A nonlinear controller for parabolic flight

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TA1 11:20

A Nonlinear Controller for Parabolic Flight L. D’Antonio and S. Monaco Dipartimento di Informatica e Sistemistica Universith di Roma “La Sapienza” Via Eudossiana 18, 00184 Roma, Italy

Abstract

flight conditions. B d on this last result the same control technique is here used to assure parabolic flight conditions.

The paper deals with the design of a nonlinear controller for asymptotic tracking of a given parabolic trajectory in flight dynamics. The problem here addressed is related to the possibility of maintaining microgravity conditions. Simulation results show the effectiveness of the proposed controller.

The paper is organized as follows: in section 2 the mathematical model is introduced. Moreover the reference trajectory to be tracked is computed starting from some usually accepted constraints from microgravity experimentation. In section 3, after some recalls on feedback linewiz% tion and asymptotic tracking, the application t o our case study is developed. The effectiveness of the control scheme is shown by the simulations, in section 4.

1. Introduction.

2. The Mathematical Model and the Reference Trajectory.

Parabolic flight represents an appealing feature for realizing microgravity conditions. As well known different frameworks are actually used to get microgravity experiments, like drop towers, sounding rockets, spacecrafts. Parabolic flight is today considered as one of the most promising because of its low cost realization.

2a Mathematical model. Our purpose is the synthesis of a dynamic controller which is able not only to force the aircraft to track a precise trajectory, but also to maintain pre-established attitude conditions. In this sense the stability of the closed loop system is a request not less importrant than asymp totic tracking. For this r e m n and because of the high performances requested, one must use an accurate control model. The only simplifying hypothesis here assumed are the following:

nee floating inside an aircraft in parabolic flight poses several problems either for the technological side or for the methodologies involved. Far from proposing a reply to the problem of automatically maintaining free floating conditions, a problem which is strongly conditioned from safety requirements, we propose hereafter a control design p r o w dure for achieving asymptotic tracking of a parabolic flight for the center of mass of a given aircraft.

0 the aircraft is a rigid body: this allows to describe completely the motion of the aircraft simply in terms of translations of its center of mass and of rotations around it; 0 the aircraft has a plane of symmetry XZ: this permits to annihilate the products of inertia I,@ = I,,, I,* = I,, in the expression of the inertia matrix of the aircraft because this matrix is calculated referring to a reference frame in which the X and the Z axes lie on the plane of symmetry of the aircraft.

This can be considered a first step in the design of a safe free floating controller. In fact such a controller can be thought at a higher level, while leaving the maintaining of free floating conditions to a safe incremental controller based on measurements of the position of the laboratory inside the aircraft w.r. to an inertial reference frame. The design procedure for the controller here proposed is based on dynamical feedback linearization and asymp totic tracking of the given process representing the dynamics of the aircraft. The nonlinear model of the dynamics p r o p 4 in [2],with the usual simlifying assumptions, but preserving the nonlinearity of the dynamics itself, is here used for the design. Nonlinear control was applied in several flight applications starting from [7], (81. More recent applications are in 191 and 131 where, starting from a complete representation of flight dynamics, feedback linearization is applied for the design of a controller working in extreme

0 the following aerodynamical derivatives are neglectable: CO,, 1 C L S9 ~ c L q 9 CLL,C M h Moreover the shortness of the experiment allows us to do two further assumptions: 0 the mass of the aircraft doesn’t vary in time, as also the inertia moments and products (in an appropriate reference frame). 0 the Earth is fixed in the space, i.e. we ignore its angular velocity ( X 7.27 X w5). In this way we assume as inertial the Earth reference system.

This work was supported by A. S. I. under study con-

tract Rs 157/91 0191-2216/93/$3.00 Q 1993 IEEE

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With these assumptions and following 121, we obtain the control model starting from the moments equation and from the Newton's second law. This is a 80 called "mixed axes" model, because the state variables are not expressed in the same reference frame (the body axes reference frame), but in two, (namely the body and the wind axes reference frame). The introduction of the wind axes reference frame is useful in expressing the desired trajectory with our state variables set. After rearranging we get:

r; = i(IZ + I& iIx,L

D e=--m

G s i n 0 = fv +gvbp

(2.la)

e=-(L-mGcos0cos@)mV

+ iLN =

fr

4-gr6,,6,

+ g&br

8,Q ,@ are the Euler angles which rotate the inertial reference frame on the wind axes reference frame

zw .

a,fl are the attack and sideslip anglea. The Euler an-

cos@

Sin Q (-S + mG cos 0sin @) = fe + gebP mV

(2.1b)

glea which rotate the wind axes reference frame in the body mea reference frame are ordinately (-P, CY, 0). Moreover V, a,P represent a basis in the space of the linear velocities of the aircraft. p, q, T are the angular rates of the aircraft w =

ij=

(2.12)

where:

x w , yw,

.

- IxI,)qp - iIxz(Ix + I, - I&+

sin Q sec 0 (L-mGcosecos@)+ mV

expressed in the body axes reference frame.

cos9Sece mV (-S+mGcosQsin@) = f,p +gubp (2.lc)

(9

D , L, S are respectively the drag, lift and side f o r m and their analitical expressions are:

b = p c o s a cosp + sin /3 sec P (L - ~GCOSQCOSQ)+ mV

sin P tan P(p cos a + r sin a)+ rsinacosP+ cos @tan0

mV

sin @tan9 (L - mGcos@cos@)+ mV

(4 + mG cos 0sin a) = f*

+ g*SP (2.ld)

sec P dr=q--(L-mGcos0cosQ)mV tanP(pcosa + r s i n a ) = f a + g a b p

(2.1e)

L w are the external torques acting on the aircraft expremed in the body axes reference frame. Their analitical expressions are:

(5) ( V2.

(

=

+ +

+ + +

allV2sin P a12rV al3pV a21V2 azzv2sin a a23qV a31V2sin p a32~V a33pV

b11-P 0 0

+

0

The inputs are:

a = p sin a-r cos a+ p = iIx,(Ix

-S + mG cos0 sin Q mV

= fP+gPbp

(2.V)

- lU + I,)qp - i ( ~ ,+"I:, - IvI,)qr+

iIzL + iIxzN = fp

+ gfi,ba + gp&&

(2.19)

:6, throttle setting 6:, aileron deflection angle

6,: elevator deflection angle 6,: rudder deflection angle G, m, P are the gravity acceleration, the mass of the aircraft, the maximum thrust, while I k j , Ijj Ij k,j = X,Y, 2 are the moments and products of inertia of the aircraft, and i = X I 1. Moreover J"-I:Z akj,bkj,Ckj k , j = 1,2,3are parameters related to the

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V, = - V s i n e

aerodynamical derivatives of the aircraft and they allow to model forces and torques acting on the aircraft. The rel& tionships between parameters and derivatives are:

a12 = %,,SE 2

a13 = PC~,SC 2

a21

(2.34

Inverting these relationships we get

If now we substitute the (2.2) into the (2.4),we get the exprmions of the desired outputs:

= $cM,,sE

(2.5~) The simplicity of these relations, beeides stability arV, as outputs. As far as the fourth outputs is concerned, we chase for obvious reasons 9 with @ d ( t ) = 0. A fully linearizing controller will be obtained with such a choice.

e,@

guments, justifies the choice of

where p is the air density, S the wing surface and E the aerodynamical chord.

2b The desired trajectory. 3. The Control Scheme.

Let us now consider the problem of determining the desired trajectory of the center of mass of the aircraft. It will be expressed in terms of the outputs (V, 8, Q, a) of the system.

3a Asymptotic tracking under feedback linearization.

The desired motion L that of a body only subjected to the pavity force, whose expressions in an inertial frame are:

fillowing [I] we refer to a system of the form

& =) .(f

+ g(x)u

v =h(4 where x E R", y E IEP, output and input vectors.

(2.2a)

= &(tin)

(2.26)

Vg(t)

VZ(t)= Gt

+ V.(tin)

The relationships between Vz(t),vu(t), outputs are:

...

Let us also introduce Q(X) (an m components vector) and b(z) = p'(x)if"'(z)] the following equations:

(2.2c)

vz(t) and the

v, = V ~ 8 C O S \ V

(2.3a)

Vv = Vcos0sinq

(2.3b)

(3.1) E R'" represent state,

Let define ri as the minimum integer such as in the rf" order derivative of y, at least one input appears explio itly. Moreover let q, 0 < q 5 m, the number of different inputs we get by deriving each output T j times, and let us suppoee that such inputs are q , ,'IL~,and that they a p pear deriving the first q outputs (this is always possible by numbering appropriately inputs and outputs).

Actually we are not interested in the initial position of the aircraft, but it is only necessary that the trajectory is parabolic.This can be simply accomplished with constraints on the components of the linear speed. The constraints are:

V=( t ) = v z (tin)

U

(an m x m matrix) W M C ~solve

aLri-1

axhi(x).(f(x) + g(x)a(z)) = 0

aLri -1 f h'(x).(g(x)@(x)) 8x

= 6,,

(3.2~)

1 5 i,j 5 q (3.2b)

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I

.

where L f h i ( z )is the Lie derivatives of hi(x) along f ( x ) , which is defined as follows:

steps the rank of A ( z ) is still q, then it is impossible to find any dynamic feedback which can increase the rank of A ( z ) (and in particular can render A ( z ) invertible). If instead the rank increases, then we need to calculate again & ( E ) and p(z) and iterate the procedure with Q > q. However the feedback could render unobservable a part of the system. This happens if T I . .. r,,, < n and in this case the dimension of the unobservable state is q u a i ... rm). This unobservable part plays a to n - ( T I crucial role on the stability of the system. If, on the other rm = n, then the system hand, it happens that T I is completely linearizable under feedback.

+ +

kh, x LF’hi(z) = dLf ( ) . f ( x ) ax

+ +

In this case the static feedback U

= a(.)

+ P’(X)< +P”(X)W”

+ .. .+

(3.3)

If when such an algorithm is applied, the resulting system is fully linearized under feedback, then asymptotical tracking can be obtained by assuming

tranforms the system in such a way that:

y(‘) = f h ; )

= (x)+A(x).w

where %ribid

(XI

- yi), 1 5 i 5 m

(3.4)

where kij are coefficents which determine the time b e haviour of the trajectory error e = y - P d . Namely the error evolves according to the equation

=

3b Application of the control to the case study. With reference to the system (2.1) we easily verifies that at the first step q = 1, and after the first iteration of the algorithm, by inserting an integrator on the 6, input ye have q = 2. Instead of calculating again &(z) and P(z), it is enough to add a second integrator on-the first input getting an invertible decoupling matrix A(3);i.e. rank[A(k)]= 4 a.e. with k in R”. Moreover we have ~1 = r2 = r3 = 3,r4 = 2 and r1 r2 r3 r4 = 1 1 =state space dimension.

If A ( x ) is invertible (i.e. q = m ) then the system results to be linearized by means of the control law (3.3). If the matrix A ( z )is singular, i.e. one has q < m, one can put one integrator on each of the first q inputs in such a way to “delay” in a differential sense the appearance of those inputs. The dynamic feedback corresponding tp what we said till now is U = a ( x ) P’(x)e p”(z)d’, ( = w‘. In this way we need to derive all the outputs once more to get the inputs V’. In this further derivation it is possible to get (in whatever outputs) at least one input of the vector w’‘. Assuming that s new inputs appeared in the (r, 1 )th

+

+ + +

+

The resulting control law obtained by solving equations (3.2) and by adding integrative actions has the form:

+

derivatives of s among the last (m - q) outputs, it is p o s sible iterate the algorithm, calculating again &(z)and &z). (it)and A(%)have the same form as before but the dimension of the state s p ~which , is dim[%]= dZm[z]+q and Q = q s = rank[A(Z)].

e

+

If the rank of A ( x )is still q even after the introduction of the first q integrators, then the next step is to put other q integrators in cascade to the first ones and so on. The number of such steps has an upper bound. In fact one can prove that (11 : 0

where

Ml(k)=

given q, if after

(3.7a)

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I, = io7 iV = io5 I , = io6 I , = ~ ioooo m = lo6

P = lo8

s = 200

Tab.2 state variables initial values:

U=(;)

V, (tin) @ ( t i n )= - arcsin V(tin)

In (3.6)v takes the form:

v=A-’(q(-

(2)+w)

*(tin) = 0

r(tin) = 0.1 Vs(tin)= 76.7

+

The control law (3.6),computed by means of the symbolic manipulator MATHEMATICA. has been imple mented to get simulation results on the control system. For this purpose we made a program in FORTRAN which accomplishes the numerical integration of a system of 9 differential equations for 15 seconds. These equations r e p resent the simulation model for the aircraft, on which the controller acts. For the integration we applied the Runge Kutta fourth order algorithm, and we accompished calculations in double precision (16 digits). The values to the parameters in the simulation model are reported in Tab.1, while the initial conditions are in Tab.2 below.

q(tin) = 0

=0

Vg(tin)= 0

Vz(tin)= -64.3

+

+

From these figures we can conclude that the position error of the center of mass of the aircraft is less than 0.1 meters in every direction, which specifies the quality of the results obtained.

Conclusions.

Tab.l: numerical values of the parameters

A nonlinear controller for asymptotically tracking a parabolic trajectory in flight dynamics was designed based on nonlinear methods. The results obtained are preliminary w.r. to the problem of maintaining free floating conditions of a laboratory for microgravity experimentations

in the simulation model.

a13 = -54

((tin)

The coefficents of the error equation (3.4)are fixed according to the equation of the Butterworth filter: s3 2rs2 2r2s r3 = 0. The graphics of interest are in Figs.1 to 10. It is readly verified from Figs.4 and 5 that the rad w.r.to the parabolic errors are small, less than 3 . flight. Moreover rolling stabilization is assured (Fig.6). An error of less than 10-3m/s is mantained on the velocity, which is due to numerical problems.

4. Simulation Results.

1000 all = -150

a ( t i n ) = 0.1

= 0.1

tin) = 0.1 q(tjn) = 0.1

P(tin) = 0.1

where the entries of w take theJorm (3.4)and the terms in (3.7)are obtained from (2.1).A-’(%) and (2)take quite complicate expressions which were computed by a symbolic tool and are not given here.

~ 3 = 1

+(tin)

References

a12 = 27

[I]A.Isidori: Nonlinear Control Systems, 2nd edition, Springer Verlag, 1989.

a13 = -54

a23 = -54

azl = -300

[2]B.Etkin: Dynamics of Atmospheric Flight, John Wiley & Sons, Inc., 1972.

a22 = -150

a31 = 100 a32 = -13.5

[3]S.H.Lane, R.F.Stenge1: Flight control design using nonlinear inverse dynamics, Automatica 24,pp. 471-483, 1988.

a33 = -3

[4]D.McRuer, I.Ashkenas, D.Graham: Aircraft Dynamics and Automatic Control, Princeton University

bll = -100

b13

= -50

b22

= -200

b33

= -50

Pres, 1973. 1517

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151 J.Roakam: Methods for Estimating Stability and Control Derivatives of Conventional Subsonic Airplanes, 1971. Published by the author: 519 Boulder, Lawrence, Kansas 66044. [6] D.McLean: Automatic Flight Control Systems, Prentice Hall, 1990

0.002

[7]G.Meyer L.Cigolani: Application of nonlinear s y s tem inverses t o automatic flight contrd design-system concepts and flight evaluations. In Theory and Application of Optimal Control in Aerospace Systems, AGARDAG251, pp. 10.1-10-29(1981).

-0.004

-0.006

(81 G.Meyer, R.Su, L.R.Hunt: Application of nonlinear transformations to automatic flight control. Automatim 20, pp. 103-107.

o.l~

*FIG.6:

P H I ERROR

[9]J.J.Romano, N.S.Sahjendra: 1/0Map Inversion, Zero Dynamics and Flight Control. IEEE %w. Aero. El=. SySt. AES26.6, pp.1022-1028.

[lo]S.Monaco, D.Normand Cyrot, S.Stornelli: Sampled Nonlinear Control for large angle maneuvers of flexible spacecraft, ESA S.P. 255,pp. 31-38,1986. FIG.1:

SPEED

110 105

2000 4000 6000 800010000

.

FIG.2:THETA

FIG,3:

SPEED ERROR

-0.142 -0.143 -0.144 -0.145 -0,146

0.002

0.001 -0.001 -0.002

FIG.8:

SECOND INPUT

FIG,9:

THIRD INPLIT

2000 4000 6000 800010000

-0.025 -0.05

-0.0002

-0,075l11

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