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N A S A TECHNICAL NOTE

NASATN D-6218
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A METHOD FOR REDUCING THE SENSITIVITY OF OPTIMAL NONLINEAR SYSTEMS TO PARAMETER UNCERTAINTY

bY JurrellR.Elliott Lungley ReseurchCenter U?Zd Willidm F. T e u p e University o f Kunsus
N A T I O N AALE R O N A U T I CASNSDP A CAED M I N I S T R A T I O N

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W A S H I N G T ODN.,

C. JUNE 1971

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TECH LIBRARY KAFB. NM

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1. Repwt No.
NASA TN D-6218

1 2. GovernmeRnetcAipcNiecoen.st'sion

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4. Title and Subtitle

0233005
Catalog No.
1 5. Report Date

II A METHOD FOR REDUCINGTHESENSITIVITY O F OPTIMAL
NONLINEAR SYSTEMS TPOARAMETER UNCERTAINTY

June 1971
6. Performing Organization Code

. -
7. Author(s1
J a r r e l l R. ElliottandWilliam

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F. Teague(University

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I 8. Performing Organization Report No.
of Kansas)L-7485 ~ . . . - . - ..

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rO:-WorkUnit No.

9. Performing Organization Name and Address

NASA Langley Research Center

HamptonV, a2. 3365

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2. SponsoringAgencyNameandAddress

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National Aeronautics and Space Administration

Washington, D.C. 20546

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125-17-06-03

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I11. Contract or Grant No.

13. Type of Report and Period Covered

". T e c g i c a l N o t e
L 14. SponsoringAgencyCode . . " " "

5. SupplementaryNotes

6. Abstract

___

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Mathematical relationships are derived and used to establish a procedure for reshaping

the optimal solution so as to reduce the statistical uncertainty in the terminal conditionsof the

system due to system parameter uncertainties of known statistical properties. The procedure

introduces the use of an augmented performance index which contains a scalar measure of the

system sensitivity partial derivatives. A nonlinear multiparameter optimal-rocket-trajectory

problem was solved by using an algorithm based on the method of steepest descent to illustrate

the procedure.

" ~

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17. Key Words(Suggested by Authoris) )

Sensitivity Optimal control Nonlinear systems

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18. Distribution Statement
Unclassified - Unlimited

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Security Classif. (of this page)

21. NO.of Pages

22. Price'

Unclassified

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Unclassified
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'For sale by the National Technical InformatioSnervice, Springfield, Virginia 22151

A METHOD FOR REDUCING THE SENSITIVITY
OF OPTIMAL NONLINEAR SYSTEMS
TO PARAMETER UNCERTAINTY
By Jarrell R. Elliott Langley Research Center
and William F. Teague University of Kansas
SUMMARY
The parameters of a nonlinear dynamical system which is to be controlled optimally a r e not always accurately known. It may therefore be desirable to accept a reduction in the predicted nominal performance of the system in exchange for the ability to better predict the outcome of the system, or plant, operation.
In this paper relationships to predict mathematically the sensitivity of the system to parameter errors are derived and used to establisha procedure for reshaping the optimal solution to reduce the statistical uncertainty in the terminal conditions of the system due to known statistical characteristics of the system parameters. The procedure requires the introduction of an augmented performance index which is a linear combination of the original performance index and a positive scalar measure of the system sensitivity composed of the weighted s u m of the variances of the performance index and terminal constraints. The augmented performance index, the sensitivity partial derivatives, and the original state variables are used inthe formulation of a new, higher dimensioned optimization problem of the same form as the original problem. The new problem introduces certain weighting factors which permit different relative importance to be attached to different types of sensitivity, such as position relative to velocity, and which allow for adjustment of performance degradation and of sensitivity reduction.
The procedure developed was illustrated by solving a nonlinear multiparameter rocket-trajectory problem. An algorithm based on the method of steepest descent was used to solve the problem because of the widespread u s e and proven versatility of this numerical technique. The example solutions serve to show the tradeoffs made possible by changing the weighting factors and to illustrate the radically different solutions one can obtain when sensitivity considerations are included in the problem formulation.

INTRODUCTION
In calculating the open-loop control time history of a physical process or plant, it is common to assume that the plant behavioor r outcome can be predicted with the aid of a mathematical model of the plant with known or deterministic values of the plant parameters. More often than not, however, the plant parameters are stochastic, not deterministic, and serious discrepancies between predicted outcome and actual outcome may occur as a result of plant-parameter variations of one of two types: (1)parameter variations during plant operation, o r (2) errors in the estimates of fixed plant parameters. These effects (that is, the sensitivity of a dynamical process to parameter variations) should be a consideration in the design of any control system.
While many studies related to plant sensitivity have been reported in the literature (see refs. 1 to 14 for a nonexhaustive list), most of them deal with linear problems. It also appears that little effort hasbeen devoted to computational aspects (a notable exception is ref. 14). The present paper deals with the stochastic nature of the parameters in nonlinear problems and shows how a well-known computational algorithm in optimization theory may be applied to obtain numerical results.
The problem considered is a multivariable, multiparameter optimal control problem (with terminal constraints) whose mathematical model consists of a s e t of ordinary firstorder nonlinear differential state equations. The analysis is limited to parameter variations of type (2). The sensitivity of the performance index and the sensitivity of terminal constraints to parameter variations, that is, the partial derivatives of these quantities with respect to the parameters, areused to construct a s c a l a r m e a s u r e of plant sensitivity. This sensitivity measure, which is the expected value of the weighted s u m of the s q u a r e s of the sensitivity partials, is multiplied by a weighting factor and added to the original performance index to form an augmented performance index. Minimization of this augmented performance index, with appropriate weighting factors, results in solutions which are less sensitive to parameter variations.
A simple nonlinear example problem, representative of rocket flight in a uniform gravitational field, is worked out in detail to show the steps required in setting up and solving plant sensitivity problems and to illustratehow reducing the sensitivity of a plant to parameter variations canmodify the open-loop control time history and state-variable time history of the plant. The example also serves the purpose of demonstrating the use of the cumputational algorithm in solving a typical problem.
This research was conducted a t NASA Langley Research Center, with William F. Teague in residence under a grant arrangement with the University of Kansas.
2

SYMBOLS n X q sensitivity coefficient matrix, a" f/aa
partition of matrix A (i=1,2, ...,n)
system parameter vector with components al,.. .,aq
mean (nominal) value of a-
aerodynamic drag coefficient aerodynamic drag "length" of control step control step-size measure
expectation of random variable ()
n X n state coefficient matrix
partition of matrix F (i=1,2, ...,n)
components of -f governing equations of state governing equations of augmented state gravitational acceleration
vector elements of augmented-state differential equations (i=1,2, ...,n)
I$+,IQ$,IQQ constant matrices used in steepest-descent algorithm and defined by equations (A16)

JA

augmepnetrefdormance

index

3

KD

modified drag coefficient

- L

vector set of terminal constraint values (in the example problem, L1 and

L2 are the values for altitude and vertical velocity component, respectively)

1

number of terminal constraints

m

dimension of control vector u-; vehicle mass

m0

initial vehicle mass

m

time rate of change of m a s s

dimension of state vector -x

number of augmented-state variables

covariance matrix

dimension of system parameter vector

S

/ sensitivity matrix, a" x aa; frontal surface area

Si

partition of matrix S (i=1,2, ...,n)

T

vehicle thrust

t

time (independent variable)

tf

final time

t0

initial time

U

horizontalvelocitycomponent, 43

U -

m-dimensional control vector

V

total velocity magnitude

V

verticalvelocitycomponent, X4

W

m X m (symmetric) control weighting matrix

relative weighting factors(i=1,2, ...,Z+l)

4

sensitivity weighting factor n-dimensional state vector fi-dimensional augmented-state vector
components of -ii.
range (with numerical subscript denoting a specific state variable) altitude flight-path angle thrust-attitude control angle n-dimensional adjoint vector associated with payoff
(2 + 1) X n variable matrix integrating factor
2 X n matrix of adjoint variables associated with terminal constraints constant Lagrange multiplier I-dimensional constant Lagrange multiplier vector atmospheric density sensitivity performance index performance index (not including sensitivity) sensitivity measure 2-dimensional terminal constraint vector
components of -\I/ (i=1,2, ...,2 )
transpose of matrix ( ) A dot over a symbol indicates a derivative with respect to time. The symbol 6 denotes a variation in a quantity, and A denotes a first-order perturbation.
5

PROBLEM STATEMENT
Consider the following fixed- time open-loop optimal control problem: Minimize

for the system

with terminal constraints

where

+-L=O
”

@p is a scalar performance index,

-x is an n-dimensional state vector,

-f is an n-dimensional state derivative vector,

-u is an m-dimensional control vector,

(q.> -a is a q-dimensional parameter vector, whose mean and covariance matrix

a r e known

= 2 ; E(6a 6aT} = P), and

+- is an 2-dimensional terminal constraint vector.

(All v e c t o r s a r e column vectors unless superscripted by T which indicates vector o r
matrix transpose, except -.a vector)’ which is a row vector.) It is desired to combine some measure of sensitivity G S to GP such that when the
+ combination is minimized, the performance index qP and *.heterminal constraints -
will be less sensitive to perturbations in the parameters -a. It should be realized that
this reduction in sensitivity will degrade the performance index qP. However, since
a system may have more performance capability than is needed, a system user may be
willing to sacrifice some performance in order to reduce terminal perturbationosr

6

$b. e r r o r s i n @P and - In ordertoprovidedifferentdegrees of performancelossand
sensitivity reduction, a weighting constant is put on the sensitivity measure @s,and the sensitivity performance index to be minimized is then defined as

The sensitivity measure should reflect the fact that the system is stochastic. The sensitivity measure will therefore be taken to be an expected value of the weighted sum
of the squares of theperturbationsin @D andeach Gi due to parameter variations.
(It is necessary to square the perturbation in GP and in each +i (i=1,2,...,Z) to insure
that the measure is positive)., Now since the parameter variations are of type (2) described in the introduction, the perturbations to first o r d e r are

and The squares of these perturbations may be expressed as

(i=1,2, ...,Z)

and
(i=l,2, ...,Z)
The sensitivity measure contains certain weighting factors, to be called relative weighting factors, which are required because one A2 quantity may be considered more, o r less, important than another (for example, one may be more concerned with velocity errors than position e r r o r s in some application although both are used as terminal constraints). Finally, the sensitivity measure is

On a trajectory defined by some open-loop control time history, the partial derivatives inside the expected value are constant. By well-known identities involving the expected value (see ref. 15), equation (1) may be rewritten as

Since E (6. haT) P, thisequationbecomes

where P is the covariance matrix of the random parameters which is assumed to be known along with the mean value, that is
This expression for & (eq. (2)) may be recognized as the weighted sum of the vari-
ances of the perturbations in performance and constraints. Finally the sensitivity performance index for minimizationis

Now returntotheproblem of computing A+P and Aqi. Sincetheseperturbations

explicitly functions of the terminal state, they may be written as

]

are (4)

(i=1,2,...,2)

- The remaining problem is that of determining

ax a-a

(tf).

In reference 1 6 it is shown that

for parameter variations of type (2) and f o r the system differential equations

ax -x = f(x,s(t),a,t) = f(x_,a,t), $t) is the solution of

8

By defining

and
aa equation (5) may be written as
-ddt S = FS + A ; S(t0) = 0
Then @s becomes

A simple, easy-to-work-with form for @s may be determined by reordering and possibly introducing some new variables such that

and

This procedure may, i f new variables are required, introduce a new system of differential

equations. The notation used will remain unchanged, however, and the system

of differ-

ential equations will continue to be called ir = f. " ..., Now partition S, whereSiT (i=1,2, n)denotesthe

rows of S, as follows:

9

Then
Notethat that is,

( I $I= x1 + WISITPSl + w2s2TPS2 + . . . + wl+lsz+lTPSkl) tc 1
SiT = FiTS + AiT; F and A have been partitioned in the same manner

as S,

FIT
""
FaT F = ""

; A=

AIT
""
A2T
""

""
FnT

""
AnT

Observe that this optimization problem involving the sensitivity index has the same form as the original problem involving only the performance index (but with more dimensions) and may be solved by any of several algorithms. Restated, the problem is to minimize

subject to the differential constraints
-2 = f = fG&2,t) ; x(to) = go SI = STF1 + A1 = gl(Z,Si7g,B,t) ; Sl(to) = 0

10

and the terminal conditions
k(Z(tf)) = 4
The augmented-state terms are defined as follows:
4

Then the differential constraints

are

= dx
dt

=

-f(i-i,-u,-Z,t).

Thus

it may be seen that the only

difference between this problem and the original problem is that it is now possible to

control, at the expense the problem is now a n

of n(

performance
l + q)-dimens

index ional

CpP' the sensi state-variable

tivity probl

of em

the trajectory, and rather than an

n-dimensional one. Also several weighting factors (ws and ~ 1 . ~..2.,W, Z + ~ ) havebeen

introduced into the problem. These weighting factors are to be used in applications to

control the loss in the performance index

C#J P and to properly emphasize the importance

of one type of t e r m i n a l e r r o r r e l a t i v e t o a n o t h e r type. The contribution and effect of

these weighting factors will be illustrated in an example problem.

It should be pointed out that solving the differential equations for the sensitivity partials gives all the information needed to perform a f i r s t - o r d e r e r r o r a n a l y s i s on any of the state variables of the original system. In the example problem, a comparison will be made between first-order error analysis results obtained by using the sensitivity partials and first-order error analysis results for which one-sigma errorws ere introduced one a t a time into the equations of motion.

The discussion herein deals with the fixed-time problem. However, fixed time was used for convenience and clarity only and is not a limitation of the formulation.

While almost any of the algorithms for solving optimization problems would be

applicable here, it was decided to use the steepest-descent algorithm as outlined in ref-

erence 16 because of its widespread use and proven versatility. The procedure used in

this algorithm is to linearize about a solution -?*(t) provided by some reasonably chosen

controltimehistory -u*(t)(which ingeneralneitherminimizes Cp norsatisfies

-@ - 4 = 0) andtosolvefor6u(t)(-whichimproves

Cp(tf) and Ic/(tf)). A new solution

provided by -u"(t) + 6-u(t) is obtained and the procedure is repeated successively until

-@ - 4 is sufficiently close to zero and @(tf) can no longer be decreased. This final

solution is said to be optimal although no necessary conditions for optimality have been

11

satisfied. A brief derivation of the necessary relationships with notation common to many steepest-descent programs is provided in appendix A. The algorithm was applied to the following example problem.

A NUMERICAL EXAMPLE

Problem Statement
The example problem is a fixed-time problem in which it is required to determine the thrust-attitude program of a single-stage rocket vehicle starting from rest and going to specified terminal conditions of altitude and vertical velocity which will maximize the final horizontal velocity. The idealizing assumptions made a r e the following:
(1)A point-mass vehicle (2) A flat, nonrotating earth (3) A constant-gravity field, g = 9.8 m/sec2 (32.2 ft/sec2) (4) Constant thrust and mass- loss rate (5) A nonlifting body in a nonvarying atmosphere with a constant drag parameter
KD = -1PC$, where S is the frontal surface area. 2
The coordinate system and pertinent geometric relations and terms are shown in figure 1. The differential equations of motion needed in the algorithm setup a r e

7 ” du = -(T cos 8 - K p V ) = 21 = f l
md t

dY
”

- -

v

=

x

2

=

f

2

* J dt = -1(T dt

sin e - KDUV) - g = k3 = f 3

wherem is thevehiclemassandwhere

and
m(t) = mo + mt

An equationfordx/dt is notincludedintheequations of motionbecuasexdoes

not enter into the problem. However the equation

k = u was integrated separately to

obtainrange.Theequationfordm/dt

is analyticallyintegrable,sincedm/dt

is a

constant, and is therefore not included in the set of differential equations.

12

The parameters of the problem, which will be considered fixed constants whose precise values are unknown, are thrust level, mass-loss rate, initial vehicle mass, and modified drag coefficient. The parameter vector is defined as

These parameters are assumed to be statistically independent and to havea normal distribution function with mean and one-sigma values as shown in table I.

TABLE I.- SYSTEM PARAMETER VALUES

Parameter
ThrustT, , kN (Ibf) . . . . . . . . . . . . . . . . . . . . Mass-lossrate, m, kg/sec(slugs/sec) . . . . . . . . . . Initialmass, m,, kg (slugs) . . . . . . . . . . . . . . . . Modified dragcoefficient, KD, kg/m(slugs/ft) . . . . . .

Mean value
17.8(4000) - 9(.-10 . 6 2 ) 1433.6(98.259)
0.048(0.001), 0.479 (0.01)

__ One-sigma value, fraction of mean valu
0.0067
0.0167
0.010
0.0167

These values are considered representative of the current state of knowledge i n

( solid rocketry. The mean values used were chosen to conform to

a flight of 100 seconds

for a rocket with specific impulse of 200 seconds, an average acceleration

x l J l o o dt of 2g, and a ratio of initial to final mass m(to)/m(tf) equal to e, the

100 t=O m(t>

\

base of the natural logarithm.

Theboundaryconditions a t t = 0 andt = 100 seconds are shown in table II.

TABLE 11.- BOUNDARY CONDITIONS

Variable
u,m/sec(ft/sec) . . . . . . . . . . y, m (ft) . . . . . . . . . . . . . . . v, m/sec(ft/sec) . . . . . . . . . .

Initial conditions (t = 0)

Terminal conditions (t = 100 seconds)
i Maximum
15 240 (50 000)

In the notation previously introduced 13

I

@1 -- y(tf) - L1 = 0 ; L1 = 15240m(50000ft)

and the vector control variable -u in the general formulation is the thrust-attitude angle 8. Thiscompletestheproblemstatement (without trajectory-sensitivity considerations).

Sensitivity Relations The sensitivity-matrix differential equation is

@=FS+A dt

where

iS =

au
"
am

- avT

and fl 0
if3 9) +
14

T dSiT
-=

Fi S + A iT

dt

or in column vector form

d-dSt=iS TF i + + The sensitivity measure is

qs = E {wl AU2(Q) + w2 AY2(tf) + w3Av2 (Qj)

which may be written as

(i=1,2,3)

(i=1,2,3)

(16)

where
1 0 0

-

0

0

0

R 2 0

0

0

0

Problem Statement Including Sensitivity The augmented-state vector is

15

with the initial conditions
.,q -';T(to) = Eta,. .
The problem restatement with sensitivity considerationsis to minimize

subject to

and
(4) Ql = fr2 - L1 = 0 ; L1 = 15 240 m (50000 ft)

q2= ii.3(tf) - L~ = o ; L~ = o m/sec (0 ft/sec)

Numerical Results

As previously mentioned, the steepest-descent algorithm of reference 17 was imple-

mented to obtain numerical results. These results were computed with the use of a

fourth-order integration subroutine with a fixed step size of 0.5 second. For this example

problem each iteration in the solution required integrationof 15 augmented state differen-

tial equations plus the forward integration of the range equation and backwards integration

of 45 adjoint variable differential equations. Each iteration required about

22 seconds on

the Control Data 6600 computer system used with no special effort having been made to

keep computer run time down. Several check solutions were computed with the use of a

0.125-second fixed step size. These check solutions always agreed to at least five, and

usually to at least seven, significant figures with those computed with the use of the

0.5-second step size.

Numerical results for a variety of cases were obtainedby changing the values of

KD (themodified dragcoefficient), ws (thesensitivityweightingfactor),and

w1,w2,w3

(the s e t of relative weighting factors). The different combinations for which results were

obtained a r e shown in table III.

16

TABLE ID.- CATALOG OF CASES

I
I

Relative-weighting-factor sets Set (1): w1 = w3 = I; w2 = 10-2
Set (2): w 1 = w3 = 1; w2 = 10-4

I{II KD 0 :.ool

I 1 I

WS
0, 0.1, 0.396, 1.0, 10.0 0, 0.1, 0.3, 1.0,10.0

0, 0.1,1.0,10.0

The relative-weighting-factor sets were chosen on the basis of the relative importance attached to velocity (wl applies to horizontal velocity and w2 applies to vertical
. velocity) and position (w3 applies to altitude errors) Set (1) gives equal weight to a
0.305-m/sec (1-ft/sec) velocity error and a 3.05-meter (10-foot) altitude error, while set (2) gives equal weight to a 0.305-m/sec (l-ft/sec) velocity error and a 30.5-meter (100-foot) altitude e r r o r .
The modified drag coefficient KD was set at zero, simulating vacuum flight, and a t 0.001, a representative value for small rockets in the earth's atmosphere. Setting KD at zero reduced the number of parameters to three. It also made possible, for ws = 0, an analytical calculus-of -variations solution for the optimal thrust-attitude time history; namely, the well known linear tangency law (ref. 18).
tan O(t) + = (Y pt

where CY and p a r e constantsdetermined by theboundaryconditions of theproblem.
This calculus-of -variations solution was used to validate the steepest -descent-algorithm programing and solution. The comparison of results showed negligible differences.
Also, for KD = 0, it can be shown that

and
- av = -1 V(tf) +tfg
aT T
Therefore, these partials will not change with ws f o r KD = 0. As an independentcheck on the sensitivity partials, perturbations in u, v, and y
a t tf due to plus and minus one-sigma errors in each parameter, taken one at a time,
werecomputed. For example,theone-sigma (lo) perturbationinuduetoparameter errors was computed by using the relation
17

wherue (9m) eanuas t +loT

tf on a trajectorywith T increasedbyitosne-sigma

value,whereut ( f > -I*& meansuat

tfon a trajectorywith

111

sigmavalue,and s o forth.Similarcomputationsweremadefor

& decreased by itsone-
Av!? andAylo.O(2n) e-

sigma perturbation values computed by this method were compared with one-sigma values

computed by using the sensitivity partial derivatives. For example, the one-sigma (lo)

perturbation in u was computed as follows by using the sensitivity partials:

where6Tlo,6mloand

so forth are one-sigmaperturbationsintheparameters.Simi-

lar computationsweremadefor Avf? andAylo(.S11uperscript

(1) refers to 10.pertur-

bations obtained by using sensitivity partials, and superscript (2) indicates that the method

of computation was to obtain an average perturbation value by assuming first plus and then

minus la variations in each parameter.

Discussion of Numerical Results
Numerical results for the converged trajectories are summarized in figures 2 to 11, which give control time histories and trajectory plots, and in tables that give sensitivity partials,one-sigmaperturbationvalues, and @ s and @p.
Figure 2 shows the control time history used initially and the converged control time history after 10 iterations for the conditions indicated (KD = 0; ws = 0) to illustrate how a reasonably chosen control time history is modified by the algorithm to obtain an "optimal" solution. Optimal here means that a gradient function has been reduced several o r d e r s of magnitude o r to some reasonable value. A plot of tan O(t) as a function of time was made for this converged trajectory, and the intercept and slope of a straight line

1%

fairing to that plot are compared in table IV with the values of the constants CY and p
required in the calculus-of -variations (C.O.V.) solution.

TABLE 1V.- LINEAR-TANGENCY-LAW CONSTANTS

Constants
..
CY.....

I I Faired values

C.O.V. values

I

--

3.49

p .....
"

-0.0401 ._

The agreement is considered to be good, in view of the crude fairing method and
other factors relatingto how theC.0.V.values of a, and p wereobtained.Alsothe
trajectories showed good agreement.

Table V(a) and figures 3, 4, and 5 summarize converged numerical results for

KD = 0 and s e t (1) relative weighting factors for several values of the sensitivity

weighting factors. The sensitivity partials of table V(a) show consistent trends, with

altitude and horizontal-velocity sensitivity partials decrea.sing in magnitude while

vertical-velocity sensitivity partials are increasing in magnitude with increases in ws.

These trends are also reflected in the root-sum-square perturbations; one-sigma (la)

perturbation values of altitude and horizontal velocity decrease while thoseof vertical

velocity increase. The percentage of the original value (value with ws = 0) of Aula,

Aylo,Avla, C$s, and C$p is plottedinfigure 3 againstsensitivityweightingfactor.

For values of ws largerthanabout 0 . 3 , littlechangetakesplaceinAvla,

AylU, and

( qS, but theperformance

C$ p)

continuestodegradealongwith

a decreasein Aula.The

control time histories for these cases, shown in figure 4, exhibit an interesting charac-

teristic. As ws becomes larger, the control tends toward a bang-bangtype of control

where the thrust is either directed straight up (vertical) o r straight down. While it

appears that the steepest-descent program indicates the existence of a bang-bang optimal

control law, attempts to predict this behavior analytically have been unsuccessful. Alti-

tude is plotted against range in figure5 where it may be observed that the trajectory

becomessteeper as ws increases.

Table V(b) and figures 6 to 9 summarize the results for KD = 0 and s e t (2) relative weighting factors. Set (2) puts less emphasis on the altitude sensitivity partials than s e t (1). Thus altitude sensitivity partials increase for set (2) rather than decrease as they did for set (l), and both horizontal and vertical-velocity partials decrease. Figure 6 clearly illustrates these results in the plots of Ayla,Avla, and Aula. By comparing this figure with figure 3, it may be seen that Ayla and Avlo have switched positions on the plots. Also it appears that $s begins to level off and remain essentially con-
stant at ws = 10.0 in figure 6 whereas it leveled off and became essentially constant at

19

TABLE V.- SENSITIVITY PARTIALS AND RELATED INFORMATION

(a) KD = 0; w1 = w3 = 1, w2 = 10-2(Set (1))

t. ~~ " "

Sensitivity partials

and related terms

(*I

.

au/aT

-

-

_.

Values for sensitivity weighting factor ws, of -

-

"

. -

0

0.1

0.396

1.0

10.0

.

-

1.106

0.973

0.372

0.163

0.02c

au/alin

-6 322

-5 970

-2 044

-838

-101

au/amo

-84.98

-77.34

-28.07

-11.93

-1.44

(1) Au l a
(2)

110.1

101.3

36.1

15.2

1.8

110.2

101.4

36.2

15.3

1.8

ay /aT ay/alin
ay/amo
g AY

52.75 -96 140 -2 755
3 209

52.75 -82 950 -2 672
3 100

52.75 -7 3 640 -2 613
3 025

52.75 -72 790
-2 608
3 019

52.75 -72 610 -2 606
3 017

AY

3 210

3 100

3 025

3 019

3 017

av/aT

0.805

0.805

0.805

0.805

0.805

%/a&

-1 419

-1 646

-1 805

-1 820

-1 823

av/amo

-41.74

-43.17

-44.18

-44.27

-44.29

(1) Av la
(2) Av 10.

48.57 48.57

50.50 50.51

51.91 51.93

52.04 52.12

52.07 52.18

@P

4 423

3 892

1489

651

79

@S

117 500

108 900

95 500

94 100

93 700

*

"

" ~

- -

Superscripts (1)and (2) refer to one-sigma (lo) perturbations obtained by using the

sensitivity-partial-derivative method and the parameter method, respectively.

20

TABLE V.- SENSITIVITY PARTIALS AND RELATED INFORMATION - Continued

.- ~
Values for sensitivity weighting factor, ws, of -

~

0

0.1

0.3

1 .o

10.o

1.106

1.084

0.723

0.448

0.104

-6 322 - 5 869

-2267

-103

174

-84.98

-81.22

-43.76

-18.88

-3.40

110.1

104.4

52.6

22.1

4.5

110.2

104.4

52.6

22.1

4.5

52.75

52.75

52.75

52.75

52.75

-96 100 -102 100 -108 500 -112100 - 126 400

-2755

-2793

-2 833

-2 856

-2946

3 209

3 260

3 315

3 347

3 474

3 210

3 260

3 316

3 347

3 475

0.805 -1 419
-41.74

0.805 -1 316
-41.09

0.805 - 1 206
-40.40

0.805 -1 146
-40.02

0.805 -899 -38.46

48.57

47.71

46.82

46.35

44.45

48.57

47.72

46.82

46.33

44.40

4 423

4 336

2 891

1791

4 17

15 510

14 240 6 060

3 760

3 200

*Superscripts (1) and (2)refer to one-sigma (la)perturbations obtained by using

:he sensitivity-partial-derivative method and the parameter method, respectively.

21

ws = 1.0 in figure 3. Part of the reason is the change in the magnitude of @s due t o

changing only the relative valuesof the relative weighting factors without regard to their

magnitude. This is clearly illustrated by the near-order-of-magnitude difference in G S

f o r set (1) and set (2)at ws = 0. (Compare tables V(a) and V(b).)The trajectories for

the two sets are the same; @s changes with the change from set (1) t o set (2). Reducing

the values cases have

of the

@s in same

table
GS

V(a)by at ws

dividing by the constant = 0, and incorporating this

'1175 5'01°0 constant

= 7.57 into the

so that both weighting

factor by multiplying each wsof table V(a) by 7.57, allows a more reasonable compari-

son. This comparison is shown in figure 7 where the ordinate is called ws (adjusted).

This figure shows the differences which come about due to different relative-weighting-

factor sets.

The control time histories for relative-weighting-factor s e t (2) are shown in figu r e 8. Again there is a tendency toward a bang-bangtype of control, as ws increases, which may be noted by observing that the angle difference between the nearly constant attitude portion of the control time history at the beginning and near the end of flight is about 180° f o r both ws = 1.0and ws = 10.0.Figure 9 shows that in comparison with the set (2) trajectory for ws = 0, the other set (2) trajectories are generally less steep. The opposite result is shown for set (1)in figure 5; in comparison with the set (1) trajectory for ws = 0, the other set (1)trajectories are more steep. These results indicate the importance of the relative weighting factors in shaping the trajectories.

Data for a nonzero value of KD,KD = 0.001,and relative-weighting-factor s e t (2)

a r e shown in table V(c) and figures 10 and 11. Data f o r ws = 0.1 a r e shown in table V(c)

but not in figures 10 and 11 because the control time histories and trajectories essen-

tially coincide with those for ws = 0. In table V(c) it may be observed that while consis-

tent data trends are shown in the first three ws columns, the data in the column for

ws = 10.0 are not consistent. The consistent data trends in the first three columns are

very similar to those in table V(b). The inconsistency of the last column may be explained

by examination of figures 10 and 11where the radically different character of the control

timehistory and trajectory for ws = 10.0maybeseen.Thisresultfor

ws = 10.0was

so unusual that it was believed necessary to verify the answer. Accordingly, verification

was obtained by iterating to the same result (essentially) from an additionaltwo different

nominal trajectories. These results therefore appear correct. It

is believed that the

data inconsistency results from the different character of the trajectory - that is, the

bending back of the trajectory as seen in figure 11. With the exception of this ws = 10.0

case, the trajectories are l e s s s t e e p with increasing ws just as they were in figure 9

for a similar case with zero drag.

For all of the cases discussed herein, good agreement was observed (see tables V(a), V(b), and V(c)) between the one-sigma perturbations computed by using sensitivity partials and those computed by using the parameter-perturbation method.

22

TABLE V.- SENSITIVITY PARTIALS AND RELATED INFORMATION - Concluded

(c) KD = 0.001; w1 = w 3 = 1, w2 =

(Set (2))

Sensitivity partials and related terms
(*I
&/aT
au/a&
au/amo

Values for sensitivity weighting factor, ws, of -

1

~

0

0.1

1.0

10.0

~

25.35

25.24

24.35

12.07

-756.9

-747.1

-668.1

-002.8

-8.11

-7.99

-7.04

-1.27

au/aK,,

-687000

-6 88 900 -697 500 -359 900

(1)
(2 1
Au la

17.37 17.37

17.28 17.28

16.52 16.52

6.61 6.61

ay/a T

31.94

31.96

32.11

31.81

ay/al;l ay/am,

-44 640 -1 438

-44 840
- 1 440

-46 037 -1 456

-44 800 -1 428

ay/aKD

14 109000

-14 020000 -14 020200

-14 03050

1730 1729

1732 1732

1751 1751

1712 1712

16.18 -7.8 -3.80 -278 700

16.13 4.6 -3.68 -280 400

15.81 101.8
-2.81 -293 500

17.01 34.5 -3.52 -313 700

7.358

7.309

7.102

7.766

7.357

7.311

7.100

7.766

@P

1 824

1824

1811

91 1

@S

655

652

630

40 3

*Superscripts (1) and (2) refer to one-sigma (lo)perturbations obtained by using

the sensitivity-partial-derivativemethod and the parameter method, respectively.

23
I

CONCLUDING REMARKS It has been shown how a sensitivity measure, composed of the weighted sum of the variances of the performance index and terminal constraints, may be added to the performance index of a stochastic optimal control problem to achieve a reduction in performance and constraint sensitivity due to parameter perturbations. It was necessary to assume that the parameters of the system remained fixed during system operation and that the stochastic nature of the problem came about because of inexact knowledge of these fixed values. It was shown how this technique increased the dimensionability of the optimization problem and introduced weighting factors o r constants for use as design parameters. These weighting factors permit different relative importance to be attached to different types of sensitivity, such as position relative to velocity, and allow for adjustmentof performance degradation and of sensitivity reduction. The feasibility of solving nonlinear multiparameter problems by using this technique was illustrated by solving an example rocket-trajectory problem through application of a steepest-descent algorithm. The example solutions also served to illustrate the tradeoffs made possible by changes in the weighting factors. Langley Research Center, National Aeronautics and Space Administration,
Hampton, Va., April 14, 1971.
24

APPENDIX A STEEPEST-DESCENT DERIVATION
The problem is to find the control time history u(t), to 5 t 6 tf, which minimizes
subject to the differential constraints
zo -2 = f(z,z,t) ; x(to) = (given)
and the terminal constraints
Thesolution is obtainediteratively.Choose a reasonabletimehistoryu*(t-) and obtain x* (t), a solution to equation (A2). This solution, in general, is such that neither @ is minimumnor Q = L.
"
Linearizing about this solution gives
or
Now, forconvenience,let A(t)bethesolution of
25
Il I I11l1l1I

APPENDIXA - Continued
with
where
];[ N
-Q =
Then equation (A5) becomes
or, letting b-+ = - a82x 6-x(tf>,

Partition A (t) as-follows :

Then, since -x (to) is given and 6-x (to) = 0,

and

-

Now for some measure of allowable control perturbation

ltf (dP)2 =

6gTW

6-u dt

;

W

=

T W

> 0

t0

(A13

minimize 64 and choose 6-+ such that " +( x(tf)) = L will be satisfied (or more nearly

so). Form an augmented function to be minimized

26

APPENDIX A - Concluded

where p and -v a r e constantLagrangemultipliers.

For JA tobeanextremum,its

variation with respect to 6-u must be zero - that is,

This requirement implies that

for all t. Solving for 6-u,

In o r d e r t o solvefor p and -v thisexpressionfor
tions (A10) and (All) to obtain

6-u is substitutedintoequa-

where the sign on the radical is minus because @ is beingminimized and where
\

Now let
u-(t) = u-*(t) + 6-u(t)
be the new control time history used to obtain a solution to equation (A2) and repeat the
same procedure until -I) - -L and the gradient of the payoff-constraint surface
a r e sufficiently close to zero. 27

REFERENCES
1. Dorato, P.: On Sensitivity in Optimal Control Systems. IEEE Trans. Automat. Contr., vol. AC-8, no. 3, July 1963, pp. 256-257.
2. Pagurek, Bernard: Sensitivity of the Performance of Optimal Control Systems to Plant Parameter Variations. IEEE Trans. Automat. Contr., vol. AC-10,no. 2, Apr. 1965, pp. 178-180.
3. Witsenhausen, H. S.: On the Sensitivity of Optimal Control Systems. IEEE Trans. Automat. Contr., vol. AC-10, no. 4, Oct. 1965, pp. 495-496.
4. Sinha, N. K.; and Atluri, Satya Ratnam: Sensitivity of Optimal Control Systems. Proceedings Fourth Annual Allerton Conference on Circuit and System Theory, William R. Perkins and J. P. Cruz, Jr., eds., Univ. of Elinois and IEEE, 1966, pp. 508 -5 16.
5. Breakwell, John V.; Speyer, Jason L.; and Bryson, Arthur E.: Optimization and Control of Nonlinear Systems Using the Second Variation. J. SOC. Ind. Appl. Math. Contr., ser. A, vol. 1, no. 2, 1963, pp. 193-223.
6. Kreindler, 'Eliezer: Synthesis of Flight Control Systems Subject to Vehicle Parameter Variations.AFFDL-TR-66-209, U.S. AirForce,Apr. 1967.(Availablefrom DDC as AD 653 600.)
7. Higginbotham, Ronald E.: Design of Sensitivity-Constrained Optimal Linear StateRegulator Control Systems. Conference Record of Second Asilomar Conference on Circuits & Systems, 68 C 64-ASIL, IEEE, 1968, pp. 330-335.
8. H a a s , V. B.; and Steinberg, A.M.: Minimum Sensitivity Optimal Control f o r Nonlinear Systems.TR-EE67-10 (NGR 15-005-021),SchoolElec. Eng., Purdue Univ., Aug. 1967.(Available as NASA CR-88863.)
9. Rohrer, R. A.; and Sobral, M., Jr.: Sensitivity Considerations in Optimal System Design. IEEE Trans. Automat. Contr. vol. AC-10,no. 1, Jan. 1965,pp.43-48.
10. Chan, S. Y.; and Chuang, K.: A Study of Effects on Optimal Control Systems Due to Variations in Plant Parameters. Int. J. Contr., First ser., vol. 7, no. 3, Mar. 1968, pp. 281-298.
11. Patrick, L. Benjamin; and D'Angelo, Henry: Optimal Control of Plants With Random Slowly-Varying Parameters. Proceedings Fifth Annual Allerton Conference on Circuit and System Theory, J . B. Cruz, Jr., and T. N. Trick, eds., Univ. of Illinois and IEEE, 1967, pp. 711-720.
28

12. Holtzman, J. M; and Horing, S.: The Sensitivity of Terminal Conditions of Optimal Control Systems to Parameter Variations. IEEE Trans. Automat. Contr., vol. AC-10, no. 4, Oct. 1965, pp. 420-426.
13. Kokotovic, P. V.; and Rutman, R. S.: Sensitivity of Automatic Control Systems (Survey). Automat. Remote Contr., vol. 26,no. 4, Apr.1965, pp.727-749.
14. Rosenbaum, Richard C.; and Willwerth, Robert E.: Launch Vehicle Error Sensitivity Study. NASA CR-1512,1970.
15. Lindgren, B. W.; and McElrath, G.W.: Introduction to Probability and Statistics. Second ed., Macmillan Co., 1966.
16. Coddington, E a r l A.; and Levinson, Norman: Theory of Ordinary Differential Equations.McGraw-Hill BookCO., Inc.,1955.
17. Bryson, A. E.; and Denham, W. F.: A Steepest-Ascent Method for Solving Optimum Programming Problems. Trans. ASME, s e r . E: J. Appl.Mech., vol. 29, no. 2, June 1962,pp.247-257.
18- BrYson, Arthur E., Jr.; and Ho, Yu-Chi:Applied Optimal Control. Blaisdell Pub. CO., c.1969.
29

T
X
Figure 1.- Coordinate system and geometric relations.
30

I

-1- I I . _ L - .. .

I

0 20 40 60 80

\
I 100

Time from lift-off, t, sec

Figure 2.- A comparison of the initial guessed thrust-attitude control time history with the steepest-descent generated optimal thrust-attitude control time history. KD = 0; ws = 0.

31

w
N
I20

IO0

80

60

40
c I
20I

.0o - I

.I

I

IO

WS

Figure 3.- One-sigma (lo) e r r o r components and sensitivity and performance indices

plotted against sensitivity weighting factor.

KD = 0; w1 = w3 = 1, w2 = 10-2.

\

-c
F -40

-7 0

-80
-9oL

L

1 __I"L_ 1- 1~ .I"1 - 1

0 I O 20 30 40 50 60 70 80 90 100

Time from lift-off, t, sec

Figure 4.- Comparison of optimal control time histories for different sensitivity
weighting factors KD = 0; w1 = w3 = 1, w2 =

33

I< lo3
ws = 10.0

1 25 50 75

i
I50x103

Range, x, ft

J

I

1

I

I

1

0

IO

20

30

40

50

Range, x, km

Figure 5.- Comparison of optimal trajectories for different sensitivity
. weighting factors. KD = 0; w1 = w3 = 1, w2 = 10-2

34

120-
0
“m 80
3
4-
0
-0)
3
> 60
Y-
O
t ca 40
a
20
Figure 6. - One-sigma (lo) error components and sensitivity and performance indices
plotted against sensitivity weighting factor. KD = 0; w1 = w3 = 1, w2 =

\\

\

\\

\
tP

0 >

\ \*

\ - - - - - Set( I): WI = w 3 = I, w2=

\

Set (2):WI = w 3 = 1 , w = IO"'

01

I

I ,I I I I I1

I I I I1111

I

I I 1 IIll

-

I

1"-~

II

.01

.I

I

10

100

ws (adjusted)

Figure 7.- Set (1)and set (2) sensitivity and performance indices plotted against a sensitivity weighting factor adjusted for equal sensitivity at w2 =O. KD=O.

\
-

+OL -40-
e2v, -50-60-

I
/

-I 10 -120 -I 30
-140

t IllIlllllI
102030405060708090 100
Time from Iift-off,t, sec

Figure 8.- Comparison of optimal control time histories for different sensitivity weighting factors. KD = 0; w1 = w3 = 1, w2 =

37

40 -

i

1

IO0

I25

l50x IO3

Range, x, ft

0

'0

I
20

I

I"

.J

~

30

40

50

Range, x, km

38

I 40r

- 1801

-225

\ws = 10.0

I"lIII1lIII~

0 20 40 60 80 100

Time from lift-off, t, sec

Figure 10.- Comparison of optimal control time histories for different
. sensitivity weighting factors. KD = 0.001; w1 = w3 = 1, w2 = 10-4

i
0

16

5 0 X lo3

rws = ‘Oa0

/ I 40-

L ws = o

c
w-r: 30-
c Q)
U
3
-E 20-
a
4

0

-20 -10

0

IO

20

30

40

50

60x IO3

Range, x, ft

I

I

I

I

1

I

I

-4

0

4

8

12

16

20

Range, x, km

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