EXAMPLE—ALTITUDE AND GLIDE-PATH CONTROL

One of the most important problems in the control of flight path is that of following a prescribed line in space, as defined for example by a radio beacon. This is crucial in the landing situation under poor visibility when the airplane flies down the ILS glide slope. We shall discuss this case by considering first a simple approximate model that reveals the main features, and then examining a more realistic, and hence more complicated case.

FLIGHT AT EXACTLY CONSTANT HEIGHT—SPEED STABILITY

The first mathematical model we consider can be regarded as that corre­sponding to horizontal flight when a “perfect” autopilot controls the angle of attack in such a way as to keep the height error exactly zero. The result will show that the speed variation is stable at high speeds, but unstable at speeds below a critical value near the minimum drag speed. Neumark (11.2) recounts that this criterion was first discovered in 1910 by Painleve, and that it was at first accepted by aeronautical engineers and scientists, but later, on the basis of the theory of the phugoid which showed no such effect, was rejected as false. In fact, to the extent that a pilot can control height error by elevator control alone, i. e. to the extent that he approximates the ideal autopilot we have postulated, the instability at low speed will be experienced in manual flight. Since speed variation is the most noticeable feature of this phenomenon, it is commonly referred to as speed stability.

The analysis that follows is essentially that of Neumark, but adapted to the notation and methods of this book. The basic assumption that the flight path is exactly horizontal implies у = 0, or в = ctx (see Fig. 4.4.), whence Ад = Да. An exactly horizontal flight path also implies L = W. The pitching moment equation is specified to be identically satisfied by means of an appropriate but unspecified control device that supplies the needed pitching moment as required. The system equations are then (5.13,19) with Act. = A6, ye = 0 and the third equation missing. We further specify that a. T = 0. The equations are then

DV	=	і <6V – cv		0	_<V« 2fl	Av Да Л 4
Da.	Clv + 2CVe 2/г – f – Cl&	о La + CDf	2/* – 2/i + Ci&	0
Da	0	0	1	0	Да

(11.5,1

We now make some simplifying approximations, i. e. that the speed deriv­atives CDv and CLf are negligible and that 2ц > CL&, CL^. Actually these are very weak approximations for a conventional airplane in cruise con­figuration. On combining the Да terms of the first two equations, eliminating q by means of the third, and observing that CLe = CWg, we get

2fiDV = CTv AV – CD Да

v

0 = 2CL AV + Gjr Да (11.5,2)

e

Elimination of Да yields the first-order speed equation

2fiDV = (cTv + 2Cd G-^ At (11.5,3)

The speed variation following an initial speed error AF0 is clearly expo­nential,

AV = AV0e}>f

with time constant given by

2M OlJ

We must now specify a propulsion system in order that CTfr may be deter­mined. The result finally obtained depends on this choice, but only in the actual value of the critical speed, not its existence. We arbitrarily choose a constant-thrust engine, for which (see Table 7.1)

®ту — 2<^те — —D)S Equations (11.5,4) then yield

(11.5,5)

The factor in the inner parentheses can be rewritten as

where dCLjdCD is the slope of the tangent to the drag polar, and CLjCD is the slope of the secant, see Fig. 10.2. Just as in Sec. 10.2, Eq. (10.2,17), this factor passes through zero at the point C’L, С’р where LjD is a maximum. It is positive for CL > C’L and negative for CL < C’L. If V be the speed
corresponding to (L/D)max, then the speed variation is seen to be stable for V > V, but unstable for V < V. That is, speed errors will die out at high speeds, but grow at low speeds. This phenomenon is seen to be related to the change of sign of KyS that occurs at the same critical speed (Sec. 10.2).