[american institute of aeronautics and astronautics aiaa guidance, navigation, and control...

American Institute of Aeronautics and Astronautics

1

Nonlinear robust autoland

Fabien LAVERGNE*† Airbus-France SAS / LAAS-CNRS, Toulouse, France

Fabrice VILLAUME and Matthieu JEANNEAU* Airbus-France SAS, Toulouse, France

and

Sophie TARBOURIECH and Germain GARCIA† LAAS-CNRS, Toulouse, France

In this paper, an innovative autoland system is detailed based on an original nonlinear robust control technique: Robust Multi-Inversion (RMI). In a first part, we present the RMI technique as an improvement of the feedback linearization. Then, the autoland design process using the RMI control is described for the inner and then the outer loop. Finally, a couple of simulations showing the robustness properties of the RMI techniques as well as the behavior of such autoland system in an operational situation is displayed.

Nomenclature � = Aircraft angle of attack � = Sideslip

CAS = Calibrated Air Speed Cl = Roll coefficient Cm = Pitch coefficient Cn = Yaw coefficient Cx = Drag coefficient Cy = Side force coefficient Cz = Lift coefficient δl = Ailerons deflection δm = Elevator deflection δr = Rudder deflection δTHS = Trimmable horizontal stabilizer deflection ∆� = Heading deviation ∆Y = Metric ground track deviation d = Ground curvilinear abscissa deg = Degree FMS = Flight Management System FPA = Flight Path Angle h = Altitude hTP = Main landing gear height above terrain

� = Flight path angle g = Gravity acceleration Ix = Roll inertia Ixz = Cross inertia Iy = Pitch inertia Iz = Yaw inertia

* Airbus-France SAS, EYCDR, 316 route de Bayonne, 31060 Toulouse. † LAAS-CNRS, MAC, 7 avenue du Colonel Roche, 31060 Toulouse cedex 4.

AIAA Guidance, Navigation, and Control Conference and Exhibit15 - 18 August 2005, San Francisco, California

AIAA 2005-5848

Copyright © 2005 by Airbus France SAS. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.


2

kt = Knot l = Reference length m = Weight MLS = Microwave Landing System N1 = Fan angular speed NDI = Nonlinear Dynamic Inversion Ny = Lateral load factor Nz = Vertical load factor p = Roll rate

� = Roll angle PID = Proportional Integral Derivate

� = Heading ρ = Air density q = Pitch rate QFU = Runway orientation r = Yaw rate RMI = Robust Multi-Inversion RNP = Required Navigation Precision s = Second S = Reference surface

� = Engine angle of attack � = Pitch angle

t = Time T = Engine net thrust TCAS = Traffic Collision Avoidance System THS = Trimmable Horizontal Stabilizer TK = Track V = True air speed Vz = Vertical speed XTP = Mean longitudinal position of the main landing gears (0 above the runway threshold) XTP = Mean lateral position of the main landing gears (0 above the runway axis)

I. Introduction ince the beginning of human flight, engineers and researchers have always tried to simplify pilot’s tasks by providing autonomous systems aimed at treating a part of the pilot’s workload. This started a hundred years

ago with basic systems such as mechanical stabilizer. Today, instead of executing raw piloting tasks, pilots can rather focus their attention on flight management thanks to integrated systems such as FMS and TCAS for instance.

A great amount of such stress-relieving system is based upon autopilot systems. Indeed, from the very beginning of the autopilot activity to nowadays, accuracy and robustness of such systems improved dramatically. This has led to very reliable, everyday-life systems. The spread of autopilots translates directly progresses achieved in the automatic control domain for sixty years now.

Historically, autopilot roots are to be found in very empirical PID control. By the time, autopilot were assuming very lazy attitude hold. The system was only about regulation and not tracking, since a dynamics guarantee was neither of interest nor achievable.

Then, autopilot function improved with pole placement and optimal techniques, turning it from a comfort function to a necessary and crucial function. Today, a lot of airspaces are only accessible to aircraft provided with autopilots challenging Required Navigation Precision (RNP). Thus a “simple” system has highly contributed to absorb the fast-growing traffic flow of the last sixty years. This would not have been possible without automatic control techniques improvement.

Another dramatically successful application is the autoland. As the landing is the more crucial and weather-dependent flight phase, the operational gains brought by the all-weather autoland are invaluable.

This is not sufficient. Since the aircraft industry is driven by both safety and economics, flight control laws have to be improved every day to tackle new challenges. For instance, recent H2 and H∞ techniques enabled to achieve flexible aircraft control and then to design lighter aircraft. Other techniques such as Nonlinear Dynamic

S


3

Inversion (NDI) can be used to better cope with Microwave Landing System (MLS), steep approach or to get more standard, portable flight control laws1-3, 6-26.

Contrary to their linear counterparts, nonlinear flight control laws are portable as they take directly into account the aerodynamic coefficients, by integrating explicitly the flight mechanics equations. Furthermore, they enable to set up flexibly new control strategies. Nevertheless, in the nonlinear case, there are only few robustness results, mainly based on robust backstepping. One of the reasons is that nonlinear differential algebra is far more complex than Laplace and other linear tools.

Still, as nonlinear control laws present a potentially wide area of application, we propose a way to tackle the robustness issue by an original nonlinear robust control technique: Robust Multi-Inversion control. Then we propose flight control laws that can be used either by a pilot or an autopilot. In another part of this article, we present an autoland autopilot loop that can be connected to the flight control laws described before. Then we give some simulations of these control laws for an Airbus aircraft.

II. RMI control: a new nonlinear robust control technique RMI control is a control technique based on a nonlinear control technique. Indeed, nonlinear control is

greatly simpler than linear control in its application principles (no gain scheduling), yet nonlinear theory is far less completed and more complex than in the linear case.

Generally, nonlinear control techniques enable to tackle explicitly form nonlinearities, i.e. nonlinearities in the own dynamic system behaviour. Nevertheless, it can be noticed that actuators nonlinearities, such as actuators saturations, are not taken into account in classical nonlinear control synthesis. Thus, form nonlinearities of the system are expressed through a fully nonlinear state-space representation, such as:

��

��

==

)(

),(.

XCY

UXAX (1)

or more often through a nonlinear affine state-space representation, as follows:

��

��

=+=

)(

).()(.

XCY

UXBXAX (2)

Nonlinear affine representations are usually associated with a control technique named dynamic inversion or feedback linearization3. The first idea of this control technique comes from a fundamental property of the nonlinear affine representations: they express the regulated output as a function of the state and the successive derivatives of the state as a function of the control variables. Then it seems possible to express directly the successive output derivatives as a function of the control. So the first step of the dynamic inversion process is to iteratively derive the output variables until the control variables appear in their expression. Let us note:

UXGXF

Uy

Uy

Ypd

p

d

).()(

)(

)(

)(

)(1

~

1

+=��

�

�

��

= � (3)

with dk the relative degrees of the system. Then the second fundamental idea of the dynamic inversion is naturally to inverse this input-output system. This would lead to express the control values to give to the system as a function of the desired derivative of the regulated output. In that goal, a state feedback is synthesized by defining a new control variable V. Hence one considers:

VXXU ).()( βα += (4)


4

with:

)()( 1 XGX −=β (5)

and:

)().()( XFXX βα −= (6)

Then the inverse system presents the following dynamics:

VY =~

(7)

i.e. the resulting system is linear and decoupled. Then it is possible to place the desired dynamics by classical linear techniques.

Before to conclude, please note that dynamic inversion is only possible if G is invertible (see Eq. (5)) or pseudo-invertible. Another limitation appears when:

ndp

kk <�

=1

(8)

Indeed, in this case, the inversed system contains some internal dynamics or hidden dynamics. If these dynamics are stable, the inversion is valid. Nevertheless, if these dynamics are unstable then the whole system will be unstable anyway, so the dynamic inversion is impossible.

Compared to the linear control techniques, the nonlinear ones enable to express explicitly the system dynamics through the fully nonlinear or nonlinear affine state-space representation. This is of particular interest for industrial applications, where the encountered systems differ often only parametrically from each other. Then the synthesized control remains formally valid when applied to another functionally comparable system. It is only necessary to update the system parameters in the control computation algorithm. Dynamic inversion control laws are then said to be “portable”.

With dynamic inversion the system is exactly (or “formally”) linearized. Furthermore, as long as the nonlinear affine model is valid along the whole operating domain of the system, this linearization is also valid along the whole operating domain and the resulting system (7) is unique. A single controller is thus sufficient to control properly the system. Generally speaking, the dynamic inversion control loop can be presented as follows:

Once again, it can be noticed that this unique architecture is portable. Nevertheless, stability and robustness proofs are harder to obtain for nonlinear systems than for linear systems. As a consequence, nonlinear robust control techniques hardly exist. That is why, for the time being, nonlinear control techniques have not been as extensively used as linear ones.

We choose to base RMI control on NDI. The aim of this original technique is to bring to nonlinear control the robust performance and stability. Indeed, nonlinear and linear models of systems always contain a certain degree of uncertainty. And this uncertainty has a negative influence on the resulting dynamics of the system.

Figure 1. Academic dynamic inversion control loop with pole placement.


5

Take the following model of the system:

��

∆+=∆+=

)()()(

)()()(

XGXGXG

XFXFXF

mm

mm (9)

with ∆ representing the uncertainties of the model. Then the real feedback is:

( ))().(1 XFVXGU mm −= − (10)

Let us take the following augmented system:

��

��

+=

+=

UXGXFY

UXGXFY

mmm ).()(

).()(~

~

(11)

The consequence of the feedback (10) is the inversion of the model and not the inversion of the system itself (Eq. (7) is no longer valid). Thus, one gets:

( )

��

��

=

−+= −

VY

XFVXGXGXFY

m

mm~

1~

)().().()( (12)

The overall control loop is then:

To overcome this problem, we propose to define a new augmented system, based on the system (11) and added with a system, called RMI system:

��

�

��

�

�

∆−+=

+=

+=

RMImmRMI

mmm

UXGXFY

UXGXFY

UXGXFY

).()(

).()(

).()(

~

~

~

(13)

Figure 2. “Real-world” dynamic inversion control loop with pole placement.


6

which can be expressed differently as follows:

��

�

��

�

�

∆−∆++=

∆++=

+=

RMImRMI

mm

UXGXFY

UXGXFY

UXGXFY

).()(

).()(

).()(

~

~

~

(14)

with:

UXGXF mmm ).()( ∆+∆=∆ (15)

Then let us define:

( )�

�

∆−∆ �

��

�=

�

��

� − �

��

�=∆

≤≤

≤≤

dtdiag

dtYYdiag

RMIm

pkk

RMI

pkkRMI

..1

..1

1

~~

1

τ

τ (16)

with the notation:

��

�

�

��

= �

��

�

≤≤

p

pkk

diag

τ

τ

τ/100

0

0

00/1

1

1

1

�

��

��

�

(17)

with � k>0. From Eqs. (16) and (17), we can verify the following exponential convergence properties:

mRMI ∆→∆ exp (18)

and:

~

exp~

YYRMI → (19)

Now it is possible to inverse the RMI system. By definition, the linearizing feedback of the RMI system is:

( )RMImm XFVXGU ∆+−= − )().(1 (20)

Then we reach linearization of the RMI system:

VYRMI =~

(21)


7

Thanks to Eqs. (19) and (21), the real system is thus asymptotically linearized:

~

exp YV → (22)

So the Eq. (7) is asymptotically valid. With this asymptotical linearization, robust performance and stability are achieved. The RMI control loop is then:

Because the inversion is asymptotically exact, precision integrators are no more useful within this control loop. Control laws are then less sensitive to actuators nonlinearities, delays, and thus gains can be boosted.

With this Robust Multi-Inversion, it becomes possible to consider real application of nonlinear flight control laws. We propose to use a common RMI inner loop for the manual and automatic pilot for our autoland application. This is the first step of the autoland synthesis described in this paper.

III. Inner loop synthesis

A. Longitudinal inner loop synthesis The simplified longitudinal flight dynamics are given below (without wind):

��

�

��

�

�

==

==

=

+=

−++=

−−+=

)cos(.

)sin(.

)cos(...1

..²...2

1.

1

)cos(..

)sin(.....

21

)sin(..²...2

1)cos(.

.

.

.

.

.

.

γ

γ

θ

τρ

γταργ

γρτα

VVxd

VVzh

q

bTIy

CmlSVIy

q

V

g

Vm

TCzSV

m

gCxSVmm

TV

(23)

Figure 3. “Real-world” Robust Multi-Inversion contr ol loop with pole placement.


8

In autoland, one of the most relevant regulated output is the altitude. It is indeed one of the regulated output of classical autopilots. Now let us remark that the vertical speed (i.e. the derivative of the height) is a quasi-linear function of the flight path angle. Then it seems interesting for an autopilot to control γ and its derivatives. Here it is necessary to point out that the vertical load factor (Nz) expression is a function of the γ derivative:

)cos(.

.

γγ +=g

VNz (24)

By this way, having flight control laws controlling the load factor implies an easy trajectory tracking. It is thus a useful control variable for a pilot as well as for an autopilot.

Moreover, Nz is homogenous to a normal acceleration, while the aircraft structural strength is dimensioned in normal acceleration (-1g<Nz<2.5g). Finally, controlling the vertical load factor appears not only useful but also necessary to protect properly the aircraft structural integrity as well as to ensure easy manual and automatic trajectory tracking. That is why, twenty years ago, flight control designers decided to issue load factor control for the Airbus A320. Then it is possible to express the system (23) in a nonlinear affine way:

��

�

��

�

�

=

�

��

�

∂∂+

∂∂++=

�

��

�

∂∂+

∂∂++−+=

q

mm

CmCmCmlSV

IybT

Iyq

mm

CzCzCzSV

mV

g

Vm

T

.

.

.

....²...2

1.

1)cos(...

1

......21

)cos(..

)sin(.

θ

δδ

αα

ρτ

δδ

αα

ργταγ

(25)

It is to note that the {�m -> Nz} transfer is non-minimum phase and that dynamic inversion is not adapted to

non-minimum phase systems. Indeed, dynamic inversion inverses the transfer function of the system so as to linearized it. By this way, zeroes with positive real part are twisted to poles with positive real parts. It means that at least one of the poles of the inversed system is unstable. Then the entire system becomes unstable.

So direct dynamic inversion of vertical load factor with the elevators is impossible because it is unstable. To counter this problem, let us go back to classical pitch rate control of an aircraft.

Usually, nonlinear longitudinal control of aircraft dynamics is proposed through a pitch control law. This kind of control laws is quite natural, as elevator deflection mainly contributes to the pitch moment (Eq. 25). Moreover, it is a very simple control law. The classical dynamic inversion should be:

m

CmlSV

Iy

CmCmlSV

IybT

IyV

mq

δρ

αα

ρτδ

∂∂

�

��

�

∂∂+−−

=..²...

2

1.

1

...²...21

.1

)cos(...1

(26)

And the RMI inversion would rather be:

m

CmlSV

Iy

CmCmlSV

IybT

IyV

m

qRMIq

δρ

αα

ρτδ

∂∂

∆+ �

��

�

∂∂+−−

=..²...

2

1.

1

...²...21

.1

)cos(...1

(27)


9

with:

qRMI

qRMI

qRMI

qdtmm

CmCmCmlSV

IybT

Iy

τ

δδ

αα

ρτ −��

��

∆−

�

��

�

∂∂+

∂∂++

=∆� .....²...

21

.1

)cos(...1

(28)

Then it is possible to choose direct first order pitch rate control, by considering:

( )qqV cqq −= .ω (29)

It is also possible to control pitch angle with a second order behavior, by considering:

( )θθωω −−−= cqqqq qzV 2...2 (30)

Given these equations, there are two ways of controlling the vertical load factor through a pitch control loop: an approximate one and an exact one. Both are based on the following equations:

γαθ += (31)

and:

..

γα +=q (32)

The approximate approach is to simplify the expression (24) to:

)cos(. γ+=g

qVNz (33)

Indeed, a stabilized Nz corresponds to an almost fixed angle of attack (see Eq. (25)). Then it is possible to set:

( ))cos(. γ−= cc NzV

gq (34)

where the extension “c” is related to the commanded value of Nz. Such a control law is simple but unadapted to perform accurate tracking. The convergence time of Nz cannot be guaranteed. It is the reason why we propose an exact approach. It is possible to inverse Eq. (25) by considering:

αρ

ρ

αγ

∂∂

∆+−=

CzSV

m

CzSVm

NzV

gRMIc

c

....21

....21

. (35)


10

and:

γ

γ

γ

τ

γδδ

αα

ργτα

RMI

RMI

RMI

dtmm

CzCzCzSV

mV

g

Vm

T −��

��

∆− �

��

�

∂∂+

∂∂++−+

=∆� .......

21

)cos(..

)sin(.

(36)

In Eq. (35), we choose to suppress the term in δm, as it is asymptotically equal to zero. Indeed Airbus-type commercial aircraft contain a THS, whose deflection brings automatically elevator deflection to zero. However, we keep this term in Eq. (36) so as to prevent any integrator overcharge.

Now it remains to find the way to link this desired angle of attack to a desired pitch rate or pitch angle. Indeed, we would like to have:

( )qqzq cq −= .

.

..2 ω (37)

As we do not possess any expression of the desired pitch rate, it is possible, thanks to Eq. (32), to change this equation to:

�

��

� −+= qzq ccq

..

.

.

..2 γαω (38)

With Eq. (24), this is equivalent to:

( ) ��

��

−−+= qV

gNzzq ccq .)cos(...2

..

γαω (39)

Neither do we possess any expression of the desired angle of attack derivative. Moreover, it can be noticed that Nz is proportional to � , so any discontinuity of the desired Nz implies discontinuity of desired � . Thus, in case of step solicitation, the derivative of the angle of attack does not exist. So it is necessary to skip an integration order, by considering:

( ) ( ) ��

��

−−+−= q

V

gNz

zzq cc

qq .)cos(.

.2...2

.

γααω

ω (40)

This formulation is equivalent to Eq. (30) and gives the pitch angle target to achieve desired Nz tracking:

( ) αθαγω

θ −++−= ccc V

gNz

z.)cos(.

.2 (41)

Thus the aircraft vertical load factor behaves along a second order filter with pulsation � q and damping zq. As a conclusion, this flight control laws architecture enables to control either pitch or vertical load factor with a unique pitch inner loop. We would like to point out that the intermediate control variables (Nz, α and θ) are critical flight variables. Moreover, the architecture presented above allows an easy protection of the flight domain. Furthermore, this protection is allowed without any additive flight control law setting. The gain in development time is then noticeable.

Note also that the control laws presented above and below rely on aerodynamics coefficients. These coefficients are nonlinear and multivariable functions of the aircraft states and parameters4. Embedded models of aerodynamics coefficients do rarely exist. Nevertheless, they can be appropriately modelled with guaranteed maximal error thanks to automated neural networks-based tools5.


11

B. Lateral inner loop synthesis The simplified lateral flight dynamics are given below (without wind):

( )

��

�

��

�

�

+=

++=

+−=

=

=

)cos(

)sin(.)cos(.)sin(.)cos(.).tan(

)sin(.....2

1

..²...21

.1

..²...2

1.

1

.

.

.

.

.

θϕϕψ

ϕϕθϕ

ϕρβ

ρ

ρ

qrqrp

V

grCySV

m

CnlSVIz

r

CllSVIx

p

(42)

For human pilots as well as classical autopilots, � (or its derivative) and � are the usual regulated output. The more straightforward approach here also is to consider the nonlinear affine system:

��

�

��

�

�

�

��

�

∂∂+

∂∂++−=

�

��

�

∂∂+

∂∂+=

�

��

�

∂∂+

∂∂+=

nn

Cyl

l

CyCySV

mr

V

g

nn

Cnl

l

CnCnlSV

Izr

nn

Cll

l

ClCllSV

Ixp

δδ

δδ

ρϕβ

δδ

δδ

ρ

δδ

δδ

ρ

......2

1)sin(.

....²...21

.1

....²...21

.1

.

.

.

(43)

Here the problem is the same as in the longitudinal dynamics: direct control of lateral load factor Ny as well as sideslip � is impossible due to non-minimum phase phenomena. So the first inversion to take place is angular acceleration inversion by RMI control (we assign to p and r a first order dynamics whose time constant being respectively � p and � r):

��

�

�

��

∆+−−

∆+−−

��

�

�

��

∂∂

∂∂−

∂∂−

∂∂

�

��

�

∂∂

∂∂−

∂∂

∂∂

�

��

�=�

�

��

rRMI

r

c

pRMI

p

c

CnlSVIz

rr

CllSVIx

pp

l

Cl

l

Cnn

Cl

n

Cn

n

Cl

l

Cn

l

Cl

n

CnlSV

IzIx

n

l

..²...2

1.

1

..²...2

1.

1

..

....²...2

1

.2

ρτ

ρτ

δδ

δδ

δδδδρ

δδ

(44)

This control law always exists, thanks to the classical aircraft design physics: ailerons are designed to control roll movements rather than yaw movements and the rudder is designed to control yaw movements rather than roll movements.


12

Thus we have:

��

��

�

∂∂>

∂∂

∂∂>

∂∂

l

Cn

n

Cnn

Cl

l

Cl

δδ

δδ (45)

Then:

0.. >∂∂

∂∂−

∂∂

∂∂

l

Cn

n

Cl

n

Cn

l

Cl

δδδδ (46)

Having assured that this control law always exist, it is then necessary to convert p and r control into � and � control. By taking Eq. (42) and inverting it with RMI control, we can assign to � a first order dynamics with time constant being � � :

( ))sin(.)cos(.).tan( ϕϕθτ

ϕϕϕ

qrp cc +−−= (47)

As the � inversion is only a cinematic equation inversion with no specific aircraft model, there is no need to use RMI control at this step. After this we can assign to � a first order dynamics with time constant being � � :

β

βτββϕδ

δδ

δρ

RMIc

filteredfilteredc V

gn

n

Cyl

l

CyCySV

mr ∆+−−+

�

��

�

∂∂+

∂∂+= )sin(.......

21

(48)

Filtering ailerons and rudder enables to recreate artificially what is done “naturally” in longitudinal thanks to the THS (see Eq. (35)). Indeed in both case, the direct feedback of actuator deflections would lead to an algebraic loop. In longitudinal, the THS action brings asymptotically a null elevator deflection. So we chose simply not to use elevator deflection for feedback. As there is no trim for lateral actuator, we recreate the trim function by filtering their deflection and thus the filtered actuator deflection converges asymptotically toward the actual actuator deflection without creating any algebraic loop.

C. Model enrichment for a better robustness In the previous part, we synthesized lateral and longitudinal control, based on simplified models. This allowed to follow a step-by-step procedure. However, full flight dynamics equations are a little more complex. The simulation results presented further are obtained thanks to enriched models. These enriched models are based on the same moment expressions as above:

��

�

��

�

�

=

=

=

CnlSVN

CmlSVM

CllSVL

..²...21

..²...21

..²...21

ρ

ρ

ρ

(49)


13

Lateral moments are still decoupled with longitudinal ones but there are some crossed influences. The pitch acceleration equation is thus:

( ) ( )

Iy

IzIxrpprIxzMq

−−−+= ... 22.

(50)

As for lateral angular acceleration, their expression follows:

( ) ( )[ ]( ) ( )[ ]

��

��

�

−+−++−−++=

−−−++−++=

IxzIzIx

IxzIxIyIxpIzIyIxrIxzqLIxzNIxr

IxzIzIx

IxzIzIzIyrIzIyIxpIxzqNIxzLIzp

.......

........

22.

22.

(51)

The flight control laws synthesis process is exactly the same as in part A and B but the equations are far more complex. Nevertheless, these control laws works well (see part V).

Now that the inner loop are synthesized, we can create autopilot control laws that fit well with them. We remind that we choose to synthesized the inner and outer loop separately to allow human piloting as well as automatic piloting.

IV. Autoland outer loop synthesis

Thanks to the use of RMI control, each control channel (.

γ , � , � ) is decoupled from each other. Then it is

possible to synthesize separately an outer loop for each channel.

A. .

γ computation Considering Eqs. (23), the FPA derivative computation is straightforward. A second-order filter enables Vz and h tracking without tracking error. Vzc and hc computation is obtained through a reference trajectory module which is not studied in this paper. As the {h, Vz} inversion is only a cinematic equation inversion with no specific aircraft model, there is no need to use RMI control at this step.

B. � computation The � computation we designed is quite simple. Its value is zero all flight long except in decrab phase where it becomes a frozen computed value.

C. � computation The � computation is a little bit more complex. First we need to link � computation to � . Even if this inversion is only a cinematic one, RMI control is needed because of the uncertainties introduced by the wind. Once we have synthesized a heading control law, behaving like a first order control law with time constant � � , it is necessary to link the desired heading to the metric localizer deviation. We suppose that we possess such an information. Thus the metric localizer deviation is not studied in this paper. By this way we can assign to ∆Y a first order dynamics with time constant � ∆Y. Here also, even if this inversion is only a cinematic one, RMI control is needed because of the uncertainties introduced by the wind.

It is important to note that, thanks to RMI asymptotic linearization, there is no need for precision integrator for each of the control laws presented in this article. Thus, compared to classical flight control design, gain can be boosted and stability margin are improved. Robustness to actuator saturations is also increased by the same way. In the next part we display some illustrations of the behavior of the control laws described here.


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V. Simulations All the simulations presented below are performed with the certified autopilot simulator of Airbus. The first simulation aims at illustrating the robustness ability of RMI control laws. Here we superpose sixteen simulations scanning the whole weight and balance domain of an Airbus aircraft. The aircraft is equipped with a q control law synthesized with a fixed value of weight and balance.

Figure 4. Robustness of a RMI q control loop towards weight and balance.


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The simulation presented below shows an operational autoland in the presence of wind. The aircraft is equipped with the full nonlinear robust autoland detailed above and also with a RMI autothrust which is not detailed in this paper.

VI. Conclusion Nonlinear control techniques have been quietly spreading in the flight control community for more than

twenty-five years. However, their application to commercial aircraft flight control laws is very rare due to the difficulty of guaranteeing their robustness. Meanwhile, the flight control designers community more and more appreciates their flexibility and portability.

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In this paper we proposed a new nonlinear robust control technique named Robust Multi-Inversion (RMI). This presents the advantage of keeping the flexibility of nonlinear control techniques added with interesting robustness properties. This control technique was used to synthesize a full autoland system.

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