Identification in Automatic . Control Systems

INTERNATI ONAL FEDERATION
OF AUTOMA T IC CO N TR OL
Identification in Automatic
. Control Systems
Fourth Congress of the International
Federation of Automatic Control
Warszawa 16-21 June 1969
Organized by
NaczelnaOrganizacjaTechniczna
w
Polsce
IDENTI:FICA TION IN A UTO:NLt1. TIC CONTROL SYSTEMS
By
A.
V. BALAKRIS!-JNAN't
V. Peterka
Institute of Information Theory
and Automation
Czechoslovak Academy of Sciences
.Prague. Czechoslovakia
Department of Engineerir.g
University
of California
Los Angeles, California
1.
INTRODUCTION
System Identification is
a
us.;> it in slightly different ways.
and, for the
purpose of
the determination of a
trolled.
o ti on and different authore
n
We shall not try to find a ge neral definition
,
this pape'r, we shall simply· treat the identification as
ma
.
th em atical model of the process which is to be con­
This paper cannot be u s e d as an instruction manual for the practical
appli c a tion
rather
very wide
of
is to
dif fer e nt methods- not e.ven as a reference paper.
give
Its aim
a re a sonably comprehensive picture of th e recent develop:­
ment of thi� special field, to point �out and to �ompare the pri nc ip al i deas
of d i ffe r ent approaches to the solution of the given problem.
stressed that such
a
It should be
survey cannot be complete and the expressed opinions
· are influenced by our kno\vledge and experience which are limited in many
respects.
-
The literature on system identification is now quite extensive.
�t the
end 'of the paper, we list only the recent papers which were available for us
/
and wrJ.ch app eared after the third IFAC Congress in 1966.
The anterior
literature is cited only where it is necessary for our exposition.
For a
more complete list of references. the reader may consult the s�rvey
paper by Eykhoff, Van der Grintern, Kwakernaak and Veltman (1966).
Generally ..peaking, there are two ·possi >le ways to determine the
.
math ematical model of a given physical process:
fRescarch
supported in part u nder Grant No. NGR 05-007-122, NASA.
2
1)
Mathematico-physical analysis based on general physical laws
(process dynamics) and
2)
Experimental identification where the main information about
the process is obtained by measurements.
Only the experi­
mental identification will' be considered in this paper.
How­
ever, when the experimental identification is applied, the
black box approach should be considered only as a last possi­
bility.
In practical si�uations, even a rough analysis together
with engineering experience usually can provide very viuuable
a priori information about the process studied, a.bout the most
suitable fonn of the mathematical model and perhaps even
approximate values of some parameters.
Actually, the process identification is always only the first step in
the solution of a more complex control p:roblem.
Both identification and
synthesis of the control system shoul.d. be considered together.
v�ry easy to say but much more difficult ·to realize.
This is
The princ�pal difficulty
is tflat·the mathematical description of the process must be adequate for
the conditions under which the system
will be operated but, these conditions
can be known only alter the synthesis for which the identification is required.
Perhaps this is the main reason why the identific�tion i-s so often handled as
a separate problem.
Nevertheless, the . final goal must be always considered.
Prom this -paint of view, the
a)
following situations should be distinguished:
O,peg::Loop Optimal Control
Tbe- 1lnal
so
goal
in this case is the determination of the input signal
that the resulting process is optimal with ·respect to some
critP.rion and, may be, to some r�strictions.
The task of syst"'m
identification is the determination of a mathematical model, the
output of which would be as 'close as possible to the output of the
real system for any inputs and initial conditions which come into
the consideration.
The system can be usually considered as a
3
deterministic system, the influence of noise and random
disturbances is of minor importance and often only measure­
ment errors can be considered.
On the other hand, also
knowledge of the initial conditions ; s often required and their
determination can be considered as a part of system identifi­
cation.
b)
Closed-Loop Control of Deterministic Systems
a
A typical representative of this case is
(Figure 1).
servomechanism
The problem which is usually solved- and for
which the system identification is required- is the determina­
tion of the correcting filter F so that the closed-loop is stable
and the output Siignal
possible.
x
follows the input signal w as well as
The input signal of the closed-loop w can be deter­
ministic or stochastic one.
The mathematical model of the
system S must be determined so that its behavior in closed
loop is the same as the behavior of the original system.
The
initial conditions are usually out of our interest in this case.
c)
Closed- Loop Control of Noisy Systems:
In both foregoing cases, the system could be considered as
deterministic and the only stochastic errors, which are to be
eliminated in the process of identificatioJ;J. are the measure­
ment errors.
But many industrial plants cannot be simply
considered as deterministic systems.
They are very often
influenced by stochastic disturbances which cannot be measured
and sometimes even the source of disturbances cannot be exactly
specified.
As an extreme case of this kind, the regulation of a
noisy system (in Figure 2) can be given.
The purpose of the
regulator R is to suppress the influence of the noise and to
reduce the deviation of the output to minimum.
6his situation
·is ofte;n met in mass production (chemical plants, paper making
machines, power plants, etc. ).
Such systems can be treated
as stochastic transformation of the input to the output and the
-task of system identification is to det�rmine some statistical
characteristics o'f this transformation which are necessar! for
the design of optimal controller.
We have mentioned only three �ypical examples to point out the wide
variety of real systems and different possible interpretations of system
identification.
For the simplicity, we shall consider only signle -variable systems
(systems with single input and single output).
Joing
Most methods. which we are
to c:liscuss can be - at least theoretically - extend<ld also for multi­
variable systems.
2.
CLASSIFICATION OF METHODS F'OR SYSTEM IDENTIFICATION
F rom the mathematical point of view, the experimental system iden­
tification can almost always be considered as a prt-.blem of finding· extrema
of tunctionals.
The form of the functional, the extremum of which is to be
found, is given by the criterion accepted for the system identification and
by the mathematical model of the system.
Therefore, the accepted criterion
olideDtitication and the accepted form of the mathematical model of the sys­
tem are apparently the most significant features of every method.
More -
ewer, the criterion of optimality also determines in what situations the
given method
can be used with some assurance or. success.
To achieve the desired extreme different co�puting technique can be
applieti.
From this point of view, two large groups of methods
can
_
be dis­
tinguished.
The method of the first group c a n be denoted as direct methods.
They
do not use the physical realization of the mathematical model of the system
and handle the identification simply as mat he m ati c al problem of finding an
extrem e.
According to the form of the· functional they producP. either the
5
expli c it mathemat ical relation f or the unknown para..•neters or they
apply
di ffer-ent h ill -c lim.b in g ;.;nd v ariat io nal numerical pr o cedur e s .
They
The m odel-adjusting meth ods f orm the second group.
phy s ical ·realiz<:'.t ·.on
tec hni que .
of t he m odel often in c OJm ..!ct ion
extent dep'"ndent on t he kind of the input signal which excites
Where it is p os s ib le to choo s e t he iniJ
m ar k e dly influence the accuracy
·•
the
with analog computing
T he exp erim e nt al identification of a physical system is
system.
use
to a
gt·eat
the unkno\',T!I
s ig nal, the cho ice
can
of the identification or simplify the wt-.ole
pr oc edur e .
Summarizin g what was said above, we
cr it er ia
can formulate the follo\vi.n,g
for c las sific ation of differ ent ident ific at ion methods:
I
a)
W hat cr iterion of opt im al ity for system
b)
What form · of the mathematical model of the system is contsidcroed;
identification is accepted;
c)
What computing te c hni qu e is used;
d)
What kind or form of the input signal is a pplied.
By combination of these ·p os. sibilities,
we can
diff e rent methods for sys tem ident ification.
get an immense number of
Some of them,
which seem
t o us as to be t he m ost im po rt a nt in pr e se nt de velopmen t of control theory,
will
be discussed in Section 4.
For the reason� me nt io ne d above, we shall
use the cr it e ri on (a) as the most important .
The most
frequent form s of
mathematical m odel s u s ed f or description of c ontrplled
systems will be
discussed s ep arately in t h e next section.
3. ·
MATHEMATICA L MODELS OF SYSTEMS
The choice of t he mathematical m odel. or more
precisely, the form
. of the input -output relat ion adopted, obviously plays a major role in the
i dent if ication pr ocedure .
The most suitable form of the
man y factor s among which the most imp ort ant are:
model depends on
6
a)
The purpose for
b)
The physic al nature of
c)
The prior
which
the identification is unclcrlakcn
the process.
knowledge about
the system
studied.
In this section we shall de s c ribe the mathematicr.l models which a rc most
commonly used for the desc:r;-iption ot systems in the time domain.
3. 1
Determ inistic
�ystems.
Systems with Finite Memory: Linear Systems.
For
a
linear time­
invariant cont rollable system, the output v (t) is given in terms of the input
r
u (t) by the convolution integral :
v (t)
=
=I
�
t
h (t - a ) u <a>
.
-eO
da
0
h (cr) u (t -a) da
(3. 1)
where h (t) · is the impulse response (weighting function) of the system.
j
It
is assumed that the system is stable :
ao
. o
I h <t> I dt
< o0
(3. 2)
In the time -discrete case (systems with discrete input and output sampled
.
with constant sampling period) similarily:
GO
v(k)
=
}:
0
(3. 3)
h (i) u (k-i)
and in ( 3. 2) sum instead of integral appears.
of a
The
impulse response
h(·
) fully determines the dynamical properties
controllable linear system and when this function is determined, the
system is clearly identified.
For inputs With a given bond M, given " arbitrarily small it is clearly
possible to find T
M
such that
I
7
l
00
T
I
h (a) u (t -a) dv
.
M
<
�;
,
t >T
.
/
M
This fact enable:.. us to approximate a linear E-y stem (with infinite
memory) by a system with
1
finite' memory and to assume that
h (t) = 0 for t > T
time'.
' Here T
is also referred to as the
M
M
In discrete case we get:
1
settling
M
v (k) =
L:
(3. 5/
h (i) u (k -i)
0
where M is the "practical length" of the impulse response (:nemory
of the system).
For computational reasons, the formula ( 3. 5) is
·often used aiso in the continuous case taking h (i) as h (i.t.T).t.T
where AT is the step of approximate integration which mu st be
sufficiently small.
By this approximation, the problem is reduced
.
to the estimation of (M + 1) unknown parameters h (i) , i = 0, 1, . . . , M .
From the computational point o f view the
verv
favorable feature of the model (3. 5) is its linearity
parameters h (i), i = 0,
.
•
•
M.
i m n:;J:-tant
111
and
the unJ\ nown
This fact significantly simpHfies
the procedure of identification and is undoubtedly the main reason
ascribable for the wide popularity of this model.
The "settling time" of the impulse response has to be a priori
known.
It
does not need to be very critical in many practical situations
where this value can be well estimated from the physical nature of
the system. (Taylor· and Balakrishnan 1 967)
As
a gene!'ali zation of the represe�tation of the impulse response
by samples, we may use the Ra:yleigh -Ritz forms :
M
h (t) =
L:
i=1
{3 : lf>. (t)
1
1
( 3 . 6)
8
Wbere the
functions �Ut) are predetermined.
1
exponentials is
to
be
u sed for this pu rpose, {3. being the parameters which have
1
.
Apparently, the class of linear system s which can be
e stimated.
described
.
Usually a set of orthonormal
by the impulse response of the type (3.6) is restricted by the
choice of the base fl.(t), i
1
=
1,
•
•
.
, N.
The disadvantage of the model ( 3. 6 )
(which i s often criticized) is the fact that even a very simple impulse response has to be
(3. 6)
if the
approximated by relatively large number of members in
This disadvantage can be
.
removed by introducing further adjustable par�meters into the base rzawa
and Furuta
ba se does not suit the given case.
)
(1967) .
(
However, it considerably complicate s the proce ss of
identification.
Nonlinear
Systems: Volterra Expansion
Any nonlinear
system that is controllable can be described by the Volterra
expansion:
CO
v(t)
GO
r
�11
n=
o
GO
.. Jr
o
.
w (t ' . . 't ) u (t -t ) . . . u (t -t ) dt
l
l
n l
n
n
•
·
.
.
• • •
dt
n
( 3 . 7)
When the system i s approximated by a system with finite memory ( m) and
_
the functi.onal power series is truncated to finite number (N) of terms and
the integration i s
The
replaced by summation, we obtain:
Kernels
h.( •
l
.
•
) can obviously be taken to be symmetric functions.
.
The Equation (3. 8) can be u sed as a mathematical model for a fairly wide
class
of nonlinecu-. systems and may tie regardt:d as generalization of ( 3 . 5)
for the
nonlinear
fication. the
are
its
-
ca se
main
gene rality
(Balakrishnan 1963).
From the viewpoint of identi­
advantage s of this type of desc ription of nonlinear systems
(the a priori knowledge of the structure is not required)
and it s linearity in the parameters h. ( . . . ) which have
.
J
ever. the dimensionality inc reases rapidly with N.
to be e stimated.
How-
9
F( (t))
- F (u (t)) , only the odd terms of expansion (3. 6) may be considered (Taylor
.
and Balakrishno.n ( 1 9 67)) . Another way to reduce the d.imcnsionality of the
problem is to introduce the adjoint space (Balakrishnan ( 19G3)) , so that the
If the
1
odd' symmetry of the system can be assum cd,
-u
=
Kernels may be assumed to have to form:
and only the single function f (.) needs to be deten11ined.
Dynamic
Systems
When a dynamic system model is available from energy /force balance
equations, the identification problem becomes a parametric one and has the
advantage in permitting the use of non-steady-state data.
The basic model
is the relation:
f
( (n) (t)
V
>
•
•
•
V
(t);
U
(m)
( t)
>
•
•
•
U
(t); J31
>
•
•
•
J3
n
)
=
0
( 3 . 10)
(i)
(i)
where v (t) and u
(t) are the derivatives of the output and input signals
and J31 .f3M are the parameters to be determined.
.
Often the more general
state space representation
Z
=
F(�. �· [}_,
t)
( 3 . 1 1)
V ':' H (Z , �· t)
can be used where Z is the vector of suitable chosen state variables and
�· v are generally ·multidimensional input/output signals.
In ( 3 . 1 0) or ( 3 . 1 1) the form of the functions f or
F
and H as well
as the order n (dimension of state space) are supposed to be a priori know .1
.
In such a case, the problem can be reduced to that of finding the initial state
of a more complicated but known system.
form (for single-input, single-output case)
,In the linear case (3. 10) takes the
•
10
( 3 . 12)
and we assume m � n .
In state spa -:e form; we have:
z = A Z +Bu
( 3 . 13)
u = C Z +Du
and a., b
1
and �he elements of the matrice s A , B arc now the parameters
•
formerly denoted as {3.
It
must be noted that in the form (3.1 3 ) the system
need not be obse rvable or controllable.
in
In the discrete case, the past values of particular signals appears
( 3 . 1) instead of the derivative s:
n
m
0
0
� ai v (k - i) = L
b u (k-i)
i
( 3 . 14)
or more generally,
Z (k+1) = A Z (k) +B u (k)
V(
k) = C
z
(k) + D u (k)
The differential/ difference models seems to be most convenient from the
view point of the number of the parameters to be estimated as well as the
relation to the control problem.
Of course the order n of the system has
to be known; experience shows tha.t most pracitcal plants can be described
·
with good approximation by the differential or difference model of third or
less order. * Such models have also been used for distributed parameter
systems ( Peterka and Vidincev - 1967).
*
u the order of th� system is not a priori kno�n, it is usually determined
in practice by making several different estimates of different orders and
choosing the minimum order which give s a suitable fit to the data. Of
course, this procedure can be misleading.
11
3. 2
N o i sy Sysiem s
By a noi sy sy st e m
we
sh a ll mean one in which tlte sy slcm i s bJ e ndcd
with an interna n oi se s our ce
errors in the in pnt -output
quite apart fr om (and to be distingui shed fr om)
Th<' noisy system may often be
measurements.
decomposed int o a det e rmini stic system and an
(such as
1
Figure 3 .
1
intc·rnal1 additive noise
load disturb ance') corrupt ing the ideal output v, as shown in
Su ch a stru cture is c ommon in
1
regul at o r ' probl em s where the
the n oise h as t o be c om pen sated by feedback l: on trol .
u n de sir ab le influence of
In the simple linear case, the determ ini sti c sy stem is linea r,
and
a
discrete
Wit h r e ference to Figure 4, the spectr al
ver si on is illustrated in Figure 4.
density of the noi se i s:
� (z)
=
a
2
E
F (z) F ( z
-1
(3. 16)
)
where a2 i s the variance of the disc rete whit e noi s e
E.
E
density (3 . 16) and th e pulse -tr ansf er function G(z) c an b e
If b oth the spectral
expressed as ratios
of polyn omials:
G (z)
1
b (z - )
n
F(z)
(3 . 1 7 )
where we have u sed the notation:
1
b (z - )
n
11
\' b
LJ k
=
0
k
z-
The noisy system as a whole can
and similarly for the other polynomials.
be described by the difference equ at ion :
N
2; Ai x (k - 1)
0
where N
=
n
of polynomial s
+m
N
2;
0
Bi u (k
- 1) +
n
2;
0
and the coeffic ients
.
ci
c
A.,
1
(3. 1 8)
(k-1 )
B.,
1
and C. are the c o e ffici ent s
1
12
1
1
1
1
l
-1
- 1
- 1
A ( z- ) = a (z - ) d (z- ) B ( z ) = d ( z- ) b ( z ) ; C ( z- } =a (z ) {3 ( z 1 )
n
n
m
' n
m
n
n
n
m
-
·
It
1 and the standard deviation er
so that also C = 1 .
o
o
e..
The remaining c oeffici ents A., C . (i = 1 , . . . N) and B. (i=O, . . . N) are enough
l
1.
1
.
.
to determine the minimum variance {steady state} control strategy (Ast rom
is useful to choose A
=
•
.
9
1 67a}.
4.
C OM PARISON OF SOME APPROACHES TO THE IDENTIFICATION
PROBLEM
According to the kind of c riterion which is used as a measure of fit
between the mathematical model and the real process, three lar�e groups
of identification methods can be distinguished:
4. 1
a)
Minimization of output error (or ' fitting' error}
b)
Minimization of equation error
c)
Statistical error criterion
Minimization of Output Error
·
The principal idea of this approach is to determine t he mathematical
model so that when excited by t he same input signal (Figure 6 ) its output
reproduces the output of the real system as well as possible.
denote the observed input and z (t) the observed output .
In
Let u (t}
the simplest
of situations, we may assume that these are stationary stochastic processes,
and that the · system is stable, and that steady state has been attained:
S
If a
linear model of the system is assumed, for example, t hen the fit error i s :
e (t) = z (t) .-
""
h ( s) u (t - s) ds
0
( 4. 11 )
and we may seek to minimise: the mean square error :
q
In
=
lim
T-oo
1
T
j_r
T
0
e (t)
2
dt
(4. 1 2 )
this case the optimal h( . ) satisfies the well-known Wiener-Hopf equation :
13
«>
1/J
It
(t)
uz
=
('
j_
0
h ( s) �
uu
(t - s) ds
t � 0
(4. 1 3)
is, of c ourse, essential in this that the observed input is also the actual
input to the system, and thus this is basically an
1
open loop' method.
In practic3, even apart from the problem of numerical solution of
the Wiener -Hopf equation (.Fortova, Kryze, Peterka, 1 9 6 6) , the re sults
using this method are not satisfactory vrhen applied to real systems under
..vrma.l ope rating conditions (using ope rating d;; �"::�., as opposed special test
inputs) .
See also Ehrenburg and Wagner ( 1 9 6 6 ) .
c onside ring the many assumptions involved.
This is not too surprising
In fact among the main reasons
for the lack of success we may m ention: a) the solution of (4. 3) is very
sensitive to e rrors in the estimated correlation functions, especially when
the autocorrelation function of the input is flat; (b) to obtain the correlation
functions with any degree of accuracy, a, long interval of obse rvation is
required which usually violates the assume d stationarity; and c) short -ti.me
correlation functions lead to error even when no noise or measurement
errors are present.
As is to be expected, the method provides good results when special
test signals are employed, such as pseudo -random input signals to excite
the system. (Briggs, Godfrey and Hammond 1 9 6 7 ; Krotolica 1 9 6 7 ; Davies
and Bruce 1 9 67 . )
When a finite settling time or finite memory c an be assumed, and the
input is noise free, it is not necessary to adopt the stochastic point of view
or to employ special input signals.
lL (
M
JM
Instead, one can directly find the best
h(.) (in the linear case) that minimizes
z ( t) -
o
h (s) u (t - s) ds
)2 dt
( 4 . 1 4)
14
where L is the total data length.
then also be used.
Sec Taylor
The nonlincar model (cf., 3. 7, 3. 8) can
196 8 for a concrete application.
The criterion (4. 14) is often re fe r re d to as
1
Least Squa res for Fit
Error' , and in th.: discrete case can be formul.aicd as that of solving the
' ill-posed' equation
Z
=
F {3
(4. 15 )
leading to the vre�J.-known
1
pseudo-inverse' solution:
( 4. 1 6 )
where Z i s the observation vector, and {3 i s the vector parameter to be
estimated, the (rectangule.r) matrix F being composed of known elements
It is essential in this method that the input be
depending on the input data.
noise free. <:ltherwise the estimate will be biassed.
or
In the discrete version
the iinc-� ·c:�!'fe· (4. 14), we have for example:
where·
f
ij
=
_
u(i -j)
,
i
=
M,
. . . L;
j
=
. .
0, . M.
and F*F' has the elements:
L
�
k;,m
u (k
- i)
u
(k - j)
0 � i, j �M
A numerical problem is the inversion o f the matrix F *F'.
(4. 1 7 )
Obviously
any method of factorization is of advantage and in particular when updating
of the estimate is required as L L11creases (as more data is available).
It should be notec: that if a pseudo random inpu' signal can be used, the
inverse matrix can be explicitly calculated, leading to tremendous sim pli­
(
fication of the iqentification algoritlun. · Briggs, Godfrey and Hammond
1 9 67
.)
Other special signals may also be used where permitted, to provide
the necessary diagonalization.
(Roberts 1 9 6 7 ; Mi:ntz and Thorp 1 9 67;
15
Egorov 1966.}
Because of its importance we shall now briefly describe an
example of a pseudorandom signal .
. Pseudorandom binary input signal
An example of pseuaorandom. binary signal is given in Figure 8 .
is a deterministic periodical signal with the following properties:
=u
u(t)
k
=
1
PP
1
pp
where
11,
p
± K
for
kT � t � ( k + 1) T ,
.=
j_Pu . u
k -1 k -J
k�1
1
K
2
u
. k
It
u
k +vp
=
for .
1 = 1 + vp
2
K
-p
.
for 1. -.t J. + vp
fu=!£
k
p
k=1
and
p· are integers and K
is either positive or negative constant.
When periods of �his input signal are applied to the unknown system
and the discrete impulse response is considered the
T
F F takes the form
F
T
F
[�
1
-1 -1
p -1
-1
1 1-
and
(
1
�T_f )
p
-1
1
2
K (p+l)(p-M)
-]
(<M+ 1)
x
(M+ 1)) matrix
-1
-1
p
p+1-M
1
1
1
1
p+l-M
p+l-M . ..
1
1
The pseudorandom binary signal can be very easy generated and
·simplifies the whole procedure of identification.
characteristics,
the pseudorandom binary input signal can be recommended
also for other types of models,
utilized.
Because of its spectral
when its other properties cannot be fully
16
Dynamic System (Differential Equation) Models
The minimiz ation of fit error when the system is dynamic (including
distributed parameter systems) leads to a nonlinear problem, e;·"n when
For a general iormulation see Balakrishno.n
the unknown sysh.:m is linear .
1968a.
It is convenient to use the state space formulation
generality.
because of its
We shall first consider: the case of an unknown linear system.
Thus let the system be represented by:
x(t) = A x(t) + B u(t)
where
u(.)
is the known noise-free input,
of measurement errors,
n(t)
and the observation
is expressed:
z(t) =Cx(t)+n(t),
where
(4. 18)
O<t
is the output observation error or noise.
fit error method is to minimize:
1
a
b
z(t),
because
(4.19)
The least squares
11 cx(t) - z(t) ll 29t
(4. 20)
.
where [a, b] is the interval over which the fit error is considered and the
minimization is 'over some or all of the parameters in the matrices A, B,
andC.
In (4. 20), x(t) is determined as soon as A, B,C are specified,
along with the initial state
x(o); indeed:
x(t) = T(t) x(o) +
with T(t)
'a'
=
l
0
t
T(t - s) Bu(s) ds
(4. 21)
exp At. Often it can be assumed that the system is stable, and
is large enough so that the first term in
(4. 21) is negligible; otherwise
the initial state l·as to be another unknown also to be determined.
of the structure
In terms
(4. 19), the minimization of (4. 20) can be interpreted as a
maximum likelihood estimate for n(t) 'assumed to be white Gaussian. (In
the continuous case, strictly speaking,
the integrals involving the observed
data must then be interpreted as Ito integrals).
17
Unlike the cnsc oi
in
is
(4. 20}
no
n
linear systc1n
rith fi.nitc memory,
longer lineal· in the unkno1·:ns.
\Ye have thus
ta�ional comp�cxity associated \':ith fiJJding the minimum of
�!·adient m e t h od s appear to be slo'v,
functional.
Ncwton-Rophso
and
the functional
al l
a
the compu­
nonlinear
a ve rsion of the
reconime:nt.lcd in Balakrishnan 1968a has been employed
1
succes sfully for the determination of aircraft s t a b il ity derivatives in T ay l or
1963. For white Gaussi.an noise in
is asymptotically unbia s sed.
Of interest>
in this pr ob lem
( 4. 19},
the maximum likelihood e st imat e
is the on -line
updating method as more
01
data is taken, tiw.t is, the ·estimate as a function of the
(4. 2-0)..
Thi� is already treated i n the survey
(19£2� and t>e·ter�nce
f>roblem of
as yet .
It i s
Thus
we c
clear that (4. 20) can be
is an
=
(
F x.(t); u(t);
No actual
(4. 18) by
the
howe\'er,
are not linear.
more gcne1·al form :
(4. 18a)
\tnknown vector parameter to be determined.
�$ m.ore comp li cat ed
in
industrial
e xte nde d to sy stem s wh ic h
13)
b
paper of Cuenocl and Sage
me.de to that work.
oul d easily (in theory) replace
de(ermine
limit
any ma gnit ud e has been reported using this method,
x(t)
where ,B
PlAY be
u p pe r
The problem
in that now a nonlinear equation has to be solved to
x(t) for given (3.
Recently a meth od whe reby· one may avoid
hav ing to solve t he dynamic equ at i ons has been reported in Bal akrislman
1968b.
Paral l el adaptive models
Another method {or minimi�ing the output error is the application of
model-adjuati:ns.tedtnique (Figure 9).
The mc..in feature of this technique
i s the p.hys-it:al realiz-ation of the model and the application of the principles
of ·extremum control for adjusting of unknown p aram et er s.
dures for �djustihg of
� odels have
Reinisch and Wernstedt
bee n proposed
;Many proce­
( Kokotovic et al.
(1966 ),
'(!Mi'/), Ra ke (1966), Nor'kin ( 1967), Dym oc k et al. ,
'
18
I
)
(1967). J)oganovsky (1968). Plander (1967), Chadeev (1967) .
popular method for automatic adjustment of parameters. is the
The most
..
Sensitivity
method."
Because of theoretical dfffi_cult ies and lack of more practical experience
.
it is very difficult to compare and rate all the different existing adjusting
procedures.
An attempt in this direction has been made by Parks (1967)
for a simple example and three chosen methods ..
.
.
The following trends in model adjusting methods seem to be most
significant: .a> stability analysis of existing schemes,
c) noise influence analysis,
instrwnentation,
of adaptation_
b) simplification of
d) improvement of speed
e) increase of the sensitivity according to some parameters
by a�ation of the model to the unknown system in closed loop with known
feedback.
4.
2
Minimization ot' Equation Error
Let us suppose that the input -output relation of a dynamic system c�.n
be deseribed by the differential Equation ( 3. 10) and let us replace the output
.
v<t> and the input u(t) by their observations '; (t) and ;:i(t) and the param-
.
"
.
eters /3 by their estimates {3.
f
(... (n)(t)
V
•
• • •
...
V
(t),
"' (m)
U
'
.
The resulting unbalance in the equation
(f),
.
•
...
�U
�
(t), �
)
=
g (t)
is calle d the "equation error" and can be used as a measure· of the deviation
of the behavior of the model from the behavior of the reai system.
the· equation error
To form
q(�) requires n derivatives .of the system output and m
derivatives of the input wh�reas usually oxlly the input and output are directly
available
•
.
This difficulty can be overcome in the linear case through the use
of 11drivative filters" operating on the output c.nd the input.
However, thes�
filters must have two properties: comJI?.utativity with the operators in the
differential equation model and sufficiently wide bandwith.
An example how
this idea can be used in the case of linear lumped parameter system is
given.in Figure 10 where A(s) , B(s), A(s) , B(s) and Q(s) are polynomials
19
in s (differential operatol") . The polynom ial Q(s) ensures the physical
realizability of derivations and its degree is equal to or greater than the
order of the system . The model operating on both t he system input and
the system outpu� is sometimes cal le d the " ge.Jeralizcd model." The
minimization of equation error has also been rece ntly used for identification
of nonlinear systems (spragu e and Kohr (196�), Hoberock and Kohr {1967},
Lion (1966), B ut le r and Bohn (1966)) .
The scope and limitations of this approach can be illustrat d by
conside ring a J.inear discrete proce ss desc ribable by the diifercnce equa­
tion model ( 3 . 14). Let u.s assume that the observed output x(.)t is
similar to (4. 19) with additive noise d.):
e
x(k) v(k) E:(k)
+
=
We can then rewrite ( 3 . 14) in the form:
n
n
n
·i)- � b.u(k-i) dk) + � a. dk-i)
x(k) + � a.x(k
1
0
1
1
.
=
=
q (k)
1
1
(4. 22)
Here q (k) is the eq uation error, in our terminology, and we have asswned
the coefficient a0 ·in ( 3 . 14) to be nonzero, and taken it to be unity without
loss of generality. Suppose now that from the observations we can write
N such eq uations, k
0, 1. ... N-1, and N is s.uch that N is much larger
than 2n. Let x 0 denote the column vector with components x(i), i O , N -1,
let ' a' denote the column vector with components a1, and similarly b' the
column vector with components
b.. Then we may represent the N equa.
1
tions in matrix form as
=
·
=
• • •
'
x0
-
Xa
- u·b :: q
:: E'
a
·where X, U are rectangular matrices with components read off from (4. 22)
and q is the column, vector ,.,.ith components q(i). Let us denote the N-by­
(2n+l) matrix L
E',
twe use x(.) here to avoid confusion with the z' s in z-transform!
20
L = [ -X , U]
Then denoting by {3 the (.2n+l) column vector of unknowns :
a
b
we have that
X
0
- L {3 = q = E: a
(4. 23)
To minimize the equation error we may use the
1
weighted least squares'
criterion, namely, we determine {3 so as to minimize:
where R is a suitably chosen positive definite NxN matrix.
solution
�
The optimal
is then given by:
1\
q = ( L* R L)
-1
L*R x
(4. 2 5)
0
If we substitute into this fliOm (4. 2 3) we have that :
1\
q = q + ( L*RL)
-1
L*
€
(4. 26)
a
so that the error in q determined by the second term, whose expected
value is not zero: so that the estimate is biassed in general.
However
the identification problem is reduced to a linear one, unlike the case where
we minimize the fit e rror . Many methods of choosing R haye been pro­
posed in an effort to remove the bias and also to aid in the upd ating or op.­
line feature . The so-called
1
instrymentation variables 1 method
) may be _classified as one such .
Polak (196 7 )
(Wong and
More recently Peterka ( 1 968)
has int..·oduced an on-line technique in which the R matrix is determined by
the input sequence.
It
would appear safe to say that one of the current
needs is find a method which is bias free and simple to instrument on line.
Stochastic Approximation
When the input can be assumed to be a stationary stochastic process,
stochastic approximation is an attractive on -line m e thod for minimizing
21
the equati on error:.., for example
q in (4. 23).
It has also the advantage
that no kr10wJ.edge of t)le input or output pr.oce s s statistics is requi red.
However, the re sults ob�ained using the: m ethod, which is e ss entially a
gradient m ethod with predetermined reJ axatioJl factors, have not lived
up to earlie r claim s .
the slow convergenc e .
A
{3n+l
Among t!1e maj or difficulties would appear to be
The general form of the algorithm is :
1\
=
{3 n+ ,..n
'1
1\
{3
( 4. 27)
Q (;3 )
n
where {3 i s the u:lknown paramete r, and
�n
tht estimate at the nt h stage,
Q( . ) is the equation error ( a s suming f3 t o b� equal to
�n) at the nth sample .
The relaxation factor -y
is a predetermined sequence, usually
n
n -y
n
=
(4. 28)
c onstant
Tsypkin ( 1 966), Holme s ( 19 67), Zhivagladov and Kaipov ( 1 9 67), Chadcev
( 1 967) lme zadze and Lelashvili ( 1 9 6'1), Panuska ( 19 6 8) .
The m ethod ca:-t
also be applied to sys�ems with finite mem ory u s ing a V olterra type e x ­
pansion.
H olm e s ( 1 9 67), Roy and Sherman ( 1 9 67 ) , Nagurn o and Noda ( 1 9 67 ) .
The socalled ' learning te chnique' i s very similar to the stochastic approxi ­
mation schem e .
The theory of convergence is rather involved, and the
many assum p tions to ensure it difficult to verify i.n practice.
Sakrison
( 19 67), V alis ( 19 68) .
C oncluding Remarks
It is apparent that the
1
Least Square s -Equation Error' techniques
have many advantages in practice such as high parameter sensitivity,
cc.. s e of on -line use, and above all simplicity.
Simplicity i s particularly
significant in the case of linear systems, where the equation error i s
actually linear i n the unknown coeffic ient s .
drawbacks.
However there are many
Param eter fit does not nece ssarily ensure the output fit
even using t he same input as that in the identification mode, e specially
when the model a ssumed is not accurate .
This is particularly serious
22
in open loop control.
control.
But the situation may be different in closed loop
For instance
a
discrete controller operates only on a finite set
of input -output data of the system ; and the relation between these neigh ­
bouring data can
be
the most important inform ':l.tion about the system which
From this point of view the minimization of equation error
is required.
may well be reasonable.
4. 3
STATISTICAL ERROR CRITERIA
As we have seen, the identification problem is essentially an esti ­
mation problem.
Thus when the necessary statistical description i s
available, optimal estimates can b e based on statistical criteria.
a procedure not only frees us from ad hoc methods.
It
Such
also provides us
with a quantitative numerical evaluation of the error.
.
Mean Square Error Criteria :
Let the vector X denote the total observation (including input and
output) , and let
{3, as before, denote the system parameter vector to be
If the situation is such t hat the j oint distribution of X and
determined.
{3 can be specified explicitly or implicityly, then a statistical criterion
that has been much studied is the mean square error c riterion
E
( 11 � - f3 !1 2 )
1\
where E denotes expected value, and {3 is the estimate, a "function of X.
The associated theory is well -known when the estimates are constrained to
be
·linear, but considerably more complex when no such restriction is place1
Stratanovic h ( 91 6 7 ) ; Kushner ( 1 96 7 ) .
Unfortunately, in the identification
problem, the estimates are nonlinear even when the system is linear.
In
fact the identification problem for dynamic systems is no less difficult
than the general nonlinear filtering problem for Markov processes.
This
becomes clear when we note that every such identification problem can be
formulated as the problem of estimating the initial state of a known, but
albeit more complex, system .
Balakri shnan 1 9 6 8) .
( For example see Cuenod and Sage ( 1 9 6 7 ) ;
Thus suppose as i n ( 4 . 1 8a) we have :
23
x (t)
=
v (t)
=
(
)
F x (t) ; u (t); J3
( 4 . 3 1)
Cx (t) + n (t)
where U:( . ) is th.·; input and v(. ) the observed :rutput.
If we form a new
vector
Y
[x, 131
=
and add the equation
�
0
=
we can rewrite ( 4. 3. 1) in the form :
Y = F (Y ; u(. >)
v (t)
=
(4. 32)
H(Y)· + n (t)
and the problem of determining J3 is the same as that of determining
from v ( s), 0 < s < t.
E [Y(O) I
Y 0)
(
Of course, it is also well known that the optim al
]
mean square estimate is given by; .the conditional expectation:
v ( s) , 0 < s < t
But the problem of . evaluating the right side is a complex one and at present
no effective constructive method has yet been found.
Many approximations
have been suggested but without any evaluation '!f the associated error.
Kushner (lg67 ) .
Method o f Maximum Likelihood
p (x
I
When (in the terminology above) the conditional probability density
J3) of X given J3 is known, (or calculable) , we can base the optimi ­
zation on the me �hod of.maximum likelihood. Thus we define the optimal
(:J
as one that maximizes for given x
I
I
p (x J3) or log . p (x J3)
24
unii ke the mean squar� error me thod, t his crite rion lead s in m any ca s es
to a v ariational proble m (of the ty pe m et wit h in opt ima l cont rol) .
not necess arily confi ned t o this cas e,
A ltho ugh
it is m ost use fu l w hen t he inp ut is
nois e free, and the out put is cont am inat ed by addit iv e G at� s s ian noise,
b ecaus e in this cas e it reduces t o a !'east s q ua re s crit erion
(4. 2 0} , (4. 14) .
T hus the minimum of t he functional has a direct int erpret at ion in te rm s of
the error in· the res pons e to t he giv en input, when it can be assum ed t hat
the input is nois e free .
T ay lor ( 1 968) .
Astrom
it for the l inear noisy sys tem de s cribed by
conditions .
( 1 967) has als o em ploy ed
( 3 . 1 8), ass um ing st eadyst ate
T he maxim um l i kelihood est im at e has als o the des irabl e pr ope rty
that it is a sym pt ot ically · unbias s ed.
tribution of
.Moreov er when t he condit ion al dis ­
X giv en {3 is known, one can us e the well- knownCramer-R ao
i nequality for the es timat e v ariance:
.
1
where J - is calcul able from the condit ional dist ribut i on ,
and t he right ­
s ide can be us ed as a meas ure of t he achiev able accura cy. It is k nown
.
that the maximum li kel ihood es tim ate achi ev es this lower bound asy mp­
totically under the conditions us ually s atis fied by ident ificat ion problem s .
S uch cal c ul ations (in the s teady s tate cas e ) hav e been pres ent ed by
As trom (1 967 ) for
5.
( 3 . 1 8 ).
AC
C
URAC
Y O F IDENTI FIC
A TION
Th e problem of prov iding s am e est imat e of t he accuracy of dete r­
mination is obv ious ly an import ant one, and cont inu es t o att ract att ent ion.
T he us e of C
arme r�Rao ineq uality has already been not ed.
hav e als o been s uggest ed.
Ot he r approac hes
The relation bet ween the errors in t he t im e do­
main and freq uency domain for a l ine ar sys tem has been inv est ig at e d by
U nbehauen and S chl egel (1967) and als o. by St robe!
( 1967).
It was s how n
that ev en s mal l errors in st ep res pons e can ca use s erious err ors in t he
frequency res pons e.
The accuracy of ide nt ificat ion has been stu died in
detail by S tepan ( 1 9 6 7 ) from the v iew point of the be havior of th e m odel and .
· the system in closed loop with a proportional controller.
6.
C ONCLUDING REMARKS
. M any remark able resul ts hav e be en achieve d in the fie ld of system s
identification but still much remains t o b e done.
The best results ·O f experimental identifi cation can be expe cted whe n
the unk nown system is inv estigated in conditions under which it will be
controlled.
From this point of v iew the elaboration - of methods suitable
for application in closed loop seems to be v ery impor tant.
·
The significance of accuracy analysis of different methods for sys­
tern s identifications is indisputable. On the other hand the question what
accuracy is actually required remains still unanswered.
The stability of many schemes for mode l adj ustin g has not yet been
cleared.
methods.
The same holds for some iterativ e procedures and hil l climbing
The ident ification of nonlinear, systems and also the suitable de scription
of nonlinear noisy system is also_ a question which is still open.
In foregoing sections we hav e tried to outline the. ideas and trends in
·
t he present dev et opment of system id entifi cation. The limited space did
not permit inclusion of much recent wor k in the fi eld. S ome fu rther re­
ferences are listed below.
We cannot hope that all important pape rs on
this subj ect are known to us and that some of t hem undoubtedly hav e not
·
been omitted, albeit inadv ertently.
To facilita te the reader' s orientation we hav e keye d the references
in the foll owing way:
( a)
criterion using o utput error
( c)
maximum lik elihood
( b)
criterion using equ ation error
26
(d) frequency methods
(e) impulse response model
(!) Volterra expansion model
( g) application of orthogonal functions
(h) differential equation model
(i)
difference equation model
(j)
noisy system desc ribed by difference equation model ( 3, 18)
(k) correlation technique
(1)
stochastic approximations
(m) learning technique
(n) .model adjusting technique
(o) pseudorandom binary input
(p) analysis of identification accuracy
Refe rences
(p, j)
Astrom K. J. :
(c, j)
Astrom K. J. :
(a, h)
Balakrishnan A. V . :
On the Achievable Accuracy in Identification
· Problems.
IFAC Symposium on Identification in Automatic
Control Systems, P..ague, June 1967, pp. 1. 8.
Computer Control of a Paper Machine - an
Application of Linear Stochastic Control Theory.
IBM Journal of Research and Development,
Vol. 1 1, 1967, No. 4 (July), pp. 389 -405.
�nWJ,cat:iQll of Control SyStems from Input ­
.OUtput flata. .
aAC �asium on Identification in Automatic
Control Systems, .tune 1967, Pr�gue, p. 1. 1 .
Balakrishnan A . V . :
Stochastic System Identification Techniques.
Chapter in " Stochastic Optimization and Central, "
John Wiley and Sons, 1 9 6 8.
�ii:bnin � v.. ;
n A. New Computing Technique in System Identifi ­
.
etittl5n, u Journal o__t Computer and System Sciences,
Vol 2, No. 1, 1968.
(m, 1)
Bar�t J. , Muszely Gy:
Pattern Recognition Algorithm Adapting with
Searching Method.
IFAC Symposiwn on Identification in Automatic
Control Systems, · �une 1967, Prague, p. 1 . 4.
Bhattacharya B. P. :
An Approach to Identification of Piecewise Linear
Control Systems .
IEEE Trans . o n Aut. Control (Correspondence),
. vol. AC - 1 1, January 1 9 66, No. 1, pp. 1 3 1 -132.
(m)
. Biglow J. w.·, McVey E. S. , Moore J. W. :
Pattern Recognition in a Learning Automatic
Control System.
IFAc· Symposiwn 011 Identification in Automatic
Control Systems, June 1967, Prague, p. 1 . 5 .
(b, i, l) Bojanov E. S. :
Application of stochastic approximation method
for object characteristic regeneration.
Automation- and Remote Control, · 1967 , pp. 9 5 103.
I
28
(k)
Borisova R. W. , Ulanov G. M. :
Evaluation of Paramete r s of s'cat i on a t'y control l e d
plants in the identifi cation probl em . Automation
and Remote Control, 1 9 66, Ko. ]. l .
Van den Bos A. : Construction of Binary Ml• l t ifrequcncy T e s t Signal s .
IFAC Symposium on Idcnti.fication in Autoinatie
Control System s, June 1 9 6 7 , Pragu<:: , P. 4. 6 .
The Ide nt i fic at ion of Random Paramet e r s .
IFAC Symposium o n Identification in Autom a tic
Control Sy stem s , June 1 9 6 7 , Prague, p. 4 . 6 .
(a, e, k, o) Driggs P. A . N., Godfrey K. R. , Hammond P. H . :
Estimation of Proce ss Dynamic Characteristi c s
by Correlation Methods Using Pseudo Random
Signal s.
lFAC Symposium on Identification in Aut omatic
Control Syst em s , June 1 9 67 , Prague, p. 3. 1 0 .
Brainin S. M. :
Brigg s P. A. N. , Clarke D . W. , Hammond P. H. :
Introduction to Statis ::ical Identification Method s
in Controi System s Control , Vol. 1 2 , 1 9 6 8, 1 1 7
( Muc h) .
(a, m)
Cha de ev V. M.
(b, i)
Clarke D. W. :
(k, p)
. .,
Cshld R. , Gyiirki J :
Some Questions
:·
Adaptive Models of Plants
IFAC Syrnposiu1n on Identifi cation in Automatic
Control System s, Prague, 1 9 6 7 { June), p. 5. 8 .
Generalized -Least -Square s E stimation of the
�ramete rs of a Dynamic Model
· tFA'C Symposium on Identificat i o n in �1.::0:1\atic
i;otit rol Syst <?.ms, Prague, 1 9 67 ( J�e ) , ):: . 3 . 1 7 .
o f the Sy�tem Ident i fi cat i on by
Stochastic Signals
IFAC Symposium , on Identification in Automatic
Control Systems, Prague , 1 9 6 7 { June) , p. 3. 2 1 .
Cuenod M . , Sage A. P. :
Comparison of Some Methods Used for Proce ss
Identification
IFAC Symposium on Identification in Automatic
Co�rol Sy stem s, Prague, 1 9 6 7 ( June), p. 1 .
(a, k, c, o) Da:Vies W� ·};}. T, �-e:�· '1. L. :
On -Line System ldentification in the Pre s ence
Drift
of
IFAC Symposium on Identification in Automatic
Control Systems, Prague, 1 9 67 { June), p. 3. 1 2 .
29
Detchm endy D. M . , Bridhar R.
:
Sequential E s t i m at ion of Stat�s and Parameters
in Noisy Nonlinear Dynami cal System s
Trans. ASME se r . D ( Journal of Basic E ngi ­
nee ri ng) Vol. 88, ser. D, 1 9 6 6, 2 ( June)
pp. 3 6 2 -368 .
(a, h, n) Doganovsky S. A. :
·
Conce rning Identification Algorithm s Realized
by Param etric Control System s
Automation and Remote Control, 19 68, 4, pp.
1 5 5 -165.
( a , n)
Dymock A . J. , Helps K. A. , Meredith J. F. :
The Identification of Three Param eters Using a
Linear Plant Model IFAC Symposium on Ident i ­
fication in Automatic Control Systems, Prague,
1 9 6 7 (June), p. 5 . 7 .
(g)
Sgorov C. V. :
(k, p)
Ehrenberg L. , Wagner M. :
Erprobung der Methode der Korrelationsfunktioncn
fiir die Bestimmung der Dynamik ind�1strieller
Anlagen
· Messen - Steuern - Regeln, Vol. 9, 1966, 2 ,
pp. 4 1 -45 .
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Automation and Remote Control, 1 9 66, 1 2 ,
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Eykhoff P. , Van 'der Grinten P . M. E. M. , Kvakernaak H . ,
Veltman B. P. Th. :
System Modelling and Identification ( Survey paper) .
Proceedings of the 3r'd Congress of IFAC, London,
1967 ( June) .
Eykhoff P. :
Proce ss .Paramete r and State Estimation ( survey) .
IFAC Symposium on Identification in Automatic
Control Systems, Prague, 1967 ( June) , P.· 2 .
•
Farison J. B. :
Statistical Lineari zation and Variance Estimates
of Products of Functions of Random Variable s
with Small Variati0n.
IEEE Proc. , 54, December 1966, pp. 197 1 - 1 9 7 2 .
Farison J. B . , Graham R. E . , Shelton Rey C . :
Identification and Control of Linear Discrete
Systems
IEEE Trans. on Aut. Control, AC 1 2 , 1967,
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30
Fariso n J.
B. :
Fei kema H . :
(a, b)
B. R. :
A Si mple M ethod for A naly z i ng and Sy nthesi z i ng
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Kybernetika, 2 , 1 9 66, N o . 4, pp. 3 3 1 -346 .
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Co mputer.
I FAC Symposium o n Identification in A uto mat ic
Co ntrol Sy st e ms, Pragu e
, 1 9 67 ( J un e)
, p. 6. 1 .
D. , C assir G.R . :
Hoppe S . G. :
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109.
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! ( F eb ruary), p.
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Graup e
( a,
IEEE T rans . o n A to m. C o ntrol ( C o rre spo nd ence ,
)
. AC - 1 2, 1 967 , No .
I FAC Sy mposiu m, o n Id ent ifi cat io n in Aut o matic
Control Sy st ems, Prague, 1 967 ( J une). p. 4. 5 .
G ai nes
(a,
Paramet er Ide nt ificatio n for a Cla s s o f Li nea r
D isci· et e Sy stems.
H sieh H. C. :
A Pred ict iv e Id ent ific<,tio n and Mod e-R eco gnit io n
T ech nique fo r Ada pi iv e Control and T rac ki ng
Sy st ems.
IFAC Symposium on Id entific atio n in Automatic
Co ntrol Sys tems
, P rag ue , 1 96 7 ( Jun e). p. 4. 1 .
Sy stem Id entificat ion wiih Noi sy :ivf ea su r emc nts.
I FA C . Symposium on Id enti fi c «.t io n in Automatic
C o nt rol sy st ems, Prague, 1 9 6 7 ( J une). p. 3. 3.
A L ease Sq uare T ech ni q ue for the id enti fi cat io n
of l inear time - inv ari ant pl an t i n a s am pl e d - d at a
sy st em.
IEEE T rans. o n All t. ControJ ( Cor r espo nd ence),
AC -10, 1965 (Oct o ber ), pp. 4 90 -40 1 .
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No nl inear Sy st ems.
Co mp ut ing Me thod s in Opt im1z at i • ,n Pro bl ems
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Hugh es M .T . G. , M ay R .:
Est imatio n o f
Data
Frequency
H rspoPs•
fl·om 'R .,.> 1dom
IFAC Symposium or, Identi ficm .inn in :\u; ..,, . . atiC'
Cont rol Sy st em s, P c a gu e.
.
I 967 ( .h;ne) , ;:•.
!' . 4 .
31
(a,.k)
Imedadze V. V . , Lelashvili Sh. G. :
Investigation of Some Algorithm of Stochastic
Approxim ation in Identification of Multidim en ­
sional Plant s
IFAC Symposium on Identification in Automatic
Control Systems, Prague, 1967 ( June) , p. 3. 1.
·
(d)
Iserman R . :
, Optimal Series of Determined Test Signals to
Measure the Dynamics of Plants Under Small
Distrubances
IFAC Symposium, on Identification in Automatic
Control Systems, Prague, 1967, (June), p. 4. 7 .
(k,u, e) Isobe T . ,. Idogawa T. , Kashiwagi H . :
Impulse Re sponse Determination with an On­
Line Cross -Correlator Automatic and Remote
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of IFAC, London, 1 9 6 6 (June) , paper 1 5D.
Ivanenko V. I . , Hohel 0. A. :
Stabilization Problems of Parameters of Arti ­
ficially Generated Random Proce sse s
IFAC Symposium on Identification in Automatic
Control Systems, Prague, 1 9 6 7 ( June) , p. 5. 1 3 .
(k, u , e) Izawa K. , Furuta K.·:
Appatur to Measure Dynamic Characteristics
Proceedings of the Third Congre ss of IFAC,
. London, 1 966 (June), paper 15C.
(k, u , g, n) Izawa K. , Furuta K. :
Process Identification Using Correlation
Coefficient
IFAC Symposium on Identification in Automatic
Control Systems, Prq.gue, 1967 ( June) , p. 3. 9.
Jacobs 0. L. R. :
(i, 1 )
Extremum ·control and Optimal Control Theory
IFAC Symposium on Identification in Automatic
Control Systems, P_rague, 1967 ( June) , p. 5. 10.
_
Jury E . I . , Oza K. G.
System Identification and the Principle of
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Klein V . : .
The Identification o.f Linear System with a Given
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Automatic and Remote Control
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·
32
(n)
Kokotovi� P . V . , Med ani� J. V . , VuS'koviC' M . I . :
S ensitivity Method in the Experim ental Design of
Adaptive Control System s
Automatic and Remote Control Ill
Proceedings of the Third Congres s of IFAC,
London, 1 96 6 (June), paper 45 B .
Identification of Operators via Static Programming
IFAC Symposium on Identi fication in Automatic
C ontrol Systems, Prague, 1 9 67 (June) , p. 1. 3.
Kovanic P . :
(a)
Kulick V. T .
{h)
Kushner H . :
Approximations to Nonlinear Filters, JACC
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Lekoff K. :
On the Problem of Equ ivalence of Model and Object
with Regard the Control of the Obj ect
IFAC Symposium on Identification in Aut omatic
Cont rol Systems, Prague, 1 9 67 (June), p. 5. 2 .
,
Sbitnev A. I. :
Int erpolating and Extrapolating Methods of the
Statistical Analy sis of Dyn().mic System.s
IFAC Symposium on Identification in Automatic
C ontrol Systems, Prague, 1 9 6 7 "( June) , p. 3. 1 8 .
·
(a, a, b, h) Leonov Yu. P. :
On Estimate of Parameters of Linear Dyna mi_c Systems
IF'A C Symposium on Identification in Automatic Control
Syste�s. Prague, 1 9 6 7 (June ). p . 1 . 2 .
(k)
Liewers P : , Buttler E . :
Measurement of Correlation Functions of Reactor
Noise by Means of the Polarity Correlation - M ethod
IFAC Symposium on Identification in Automatic
C ontrol Systems, Prague, 1 9 67 ( June), p. 3 . 7 .
Liff A. I. :
(k)
System Identification in the Pre sence of Noise by
Digital Techniqu e s
IEEE International Cov. Rec . , Vol. 14, pt. 6, 1 9 6 6 ,
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Maer S. :
On the Uniquenes s of the Adjustment in Defining t he
Dynamic Characteristics with the Help of Self­
adjusting Model
IFAC Symposium on Identification in Automatic
C ontrol Systems, Prague, 1 9 67 ( June), p. 5 . 6
·
33
(b
�)
'n
Marsik J:
Quick -He sponse Adaptive Identification
IFAC Symposium on Identification in Automatic
C ont rol Systems, Prague, 1 9 67 Uune), p . 5 . 5,
., ,
Versuche m it eincm selbsteinstcllenden Modell
zur automatischen Kennwertem ittlung von Regel strecken
Messen - Steurern - Regeln, V ol. 9, 1 9 6 6 , No. 6
.
Maslov �. P. : On one Estimate of Id entification Accuracy Automation
an� �mote Control, 19 66, 1 0, pp. 5 8 -G l .
Marsik J. :
(p)
im
Mas1ov E. P, : An Qptimal Statistical Adaptive Model
IFAC Symposium on Identification in Automatic
Control Sy stems, Prague, 1967 (June), p. 3. 4.
{j)
Mayne D. Q. :
A method for estimating discrete transfer functions.
In " A dvance s in Computer C ontrol second U. K. A. C.
Control Convention, The University of Bristol, April
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Parameter e stimation
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McGhee R . B . , Walford R . B . :
A Monte Carlo Approach to the Evaluat�on of Concli ­
tional Expe ctation Parameter E stimates for Non ­
linear Dynamic Systems ..
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{n)
Meyerhoff H . J. :
Automatic process identification.
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'
Vol. 1 book 3, paper 36 F.
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(a, e, m) Nagumo Jin -Ichi, Noda Atsuhiko :
A Learning Method for System Identification.
IEEE Transactions on Automatic Control, AC - 1 2 ,
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34
Narendra K. S . , Gallrnan P . G . :
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Nikiforuk P . N. , Pronovost R . .J . :
A Three Variable Self -Adaptive Flight Control
System Using Di sc rete Inter\'al Binary Noise
Perturbation Signals.
IFAC Symposium on Identification in Automatic
Control Systems, Prague, 1 9 6 7 ( June) , p. 3. 2 .
( a, b, n) Norkin K. B . :
Proc e s s Identification with the Use of Search
Controlled Adaptive Model.
IFAC Symposium on Identification in Automatic
Control Systems, Prague, 1 9 5 7 ( June), p. 5. 1.
(j, 1)
Panuska V. :
A Stochastic Approximation Method for Ident i ­
fication o f Linear System Using Adaptive
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Preprint s of Ninth Joint Automatic Control
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Ann Arbor Michigan, June 26 - 2 8 .
: n)
Parks P. C . :
Stability Problents o f Model -Refe rence and
Identification System s .
IFAC Symposium o n Identification i n Automatic
Control System s, Prague, 1 9 6 7 ( June) , p. 5 4 .
( a , e , m ) Pear son A . E. :
A Regression Analysis in Function Space for the
Identification of Multivariable Linear Systems
Automatic and Remote Control liJ
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London, 1966 ( June), paper 46B.
Perdreatlville F. J., Goodson R. E. :
Identification of Systems Described by Partial
Diffe rent ial Equat ions Trans. ASM E , ser. D. ,
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(p, k , e ) Perelman I. I. :
Estimation Accuracy of Impulse Response of a
Linear Dynamic Plant
IFAC Symposium, on Identification in Aut omatic
Control System s, Prague, 1 9 6 7 ( June), p. 1 . 9 .
Pcterka V . , Vidincev P . :
Rational -Fraction Approximation of Transfer
Functions
IFAC Symposium on Identification in Automatic
Control Systems , Prague, 1 9 6 7 ( June) , p. 2 . 1 .
35
Petcrka V . :
(b, i)
v
A New Appro;:�ch to Identification of Discrete
Dynamic Systems (in C zech )
Kybernetika, 4 ( 1 96 8 ), 8 .
Pete�ka V . , Smuk K. :
On -Line Estim ation of Dynamic M odel Param et ers
from Input -Output lJo.ta
Paper submitted for the 4th Congress of IFAC .
(a, k, n) Plander I. :
Automatic Iteration l\1 ethod of Proce ss Ident i fi ­
cation with Cyclic -Adjustment of Parameters
Using Analog and Hybrid Computers
IFAC Symposium on Identification in Automatic
Control System s, Prague, 1 9 6 7 (June), p. 5. 9 .
(a, f, 1) Popkov Yu. S. :
Re sotration of Dynamic Characteristic s of Non­
linear Plant s in the Presence of Noise
IFAC Symposium on Identification in Automatic
Control Systems, .?rague, 1 9 6 7 ( June) , p. 3. 1 5 .
Radtke M . , Seidel H . P. · :
On Estimation of Static and Dynamic Parameters
of Nonlinear Transfer Elements from Step Responses
IFAC SJ"'llposium on Identification in Automatic
Control Systems, Prague, 1 9 6 7 ( June), p. 4. 3 .
Rajbman N. S. , Chadee\' V . M. :
A daptive Models in Control Systems ( in Russian) ,
Moscow 1 9 6 6 .
(a, h, n) Rake H. :
Automatische Prozessidentifizierung durch ein
selbsteinsteltendes System
Messen -Steuern - Regeln, 9, 1 9 6 6 , 6, pp. 2 1 3 -2 1 6 .
(a, n)
Reinisch K. , Wernstedt J. :
Model Method of Analyzing Disturbed Linear and
Special Nonlinear Systems by Means of Aperiodic
Test Signals
IFAC Symposium on Identification in Automatic
Control Systems, Prague, 1 9 6 7 ( June), p. 4. 2 .
(<:.., g)
Roberts P. D. :
(m, f)
Roy R . , Schley C. :
Process ldentificati!)n Using a Mode Learning
Machine
Proceedings of the 7th Joint Automatic Control
Conference, Univ. of Washington, Scattle, 1 9 6 6
(August) , pp. 6 39 -6 4 8 .
Orthogonal Transformations Applied t o Control
System Identification and Optimization
IFAC Symposium on Identification in Automatic
Control Systems, Prague, 1 9 6 7 ( June), p. 5. 1 2 .
36
(m, f) Roy R. T
. •
Sherman J. :
System Id en ti fi c at i on and Patt ern Recognition
IFAC Symposium on Id€mtificaiion in Automatic
Control System s, Prague , 1 9 6 7 ( June) . p. 1 . 6 .
( m , f) Roy J. , Sh.crman J. :
Learning . Technique for Volt e r ra Serie s
Repre sentation
IEEE Trans . of Aut . Cont rol, AC - 1 2, 1 9 6 7 , 6
(December), pp . 7 6 1 - 7 6 4 .
A
_( a, h, l) Sakrison r 1• J. :
The Use of St o c ha s t i c A pp : · oxim ation t o Solve
the System Identifi cation Problem
IEEE Trans. of Autornaiic Control, AC - 1 2 ,
1 9 6 7 , 5 ( October) , pp. 5 6 3 -5G 7 .
(p, h) Schm idt H. :
den Einfluss von Stol un gen und M e s s fehlern
der Auswe rtung von g0me s s e ne n Sprungant ­
worten Messen -Steuern - R e geln, 1 1 , 1 9 6 8, 5
(Mai), pp. 1 5 8 - 1 6 3 .
Ube r
'
bei
Shaw
L.
•
Robinson G. :
Invariant Estimation of
eters
IFAC Symposium
Control Syst em s ,
(a,
n)
or:
Stochasti.c
System Param ­
Ident i fi cation in
Pragu e ,
Automatic
1. 9 6 7 ( Jun e), p. 3. 1 6 .
Skakala J . , Sutek L . :
Adaptive Model for Solving Iclenti fication Problem
Automatic and Remote Cont rol Ilf ( Proceedings
of the Third Congre s.:; of IFA C , Lundon, 1 9 6 6
(June) vol. 1 , book 3 , paper 4 6 E .
(a, n) Skala K. :
Smith F. W. :
Experimental Identification o f S)·stems by Adapting
of the Parallel Model
IFAC S;rmposium on Identification in Automatic
Control Systems, Prague, 1 9 67 ( .J'une) , p. 5. 3 .
System La place -Transform E stimation from
Sampled Data
IEEE Trans. on Aut . Control, AC - 1 3 , 1 9 6 8, 1 ,
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(a, g) Smith G. W . , Clement P. R. :
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Exponential Spectral Coefficients .
Journal of thc ' Franklin In s t lttt t e , 2 R l , H l ci u ,
No. 2 ( F e br ua r y} , p_l). 1 n · 1 � ·�
. 37
(b, i, p) Smith F. W. , Hilton W. B. :
Monte Carlo Evaluation of Methods for Pulse
Transfer Function Estimation.
IEEE Trails. on Aut. Control, A C - 1 2 , 1 9 67,
October, pp. 568 -57 6 .
(p)
(h)
Stepan J. :
The Identification o f Controlled Systems and
the Limit of Stability.
IFAC Symposium on Identification in Automatic
Control Systems, Prague, 1 9 6 7 ( June), p. 1 . 1 1 .
Stratonovich R. L . :
Conditional .M arkov Processes and Their Appli­
c ation to the Theory of Optimal Control, Ame rican
E lsevier Publishing C o . , 1 9 6 8 .
( d)
Strobel H. :
On a New Method o f Determi ning the Transfer
Function by Simultaneou s Evaluation of the
Once Imaginary Parts of the' Measured Frequency
Response.
Proceedings of the Third· Congress of IFAC, Vol.
1 , book 1, 1 9 6 6 ( June) , paper 1 F.
(p)
Strobel H. :
On the Limits Imposed by Random Noise and
Measurement Errors Upon System Ident i fication
in the Time Domain.
lFAC Symposium on Identification in Automatic
Control System s, Prague, 1967 (June), p. 4. 4 .
(p)
Strobe! H. :
Uber die durch stochastische Storsignale und
M e ssfehler bedingten Greuzen der Kenuwerter ­
mittlung im Zeitbereich.
Me ssen -Steuern -Regeln, 1 1, 1 9 68, No. 2
(February), pp. 54 - 5 () .
(a, f)
Taylor L. W. , Balakrishnan A. V . :
Identification of Human Response M odels in Manual
Control System s.
IFAC Symposium on Identification in Automatic
Control System s, Prague, 1 9 6 7 , June� p. · 1 7 .
(h)
Taylor L. R. and K. W. llift :
A Modified Newton -Raphson method for deter ­
mining Stability Derivatives from Flight Data,
Proceedings of the Contemcean Computing
Methods in Optimi zation Problem, San Remo,
1 9 6 8, (Academic Pre ss) .
(m)
Tsypkin Ya. Z . :
Adaptation, Learning and Selflearning in Control
System s.
Third IFAC Congress, London, June 1 9 6 6 .
38
Unbchauen H. :
( p)
Unbehauen H.
,
Kennwertcrm ittlung von Regclsystemen an H<2nd
des geme sscnen Verlau-fs der Ubergangsfunktion.
Me ssen -Steuc rn -Regeln, 9, 1 9 6 6 , No. 6, p. 1 8 8 191.
Schlcgel G. :
Estimation . of the Accuracy in the Identification
of Control System s Using Dete rmini stic Test
Signal e .
IFAC Symposium on Identification in Automatic
Control System s, Prague, 1 9 6 7 ( June). p. 1. 1 2 .
Some Methods of Controlled Plants Dynamic
Characte ristics Dete rmination and Inve stigation
of Their Accuracy.
IFAC Symposium on Identification in Automatic
Control System s, Prague, 1 9 6 7 , June, p. 3. 1 9 .
V oronova L. I. :
(q, k) Welfonder E. :
Correlation Method for Identification of Disturbed
Control Syst ems by Periodica} Test Signal.
IFAC Symposium on Identification in Automatic
- Control System s, Prague, 1 9 6 7 , June, p. 3. 8 .
Zur Erm ittlung d e r Kennwerte gertorter Systeme
durch Mode 11 methoden unter Verwendung
aperiodisher T .:- st signale.
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pp. 1 6 4 - 1 6 9 .
(a, h, p) Wernstedt J. :
Off -Line Statistical Identification o f the Kinetics
of a Nuclear Reactor (A Comparison of Digital
Models) .
!FAC Symposium on Identification in Automatic
Control _ Systems, Prague, 1 9 6 7 , June, p. 3. 2 0 .
Wilkic J. D. F. :
( b , i ) Wong K. Y . , Polak E . :
Identification of Linear Disc rete Tim e Systems
Using the Instrumental Variable Method.
IEEE Trans. on Aut. Control, AC 1 2, 1 9 67.
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(b, k) Zhiroglyadov V . P
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Kaipov V . K. :
On Application of Stochastic Approximations
Methods in the Identification Problem.
Automation and Remote Control, 1 9 66, No. 10,
p. 54.
(b, k) Zhivoglyadov V . P
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•
Kaipov V. K. :
_Identification _on Distribut e d Parameter Plants in
the Presence of Noises. - IFAC Symposium on
Identification in Automatic Control System s,
Prague, 1 9 6 7 , June, p. 3. 5 .
39
(c, j)
Astrom K .
J. , Bohlin
T. :
Numerical Identifi c ation
Ope r ating
R e cor d s .
Proc. IFAC Symposium
Septemt e r 1 9 6 5 .
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Linear
Dynmn ic
System s
on Se l f -Addptive Control
from
Kormal
System,
Teddingt on,
(a, g , n) Barker H. A . , Hawley D. A. :
0 rthogonali zing Technique for Systems Identification and Optimi ­
zat ion.
Proceedings 3rd Congress of IFAC, London, June 1966,
(j)
paper
36A.
Bohlin T. :
Real -Time E st imation of Time -Variable Process Characteristics .
IBM System Development Division, Nordic Laboratory, Sweden,
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(b, h)
Butler, R. E. and Bohn E. V. :
An Automatic Identification Technique for a Class of Nonlinear
Systems.
IEEE Transaction on Automatic Control, AC - 1 1 , 1 9 6 6 , pp. 2 9 2 -2 9 6 .
( b , h)
Hob erock L. L.: and Kohr R. H. :
An Experimental Deterrninat�on of Differential Equations to De scribe
Simple Nonlinear System s .
ASME Transactions Ser. D. , Vol. 89, 2, June 1 9 6 7 , pp. 39 3 - 39 8 .
·
Koivo A. J. , Stoller R. L. :
pn Least Square s Estimation in Nonlinear Dynamic Systems with
Time Delay.
Preprints 9th Joint Automatic Control Confe rence ·1 9 6 8, Univers-ity
of Michigan, pp. 1 1 6 -122. ·
( b, i)
Levin M. J. :
I
.
E st imation of System Pulse Transfer Function in the Presence
of Noise.
IEEE Trans. Automatic Control, AC -9, .July 1 9 64.
( b , h)
Lion P, M. :
Rapid Iclentification of �inear and Nonlinear Systems.
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(c, i)
Rogers A. E . , Steiglitz K. :
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IEEE Trans. Automatic Control, AC - 1 2, 5, October 1 9 6 7 , pp. 5!14 597 .
40
( a, n)
Rose R. E . , Lance G. M. :
Analysis of the Behavior oi Continuous Param eter Estimation
Methods.
Preprints 9th Joint Automatic Control Conference 1 9 6 8 ,
Univer�;.ty o f Michigan.
(b, h)
Sprague C. H. , Kohr R. H. :
The Use of Piecewise Continuous Expansions in the Identification
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\
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University of Michigan.
·
(b, i, 1) vali s J. :
An Ite rative Method for Experimental Identification of Discrete
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CSAV, Prague 1 9 6 7 .
·,
41
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