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H. Gorecki
A. Korytowski
Technical University of Cracow
Institute of Control, System Engineering
and Telecommunication
Cracow, al. Mickiewicza 30, Poland
ON RELATIONS BETWEEN FEASIBLE OBSERVATIONS AND DECISIONS
Abstract
The paper is an attempt at a theory of relations connecting
feasible observations /or measurements/ and feasible decisions
/or controls/ in general cybernetic systems. The theory gives
a formal framework and a tool for quantitative analysis of the
following facts:
41. An increase in observation possibilities, e.g. an increase
of the precision of measurement, enlarging the scope of obser-
vation etc., results in an increase in decision possibilities
by making more effective decisions possible. This works also in
the other direction: if there are more feasible decision, new
observations or measurements become available.
2. In the framework of a cybernetic model no decisions and/or
observations which generate antinomies can be simultaneously
feasible. This creates interesting and important constraints on
measurements and decisions in systems which include man or where
a buman or automatic decision maker is an object of observation,
and where the results of observation may be known to this
decision maker.
3. The observation /measurement/ takes time and changes its
object and thus the result of observation always refers to the
past rather than to the present. This normally is due to
physical effects though other phenomena, like psychological,
may also be important depending on the nature of the object.
The facts of group 1 are in a sense opposite to those of
groups 2 and 3. This leads to the existence of optimum degision~
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-measurement possibilities. Conditions for this optimum to
exist together with its significance for biological and
technological systems will be discussed.
The subject of this paper is of interdisciplinary interest
and has been studied, partially and from particular angles,
within the framework of control theory /facts of group 1/,
mathematical logic /theory of antinomies, principles of mathe-
matics - mainly facts of group 2/, physics /theory of measure-
ment, principles of quantum machanics - mainly facts of group 3/
and philosophy /the classic problems of free will and con-
sciouness/. The relevance of the presented theory to these
fields will also be discussed.
List of Figures.
Fig. 1. Control system.
Fig. 2. Minimal value of the functional 8 according to /16/.
Fig. 3. Minimal value of the functional S according to /20/.
Fig. 4. Control system with time-delay in observation.
Fig. 5. Minimal values of the functional 8 as the function of
$ 5 (§)-s8 parameter, according to /40/.
Fig. 6. Relative time-delay (3)" as the function of (E)
according to /41/.
Fig. 7. Minimal value of the functional s* as the function of
(E)accoraing to /42/.
Fig. 8. The dependence of the functionals Jo on relative time-
delay (#). for the system without delay curve Fas and
with time delay curve F,.
Fig.1.
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Henryk Gorecki
Adam Korytowski
ON RELATIONS BETWEEN FEASIBLE OBSERVATION AND DECISIONS
4. Introduction
Fhe goal of this paper is to formulate some basic questions
or problems connected with dynamic systems. We will consider
these systeas from the control-theoretical point of view.
The first question is to establish how the actual state
of knowledge influences as a feedback on the foundations.
Three classes of the systems will be discussed:
1. The systems for which there exists the concordance between
identification and control,
2. The systems for which there exists antinomy between iden-
tification and control /decision/,
3. The systems with time delay between observations and
controls.
2. Problems and examples
2.1. Antinomy between energy and accuracy
Let us consider the systems presented in Figure 1.
We assume the model of the system as a deterministic one
described by the equation
yepu V4
y0) = ¥9
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r—>—_—Ss«~PProccesss
t~<— Controller -~<—
Fig. 1.
20
01 02 03 0.4 05 06 07 08 os
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where y — output
u - decision
and the criterion function as
co
Is I (+ A®u? at /2/
)
where A - constant coefficient
Application of the Bellman equation leads to the determination
of the optimal controller in the sense of minimization of
functional /2/.
In the general case, the system is described by the vector
equation
y= £(z,u,t) /3/
and the criterion function is
te
I= G(yp) + J £4 (yu, %) at /4/
°
We denote the minimal value of I by:
t
s(t) = min 6(y»)+ ¢ £,(y,¥,8) ds /5/
veU ~*~ % 7”
U admissible region of decision
Then the Bellman equation is /-1_7
_ 28. = 28
Be Bey Hoes) +L SF aah oy
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From /5/ we have for t = ty
8(y— 1 te) = Cys) /7/
Assuming that the minimum of S with respect to u is
inside the region U_ we can search for it using the equation
obtained by the differentiation of equation /6/
Of, (yu st) + 2s 3 £(y*,2*, 4) =6
ou oy oe
In our case, from equations /1/ and /2/ we see that:
G(¥~g) = 0
£,(ysast) = y? + Xu? /9/
£(y,u,t) = - 7,8
From /6/ we have
o= (y+ PuRe + a oF ut] /10/
and from /8/
anak s £5 ris ) /
Elimination of = from /11/ and /10/ leads to:
vy
0 = y - Aye? /12/
Fig.2.
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The optimal decision in the feedback form is
We - te /13/
returning to /1/ gives
ye-+y¥* /14/
eA.
My
the solution of which is
t
~ Ey
y(t)=y,e * /15/
and the minimal value of the functional is
oo
S= {@ + 22] dt = Any? = ze . ye /16/
o t
Following the same way we find for the process described
by the equation
y(t) =-fy+hu /17/
¥(0) = Yo
i “y t | |
F(t) = ¥Qe “G) 18/
u¥(t) = 4 - (5 r /9/
and the minimal value of the functional is
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1- (Ey y
,.@ + ( | x i
2 2
K K
(j) -()
There exists concordance between attainable accuracy and
control. If the action ih vises,then accuracy also rises.
120/
Fig.3.
The total ISE decreases. See Fig. 2 and Fig. 3.
2.2. Antinomy between identification time and control time
Let us assume that our process is a priori not known, and
for its recognition we need an interval of time equal v .
After that we can control by the time equal t) which in
particular may be equal to infinity. Now we analyse our
system /see Fig. 1/, but we must take into account that
between the process and the controller there exists some
Fig. 4. delay /see Fig. 4/.
The first interval /Identification/
Let us assume that during the interval V of time we
have recegnized that our process can be described by the
equation
$(t) = + tn u(t-T) /24/
y(o) = vy) = y,
and
u(t) = 0 for t<T
As the criterion functional we assume
B(u) = i ‘Ty? (+) + Deu? (+)) at /22/
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28
Ty
10
08
06
O44
0,2
0 1 2 & 5 6 . i
+
Fig. 3.
(—>— Process
(-)
Uls)
L__<=—| Controller
Fig. 4.
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During the interval of identification, according to /21/
we have
x
8(u) = | y(t) ab = y2 /23/
3°
After this time the controller can be informed about the
process and we have
8(a) = y2T+ J (v(t) + Mu (er) at . /24/
Applying Bellman’s procedure leads to the relations
0 = y@(t) + Mu? (tt) + ae (te u (--t)) /25/
and
o = 2 Mu(e-t) + + 28 /26
u(t ) + rT Dy /
Elimination of 28 from equations /25/ and /26/ gives:
y
3
y@(t) - APu® (tr) = 0 /27/
We realize negative feedback so we choose
u(t-t) =- fy /27/
which inserted to equation /21/ yields
5(t) = - 1+ y(t)
Any /28/
¥ #0
The solution of /28/ is
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t
~ Wy
y(t) =y,e * /29/
= -
Yo i
u(t-1) = - ye /30/
The value of the functional is obtained from /84/, /29/ and /30/
of
s+ sn, (<8 +e + ) 1314
Ty
Minimum of S as a function of VY is attained for
Dt,
v* = —+ 1ne 132/
and
AT
s*. —i v4 + 1n2) /33/
2
It is evident from the relations /32/ and /33/ that there
is an antinomy between the time of identification T and
control action rr -« For that reason it is not possible to
Be
obtain an arbitrarily high accuracy. The product of time delay
‘T and control Tr is constant
4 4
_—_= 1
% Tp Bm 134/
i
Similarly for process
x(t) = -py(t)+ BF u(t) /35/
y(0) = y(t) = ¥, , and u(t) =0 for t<t
Pig.5.
Fig. 6.
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we have
u(t-T] = (1 - (8) ule) /36/
which with the equation /35/ gives
e
x(t) = - 3 I + &) y(t) /32/
The solution of it is
yt) = y, e “+(§) t
u(t-t) = Bey ( - I, +(§)° . e. (5) t 139/
Finally using /38/ and /39/ we can calculate the minimal
138/
and
value of the functional:
, } K\e'¢
: ne +() i 4 \2 m2 2
S(u)= Re =F ; &) +h- a+(§) TYG /40/
K kK
a(§) (8)
The functional S(u) as the function of relative time delay
Fr attains its minimum for
~ 3, @ hPL
(F) ah a Th //
see Fig. 6,
The minimal value is equal to:
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05
04
03
Fig. 5.
02
4c
10
Cal, (Pal, (af, °*
al
oO
004
003
0,02
Fig. 6.
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*. (- L.(6) 2
s¥(u) =|1 + In = Yo /42/
(5)
The dependence of (ey and s*(u) as the functions of
& is shown in Fig. © and 7.
The conclusion is evident.
Fig.7.
In the presence of time delay it is impossible to obtain
an arbitrarily small error, even using the optimal controller.
The analysed controller has the disadvantage that in the
case of over estimation of the value of time delay the whole
closed system may be unstable. To prevent this it is necessary
to maintain stability condition.
Let us finally analyse the same process, but using a
conventional controller.
We have in the interval of identification T
y(0) = y(t)= Yo
and f43/
u(tt)=0, t<T
In the period of control we have the equation
3(t) = - 2 u (Tt) /4444/
i
The functional see (3,37 is equal to
He
&
2, cos rm x Teos
— 2 (i-sin ) (Mol
d.
°
1
nD
B.
is)
ra
Su)
12:
Qs:
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>i
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The minimum of this functional with respect to the parameter
Tr of the controller is at the point for which
> ~
t
l
cos — = /46/
uO
This optimal value is
Fe = 0,739 /47/
pa
fhe minimal value of
er
J,
Tne 2 4 /48/
ane G nie ose
1- 1 = 0,739
The condition of stability is
14-0
or ce <> /49/
Fig. 8. In figure 8 there is shown the dependence of the functional
on x= T- xr for the system with time delay /curve Fp/,
i
and for the system without time delay /curve B,/
8
ees
aay 2a
It is evident that the antinomy between time of identi-
fication and control yields the limited accuracy given by
equation /48/.
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2
|
2 _
So 5
is -
BI
- tt
“w : 3
g =
ia 1S oa
iro ” :
u u
u- ous
ape 3 Q e 3 3
hg o a g g g
160
120 4
0,80 4
0.40 4
0,00
Fig. 8.
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References
41. Bellman R.: Dynamic Programming, Princeton University '
Press, New Jersey 1957.
2. Gorecki H., Popek L.: Control of the System with Time Delay,
3-rd Symposium Control of Distributed Parameter Systems
1982 Toulouse, France.
3. Gorecki H., Popek L.: Parametric Optimization Problem for
Control Systems with Time-Delay, Proc. of IXth Congress
of IFAC, Budapest 1984.
