Pole assignment for systems over rings

Pole assignment for systems over rings

Citation for published version (APA):

Eising, R. (1982). Pole assignment for systems over rings. (Memorandum COSOR; Vol. 8216). Technische Hogeschool Eindhoven.

Document status and date: Published: 01/01/1982

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Department of Mathematics and Computing Science

Memorandum COSOR 82 - 16

Pole assignment for systems over rings by

Rikus Eising

Eindhoven, the Netherlands September 1982

Abstract.

POLE ASSIGNMENT FOR SYSTEMS OVER RINGS by

Rikus Eising

Eindhoven University of Technology Dept. of Mathe~atics and Computing Science

Den Dolech 2, Eindhoven, the Netherlands.

A very simple proof of the pole assignment theorem for systems over a principal ideal domain (and other rings) is given. Furthermore, an algo-rithm is presented. Extensions are also indicated.

Keywords. Pole assignment theorem, systems over a principal ideal domain, Pole assignment alsgorithm.

In [2J Morse gave a proof of the pole assignment theorem for systems over the ring JR[zJ (the ring of polynomials in the variable z with coefficients in the field of real numbers). In [3J Sontag observed that the proof by Morse in fact proved the pole assignment theorem for systems over a

prin-cipal ideal domain.

In this paper we present a very simple proof for the pole assignment theorem for systems over a principal ideal domain. In fact this proof extends to the case of systems over many other rings (as will be shown). Furthermore, we will present an algorithm which gives the feedback gains in a

straight-forward way.

Let R be a principal ideal domain. Let a system (A,B) over R be given where A is an n x n-matrix over Rand B is an n x m-matrix over R. Reachability of

(A,B) means that

(1) [n-A., BJ

has a right inverse over the polynomial ring

R[AJ.

The following Lemma is the basis for our results.

(2) Lemma. Let a system (A,B) be given in the form

(3)

where BI is unimodular. Then the properties

(i) (A,B) is reachable (H) (F 1 ,[G

I ,G2J) is reachable are equivalent.

3

-Proof.

Right invertibility (over R[A]) of

[AI - A , B]

is equivalent to right invertibility of

-G

1

o

which is equivalent to right invertibility of

Next we show that the form (3) can be obtained for any reachable system.

o

(4) Lemma. If (A,B) is reachable then there exists a state space isomorphis~

-I

U₁ and an input space isomorphism V_t such that (UIAU₁ ,UtBV_t) has the structure (3) with B

t unimodular (B) is not empty).

Proof.

Because B has at least one unit invariant factor it is immediately

clear that a factorization of 'B as required in (3) is possible. Such a

factorization can be obtained using for instance the Smith form of B.

0

Now we present the proof of the pole assignment theorem. (We will use O'(M) for the set of zeroes of the characteristic polynomial of M).

(5) Theorem. Let (A,B) be a reachable system. Let A

=

{AI, .•. ,A

n} be a set of n elements of

R.

Then there exists a feedback matrix K such that

Proof. We may suppose that (A,B) has the structure (3). Suppose that B

J

is an rx r-matrix. Let A) .., {A), ••• ,A

r} and AZ = {Ar+1, ••• ,An}. Suppose that K) and Kl are matrices such that (assumption for induction)

Next we perform a state space isomorphism

which gives

and we apply a feedback

to the latter pair of matrices, This gives

[

F)+G)K)+G2K2

HI + AIKt - Kt F 1 - K1G)K 1 - K1G2K2

I

-K _I

Because B} is unimodular it is easy to find matrices K3 and K4 such that

HI

+ B1 K3 = 0 and a(A₁ + B₁K₄) = At' These K3 and K4 can be used in a feed-back matrix

5

-This proves the pole assignment theorem by induction if we can prove a starting point for the induction argument. Lemma (2) and Lemma (4) show that such a starting point consists of a system (A,B) where B is a right invertible matrix. Such a case can be handled easily.

o

Remark 1. The ring R does not need to be a principal ideal domain. The only thing we need for our proof is the factorization as in (3). This can be obtained for various rings (for instance valuation domains).

Remark 2. The proof of the pole assignment theorem is closely related to [1].

We will now describe a method which transforms (A,B) into a pair of struc-tured matrices which is very useful for the actual calculation of the feed-back matrix.

Suppose that we have obtained

(6)

by applying a state space transformation and an input space transformation. Next we start the same procedure for

using a state space transformation U

where B2 is unimodular. The matrices U

2 and V2 are used for a state space transformation and an "input space transformation" on (6) in the following

way ][

u~J

I 0 0

([ :2

o ][ FJ G)

:

]

[ :2 : ][:2

0

])

0 V 2 0

=

I HI Al BI 0 0 I F3 G₃ G₄ 0

0

H2 A2 0 B2

---H] All AI2 I B]

Continuation of this process gives us via a sequence of state space iso-. morphisms and input space transformations (at the various levels) a struc-iso-."

tured pair (equivalent to (A,B».

U[A, B]U V

=

(7) AQ,

*-

I

_BQ,

₀

0

'*

•

_'*

•

₀

*

•

*

•

₀

*

A_Z

*

B2

*

At

*

B] n m

7 -Here U =

[U.

0]

_{• • •}

[U2

0]

_U]

o

I , 0 I -1 0 0 -1 0 0 I 0

[U~l

0 ]

U 2 UJ/,

[U-

1

oW

01

U

_v

= 0 V 2 0 0 VJI, 0 = 0 VJI,

o

VI

o

I 0 VI 0 0 I 0 0 I 0 0

and B₁, ••• ,BJI, are unimodular matrices. The number of coluIIms in [*BiJ is the dimension of V.: The dimension of U. equals the dimension of

1 1 '

diag{AJ/" ••• ,AiJ·

The structured matrix (7) can easily be used for pole assignment. A feed-back matrix can be found by applying the method, . used in the proof of the

theorem, recursively. In order not to become involved in many details we will show this for the case where (7) has the following form

A2

*

B2

0

(8)

*

Al

*

B1 Observe that a

[~

-1 0

][

I 0

] =

(UAU-I,UB) ~ B2]V2 -1 -I [AI ~JV 2 B] 0 VI

([

~

G 1

]

[ G2 0 _]

V~l

₎ Al H2 B] 0 0 I

Suppose that A

=

A2 u Al and that K2 is such that cr{A

2 + B2K2)

=

A2• Such

a K2 can easily be obtained because B2 is unimodular. This means that

=

A

2

(we need a zero matrix here because of the"* part in

[*

B₂

J).

Let

Then we have

Next we perform a state space transformation and we apply a feedback in the following way

(9) [ I

-K 22

Then we obtain

Now we choose a feedback matrix

[ K:2

0

1

VI

Kil

where KI2 is chosen such that Xl _{+B I}K

12 = 0 and KII

cr(A

_I + B] KIt)

=

A],

9

-By adding these two feedback matrices we obtain a feedback matrix (for a state space isomorphic system) which gives as the new spectrum the set A. (The A-matrix of this isomorphic system has a block triangular

struc-ture). Now it will be clear how to obtain the feedback matrix for the system (A,B).

Remark

3. The matrices G

2 and H2 (and thereby also K21) are empty if the last column of B

Z has index n (less than n is not possible in this example). In this case (9) consists only of a state space transformation.

Same observations.

- The algorithm used for the construction of (7) can also be used as a test for reachability (B~ has to be unimodular for reachability).

-"If B~ is not unimodular, the algorithm enables one to determine the part of the spectrum of A that cannot be changed (B

t

=

0). If (A,B) is

reach-able over the quotient field then the invariant factors in B

t indicate

which elements in R can be assigned.

- All matrices B. (i

=

1, ••• ,t) can be taken diagonal.

~

- The algorithm can be used for systems over ~ in a very useful way if one takes U1 and VI (in Lemma (4» to be unitary and such that BI contains "good" (possibly not "small") singular values of B. A somewhat modified Singular Value Decomposition can be used here. The same idea can also be used for B

2 •••• ,Bt. This algorithm is closely related to [IJ.

References.

[I] Eising. R.; Pole assignment, a new proof and algorithm. Systems and Control Letters, Vol. 2, (1982), 6 - 12.

[2] Morse. A.S.; Ring Models for Delay Differential Systems. Automatica, Vol. 12, (1976),529-531.

[3J Sontag. E.D.; Linear Systems over Commutative Rings: a Survey. Ri-cerche di Automatica, Vol. 7, (1976), 1- 34.