AIAA Guidance Navigation and Control Conference, 15-18 August 2005, San Francisco, CA

Robust and Scheduled Flight Control with Handling

Quality Requirements

David Saussie¤

Ecole Polytechnique de Montreal, Quebec, H3T 1J4, Canada

Ouassima Akhrify

Ecole de Technologie Superieure, Montreal, Quebec, H3C 1K3, Canada

Lahcen Saydyz

Ecole Polytechnique de Montreal, Quebec, H3T 1J4, Canada

This paper deals with longitudinal flight control design for commercial aircraft. Thecontrol law is a fixed-structure C∗ law with five gain parameters to be tuned in orderto fulfill a number of handling quality requirements. A method is put forth for findingwhole regions of gains in the gain parameter space for each flight case. These regions havethe property that any choice of the parameters is guaranteed to satisfy specific handlingqualities. This paper puts forward a set of guidelines to use these parameter regions toaddress both the robustness of the control law and the gain scheduling issues. This hasbeen applied to the full flight envelope and the robustness aspect is illustrated on a givenflight case for a business jet aircraft.

Introduction

The development, integration and flight testing of flight control systems are costly and time-consuming.Although modern techniques are providing effective and robust controllers,1,2 classical flight control systemsare still widely used because of a well-studied and understood architecture. However, they have to deal withstringent performance and robustness requirements over the full flight envelope.3–5

This article expands upon the work initiated in Refs. 6, 7 and 8. In this work, the authors proposea method to efficiently choose the varying parameters of two fixed architecture classical control laws. Forany given flight case, the guidelines provide gain regions that fulfill the handling quality requirements. Thismethod relies in part on the Guardian Map approach developed in Ref. 9, to determine the acceptable gainparameter regions. Thus, for each flight case, such regions can be defined and the question in this paper isto see how they can be used to tackle the gain-scheduling and robustness issues.

Section 1 presents the C∗ controller and the handling qualities that must be satisfied. Section 2 showshow the suggested guidelines can be used to fulfill most criteria. In section 3, a robust controller is designedto satisfy all criteria for varying masses and centers of gravity at a given flight case. In section 4, a gain-scheduling method is proposed.

I. Flight Control System Architecture and Handling Quality Requirements

Flight control systems aim at improving the aircraft characteristics via control in order to achieve certainlevel of handling qualities (HQ) when the pilot “interfaces” the aircraft.

Figure 1 shows the C∗ longitudinal flight control system consisting of a stability augmentation system(SAS) and a control augmentation system (CAS).

∗Ph.D. Student, Electrical Engineering Department, 2500 Chemin de Polytechnique.†Professor, Electrical Engineering Department, 1100 Notre-Dame W.‡Professor, Electrical Engineering Department, 2500 Chemin de Polytechnique.

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AIAA 2005-5978

Copyright © 2005 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.

Stick

measZn

measq

ref eq

S.A.S.C.A.S.

+

-

+ ++

+ -

+

+

*C

ffK

s

K i

pK

Actuator Aircraft

Sensors

wozs

sqK

12.4

Noise

Filter

Figure 1. Architecture of C∗ Longitudinal Flight Control System.

The variable parameters are the wash-out filter pole, zwo, the feedback gain, Kq, the PI gains, Kp andKi, and the feedfoward gain Kff .

The overall performance objective is to track pitch rate commands with predicted Level 1 handlingqualities and desired time domain response behavior. The handling quality criteria considered in this articleare short period mode damping ratio ζsp , pitch attitude bandwidth ωBWθ

, phase delay τp, path bandwidthωBWγ

, Gibson dropback Drb, gain margin MG and phase margin Mϕ. The boundaries of these criteria aredefined by military standards.10 Table 1 summarizes the handling quality boundaries and the specificationsof time domain response behaviors.

Table 1. Handling Quality Boundaries

HQs L.1 Boundaries

ζsp 0.35 < ζsp < 1.35

ωBWθ> 1.5 (rad/s)

τp < 0.2 (s)

ωBWγ > 0.6 (rad/s)

Drb −0.2 < Drb < 0.5

MG > 6 (dB)

Mϕ > 45 (◦)

S.T. 2% within 3 (s)

II. Suggested Methodology For a Flight Case

This section presents the method suggested in Ref. 7. The control structure being fixed, the values ofthe 4 gains (Kq,Kp,Ki,Kff ) and the wash-out filter pole zwo are to be chosen in order to satisfy the abovespecifications. Modal and frequency criteria are set up in two different ways. Model order reduction andGuardian Maps are used to set Short Period damping whereas the frequency criteria are dealt with in directcomputational manner using appropriate formulae developed in Ref. 7.

The SAS loop is first tuned with the objective to make the Short Period modes sufficiently damped;this entails choosing appropriate values for Kq and zwo. Figure 2 shows for a particular case the kind of

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region that can be obtained in the (Kq, zwo) parameter space. The region marked with * guarantees that allcomplex poles have a damping ratio of 0.7 or more for any choice of (Kq, zwo) in this region.

0 0.2 0.4 0.6 0.8 1−5

−4.5

−4

−3.5

−3

−2.5

−2

−1.5

−1

−0.5

0

Kq

z wo

ζ=0.35

ζ=0.7

*

Figure 2. Guardian Map Zone for SAS Loop

0 0.5 1 1.5 20

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

Kff

Ki

Drb=0

Drb=0.5

Drb=0.3

Drb=−0.2

Figure 3. Dropback Zone

Once a (Kq, zwo) pair is chosen, the designer can proceed to the choice of the three remaining gains inorder to finish the CAS design. As proven in Ref. 7, the dropback only depends on the Ki and Kff gains.Figure 3 illustrates the valid region to fulfill the drobpack requirement (e.g. every choice (Kp, Ki) below theline Drb=0.3 guarantees a dropback which is no greater than 0.3).

For a given Kff , a (Kp, Ki) pair can then in turn be elected in the marked region of Figure 4. Valid gainscan therefore be found in this way so that all the modal and frequency criteria are satisfied. A GUI basedupon this method has been developed; for each choice of the flight case, all graphics are updated allowingone to find better gain choices in order to satisfy the settling time constraint.

0 0.5 1 1.5 2 2.5 3 3.5 40

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

Kp

Ki

ζ=0.7

τd=0.14 s

ωBW

θ

=1.75 rad/s

PM=45°

GM=6dB

*

Figure 4. Handling Quality Drawing

0 1 2 3 4 50

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

Time (s)

θ (°)q (°/s)C*

Figure 5. Time-Responses to a Stick Unit Step

The authors are presently working on an optimization procedure which, from an initial feasable guessfor the gains obtained as described above, would improve the results further. Good time-domain responsesto a stick unit step can nevertheless be easily found as is shown in Figure 5 (time responses have beennormalized).

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III. Robust Design

Our first aim is to conceive a robust controller to varying masses and/or centers of gravity at a fixed flightcondition. A controller (i.e. a unique choice of the gains) is sought that remains effective (that is satisfiesall the HQs) even if the aircraft mass or its center of gravity vary.

A. Uncertain Models

For a given flight condition, we have 6 different configurations; the Table 2 shows the different cases for oneflight condition, namely that of Table 3.

Table 2. Grid Points for varying m and xcg

m (lbs) \ xcg (% chord) 16 20 35 3830,000 × ×32,00039,000 × ×46,000 × ×

Table 3. Flight Condition

Mach Number Altitude Dynamic Pressure0.4 5000ft 197.5psf

B. Robust SAS Loop

For each configuration, we can then draw the 0.7 damping ratio zones and superpose them as illustrated inFigure 6. Any pair (Kq, zwo) chosen inside the dashed region yields complex poles with a damping ratiogreater than 0.7, for all the configurations under consideration.

0 0.2 0.4 0.6 0.8 1−5

−4.5

−4

−3.5

−3

−2.5

−2

−1.5

−1

−0.5

0

Kq

z wo

Figure 6. Superposition of Different Cases Damping Ratio Zones - SAS Design

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C. Robust CAS Loop

For a choice of a (Kq, zwo) pair, we proceed to the CAS design. By drawing for each configuration the validdropback zones, we can again find a common region that ensures that dropback is at least Level 1 for eachconfiguration (Fig. 7).

0 0.5 1 1.5 20

1

2

3

4

5

6

7

Kff

Ki

Figure 7. Common Region for Level 1 Dropback

0 0.5 1 1.50

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

Kp

Ki

Figure 8. Common Region for Level 1 HQs - CASDesign

For a given Kff ,the dashed region in Figure 8 shows the common (Kp, Ki) region that will satisfy all thehandling qualities at Level 1 for each configuration.

For a particular choice in the dashed region, we obtain the time-domain responses of Figure 9. Asdesigned, all the handling qualities are Level 1.

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

1

2

3

4

5

6

Time (s)

q (r

ad/s

) | θ

(ra

d)

θq

Figure 9. Pitch Attitude and Pitch Rate Responses to a Stick Unit Step for the 6 Different Configurations

For the 3 lower values of xcg , the time responses are the better ones and they get worse for increasingvalues of xcg .

After finding continuous uncertain models depending on the mass and the center of gravity11 (via inter-polation), it is possible to rigorously analyse the robustness of the design by using the guardian map theory.This is presently being pursued by the authors.

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IV. Gain-Scheduling

The flight envelope is composed of 20 linearized flight cases, and for each flight condition, owing to theabove guidelines, it is possible to find valid gains that satisfy all the handling quality requirements. Sincea single controller which fulfills all criteria for every flight condition cannot be found, one has to resort togain-scheduling. It consists of finding a function depending on a flight condition variable (Mach, altitudeand/or dynamic pressure) that interpolates all the individual gain values found for each flight case. Theapproach can help solve this problem in principle, or at least help achieve what might be referred to aspre-gain-scheduling.

A. Flight Envelope

Experiments have shown that dynamic pressure could be a good candidate as the scheduling variable in ourstudy. The full flight enveloppe we are using covers altitudes from 5000 to 45000 fts and Mach numbers from0.2 to 0.9.

B. Scheduling the SAS Loop

This subsection outlines how the parameters Kq and zwo can be interpolated in terms of the dynamicpressure.

In Figure 10, guardian map regions that guarantee the 0.7 damping ratio are plotted for 8 flight conditions.It is seen that as the dynamic pressure increases, the loop moves and shrinks. The goal is then to find acurve depending on the dynamic pressure that passes through those loops.

0 0.5 1 1.5−5

−4

−3

−2

−1

0

DP=77.10 0.5 1 1.5

−5

−4

−3

−2

−1

0

DP=133.9

0 0.5 1 1.5−5

−4

−3

−2

−1

0

DP=197.5z wo 0 0.5 1 1.5

−5

−4

−3

−2

−1

0

DP=209.3

0 0.5 1 1.5−5

−4

−3

−2

−1

0

DP=301.40 0.5 1 1.5

−5

−4

−3

−2

−1

0

DP=308.6

0 0.5 1 1.5−5

−4

−3

−2

−1

0

DP=410.20 0.5 1 1.5

−5

−4

−3

−2

−1

0

DP=444.3 K

q

Figure 10. Guardian Map Zones for 8 Flight Conditions - SAS Loop

Utilizing the Matlabr Curve Fitting toolbox,12 we obtained the following interpolating curves:

zwo = −0.01026p + 0.03885 (1)

Kq = 0.7391e−0.006303p + 0.02921 (2)

where p is the dynamic pressure. Forms other than the exponential one could of course be chosen for theKq gain.

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C. CAS Scheduling and Results

Once the appropriate curves have been chosen for the SAS loop parameters, three more curves have to beenfound for the CAS loop. As before, by using the method and considering the former choices for Kq and zwo ,one can plot the dropback and the handling quality criteria for each flight condition. Dynamic pressure isagain chosen as the scheduling variable.

After a curve is chosen for Kff using the dropback formulae, one would find the (Kp,Ki) regions shownin Figure 11.

0 1 2 3 40

5

10

0 1 2 3 40

5

10

1 2 3 40

5

10

0 1 2 3 40

5

10

0 1 2 3 40

5

10

0 1 2 3 40

5

10

0 1 2 3 40

5

10

0 1 2 3 40

5

10

* *

* *

*

*

*

*

DP=77.1 DP=133.9

DP=197.5 DP=209.3

DP=301.4 DP=308.6

DP=410.2 DP=444.3 K

p

Ki

Figure 11. HQ Zones for 8 Flight Conditions - CAS Loop

The Curve Fitting toolbox can once more help find suitable functions. The following forms were chosenfor Kp, Ki and Kff :

Kp =1184000

p3+ 0.1383 (3)

Ki = 2.331e−0.003285p + 1.798 (4)

Kff =349400

p3+ 0.1826 (5)

Figures 12 and 13 show the time-domain pitch responses to a stick unit step after gain-scheduling. A fewtrials were necessary to find good time responses but this new method guided us well in finding worthwhilescheduling functions. For all flight conditions, we are insured that all the handling quality requirements arefulfilled.

Conclusion

Classical flight control system architecture are still widely used in industry. A new method based upon amodern technique has been proposed to choose efficiently the variable control gain parameters. The presentarticle shows how this simple new method can help solve the robust and gain-scheduling problems, or at leastguide the resolution of these problems. These results are being improved upon via optimization. Althoughthe robust and gain-scheduling issues have been addressed separately, they could both be treated at thesame time in order to find a robust gain-scheduled controller.

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02

46

810

100

200

300

400

0

0.5

1

1.5

2

Time (s)Dynamic Pressure (kpa)

q (r

ad/s

)

Figure 12. Pitch Rate Responses to a Stick UnitStep versus dynamic pressures

0 1 2 3 4 5 6 7 8 9 100

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

Time (s)

q (r

ad/s

)

Figure 13. Pitch Rate Responses to a Stick UnitStep

Acknowledgments

This work was funded by NSERC and Bombardier Aerospace Inc. under Grant CRD 215 465-98. Theauthors would like to thank Fraser MacMillen, David Reitz and Navid Niksefat of Bombardier AerospaceInc. for their constructive remarks

References

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2TISCHLER, M. B., Advances in aircraft flight control , London : Taylor & Francis, 1996.3GIBSON, J. C., The definition, understanding and design of aircraft handling qualities, Delft, Netherlands: Delft

University Press, 1997.4HODGKINSON, J., Aircraft handling qualities, Reston, Va : American Institute of Aeronautics and Astronautics, Inc.;

Oxford, United Kingdom : Blackwell Science Ltd., 1999.5MITCHELL, D. G., DOMAN, D. B., KEY, D. L., KLYDE, D. H., LEGGETT, D. B., MOORHOUSE, D. J., MASON,

D. H., RANEY, D. L., and SCHMIDT, D. K., “Evolution, revolution, and challenges of handling qualities,” Journal of Guidance,Control, and Dynamics, Vol. 27, No. 1, 2004, pp. 12–28.

6SAUSSIE, D., SAYDY, L., and AKHRIF, O., “Flight control design with robustness and handling qualities requirements,”CCECE 2003 - Canadian Conference on Electrical and Computer Engineering., Vol. 3, IEEE, 2003, pp. 1749–1752.

7SAUSSIE, D., AKHRIF, O., and SAYDY, L., “Longitudinal flight control design with handling quality requirements,”Tech. Rep. EPM-RT-2005-01, 2005.

8SAYDY, L., AKHRIF, O., and ZHU, G., “Handling quality characterization of flight system controller gains,” ICECS2000. 7th IEEE International Conference on Electronics, Circuits and Systems, Vol. 2, IEEE, 2000, pp. 721–724.

9SAYDY, L., TITS, A., and ABED, E., “Guardian maps and the generalized stability of parametrized families of matricesand polynomials,” Mathematics of Control, Signals and Systems, Vol. 3, 1990, pp. 345–371.

10US Department of Defense, “Flying qualities of piloted aircraft,” MIL-STD-1797A, 1990.11ZHU, G., AKHRIF, O., SAYDY, L., and HENTABLI, K., “Robustness augmentation of fixed structure flight control

systems via µ,” AIAA paper 2000-4366, Aug. 2000.12MATLABr, Curve fitting toolbox user’s guide, The MathWorks, Inc., 2002.

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