Gust Alleviation

For the final example in this chapter, we turn to a study of the application of auto­matic controls to reduce the response of an airplane to atmospheric turbulence (Byrne, 1983). This is obviously a useful goal for many flight situations, the benefits including increased passenger comfort, reduction of pilot workload, and possibly re­ductions in structural loading and fatigue, and in fuel consumption. The case reported here is for a STOL airplane, which is especially vulnerable to turbulence, since the relatively low operating speed makes it more responsive to turbulence, and because its duty cycle requires it to spend relatively more time at low altitudes where turbu­lence is more intense.

The numerical data used in the study were supplied by the de Havilland Aircraft Co., and although it does not apply to any particular airplane, it is representative of the class.

In this situation, where random turbulence produces random forces and moments on the airplane, which in turn result in random motion, the methods of analysis we have used in the foregoing examples, being essentially deterministic, are not applica­ble. Random processes have to be described by statistical functions. Let f(t) represent such a random function. Two of the key statistical properties that characterize it are the spectrum function, derived from a Fourier analysis of f{t) and the closely related correlation function (Etkin, 1972). The spectrum function or spectral density, as it is frequently called, is denoted Фц(ш). The area under the Ф^(ш) curve that is con­tained between the two frequencies or, and w2 is equal to the contribution to f2 (where f2 is the mean-squared-value of /)3 that comes from all the frequencies in the band wі => o)2 that are contained in the Fourier representation of /.

A BASIC THEOREM

When a system with transfer function G(s) is subjected to an input with spectrum function <3>„(ft>) the spectral density Фгг(ш) of the response r(t) is given by

Фгг(ш) = d>,,(ft))|G(/ft))|2 (8.9,1)

where G(iw) is the frequency response function defined in Sec. 7.5. There is a gener­alization of (8.9,1) available for multiple inputs (Etkin, 1972, p. 94). Figure 8.29 shows the relationships expressed in (8.9,1) for a second-order system of moderate damping.

In the case at hand the motion studied is the lateral motion, and the forces needed are Y, L, and N. The gust vector g (Fig. 8.1) that is the source of these forces has ele­ments that represent aspects of the motion of the atmosphere. It has four components

g = К Ps rig r2g]T

’The factor comes from the use of two-sided spectra.

be successful in doing so. In the cited study, the output chosen to be minimized was a passenger comfort index (see below).

A block diagram of the system considered is given in Fig. 8.30. The state vector is the set x = [v p г ф]т and the control vector is с = [Д, <5,.]T. The model includes full state feedback via the (2X4) gain matrix Kx, control servo actuators described by the (2X2) matrix J, and also includes the possibility of using measurements of the turbulent motion to influence the controls via the (2X4) gain matrix Kg.

We now proceed to complete the differential equation of the system. We start with the servo actuator transfer function J(.v), which is given, as in our previous ex­amples, by

c = PK x – Pc + PKjg

We can now combine (8.9,2) and (8.9,7) into the augmented differential equation of the system:

(8.9,8)

This can be written more compactly, with obvious meanings of the symbols, as

z = Az + Tg (8.9,9)

In the cited study, various control strategies were examined, differentiated primarily by whether or not gust “feedforward” was included (i. e., Кл Ф 0). When only state feedback was employed, (Kg = 0) linear optimal control theory was used to ascertain the optimum values of the gains in Кл. To this end, a function has to be chosen to be minimized. The choice made was a passenger comfort index made up of a linear combination of sideways seat acceleration along with angular accelerations p and r. The seat acceleration depends on how far the seat is from the CG, so an average was used for this quantity. The optimum that resulted entailed the feedback of each of the four state variables to each of the two controls, a very complicated control system! However, it was found that there was very little difference in performance between

this optimum and a simple yaw damper. The result is shown on Fig. 8.31, in the form of the spectral density of sideways acceleration of the rearmost seat. This form of plot, /Ф(/) vs. log /, is commonly used. The area under any portion of this curve is also equal to the mean-square contribution of that frequency band, just as with Ф(/) vs. /. Results are shown for three cases—the basic airframe with fixed controls, a conventional autopilot, and the selected yaw damper. Very substantial reduction of re­sponse to turbulence has clearly been achieved with a relatively simple control strat­egy-

An alternative to conventional linear optimal control theory was found to be bet­ter for the case when gust measurement is assumed to be possible. It stems from a theorem of Rynaski et al. (1979). It is seen from (8.9,2) that if one could make Be + Tg = 0 then one would have completely canceled the gust input with control action, and the airplane would fly as if it were in still air! This equality presupposes that the control is given by

c = —B’Tg

that is, that В has an inverse. This would require В to be a square matrix [i. e., to be (4X4)], which in turn would require that the airplane have two more independent

 Figure 8.32 Lateral acceleration spectra. Rearmost seat, with yaw damper and gust feedforward.

(and sufficiently powerful) controls than it actually has. Although this is not beyond the realm of imagination, it was not a feasible option in the present study. However, there is available the “generalized inverse,” which provides in a certain sense the best approximation to the desired control law. The generalized inverse of В is the inverse of the (2X2) matrix BTB. This leads to the control law

c = —(BTB) !BTTg

(The second BT is needed to yield a (2X1) matrix on the right-hand side). This law still requires, however, that all four components of g be sensed in order to compute c. Sensing all components of g is not impossible, indeed it may not even be impractical. However, a good result can be obtained with a subset of g consisting only of vg and r2g, both of which can be measured with an aerodynamic yawmeter, a sideslip vane or other form of sensor. The end result of combining gust sensing in this way with the yaw damper, with the gust sensor placed an optimum distance forward of the CG, is shown in Fig. 8.32. It is clear that this control strategy has been successful in achiev­ing a very large reduction in seat acceleration.