Phugoid Suppression: Pitch Attitude Controller
The characteristic lightly damped, low-frequency oscillation in speed, pitch attitude, and altitude that was identified in Chap. 6, was seen in Chap. 7 to lead to large peaks in the frequency-response curves (Figs. 7.14 to 7.18) and long transients (Fig. 7.20). Similarly, in the control-fixed case, there are large undamped responses in this mode to disturbances such as atmospheric turbulence. These variations in speed, height, and attitude are in fact not in evidence in actual flight; the pilot (human or automatic) effectively suppresses them, maintaining flight at more or less constant speed and height. The logic by which this process of suppression takes place is not unique. In principle it can be achieved by using feedback signals derived from any one or a combination of pitch attitude в, altitude h, speed v, and their derivatives. In practice, the availability and accuracy of the state information determines what feedback is used.
Since the phugoid oscillation cannot occur if the pitch angle в is not allowed to change (except when commanded to), a pitch-attitude-hold feature in the autopilot would be expected to suppress the phugoid. This feature is commonly present in airplane autopilots. We shall therefore look at the design of an attitude hold system for the jet transport of our previous examples. Pitch attitude is readily available from either the real horizon (human pilot) or the vertical gyro (autopilot). Consider the controller illustrated in Fig. 8.5. From (8.2,1) we see that the overall transfer function is
Я" = , ( І(Л (8-3,1)
0C 1 + GHSe(s)J(s)
If we write GeSe(s) = N(s)/D(s), and J(s) = N'(s)/D’(s), then the characteristic equation is
D(s)D'(s) + N(s)N'(s) = 0 (8.3,2)
To proceed further we need explicit expressions for the above transfer functions. Since в is an important variable in both the short period and phugoid modes, it might be expected that neither of the two approximate transfer functions for GeSe derived in Sec. 7.7 would serve by itself. We therefore use the exact transfer function derived
Figure 8.6 Root locus of pitch controller with proportional control.
from the full system of linearized longitudinal equations of motion. Then N(s) is given by (7.7,2) and D(s) by (6.2,2). The result is
^ -(1.158s + 0.3545s + 0.003873)
GV – 54 + 0_750468s3 + 0.935494s2 + 9.463025 X 10~s + 4.195875 X 10~3
As to J(s), a reasonable general form for this application is
J(s) = — + k2 + k3s (8.3,4)
For obvious reasons, the three terms on the right hand side are called, respectively, integral control, proportional control and rate control, because of the way they operate on the error e. The particular form of the controlled system, here GeSe(s), determines which of jk„ k2, k3 need to be nonzero, and what their magnitudes should be for good performance. Integral control has the characteristic of a memory, and steady – state errors cannot persist when it is present. Rate control has the characteristic of anticipating the future values of the error and thus generates lead in the control actuation. In using (8.3,4), we have neglected the dynamics of the elevator servo actuator and control surface, which would typically be approximated by the first-order transfer function 1/(1 + rs). Since the characteristic time of the servo actuator system, r, is usually a small fraction of a second, and we are interested here in much longer times, this is a reasonable approximation.
For the example airplane at the chosen flight condition it turns out that we need all three terms of (8.3,4) to get a good control design. This might not always be the case. Let us first look at the use of proportional control only, in which case J is a constant gain, k2. To select its magnitude, we use a root locus plot2 of the system, Fig. 8.6, in which the locus of the roots of the characteristic equation of the closed loop system are plotted for variable gain k2. We see that at a gain of about —0.5 the phugoid mode is nearly critically damped, that is, it is about to split into two real roots. At this gain, the phugoid oscillation is effectively eliminated. We note that at
the same gain the short period roots have moved in the direction of lower damping. The response of the aircraft to a unit step command in pitch angle with only proportional control is shown in Fig. 8.7a. It is clear that this is not an acceptable response. There is a large steady-state error (steady-state error is a feature of proportional control) and the short-period oscillation leads to excessive hunting. The steady-state error could be reduced by increasing k2 (see Exercise 8.2), but this would further decrease the short period damping.
We digress briefly to explore the reason for the damping behavior. It was noted previously (Sec. 6.8 and Exercise 6.4) that the term of next-to-highest degree in the characteristic equation gives “the sum of the dampings.” That is, the coefficient of s3 in (8.3,3) is the sum of the real parts of the short-period and phugoid roots. Now when J = k2 the closed loop characteristic equation (8.3,2) becomes D(s) + k2N(s). That is we add a second degree numerator to a fourth degree denominator, leaving the coefficient of s3 unchanged. Thus any increase in the phugoid damping can only come at the expense of that of the short-period mode. This is exactly what is seen in Fig. 8.6. The shifts of the two roots in the real direction are equal and opposite.
To eliminate the steady-state error, we use integral control and choose
1 + (8.3,5)
The result is shown in Fig. 8.7b. The steady-state error has been eliminated, but the short-period oscillation is now even less damped. Now the damping of the short – period mode is governed principally by Mq [see (6.3,14)], so in order to improve it we should provide a synthetic increase to Mq. A signal proportional to q is readily obtained from a pitch-rate gyro. Since q = в in the system model we are using, we accomplish this by adding a third term to J:
s + 1 + у j (8.3,6)
The result, shown in Fig. 8.7c is an acceptable controller, with little overshoot and no steady-state error. A commanded pitch attitude change is accomplished in about 10 sec. Note that in this illustration, all the constants in (8.3,4) are the same, that is, -0.5. Fine-tuning of these could be used to modify the behavior to reduce the overshoot or speed up the response. Throughout this maneuver, the elevator angle remains less than its steady-state value (which it approaches asymptotically), so that the gains used are indeed much smaller than the elevator control is physically capable of providing (see Exercise 8.2).
The preceding analysis does not reveal the underlying physics of why k2 damps the phugoid. This can be understood as follows. An angle в in the low frequency phugoid implies vertical velocity (i. e., h = VO). Now a positive elevator angle proportional to a slowly changing в implies a negative increment in angle of attack and hence in the lift as well. Thus k2 leads to a vertical force (downward) 180° out of phase with the vertical velocity (upward), exactly what is required for damping.
Although the phugoid oscillation has been suppressed quite successfully by the strategy employed above, it should be remarked that in the example case neither the speed nor the altitude has been controlled. As a consequence, the speed drifts rather slowly back to its original value, and the altitude to a new steady state.
Finally it should be noted that a controller design that is correct for one flight condition, in this case high speed at high altitude, may not be acceptable at all speeds and altitudes, for example, landing approach. In the real world of AFCS design, this problem leads, as in most engineering design, to compromises between conflicting requirements. If the economics of the airplane justifies it, gain scheduling can be adopted; that is, the control gains are made to be functions of speed, altitude, and configuration.