Multilevel Hierarchical Optimization of Aircraft Trajectories

Copyright © IF AC Distributed Intellige nce Systems, Varna, Bulgaria. 1988

MULTI LEVEL HIERARCHICAL OPTIMIZATION OF AIRCRAFT TRAJECTORIES L. Mikhailov and

J.

Zaprianov

Complex Automa/ion and Systems, Institute of Information, Chapaev str. 55, Sofia 1574, Chapaev sir. 55, Bulgaria

Abstract. The problem of optimal aircraft control is considered. A multilevel hierarchical method for synthesis of optimal trajectories is proposed, based on subarc decomposition of the overall flight and coordination procedure of the interface conditions between the subarcs. The hierarchical method is realized by distributed computational two-level structure. Independent optimization problems for each subarc are solved on the first level by local controllers and coordination of the local optimal solutions towards the global optimum is carried out on the second one. Digital simulation of the proposed hierarchical method for a big transport aircraft is presented. Keywords. Optimal control; hierarchical systems; aircraft; trajectory planning; subarc decomposition.

INTRODUCTION

dynamical equations of motion for each subarc are derived and an appropriate additive performance index is chosen, which gives the fuel consumption for each subarc and for the overall flight. Independent local optimization problems are solved on the first level of hierarchical structure. The boundary terminal conditions between the subarcs are taken into consideration via an additional second-level controller, using simple gradient coordination procedure, which ensures fast quadratic convergence of the iterative solutions at the first level.

The synthesis of optimal aircraft trajectories has been a subject of continuous theoretical and practical interest. Usually, the optimal control problem is formulated and solved by means of the Pontryagin's minimum principle. The general system of dynamical equations of mot ion, commonly employed for computation of overall aircraft optimal trajectories is strongly nonlinear. The synthesis of optimal trajectories results in nonlinear twopOint boundary value problem (TPBVP), which leads to formidable computation difficulties (Gordon, 1983). Presently, the overall trajectory planning, including optimal flight profile, optimal velocity and cruise altitude etc., is performed on big ground dispatch computers, due to the great computational expences (Schultz, 1985).

The proposed method is tested by digital simulation for a big transport aircraft and some simulation results are presented.

HIERARCHICAL CONTROL SYNTHESIS

The significance of sophisticated preflight planning is reduced however, when some unforeseen circumstances or disturbances arise during the flight. In such cases it is desirable a new optimal trajectories to be generated on-line. To overcome the computational difficulties solving the above problem, simplification techniques to produce results which are meaningful and attainable at reasonable cost are used usually.

The nonlinear mathematical model of an aircraft in general form is given by f (x,u,t),

x(t ) = x , o 0

(1 )

m where X€:R is the state vector, U€lR is the control vector and f(.) is known vector function, n ff lR • n

The performance index we use ratic form:

Singular perturbation techniques provide a powerful approach to simplify the solution of nonlinear optimal control problem. By means of singular perturbations, the system dynamics is separated into slow and fast modes and the synthesis is carried out for slow and fast subsystems of lower orders (Calise, 1981; Ardema, 1985). The use of matched asymptotic expansions and the energy state approximation for synthesis of fuel-optimal aircraft trajectories with fixed terminal time is presented in (Burrows, 1983) . The solution of the optimal problem is approximated with series of independent optimal subproblems, which are solved for all phases of the flight, namely climb, cruise and descent. Howe~er, the solution of the overall problem is suboptimal.

t

is in standard quad-

f

J F(x,u,t)dt,

J t

(2)

o

where F(.) is scalar function, giving the concrete control purpose, to and t are initial and final f times of the overall flight. The control problem is to find such controls, which minimize Eq. 2 under restriction, given by Eq. 1. To solve this problem, the overall flight trajectory is decomposed into N subarcs with time intervals (t!, t;), i = 1, ••• ,N, where t! and t; denote the initial and final times of the i-th subarc,

An hierarchical approach for synthesis of optimal

l

aircraft trajectories with reduced computatonal expences is proposed in this paper. The overall flight trajectories are decomposed into subarcs, corresponding to the different flight phases. The

to

N

= t · These subarcs correspond to the f f main flight phases, as climb, descent , cruise, landing etc.

181

to' t

182

L. Mikhailov and

Thus, the initial system, Eq. 1 can be described by a set of N subsystems

J.

Zaprianov

From the transversality conditions it follows, that the terminal condition

.i

o

x

i = 1, ... IN

(11)

(3)

has to be fulfilled. and the performance index is given by t

J

i

N f L f Fi(xi,ui,t)dt. i=l ti

(4)

On the first level of the hierarchical structure local two-point boundary value problems, Eq. 8 Eq. 10 have to be solved, where the initial conditions for the state vector Xoi are

o

,

g~ven

by t h e se-

cond level coordinator and the terminal conditions Assuming smooth flight trajectories, the interface constraints between the state variables and time can be written i+l Xo

= 1, •.•

i

(5)

,N~i.

By means of the above sub arc decomposition, the control problem is to minimize the performance index Eq. 4 subject to restrictions Eq. 3 and Eq. 5. Introducing Lagrange's multipliers, this problem is reduced to unconstrained optimization of the following index i t f

N

J

o

(6)

When the initial conditions xoi, t

i 0_

=

Second level coordination On the second level the initial conditions for each subarc are to be calculated. The coordination procedure is based on the prediction principle and it is performed iteratively (Bauman, 1968): z

(k)

z (k-l)

+ /:;z,

( 12)

where z (k) = {zi+l} (k), i = 1, ... ,N-l are the coor i+l i+l"\' i+l T , dination variables, z (x t ) , and k ~s o 0 , +1 the iteration index. The increments Dz = {/:;z~ }, i 1, ... ,N-l are computed by gradient search for

L {f [F i + (Ai) T (fi_xi)] dt} + i=l ti

where A\ i = 1, •. ,N, pi and li, i Lagrange's multipliers.

for the costate vector Ai(t!) are calculated from Eq. 11.

a local minimum of the function p(z) = {pi+l(zi+l)}, i = 1, .•. ,N-l, (Zaprianov, 1987a), where lllAH1_ pi

1, ... ,N-l are

211-FHl + for each subarc

are fixed, i = 1, .•. ,N, the index J is additively separable. This means, that the minimization can be carried out via solution of N independent optimization problems. Thus the global optimal control problem can be solved via an hierarchical distributed structure, consisting of local subcontrollers on the first level, dealing with independent optimization subproblems for each subarc. The predicted values of the initial conditions are computed on the second level by coordinator and are set to the local subcontrollers.

112

(AH1)TfHl_1~ 11

(13)

The coordinatior task is to find such prediction z*, which ensures local minimum of p(z). Second derivative Newton method is used, which generates a sequence of admissible pOints z(k+l): 7.(k+l) = z(k) + a(k)dz(k), p(z(k +l))
Local optimization

AO. (Initial prediction). The initial value z(O) is chosen; k = O. Go to Al, .

The local subproblems are solved using Pontryagin's minimum principle (Bryson, 1975). The local Hamil-

Al. (Convergence test). On the base of the local

tonian 3(i

TPBVP solutions the vector p(z(k)) is computed,

~ for the i-th subarc can be written Fi + (Ai)Tfi + (pi) T (xi_xi+l) + f 0 1 i( i_ i+l) t f to '

Eq. 13. The gradient a(k) = 9p(z(k )) is found. If a(k) < E, E ~ 0 is sufficiently small num(7a)

ber, the search is terminated and z(k) is the optimal prediction. Else go to step A2.

i = 1, . . . ,N-l, A2. (Computation of search direction). The Hessian

3(N = FN + (AN)TfN, i = N.

(7b) matrix A

The necessary conditions for minimum are .i x

~c'aA

>.i

i

-

i aJt i ax

fi(xi,ui,t); i _(~)T ax~

-

_

-

<7

v

2

p(z

(k)

) is calculated and the

search direction dz(k) is determined from (8)

Go to step

(af\ TAi; ax~

A~.

(9)

A3. (Computation of step length). The positive scalar a(k) is calculated, so that

1, ... ,N.

The optimal control u equation

(k)

i

can be obtained from the Go to step A4. 0, i

1, ... ,N.

( 10)

Multilevel Hierarchical Optimization of Aircraft Trajectories A4.

183

i timum thrust (T )* for each subarc is found:

(Computation of the new prediction) .

Tmax' if Ti ~ Tmax -i T

Go to step Al.

< T. < T

if T.

ml.n

1.

(19)

max

The second level coordinator determines the initial conditions (zi+l) (k), i = 1, ••• ,N-1 for each local subcontroller for the k-th iteration step, while the initial conditions for the first subarc are constant. After solving the local TPBVP, the

where

,,~g

+ (1-"!)C (Hi,Vi)W 1 i i i 2C (H ,V )W ("!-1) 2

coordinator forms the function p(k) , Eq. 13 and computes new initial conditions (zi+1) (k+1) , i = 1, ... ,N-1, in accordance with the above coordination algorithm. The proposed gradient search has fast quadratic convergence and a few iterations are needed in practice, the optimum to be found.

(21) i

The optimal path angle (0 )* is found in an analogous way:

e

~ 0max

i

omax , i f i

0 , if 0

The pOint-mass equations of motion for two-dimensional flight in the vertical plane we employ are: V

H

v0

V

g(T-D-w0)/w

M

-F,

( 14)

(15) The aircraft model involves the drag D(H,V) and the fuel flow rate F(H,V,T). The performance index used gives the fuel consumption over the whole flight: f

f

=

J

t

F(H,V,T)dt.

The overall flight is decomposed into 3 phases: climb, cruise and descent and the corresponding subarcs are determined. The initial values for the first subarc are given, the remaining are computed by the coordinator. The Hamiltonians ~, Eq. 7 are formed and the costate equations are obtained: .i = Xi X X .i = 3(i -"H H

-"

~i = ;i{ i

_~! where

~

V

=

X! =

0 Fi H

-

"i Di/wi vg H

-

" iFi MH

Fi(l_"i) + "i + "i0i _ "i Di / Wi V M X H vg V ,,~g2 (T i _Di) / (W i ) 2 ,

a;0 /dX i , ~

=

a;0 /d Hi

3ei = 0 T

i i i F (l_"i) + "v g / w T M

Xi 0

,,~vi

(17)

(22)

max

0IDl.n .

Due to the limited space, the exact expression for i the is not given here, but it can be found in (Zaprianov, 1987a).

e

i "vg

A big transport subsonic aircraft is simulated and the proposed hierarchical control is applied. The following values for the initial conditions, Zi = (Xi Hi vi Mi tilT i = 1,2,3 are accepted: o 0 0 o 0 ' 1 Z (0 2 150 100 O)T Z

2

z3

(200

10

250

96

1000) T

(800

10

250

90

4000)T,

where the dimensions of the state components are as follows: X in [km], H in [km], V in [m/sec], M in [ tJ, t in [ sec ].

0

The local TPBVP for each subarc, Eq. 14, Eq. 17, Eq. 19 and Eq. 22 are solved by means of the shooting method (Speyer, 1985) on each iteration by the local subcontrollers. The above initial conditions are used as initial prediction of the coordination variables. The results of the second-level coordinator are given in Table 1.

Iteration 1 2 3 4 5 6

x2

H2

0

0

200 245.1 239.4 234.8 235.2 235.3

10 12.66 11.42 11.38 11. 35 11.35

v

2

M2

0

0

250 248.2 239 . 0 237.7 238.1 238 . 1

96 95.8 95 . 6 95.3 95.2 95 . 2

t

2

x3

H3

0

0

0

800 841 824 792 784 782

10 12.66 11.42 11.3 8 11. 35 11.35

1000 1340 1280 1234 123 8 1137

v

3 0

250 248.2 239.0 237.7 238 .1 238.1

etc.

= 0

The obtained optimal flight trajectories and corresponding controls are shown on Fig . 1 and Fig. 2. (18a) (18b)

From the first equation (18a), taking into consideration the restriction, given by Eq. 15, the opD.I.S.M.-G

~

< 0

TABLE 1. Coordination results

The controls satisfy the equations

0

0

if

ei

(16)

o

- V

oml.n . ,

<

min

Digital simulation

where X is the range, H is the altitude, V is the airspeed, 0 is the flight path angle, T is the thrust, vi is the weight, D is the drag, M is the aircraft mass, F is the fuel flow rate and g is the acceleration due to the gravity . The control variab les T and 0 are constrained as follows

t

(20)

The coefficients Cl and C are taken from the fuel 2 rate model:

FLIGHT HIERARCHICAL CONTROL

x

i

In comparison with the global solution, the hierar chical approach to optimal trajectories planning offers some advantages. At first, it takes into consideration the physical peculiarities of the flight. The proposed subarc decomposition leads to simplification of the equations of motion and lightens the solutions of the optimal subproblems.

J.

L. Mikhailov and

184

For example, during the cruise flight, H = const, and T = D. Thus the local optimization for the second subarc is solved by explicit equations (Zaprianov, 1987b).

e =0

Zaprianov

't~O ~ ' _':·m::" ::: /' ::,: · .:: ·':.. I

_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _,

1"

Further, the computations on the first level are carried out in parallel, which gives a possibility they to be performed by smaller computers in distributed structure, which reduces the computational time significantly. CONCLUSION hierarchical method for synthesis of optimal aircraft trajectories is presented, realized by distributed computational structure on the first level. Due to the parallel operations of subcontrollers, significient reduction of the computational time can be achieved, which gives an opportunity for on-line implementation. An

HO

100 "

)0,

,;,"

",'

",0 ,;00

WOO 1100 1400 1700 3000 );00 )"0 3900 4200 4500 TIME (~t.:cl

Fig. 1 (b). Optimal flight velocity REFERENCES Ardema, M.D. (1985). Separation of time scales in aircraft trajectory optimization. J. GU.i dance, ~, 275-278. Bauman, E. (1968). Multilevel optimization techniques with application to trajectory decomposition. In C. Leondes (Ed.), Advances in Control Systems, vol. 6, Academic. Bryson, A.E. and Y.C. Ho (1975). Applied optimal control. John Wiley and Sons, N.Y. Burrows, J.W. (1983). Fuel-optimal aircraft trajectories with fixed arrival times. J. Guidance, .§., 14-19. Calise, A.J . (1981). Singular perturbation techniques for on-line optimal flight-path control. J. Guidance, 4, 398-405. Gordon, C. (1983). Flight software for optimal trajectories of transport aircrafts. AlAA paper 83-2241. Schultz, R. and N. Zagalsky (1985). Aircraft performance optimization. J. Aircraft, 14, 108-114. Speyer, L.J., D. Dannemiler and D. Walker (1985). Periodic optimal cruise of an atmospheric vehicle. J. Guidance, 8, 31-38. Zaprianov, J.D. (1987a)~ Synthesis of multilevel decentralized control systems. D.Sc . Thesis, (in Bulgarian) • Zaprianov, J.D., D.I. Boyadjiev and L.M. Mikhailov (1987b). Optimal range performance control of aircraft. In Preprints 10th IFAC World Congress, .§.. 253-256.

.. 15

H

Ikol

I\IU

"

It I

9~

"" ;,

""

~l:.i ~·I

", :I;~

"U U~J

tui

bl uti ,,'j L......J 'u..,. ,-

o:'·,'C' J -".., UO-I:::'L.OO:-:I-:"U:::O~lb;':0::-0-:;:21'::00;-:2~'0;:;O:-:;2~10:;:0-:;;30~00;-:JJ ;:;·:;;OO;--:;37,JO:;:0~3,;;;00;-:4;:;,~00;-;4;.500 T[ME

(sec)

Fig. 1 (c). Optimal flight weight

1000 X.

Ikml

900

13 BOO

12

11

700

10 600 500 400 300 200 lVO

o0

JOO

GOD

900

1200 "iOO 1000 2100 2400 2700 JOOO J300 3600 J:lJO mo 01'.;00 0 0

THE (secl

Fig . 1

(a). Optimal flight altitude

300

600

900

1200 1500 IBOO 2100 2400 2700 3000 3300 3600 3900 4200 4500 TlHE

(secl

Fig. 1 (d). Optimal flight range

Multilevel Hierarchical Optimization of Aircraft Trajectories

12

~T~I~tl~

______________________________________________,

10

5 -

3 .

'J ,L-.,,-' 3Q"Q-::'u,", ""'!,.. ]v.. ":. -""'I"-''' ':-:-:-I;'=UU:-:::!O':-::: OO""''~IOO:;;O-:':-:';'0:::-0-::2~IOO::-;:30::::00;-:;;3]~OO-3~GO;;;:O-:]::::"u:::-o-;'~ 20;;0 --;;;'5'00 TIME

(SCt;)

Fig. 2 (a). Optimal thrust - 1st control

9 Id,gl

·1

·2 ·3

-, -50~~3~00~~60~0~9~JO~I~2~00~15~O~ O -I~80~0~2~IO~ O~2~'O=O~27~O~O~3~OO=O~33~OO~3~60=O~3~~OO~'~10=O~'500 TIME Is,cl

Fig. 2 (b). Optimal heading - IInd control

185