A method for identifying non-linear systems with applications to vehicle dynamics and chemical kinetics

RCH1EF

I.

INTRODUCTT

An algorit

_{was presented in (1]}

for determining

_{unknown parameters}

_{in an}

otherwise known dynamical system from

measurements of the time

_{history of one}

state variable.

_{The system}

_considered

could be nonlinear and of any order

_and

the unknown

_{parameters could appear}

non-linearly.

_{The method determines}

_the

parameters by seeking

_{the least}

_square

error between the calculated and

Observed time history.

_{A Somewhat}

different technique

_{that bears a}

_strong

rese1lance to this has

_{recently been}

presented by Chapman

_{and Kirk (2].}

A computer

_{program based on the}

algorjt

_{of [1] was}

_{developed in which}

only the dynamical equatio

(and the

equations of

_differential

_corrections)

need to be

_supol.je

_{In applying the}

program to identify systems with a large

ntmther of unkø

_paraxfleters

itwas-found that

_{sometimes the}

parameters were

not determined uniquely and

_sometimes

convergence was imoossiblo

_{to achieve.}

It was felt

_that

_ifficultjes

could be eliminated by basing the

identi-fication on

_{measurements of more than}

one state variable, and so the algoritJ

and computer program have been

_generajjz.

ed to include

_{any number of}

measured

records.

_{Subsequent application}

_{of this}

program to several

_{Systems has}

sub-stantiated the capability of the

genera'-ized algorjt

_{to identify}

_{Inany unknown}

parameters in large

_Systems.

Further-more, because the

_{method is of the}

hoOtj

_t

_{e, memory}

are

relat.ve y small so that the

identi-fication can be

_{accompijse on a}

computer of moderate

_size.

The generalized

_{algorithm is}

presented below together

_{with examples}

work was

_{supported by the Naval}

Ship Research

_{and Development}

_Center,

ava1 Air Systems

_{Command and Oceazijcs.}

internal research.

1

?IL4U LZt5'UC_AL KINETICS*

Theodore R. Goodman

_{and Theodore P.}

_Sargent

OCEANIcs, Inc., Plainview, New York

ABSTRACT

.-r

Lab. v.

ScheepsbouwkuncJe

Technische Hogesthool

De1IL

A METHOD FOR

IDENTIFYING NONLINEAP. _{SYSTEMS WITH APPLICATIONS} TO VEHIcr.1

A mathematical

_{method is presented for determjnjn}

unknown parameters in

an

otherwise known dynamical system from

_{measurements of the time}

history of one or

more

of the state

_variables.

_{The system considered}

can be nonlinear and of any order and

the unknown

_{parameters can appear}

nonlinearly.

_{A computer}

_{program was developed in}

which only the dynamical equations

_{and the equations}

_{of differential}

corrections need

to be supplied in order to carry

_{out the identification}

The results of applying

_the

program to five distinct

_{syStems are presented.}

The fiVe systems

_are:

_{a surface}

ship, a hydrofoil

craft, a V/STQi. craft, a conventional airplane

_{and a chemical}

kinetic process.

of its application to five different

Systems.

_{The examples presented}

demonstrate the

_{versatility of both the}

algorjt

_{and the computer program in a}

variety of different situations.

II.

DERIV7 _{OF THE BASIc ALGORITHM}

The dynaxnicaj.

_{equations describing}

the system

_{are assumed to be given in the}

form

- V ? k.

= g

(yj,

c

ic);

y(0)

C(1)

where the dot denotes

_{differentiation}

with respect

_{to tine t, a denotes}

_the

unknown parameter

_{vector and c denotes}

the initial value

of the Solution

_vector

y and, may or may not

_{be totally known.}

SUppoSe measuremants

_{b1mg b2,}

have been taken

_{of the state}

_var.ablas

y1,

_{21 .. at times t}

_{It is required}

to find an initial

_{vecor c together with}

a Parameter

_{torawiicl minimize}

_the

s

_{of the squares of the deviations:}

+ r

best agreem

_{with the}

_measuremen

_in

a least square

_sense.

Solutjo

_{of (1) is sought}

_{which is in}

where the weighting

_{factors w are chosen}

the same order of magnitude.

to make each sum nondimensional and

_{Thus, the}

of

m=l

V

[Y2(tm)_b2ij2

+ ...

(2)

The parameter

_{vector, a, will be}

Suppressed in (1) by

_{considering its}

components to be additional

_state

variables hto

the equation

1 _V

The number n is

_{thereby increased to}

include the

addjtic)

_{state variables}

j

determined in the following

way; The

'" ,initial vector is estimated and (1)

_{Iii"ntegrated.}

is

The esIiiid initial vector

is denoted by

and the resulting

solution of (1) by3. The deviation

can

then

_{e calculated 'hd its value denoted}

by c*

_{Suppose the initial vector to be}

changed by an increment óc; this would

cause the solution vector to be changed

by an increment 6y and the deviation

_by

an increment 6c.

_{From (2) it is seen}

that

S

S

= v1

_{[Yl(tm)_bn]6Yi(tm)}

_{+ ...}

₍₄₎

The equations which the incremental

solution vector satisfies are called the

equations of differential corrections

and are obtained by expanding

_{(1) in a}

Taylor series and retaining only linear

terms:

n

*

d;(t)

6Y

(5)

j=1.

The asterisk means that the coefficients

are calculated using solution

y.

Eguatiàn (5) is now integrated

_{thejth tine the integration}

n times;

is performed.

the initial conditions are

_{that óy.(0)=l}

and all the other dy. (0 's vanjsh.

This

s ecia]. solution is enoted by

and

the genera

_{so ution can then}

r

tten,

by superposition:

n

=

j

(Y

ii

Ct))

(6)

This incremental solution

_{vector is used}

to express 6y1, 5y2,

_{... in terms of óc,}

and upon substituting into (4) and

inter-changing the order of summation

_the

variation of the deviation

becomes

mation on j from 1 to

n.

The variation

where the repeated suffix ianlies

sum-of the deviation has thus been

_expressed

directly in terrs of the variation

of

I.'.

=

6c{w1

_{[Y1(tm)_bl]45Yl(tm)}

+

...

}

(7)

each of the initial conditions.

_In

order for s to be minimum 6c

_{must vanish}

for an arbitrary variation in

_{the initial}

conditions.

This meais that if

C

and the

_{xt'nded c vectorjnr!1,s the}

_U.

unknown parameter vector

₍

_additi

the state variable initial conditions.

The parameters of. the system will be

is defined to be

=

1

:

[Yltm_bjjYijtm+

then the error will be minimized with

respect to the cIs whenever

0, j = 1,

..

In general, using the estimated

vector c

and the resulting solution

vector y* the values of U. will not

vanish.

_{Denote the valueof U. as}

calculated in this way by U.

The

objective is to make the U.3vanish by

an iteration procedure.

_{Cnsider the}

increment in U1 caused by the increment

my.. From () there is obtained

6U v1

6y1(t)

m=l

6Yij(tmH

:"

(10)

In order for each U. and hence 6c to

vanish the conditjo2

Uj =-.0

must be imposed.

_{Upon substituting (6)}

into (10) and interchanging the order of

summations there is finally obtained

=

{wi

_{y13 (tm) 45Y1i(t)+.}

(12)

Equations (8) with y = y, together

with

Equations (11) and (12), constitute

n

simultaneous linear algebraic equations

for the n unknowns

_{Upon adding the}

incremental values to the estimated

values of c, improved estimates of the

c4 are obtaj.ned, and the procedure is

then repeated until

_{convergence is}

achieved.

A modification of the above algorithm

which at tines is found to be

useful is

to introduce some or all of the bs

_into

the right hand sides of Equations

_{(1) and}

(6) in place of the respective

_ye's.

III.

GUIDELINES ON USING THE PROGRJM

The algorithm presented in Section

II is in the form of an iteration

procedure, and as with most iteration

procedures convergence is not assured.

The principle

_{purpose of this section}

will be to show how convergence can be

achieved even with a

_{poor initial guess}

for the unknown parameters (c*Is)

_{and how}

confidence in the final results

_{can be}

established.

(9)

2

'-

-I,',

[

---/

Convergence depends upon a number of

factors, the most evident being the compatibility between the mathematical model and the physical system. Assuming such compatibility to exist, the next most evident factor is the quality of

the initial guess of the unknown para-meters. Short of saying that they should

represent the best available guess, no further prescription is offered.

Fre-guently, however, even the best-initial guess is not good enough to achieve

convergence with a given set-of records. -When this situation occurs, recourse may

be made to a number of different

strat-egies. The first of these has already been mentioned in the introduction,

namely, using the measured records of

additional state variables if available.

The second strategy is to use a

gain' on the óc vector. This can be

explained by considering that the 6c vector when added to the c* vector should move the updated c vector in the direction in c-space such that c/tends towards its minimum. Sometimes, however,

6c may be so large that the updated value

of c lies outside the domain of convergence even though convergence might have been achieved by preserving

the di1rectjori in c-space indicated by óc but reducing its size. This can be

accomplished simply by multiplying the 6c vector by a scalar less than unity (the gain) before adding it to c. Thus,

the gain can be considered as a safety

valve to inhibit an overcorrectjon.

The most powerful strategy for inducing convergence is by varying the

length of record. If the record is long

and

the guess for the cs is poor, then

the trajectories calculated from Equatior. t Cl) may bear no resemblance to the

measured records and the ensuing

iter-ations will diverge. Difficulties can also occur in the incremental solution vector, Equation (5), because of a

resonance phenomena. Resonance can be

demonstrated explicitly when the

dynamical equations are linear, for then the equations of differential correction

-. are of the same form as the dynamical

equations except that they have

resonating forcing functions. It might

be expected that even for nonlinear systems a kind of

resonance

may also occur, in which case the incremental

solutions will become large over a

sufficiently long time interval. This

behavior is clearly opposite to the ultimate behavior required for con-vergence, which is that the incremental

solutions vanish identically. Thus, if

too long a record is used then such a record, when combined with a poor guess

3

OCEANICS

for the c*ls, may cause the iteration

procedure to diverge. The remedy for this is to shorten the record so that

such divergence is inhibited. Hence, a

practical strategy for achieving con-vergence with a poor guess for the cs is to start with a short record and use the converged output as the guess for a

record that is somewhat longer. This

procedure is repeated until the complete

record is processed. This strategy is similar to and, indeed, motivated by the one used by Roberts and Shipman (3) in applying shooting methods to the numerical integration of two-point

boundary value problems.

-Assuming that sufficient measured data is used and that the calculated and measured trajectories are well matched, it might be presumed that the values obtained for the unknown parameters are a

good approximation to the true values.

However, it can only be said that this set of values minimizes the mean square error between measured and calculated trajectories, while it is still

con-ceivable that another set of values, grossly- different from the converged set,

and which produces a somewhat larger mean sua1e' error, still results in an accept-able fit between the measured and

-calculated trajectories. Indeed, it might be the latter Set that best

represents the physical system. This

could happen, for example, when a

parameter is sought that, plays a minor role in the data that is being analyzed,

in which case virtually no information abtit that parameter is contained in the dta and so the value determined for it. by system identification will be spurious.

The spurious value could even contaminate the values determined for the other un-known parameters.

Procedures will now be presented

which will aid in establishing

confidence-in the results and which use the method

of system. identification itself. These procedures, of course, should be

supplemented by available engineering estimates of the parameters obtained

from other sources.

The first factor that can affect confidence is the number of measured variables used as the basis of the

identification. Generally, the more,

records that are fitted the more

confidence

there is in the identified

parameters. Thus, the record of every state variable that has been measured

is normally used.

It has already been -indicated that

gradual lengthening of the record is an

aid in promoting convergence. At the caine time, by monitoring the converged

c*Is after each lengthening, this procedure can also be.used to indicate

confidence in the identified parameters.

It would be expected that a short record would not contain sufficient information

to properly define the ce's. However,

as more information is introduced into the trajectory by lengthening the record, the c1s should stabilize, after which no further lengthening necessary.

Failure to stabilize is an indication of

low confidence.

Another means of establishing confidence is to vary the weighting factors from the values prescribed in

Section II. Clearly, if changing the weighting factors has no apparent effect

on the results, confidence is increased. If confidence cannot be established using the techniques just presented, this may be due to one of several causes. One possibility is that the data may be inadequate in one or more of the

follow-ing ways: too short a record, too large a sampling time, measurement of map-'propriate state variables, measurement

of too few state variables, lack of accuray in the data. All of these possibilities should be examined before-hand by inputting computer generated

data to the progrith isumcd values

parameters. To simulate reality 'Ei11data should be contaminated with

noise. By attempting to retrieve the assumed parameters from this data the measuring requirements for the actual

experiment can be established.

IV. APPLICATIONS 1) Surface Ship

The nonlinear three degree of

freedom system of 'equations describing

the steering and maneuvering of a surface ship are

ia -f(u) + C1v2 + C2r2 + C3u26.

= C4uv + C5ur + C6 £. + C7u26 (13)

r - C8UV

+ C9ur + c10 + C11u26

where u = surge veloOity,

V = sway velocity,

r =

yaw rate,

6 rudder deflection,

f Cu) known function of u that represents the difference between thrust and resistance,

C1 coefficients to be determined.

.The,umeasuredM input to the system identification program is provided by a maneuver generated on the computer with parameters typical of a conventional

.4

OCEAMCS

cargo

shiv

[4J. The maneuver chosen is a turning circle initiated by commanding

35 degree rde.r angle. -

-Difficulty in converging to the known set of eleven coefficients was encounter-ed with the algorithm as formulatencounter-ed even

though the predicted time histories of u,

v and thcxrselves became indistinguish-able from their respective input records.

Efforts to improve the results by

sampling at a higher rate or by taking a longer record proved fruitless and only by retaining an inordinate number of significant digits in the input data did

the converged coefficients agree with

their known values. Unfortunately, the

accuracy of data measured by real sensors is limited. The difficulty in obtaining good coefficients was

elim-inated by applying the modification to the algorithm suggested at the end of

Section II. Using the modified algorithm

the effect of varying the sampling

period, length of record, number of records, and accuracy of the data was

investigated to indicate the measuring requirements for the identification of a

real ship or model from a maneuver. It

was found that a 'sampling period of from

one to two seconds and a record length of from one-half to one minute was adequate

for, successful identification. However,

it was found necessary to have measured all three variables, namely u, v and r,

to identify the eleven coefficients in' Equation (13). In order to assess the

accuracy requirements, the input was

contaminated by adding Gaussian noise of

2 and 5% error magnitudes, where this

error is based on the indicated per-centage of the maximum value of the

variable. The results of identification with these inputs are shown in Table 1.

Table 1. Results of System

Identif

i-cation of Surface Ship

4 Dig.

True Digit 2% 5%

Values Data Noise Noise C1 x102 -.203 -.202 -.071 .127 C2 x10 .113 .109 -.589 -1.64 C3 x103 -.214 -.213 -.164 -.091 C4 xl0 -.146 -.149 -.185: -.231 C5 -.334 -.337 -.370 -.415 C6 x1D .958 .942 .796' .611 C7 x103 .323 .322 .321 .318 x10 -.122 -.114 -.072 -.012 C9 x102. -.397 -.391 -.360 -.315 C10 -.120. -.116 -.097 -.070 C11x105 -.584 -.583 -.586 -.591

It is seen that as the noise level is

increased some coefficients remain close to their true values while others drift away and still others lose all

signif-icance. The coefficients C1 and C are

seen to deteriorate the most rapidy.

These coefficients have been shown in (4] to be of minor importance.

2) Hydrofoil Craft

The nonlinear longitudinal

equa-tioris of motion for a typical hydrofoil

craft under autopilot control are

Normal Force Equation1

h=3217

L LR

Pitching Moment Equation,

ei(ciwp)

r2

(+d2'R)

The lift on the forward foil is

y5+C5 ci m 3

[

+ C3 6+T4] [Y6+C6 _CpJ.

where

2j CF0 + h - ze

Similar expressions are given for the lift LR on the rearward foil, where the unknown parameters C31C5,C61C9 are

replaced by C4I,C71C8 and C10.

Autopilot Transfer Functions,

4

I

(ra

[_ts+1 + - l3 rs+lj

Flap Actuation System Transfer Functions,

4 ls

6fs+')4f

e (r2e+1, ecosunand command (14) where h heave, 0 pitch angle,

v

.craft velocity (assumed constant),

.5

1JCEANICS

LF,XR,&HS = lengths from c.g. to forward foil, roal foil and height sensor, = given constants,

= vertical distance from

height sensor to un-disturbed water surface at design foilborne condition,

s Laplace transform C's = unknown parameters.

To demonstrate the feasibility of using system identification to estimate

the C's from full-scale tests1 comPuter trajectories were first used as trial inputs. The

sponsoring

agency supplied

the computer generated trajectories but withheld the values of the C's used in

the generation until the results of system identification became known.

Three sets of trajectories for each of

the three different configurations, termed A, B, and C (i.e., for three different sets of C's) were suDplied. A

run in each case consisted of a step

change in the commanded height after the craft had reached equilibrium.

Altogeth-er, there were 9 sets of trajectories

supplied.

Even

though there was no prior

information given as to the proper values for the C 's Other than the crudest order

of magnitude estimates, no difficulty was

encountered by the system identification

program in converging to a set of C's for

each configuration and run. Confidence

in the converged results was investigated

by varying the record length as described

in Section III, and also by comparing the

results from different runs of the same

configuration. In all cases high con-fidence in the results were indicated.

The estimates from system identification

are given in Table 2.

Table 2. Hydrofoil System Identif-ication Summary Configuration True (Estimated) A B C .0450(.0439) .0450(.0439) .131 (.131) .131 (.129) C3 .17941.179) .17941.179) C4 .32761.327) .4446(.445) C5 1.912(1.97) 1.912(1.97) C6 .5097L53) .50971.53) C7 2.668(2.67) 2.053(2.06) C8 .44271.45) .4427(.45) C9 0 (0(.2)) 10.0 (10.3) C 0 (01.1)) 0. (01.2)) .09501.0942) .131 (.129) .1906(.l90) .4446 (.445) 2.114(2.18) .55791.58) 1.806(1.87) .51261.51) 0 (0(.2)) 0 (0(.l)) eco +

(Tis+l)

Wcommand - CHS) connand l5 'i + '16 CBS CR$O - h + _HS0

The parameters C and C0 represent

coefficients of nonlinear lift. Non-linear lift appeared to play an insig-nificant role in almost all the trajec-tories making it impossible to obtain firm estimates of these parameters. It

was only possible to estimate the order of magnitude of these parameters. This has been indicated in the table by use of an order symbol. As an example1 the entry DC. 2) is to be interpreted to mean that the value of the parameter is no greater than ±.2. Th the case of C9 for configuration B, the nonlinear lift term had been "beefed up so that it played a more significant role in the trajectory, and was therefore detectable.

Also presented in Table 2 are the true values of the C's disclosed-after

the results of identification became known. The comparison shows a remark-able agreement between the values

esti-mated by system identification and their

respective true values. 3) V/STOI, Model Tests

The next case is the determination of longitudinal dynamic stability

derivatives from model tests of a quad

configuration, ducted-propel],er V/STOL

aircrft flying at low speeds and high

duct incidence (5]. The model closely resembles a 0.145 scale dynamic model of the Bell X-22A V/STOL aircraft. The tests were performed at the Princeton Dynamic Model Track. A character.stic of this facility is its ability to

re-strict the degrees of freedom of model motion. The data presented in (5] are plotted time histories of motion in various longitudinal degrees of freedom when the model is disturbed from trimmed

level flight. This data was analyzed

in

(6] on the assumption that linearized small perturbation equations are valid.

The procedure used in (63 to determine stability derivatives was first to

analyze the one-degree-of-freedom (pitch

angle only) tests to give values for pitch damping (Me) and angle-of-attack stability (M) and then, knowing these quantities, to analyze the

two-degree-of-freedom tests (pitch angle and horizontal velocity) to give the speed stability

(Mu) and the rate of change of horizontal

force with velocity (X). Finally, the

three-degree-of-freedom (pitch angle. horizontal velocity and vertical velocity)

tests were analyzed to give the lift curve slope (Z) and the rate of

change

of lift

with velocity (Zn). Al]. of these

analyses were carried out using the root

locus method (see e.g. [73).

Analysis of this model is complicated

6

OCEANICS

by the fact that it is. dynamically un-stable for all but the lowest duct angle

tested. The resulting divergent node dominates the motion, thereby making it difficult to measure the characteristics

of the othermodes. To make the data more tractable for analysis, known

amounts of augmented stability were introduced by adding a mechanical spring

(ke

)

in

the one-degree-of-freedom tests

rn

and

various amounts of damping feedback (sMê) in the two- and three-degree-of-freedom tests. This caused the time histories to be longer

before

the model

hit the stops and, in addition, produced

oscillatory behavior.

It wa5 originally thought that

system

identification

could estimate all. the stability derivatives simultaneously

from the complete three-degree-of-freedom

tests. However, the amount of pItch rate feedback introduced into the model to permit sufficiently long test runs for

analysis purposes also produced a highly

damped mode so that the characteristic roots of the response modes of the model were

widely

separated. Even though the modes are separated, they are not suE f i-ciently far apart to permit the

replace-ment of the equations of motion by an

approximate set of reduced equations. Thus,' in order to estimate all of the stability derivatives from measured data, it is necessary first to determine both

and

M from the one-degree-of-freedom

tests an

then

to use the known values of

and M in two- or three-degree-of-freedom tests to

find

the other para-meters.

To simplify, the presentation, certain known additional terms in the equations of motion have not been shown. These include -those due to the

displace-ment of

the model center of gravity from

the pivot axis and the mounting linkage mass of the model. On the other hand, these terms have been accounted for in

determining the aerodynamic

character-istics from the analyzed results. The same nomenclature used in (5] and (6] has been adopted. In addition, the downwash lag (M) and rate of change of horizontal force with velocity (X) are assumed to be negligible, which is consistent with

The following is a

smunary of the equations used in (6] for analysis of the

model test data;

One degree of freedom

V

B f decrees of freedom

- - Xu

+ g8 0 Uf + B e

ree decrees of freedom

- _{X, u}

_{+ - gB}

₀

Uf+WfZWf-Z U eO

W

- M

Wf +B -Mj

- M U09 = 0

here

pitch angle,

model horizontal perturbation

velocity,

model vertical perturbation velocity,

ratio of vertLcal to horizontal

masses,

model initial velocity along flight

f path,

g = acceleration due to gravity,

= + ,

mechanical spring constant,

model pitch moment of inertia about

'

pivot axis.

The quantities -, k , I _{and th}

t In

y

are known for each experiment and have uen accounted for in present.thg the inal results. _{The two unknown} deny-tives, N and iç,,

ifl

_the one-degree-of-reedom tests can be determined by measur-ng the oscillation period and dampimeasur-ng,

ut for completeness, they _{are detemined} ing the system identification _program. e remaining stablitv derivatives are

hen found from the higher degree _of

reedom tests with Me and assumed

own. Since there _{.s no inherent} re-tniction on the number of parameters

at can be obtained by the _system iden-ification program, it is possible to

etermine all four remaining derivatives

'

X, Z

and Z, directly from the

three-egree-of-frea -tests using Iguation

17), without tho intermediate step of étermining Mu and X from the two-degree-f-freedom tests using Equation (16).

th approaches have been taken here and

he results for both are presented in

Yable 3. Typical matches between bserved time histories and

_those

pre-acted by the system identification

pro-ram are shown in Figure 1.

Well-defined values for Z, and

ould not be found even thoug _the pre-icted time histories matched well with

he observed time histories, _{and it is} (16)

believed that any attempt to do so

amounts to an overrefined analysis of the-available data. The reason that and

are indeterminate is probably because

a they play an insignificant role in the

measured three-degree-of-freedom

trajec-tories. There is a

discrepancy

in the values of and between the results

obtained from the two-degree-of-freedom

tests and the results obtained from the (17) three-degree-of-freedom tests. More

confidence is placed on the two-degree-of-freedom results because the deriv-atives and have been inhibited in

these tests.

Table 3. esu1ta of V/STOL Model

Identification - Duct Angle of

.70C

xu

Zn Zw 7

OCEANICS

Present Method Reference (6]

_955a

.297a _,700b 484b -.lgc - 14c

Also

presented in Table 3 for

compar-ison purposes aestimates of the

stability derivatives presented

in (61

as obtained by the root locus method. The

input to this

method

is the damping and period measured from the data records.

For real data these inputs constitute

only

a portion of the information

avail-able in the cornolete records and only define overall features of the data. As a consequence the precision with which

the _{unknown parameters are determined} using the method -may be illusory and

_no

criterion is provided to evaluate the

accuracy of the results. By contrast, the method presented here uses complete records of all measured state variables and not merely some of their overall

features.

4) T-33 Flight Test

The fourth ease is the determi-nation of the lateral stabilityand control derivatives of a T-33 aircraft

from flight test data. _{The equations of} motien* appropriate to this maneuver are

*Supplied by the sponsoring agency.

aobined

_{from analyzing}

of-freedom test.

a

One-degree-b

obtained from analyzing a

two-degree-of-freedom test.

C

obtained from analyzing. a three-degree-of-freedom test.

0

100

0

_LpL8Lr

.05467 a0 Y8 -l

e

. 0

NpNNr

I

The data consisted of digitized time histories of p. 8 and

r

sampled every

twentieth of a second for 4.7 seconds.

It can be seen from 5,ation (18)

that there are eleven unknown parameters to be estimated from the data. thile it

is not expected that all eleven

para-meters

play

equally

significant

roles in

the trajectory, no attempt was made

beforehand

to eliminate any from the equations, but rather the system

identi-fication

program itself was used to

indicate the confidence to be placed on

the estimated results by varying the length of record analyzed.

Table

4.

_{Results of T-33}

_{.Xdentifjcatjon}

.8

10EA'IICS

estimaths of some of the parameters.

This disagreement can probably be

attrib-uted to a difference in weighting factors and not to the method used. The

para-meters with the greater discrepancy are

those with the lesser confidence, and so the parameters near the bottom of both lists should, at best, be considered approximate.

5) Chemical AdsOrptIon

The last case is a problem from chemical kinetics chosen to show the application of the present methdd to identify the parameters in a

differ-ential equation of a very

different

form

from those previously considered. The

problem is to fit chemisorption rate data

to the Elovich equation:

ae (19)

where q is

the volume of material adsorbed

and a and a are the parameters sought.

The Elovich equation was chosen

be-cause it is typical of the kind

encounter-ed in chcmisorption processes and also because measured data (9] and estimates

of a and

a using this

data are both

avail-in

the

open

literature. In fact, there are two independent estimates of these parameters from this same set of data,

one obtained by graphical means (10] and

the other by analog matching (11]. The values of a and determined by

the present method are given

in

Table 5

along with the values presented in (10] and [11]. Time histories of q calculated on the basis of the present values of a and a are shom in Figure 3 together with

measured data points taken from (9].

Although not

shown, time histories of 'q

using values for a and a taken. frem (10]

and (111 also agree well with the measur-ed data points even though their

respec-tive mean-square errors are not minimal.

V.

CONCLUSIONS

A technique has been presented for

identifying many unknown parameters

appearing in large dynamical systems from

measurements

of the time history of one or more of the state variab1s. As a demonstration of the versatility

_and

validity of the

technique,

it has been successfully applied

to

five distinct systems whose unknown parameters were

Table 5. Results of Identifying

Chemisorption Parameters Present Temp. Method [10] (11] 100°C a .0146 .0164 .0167 s 7.79 8.50 8.37 132°C a .0388 .036 .0364 a 7.09 7.00 6.92 Present Quasi-Method Linearization -,991 -21,1 -20.0 a -2,67 -2.51 -10.4 -9.40 Lr 2.07 _1.59 N3 6.87 6.61 -.299 - .377 .974 .694 .776

_.0904

a0 .102

_.0148

N8 -.860

-1.37

T a est.mated resu ts are presented

in. Table 4 in order of their confidence,

and the trajectories predicted on the basis of these results are shown in

Figure 2. Also shown in Table 4 and Figure 2 are

a

determination of the para-meters and the corresponding trajectories

as obtained using the method of

quasi-linearization (81*. Both methods

dupli-cate the measured trajectories very well, even though they do not agree on their

tAnalyaie of the date using quasi-linearization was carried Out at the Cornell Aeronautical Laboratory and Supplied by the 5pOnsoring agency.

where

bank

angle, roll rate,

8=

sideslip angle, yaw rate,

,ither givcn

ifl

advance or had been

deter-med

by an earlier Lavestigator using a

re established method. A feature of

.c present

technique,

believed to be

iquc, is the capability of determining level of confidence -for each of the

verged pararetere while in the curae

f performing the identification. urthermOrc, the system identification omputer program is structured in euch a

ay that

the user need only supply state-ts describing the system: then, with

o requirement for any apptoximatioa to

e system solution and, in fact,

no

re-irement for any additional mathematical

nalysis whatsoever, the user can expect o obtain estimates of the

parameter;,

e auality of which will be as high as be sustained by the data.

FERENCS

GOODMAN, T.R., "S::stiem Zdentiuication and Prediction An

Algorithm

tsing a Newtonian Iteration Procedure,' Quart.

Applied Math, XXIV, _{L-.!! £49-255.} CHAPMAN, C.?. and

RIF!c, D.B., 'A

Method for Extractina Aerodynamic Coefficients from Free-Flight Data," AIAA Journal, 8, 4, 1970, 753758.

ROBERTo, S.M. and S!IPMAN, J.6., "Continuation in Shooting Method; for

Two-Point Boundary Value Problem;,"

J. Math. Anal. and Appi. 18, 1, 1967, 45-58.

STRQM-EJSEN, J., "A DLi.tal Computer Technique for Prediction of Standard

Maneuvers of Surface Ships," DTIIB Report 2130, Dcc. 1965.

PUTNAM, F., TRAYBAR, J.,, CURTISS, H.C.

JR., and <ukon, J.P., "An Investiga-tion of the Dynamic Stability

Char-aoteri;tic; of a Quad Configuration,

Ducted Propeller V/S?OL Model, Volume

II, Phase II - Longitudinal Dynanics

at High Duct Inc1enc Data Report,"

USAAVLABS T.R. 68-493, 1968.

CUR'PISS, H.C., JR., "An Investigation

of the Dynamic Stability

Charnctor-iStica

of a Quad Configuration

Ductad Propeller V/STO Model,' Vol. IV,

USMVLABS T.R. 68-49D, May 1969.

SAV7T, C.J., 'Central System Design," McGra-Hi11, New York, 1958.

BELLMAN P.., AC-IWAi)A, ., KAIJ.DA, R.,

"Quasi-Linearisation, System

Identi-fication, and Prediction," p.and Corp. Memo P.M 3812-PR, Aug. 1963.

TAyLOR, H.S. and WILLIAMSON, A.T., "Molecular and Activated Adsorption of Hydrogen on Mangancus Oxide Surface;," J. Mier. Chem. Soc., 53,

1931, 2166-2180.

0. TAYLOR, H.A. and THOM, N., "Kinctic of Chcaisorption," J. Anar. Chem.

9

OCEANICS

Soc., 74, 1952, 4169-4173. 11. SOt4EP.FIELD, 3.?., "Solving the

Elovich Equation by Analog Computer,"

Instrument and Control Systems, 41, 1968, 137-138.

..

10

2.0

Figure 1

V/STOL MODEL TEST

TII4E HISTORIES

OCEAMCS

time-sec.

Data,Ref. (5),Run 980

Fit using present method

\

/

time - sec.

3

C

4

5

S

I

-25

C)

-30

no

-4

-6

-8

-10

time - Sec.

2

\

timo - sec.

Figure

T-33 FLIGHT TEST TIME HISTORIES

Fit using qu1-arizatñ

/

/

1 \2

.3

time - soc

.2

-.2

S

-.4

a)

V

-.6

S S U S

.

S S S S

40

20

80

time-mm.

40

time

-mm.

OCEANICS

120

60

160

80

Figure 3

TIME HISTORIES OF VOJUNE ADSORBED

O

Data (9)

Fit using present

method

temp.

100°C