There are five lateral state variables that can be used readily as a source of feedback signals—{v, p, г, ф, ф); v from a sideslip vane or other form of aerodynamic sensor, p and r from rate gyros, and ф, ф from vertical and directional gyros. Lateral acceleration is also available from an accelerometer. These signals can be used to drive the two lateral controls, aileron and rudder. Thus there is a possibility of many feedback loops. The implementation of some of these can be viewed simply as synthetic modification of the inherent stability derivatives. For example, p fed back to aileron modi
(M
Figure 8.18 Altitudehold controller, (a) Height, speed, and pitch angle, (h) Elevator and throttle controls.

ftes Lp (roll damper), r to the rudder modifies Nr (yaw damper), and v to rudder modifies the yaw stiffness Nv, and so on. It is a helpful and instructive preliminary to a detailed study of particular lateral control objectives to survey some of these possible control loops. We could do this analytically by examining the approximate transfer functions given in Chap. 7. However, we prefer here to do this by way of example, using the now familiar jet transport, and using the full system model. We treat each loop as in Fig. 8.19, as a negative feedback with a perfect sensor and a perfect actuator, so that the loop is characterized by the simple constant gain K. For each case we present a root locus plot with the gain as parameter (Fig. 8.20) (All the root loci are symmetrical about the real axis; for some, only the upper half is shown). As is con
Figure 8.19 Representative loop.

ventional, the crosses designate the open loop roots (poles) and the circles the open loop zeros. The pair of complex roots corresponds to the Dutch Roll oscillation; the real root near the origin is for the spiral mode; and the real root farther to the left is that of the heavily damped roll mode.
Since the root loci always proceed from the poles to the zeroes as a] increases, the locations of the zeros can be just as important in fixing the character of the loci as the locations of the poles. The numbers on the loci are the values of the gain. Zero gain of course corresponds to the original open loop roots. The objective of control is to influence the dynamics, and the degree of this influence is manifested by the amount of movement the roots show for small changes in the gain. We have not included root loci for acceleration feedback, and of the remaining ten, two show very small effects, and are therefore not included either. These two are the aileron feedbacks: v —*■ 8a and r —» 8a. Each of the other eight is discussed individually below.
ф—* 8a It was pointed out in Chaps. 2 and 3 that airplanes have inherent
aerodynamic rotational stiffness in pitch and yaw, but that there is no such stiffness for rotations about the velocity vector. This fundamental feature of aerodynamics is responsible for the fact that airplanes have to sideslip in order to level the wings after an initial roll upset. This lack can be remedied by adding the synthetic derivative
— LSa (18а/йф = ~KLSa
We might expect that making such a major change as adding a new aerodynamic rotational stiffness would have profound effects on the airplane’s lateral dynamics. Figure 8.20a shows that this is indeed the case. The time constants of the two nonperiodic modes are seen to change very rapidly as the gain is increased, until with even a small gain, if < 1, that is, less than 1° of aileron for 1° of bank, these two modes have disappeared, to be replaced by a low frequency, heavily damped oscillation. The Dutch Roll remains virtually unaffected by the aileron feedback for any modest gain.
p —» 8a This root locus is shown in Fig. 8.20b. The largest effect is on the
roll mode, as might be expected, where a positive gain of unity (corresponding to a decrease in Lp) results in a substantial reduction in the magnitude of the large real root. This is accompanied by an increase in the spiral stability and a slight reduction in the Dutch Roll damping. A negative gain, (an increase in Lp) increases the Dutch Roll damping, shortens the roll mode time constant and causes a slight reduction in the magnitude of the spiral root (the latter not visible in the figure).
i//—> 8a Because ф is the integral of r, the transfer functions for ф have an s
factor in the denominator, and hence a pole at the origin. This is
seen in Fig. 8.20c. The expansion theorem (A.2,10) shows that the zero root of the characteristic equation leads to a constant in the solution for ф. This is consistent with the fact that the reference direction for ф is arbitrary.
The feedback of ф to aileron has little relative effect on the Dutch Roll and rolling modes. Its main influence is seen on the spiral and zero roots, which are quite sensitive to this feedback. For negative gain (stick left for yaw to the right) these two modes rapidly combine into an oscillatory mode that goes unstable by the time the gain is 0.5 (0.5° aileron for 1° yaw). For all positive gains, there is an unstable divergence.
v —>■ 8r This feedback (Fig. S.20d) represents rudder angle proportional to
sideslip, with positive gain corresponding to an increase in Nv. Note
that a gain of 0.001 for v corresponds to 8r/(3 = 0.774. The principal effect is to increase the frequency of the Dutch Roll while simultaneously decreasing the spiral stability, which rapidly goes unstable as the gain is increased. The reverse is true for negative gain. The roll mode remains essentially unaffected.
p —» 8r Roll rate fed back to the rudder has a large effect on all three modes.
For positive gain (right rudder for roll to the right) the damping of the Dutch Roll is increased quite dramatically—it is quadrupled for a gain of about 0.2° rudder/deg/s of roll rate. This is counterintuitive (see Exercise 8.10). At the same time, the damping of the roll mode is very much diminished, and that of the spiral mode is increased. With further increase in gain the two nonperiodic modes combine to form an oscillation, which can go unstable at a gain of about 0.4.
r—» 8r The large effects shown in Fig. 8.20/for the yaw damper case are
what would be expected. As an aid in assessing the damping perfor
mance, two lines of constant relative damping £ are shown on this figure. Negative gain corresponds to left rudder when yawing nose right. A very large increase in Dutch Roll damping is attained with a gain of 1, at which point there is a commensurate gain in the spiral damping. There is some loss in damping of the roll mode. The beneficial effects of yawrate feedback are clearly evident from this figure. The behavior for larger negative gains, beyond about — 1.4, is especially interesting. For this airplane at this flight condition, the two real roots combine to form a new oscillation, the damping of which rapidly deteriorates with further increase in negative gain. This feature complicates the choice of gain for the yaw damper. The pattern shown is not the only one possible. Figure 8.20г is the corresponding root locus for the STOL airplane of Sec. 8.9, flying at 10,000 ft and 200 k. It illustrates the importance of the location of the zeroes of the closed loop transfer function. For the jet transport the real zero, z is to the left of the real roll mode root p. For the
STOL airplane the reverse is the case. The direction of the locus emanating from this root is therefore opposite in the two cases, with a consequent basic difference in the qualitative nature of the dynamics. For the STOL airplane the Dutch Roll root splits into a real pair, one of which then combines with the spiral root to form a new low – frequency oscillation. In viewing Fig. 8.20/ it should be noted that it was drawn for the nondimensional system model, and hence the numerical values for the roots and the gains are not directly comparable with those of Fig. 8.20/.
8r Feeding back bank angle to the rudder produces mixed results (Fig. 8.20g). When the gain is negative, the spiral mode is rapidly driven unstable. On the other hand if it is positive, to improve the spiral, the Dutch Roll is adversely affected.
ф—* 8r The consequence of using heading to control the rudder is also
equivocal. If the gain is positive (heading right induces right rudder) the null mode becomes divergent. If the gain is negative the two nonperiodic modes form a new oscillation at quite small gain that quickly becomes unstable.