Robust Gust Load Alleviation for a Flexible Aircraft

Nabil Aouf, Benoit Boulet Ruxandra Botez Centre for Intelligent Machines Département de Génie de la production automatisée McGill University Ecole de technologie supérieure

3480 University Street, Montréal, Québec, 1100 Notre-Dame Street West, Montréal, Québec, Canada H3A 2A7 Canada H3C 1K3

Tel: (514) 398-1478, Fax: (514) 398-4470 Tel: (514) 396-8560, Fax: (514) 396-8595 E-mail: aouf@cim.mcgill.ca, boulet@cim.mcgill.ca, E-mail: ruxandra@gpa.etsmtl.ca

AbstractAn ∞ controller design is presented to establish a nominal performance baseline for the vertical

acceleration control of a B-52 aircraft model with flexibility. µ-synthesis and analysis is used to study therobust performance of the aircraft taking into account specific input and output multiplicative uncertaintymodels. The aircraft is assumed to be subjected to severe wind gusts causing undesirable vertical motion.We use the Dryden gust power spectral density model to guide the performance specifications andcontrol designs, as well as for time-domain simulations. Motivation for the use of an ∞ nominalperformance baseline specification is given in terms of a new interpretation of the Dryden model. The

∞ optimal controller is shown to reduce dramatically the effect of wind gust on the aircraft vertical

acceleration. Robust performance is achieved with a µ-synthesis controller design using a D-K iterationprocedure.

I- INTRODUCTION

Gust load alleviation (GLA) systems can be usedto reduce the effects of wind gusts on verticalacceleration of aircraft. Their purpose is toreduce airframe loads and improve passengercomfort. In this paper, the longitudinal dynamicsof the B-52 bomber are studied [6]. The dynamicmodel of the aircraft includes structuralflexibility. Such a model is more realistic than arigid-body model, but it can also make feedbackcontrol design for gust load alleviation morechallenging.

We present ∞ and µ controller designs for amodel of the B-52 aircraft with flexible modes.The gust is generated with the Dryden powerspectral density model. This kind of model lendsitself well to frequency-domain performancespecifications in the form weighting functions.The ∞ and µ controllers are shown to meet thedesired nominal performance and the robustperformance specifications with reasonablysmall control surface deflection angles.

Previous research on GLA control systemsreported in [2], [5], [7] take approaches differentto ours, notably LQG and ∞ for a rigid-bodyaircraft .

II- GUST MODEL

Two classical analytical representations for thepower spectral density (PSD) functions ofatmospheric turbulence were given by VonKarman and Dryden [6]. As the Dryden PSDfunction has a simpler form than Von Karman'swe chose to use the former. It can be written as:

Φ w

w ww

w

LL

U

UL

U

( )ω

σ ω

πω

=

+

!

"

$##

+

!

"

$##

2

0

2

00

2 2

1 3

1

(1)

where:σw is the RMS vertical gust velocity (m/s),Lw is the scale of turbulence (m), andUo is the aircraft trim velocity (m/s).

The scale length Lw is dependent on the aircraft'sheight h when atmospheric turbulence isencountered, as follows:At h >580 m (1750 ft), Lw = 580 m,At h < 580 m, Lw = h m .For thunderstorms, at any height:Lw = 580 m (1750 ft), σw = 7 m/s (21 ft/s).

Gust signals have to be generated with therequired intensity, scale lengths and PSDfunctions for some given flight velocity andheight. In order to generate these gust signals, anoise source with PSD function Φn ( )ω = 1 inthe frequency band of interest is used to providethe input signal to a linear filter G sw ( ) chosensuch that the squared magnitude of its frequencyresponse is the PSD function Φw ( )ω . The gustgenerator setup is as follows:

Linear filter n(t)

Fig. 1: Gust signal generator

where n t N( ) ~ ( , )0 1 is a Gaussian white noiseprocess of unit intensity and zero mean, andwg1(t) is the random continuous vertical gust, sothat, formally, 1g ww G n= . The PSD of the

output signal is related to the PSD of the inputsignal as follows:

G j G jw w N wΦ Φ( ) ( ) ( )ω ω ω ω= =2 21 6 (2)

An expression for the Dryden filter can be foundthrough spectral factorization of wΦ ( )ω ,which yields

(3)

A proposed alternative use of the Dryden modelis to consider the noise n to be anydeterministic finite-energy signal in

2 2: [0, ): 1n n= ∈ ∞ ≤$1 . The gust signal

lives in 2: : [0, )wG n n= ∈ ⊂ ∞$: 1 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 representativeof real wind gusts acting on an aircraft passingthrough a turbulence. Although the stochasticnature of the signal is lost, the resulting set ofbounded-energy gust signals can be used for aworst-case ∞ design, which is desirable in asafety-critical application such as GLA.

III- FLEXIBLE AIRCRAFT MODEL

The short-period approximation for the rigid-body motion of the B-52 aircraft is considered.The aircraft's rigid-body dynamics equations areaugmented by adding to the state variables a setof generalized coordinates associated with thenormal bending modes. Structural displacementwas considered small compared to the whole

aircraft structure. The j th flexible mode isrepresented by the following second-order linearconstant-coefficient differential equation interms of its modal coordinate η j :

22j j j j j j j jη ζ ω η ω η ρ φ+ + = .(4)

where , ,j j jς ω ρ are the damping ratio, frequency

and gain of the thj flexible mode, and jφ is its

corresponding generalized force. Thus, the rigidaircraft dynamics may be augmented with pairsof first-order equations corresponding to eachflexible mode considered. Five structuralflexible modes were considered significant andwere kept in the longitudinal dynamic model ofthe B-52 aircraft [5].

The control inputs for the longitudinal motionare the deflection angles (in radians) of theelevator δ el and the horizontal canard δ hc . Thelongitudinal dynamics of the flexible aircraft interms of the state variable representation is:

g gx Ax Bu B w= + + (5)

y Cx Du= +

G sw ( )

2

0

0

20 33

)(

+

+=

sL

U

sL

U

L

UsG

w

w

w

w

w πσ

1( )gw t

where 12x∈ is the state vector,x

q

T =[ ]α η η η η η η η η η η1 1 5 5 7 7 8 8 12 12

2u∈ is the control vector, u = [δel δhc]T

(radians), y∈ is the vertical acceleration (g),3

gw ∈ is the vertical gust velocity at three

stations (m/s) , α ∈ is the angle of attack(radians) and q∈ is the pitch rate(radians/s).

The form of the A matrix in state-space equation(5) shows the couplings between the aircraft'sflexible structure and rigid-body dynamics:

AA A

A Arr ra

ar aa

=

!

"

$# (6)

where Arr are the rigid-body terms, Ara therigid/aeroelastic terms, Aar the aeroelastic/rigidterms, and Aaa the structural flexibility terms.The eigenvalues of A corresponding to the rigid-body mode are λ1 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 (rd/s) 7.60 15.22 19.73 20.24 38.29

ς i0.393 0.056 0.011 0.067 0.023

Input terms associated with the effects of windgust acting at three different body stations arealso included in (5). They appear as threedifferent gust signals wg1, wg2, and wg3 acting inthe longitudinal dynamic equations. Thus, thegust vector is defined as wg=[wg1 wg2 wg3]

T.

The second and third gust signals wg2 and wg3 aretime-delayed versions of wg1. The second gust isdelayed by a time τ1 = U0/x1 = 0.06 s, where x1 isthe distance from the first body station, whichencounters the gust first. The third input isdelayed by time τ2 = U0 /x2 = 0.145 s. First-orderlags are used to model the delays. The formalgeneration of the gust vector is shown in Figure2.

Fig. 2: Gust signals at three body stations

IV- ∞ -OPTIMAL CONTROL

A- Problem Setup

A block diagram of the closed-loop gustalleviation design problem with weightingfunctions is shown in Figure 3 below:

Fig. 3 Setup for ∞ control

where 3gw ∈ is the gust disturbance,

1nw ∈ is an acceleration measurement noise,

0r = is the vertical acceleration setpoint,

1z ∈ is the weighted measured error, and2

2z ∈ is the weighted controller output. The

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

to y is given by

[ ]

( )[ 0]

gA B BG s

C D

=

. (7)

The signal wn1 is a small disturbance that has a

role to play in regularizing the ∞ designproblem. It can also be seen as a realmeasurement noise. Its amplitude is specified by

1

0 06 1. s+

1

0145 1. s+wg3

wg2

DrydenFilterGw(s)

+

++

-

wg

wn

r

W s1( ) W su ( )

G s( )K s( )

ε

z1 z2

yue

wn1

n1gw

410ε −= as nw is assumed to have a maximum

amplitude of 1. For convenience, we will use thenotation T x yxy:= 6 for closed-loop transfer

matrices mapping signal x to signal y .

B- Weighting Functions for NominalPerformance

Choosing 7 / , 580w wm s L mσ = = in the

Dryden model, we obtain a gust that has most ofits power concentrated in the frequency band[ . , ]01 6 Hz. The specification is that ourcontroller has to be able to regulate the verticalacceleration in this interval with an amplitudeattenuation of at least 500 (-54 dB).

The closed-loop vertical acceleration of theaircraft can be written in terms of the gust vector

wg and the disturbance nw as follows:

g nw y g w y ny T w T w= + , (8)

where Tw yg and

nw yT are the transfer matrices

mapping wg and nw to y respectively. Thus,

the controller has to minimize T jw yg( )ω and

( )nw yT jω over [ . , ]01 6 Hz. 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 500over [ . , ]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. (9)

The weighting function is

W sk

a s1

1

0 1( ) =

+(10)

with k1 500= , a0 0 05= . . A plot of

W j1

1( )ω −

is shown in Figure 5.

The controller outputs consist of deflectionangles (in radians) of the aircraft's elevators andhorizontal canards. In order to make sure that

these angles will remain within acceptablelimits, we took the output of the controller u asone of the controlled variable z2 . Define the

input vector w w wgT

nT: [ ]= .The use of a

suitable weighting function Wu on u such that

W Tu wu ∞ < 1 in closed loop minimizes actuator

travel while meeting the other performance spec.The weighting function Wu is a diagonal

transfer matrix W

Wu

u

1

2

0

0

!

"

$# so that the above

∞ -norm condition implies

T j W jwu u1 1

1( ) ( )ω ω< −

(11)

and

T j W jwu u2 2

1( ) ( )ω ω< −

, (12)

where TT

Twuwu

wu

=

!

"

$#

1

2

. In our study, we tried two

different types of control weighting functions.The first type is composed of two first-orderbiproper filters

W s

s l

ss l

s

u

c u

c

c u

c

1

1 1

1 1

2 2

2 2

0

0( ) =

++

++

!

"

$

####

ωε ω

ωε ω

(13)

where ε ε1 2, are very small. The parameters lu1

and lu2 represent the maximum controller gains

at frequencies below the cutoff frequencies ω c1

and ω c2 . The second type of control weightingfunction that we tried is constant:

Wk

kuu

u

2 1

2

0

0=

!

"

$# . (14)

Here 1 1ku and 1 2ku represent the maximumcontroller gains at all frequencies in closed loop.For both types of weighting functions, we took

k lu u11

1 0 2− = = . , k lu u21

2 0 5− = = . and

ε ε1 2 0 0001= = . .

The weighting function W su1( ) gives us more

degrees of freedom to constrain the controlsignals in order to satisfy physical actuatorsaturation and bandwidth constraints. In our casestudy, the filter cutoff frequencies were selected

as 1 212, 13c cω ω= = rd/s. However, it finally

proved preferable to use the constant weighting

matrix Wu2 for our application, since it did not

increase the order of the generalized plant model

P s( ) by 2 like W su1( ) would, (and hence the

resulting ∞ controller would have a lowerorder). Moreover, the resulting closed-looptransfer functions still satisfied inequalities (11)

and (12) with W su1( ) . Figures 6 and 7 show the

magnitude of Wu with both forms.

C- ∞ Controller

The GLA problem of Figure 3 can be recast intothe standard ∞ -optimal control problem [4] ofFigure 4 below.

Fig. 4: Standard setup for ∞ control

The nominal generalized plant model

P sP s P s

P s P s( )

( ) ( )

( ) ( )=

!

"

$#

11 12

21 22

(15)

has a minimal state-space realizationx 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

(16)

which combines the aircraft model andrealizations of the weighting functions. The

vector of exogenous signals is [ ]T Tg nw w w=

and the signals to be minimized are collected in

: [ ]T Tz y u= .

As gusts act over a relatively short period oftime, they can be considered as signals withfinite energy. Such signals can have spectralcontents similar to the PSD of stochastic Dryden

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

remark provides motivation for ∞ GLAcontrol design as

2 2min max min max

min minsup ( ) ( )

wz wz wK Kw n

wz w wz wK K

T w T G n

T G T j G jω

ω ω∈ ∈∈ ∈

∞∈ ∈ ∈

=

= =+ +/ &

+ +

(17)where 6 is the set of all finite-dimensional,causal linear time-invariant stabilizingcontrollers. Note that we chose to use theperformance weighting function 1( )W s instead

of )(sGw in our design, but

1( ) ( )wW j G jω ω> at all frequencies, which

leads to better performance.

The overall objective in this ∞ -optimal

controller design was to minimize Twz ∞ over

the set 6 , in order to get Twz ∞ < 1. This

would guarantee that the performancespecification is satisfied on the nominal model.Following [4], an ∞ controller K s( ) of order

13 was designed using the Matlab µ-Synthesisand Analysis Toolbox [1] that achieved a norm

of Twz ∞ = 0 68. . Figure 5 shows that our ∞

controller meets the gust alleviation performancespecification given above. We can see that themaximum singular value of Tw yg

is well below

10-4 over 2 01 6π[ . , ] rd / s.

Fig. 5: Achieved performance

P s( )

wz

y u

K s( )

10-2

100

102

104

106

108

10-8

10-6

10-4

10-2

100

102

104

( )wg yT jω

( )wny

T jω

W j1

1−( )ω

The norms (maximum singular values) of thefrequency responses of Twu1 and Twu2 shown inFigures 6 and 7 satisfy the constraints of (11)and (12) for both control weighting functions asmentioned above.

Fig. 6: Constraint on elevator angle

Fig. 7: Constraint on horizontal canard angle

V- µ − CONTROLLER

The ∞ controller design of the previoussection provides nominal performance. That is,performance is guaranteed only if the modelrepresents the aircraft's dynamics perfectly,which is clearly too optimistic. In this section,uncertainty in the frequency responses of theactuators and sensors is taken into account in themodel and the controller design. Note that thisuncertainty may also include variations in theaerodynamics of the control surfaces which may

be caused by changes in altitude and velocity ofthe aircraft. As shown in Figure 8, we includetwo complex multiplicative uncertainty blocks inthe model: 1 2 1 2: diag , , ,u u u u u∆ = ∆ ∆ ∆ ∆ ∈and y∆ ∈ . These perturbations represent the

uncertainty in the frequency responses of theactuators and the sensor, respectively. Thecorresponding weighting functions are

0.75 500

4000.75 22.5

0400

uncu

s

sWs

s

+ +=

+ +

, (18)

0.15 30

100uncy

sW

s

+=+

(19)

Fig. 8: Setup for robust control design

This design can be recast into the generalµ − synthesis setup as given by Figure 9. Definethe complex structured uncertainty set

1 2

3 31 2

: blockdiag , , :

, ,

u u y

u u y×

Ω = ∆ = ∆ ∆ ∆

∆ ∆ ∆ ∈ ⊂

(20)

and the augmented structured uncertainty set

4 3 7 6

: blockdiag , :

,

s p

p× ×

Γ = ∆ = ∆ ∆

∆∈Ω ∆ ∈ ⊂

(21)

where 4 3p

×∆ ∈ is a fictitious uncertainty

linking the exogenous inputs T

g nw w to the

output variables [ ]1 2

Tz z . This fictitious

perturbation is included to transform a robustperformance design problem into an equivalentrobust stability problem, which is easier to solve[1]. The inputs and outputs of the structureduncertainty ( ) , ( )s ss jω∞∆ ∈ ∆ ∈ Γ in Figure

9 are, respectively, the vectors

10-2

100

102

104

106

108

10-14

10-12

10-10

10-8

10-6

10-4

10-2

100

W ju11 1

( )ω −T jwu1

( )ωku1

1−

10-2

100

102

104

106

108

10-14

10-12

10-10

10-8

10-6

10-4

10-2

100

W ju21 1( )ω −T j

wu1( )ω

ku21−

0r =

4w3w

++

wn

εwn1

+-

W s1( )W su( )

K s( )

z1z2

ue

wg

G s( )y

+ +

3z 4z

+ +

u∆uncuW uncyW ∆y

63 4 1 2

TTTsz z z z z = ∈

and

73 4

TT Ts g nw w w w w = ∈ .

The structured singular value µΓ of a complex

matrix 6 7M ×∈ is defined with respect to thestructured uncertainty set Γ as follows:

( ) 1( ) : min : ,det 0M I Mµ

−Γ = ∆ ∆ ∈ Γ − ∆ =

(22)unless no such perturbation exists, in which case

( ) 0MµΓ = .

Fig. 9: Standard setup for µΓ design

A robust controller design based on µΓ is less

conservative than a robust ∞ design (not to be

confused with our ∞ controller of the previoussection which is optimal for the nominal model,but was not designed to be robust to modeluncertainty). This is because the structureduncertainty s∆ is taken into account as a full

block of uncertainty in a typical robust ∞

design.

A µ −synthesis consists of finding the optimalcontroller that minimizes the peak value of

[ ( )]wzT jµ ωΓ over all frequencies. That is,

( ) Smin sup [ ( )]wzK s

T jω

µ ωΓ∈ ∈. (23)

Compare this with the optimization problem forthe ∞ controller design in (17). The mainbenefit offered by µ − synthesis is robustperformance. That is, according to the MainLoop Theorem [1], if a controller achievessup [ ( )] 1

s sw zT jω

µ ωΓ∈

<

, then stability and the

performance specification 1wzT ∞ < hold for

all ( ) , 1, ( )s jω∞ ∞∆ ∈ ∆ ≤ ∆ ∈ Ω .

It is well known that no algorithm is yetavailable to compute ( )MµΓ in the generalcase (including our case). Thus, the optimizationproblem (23) cannot be solved directly.However, the so-called D-K iteration algorithm[1] has been proposed to minimize an upperbound for µΓ . The D-K iteration is an attemptto solve

1

,min

s sl r

l w z rK D D

D T D−

∈ ∞6(24)

where the so-called left and right D-scales( ), ( )l rD s D s ∞∈ are minimum-phase and

have frequency responses of the form

1 2 3 4 1 2 3( ) : diag , , , : , ,l lD j d d d I d d dω +∈ = ∈

1 2 3 3 1 2 3( ) : diag , , , : , ,r rD j d d d I d d dω +∈ = ∈

This algorithm involves an iterative sequence ofminimizations over ( )K s ∈6 (holding the D-

scales fixed) using the ∞ technique, then over

the D-scales (holding ( )K s fixed).

VI- SIMULATION RESULTS

In this section, we use the Dryden model withparameters σ w =7m s, L mw = 580 to generatesevere wind gusts for simulation purposes.Figure 10 shows the gust vector used in oursimulations.

Fig.10: Gust signals for simulation

0 5 10 15-5

0

5

10

time(s)

verti

cal c

ontin

uous

gus

t wg1

0 5 10 15-5

0

5

10

time(s)

verti

cal c

ontin

uous

gus

t wg2

0 5 10 15-5

0

5

10

time(s)

verti

cal c

ontin

uous

gus

t wg3

Gust vector actingon the aircraft

mLsm ww 580,7 ==σ

y u

( )s s∆

( )sP s

( )K s

sz sw

)( sm )( sm

)( sm

Figure 11 shows a magnitude plot of thefrequency response of the Dryden filter forsimulation. We can see that the performancespecification enforced by the weighting functionshould result in efficient GLA.

Fig. 11: Dryden filter and performance weighting

Time-domain simulations were conducted andresults are presented below. The results confirmthat the ∞ controller can dramatically reducethe effect of wind gusts on the verticalacceleration of the aircraft for the nominalmodel comparing to the results of different 2 (LQG) controllers [2],[3].

The goal of the simulation with the nominalmodel (without uncertainties) is to show that our

∞ controller can deal with strong turbulencewithout exciting the flexible modes orgenerating large control angles that wouldsaturate the control surfaces. Figure 12 and 13show respectively the effect of the gust

1 2 3[ ]Tg g g gw w w w= on the B-52 aircraft,

without using a feedback controller (Fig. 12),and with the ∞ controller (Fig. 13).

Fig. 12: Open-loop Fig. 13: ∞ control

These plots show a dramatic improvement inflight comfort. Figures 17 and 18 show thecontrol angles. Notice that the angle swings ofthe elevator and the horizontal canard controlsurfaces were reasonable.

Fig. 14: Elevator Fig. 15: Horizontal canard

Figure 16 below shows the magnitude of theweighting functions

1

2

( ) 0( )

0 ( )uncu

uncuuncu

W sW s

W s

=

, ( )uncyW s .

These weighting functions specify the amount ofuncertainty in the actuators and the sensor,respectively. We specified nearly 20% ofuncertainty at low frequencies for the firstactuator (elevator) and around 8% of uncertaintyfor the second actuator (horizontal canard). Forthe sensor we assumed an uncertainty of 3.5% atlow frequencies. These uncertainties grow withfrequency until they reach a constant level athigh frequencies.

Fig. 16: Norm bounds for uncertainties

Our µ controller obtained using a D-K iterationreached the robust performance specified. Figure17 shows the µ − bounds for the controller

10-2

10-1

100

101

102

103

104

10-4

10-3

10-2

10-1

100

101

)( ωjGw

W j1

1−( )ω

10-2

100

102

104

106

10-2

10-1

100

weighting uncertainties

( )uncyW jω2 ( )uncuW jω

1( )uncuW jω

obtained in the second D-K iteration. Themaximum of the upper bound for µ acrossfrequencies is equal to 0.792, and thereforerobust performance was achieved.

Fig. 17: Upper and lower bounds onµ forµ design

An ∞ controller designed for the model withuncertainties led to an maximum norm of thefrequency response of the closed-loop systemequal to 15.35. This is too high and henceunacceptable from a robust performance point ofview. A µ − analysis was done for the ∞

controller and the results are shown in Figure 18.It is seen that the maximum of µ obtained with

the ∞ controller is equal to 3.1. This valuebeing much larger than one confirms the loss ofrobust performance.

Fig. 18 Upper and lower bounds onµ , ∞ design

This was expected because the ∞ design isunable to take into account the structure of theuncertainty as opposed to the µ − design.

VII- CONCLUSIONWe presented a GLA ∞ -optimal controllerdesign for a B-52 aircraft model with flexiblemodes. The gust was generated with a Drydenpower spectral density model. This kind ofmodel lends itself well to frequency-domainperformance specifications in the form ofweighting functions. The ∞ controller wasshown to meet the desired performancespecification with reasonably small controlsurface deflection angles. A µ design was thendeveloped for a perturbed model includingmultiplicative uncertainties in the actuators andthe sensor. The µ controller reached the robustperformance level specified. We also comparedan ∞ controller design for this uncertainmodel with the µ design. We pointed out the

loss of robust performance of the ∞ designcompared to the µ design. Future research willfocus on the issues related to uncertain modalparameters and different flight envelopes.

VIII- REFERENCES[1] Balas, G.J., Doyle, J.C., Glover, K., Packard, A.,Smith, R., 1995 µ -Analysis and synthesis toolbox

MUSYN Inc, and the MathWorks, Inc.[2] Botez, R.M., Boustani, I. and Vayani, N.,"Optimal control laws for gust load alleviation",1999, Proceedings of the CASI 46th AnnualConference, pp. 649-655, May 3-5, Montréal,Québec, Canada.

[3]- N.Aouf, B.Boulet, R.Botez. 2 and ∞

optimal gust load alleviation for a flexible aircraft. Toappear in Proc. of 2000 American ControlConference, Chicago.[4] Doyle, J.C., Glover, K., Khargonekar, P., Francis,

B., 1988. State space solutions to standard 2 and

∞ control problems. IEEE Trans. AutomaticControl AC-24(8), 731-747.[5] Mclean, D, 1978, Gust load alleviation controlsystems for aircraft. Proc. IEE, Vol.125, No.7, July,pp. 675-685.[6] Mclean, D, 1990. Automatic flight controlsystems. Prentice Hall International, Series inSystems and Control engineering.[7] Niewoehner, R.J., and Kaminar, I.I., 1996 Designof an autoland controller for an F-14 Aircraft using

∞ synthesis. Journal of Guidance, Control, and

Dynamics, Vol. 19, No 3, May, pp. 656-663.

10-2

100

102

104

106

0.6

0.65

0.7

0.75

0.8CLOSED-LOOP MU: CONTROLLER #2

FREQUENCY (rad/s)

MU

10-2

100

102

104

106

0.5

1

1.5

2

2.5

3

3.5CLOSED-LOOP MU: CONTROLLER #1

FREQUENCY (rad/s)

MU