design of nonlinear adaptive flight control system …...nonlinear adaptive flight control system...

American Institute of Aeronautics and Astronautics

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Design of Nonlinear Adaptive Flight Control System based on Immersion and Invariance

Yuta Kobayashi1 and Masaki Takahashi.2 Keio University, Yokohama, 223-8522, Japan

One of the most important issues in the flight control system of a space plane is how to overcome various uncertainties. This study aims to design a nonlinear adaptive flight control system based on the Immersion and Invariance (I&I) method to tolerate the uncertainties of parameters. In general, it is difficult to acquire several exact parameters, such as stability derivatives, of a space plane because of its wide flight envelope. Therefore, it is necessary to design a nonlinear fight control system under the condition where uncertainties of parameters exist. This study focuses on the I&I which is a new adaptive control method. In this study, the unknown parameters are considered to be multiplicative uncertainties toward the rigorous dynamics model because the controller designed by this kind of method can tolerate not only the specific disturbance but also various uncertainties. The flight control system based on the I&I is applied to space plane model and the availability of the proposed flight control system is verified by the six-degree-of-freedom nonlinear simulation.

Nomenclature g = Acceleration of gravity m/s2 ρ = Density kg/m3 m = Mass of aircraft kg Ixx,Iyy,Izz = Moment of inertia kgm2 Ixy,Iyz,Izx = Product of inertia kgm2 u = Aircraft velocity around the x body axis m/s v = Aircraft velocity around the y body axis m/s w = Aircraft velocity around the z body axis m/s p = Aircraft angular velocity around the x body axis rad/s q = Aircraft angular velocity around the y body axis rad/s r = Aircraft angular velocity around the z body axis rad/s X = External force around the x body axis N Y = External force around the y body axis N Z = External force around the z body axis N L = Moment around the x body axis Nm M = Moment around the y body axis Nm N = Moment around the z body axis Nm φ = Euler angle around the x body axis rad θ = Euler angle around the y body axis rad ψ = Euler angle around the z body axis rad x = Forward directional distance m y = Starboard directional distance m z = Down directional distance m

, ,a e rδ δ δ = Aileron angle, Elevator angle, Rudder angle rad

tδ = Thrust Nm ,el erδ δ = Elevon angle (left, Right) rad

1 Master’s Student, School of Science for Open and Environmental Systems, 3-14-1 Hiyoshi Kohoku-ku, [email protected].

2 Assistant Professor, Department of System Design Engineering, 3-14-1 Hiyoshi Kohoku-ku, [email protected].

AIAA Guidance, Navigation, and Control Conference10 - 13 August 2009, Chicago, Illinois

AIAA 2009-6174

Copyright © 2009 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.


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I. Introduction PACE plane has a characteristic that it is difficult to acquire several exact parameters, such as stability derivatives, because of its wide flight envelope. In the past, a control system based on gain scheduling has

been used to overcome the variation in dynamic characteristic1. However, because of design cost and complexity, many new control methods have been actively researched, especially nonlinear control systems 2,3. In general, nonlinear control systems have drawbacks such as vulnerability to modeling errors. Therefore, it is necessary to design a nonlinear flight control system under the condition where uncertainties of parameters exist.

Against these backgrounds, this research focuses on a nonlinear flight control system based on the Immersion and Invariance (I&I). The method based on the I&I attracts as much attention as the cancellation and domination design methods. In the cancellation and domination design method, the controller is designed on the basis of a Lyapunov function to stabilize the total system. On the other hand, the I&I method can lead a control algorithm based not on a Lyapunov function but on the I&I. In addition, a Lyapunov function, which has interference of state and parameter estimation, will be derived when system stability is analyzed. As described above, the I&I method has the advantage that the control algorithm and Lyapunov function can be spontaneously led, and it is said that the I&I method is effective for nonlinear systems4.

We apply a nonlinear adaptive flight control system based on the I&I method to an airframe model of a space plane to reveal the feasibility of a horizontal take-off and landing space plane assisted by a sea ship5. In designing a control system, the controller which can achieve desired functions is needed even though there are unknown aerodynamic moments and unknown aerodynamic forces. Therefore, the proposal control system describes these uncertainties as unknown parameters and adapt estimated parameters to unknown ones on the basis of the I&I adaptive law. In this study, the unknown parameters are considered to be multiplicative uncertainties toward the rigorous dynamics model when the controller is designed. The controller designed by this kind of method can tolerate not only the specific disturbance but also various uncertainties.

Besides, in this research, the control gains are acquired on the basis of genetic algorithm (GA) in order to design the control system systematically because there is no effective proposal which can acquire the exact control gains. The availability of the flight control system is verified by the six-degree-of-freedom nonlinear simulation. In the simulation the nonlinear adaptive flight control system based on the I&I and the robust control system called H∞ ⋅MDM/MDP (applied to the Automatic Landing Flight Experiment (ALFLEX)) are compared using the Monte Carlo approach. The contents of simulation are nominal flight, unknown aerodynamic moments acting on the body, and gust.

The paper is organized as follows: First, the concept of the space plane is briefly discussed in Sec.II. Sec.III is devoted to the Immersion and Invariance. In Sec.IV the nonlinear adaptive flight control system based on the I&I is shown. Simulation results are shown in Sec.V. Finally, the conclusions can be found in Sec.VI.

II. Concept of a horizontal take-off and landing space plane assisted by a sea ship In our laboratory, a concept of a future generation space transportation system has been proposed and

studied. At the time of proposing the concept, the following three requirements were given: i. High level of security

ii. Early realization using existing technology iii. Developable in Japan overcoming the geographical constraint

To satisfy these requirements, a concept of a horizontal take-off and landing space plane assisted by a sea ship has been proposed. The concept has three major aspects. First, the space plane takes off horizontally from the ship. Second the space plane carries out a suborbital flight mission. Finally, it lands on the same ship. The outline drawing of this concept is represented in Fig.1.

S

Figure 1. Concept of horizontal take-off and landing space plane assisted by sea ship.


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The concept is different from not only conventional space transportation systems but also general aircraft in taking-off and landing from the same scale and high speed platform. Therefore, our laboratory has been verifying the operability of the take-off and landing phase using the autonomous flight of a proof of concept (POC) airframe model. The appearance and the specification data of POC are shown in Fig.2 and Table1. In this research, the nonlinear adaptive flight control system based on the I&I method is applied to the airframe illustrated in Fig.2.

III. Immersion and Invariance4,6 The I&I is composed of immersing the system and invariance of the manifold. The basic scheme of

immersion is the action to immerse the plant dynamics into a target system that captures the desired characteristic. In contrast, the invariance of the manifold is the characteristic that a trajectory which started at a random point in a manifold remains in the manifold. The I&I method aims to stabilize the system by using these aspects. The characteristics of I&I are shown in Fig.3.

On the other hand, the adaptive control method based on the I&I is a new adaptive control scheme proposed by Ortega and Astlfi4. It differs from most of the adaptive controller design methodologies because it can naturally lead the control input and tuning rule of parameters by designing the adaptive control system on the basis of I&I. In addition, a Lyapunov function, which has a state and parameter estimation interference, is derived when system stability is analyzed. The procedure of the control design is as follows:

i. Select a closed loop system which is generated when an input is made by using an unknown parameter as a known one

ii. Think of the selected closed loop system as a target dynamics iii. Add a new term to the tuning rule of parameters iv. Form a manifold by using the new term

The adaptive control method based on the I&I is well-suited to situations where a stabilizing controller for a nominal reduced-order model is known, and we would like to robustify it with respect to higher-order dynamics.

Figure 2. Airframe model (POC).

Table 1. Airframe model data. Parameter Value

Weight [kg] 3.649 Length [m] 1.360 Span [m] 1.132

Height [m] 0.355 Area of Main Wing [m2] 0.659

Area of Vertical Tail Fin [m2] 0.0538 Area of Rudder[m2] 0.0124

Ixx = 0.0716 Moment of Inertia[ 2kg m⋅ ] Iyy = 0.2671

Izz = 0.3241 Aspect Ratio 1.945

Thrust System Ducted Fan Max Thrust [kgf] 1.7


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IV. Nonlinear adaptive flight control system based on I&I5

A. Definition of the coordinates The definition of the coordinates with respect to the aircraft model of a space plane is shown in Fig.4. The

velocity, attitude angle, angular velocity, aerodynamic force, and moment are described as follows:

[ ] [ ] [ ][ ] [ ]

, , , , , ,

, , , , ,

T T T

T T

u v w p q r

F X Y Z L M N

ν ϕ φ θ ψ ω= = , , =

= Μ = (1)

Next, the six-degree-of-freedom nonlinear equations of motion are given by

( )( )( )

( )( ) ( )

( )

2 2

sin

cos sin

cos cos

xx xz zz yy xz

yy xx zz xz

zz xz yy xx xz

X m u qw vr g

Y m v ur pw g

Z m w pv uq g

L pI rI qr I I pqI

M qI pr I I p r I

N rI pI pq I I qrI

θ

θ φ

θ φ

= + − +⎧⎪

= + − −⎪⎪ = + − −⎪⎪⎨ = − + − −⎪⎪ = + − + −⎪⎪ = − + − +⎪⎩

(2)

CG

Lp,

Mq,

Nr,x

y

z

u

v

w

Vc

rδ

elδ

erδ

CG

Lp,

Mq,

Nr,x

y

z

u

v

w

Vc

rδ

elδ

erδ

Figure 4. Body Axis.

ξ1

ξ2

π

π

x1

x2

x3ζ

ζ

S

Target dynamics(Asymptotically stable in a large)

Mappingx=π(ξ)

Manifold

To give an attraction toward the manifold by control algorithm

Point on the manifold converges to the origin(∵Target dynamics is asymptotically stable)

① Point on the manifold is asymptotically stable

② Point not on the manifold is also asymptotically stable

Figure 3. Immersion and Invariance.


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B. Outline of flight control system The nonlinear adaptive flight control system based on the I&I has three subsystems, guidance, attitude

control, and velocity control. A block diagram is displayed in Fig.5.

C. Guidance system The Guidance system calculates the reference attitude angle for attitude control system for obtaining the

desired trajectory. The navigation equations are given in (3).

( ) ( )( ) ( )

( )

cos sin sin cos ssin cos cos sin sin

cos sin sin cos sin sin cos sin cos

sin sin cos cos

x u v w v w

y u v w v w

z u v w

θ φ θ φ θ ψ φ φ ψ

θ φ θ φ θ ψ φ φ ψ

θ φ φ θ

= + + − −⎧⎪

= + + + −⎨⎪ = − +⎩

(3)

Using the identity of trigonometric function, the equations can be rewritten as

2

2

1

cossinsin

x Vy Vz V

λλγ

=⎧⎪ =⎨⎪ =⎩

(4)

where

( )

( ) ( )

221

2 22 1

1

sin cos

cos cos sin

cos sinarctancos

sin cosarctan

V u v w

V V v w

v wV

v wu

φ φ

γ φ φ

φ φλ ψγ

φ φγ θ

= + +

= + −

⎛ ⎞−= + ⎜ ⎟

⎝ ⎠+⎛ ⎞= − ⎜ ⎟

⎝ ⎠

(5)

Next, both present position and next target point are used for determining the desired trajectory, straight line or circle.

Straight line trajectory

0se ax by c+ + =

( )arctan arctans p sa k eb

λ λ ⎛ ⎞= = − −⎜ ⎟⎝ ⎠

AttitudeControllerGuidance

AirspeedController

Flight Model

ωϕν ,,, ,x y z

dϕ

dV

δ

Τ

Figure 5. Block diagram of nonlinear adaptive flight control system based on I&I.


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( ): integer numbern nφ π= 　 (6)

Circular trajectory

( ) ( )2 2 2 0ce x a y b c− + − − =

( )arctan arctanc p cx a k ey b

λ λ⎛ ⎞−

= = − −⎜ ⎟−⎝ ⎠

cosarctancos sin sin

GG

βφα α β

⎛ ⎞= ⎜ ⎟−⎝ ⎠

2

: centripetal accelerationVGgc

= 　 (7)

where 0pk > . Furthermore, utilizing present altitude and reference altitude, the equation given in (8) is led.

( )( )arctan z dk z zγ = − − (8)

with 0zk > . The target value of attitude angle can be calculated by using the expressions given in (5) to (8).

D. Attitude control system The attitude control system calculates the actuator commands to obtain the desired attitude angle given by

the guidance system under the condition where the unknown aerodynamic moments act upon the aircraft. First, the equations of rotational motion are given by

( )Rϕ ϕ ω= (9)

( )I S Iω ω ω= +Μ (10)

where

1 2 3

0, , 0 0

0

xx xzTT T T

yy

xz zz

I II I I I I

I I

−⎡ ⎤⎢ ⎥⎡ ⎤= =⎣ ⎦ ⎢ ⎥⎢ ⎥−⎣ ⎦

( )1 sin tan cos tan0 cos sin

sin coscos cos

Rφ θ φ θ

ϕ φ φφ φ

θ θ

⎡ ⎤⎢ ⎥⎢ ⎥= −⎢ ⎥⎢ ⎥0⎢ ⎥⎣ ⎦

( )0

00

r qS r p

q pω

−⎡ ⎤⎢ ⎥= −⎢ ⎥⎢ ⎥−⎣ ⎦

In expression (10), [ ], , TL M NΜ = are the unknown aerodynamic moments acting on the body. For simplicity it is assumed that the aerodynamic moments can be described by


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( )( )( )

1

2 2 2

3 3 3

T

T

T

I

I B

I

ϑ ρ ω

ϑ ρ ω δ

ϑ ρ ω

1 1⎡ ⎤⎢ ⎥⎢ ⎥Μ = +⎢ ⎥⎢ ⎥⎣ ⎦

(11)

where [ ] 3, , Ta e r Rδ δ δ δ= ∈ are the control inputs,

( )i iIρ ω are the rigorous model which uses the exact wind tunnel data, iϑ are unknown parameter vectors. and

0

0 00

a r

e

a r

L LB M

N N

δ δ

δ

δ δ

⎡ ⎤⎢ ⎥= ⎢ ⎥⎢ ⎥⎣ ⎦

Next, the control law δ which aims to make the attitude angle ϕ follow a reference signal dϕ from the guidance system will be led. Define the tracking errors

( ) 1 , d dRϕ ϕ ϕ ω ω ϕ ϕ−= − = −　 (12)

and the energy function

( ) 21

12

T TH K Iϕ ω ϕ ϕ ω ω1, = +

2 (13)

where 1 1 0TK K= > is a constant matrix. Then, the system of (9), (10) can be written in the perturbed Hamiltonian form

( )( ) ( )

( )( )

1

1

0 0

T

T T

HR I

tHI R SII

ϕ ϕϕκ ϕ ωϕ ωω

ω

−

−

⎡ ⎤∂⎢ ⎥⎡ ⎤⎡ ⎤ ∂ ⎡ ⎤⎢ ⎥= +⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ Μ + , ,∂−⎢ ⎥ ⎢ ⎥ ⎣ ⎦⎣ ⎦ ⎣ ⎦ ⎢ ⎥∂⎢ ⎥⎣ ⎦

(14)

where

( ) ( ) ( ) ( ) ( ) ( )1 1 111

Td d dt I R K S IR IR IRκ ϕ ω ϕ ϕ ω ϕ ϕ ϕ ϕ ϕ ϕ− − −−, , = + − − (15)

Consider now the parameter estimation errors

( )ˆi i i i iz Iϑ ϑ β ω= − + (16)

for 1,2,3i = , where iϑ are new states and ( )iβ ⋅ are continuous functions to be defined. A control law which drives to zero the Hamiltonian function of (13) along the trajectories of (14) when 0iz = is given by

( )( ) ( )

( )( ) ( )

( )( ) ( )

( )( )1 1 1

12 2 2 2 2 2

3 3 3 3 3

ˆ

ˆ

ˆ

T

T

T

I I

B I I B t K I

I I

ϑ β ω ρ ω

δ ϑ β ω ρ ω κ ϕ ω ω

ϑ β ω ρ ω

1 1

−1 −

⎡ ⎤+⎢ ⎥⎢ ⎥

= − + − , , +⎢ ⎥⎢ ⎥⎢ ⎥+⎢ ⎥⎣ ⎦

(17)

where ( )2 2K K ϕ ω= , is a positive-definite matrix-valued function. The resulting closed-loop system can be written in the perturbed Hamiltonian form


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( )

( ) ( )( )

1

12

0 0

T

T T

HR I

HI R S KII

ϕ ϕϕ

ϕ ωωω

−

−

⎡ ⎤∂⎢ ⎥⎡ ⎤⎡ ⎤ ∂ ⎡ ⎤⎢ ⎥= −⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ∆∂− −⎢ ⎥ ⎢ ⎥ ⎣ ⎦⎣ ⎦ ⎣ ⎦ ⎢ ⎥∂⎢ ⎥⎣ ⎦

(18)

where each element of the perturbation vector is given as follows:

( )Ti i i iz Iρ ω∆ = (19)

Finally, the update laws iϑ and the functions ( )iβ ⋅ will be designed so that the perturbation in (19) is driven asymptotically to zero. Consider the estimation errors given in (16) and the dynamic update laws

[ ]ˆ ii i i

i

IIβ

ϑ ωω

∂= − + ∆

∂ (20)

Note that the term in brackets is a function of ,ϕ ω and the first and second derivative of dϕ , therefore is measurable. Using expression (20), the dynamics of (16) along the trajectories of (18) are given by

( )Tii i i i

i

z I zIβ

ρ ωω

∂= −

∂ (21)

The function ( )iβ ⋅ is selected as follows:

( ) ( )0

iI

i i i iI y dyω

β ω γ ρ= ∫ (22)

where 0iγ > , which implies that ( ) ( )i

i i ii

IIβ

γ ρ ωω

∂=

∂, yielding the system

( ) ( )Ti i i i i i iz I I zγ ρ ω ρ ω= − (23)

which has a uniformly globally stable equilibrium at zero. In addition, global stability of the equilibrium ( ) ( )0,0,0zϕ ω, , = and convergence of ϕ and ω to zero follows by considering the Lyapunov function

( ) ( )H W zϕ ω, + . where

( ) ( )3 2 2

1

1 , 2iii

W z z Wγ=

= = − ∆∑ (24)

E. Velocity control system The Velocity control system asymptotically regulates the aircraft speed to a desired set-point in the presence

of unknown aerodynamic forces acting on the body. First, the translational dynamics of the aircraft are described by the equation

( ) ( )m S m mg Fν ω ν ε φ θ= + , + + Τ (25)

where m is the aircraft mass, g is the gravitational acceleration, [ ],0,0 TxTΤ = is the thrust and

( ) [ ]sin sin cos cos cos Tε φ θ θ φ θ φ θ, = − , , .

Next, the control law for the thrust xT that regulates the total airspeed V ν= to a desired constant dV will be led.


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To begin with, the kinetic energy error is defined

( )2212 dE m V V= − (26)

and note that, from expression (25), the dynamics of E are given by

( )T TxE mg F uTν ε φ θ ν= , + + (27)

The estimation error is then defined as

( )ˆz F F β ν= − + (28)

where F is a new state and ( )β ⋅ is a continuous function to be defined, and the control law is

( )( ) ( ) ( )1 ˆT TxT F mg V E

uν β ν ν ε φ θ κ⎡ ⎤= − + + , +⎣ ⎦ (29)

where ( )Vκ is a positive function. By selecting the update law

( ) ( ) ( )( )1ˆ ˆF S g Fm

β ω ν ε φ θ β νν∂ ⎡ ⎤= − + , + + + Τ⎢ ⎥∂ ⎣ ⎦

(30)

yields the closed-loop system

( )

1

TE V E z

z zm

κ νβν

⎧ = − −⎪⎨ ∂

= −⎪ ∂⎩

(31)

Finally, an appropriate selection of ( )Vκ and ( )β ν ensures that the cascaded system of (31) has a globally stable equilibrium at the origin with E converging to zero. Two such selections are

( ) ( ) 3, i i iV kκ β ν γ ν= = (32)

and

( ) ( )2 , i i iV kVκ β ν γ ν= = (33)

where 0, 0ik γ> > .

F. Obtainment of control gains using genetic algorithm The control gains of flight control system are acquired systematically by using a learning method based on

evaluation functions because there is no unified suggestion which can obtain the control gains. In fact, the gains with respect to the guidance, attitude control and airspeed control shown in the previous section are acquired using the genetic algorithm based on evaluation functions. The simulation cases in designing control gains and the evaluation functions are as follows: <Simulation cases> • Take-off phase (both nominal and under disturbance conditions) • Steady flight phase (both nominal and under disturbance conditions) • Turning phase (both nominal and under disturbance conditions) • Landing phase (both nominal and under disturbance conditions)


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<Evaluation functions> • Following capability of altitude

1 11exp d

tJ a z z

t⎛ ⎞= − −⎜ ⎟⎝ ⎠

∑ (34)

• Following capability of target trajectory

2 21exp d

tJ a y y

t⎛ ⎞= − −⎜ ⎟⎝ ⎠

∑ (35)

• Stability of actuators

( )323 31 exp

3 el er rt

aJ a

tδ δ δ⎛ ⎞= − + +⎜ ⎟

⎝ ⎠∑ (36)

• Stability of thruster

( )424 41 exp t

t

aJ at

δ⎛ ⎞= −⎜ ⎟⎝ ⎠

∑ (37)

• Vibratility of state vectors

( ) ( )52 535 51 exp exp

3 3t t

a aJ a u v w p q r

t t⎧ ⎫⎛ ⎞ ⎛ ⎞

= − + + + − + +⎨ ⎬⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠⎩ ⎭

∑ ∑ (38)

• At-end condition

( )( )6

0 fall,attack angle, sideslip angle, storage

1 other than those aboveJ

⎧⎪= ⎨⎪⎩

(39)

The results of each simulation based on the above-mentioned functions are evaluated as follows:

( )6 1 2 2 4 5iJ J J J J J J= + + + + (40)

V. Numerical Simulation

A. Conditions of Simulation The availability of the flight control system was verified with a six-degree-of-freedom nonlinear simulation.

In the simulation, the nonlinear adaptive flight control system based on the I&I and the robust control method named H∞ ⋅MDM/MDP (applied to ALFLEX) were compared using the Monte Carlo approach. The airframe model, external environment model, and guidance/control law were considered as a mathematical model in the simulation. In the airframe model, the actuator characteristic was expressed using the second order time delay model with restrictions of position and velocity. In addition, the characteristic of sensor was assumed to be ideal that there were no errors in both static and dynamic conditions. As the external environment model, only wind was used. The constant wind and gust models were constructed using the MIL-F-9490D method applied to the ALFLEX simulation7.


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The contents of simulation are shown as follows: (a) Nominal (Taking-off → Steady flight → Up and Down → Landing) [Details]

• Way Point [x, y, -z] T [0, 0, 0] T → [200, 0, 30] T → [800, 0, 5] T → [1200, 0, 20] T → [1600, 0, 40] T → [2000, 0, 15] T → [2400, 0, 0] T • Velocity reference [V ref] = 13 m/s (at Landing phase) = 20 m/s (at the other phases)

• Height reference [height ref] (at Landing phase) = ( ){ }exp 'h a t t′ ⋅ − − where, h’ is the height reference before landing phase and t’ is the time when landing phase started. • Disturbance : none • The number of simulation : 1

(b) Random Disturbances of Aerodynamic moment (Monte Carlo approach) [Details]

• Way Point : [0, 0, 30] T → [500, 0, 30] T → [1000, 0, 30] T → [2000, 0, 30] T (Steady Flight) • V ref = 20 m/s

• height ref = 30 m

• Random Disturbance (Unknown Aerodynamic Moment)

Equation (11) is used as the random disturbance (unknown aerodynamic moment). In nominal flight, the unknown parameter T

iϑ is [1, 1, 1] T. The random disturbance (unknown aerodynamic moment) occurred twice in this each simulation. The details are as follows: (First) Time : 25 ~ 35 s Unknown parameter : -40 < T

iϑ < 0 (random) (Second) Time : 35 ~ 50 s Unknown parameter : 0 < T

iϑ < 600 (random)

• The number of simulation : 100 (c) Random Disturbances of Gust (Monte Carlo approach) [Details]

• Way Point : [0, 0, 30] T → [500, 0, 30] T → [1000, 0, 30] T → [2000, 0, 30] T (Steady Flight) • V ref = 20 m/s

• height ref = 30m

• Random Disturbance (Gust)

The Dryden wind turbulence model8 is used as the random disturbance (Gust). Only head-on wind is considered in the simulation. Therefore, the gust disturbance affects the velocity u , w and angular velocity q (pitch rate). The random disturbance (Gust) occurred twice in this each simulation. The details are as follows: (First) Time : 20 ~ 25 s (Weak Wind (random)) (Second) Time : 40 ~ 45 s (Strong Wind (random))

• The number of simulation : 100


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In these simulations, only longitudinal motion is considered. However, it goes without saying that the vertical and lateral motion can be treated in the same way.

B. Results and Discussions In this section, the simulation results of three different conditions as referred to above are represented. First, the time histories of height, velocity, pitch rate, actuator steerage and attack angle are respectively

shown as the result of content (a) in fig. 6 - fig. 10. In each figure, the left-hand chart is the result of proposed I&I adaptive control. In contrast, the right-hand chart is the result of robust control ( H∞ ⋅MDM/MDP).

Second, as the result of content (b), hundred-time height and velocity are displayed in Fig.12 and Fig.13. The bar chart of total flight results using the Monte Carlo approach is shown in Fig.11. In this regard, the bar chart expresses the result of success, fall, divergence of attack angle, divergence of sideslip angle and divergence of actuator steerage respectively. In addition, the time history of error of parameter estimation is represented in Fig.14

Finally, hundred-time height and velocity are shown in Fig 16 and Fig.17 as the result of content (c). The bar chart of total flight results using the Monte Carlo approach is shown in Fig.15. In addition, the time history of error of parameter estimation and Gust are represented in Fig, 18 and Fig.19.

In what follows, discussions obtained from these results are described. (a) The results plotted in Fig.6 and Fig.7 indicate that the I&I adaptive control has better readiness and following capability than the robust control ( H∞ ⋅MDM/MDP). In addition, Fig.8 to Fig.10 shows following two remarks. One is that each values of H∞ ⋅MDM/MDP are more vibrationally than those of I&I adaptive control. The other is that the results of I&I adaptive control moves more steeply. However, the vibrations and steep responses are caused by the characteristic of actuator; therefore these two points are not as important in considering the quality of both controllers. Additionally, in this time the trajectory tracking is emphasized in designing I&I adaptive control using Genetic Algorithm. If the other point such as restraining vibration or steep response is valued, I&I adaptive control can constrict the vibration or steep response. As presented above, the results of simulation (a) can be summarized as bellow: • I&I adaptive control can be designed just as the designer designed by using the proposed evaluation

functions. • I&I adaptive control has better capability especially in trajectory tracking and readiness. (b) In this simulation, the value of moment is fluctuated suddenly as an error of stability derivatives or a damage of wing surface. First, the bar chart in Fig.11 represented that I&I adaptive control overcomes all of the one-hundred-time flight. On the other hand, H∞ ⋅MDM/MDP has much failure times than those of success. This is because that the given parameter fluctuation is severer than supposed robust area of H∞ ⋅MDM/MDP when it was designed. In Fig.12 and Fig.13 shows that I&I adaptive control is more stable than H∞ ⋅MDM/MDP in both height and velocity. In addition, the result in Fig.14 indicates that I&I adaptive control can adapt the suddenly parameter fluctuation. As presented above, the results of simulation (b) can be summarized as bellow: • I&I adaptive control has better capability in random disturbances (unknown aerodynamic moment) than

H∞ ⋅MDM/MDP (c) Figure. 19 shows that this simulation gives exceptionally-strong wind. However, from Fig.15 to Fig.17, both controllers can tolerate the influences of gust. In, addition, Fig.18 indicates that I&I adaptive control can adapt the gust. This result gives rise to that I&I adaptive control is more stable flight than H∞ ⋅MDM/MDP. As presented above, the results of simulation (c) can be summarized as bellow: • Both I&I adaptive control and H∞ ⋅MDM/MDP are probably same performance with regard to the gust

model.


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Figure 6. Time history of height.

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Figure 7. Time history of velocity.

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Figure 8. Time history of pitch rate.


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Figure 12. Time history of height (100 times data).

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Figure 14. Time history of Error of parameter estimation (Pitch).


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Figure 16. Time history of height (100 times data).


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Figure 19. Time history of Gust.

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VI. Conclusion This research focused on the nonlinear adaptive flight control system based on Immersion and Invariance. The control system was applied to the airframe model of space plane concept proposed in our laboratory whose name is a horizontal take-off and landing space plane assisted by a sea ship. In designing a control system, the proposal control system described aerodynamic uncertainties as unknown parameters and adapted estimated parameters to unknown ones on the basis of I&I adaptive law. In this study, the unknown parameters as mentioned above were considered to be multiplicative uncertainties toward the rigorous dynamics model because the controller designed by this kind of method can overcome not only the specific disturbance but also various uncertainties. The availability of the control system was verified by the six-degree-of-freedom nonlinear simulation. As the results of simulation, it was confirmed that the nonlinear adaptive flight control system based on I&I has better performance than the robust flight control system ( H∞ ⋅ MDM/MDP) in both nominal and disturbance conditions. For the future, we will improve the performance of the flight control system in order to verify the value by using the actual equipment.

Appendix In this section, the past achievement with regard to a step-by-step development of the experimental plane whose purpose is to reveal the availability of proposal concept of space transportation system is represented. In what follows, the results of experiment which was held in January, 2009 are shown. First, the data acquired by GPS (Global Positioning System) and IMU (Inertial Measurement Unit) during the flight is displayed in Fig.20 and Fig.21. Second, the online actuator commands for horizontal steady flight calculated by I&I adaptive control are shown in Fig.22. These results give rise to following remarks: • On-Board System can get the both GPS and IMU data and estimate the acquired data by Kalman Filter. • The system can calculate the actuator commands for horizontal steady flight. In addition, what remains to be done are summarized as below: • Accuracy improvement of the estimate data using Kalman Filter • Development of driver system which can actualize the actuator commands • Reduction of computational load and performance upgrade of the On-board Computer

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Acknowledgments

This research is supported by the Research Grant of Keio Leading-edge Laboratory of Science & Technology. The authors gratefully acknowledge this support.

References

1Atsushi Fujimori, et al, “Flight Control Design of an Unmanned Space Vehicle Using Gain Scheduling”, AIAA JOURNAL OF GUIDANCE, CONTROL, AND DYNAMICS, Vol.28, No.1, 2005

2Sakushi Sunakawa and Hirobumi Ohta, “Nonlinear Flight Control for a Reentry Vehicle Using Inverse Dynamics Transformation”, AERONAUTICAL AND SPACE SCIENCE JAPAN, Vol.45, No.516, 1997

3L. Sonneveldt, et al, “Nonlinear Flight Control Design Using Constrained Adaptive Backstepping”, AIAA JOURNAL OF GUIDANCE, CONTROL, AND DYNAMICS, Vol.30, No.2, 2007

4Alessandro Astlfi, and Romeo Ortega, "Immersion and Invariance: A New Tool for Stabilization and Adaptive Control of Nonlinear Systems”, IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL.48, NO.4, APRIL 2003

5Y.Ohkami, et al, “A Proof-of-Concept Approach for Horizontal Take-off/Landing Rocket Plane”, 16 IFAC Symposium on Automatic Control in Aerospace, 2004

6Alessandro Astlfi, Dimitrios Karagiannis, and Romeo Ortega, “Nonlinear and Adaptive Control with Applications”, Springer, 2008

7NAL/NASDA ALFLEX Group, “Flight simulation model for Automatic Landing Flight Experiment (Part I: Free Flight and Ground Run Basic Model)”, TECHNICAL REPORT OF NATIONAL AEROSPACE LABORATORY, Vol. 1252, 1994

8HL Dryden, “Measurements of Intensity and Scale of Wind-tunnel Turbulence and Their Relation to the Critical Reynolds Number of Spheres”, National Bureau of Standards, 1946

9Kanichiro Kato, Akio Oya and Kenzi Karasawa, “Introduction of Aircraft Dynamics”, University of Tokyo Press, 1982 10Brian L. Steavens and Frank L. Lewis, “Aircraft Control and Simulation 2nd Edition”, JOHN WILEY & SONS,

INC,1986.

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Figure 22. Actuator commands calculated by I&I adaptive control.

design of nonlinear adaptive flight control system …...nonlinear adaptive flight control system...

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