A Hybrid-Type Active Vibration Isolation System Using ... - Science Direct

Journal of Sound and Vibration (1996) 192(4), 793–805

A HYBRID-TYPE ACTIVE VIBRATION ISOLATION SYSTEM USING NEURAL NETWORKS K. G. A Department of Mechanical Engineering, Pohang University of Science and Technology, Pohang 790-784 , Korea

H. J. P Department of Mechanical Design and Production Engineering, Seoul National University, Seoul, Korea

M. Y. J Department of Industrial Engineering

 D. W. C Department of Mechanical Engineering, Pohang University of Science and Technology, Pohang 790-784 , Korea (Received 9 August 1994, and in final form 4 September 1995) Vibration isolation of mechanical systems is achieved through either passive or active vibration control systems. Although a passive vibration isolation system offers simple and reliable means to protect mechanical systems from a vibration environment, it has inherent performance limitations, that is, its controllable frequency range is limited and the shape of its transmissibility does not change. Recently, in some applications, such as active suspensions or precise vibration systems, active vibration isolation systems have been employed to overcome the limitations of the passive systems. In this paper, a hybrid-type active vibration isolation system that uses electromagnetic and pneumatic force is developed, and a new control algorithm adopting neural networks is proposed. The characteristics of the hybrid system proposed in the paper were investigated via computer simulation and experiments. It was shown that the transmissibility of the vibration isolation system could be kept below 0·63 over the entire frequency range, including the resonance frequency. 7 1996 Academic Press Limited

1. INTRODUCTION

Vibration isolation of mechanical systems is crucial to higher productivity, accuracy in precision measurement, and the improvement of comfort and safety—for example, in cars. Vibration isolation systems may be categorized as active or passive, depending on whether or not external power is required for the isolator to perform the function [1]. Although passive vibration isolators offer simple and reliable means to protect mechanical systems from a vibration environment, they have well-known inherent performance limitations that cannot be overcome. On the other hand, active vibration isolation systems, with parameters that change according to excitation and response characteristics of the system (by using active elements that operate with pneumatic, hydraulic or magnetic force and so on), can provide significantly enhanced vibration isolation performance [2]. 793 0022–460X/96/190793 + 13 $18.00/0

7 1996 Academic Press Limited

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Schubert [1] constructed a 1-DOF active vibration isolation system by employing hydraulic equipment, and established the fundamentals of a hydraulic isolation system through theoretical analysis and experiments. Although this method produces a decrease in transmissibility at the desired frequency, it suffers from the increase of transmissibility just before that frequency. Hong Su [2] proposed several active control schemes. They are based upon absolute displacement, velocity and relative response variables, and deal with the dynamics of an electromagnetic actuator applied to a 1-DOF active isolation system via computer simulation. Bozich [3] developed a vibration isolation system based on neural network control architecture. However, no results were presented at low frequency which, in reality, can have a significant influence on the system; that is, experiments were conducted only in a frequency range higher than 20 Hz. In general, passive vibration isolators have low pass filter properties and active ones have high pass filter properties. Therefore when constructing an active isolation system using an actuator, it is possible to provide an effective system by adding an actuator to a passive isolator. In this way, one can enhance the vibration isolation performance by simultaneously using an air spring as a passive element and an electromagnetic actuator as an active element. Closed loop control of physical systems compares the current state of the system to the goal state and gives commands that will cause the system to move toward the goal state. The controller must therefore know the properties of the system for optimal control. However, there are a number of situations in which this cannot be done; no model of the system may be available, or the model may be so complex that it is impractical to use it for control purposes, or the system may change over time, or the controller itself may change over time. In these situations, the vibration isolation system to be described in this paper requires an adaptive controller that learns to control the system during operation. Neural networks provide a fast method for autonomously learning to produce a set of output states, given a set of input states. Therefore, in the work described in this paper, first, a l-DOF active vibration isolation system was constructed employing air springs as passive vibration isolation elements and an electromagnetic actuator as an active element. Second, a theoretical system modelling was obtained under the linear system assumption, and then simulations were performed based on it. Third, a neural network control architecture including a linear controller, which can autonomously learn to control the incompletely modelled system, was developed. Finally, experiments were conducted to show the characteristics of the proposed active vibration isolation system. 2. MODELLING OF THE VIBRATION ISOLATION SYSTEM

In this section, each component of the system is modelled through linearization near the reference position under the small variation assumption in order to perform simulation of the linear controller, which is a part of the proposed control algorithm. As shown in Figure 1, the vibration isolation system is comprised of a mass, a base, a reservoir, and one active and three passive vibration isolation elements. The equation of motion for the mass can be written as Mx¨1 + fp + fa = 0,

(1)

where x1 is the displacement of the mass M, and fp and fa are the forces exerted through passive and active elements respectively. First, the passive term can be written as [4] fp = C0 y˙ + K0 y + (PR − PS )AReq ,

(2)

    

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Figure 1. A schematic diagram of the vibration isolation system.

where C0 is the damping coefficient of the passive vibration isolation system, y = x1 − x0 is the relative displacement, x0 is the displacement of the base, PR is the reference pressure in the air spring and the surge tank in an equilibrium state, PS is the pressure in the air spring, AR is the area of the air spring at the reference pressure, and AReq = 3AR is the equivalent area of the air spring at the reference pressure. Assuming that the pressure in the air spring is equal to the pressure in the surge tank, i.e., P = PS = PT ,

(3)

where PT is the pressure in the surge tank, and substituting equation (3) in equation (2) yields fp = C0 y˙ + K0 y + (PR − P)AReq .

(4)

With the assumption of an adiabatic process, the state equation inside the air spring is P = PR (VR /V)m = PR {VR /(VR + AVeqy)}m,

(5)

where VR is the volume of the air spring and surge tank under standard pressure, AV is the area of the air spring, AVeq = 3 AV is the equivalent area of the air spring and m is the ratio of specific heats. Substituting equation (5) in equation (4) yields fp = C0 y˙ + K0 y + PR [1 − {VR /(VR + AVeq y)}m]AReq .

(6)

This equation can be linearized around the reference position upon assuming small variation, and, since the relative displacement at the reference position is zero, it becomes; fp = C0 y˙ + (K0 + PR AReq mAVeq /VR )y,

(7)

fp = C0 y˙ + Keq y

(8)

Keq = K0 + PR AReq mAVeq /VR .

(9)

or

where

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Taking the Laplace transform of equation (8) gives fp (s) = (C0 s + Keq )y(s),

(10)

Second, the active control force fa is proportional to the product of armature current ia , the magnetic flux strength of the field, and the gap between the permanent magnet and the electromagnet. The relationship has strongly non-linear properties, but upon assuming that the system works at small variation and at low frequency range below the cut-off frequency of the actuator, fa can be represented as fa = KE ia + Ky y,

(11)

where KE is the current stiffness defined as the varying rate of the actuator force to the armature current change in the electromagnet when the gap between the permanent magnet and the electromagnet is constant, and Ky is the displacement stiffness defined as the varying rate of the actuator force according to the variation of the relative displacement between the mass and the base when the current in the electromagnet is constant. Taking the Laplace transform of equation (11) yields; fa (s) = KE ia (s) + Ky y(s).

(12)

The armature current ia is a function of the control voltage v(t), armature resistance R, armature inductance L, and counter emf e(ia , y˙ ) [2]: L dia (t)/dt + Ria (t) + e(ia , y˙ ) = v(t).

(13)

Assuming the influence of eddy currents due to the counter emf to be negligible [2], one can neglect the dynamic characteristics of the amplifier and hence obtain fa (s) = {KE /(Ls + R)}v(s) + Ky y(s).

(14)

Substituting equations (10) and (14) into the Laplace transform of equation (1) yields the transfer function that represents the relationship between the mass displacement as an output and base displacement with control voltage as input: x1 (s) =

C0 s + K 1 x (s) + v(s). Ms 2 + C0 s + K 0 Ms 2 + C0 s + K

(15)

Here K = Keq + Ky .

(16)

Also one can obtain the transfer function for acceleration from equation (15): x¨1 (s) =

C0 s + K 1 x¨ (s) + v(s). Ms 2 + C0 s + K 0 Ms 2 + C0 s + K

(17)

Here K = Keq + Ky = K0 + PR AReq m(AVeq /VR ) + Ky .

(18)

3. CONTROL ALGORITHM

Error back-propagation, which is to be used here, is currently the most widely used architecture in neural network applications. The network is capable of arbitrarily accurate approximation to arbitrary mapping with a sufficiently high number of hidden units. The architecture shown in Figure 2 provides a method for training the neural network controller that minimizes the overall error.

    

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Figure 2. The structure of the controller.

The desired output of the system for the present case is X1d = X 1d = X1d = X 1d = 0. According to the difference between these values and those measured on the system mass and base (e in Figure 2), the linear controller generates an output (VL ) which is fed into the power amplifier that operates the magnetic actuator. At the same time, the neural controller also generates an output (VN ) that goes into the power amplifier along with VL by inputting the state values (X0 , X 0 , X0 , X1 , X 1 , X1 , X 1 ) to the proposed back-propagation neural network. These values are obtained through differentiation or integration of the measured acceleration signals. The neural controller uses the already stored data as input to the neural network as well as the newly imported data measured on the mass and base. By doing so, not only sign and magnitude but also time history can be constructed in order to provide relevant information about the signal pattern. A more specific explanation is as follows. Every time new data comes in, the previous data is pushed to another place for storage. When the total number of data becomes N, the oldest data is removed by the new data being imported. Thus the time history of the vibration pattern is progressively constructed by simultaneously using the past data as well as the current data as input to the neural network. The linear controller output (VL ) is calculated from the measured values on the mass and base (X0 , X 0 , X0 , X1 , X 1 , X1 , X 1 ). This output (VL ) serves as an error that trains the neural network; that is, updates the weights by back-propagation. This process is repeated until the error falls within the desired error boundary. The iteration is performed only once at each sampling instant so that the best weight values can be obtained through on-line learning. The simultaneous use of the linear and neural controllers prevents, to a

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considerable extent, the instability that can be caused by the unstable neural controller output at the initial learning stage. After learning is completed, most of the controller output (V) is generated by the neural controller. A sudden disturbance reactivates the linear controller, which then takes over the control process until the neural controller learns the new environment again. In general, back-propagation is known to be supervised learning algorithm because a target value should be given beforehand. However, in the sense that the proposed algorithm does not need to be given a target value for training, it can be considered as a semi-unsupervised learning algorithm. This algorithm needs short computational time because of the simple learning algorithm, and can overcome the instability of the linear controller caused from the frequency variation of the disturbance. It is also adaptable to the variations of the system parameters. The general back-propagation learning rule is described in detail in numerous references [5–7].

4. SIMULATION OF THE VIBRATION ISOLATION SYSTEM CHARACTERISTICS

Simulation was performed with the models derived by the methods described in section 2, in order to investigate the system behavior according to various control gains. 4.1.      When the displacement, velocity, acceleration and jerk of the mass are used for feedback control, the control input voltage to the power amplifier of the actuator is expressed as v(s) = −(Kp + Kv s + Ka s 2 + Kj s 3 )x1 (s).

(19)

Substituting equation (19) into equation (15) yields the transfer function x1 (s) = C0 Ls 2 + (C0 R + LK)s + KR x (s). (LM + KE Kj )s 3 + (MR + C0 L + KE Ka )s 2 + (C0 R + LK + KE Kv )s + KR + KE Kp 0 (20) Equation (20) shows that the denominator of the closed loop transfer function changes according to the feedback of the mass signal, which makes it possible to control the resonance frequency and transmissibility at a resonance frequency by using the mass signal feedback control. After substituting the feedback gains into equation (20), the transmissibility curve was plotted when each feedback gain was varied respectively. Figure 3(a) shows a trend that an increase in the acceleration feedback gain lowers the resonance frequency, while in Figure 3(b) it is shown that an increase in the velocity feedback gain brings down the transmissibility at the resonance frequency until the transmissibility becomes lower than unity over the entire frequency range. Little change occurs with increasing the displacement feedback gain, as shown in Figure 3(c). It is shown in Figure 3(d) that, with an increase in the jerk feedback gain, the resonance frequency, as well as the transmissibility at that frequency, decreases and finally the transmissibility becomes lower than unity over the entire frequency range.

    

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Figure 3. The transmissibility curve for mass signals feedback control. (a) Mass acceleration feedback: ——, Ka = 0; · · · · · · · , Ka = 5; – – – –, Ka = 10; – · –, Ka = 20. (b) Mass velocity feedback: ——, Kv=0; · · · · · · , Kv = 20; – – – –, Kv = 100; — · —, Kv=500. (c) Mass displacement feedback: ——, Kp=0; · · · · · · , Kp = 20; – – – –, Kp = 100; — · —, Kp=200. (d) Mass jerk feedback: ——, Kj=0; · · · · · · , Kj = 1; – – – –, Kj = 5; — · —, Ka = 50.

4.2.      When active control is conducted by using base signals, the control input can be expressed as v(s) = −(Kpb + Kvb s + Kab s 2 + Kjb s 3 )x0 (s).

(21)

Substitution of equation (21) into equation (15) gives the transfer function x1 (s) =

−KE Kjb s 3 + (C0 L − KE Kab )s 2 + (C0 R + LK − KE Kvb )s + KR − KE Kpb x0 (s). LMs 3 + (MR + C0 L)s 2 + (C0 R + LK)s + KR

(22)

It can be seen from equation (22) that feedback of base signals changes the numerator of the closed loop transfer function. Therefore, theoretically it is possible to make the transmissibility zero with proper adjustment of the feedback gain of the base signals [8]. The effect of the various feedback gains of the base signal on transmissibility is illustrated in Figure 4. 4.3.        From the above simulation results, it was realized that base jerk feedback causes the vibration isolation system to become unstable. Therefore, the four state variables of mass and the three variables of base, except jerk, were used for feedback control. The control input voltage to the power amplifier of the actuator is expressed as v(s) = −(Kp + Kv s + Ka s 2 + Kj s 3 )x1 (s) − (Kpb + Kvb s + Kab s 2 )x0 (s).

(23)

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Figure 4. The transmissibility curve for base signals feedback control. (a) Base acceleration feedback: ——, Kab = 0; · · · · · · · , Kab = 10; – – – –, Kab = −3; – · –, Kab = −10. (b) Base velocity feedback: ——, Kvb=0; · · · · · · · , Kvb = 10; – – – –, Kvb = 30; – · –, Kvb = 60. (c) Base displacement feedback: ——, Kpb=0; · · · · · · · , Kpb = 1000; – – – –, Kpb = 5000; – · –, Kpb = 10 285. (d) Base jerk feedback: ——, Kjb=0; · · · · · · · , Kjb = 0.1; – – – –, Kjb = 0.5; – · –, Kjb = 1.

Substituting equation (23) into equation (15) yields the transfer function x1 (s) = (C0 L − KE Kab )s 2 + (C0 R + LK − KE Kvb )s + KR − KE Kpb x (s). (LM + KE Kj )s 3 + (MR + C0 L + KE Ka )s 2 + (C0 R + LK + KE Kv )s + KR + KE Kp 0 (24) With the feedback gains determined from the preceding simulation results, the transmissibility of the vibration isolation system was obtained as shown in Figure 5.

Figure 5. The transmissibility curve for seven signals feedback control Ka = 10; Kv = 1000; Kp = 1; Kj = 50; Kab = −10; Kvb = 10; Kpb = 10 285.

    

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Figure 6. The overall experimental set-up: (a) the vibration isolation system; (b) the equipment of the system. Key for (a): 1, mass; 2, base; 3, surge tank; 4, accelerometer on mass; 5, accelerometer on base; 6, air spring; 7, electromagnetic actuator; 8, linear guide; 9, proportional control valve; 10, regulator; 11, exciter. Key for (b): 1, filter; 2, oscilloscope; 3, accelerometer amplifier; 4, terminal board; 5, control valve amplifier; 6 units for power amplifier and operational amplifier; 7, function generator; 8, power supply (24 V); 9, power supply (40 V); 10, power supply (15 V).

5. APPARATUS

The vibration isolation system shown in Figure 1 consists of five parts; that is, a mass, air springs (three), the actuator comprised of a permanent magnet and electromagnet, the surge tank connected to the air springs and to the air supply through a regulator, and a base. Linear motion guides were set up between the base and the mass at three points for unidirectional motion of the mass. Accelerometers were used to measure the vibrations of the mass and base. The measured signals were passed through filters and differentiating amplifiers, integrating amplifiers, and an A/D converter and then fed into a computer. After evaluating control inputs based on the digitized values according to the preceding control algorithms, the computer sends out the corresponding voltage to the power amplifier of the electromagnetic actuator

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Figure 7. A schematic diagram of the electromagnetic actuator.

through a D/A converter. The power amplifier gives the current, which is proportional to the voltage, to the electromagnetic actuator. In Figures 6 and 7, respectively, are shown photographs of the overall experimental set-up and a schematic diagram of an electromagnetic actuator.

6. EXPERIMENTS

Before conducting the real experiments, some preparations and preliminary experiments were performed. The reference air pressure was adjusted to 1·2 kgf/cm2 for both the air springs and the surge tank, and DC 1·5 V was applied to the power amplifier of the electromagnetic actuator. The base was excited by the exciter generating sinusoidal motion and the frequency was increased by 0·5 Hz at each step; in this manner the vibration transmissibility as a function of the excitation frequency was obtained, as shown in Figure 8. In this figure, two peaks can be noticed. These are caused by the air springs mounted underneath the base for base excitation; that is, the base is pressed down when the actuator exerts upward control force. Therefore the system appears to be 2-DOF in Figure 8, with the resonance frequency located between 5 Hz and 6 Hz. The maximum transmissibility was 4·054 at 5·5 Hz.

Figure 8. The transmissibility curve for the vibration isolation system (uncontrolled).

    

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Figure 9. The acceleration signal of mass: (a) 5 Hz; (b) 10 Hz.

Seven feedback signals were selected from the simulation results. For mass, they are acceleration, velocity, displacement and jerk signals; for the base, they are acceleration, velocity and displacement. The characteristics of the system response when the base is excited at 5 Hz and 10 Hz respectively are shown in Figures 9(a) and (b). The measured transmissibility in three different cases is shown in Figure 10; when the system is uncontrolled, controlled by only the linear controller, and controlled by the proposed controller, where the fixed gain tuned at 10 Hz was used. In the case in which only the linear controller is used, the overall vibration isolation performance does not seem to be so good except near the frequency to which the gain is tuned, where vibration is suppressed to a considerable extent. On the other hand, transmissibility of the vibration isolation

Figure 10. The transmissibility curve for the vibration isolation system. – – – – –, Uncontrolled; · · · · · , linear controller; ——, proposed controller.

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Figure 11. The output of the linear and neural network controller. ——, Linear controller; – – – –, neural network.

system obtained from the proposed control algorithm was below 0·63 over the entire frequency range, including the resonance frequency. The controller outputs—that is, the control input to the plant—are shown in Figures 11 and 12. Only the linear controller was applied until one second from the start, at which point the proposed control scheme became active. The output of the proposed controller (V) is divided into two components in Figure 11 for comparison: namely, the contribution of the linear controller output (VL ) and the contribution of the neural network controller output (VN ). With the proposed controller switched on, the neural network takes over a major portion of the output and the linear controller output is reduced. This implies that the proposed scheme can expect to show adaptability. On the other hand, in Figure 12 is shown the behavior of the added controller output (V). In less than half a second, the output of the proposed controller comes to have a shape similar to that of the linear controller only, which proves that the neural network learns efficiently from the linear controller. 7. CONCLUSIONS

A hybrid-type active vibration isolation system that uses air springs and an electromagnetic actuator has been described. The air springs were used as passive elements and the electromagnetic actuator as an active element. In addition, a new control scheme,

Figure 12. A comparison between the output without a neural network and that with a neural network.

    

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where a linear controller and a neural network controller supplement each other, has been proposed. Our conclusions are as follows. (1) The electromagnetic actuator gives a fairly fast control response for vibration isolation. (2) The hybrid-type system made up of passive and active elements provides a better vibration isolation performance. (3) The proposed control scheme efficiently learns from the linear controller. (4) The proposed control scheme can suppress the transmissibility of the vibration isolation system to below 0·63 over the entire frequency range, including the resonance frequency, without complex calculation or prior manipulation. REFERENCES 1. D. W. S and J. E. R 1969 Transactions of the American Society of Mechanical Engineers, Journal of Engineers for Industry, 981–990. Theoretical and experimental investigation of electrohydraulic vibration isolation systems. 2. H. S, S. R and T. S. S 1990 Transactions of the American Society of Mechanical Engineers, Journal of Vibration and Acoustics 112, 8–15. Vibration-isolation characteristics of an active electromagnetic force generator and the influence of generator dynamics. 3. D. J. B and H. B 1991 MacKay International Joint Conference on Neural Networks 2, 775–780. Vibration cancellation at multiple locations using neurocontrollers with real-time learning. 4. Y. I, M. O, N. S, K. U and K. K 1991 Transactions of the Japan Society of Mechanical Engineers 57, 179–184. Active control of precision vibration isolation system. 5. B. W 1990 Proceedings of the IEEE 78, 1415–1442. 30 years of adaptive neural networks: Perception, Madaline, and Back-propagation. 6. R. K. E 1988 Proceedings of IEEE Conference on Neural Networks, 587–594. A learning architecture for control based on back-propagation. 7. H. M, M. K, T. S and R. S 1988 Neural Networks 1, 251–265. Feedback-error-learning neural network for trajectory control of a robotic manipulator. 8. J. W. B, D. W. C and H. J. P 1993 Korea–U.S. Vibration Engineering Seminar, Taejon, Korea, 305–322. A comprehensive investigation of active vibration isolation systems using an air spring.