Signal Processing 77 (1999) 261}274
Orthonormal basis functions for modelling
continuous-time systems
HuK seyin Akc7 ay *, Brett Ninness
Institute for Dynamical Systems, Bremen University, Postfach 330440, 28334 Bremen, Germany
Centre for Integrated Dynamics and Control (CIDAC), University of Newcastle, Callaghan, NSW 2308, Australia
Department of Electrical and Computer Engineering, University of Newcastle, Callaghan, NSW 2308, Australia
Received 14 May 1998; received in revised form 24 February 1999
Abstract
This paper studies continuous-time system model sets that are spanned by "xed pole orthonormal bases. The nature of
these bases is such as to generalise the well-known Laguerre and two-parameter Kautz bases. The contribution of the
paper is to establish that the obtained model sets are complete in all of the Hardy spaces H (P), 1(p(R, and the
N
right half plane algebra A(P) provided that a mild condition on the choice of basis poles is satis"ed. A characterisation of
how modelling accuracy is a!ected by pole choice, as well as an application example of #exible structure modelling are
also provided. 1999 Elsevier Science B.V. All rights reserved.
Zusammenfassung
In diesem Artikel werden Modellmengen fuK r Systeme in stetiger Zeit betrachtet, wobei diese Mengen von orthonormalen Basen mit "xierten Polen erzeugt werden. Diese Basen verallgemeinern die wohlbekannten Laguerre-Basen
und die zweiparametrigen Kautz-Basen. In dieser Arbeit wird gezeigt, dass die erhaltenen Modellmengen in allen
Hardy-RaK umen H (P), 1(p(R, und in der Algebra A(P) der rechten Halbebene vollstaK ndig sind, vorausgesetzt, dass
N
eine schwache Bedingung an die Pole der Basis erfuK llt ist. Eine Charakterisierung des Ein#usses der Polvorgabe auf die
Modellgenauigkeit, sowie ein Anwendungsbeispiel der Modellierung einer #exiblen Struktur werden gegeben. 1999
Elsevier Science B.V. All rights reserved.
Re2 sume2
Cet article eH tudie des ensembles de modèles de systèmes à temps continu qui sont engendreH s par des bases
orthonormales à po( les "xes. La nature de ces bases est telle qu'elles geH neH ralisent les bases bien connues de Laguerre et de
Keutz à deux paramètres. La contribution de cet article est d'eH tablir que les ensembles de modèles obtenus sont complets
dans tous les espaces de Hardy H (P), 1(p(R, ainsi que l'algèbre du demi-plan de droite A(P), pourvu qu'une
N
condition douce sur le choix des po( les des bases soit satisfaite. Nous fournissons eH galement une caracteH risation de la
fac7 on dont la preH cision du modèle est a!ecteH e par le choix des po( les, ainsi qu'un exemple d'application de modeH lisation de
structures #exibles. 1999 Elsevier Science B.V. All rights reserved.
Keywords: Rational basis functions; Orthonormal; Completeness; Continuous-time systems
* Corresponding author. Tel.: #49-421-218-9394; fax: #49-421-218-4235.
E-mail address: [email protected] (H. Akc7 ay)
0165-1684/99/$ - see front matter 1999 Elsevier Science B.V. All rights reserved.
PII: S 0 1 6 5 - 1 6 8 4 ( 9 9 ) 0 0 0 3 9 - 0
H. Akc7 ay, B. Ninness / Signal Processing 77 (1999) 261}274
262
Nomenclature
C
R
P
PM
D
T
H (P)
N
A(P)
A(D)
sp A
a
"eld of complex numbers
"eld of real numbers
open right half-plane +s3C: Re+s,'0,
closed right half-plane +s3C: Re+s,*0,
open unit disk +z3C: "z"(1,
unit circle +z3C: "z""1,
Hardy spaces of functions f (s) analytic
on P and such that "" f ""N"
N
(1/2p)sup " f (x#jy)"N dy (R, 0
V \
(p(R and "" f "" "sup P " f (s)"(R
QZ
right half-plane algebra + f: f3H (P) and
continuous on PM ,
disk algebra + f: f analytic on D and continuous on DM ,
linear span of A
complex conjugate of a
1. Introduction
The use of orthonormal bases for the purposes of
approximation and analysis is fundamental to
many areas of applied mathematics. In particular,
in the areas of control theory, signal processing and
system identi"cation, there has long been interest
in the use of the trigonometric (FIR), &Laguerre',
and &two-parameter Kautz' bases [17,20,21].
More recently, in a discrete-time setting, this interest has been revived in a string of works
[9}11,30,32,41,42,44] and this has led workers to
consider orthonormal constructions which generalise the Laguerre and two-parameter Kautz cases
[7,8,12,18,34]. One of these e!orts presented in
[3,31] considers the orthonormal basis functions
de"ned on D6T by a choice of numbers m 3D,
L
n"1,2,2, as
(1!"m "
L (z),
B (z)O
L
L\
1!m z
L
(1)
L z!m
I , (z)O1.
(z)O “
L
1!m
z
I
I
In the special case of m "m3R this becomes the
L
discrete-time Laguerre basis, and in the special case
of m "m3C this provides the discrete-time twoL
parameter Kautz basis; see [31] for more details on
the generalising aspects of de"nition (1).
In the case of considering continuous-time
model descriptions, several important works have
also recently appeared that employ continuoustime Laguerre and two-parameter Kautz bases
[24,25,43] and rational wavelet bases [13]. The
purpose of this paper is to make some further
contribution in this area by considering some issues
related to a natural generalisation of continuoustime Laguerre and Kautz bases.
The generalisation to be considered is analogous
to the extension presented in (1). Namely, a set of
basis functions +B (s), are treated which are de"ned
L
by a choice of numbers a 3P, ∀n as
L
(2 Re+a ,
L u (s),
B (s)O
L
L\
s#a
L
(2)
L s!a
I, u (s)O1.
u (s)O “
L
s#a
I
I
We set B (s),1. The rational basis functions
+B ,
are orthonormal in H (P) with respect to
L LV
the inner-product (see Section 3)
1 1B ,B 2O
B ( ju)B ( ju) du
L
K
L K
2p
\
1; m"n,
"
0; mOn.
Analogous to the discrete-time case, the continuous-time Laguerre basis (studied, for example, in
[23,33,43]) is obtained as a special case of (2) by the
choice a "a3R and the continuous time twoL
parameter Kautz basis (studied in [43]) by the
choice a "a3C.
L
It should be acknowledged that both the continuous and discrete-time generalisations shown in
(2) and (1) enjoy a long history in both the pure
mathematics [14,26,40] and engineering literature
[19,28,36].
2. Main result
As mentioned in the introduction, an important
motivation for the consideration of orthonormal
H. Akc7 ay, B. Ninness / Signal Processing 77 (1999) 261}274
parameterisations is for approximation purposes.
In this setting, a dominant question must arise as to
the quality of the approximation. Pertaining to this,
one of the most fundamental properties that might
be required is that linear combinations of the basis
elements be capable of arbitrarily good approximation.
Put more precisely in the context of systems
theory, this involves considering an element f (s)
living in a normed linear function space (X,"" ) "" ),
6
and for arbitrary e'0 and for su$ciently large
n being able to "nd an element g(s)3sp+B (s),L
I I
such that "" f!g"" )e. Here X is a complex vector
6
space and the linear span is with respect to the "eld
of complex numbers.
If this is in fact possible for arbitrarily small e,
then sp+B (s),
is said to be &complete' in X. The
I IV
choice of the function space depends on the
application of the approximate model, but for
quadratic optimal control purposes or mean square
optimal prediction purposes, the choice H (P)
would be appropriate, while for robust control or
estimation purposes, the choices A(P) or H (P) (for
N
large p) would be suitable.
With regard to the discrete-time basis (1), the
approximation issues have been addressed in [3]
where the following result was obtained.
Theorem 1 (Akc7 ay and Ninness [3, Theorem 6 and
Corollary 7]). Consider the set of functions +B (z),
I
dexned by (1). Then the set X"sp+B (z),
is
I IV
complete in A(D) and H (T ) for all 1)p(R if and
N
only if
(1!"m ")"R.
I
I
(3)
The main result of this paper is to establish an
analogous result for the continuous-time basis (2)
as follows.
Theorem 2. The model set spanned by the basis functions +B (s),
is complete in all of the spaces
L LV
H (P), 1(p(R, and A(P) if and only if
N
Re+a ,
L "R.
1#"a "
L
L
(4)
263
Condition (4) is satis"ed by the Laguerre and
two-parameter Kautz bases and the rational
wavelets in [13]. In fact, it is a very mild condition.
For example, when a sequence of poles with "xed
real part is chosen, the only way it could be violated
is that the imaginary parts of the poles must diverge
to in"nity faster than the linear rate. If a sequence
of magnitude bounded poles are chosen, then in
order to violate (4) the real parts of the chosen basis
poles must approach to the imaginary axis very
rapidly.
In contrast to the Laguerre and two-parameter
Kautz bases, where all the poles are "xed at the same
value, the general basis (2) enjoys increased #exibility
of pole location. For example, slow and fast modes
may coexist in the model structure. As a result,
a fewer number of basis functions (and hence, for
system identi"cation applications, a fewer number of
data) may be used without sacri"cing modelling
accuracy. This feature is quanti"ed in Section 4.
Note that the conclusion of Theorem 2 may be
extended to H (P). Moreover, it is possible to con
struct orthonormal model sets that are norm dense
in H (P), 1)p(R, and have a prescribed
N
asymptotic order [4]. In [4], it is also shown that
the Fourier series formed by the general basis functions converge in all spaces H (P), 1(p(R.
N
3. Proof of Theorem 2
Before proceeding to the proof of Theorem 2, we
shall demonstrate that the basis functions in (2) are
indeed orthonormal. When n'm, we have by Cauchy's Integral Theorem [37]
1B , B 2
L K
1 (4 Re+a , Re+a , u ( ju)
L
K L\
"!
du
2p
(ju#a )( ju#a ) u ( ju)
\
L
K
K
1
(4 Re+a , Re+a , u (s)
L
K L\ ds
"!
2pj R (s#a )(s#a ) u (s)
L
K
K
1 P (4 Re+a , Re+a , u (s)
L
K L\ ds
" ! lim
2pj
(s#a )(s#a ) u (s)
\
P
L
K
K
P
# O(r\)
H. Akc7 ay, B. Ninness / Signal Processing 77 (1999) 261}274
264
Then provided Re+a ,'0 for all n, m 3D for all n.
L
L
Also
1
(4 Re+a , Re+a , u (s)
L
K L\ ds
" lim 2pj CP (s#a )(s#a ) u (s)
L
K
K
P GFFFFFHFFFFFI
# O(r\) " 0,
(5)
where the closed path C consists of a segment of
P
the imaginary axis and an r radius semicircle in
P centred at the origin and is traversed counter
clockwise. When n"m, we have
1 Re+a ,
L
1B ,B 2"
du"1,
L L
p
u#2u Im+a ,#"a "
\
L
L
where the second equality follows from the formula
3.3.16 in [1]
1
2
2ax#b
dx"
tan\
.
ax#bx#c
(4ac!b
(4ac!b
Now we return to the proof of Theorem 2. Consideration is "rst given to the underlying space being
A(P).
Lemma 3. The linear span of the set +B (s),
with
L LV
B (s) dexned in (2) is complete in A(P) if (4) holds.
L
Proof. This will be established by "rst addressing
the completeness of sp+u (s),
in A(P). Notice
L LV
that since the bilinear map
1!z
s"
1#z
preserves the supremum norms between A(P) and
A(D), the question of whether sp+u (s),
is comL LV
plete in A(P) is equivalent to the question of the
completeness of
1!z
u
L 1#z
LV
in A(D). Let
1!a
L,
mO
L 1#a
L
L 1#a
J.
a O(!1)L “
L
1#a
J
J
(6)
1!z
L 1#a L z!m
J“
I
u
"(!1)L “
L 1#z
1#a
J I 1!mIz
J
"a (z).
L L
Therefore, it is su$cient to establish the completeness of sp+
(z),
in A(D). To achieve this, conL LV
sider the functions +B (z), de"ned in (1). Then
I
m (z)#
(z)
L\ ; n*1
B (z)" L L
L
(1!"m "
L
and hence sp+B (z),L Lsp+
(z),L . However
I I
I I
by Theorem 1, sp+B (z),
is complete in A(D) if
I IV
and only if (3) is satis"ed.
Now, since
1#"a "*"1!a"""1!a ""1#a "
L
L
L
L
then by the de"nition in (6)
1!a
"1!a "
L*
L
"m ""
L
1#"a "
1#a
L
L
so that
"1!a "
L
(1!"m ")) 1!
L
1#"a "
L
L
L
Re+a ,
L .
"2 1#"a "
L
L
Conversely since
1!a
1!a 1#a
"1!a"
1#"a"
L )
L
L"
L
L"
1#a 1#a
"1#a " "1#a "
1#a
L
L
L
L
L
then
1#"a"
L
(1!"m ")* 1!
L
"1#a "
L
L
L
Re+a ,
L .
"2 "1#a "
L
L
As well,
"1#a ")2(1#"a ")
L
L
H. Akc7 ay, B. Ninness / Signal Processing 77 (1999) 261}274
so that
and so on. Let
Re+a ,
Re+a ,
L .
L ( (1!"m "))2 L
1#"a "
1#"a "
L
L
L
L
L
QO 8 Q .
L
L
Since
Therefore, (3) holds if and only if (4) holds implying
that under the de"nition (6), then (4) is necessary
and su$cient for sp+B (z),
to be complete in
I IV
A(D) and hence for sp+u (s),
to be complete in
L LV
A(P). Summing the identity
(2Re+a ,B (s)"u (s)!u (s); k*1
I I
I\
I
over k"1,2,n then provides
L
(2Re+a ,B (s)"1!u (s)"B (s)!u (s).
L
L
I I
I
Hence sp+u (s), Lsp+B (s), . This completes
L LV
L LV
the proof. 䊐
Next, the question of the H (P) su$ciency of (4)
N
is considered.
Lemma 4. Suppose that (4) is satisxed. Then
sp+B ,
is complete in H (P) for all 1(p(R.
L LV
N
Proof. Let m denote the multiplicity of a . Suppose
"rst that m is "nite. Then reorder the basis poles so
that a "a "2"a . We rede"ne the basis
K
functions in (2) by
BI (s)O
L
B (s),
L
s#a
B (s),
L>
s!a
n(m,
(7)
n*m.
This modi"cation of basis functions removes
a from the pole parameter set +a , .
K
I IV
Let Q denote the set containing all partial fracL
tion expansion terms of BI . For example, when
L
m'1
1
1
1
Q O
, ,
,
,
K
s#a 2 (s#a )K\ s#a
K>
265
Re+a ,
I "R,
1#"a "
I
I$K
sp+BI ,
is a complete set in A(P) by Lemma 3,
L LV
and so is sp+B 6Q, since
sp+BI , Lsp+B 6Q,.
L LV
Let f3H (P) and e'0. Approximate f by a funcN
tion g3A(P) that has the properties
lim "s""g(s)""0, s3P,
Q
and
"" f!g"" (e.
N
This is possible since such functions form a dense
subset of H (P) (see for example, Garnett [15, CoN
rollary 3.3 in Chapter II]). Let h(s)"(s#a )g(s).
Then h3A(P). Since sp+B 6Q, is a complete set in
A(P), there exists a u3sp+B 6Q, such that
""h!u"" (e
or
u(s)
e
g(s)!
(
, s3P.
s#a
"s#a "
Hence
u
1
f!
(e#
e.
s#a
s#a
N
N
We have shown that the set
v(s)
: v(s)3sp+B 6Q,
(8)
s#a
is a dense subset of H (P) for all 1(p(R. It
N
remains to show that PLsp+B , . This will
L LV
imply that sp+B ,
is a complete set in H (P) for
L LV
N
all 1(p(R. To this end, let v3sp+B 6Q,. Since
v3sp+B 6Q,, it can be written uniquely as a linear
PO
H. Akc7 ay, B. Ninness / Signal Processing 77 (1999) 261}274
266
combination of the elements in Q and B as follows:
c
I
,
v(s)"c # (s#a ),I
I
I
where N(k) denotes the multiplicity of a in the
I
set +a ,a ,2,a , and only a "nite number of the
I
coe$cients c are nonzero. Notice that c "0.
I
K
Then
v(s)
c
c
" #2# K\
s#a
s#a
(s#a )K
c
I
# .
(s#a )(s#a ),I
I
IK>
Since a Oa for k'm, the terms c /(s#a )
I
I
(s#a ),I admit further expansions
I
c
I
(s#a )(s#a ),I
I
d
d
d
,I .
" # #2#
s#a
s#a
(s#a ),I
I
I
Hence
1
1
1
v(s)
3sp
, ,
,
,2
s#a
s#a 2 (s#a )K s#a
K>
"sp F,
where
FOQ6
1
.
(s#a )K
(9)
Thus PLsp F. To complete the proof for m(R,
we need to show that sp FLsp+B , . Let n be an
L LV
arbitrary positive integer. Write the partial fraction
expansions of the basis elements B ,B ,2,B in the
L
following linear equation form:
B
a
B
a
" $
$
a
a
L
a
B
L
0
$
L
2
0
2
0
\
$
2
a
LL
1
s#a
1
(s#a ),
;
.
$
1
(s#a ),L
L
The degree of B is k, which implies that a O0 for
I
II
all k)n and thus the lower triangular matrix
above is invertible. Hence, for i"1,2,n
1
3sp+B ,k"1,2,n,Lsp+B , .
I
I IV
(s#a ),G
G
Consequently sp FLsp+B , .
L LV
Suppose now that m"R. Then for each n,
de"ne Q as the set containing all partial fraction
L
expansion terms of B . In this case, the proof above
L
still applies with great simpli"cations since P de"ned in (8) equals to sp Q, and F de"ned in (9)
equals to Q for all m.
Proof of Theorem 2. It only remains to establish the
necessity of (4) for completeness in H (P) spaces.
N
Suppose then that (4) fails to hold. Then in this case
the "nite Blaschke products j (s) de"ned by
L
L "1!a "
I , b (s)O1
j (s)Ob u (s), b O “
L
L L
L
1!a
I
I
converge (as nPR) uniformly on P to a nonzero
function j(s)3H (P) which has zeros precisely at
the points a ; see, for example, Garnett [15, ChapL
ter II]. Therefore, the linear functional F de"ned on
H (P) for all 1)p(R and A(P) by
N
1 j( ju)
F(h)O
h( ju)
du
(!ju#1)
2p
\
is nontrivial and bounded. However, by Cauchy's
Integral Theorem, it vanishes for any B of the form
L
(2) since
1 !(2 Re+a ,
L
F(B )"
L
2p
(
ju#1)(
ju!a
)
\
L
L\ ju#a "1!a " ju!a
I “
G
G du
;“
ju#a
G
I ju!aI G 1!aG
H. Akc7 ay, B. Ninness / Signal Processing 77 (1999) 261}274
1 W(ju)
"!
du
2p
( ju#1)( ju#a )
\
L
1
W(s)
"!
ds
2pj R(s#1)(s#a )
L
W(s)
1 P
"!lim
ds
(s#1)(s#a )
2pj
\
P
L
P
#O(r\)
1
"lim
2pj
P
GFFFFHFFFFI
"0,
W(s)
ds#O(r\)
(s#1)(s#a
)
C
P
L
where
L "1!a " "1!a " s!a
I “
G
G
W(s)O(2 Re+a , “
L
s#a
G
I 1!aI GL> 1!aG
is analytic on P and the closed path C is as in (5).
P
Similarly, F(B )"0. (Note that the remainder term
above vanishes as O(r\) in this case). Hence by an
application of the Hahn}Banach Theorem (see, for
example [2, Section 30]), sp+B ,
de"ned by (2)
L LV
is not dense in any of the spaces A(P) and
H (P), 1)p(R. This concludes the proof. 䊐
N
It should be pointed out that the su$ciency part
of Lemma 3 together with Lemma 4 gives the half
of the su$ciency condition in Achieser [2, Section
A.4] for the completeness of the model sets spanned
by B and the Cauchy kernels
1
B (s)O
, a3+a ,a ,2,
?
s#a
in the Lebesgue spaces ¸ , 1(p(R, and in the
N
space of complex functions continuous on the imaginary axis including R. The linear span of the Cauchy kernels does not contain systems which have
repeated poles whereas with the bases (2), a greater
#exibility is utilised on the choice of basis poles.
3.1. Ensuring real-valued impulse response
Up until this point, the basis (2) has been considered with complete generality of pole location
267
save for the condition (4). However, in any application involving the modelling of a physical process,
it is necessary to ensure that the underlying
modelled impulse response is real valued. If complex valued choices for +a , are made in order to
I
accommodate resonant characteristics, then this
realness of impulse response is lost unless some
restriction is placed on how linear combination of
the basis functions is taken.
The purpose of this section is to illustrate how to
use the basis formulation (2) in such a way that
imposing realness of the weightings in the linear
combination ensures realness of the underlying impulse response. This is achieved by requiring that if
a set of numbers +a ,a ,2,a , used to de"ne bases
L
via (2) contains a complex valued element (say a ),
I
then it always also includes its conjugate a .
I
To be more explicit on this point, suppose that
the numbers a ,2,a
are real so that the basis
L\
functions B ,2,B
have real-valued impulse re
L\
sponses. We now wish to include a complex pole at
!a . Then two new basis functions B , B with
L
L L
real impulse responses should be formed as a linear
combination of B and B
generated by (2). These
L
L>
new functions then replace B and B
in any
L
L>
modelling applications that require a real-valued
impulse response.
The linear combination we are suggesting can be
expressed as
B
c
L " B
c
L
c
c
B
L ,
B
L>
c , c , c , c 3C.
(10)
Therefore, considering only B for the moment, if
L
we choose complex poles in conjugate pairs as
!a "!a then
L>
L
(2 Re+a ,(bs#k)
L
B (s)"
u (s),
(11)
L
s#(a #a )s#"a " L\
L
L
L
where u (s) has real-valued impulse response and
L\
the real coe$cients b, k are related to the choice of
c , c by
a b#k
c " L
,
2a
L
a b!k
c " L
.
2a
L
H. Akc7 ay, B. Ninness / Signal Processing 77 (1999) 261}274
268
Therefore, to ensure a unit norm for B we must
L
choose b and k according to the constraint that
"c "#"c ""1 which becomes
x2Mx"2"a ",
(12)
L
where
xO(b,k)2,
MO
"a "
L
0
0
1
.
Now, suppose we make two pairs of choices:
x"(b,k)2 giving a basis function B and y"(b,k)2
L
giving another basis function B. These two choices
L
correspond to two pairs of complex numbers
+c , c , and +c , c ,. The requirement c c #
c c "0 ensuring orthogonality of B and B can
L
L
be expressed as needing
x2My"0
(13)
to hold, and in fact many solutions x and y to (12)
and (13) will exist. To formulate them, suppose we
begin by choosing any x satisfying (12). All solutions to (12) are given by b"(2 cos h and
k""a "(2 sin h, 0)h(2p. Then for a "xed h,
L
a unique y that satis"es (12) and (13) is found by
rotating x ninety degrees in the normalised eigenspace of M:
y"M\
0
!1
1
0
Mx
or, to be more explicit
y"(2(!sin h, "a " cos h)2.
(14)
L
To summarise this discussion, if we want to include
complex modes in a model structure, then we obtain two basis vectors B and B from two linear
L
L
combinations of B and B
that come from the
L
L>
unifying construction (2). Let 0)h(2p. Then the
basis functions B and B are found as
L
L
(4 Re+a , (s cos h#"a " sin h)
L
L
B (s)"
u (s),
L
L\
s#(a #a )s#"a "
L
L
L
(4 Re+a , (!s sin h#"a " cos h)
L
L
B(s)"
u (s).
L
L\
s#(a #a )s#"a "
L
L
L
These real-valued impulse response basis vectors
B and B are then used for modelling instead of
L
L
B and B . If we require further basis functions
L
L>
with complex modes then we repeat the process in
(10) by forming B
and B from linear combiL>
L>
nations of B
and B
and so on, and in this
L>
L>
way arbitrary complex pole con"gurations may be
accommodated. For example, when a "a "
L
L>
2"a
"a
, with chosen h"0 the
L>K
L>K>
above basis construction process yields for
k"0,2,m
(2b s
s!bs#c I
B (s)"
u (s),
L>I
L\
s#bs#c s#bs#c
(2bc
s!bs#c I
B (s)"
u (s)
L>I
L\
s#bs#c s#bs#c
where b"2 Re+a , and c""a ". With n"1
L
L
plugged in, this is the de"ning formula for the
two-parameter Kautz functions in [43].
Having now illustrated how the constraint of
realness of impulse response may be easily accommodated via constraining realness of linear combination weights, it remains to establish that this
latter restriction does not destroy completeness
properties. For this purpose, note that the basis
functions of the form described above and the basis
functions generated by (2) are related by a unitary
block diagonal matrix of the form
c
c
,2 .
c
c
Let ; 3CL"L denote truncations of ;. For a given
L
G3H (P) and e'0, via Theorem 2 there exists
N
a G 3H (P) given by
L
N
L
G (s)" a B (s), a 3C
L
I I
I
I
such that ""G!G "" )e/2. Then G can be written
L N
L
as
;Odiag 1,2,1,
G (s)"a2 ;\ W (s)
L
L
L
L
" h B (s), h Oa #jb ; a ,b 3R,
I I
I
I
I
I I
I
where h2Oa2 ;\ and here W2O(B ,2,B ) refers
L
L
L
to a set of basis functions that have real-valued
impulse responses.
H. Akc7 ay, B. Ninness / Signal Processing 77 (1999) 261}274
Now, assume that G(s) has a real-valued impulse
response so that G( ju)"G(!ju). Then, using the
fact that "" f """"" f """"" f (!ju)"" for any f3H (P)
N
provides
""G!G "" )e
L N
Proof. The proof will be by induction. First, when
n"1
a #a
B (k)B (s)"
(k#a )(s#a )
while
a #a
"
(k#a )(s#a )
L
N G(!ju)! h B (!ju) )e
I I
N
I
so that the result holds for n"1. Now suppose it
holds for n'1. Then
L
N G! h B )e
I I
N
I
K (s,k)"K (s,k)#B (k)B (s)
L
L
L\
L
and hence via the triangle inequality
1!u (k)u (s)
(k!a )(s!a )
1
" 1!
s#k
(k#a )(s#a ) s#k
L
N G! h B )e
I I
N
I
269
L
L h #h
IB
G! a B " G! I
I I
I
2
N
N
I
I
e e
) # "e
2 2
so that indeed, arbitrarily accurate modelling of
real impulse G(s) is possible by taking real linear
combinations of real impulse versions of the basis
+B ,.
I
4. Approximation of 5nite-dimensional systems
While the completeness result of Theorem 2 provides a theoretical pedigree for considering bases (2)
for system approximation purposes, it leaves open
the question of the quality of approximation for
a "nite number of bases. Addressing this will be the
concern of this section, where a useful tool is to use
the so-called &reproducing kernel' K (s,k) associated
L
with the linear space sp+B (s),L .
I I
Lemma 5. Consider the basis functions +B ,L deI I
xned by (2). Then
1!u (k)u (s)
L
L
L .
K (s,k)O B (k)B (s)"
L
I
I
s#k
I
1!u (k)u (s)
L\
L\
"
s#k
a #a
L
L
#
u (k)u (s)
L\
(k#a )(s#a ) L\
L
L
1
"
s#k
!
(k#a )(s#a )!(a #a )(s#k)
L
L
L
L
(s#k)(k#a )(s#a )
L
L
;u (k)u (s)
L\
L\
1!u (k)u (s)
L
L .
"
s#k
Therefore, by induction the result holds for
all n. 䊐
The utility of this result becomes apparent in the
derivation of the following expression for the
"nite-order approximation error.
Lemma 6. Suppose f (s) is analytic on P and has
a partial fraction expansion
K
c
I .
f (s)" s#c
I
I
H. Akc7 ay, B. Ninness / Signal Processing 77 (1999) 261}274
270
Dexne f (s) as an approximation to f (s) obtained by
L
projection onto sp+B (s),L :
I I
L
f (s)O 1f,B 2B (s).
L
I I
I
Then
K
c
L
I
" f (ju)!f (ju)") “
L
ju#c
I J
I
c !a
I
J.
c #a
I
J
(15)
Proof. By the de"nition of f (s) and for any k3P
L
L
1
f (k)" f (s)B (s) ds B (k)
L
I
I
2pj R
I
1
"
f (s)K (s,k) ds,
L
2pj C
where C consists of the imaginary axis and an
in"nite radius semi-circle in the open right halfplane; it is traversed clockwise. Using this de"nition
and Cauchy's integral formula gives, for k3P
arbitrary
1
f (s)
ds.
f (k)"
2pj Ck!s
Therefore, by Lemma 5 and using the fact that
s"!s for s3jR
" f (k)!f (k)"
L
1
f (s)
"
u (k)u (s) ds
L
2pj Ck!s L
u (k) K
1
L s#a
J ds
" L c
“
I C(s#c )(k!s)
2pj
I
I
J s!aJ
K
1
L c !a
J,
""u (k)" c
“ I
L
I(k#c )
I J cI#aJ
I
where in moving to the last line Cauchy's residue
theorem was used to evaluate the integral after
performing the change of variable s C !s. Taking
the limit as Re+k, P 0 then gives the result.
The result exposes the dependence of the approximation error on the choice of poles +!a , in the
L
base B (s). Namely, the closer the poles +!a , are
L
L
chosen to the poles +!c , of the function f (s) being
I
approxiated then the more accurate the approxi-
mation of f (s) will be, and in such a way as to
decrease exponentially with increasing n.
Certainly the error bound (15) gives strong motivation for the consideration of the general basis
(2), since (in contrast to the Laguerre and Kautz
cases where all the poles are "xed at the same value)
the increased #exibility of pole location +!a , will
L
increase the possibility of making "c !a " small (for
I
J
some l) for every k, and hence making the total
product “L "c !a ""c #a "\ as small as posJ I
J I
J
sible.
5. Application example
We conclude the paper by presenting an application example that illustrates the utility of the basis
(2) for modelling purposes. The example involves
measurements of the frequency response of a 58 cm
long, 5 mm wide cantilevered piezo-electric laminate beam (for further details, see [29]). These
measurements are shown as dots in Fig. 1. For the
purposes of control (sti!ness compensation using
the piezo-electric actuators) a transfer function
model that explains this frequency response is required. There are many ways in which this may be
achieved [6,27,35,38,39], but for the purpose of
illustrating the e$cacy of the basis (2), the simple
least-squares method introduced by Levy [22] to "t
a model
N(s) b sL#b sL\#2#b s#b
L\
G (s)"
" L
L
D(s)
sL#d sL\#2#d s#d
L\
to the frequency response measurements +G ,, at
I I
the frequencies +u ,, by means of minimising the
I I
cost
,
< " "D( j u )G !N( j u )"
,
I I
I
I
will be studied. Our purpose is not to present best
possible results on this data set, but to illustrate by
an example that the model parameterised by the
orthonormal basis (2) should be preferred to polynomial models when one is concerned with numerical conditioning.
H. Akc7 ay, B. Ninness / Signal Processing 77 (1999) 261}274
271
Fig. 1. Estimation using the polynomial and the orthonormal bases. The dots are the measurements, the solid line is the estimate using
the basis (2) to parameterise the model, the dash}dot line is the estimate using the Tchebychev polynomials to parameterise the model.
The Tchebychev and the polynomial estimates are identical.
As is well known [35], "nding this estimate involves solving the so-called &normal equations'
( ju )LG
$
( ju )LG
, ,
!( ju )L\G , 2, !G , ( ju )L, 2, 1
"
$ $
!( ju )L\G , 2, !G , ( ju )L, 2, 1
,
,
,
,
GFFFFFFFHFFFFFFFFI
U
d
L\
$
;
d
b
L
$
b
,
for which the numerical stability of the solution is
highly dependent [16], on the conditioning of the
matrix U2U. However this can be altered via reparameterisations of the model G(s). For example,
in [5] the parameterisation
L
N(s)"b # b p (s),
I I
I
(16)
L\
D(s)"sL# d p (s),
I I
I
where each p (s) is an order k &modi"ed
I
Tchebychev' polynomial (p "1) is suggested as
a means of improving numerical conditioning.
In Fig. 1, the dash}dot line shows the results of
using the above Tchebychev parameterisation to "t
an n"18th-order model to the observed frequency
response. Note that the second resonance peak is
completely missed, and that the resonance frequencies from the 3rd resonant mode onwards are signi"cantly shifted.
272
H. Akc7 ay, B. Ninness / Signal Processing 77 (1999) 261}274
Fig. 2. The singular values of U using the polynomial (natural and the Tchebychev) and the rational orthonormal basis (2).
However, if the model is parameterised using the
orthonormal basis (2) as
L
N(s)"b # b B (s),
I I
I
(17)
L
D(s)"sL# d B (s),
I I
I
with the pole choice !a "!a"!2u , then
I
,
the ensuing 18th-order least-squares estimate is the
solid line shown in Fig. 1, which now captures the
second resonance peak, and correctly matches the
resonance frequencies from the 3rd resonant mode
onwards.
Since the model structures (16) and (17) both
span the same manifold of rational models, the only
explanation for the di!erence in results is that of
di!erences in numerical conditioning. Fig. 2 shows
the singular values of U for the three model parameterisation choices. (There are 36 singular values
of U since the chosen model order is 18). Considering the log-scale employed, the parameterisation
using the basis (17) enjoys a two order of magnitude
better conditioning (the ratio of the largest to the
smallest singular value) than either a Tchebychev
polynomial, or conventional polynomial parameterisation.
As a consequence, for such applications of modelling resonant structures over large bandwidths,
we suggest that the basis (2) should be employed in
the interests of the resultant frequency response
having no artifacts due to poor numerical conditioning.
6. Conclusion
This paper has provided a preliminary study of
the approximation properties of a particular class
of rational orthonormal bases that are suitable for
continuous-time system modelling. The main result
was to establish that the bases were capable of
arbitrarily good approximation with respect to
a wide variety of norms employed in the systemtheoretic analysis of stable systems. The utility of
H. Akc7 ay, B. Ninness / Signal Processing 77 (1999) 261}274
the generalising nature of the particular bases considered here was also exposed by establishing that
signi"cantly improved "nite-order approximation
accuracy was possible by exploiting the #exibility
in allowed pole position. This is in contrast to the
more well known Laguerre and two-parameter
Kautz bases, which are obtained as special cases of
the bases considered here by choosing all the poles
"xed at one location.
Acknowledgements
The authors would like to thank Dr. Reza
Moheimani for providing the experimental data
used in the application example. Akc7 ay gratefully
acknowledges the partial support of Alexander
von Humboldt Foundation. Ninness gratefully acknowledges the support of CIDAC and the Australian Research Council.
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