IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 54, NO. 5, MAY 2009
A Sampled-Data Servomechanism
for Stable Well-Posed Systems
Zhenqing Ke, Hartmut Logemann, and Richard Rebarber
Abstract—In this technical note, an approximate tracking and disturbance rejection problem is solved for the class of exponentially stable wellposed infinite-dimensional systems by invoking a simple sampled-data lowgain controller (suggested by the internal model principle). The reference
signals are finite sums of sinusoids and the disturbance signals are asymptotic to finite sums of sinusoids.
Index Terms—Disturbance rejection, infinite-dimensional systems, lowgain control, sampled-data control, tracking.
I. INTRODUCTION
There has been much interest in low-gain integral control over the
last thirty years. The following principle (tuning integrator) has become well established (see, for example, Davison [1], Lunze [6] and
Morari [7]): closing the loop around an asymptotically stable, finite-dimensional, continuous-time plant, with square transfer function matrix
, compensated by an integral controller ("=s)I , will result in a stable
closed-loop system that achieves asymptotic tracking of arbitrary constant reference signals, provided that the gain parameter " > 0 is sufficiently small and the eigenvalues of the steady-state matrix (0) have
positive real parts. This principle has been extended to various classes
of infinite-dimensional systems (see Logemann and Townley [5] and
the references therein). Moreover, discrete-time and sampled-data versions of the tuning integrator have been developed (for infinite-dimensional systems) by Logemann and Townley in [4].
Hämäläinen and Pohjolainen [2] succeeded in generalizing the above
principle to the multi-frequency case in which the reference and disturbance signals to be tracked and rejected, respectively, are (finite) linear
combinations of sinusoids having prespecified frequencies. Their solution, inspired by the internal model principle, is a simple low-gain
tuning controller and it is shown to work for exponentially stable infinite-dimensional systems with impulse response in the Callier-Desoer
algebra. Rebarber and Weiss [8] proved a similar result for the (more
general) class of exponentially stable well-posed systems. Ke, Logemann and Rebarber [3] developed a sampled-data version of the tuning
regulator presented in [2], [8]. The main result in [3] guarantees approximate tracking and disturbance rejection for stable infinite-dimensional
systems which have the property that their impulse responses are exponentially bounded matrix-valued Borel measures. We mention in this
context that systems with measure impulse responses are necessarily
regular (see [9] and the references therein for details on well-posed
and regular systems): whilst verification of the regularity property can
sometimes be difficult, it is usually even more difficult to show that the
impulse response is a measure.
In this technical note, we consider essentially the same problem as in
[2], [3], [8], but we seek a sampled-data solution which applies to expo-
G
G
Manuscript received August 19, 2008; revised December 05, 2008. Current
version published May 13, 2009. This work was supported by EPSRC Grant
GR/S94582/01, LMS Scheme 2 Grant 2515, and NSF Grant DMS-0606847.
Recommended by Associate Editor K. A. Morris.
Z. Ke and H. Logemann are with the Department of Mathematical Sciences,
University of Bath, Bath, BA2 7AY, U.K. (e-mail: [email protected];
[email protected]).
R. Rebarber is with the Department of Mathematics, University of NebraskaLincoln, Lincoln, NE 68588-0130 USA (e-mail: [email protected]).
Digital Object Identifier 10.1109/TAC.2009.2013032
1123
nentially stable well-posed systems (avoiding the assumption that the
impulse response of the system is a measure). Adopting an input-output
approach, we do not invoke any results from the state-space theory of
well-posed systems, so that, for the purposes of this technical note, an
exponentially stable well-posed system is simply a system with the
property that its transfer function is holomorphic and bounded in a
: Re s > g for some < 0. The essence
half-plane fs 2
of the main result of the technical note can be described as follows:
low-gain sampled-data control based on a discrete-time version of the
continuous-time controller given in [2], [8], in conjunction with suitable low-pass filters, achieves approximate tracking and disturbance
rejection for exponentially stable well-posed systems.
The technical note is structured as follows. In Section II, we state a
number of preliminary technical results used in the technical note. In
Section III, we first prove a discrete-time result which is a crucial tool
for the proof of the main result of the technical note. We consider a
feedback controller with transfer function of the form
"
K0(z) +
N
Kj
z
0
j
j =1
(1.1)
K
where 0 is holomorphic and bounded on fz 2
: jz j > g for
some 2 (0; 1), Kj 2 m2p and j 2 with jj j = 1. Applying
this controller to a discrete-time plant with transfer function which
is holomorphic and bounded on fz 2 : jz j > g, we show that the
transfer function of the closed-loop feedback system is holomorphic
and bounded on fz 2 : jz j > g for some 2 (; 1), provided that
(i) all the eigenvalues of j (j )Kj have positive real parts, and (ii)
the gain parameter " is sufficiently small. This result is an extension of
a result in [4] on low-gain discrete-time integral control.
In Section IV, the main result of Section III is then used in the context
of approximate tracking and disturbance rejection for infinite-dimensional sampled-data feedback systems. The continuous-time plant is assumed to have a transfer function which is holomorphic and bounded
on fs 2 : Res > g for some < 0. The sampled-data servomechanism consists of a discrete-time feedback controller of the form (1.1)
with j = e , where j 2 i for j = 1; . . . ; N and > 0 is the
sampling period, in conjunction with two filters with transfer functions
N e t j ,
1 and 2 . The reference signal r is given by r (t) =
j =1
p
j 2 and the disturbance signals are assumed to be asymptotically
equal (in a suitable sense) to signals of the same form. If all the eigenvalues of (j )Kj have positive real parts and 1 (j ) and 2 (j ) are
equal to the identity for all j = 1; . . . ; N , then it is shown that, for
every > 0, there exists > 0 such that, for every sampling period
2 (0; ), there exists " > 0 such that, for every " 2 (0; " ), the
output y of the closed-loop sampled-data system can be decomposed
as y = y1 + y2 , where y1 (1)e01 2 L2 ( + ; p ) for some < 0 and
y2 satisfies lim supt!1 ky2 (t) 0 r(t)k .
Notation: For > 0, 2 and 2 , define := fz 2 :
jz j > g, := fz 2
: Re z > g and (; ) := fz 2
:
jz 0 j < g. For a set U , let cl(U ) denote the closure of U .
In the following definitions, let be open and let X = m or
X = p2m . We define
P
P
G
F
F
G
F
F
H 1 (
; X ) :=ff : ! X jf is holomorphic and bounded g
1 ( 1 ; X ) :=
H<
H 2(
0<<1
H 1(
; X)
; X ) := ff : ! X jf is holomorphic and
1
sup
x>01
kf (x + i )k2 d < 1
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:
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For 2 and 1 q < 1, we define the exponentially weighted
Lq -space Lq ( + ; X ) by
q
( + ; X ) : f (1)e01 2 Lq ( + ; X ) :
Lq ( + ; X ):= f 2 Lloc
We write H 1 (
) := H 1 (
; ), H 2 ( ) := H 2 ( ; ) and
Lq ( + ) := Lq ( + ; ). For A 2 m2m , let (A) denote the spectrum of A. For N 2 , set N := f1; 2; . . . ; N g.
II. PRELIMINARIES
In this section, we present some technical results for both discretetime and continuous-time systems. The proofs of Lemmas II.1, II.2 and
II.3 are straightforward: they can be found, for example, in [3].
Lemma II.1: Assume that H is a discrete-time input-output operator
with impulse response in `1 ( + ; p2m ) and transfer function . If
v : + ! m satisfies limk!1 (v(k) 0 k ) = 0, where 2 1
and 2 m , then limk!1 ((Hv )(k) 0 k () ) = 0.
Lemma II.2: Assume that H is a continuous-time input-output operator with impulse response h 2 L1 ( + ; p2m ) and transfer function
. Let u : + ! m be measurable. If u is bounded, then
H
H
H
lim sup k(Hu)(t)k khkL lim sup ku(t)k :
!1
!1
t
t
Moreover, if u satisfies limt!1 (u(t) 0 et ) = 0, where 2 m , then
H
lim (Hu)(t) 0 et ( )
!1
t
The (zero-order) hold operator
defined by
(H v )(t) := v (k);
m
)
! F ( +;
m
) is
H
H
H
!0 H (e ) = H( ); 8 2 0 :
lim
For the purposes of this technical note, it is convenient to the define
the concept of a (finite-dimensional) filter as follows.
Definition II.4: A (finite-dimensional) filter is an exponentially
5
stable, strictly causal, finite-dimensional system.
We note that a filter has impulse response of the form t 7! CeAt B ,
where A 2 n2n , B 2 n2m , C 2 p2n and all eigenvalues of A
have negative real parts.
Lemma II.5: Let H be a continuous-time input-output operator with
2 H 1 ( ) for some < 0, and let F be a
transfer function
single-input-single-output filter. Then there exists 2 (; 0) such that
the impulse response of HF is in L1 ( + ).
H
H
NH(z)NK (z) + DH (z)DK (z))j > 0
inf jdet (
z
2
then (I +
)01 2 H 1 (
; p2p ).
The proof is straightforward and is therefore omitted (see also [10,
Lemma 3.1], of which Proposition II.7 is a special case).
III. A DISCRETE-TIME RESULT
The following proposition will be crucial in the proof of Theorem
IV.1, the main result of this technical note. It is also interesting in its
own right.
and let j 2 , jj j = 1 for all
Proposition III.1: Let N 2
j 2 N be such that j 6= k for all j; k 2 N , j 6= k . Assume that
2 H<1 ( 1 ; p2m ) and that there exist Kj 2 m2p such that
P
8t 2 [k; (k + 1) ) :
Lemma II.3: Assume that H is a continuous-time input-output op1 ( + ; p2m ) for some 0 and
erator with impulse response in L
let H be the sample-hold discretization of H , defined by H :=
S H H . Set := e and let and denote the transfer functions of H and H , respectively. Then 2 H 1 ( ; p2m ) and
H
DN
D
H D N DN
X
Y
DX NY
H
ND
D H ND
ND
Y
X
XN YD
K
H
D N
N D K
N N D D
HK
= 0:
8k 2 + :
H : F ( + ;
F
HF
2 0,
Let > 0 denote the sampling period and let F ( + ; m ) and
F ( + ; m ) denote all m -valued functions defined on + and + ,
respectively. The ideal sampling operator S : F ( + ; m ) !
F ( + ; m ) is defined by
(S u)(k) := u(k );
F
Proof: Since the transfer function of F is a strictly proper stable
rational function, there exists 2 (; 0) such that 2 H 2 ( ), and
2 H 2 ( ). Let g denote the impulse response of HF . By
hence,
the Paley-Wiener Theorem, g 2 L2 ( + ). Therefore, it follows easily
from Hölder’s inequality that g 2 L1 ( + ) for every 2 (; 0).
We close this section with the statement of a result from the fractional
representation theory of feedback systems. To this end, let be
open and let Q denote the quotient field of H 1 (
), i.e., Q = fn=d :
n; d 2 H 1 (
); d 6= 0g.
Defintion II.6:
(i) A left-coprime factorization of
2 Qp2m (over H 1 (
)) is
a pair ( ; ) 2 H 1 (
; p2p ) 2 H 1 (
; p2m ) such that
det
6= 0, = 01 and , are left coprime, i.e.,
2 H 1 (
; p2p ), 2 H 1 (
; m2p ) satisthere exist
+
= I.
fying
(ii) A right-coprime factorization of
2 Qp2m (over H 1 (
)) is
p2m
1
) 2 H 1 (
; m2m ) such that
a pair ( ; ) 2 H (
;
0
1
det 6= 0, =
and ; are right coprime, i.e., there
exist
2 H 1 (
; m2p ), 2 H 1 (
; m2m ) satisfying
+
= I.
5
Proposition II.7: Let
2 Qp2m and 2 Qm2p . Assume that
there exist a left-coprime factorization ( H ; H ) of and a right-coprime factorization ( K ; K ) of (both over H 1 (
)). If the matrix
H K + H K has an inverse in H 1 (
; p2p ), i.e.,
P
j (j )Kj 0 ;
Let
K0 2 H<1(
1;
8 j 2 N:
(3.1)
2 ) and set
m p
K" (z) := " K0 (z) +
N
Kj
z
0
j
j =1
:
(3.2)
Then there exists "3 > 0 such that
K" (I + PK" )01 2 H<1(
1;
2 ); 8" 2 (0; "3 ):
m p
Although Proposition III.1 is contained as a special case in [3, Theorem 3.1], we prove this result to make the technical note self-contained. We emphasize that the proof given here is new, with coprime
factorizations playing a pivotal role and thereby providing an alternative approach to that developed in [3]. It is convenient to first state and
prove the following lemma which will facilitate the proof of Proposition III.1.
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Lemma III.2: For > 0, set B := (1; ) \ 1 and let U 2 H 1 (U; m2p ) and
cl(B ) be open. Let 2 H 1 (U; p2m ),
m
2
p
K 2
. If
Q
H
Q
( (1)K) 3
z
Q
7!
I + " (z)
H(z) + z K0 1
01
2 H 1 (B ; p2p ):
3
D(z) := z 0z 1 Ip ; N(z) := H(z)D(z) + z1 K
N D0
H
(K 3 K)01 K 3 (z) + Ip 0 (K 3 K)01 K 3 (z)
N
H
3
Q N
inf jdet (" (z) (z) +
2
8" 2 (0; "3 ):
D(z))j > 0;
3
H
det ("n (zn 0 1) (zn ) (zn )+
Q
"n (zn )K + (zn 0 1)Ip ) = 0;
8n 2
+:
(3.4)
PK 0
2
1
cl ( (j ; )) \ cl ( (k ; )) = ;;
Setting j :=
function
1
\
z
7! P(z)
8j; k 2 N ; j 6= k:
N
j =1 j , it is clear that the
(j ; ) and :=
K0 (z) +
N
Kj
z 0 j
j =1
1 > 0 such that
is bounded on 1 n . Thus, there exists "
(I +
Seeking a contradiction, suppose that such a constant " does not exist.
Then there exist "n # 0 and zn 2 cl(B ) such that
Q
By (3.5), Rezn ! 1 as n ! 1, and thus, (1 0 Rezn )="n ! 0 as
n ! 1, contradicting (3.7).
We are now in the position to prove Proposition III.1.
1
2
Proof of Proposition III.1: We first show that (I +
")
H ( 1 ; p p ) for sufficiently small ". Since j 6= k for all j; k 2
N , j 6= k , we can choose > 0 sufficiently small such that
D(z) = Ip:
By Proposition II.7, it is sufficient to show that there exists " > 0 such
that
0 Rezn )
M (1
j1 0 zn j
M (1 0 Rezn )
=
2(1 0 Rezn ) + jzn j2 0 1
Mp
1 0 Rezn ; 8n n0 :
p
2
H
we conclude that ( ; ) is a right coprime factorization of (z) +
K=(z 0 1) over H (B ), since (z) 1 (z) = (z) + K=(z 0 1)
and
z B
1 0 Rezn
"n
2 (0; "3 ),
Proof: Note that, by (3.3), rkK = p, so that K K is invertible.
Setting
1N D
As a trivial consequence of (3.7), Rezn < 1 for n n0 . Invoking this,
together with (3.8) and the fact that jzn j 1 for all n 2 + , we obtain
(3.3)
0
then there exists " > 0 such that, for all "
1125
Fix j
PK" )01 is bounded on
2N
1
n for all " 2 (0; "1 ):
and set
H(z) := K0(z) +
Kk
:
z
0
k
k2N;k6=j
Then there exists an open set Vj
H (Vj ; m p ) and, furthermore,
2
1
(3.9)
cl(
j ) such that
H
2
P(z)K" (z) = "P(j w) H(j w) + wj0Kj1
!1 "n = 0, we may conclude from (3.4) that
Since limn
where w := j z . Setting
lim zn = 1:
!1
(3.5)
n
H~ (w) := H(j w); Q(w) := P(j w);
j Kj
~ j := K
Moreover, we obtain from (3.4) that
1 0 zn
"n
2 ((zn 0 1)Q(zn )H(zn ) + Q(zn )K) ; 8n 2
it follows that
+:
~
P(z)K" (z) = "Q(w) H~ (w) + wK0j 1
(3.6)
Consequently, by (3.3) and (3.5), there exists > 0 and n0
that
1 0 zn
"n
2
;
2
8n n0 :
Q H
such
n
+:
2
K1
1
2
Since 2 H< ( 1 ; p m ) and 0 2 H< ( 1 ; m p ), we see that
2 H< ( 1 ; p m ) and ~ 2 H (Uj ; m p ), where
H
2
(3.7)
Q
Furthermore, since the function z 7! (z 0 1) (z) (z) + (z)K is
bounded on cl(B ), it follows from (3.6) that there exists a constant
M > 0 such that
j1 0 zn j M; 8n 2
"
P1 1 2
Q
:
(3.8)
j Vj
Uj := j cl(
j ) = cl ( 1 \
(1; )) = cl(B )
with B := 1 \ (1; ) (as in Lemma III.2). Moreover, by (3.1)
Q(1)K~ j
P
j (j )Kj
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8j 2 N :
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The following theorem is the main result of this technical note.
Theorem IV.1: Let N 2 and let j 2 i for all j 2 N be such that
j 6= k for j; k 2 N , j 6= k . Assume that the transfer function
of
p m
m p
G is in H ( ;
) for some < 0 and there exist Kj 2
such that
1
G2
2
G( )K ) (
Fig. 1. Sampled-data low-gain control with filters.
It follows from Lemma III.2 that, for every j 2 N , there exists "j
(0; " ) such that, for all " 2 (0; "j ), the function
1
w
1
is in H (B ;
01
Q(w) H~ (w) + wK0 1
7!
~j
I +"
2 ). Consequently,
3
2
PK )01 2 H 1(
"
N
j
1
! z 0 j
lim
z
P( )K
j
2 ); 8" 2 (0; "3 ):
p p
is invertible for all j
(I +
(3.10)
j
P K 00
K PK
PK 0
K 2
PK 0 1
K
j
1
2
Consider the sampled-data system shown in Fig. 1, where G is the
input-output operator of the continuous-time plant, K;" is the inputoutput operator of the discrete-time controller, F1 and F2 are filters,
r is a reference signal and d1 and d2 are disturbance signals. We assume that the transfer function of G is in H ( ; p m ) for some
< 0, or equivalently, that G is the input-output operator of an exponentially stable well-posed state-space system. Mathematically, Fig. 1
can be expressed as
y = G(F2
H yc + d1 ) + d2 ;
yc = K;"
2
S (r 0 F1 y):
(4.1)
and
N
Kj
z
j =1
0 e F2( ) = I
N
N
j =1
e
1j
2
t
j,
t
1j + p1 (t);
t
2j + p21 (t) + p22 (t);
e
j =1
N
d2 (t) :=
8j 2 N :
m;
j
Suppose that r is given by r(t) :=
are given by
(3.11)
IV. A LOW-GAIN SAMPLED-DATA CONTROLLER
1
K0(z) +
2
p
e
j =1
m
j
2
p
F1
(4.3)
, and d1 , d2
;
(4.4a)
2j
2
p
;
(4.4b)
where p1 2 L2 ( + ; m ), p21 2 L2 ( + ; p ) for some 2 (; 0),
1
p
( +;
) with limt
p22 (t) = 0. Then, for every
and p22 2 Lloc
> 0, there exists > 0 such that, for every sampling period 2
(0; ), there exists " > 0 such that, for every " 2 (0; " ), the output
y of the sampled-data feedback system (4.1) can be decomposed as
y = y1 + y2 , where y1 2 L2 ( + ; p ) and y2 satisfies
!1
2 N . Therefore, using
"
="
1
K
d1 (t) :=
P(z)K (z))01 = ("P( )K )01 :
G
;" (z )
j
Hence, " (z )(I + (z ) " (z )) 1 has a finite limit as z ! j for
1
is bounded on a neighbourhood
every j 2 N , so that " (I +
")
1
3 of the set fj : j 2 N g. Since (I +
2 H< ( 1 ; p p )
")
and, for some 2 (0; 1), " is bounded on n 3, it follows that
1
2 H< ( 1 ; m p ).
" (I +
")
K
K
F1 ( ) = I
1( 1 ; p2p ) for all " 2
Next we prove that (I + PK" )01 2 H<
3
1
p2m
0
1
m2p
) and K 2 H< ( 1 ;
),
(0; " ). Since P 2 H< ( 1 ;
0
1
it is clear that (I + PK" ) is meromorphic on for some 2
01 has at most
(0; 1). Letting 2 (; 1), it follows that (I + PK" )
finitely many poles in the compact annulus cl( ) n 1 . By (3.11),
01 does not have any poles on the unit circle @ 1 and so
(I + PK" )
there exists 2 (; 1) such that (I + PK" )01 2 H 1 ( ; p2p ),
where depends on ".
1( 1 ; m2p ), note
Finally, to show that K" (I + PK" )01 2 H<
that, by (3.1),
(3.2)
Let > 0 be the sampling period and assume that the transfer function
;" of K;" is of the form
g and invoking (3.9) and (3.10), we
1;
(4.2)
K
F
p p
Setting " := minf"j : j
conclude that
8j 2 N :
0;
j
where 0 2 H< ( 1 ; m p ). Assume that the transfer functions
and 2 of the filters F1 and F2 satisfy
01 2 H 1 (
j ; p2p ); 8" 2 (0; "j ); 8j 2 N :
(I + PK" )
(I +
2
j
!1 k
lim sup y2 (t)
t
0 r(t)k < :
Before we prove the theorem, we provide some commentary in the
following remark.
Remark IV.2:
(i) Theorem IV.1 says that the output y can be decomposed in the
form y = y1 + y2 , where
• the signal y1 is “small” in the sense that y1 2 L2 ( + ; p ),
implying that the “energy” of the restriction y1 j[t; ) converges to zero exponentially fast (with exponential rate ) as
t ! 1.
• the signal y2 is “persistent” and, for all sufficiently large t 0,
y2 (t) is close to r (t) in the sense that ky2 (t) 0 r (t)k < .
(ii) Denoting the Lebesgue measure on + by L , the conclusions
of Theorem IV.1 imply that
1
!1
f T : ky(t) 0 r(t)k g) = 0
lim L ( t
T
that is, as t ! 1, the error y (t) 0 r(t) is “bounded in measure”
by .
(iii) An inspection of the proof of Theorem IV.1 (see (4.22)) shows
that if the impulse response of G is a p m -valued Borel measure on + , p1 (t) ! 0 and p21 (t) ! 0 as t ! 1, then
ky(t) 0 r(t)k < .
y1 (t) ! 0 as t ! 1, so that lim supt
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(iv) One of the motivations for including the term p21 2 L2 ( + ; p )
in the disturbance d2 is that it can be used to model non-zero initial conditions in an exponentially-stable well-posed state-space
realization of G.
(v) An inspection of the proof of Theorem IV.1 (see the argument
guaranteeing the existence of ) shows that, for given fj : j 2
N g, and " can be chosen to be uniform for all signals r , d1
and d2 with j , 1j and 2j , j 2 N , satisfying a pre-specified
bound.
(vi) A filter with transfer function F satisfying F(j ) = I for all
j 2 N can be constructed in the following way:
F(s) := h(1s)
N
h(l )
l=1
0 k )
l 0 k
(s
k2N;k6=l
:=
F1 GF2 ;
H
:=
S H H
=
S F1 GF2 H :
H (e ) = H(j ) = G(j ); 8j 2 N :
By (4.2) and (4.5), there exists 1
H (e
e
)Kj
(4.5)
2 (0; 0 ) such that
0;
8(; j ) 2 (0; 1 ) 2 N :
for j
2 N , a routine calculation yields
kv1 (t) 0 (H v1 ) (t)k
N
G(j )Kj )01 0 H (e )Kj
(
j =1
N
+
max
s2[0; ]
j =1
01
kKj kk j k
je s 0 1jkLj j k; 8t 0:
Let > 0. By (4.5) and (4.8), there exists ~
kv1 (t) 0 (H v1 ) (t)k The transfer functions of H and H are denoted by H and H , respectively. By Lemma II.5, there exists 2 (; 0) such that the impulse responses of H , F1 G and GF2 are in L1 ( + ; p2m ). Hence,
1 ( 1 ; p2m ) and
by Lemma II.3 and (4.3), H 2 H<
lim
!0
2i
Since j
I
where h(s) is a real Hurwitz polynomial, the degree of which is
greater or equal to N . It is clear that F is a strictly proper stable
rational function. Moreover, if the j occur in complex conjugate
5
pairs, then it is easy to see that F has real coefficients.
Proof of Theorem IV.1: Setting 0 := 2= supfjj 0 k j : j; k 2
N ; j 6= kg, it follows that if 2 (0; 0 ), then e 6= e for all
j; k 2 N , j 6= k . Define
H
1127
4M
(4.8)
2 (0; 1 ) such that
8t 0; 8 2 (0; ~ )
;
(4.9)
where M denotes the L1 -norm of the impulse response of GF2 . Similarly, there exists 2 (0; ~ ) (0; 1 ) such that
kv2 (t) 0 (H v2 ) (t)k ; kv3 (t) 0 (H v3 ) (t)k 4M
;
8t 0; 8 2 (0; ):
(4.10)
In the following, let 2 (0; ). Invoking the fact that H 2
1 ( 1 ; p2m ) together with (4.6) and Proposition III.1, we conH<
clude that there exists " > 0 such that
K;" (I + H K;" )01 2 H<1 (
m2p
1;
);
8" 2 (0; " ):
(4.11)
Let " 2 (0; " ). Using (4.7) and exploiting the structure of K;" , we
obtain
K;" (I + H K;" )01
(e
)=
H (e )Kj
Kj
01
: (4.12)
The output yc of the discrete-time controller (see (4.1)) is given by
(4.6)
yc
S 0 F1 (GF2 H yc + Gd1 + d2 )]
= K;" S r 0 K;" H yc 0 K;" S F1 Gd1 0 K;" S F1 d2
= K;" [r
In particular,
and thus,
H (e
)Kj is invertible;
8 (; j ) 2 (0; 1 ) 2 N :
(4.7)
For j 2 N , set Lj := Kj (G(j )Kj )01 , where we have used that,
by (4.2), G(j )Kj is invertible for every j 2 N . Define the functions
v1 ; v2 ; v3 2 F ( + ; m ) by
v1 (t) :=
v3 (t) :=
N
j =1
N
j =1
e t Lj j ;
e t Lj
v2 (t) :=
N
j =1
e t Lj G(j ) 1j
v3 (k) :=
N
j =1
N
j =1
e k Lj j ;
e k Lj
=
K;" (I + H K;" )01 (S r 0 S F1 Gd1 0 S F1 d2 ): (4.13)
Since p1 , p21 and the impulse responses of F1 G and F1 are L2 -functions, we conclude that
lim (F1 Gp1 )(t) = 0;
t!1
2j :
v2 (k) :=
N
j =1
e k Lj G(j ) 1j
lim (F1 p21 )(t) = 0:
t!1
(4.14)
Invoking the fact that the impulse responses of F1 G and F1 are
L1 -functions, together with Lemma II.2, (4.3) and (4.14), we obtain
2j :
Let 2 (0; 1 ) and set Lj := Kj (H (e )Kj )01 for
j 2 N (H (e )Kj is invertible by (4.7)). Define the sequences
v1 ; v2 ; v3 2 F ( + ; m ) by
v1 (k) :=
yc
lim
t!1
(F1 Gd1 )(t)
lim
t!1
(F1 d2 )(t)
0
lim
k!1
( F1 Gd1 )(k )
0
lim
k!1
( F1 d2 )(k )
N
0
e t G(j ) 1j
j =1
N
t
j =1
e
2j
=0
=0
showing that
S
S
0
N
e k G(j ) 1j
j =1
N
k
j =1
e
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2j
= 0:
= 0;
(4.15)
(4.16)
1128
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 54, NO. 5, MAY 2009
By (4.11), the impulse response of ;" ( + ;" )01 is in
1
m2p
( +
), so that it follows from Lemma II.1, (4.12), (4.13),
(4.15), and (4.16) that
`
K I
;
HK
y k 0 v1 (k) + v2 (k) + v3 (k)) = 0:
lim ( c ( )
k!1
(4.17)
it follows from (4.19)–(4.21) that
lim sup
t!1
Then, by (4.9), (4.10) and (4.17),
k H y t 0 v t) + v2 (t) + v3(t)k
k H y 0 v1 + v2 + v3 )) (t)k
k H v t) 0 v1(t)k
kv t 0 H v2 ) (t)k
kv t 0 H v3 ) (t)k
lim sup ( c )( )
1(
t!1
lim sup ( ( c
t!1
+ lim sup ( 1 ) (
t!1
(
+ lim sup 2 ( )
t!1
(
+ lim sup 3 ( )
t!1
3
GF
ky2(t) 0 r(t)k
M lim
sup k(H yc )(t) 0 v1 (t) + v2 (t) + v3 (t)k
t!1
(4.18)
N
GF2 v2 )(t) 0
(
j =1
34 < completing the proof.
GF2 v1 )(t) 0 r(t)) = 0;
lim
t!1
L
lim sup
t!1
Moreover, we conclude from Lemma II.2 and (4.3) that
!1
M
Finally, recalling that
denotes the 1 -norm of the impulse response
of
2 , Lemma II.2 and (4.18) yield
4M :
lim ((
t
ky2 (t) 0 r(t)k
lim
sup k(GF2 (H yc 0 v1 + v2 + v3 )) (t)k :
t!1
(4.19)
e t G(j )
;
=0
1j
(4.20)
REFERENCES
and
GF2 v3 )(t) 0 d2 (t) + p21 (t))
lim ((
t
!1
= lim
t!1
GF2 v3 )(t) 0
N
(
j =1
e t 2j 0 p22 (t)
:
=0
(4.21)
Setting
y1 (t) := (Gd1 )(t) 0
N
j =1
G(j )e t
p t
1j + 21 ( )
(4.22)
and
y2 (t) := (GF2 H yc )(t) +
N
j =1
G(j )e t
d t 0 p21 (t)
1j + 2 ( )
y y y
it follows that = 1 + 2 . Denoting the Laplace transform by L and
invoking (4.4), we obtain that
Ly
s
( ( 1 )) ( ) =
N
G(s) 0 G(j ))
(
s 0 j
j =1
1j
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G(s)(Lp1)(s) + (Lp21)(s):
+
G p2 H 1( ;
p1 2 L2 ( + ; m ) and
p21 2
L ; ) with < < 0, it follows that L(y1) 2 H22 ( ; pp ).
Hence, the Paley-Wiener Theorem implies that y1 2 L ( + ; ).
Since
2
p m
2
( +
),
Furthermore, since
ky2 (t) 0 r(t)k k(GF2 (H yc 0 v1 + v2 + v3 )) (t)k
+ k(GF2 v1 )(t) 0 r (t)k
N
+
j =1
+
G(j )e t 1j 0 (GF2v2)(t)
kd2 (t) 0 p21(t) 0 (GF2 v3 )(t)k ; 8t 0
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