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On the limiting zeros of sampled multivariable systems
Systems & Control Letters 2 (1983) 292-300 North-Holland Publishing Company
February 1983
On the limiting zeros of sampled multivariable systems Y o s h i k a z u H A Y A K A W A , Shigeyuki H O S O E , a n d M a s a m i ITO Automatic Control Laboratory, Faculty of Engineering, Nagoya University, Furo-cho, Chikusa-ku, Nagoya 464, Japan Received 24 August 1982 For time-invariant linear systems with single input - single output, it has been shown [3] that as the sampling period goes to 0, the zeros of the corresponding discrete time system obtained by sampling converge to some specific constant values determined only be the degrees of the finite and the infinite zeros of the original continuous time system. It is natural to inquire the extent to which this result may be carried over to the case of multivariable systems. In this note it is shown that the result is true also for muhivariable systems if the difference a m o n g the degrees of infinite elementary divisors of the pencil corresponding to the original continuous system is less than two. When the last condition is not satisfied, the limiting values of the zeros of the discrete system depend, in general, not only on the integers cited above but also on system parameters.
Keywords: Continuous system, Sampled system, Limiting zero, Canonical form of matrix pencil, Infinite elementary divisor.
1. Introduction Consider a time-invariant controllable and observable linear system Sc:
3c=Ax+Bu,
(1)
y=Cx,
with x ~ R" and u, y ~ R " , and the corresponding discrete time system
So:
xk+t=F(h)xk+G(h)U~.,
yk=H(h)xk,
(2)
where h is the sampling period and F ( h ) = e Ah,
h A
G ( h ) = f o e "Bd'r,
H ( h ) = C.
(3)
It is assumed that Sc is invertible, i.e. det C ( s l - A ) - I B ~ O.
(4)
Observe that under the assumptions of the completeness of Sc and (4) the definitions [1] of system zeros, invariant zeros and transmission zeros coincide, and these zeros are the complex roots, including multiplicities, of det F ( s ) = 0
(5)
where F(s) represents the pencil [2] corresponding to S¢ defined as
Hereafter we shall call these zeros simply the zeros of Sc. Also observe that the completeness of Sc implies that of S D for sufficiently small h [5]. Moreover it can be seen from the discussion in Section 2 that (4) 292
0167-6911/83/0000-0000/$03.00 © 1983 North-Holland
Volume 2, Number 5
SYSTEMS & CONTROL LETTERS
February 1983
implies that det H ( h ) ( z I , , -
F(h))-'G(h)
~ 0
(6)
for small h. Therefore the zeros of S o are the complex roots of det Fl,( z ) = 0 where
(F(h)-zI,, r,,(z) = I 14( )
G(h)) o
(7 /
N o w questions of obvious interest are the relation of the zeros and the poles of S D to those of Sc. For the poles, the situation is quite simple, since there is one-to-one correspondence expressed as p ~ e ph a m o n g the poles p of Sc and e ph of S o. The zeros, however, do not have such simple relation. Recently Astr/Sm and et al. [3] showed the following interesting result: If Sc is a single input - single output system and the degree of the numerator polynomial of the transfer function C ( s I - A ) - IB of Sc is r, then as the sampling period h ~ 0, r zeros of S o converge to 1 and the remaining n - r - 1 zeros of S D go to the zeros of some definite polynomial that is uniquely determined by the integer n - r. N o w it is natural and non-trivial to ask: Can AstrOm's results be extended to the case of multivariable systems? To answer this, denote by d l + 1. . . . . d,,, + 1 (d~ <~ • • • <~ d,,,) the degrees of the infinite elementary divisors of F(s) [2], [4]. Notice that when m = 1, we have d I = n - r. In this brief note it will be shown that the answer to the above question is yes when d,,, - d~ < 2. On the other hand if d,, - d~ >_-2, we shall show that the location of zeros of discrete system S o when h ~ 0 depends, in general, on the parameters of A, B, C and thus it can not be determined only by the integers n, d~ . . . . . d,,.
2. Limiting zeros Assume that Sc has r zeros z , , z2 . . . . .
zr.
(8)
Since det F ( s ) = det(s/,, - A ) det(C(s/,, - A ) - I B ) :~ 0 by assumption (4), pencil F(s ) is regular. For regular pencils, it is k n o w n [4] that there exist nonsingular matices P and Q of the form t/
2)n
m
P22
m
which transform F ( s ) to
P(s) = pr(s)a
=
0)n
n
m
Q2,
:l
L
Q22
,9,
m
0
0
A ~ - sI,, _ r
B~ o
si r
"
I"
a= I A - sl,,
B]O
(lOa)
[
where A ~ = block-diag( Na,, . . . . Nd.,, ),
B~ = block-diag( ba,, . . . . ba. ,), (lOb)
C.~ = block-diag( Ca, . . . . . Ca. ,), 0
1
0
iXi,
N,=
bi =
: i × 1,
( 7 , = ( 1 , 0 . . . . . 0): l × i ,
(lOc)
1 0
0 293
Volume 2, Number 5
SYSTEMS & CONTROL LETTERS
February 1983
and A/is any normal form such that the polynomial d e t ( A / - sir) has the zeros zj . . . . . z . Now to consider the zeros of S o, expand det Fh(z) as det r , , ( z ) = a,,_,,,(h)zt'-'" + a , , _ , , , _ , ( h ) z " - " ' - '
+ ... + a,(h)z
+ ao( h ) .
(11)
In general the coefficients a,(h)'s are functions of h and in particular a,,_,,,(h) = ( - 1 ) " det H ( h ) G ( h )
= ( - 1)" det CfoheA'Bdr.
(12)
For a while assume that a , _ , , ( h ) is not identically zero. Then, since a,_,,,(h) is analytic in h, it is not zero for any h (> 0) sufficiently small. In this case (1 1) can be written as det Fh(z ) = a t , _ , , ( h ) ( z - 3 , ( h ) ) ( z - e 2 ( h ) ) . . .
( z - e, ..... ( h )).
(13)
We are concerned with the limiting values of 3~(h)'s as h ~ 0, which will be called the limiting zeros of S o. Notice that the limiting zeros are not the zeros of lim h ~0 det Fh(z) but the zeros of polynomial f ( z ) where f ( z ) = lim
1
h - 0 a ...... (h)
det Fl,(z ).
(14)
Our first assertion is the following. Proposition. Assume that Sc satisfies (4), (8) and that the difference between the largest and the smallest degrees of the infinite elementary divisors o f F ( s ) is less than 2, say d I . . . . . d t = d and dr+ t . . . . . d,,, = d + 1. Then S D has n - m limiting zeros. Furthermore r limiting zeros among them are located at 1 and the m-I remaining n - r - m limiting zeros coincide with the complex roots of B~( z )B~+ t ( z ) - 0 where
B'( z ) = det( eN' C~ - zI'
f°'eU:b'd~ 0 )
(15)
Remark 1. The polynomial Bi(z ) in (15) is equal to a constant multiple of the polynomial defined in [3. Lemma 1]. Proof. Firstly let/5 = block_diag(Pll ' P22) and Q = block-diag(P~ l, Q22) where Ptl, P22 and Q22 a r e the submatrices in (9) and let /~h(z ) =/sFh(z)Q.
(16)
Then from (10) it is easy to see that Ph(z) has the form
rh(z)=(P(z)-zIn :l( h )
O(h) o
(17)
where
P ( h ) = e ~-h,
0 ( h ) = f0he£'bdz,
d = d -/~QS'Q2, - / ' , 2 P S ' d ,
/~(h) = (7,
(18) (19)
and Pi2, Q21 and A, /~, C are the submatrices given in (9) and (10) respectively. Moreover, substituting 294
SYSTEMS & C O N T R O L LETTERS
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February l983
(10c) into (19), A has the form dj
r
d2
d,,, ×
X
0
0
A/
r
X
X
I
0
0
l
x
dL
X
4=
......
x
X
x
X
x
I
0
0
l
d2 X
. . . . . .
X
X
.
.
.
.
.x
.
X
x.
.
.°, .
X
X
X
. . . . . .
X
x
I
0
•
0
I
X
. . . . . .
dm
X
X
X
X
x
(20) where ×'s represent the entries which are non-zero in general. Next, define
(21)
L,(z) = ~(h)r~(z)f'(h) where
17"(h ) = block-diag( V( h ), h - lI,,, ),
U(h) = block-diag(V-'(h), U ( h ) ) ,
V( h ) = block-diag(Ir, Va,( h ), Vd2(h ) .... ~ Vd.( h )),
V/(h) = diag(h i-l, h ,-2 ..... h, 1),
(22) (23)
and U ( h ) = diag( h - t d ' - l), h-Ca._-,) . . . . . h - t d . , - ,) ).
(24)
Then, describing -f'h(z) as
P~(~) =
P(h)-zI.
~(h)
6(h) l 0 ]'
(25)
we obtain ^
F( h ) = e A¢'~,
(26a)
(26b) 295
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H( h ) = U( h )CV( h ) = C,
(26c)
where
A(h) = V - ' ( h ) A V ( h ) h .
(27)
Notice that the second equality in (26b) was obtained by substituting a new variable o = ¢/h and by the fact that V - I ( h ) / ~ = )~. It is easy to verify that A ( h ) has the form r
dl
d2
A/h
Ja,(h)
Jd'-(h)
...
Ja.,(h )
dn, r
Jd,(h)
Kd,(h)
Laud(h)
...
La;,,(h)
dl
A(h)= J.:(h)
_ La'(h)
_ Kd,(h )
•"
L,,,'(h)
d2
Ka,,.(h)
d.,
LZ(h)
J..,,(h) where
,,,,
(28)
i
J'(h)=Ix
0 • h i,
0
(29a)
.h, X
i
xh
0
i
Xh/-~+l
xh 2
Xh/-1+2
Ki(h)=i
(29c)
L/(h)=i
1
Xh i
Xh 2
. . . . . .
Xh I
•..
Xh 2
xh
Xh
N o t e that lim jr(h) = O,
lim J,.(h) = O,
lim K, ( h ) = N~
h---,0
h---, 0
h~0
for i = 1, 2 ....
(30)
and ro
forj > i-
1,
J
X
lim L / ( h ) = h~0
0
..-
0
0 i
f o r j = i - 1.
0
(31)
0
Therefore, since d~ . . . . .
d I = d and d/+ ~ . . . . . ,
(d+l)
296
(m-t)LO riO dl 0
limA(h)= h--,O
dl
d,,, = d + 1 by the assumption, we get
(d+ l)(m-O
Nd.~
0
0 0
M
Nd+j ..... t
(32a)
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February 1983
where
N,. 4 = block-diag ( N; . . . . . N,. )
(32b)
t
J and d X
d+l
X
0
0 X
M=
d
X
0
X
d+l
X
0
0
!
X
i
d+l
0
X
(d+ l ) ( m - / ) ,
(32c)
X
0 d/
and thus
lim h~O
F(h) =
e limh-°2(h) ~-
:]
eNa.t X
(33a)
e Na+~.... ;
[o][ o ]
Furthermore, decomposing B as
j~
B~
B (j)
0
0
Bt21
where " ' - block-diag ( b a, ha,.
b a)
and
B~2~ = block-diag(ba+ l, ba+, . . . . . ha+ , ) ,
we get lim ( ~ ( h ) = f ' e ' h~0 - aO
......
aq,,,Oj~da /
r[f° 0 dl
l Ndto ( I ) e • B~ do
(d+ l) (m-;)[
0
m-I
1.
(33b)
N o w summarizing the discussions we can obtain: -m d (i) F r o m (16) and (21) and since det /.)(h) det 17(h)= h r . . . . , it follows that det Fh(z) = d e t / 5 det {~
det Fh( z )
(34)
h r",'_,d,
297
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SYSTEMS & CONTROL LETTERS
February 1983
and d e t ( [ / ( h ) G ( h ) ) = det P22 det Q22
det(H(h)G(h)}
(35)
hr; "_,d,
(ii) Also lim
detf',,(z)=det(
h~O
''-I[ z , l i m ~ , ( z ) ] = ( 1 - z j "fin/t ~,/~ z ~B J ,,+,~
', h - , . O
(36)
"
and lim det(/2/(h)(~(h)} = det{ lim [ I ( h ) ( ~ ( h ) } & a ( = 0). h~0 h~0
(37)
The interchange of operations lim and det is permitted because/~h(z) and H ( h ) ( ~ ( h ) are continuous in h. The second equalities in (36), (37) come from (26c), (33a) and (33b). F r o m (12), (35) and (37) we get
..... (h)
lim - -
= (-
lim
1)"
h-.O
hy.7.,d '
h--,O
det(H( h )G( h )) h rT ,d, (38)
= ( - 1)" lim d e t { I 2 I ( h ) G ( h ) ) / d e t P22 det 022 = ( - 1)" a . h~0 det P22 det Q22 This shows that a ...... (h) is not identically zero, which was a tentative assumption. Therefore limiting zeros. On the other hand (34) and (36) imply that lira h~o
det F,,( z )
hC7._,d,
1 (1 , t ,,,- i det/Sdet 0 - z ) B d ( z ) B d + , ( z ).
SD
has n - m
(39)
Therefore from (38) and (39) we obtain f(z)=
lim h~O a,,_m(h ) =
lim
(_l),,detP22deta22(1 -z a det/5 det Q
This is the desired result.
/
h~7,.,a '
h~O
-z)
lim h~O
hrT.,,i,
, ,
,,-, B~(z)B,~+, ( z ) .
(40)
[]
Remark 2. It is possible to show that the following holds for r zeros 2,(h) (i = 1. . . . . r ) of S D that converge to 1 as h ~ 0 : lim h~0
- , -,Lth~-1 h
z,
f o r i = 1 . . . . . r,
(41)
where zi's are the zeros of S¢. Thus if Re z i < 0, %(h) remains incide the unit circle for sufficiently small h ( > 0) (i = 1. . . . . r). In fact, (13) gives that
det-F'h(l+hz)= fi (1-z.i(h)+hz) a, -- ,,(h) i=l
nI-I --/91 (l-zj(h)+hz) j=r+
(42)
1
and thus we obtain
nfim ( 1 - ~ j ( h ) + h z ) .
detFh(1 + h z ) = f i ( z . i ( h ) - l )
hran - - re(h) 298
i=1
Z --
h
/j=r+l
(43)
Volume 2, Number 5
SYSTEMS & CONTROL LETTERS
February 1983
On the other hand from (3) and (5) it is easy to see that det Fh(1 + h z )
lim h--,0
= det F ( z )
h"
(44)
where F ( z ) is the pencil corresponding to Sc. M o r e o v e r since n = r + E'i'= idi, (38) gives lim
det Fh(1 + hz)
l i m h _ o ( d e t Fh(1 + h z ) / h " )
hra,_,,(h)
l i m h _ 0 ( a ..... (h)/h~:,,.,d,)
h--O
= ( - 1)" d e t / 2 2 det 022 lim det Fh(1 + h z ) h~o h"
(45)
Finally observe that n--m
Ba+ .... ,/ ( 1 ) • 0
I--I ( 1 - ~ . j ( h ) + h z ) = f l B ~ ( 1 )
lim
h~O j~r+
I
with/3 = ( - 1)" det P22 det Q22/a det/3 det Q. Thus there follows from (43)-(45) d e t / ' ( z ) -- "~
z - lim
i=l
(46)
h
h -"~ 0
where y is a non-zero constant value. Since det F(z) has r zeros zl . . . . . zr by assumption, this proves our assertion. In the remaining part it will be shown that if the difference between the greatest and the smallest degrees of the infinite elementary divisors of F(s) is greater than one, A s t r 6 m ' s result can not be extended, i.e. in general the limiting zeros of S o can not be determined only by the integers n, d t . . . . . d,,,. For this it is enough to show an example. Consider a continuous system Sc where
.4=
a
1
1 1
1 1
1
0
0 1
1 1
B--
'
0
0 0
C=
'
1
0
0
0
I
0
0
0
"
Observe that 0 14
['(s)=
0 1
--a
--
-1
-1
- 1
- 1
0
14
T(s)
- 1
-1
0
-1
-1
= 12
12
0
0
-1
-1
-s0 0 ................... 0 -s 1
0
1
0
0
0
"6
0
0
-s
1
0
0
0
0
0
-s
0
1
1 0
0 1
0 0
0 0
0 0
0 0
Therefore the pencil F(s) has two infinite elementary devisors of degrees two and four, irrespective of the value of p a r a m e t e r a. On the other hand, the limiting zeros of Fh(z) are obtained by solving f(z)=
z2 + ~8 + 2 a z + 1 = 0
(a*2),
(47)
for I-.-+h+ ... nh+ l+2ah2+ 2
h+2h~+ ... -..
rh~=)= h + 2 + % 2 + " " 2
h+3+ah2+
...
I-z+h+
...
h+~ h~+''" h+2h3+
...
h+~h~. . . . h+
I+ah2+ 2
,, +~h2 + .-.."
2
I-=+~h~+"" h+~h2+
I+--"~ah2+3+4ah3+ ""
-.-
6
h+h~+"" l-z+h+
...
2
h +'h:+ .-ah2+ 2
1+2ah3+ 6
~h~+~h' . . . . "'"
~h"+2+%~+"" 6
~I h ~ '+
3+ah2 + -'.
I+ah3+3+4aha+ 6
"'"
24
~J'~+~+"" h +~h2+
"'"
6 I 0
0 I
0 0
0 0
0 0
0 0
299
Volume 2, Number 5
SYSTEMS & CONTROL LETTERS
February 1983
by expansion and thus det Fl,(z) = / -t 2- i-~a h4 0 ( h , ) )+z 2
+ ( 44-+-a- - ~ h + O ( h S ) } z + ( --~h2-a 4 + O ( h S ) ) •
Clearly the roots of (47) depend on parameter a.
References
[1] A.G.J. Macfarlane and N. Karcanias, Poles and zeros of linear multivariable systems: a survey of the algebraic, geometric and complex-variable theory, Int. J. Control 24 (1976) 33-74. [2] F.R. Gantmacher, The Theory of Matrices, Vols. I and II (N.T. Chelsea, 1959). [3] K.J. ,~.str6m, P. Hagander and J. Sternby, Zeros of sampled systems, in: Proc. 19th IEEE Conf. Decis. Control Syrup. Adapt. Processes (1980) pp. 1077-108 I. [4] N. Suda and E. Mutsuyoshi, Invariant zeros and input-output structure of linear, time-invariant systems, Int. J. Control 28 (1978) 525-535. [5] H.H. Rosenbrock, State Space and Multivariable Theory (William Clowes and Sons, London. 1970).
300
February 1983
On the limiting zeros of sampled multivariable systems Y o s h i k a z u H A Y A K A W A , Shigeyuki H O S O E , a n d M a s a m i ITO Automatic Control Laboratory, Faculty of Engineering, Nagoya University, Furo-cho, Chikusa-ku, Nagoya 464, Japan Received 24 August 1982 For time-invariant linear systems with single input - single output, it has been shown [3] that as the sampling period goes to 0, the zeros of the corresponding discrete time system obtained by sampling converge to some specific constant values determined only be the degrees of the finite and the infinite zeros of the original continuous time system. It is natural to inquire the extent to which this result may be carried over to the case of multivariable systems. In this note it is shown that the result is true also for muhivariable systems if the difference a m o n g the degrees of infinite elementary divisors of the pencil corresponding to the original continuous system is less than two. When the last condition is not satisfied, the limiting values of the zeros of the discrete system depend, in general, not only on the integers cited above but also on system parameters.
Keywords: Continuous system, Sampled system, Limiting zero, Canonical form of matrix pencil, Infinite elementary divisor.
1. Introduction Consider a time-invariant controllable and observable linear system Sc:
3c=Ax+Bu,
(1)
y=Cx,
with x ~ R" and u, y ~ R " , and the corresponding discrete time system
So:
xk+t=F(h)xk+G(h)U~.,
yk=H(h)xk,
(2)
where h is the sampling period and F ( h ) = e Ah,
h A
G ( h ) = f o e "Bd'r,
H ( h ) = C.
(3)
It is assumed that Sc is invertible, i.e. det C ( s l - A ) - I B ~ O.
(4)
Observe that under the assumptions of the completeness of Sc and (4) the definitions [1] of system zeros, invariant zeros and transmission zeros coincide, and these zeros are the complex roots, including multiplicities, of det F ( s ) = 0
(5)
where F(s) represents the pencil [2] corresponding to S¢ defined as
Hereafter we shall call these zeros simply the zeros of Sc. Also observe that the completeness of Sc implies that of S D for sufficiently small h [5]. Moreover it can be seen from the discussion in Section 2 that (4) 292
0167-6911/83/0000-0000/$03.00 © 1983 North-Holland
Volume 2, Number 5
SYSTEMS & CONTROL LETTERS
February 1983
implies that det H ( h ) ( z I , , -
F(h))-'G(h)
~ 0
(6)
for small h. Therefore the zeros of S o are the complex roots of det Fl,( z ) = 0 where
(F(h)-zI,, r,,(z) = I 14( )
G(h)) o
(7 /
N o w questions of obvious interest are the relation of the zeros and the poles of S D to those of Sc. For the poles, the situation is quite simple, since there is one-to-one correspondence expressed as p ~ e ph a m o n g the poles p of Sc and e ph of S o. The zeros, however, do not have such simple relation. Recently Astr/Sm and et al. [3] showed the following interesting result: If Sc is a single input - single output system and the degree of the numerator polynomial of the transfer function C ( s I - A ) - IB of Sc is r, then as the sampling period h ~ 0, r zeros of S o converge to 1 and the remaining n - r - 1 zeros of S D go to the zeros of some definite polynomial that is uniquely determined by the integer n - r. N o w it is natural and non-trivial to ask: Can AstrOm's results be extended to the case of multivariable systems? To answer this, denote by d l + 1. . . . . d,,, + 1 (d~ <~ • • • <~ d,,,) the degrees of the infinite elementary divisors of F(s) [2], [4]. Notice that when m = 1, we have d I = n - r. In this brief note it will be shown that the answer to the above question is yes when d,,, - d~ < 2. On the other hand if d,, - d~ >_-2, we shall show that the location of zeros of discrete system S o when h ~ 0 depends, in general, on the parameters of A, B, C and thus it can not be determined only by the integers n, d~ . . . . . d,,.
2. Limiting zeros Assume that Sc has r zeros z , , z2 . . . . .
zr.
(8)
Since det F ( s ) = det(s/,, - A ) det(C(s/,, - A ) - I B ) :~ 0 by assumption (4), pencil F(s ) is regular. For regular pencils, it is k n o w n [4] that there exist nonsingular matices P and Q of the form t/
2)n
m
P22
m
which transform F ( s ) to
P(s) = pr(s)a
=
0)n
n
m
Q2,
:l
L
Q22
,9,
m
0
0
A ~ - sI,, _ r
B~ o
si r
"
I"
a= I A - sl,,
B]O
(lOa)
[
where A ~ = block-diag( Na,, . . . . Nd.,, ),
B~ = block-diag( ba,, . . . . ba. ,), (lOb)
C.~ = block-diag( Ca, . . . . . Ca. ,), 0
1
0
iXi,
N,=
bi =
: i × 1,
( 7 , = ( 1 , 0 . . . . . 0): l × i ,
(lOc)
1 0
0 293
Volume 2, Number 5
SYSTEMS & CONTROL LETTERS
February 1983
and A/is any normal form such that the polynomial d e t ( A / - sir) has the zeros zj . . . . . z . Now to consider the zeros of S o, expand det Fh(z) as det r , , ( z ) = a,,_,,,(h)zt'-'" + a , , _ , , , _ , ( h ) z " - " ' - '
+ ... + a,(h)z
+ ao( h ) .
(11)
In general the coefficients a,(h)'s are functions of h and in particular a,,_,,,(h) = ( - 1 ) " det H ( h ) G ( h )
= ( - 1)" det CfoheA'Bdr.
(12)
For a while assume that a , _ , , ( h ) is not identically zero. Then, since a,_,,,(h) is analytic in h, it is not zero for any h (> 0) sufficiently small. In this case (1 1) can be written as det Fh(z ) = a t , _ , , ( h ) ( z - 3 , ( h ) ) ( z - e 2 ( h ) ) . . .
( z - e, ..... ( h )).
(13)
We are concerned with the limiting values of 3~(h)'s as h ~ 0, which will be called the limiting zeros of S o. Notice that the limiting zeros are not the zeros of lim h ~0 det Fh(z) but the zeros of polynomial f ( z ) where f ( z ) = lim
1
h - 0 a ...... (h)
det Fl,(z ).
(14)
Our first assertion is the following. Proposition. Assume that Sc satisfies (4), (8) and that the difference between the largest and the smallest degrees of the infinite elementary divisors o f F ( s ) is less than 2, say d I . . . . . d t = d and dr+ t . . . . . d,,, = d + 1. Then S D has n - m limiting zeros. Furthermore r limiting zeros among them are located at 1 and the m-I remaining n - r - m limiting zeros coincide with the complex roots of B~( z )B~+ t ( z ) - 0 where
B'( z ) = det( eN' C~ - zI'
f°'eU:b'd~ 0 )
(15)
Remark 1. The polynomial Bi(z ) in (15) is equal to a constant multiple of the polynomial defined in [3. Lemma 1]. Proof. Firstly let/5 = block_diag(Pll ' P22) and Q = block-diag(P~ l, Q22) where Ptl, P22 and Q22 a r e the submatrices in (9) and let /~h(z ) =/sFh(z)Q.
(16)
Then from (10) it is easy to see that Ph(z) has the form
rh(z)=(P(z)-zIn :l( h )
O(h) o
(17)
where
P ( h ) = e ~-h,
0 ( h ) = f0he£'bdz,
d = d -/~QS'Q2, - / ' , 2 P S ' d ,
/~(h) = (7,
(18) (19)
and Pi2, Q21 and A, /~, C are the submatrices given in (9) and (10) respectively. Moreover, substituting 294
SYSTEMS & C O N T R O L LETTERS
Volume 2, N u m b e r 5
February l983
(10c) into (19), A has the form dj
r
d2
d,,, ×
X
0
0
A/
r
X
X
I
0
0
l
x
dL
X
4=
......
x
X
x
X
x
I
0
0
l
d2 X
. . . . . .
X
X
.
.
.
.
.x
.
X
x.
.
.°, .
X
X
X
. . . . . .
X
x
I
0
•
0
I
X
. . . . . .
dm
X
X
X
X
x
(20) where ×'s represent the entries which are non-zero in general. Next, define
(21)
L,(z) = ~(h)r~(z)f'(h) where
17"(h ) = block-diag( V( h ), h - lI,,, ),
U(h) = block-diag(V-'(h), U ( h ) ) ,
V( h ) = block-diag(Ir, Va,( h ), Vd2(h ) .... ~ Vd.( h )),
V/(h) = diag(h i-l, h ,-2 ..... h, 1),
(22) (23)
and U ( h ) = diag( h - t d ' - l), h-Ca._-,) . . . . . h - t d . , - ,) ).
(24)
Then, describing -f'h(z) as
P~(~) =
P(h)-zI.
~(h)
6(h) l 0 ]'
(25)
we obtain ^
F( h ) = e A¢'~,
(26a)
(26b) 295
February 1983
SYSTEMS & CONTROL LETTERS
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H( h ) = U( h )CV( h ) = C,
(26c)
where
A(h) = V - ' ( h ) A V ( h ) h .
(27)
Notice that the second equality in (26b) was obtained by substituting a new variable o = ¢/h and by the fact that V - I ( h ) / ~ = )~. It is easy to verify that A ( h ) has the form r
dl
d2
A/h
Ja,(h)
Jd'-(h)
...
Ja.,(h )
dn, r
Jd,(h)
Kd,(h)
Laud(h)
...
La;,,(h)
dl
A(h)= J.:(h)
_ La'(h)
_ Kd,(h )
•"
L,,,'(h)
d2
Ka,,.(h)
d.,
LZ(h)
J..,,(h) where
,,,,
(28)
i
J'(h)=Ix
0 • h i,
0
(29a)
.h, X
i
xh
0
i
Xh/-~+l
xh 2
Xh/-1+2
Ki(h)=i
(29c)
L/(h)=i
1
Xh i
Xh 2
. . . . . .
Xh I
•..
Xh 2
xh
Xh
N o t e that lim jr(h) = O,
lim J,.(h) = O,
lim K, ( h ) = N~
h---,0
h---, 0
h~0
for i = 1, 2 ....
(30)
and ro
forj > i-
1,
J
X
lim L / ( h ) = h~0
0
..-
0
0 i
f o r j = i - 1.
0
(31)
0
Therefore, since d~ . . . . .
d I = d and d/+ ~ . . . . . ,
(d+l)
296
(m-t)LO riO dl 0
limA(h)= h--,O
dl
d,,, = d + 1 by the assumption, we get
(d+ l)(m-O
Nd.~
0
0 0
M
Nd+j ..... t
(32a)
Volume 2, Number 5
SYSTEMS & CONTROL LETTERS
February 1983
where
N,. 4 = block-diag ( N; . . . . . N,. )
(32b)
t
J and d X
d+l
X
0
0 X
M=
d
X
0
X
d+l
X
0
0
!
X
i
d+l
0
X
(d+ l ) ( m - / ) ,
(32c)
X
0 d/
and thus
lim h~O
F(h) =
e limh-°2(h) ~-
:]
eNa.t X
(33a)
e Na+~.... ;
[o][ o ]
Furthermore, decomposing B as
j~
B~
B (j)
0
0
Bt21
where " ' - block-diag ( b a, ha,.
b a)
and
B~2~ = block-diag(ba+ l, ba+, . . . . . ha+ , ) ,
we get lim ( ~ ( h ) = f ' e ' h~0 - aO
......
aq,,,Oj~da /
r[f° 0 dl
l Ndto ( I ) e • B~ do
(d+ l) (m-;)[
0
m-I
1.
(33b)
N o w summarizing the discussions we can obtain: -m d (i) F r o m (16) and (21) and since det /.)(h) det 17(h)= h r . . . . , it follows that det Fh(z) = d e t / 5 det {~
det Fh( z )
(34)
h r",'_,d,
297
Volume 2, Number 5
SYSTEMS & CONTROL LETTERS
February 1983
and d e t ( [ / ( h ) G ( h ) ) = det P22 det Q22
det(H(h)G(h)}
(35)
hr; "_,d,
(ii) Also lim
detf',,(z)=det(
h~O
''-I[ z , l i m ~ , ( z ) ] = ( 1 - z j "fin/t ~,/~ z ~B J ,,+,~
', h - , . O
(36)
"
and lim det(/2/(h)(~(h)} = det{ lim [ I ( h ) ( ~ ( h ) } & a ( = 0). h~0 h~0
(37)
The interchange of operations lim and det is permitted because/~h(z) and H ( h ) ( ~ ( h ) are continuous in h. The second equalities in (36), (37) come from (26c), (33a) and (33b). F r o m (12), (35) and (37) we get
..... (h)
lim - -
= (-
lim
1)"
h-.O
hy.7.,d '
h--,O
det(H( h )G( h )) h rT ,d, (38)
= ( - 1)" lim d e t { I 2 I ( h ) G ( h ) ) / d e t P22 det 022 = ( - 1)" a . h~0 det P22 det Q22 This shows that a ...... (h) is not identically zero, which was a tentative assumption. Therefore limiting zeros. On the other hand (34) and (36) imply that lira h~o
det F,,( z )
hC7._,d,
1 (1 , t ,,,- i det/Sdet 0 - z ) B d ( z ) B d + , ( z ).
SD
has n - m
(39)
Therefore from (38) and (39) we obtain f(z)=
lim h~O a,,_m(h ) =
lim
(_l),,detP22deta22(1 -z a det/5 det Q
This is the desired result.
/
h~7,.,a '
h~O
-z)
lim h~O
hrT.,,i,
, ,
,,-, B~(z)B,~+, ( z ) .
(40)
[]
Remark 2. It is possible to show that the following holds for r zeros 2,(h) (i = 1. . . . . r ) of S D that converge to 1 as h ~ 0 : lim h~0
- , -,Lth~-1 h
z,
f o r i = 1 . . . . . r,
(41)
where zi's are the zeros of S¢. Thus if Re z i < 0, %(h) remains incide the unit circle for sufficiently small h ( > 0) (i = 1. . . . . r). In fact, (13) gives that
det-F'h(l+hz)= fi (1-z.i(h)+hz) a, -- ,,(h) i=l
nI-I --/91 (l-zj(h)+hz) j=r+
(42)
1
and thus we obtain
nfim ( 1 - ~ j ( h ) + h z ) .
detFh(1 + h z ) = f i ( z . i ( h ) - l )
hran - - re(h) 298
i=1
Z --
h
/j=r+l
(43)
Volume 2, Number 5
SYSTEMS & CONTROL LETTERS
February 1983
On the other hand from (3) and (5) it is easy to see that det Fh(1 + h z )
lim h--,0
= det F ( z )
h"
(44)
where F ( z ) is the pencil corresponding to Sc. M o r e o v e r since n = r + E'i'= idi, (38) gives lim
det Fh(1 + hz)
l i m h _ o ( d e t Fh(1 + h z ) / h " )
hra,_,,(h)
l i m h _ 0 ( a ..... (h)/h~:,,.,d,)
h--O
= ( - 1)" d e t / 2 2 det 022 lim det Fh(1 + h z ) h~o h"
(45)
Finally observe that n--m
Ba+ .... ,/ ( 1 ) • 0
I--I ( 1 - ~ . j ( h ) + h z ) = f l B ~ ( 1 )
lim
h~O j~r+
I
with/3 = ( - 1)" det P22 det Q22/a det/3 det Q. Thus there follows from (43)-(45) d e t / ' ( z ) -- "~
z - lim
i=l
(46)
h
h -"~ 0
where y is a non-zero constant value. Since det F(z) has r zeros zl . . . . . zr by assumption, this proves our assertion. In the remaining part it will be shown that if the difference between the greatest and the smallest degrees of the infinite elementary divisors of F(s) is greater than one, A s t r 6 m ' s result can not be extended, i.e. in general the limiting zeros of S o can not be determined only by the integers n, d t . . . . . d,,,. For this it is enough to show an example. Consider a continuous system Sc where
.4=
a
1
1 1
1 1
1
0
0 1
1 1
B--
'
0
0 0
C=
'
1
0
0
0
I
0
0
0
"
Observe that 0 14
['(s)=
0 1
--a
--
-1
-1
- 1
- 1
0
14
T(s)
- 1
-1
0
-1
-1
= 12
12
0
0
-1
-1
-s0 0 ................... 0 -s 1
0
1
0
0
0
"6
0
0
-s
1
0
0
0
0
0
-s
0
1
1 0
0 1
0 0
0 0
0 0
0 0
Therefore the pencil F(s) has two infinite elementary devisors of degrees two and four, irrespective of the value of p a r a m e t e r a. On the other hand, the limiting zeros of Fh(z) are obtained by solving f(z)=
z2 + ~8 + 2 a z + 1 = 0
(a*2),
(47)
for I-.-+h+ ... nh+ l+2ah2+ 2
h+2h~+ ... -..
rh~=)= h + 2 + % 2 + " " 2
h+3+ah2+
...
I-z+h+
...
h+~ h~+''" h+2h3+
...
h+~h~. . . . h+
I+ah2+ 2
,, +~h2 + .-.."
2
I-=+~h~+"" h+~h2+
I+--"~ah2+3+4ah3+ ""
-.-
6
h+h~+"" l-z+h+
...
2
h +'h:+ .-ah2+ 2
1+2ah3+ 6
~h~+~h' . . . . "'"
~h"+2+%~+"" 6
~I h ~ '+
3+ah2 + -'.
I+ah3+3+4aha+ 6
"'"
24
~J'~+~+"" h +~h2+
"'"
6 I 0
0 I
0 0
0 0
0 0
0 0
299
Volume 2, Number 5
SYSTEMS & CONTROL LETTERS
February 1983
by expansion and thus det Fl,(z) = / -t 2- i-~a h4 0 ( h , ) )+z 2
+ ( 44-+-a- - ~ h + O ( h S ) } z + ( --~h2-a 4 + O ( h S ) ) •
Clearly the roots of (47) depend on parameter a.
References
[1] A.G.J. Macfarlane and N. Karcanias, Poles and zeros of linear multivariable systems: a survey of the algebraic, geometric and complex-variable theory, Int. J. Control 24 (1976) 33-74. [2] F.R. Gantmacher, The Theory of Matrices, Vols. I and II (N.T. Chelsea, 1959). [3] K.J. ,~.str6m, P. Hagander and J. Sternby, Zeros of sampled systems, in: Proc. 19th IEEE Conf. Decis. Control Syrup. Adapt. Processes (1980) pp. 1077-108 I. [4] N. Suda and E. Mutsuyoshi, Invariant zeros and input-output structure of linear, time-invariant systems, Int. J. Control 28 (1978) 525-535. [5] H.H. Rosenbrock, State Space and Multivariable Theory (William Clowes and Sons, London. 1970).
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