Vibration Control Strategies

A particular field of mechatronics is the active damping of mechanical vibrations. There are two radically different approaches to disturbance rejection: feedback and feedforward.

Feedback

The principle of feedback is presented in Fig. 1; the output y of the system is compared to the reference input r and the error signal, e = r - y, is passed into a compensator H(s) and applied to the system G(s).

The design problem consists of finding the appropriate compensator H(s) such that the closed-loop system is stable and behaves in the appropriate manner.

In the control of lightly damped structures, feedback control is used for two distinct and somewhat complementary purposes: active damping and model-based feedback.

The objective of active damping is to reduce the resonant peaks of the closed-loop transfert function

In this case F(s) is very close to G(s), except near the resonance peaks where the amplitude is reduced. Active damping can generally be achieved with moderate gains; another nice property is that is can be achieved without a model of the structure and with guaranteed stability, provided that the actuator and sensor are collocated and have perfect dynamics. Of course actuators and sensors always have finite dynamics and any active damping system has a finite bandwidth.

The control objectives can be more ambitious and we may wish to keep a control variable (a position, or the pointing of an antenna) to a desired value in spite of external disturbances d in some frequency range. From

we readily see that reducing the effect of external disturbances requires large values of GH in the frequency range where the disturbance is significant. From Equ. 1, we see that GH >> 1 implies that the closed-loop transfer function F(s) is close to 1, which means that the output y tracks the input r accurately. In general, to achieve that, we need a more elaborate strategy involving a mathematical model of the system which, at best, can only be a low-dimensional approximation of the actual system G(s). There are many techniques available to find the appropriate compensator and only the simplest and the best established will be reviewed in this text. They all have a number of common features:

The bandwidth w_c of the control system is limited by the accuracy of the model; there is always some destabilization of the flexible modes outside w_c (residual modes). The phenomenon whereby the net damping of the residual modes actually decreases when the bandwidth increases is known as spillover (Fig. 2).

The disturbance rejection within the bandwidth of the control system is always compensated by an amplification of the disturbances outside the bandwidth.

When implemented digitally, the sampling frequency w_s must always be two order of magnitude larger than w_c to preserve reasonably the behavior of the continuous system. This puts some hardware restrictions on the bandwidth of the control system.

Feedforward

When a signal correlated to the disturbance is available, feedforward adaptative filtering constitues an attractives alternative to feedback for disturbance rejection; it was originally developed for noise control, but it is very efficient for vibration control too. Its principle is explained in Fig. 3 . 

The method relies on the availability of a reference signal correlated to the primary disturbance; this signal is passed through an adaptive filter, the output of which is applied to the system by secondary sources. The filter coefficients are adapted in such a way that the error signal at one or several critical points is minimized. The idea is to produce a secondary disturbance such that it cancels the effect of the primary disturbance at the location of the error sensor. Of course, there is no guarantee that the global response is also reduced at other locations and, unless the response is dominated by a single mode, there are places where the response can be amplified; the method can therefore be considered as a local one, in contrast to feedback which is global. Unlike active damping which can only attenuate the disturbances near the resonances, feedforward works for any frequency and attempts to cancel the disturbance completely by generating a secondary signal of opposite phase.

the method does not need a model of the system, but the adaptation procedure relies on the measured impulse response. The approach works better for narrow-band disturbances, but wide-band applications have also been reported. Because it is less sensitive to phase lag than feedback, feedforward control can be used at higher frequency (a good rule of thumb is w_c  =  w_s /10 ); this is why it has been so successful in acoustics.

The main limitation of feedforward adaptive filtering is the availability of a reference signal correlated to the disturbance. There are many applications where such a signal can be readily available from a sensor located on the propagation path of the perturbation. For disturbances induced by rotating machinery, an impulse train generated by the rotation of the main shaft can be used as reference. Table 1 below summarizes the main features of the two approaches.