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
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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