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+2 and +∞ -Optimal Gust Load Alleviation for a Flexible Aircraft

Nabil Aouf, Benoit Boulet, and Ruxandra Botez*

McGill Centre for Intelligent MachinesMcGill University

3480 University Street, Montréal, Québec, Canada H3A 2A7

*Département de Génie de la Production AutomatiséeEcole de technologie supérieure

1100 Notre-Dame Street West, Montréal, Québec, Canada H3C 1K3

Abstract

+2 (LQG), weighted-+2 and +∞ techniques are usedto establish a nominal performance baseline for the verticalacceleration control of a B-52 aircraft model withflexibilities. The aircraft is assumed to be subjected tosevere wind gusts causing undesirable vertical motion. Thepurpose of the controllers is to reduce the transient peakloads on the airplane caused by the gusts. Our designsaccount for the flexible modes of the aircraft model. Weuse the Dryden gust power spectral density model to guidethe performance specifications and control designs, as wellas for time-domain simulations. Motivation for the use ofan +∞ performance specification is given in terms of anew interpretation of the Dryden model. Both weighted-+2 and +∞ optimal controllers are shown to reducedramatically the effect of wind gust on the aircraft verticalacceleration when compared to the regular +2 controller.

I- INTRODUCTION

Gust load alleviation (GLA) systems use motion sensorfeedback to drive the aircraft's control surfaces in order toattenuate aerodynamic loads induced by wind gusts. Thispaper compares three nominal baseline GLA controllersdesigned using +2 (LQG), weighted-+2 and+∞techniques for the longitudinal dynamics of a B-52bomber. Motivation is given for each approach in terms ofgust models and performance specifications. The dynamicmodel of the aircraft used for control design includes fiveflexible modes. Such a model is more realistic than a rigid-body model, but it can also make the GLA feedbackcontrol design problem more challenging [2]. Previousresearch on gust load alleviation reported in [3], [5] usedan LQG approach without performance weights forflexible aircraft models. Recent works [1], [7] reported onapplications of +∞ techniques to flight control problems,but rigid-body models of the aircraft were used. Theneglect of flexible modes may lead to robustness problemsduring and after system commissioning.The gust signals are assumed to be generated by theDryden power spectral density model. This model lendsitself well to frequency-domain performance specifications

in the form of weighting functions. Since the Dryden gustmodel is simply a white noise driving a stable real-rationallinear time-invariant filter, it is natural to consider thedesign of an LQG controller, or its equivalent deterministic+2 counterpart. However, as a bumpy flight experiencewould suggest, wind gusts are acting only over briefperiods of time. Air turbulence in a localized area may lastfor a time long enough to warrant the Dryden colorednoise approximation. But the point here is that an aircraftmomentarily passing through the turbulence will onlyexperience its effect for a brief period of time. Thus,rather than using signals of finite average power for LQGGLA control design, we suggest that such behavior may bebest captured by bounded-energy signals with spectralcontents given by the magnitude of the Dryden filter. Inthe deterministic, worst-case L2 setting, this remark givessome motivation to design a baseline GLA controller withan +∞ performance specification.

II- VERTICAL GUST MODELS

The classical Dryden representation of the power spectraldensity (PSD) function of atmospheric turbulence can bewritten as:

Φ w

w w w

w

L L U

U L U( )ω

σ ω

π ω=

+

+

20

2

0 0

2 2

1 3

1

1 6

1 6

(1)

where σw is the RMS vertical gust velocity (m/s), Lw is thescale of turbulence (m), and Uo is the aircraft trim velocity(m/s). For thunderstorms, typical values used are Lw = 580m (1750 ft), σw = 7 m/s (21 ft/s). Formally, the Drydenmodel generates gust signals through the relationshipw G ng w= , where n t( ) is a zero-mean, unit-intensity

gaussian white-noise process with PSD Φn ( )ω = 1, andwg(t) is the random vertical gust process. An expression forthe Dryden filter Gw can be found through spectral

factorization of wΦ ( )ω , which yields

. (2)2

0

0

20 33

)(

+

+=

sL

U

sL

U

L

UsG

w

w

w

w

w πσ

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A proposed alternative use of the Dryden model is toconsider the noise n to be any deterministic finite-energy

signal in 1: [ , ) := ∈ ∞ ≤n L n2 20 1< A . The gust signal

then lives in : 1: : [ , )= ∈ ⊂ ∞G n n Lw; @ 2 0 , and its

energy is bounded by w GL

Ug ww w

20

9 2

10≤ =

∞

σπ

.

Furthermore, such signals taper off at infinity in the timedomain. Hence, they may be more representative of realwind gusts acting on an aircraft passing through aturbulence. Although the stochastic nature of the signal islost, the resulting set of bounded-energy gust signals canbe used for a worst-case +∞ design, which may bedesirable in a safety critical application such as GLA.

III- FLEXIBLE AIRCRAFT MODEL DESCRIPTION

The short-period approximation for the rigid-body motionof the B-52 aircraft is considered. The rigid-bodydynamics are augmented by a set of modal coordinatesassociated with the normal bending modes of the B-52.

The i th flexible mode is represented by the following

equation in terms of its modal coordinate η j :

�� �η ς ω η ω η ρ φi i i i i i i i+ + =2 2 , (3)

where ς ω ρi i i, , are the damping ratio, frequency and

gain of the i th flexible mode, and φ i is its corresponding

generalized force. Thus, the rigid-body dynamics may beaugmented with pairs of first-order equationscorresponding to each flexible mode considered. Fivestructural flexible modes were considered significant andwere kept in the B-52 aircraft longitudinal dynamic modeltaken from [5]. The control inputs for the longitudinalmotion are the deflection angles (in radians) of the elevator

δ el and the horizontal canard δ hc . The longitudinaldynamics of the flexible aircraft are given by:

�x Ax Bu B wg g= + + (4)

y Cx Du= +where x R∈ 12 is the state vector given by,

x qT = [ � � � � � ]α η η η η η η η η η η1 1 5 5 7 7 8 8 12 12

u Rel hc

T= ∈δ δ 2 is the control vector (rad)

y R∈ is the vertical acceleration (g),

w w w w Rg g g g

T= ∈1 2 3

3 is the vertical gust velocity

at three stations along the airplane (m/s), α ∈R is theangle of attack (rad), q R∈ is the pitch rate (rad/s).There are couplings between the flexible modes and therigid-body mode. The eigenvalues of A corresponding tothe rigid-body mode are λ 1 2, = ±-1.803 2.617j . The

five flexible modes are listed in Table 1 below.

Table 1: flexible modes1 2 3 4 5

ω i 7.60 15.22 19.73 20.24 38.29

ς i 0.393 0.056 0.011 0.067 0.023

Note that the second and third gust signals wg2 and wg3 areactually delayed versions of wg1. The second gust isdelayed by τ1 = U0/x1 = 0,06 s, where x1 is the distancefrom the first body station to the second. The third gustinput is delayed by τ2 = U0 /x2 = 0,145 s. First-order lagapproximations of the time delays are used:

ws

wg g2 1

1

0 06 1=

+., w

swg g3 1

1

0145 1=

+..

IV- PROBLEM SETUP

A block diagram of the closed-loop GLA design problemwith weighting functions is shown in Figure 1 below:

Fig. 1: problem setup

where w Rn1 ∈ is an acceleration measurement noiseused to regularize the problem, with amplitude specifiedby 410−=ε as wn is normalized, r R∈ is the vertical

acceleration setpoint (r = 0), e R∈ is the measurederror, z R1 ∈ is the weighted measured acceleration,

z R22∈ is the weighted controller output, and F s( )

contains the lags approximating the delays. The plant

transfer matrix G s( ) mapping [ ]u wgT T

to y is given

by

G sA B B

C Dg( )

[ ]

[ ]=

�!

"$#0

. (5)

The GLA problem of Figure 3 can be recast into thestandard +2 and+∞ optimal control problem of Figure 4below.

Fig. 2: standard control problem setup

The vector of exogenous inputs in Figure 4 is

w n wnT: [ ]= for the +2 controller designs, or

P s( )wz

−e u

K s( )

+

+ +

-

wg

wn

r = 0

W s1( )W su ( )

G s( )K s( )

ε

z1z2

yu

e

wn1

F s G sw( ) ( )n

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w w wgT

nT: [ ]= for the +∞ controller design. The

signals to be minimized are collected in zW y

W uu

:= �

!

"

$#

1 .

The nominal generalized plant

P sP s P s

P s P s( )

( ) ( )

( ) ( )=

�!

"$#

11 12

21 22

(6)

has a minimal state-space realization�x A x B w B u

z C x D w D u

y C x D w D u

P P P

P P P

P P P

= + += + += + +

1 2

1 11 12

2 21 22

(7)

which combines the aircraft model, the Dryden filter (forthe +2 controller) and the weighting functions. For

convenience, we will use the notation T x yxy:= � for

closed-loop transfer matrices.

V- +2 CONTROL

Assuming that the origin of the gust is a zero-meangaussian white noise of unit intensity, we first design anunweighted +2 controller. This control design is basedon the generalized plant with exogenous inputs

w n wnT: [ ]= and outputs z z zT

T= 1 2 . Thus, the

gust vector generator FGw is embedded in the

generalized plant P . Note that all our control designswere carried out following [5] and using the Matlab µ-Synthesis and Analysis Toolbox [3]. For regular +2

control design, we used the weighting functions W1 1=and Wu = 1. Closed-loop performance is indirectly

specified by wG . For instance, choosing σ w m s= 7 /

and L mw = 580 in wG for simulation purposes, we

obtain a gust that has most of its power concentrated in thefrequency band [ . , ]01 6 Hz. The objective of the +2

controller design is to minimize

T T j T j dwz wz wz2

1

21

2= �

! "

$#∗

−∞

∞Iπω ω ωTr{ ( ) ( )} , (8)

the +2 norm of Twz , over all finite-dimensional, linear

time-invariant stabilizing controllers K s( ) . Assuming

that z t( ) is ergodic, its average power is then minimizedas

lim ( ) ( )T T

T

wzTz t dt E z t T

→∞ −= =I1

22 2

2

2> C . This has

the effect of decreasing the +2 norm of submatrices of

Twz by virtue of the fact that

G G G G1 2 2

2

1 2

2

2 2

2= + . For example, the closed-

loop vertical acceleration of the aircraft can be written interms of the uncorrelated white noises n and wn asfollows: y T w T n T wwy ny w y nn

= = + . (9)

The +2 norms of the transfer matrices Tny and Tw yn

satisfy T T Tny w y wzn2

2

2

2

2

2+ ≤ . The +2 controller

obtained had 20 states and achieved 2wzT = 0.2. Figure

3(a) shows the spectral norms of T jny( )ω and

T jw yn1( )ω with the +2 controller for comparison with

the other controllers.

Fig. 3: Norms of closed-loop transfer matrices

VI- WEIGHTED +2 AND +∞ -OPTIMAL CONTROL

A- Weighted +2 controller

In order to improve the performance obtained with theregular +2 controller, we introduce a performanceweighting function

W sk

a s11

0 1( ) =

+(10)

on the acceleration y , with k1 500= , a0 0 05= . . This

weighting function is the same as the one used for the +∞design in order to compare the results obtained for both

controllers. We obtained Twz 215= . with the weighted-

+2 controller, which means that

W T W Tny w yn1 2

2

1 2

22 25+ ≤ . . Figure 3(c) shows that the

weighting function led to much better GLA performance at

10 3−

108

10 3−

108

10 3−

108

( )a ( )b

( )c

ω (log) ω (log)

ω (log)

T jw yn1( )ω

T jny ( )ω

G jw ( )ω

T jw yn1( )ω

T jny ( )ω

W j11− ( )ω

T jw yg( )ω

T jw yn1( )ω

W j11− ( )ω

W −+2

+2 +∞

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low frequencies by trading off sensor noise rejection,which deteriorated slightly, but remained acceptable.

B- +∞ -Optimal Design

As gusts act over a relatively short period of time, they canbe considered as signals with finite energy. Such signalscan have spectral contents similar to the PSD of stochastic

dryden gust signals by using )(sGw as a filter. This

remark provides motivation for +∞ GLA control design

as min max min max minK w

wzK n

wz wK

wz wT w T G n T G∈ ∈ ∈ ∈ ∈ ∞

= =6 : 6 1 62 2

where 6 is the set of all stabilizing controllers.

B1- Weighting Functions for Nominal Performance

The specification is that our controller has to be able toregulate the vertical acceleration in the gust bandwidthwith an amplitude attenuation of at least −54 dB(500).The closed-loop vertical acceleration of the aircraft can bewritten in terms of the gust vector wg and the noise wn

as follows:

y T w T ww y g w y ng n= +

1, (11)

The gust alleviation performance specification on

T jw yg( )ω can be enforced through the use of a

weighting function W1 of amplitude at least 500 over

[ . , ]01 6 Hz, as long as we get W Tw yg1 1∞

< with the

controller K , which implies

T j W jw yg( ) ( )ω ω< −

1

1. (12)

The weighting function is as given in (10). A plot of

W j1

1( )ω −

is shown in Figure 3(b)(c). The controller

outputs consist of deflection angles (in radians) of theaircraft's elevators and horizontal canards. In order to keepthese angles within acceptable limits, we used a suitable

weighting function Wu on u such that W Tu wu ∞ < 1.

The weighting function Wu is a constant diagonal matrix

W k ku u u= diag 1 2,; @ so that the above +∞ -norm

condition implies

T j kwu u11

1( )ω < −

(13)

and

T j kwu u21

2( )ω < −

. (14)

where TT

Twuwu

wu

=�!

"$#

1

2

. Here ku11 0 2− = . , ku2

1 05− = . are

the maximum allowed control gains in closed loop. It ispreferable to use a constant weighting matrix for thisapplication, since it does not increase the order of P s( ) .

Hence, the resulting +∞ controller has a lower order.

B2- +∞ -Optimal Controller

The overall objective in this +∞ controller design was to

minimize Twz ∞ over all finite-dimensional, linear time-

invariant stabilizing controllers K s( ) , in order to get

Twz ∞ < 1. This would guarantee that the performance

specification is satisfied. A norm of Twz ∞ = 0 68. was

achieved. Figure 7 shows that our +∞ controller meetsthe GLA performance specification given above. We cansee that the maximum singular value of Tw yg

is well

below 10-5 over 2 01 6π[ . , ] rd / s. The spectral norms of

the frequency responses of Twu1 and Twu2 satisfy theconstraints of (13) and (14) respectively.

VII- SIMULATION RESULTS

In this section, we use the Dryden model with parameters

σ w =7m s, L mw = 580 to generate severe wind gustsignals for simulation purposes. Figure 4(a),(b),(c) showsthe gust signals used in our simulations. The open-loopvertical acceleration response of the B-52 to these gustsignals is shown in Figure 4(d).

Fig. 4: Gust signals and open-loop response

Figure 3(a) shows a magnitude plot of the frequencyresponse of the Dryden filter for the simulation. We seethat the performance specification enforced by theweighting function W1 used for the weighted-

+2 and+∞ controllers (Figure 3(b)(c)) should result inefficient gust alleviation. Time-domain simulations wereconducted and results are presented below. The resultsshown in Figure 5(a)(b) confirm that the +∞ controllercan dramatically reduce the effect of wind gusts on thevertical acceleration of the aircraft for the nominal model.This controller also seems suitable from the point of view

)( sm )( sm

)(

0 5 10 15-5

0

5

10

time(s)

vert

ical

con

tinuo

us g

ust

wg1

0 5 10 15-5

0

5

10

time(s)

vert

ical

con

tinuo

us g

ust

wg2

0 5 10 15-5

0

5

10

time(s)

vert

ical

con

tinuo

us g

ust

wg3

( )a ( )b

( )c ( )d

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of control effort. From Figure 5(a), it can be seen that the

elevator angle remained within ± ±0 25. ) radians ( 14$ in

the simulation. The +∞ controller gave the best GLAperformance without exciting the flexible modes of themodel or generating large control angles that couldsaturate the control surfaces. A comparison of peakvertical accelerations with the +∞ controller (Figure 5(b))and without any feedback control (Figure 4(d)) indicatesthat the +∞ controller reduced the acceleration by a factor

of 105 . This could translate into dramatic improvementsin flight comfort and reduced airframe loads.

Fig. 5: Simulation results

In contrast to the +∞ controller, the +2 controller(similar to the one proposed in [4]) only reduced the peakacceleration by a factor of 14 as seen in Figure 5(f). Thiscontroller is also of higher order than the +∞ controller,which can be a disadvantage for implementation.A significant performance improvement was obtained overregular +2 by using a weighted-+2 control approach, asevidenced by Figure 5(d). The performance obtained withthe weighted-+2 controller is more comparable to the

performance of the +∞ controller, although still about 20

times worse in terms of peak acceleration. The +∞controller appeared to control the flexible modes better,which is not surprising in view of its low closed-loop

T jw yg( )ω in the frequency range of the flexible modes

(see Figure 3(b)).

VIII- CONCLUSION

We presented a comparison of +2 , weighted-+2 and

+∞ GLA controller designs for the nominal model of aB-52 aircraft with flexible modes. The gust was modeledby a Dryden filter whose input was assumed to be whitenoise for the two +2 designs. We argued that gust signalsmay be better represented by a set of filtered deterministic,bounded-energy signals, with the Dryden transfer functionused as the filter. This motivated the use of +∞ controlfor GLA. This kind of model lends itself well tofrequency-domain performance specifications in the formof weighting functions. The +∞ controller was shown tomeet the desired performance specification withreasonably small control surface deflection angles. Itoutperformed both the +2 and weighted-+2 controllersin the simulation. It was also shown that it may provebeneficial to weight the outputs in an +2 GLA controldesign. Future investigations will address the robustnessissues related to uncertain modal parameters and differentflight envelopes.

IX- REFERENCES

[1] Vincent, J.H., Emami-Naeini, A., Khraishi, N.M., CaseStudy Comparison of Linear Quadratic Regulator and +∞Control Synthesis, AIAA Journal of Guidance, Control,and Dynamics Vol. 17, No. 5, Sept.-Oct. 1994, pp.958-965.[2] Newman, B., Buttrill, C., Conventional Flight Controlfor an Aeroelastic Relaxed Static Stability High-SpeedTransport, Proc. AIAA GN&C Conf., Aug. 1995,Baltimore, MD, pp. 717-726.[3] Balas, G.J., Doyle, J.C., Glover, K., Packard, A.,Smith, R., µ -Analysis and synthesis toolbox, MUSYN

Inc, and the MathWorks, Inc, 1995.[4] Botez, R.M., Boustani, I. and Vayani, N., Optimalcontrol laws for gust load alleviation, Proceedings of the1999 46th CASI Annual Conference, May 3-5, Montréal,Québec, Canada, pp. 649-655,[5] Doyle, J.C., Glover, K., Khargonekar, P., Francis, B.,State-space solutions to standard +2 and +∞ controlproblems, IEEE Trans. on Automatic Control AC-24(8),1988, pp. 731-747.[6] Mclean, D, Gust load alleviation control systems foraircraft, Proc. IEE, Vol.125, No.7, July 1978, pp. 675-685.[7] Niewoehner, R.J., and Kaminar, I.I., Design of anautoland controller for an F-14 Aircraft using +∞synthesis. AIAA Journal of Guidance, Control, andDynamics Vol. 19, No. 3, May 1996, pp.656-663.

+∞

W −+2W−+2

+2+2

( )a ( )b

( )c ( )d

( )e ( )f

+∞