Pole assignment : a new proof and algorithms

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Eising, R. (1981). Pole assignment : a new proof and algorithms. (Memorandum COSOR; Vol. 8110). Technische Hogeschool Eindhoven.

Document status and date: Published: 01/01/1981

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STATISTICS AND OPERATIONS RESEARCH GROUP

Memorandum COSOR 8)-10

Pole assignment, a new proof and algorithms

by Rikus E:l:s.ing

Eindhoven, The Netherlands August ] ~8) ,

Rikus Eising

Eindhoven University of Technology

Department of Mathematics and Computing Science P.O. Box 513

5600 MB Eindhoven The Netherlands

Abstract

In this paper a new proof of the pole assignment theorem is given. This proof is a very straightforward one. It is not based on canonical forms and also the reduction to the single input case (Heymann's Lemma) is not used. Furthermore, an algorithm is given which is based on structural properties and also an algorithm is presented which allows to take into account numerical aspects. This latter algorithm uses only unitary matrices as transforming matrices preceding the pole assignment.

I. Introduction and preliminaries

Proofs of the pole assignment theorem are mostly based on Heymann's Lemma [3], thereby reducing the mUlti-input case to the single input case. Then some canonical form for the single input case is used in order to be able to specify the feedback matrix which assigns the poles of a system as specified beforehand. Other proofs use a canonical form for the multi-input case. The proof in this paper is based on a completely different approach. It is a straightforward and constructive proof. One of the very nice aspects is that the mUlti-input case is easier than the single-input case (as it should be of course, because there is more freedom). This is very easily seen from the algorithms. The algorithms which we will describe allows to take into account numerical considerations (based on singular value decompositions). The transformations, in order to reduce

the system to.a formwhich is

well

apt for pole assignment, are all unitary. A very recent algorithm also using such transformations can be found in

[4]. An advantage of our approach as compared to [4J is that the sequence of unitary matrices is generally rather short and the transformations do not interfere with the pole assignment process itself.

The following lemma is basic in our approach.

(1) Lemma. Let (A,B) be a controllable pair. Suppose that this pair has the form

(2)

([F G],[O])

H A] B

where B] has full row rank. (This form may be obtained via a (unitary) state space transformation). Then coptrollability of (A,B) is equivalent to controllability of (F,G).

Proof. If

nF

=

An, nG

=

0, n ~

°

and

A

is an eigenvalue of

F,

then

[n,O] ACn,O]

On the other hand, if [n 1,n2

J

[~

1

1

]

=

~[nl,n2]

, n2B]

=

0 and [n),n2

J

~ 0, then _n2

=

°

and we have n}F

=

~nl'

n}G

=

_{0, n1} ~ 0 • In the pole placement part of the algorithms and also in the theorem

(thus after having applied the preliminarY unitary transformations) we will use state space transformations like

on a structured pair (A,B) as

([F G], [0] )

H A1 B]

Thus we have

This kind of transformation performs a feedback on the pair (F,G) •. Jhe

most left upper block becomes F + GK. Such a state space transformation will be called an .F-type transformation (based on K). We use JR ixj as

the notation for the set of real ixj-matrices and a(A) will denote the spectrum of the matrix A. The set of complex numbers will be denoted by ¢ , If A E ¢ then A denotes the complex conjugate of A and ReA denotes

the real part of A. The number of elements in a set A is denoted by =If A.

II. The res ul ts

(3) Theorem. (Pole assignment). Let (A,B) be a controllable pair where A E JRnxn and B € JRnxm • Let A = {Al, ••• ,A

n} be a symmetric set of com-plex numbers. (If A E A then

I

E A). Then there exists a feedback matrix

K € JRmxn such that a(A +BK)

=

A.

Proof. We may assume that (A,B) has the form (2). Let the rank of B} be r. Now we have two possibilities: a and

S.

a : There exists a partition of A: A == A} u A2 such that =If Al == r,

:#

A2 == n-r, A} and A2 are symmetric.

S

Case a is not possible. Now there exists a partition of A : A ==

are symmetric.

First we consider case a. Assume that we can find Kl such that a(F +GK₁)

= A₂, Apply an OF-type transformation based on K]. The last r rows in the transformed matrix A can be chosen arbitrarily using a feedback matrix because BI is right invertible. Therefore, we can construct a marix K E JRmxn such that a(A +BK) == A • (The block corresponding to H can be made zero and the block corresponding to Al can be given eigenvalues AI)'

Next we consider case ~. Assume that we can find K} such that cr(F +GK})

=

=

A4 U {ReXL We may assume that F + GK

1 has the form

[

FI

f]

o

ReA

and that the lower left element y of G is unequal to zero. (Both can be obtained via a state space isomorphism.) Because B} has full rank we can choose a feedback matrix such that the block corresponding to Hhas the form

o . . . .

0 <5

o

o

o

and the block corresponding to Al has the form

[

Rei;

a]

_ 0 All

",([ReX

y]\

=

where cr(Al})

=

A3 and v }

<5 ReA

This proves the pole assignment theorem by induction if we can prove the case where B has full row rank. This case plays the role of a starting point for the induction argument and it is in fact

a

trivial case to prove.

Remark

If A E 1R then we only have to deal with case a which gives a very si~

pIe proof of the theorem.

Remark

If we allow complex feedback then we also have only case a because f..,

and thereby hI and

hZ'

does not have to be symmetric anymore.

Remark

Obviously, it is possible to make a different choice for ReA in the proof above. It will also be clear that the proof can be given without choosing some specific values in the small Z x 2-matrix by just requir-ing that the 2 x Z-matrix has A and ~ as its eigenvalues. For instance,

one could give this 2 x 2-matrix a companion structure.

Next we describe a pole assignment algorithm which is directly based on the induction argument as used in the theorem. This algorithm does not depend on numerical properties of the pair (A,B).It is presented here in order to obtain a better understanding of the induction process as is used in the theorem. It also provides a Brunovski like canonical form for (A,B).

Let (A,B) be a controllable pair. Then there exists a state space

iso--]

where B} has full row rank. By Lemma (1) (F,G) is a controllable pair. Thus there exists a state space isomorphism (for (F,G» such that

-1

(T₂FT₂ ,T₂G) has the form

and we have that B2 has full row rank. Furthermore,

FI G₁ 0

o

HI A2 B2

o

--- r

HT- 1

2 Al

Again by Lemma (I) we have that (F],G}) is a controllable pair. We can continue with this procedure and eventually we obtain a state spaceequi-valent form for (A,B) with the structure

~

Bk 0

0

•

•

•

(4} (A,S)

=

•

•

•

0

o

X₂ A2 B2 Xl _A}

where B. has full row rank for i

=

1, .. "k,

1.

The structure

(4)

is very useful for pole assignment purposes, We will demonstrate this for the case where

(A,S)

has the structure

(k

=

3)

(A,S)

=

(

A3 B3 0

X_{2 A2 B2}

X) A)

We will suppose that the desires eigenvalue set A can be partitioned as

d .. xd.

A = Al U A2 U A3 where Ai is s·ymmetric,

#

Ai

=

di and Ai E 1R. 1. 1.

i

=

1,2,3, If A cannot be partitioned in this way we can obtain pole assignment using a construction as in case S of Theorem (3), Because B3 is right invertible it is easy to obtain K3 such that cr(A₃ + B3K3)

=

A₃• Apply F-type transformation based on K3'

I

o

o

(

o

o

o

I A3 + B3K3

(

-X₂ XlI

A:3

B3 0 X_{2 A2} B2 XlI X_{12 Al} B3 0 A2 B2 X_{I2 Al} I

o

o

I

o

o

o

o

o

) =

o

o

I

o

0 I 0 0

)

_, BI Then construct K2

=

[K₂₁ ,K

ZZ J such that X2 + B2K21

=

0 and such that cr[A2 + _B2KZZ

J

=

A₂, This can easily be done becauseK

2 is right i~vert­ ible. Next we apply an F-type transformation based on K2'

I 0 0 A +B K _B3 0 I ,0

ol

I 0 0 0 3 3)

:J

(

0 I 0 X_{2 A2 B2} 0 I 0 I 0 0 -K -K I _Xll X_{12 A1 K21 K22} -K -K I _B1 21 22 21 22 A +B K _B] 0 0 3 3 3

r

0 A Z+B2K22 B2 ' 0

\

\

}

....

X ll X_1Z Al

-

B1 Finally we construct K1

=

[K 13 ,K1Z ,Kn ] such that

Again we use the right invertibility of B

t• We have obtained a feedback

Watrix

K

such that

A

+

BK

is similar to an upper block triangular matrix having the spectrum that was specified beforehand.

This was obtained. using f-type transformations which did not affect

B

and at the end of the procedure the construction of the feedback matrix was almost a trivial step. An analogous procedure can be applied to the more general structure (4).

Consider again

(A,B)

as in (4). Now take the controllability matrix

n-I

[B.AB •... ,A

BJ.

This matrix has the following structure

0

B_k ••• B,

0

*

0

•

*

(5) _B2Bl

*

Bl

*

The matrices Bl.B2Bl, •• ~.~ ••• Bl all have full row rank. This structure shows that the row rank of B. is equal to the dimension of the factor

1.

space S./S. 1 where S. is the space generated by th.e columns of

1. 1.- 1.

i-I

[B,AB ••..• A BJ,

and

So

=

O. Here i

=

I, •••

,k. This shows that the Kronecker indices are also closely related to the structure (4). In fact, we even obtain a new canonical form under the transformation group given by state space isomorphisms, feedback and input transformations. This canonical form which is very much related to th.e Brunovski canonical form is given by

the following:

In (4) all block matrices are zero matrices except for the matrices B. which have the structure [I.,OJ where the dimension of the

iden-1 ' 1.

tity matrix L is equal to the number of rows of B •• The zero

1. 1

matrix in this structure may be empty.

The structure (5) also reveals that for the construction of (4) we only used the columns of

k-1 [B,AB, •.• ,A B ] .

The remaining columns in

k n-1 [A B, ..• ,A B ] ,

did not playa role in this process. However, among these latter vectors there may be vectors which, together with the previous ones, comprise

k-l a "better" independent set than only those in [B.,AB, •.• ,A BI.

We will illustrate this rough terminology with the following example. Take a pair (A,B) where B is the matrix

0

.

.

0 €: n-1 0

0 _€:l

a

. .

.

a

and ~l""'€:n-I are small nonzero numbers. Now (A,B) is clearly a con-trollable pair. However the construction of a feedback matrix hased on the inverse of B will almost inevitably result in a matrix with large numbers. On the other hand, suppose that (A,b

l) is: a controllable pair, where b_l is the first column of B. The construction of a feedback matrix based on A and b_I may result in a matrix with more reasonable numbers. Whether or not this will be possible can be decided after having taken

into account all columns of the controllability matrix. This motivates us to modify the above algorithm, which is somewhat naive with respect to these problems, in such a way that also numerical considerations can be taken into account. This new algorithm will be described in the next part of this paper.

In the algorithm to follow we use Singular Value Decomposition of a ma-trix. Let M be an n xm-matrix over JR. Then there exist unitary matrices

U and V such that

M " UDV,

where

o

o

0

~ d > 0 and some of the zero matrices may be empty

r

depending upon n,m and r which is the rank of M.(See [5].)

We use this S V D Theorem in th.e following way. Let M be an n xm-matrix over JR. Then there exist unitary matrices U

where

0

0

-

- -

-

I-

-

-

-

-Dl

...

.

d _r I I

0

• L d 1

.

_L L Here d] ~ ••• ~ d

r > 0 and some of the zero matrices may be empty de-pending upon n, 111 and r. This latter factorizat;i.on of M will also be called S V D of M.

Next we consider a controllable pair (A,B). Let

where U

t and VI are unitary and

Here d

1

,

1 ~ ••• ~. d1

,r

> 0 (S V Do of B).

1

Partition D} as

o

Q

o

o

D 1,g 0 0 where d

l

0

d

0

• 1, gl • 1, b 1 D = _{D} b}

₌

1,g _•

,

0

.

0

.

J

d d} 1 1,gl+}

-

,

Here the breakpoint is chosen according to some criterion such that D}

,g

contains only "good" singular values and D contains "bad" singular 1,b

values. However, this has to be done in such a way that

0 0 0

(6)

(

_\

A 0 0 0 VI

)

D _{], g}

0 0

is still a controllable pair. (This can always be obtained.) Observe that this controllability condition is a limiting factor for the break-point between "good" and "bad" s'ingular values. Whether or not a singu-lar value is "good" or "bad" depends, upon the application one has in mind. Therefore we will not specify "goodness" and "badness". However, generally one would like D₁,g only to contain singular values which are not very small. This is because D1 will have to be inverted for the

construction of a feedback matrix and we do not want this feedback ma-trix to contain very large elements. Also, with respect to round off errors, the occurrence of small singular values constitutes a very unfa-vourable situation. Concerning this topic we refer to [2J where numeri-cal properties of-the algorithm to follow are reported.

(7)

([F G], [B'l,h]) .

H At BI - ,g Here [ 0 B -1 ,b - 0 B

=10

I,g [I,g

o

Observe that because

o

o

)

is controllable we have that (F,G) is also a controllable pair by Lemma (1) •

Let

be the S V D of G. Apply the state space transformation

(8) =

As in the former stage we partition D2 as

o

o

o

o

D2 b 0

_,

D2,g 0 0

Here D

2,g contains only "good" singular vlaues and D2,b contains "bad" singular values. Again the breakpoint is such that

0 0 0

(F

0 0 0

v

2_/ ) .,

D

2,g 0 0

is still a controllable pair. This results in an analogous partition of

(F ,

D2 V

2) as in (7). The effect of the unitary transformation in (8)

on

(A ,

D} VI) is given by

([u;

0] [F G] [U2 0],

[u~

0]

\.. 0 I H A} 0 I 0 I

[:l.b])=

FI G₁ X 2 A2 Xl B2,b B 2,g A1 B l,g

Here (F ,G) is still a controllable pair by Lemma (1). The matrices B

1 1 2,g

and. B₂,b are defined as

B2 b

=

[0

' 0

B

2,g

=

[D

2,g

°

Continuing this process we eventually obtain a pair of matrices, which is state space isomorphic to (A,B) by means of a unitary transform, with the following structure

~Q,

I

~g;gl

•

•

• •

•

(9)

•

X3 X 2

•

B4 ,b

-

_•

•

B _4,g A3 Xl B3,b B 3,g A2 B2 b

•

B 2,g Al B 1,g

This is a very useful structure as we will show in the next because of the structure of the matrices B. and B. b for i == 1, ... ,t-l. (The

1,g 1, matrix BQ"b is empty.) B. b _1, 0 Y. b

-

1,

-

-

-

-

'"

-

-

-

-

V. ₁ B. _1,g

Y.

_1,g 0 See also (7).

Having obtained this structure, using only unitary matrices we can easily obtain a pole assignment algorithm. As in th.e case of the naive algorithm we will demontstrate this for a special case where the s'tructure (9) is

A3 _{B3 ,g} X 2 A2 (10)

\

Xl R2 b

_,

B _2,g Al B _1,g

Observe that B. has full row rank for i = 1,2,3. We will suppose that 1,g

the desired spectrum A can be partitioned as A = Al U A2 u A3 where Ai

d. xd.

is synnnetric

:fI:

A _{i i ' i}

=

d A lR 1 1 . If this cannot be obtained

EO:

then we can proceed as in case

B

of Theorem (Z). First we construct K3 such that a(A

3+B3,gK3) = A3 and we apply an r-typetransformation based on K 3• Then we obtain A3+B3 K3 _,g B _{3,g B2,b} BI b .

,

X₂ A _{B2 -K3 B2,b} 2 ,g X_{l2 XlI Al} B l,g

Observe that the matirx B2,g is transformed into B2,g - K3 BZ,b • This matrix has to be used for the next feedback (r-type transformation). We have to do this in a way such that ~+B3,g K3 is not changed.

be done because the structure of B2,b and BZ,g

-IS

BZ,b. is

Using this structure we can easily construct K2

=

[K~, K

Z2

J

such that

X2 + BZ,g K21

=

0

cr(A_Z + _{B2,g K12 )} a 11.2 '

The latter condition ensures that A3 + B

3,g K3 is not changed if we use an F-type transformation based on KZ"

1 0 0

1

3

+:

3

_,b

B B2 b -1 _{0 0} 3,g

,

0 1 0 X z AZ BZ -K3BZ b 0 1 0

l

X

l2 ,g ,

L

-K ZI -K 12 1 Xn Al K21 K12 I A3 + B),gK3 B _3,g BZ

_,

b 0 _{AZ+B Z ,g} K12 BZ -K3B2 _{,g ,} b ,.... ,.., X I2 XII At

[:2

0]

l>b]

=

l HI,b _

l

BI _-K2B1 bJ 1 _ l,g _{,g ,}

The structure of this latter matrix is

Finally we compute a feedback matrix (using the above structure) such that

=

o

... XlI +B 1,gK12

=

o

cr(A_l + B],g K 11}= A] , B] b K}

_,

=

0 •

Now we have obtained

o

B 3,g

o

B -LB 2,g --3 2,b

This matrix clearly has the desired spectrum because of the upper block triangular structure. This concludes the pole assignment algorithm. It will be clear that the same kind of construction can be done for the

general structure (9).

We will now consider the controllability matrix for a pair of matrices with the structure (9). It is easily seen that if it is not possible

to choose B. such that they only contain "good" singular values, we 1.,g

have that the controllability matrix is almost singular. Of cours-e, B. 1.,g can always be chosen to be right invertible, but it may be necessary that B. contains a small singular value for some i. This may happen

because the partition in "good" and "bad" singular values is restricted by a controllability condition (see (6». The relation between the singu-lar values of the controllability matrix and the singusingu-lar values of the matrices B. is not clear at the moment.

1,g

Some remarks towards a possible definition of the condition of the pole assignment problem. I f it is not possible to choose B. and B. b such

1, g 1,

that we still have controllability with

instead of controllability with

and such that B. contains only good singular values (see (6)), then 1,g

the problem should be called ill-conditioned.

I f during the construction of (9) it is indeed possible to ch.oose B. b

1,

and B. such that B. contains only good singular values for all i,

1,g 1,g

then the problem might be called well-conditioned.

As a final remark we will stress the fact that the construction of (9) uses more columns of the controllability matrix than the construction of (4). Numerical properties of this pole assignment algorithm will be reported in [2J.

After having completed this manuscript the authors attention was drawn to a paper (P. van Dooren, IEEE Trans. Autom. Control, Vol. AC-26, pp. 111-130) by Dr. Paul van Dooren in a private discussion. In that paper the structure (4) also appears.

References

[1J P. Brunovski, A classification of Linear Controllable Systems, Kyber-netica, Vol. 3, ]970.

[2J R. Eising, Numerical Aspects of a Pole Assignment Algorithm, in prepara-tion.

[3J M. Heymann, Comments on Pole Assignment in Multi-Input Controllable Linear Systems, IEEE Trans.Autom.Control Vol. AC-13, 1968.

[4J A. Varga, A Shur Method for Pole Assignment, IEEE Trans. Autom. Control, Vol. AC-26, ]981.

[SJ J.H. Wilkinson, The Algebraic Eigenvalue Problem, Clarendon Press. Ox-ford,1965.