Yaw dampers are widely used as components of stability augmentation systems (SAS); we saw the potential beneficial effects in Fig. 8.20/. At first glance the yaw damper would appear to be a very simple application of feedback control principles—just use Fig. 8.20/as a guide, select a reasonable gain, and add a model for the servo actuator/control dynamics. However, it is not really that simple. There is another important factor that has to be taken into account—namely that during a steady turn, the value of r is not zero. If, in that situation, the yaw damper commands a rudder angle because it senses an r, the angle would no doubt not be the right one needed for a coordinated turn. In fact during a right turn the yaw damper would always produce left rudder, whereas right rudder would usually be required (see Fig. 7.24). This characteristic of the yaw damper is therefore undesirable. To eliminate it, the usual method is to introduce a high-pass or “washout” filter, which has zero gain in the steady state and unity gain at high frequency. The zero steady state gain eliminates the feedback altogether in a steady turn. The system that results is pictured in Fig. 8.21, where the meaning of the filter time constant is illustrated. For the servo actuator/rudder combination of this large airplane we assume a first order system of time constant 0.3 sec.
The closed loop transfer function for Fig. 8.21 is readily found to be [see (8.2,1)]
1 + WJGrSr
This transfer function was used to calculate a number of transient responses to illustrate the effects of J(s) and
As a reference starting point, Fig. 8.22 shows the open loop response (W = 0) to
Figure 8.21 Yaw damper.
a unit impulse of yaw rate command rc. It is evident that there is a poorly damped oscillatory response (the Dutch Roll) that continues for about 2 min and is followed by a slow drift back to zero (the spiral mode). Fig. 8.22a shows that the control dynamics (i. e., J(s)) has not had much effect on the response.
Figure 8.23 shows what happens when the yaw damper is turned on with the same input as in Fig. 8.22. It is seen that the response is very well damped with either of the two gains shown, which span the useful range suggested by Fig. 8.20/, and that the spiral mode effect has also been suppressed.
It remains to choose a time constant for the washout filter. If it is too long, the washout effect will be insufficient; if too short, it may impair the damping performance. To assist in making the choice, it is helpful to see how the parameter a = 1/tw0 affects the lateral roots. Figure 8.24 shows the result for a gain of К = —1.6. The roots in this case consist of those shown in Fig. 8.20/ plus an additional small real root associated with the filter. It is seen that good damping can be realized for values of a up to about 0.3, that is, for time constant т down to about 3 s. This result is very dependent on the gain that is chosen. While the oscillatory modes are behaving as displayed, the real roots are also changing—the roll root decreasing in magnitude from —2.31 at a = 0 to —1.95 at a = 0.32. The new small real root starts at the origin when a = 0 and moves slowly to the left, growing to -0.00464 at a = 0.32. When the fdter time constant is 5 s the small root is —0.0038, corresponding to an aperiodic mode with rhalf = 182 s. It is instructive to compare the performance of the yaw damper with and without the filter for an otherwise identical case. This is done in Fig. 8.25. It is seen that the main difference between them comes from the small real root, which after 5 min has reduced the yaw rate to about 5% of its peak value. This slow decay is unlikely to present a problem since the airplane heading is inevitably controlled, either by a human or automatic pilot. In either case, the residual r would rapidly be eliminated (see Sec. 8.8).
(a) Initial response, 0-60 s
Figure 8.22 Yaw rate impulse response—open loop, W = 0. (a) Initial response, 0-60 sec. (b) Long-term response.