An efficient algorithm for optimum trajectory computation

by KENTON S. DAY ESL Incorporated

Sunnyvale, California

INTRODUCTION

This paper describes a variation to the steepest-descent method for generating optimum trajectories.

The steepest-descent approach to trajectory optimiza-tion was formulated by Kelley,l,2 Bryson et aI., 3-5 for numerically solving a variety of two-point boundary-value problems. The procedure is iterative, requiring repeated forward numerical integrations of the state differential equations and backward integrations of the adjoint equations. In many applications, however, convergence was slow; thus, several techniques for speeding convergence were devised.6 ,7,8

The computational algorithm presented in this paper requires that a perturbation of fixed size along one segment of the trajectory effect the same diminution of the penalty function as along any other segment.

This essentially compensates for differences in sen-sitivity over various parts of the trajectory. An ad-vantage of choosing the direction of uniform sensitivity is that a priori knowledge of the system behavior can be used effectively to speed convergence without in-creasing computational effort.

In the proposed method the nominal trajectory must be computed and stored, and the adjoint equations integrated backward. The uniform sensitivity direc-tion is searched to determine the optimum step size corresponding to the greatest diminution of the penalty function. Only the unconstrained optimization problem is considered in this paper, though a general formula-tion of optimum programming requires terminal con-straints on the state and inequality concon-straints on both the state and control variables all along the trajectory. (Constraints of this nature are often con-verted to unconstrained problems by judicious selection of a penalty function. 2,5,6)

The proposed method is compared to the conven-tional steepest-descent algorithm by an example of the interceptor missile control problem. In all cases studied, the proposed method converged immediately whereas

129

conventional steepest-descent was oscillatory and slow.

Neither method, however, can distinguish between local and global minima as with most other iterative algorithms. The usual procedure is to rerun the method with different starting conditions.

PROBLEM FORMULATION The problem is to minimize

fP = fP[x ( T), TJ subject to constraints

dx/dt=f(x, a, t) x(to) =Xo

(1)

(2) (3)

W[x(T), TJ=O (4)

where X=(Xl···Xn)'= state vector, f=(fl···fn)'=

velocity vector, fP and Ware scalar functions, and aCt) is a scalar control variable. The terminal time T, fixed or free, is determined from (4). Constraints at x (T) as well as on a (t) are assumed to be accounted for in (1). Furthermore,

f

is assumed to be differentiable in x and a.

STEEPEST-DESCENT

The method of steepest-descent requires guessing an initial control a(t), and integrating (2) from initial state (3) until W =0 to obtain the first nominal tra-jectory x(t). A small control variation oa(t) causes a corresponding trajectory variation oX (t) that must satisfy (to first order)

(d/dt)ox=F(t)&c+G(t) oa (5) where

[

dh ... /da]

G(t) =

dfn/da

130 Fall Joint Computer Conference, 1970

a( t}

100

T

Figure 1-Control perturbations

and the partial derivatives are evaluated along

x

(t) . Perturbations in <p and W at the terminal time due to oa(t) and ^dT are

d<p= [a<p(T)/ax], dx(T)

+

(a<p/aT) dT and

dW =0= [aW(T)/ax], dx(T) +[aW(T)/aT] dT where

and

a<p/ ax = (a<p/ aXle " "a<p/ axn )'

aw/ax= (aw/aXI" " ·aw/axn )'

dx(T) =ox(T) +[dx(T)/dt] dT These equations combine to give

d<p= {a<p/ax- [(ci>/W) (aW lax) ]}'t=Tox(T) ="-k'ox(T) (6)

where k is a vector constant (defined explicitly) for the nominal x (t), and

ci> = a<p/ aT

+ [(

a<p/ ax)' (dx/ dt) ]t=T

TV

=aw/aT+[(aw/ax)'(dx/dt) ]t=T

In place of (6), the penalty function variation can be expressed as an integral

d<p= fT p'(t)G(t)&x(t) dt

to

where pet) = (PI"" "Pn)' is defined by the differential equation

dp/dt= -F'(t)p

with boundary conditions

peT) =a<p(T)/ax- {(ci>/W)[aW(T)/ax]}

The usual procedure is to choose ^{oa (t)} so as to cause the greatest diminution in <p for a fixed

II

^oa

II

defined by

/I

^oa

/I

= fT oa²(t) dt

to

(7) The gradient direction is oa(t) = -p'(t)G(t) and the optimum step size "A minimizes <p[a- "Ap'G] for the present iteration. Thus, the best control for the next iteration is

anew(t) =a(t) +"Aoa(t) UNIFORM SENSITIVITY

Uniform sensitivity varies from the method just described in the choice of oa. Instead of (7), the norm is taken as

/I

^oa

/I

= fT a(t)oa²(t) dt

to

aCt) >0 (8)

and the uniform sensitivity direction is

oa(t) = -p'(t)G(t)/a(t) (9) It remains to determine a (t) such that the sensitivity of (1) to oa(t) is uniform over the entire trajectory.

Assuming that a nominal control aCt) and trajectory x(t) on [to, T] are available, the time interval [to,·T]

is partitioned into small increments of width fl.t. Any small control perturbation oaT at time r, to~r~ T, of the type shown in Figure 1, with amplitude A (r), duration fl.t, and fixed norm

II

^oaT

II

is required to effect an equal change in <p as a control variation oaT' at r' of the same size (i.e.,

/I

^oaT' II = II ^oaT II implies d<pT=

d<pT' ). The effect of oaT on x (T) is

J

^T+at

ox(T) = if>(T, s)G(s)A(r) ds

T

where if> is the transition matrix of (5). The norm of oaT, from Figure 1 and (8) is

From (6), <p is uniformly sensitive if k' OX (T) is constant, so for d<pT=d<pT', it follows that

[ J

^{T+at ]}

d<PT= k' T if>(T, s) G(s) ds A (r) = constant (11)

Eliminating A (r) from (10) and (11) gives

[ _f

^T+~t

^]2

X k' ^T cf> ( T, s) G (s) ds ⁽¹²⁾ The original steepest-descent gradient direction -p'(t)G(t) is modified by aCt) (see (9)) at r=t to produce uniform sensitivity in the penalty function.

The optimum step size A, as before, minimizes

<I' (a - AP' (t) G (t) / a (t) ). The method always generates directions of descent whenever steepest-descent does.

Although (12) is an integral equation, the mean-value theorem can be invoked for both sides of the equation giving

forr ~~,

r

~ r+ At. It is irrelevant to let At~O since control perturbations vanish in the limiting case. How-ever, the right hand side of (12) is smooth, and in keeping within the framework of practicttlity, it should suffice for small but finite At to substitute r for ~ and

r,

i.e.,

a(r) ~C[k'<p(T, r)G(r)

J2

⁽¹³⁾

where C is a positive constant and r is any time on [to, TJ.

Equation (13) calls for computing a transition matrix, but this is not always necessary for the iterative

VELOCITY

DRAG

GRAVITY

GROUND RANGE

Figure 2-Vehicle nomenclautre and coordinate system

Trajectory Computation 131

I

T

1~

^l--.1t

A (7"')

I

~.

I

·t ₀ 7"' T

Figure 3-Approximation to equation 12 for the interceptor missile example

process to be efficient. It may suffice to know gross behavior of aCt) on [to, TJ. Trends can often be ob-tained by treating F(t) and G(t) as constants or by estimating aCt) a priori as in the following example.

Example: the interceptor missile

This example considers optimal guidance of an inter-ceptor missile to a fixed space point in minimum time with the additional terminal constraint that the inter-cept occur with a specified flight path angle. The dy-namic equations for the nomenclature of Figure 2 are

dr/dt= V cosO dhjdt= V sinO

dV/dt= [F(t)/m(t) J cosa-gsinO

- [p(h) V2CD(a, M) S/2m(t)

J

dO/dt= [F(t)/m(t) J(sin a)/V - (g/V) cosO

+[p(h) V2CL(a, M) Sj2m(t)

VJ

where r=range, h=altitude, V = velocity, O=flight path angle, a = angle of attack (control variable), g = gravity, F=thrust, m=mass, p=air density, M = mach number, and S=reference area. The stopping condition and penalty function are

W=Y2[r(T) -rrJ2=0

<1'= T+Y2R1[h(T) -hrJ2+~R2[O(T) -OrJ2

where (r rh r VO r)' defines the terminal state vector

132 Fall Joint Computer Conference, 1970

,\ ~ STEEPEST - DESCENT 10.. ',"\. II'

~'

'"', '''.' _\ ,,".', " " , , , "., _'.'

^{/ \}

_{\ , '}

^;\

\

\

'-' \

\ Y

UNIFORM SENSITIVITY

\.

....

O~_~I~I~"_~~~~~~.~ ^___ ~.~. ^__ ~.~_~.~.~_~.

1 2 3 4 5 6 1 8 9

ro

NUMBER OF IlERATIONS

en ~ ~'L

V

STEEPEST - DESCENT

Q ~, " A

4C \ ' , . , ' '/~" ^{/ '}

i!:

~-o.2 -\ " ' , / '\ ~\

f

~

^{\ ' . / \ /.}

~ ^{I \ \ '}

~

s \ \,'

; t-O·

^{1 -}

^{\ r}

UNIFORM SENS ITIVITY :e e5 \ ..

m ~

_~ ⁰ ₁ ^• ₂ ^.'",,-_{3 4} ^• ₅ ^• ₆ ^{• •} _{7 8} ^{• •} _{9 10} ^•

4C

NUMBER OF ITERATIONS

1 0 K - - - .

~--.---

. __ ... .r

STEEPEST - DESCENT

" _". --

^"--

^{... ---} _--... ...

\ ....

,

\

".,

5K -

\r

UNIFORM SENSITIVITY ... " .

\

.,

"

o I • ' J , t I • I I I I

1 2 3 4 5 6 7 8 9

ro

NUMBER OF ITERATIONS

Figure 4-Illustration of convergence for the interceptor missile example

[VeT) is unspecified], and Rl = 10-⁴ and R2= 10⁴ are constant weighting factors selected. to equalize each term in cp when equalities in the convergence criteria are satisfied. Iteration terminated when the following conditions of convergence were simultaneously achieved: reT) =rI,

I

^{h(T) -hI}

I

~100 feet,

IO(T) -fh

I

^{~0.02 rad,}

and ~t~O.Ol seconds = difference between successive iterations.

The function for a (t) shown in Figure 3, was

esti-mated from the linearized state equations and used as an approximation to (13). This type of result for aCt) might be expected since a control variation 5a early in flight, when the velocity and atmospheric density are high, has much greater effect on the terminal state (and consequently cp) than the same 5a later in flight at a higher altitude. The curves in Figure 4 illustrate convergence for steepest-descent and uniform sensi-tivity for an intercept at 100,000 feet range, 20,000 feet altitude, and zero flight path angle, and are repre-sentative of convergence behavior for other space point intercepts that were studied .

CONCLUSION

A practical extension to the method of steepest-descent has been developed which seems to enhance conver-gence properties without significantly increasing com-putational effort. Based on the requirement that the penalty function be equally sensitive to a fixed control perturbation along any segment of the trajectory, the direction of uniform sensitivity was determined. The uniform sensitivity direction essentially represents a modified steepest-descent gradient. Therefore, the basic steepest-descent algorithm is unaltered except at the stage where the control perturbation is computed.

The proposed method always generates directions of descent whenever steepest-descent does, and has con-vergence properties superior to the latter-at least when applied to optimal interceptor missile control. In fact, the development evolved out of difficulty in forcing steepest-descent convergence in application to optimal missile guidance. Figure 4 typifies the level of improvements in convergence that were realized for two types of missiles and over a wide range of atmos-pheric intercepts.

REFERENCES

1 H J KELLEY

Gradierd theory of optimal flight paths ARS J Volume 30 pp 947-953 1960 2 H J KELLEY

Method of gradients

Optimization Techniques G Leitmann Ed Academic Press New York New York Chapter 61962

3 A E BRYSON et al

Determination of lift or drag programs that minimize re-entry heating

J Aerospace Sci Volume 29 pp 420-430 1962 4 A E BRYSON W F DENHAM

A steepest-ascent method for solving optimum programming problems

J Appl Mec Volume 29 pp 247-257 1962

5 W F DENHAM A E BRYSON

Optimal programming problems with inequality constraints-II: Solution by st~epest-ascent

AIAAJ Volume 2 pp 25-341964 6 L S LASDON et al

The conjugate gradient method for optimal control problems IEEE Trans on Automatic Control Volume AC-12 pp 132-138 1967

Trajectory Computation 133

7 W E WILLIAMSON W T FOWLER

A segmented weighting scheme for steepest-ascent optimization AIAA J Volume 6 pp 976-9771968

8 R ROSENBAUM

Convergence technique for the steepest-descent method of trajectory optimization

AIAA J Volume 1 pp 1703-1705 1963

Hybrid computer solutions for optimal control of

Im Dokument FALL JOINT (Seite 140-146)