The phugoid makes its presence known not only in the form of transient perturbations from a steady state, but also in maneuvers, as illustrated in Sec. 7.7. We saw there for example that in changing from level to climbing flight by opening the throttle (Fig. 7.21) there results a protracted, weakly damped approach to the new state that would take more than 10 min to complete. Transitions from one value of у to another are obviously not made in this manner, and the pilot suppresses the oscillation in this case as well. Provided that the correct в is known for the climb condition, the same technique as discussed above would work, that is, control operating on pitch – attitude error. We illustrate an alternative concept that does not require any knowledge of the final correct pitch attitude, but that uses speed error alone. It is not selfevident how speed should be controlled, in the light of the discussion in Sec. 7.1. We saw there that both elevator and throttle influence the speed, but that the short – and long-term effects of each of these controls are quite different—the throttle principally affects the speed only in the short term. For a change of steady-state speed, the elevator must be used. Clearly, a sophisticated speed control might use both. We shall see in this example, however, that when the primary aim is to suppress the phugoid, which is a very long period oscillation, the goal can be achieved with the elevator alone. Figure 8.8 shows the system.
The command is the speed uc and the feedback signal is the actual speed u. For output we choose speed and flight-path angle, that is, у = [и у]7. The control vector is c = [Se 8p]r of which only the elevator is in the feedback loop. Since the controlled variable is u, which does not change appreciably in the short-period mode, we can use the phugoid approximation for the aircraft transfer function matrix G(s), which is the (2X2) matrix of transfer functions from c to y:
Two of the elements of G are implicit in (7.7,7), since GyS = Ges — GaS where 8 stands for either 8e or 8p. The remaining two are (see Exercise 8.4)
Ultimately we shall want to calculate the time responses of и and у to a throttle input Sp. So the transfer functions we need are the two corresponding closed loop transfer functions. If we denote these by GuSp(s) and GySp(s), respectively, we find that they are given in terms of the aircraft transfer functions by (see Exercise 8.4)
Each of the aircraft transfer functions in (8.4,3) can, as usual, be expressed as a ratio of two polynomials, for example:
When this is done (8.4,3) becomes
We know that the denominator of a transfer function is the characteristic polynomial. We also know that a linear invariant system of the kind under discussion can have only one independent characteristic equation. Thus we have an apparent paradox, since the denominator of (8.4,6) is not the same as that of (8.4,5), having the extra factor f(s). Now it can be shown (see Exercise 8.6) that f is a factor of the bracketed term in the numerator of (8.4,6), and hence that it divides out of the right side and leaves the same characteristic polynomial as in (8.4,5).
As indicated above, the second-order phugoid approximation should be expected to be reasonable for this case. We shall therefore use it to choose the gains in J(s), but at the end will check the solution for suitability with the exact fourth-order equations. To this end we examine the effect of J(s) on the characteristic equation, that is, on
f(s) is given by (6.3,9):
f(s) = As2 + Bs + C
NllSr is given by (7.7,7):
NuSe = as + ao
and for J(s) we use
J(s) = kt + k2s
so that Dj = 1 and A, = kt + k2s. Note that the k2s term implies a signal proportional to acceleration. Such a signal could be obtained from an x-axis accelerometer or by
Figure 8.9 Speed controller. Root locus plot of GuSe. Phugoid approximation.
differentiating the signal from the speed sensor. The closed loop characteristic equation then becomes:
A’s2 + B’s + C = 0 (a)
where A’ = A + axk2 (b) (8.4,9)
В’ = В + alkl + aji2 (c)
С = C + ajcx (d)
The numerical values of the constants for the example jet transport are
A = 2.721 X 107 В = 2.633 X 105
C = 1.376 X 105 (8.4,10)
a, = 8.218 X 108 a0 = 3.653 X 108
To assess what range of values of k, and k2 would be appropriate, we use three guides:
1. A reasonable elevator angle for, say, a 10 fps (3.048 m/s) speed error
2. The root locus plot for GuSe
3. The graph of k2 vs. kt for critical damping
(1) The first of these is arrived at by noting that 1° of elevator for 10 fps speed error gives a ky of 0.0017 rad/fps. (2) The root locus plot is shown on Fig. 8.9 and indicates that the open loop roots can be moved very appreciably with a proportional gain as low as 0.005. (3) For the third guide, we note that critical damping corresponds to B’2 – AA’C’ = 0. With the aid of (8.4,9) and (8.4,10) this leads to an algebraic relation between к у and k2 that is solved for the graph shown on Fig. 8.10. The useful range of gains is the space below the curve, which corresponds to damped oscillations. The farther from the curve, the more overshoot would be expected in the response. We have for illustration arbitrarily chosen the gains indicated by the point marked on the graph, without regard for whether it is optimum. When used to calculate the response of airplane speed to application of a negative step in thrust, with the phugoid approximation, the result is as shown in Fig. 8.11. The throttle input corre-
Figure 8.13 Speed controller—exact equations. Gamma response.
sponds to a steady-state descent angle of a little less than 3°. The maximum speed error, which is seen to be less than 3 fps at an initial speed of 774 fps, would probably not be perceptible to the pilot. This suggests that the chosen gains are probably not too small. The maximum elevator angle during this maneuver is less than 2° (see Fig. 8.14) so the gains are not excessive either.
To assess the performance of the controller with certainty, it is necessary to use the exact equations. The full matrix A for this example is (6.2,1), and В is (7.6,4). The most important elements of the solution are displayed in Figs. 8.12 to 8.14. The result for the speed in Fig. 8.12 confirms that the phugoid approximation is indeed good enough for preliminary design. Figure 8.13 demonstrates that the steady-state flight path angle is reached, with a small overshoot, in about 20 s. Figure 8.14 demonstrates that the elevator angle required to achieve this is small. To understand the physics of the maneuver, it is helpful to look at the angle of attack variation, graphed in Fig. 8.15. It shows that there is a negative “pulse” in a that lasts about 10 s. This causes a corresponding negative pulse in lift, which is the force perpendicular to the flight path that is required to change its direction.
Finally, these graphs should be contrasted with those of Fig. 7.21, which show the uncontrolled response to throttle. Feedback control has made a truly dramatic difference!
Figure 8.15 Speed controller—exact equations. Angle of attack response.