Polynomial matrices and feedback Citation for published version (APA): Eising, R. (1981). Polynomial matrices and feedback. (Memorandum COSOR; Vol. 8104). Technische Hogeschool Eindhoven. Document status and date: Published: 01/01/1981 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: 23. Jul. 2021
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Polynomial matrices and feedback · Polynomial Matrices and Feedback by Rikus Eising Memorandum COSOR 81-04 Eindhoven, March 1981 The Netherlands - 1 ":" Introduction In this paper
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Polynomial matrices and feedback
Citation for published version (APA):Eising, R. (1981). Polynomial matrices and feedback. (Memorandum COSOR; Vol. 8104). TechnischeHogeschool Eindhoven.
Document status and date:Published: 01/01/1981
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.
ROBABILITY THEORY, STATISTICS, OPERATIONS RESEARCH AND SYSTEMS THEORY GROUP
Polynomial Matrices and Feedback
by
Rikus Eising
Memorandum COSOR 81-04
Eindhoven, March 1981
The Netherlands
- 1 ":"
Introduction
In this paper we describe the use of feedback with respect to some
polynomial matrix constructions.
Given a polynomial matrix P(z) € ~[zJPxq (the set of p x q-matrices
with entries in ~[zJ (the set of real polynomials in z», we construct
a matrix QCz) € ~ [zJ (q-p) Xq such that
[PCZ) ]
QCz)
is unimodular (q ~ p and P(z) has full rank for all z € ~ (the set of
complex numbers». The method we use also gives the inverse of this
matrix. A number of applications can be found in [1J.
Of course this problem has a well known solution. Algorithms providing
us with Q(z) are mostly based on elementary row (column) operations
thereby reducing P(z) to some simple form. Having obtained such a simple
form (for instance lower triangular, Hermite form (like), Smith form
(like» the construction of Q(z) is straightforward. See [9J. Our con
struction is not based on elementary row (column) operations. We work
on real matrices directly.
One of the main problems concerning the methods based on elementary
row (column) operations, which in turn are based on the Euclidean
algorithm is their numerical behavior. Our method allows pivoting
techniques based on the values of the coefficients of the occurring
polynomials (the traditional methods use a "pivoting technique"
based on the degree of the polynomials). Another basic feature of
our method is rank determination. This can be done properly using
singular value decomposition. See [6J.
Preliminaries
Consider a pair of real matrices (A,B) such that the nonzero eigenvalues
of A are controllable, i.e. [zI - A,BJ is right invertible for z + o.
- 2 -
An equivalent condition for this is:
(1) [I - zA,zB]
~s right invertible for z € ~ or equivalently right invertible over
R[zJ. This can be seen as follows. Suppose A € cr(A) ~ ~ O. Then I 1
[~I - A,BJ has full rank. Therefore [I - r A, r B] has full rank.
On the other hand we have that (I) has full rank for z = 0 and if 1
z ~ 0 then [z1 - A,B] has full rank. We will need a solution for
the problem as posed in the introduction for the case.F(z) = [I - zA,zB].
Let F be such that A + BF is nilpotent. Such an F exists by the pole
placement theorem because all eigenvalues of A, unequal to zero, are
controllable (see [4]). The generally non unique matrix F can be con
structed using one of a large variety of algorithms (see [7J).
Now we have that
is unimodular because
Here U(z) = (I - z[A + BF])-I is a unimodular matrix because A + BF is
nilpotent. Observe that the computation of U(z) can be done very easily
because
n-l n-l U(z) = I + z[A + BFJ + ... + z [A + BFJ
Here A € R nxn.
- 3 -
It can easily be shown, using the Brunovsky canonical form (see [3J),
that F can be chosen in such a,way that the degree in z of U(z) does
not exceed ci - I + c where ci is the controllability index (see [7J) of
the controllable part of (A,B) and c is the dimension of the non con
trollable part of (A,B). Concerning the degree of U(z) we have the nxn nxm
following in the generic case. Let A E lR ,B E lR • Let kl be
the smallest integer not less than n/m. Then F can be chosen such that
degz U(z) S kl - 1.
The results
Let P(z) E lR [zJPxq be such that P(z) has rank p for all zEit (again
q ~ p). Then Po has rank p as a real p x q-matrix.
Here
n P(z) = P + P z + ... + P z o 1 n
and n is the degree of P(z). Multiply P(z) by P such that P is an inver
tible real q xq-matrix and
If q m p then 0 is the empty matrix.
Partition P(z)P as
P(z)P = [Pa(z),Pb(z)]
such that Pa(z) E lR [z]qxq. Observe that P (0) = I. a
Define P . by a~
P (z) I + P al z + ••• + P n = z a an
and Pbi by
Pb(z) Pblz + ••• + n
= Pbnz •
- 4 -
Now consider matrices A and B defined by
o
I (4) A=
-P .an
I - p al
rp I .bu
, B '"
It can easily be shown that [I - zA,zBJ has full rank for all z E ~.
Consider a feedback matrix F E ]R n(q-p)x'np such that
is unimodular. See (2).
Next we observe that
I
zI I
'n-l Z I. . '. zI - - - - -
0
I
'0 I I
I I - -I I
I o : ..... 0 I X (z) n
0 . 0
. I X2(z)
o . : 0 P (z) a
F-1 FZ
r I zp I ZPb an n
I .. zI
•
I -zI I + zp all zPb 1
- - - - - - -F I
y (z) n
def. ~
'" P(z) • . Y2(z)
Pb(z) I-
I I
==
- 5 -
Here
. 2 .
X.(z) = zp . + z p . 1 1 a1 a,l+ + ••• +
n-i+l p z . ( . ) a,l+ n-l
for i := 2, ••• ,n and Y.(z) is defined analogously for i = 2, ••• ,n. Further-1
more F 1: and F 2 stem from a partition of F according to the dimension
of the other matrices involved.
Observe that P(z) is unimodular because both factors 1n the left hand
side are unimodular. ~
The next transformation on P(z) we need is
I
o
. I I
o· ... ·0 o - '-
-F 1 I 0 I
P(z) :=
I I X (z) n
I Y2
(z)
- -o . . . . . 0 I P a (z) I P
b (z)
- - - - - - - -1- - -o ..... 0 IF 2 - F IX I I - FlY
This latter matrix is still unimodular. Therefore
(6)
T T T T T T is a unimodular matrix. Here X := [Xu , ••• , XZ] and Y = [Yn , ••• , Y2] and T denotes transposition. Thus we have obtained a matrix Q(z) such that
[
PCZ) ]
Q(z)
is unimodular, namely Q(z) = [F2 - FIX,I - FIYJ. Now it is clear, see (2),
how the inverse of (6) could be computeEL ..
- 6 -
We will describe part of this construction in more detail. Let M(z) be
such that
[I - zA,zB]M(z) = I.
For instance take
M(z) ... [ U(Z)]
-FU(z) •
Again U(z) = (I - z[A + BFJ)-l and A + BF is nilpotent. Then we have