESIGNED PARAMETERS

Fig. 8.(a) Bode diagram of the speed frequency response. (b) Bode diagram of the disturbance frequency response.

In conclusion, the bode diagrams of the command response and the disturbance response imply the principle of vibration suppression of the proposed method.

 

V. SIMULATION AND EXPERIMENTAL RESULTS

A. Simulation

Simulations of the PI controller, the PI + disturbance observer controller, and the proposed method controller are conducted. The simulation parameters are the same as those in Table I.

Figs. 9(a)-9(d) show the load speed response, the torsional torque response, the disturbance response, and the position tracking response with three different control systems, respectively. From the curves in Fig. 9(a), a large overshoot (55%) can be found with the PI controller. The vibration in speed lasts for 4 cycles to 5 cycles. The disturbance observer can reduce the vibration (3 cycles) and overshoot (45%), but its performance is not as good as that of the approach proposed in this study. The overshoot is only 12%, and the vibration is almost completely suppressed with the proposed method. The results in Fig. 9(b) indicate that torsional torque is remarkably suppressed by the proposed method, whereas vibration remains for a long time both in the PI controller and the PI + disturbance observer. Fig. 9(c) shows that the disturbance rejection with the PI controller is not as good as those with the other two ways. This result is consistent with the bode diagram in Fig. 8(b). The proposed method has the same disturbance response as the PI + disturbance torque observer because the command torque compensator has no effect on the disturbance rejection capability. Fig. 9(d) describes that the proposed method performs best. The proposed method effectively suppresses vibration and shortens the positioning time in contrast to the PI controller and the PI + disturbance torque observer.

Fig. 9.(a) Speed response under three different control systems. (b) Torsional torque under three different control systems. (c) Disturbance response under three different control systems. (d) Position tracking under three different control systems.

B. Experimental Results

A general picture of the proposed control algorithm is illustrated in Fig. 10 to validate the proposed method. The experimental platform is set up as in Fig. 11. The platform consists of a permanent-magnet synchronous motor (PMSM), a servo driver, a ball screw, and a coupling device, among others. The PMSM parameters are shown in Table III.

Fig. 10.Diagram of the software control unit.

Fig. 11.Experimental platform.

TABLE IIIPMSMPARAMETERS

The software algorithm of the drive system is designed based on STM32f407. The three different control systems in section V. A are tested for further comparison. The command speed is given in Fig. 12.

Fig. 12.Command speed wave.

When the motor outputs the positioning signal, the sampling of command torque, motor feedback speed, and position error begins. Results are shown in Figs. 13-17. The three signals from up to down are command torque, motor feedback speed, and position error, respectively. From Fig. 13, the vibrations in position, torque, and speed last for a long time (t > 125 ms) with the PI controller. Fig. 14 shows that the disturbance observer suppresses the vibration to a certain extent, but its performance is not as promising as the proposed method in Fig. 15. The vibration with the proposed method in Fig. 15 is nearly suppressed, and the positioning time is only t ≈ 69 ms. These experimental results have the same tendency as the simulation results. Disturbance rejection experiments are also performed. We apply inconsiderable disturbance torque on the load side suddenly after the fixing position. The experimental results are described in Figs. 16 and 17. Fig. 16 shows that, with the PI controller, the torque, speed, and position vibrate because of the disturbance. By contrast, the corresponding signals rapidly return to the stable value with the proposed method in Fig. 17. This result is in accordance with the simulation results in Fig. 9(c). In conclusion, the proposed method performs better than the PI controller in command response and has better disturbance rejection capability.

Fig. 13.Results of the PI controller.

Fig. 14.Results of the disturbance torque observer.

Fig. 15.Results of the proposed method.

Fig. 16.Disturbance experimental results of the PI controller.

Fig. 17.Disturbance experimental results of the proposed method.

 

VI. CONCLUSION

Vibration suppression is crucial for industrial applications. This paper presents a vibration suppression strategy based on torque feed-forward compensation and disturbance observer for a two-mass system. The control parameters KP, KI and ωcf, KG are designed by using a few system parameters (Jm, JL, Ks) based on the CDM. A low-order Luenberger disturbance observer is applied to improve the disturbance rejection capability. The parameters of the disturbance observer are designed by analyzing the transfer function between motor speed and disturbance torque. We also analyze the transfer functions between load speed and speed command under both PI control and the proposed method. The bode diagrams explain how the proposed method can suppress vibration in theory. The proposed method offsets the resonance peak. Hence, the system should be stable at the resonance point. The simulation analysis and experimental results verify the good performance of the proposed method.