外文翻譯---反饋控制消除機械振動（英文）

1、Feedback Control for Canceling Mechanical Vibrations James S . Montanaro and Guy 0. Beale Electrical and Computer Engineering Department George Mason University Fairfax, Virginia gbeale @ gmu.edu Abstract - Line

2、ar Quadratic Gaussian (LQG) control is applied to the application of active vibration cancellation. The virtual mass of a vibration motor is controlled based on measurements of acceleration. The final controller is

3、a modified LQG design developed for an augmented model of the vibration motor. This paper describes the modeling and parameter identification of the vibration motor, development of performance specifications for th

4、e controller, design of the LQG controller, and experimental testing of the resulting control system. Modeling of the vibration motor was based on both theoretical derivation and experimental data collection. Devel

5、opment of the specifications for the controller was done primarily in the frequency domain, using the return ratios of two different transfer functions. Controller design was done using the continuous-time LQG algori

6、thm. The controller was implemented as an analog circuit with operational amplifiers. Some closed-loop stability issues were observed during the experimental testing, and an explanation is offered for these problems.

7、 I. INTRODUCTION This paper describes the design of a feedback control system that may be used to cancel mechanical vibrations on the surface of an object. The cancellation is accomplished by applying an alternating f

8、orce to the surface through a vibrating motor. Applications for which it is desired to reduce vibrations in an object range from home appliances and automobiles to aircraft and high-speed trains [11,[21,[31. Reducing

9、 mechanical vibration provides for improved user comfort and safety, and it increases product reliability and durability by reducing wear. Four tasks were undertaken as part of this design project. First it was nece

10、ssary to determine the properties and mathematical model of a commercially available vibrating motor which would be used to apply the canceling vibration. The second task was to develop a set of performance specific

11、ations for the system to be used during design of the controller. The third task was to use feedback control design techniques to design a controller which, given an input signal from a motion transducer, will gene

12、rate a control signal for the motor to counteract that motion. Both H ,and Linear Quadratic Gaussian (LQG) techniques were investigated. The final controller is a modified LQG design developed for an augmented model o

13、f the vibration motor. The fourth task was to implement the controller in analog circuitry and to test the closed-loop system. The first three tasks are described in this paper. Results from implementing the control

14、ler and testing the complete system are also presented, along with an interpretation of the some of the experimental results. 11. MODELING AND IDENTIFICATION The major component in the vibration cancellation system

15、is a vibration motor. This motor would be attached to the surface of whatever object that has the vibrations that are to be cancelled. Mounting an accelerometer on the motor casing senses the vibrations of the objec

16、t. The acceleration of the motor casing is sensed, and the motor is controlled in such a way as to produce its own vibrations that cancel those from the external disturbance. Canceling that vibration is accomplished

17、 by stilling the motor. That is, by crafting a control signal to hold motor acceleration to zero (or nearly so), the disturbance vibration is counteracted. The vibration motor (Fig. 1) is modeled by a mass ml, repre

18、senting the outer motor casing, and an inner suspended mass mz. A spring and damper, which model the suspension, connect the two masses. Motor force is electromagnetic and acts between the two masses, pushing them ap

19、art or pulling them together depending on the direction of the drive (control) current Z , .The motor force is the control force F,. Motor L b Disturbance Force I I L 2 IController Fig. 1. Schematic of vibration c

20、ontrol system. The motor response to an alternating disturbance force applied to m, with no feedback will be frequency dependent. At very low frequency, the suspension will hold m land m2 in relative equilibrium. T

21、he masses will move in unison as the motor accelerates back and forth. The effective mass of the system is ml + m2. At very high frequency, the magnitudes of the velocity and displacement of ml approach zero, and

22、no force is transmitted through the suspension to m2. In effect, the disturbance force acts only on mI, so the effective mass is ml. Over some frequency range there must be a transition between low frequency and h

23、igh-frequency behavior. Clearly the path through the suspension is frequency dependent. At high frequency, this path vanishes, since displacement and 0-7803-4503-7/98/$10.00 1998 IEEE 1433 frequencies. That is, rathe

24、r than account for the resonances in the model, we can guarantee stability by making the return ratio less than one at those frequencies. 111. CONTROLLER DESIGN A. Pelformance Criteria Several criteria were selected fo

25、r evaluating controller designs. First, the closed-loop disturbance response should be as small as possible over some useful frequency range. This is equivalent to making the virtual mass as large as possible in that

26、 range. One goal in this performance measure was to attain a virtual mass at least ten times the motor's open-loop mass. Second, motor excursion (xI) should not be excessive at any frequency, so as to avoid the l

27、imits of suspension travel. Third, control force should not be much greater than the disturbance force at any frequency. This ensures that control energy is not being wasted at a frequency where the motor cannot resp

28、ond adequately. Fig. 3 shows the frequency response magnitude from the control input to the output for the system model of eqn. 2. From inspection of the figure, it is apparent that the controller should not waste en

29、ergy by trying to counteract very low frequency disturbances. Open-Loop Control Response Magnitude (U toy) 40 30 20 10 z o - 2 -10 P = - 2 0-30 -40 -50 4 0 10“ 10' 1 0 21 0 'I O 'Frequency (Hz)

30、 Fig. 3 Open-loop control response. Fourth, sensor noise imposed on the plant by the control loop should not be amplified excessively. Fifth, loop gain should be less than Yi at all high-frequency plant resonances.

31、This leaves a factor-of-two margin for error. Finally, closed-loop stability should be robust to reasonable changes (or modeling errors) in the plant's physical parameters and to the attachment of the plant to a

32、load mass. B. Structure of the Controller C and F matrices, respectively. The convergence of this dummy state forces the actual and estimated outputs to be driven to zero. By this mechanism, the plant output can be r

33、egulated. In the final controller design, the observer and control gains were computed using the Linear Quadratic Gaussian technique with Loop Transfer Recovery (LQGLTR). In order to improve disturbance rejection, th

34、e open-loop system model was augmented by placing an additional dynamic system in series with the control input. The augmentation is described in more detail in the next section. The Kalman filter and linear regulato

35、r gains were calculated for this augmented system model. The weighting matrices were tuned manually to obtain good performance of the closed-loop system, relative to the performance criteria mentioned above. C. Desig

36、n o fthe Controller With the pole placement design, the observer poles initially were placed much faster than the plant poles. This approach did not give adequate attenuation at the high-frequency resonances. Attempts

37、 at improving performance by moving the poles locations were marginal at best. The increase in effective mass occurred only over a very small frequency range, and the control energy was considered too high for the am

38、ount of disturbance rejection achieved. In an attempt to improve performance with an efficient design procedure, the Linear Quadratic Gaussian technique was used with Loop Transfer Recovery [4], [5]. The Kalman filt

39、er design parameters r , W, and V (the disturbance input matrix and the disturbance and noise process covariance matrices) were initialized to G, 1, and 1, respectively, where G is the disturbance input matrix. Sin

40、ce both W and V are scalars, it is their ratio that matters and not their individual values. Therefore, only W is adjusted. For LQR state feedback design, the following performance index, which weights the costs of o

41、utput and control energy, was used. J = lom ( y ' + ru')dt =lom ((cx + DUI' + ru ')dt = lom ( x*Qx + uT Ru + 2x'Nu)dt (6) Q = CTC, R = r + D2, N = CTD The system's closed-loop be

42、havior was manipulated by adjusting the scalar parameters W and r. Since the key performance criteria were all responses to the disturbance input, closed-loop performance cannot be seen directly in the Kalman filter

43、 response and the response recovered by the LQR. Nonetheless, the influences of Wand r were consistent with LQGLTR principles. For example, increasing W increased the Kalman filter bandwidth, as evidenced by increase

44、d high-frequency noise response. Decreasing r (the control energy cost) also increased noise bandwidth by recovering of the Kalman filter,s response. Experimentation proceeded since there were only two The initial

45、 structure for the controller consisted 6f an observer and feedback gain matrix, both designed with pole placement Since there is a direct feedthrough parameters to adjust. The results achieved, however, were term f

46、rom the process disturbance to the Output y' the observer's estimated state does not converge to the true state. of the true state and the process disturbance, multiplied by the virtually identical to those ob

47、tained using pole placement. It was apparent that the desired results could not be obtained Rather, it 'Onverges to a dummy state that is a 'Ombination solely by feeding back an estimated plant state (dummy st