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Pole Reconstruction of Systems from LaguerreBasis Representations
ALEXANDROS SOUMELIDISComputer&Automation Res. Inst.Systems and Control LaboratoryKende 13–17, H-1111 Budapest
FERENC SCHIPPEotvos Lorand University
Department of Numerical AnalysisPazmany Peter 1/C, H-1117 Budapest
JOZSEF BOKORComputer&Automation Res. Inst.Systems and Control LaboratoryKende 13–17, H-1111 Budapest
Abstract:This paper investigates an opportunity to identify poles of systems with unknown structure on the basis ofdiscrete-time spectral signal representations generated on the basis orthogonal Laguerre systems. The convergenceof the Laguerre representation coefficients is analyzed within the framework of a hyperbolic metrics generatedby the Blaschke functions associated with the Laguerre system, and a conceptual base is given to solve the taskof finding the poles of the system. An algorithm is proposed to find the poles, and a discussion is given uponthe criteria of finding all of them. An efficient implementation of the algorithm based upon frequency domainmeasurement is available for practical computations.
Key–Words:Systems theory, System identification, Pole structure estimation, Frequency domain methods, Rationalorthogonal bases
1 Introduction
Determining the pole locations associated to signalsand systems modeling is needed in many applicationsand several approaches are known to cope with thisproblem. Detection of spectral peaks in noisy sig-nals – a frequent task in analyzing vibrating mechani-cal systems, electrical rotating machines, or industrialplants (such as nuclear power plants) – is in essence apole-identification problem. The phase angle associ-ated with the pole can be identified as the frequency,and its absolute value – corresponding to the attenua-tion represented by the pole – can be connected withthe height and width of a spectral peak. The spectralpeak-analysis can be performed by Fourier-analysis ofthe signals that can efficiently be realized by usingthe Fast-Fourier-Transform (FFT) algorithm. How-ever the exact pole locations cannot be determined bythis way, spectral peaks cannot be separated in manycases, as well as analyzing the height and width of thepeaks does not result in unique solutions for determin-ing the poles attenuations.
For use of parametric methods in time domain onecan cite variations of Prony–methods [5] and thoseassuming linear signal- and system-models, i.e. ap-plying autoregressive (AR) or autoregressive moving-average (ARMA) model identifications and associatedspectral analysis [4], as well as identification of matrixpartial fraction models [3], and subspace approaches[6]. Disadvantage of these approaches is that associ-ated to the parametrization problem, both the structure
and the parameters have to be estimated, leading to, inmany situation, unreliable results.
Another approach is the use of rational orthogo-nal bases [2] that needsa priori knowledge upon thepole locations. Special attention paid on the problemsof pole selection and validation [1]. There exist meth-ods to refine the pole locations starting from an ap-proximate placement of poles [7], however the generalidentification problem has not been solved so far.
This paper proposes a new approach that isclosely related to signal and system representationsusing discrete rational Laguerre–basis inH2(D).Based upon the analysis of the Laguerre–coefficientsof the rational transfer function – that can be com-puted using frequency domain data obtained fromnon-uniformly distributed measurements – by ap-plying a specific hyperbolic transform originatedfrom the Blaschke–function inherent in the Laguerre–representation, the poles of the system can be identi-fied. The transform is related to the Poincare unit discmodel of the hyperbolic geometry. This will allow anice geometric interpretation of the resulting identifi-cation algorithm.
2 The discrete Laguerre–systemDefinition 1 The discrete Laguerre system basedupon parameterb (b ∈ D) is defined as:
Φn(z) =
√1− |b|2
1− b zBn
b (z) (n = 0, 1, 2 . . . ), (1)
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where
Bb(z) = eiδz − b
1− b z
is the Blaschke function with the arbitrary constantδ ∈ [0, 2π).
It is well known – see [9] – that the discreteLaguerre system forms an orthonormal basis in theHardy spaceH2(D). Any function F ∈ H2(D) canbe expressed by the representation:
F (z) =
∞∑n=0
lnΦn(z), (2)
where the coefficients{ln} – the so-called Laguerrecoefficients – can be computed by using the inner-product belonging to the spaceH2(D) as:
ln = 〈F,Φn〉.
By substituting the expression of the Blaschke func-tion in (1) by selecting0 for the value of the arbitraryparameterδ, the elements of the Laguerre system getthe form
Φn(z) =√
1− |b|2(z − b)n
(1− b z)n+1(n = 0, 1, 2 . . . )
The representation of any functionF ∈ H2(D) canbe expressed on the basis of an orthogonal projectionupon the Laguerre system components, i.e. the repre-sentation coefficients can be expressed as:
ln =⟨F (z),
√1− |b|2
(z − b)n
(1− b z)n+1
⟩=
=
√1− |b|2
2π
∫ π
−π
F (eit)(e−it − b)n
(1 − b e−it)n+1dt =
=
√1− |b|2
2π
∫ π
−π
F (eit)(1− beit)n
(eit − b)n+1eit dt
Rewriting this form into a complex contour integralupon the unit circle, i.e. by applying the substitutionz = eit and consideringT := {z ∈ C : |z| = 1},
ln =
√1− |b|2
2πi
∮
T
F (z)(1− bz)n
(z − b)n+1dz =
=√
1− |b|21
2πi
∮
T
F (z)(1 − bz)ndz
(z − b)n+1.
According to Cauchy’s integral formula the Laguerrecoefficients can be expressed in the form
ln =
√1− |b|2
n!
[dn(1− bz)nF (z)
dzn
]
z=b
(n = 0, 1, 2 . . . )
(3)
As an example letF be considered as
F (z) =1
1− a z(a ∈ D). (4)
It is obvious thatF ∈ H2(D) and a single pole of thefunction resides outside the unit circle. The parametera can be referred as aninverted polebelonging toFwith the exact definitionp = 1/a.
The coefficients belonging to the Laguerre–representation ofF according to (3) are given as
ln =
√1− |b|2
n!
[dn
dzn(1− bz)n
1− a z
]
z=b
. (5)
In the special case when the parameterb belonging tothe Laguerre–representation is selected to be equal toa, i.e. b = a,
ln =
√1− |a|2
n!
[dn
dzn(1− az)n−1
]
z=b
.
The function to be differentiatedn-times is a(n− 1)-degree polynomial ofz, hence the derivatives equalto zero except ifn = 0. This means that only onecoefficient differs from zero, namelyl0, it is actually
l0 =√
1− |a|2[
1
1− az
]
z=b
=1√
1− |a|2
The expansion (2) contains only one term,
F (z) = l0Φ0(z) = l0
√1− |a|2
1− a z=
1
1− a z,
i.e. the function (4) is represented exactly.
In the case ifb 6= a the Laguerre–coefficients canbe computed by the formula (5). The coefficientl0 isgiven immediately as
l0 =√
1− |b|2[
1
1− az
]
z=b
=
√1− |b|2
1− ab
The coefficientl1 can be computed from formula
l1 =√
1− |b|2[d
dz
1− bz
1− az
]
z=b
.
By realizing the derivation
d
dz
[(1− bz)(1 − az)−1
]=
= −b (1− az)−1 + (1− bz) a (1− az)−2 =
=a (1− bz)− b (1− az)
(1− az)2=
a− b
(1− az)2,
the Laguerre–coefficient is given as
l1 =√
1− |b|2a− b
(1− ab)2.
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To obtain a general formula for everyln (n =0, 1, . . . ) Laguerre–coefficient, consider the casewhen the functionF (z) possesses a single invertedpolea ∈ D of multiplicity m (m = 1, 2, . . . ), i.e.
F (z) =1
(1− a z)m(a ∈ D m ∈ N).
The Laguerre-coefficientsln (n = 0, 1, 2, . . . ) can becomputed by the form
ln =
√1− |b|2
n!
[dn
dzn(1− bz)n
(1− a z)m
]
z=b
. (6)
By denoting the function to be differentiated asf(z),i.e.
f(z) = (1− bz)n(1− a z)−m,
it is clear that then-th derivative can be computed byapplying the generalized Leibnitz rule
(uv)(n) =
n∑k=0
(n
k
)u(n−k)v(k).
The (n − k)-th derivative (k < n) of the termu =(1− bz)n and thek-th derivative ofv = (1 − a z)−m
are given as
u(n−k) = (−1)n−kn(n− 1) . . . (k + 1)bn−k
(1− bz)k
v(k) = m(m+ 1) . . . (m+ k − 1)ak(1− az)−(m+k),
respectively. Since
n(n− 1) . . . (k + 1) =n!
k!, and
m(m+ 1) . . . (m+ k − 1) =(m+ k − 1)!
(m− 1)!,
then-th derivative off can be expressed as
f (n)(z) =n!
n∑k=0
(−1)n−k
(n
k
)(m+ k − 1)!
k!(m− 1)!·
· bn−k
(1− bz)k ak(1− az)−(m+k).
By applying the identity
(m+ k − 1)!
k!(m− 1)!=
(m+ k − 1
k
),
setting out the common factor(1 − az)−(m−1) fromthe sum, and bringing it to the common denominator(1− az)n+1,
f (n)(z) =n!
(1− az)m−1·
1
(1− az)n+1·
·
[ n∑k=0
(−1)n−k
(n
k
)(m+ k − 1
k
)· (7)
· bn−k
(1− bz)k ak (1− az)n−k
]
In the case ifm = 1, i.e. a pole with no multiplicity
is considered,n∑
k=0
(−1)n−k
(n
k
)bn−k
(1− bz)k ak (1− az)n−k
– according to the binomial theorem – forms then-thpower of the expression
a(1− bz)− b(1− az) = a− b.
Hence, according to (6) then-th Laguerre–coefficient– in the case when the functionF to be representedcontains a single pole with no multiplicity – can beexpressed as
ln =√
1− |b|2(a− b)n
(1 − ab)n+1. (8)
In the cases of poles with multiplicitym > 1 thebinomial theorem cannot immediately be applied be-cause of the presence of the factor
(m+ k − 1
k
)=
m(m+ 1) . . . (m+ k − 1)
k!
in the sum. However it can be applied in an iterativeway such that – as it can be shown – a sum containingm terms with decreasing powers ofa−b
1−abis given for
then-th Laguerre–coefficient:
ln =
√1− |b|2
(1− ab)m
{[a− b
1− ab
]n+ (9)
+m−1∑k=1
Ckn(n− 1) . . . (n − k + 1)
[a− b
1− ab
]n−m+k}.
whereCk (k = 1, 2, . . . , (m−1)) are constant factors.The exact form of these factors is indifferent in respectto the subject of this paper.
3 Convergence of the Laguerre–coefficients
In respect of convergence the several terms ofthe Laguerre–coefficients belonging to multiplicitiesgreater than1 can be analyzed separately. Two casesshould be discriminated, form = 1 andm > 1, re-spectively
λ1(n) =
[a− b
1− ab
]n
λm(n) =n(n− 1) . . . (n −m+ 1)
[a− b
1− ab
]n,
where the constant factors has been disregarded.The conditions of convergence ofλ1 is obvious,
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sinceλ1 forms a geometrical sequence with quotient
q =a− b
1− ab,
and as it is well known, the condition of the absoluteconvergence is|q| < 1. This condition can be verifiedby using the following proposition:
Lemma 1 If a, b ∈ D,∣∣∣∣a− b
1− ab
∣∣∣∣ < 1. (10)
Proof: By considering the square of (10)
a− b
1− ab
a− b
1− ab=
|a|2 − (ab− ab)− |b|2
1− (ab− ab) + |a|2|b|2< 1.
The denominator is positive, hence both sides is mul-tiplied by it, than(ab− ab)-t is added:
|a|2 + |b|2 < 1 + |a|2|b|2
With some manipulations
0 < (1− |a|2)(1− |b|2)
0 < (1 + |a|)(1 − |a|)(1 + |b|)(1− |b|).
Since|a| < 1 and|b| < 1, the expression in the right-hand side is positive, that proves the proposition (10).⊓⊔
Convergence ofλm is also based upon the condi-tion |q| < 1.
Lemma 2 If q ∈ D, for any numberm ∈ N the se-quence
n(n− 1)(n − 2) . . . (n−m)qn (n ∈ N > m)
absolutely converges.
Proof: By applying the quotient criterion, the
(n+ 1)n(n− 1) . . . (n −m+ 1)
n(n− 1)(n − 2) . . . (n−m)|q| < 1
condition should be satisfied to achieve absolute con-vergence. After simplification
n+ 1
n−m|q| < 1,
i.e. for indicesn > m
|q| <n−m
n+ 1= 1−
m+ 1
n+ 1< 1. (11)
As n increases toward infinity, n−mn+1 forms a
monotonously increasing sequence toward1, hencethere exist an indexnq that makes (11) true for anyn > nq , which proves the proposition. ⊓⊔
In both cases the quotientq = a−b1−ab
plays crucialrole. By comparing its form with the Blaschke func-tion introduced in definition (1) some correspondencecan be observed. To elaborate this correspondence,some group-theoretic considerations are required inassociation with the Blaschke function.
The Blaschke function upon parameters(b, ǫ) isdefined as
Bb(z) := ǫz − b
1− bz(12)
(z ∈ C, b = (b, ǫ) ∈ B := D×T).
If b ∈ B, thenBb is an1 − 1 map onT andD, re-spectively, which means that the Blaschke function isan inner function in the spaceH2(D). The restrictionsof the Blaschke functions on the setD or T with theoperation of function–composition
(Bb1 ◦Bb2)(z) := Bb1(Bb2(z))
form agroup. A group(B, ◦) defined in the set of pa-rametersB := D×T can be obtained if the operation
Bb1 ◦Bb2 = Bb1 ◦ b2
is considered, which is isomorphic to the group({Bb, b ∈ B}, ◦). Either of these isomorphic groupscan be referred by the nameBlaschke–group.
The neutral and the inverse element of the group(B, ◦) can be obtained as
e := (0, 1) ∈ B and b−1 = (−bǫ, ǫ)
respectively, ifb = (b, ǫ) ∈ B.It can be proved that the map
ρ(z1, z2) :=|z1 − z2|
|1− z1 z2|= |Bz1(z2)| (13)
(Bz1 := B(z1,1), z1, z2 ∈ D)
is a metric onD. Moreover the Blaschke functionsBb
(b ∈ D) are isometries with respect to this metric, i.e.
ρ(Bb(z1), Bb(z2)) = ρ(z1, z2) (b ∈ D, z1, z2 ∈ D).
The lines in this model are the sets
Lb := {Bb(r) : −1 < r < 1} (b ∈ B),
i.e. circles crossing perpendicularly the unit circle.This model is known in the hyperbolic geometry asthe Poincare model of the unit–disc.
Any element of the Blaschke–groupB can becharacterized as a transform that mapsT andD ontothemselves, respectively. which can be consideredas a hyperbolic transform upon the unit disc. The”hyperbolic” nature of this transforms arises from itsdistance–preserving property with respect to the hy-perbolic metric according to (13).
By returning to the expression of the quotientq, it
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a
b1
b2
Figure 1: Hyperbolic circles belonging to Laguerre–representations.
can be observed that its conjugateq can be expressedas a hyperbolic transform related to the Blaschke–group elementb = (b, 1), i.e.
q = Bb(a).
This relation has great significance in respect to iden-tifying poles: in the case if the quotientq – by assum-ing a Laguerre representation parameterb ∈ D – is inany way estimated (the opportunities to do this will bediscussed later in this paper), on the basis of this es-timate, the location of the polea within the unit disccan be derived. The following proposition – that hastrivially been proved by the above discussion – hascrucial significance:
Proposition 1 Let q be the quotient belonging to theLaguerre representation based on parameterb ∈ D
of a functionF ∈ H2(D) containing a single inversepolea ∈ D. A hyperbolic transform belonging to theinverse group elementb−1 = (−b, 1) reconstructsa,i.e.
a = Bb−1(q). (14)
This proposition directly gives an opportunity toidentify a single pole on the basis of the Laguerre–representation, sinceq can immediately be derivedfrom the Laguerre–coefficients. An extension to mul-tiple poles will be constructed in the subsequent sec-tions.
Fig. 1 illustrates the significance of the quotientq associated with Laguerre–representations. One poledenoted bya has been represented by applying twodifferent Laguerre–parameters,b1 and b2. Two hy-perbolic circles has been drawn (the image of a circle
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1
Figure 2: Hyperbolic circles indicating the pole posi-tions.
will also be circle with a displacement on its center)
aroundb1 andb2 with hyperbolic radii|q1| =∣∣∣ a−b11−ab1
∣∣∣and |q2| =
∣∣∣ a−b21−ab2
∣∣∣, respectively. The two circles cross
each other just in the position of polea.In Fig. 2 the hyperbolic circles belonging to
the quotientsq of the Laguerre-representations of thepole-seta = [0.6ei
π2 0.25ei
3π2 0.45ei
π8 ] with respect
to an arbitrarily selected set of parametersbi are pre-sented. The sections of the hyperbolic circles indicatethe pole positions.
4 Convergence in the case of multiplepoles
Any functionF ∈ H2(D) can be expressed in partialfraction form, i.e. for a set of inverse polesak ∈ D andtheir multiplicitiesmk ≥ 1 (k = 1, 2, . . . , N , N ∈ N)
F (z) =
N∑k=1
Ak
(1− akz)mk
with constant coefficientsAk ∈ R. Hence theLaguerre–coefficients belonging to a parameterb ∈ D
associated to functionF can be expressed term-by-term, i.e. they can be expressed as a linear combina-tion of partial Laguerre–coefficients belonging to ev-ery term of the partial fraction.
Concerning the convergence of the Laguerre–coefficients let the case of two identical poles with nomultiplicity be considered, i.e. letF has the form of
F (z) =A1
1− a1z+
A2
1− a2z(a1,2 ∈ D, A1,2 ∈ R)
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The Laguerre–coefficients belonging to parameterb ∈D get the form
ln = A1c1qn1 +A2c2q
n2 (n = 0, 1, 2, . . . )
where
cj =
√1− |b|2
1− ajbqj =
aj − b
1− ajb(j = 1, 2)
By introducing the notationscj = djeiδj , qj = rje
iϕj ,and fj = djAj (j = 1, 2),
ln = f1rn1 e
i(nϕ1+δ1) + f2rn2 e
i(nϕ2+δ2).
In respect to convergence the absolute value ofln hasgreat significance, let it be denoted byℓn. By express-ing ℓ2n
ℓ2n =f21 r
2n1 + f2
2 r2n2 +
+ 2f1f2rn1 r
n2 cos[n(ϕ1 − ϕ2)+ (δ1 − δ2)]
Let this sequence be substituted by a continuous func-tion of parametert ∈ R
+ with the property for anyinteger valuet = n let ℓ(t) = ℓn,
ℓ2(t) = f21 e
2α1t + f22 e
2α2t+
+ 2f1f2e(α1+α2)t cos[t(ϕ1 − ϕ2) + (δ1 − δ2)],
whereαj = log rj , αj < 1 (j = 1, 2). Furthermore,to consider convergence let the majoring function –that is given by substituting thecos function by con-stant1 –
ℓ2(t) = f21 e
2α1t + 2f1f2e(α1+α2)t + f2
2 e2α2t
be used instead, which can be expressed in the simpleform
ℓ(t) = f1eα1t + f2e
α2t.
It is clear thatℓ(t) <= ℓ(t) at least att > t0 foran adequately selectedt0 ≥ 0, hence convergence ofℓ(t) involves also convergence ofℓ(t) in t → ∞. Thefunction ℓ(t) is obviously converges to0 if t → ∞,sinceαj < 0 (j = 1, 2).
The convergence rate – in comparison with a sin-gle exponential corresponding to a geometrical se-quence – can be characterized by an equivalent decayrate defined by the function
α(t) = ℓ(t)/ℓ(t), (15)
wheredot denotes the derivation by parametert. Theconvergence of the Laguerre–coefficients can be char-acterized byα(t) values in larget values. Anequiv-alent convergence quotientQ – associated with anequivalent geometrical sequence – can be obtained asthe exponential function of the limit ofα(t) att → ∞,i.e.
Q = eα∞ α∞ = limt→∞
eαt.
Applying function (15) toℓ(t),
α(t) =f1α1e
α1t + f2α2eα2t
f1eα1t + f2eα2t
is given. The limit of this function att → ∞ – asit can easily be verified by factoring outeα1t or eα2t
from the sum – is given as
limt→∞
α(t) =
{α1 if |α2| > |α1|,α2 if |α1| > |α2|.
This result means that in the respect of convergencein the infinity, the exponential with the smallest decayrate will be dominant. Translating it to the sequenceof the Laguerre–coefficients: the term consisting ofthe geometrical sequence with the greatest quotientwill dominate in the convergence of the complete se-quence.
This assertion can be extended to the case whenthe functionF contains any number of poles with nomultiplicity, since the considerations taken above caneasily be expressed for multiple terms in the expres-sion of the Laguerre–coefficients. The exact statementfor this case can be expressed in the following form:
Proposition 2 Let F a function belonging to thespace H2(D) with N ∈ N number of inversepoles ak ∈ D (k = 1, 2, . . . , N of multiplicity 1.The sequence of the coefficientslk belonging to theLaguerre–representation ofF based upon parameterb ∈ D
(i) absolutely converge to zero, and
(ii) its convergence rate is equal to
max
{∣∣∣∣a1 − b
1− ba1
∣∣∣∣ ,∣∣∣∣a2 − b
1− ba2
∣∣∣∣ , . . . ,∣∣∣∣aN − b
1− baN
∣∣∣∣}
In the case when poles with multiplicity greater than1 occur, the situation is more complicated. On thebasis of the expression of the Laguerre–coefficients(9) it can be shown that an analogousα(t) functionconverges to one of theαk values belonging toak thatcan be that possessingαk of minimal absolute value,or has the maximal multiplicitymk – depending uponwhethereαk or t−(mk−1) converges faster to zero.
5 Identifying polesThe equivalent convergence ratio found on the basisof the limit of α(t) depends on the selection of thebparameter of the Laguerre–representation. Differentbvalues can result in differentQ values of the variety
{a1 − b
1− a1b,a2 − b
1− a2b, . . . ,
aN − b
1− aNb
},
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3
Figure 3: Identifiability domains belonging to pole-set
a = [0.8 0.8eiπ4 0.8e−i
π4 ].
whereak ∈ D (k = 1, 2, . . . , N ) are the inverse polesof a functionF ∈ H2(D). This fact can serve as thebasis of a method of finding the poles: a principal pro-cedure is specified as follow:
Procedure 1 By selecting a parameter valueb ∈ D,
1. Estimation of the coefficients of the Laguerre-representation based upon specificb.
2. Estimation of the equivalent convergence rateQaccording to the Laguerre–coefficients obtained.
3. Applying the hyperbolic transform(14) with theparameterb on Q results in the estimation of aparticular poleak.
4. Steps 1 to 3 are repeated for a new value of pa-rameterb.
The values of parameterb involved in the procedurecan be selected either arbitrarily, randomly, or accord-ing toa priori considerations upon the pole locations.
This procedure can be realized in practical cases,since a discrete algorithm to estimate Laguerre–parameters is available [8]. This is an FFT-based ef-ficient algorithm working on non-uniformly sampledfrequency domain measurements.
The identifiability problem of the poles can bediscussed on the basis of the hyperbolic transform re-alized by the Blaschke–functions that belong to pa-rameterb ∈ D, and represent points in the unit disc.According to Proposition 2 from a specificb ∈ D thepole that is located in the maximal distance with re-spect to the hyperbolic metrics can be reached. Hence
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Figure 4: Identifiability domains belonging to pole-set
a = [0.75 0.8eiπ4 0.8e−i
π4 ].
the unit disc can be divided to regions with respect toparameterb this determines which pole can be reachedfrom one particular region. Aidentifiability mapbe-longing to the pole-set, containing a real pole and a
conjugated complex pair,a = [0.8 0.8eiπ4 0.8e−i
π4 ]
is presented in Fig. 3. All the poles can be asso-ciated with particular regions, which means that allof them are reachable, i.e. by an adequate allocationof b parameters all the poles are identifiable. In Fig.4 the identifiability domains belonging to a slightly
modified pole-seta = [0.75 0.8eiπ4 0.8e−i
π4 ], where
the real pole has been displaced to0.75 can be seen.However the structure of the domains has dramaticallybeen changed: there is no domain belonging to poleNo. 1, i.e. this pole is unidentifiable.
The poles that are unidentifiable in the contextof the current discussion can be handled by the ad-ditional use of rational orthogonal bases, however thistopic spreads beyond the subject of the current paper.
The method is illustrated by an example: afunction F ∈ H2(D) possessing 3 poles, onereal pole and a conjugated complex pair,a =
[0.8 0.8eiπ4 0.8e−i
π4 ]. The parameterb belonging to
the Laguerre–representation selected to beb = 0.7eiπ2
that falls in the identifiability region of the third pole,
a3 = 0.8e−iπ4 , as it can be verified in Fig. 3. Fig. 5
presents the absolute value of the estimated Laguerre–coefficientsln, as well as the absolute value and thephase of the estimated convergence rateqn. The con-vergence ofqn can be verified both on the latter twodiagrams, and in Fig. 6, where the locations ofqn
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0 50 100 150 200 250−2
−1
0
1
2
3
0 50 100 150 200 2500.5
1
1.5abs(Q) = 0.955329 / err = 7.99161e−005
0 50 100 150 200 2500.1
0.2
0.3
0.4
0.5
0.6arg(Q)=0.454180
Figure 5: Identifying polea3: |ln|, |qn|, andarg |qn|.
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1
−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
error=0.000352892
b →
a1
a2
a3
Figure 6: Identifying polea3.
on the complex plane have been presented by whitepoints. The sequenceqn converges to the polea3 byforming a spiral in the complex plane. The error be-tween the real and the estimated pole locations falls inthe magnitude of10−4 in root mean square sense.
6 ConclusionThis paper discussed the problem of identifying polesof systems with unknown structure on the basis ofdiscrete-time spectral signal representations generatedon the basis orthogonal Laguerre systems. After an-alyzing the convergence of the Laguerre representa-tion coefficients by using the concept of a hyperbolicmetrics generated by the Blaschke functions associ-ated with the Laguerre system, the conceptual frame-work has been presented to solve the task of finding
the poles of the system. An algorithm has been pro-posed to find poles with multiplicity one, and criteriaare given to find all of them.
Acknowledgements: This work was supported bythe European framework project ADDSAFE (Grantagreement no.: FP7-233815), and the Control Engi-neering Research Group of HAS at Budapest Univer-sity of Technology and Economics, which are grate-fully acknowledged by the authors. The EuropeanUnion and the European Social Fund also providedfinancial support to the project under the grant agree-ment no. TAMOP-4.2.1./B-09/1/KMR-2010-0003.
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