A new method to determine the infinite zero structure of a linear time-invariant system

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linear time-invariant system

Citation for published version (APA):

Geerts, A. H. W. (1988). A new method to determine the infinite zero structure of a linear time-invariant system. (EUT-Report; Vol. 88-WSK-01). Technische Universiteit Eindhoven.

Document status and date: Published: 01/01/1988

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INFORMATICA COMPlJI1NG SCIENCE

A new method to detennine the infinite zero structure of a linear time-invariant system

by

A.H.W. Geerts

AMS Subject Oassifications: 93B25. 93B27

EUT Report 88-WSK-Ol ISSN 0167-9708

Coden: TEUEDE

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ABSTRACT

A new and straightforward method is proposed to compute the Smith-McMillan factorization at infinity of the transfer function that corresponds to a general linear time-invariant system. This technique is based upon a dual version of Silverman's structure algorithm.

KEYWORDS

Linear system, structure algorithm, Smith-McMillan factorization, transfer function, infinite zeros.

January 1988

Research supported by the Netherlands Organization for the Advancement of Pure Scientific Research (Z.W.O.>.

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1. Introduction

Consider a general linear time-invariant system

x,

given by the quadruple (A, B, C, D) with D not left invertible. Then the transfer function T(s) :: D + C(sl - A)-'B (s E c) may have zeros at infinity. If ker(D) :: 0 then T(s) will have no infinite zeros since D

=

lim T(s) is of full (column) rank ([1]).

8~

These infinite zeros are strongly related to the so-called ~

infinite dYnamics of the output nulling trajectories ([2]). lore precisely, let ,,(x) denote the subspace of initial conditions

for which there exists a regular input (see e.g. [5]) such that the resulting output is identically zero and let "d(X) denote the subspace of initial states tor which there exists an iapulsive-smooth distribution ([5]) which yields an output equal to zero. Then the factor space "d(X)/".(Z) is associated with the dynamics mentioned above ([2] - [3]). In the case that ker(D) =

lOt it is known that "d(Z) :: ",(z) ([5]' [1, Sec. 4])' i.e.

"'d(Z)/",(Z) ::

lot

and we have only fixed finite dynamics of the output nulling trajectories «(2]). There have appeared a number of papers (e.g. [1] - [4]) concerning the computation of the infinite zero structure for T(s}, thus aaking it clear that any other article on the same subject is not likely to contain substantial new results on this topic. But then this is not our objective. OUr only claim is to demonstrate that the concept of structure algoritha ([6J) indeed provides all information on the Smith-McMillan form of T(s} at infinity ([4]), but that this information can be gained in an elegant and transparent manner as well. Instead of a primal version of the structure algorithm (see e.g. [4J) we will aake use of its dual counterpart, first introduced in [5]. The structure at infinity of T(s) then follows immediately from results due to a generalization of the algorithm in [5], displayed in detail in [7, Sec. 4].

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2. Preliainaries

Ve consider the system x described by the matrix quadruple (A, B,

c,

D), with respective dimensions n x n, n x m, r x n, r x m. For all details concerning the aeneralized ~ structure algorithm we refer to [7, Sec. 4]. There it is sbown that the algorithm transforms (A, B, C, D) into

- c * c

*

(A, n~.

,

B , 0 ], C, [IL , 0 , 0 ])

~ a 0: 'i> a a (2.1a)

(on and a integers such that 0 SonS a, a ~ 1), where tbe two matrix partitions (both counting m columns) are such that:

(2.1b) is left invertible with rank ~ = qo + ql + ••• + qan and qo

=

rank (D) c -- B = V H, a a

...

,

(2.1c) B P V ]

a-1 a-1 a-1 (2.1d) left invertible, rank ~1 = ro + r1 + ••• + ra-1 (r o = m

-qo), and such that im(W ) • w(.t}, the stronalY reachable

a

subspace ([5, Def. l.13]); the matrix HI is an upper triangular matrix of dimension "a-1 x~, ~ • PI + Pa + + P with p.

=

rank (B,P.V.) • number of coluans of B

1,P1,V1

a l l 1 1

(Va the identity matrix) while

(2.1e) - the zero matrix block

0:

counts ~ columns, the zero blocks

*

0a have 2a

=

0, + 01 + ••• + 0a columns (oi ~ 0) and ~ + ~

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- the matrix ~ has a partition like ~: ~

=

[Bo, B1 , • • • , B~] • Now define i~(s} • [To(s), 'l(8), ••• , '~(s)] , (2.1£) (2.2a) with Ti(S)

=

Di + C(sl - A)·'B

i (i

=

0, 1, ••• ,

on)·

Next, let C1 be the (unique) solution of the equation

(2.2b)

CWa • ~Cl (2.3)

then it is easy to show that ([7, App. 1, Lemma 1, (iii)])

-c.

=

*

-1 R, 1

*

.d,

* -

lie K2 Ra· •• ·.····K R' 10 ~10 ~

* -

lie K2 Rz···K R' 11 ~ll ~

*

lie .d_a ••••••••• K R' ~12 ~ III

*

.d, ••••• K R'

'1>11

~

*

K R'

'1>&'1>-~

*

.d ~

*

K +1 ••••••

on 10

III K +1 •••••• ~ 11

*

Ie +1 •••••• ~ 12 lie K +1 ••...•

'1> 11

*

K~+l 1~-1

*

K +l

~ l~

*

K _a 10 III K a l l lie K a'2

*

K aI,

*

Ka 1~-1

*

K a

Ion

(2.4) In every block column the matrices K. ( j . 1 •••• , a, i

=

0,

JU

lie

••• , min(j - 1,

cx..»

form row partitions of the matrices K.

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*

which appear in part 1 of algorithm steps 1 - a (recall that K.

J₁

has a . (. _-m1n 1 )

=

qo + ••• + q . (. 1 ) rows). Further the

J- ,

«0

mln l- ,

«0

.

*

matrlces Aj (j

=

1, ••• ,

«0)

are such that

I

~;

1. [I.

R

~]

-,

=

S

~

-1

R.' J J J

J

(2.5)

* (see again the first parts of the algorithm steps). In fact, Aj

* -

-=

K. R.· + R.·. Note J a J l

*

that "'j has qj rows and that

c

1 is a

~ X ~1 matrix.

Then, recalling the 'a-1 x 'a-I matrix All in [7, App. 1, Lemma 2], we have the result below following readily frOll straightforward calculations.

Leua 2.1.

Due to the special block decompositions of C" All and HI it is possible to show that for large Is

I

C 1 (sI - A 11) -'H 1 has the

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Lemma 2.2.

For

I!I

largeNwe ha!e C,(sl - A,,)-1H,

=

O(S -1) O(s -I) 0(8-') •••••••••••••••••••••• 0(S·I) O{s·') O(s-') 0(S-I) •••••••••••••••••••••• 0(8-1

)

O(S-I) O(s -1) 0(.-1) .••••..•.•...•..•••... O(s-')

O(s -.) 0(8-1 ) O(S -1) •••••••••••••••••••••• 0(8-')

0(.·4) 0(. -3) O(S-I) •••••••••••• O(.-I)

O(s

-~)

_O(s-~ + 1 ₎ 0(.·') •••• 0(S-I)

Here, Landau's O-symbol is interpreted in the usual way. Proof. Appendix.

The Lemmas stated here are basic tools for obtaining the infinite zero structure for T(s). This will be shown in the final Section.

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3. The infinite zero structure for the transfer function

The techpique to uncover the zero structure of T(s} at infinity to be treated in this paper has already been displayed before in [7, Sec. 4], in a hidden fashion. Since this will become clear in the sequel, we will keep our computations sketchy in this Section and leave the obvious reformulations of certain calculations in [7, Sec. 4] to the reader. To start, consider T(a) - D + C(sl - A}-'S and recall step 0 in [7, Sec. 4]. Then

( (2.2b»

T(s)So - [To(s), To(s)]

-

(3.1> _(3.2)

Next, observe the difference between S, in [7, (4.10)] and

*

.

[1

[0 -s -'I( ]

1

S,(s):- qo ,. 1, ,

o

S,

and note that SICS) is hieausal. Then «2.2b» :(S)S08"S) == [To(S~, .·'T1(s), a-'T,(s»)

(3.3)

with T,(s) • ctsl - A)-18, • (3.5)

Consider st~p 1 .i~ [7, Sec. 4]. Then it is easily established that T(S)SOSl(S)P,V, :==

. r

_:J

rq:+q,

0 T(s)80S,(s) q;+q, V, 0

-[To(s), s-'T,(s}' s-a(CB,p,V1 + C(s! - A)-llB,P,V,),

-a·IC(sl - A)-IB,P,V" 0], (3.6) compare [7, (4.24)] without the last block. Next, instead of S. in step 2, part 1 of the algorithm ([7, (4.29)] with k • 1) we take the bicausal matrix SaCs) :==

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*

o

0 Sa

and

PO[.~:~~:iPlfi.g ~3'l6)

by Sa(s)

:-o

I

f1

0

o

I

a,

then yields

tha~

T(S)SoS:<s)P,v,s:(s) equals [To(s), S-rTI(S)'

s-·T.(~):

!-aTa(S),

rIC{SI - A) -18₁P,V" 0] ,

with Ta(s),- C(,l - A)-lB ••

I

(3.7)

If we continu, in this way, following the steps of the I

algorithm, we ftnallY arrive at the next situation. Define (i

=

1, r'" a)

Ti (8) -

erst -

A)-'B.f.V.

1 I" 1 1 1

(3.8) then there is albicausal matrix N(S) such that T(a)w(s) •

-

1-

_

-~_

[To(s), S-iT1(8), .-2T.(s), •••• , . T~(.)

S -lTll (s), s

-.IT,>

(S), •••• , 8

-a:,r

(a)

I

f

61 0,

I 0, 0, ••• , 0] (3.9)

where the zero

~lOCkS

at the end of the partition bave a" 0 ••

•••• 0

cOlumn.~

respectively.

o

.I-Next, if H,

=

[$".

B,a' ••• , H, ], then from Lemma 2.1

o

I

T_i1(a) •

(~(S»Cl(SI

- A1l)-IHi, (3.10) and thus T(s)w(s)

=

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-[8- 1B,1' s·2B_{u '}

...

,

0, ••• , 0 -: 0, ••• , 0

-~ (s) [A(S) Cl(al - 1u) -'B,A₁(S)

I

0] (3.11) with the zero block at the end counting ~ coluans and A1(S) ...

8 -'I p, Now define

-

-",(s) := -A-1_(S)C 1(sI - 1,,)-lR,A,(S) and set ,,(s) ... I ", (s)

g'1>

o

I

eO.

o

0

o

(3.12) (3.13) (3.14)

then T(S)H(S)"(S)

_I

₀

I

0] and 1t(a},,(s) is

bicausal since ", (s) vanishes for

18 I ...

co (coabine Leua 2.2,

(3.1l) and (3.11) - (l.12». Finally, write ~ == U G with

u

'1><;'

U left orthogonal, G invertible and let U be such that U_c

<;,

~ c

is left orthogonal and U := [U , U

1

is orthogonal and

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invertible ([7, (4.59)]). Then "Ids) := [i~(s), Uc] is bicausal and we establish that

~-l(s)T(s)N(S)~(S) ..

[A~.)

:

1

(3.15)

which is the (unique) SlIIith·lfclfillan for. (e.g. [1, Def. 3.1]) of the transfer function T(s).

The total number of infinite zeros n then is 00

n

=

(O.qo) + 1.q, + 2.qa + •••• + a-.q

00 u ~

(3.16) and it holds indeed that

n .. dim (~ ) - dim (~)

00 d (3.17)

see e.g. [2] - [3].

Let us consider the last two relations again.

In our dual version of the structure algorithm we have introduced the two integers

<i>

and a, see Sec. 2. Although in some other papers concerning the structure algorithm only one integer is appearing instead of two, one good reason (there are several) for the proposed distinction between

<i>

and a ([7, (4.47) - (4.52)]) is the following observation. From (3.16) it

is clear that noo .. 0 if and only if

<i> ..

0, regardless the value

*

of a! (Note that in this case «2.4)) C, .. [K

1 , ' '2 ' •••• , '0

* • - •

K ], therefore (Lemma 2.2) C,('I - A,,)-'8, • rots-I), O(s·'), a, 0

••• , O(s-')] and A(S) .. I (3.11).) Also, from (3.17), ncO

qo 00

~ ~d .. ~ c:::t , c ~ c:::t S .. ~ n , .. " where • denotes the controllable output nullina subspace (see e.g. [3]).

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Bence we conclude that aD

=

0 ~ n.

=

0 ~ s

=

w.

Also observe the following. Recall that DS o == [Do' 0] and BS o ==

[8

0 , Bol with Do left invertible and rank qo • rank (D) ([7,

(4.2) - (4.3)], see also (2.lb» and define 10 :-= 1 -

8

0(Do'Do) -lD o'C

Co := (I - Do(Do'Do) -'Do')C •

Proposition 3.1.

(3.18a) (3.lSb)

n. == 0 ~ Cow == 0 and Ao(w) c:,. and., == < Jter(Co} lAo>

Proof. =- If e;, == 0 then the "preliminary closed-loop" matrix 1e;,

• A - ~(~'~e;,)-l~e;,'C ([7, (4.63)], see also (2.lb), (2.1f» equals Ao «3.18a» and hence from [7, Le . . as 4.2 - 4.4] Ao(") c: wand < _{ker(C o) lAo>} == "d == ., since,. c: .,. Fro. (2.3), (2.lb)

and ().l8b) we establish that Co" •

o.

~It follows that w c:., and therefore n == O •

•

From Proposition 3.1 it is readily established that if n • 0

_•

*

then the set of invariant zeros 0

ez)

o(A o

I'"

s) == O(A~

1"1,,'

(note that Ao (S)

.

*

show that II aeneral 0 (z) == o(A~I"I">.

(see e.g. [3]) equals c: s}. Blaewhere we will

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We conclude this Section with tbe observation tbat, even if D in T{s)

=

D + C(sl . A)-'B is not left invertible. then still it is possible that there are QQ infinite zeros. This is the case jf

and only i f "

=

(ker{CoJ lAo > and Ao(fI) c: fI and Cofl =

o.

Observe that it ker(D) • 0 then fI • 0 (and conversely). Bence two of tbese conditions are trivially satisfied and indeed " • ( ker (Co) 110 ).

lenrt

Note that the system E in [1] (see [1, (4.53)]) is sucb that a

tbe corresponding transfer function T (s) bas no infinite zeros a

and indeed S(E) • fI'E }, "d(E) • "U ) I 1 (fI(E» c: fI(E ),

a a a a ~ a ex

Ca-(fI(E_a» • 0 and ,.(z) • (ker(C ) 11 ) with C •

u ex ~ ~ ~

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Appendix

We are confident that the next sketchy proof of Lemaa 2.2 will

-

-satisfy the reader. Write Ci(sl A,,)-lB, =- s·'CIB, + +

-~ + 1_ _ ~ - 2 _

S C I (Au )B, +

+ ••••

Recall the foras of C, «2.4) ), _{Au ([7, App. 1. Lemaa 2]) and} that Bl is an upper triangular aatrix. Then C,B, looks like

qo x x x x X ql X X X X X qa 0 X X X X q. 0 0 x x x 0 0 0 x x q~ 0 0 0 0 0 x x ro r1 ra r. r ~-2 ra-l

i.e. the qa x r 0' q. x (ro+r,),

...

; q~ x (ro+r,+.".+r~_2) lower triangular blocks of C,B, are necessarily zero.

Next, consider CIA"B'" For this aatrix it is easy to see that its q. x r o, q .. x (ro+r,), ••• , q~ x (ro+rl+ ••• +r~_3) lower triangular blocks are necessarily zero. In this way we can continue; finally we arrive at

cl(i,,~-

2)8& which has the fora below: qo x x x x X q ... X X X X X qa X X X X X q. X X X X X X X X X X X X X X q~ 0 x x x x r 0 r 1 fa f. r _0.-1

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i.e. only the q x ro lower triangular block aust be zero. For

~

the next coefficient in the Laurent expansion of

N _ _ _ ~-1_

C,(sI - .1.₁₁)-'8" C,(Al l )8" there are no blocks anyaore

that must be zero. Combining' the knowledge of all zero blocks appearing in the Laurent coefficients then yields that Ca(aI

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-References

[1] J.K. Dion & C. Commault, "Smith-McMillan factorizations at infinity of rational matrix functions and their control interpretation", Wh i Contr. 1dtU.:.. vol. 1, pp. 312 -320, 1982.

[2] C. Commault , J.K. Dion, "Structure at Infinity of Linear Kultivariable Systems: A Geometric Approach", ~ Trans. Automat. Contr., vol. AC - 27, pp. 693 - 696, 1982.

[3] B. Aling & J .M. Schumacher, "A nine-fold canonical decomposition for linear systems", !Db..

!L.

CORtr., vol. 39, pp. 779 - 805, 1984.

[4] L.K. Silverman & A. Kitap¢, "System structure at infinity", Syst. i. Contr. 1dtU.:., vol. 3, pp. 123 - 131,

1983.

[5] K.L.J. Bautus , L.K. Silverman, "System structure and singular control",

Lin.:.

AlL. i

MlR.L..,

vol. 50, pp. 369 -402, 1983.

[6] L.K. Silverman, "Inversion of multivariable linear systems", ml T,ans. Automat. Contr., vol. AC - 14. pp. 270 - 276, 1969.

[7] A.B.W. Geerts, "All optimal controls· tor the singular linear-quadratic problem without stability: a new interpretation of the optimal cost", Memorandum COSOR 87 -14, Eindhoven University of Technology, 1987.