AD-AD91 655 SCHOOL. OF AEROSPACE ME01CINE BROOKS AFS TX F/6 12/1F A CLASS OF SEQUENTIAL INPUT ADAPTIVE SYSTEMS. (U)I DEC 790D E GREENE

7 UNCL ASS IF IED SANTR79-38

mEEEh~h~mEND~

I 280

Report SAM.TR. 79-38

A CLASS OF SEQUENTIAL INPUT ADAPTIVE SYSTEMS

David E. Greene, Major, USAF

December 1979

Final Report for Period 1 January 1979 - 1 July 1979

Approved for public release, distribution unlimited.

USAF SCHOOL OF AEROSPACE MEDICINEAerospace Medical Division (AFSC)Brooks Air Force Base, Texas 78235

C8 .1 1n n13-1

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Adaptive controlan-machine tracking systems

20 ABSTRACT (Cont .e n reverse side if necessary and Identify by blotk n, ber, A mathematical theory forfirst, second, and third order sequential input adaptive systems is presented. Inthese systems, a theoretical controller predicts the input at instants by n termsof a Taylor series representation of the input and effect open-loop control overintervals to obtain n4

h order projected error responses. The second order systemis the lowest order system of this class that describes the mean tracking behav-ior in a given antiaircraft artillery man-machine tracking system. Further, thesecond order system is a most plausible system consistent with limitations of thehuman controller and the manner the subjects were trained.

DD 1JAN73 1473 ARASSIFIEDSECURITY CLASSIFICATION OF THIS PAGE (%'hen Data Fnt

A CLASS OF SIQULMTIAL INPUT ADAPTIVE SYSTEM1S

I NTRODUCT ION

The sequential input adaptive system theory introduced in a previous

publication (2) is generalized to nth order systems in this paper. A comn-

parative study is made of the first, second, and third order systems.

In the new~ theory it is assumed that a living controller, through condi-

tioning, accommodates to the input and the controlled plant so that the total

system has basic mathematical properties. In this controller-centered theory,

* predictions are made instantaneously and tracking movements are governed by

the strategy.

The sequential theory describes and predicts, manual control -tracking

behavior in a complex antiaircraft artillery (AAA) man-machine system (2).

For this system, the sequential theory gives a description of mean tracking

behavior that has a closer correlation with the experimental data than does

the description given by the optimal control approach of Kleinman et al. (4).

Furthermore, the sequential theory applies to the human eye tracking system

(3).

In this paper the basic optimal process is first presented. This process

is then sequentially appl iedl to represent first, second, and third order input

adaptive systems. Finally, the theoretical tracking descriptions are compared

with tracking data from an AAA man-machine system.

THE flASIC OPTIMAL PROCESS

The basic optimal process is defined by an nth order differential equa-

tion in projected error with parameters determined such that a cost functional

is minimized. Properties of the basic optimal process are presented for

first, second, and third order systems.

The input adaptive system is represented in Figure 1.

: _e~). C-Pm(t

Figure 1. The input adaptive system.

The internal system C-P is the controller-plant. It is assumed that in the

nt h order system the controller

1. estimates the system error state at discrete times ti (to < t1 <

< tr) of the tracking interval

e(ti -i(t+) - -(t-),l1 1

2. predicts the input at each t i by n terms of a Taylor series repre-

sentation of the input, and

3. effects systematic open-loop control over the intervals (ti, ti+j)

to reduce the error relative to tile predicted input.

There is no time delay in the prediction-control process. The input is

assumed to he piecewise n-I times diffirentiable and such that the one-sided

limits in equation I exist.

The prediction-control process is represented by the repeated application

of a basic optimal process. In the basic optimal process, the system input

i(t) is predicted at time t i by

n-1 i(J)(ti)i(t) ( (t - ti)J t > ti (2)

j=O j!

and an error response to this predicted input, e(t) i(t) - m(t) where n(t)

is the system output, is determined by

ne(n)(t) + a ce(n-J)(t) = 0 t > t, (3)

j=1

e(J)(ti) = i(J)(t+) 0 <()t) O j < n-I

where (aij) are constants that minimize the associated cost functional

nJ nt)2+ V a nj)t)d (4)ti [jjl

for given nonnegative constants (Oj) with On > 0. The constants (aj)

are denoted the strategy.

Justification for the forms in equations 3 and 4 follows from that given

in reference 2. The basic operation in defining the variational problem is

the extension of the upper limit in the cost functional and the continuation

of the projected error through all future time.

3

In the basic optimal process for n = 1, 2, and 3, the parameters (a,)

are uniquely determined by the strategy (0j). Cost functional JI has a

unique minimum when

al = 01. (5)

Cost functional J2 has a unique minimum when

cI (0I2 + 202)1/2 (6)

a2 a2. (7)

Cost functional J3 has a unique minimum when

aI = (p+A+B) I/2 + {2p - (A+B) + 2 p - 1(A+B))2 + 3 (A-R) 112 1112 (q)

K 2 4 11p (^+,^))2 + 3 (AB 112111/2 (q)

a2 (p+A+9)l/2 + {2p- (A+B) + 2 - A 9]

' 93 = a3 (10)

where

A b + +b a3) 1/2] 1/3

4

b !b2 a3)1/2] 1/3

a- 1 322-04

3

b = 1 -~2016 + 98128022 - 278032]

and

+ + .3) 1/2]1/3 rRjI I(*

27uDitrjicltiOnl/ --

3- 2 a 3 12 1/

7 . [ - T + 98282 3 - 28

The rots f comlex umber z =reje n eqation 8 a d ar aenaDit;7pca

1 ri/ l2 if

35

Critical points for J1, J2, and J3 may be obtained by direct com-

putation (evaluation and differentiation of the cost functional) or by other

variational methods (1). The cost functionals may be shown to have minimums

at the critical points defined hy equations 5-10 through a consideration of

quadratic forms (see reference 5).

Cost functionals Jl, J2, and J3 , evaluated for unit step inputs

applied at t = 0 with zero initial conditions on the system output, are given

For reference in the Appendix.

The strategy parameters (aj) have physical meaning: they indicate the

importance the controller gives to each quantity in equation 4. The manner in

which the strategy affects the system response is clearly illustrated in the

second order system when the input is a step function. For this input, the

system output response is overdamped when I >> 02 (so that I2 >>

22) and is underdamped when a2 >> I (so that i2 << 2B2).

Two additional properties of the variational problems are given. First,

any solution to equation 3 generated by the variational process tends to zero

as t becomes large. This follows from the existence of the cost functional

(equation 4). Second, the relationships that minimize Jn (n = 2 or 3) also

riinimize Jn-1; for if 1n is set equal to zero in these relationships, then

the parameters (j.j) that minimize Jn-1 are obtained. (For n = 3, this is

most easily shown using the necessary conditions for J3 to have a minimum as

given in the Appendix.)

In the basic optimal process the input is predicted hy n terms of a

Taylor series representation of the input. An error response to the predicted

input is determined by the strategy and the initial conditions through an nth

order differential equation.

6

TIE[ ADAPTIVE PPOGPAM

The basic optimal process is sequentially applied in the tracking

alqorithm called the adaptive program. In option 1) of the adaptive program

the basic optimal process is applied at constant increments of time Atc. In

option 2) of the adaptive program the basic optimal process is applied when

the absolute value of the system error exceeds the system error threshold

rT, but is applied such that ti+ 1 - ti > TS where TS is the minimum

period for a sequential problem. (The criteria in option 2) can be gener-

alized to include threshold conditions on derivatives of the system error.)

The adaptive program predicts the input over successive time intervals by a

sequence of steps, ramps, or parabolas (for n = 1, 2, or 3, respectively) and

produces nth order error responses (dependent upon the strategy) to the

components of the predicted input. The system output and the system error are

defined over each suhinterval (ti, ti+l) by m = i - e and e = i - m,

respectively.

CONTINUOUS MODELS

Suppose that the input i(t) is n times differentiable for t > t o and

the strategy is constant. If the time intervals between successive applica-

tions of the basic optimal process are small, then the sequence of differen-

tial equations 3, solved in the adaptive program, is approximated by

e(n)(t) + n Oje(n-J)(t) = i(n)(t) t > to (11)

e(to) = i(t0) - M(to)

e(n- )(t : i(n- )(t ) -

This result uses the identity e E i - i + e. The initial value problem (11)

is an approximate continuous representation for the sequential theory.

Associated with eqLuatirn 11 is a feedhack control system with unity feed-

back and open-loop transfer function ) zjs-J. This feedback control systemj=1

is another continuous model.

Fquations 5-10 qualitatively describe how the strategy ( ij) regulates

the parameters (cj) in these models.

The transition fron a higher order system to a lower order system is

examined for the special case where the input is n times differentiahle for

t > t 0 . Consider the system (equation 11) or the associated feedback con-

trol system and n = 2 or 3. Let (aj) he formally defined in terms of (nj)

Dy equations 6-10. As n tends to zero, the steady-state solution to the

nt h order system aplproaches the steady state solution to the n- 1th order

system with the samie input. (This is 'lost easily seen using Laplace trdns-

forms.) Therefore, there is a regular transition in the steady-state solution

through the strategy from a higher order system to a lower orler syste whel

the input is properly differentiable.

ANT[AIPCPAFT APTILLEPY AM-ACHIML VPACrIN , Y)ST

The sequential theory is applied to maniual control tracking in an AA

system. The theoretical trackinq descriptions provided by the first, second,

and third order input adaptive systemn are compared with the experimontal

data.

The tracking experiment is described in a previous piblication (2). Two

well-trained operators manually tracked tarpets nn a sirlulated AAA syste"-'.

!)ne operat.,)r control led the systeti in azimuth and the other operator con-

trolled the system in elevation.

In the present comparative study, only the Trajectory I tracking task is

considered and only option 1) of the adaptive program is used. The azimuth

and elevation components of Trajectory I are given in Figure 2. Time his-

tories of ensemble averages (15-1S runs) of the experimental azimuth and elo-

vation tracking errors are given in Figure 3.

In the analysis it is assumed that the strategy of each operator is

constant throughout the tracking task. The adaptive program is applied sepa-

rately to each azimuth and elevation trackinq task. For convenience, the

initial conditions used in the adaptive program are e(fl) = e (n-l)(f) =

p.

The parameters used in the adaptive program for the second order system

are those which were previously identified from the AAA tracking data (2).

For the purpose of comparison, the n component of the strategy is held

fixed for the first, second, and third order systems. The parai~eters used in

the adaptive prograii for each nth order system are

( 1<j<n

ij = ,

10 n

and At. = .1 s. For each system the strategy is primarily to minimize the

projected error.

The adaptive program system tracking errors on the azimuth and elevation

components of Trajectory I are given in Figure 4 for the first, second, and

-- . . .. . . . . . . .. .. - il n li l l .. .. . . . . . . I Inlm n li i nl . . ...9

0.0 -. 10.0 tso S.C 5.0 ,,0 C O00 0. 00 IO t t 0 50 00 tTIr ~fI TIEC ICS

La) (b)

Fiqure 2. Trajectory I components: (a) azimuth, (h) elevation.

iI

0 0.

:t S E C, TIE ism )

0'70rT. [.C. I a.CVRION r am I

(a) (b )

Figure 3. Experimental mean tracking errors on the azimuth and elevation om-

ponents of Trajectory I: (a) azitmjh, () elevation.

T y

! :4

(a) _

6164"

TISI. 'IEI

;4I

TI'. a)'I.S 'Cl

'7"ERMiI~m (W {S ORWCI EVRTJn¢ -'. l I lIMO ORMI

Fiur 4 datie rgrm pto )150 trakin errorsl QLonI the1 imt and1 ele

10

(atio con-nt ofTajcor : a irtore ssem h

scn o(

O 3\

00 5. 10. 5. , 50 00 00 C . . .0 1. . . . . C 00 C

.. .. ,1ot~ '(Cl 0.5,11 ,0- ' (oi , ., ,.

Figure 4. Adaptive program option 1) tracking errors on the azimuth and ele-

vation components of Trajectory I: (a) first order system, (h)

second order system, (c) third order system.

.............................:.... , 0 c,,,. r .AW'{ cl,.;-. bJz k

,' ... 1'. .... . ,.i olJ .,(

third order systes. The option 1) tracking errors for the qiven strategies

hecom:ie smaller as n increases. This reduction in tracking error is consisten

with the improved predictions of the input (equaLi on 2) as n increases. The

nth order system tracking errors are proportional to the nth derivatives

of the input. The second order system tracking errors are characteristic of

the experimental data. As is shown in reference 2, the option 1) second order

system tracking errors on all trajectories have a closer correlation with the

experimental data than those of the optimal control approach (4).

The first order syste with its single strateqy paramleter does not

represent the tracking hehavior in the AAA system. First, the firsL order

system tracking errors considerably differ in amplitude and profile from the

experimental trackinq errors. Second, the identified parameter B2 in the

second order system is such that 2 = aq > > 0 and therefore the tracking

response is characteristic of at least a second order syste, .

Properties of the continuous model (equation 11) are used to classify the

AAA tracking response. Although the tracking responses in Figure 4 were

computed as sequences of transient responses, they essentially represent

steady-state solutions to equation 11. (W1hen the exact initial conditions are

used or when the input is discontinuous, the transient response may dominate

the steady-state response (2).) As noted earlier, there is a reqular transi-

tion in the steady-state solution through the strategy fromi a higher order

system to a lower order system. Therefore, a third order systei:i response

corresponding to strategy (1, 10, F) where c is positive and sufficiently

small approximates the second order system response corresponding to strategy

(1, 10) (provided i(t) is properly differentiable). This derionstrates that a

third order system can also represent the trackinq behavior in the AAA sysLem.

1 2

F-urLher infjrm(O1on is known about the AAA sysLem. The subjects were

trained to minimize mean-square tracking error (2). This suggests that the

n component of the strateqy should he large. Thus, a third order system

with strategy (1, 10, c) and e positive and small is not appropriate. It is

further noted that a third order system with small a3 strategy component

would be ineffective in reducing the tracking error associated with the

sudden, initial appearance of the target. In addition, third order system

descriptions with 03 " al and 03 >> 82, as illustrated in Figure 4,

are not characteristic of the tracking data.

The second order system is the lowest order system of this class that

describes the mean tracking behavior in the AAA system. The second order

systeia description with strategy 82 >> al is consistent with the manner

the subjects were trained.

DISCUSSION

A mathematical theory was presented for first, second, and third order

se(piential input adaptive systems. In these systems, a theoretical controller

predicts the input at instants by n terms of a Taylor series representation of

the input and effects open-loop control over intervals to obtain nth order

projected error respenses.

The second order system is the lowest order system of this class that

prodluces a description characteristic of the mean tracking behavior in the AAA

ian-machine system. Higher order systei;i descriptions may not be appropriate

in this application. In a third order system, for example, the controller

would have to regularly infer the target acceleration from the visual display

and control using this information, stressing or exceeding human capabili-

ties. Third order system descriptions with small 83 strategy component are

not appropriate because the subjects were trained to minimize mean-square

13

tracking error. Therefore, the second order system is the most plausible

system consistent with 1 imitations of the human controller and the manner the

subjects vere trained.

A distinctive feature of the new theory is that it represents the overall

system tracking response by a sequence of transient responses. The sequential

theory provides a broad nonlinear description for tracking systetis with living

control lers.

REFEP,-NCES

I. Gelfand, I. ' and . V. Fomin. Calculus of variations. Englewood

Cliffs, N.J.: Prentice Hall, 1963.

2. Creene, P. F. A mathematical theory for sequential input adaptive systemIs

with applications to iian-machine tracking systems. IEEE Trans Syst !lan

Cybern SMC-8:498-507 (1979).

3. Greene, 0. E., and F. E. Ward. Human eye tracking as a sequential input

adaptive process. Accepted for publication in Riological Cybernetics.

4. Kleinman, 0. L., S. Baron, and 11. 14. Levison. A control theoretic

approach to manned-vehicle systems analysis. IEFE Trans Aut Control

AC-16:824-832 (1971).

5. Wylie, C. R. Advanced engineeringq mathematics. New York: fcGraw-Hill,

1966.

14

APPENDIX

Cost functionals Jl, 32, and 03, evaluated for unit step inputs

applied at t = 0 with zero initial conditions on the system output, are

dl = 2a 1--+-i1 [

J = 1 [23 + 12 22 + a2 K 2 +2ic 23 2 2 2 J

3 - 3 E3 (12 + 812133 + 2a13 +3 2 [ 2 +3( 1 2- 3)

Necessary conditions for J3 to have a minimum are

3 = '3

with al and a2 such that

2 + 1312 2) + 2+ a2 1- 11J + 2 3 + 0

8 133 + I a 2 -22j + 22a3 + 83a12 0.

15