Polynomial matrices and feedback

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

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Department of Mathematics

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.

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 deg_z 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

_o

+ P z + ... + P z

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) _a

=

I + _{P al z + ••• +} P z n an and Pbi by Pb(z) = Pblz + ••• + _{Pbnz •} n

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 X 2(z)

o .

: 0 P (z) a F-1 FZ

r

I zp I ZP b 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 Y 2 (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 , ••• ,

X

Z

]

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)

=

[F

2 - FIX,I - FIYJ. Now it is clear, see (2), how the inverse of (6) could be computeEL ..

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

n-l n-)

[z 1, ••• ,zI,1J[I - zA,zBJM(z)

=

[z 1, ••• ,z1,1J

or

n-I

[O, ••• ,O,Pa(z),Pb(z)JM(z)

=

[z I, ••• ,z1,1J

which shows that the appropriate lower right part of M(z) constitutes the right inverse for P(z).

Remark

Because the degree of U(z) depends on P we will not give an upper bound for deg U(z). Observe that

Concerning U(z) we have in the generic case

7

-The method of adding rows to P(z) ~n order to obtain a unimodular matrix can also be used for two-vari~le polynomial matrices.

Let P(s,z) be a two-variable polynomial matrix such that PCs,z) has full rank for all (s,z) €

~2.

Then it is know that P(s,z) has a polynomial inverse (see [9J) and that there exists a polynomial matrix Q(s,z) such that (7) [ P(S'Z) ] QCs, z) is unimodular. Define P. (s) by ~

Then PO(s) has full rank for all s € ~. Therefore there exists PCs) such that

such that

n P (s)z

an

pes)

can be constructed using the method described for the one-variable

case.

Now matrices A(s), B(s) can be defined as in (4). In order to be able to construct FCs) such that

is nilpotent we have to assume that (A(s),B(s» is a controllable pair over R[sJ. This a because the, only pole placement algorithm for systems defined over R[sJ (generally systems defined over a principle ideal domain) which is available at the moment uses controllability in a cru-cial way (see [8J).

Now controllability of (A(s),B(s» is equivalent to P (s) having full n

rank for all s €

~

and P(s,z) having full rank for all (s,z) €

~2.

This

is the so called regular case (the highest coefficients matrix has full rank). With the usual interpretation this means that P(s,"") has full-' rank for s € ~.

Summarizing we have:

If P(s,z) is a two variable polynomial matrix such that P(s,z) has full rank for all (s,z) E

~2

and P (s) has ,full rank for all s €

~

then we

n

can construct Q(s,z) such that (7) is unimodular using the "feedback based" method described above.

An

example which shows that this construction fails in case P (s) does _n not have full rank for all s € ~. is given by

[ 2 2:1

P(s,z)

=

1 + sz ,s ~

P(s,z) has full rank (one) for all (s-,z) € 1&2 but P (s) = [s,OJ does not

n

have full rank for all s €:~.

According to (4)

It is easily seen that I - z[A + BFJ is not unimodular for any F.

Remark

If a matrix P(s,z) does not satisfy the conditions for the construction above then pez,s) might satisfy these conditions. In this case an obvious modification of the algorithm performs the task.

9

-Remark

If we consider a matrix P(zl,:",zm) such that P(zl"",zm) has full rank for all (zl"",zm) E ~m then it is possible to add rows to P

(given that P has more columns than rows) in order to make the new matrix unimodular. See [9J. The technique described in this paper can only provide a solution to this problem in the case where the number of columns is one more than the number of rows. Furthermore we need that P is regular. (P(zl""'z ) has full rank for some

m m-l

z;

=

dO and (zl""'z, I'z, I""'z ) E IC ). In the case where the

... ~- ~+ m

number of columns is one more than the number of rows we deal with a single input system after the construction of A and B, according to.(4). Now (A,B) is controllable and therefore we have coefficient assignability using the controllable canonical form for (A,B).

Many times the matrix

[zI - A,BJ

occurs in system theory. The construction of a right inverse for this matrix has a lot of applications. See for instance [5J where the dual

case is used for the construction of observers.

First we consider the single input case such that the matrix [zI - A,BJ has full rank for all Z E IC. Thus (A,B) is a controllable pair. We

trans-form this pair into controllable canonical trans-form and then [zI - A,BJ becomes z -1 0 .. ·· ·0

_.

0 0 . 0 0 -1 0 -a z -a n 1

Now it is easily seen that (because all but the first column constitute a uni-modular matrix) a right inverse M(z) of [zI - A,BJ can be constructed such

that

Here

and M(z) is partitioned such that M

2(z) is just a row vector. From the Brunovsky canonical form for (A,B) we can conclude that

[zI - A,BJ has a right inverse M(z) such that

Here

M(z) = [Ml (z) ] M

2(Z)

is partitioned according to the dimensions. of A and BJand ci is the con-trollability index of (A,B).

A well know result (see [2J) is

Here n is the dimension of A and m is the rank of B. This result can be considered a worst case upper bound for the degree of M(z) because we

- 11

-have that

ci~n-m+l.

A generic bound for deg M(z) is given by z

where k} is the smallest integer not less than n/m. This can easily be seen. We will now describe how the method presented above can be used for the determination of a greatest common divisor of a number of polynomials and thereby also for the construction of a (right or left) GCD of polynomial matrices. Also the linear combination of the polynomials which adds ~p to

the GCD will be found (the so called Bezout parameters).

Let PO(z)'Pt(z), ••• ,Pm(z) be P?lynomials. Suppose that there exists an i such that p.(O)

#

0. If not then we can extract a power st from p.(z)

~ J

j

=

O, ••• ,m. Now we may assume that POCO)

=

1 and PjCO)

=

°

for j

=

l, ••• ,m. This can be obtained as in (3). Now build A and B as in (4). The nonzero eigenvalues of A may not be controllable. In this case transform (A,B) by means of a state space isomorphism T into

such that the nonzero eigenvalues of Al are controllable and A4 is inver-tible. Let [O, ••• ,O,IJT- 1 =[t

l,t2

J,

partioned according to the partitioning -1

of TAT • Then (tl,A_l) is an observable pair. Transform this pair into observable canonical form by means of a state space isomorphism T_t• Thus. we have

as A and B are related to PO(z)"",Pm(Z) The polynomials PO(z), ••• ,Pm(z) do not have a common zero. The.method described earlier provides us

with AO(Z), ••• ,Am(Z) such that

+ ••• +

P

(Z)A (z)

=

m m

and it is easily seen that

PO(Z)AO(Z) + ••• + P (Z)A (z) _{m m}

=

GCD(p (z), ••• ,p (z».

0 ~ m

Remark

At the moment we do not have proven results concerning the. numerical proper-ties of the methods described in this paper. The choice of

P

in (3) is im-portant with respect to this matter. Also the construction of the feedback matrix F should be done in a stable way. These points are currently under

13

-References

[IJ Antsaklis, P.J.; Some Relations Satisfied by Prime Polynomial Matrices and Their Role in Linear Multivariable System Theory. IEEE Trans. Autom. Contr. Vol AC-24, August 1979.

[2J Barnett, S.; Matrices in Control Theory. Van Nostrand Reinhold, 1971. [3J Brunovsky, P.; A Classification of Linear Controllable Systems,

Kybernetika, Vol 3, 1970.

[4J Hautus, M.L.J.; Controllabil,ity and Obsservability Conditions of Linear Autonomous Systems .• Proe. Koninklijke Nederl Akademie van Weten-schappen, Series A, 72, no. 5 1969.

[5J Hautus,. 1.f~L.J •. , and Scntag, E.D .. ; An approach to detectability and

observers. MEMO-COSOR 79-08. Eindhoven University of Technology 1979. [6J Jong, L.S. de; Numerical aspects of realization algorithms in linear.

systems theory Ph.D. thesis Eindhoven University of Technology 1975.

[7J Kailath~ T.; Linear Systems, Prentice Hall 1980.

[8J Morse, A.S.; Ring l:odels for Delay Differential Systems. Automatica, Vol. 12, 1976.

[9J Sontag, E.D.; On ge~eralized inverses of polynomial and other matrices, IEEE Trans. Autom. Contr. Vol AC-25 June 1980.