On Distributed Order Lead-Lag Compensator* - mechatronics - Utah ...

Proceedings of FDA’10. The 4th IFAC Workshop Fractional Differentiation and its Applications.
Badajoz, Spain, October 18-20, 2010 (Eds: I. Podlubny, B. M. Vinagre Jara, YQ. Chen,
V. Feliu Batlle, I. Tejado Balsera). ISBN 9788055304878.
On Distributed Order Lead-Lag Compensator ⋆
Yan Li ∗,∗∗∗ Hu Sheng ∗∗,∗∗∗ YangQuan Chen ∗∗∗
∗ School of Control Science and Engineering, Shandong University, Jinan,
Shandong 250061, P. R. China. (e-mail: liyan cse@sdu.edu.cn)
∗∗ Department of Electronic Engineering, Dalian University of Technology,
Dalian 116024, P. R. China. (e-mail: shenghu 01@163.com)
∗∗∗ Center for Self-Organizing and Intelligent Systems (CSOIS), Electrical
and Computer Engineering Department, Utah State University, Logan, UT
84322, USA. (e-mail: yangquan.chen@usu.edu)
Abstract: In this paper, we derive the impulse response of the distributed order lead-lag compensator.
Based on the derived analytical impulse response, we present how to compute the distributed order leadlag
compensator in Matlab. The analytical method discussed in this paper is compared with the numerical
inverse Laplace transform (NILT) method which shows two advantages of the analytical one. The illustrated
figures in time domain are provided as proof of concept.
Keywords: Distributed order, Lead-lag compensator, Controller design, Fractional Calculus.
1. INTRODUCTION
Fractional calculus is a mathematical discipline which deals with
derivatives and integrals of arbitrary real or complex orders Oldham
et al. (1974); Miller et al. (1993); Podlubny (1999); Kilbas
et al. (2006). It was proposed more than 300 years ago and the
theory was developed mainly in the 19th. Several books Oldham
et al. (1974); Miller et al. (1993); Podlubny (1999); Kilbas et al.
(2006) provide a good source of references on fractional calculus.
It has been shown that there are a growing number of physical
systems whose behavior can be compactly described using
fractional-order system (or systems containing fractional derivatives
and integrals) theory Hartley et al. (1999, 2003). Moreover,
fractional calculus has also been applied to almost every direction
of control theory and its applications Podlubny (1999); Kilbas
et al. (2006); Oustaloup (1991); Podlubny (1999); Sabatier et al.
(2007).
The idea of distributed order equation was first proposed by
M. Caputo in 1969 Caputo (1969) and partly solved by him
in 1995 Caputo (1995). Those distributed order equations were
introduced in the constitutive equations of dielectric media Caputo
(1995) and in the diffusion equation Bagley et al. (2000).
In Lorenzo et al. (2002), the authors studied the rheological
properties of composite materials. The distributed order fractional
kinetics was discussed in Sokolov et al. (2004). In Umarov
et al. (2006), the multi-dimensional random walk models were
governed by distributed fractional order differential equations.
The ultraslow and lateral diffusion processes were discussed
in Kochubei (2008). For the theories of distributed order equations,
we cite Hartley et al. (1999, 2003); Caputo (1995); Bagley
et al. (2000); Lorenzo et al. (2002); Sokolov et al. (2004);
Umarov et al. (2006); Kochubei (2008); Lorenzo et al. (1998);
Caputo (2001); Tsao (1987); Bohannan (2000); Connolly (2004);
Atanackovic et al. (2005); Mainardi et al. (2007a,b); Srokowski
(2008); Sun et al. (2009, 2010); Chen et al. (2009); Diethelm
⋆ This work is partially supported by the Projects of National 863 Program(Grant
No. 2009AA04Z220 and 2006AA040206) and the Shandong University Research
Startup Fund to Yan Li.
et al. (2009); Atanackovic et al. (2009a,b,c). It can be proved
that both integer and fractional order systems are special cases
of distributed order systems Lorenzo et al. (2002). Particularly
when the complexity, networks, nonhomogeneous, multi-scale
and multi-spectral are considered, the distributed order operator
becomes a more precise tool to describe the above phenomena
Hartley et al. (1999, 2003); Caputo (1995); Bagley et al.
(2000); Lorenzo et al. (2002); Sokolov et al. (2004); Umarov et al.
(2006); Kochubei (2008); Lorenzo et al. (1998); Caputo (2001);
Tsao (1987); Bohannan (2000); Connolly (2004); Atanackovic
et al. (2005); Mainardi et al. (2007a,b); Srokowski (2008); Sun
et al. (2009, 2010); Chen et al. (2009); Diethelm et al. (2009);
Atanackovic et al. (2009a,b,c); Adams et al. (2008); Atanackovic
et al. (2007); Mainardi et al. (2007). Therefore, motivated by
the previous references Goodwin et al. (2000); Nise (2004) and
the applications of the fractional distributed operators to control,
filtering and signal processing, a distributed order lead-lag
compensator is derived step by step in the following. Firstly, the
classical integer-order lead-lag compensator can be rewritten as:
s + µ
s + λ =
∞
−∞
δ(α − 1)
α s + µ
dα,
s + λ
where λ = µ are arbitrary positive real constants, δ(·) denotes
the Dirac-Delta function and
s+µ
s+λ
α
is a fractional-order lead-
lag compensator with order α ∈ R. Moreover, the summation
of a series of fractional order integrators/differentiators can be
expressed as:
k
s + µ
s + λ
αk
=
∞
−∞
s α + µ
δ(α − αk) dα,
s + λ
k
where k can belong to any countable or non countable set. Now,
it is straightforward to replace δ(α − αk) by a weighted
kernel w(α). It follows that the right side of the above equation
becomes
∞ α s + µ
w(α) dα,
s + λ
Page 1 of 5
−∞
k
Article no. FDA10-137

where w(α) is independent of time and the above equation
defines a distributed order lead-lag compensator. Particularly,
when w(α) is a piecewise function,
∞ α s + µ
w(α) dα =
s + λ
−∞
l
bl
w(αl)
al
α s + µ
dα,
s + λ
where l belongs to a countable set, al, bl are real numbers, αl ∈
(al, bl) and w(α) is a constant on α ∈ (al, bl). Based on the
above discussions, without loss of generality, we focus on the
discussions of the uniform distributed order lead-lag compensator
where 0 < a < b ≤ 1.
b
a
α s + µ
dα,
s + λ
In this paper, we first focus on the inverse Laplace transform of
the uniformly distributed order lead-lag compensator by cutting
the complex plane and computing the complex integrals. The
derived results can be easily computed in Matlab and applied
to obtain the asymptotic properties of the continuous impulse
responses. Moreover, the advantages of the derived results are
also discussed. Lastly, several figures are provided as proof of
concepts.
2. MATHEMATICAL PRELIMINARIES
2.1 Laplace transform
The Laplace transform of a function f(t), defined for all real
numbers t ≥ 0, is the function F(s), defined by:
F(s) = L{f(t)} =
∞
0
e −st f(t)dt. (1)
The inverse Laplace transform is given by the following complex
integral:
f(t) = L −1 {F(s)} = 1
2πi
σ+i∞
σ−i∞
e st F(s)ds, (2)
where σ is a real number so that the contour path of integration
is in the region of convergence of F(s) normally requiring σ >
Re{sk} for every singularity sk of F(s) and i 2 = −1 Davies
(2002).
2.2 Fractional Calculus
Fractional calculus plays an important role in modern science
Podlubny (1999); Sabatier et al. (2007); Xu et al. (2006);
Chen et al. (2002); Podlubny (1999). The uniform formula of
the fractional integral (the Riemann-Liouville fractional integral)
with α ∈ (0, 1) is defined as
aD −α
t f(t) = 1
Γ(α)
t
a
f(τ)
dτ, (3)
(t − τ) 1−α
where f(t) is an arbitrary integrable function, aD −α
t is the
fractional integral of order α on [a, t], and Γ(·) denotes the
Gamma function. The Laplace transform of 0D −α
t f(t) is
L 0D −α
t f(t) = s −α F(s),
where F(s) = L{f(s)}. Moreover, for an arbitrary real number
p, the Riemann-Liouville and Caputo fractional derivatives are
defined respectively as
aD p
t f(t) = d[p]+1
dt [p]+1 [ aD −([p]−p+1)
t f(t)] (4)
and C a D p
t f(t) = aD −([p]−p+1)
t
f(t)], (5)
dt [p]+1
where [p] stands for the integer part of p, D and CD denote
the Riemann-Liouville and Caputo fractional derivatives, respectively.
Lastly, the geometric and physical interpretation of
fractional-order integration and differentiation was suggested in
Podlubny (2002) and in Podlubny et al. (2006). It is shown that,
when the Caputo derivatives are used, the interpretation of initial
values is the same as in the classical integer order case.
[ d[p]+1
3. IMPULSE RESPONSE OF THE UNIFORM
DISTRIBUTED ORDER LEAD-LAG COMPENSATOR
In this Section, the inverse Laplace transform of b
a
a
α s+µ
s+λ dα
is derived which can be expressed as a finite integral. It follows
from the shifting property of Laplace transform that
L −1
b
α −µt −1 s + µ e L {g1(s)} , (λ > µ),
dα =
s + λ e −λt L −1 {g2(s)} , (λ < µ),
where 0 < a < b ≤ 1, γ = |λ − µ| > 0,
and
g1(s) =
g2(s) =
b
a
b
a
s
s + γ
s + γ
s
α s ( s+γ
dα = )b − ( s
ln( s
s+γ )
s+γ )a
α s+γ
( s dα = )b − ( s+γ
ln( s+γ
s )
s )a
It follows from 0 < a < b ≤ 1 and the singularity of the
logarithmic function that there are three branch points (s =
{0, −γ, ∞}) of g1(s) and g2(s). Therefore, in order to find the
single valued analytical domain, we need to cut the complex
plane by connecting the branch points. When λ > µ, the inverse
Laplace transform of g1(s) is equivalent to the following path
integral.
L −1 {g1(s)} = 1
2πi
= 1
2πi
σ+i∞
σ−i∞
H
s (
st s+γ
e )b − ( s
ln( s
s+γ )
s (
st s+γ
e )b − ( s
ln( s
s+γ )
s+γ )a
s+γ )a
where σ > 0 and the complex integral path H starts from −∞
along the lower side of real axis, goes around the lower half
circular of |s + γ| = ǫ → 0, then goes to the origin along the
lower side of the negative real axis, encircles the circular disc
|s| = ǫ → 0 in the positive sense, then goes to s = −γ along
the upper side of real axis, goes around the upper half circular of
|s + γ| = ǫ → 0, and ends at −∞ along the upper side of the
negative real axis. It follows that
L −1 {g1(s)} = 1
s (
st s+γ
e
2πi H
)b − ( s
s+γ )a
ln( s
s+γ )
ds.
It then follows from the residues of estg1(s) equal zero at s = 0
and s = −γ that
L −1 {g1(s)} = 1
s (
st s+γ
+ e
2πi lo up
)b − ( s
s+γ )a
ln( s
s+γ )
ds,
where “lo” denotes the straight line starts at −∞ and ends at the
origin along the lower side of the negative real axis, and “up”
Page 2 of 5
ds,
,
.
ds

denotes the straight line starts at the origin and ends at −∞ along
the upper side of the negative real axis.
It follows that
and
s (
st s+γ
e
lo
)b − ( s
s+γ )a
ln( s
s+γ )
ds
0
= e
γ
−xt xbe−ibπ (γ − x) −b − xae−iaπ (γ − x) −a
ln( xe−iπ
γ−x )
d(−x)
=
+
γ
∞
γ
=
0
∞
+
e −xt xb (x − γ) −b − xa (x − γ) −a
ln( x
γ−x )
d(−x)
e −xtxbe −ibπ (γ − x) −b − xae−iaπ (γ − x) −a
ln( x
dx
) − iπ
γ
up
γ
γ−x
e −xt xb (x − γ) −b − xa (x − γ) −a
ln( x
γ−x )
dx,
s (
st s+γ
e )b − ( s
ln( s
s+γ )
0
∞
+
= −
−
Therefore,
s+γ )a
ds
e −xtxbeibπ (γ − x) −b − xaeiaπ (γ − x) −a
ln( xeiπ
γ−x )
d(−x)
γ
γ
0
∞
γ
e −xt xb (x − γ) −b − xa (x − γ) −a
ln( x
γ−x )
d(−x)
e −xtxbeibπ (γ − x) −b − xaeiaπ (γ − x) −a
ln( x
dx
) + iπ
γ−x
e −xt xb (x − γ) −b − xa (x − γ) −a
ln( x
γ−x )
dx.
where Im{·} denotes the imaginary part of ·.
Moreover, by using the same procedure of the above derivation,
we have
g 1 (t)
0
−0.05
−0.1
−0.15
−0.2
−0.25
−0.3
−0.35
−0.4
−0.45
NILT
Analytical Result
−0.5
0 1 2 3 4 5
Time t
6 7 8 9 10
Fig. 1. The plots of g1(t) using (6) and NILT.
L −1 {g2(s)} = 1
s+γ
st ( s + e
2πi lo up
)b − ( s+γ
ln( s+γ
s )
γ
e
=
−xt
+ π2
0
π ln 2 γ−x
x
a
s )a
γ b
− x
γ − x
·
−π cos(bπ) + sin(bπ)ln
x
x
a
γ − x
γ − x
−
−π cos(aπ) + sin(aπ)ln dx.
x
x
(7)
Theorem 1. The inverse Laplace transform of
b
α
s+µ
a s+λ dα can
be expressed as:
L −1
b
α −µt −1 s + µ e L {g1(s)} , (λ > µ),
dα =
s + λ e −λt L −1 {g2(s)} , (λ < µ),
where L −1 {g1(s)} and L −1 {g2(s)} are shown in equations (6)
and (7), respectively.
4. IMPLEMENTATION OF THE DISTRIBUTED ORDER
LEAD-LAG COMPENSATOR
In this Section, two distributed order lead-lag compensators are
illustrated as the proof of concept. The analytical results derived
in this paper are compared with the numerical inverse Laplace
transform (NILT) algorithm Brancik (1999, 2001), where the
“quadgk” function in Matlab is applied to compute the finite
integral.
L −1 {g1(s)} = 1
s (
st s+γ
+ e
2πi lo up
)b − ( s
s+γ )a
ln( s
s+γ )
ds
γ
e
=
0
−xt
π Im
xbe−ibπ (γ − x) −b − xae−iaπ (γ − x) −a
ln( x
dx
γ−x ) − iπ
γ
e
=
0
−xt
π ln 2
x
γ−x + π2 Let
0.9 α s
g1(s) =
dα,
0.1 s + 1
we have
g1(t) = L
b
x
x
· π cos(bπ) − sin(bπ)ln
γ − x
γ − x
a
x
x
− π cos(aπ) − sin(aπ)ln dx,
γ − x
γ − x
(6)
−1 {g1(s)}
1
e
=
0
−xt
π ln 2
x
1−x + π2
0.9
x
x
·
π cos(0.9π) − sin(0.9π)ln
1 − x
1 − x
0.1
x
x
− π cos(0.1π) − sin(0.1π)ln
1 − x
1 − x
dx.
Page 3 of 5
The plot of g1(t) is shown in Figure 1.
Moreover, let
0.9 α s + 1
g2(s) =
dα,
s
0.1
ds

g 2 (t)
0.5
0.45
0.4
0.35
0.3
0.25
0.2
0.15
0.1
0.05
NILT
Analytical Result
0
0 1 2 3 4 5
Time t
6 7 8 9 10
Fig. 2. The plots of g2(t) using (7) and NILT.
it can be proved that
g2(t) = L −1 {g2(s)}
1
e
=
0
−(1−x)t
π ln 2
x
1−x + π2
0.9
x
x
·
−π cos(0.9π) + sin(0.9π)ln
1 − x
1 − x
0.1
x
x
−
−π cos(0.1π) + sin(0.1π)ln
1 − x
1 − x
dx,
where the commutation law of convolution is applied in the above
equation. The plot of g2(t) is shown in Figure 2.
Remark 2. In the above two cases, the intervals of the integrals
are [0.1, 0.9], which show that the discussed algorithm is applicable
for all 0 ≤ a ≤ b ≤ 1. Therefore, our discussion can be
extended to the whole time domain, i.e. −∞ < a < b < ∞.
Remark 3. It can be seen from the above two figures that the
analytical results derived in this paper can accurately compute
the values of the two distributed order lead-lag compensators.
Moreover, there are two features of the analytical results:
• The convergence speed of the finite integral is getting faster
for large t.
• The impulse response can be computed accurately on the
whole time domain.
5. CONCLUSIONS AND FUTURE WORKS
In this paper, we derived the impulse response of the distributed
order lead-lag compensator. Based on the derived analytical impulse
response, we presented how to compute the distributed
order lead-lag compensator in Matlab. The method discussed
in this paper was compared with the numerical inverse Laplace
transform (NILT) method. It was shown that the convergence
speed of the finite integral is getting faster for large t and the
initial value can be computed accurately. The illustrated figures
in time domain were provided as proof of concept.
Our future works include the discretization of the distributed order
lead-lag compensator and its applications to signal processing
and control.
ACKNOWLEDGEMENTS
The authors would like to thank the chairs and reviewers for their
valuable comments to improve this paper.
Page 4 of 5
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