Category Dynamics of. Atmospheric Flight

To be able to assess aircraft handling qualities one must have a measuring technique with which any given vehicle’s characteristics can be rated. In the early days of aviation this was done by soliciting the comments of pilots after they had flown the aircraft. However, it was soon found that a communi­cations problem existed with pilots using different adjectives to describe the same flight characteristics. These ambiguities have been alleviated considerably by the introduction of a uniform set of descriptive phrases by workers in the field. The most recent set (ref. 12.12) is referred to as the “Cooper-Harper Scale” where a numerical rating scale is utilized in con­junction with a set of descriptive phrases. This scale is presented in Table 12.4 and is similar but not identical to previous scales developed separately by Cooper and Harper. Care must be taken in interpreting past research, to determine which scale the results are based on. To apply this rating technique it is necessary to describe accurately the conditions under which the results were obtained. In addition it should be realized that the numerical pilot rating (1 to 10) is merely a shorthand notation for the descriptive phrases and as such no mathematical operations can be carried out on them in a rigorous sense. For example a vehicle configuration rated as 6 is not necessarily

Cooper-Harper Bating Scale (Ref. 12.12)

twice as bad as one rated at 3. The comments from evaluation pilots are extremely useful and this information will provide the detailed reasons for the choice of a rating.

Other techniques have been applied to the rating of handling qualities. For example, attempts have been made to use the overall system performance as a rating parameter. However, due to the pilot’s adaptive capability, quite often he can cause the overall system response of a bad vehicle to approach that of a good vehicle, leading to the same performance but vastly differing pilot ratings. Consequently system performance has not proved to be a good rating parameter. A more promising approach involves the measurement of the pilot’s physiological and psychological state. Such methods lead to objective assessments of how the system is influencing the human controller. The measurement of human pilot describing functions is part of this technique.

The assessment of handling or flying qualities of airplanes depends in the final analysis on pilot opinion. The earhest requirement (ref. 12.30) simply stated, “During this trial flight of one hour it (the airplane) must be steered in all directions without difficulty and at all times be under perfect control and equilibrium.” From this simple but hard-to-interpret statement has evolved a much more quantitative and sophisticated set of criteria. These are still far from perfect, and the introduction of each new class of vehicle, STOL (ref. 12.30), rotorcraft, SST, etc., requires a reassessment of the existing criteria for application in the new situation.

When a pilot flies an aircraft he forms subjective opinions concerning the suitability of the man-machine system for performing the assigned task. In arriving at an assessment he is influenced by many parameters. These

508 Dynamics of atmospheric flight range over a wide spectrum and include:

1. Aircraft stability; response to external disturbances such as turbulence.

2. Aircraft controllability; the response of the aircraft to actuation of the controls.

3. Cockpit design; the ease with which instruments can be read; the comfort of the seat.

4. View from the cockpit; on landing approach is a sufficiently clear view of the ground provided?

5. Mission; e. g. high-altitude cruise, landing approach in a crosswind.

6. Pilot’s background and emotional and physical state; the familiarity of the pilot with the present aircraft and mission; impaired functioning arising from emotional and physiological factors.

7. External environment; visibility and weather conditions.

The term handling qualities is used to refer to those characteristics of the aircraft which the pilot considers to influence the ease of performing the mission. Much of the work in the area of handling qualities has centered on the determination of the influence of aircraft stability and control. It is the aim of this research to establish general specifications, to ensure that future vehicles can complete their intended missions safely, efficiently, and with a minimum of pilot fatigue.

In an age where more and more of the aircraft control task is being devolved to automatic equipment (e. g. autopilots, blind landing systems, stability augmentation systems) the role of the human pilot will perhaps slowly change from that of an active element in the man/machine system to that of a manager overseeing the operation of the automatic controls. In this situation the pilot must monitor the performance of the equipment and be prepared to take over in the event of a failure. This philosophy quite rightly predicates that the human pilot should make the final decisions that determine the fate of the craft under his command. Moreover, human pilots are uniquely capable of assessing the meaning of complex data patterns which indicate the state of the vehicle under conditions that the automatic equipment has not been designed to handle (witness Apollo 13!). On the other hand, this modus o’perandi poses a serious problem for the pilot, for he is then expected to assume manual control of a vehicle at a critical time, following a system failure. If he is unable to make the transition from passive to active control with sufficient speed and precision, disaster could well be the result.

Thus it appears that if the pilot is expected to assume manual control at any time, the system should be so structured that he is either kept actively in the control loop at all times or is constantly made aware of the feel of the present aircraft configuration through some auxiliary task which he can practice on during critical phases of the flight. Research on the ability of pilots to control vehicles following stability augmentation system (SAS) failures has indicated that the resulting step change in vehicle dynamics can lead to an unstable man-machine system and loss of control (ref. 12.29). The mechanism behind this problem is as follows. With the SAS operating properly the vehicle dynamics are satisfactory and the pilot adopts a control technique to suit. The sudden SAS failure results in less satisfactory vehicle dynamics, which demands a much more concentrated effort on the part of the pilot in order to maintain control. Immediately following SAS failure, however, the pilot attempts to continue to employ the control technique he has been using previously with the SAS operative. This combination of man-machine dynamics can lead to an unstable system. If the system is to be fail-safe the pilot must be able to detect the change quickly and alter his control technique in time to recover from the upset. Consequently the advent of more automatic equipment does not diminish the need to study the role of man in the vehicle control loop. On the contrary, it generates new and more difficult problems requiring an even better understanding of the human pilot.

Pilot models are being developed to describe the control situation when displays other than the compensatory type are utilized. An example of this is the pursuit display. The single-degree-of-freedom tracking task with a pursuit display is identical to the compensatory task of Fig. 12.3a except that the displayed variables are different—i. e. the pilot has different infor­mation. In the compensatory task only e(t) is displayed (Fig. 12.3) whereas in the pursuit task both i(t) and m(t) are separately displayed. Figure 12.6 illustrates the difference between the two displays for the same system state. It can be seen that additional information is presented to the pilot on the pursuit display. Although e(f) is available in both cases, only the pursuit display separates the error into its components and conveys this information to the pilot. For example, a pursuit display tells the pilot whether his tracking error is due to a difficult input signal, i(t), or due to erratic pilot control of the aircraft, m(t), which in turn can affect his strategy in bringing the tracking

Fig. 12.6 Displays, (a) Compensatory. (6) Pursuit.

error under control. Whether the one display or the other is best for the mission at hand is a complex function of the task performed.

A technique for measuring human-pilot describing functions has been developed for the single-degree-of-freedom tracking task with a pursuit dis­play for situations where a secondary disturbance signal is present. This task is shown in Fig. 12.7. It might for example represent a mid-air refueling task where i(t) represents the tanker’s altitude and the secondary disturbance g(t) represents turbulence acting on the controlled aircraft. The model of this task can be formulated in several ways. Figure 12.8 shows two useful forms of the model. The pilot is represented by a pair of describing functions (F2(s), Y2(s)) or (Ys(s), Yfls)) since the pilot is considered to have

two inputs and one output. [Since e(t) = i(t) — m(t) it is redundant to con­sider a case with three inputs.] Again n(t) represents the remnant. The describing function pairs are chosen to minimize the root mean square of that part of the vehicle input signal o(t) which is accounted for by n(t) (as was done for the compensatory display). The describing functions that result are [where the aircraft transfer function is A(s)]:

Гі(г’со) = (77— + A(ia>)

Г2(гш) = (1 — Y1(ico)A(ia))) ■ ф2(а>)

Y3(ia>) = Y2(ico)

Y^ico) = Y^ico) – f – Гг(т>)

where фАсо) =

Ф«(ю)Ф„(й>)-Фл(®)Ф«(«>)

__ Ф»ФИ(<») – Og0(tt))O,-g(ft>)

2 Фц(о>)Фдд(а>) — Фід(а>)ФЯі(о>)

The denominators of (12.3,1) both vanish if either (i) g(t) = 0 or (ii) g(t) = const X i(t). In either of these cases the measurement of фг and ф2, and hence of the describing function pairs, would not be possible. In addition the following are found to hold:

ФшМ = Фдп{ы) = 0

Ф0»

X {|Т’2(іса)|2фіі(й)) + І Т1(г<ы)|2Ф!,д(а)) + 2 Re [F|(*’o)) (12.3,3) X Г1(*со)фг9(с»)]}

As yet no general set of rules comparable to those for the compensatory task has been developed to cover this model. A typical measured pursuit model is shown in Fig. 12.9. It was found that the measured data could be fitted quite well by describing functions of the form (12.2,2). The task in this example was the same as the one used for Fig. 12.5, except that a pursuit display was used and a secondary disturbance added. If g{t) is made very small it is assumed that such models will also approximate pursuit tasks with no secondary disturbances.

Fig. 12.9 Experimentally measured pilot model; pursuit display, (from ref. 12.5) 504

Many critical tasks performed hy pilots involve them in activities that resemble those of a servo control system. For example, the execution of a landing approach through turbulent air requires the pilot to monitor the aircraft’s altitude, position, attitude, and airspeed and to maintain these variables near their desired values through the actuation of the control system. It has been found in this type of control situation that the pilot

492 Dynamics of atmospheric flight Table 12.3

Hand-Operated Control Forces (From Flight Safety Foundation Human Engineering Bulletin 56-5H) (see figure on page 495)

can be modeled by a set of constant-coefficient linear differential equations (termed “human-pilot describing functions”). Much of the original research in the field of human-pilot describing functions has concentrated on the pilot’s performance in a single degree of freedom compensatory tracking task with random-appearing system inputs. In a single-degree-of-freedom task the pilot controls a single state variable through the actuation of a single control. A compensatory display is one in which the tracking error is pre­sented, regardless of the source of the error. Fig. 12.1 shows the block diagram for such a task. Here a pilot is concentrating on controlling the pitch attitude

of the aircraft through the use of the artificial horizon display. The system input in this case is turbulent air which produces random pitching motion of the vehicle.

The pilot model used in compensatory tasks consists of the describing function and the remnant as shown in Pig. 12.2. (See also Sec. 3.5.) Here the task is the same one presented in Fig. 12.1, but the human pilot has been replaced by a mathematical model. The model consists of two parts, as shown: Y(s), the linear describing function (written in Laplace transform notation), and n(t) the remnant. Since a linear model is never able to describe the pilot completely, Y(s) is insufficient by itself, and it is necessary to include the remnant n(t), which is that signal that must be added in order to have all the time signals circulating in the system of Fig. 12.2 correspond exactly to those of Fig. 12.1 when the identical input is present. The Y(s) selected to describe the pilot in any particular task is chosen so as to minimize that part of the input signal to the aircraft which arises from n(t). Thus the linear pilot model that results is that which accounts for as much pilot input to the aircraft as possible, and a measure of its adequacy is the fraction of the pilot input to the aircraft accounted for by Y(s).

The human-pilot describing function is useful in studying two classes of problems. In the first the describing functions derived from previous re­search are utilized to aid the systems designer. With a mathematical de­scription of the pilot at hand he can close the loop around the mathematical description of the proposed vehicle in order to predict the overall system response. The second type of study involves the measurement of actual human-pilot describing functions as the pilot flies a particular vehicle in order to obtain an objective measure of how the task affects the pilot.

Due to the complex nature of the situation it is possible to model the pilot in many ways and to measure the model by employing a variety of techniques. One of the most successful approaches to the measurement problem utilizes power-spectral-density measurements of signals circulating in the control loop. The general case of a tracking task of one degree of freedom with a compensatory display is illustrated in Fig. 12.3a. In this task the pilot must control the aircraft response m(t) in such a fashion that it matches as closely as possible the desired aircraft response i(t). The pilot does this by viewing the instantaneous error e(t) and altering his input o(t) to the aircraft. It is found that the pilot’s control technique is primarily influenced by the type of input i(t), the dynamics of the control system, the type of display and the dynamics of the aircraft. Any useful pilot model must reflect these influences.

Past research in this field has concentrated on tasks with random appearing input signals i(t) because so many real-world situations involve this type of disturbance. Thus the pilot models that have been developed apply strictly only to tasks with the above type of input. The system of Fig. 12.3a is modeled by that of Fig. 12.36. Note that the model includes the dynamics of the control system and that the signal o(t) corresponds to the position of the control column. It has been found that in the frequency band of primary interest and for the type of controls normally found in aircraft, such a model is fairly insensitive to the exact control system used and that pilot models developed on this basis are quite general. Now the linear system of Fig. 12.36 can be redrawn as the point by point sum of the two linear systems of Fig. 12.4 (if the aircraft is assumed to be a linear system). It follows that

e(t) = eft) + eft)

o{t) = off) + oft)
m(t) = mft) + m2(t)

The describing function Y(s) is chosen to minimize the r. m.s. value of oft). Note that this is not the same criterion as used in defining the open-loop describing function in Sec. 3.5, where the mean-square-remnant was mini­mized in the presence of a fixed input. The difference of course is that we are dealing here with a closed-loop system, in which signals derived from the remnant circulate the loop and appear at the input to the pilot. The process of minimizing o22(t) can he carried out in a manner basically similar to that used in Sec. 3.5 with the result (ref. 12.2)

Y(ico) = Фі0(а>)ІФіе(а>) (12.2,1)

where Фг0(<ы) is the cross-spectral density of i(t) and o(t) etc. (See Sec. 2.6). We also find that Фіп(т) = 0,—i. e. the remnant is uncorrelated with the input signal i(t). This linear model, T(-s), is a best fit in the root-mean-square

Fig. 12.5 Experimentally measured pilot model, compensatory display, (from ref. 12.5)

1.0

~l I I I j I I I I Bandwidth of i(f)-2.8 rad/sec Aircraft dynamics-if/s

This form is preferred for measurement because it is not possible to measure o2(t) directly. The remnant n(t) exists because in actuality the human pilot is not operating exactly as a linear/invariant mathematical system. The signal n(t) is a random-appearing variable and hence is not predictable. However, some measurements have been made of its statistical properties (ref. 12.6) over a range of task variables. Figure 12.5 shows a typical experi­mentally measured pilot describing function together with p2. In this task the input to the pilot was the deflection of the artificial horizoil display (in inches) and his output was control column deflection (in degrees). It is seen from the plot of p2 that the describing function models the low-frequency performance of the pilots quite well, but is less satisfactory for a> > 5 rad/sec.

The following form for the describing function has been developed to cover the single-degree-of-freedom compensatory tracking task with a random-appearing input (ref. 12.3):

(12.2,2)

In this formulation e~TS represents the pure transmission time delay within the pilot associated with nerve conduction and stimulation, т is estimated to range from.06 to.10 sec. The factor in curly brackets is a reasonable representation of the dynamics of the neuromuscular system of the arm with typical values: 1/TN = 10 sec-1, ojn = 16.5 rad/sec, and £jV = .12. (TKs + 1)/(T’Ks – j – 1) represents a very low frequency lag-lead component. The remaining terms Kp[(TLs – f 1)I(TjS -j – 1)] are the adaptive portions of the model; the values of Kv, TL, and Tj are altered by the pilot to suit the particular system being controlled. It is found that for most engineering applications, in which an exact pilot model is not required at very low and very high frequencies, an adequate approximation is

{TLs + 1)

{T f + 1 )(2> + 1)

The following set of adjustment rules for the pilot model have been developed by McRuer et al. (ref. 12.3).

1. Stability: The human adopts a model form to achieve stable control— i. e. one that produces a stable closed-loop system.

2. Form selection—Low frequency: The human adopts a model form to achieve good low-frequency closed-loop system response to the input signal. A low-frequency lag, Tz, is generated when both of the following conditions apply:

(a) The lag would improve the low-frequency characteristics of the system.

(b) The aircraft dynamics are such that the introduction of the low- frequency lag will not result in destabilizing effects at higher fre­quencies that cannot be overcome by a single first-order lead, TL, of somewhat indefinite but modest size.

3. Form selection—Lead: After good low-frequency characteristics are assured, within the above conditions, lead is generated when the aircraft dynamics together with the pilot time delay are such that a lead term would be essential to retain or improve high-frequency system per­formance.

4. Parameter adjustment: After adoption of the model form, the describing

function parameters are adjusted so that:

(a) Closed-loop low-frequency performance in operating on the input signal is optimum in some sense analogous to that of minimum mean-squared tracking error.

(b) System phase margin, фм (see Sec. 11.6), is directly proportional to а>{, the input signal bandwidth (loosely defined as the frequency above which the input spectrum decreases rapidly), for values of ojj less than about 2.0 rad/sec. The strong effect of forcing-function bandwidth on the phase margin is associated with the variation of TN with со{.

. (c) Equalization time constants TL or Tp. when form selection requires 1 jTL or 1 jTj << a>c, the system crossover frequency (the frequency at which G(ico) H(ico) equals unity—see Eig. 11.15), it will be adjusted such that low-frequency response will be essentially insensitive to slight changes in TL or Tx (for co{ << oof).

5. ooc Invariance properties:

(a) Independence of coc w. r.t. Kc: Let the aircraft static gain be Kc, and that of the pilot be Kv [see (3.2,4) and (3.4,26)]. After initial adjustment, changes in Kc are offset by changes in the pilot gain, Kp; i. e. system crossover frequency, coc, is invariant with Kc.

(b) Independence of coe w. r.t. co{: System crossover frequency depends only slightly on the input bandwidth for ші < 0.8caco. (coeo is that value of u)c adopted for ooi « coe.)

(c) ct)c Regression: When oji nears or becomes greater than 0.8coco, the crossover frequency reduces to values much lower than ooco.

Although the above pilot model was developed to describe the single – degree-of-freedom compensatory tracking task, it is finding more and more use in the general situation of the multiple-loop tracking task. In such a task the pilot controls a number of vehicle variables simultaneously. It has been found that the same basic form of pilot model can be applied in many cases with slight modification to the values of some of the parameters (such as the time delay t) to account for the additional complexities of the task (such as visually sampling the outputs of several instruments). In this application a single describing function is used to close each control loop actually closed by the pilot. For example, if the task is to control both the pitch and roll attitudes (assuming the pitch and roll modes to be uncoupled), one describing function would close the roll loop while a second would close the pitch loop.

CHAPTER 12

By L. D. Reid and B. Etktn

12.1THE HUMAN PILOT

Although the analysis and understanding of the dynamics of the airplane as an isolated unit (which has been the burden of the preceding chapters) is extremely important, one must he careful not to forget that for many flight situations it is the response of the total system, made up of the human pilot and the aircraft, that must be considered. It is for this reason that the designers of aircraft should apply the findings of studies into the human factors involved in order to ensure that the completed system is well suited to the men who must fly it.

Some of the areas of consideration include:

1. Cockpit environment; the occupants of the vehicle must be provided with oxygen, warmth, light, etc., to sustain them comfortably.

2. Instrument displays; instruments must be designed and positioned to provide a useful and unambiguous flow of information to the pilot.

3. Controls and switches; the control forces and control system dynamics must be acceptable to the pilot, and switches must be so positioned and designed as to prevent accidental operation. Tables 12.1 to 12.3 present typical pilot data concerning control forces.

Table 12.1

Estimates of the Maximum Rudder Forces that can be Exerted for Various Positions of the Rudder Pedal (Ref. 12.1)

Typical Rates of Stick Movement in Flight Test Pull-ups Under Various Loads for 6 in. to 8 in. Deflection (Ref. 12.1)

4. Pilot workload; the workload of the pilot can often he reduced through proper planning and the introduction of automatic equipment.

The care exercised in considering the human element in the closed-loop system made up of pilot and aircraft can determine the success or failure of a given aircraft design to complete its mission in a safe and efficient manner.

The above consideration of pulse trains (which can be so easily analyzed) has shown the important effects of loop gain and phase lag on system stability.

These concepts are brought into a somewhat more useful perspective when we consider a sinusoidal input, for all inputs to linear systems can be Fourier – analyzed into separate sinusoids, the individual responses to which can be linearly superposed to construct the output. Suppose then that there is a steady sinusoidal input represented by

x = X0eiat

and a steady sinusoidal output

У = Y0ei<at

(This implies of course that the system is stable.) The error is e = x — у = e0eim*, where e0 = X0 — Y0. Now we recognize that the critical phase lag is 180°, since this generates the maximum error signal, just as in the case of the pulse train. So let

Y0 = Ке0е-™ = – K4

Then we get

= ea + Гв — r0 — YJK

The input required to maintain a steady oscillation of given amplitude is seen to diminish as К increases until it vanishes altogether at К = 1, i. e.

at К — 1 and phase lag of 180°, the system oscillates steadily with no input. This situation clearly represents a stability boundary; further increase in gain corresponds to instability.

The Nyquist criterion (11.4) rigorously derived from a theorem of Cauchy contains the conclusion derived somewhat heuristically above. It uses the frequency response curve for the open-loop system, i. e. G(ioo), and its relation to the point (—1, 0) of the complex plane, to assess stability. The amount by which the frequency response curve “misses” the critical point (—1, 0) leads to the concepts “gain margin” and “phase margin” illustrated in Fig. 11.15.

The effect of gain is well illustrated by the familiar public-address acoustic system, in which “whistling” or oscillation occurs when the volume control is set too high. As a model for this case, consider the transfer function Ke~Ts, a simple gain with time delay.

If the system input were a single short pulse (of duration < < T) as in Fig. 11.146, the signals in the e and у channels would be as shown, a sequence of alternating pulses at time interval T, all of the same width, but with magnitudes 1, if, A2 …. It is clear that if А < 1 the pulses form a dimin­ishing sequence that ultimately dies out, and that if A > 1, there is an in­creasing series which is a divergent, or unstable situation. This would correspond in the case of the P. A. system to an acoustic pulse travelling from the loudspeaker to the microphone and arriving there stronger than the one originally fed in.

EFFECT OF PHASE LAG

Suppose now that the input is a series of pulses, equally spaced but alternating in sign. If the time lag T is such that the feedback pulses fall in the “empty spaces” between the input pulses there is no interference of the pulses, each input can be considered individually, and the criterion for divergence is the same as above, i. e. A > 1. If, however, the time lag is such that each return pulse coincides exactly with the next input, as illus­trated by the dotted pulses in Fig. 11.146, then the error signal and the ouput form the sequences

e: 1 -(1+A) (1+ A + A2) -(1+ A + A2 + A3)— у: A – A(l + A) K(l + K + Kz) ••• A(1 + A +A2 + —)

The output is seen to contain the sum of a geometric progression of factor A, which is divergent if А > 1 and converges to the limit (1 — A)-1 if A < 1. Thus in the case of the alternating input we find again that the stability criterion is A < I. This is clearly the “worst” phase lag for a pulse train since each return pulse arrives at such a time that it provides the maximum reinforcement to the next input.

We have seen in previous examples how “closing the loop” can modify the basic stability of an airplane. In Sec. 11.4 feedback was used to stabilize an unstable vehicle, and in Sec. 11.5 the addition of a feedback loop to lock on to an altitude or glide reference made a stable vehicle go unstable. We have also seen in the examples how the stability of a linear feedback system can be calculated by formulating the appropriate system matrix and treating it as we would any other linear system.

(b)

Fig. 11.14 (a) Simple feedback system. (6) Single-pulse input: G = Ke~Ts.

For complicated multiloop systems there is relatively little that can use­fully be said in a general way about closed-loop stability. For simple systems, however, as in Fig. 11.14 we can arrive at some general conclusions about the effect of loop gain and phase lag on stability.

CHARACTERISTIC EQUATION

As has been seen in the examples treated, the addition of a feedback loop modifies the characteristic equation, and hence the stability of a system. If the transfer function of Fig. 11.14 is a ratio of two polynomials

N(s)

D{s)

then the overall system transfer function is

which is to be eonstrasted with the open-loop equation D(s) = 0. Thus the change in the characteristic equation is produced by the numerator N(s), and the least possible change is the addition of a constant.

Computations of the stability and performance were carried out with (11.5,12) for the same jet transport airplane used in preceding examples, flying at sea level. The drag polar is (11.5,6) corresponding to the landing configuration. The data that differ from those of Sec. 9.1 are as follows:

CDx = .959, t* = .0460 sec, fi = 101.8,

Ve = 167.4 fps, CWt = 1.8, CDe = .377

The eigenvalues corresponding to a range of Kx and K2 are shown on Fig. 11.12 in the form of root loci. Point A corresponds to the uncontrolled phugoid, and increasing proportional gain K1 with zero rate gain produces the branch A В of the locus. The system rapidly goes unstable without error-rate control, but is easily stabilized with a modest value of K2. For example, at point C on Fig. 11.12, with Kr = .002 (about 12° elevator per 100 ft of height error) and K2 = .010 (about 12° elevator per 20 ft/sec height error-rate), the eigenvalue characteristics are:

Phugoid: period = 10.4 sec ^=.54

Three real roots: th&lt = 94.0, 1.68, 0.86 sec

(а)

The short-period mode has disappeared, being replaced by a pair of real roots, and the third real root is associated with the extra degree of freedom.

The performance of the system, i. e. its ability to track the glide slope, can be in part inferred from the frequency response associated with zEi input and zE output. This is computed by taking the Laplace transform of (11.5,12) (which simply changes D to s wherever it occurs), replacing s by id), and solving the resulting complex algebraic equations for the ratio zEjzE. as a function of a). The result is shown on Fig. 11.13. The system is seen to be able to follow waves in the ILS beam fairly closely down to wavelengths of the order of v,- mile (со = 2 X 10~2) at which point a phase lag of 40° has developed. This calculation is not, of course, sufficient to decide on the acceptability of the chosen gains. For that purpose one should calculate actual flight paths in the presence of wind shear and turbulence, and relate the dispersions to what is acceptable for a given mission.