Linear multivariable systems : preliminary problems in mathematical description, modelling and identification Citation for published version (APA): Hajdasinkski, A. K. (1980). Linear multivariable systems : preliminary problems in mathematical description, modelling and identification. (EUT report. E, Fac. of Electrical Engineering; Vol. 80-E-106). Technische Hogeschool Eindhoven. Document status and date: Published: 01/01/1980 Document Version: Publisher’s PDF, also known as Version of Record (includes final page, issue and volume numbers) Please check the document version of this publication: • A submitted manuscript is the version of the article upon submission and before peer-review. There can be important differences between the submitted version and the official published version of record. People interested in the research are advised to contact the author for the final version of the publication, or visit the DOI to the publisher's website. • The final author version and the galley proof are versions of the publication after peer review. • The final published version features the final layout of the paper including the volume, issue and page numbers. Link to publication General rights Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights. • Users may download and print one copy of any publication from the public portal for the purpose of private study or research. • You may not further distribute the material or use it for any profit-making activity or commercial gain • You may freely distribute the URL identifying the publication in the public portal. If the publication is distributed under the terms of Article 25fa of the Dutch Copyright Act, indicated by the “Taverne” license above, please follow below link for the End User Agreement: www.tue.nl/taverne Take down policy If you believe that this document breaches copyright please contact us at: [email protected]providing details and we will investigate your claim. Download date: 06. Aug. 2021
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Linear multivariable systems : preliminary problems inmathematical description, modelling and identificationCitation for published version (APA):Hajdasinkski, A. K. (1980). Linear multivariable systems : preliminary problems in mathematical description,modelling and identification. (EUT report. E, Fac. of Electrical Engineering; Vol. 80-E-106). TechnischeHogeschool Eindhoven.
Document status and date:Published: 01/01/1980
Document Version:Publisher’s PDF, also known as Version of Record (includes final page, issue and volume numbers)
Please check the document version of this publication:
• A submitted manuscript is the version of the article upon submission and before peer-review. There can beimportant differences between the submitted version and the official published version of record. Peopleinterested in the research are advised to contact the author for the final version of the publication, or visit theDOI to the publisher's website.• The final author version and the galley proof are versions of the publication after peer review.• The final published version features the final layout of the paper including the volume, issue and pagenumbers.Link to publication
General rightsCopyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright ownersand it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights.
• Users may download and print one copy of any publication from the public portal for the purpose of private study or research. • You may not further distribute the material or use it for any profit-making activity or commercial gain • You may freely distribute the URL identifying the publication in the public portal.
If the publication is distributed under the terms of Article 25fa of the Dutch Copyright Act, indicated by the “Taverne” license above, pleasefollow below link for the End User Agreement:www.tue.nl/taverne
Take down policyIf you believe that this document breaches copyright please contact us at:[email protected] details and we will investigate your claim.
Furuta assumes further the following model of the transfer function matrix
!.(z ,~)
Q(z,~)
r. -1
T ~il
-87-
----'-_ [ Il z -I +
Q(z,~)
+ ••• +
(q X p) matrix ~ .. 1]
(q x I) vector
(215)
which corresponds with (209). Using (215) and (209) the vector of parameters
is identified minimizing J with respect to ~.
The parameter estimation procedure is described in Furuta(1973). Assuming
the transfer function matrix has already been identified, the nonminimal
realization of the transfer function matrix can be found as :
~(k+1 ) = ~(k) + £':: (k)
Rn (216) I.(k) = ~(k) ~(k) E
where 0 •..... 0 -a I
R,:;. !~ I -I
F = G = b 0
0 .... 0 I -a I II
1-
H = [Q " .. . ,Q, .!.J
This is a very well known Frobenius canonical form ( see for ex. Hajdasinski(1976»,
and it is seen Q(z,~) - is the anihilating polynomial of F. From this form
Ho and Kalman(1966) extracted a controllable and observable realization,
-88-
called the minimal ~ealization. It is based on the Hankel matrix, which can be
derived as the product of the controllability and observability matrices.
w
(217)
H = [vT w1 -n --
(218)
Furuta introduces the concept of ~ - practical controllability and
observability, which gives a solution to the order test and further to the
£ - minimal realization :
Definition 17 The state x of the system (216) is said to be ~- practically
controllable ( observable ) if ~ is element in the space
spanned by the eigenvectors of ~T(yyT) corresponding to the eigenvalues
larger then ~,W and V denote controllability and observability matrices.
Remark It is easily seen that seeking for eigenvalues of the ~~T(~ ~T)
matrix is equivalent to the diagonal decomposition of the HT matrix, ...., which is only slightly different from the singular value decomposition of the
Hankel matrix.
Definition 18 The system represented by the states which are ~- practically
controllable and observable is said to be ~- practically
minimal realization.
The order test is made by comparison of eigenvalues of the ~~T , and
truncation of the selection matrix consisting of normal eigenvectors of
~ ~T, such that consists only of these eigenvectors which correspond to
eigenvalues larger than c.
-89-
w.(i =I, ..• ,n ) - normal eigenvectors of W WT -1 0
If £- is chosen as
v > £ > vn +1 n 0 0
where
n 0
2: v. 1
i= 1 -= n
L: v. 1
i=1
v.( i = I, ••• ,n) - eigenvalues of W wT 1
(219)
(220)
(221)
then the order test is considered as compl.eted, and £ - minimal realization
of the system is found as :
{ F -0
Remark
G -0
H = H s } -0
(222)
This refers to the notion of the numepiaal pank of the Hankel matrix
which will be discussed while considering the 8inguZaP value deaompo-
8ition of the Hankel matPix.
The Furuta's method is rather inefficient and incorporates an unnecessary step
of the transfer matrix identification with arbitrarily chosen ~. This method,
however, pointed out the way to the more advanced methods of the approximate
minimal realization and to the eigenvalue problem in determination of the system
order.
-90-
3.4. Miscellaneous order tests. The pattern recognition method by
Thiga and Gough.
The approach presented by Thiga and Gough is direct in the sense the
model parameters are not estimated at each 'step. The test is based on
a measure of the linear dependence(op independence) of features displayed by
each model opdeP.
This method is totally empirical and assumes identification of the order
(understood as a degree of the transfer function denominator) of individual
subsystems, such that the final system is described in form of the transfer
function matrix.
G = [gij 1 i=1,2, ... ,q j = 1,2, ... ,p (223)
where g .. correspond to the linear differential or difference equations 1J
n. m n.> m i=I,2 ... q
L: L: 1
d.ky. (t) c j J1.U j (t) 1 1
k=O J1. = 0 d ik " o j=1 ,2 ... p
and d ik and c j J1. are differential operators
or
y. (k) 1
9.=0
b. o u.(k-£) -J'" J
J1.=0
a. o y.(k-£) 1", L
k 0,1,2, ... , N
The order test is completed when all subsequent n. are found. 1
(224)
(225)
Authors proposed to solve the order recognition problem by testing a single-
valued function of pattePn, which will reflect a decision surface f(x) = £,
-91-
where E - is the treshold of pattern recognition. For the sake of the order
test for a noisy system, such a function must serve two following tasks :
I. help to spread the cluster of the patterns in the two classes further
apart in the pattern space.
2. reduce the dimensionality of the pattern space by combining dimensions
The input-output cross-correlation function is proposed as a one for the
training of the system :
R (k) uy =
T
M ± j=1
u(j) y(j + k) (226)
Authors are reporting quite a succes in the order discrimination during an
experiment carried on for a set of II representative systems " chosen from
amongst the vast possible combinations of characteristic roots of low-pass
filters, up to seventh order. The machine was learned to extract from such
patterns purported characteristics of a given system order and type of roots.
This type of approach was tested with different types of processes proving
its usefulness.
3.5. Akaike's FPE ( final prediction error) and AIC ( Akaike's maximum Information
Criterion) as order tests for MIMO systems.
Akaike has proposed two new approaches to the order determination and
consequently parameter estimation. These two methods - FPE and AlC are
assymPtoticall~equivalent in final results, as the distribution of the equation
error (for the AR model - considered as the prediction error ) converges to the
Gaussian one.
These two methods employ much more statistical properties of given measurements,
than some 11 a priori II assumptions about its nature . However one general
-92-
assumption, which in practical cases almost always holds, is the assymptotic
Gausseness of the time series samples • To get some more insight into these
two methods, let us study some important passages in their derivations, however
no claim for completeness is being made.
3.5.1. Statistical predictor identification - Final Prediction Error
Approach.
Let us consider first a single input-single output system and assume that the
output of this system is a stationary and ergodic process Y(n). In practice
Y(n) is given as a function of the recent values of Y(n) and the the structure
or the parameter of the function is determined. There is considered the situation
where the structure is identified using an observation of a process X(n) and,
using the structure, the prediction is made with another process Y(n) which is
independent of X(n), but with one and the same statistical property as X(n).
The FPE is defined as the mean square prediction error
FPE of X(n) - 2 = E {(X(n) - X(n)) } (227)
for X(n) being the predictor of X(n)
When the process X(n) is stationary and the predictor Y(n) of Y(n) is linear
and given by
M
Y(n) = ~ ~(m) Y(n-m) + ~(O)
m=1
where ~(m) is a function of X(n)
Using (227) and (228) we have
FPE of ~(n) = a2(M) +
M ~ ~=O
(228)
(229)
where
(i(M)
-93-
M
E {(yen) - L: "M(m) Y(n-m)
m=1
2 - "M(O) ) }
where "M(m) denotes the member of the set of parameters {a(m)} giving.
in the sense of mean squares, the best linear predictor ie.
M
min E {(yen) - ~ a(m) Y(n-m) - a(0»2}
I a (m) ( m=1
ll"M(m) ~(m) - "M(m) m 0.1.2 .... M
and
VM+ 1 (i.m) = E(Y(n-i)Y(n-m» i.m 1 ,2 , .•• M
VM+1(0.m) VM+ 1 (m. 0) = E(Y(n» m 1,2, ••. M
VM+1(0.0) =
(230)
(231 )
(232)
(233)
(234 )
In the relation (229) all components containing expectations of products
of noncorrelated quantities are neglected.
It is seen that FPE is composed of two components: prediction for a given M
a2(M) • and the second due to statistical deviation of ~(m) from "M(m).
Behaviour of the FPE .namely decrease of a2(M) for increasing M and increase
of the second term for increasing M. suggests that there exists an optimum
for a certain M. which is nothing but the order of the autoregressive model(228). 'TPE
x "'F-'\>E. :E x
. * G'l(IM)
0--0 '[~~-
z..
-94-
This idea is worked out further in Akaike(1970). Following this idea we get:
M
X(n) = L a(m) X(n-m) + a(O) + c(n)
m=1
(235 )
where c(n) are the samples of the" white noise" • uniformly distributed and
E(£(n)) o (236)
If there is a collection of data avaliable. { X(n) ; n = -M+l. -M+2 •••• N}.
the parameter ~(m) is defined as the least squares estimate of a(m) ( ~(m)
is the parameter of the predictor (228) and we are going to find it basing
on observations of X(n)).
Defining
where
C (m.i) xx =--N
n=1
N
x = m
N L. X(n-m)
n=1
M
C (m.i) a..(m) xx M = -- L N n=1
M
(237)
m=O,I,2, •.• M (238)
=> L. Cxx(m.i) ~(m) M M
_1_~ ( L ~(m)X(n-m) - ~(m) Xm) ( X(n-i) - Xi )
N n=1 m=1 m=1
M M
L C (m.i) a..(m) xx M 1 '"" -= -.L-, (X(n) N
n=1
(239)
m=1
-95-
where
M N
L 2: ~(m)X(n-m) m=1 n=1
(240) N
Thus for N large enough we can write :
M
L e (m,"') ~(m) = e (0,"') xx xx (241 )
m=1
{ '" = 1,2,3, ... ,M for
m = 0, I ,2 , •• • ,M
according to (237) and (238). Relation (241) can be rewritten in the form
e (I , I) e (2, I ) xx xx
e (1,2) e (2,2) xx xx
e (I,M) e (2,M) xx xx
or
if e 1S nonsingular -xx
~= e -Ie -xx -M
e (M,I) xx
e (M,2) xx
e (M,M) xx
if e is singular or ill conditioned -xx
~= + c e -xx -M
~(I)
~(2)
~(M)
e (0,1) xx
e (0,2) xx
e (O,M) xx
where + C - is the Moore-Penrose pseudo inverse of C . -xx -xx
(242)
(243)
(244 )
(245)
-96-
The" zero" coefficient ~(O) is estimated basing on (235)
M
~(O) Xo - .L. ~(m) Xm m=1
Following the definition given previously - (228) using (246), we get
M
yen) =2: m=1
~(m) ( Y(n-m) - Xm ) - x o
the relation for the predictor yen) of yen).
Assuming that yen) is generated by relation
M
yen) L a Yen - m) + a + 6(n) m 0
m=1
where
6(n) has exactly the same statistical properties
M
as E(n), we get
M
(246)
(247)
(248)
yen) - Yopt(n) = 6 (n) - L lI~(m)y(n-m) - ( lIXo - L ~(m)lIXm )
for a m
m=1 m=1
~(m) are coefficients of the optimal predictor Yopt(n).
(249)
Relation (249) is somewhat abstract and refers to statistical properties of
the optimal predictor. It is assumed here,the process yen) is stationary and
ergodic. thus defining the following:
lI"M(m) ~(m)
yen) yen) - E(Y(n» (250)
lIX~ X~ - E(X(n»
-97-
basing on (247) and(248) ,(249) we get two relations
M
yen) - Y (n) opt
M
o(n) - (L. M
~(m)Y(n-m) -~ ~(m)Y(n-m) - 2. ~(m)E(Y(n-m) m= I m= I m=1
M
+ L ~(m)E(Y(n-m)) ) - (X - E(X(n» ) + o
m=1
M
+ Z ~(m)( Xm - E(X(n»
m= I
from (249)
M
yen) - yen) = o(n) - L ~(m) (Y(n-m) - Xm) -
m=1
M
ao = Xo - L ~(m)Xm m=1
thus
M ~
yen) - yen) 8(n) -~ ( ~(m) - am)Y(n-m) =
m=1
M
= o(n) - Z « ~(m) - am(m» + ( ~(m) - am» Y(n-m) =
m=1
M
= o(n) - L. lI~(m) Y(n-m) +
m=1
M
2. (~(m) - am) Y(n-m)
mel
a Y(n-m) + a m 0
(251 )
(252 )
(253)
(254)
From (251) and (254) we see that ( yen) - yen»~ will statistically be equal
( yen) - Y t(n» if : op
+
M z:. ( "M(m)
m=1
? - a ) Y(n-m) =
m
-98-
M
L (~(m) - "M(m) ) y(n-m)
m=1 M
( Xo - E(X(n))) + ~ ~(m)(Xm - E(X(n)) )
m=1 (255)
From (255) it is seen that statistically left and right sides are equivalent
if am ~ "M(m) , thus (249) is proven. Taking into account independence of
yen) of 6"M(m) and 6X ~ , we get :
M M
FPE of yen) E( yen) - yen) )2 = (52 + L Z E(6"M(m)6"M(~)Rxx(~-m) +
m=1 ~=I
where
+ E( 6X o
M
2: ~(m)6Xm)2 m=1
R (~- m) xx
2 E( X(n-~)X(n-m) ) - (E( X(n) )
(256)
(257)
For the sake of numerical solution we have to be interested in assymptotic
properties of the FPE, which hopefully can be a lot simpler to handle than
statistical evaluation of all subsequent quantities of the relation (256).
For this sake quantities 6Xo and 6"M(m) have been introduced.
For the asymptotic evaluation of the FPE we will need the following theorem
Theorem 9: Under the assumption of the stationarity and ergodicity of X(n),
the limit distribution iN 6X =.(J;i1 (X - E(X(n)) ) and o 0
IN' 6"M(m) =.fN1 ( ~(m) - "M(m)) for m - 1,2, ••. M,
when N tends to infinity, is (M+I) - dimensional Gaussian with zero mean and
the varianae matrix :
-I !lr.!
where
and
M
1 ... L "M(m)
m=1
-99-
!M - (M x M) matrix of R(i,m)
o - denotes a zero column vector
R (i - m) xx
( this theorem is a special case of the limit theorem as presented in the
book of Anderson T. w. (1971) ).
From the ergodicity of the process it is clear that C (i,m) converges to xx
R (i- m) as N+ 00 , with probability one. Thus !. is a consistent estimate xx -M
of ~ ( a vector of parameters of the optimal predictor) with convergence
thus the choice fi = 3 assures the numerical rank of the ~. matrix equal to:
(E,o,fi)2 = (0.05; 0.1; 3)2
and provides the relative error with respect to the Euclidean norm
tt~.-~Ile
U ~. II e
= 0.03829
The system order r > fi min(p,q)., thus it is the smallest integer fulfilling
r> 1.5. And again, we have as the result of the order test r = 2, which is
sufficient to provide fi = 3.
It is demonstrated then, that all three order tests provide us with the same
result r = 2 and fi = 3.
Conclusions
There exists already a quite extended class of order tests. Certainly this
choice of order tests mentioned here is one of many possible choices. However,
it shows quite well the mainstream in the methodology of model building.
The Guidorzi's order test has been applied for many practical problems
providing quite satisfactory results.
-124-
However, in many cases it was possible to find lower order approximations of
investigated systems. Such a comparison between the Guidorzi ' s method,
Ho-Kalman's algorithm and the partial minimal realization by Tether (Tether A.
(1970)) was made by A. Krause (1976), showing that it was always possible,
for noisy dat~, to find a lower order model for Tether's and Ho-Kalman's methods,
whereas for Guidorzi's methods a higher order model was found.
Tse-Weinert's methods, being a special case of the determinant test for the
Hankel matrix, gives very good results. Examples can be found in an original
publication by Tse and Weinert, (1975).
Furuta's approach is rather unnecessarilly complicated, but leading to quite
good results for low dimensional model.
Akaike's methods have already been discussed. They are very interesting from
a theoretical point of view. However, in practical applications demand a large
amount of data.
Order tests proposed for the Hankel model are rather simple and efficient.
Especially good results are with use of the s.v.d. method. This is, however,
the beginning of research in this field and many unexpected events may occur
during application of these methods in practice.
4. Multivariable system identification
There has already been much said about identification and parameter estimation.
However, the general definition of the identification has not yet been given.
According to Zadeh (1962), identification is the determination, on the basis of
input and output, of a system within a specified class of systems, to which the
system under test is equivalent.
-125-
This short definition covers a huge range of knowledge about the modelling and
model building basing an input/output data. A good survey of problems
concerning modelling and identification of single input - single output systems
has been written by Fasal K.H.,and Jorgel H.P. as the theoretical lecture for
the 5th IFAC Symposium on Identification and Parameter Estimation - Darmstadt
1979. For there is no substantial difference in general classification of
MlMO and SISO systems it is fair to quote this tutorial lecture as a good
reference.
Of our particular interest, there is an expePimentaZ anaZysis of a process,
which in general is considered just as the "identification". If the structure
of the model is known in advance, or at least can be estimated (see the previous
chapter), parametPic methods can be used for identification of the mathematical
model. In cases where the structure cannot he estimated non-parametric
techniques have to be applied.
Among the various goals of identification the below mentioned are most
frequently quoted:
deeper insight into the nature of the process
prediction of the dynamical behaviour of the process
verification of theoretical models
computation (or better estimation) of non-measurable variables
design of the controller
optimization of the-process.
Most of all, our interest will be attached to discrete time modeZs which will
be split into stochastic modeZs and detenninistic modeZs. However, the
deterministic models, as purely conceptual ones, usually serve as the prelim-
inary information about the process.
A survey of methods for identification of multivariable systems has recently /
been written by Niederlinski A. and Hajdasifiski A. (1979). This paper discusses
-126-
multi variable systems structures, parametric and non-parametric identification
techniques for mu1tivariab1e systems.
In this report we will present only the outline of a few of the most important
techniques, which are related to already discussed order tests. Also more
attention will be given to the Markov-parameter approach.
It should also be noted that the parametric identification of canonical models
by Guidorzi and Tse-Weinert has already been presented in chapters 3.1 and 3.2,
(however briefly), the Akaike's information criterion approach has also been
presented in quite SOme detail. Thus the following algorithms, as in the
illustrative examples, will be discussed further;
Tether's minimal partial realization algorithm
Gerth's algorithm
The Approximate Gauss-Markov Scheme with the s.v.d. Minimal Realization
algorithm
4.1. The Tether's minimal partial realization algorithm
The problem of minimal partial realization is formulated as follows:
1. Given an infinite sequence of Markov-parameters {M.},.which does not 1
have a finite dimensional realization, it is desired to find such a finite
dimensional realization so that its first No Markov-parameters are
correspondingly equal to the first No Markov parameters of the first
sequence, or
2. Given a finite sequence of Markov-parameters M , M, ..•. M __ I' find a -0 - ~o-
finite - dimensional realization whose first No Markov-parameters are
correspondingly equal to the Markov-parameters of the given finite sequence.
-127-
The second alternative is much more suited to the purpose of identification
and gives a powerful tool for handling the noisy case. It does not, however,
incorporate any stochastic considerations, assuming that only some extraction
of the Markov-parameters from the noise corrupted systems is prior to the
minimal partial realization procedure. The following two definitions are
useful:
Definition 20: {~,!,~} is said to be a partial realization of order{Nolof the
sequence {M.}, if an only if M. = C AiS holds for i = 0,1,2, -1 -l. - --
N - 1. o
Definition 21: {~,!,~} is said to be a partial minimal realization of order
{No)if and only if the dimension of A is minimal among all other {~;!;~I}
satisfying definition 20.
Tether (1970) proved that for every finite series of Markov-parameters {M.}, -1
i=O,J,2, .•.. No 1 there exists an extension sequence {~o+l""'}' for
which a completely controllable and observable partial realization exists via
the Ho-Kalman's algorithm.
It can also be demonstrated that among all possible partial realizations must
exist at least one minimal partial realization which is unique if and only if
the extent ion sequence is unique. Defining
M !:!.1 ........ !:!.r,' -~l N
-0
I (339)
Mi ~
~2.' ••••••• ~,
!:!.r,-I .•...•.•.... !:!.r,' + N - 2
-128-
it is possible to prove that given a finite sequence of r x m matrices
{M , M1 , •••••• M. I} satisfying ~ - ~o-
rank ~l ,N rank ~'+I ,N rank ~',N+I
for some N,N' such that N'+ N = N the extension of the sequence o
(340)
{M , M' ....... M.. I} to {M ••••• M.. I' •••• M k I} with 0 ... k < '" for which --0 - ~o- -0 ~o- ~o+ -
rank !!(m' ,m) = rank ~l ,N
where ro 1+ m = N + k, is unique. o
This result proven by Tether (1970) leads to the following reaZizabiZity
criterion:
(341 )
Let {~, ~l, •••.•. ~O-I} be an arbitrary finite sequence of q x p real matrices.
let H. . for i + j ... N be a corresponding Hankel matrix. Then a minimal partial 1; J 0
realization {A, !, ~} given by the Ho-Kalman algorithm is unique and realizes
the sequence up to and including the Noth term if and only if there exists
positive integers Nl and N such that:
I. N' + N = N o
2. rank ~lN rank ~'+I ,N rank ~l ,N+I
(342)
(343)
If the realizability criterion is satisfied then the Ho-Kalman algorithm can be
applied with
n = rank ~l ,N
using for the Ho-Kalman algorithm ~'N and a~' N' , The resulting minimal partial realization is unique because the extension
(344)
k sequence ~k = ~!! for k No,No + 1, generated by the realization satisfies
rank .!!--l + :...~ j, N+i = rank~, ,N (345)
for all i,j ?O
-129-
If 1 and 2 are not satisfied, then a minimal partial realization - if it exists -
may not be unique, for in order to use the Ho-Kalman algorithm new matrices
{~o' .•...• ~o-I} must be found until
rank ~1+I,M = rank ~I,M+1 = rank ~l M , (346)
where Ml + M = Po These matrices can be partially or completely arbitrary.
since (~,!,~) are functions of the sequence {~o .••• ~O-I} they may also not
be unique.
One of the most important results of Tether (1970) is the determination of a
lower bound for the minimal partial realization dimension. This is formulated
as the following Lemma.
Lemma 2 Let (~,!,~) be a partial realization where first No Markov-parameters
are equal to the given sequence {M , Ml , ••••• M I}' Then the dimension of --0 - ~o-
the minimal partial realization satisfies the following inequality:
No n ~ n(No) = L
j=1
rank H. N 1 . J' 0+ -J
No
2: j=1
rank H. N • -.J, O-J
This lower bound can be achieved for a "suitably chosen extension!!. This
(347)
"suitable choice lt is the subject of the fundamental theorem of minimal partial
realization given by Tether (1970). In order to state this theorem, it is
necessary to formulate one more definition and one more lemma.
Definition 22: Let NI(No) equal the first integer such that every row of the
block row [~l (No) ...... M ] is linearly dependent on the rows of the 410-1
Hankel matrix ~l(No), No-NI(No)' let N(No) equal the first integer such that
every column of the block column [~(NO) .....• ~0_11T is linearly dependent
on the columns of the Hankel matrix ~O-N(No),N(NO)'
Lemma 3 Let n(No), NI(No) and N(No) be integers defined previously. Then
any extension {~o'~o+1 ....• } of {~o, ~l ••••• ~o-I} whose realization
-130-
achieves the minimal lower bound n(No) for its dimension also satisfies
rank ~(No),N(No) = rank ~l(No)+I,N(No) =
= rank ~l(No),N(No)+1 (348)
for that extension.
From this Lemma follows the Minimal Partial Realization Theorem, which is central
in Tether's algorithm:
Theorem II Let {~, ~1 •••••• ~0~1} be a fixed partial sequence of q x p
matrices with real coefficients and let n(No),N(No) and N1 (No) be integers
defined previously. Then:
I. n(No) is the dimension of the minimal partial realization
2. N(No) and N1 (No) are the smallest integers such that (348) holds
simultaneously for all minimal extensions
3. There is a minimal extension of order P(No) = N(No) + N1 (No) for which
n(No) is the dimension of the realization computed by the Ho-Kalman
algorithm, but which in general is not unique.
4. Every extension which is fixed up to P(No) is uniquely determined thereafter.
It must be stressed that Tether's algorithm does not explicitly recognize the
stochastic nature of experimentally determined Markov-parameters. Nevertheless
it proves useful for real-life noisy and incomplete data because it can
generate an approximate model which agrees with the data. A number of successful
applications of this algorithm have been described (see Rossen-Lapidus (1972),
Hajdu (1969), Roman (1975)).
4.2. Gerth's algorithm
Tether's algorithm was assuming the availability of Markov-parameters. However,
they must be estimated. Therefore, an asymptotically unbiased and efficient
estimation of Markov-parameters is an essential step for the successful
identification of multivariable systems via the realization theory.
-131-
Gerth (1971,1972) proposed a multistage procedure for the initial estimation
of Markov-parameters and further refinement of the finite series {M.} in a way -:L
that assures the linear dependence of the extension series. It works on a
noise-corrupted data set of input-output pairs, assumes a ppeseZeated degree r
of the minimal polynominal and applies the Ho-Kalman algorithm to determine
a realization.
Gerth's algorithm consists of two separate steps:
I. Estimation of the finite series of Markov-parameters {N.} for i c O.I.Z •••• k -1
2. Determination of the minimal polynominal coefficients ai and matrices M. --:L
for i = 0.1 ••••• r - I. which are as "close as possible" to N .• for the -1
)
same i and specify an infinite realizable sequence via the relation
r
Z i.:~
"'i'M' • <\;1 _ )-t. (349)
with ~) = -a . being the coefficients of the minimal polynominal with a = I. ~1 r
The Gerth's algorithm assumes r - the order equal n - the dimension of the
realization. Assuming further that:
1. initial conditions are zero
2. ncises corrupting the signals are white and zero mean
3. number of samples is sufficiently large
and using the Hankel model and the least-squares method. the algorithm results
in the estimate of (k + I) Markov-parameters:
N (350)
-132-
where NT ~ l~' ~" ...... ~1 and matrices; and Yare defined by
yT = [~(~), ~(~ + I), .•••• ~(~ + m)l
i(i) =[~ (i), ~ (i) ••.•.. y (id " q
S ""1Il
~(£ - I) u(£ + m - I)
~(~ - k - I) ••• u(£ + m - k - I)
~T(i) = [u, (i), u2 (i) .••.• up(i)]
(351 )
(352)
Using an input excitation {u.} = {I,O,O •••.• } for each of the p x q partial >
systems and using the Hankel model (8), the excitation matrices for these partial
systems will be
T '0 -UJ"
w .. J> (353)
o
for j = 1,2, ••••• q, £ = J, ••••• p, where Wj~ is the individual weighting factor
for every partial system.
For every partial system the following Hankel matrix is constructed from the
elements of {~-i} for i = 0, I, ..... k.
N '0 -o,jJV
. ~r-I,j£
N '0 ...... N '0 -jlJ" k-r oJ "
(354)
-133-
where N .. , is the jR,-th element of N .. The vector containing elements of 1,JN -~
{N.} for i = r, r + I, '" k is defined as: -1
N ." N I ., ....... Nk ., r,]J<. r+ ,]N ,]JV (355)
The estimation of the minimal polynominal coefficients is also performed using
the least-squares method and the following loss function.
obtained from (349), with the coefficients vector
and
h T --r
H
-a , I
vT = [v T ,V T ....... ~pT]
-11 -12 ."
T -u diag[T TTl ""'-'11 ' -U11.'····· -uqp
The minimization of a gives:
(356)
(357)
(358)
(359)
(360)
(361)
It can be proven that the least-squares estimation of the minimal polynominal
coefficients for a given degree r is unique if and only if rank ~ = r. Next
the finite sequence {M.}, for i = 0,1, .•••. r-I, must be determined using again -1
least-squares estimation for computing.
min m. , -J"
k
2: i=o
[ y(i).,(h ,M) - Y(i).,(N)12
J" --r - J" - (362)
-134-
for j = 1,2, •••• p, R. = 1,2, •••• q, where y(i).,(h M) is the output sequ~nce J'" -r-
generated for the jR.-th partial system by T ., for the already determined -UJ'"
h but unknown {M.}, and y(i).,(N) - is the output sequence generated for the -r -J.. Jx. -
same partial system and the same excitation, but for known {N.}. The solution -1.
to (362) is given by:
where
with
(G T T T. GT)-I G T T T , ., n., - -U-UJ'- - - -UJ "'-u -J '"
nT
., =[N ." N ." ..... Nk .,] - J'- o,J'" I oJ'- oJ'"
G
E ~ 0 I - I
== ---I r-I I
I -r-I I
I
k-r+1 1 •••••• R e - -r
~l ~T =[ 0 .... 0 .... 1. ... 0]
r-th position
(363)
(364)
(365)
(366)
(367)
(368)
If all T ., are nonsingular, the sequence {M.}, i = O,I ••••• r-l, with a given -UJ'" -,
vector h is always unique and can be used to determine the minimal realization -r
using the Ho-Kalman's algorithm.
Extensions of Gerth's idea can be found in papers by Hajdasifiski (1976,1978).
4.3. The approximate Gauss-Markov scheme with the singular value decomposition minim.al =ealization algorithm
Using the Hankel model (8) and relations (351) (352), defining the noise vector
ET = [.~(O, ~(2.+ I)
~T(i) = [ e1 (i) e
2 (i)
(369)
(370)
-135-
and
(371 )
the block vector containing estimates of the first k + 1 Markov-parameters
(372)
the block vector containing the remaining Markov-parameters and the matrix
of initial conditions ~ , the Hankel
~( -k-2) ............. ~( -k+m-2)
~( -k-3) ............. u( -k+m-3) ... ~(-I )
o
o •....•............• u,l-iJ
model, for the noisy system will be following:
Y
Assuming that input samples {u.} and the noise samples {e.} are mutually -1. -1
uncorrelated stationary processes and that E{~(i)} = 0, an estimate of the
(373)
(374)
first k + 1 Markov-parameters can be found minimizing the error function Vw
(see Hajdasinski (1976)(1978)(1979), Niederlinski-Hajdasinski (1978)).
where
Y S T N -1tl -
(375)
(376)
is the estimate of the Y. Thus applying the well known formalism, an expression
for a minimal trace of Vw is found:
= 0;
N (377)
-136-
Expressing N in terms of E
N (378)
it is seen that the term (5 W 5 T)-I 5 WE is the bias of the estimate (377). -m---m -m--
This expression depends on properties of the noise E and asymptotically vanishes
when E{e(i)} o and there is no correlation between samples of E and 5 . - """1ll
Also for the zero initial conditions the second term in the expression (377)
vanishes. In case E{!} f 0, for this type of model i.e. (374), it is very
easy to estimate the bias of the estimate - see Hajdasifiski (1978).
Considering the following expression:
as an accuracy criterion, it follows that:
E{(~ - ~)(~ -
Substituting E{E ET} = -I Rand W = R we get:
see also Hajdasifi.ski (1976 ,1978,1979)
-I Thus choosing ~ =! , this results in an efficient estimate of the Markov-
parameters, since it is easy to show that:
E{(M - N)(M - N)T} - - - - R
(379)
(380)
(381 )
(382)
This way of weighting resembles very much the classical Gauss-Markov estimation
for 5ISO systems.
Defining
T n
-137-
n. T = [e.(t), e.(9.+I), ...... e.(9.+m)]
-1 1 1 1
the noise covariance matrix will take the form:
T = E{E, E, }
It is seen that the trace of g is the following sum of squares:
m+1 trg 2:
i= I j = I
(383)
(384)
(385)
(386)
Writing the R matrix in an explicit form it can be seen that the trace of g
is equal to the trace of R. (Hajdasifiski (1978».
trg trR (387)
Treating the! matrix as the covariance matrix of a hypothetical sinie-input
single-output noise filter it is possible to reconstruct this noise and its
covariance matrix. This "composite" noise model is but a mathematical fiction,
having no strictly physical interpretation. According to (386) and (387),
minimization of the (379), which appears in the minimal tr!, results in the
minimal trace of the noise covariance matrix, attaining in this way the main
goal of an efficient estimation.
If all noises corrupting a multi variable system are stationary, the R matrix
has also a very simple structure:
R =
S 2
s 2 1
S 2 m
S 2 S 2 •••• 1 2
S 2 •••••••••
s2 m-I
S 2 S 2 m-I m
S 2 rn-2
S 2 m-I
(388)
-138-
In practice the R matrix must be estimated using a finite number of samples,
taking as 52 and 52 m
m + 1 q S2 = L ~
-2(.) (389) e i J m +
j i = 1
m-k+1 q
52 = 2: L. e. (j) e. (j+k) (390) 1 1
n m-k+1 j = 1 i = 1
where e.(.) are standing for estimates of e.(.). 1 1
Thus only the first part of the ~ matrix, being the estimate of !, can be
computed with a sufficient accuracy, for in an explicit method there are only
m + 1 input-output and residual error samples available. Higher ordered elements
in R will be increasingly less accurate. As it is demonstrated by Hajdasifiski
(1978), the quality of the ~ matrix estimation plays a key role in the convergence
of Markov-parameters estimates. The realization theory once again helps to get
improved results, providing meaDS to reconstruct the composite noise covariance
matrix. For a detailed derivation of this result the reader is referred to
Hajdasifiski (1978).
Treating E as the set of specially arranged samples of a one-dimensional noise
with the covariance matrix!, it can be assumed that this noise is generated
by a colouring filter from a hypothetical white noise. In such a caSe an
estimate R of R is given by the following expression:
-139-
R ~ee(o)
~ ee(l)
'I' ee (I ) . • • •• ~ ee (m)
~ ee(m-I)
~ ~ ee(m).............. ee(o)
where E is the estimate of the multivariable noise, calculated using the
residual error of the L.S. estimation of Markov-parameters.
(391 )
Assuming that the noise e(k) is generated from s(k) (white noise) by a moving-
average filter,
e(k) -I
+ C(z ) s(k) = (C a
(392)
a very interesting decomposition of the R matrix can be found. From (391) it
follows
with
Since
for
e = £~
~ = ~ (k) ~ (k + I) ....• ~ (k + T [T T T
[ T T T
~ = ~ (k-v) ~ (k-v+l) ..••• ~ (k
C v-I C v-2 C 0 ...... 0 0
C = 0 C v-I C C ...... 0
1 0
o ..........•..... C I ••••••••• C v- 0
E{ii T} C cTa2
E{~~T} = a2 1
(393)
(394)
(395)
(396)
(397)
(398)
Then with accuracy to a constant factor the composite noise covariance matrix
is
R T
aC C (399)
-140-
However, for the Gauss-Markov estimation it is only necessary that the
weighting matrix W is similar to the covariance matrix R.
w (400)
Knowing a finite number of initial elements in the R matrix and assuming
they are sufficiently exact, it will be possible to find, via the realization
theory, the remaining elements and reconstruct the full rank covariance matrix
R.
A comparison of three methods for Markov-parameters estimation is presented by
Hajdasifiski (1978). Sequential algorithms are also available at present. A
general background can be found in Hajdasifiski (1976). Another approach to
the identification with Markov-parameters is presented by Rissanen,Kailath
(1972h Anderson, Brasch, Lopresti (1975).
Having identified Markov-parameters {M.}, the minimal realization algorithm ~
with the s.v.d. of the Hankel matrix was proposed by Hajdasifiski and Damen (1979).
Application of the last is straighforward using results derived in the chapter
3.4.2. - (150) - (158). For the numerical example see Example 6.
- 141 -
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EINDHOVEN UNIVERSITY OF TECHNOLOGY THE NETHERLANDS DEPARTMENT OF ELECTRICAL ENGINEERING
Reports:
93) Duin, C.A. van DIPOLE SCATTERING OF ELECTROMAGNETIC WAVES PROPAGATION THROUGH A RAIN MEDIUM. TH-Report 79-E-93. 1979. ISBN 90-6144-093-9
94) Kuijper, A.H. de and L.K.J. Vandamme CHARTS OF SPATIAL NOISE DISTRIBUTION IN PLANAR RESISTORS WITH FINITE CONTACTS. TH-Report 79-E-94. 1979. ISBN 90-6144-094-7
95) Hajdasinski, A.K. and A.A.H. Damen
96)
REALIZATION OF THE MARKOV PARAMETER SEQUENCES USING THE SINGULAR VALUE DECOMPOSITION OF THE HANKEL MATRIX. TH-Report 79-E-95. 1979. ISBN 90-6144-095-5
Stefanov. B. ELECTRON MOMENTUM TRANSFER CROSS-SECTION IN CESIUM AND RELATED CALCULATIONS OF THE LOCAL PARAMETERS OF Cs + Ar MHD PLASMAS. TH-Report 79-E-96. 1979. ISBN 90-6144-096-3
97) Worm, S.C.J. RADIATION PATTERNS OF CIRCULAR APERTURES WITH PRESCRIBED SIDELOBE LEVELS. TH-Report 79-E-97. 1979. ISBN 90-6144-097-1
98) Kroezen, P.H.C. A SERIES REPRESENTATION METHOD FOR THE FAR FIELD OF AN OFFSET REFLECTOR ANTENNA. TH-Report 79-E-98. 1979. ISBN 90-6144-098-X
99) Koonen, A.M.J. ERROR PROBABILITY IN DIGITAL FIBER OPTIC COMMUNICATION SYSTEMS. TH-Report 79-E-99. 1979. ISBN 90-6144-099-8
100) Naidu, M.S. STUDIES ON THE DECAY OF SURFACE CHARGES ON DIELECTRICS. TH-Report 79-E-I00. 1979. ISBN 90-6144-100-5
101) Verstappen, H.L. A SHAPED CYLINDRICAL DOUBLE-REFLECTOR SYSTEM FOR A BROADCAST-SATELLITE ANTENNA. TH-Report 79-E-I0l. 1979. ISBN 90-6144-101-3
102) Etten, W.C. van THE THEORY OF NONLINEAR DISCRETE-TIME SYSTEMS AND ITS APPLICATION TO THE EQUALIZATION OF NONLINEAR DIGITAL COMMUNICATION CHANNELS. TH-Report 79-E-I02. 1979. ISBN 90-6144-102-1
103) Roer, Th.G. van de ANALYTICAL THEORY OF PUNCH-THROUGH DIODES. TH-Report 79-E-I03. 1979. ISBN 90-6144-103-x
104) Herben. M.H.A.J. DESIGNING A CONTOURED BEAM ANTENNA. TH-Report 79-E-104. 1979. ISBN 90-6144-104-8
EINDHOVEN UNIVERSITY OF TECHNOLOGY THE NETHERLANDS DEPARTMENT OF ELECTRICAL ENGINEERING
Reports:
105) Videc, M.F. STRALINGSVERSCHIJNSELEN IN PLASMA'S EN BEWEGENDE MEDIA: Een geometrischoptische en een golfzonebenadering. TH-Report 80-E-105. 1980. ISBN 90-6144-105-6
106) Hajdasinski, A.K. LINEAR MULTIVARIABLE SYSTEMS: Preliminary problems in mathematical description, modelling and identification. TH-Report 80-E-106. 1980. ISBN 90-6144-106-4
107) Heuvel, W.M.C. van den CURRENT CHOPPING IN SF6. TH-Report BO-E-l07. 19BO. ISBN 90-6144-107-2
108) Etten, W.C. van and T.M. Lammers TRANSMISSION OF FM-MODULATED AUDIOSIGNALS IN THE B7.5 - lOB MHz BROADCAST BAND OVER A FIBER OPTIC SYSTEM. TH-Report 80-E-l08. 19BO. ISBN 90-6144-108-0