# Category Dynamics of Flight

## Analytical Tools

A. l Linear Algebra

In this book no formal distinction is made between vectors and matrices, the former being simply column matrices, as is common in treatments of linear algebra. In par­ticular the familiar vectors of mechanics, such as force and velocity, are simply three – component column matrices. We use boldface letters for both matrices and vectors, for example, A = [a, y] and v = [u,]. The corresponding lowercase letter defines the magnitude (or norm) of the vector. The transpose and inverse are denoted as usual by superscripts, for example, AT and A-1. When appropriate to the context, a subscript is used to denote the frame of reference for a physical vector, for example, V£ = [u v wr denotes a vector whose components in frame FE are (и, v, w). The three – component vectors of physics have the following properties:

Scalar product

c = a • b = arb = + a2b2 + a3b3 (A.1,1)

c is a scalar, with magnitude ab cos в, where в is the angle between a and b

Vector product

(A. 1,2)

c is a vector perpendicular to the plane of a and b, with direction following the right-hand rule for the sequence a, b, c and has the magnitude ab sin в, where в is the angle (< 180°) between a and b

Unit vectors

The basis unit vectors are i, j,k such that

(A. 1,4)

 J*co ^ ‘ t=0

where x(O) is the value of x(t) when t = 0.1 The process may be repeated to find the higher derivatives by replacing x(t) in (A.2,2) by x(t), and so on. The result is

dn~^x dn~2x

= ~ (0) – s (0)—————— VO) + snx(s) (A.2,3)

TRANSFORM OF AN INTEGRAL

Let the integral be

and let it be required to find y(s). By differentiating with respect to t, we get

= x(t)

thus

and

1 1

y(s) = — x(s) + — y(0)

## 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.

## Roll Controller

This example is of another common component of an AFCS, a control loop that maintains the wings level when flying on autopilot, or that can be commanded to roll the airplane into a turn and hold it there. We shall see in this particular case that the resulting turn is virtually truly banked, even though no special provision has been made to control sideslip.

The block diagram of the system is shown in Fig. 8.26. It incorporates the yaw damper described in the previous section and adds two additional loops. The outer loop commands ф. The ф error is converted to a roll rate command by Jp, and it is the roll rate error that is then used to drive the aileron servo actuator. If the roll rate fol­lowed the command instantaneously, without lag, the bank angle response would be exponential (i. e., ф ф). In reality of course this ideal behavior is not achieved be­cause of the airframe and servo dynamics.

For this example we use the state vector approach to system modeling in order to provide another illustration, one that differs in detail from that of Sec. 8.5.

As usual the starting point is the basic aircraft matrix equation,

x = Ax + Be (8.8,1)

in which x = v p г ф]т and c = [5a Sr]T.

The differential equations that correspond to the various control transfer func­tions in the figure are found as follows. For the yaw damper components, we have the same form of transfer functions as previously, that is,

(8.8,2)

s +

s H—

Krshr

s + Ays + ±

TvO / t.

From (8.8,4) we get the differential equation

For Jp we use the constant Kp, and for Ja we use a first order servoactuator

s +

The relation between Sa, p, and фс is seen from the diagram to be

Ф;(Рс P) ^а^р^Фс Ф) JaP

When we substitute for Ja and Jp the differential equation that results is

Ka 1 KaK„

ф-^р–8а+~^фс

T T T T

‘ (1 ‘ П 1 n • n

Equations (8.8,5) and (8.8,8) are the additional equations required to augment the ba­sic system (8.8,1) to accommodate the addition of the two control angles as depen­dent variables. However, a little more manipulation is needed of (8.8,5). To put it in first-order form, we define the new variable

y = 8r (8-8,9)

and to put it in canonical form, we must eliminate r. This we do by using the third component equation in (8.8,1). When these steps have been aken the system can be assembled into the matrix equation

z = Pz + Q фс

where z = [u p г ф 8a Sr y]T

The matrices P and Q are:

Q = [0 0 0 0 (KaKpha) 0 0]T (8.8,12)

Equation (8.8,10) was solved by numerical integration for two cases, with the results shown on Figs. 8.27 and 8.28. The various gains and time constants used were se-

 Figure 8.27 Response of roll controller to initial ф of 0.262 rad (15°). (а) ф, p, and 8a. (b) /3, г, ф, 8r.

lected somewhat arbitrarily, as follows:

Kp = 1.5 Ku = -1.0 Kr = -1.6; r„ = .15 тг = .30 тм = 4.0

On the first of these figures, response to an initial bank error, we see that all the state variables experience a reasonably well damped oscillatory decay, and that the maxi-

 0 5 10 15 20 25 30 Time, s (fe) |3, r, V|/, Sr Figure 8.28 Response of roll controller to roll command of 0.262 rad (15°). (а) ф, p, 8a. (b) Д r, ф, 8r.

mum control angles required are not excessive—about 20° for the aileron and less than 1° for the rudder. The time taken for the motion to subside to negligible levels is equal to about two Dutch Roll periods. All the variables except ф subside to zero, whereas ф asymptotes to a new steady state. When level flight is reestablished, the airplane has changed its heading by about 1.8°.

The second figure shows the response to a 15° bank command. The new steady state is approached with a damped oscillation that takes about 15 s to decay. The steady state is clearly a turn to the right, in which r has a constant value and ф is in­creasingly linearly. All the other variables, including the two control angles, are very small. It is especially interesting that the sideslip angle is almost zero. Clearly this controller has the capability to provide the bank angle needed for a coordinated turn. (The angle of attack and lift would of course have to be increased.)

## Yaw Damper

Yaw dampers are widely used as components of stability augmentation systems (SAS); we saw the potential beneficial effects in Fig. 8.20/. At first glance the yaw damper would appear to be a very simple application of feedback control princi­ples—just use Fig. 8.20/as a guide, select a reasonable gain, and add a model for the servo actuator/control dynamics. However, it is not really that simple. There is an­other important factor that has to be taken into account—namely that during a steady turn, the value of r is not zero. If, in that situation, the yaw damper commands a rud­der angle because it senses an r, the angle would no doubt not be the right one needed for a coordinated turn. In fact during a right turn the yaw damper would al­ways produce left rudder, whereas right rudder would usually be required (see Fig. 7.24). This characteristic of the yaw damper is therefore undesirable. To eliminate it, the usual method is to introduce a high-pass or “washout” filter, which has zero gain in the steady state and unity gain at high frequency. The zero steady state gain elimi­nates the feedback altogether in a steady turn. The system that results is pictured in Fig. 8.21, where the meaning of the filter time constant is illustrated. For the servo actuator/rudder combination of this large airplane we assume a first order system of time constant 0.3 sec.

The closed loop transfer function for Fig. 8.21 is readily found to be [see (8.2,1)]

JGrSr

1 + WJGrSr

This transfer function was used to calculate a number of transient responses to illus­trate the effects of J(s) and

As a reference starting point, Fig. 8.22 shows the open loop response (W = 0) to

 a Figure 8.21 Yaw damper.

a unit impulse of yaw rate command rc. It is evident that there is a poorly damped os­cillatory response (the Dutch Roll) that continues for about 2 min and is followed by a slow drift back to zero (the spiral mode). Fig. 8.22a shows that the control dynam­ics (i. e., J(s)) has not had much effect on the response.

Figure 8.23 shows what happens when the yaw damper is turned on with the same input as in Fig. 8.22. It is seen that the response is very well damped with either of the two gains shown, which span the useful range suggested by Fig. 8.20/, and that the spiral mode effect has also been suppressed.

It remains to choose a time constant for the washout filter. If it is too long, the washout effect will be insufficient; if too short, it may impair the damping perfor­mance. To assist in making the choice, it is helpful to see how the parameter a = 1/tw0 affects the lateral roots. Figure 8.24 shows the result for a gain of К = —1.6. The roots in this case consist of those shown in Fig. 8.20/ plus an additional small real root associated with the filter. It is seen that good damping can be realized for values of a up to about 0.3, that is, for time constant т down to about 3 s. This result is very dependent on the gain that is chosen. While the oscillatory modes are behav­ing as displayed, the real roots are also changing—the roll root decreasing in magni­tude from —2.31 at a = 0 to —1.95 at a = 0.32. The new small real root starts at the origin when a = 0 and moves slowly to the left, growing to -0.00464 at a = 0.32. When the fdter time constant is 5 s the small root is —0.0038, corresponding to an aperiodic mode with rhalf = 182 s. It is instructive to compare the performance of the yaw damper with and without the filter for an otherwise identical case. This is done in Fig. 8.25. It is seen that the main difference between them comes from the small real root, which after 5 min has reduced the yaw rate to about 5% of its peak value. This slow decay is unlikely to present a problem since the airplane heading is in­evitably controlled, either by a human or automatic pilot. In either case, the residual r would rapidly be eliminated (see Sec. 8.8).

 Time, s (a) Initial response, 0-60 s

 Figure 8.22 Yaw rate impulse response—open loop, W = 0. (a) Initial response, 0-60 sec. (b) Long-term response.

## Lateral Control

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-

 Time, s (a)

 (M Figure 8.18 Altitude-hold 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 yaw-rate 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.

## Solutions

The above equations contain first derivatives on the right side as well as on the left side and hence are not in the canonical form. This is no impediment to numerical in­tegration, however, since if the derivatives are calculated in the sequence given, each one that appears on the right has already been calculated in one of the preceding equations by the time it is needed. The twelfth and final relation needed is that which describes the limiter, in the form

= f(y4)

From the values of CDo and CLa given in Sec. 6.2, we find that D0 = T0 =

0. 0657 W. In Sec. 7.6 it was given that 8p = 1 corresponds to a thrust of 0.3VT. It fol­lows that zero thrust corresponds to y4 = —0.0657/.3 = —0.219. The nonlinear rela­tionship for y4 is therefore implemented in the computing program by a program fragment equivalent to

Ь8р = y4

IFy4<-0.219 THEN A8p = -0.219 (8.5,14)

IF y4 > 0.10 THEN A8p = 0.10

where the maximum engine thrust has been assumed to be 10% greater than cruise thrust. Equations (8.5,13 and 8.5,14) are convenient for numerical integration. We have calculated a solution using simple Euler integration of the equations for the ex­ample jet transport with the matrices A and В given in Secs. 6.2 and 7.6, and with the following control parameters: те = 0.1; тр = 3.5; к = 0.0002; aQ = а{ = a2 = —0.5; b0 = 0.005; b, = 0.08; b2 = 0.16. Figure 8.18 shows the performance obtained in re­sponse to an initial height error of 500 ft.

It is seen that the height error is reduced to negligible proportions quickly, in about 20 s, accompanied by a theta pulse of similar duration and peak magnitude about 7°. Even with extreme throttle action, the speed takes more than 2 min to re­cover its reference value. This length of time is inherent in the physics of the situa­tion and cannot be shortened significantly by changes in the controller design. On the other hand, there is no operational requirement for more rapid speed adjustment when cruising at 40,000 ft.

The peak elevator angle needed is less than 3°, but the thrust drops quickly to zero, stays there for about 30 s, then increases rapidly to its maximum. Toward the end of the maneuver the throttle behaves linearly and reduces the speed error smoothly to zero.

## The Differential Equations

The basic matrix differential equation of the airframe, with 0O = 0, and ДzE = —Ah is obtained from (4.9,18) and (7.6,4). Up to this point, we have neglected engine dy­namics and in effect regarded thrust as proportional to Sp. The matrix В of (7.6,4) is structured in that way. To accommodate the facts that Sp actually represents the throt­tle setting, not the thrust, and that the two are dynamically connected, we need to in­troduce two new symbols, y5 and c*. The quantity y5, when multiplied by XSp, etc. yields the aerodynamic force and moment increments AXC, etc., and c* is defined be­low. The differential equation of the airframe is then

x = Ax + Be* (8.5,4)

where x = [Дм w q в A h]r

c* = A y5f A = [аи]

В = [bu]

To obtain the differential equations of the three control elements, we begin with their transfer functions, which are specified for this example to be

Je(s) = (a0s~1 + a, + a2s){ 1 + re5)_l

= (a0 + a{s + a2s2)(s + Tes2)-1 (8.5,5)

Cp(s) = (Vі + bt + b2s)

= (b0 + bjS + b2s2)/s (8.5,6)

Jp(s) = 1/(1 + v) (8.5,7)

The first two of these contain proportional + rate + integral controls, all of which were found to be needed for good performance. The time domain equations that cor­respond to the elements of the controller are then as follows (verify this by taking their Laplace transforms):

в, = —к Ah (a)

теА8е + A 8e = a0e0 + ахёв + а2ёв (b) (8.5,8)

У 4 = V„ + V„ + b2eu (c)

y5 = A 8p – y5 (d)

After substituting the expressions for the two error signals, (8.5,8) yield three equa­tions for the controls

теА8е + A 8e = —ка2АЇг — a26 — kaxAh — axQ — ka0Ah — а0в (a)

y4 = – b2Au – bxAit – b0Au (b) (8.5,9)

Vs = “ Л (c)

For convenient integration we want a system of first-order equations and therefore have to do something about the second derivatives in (8.5,9). Since в = q we can re­place в with q. For the other second derivatives, we introduce three new variables, as follows:

Уі = дй О)

У2 = ДА (b) (8.5,10)

Уз = (с)

With these definitions, (8.5,9a and b) can now be rewritten in terms of first deriva­tives as

теУз + Уз = ~{a2q + ka^ + axq + а0в + ka0Ah + kaxy2) } j

У4 = ~ФіУ + b0Au + bxyx) ‘ ’

The state vector now consists of the original five variables from (8.5,4) plus the two control variables ASe and A8p, plus the five y, defined above, making 12 in all. We therefore require 12 independent equations. From the foregoing equations (8.5,4)

(8.5,10) , (8.5,11), and (8.5,9c) we can get 11 of the required differential equations. That fory, is obtained from (8.5,10a) by differentiating the first component of (8.5,4) and that for y2 by differentiating the fifth. The result of that operation is

y, = aX2w + axxyx + aX4q + bxxy3 + bX2y5 y2= – w + u0q

Finally, the 11 independent differential equations are assembled as follows:

Am = у і

w = a2XAu + a22w + a23q + b2XA8e q = a3]Au + a32w + a33q + b3XA8e d = q Ah = y2 A4 = Уз

У s = (~У s + Ь8р)/тр (8.5,13)

У = аххух + aX2w + aX4q + bxxy3 + bX2y5 y2 = u0q – w

y3 = —(a2q + ka2y2 + axq + aQ0 + ka0Ah + kaxy2 + у3)/те У 4 = “ОгУ і + b0Au + bxyx)

## EXAMPLE—AN ALTITUDE CONTROLLER

In view of the above, we illustrate an altitude controller that also incorporates control of speed, using once again our example jet airplane. This time we make the system model more realistic by including first-order lag elements for the two controls: that for the elevator is mainly associated with its servo actuator (time constant 0.1 s); and that for the throttle with the relatively long time lag inherent in the build up of thrust of a jet engine following a sudden movement of the throttle (time constant 3.5 s). An­other feature that is incorporated to add realism to the example is a thrust limiter. Be­cause transport aircraft inherently respond slowly to changes in thrust, the gains cho­sen to give satisfactory response for very small perturbations in speed will lead to a demand for thrust outside the engine envelope for larger speed errors. We have there­fore included a nonlinear feature that limits the thrust to the range 0 < Г < 1.1Г0. This contains the implicit assumptions (quite arbitrary) that the airplane, flying near its ceiling, has 10% additional thrust available, and that idling engines correspond to zero thrust.

At the same time this example illustrates an alternative approach to generating the analytical model of the system, in terms of its differential equations. In the previ­ous illustrations we have, by contrast, used what may be termed “transfer function al­gebra” to arrive at transfer functions of interest, and then used these to obtain what­ever results were desired. The end result of the modeling to follow is a system of differential equations that is then integrated to get time solutions. Since the limiter is inherently a nonlinear element, it is in any case not possible to include it in a transfer function based analysis.

The system block diagram is shown in Fig. 8.17. The commanded speed and alti­tude are the reference values u0 and /z„, so that the two corresponding error signals are — Aи and — Ah. Note that h is the negative of zE used in Chap. 4. The inner loop for в is that previously studied in Sec. 8.3, with the Je(s) modified to account for the

elevator servo actuator. The logic of the outer loop that controls h warrants explana­tion. If there is an initial error in h, say the altitude is too low, then in order to correct it, the airplane’s flight path must be deflected upward. This requires an increase in angle of attack to produce an increase in lift. The angle of attack and the resulting lift could of course be produced by using an angle of attack vane as sensor, and no doubt an angle of attack commanded to be a function of height error would be very effec­tive. It might be preferred, however, to use the vertical gyro as the source of the sig­nal, and since short-term changes in в are effectively changes in a, then much the same result is obtained by using в as the commanded variable. We have chosen to use stability axes, so that in the steady state, when Ah is zero, the correct value of в is also zero. Thus, in summary, the system commands a pitch angle that is proportional to height error and the inner loop uses the elevator to make the pitch angle follow the command. While all this is going on the speed will be changing because of both grav­ity and drag changes. The quickest and most straightforward way of controlling the speed is with the throttle, and the third loop accomplishes that. (The symbols y4 and y5 denote the inputs to the limiter and the airframe, and are elements of the state vec­tor derived below.)

## 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, or when the air­plane flies down the ILS glide slope. We 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 (1950) 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 pilots can control height error by elevator control alone, that is, to the extent that they approximate 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.

We could analyze this case by applying (4.9,18) to the stated flight condition. However, it is both simpler and more illuminating to proceed directly from first prin­ciples. The airplane is flying on a horizontal straight line at variable speed V. It is im­plied that a is made to vary, by controlling 8e, in such a way that the lift is kept ex­actly equal to the weight at all times. The equation of motion is clearly

mV=T-D (8.5,1)

where T is the horizontal component of the thrust, and D is the drag. Since the speed cannot change very rapidly, then neither does a, and we can safely ignore any effects of q and a on lift and drag. Consequently, T and D are simply the thrust and drag or-

dinarily used in performance analysis, as displayed in Fig. 8.16. We denote the refer­ence thrust and drag by T0 and D0 and define the stability derivatives

Ту = дт/dv and Dv = dD/dV so that T — D = (T0 + TvAV) – (D0 + DvAV)

Since V = V0 + AV, and T0 = D0, (8.5,1) becomes

mAV = (Tv – Dv)AV (8.5,2)

This first-order differential equation has the solution

AV = aekt

with A = {Tv — Dv)/m (8.5,3)

Tv and Dv are the slopes of the tangents to the thrust and drag curves at their intersec­tion. If they intersect at a point such as P in Fig. 8.16, then Tv < Dv, A is negative, and the motion is stable. If, on the other hand, the flight condition is at a point such as Q, the reverse is the case. A is then >0, and the motion is unstable. If when flying at point Q there is an initial error in the speed, then it will either increase until it reaches the stable point P or it will decrease until the airplane stalls. The stable and unstable regimes are bounded by the speed V*, which is where the thrust curve is tan­gent to the drag curve. V* will be the same as Vmd of Fig. 7.1 if Tv = 0. Hence the appellation “back side of the polar” is used to describe the range V < V*, with refer­ence to the portion of the aircraft polar (the graph of CL vs. CD) for which C, is greater than that for maximum L/D.

Although we have analyzed only the case of horizontal flight, the result is similar for other straight-line flight paths, climbing or descending (see Exercise 8.8). Flight in the unstable regime can indeed occur when an airplane is in a low speed climb or landing approach. This speed instability is therefore not entirely academic, but can present a real operational problem, depending on by what means and how tightly the aircraft is constrained to follow the prescribed flight path. An important point insofar as AFCS design is concerned is that for speeds less than V* it is not possible to lock

exactly onto a straight-line flight path, and at the same time provide stability, using the elevator control alone, no matter how sophisticated the controller! To achieve sta­bility it is mandatory to use a second control. This would most commonly be the throttle, but in principle spoilers that control the drag could also be used.

## Speed Controller

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:

(8.4,1)

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)

XSp u0Mjs

m f(s)

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)

(8.4,3)

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

(8.4,7)

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.