NASA CR-178029 .. , 

Design and Verification by Nonlinear 

Simulation of a MachlCAS 

Control Law for the NASA TSRV 

8-737 Aircraft 

Final Report 

Kevin R. Bruce 

Boeing Commercial Airplane Company 

Seattle, Washington 

(hASA-CB- 178029) DESIGN APE VERIFICATICN BY ~a7-2~290 

LCNLIHEBB SIdULA'IICN CF B F!ACR/CAS CONTRCL 

LAY PCB TEE NASA 'ICV E737 AIGCBAPT Final 

8eEort (Boeing Ccaxriercial Airilane Co. 1 Unclas 

fie P CSCL OIC G3/08 45367 

Prepared for 

NASA TSRV 

Contract NAS1-14880 

f 

NASA 

National Aeronautics and 

Space Ad m t nis t ra t io n 

August 1986 
NASA CR-178029 

Design and Verification by Nonlinear 

Simulation of a MachlCAS 

Control Law for the NASA TSRV 

B-737 Aircraft 

Final Report 

Kevin R. Bruce 

Boeing Commercial Airplane Company 

Seattle, Was hi ng t on 
TABLE OF CONTENTS 

Page 

1.0 SUMMARY ....................................................... 1 

2.0 INTRODUCTION .................................................. 2 

3.0 SYMBOLS ........................................................ 3 

4.0 DESIGN AND PERFORMANCE REQUIREMENTSMACWCAS CONTROL LAW (LINEAR DESIGN) ........................ 5 

4.1 Inner Autopilot Loop ........................................... 6 

4.2 Velocity Hold Loop ............................................. 7 

5.0 DESIGN OF NONLINEAR FEEDFORWARD TERM ..................... 9 

6.0 WIND SHEAR AND TURBULENCE .................................. 11 

7.0 IMPLEMENTATION OF MACWCAS HOLD SYSTEM ................... 13 

7.1 Conversion to True Airspeed ..................................... 13 

7.2 Control Law Initial Conditions ................................... 14 

8.0 VALIDATION OF RESULTS ......................................... 15 

8.1 Noise Free Performance ......................................... 15 

8.2 Performance in Wind Shear and Turbulence ........................ 16 

9.0 CONCLUSIONS .................................................... 19 

1 
1.0 SUMMARY 

.. 

A MacWCAS control law was designed and developed for use on the NASA Terminal 

System Research Vehicle (TSRV) B737 aircraft to conduct research in profile descent procedures and approach energy management. The system operates by using the elevator to control 

speed; the throttle position being fixed. 

The control law was designed primarily using linear analysis techniques, although a nonlinear feedforward term was included in the system to improve response to a ramp input. The 

system was modeled on a nonlinear aircraft simulator to confirm the original linear design 

and validate the system design at additional flight conditions. 

The system satisfied all the design requirements for gain and phase margin, transient speed 

target overshoot, speed holding in wind shear, and elevator activity in turbulence. 

1 
2.0 INTRODUCTION 

This report documents the design and performance analysis of a MacWCAS control law 

for implementation on the NASA TSRV B737 aircraft. The law is designed to enable NASA to 

perform approach energy management research. For example, on starting descent to an airport, the pilot switches the autopilot from a straight level flight mode to a constant Mach 

descent mode. As the aircraft descends, air density increases and thus CAS increases. At some 

preselected value of CAS, the autopilot switches from Mach hold mode to CAS hold mode and 

continues descending. The Mach, CAS commands, and the switching point may have been 

determined by an energy management computer to satisfy target objectives of distance versus 

altitude, speed, and time of arrival. 

The system operates by using the elevator to control speed. The throttle normally would be set 

at idle thrust, although other settings may be selected to allow for operational considerations 

(e.g., pressurization or anti-icing considerations). 

The report details the linear design results and describes modifications made to the basic 

design to improve performance for ramp input demands and in wind shear conditions. Consideration is given to a practical implementation yielding transient free switching between Mach 

and CAS mode and also to minimize modification to the existing aircraft. 

The preferred design configuration was implemented on a nonlinear aircraft simulator at the 

Boeing Renton Flight Simulation Center. The nonlinear simulation was used to confirm the 

original design results and validate the design at additional flight conditions. 
3.0 SYMBOLS 

a 

VCAS 

VCASE 

Ocmd 

Mach 

ve 

VCLIM 

QF 

vwo 

Y 

VGT 

Vmwic 

v 

V 

Actuator gain (scheduled) 

Angle of attack (deg) 

Calibrated airspeed (kn) 

Calibrated airspeed error (kn) 

Commanded attitude 

Commanded calibrated airspeed (kn) 

Commanded Mach number 

Commanded true airspeed (kn) 

Commanded elevator deflection 

Conversion factor (CAS to TAS) 

Current measured Mach number 

Damping ratio 

Dryden wind spectrum (longitudinal gusting) 

Dryden wind spectrum (vertical gusting) 

Elevator deflection (deg) 

Equivalent airspeed (kn) 

Feedforward rate limit 

Filtered pitch rate (deg/s) 

Filtered velocity (kn) 

Flightpath angle 

Gains (fixed) 

Groundspeed (kn) 

Initial condition for wind velocity (kn) 

Inertial acceleration of airplane (kds) 

Inertial velocity of airplane (kn) 

3 
h 

8 

a 

U 

vMW 

FHWS 

Lu, Lw 

Vwind 

ACSL 

ATOPS 

CAS 

RFSL 

TAS 

Integrator gain 

Longitudinal gust (ds) 

Nominal height (m) 

Perturbed height (m) 

Pitch rate (rad/s) 

Pitch angle (rad) 

RMS longitudinal gust level (ds) 

RMS vertical gust velocity (ds) 

Speed of sound 

Switch velocity (CAS to Mach) 

Switch velocity (Mach to CAS) (kn) 

Time constant for complementary filter (sec) 

True airspeed (kn) 

True airspeed command (kn) 

True airspeed error 

Velocity error gain 

Velocity of airplane in longitudinal axis 

Vertical gust (ds) 

Wind shear magnitude (kn) 

Wind shear scaling factor (horizontal) (m) 

Wind turbulence characteristic lengths (longitudinal and vertical) 

Wind velocity (kn) 

ACRONYMS 

Advanced continuous simulation language 

Advanced transport operating systems 

Calibrated airspeed 

Renton Flight Simulation Laboratory 

True airspeed 

4 
4.0 DESIGN AND PERFORMANCE REQUIREMENTS 

To ensure satisfactory performance of the final control law, the following design and 

performance requirements were imposed on the design. The control law must: 

1. 

2. 

3. 

4. 

5. 

6. 

7. 

Be suitable for use at all flight conditions and throttle settings. 

Be implemented with minimum modification to the existing autopilot design (refs. 1 

through 4) commensurate with meeting all stability and performance requirements. 

Maintain approximately constant transient response performance at all flight conditions with a speed control bandwidth equal to or greater than 0.05 r/s. 

Provide satisfactory damping (approximately = 0.7) on dominant roots at all flight 

conditions and demonstrate insensitivity in damping and bandwidth to 10% variations in nominal gains. 

Limit overshoots in indicated airspeed to within 2 kn for step inputs; maintain indicated airspeed to within 2 kn for a ramp input of 1 Ws2. For wind shear of 1 kn/s the 

maximum error in airspeed should not exceed 10 kn and the steady state error should 

be zero. 

Maintain gain and phase margins equal to or greater than 6 dB and 60 deg, 

respectively. 

Elevator activity should not be greater than comparable existing control systems. 

The MacWCAS control system can be considered to consist of two control loops (fig. 1): 

1. An outer velocity hold loop using true airspeed (VT) and acceleration (v) feedback. 

The reference input, either Mach or CAS command, is converted to a true airspeed command 

that drives the outer (velocity hold) loop. As the MacWCAS conversion is outside the control 

loop, the linear analysis is restricted to considering the attitude loop and the true airspeed 

control loop. 

The reason for converting to true airspeed instead of controlling CAS or Mach directly is that 

this approach requires only one control law. Also, true airspeed is compatible with inertial 

acceleration feedback, which is used to provide damping of the velocity loop. This way, approximately constant inner loop dynamics are achieved over the entire flight envelope without the 

need for outer loop gain programmers. 

The practical implementation of generating the true airspeed command from either CAS or 

Mach is considered in a later section of this report. 

The design of the control law was accomplished by first considering the inner attitude loop in 

isolation (i.e., 8 and q feedback only). An attitude control loop was designed that gave suitable 

performance over the whole flight regime. Next, the velocity loop was added and the performance of the overall loop optimized. 

5 
4.1 INNER AUTOPILOT LOOP 

Speed W( ds) r 

The inner autopilot loop (fig. 2) was modeled using a gain (KV,) and a first-order lag at 20 

rads to represent the elevator actuator. The q feedback was obtained from the baseline control 

system and was prefiltered through a 16 rads washout filter and scaled in deg/s. The washout 

filter originally was added to eliminate steady state biases on pitch rate feedback. 

Low (120 kn) 

Mid (250 kn) 

High (320 kn) 

The inner autopilot loop was designed to give approximately constant performance in terms of 

bandwidth and transient response. The three cases selected for examination were considered 

representative of the whole flight envelope. They consisted of: 

1. High speed CAS = 320 kn, H = 6100m, flaps = 0 deg 

2. Mid speed CAS = 250 kn, H = 3048m, flaps = 0 deg 

3. Low speed CAS = 120 kn, H = 500m, flaps = 40 deg 

9.27 

16.0 

15.9 

In the current aircraft design, the forward path gain Kv, is preprogrammed to vary with VCM 

from 1.0 at 120 kn to 0.275 at 360 kn (fig. 3). It was considered desirable to maintain this gain 

scheduling. 

Root locus techniques were used to examine the low-speed and high-speed performance and 

determine acceptable values for Ke and K,. 'Ihble 1 shows the dominant complex roots for the 

nominal low-speed gain values KO = 8 and K, = 5.16. 

6 

0.98 

0.6 

0.6 

Table 1. Inner Loop Dominant Complex Roots for Nominal Gains 

The root loci for the low-speed case is shown in Figure 4 for gain variations of KO and K, 

around their nominal values. It can be seen that the system is well damped for large variations in gains Ke, K, ( r 30%). The corresponding root locus plot at high speed (fig. 5) shows 

lower damping ({ = 0.6); however, the system is not sensitive to gain variation of k 10%. The 

transient response at high and low speed (for a step change in pitch command, Ocmd) was 

considered satisfactory (fig. 6). The rise time (time of 63% of final value) was 0.8 sec compared 

with 0.75 sec at low speed. 

A summary of the design results for the inner autopilot loop is shown in Table 2. The table 

includes the design results achieved for the mid-speed case. The minimum gain and phase 

margins were 23 dB and 82 deg respectively for the three cases considered. A typical open loop 

frequency response is shown in Figure 7 (low-speed condition). 
High speed (320 kn) 

Mid speed (250 kn) 

Low speed (1 20 kn) 

Rise 

time (sec) 

to 63% 

I I I I I 

Settling Gain Phase 

margin margin 

(deg) 

time (sec) 

to 95% 

Gain 

margin 

Gains Rise Settling 

to 63% to 95% K,* $ & time (sec) time (sec) 

Rise 

I I I I .. 

I Settling I I Gain I Phase 

0.426 5.16 8 0.80 3.05 1 .o 4.5 23.5 

0.6 5.16 8 0.75 2.6 1.2 4.8 23.3 

1.0 5.16 8 0.75 2.25 1.65 3.0 23.3 

time (sec) 

to 63% 

Phase 

margin 

(deg) 

1 time (sec) 

to 95% (3 dBI 

95.2 

91.1 

82.0 

12.5 

12.5 

15.0 

Table 2. Inner Autopilot Loop Performance 

4.2 VELOCITY HOLD LOOP 

19.5 0.175 0.1 42 13.8 65.5 

19.5 0.18 0.15 12.2 66.3 

39.0 0.09 0.13 18.9 75.9 

The basic velocity hold loop (fig. 8) was designed with the same objectives as for the inner 

autopilot loop (i.e., to minimize gain variation and give similar transient performance over the 

whole flight regime). A summary of the results for the three speed cases is given in Table 3 for 

KO = 8, K, = 5.16, Kv, = 0.1, K, = 0.35. The dominant complex roots are shown in Table 4. 

Speed O( r/s) 

Low (1 20 kn) 0.33 

Mid (250 kn) 0.25 

High (320 kn) 0.23 

High speed (320 kn) 

Mid speed (250 kn) 

Low speed (1 20 kn) 

r 

0.72 

0.62 

0.70 

Table 3. Outer Loop Performance 

Considering the low-speed case the rise time (to 63% of final value) was 15.0 sec (fig. 9). 

Adequate gain and phase margins of 18.9 dB and 75.9 deg, respectively, were obtained (fig. 10) 

and the system closed loop bandwidth (to the 90 deg phase shift point) was 0.16 r/s (fig. 11). 

7 
The root locus plots show the system sensitivity to gain variations of 50% and it can be seen 

that the damping of the dominant complex pole (speed control) is sufficiently insensitive to 

gain variations (figs. 12 and 13). 

In the high-speed case, the rise time to 63% had decreased to 12.5 sec (fig. 14). Stability 

margins were again more than adequate at 13.8 dB and 65.5 deg, respectively (fig. 15). The 

root loci of the dominant complex poles are shown in Figure 16 together with the line of 

constant damping for r = 0.7. The diagrams show the insensitivity of the system dynamics to 

gain variation. 

Results at mid speed were very similar to the high-speed case for both the transient and 

frequency response (table 4). 

It was considered desirable to restrict the maximum vertical acceleration to 0.1 g,. It can be 

seen from Figure 17 that a 1-kn step velocity command (high-speed case) gave a maximum 

vertical acceleration of -0.32 kn/s (-0.017 gJ. Hence, a 6-kn change in input would just exceed 

the 0.1 g, criterion. A limiter on airspeed error was included to overcome this problem. 

8 
5.0 DESIGN OF NONLINEAR FEEDFORWARD TERM 

The control law described previously, gave good performance to commanded step changes 

in velocity. However, it resulted in poor performance for a ramp input (Le., constant acceleration demand). A 0.6 kn/s (1 Ws) ramp input produced a steady state velocity error of 6 kn (fig. 

18), whereas the design requirement allows 2 kn for this condition. 

This problem could be solved by using one of two methods: 

1. Additional forward path integrator. 

2. Nonlinear feedforward term. 

The second method, the nonlinear term, avoided the possibility of stability problems and was 

considered a simpler option to implement practically. 

The feedforward term consisted of essentially a differentiator followed by a rate limited lag. 

For implementation purposes, the differentiator was modeled by a gain and a high-pass filter 

(washout filter) having a small time constant compared with the bandwidth of the velocity 

hold loop. As the dominant loop time constant was 5 sec, a time constant at least ten times 

smaller was required and a value of 0.1 sec was nominally selected. This derivative term can 

be added to the control system error signal and can be interpreted as a shaped command 

input. 

Consider the action of the derivative term alone (fig. 191, that is, one without any rate limiter 

or lag, then for a ramp input, the output from the differentiator is a step. By selection of the 

appropriate gain, the addition of this signal to the loop error signal drives the system to null 

the velocity error due to the ramp input (fig. 20). 

However, for a step input, the output of the differentiator is an impulse, and the addition of 

this signal to the loop error signal causes an undesirable overshoot in the system response 

(fig. 21). 

A solution to the problem is the introduction of a nonlinear element, namely a lag having a 

rate limit (VCLIM) (fig. 22). This has the effect of severely attenuating the feedforward impulse 

response for a step command in velocity. It can be seen in Figure 23 that lowering the limit 

reduces the overshoot on the step response such that for V,,, s 0.6 kds, the response approximates the step response for the baseline system without the feedforward term. 

With a ramp input, the output from the differentiator is a step, and the rate limited lag only 

affects the settling time of the lag. For KF = 10.8 and washout filter time constant of 0.1, the 

feedforward gain is approximately 1. Therefore, for a ramp input of 0.6 kds, the limit only 

affects the ramp response for value of VCL, < 0.6 kds. This is verified in Figure 24 which 

shows the same ramp response for V, L 0.6 and VCLm = 0.06. 

9 
The results presented above were carried out at high (VCAS = 320 kn, H = 6100m) and the 

gain KF was optimized to null the error in TAS (K, = 10.8). However, at low speed the optimum response was obtained with the gain KF = 15. This is due to the fact that for the same 

input (whether step or ramp), the response of the feedback terms VT and v is not identical over 

the whole flight envelope, whereas, obviously the feedforward term is the same. Rather than 

have gain scheduling, a compromise value of gain was selected (KF = 13.0) that gave acceptable performance over the whole aerodynamic range. Maximum error to ramp input of 0.6 

kds was within +2 kn for both the high- and low-speed cases (figs. 25 and 26). 
6.0 WIND SHEAR AND TURBULENCE 

The effect of wind shear and turbulence on the control law was investigated using transient and frequency response techniques. 

For the control system in Figure 8, a wind shear of 1 kn/s gave rise to 10 kn steady state error 

in airspeed (fig. 27). This system used inertial acceleration to provide the derivative feedback 

of velocity. Good tracking of wind shear without the standoff error could be obtained by feeding back the derivatives of airspeed. However, this would result in excessive elevator activity 

in turbulent situations. The addition of a complementary filter in the derivative feedback path 

allows a compromise to be made between the maximum transient airspeed error in a wind 

shear situation and the elevator activity experienced in turbulence. The complementary filter 

consists of: 

(1) Differentiating and low-pass filtering the airspeed signal. This gives long-term derivative feedback of airspeed and eliminates the standoff error. The turbulence component is filtered to reduce elevator activity. 

(2) High-pass (or washout) filtering the inertial acceleration signal and adding the signal 

to the airspeed signal. The two signals complement each other giving a unity transfer 

function. The high-frequency component of the inertial signal is comparatively free of 

noise. 

The implementation of this filter is shown in Figure 28. The maximum airspeed error in a 

wind shear and the time required for the error in airspeed to reduce to zero are a function of 

the time constant However, the penalty for reducing the effect of wind shear is an increase 

in the elevator activity due to turbulence. The relationships between elevator activity and 

maximum CAS error are shown in Figure 29 for a Dryden turbulence spectrum (ref. 5) with an 

rms level of 0.3 m/s (combined horizontal and vertical). 

As the control law is not to be used for final approach, then the performance criterion in wind 

shear is not severe (i.e., 10 kn max error for 1 kds wind shear). The second constraint considered was that of elevator activity, which should not be worse than results obtained with the 

existing TSRV aircraft. As the airplane does not have a velocity hold mode that uses the 

elevator, then results were taken from turbulence results for the TSRV glideslope control law. 

The airplane does have this mode but not using elevator. Although the results were not directly comparable, they provided a guide to the levels of elevator activity that were 

acceptable. 

From Figure 29, an initial value of 7v = 10 sec was taken. For this system, the effect of 

turbulence was determined by modeling the wind using the Dryden spectrum for horizontal 

and vertical and evaluating the rms elevator activity. The results of vertical and longitudinal 

turbulence of 1.5 m/s rms are shown in Table 5 below. When compared with the results for the 

glideslope track law, the level of elevator activity was considered acceptable. 

11 
Low-speed elevator Activity (deg rms) 

Velocity hold law Glideslope track law 

Vertical gusts (1.5 m/s rms) 1.4 1.56 

Longitudinal gusts (1.5 m/s rms) 0.5 0.835 

Table 5. The rms Elevator Activity Due to Turbulence 

The effect of the complementary filter (7v = 10 sec) on the frequency response of the elevator in 

longitudinal turbulence can be seen in Figure 30. Above approximately 0.3 r/s, the complementary filter raises the frequency response by approximately 7 dB; however, this was considered acceptable. At low speed (fig. 31), the increase in the frequency response caused by the 

addition of the filter was not significant above 0.1 r/s. However, below 0.03 r/s the filter 

reduced the sensitivity to gust, i.e., improved performance to wind shear. 

In the case of vertical turbulence, the effect on the elevator activity of including the complementary filter was not significant (fig. 32) for both the high and low speed. 

The effect of the complementary filter on the response of airspeed error for longitudinal turbulence can be seen in Figure 33. The filter has the effect of reducing the sensitivity to wind 

shear (i.e., low-frequency sensitivity) without significantly increasing the sensitivity to 

turbulence. 

The sensitivity of the airspeed error to vertical turbulence was not significantly affected by 

the inclusion of the filter in both the high- and low-speed cases (fig. 34). 
7.0 IMPLEMENI'ATION OF MACH/CAS HOLD 

7.1 CONVERSION TO TRUE AIRSPEED 

Consideration has been given to the method of converting from CAS and Mach to true airspeed, which is used in the control law. Figure 1 shows a simplified block diagram of the 

control system using a true airspeed command (VTCMD) and true airspeed (VT) feedback to form 

the error signal 

'TE = 'TCMD (1) 

It is possible to compute VTCMD from the pilot selected VCAsCMD using a conversion formula. 

However, the conversion formula is complex and requires accurate measurement of several 

parameters. The risk of incurring a significant error in VTCm is therefore substantial. Such 

an error would cause an equivalent standoff between the selected VCASCMD and the actual 

VcM. This approach is therefore unacceptable. However, this problem can be eliminated by 

implementing the control loop in terms of the error in VcM. 

'CASE = 'CwMD - (2) 

VTE = F*VCASE (3) 

and conversion of this error signal into the equivalent true airspeed error signal 

In this case, a less accurate conversion factor (derived from speed conversion tables) will 

suffice. 

1 F= 

(1 - 0.394~ IO-%) (4) 

where h is in m. As VTE is controlled to zero, so VcAsE must go to zero, and an exact measurement of h is not critical to accurately controlling VCAS. 

In the Mach hold control loop, the accuracy problem can be avoided. Restating Equation 1 

above 

VTE = VT~,, - VT 

then VTcM, can be obtained from V, = MackMD * a 

where a = speed of sound and MaCkMD = commanded Mach number. 

therefore 

- MachcMD . V, 

vTCMD - Mach 

where Mach = Current Mach number. 

It can be seen that errors in measuring V, will not cause steady state errors in Mach hold. As 

Mach tends to Mach,,,, so V, tends to zero. The implementation of the equations discussed 

above is shown in Figure 35. 

13 
7.2 CONTROL LAW INITIAL CONDITIONS 

During the first section of descent, the control system is designed to hold the aircraft at 

constant Mach. When CAS equals a preselected CAS, then the system automatically switches 

to CAS hold and continues the descent by maintaining constant CAS. On ascent, the reverse is 

true. The control law initially is designed to maintain constant CAS, and at a preselected 

Mach, the system switches to holding constant Mach. 

In order to minimize transients on switching between the two modes of operation, it is necessary to set 

VCWMD = VCAS (8) 

when in the Mach hold mode (in descent), and switch to CAS hold when 

VCAS = VC&rnH (9) 

Similarly, when the airplane is climbing and in the CAS mode, the Mach command should be 

set to the current value, i.e.: 

MackMD = Mach (10) 

Mach = MachsmH (11) 

and the system switched to Mach hold when 

An additional important consideration is that the forward loop integrator should be held at its 

current value (clamped) if the elevator limits during flight. The integrator should be “unclamped” when the elevator comes of the limit or the input to the integrator changes sign. 
8.0 VALIDATION OF RESULTS 

The equations describing the MacWCAS control law were implemented on a nonlinear 

1. Validate the design results obtained using linear analysis design techniques and 

results obtained using a simplified nonlinear model with linear aerodynamics (ACSL 

Simulation (ref. 6)). 

2. Examine the noise free performance at other flight conditions not covered by the 

linear analysis. 

3. Demonstrate the ability to provide transient free switching from Mach hold to CAS 

hold and vice versa. 

4. Confirm the performance of the system in wind shear and turbulent conditions. 

aircraft simulator. This was considered desirable in order to: 

8.1 NOISE FREE RESULTS 

Initially, the free airplane responses obtained using the ACSL program were compared with 

the results obtained with nonlinear aircraft simulation. 

The ACSL simulation used derivatives obtained at two flight conditions namely: 

1. Low speed: V, = 120 kn, H = 500m, y = 0, and flaps = 40 deg 

2. High speed VCAs = 320 kn, H = 6100m, y = 0, and flaps = 0 deg 

Simulation runs were carried out at these conditions with an initial offset of 1 deg from 

trimmed condition. Figure 36a shows plots of 0 and CY for the linear simulation at the lowspeed conditions and Figure 36b shows the nonlinear simulator results for 0 and CY. 

It can be seen that the ACSL simulation gave a phugoid oscillation of 37 sec compared with 42 

sec with the nonlinear simulator. The short period response of CY crossed the trimmed steady 

state value at 1.6 sec and 4.0 sec (for the linear system) compared with 2.0 sec and 4.5 sec for 

the nonlinear aircraft simulation. 

At high-speed, the linear system took 106 sec (fig. 37a) for the phugoid oscillation compared 

with 82 sec (fig. 37b) for the nonlinear aircraft simulation. The short period crossing points (CY 

response) occurred at 0.64 and 1.8 for the ACSL system and 0.7 and 2.2 for the nonlinear 

aircraft simulation model. 

A detailed comparison of the B-737 aerodynamics on the nonlinear aircraft simulation was not 

possible within the time scale available. However, the discrepancy between the ACSL and 

nonlinear aircraft simulation was acceptable considering the difference in the levels of sophistication between the two models. 

To confirm the validity of the overall control law, the time response to step change in true 

airspeed was examined. Figure 38a shows that for the linear model at high speed, it took 20 

sec to achieve 95% of the steady state value compared with 18 sec for the nonlinear aircraft 

simulation result. At low speed, the 95% level was achieved after 36.8 sec using the ACSL 

simulation (fig. 38b) and 35 sec with the nonlinear aircraft simulation. This small discrepancy 

in the models was considered acceptable and could be explained by differences in the aerodynamics and by the additional complexity and nonlinear elements of the nonlinear aircraft 

simulation. 

15 
small discrepancy in the models was considered acceptable and could be explained by differences in the aerodynamics and by the additional complexity and nonlinear elements of the 

nonlinear aircraft simulation. 

A disadvantage of the linear simulation was its restriction to a limited number of flight 

conditions. The addition of extra flight conditions would require extensive modification to the 

program. However, the nonlinear aircraft simulation allowed the performance of the system to 

be determined at other flight conditions, namely: 

1. CAS = 350 kn, H = 6100m, flaps 0 deg 

2. CAS = 250 kn, H = 6100m, flaps 0 deg 

3. CAS = 200 kn, H = 3048m, flaps 0 deg 

4. CAS = 160 kn, H = 500m, flaps 15 deg 

The response to a step change in true airspeed at these flight conditions is shown in Figure 39. 

It can be seen that the performance is within the design requirements for all conditions. 

In addition to the step response, the effect of a ramp input (0.3 ds2) was examined. In order to 

achieve a similar response at both high and low speed compared with the ACSL results, a 

feedforward gain (KF) of 6.5 was selected (figs. 40 and 41) compared with linear design requirement of 13.0 (figs. 25 and 26). 

A primary requirement of the system is that it should provide transient free switching between the Mach and CAS hold mode. This is demonstrated in Figure 42, which shows an 

aircraft descent from approximately 6700m, at a rate of -7.6 ds. The Mach hold mode has 

been selected so that initially the system captures and holds Mach = 0.7. When holding Mach 

number and descending, CAS increased. The simulation stopped when CAS was 324.7 kn, 

although the measured value tended toward the requirement of 325 kn. Maximum transient 

error at switchover was less than 1 kn. 

Transient free switching between CAS and Mach numbers was also demonstrated. For example, Figure 43 shows the effect of Mach and CAS for an aircraft climbing from 6100m at an 

initial nominal climb rate of 7.6 m/s. Initially, the control system is in the CAS hold mode, 

maintaining 300 kn while Mach number increases. At Mach = 0.68, the Mach hold mode 

automatically is switched in and the system holds Mach with a transient at switchover of less 

than Mach 0.003. 

8.2 PERFORMANCE IN WIND SHEAR AND TURBULENCE 

Figure 44 shows the results of the ACSL simulation for a horizontal wind shear of 1 kds 

plotted in terms of Mach response. Peak error in terms of airspeed was 7 kn and the error was 

reduced to 10% of its peak value by 38 sec. Figure 44 also shows the results of the nonlinear 

simulation, which gives approximately the same peak error and settling time as the ACSL 

simulation. 

To determine the effect of wind shear at low speed, a wind shear profile was selected. The wind 

velocity is given by the equation: 

VMW = FHWS . VMW,, 

where FHWS is a scaling factor dependent on height (fig. 45) and V,, is an initial condition 
for wind velocity set to 1 kn. The velocity error (VCAsE) is shown in Figure 46 for an aircraft 

climbing at a nominal 7.6 m/s. It can be seen that the maximum error in CAS was 0.32 kn. 

Inputs rms (mk) 

Elevator rms response (deg) 

Velocity rms response (kn) 

In addition to wind shear, the effect of turbulence was examined at high and low speed. A 

Dryden wind spectrum (ref. 5) was selected, which is defined as 

1.38' 1.38' 

0.26 0.7 

2.83 0.47 

for longitudinal gusting, and 

where Lu, Lw are the characteristic lengths 

a,, a- are the rms gust velocity 

and7 - w/v. 

Typical values of scale length and rms level were selected, i.e., Lu, Lw = 183m; uu, (I- = 1.5m 

rms. A typical plot of gust is shown in Figure 47; the rms level for the 100 sec sample was 

1.4m. The effect of turbulence on the elevator is shown in Figure 48. 

A summary of the results is shown in Table 6. For a medium level of turbulence (1.5m rms) the 

elevator response was considered acceptable. 

Longitudinal Vertical 

Calculated for 100 sec time history. 

Table 6. Summary of Turbulence Results 

17 
9.0 CONCLUSIONS 

1. A Mach/CAS hold control system was designed using linear analysis techniques for 

flight conditions. 

High speed CAS = 320 kn, H = 6100m, y = 0, flaps = 0 deg 

Mid speed CAS = 250 kn, H = 3048m, y = 0, flaps = 0 deg 

Low speed: CAS = 120 kn, H = 500m, y = 0, flaps = 40 deg 

2. The basic design was modsledTo improve performance in wind shear by the addition of a complementary filter in 

the feedback path. 

To improve the response to a ramp input by the inclusion of a nonlinear feedforward 

term. 

2. The basic design was modsledTo improve performance in wind shear by the addition of a complementary filter in 

the feedback path. 

To improve the response to a ramp input by the inclusion of a nonlinear feedforward 

term. 

3. For the design cases considered, the system satisfied the design criteria. 

The minimum gain and phase margins were 12 dB and 65 deg. 

The rise time (to 63% of final value) for a step input was 13.5 sec -t 11%. 

The overshoot to a step input was less than 2 kn. 

The peak error in airspeed, due to a 1 kds wind shear, was less than 10 kn. 

The error reduced to zero in the steady state. 

Elevator activity was less than 1.5 deg rms for vertical gusts of 1.5 m/s rms. 

For a ramp input of 1 kn/s2, the error in airspeed was less than 2 kn. 

4. Consideration was given to practical implementation in order to provide transient free 

switching between the Mach and CAS hold modes. The system has been designed to be 

implemented with minimum modification to the existing design. 

5. The control system was implemented on the nonlinear aircraft simulator, which enabled 

simulation runs to be made at several flight conditions not covered in the linear design, 

namely: 

CAS = 160 kn, H = 500m, flaps = 15 deg 

CAS = 200 kn, H = 3048m, flaps = 0 deg 

CAS = 250 kn, H = 6100m, flaps = 0 deg 

CAS = 350 kn, H = 6100m, flaps = 0 deg 

6. The nonlinear aircraft simulation runs confirmed that the system satisfied the design 

requirements-for noise free, wind shear, and turbulent conditions. 

19 
REFERENCES 

1. 

2. 

lkrminal Configured Vehicles Program Test Facilities Guide, NASA SP 435, 1980. 

D. A. Wolverton and C. Stominski, Flight Control Computer Software Description, Computer Science Corporation, TAO Number 18303, June 1982. 

A. E. Goodwin, NASA 515 Flight Control System Description, RSFS Aircraft, Boeing 

Commercial Airplane Company, September 1976, NASA Contract 14070. 

L. G. Malcom and P. L. O'Toole, WCS Software Description, The Boeing Commercial 

Airplane Company, D6-42958, November 1976. 

J. Roskan, Airplane Flight Dynamic and Automatic Flight Controls Pt. 11, University of 

Kansas, 1979. 

ACSL-Advanced Continuous Simulation Language, User Guidemeference Manual 

Mitchel and Gauthier Associates 2nd Edition 1975. 

3. 

4. 

5. 

6. 

'0 
i$,T -0 

21 
“2 

+K 

6ec 

6e 1 + 0.05 sec 

1 .o 

0.75 

Gain 

KV2 

0.50 

0.25 

0. 

916.8 sec 

16 sec +1 

Figure 2. Inner Loop Autopilot Loop 

I I I 1 

0 100 200 300 400 

Velocity (VcAs), kn 

“2 Figure 3. Gain Scheduling for K 
Nominal Gains 

Ky2' 1.0 

Kq = 5.16 

KO = 8.0 

5 = 0.7 \r Kq - 

-12 -10 ' -8 -6 -4 -2 

Real 

12 

10 

8 

Imaginary 

6 

4 

2 

0 

Figure 4. Root Locus for Dominant Complex Poles Inner Loop, Low-Speed Case 

(e, 8 Feedback Only) 

23 
Nominal gains 

K = 0.426 

= 5.16 

= 8.0 

"2 

Kq 

Ke 

N = Roots for 

nominal gains 

1 I 1 I I I 

-1 2 -10 -8 -6 -4 -2 

Real 

16 

14 

12 

10 

Imaginary 

8 

6 

4 

2 

0 

Figure 5. Root Locus of Dominant Complex Poles Inner Loop, High-speed Case 

(e, 8 Feedback Only) 

24 
1 .oo 

0.80 

0.60 

0.40 

0.20 

0.00 

0.0 1 .o 2.0 3.0 

Time, sec 

Figure 6. Transient Response to Unit Step (High and Low Speed) 

4.0 5.0 

25 
Gain 

(dB) 

Gain 

0 \ < 

-20 7 

-40 

-60 

-80 

-1 00 

-120 

Phase, 

deg 

400 

300 

200 

100 

0 

. 

. . I 

I a 

I rn 

. S 

1 ~*.*~.....-----=.. =---...* -..... e-. 

0. 

0. 

-9. 

.. . Phase -0. 

0 < 

% 

t *e. 

% 

-... % 0. 

-..-... 

.*..... .** i' I If I I II I I II I Ill 

2 468 

'.. 

2 468 2 4 6a 2 4 6a 

Frequency, radls 

Figure 7. Low-Speed, Open Loop Frequency Response 

26 
27 
0.60 

0.40 

0.20 

0.00 

L 

I 

Time, sec 

Figure 9. Response of klocity Hold Loop to Unit Step (Low-Speed Case) 

28 
400 

300 

200 

100 

2 468 2 4 68 2 468 2 4 68 

0.01 0.1 1 10 100 

3 

Frequency, radls 

Figure 70. Wocity Hold Loop Frequency Response (Low Speed), Open Loop 

29 
Phase, 

deg 

400 

300 

200 

100 

0 

2 4 6a 2 4 60 2 460 2 4 60 

0.01 0.1 1 10 100 

Frequency, radls 

Figure 1 1. Wocity Hold Loop Frequency Response (Low Speed), Closed Loop 

30 
5 = 0.7 \ 

K, = 0.05 

'T 

., s2 -1.5 

= 0.525 

:, = 0.44 

Nominal gains 

K = 1.0 

"2 

$1 = o.l 

K, = 0.35 

N = Roots for 

nominal gains 

(x = 0.175 

I I I 

-0.3 

Real 

0.4 

0.3 

Imaginary 

0.2 

0.1 

0 

Figure 12. Root Locus for Dominant Complex Poles (Outer Loop, Low-Speed Case) 

31 
Nominal gains 

Ke 

Kq = 5.16 

Kg, = 8 

N = Roots for 

nominal gains 

1 I I 

-0.3 -0.2 -0.1 0 

Real 

1.3 

3.2 

Imaginary 

0.1 

0 

Figure 13. Roof Locus for Dominant Complex Poles (Outer Loop, Low-Speed Case) 
1 .oo I= 

3 

:: 

2 

C 

0 

Q 

a, 

u) 

C 

3 

U 

a, 

a, 

CI 

.- c. 

a 

.- 

Q 

a, 

s 

20 40 60 

Time (seconds) 

80 100 

Figure 14. Response of klocity Hold Loop to Unit Step (High-speed Case) 

33 
Phase, 

deg 

100 

300 

200 

100 

0 

Frequency, rad/s 

Frequency response open loop VTASIVCMD 

Figure 15. Open Loop Frequency Response (High-speed Case) 

34 
Nominal Gains 

k1 = 0.35 

= 0.1 

K, = 0.426 

2 

(b) Kq = 5.16 

KO = 8.0 

j = 0.7 \ 10.3 \ K, = 0.425 i 

- I u70 

N = 

nominal 

Roots for 

gains 

Kv2 =054 k 

K, = 0.175 

I I 1 \I YO 

-0.3 -0.2 -0.1 

Real 

0.2 

Imaginary 

0.1 

6 = 0.7 1 

KO = 4 

Nominal gain 

KO= 12.0 \ 

I I 

\ 

1 

-0.2 -0.1 

Real 

0.2 

Imaginary 

0.1 

0 

Figure 16. Roof Locus for Dominant Complex Roots (High-speed Case) 

35 
0.2 

(0.01 0) 

0.1 

(0.005) 

0 Vertical 

acceleration 

knls (9) 

-0.1 

(-0.005’ 

-0.2 

(-0.01 a 

-0.3 

(-0.01 6

-0.4 

(-0.021 

Time, sec 

Figure 17. krtical Acceleration for 1-kn Step Input (High Speed) 

36 
High speed 

490 

470 

450 

True 

airspeed, kn 

430 

41 0 

39c 

- 

Control system as shown in Figure 8 

I 1 I I 1 

00 20 40 60 80 100 

Time, sec 

Figure 18. Response to Ramp Input (High Speed) 
4 >" 

38 
High speed 

450 r 

418 

410 

True 

airspeed, 

kn 

- 

System as shown in Figure 19 

J 

I I I I 1 

Velocity 

command, 

kn 

418 

410 

- 

System as shown in Figure 19 

J 

I I I I 1 

Figure 20. Derivative Feedforward Term: Ramp Response 

39 
1 .o 

0.6 I 

0.2 

Error in 

CAS, kn 

-0.2 

-0.6 

-1 .o 

0 

High speed 

System as shown in Figure 19 

I I 1 I I 

8 16 24 32 40 

Time, sec 

Figure 2 1. Derivative Feedforward Term: Step Response 

40 
I-------- 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

rn 5 . 

F 

c+ F 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

s! 

II 

> 

IE 

.> 

t 

n 

C 

I5 

> 

41 
1 .c 

0.6 

0.2 

Error in 

'CAS 9 kn 0 

-0.2 

-0.6 

-1 .o 

High speed 

VCLIM = 0.6 knls 

Baseline system 

\ 

I No limit 

System as shown in Figure 22 

I I I I I 

0 8 16 24 32 40 

Time, sec 

Figure 23. Step Response for Variation in Rate Limit (VCLIM) 

42 
True 

airspeed, 

kn 

418 

410. 

- 

Control system as shown in Figure 22 

I I i I 1 

Figure 24. Ramp Response for Variation in Rate Limit (VCJM) 

43 
200 

180 

160 

True 

airspeed, 

kn 

140 

12c 

1 oc 

Low speed System: Figure 23 

= 13.0 

= 0.1 

= 0.1 

KF 

VCLIM = 1.0 

TWO 

TB 

1.2-kn error for 0.6-kn/s input 

- 

Control system as shown in Figure 22 

I I I I I 

40 60 80 100 0. 20 

Time, sec 

Figure 25. Final Low-Speed Response 
High speed 

490 

470 

450 

True 

airspeed, 

kn 

430 

410 

39c 

0 

I I I I I 

20 40 60 

Time, sec 

80 100 

Figure 26. Final High-speed Response 

45 
High speed 

20 

16 

12 

%Asp kn 

8 

4 

0 

Control system as shown in Figure 8 

I I I I 1 

20 40 60 80 100 

Time, sec 

Figure 27. Steady State Calibrated Airspeed Error Due to Wind Shear of 1 kn/s 
01 

5 

m 

4 

47 
0.125 r lor 

1- 48 

0.075 

Peak elevator 

activity, deg 

0.05 

CAS error due to wind shear of 1 kn/s 

Elevator activity due to turbulence of 1 ft/s2 

combined horizontal and vertical 

Dryden turbulence spectrum 

- High speed 

-------------- Low speed 

L. Elevator 

Complementary filter time constant, Tv , sec 

Figure 29. Elevator Activity and CAS Error krsus Complementary Filter Time 

Constant for Wind Shear of 7 kn/s 
-20 

-40 

-60 

-00 

I I Ill I I II 

2 4 60 2 4 60 

0.01 0.1 

I I Ill I I IIJ 

2 4 60 2 4 60 

1 10 100 

Frequency, rad/s 

-1 0 

-20 

Gain 

(dB) 

-40 

-60 

Figure 30. Frequency Response of €levator tie/ (deg) (With and Without Complementary Filter) (High Speed) U 

No filter r 

2 4 60 2 4 60 2 4 60 2 4 60 

0.1 1 10 100 

Frequency, rad/s 0 01 

Figure 31. Frequency Response of Elevator 6 (deg) (With and Without Complementary Filter) (Low Speed) e/, 

49 
Gain. 

20 

I I I1 I I II 1 I II -40 

Frequency, radls 

I I It 

1 Low speed, no filter 

2 Low speed, filter 

3 High speed, no filter 

4 High speed, filter 

Figure 32. Elevator Frequency Response in Vertical Turbulence 

0.01 0.1 

1 Low speed, no filter 

2 Low speed, filter 

1 

Frequency, radls 

10 100 

3 High speed, no filter 

4 High speed, filter 

Figure 33. Airspeed Error Frequency Response in Longitudinal Turbulence 

50 
Gain, 

1 Low speed, no filter 

2 Low speed, filter 

3 High speed, no filter 

4 High speed, filter 

figure 34. Airspeed Error Frequency Response in krtical, Turbulence 

51 
Mach 

hold select 

’ CAS 

- 1.689 y hold select 

kn 

VT 

I 1. 

Mach 

CMD- 

- 

Mac

/I a 

1.689 y 

VT 

kn 

h 

2 

0.394 x lo” 

Figure 35. Implementation of MachICAS Hold 

52 
(a) ACSL simulation 

1.0g Low speed 

0.6 

Airplane 

response 

from 

(deg) 

trim - 37 sec - I 

0.2 f 

-0.2 

(b) Nonlinear simulation 

1 .Or 

Airplane 

response 

from trim 

(deg) 

Angle of attack (Y ---------- 

Attitude e 

Trimmed 

condition 

Angle of attack (Y --------- 

Attitude e0.8 

I 

I 

I 

I 42 sec 

0.6 4. 

0.4 f 

I 

Trimmed 

0.2 

'-e 

Figure 36. Transient Response of Free Aircraft (1-deg Initial Offset on OL and e) (Low Speed) 

53 
(a) ACSL simulation 

High speed 

1.0' 

0.6 

Airplane 

response 

from 

trim 

I (deg) o.2 

0 

-0.2 

(b) Nonlinear simulation 

0.6 .. . 

-c \ 2o 

7 Altitude response 

'7 Angle of attack CY 

53 sec 7 for V2 cycle 

100 

L \ 1 

Airplane 

response 

from 0.4 

trim 

fe (deg) 

- 41 sec for 112 cycle- 

\ 40 =e, sec 

, 

I 

60 80 100' 

Figure 37. Transient Response of Free Aircraft (7-deg Initial Offset on 01 and 0) 

(High Speed) 

54 
(a) High speed 

True 

airspeed 

error 

(kn) 

0 

-1 

-2 

-3 

-4 

-5 

I I I I I I 

0 10 20 30 40 50 60 70 80 

Time, sec 

-6 

0 

-1 

-2 

True 

airspeed 

error -3 

(kn) 

-4 

-5 

-6 

(b) Low speed 

Nonlinear 

Simulation 

0 10 20 30 40 50 60 70 80 

Time, sec 

Figure 38. Step Response of klocity Hold Loop 

55 
170 

Time, sec 

- 

Figure 39. Step Response (Nonlinear Aircraft Simulation) 

56 
. 

270 

260 

250 

240 

True 

airspeed, 

WS 

230 

220 

21 0 

Feedforward gain (KF) = 6.5 

200 ~ 

0 10 20 30 40 50 60 70 80 

Time, sec 

Figure 40. Ramp Response Low Speed (Nonlinear Simulation) 

57 
C 

Q 

.- 

I I I I 1 I I I 1 

v) 

(0 

d 

0 

W 

d 

In 

In 

P 

0 v) v) Cd dXd 

UY 

N 

d 

0 UY 0 d 0 0 d d d 

a 

3 

!= 
Height x lo3, 

312 

7.0 

6.6 

m 5.8 624.0 

0.703 

Mach no. 

- 

- 

Switch from Mach 

- hold to CAS hold 

- 8 

- 

- 

0.699 

0.697 

0.695 

0.693 

0.691689 

0.687 

0.685 

59 

Captures desired J - 

- Mach no. 

- 

- Switches 

- to CAS hold A\ 

- 

I I I 
1 1 I I I I I 

Q) 

PI (v 

Q) 

w '9 

0 0 0 

I I I I I I I I 

d 

a, 

N 

d N 0 Q) (0 

Q, Q, Q, a, Q) Q) w Q, Q, N N N 2- N N N 

0 

0 

m 

E Si= 

0 

d 

0 

N 

0 

60 
0 

3 

0 

(D 

s 

0 

cu 

0 

'D 

d 

w 

5 

.a LL 

61 
2.2 

2.0 

1.8 

1.6 

1.4 

Wind shear 

scaling factor 1.2 

FHWS 

1 .o 

0.8 

0.6 

0.4 

0.2 

Figure 45. Plot of Wind Shear Scaling Factor FHWS krsus Height 

1 1 1 1 1 1 1 1 

0.5 1 .o 1.5 2.0 2.5 3.0 3.5 4.0 

Height x lo3, rn 
63 
U 

r 

I + 

I 

r- T I 

I cu 0 P 

0 

E 

0 

m 

0 

OD 

0 

b 

0 

2 

0 

P 

3 

U 

.r? 

E 

II 

3 

4 

B 

8 

o;, 

f 

0 
m 

0 

s1 

0 

Q) 

0 

aD 

0 

PI 

0 

(0 

0 In 

0 

w 

0 

m 

0 

N 

s1 

0 

L 

0 2 

2 

i= 0 

d 

C 

E 

65 
I. Report No. 

L. Title and Subtitle 

2. Government Accession No. 

NASA CR-178029 

Design and Verification by Nonlinear Simulation 

TCV B737 Aircraft 

Kevin R. Bruce 

of a Mach/CAS Control Law for the NASA 

7. Author(s) 

3. Performing Organization Name and Address 

Boeing Commercial Airplane Company 

P.O. Box 3707 

Seattle, WA 98124 

NASA Langley Research Center 

Hampton, VA 23665 

2. Sponsoring Agency Name and Address 

3. Recipient's Catalog No. 

5. Report Date 

6. Performing Organization Code 

8. Performing Organization Report No. 

December 1986 

10. Work Unit No. 

11. Contract or Grant No. 

17. Key Words (Suggested by Author@)) 

NAS1-14880 

18. Distribution Statement 

13. Type of Report and Period Covered 

19 Security Classif (of this report) 

Final Report 

20. Security Classif (of this page) 21. No. of Pages 

14. Sponsoring Agency Code 

Unclassified 

I 

5. Supplementary Notes 

NASA Technical Monitor: 

Boeing Technical Supervision: 

Boeing Contract Manager: 

Dr. J. F. Creedon 

A. A. Lambregts 

R. L. Erwin 

6. Abstract 

Unclassified 

A MacWCAS control system using an elevator was designed and developed for use on the NASA 

TCV B737 aircraft to support research in profile descent procedures and approach energy 

management. 

The system was designed using linear analysis techniques primarily. The results were confirmed and the system validated at additional flight conditions using a nonlinear 737 aircraft 

simulation. All design requirements were satisfied. 

Flight Control System 

MacWCAS Control System 

Profile Descent Procedure 

Nonlinear Simulation 

Unclassified-Unlimited 

22. Price 

.