2008 5th International Conference on Electrical Engineering, Computing Science and Automatic Control (CCE 2008)
1
Modi ed Carleman Linearization and its use in
oscillators
Joaquín Collado, Irving Sánchez *
Automatic Control Dept.
CINVESTAV
Av. IPN 2508
Col. Zacatenco
07360 México, D. F.
MEXICO
Abstract— The standard Carleman Linearization states that
every analytic n-dimensional nonlinear systems is equivalent to
an in nite dimensional linear system. In this paper we truncate
this linearization and introduce some modi cation which reduces
the error in the truncation process. We applied this Modi ed
Carleman Linearization to two van der Pol oscillators with slight
different frequencies. Each one is approximate by a 14th order
linear system; then we coupled this two linear oscillators and look
for a synchronization. We conclude that even the approximation
is very good, is not possible synchronize high order linear
oscillators as the nonlinear oscillators do.
redundancies in its application to the Linearization. Section III
presents the original Carleman Linearization, and Section IV
the modi ed version. The modi ed Carleman Linearization is
applied to the van der Pol oscillator in Section V. In Section
VI two linearized van der Pol oscillators with slightly different
frequency are coupled in a symmetric form and it is shown
that it is impossible to synchronize linear oscillators with
diffusively symmetric coupling.
Keywords - Carleman Linearization, synchronization, van der
Pol oscillators.
II. P RELIMINARIES
In this section we introduced some de nitions and preliminaries required in the sequel.
I. I NTRODUCTION
De nition 1: Given A 2 Rn
Kronecker Product [4] or [21] is
2
a11 B a12 B
6 a21 B a22 B
6
AeB = 6
..
..
..
4
.
.
.
The Carleman Linearization was proposed for autonomous
nonlinear analytic systems almost 80 years ago [6], the linearized system is equivalent to the original nonlinear system;
however the procedure gives us an in nite dimensional system,
to be used is required to truncate at some nite dimensional
point. Bellman proposed, but he did not developed [1], that
this truncation be performed in conjunction with some modi cation. This approximates the deleted terms in the Taylor
series by a linear combination of the terms that we keep in
the truncation. This paper work out this modi cation based on
sampling the limit cycle trajectory of the van der Pol oscillator
and using least squares method to calculate the parameters
introduced in this modi cation. This modi ed Carleman linearization does not guarantee any stability property, then we
reassign the eigenvalues to the j!-axis and these eigenvalues
satisfy an integer relation, see Fig. 1. This last condition is
required in order to have a periodic solution of the approximate
linear system.
an1 B
and B 2 Rs t , their
a1m B
a2m B
..
.
anm B
3
7
7
7 2 Rns
5
mt
Remark 2: The usual notation for the Kronecker Product is
, but we will reserve for a reduced Kronecker Product, as
explained in the sequel.
2
Let the vector x =
xex =
x1
x2
xn
T
, then
4x21 ; x1 x2 ; x1 x3 ; : : : ; x1 xn ; x2 x1 ; x22 ; x2 x3 ; : : : ; x2 xn ;
{z
} |
{z
}
|
n terms
n terms
3
2
x3 x1 ; x3 x2 ; x23 ; : : : ; x3 xn ; : : : ; xn x1 ; xn x2 ; : : : ; x2n 5 2 Rn
{z
}
|
{z
}
|
The organization of the paper is as follows: in section
II are described the Kronecker Product of matrices and a
Reduced Kronecker Product is introduced in order to remove
n terms
n terms
Remark 3: Note that all mixed products xi xj for all i 6= j,
appear duplicated in x e x. Let the operation x x be de ned
as x e x removing all the duplicated terms, i.e.
* E-mail: fjcollado,isanchezg@ctrl.cinvestav.mx
IEEE Catalog Number: CFP08827-CDR
ISBN: 978-1-4244-2499-3
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978-1-4244-2499-3/08/$25.00 ©2008 IEEE
an2 B
m
13
2008 5th International Conference on Electrical Engineering, Computing Science and Automatic Control (CCE 2008)
x
2
6
x = 4x21 ; x1 x2 ; x1 x3 ; : : : ; x1 xn ; x22 ; x2 x3 ; : : : ; x2 xn ;
{z
} |
{z
}
|
n terms
d h (2) i
x
dt
(n 1) terms
3
7
x2 ; x3 x4 ; : : : ; x3 xn ; : : : ; x2n 5 2 Rn(n+1)=2
{z
}
|{z}
|3
(n 2) terms
d
[x x]
dt
N
X1
=
[Ak In + In
2
=
Ak ] x(k+1)
k=1
1 term
Hence, x(2) satis es a differential relation that depends on
x ; : : : ; x(N ) but not on x(1) , however this equation has the
same structure than (3).
In order to avoid redundancy in our linearized model,we
will use through the paper x x, for this reduced Kronecker
Product. This product has been known for a long time, its use
in Control Theory was introduced by Brockett [5]. Qiu and
Davison named bialternate product in [15] and they used it
for Robust Stability.
(2)
Continuing in this form yields a differential relation for x(j)
to degree N of the form
NX
j+1
d h (j) i
=
Aj;k x(k+j
x
dt
1)
(4)
k=1
III. BASIC C ARLEMAN L INEARIZATION
where A1;k = Ak and for j > 1;
Let be a nonlinear autonomous dynamic system described
by the Ordinary Differential Equation (ODE):
Aj;k
+
x
x] +
mk
(2)
, mk ,
Then
x_ =
1
X
x
k
Ak x(k)
{z
times
x 2 Rmk .
}
6
6
x ,6
4
where
dim(x )
Ak
x
x(2)
..
.
x(N )
3
7
7
7
5
Thus we can write a nite dimensional linear differential
equation, which is an approximation of the original ODE (1)
d
x
dt
Ak x(k)
k=1
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In
= M
= n + m2 + ::: + mN
(3)
If we rewrite (1) after truncate the power series to N from
(3)
Then we obtain
In
2
is a power series representation equivalent to the original
nonlinear system (1) and (2), see [6] or [16].
N
X
In
Now we de ne
k=1
x_ =
Ak
1 Kronecker products in each term, and
Remark 4: Note that the index in Ak matrices corresponds
to the degree in x of the Taylor series and mk it is the number
of different elements in x(k) .
We introduce the notation x(k) = x
|
In + In
}
Remark 5: Note that the differential relations (4) do not
dx(j)
depends on x(k) for
constitute an ODE, because each
dt
k j.
n+k 1
,
k
k = 1; 2; :::, and means the Modi ed Kronecker product as
de ned in the section of Preliminaries.
where the matrices Ak 2 Rn
{z
+ In
where there are j
j terms.
We assume that f (x) is analytic in Rn , thus we can write
(1) in Taylor series form [16] as:
x] + A3 [x
In
|
(j 1) times
(1)
x_ = f (x)
f : Rn ! R n
f (x) = A1 x + A2 [x
= Ak
2
6
6
6
= 6
6
4
A11
0
0
..
.
A12
A21
0
..
.
:::
:::
:::
..
.
A1N
A2;N
A3;N
..
.
0
0
:::
AN 1
~
= Ax
14
1
2
3
7
7
7
7x
7
5
(5)
2008 5th International Conference on Electrical Engineering, Computing Science and Automatic Control (CCE 2008)
II) If we want to approximate a particular solution, as the
limit cycle in a van der Pol oscillator, sample this trajectory
equidistantly and again apply least squares to minimize the
same criteria J.
This state equation is called a Truncated Carleman Linearization of the nonlinear-analytic state equation (1)1 .
Remark 6: Truncation of the in nite dimensional vector
x(1) to the M -dimensional vector x , is equivalent to keep
all the terms of degree less or equal to N .
In this way, the equation (7) represents a, better in general,
nite order linearization of degree M of (1).
IV. M ODIFIED C ARLEMAN L INEARIZATION
In (5) the high order terms have been deleted, Bellman [1]
proposed to make an approximation of the deleted terms by
a weighted linear terms that depends on the variables in x ,
this improves in general the linear approximation of (1), thus
we can write the state equation of x as
d
x
dt
2
6
6
where G(x ) = 6
4
V. L INEARIZED VAN DER P OL O SCILLATOR
The Modi ed Carleman linearization can be used for any
non-linear systems whose right hand side of de ning equations
be analytic; we are going to apply the above procedure to the
well known van der Pol Oscillator [13] and [20], the equation
that describes this system is
~ + G(x )
= Ax
(6)
3
g1 (x)
g2 (x) 7
7
7 contains the terms of order
..
5
.
gN (x)
higher than N , and each gi (x) is polynomial of degree
N + 1. Partitions on G(x ) are compatible with partitions in
~
A.
If we make approximations gi (x) by
N
X
(k)
,
i;k x
z + "(z 2
where
We de ne x1 , z, x2 , z and x ,
2 Rmi mk , see [2]; then the nite order linearization of
degree M of (1) becomes:
d
x
dt
A~ + R x
=
(7)
where
6
6
R =6
4
1;1
1;2
2;1
2;2
..
.
N;1
..
.
N;2
:::
:::
..
.
:::
1;N
2;N
..
.
N;N
3
7
7
7;
5
i;k
2R
mi mk
x1
x2
x = A1 x + A2 x(2) + A3 x(3) + A4 x(4)
0
1
A1 =
; A2 = A4 = 0;
!2 "
0 0 0 0
A3 =
0
" 0 0
(8)
Not necessarily all the row blocks i; for i = 1; : : : ; N
need to be different from zero. They may contain up to M 2
parameters. There are two forms to determine this set of
parameters:
T
, then
(10)
Then using (5) the linearization of van der Pol Oscillator
for N = 4 is
I) Choosing a region D
Rn of the original state
space where our system is know to be operating, sample enough number of points in D and minimize J =
N
X
(k)
gi (x)
using least squares [3], [17], [18] and
i;k x
d
x
dt
k=1
[19] to calculate the free parameters introduced.
1 Note that the initial conditions cannot be arbitrary, because the conditions
for x(i) ; i 2; depend directly of initial conditions of x, since x(i) , (i 2)
is the Reduced Kronecker Product of x with itself.
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(9)
rewriting (9) in Taylor series form and using the Modi ed
Kronecker Product as de ned in the Preliminaries Section for
x, we have
= Ax
2
1)z + ! 2 z = 0
where " is the non-linear damping parameter (when " =
0 the van der Pol oscillator reduces to an harmonic linear
oscillator, see [20]) and ! it is the frequency in radians.
k=1
i;k
3
2
A1;1
6 0
= 6
4 0
0
~
= Ax
0
A2;1
0
0
the vector state x 2 R14 , and
15
A1;3
0
A3;1
0
3
0
A2;3 7
7x
0 5
A4;1
(11)
2008 5th International Conference on Electrical Engineering, Computing Science and Automatic Control (CCE 2008)
A1;1
A2;1
A2;3
A3;1
A4;1
=
2
= 4
2
0
= 4 0
0
2
6
= 6
4
2
6
6
= 6
6
4
0
1
; A1;3 =
!2 "
3
0
2
0
!2
"
1 5
2
0
2! 2"
0
"
0
0
!2
0
0
0
!2
0
0
0
0
0
2"
3
"
2! 2
0
4
"
2! 2
0
0
0
0
0
"
3
0 0
0 0 5
0 0
3
0
0
2
0 7
7
2"
1 5
3! 2 3"
0
0
3
0
2"
2
3! 2
3"
0
4! 2
0 0
0 0
1
7:9848 1:3921
0:0137
0:0011
0:004 1:9669
4:4836 2:3702
0:2828 0:004
0:0021 0:0056
0:0019 0:0009
=
In this form we obtain (8) for the van der Pol oscillator,
then we have a nite order linearization of the van der Pol
Oscillator, which is an unstable 14th order system, because
some eigenvalues of the matrix A = A~ + R are on the right
hand side of the complex plane.
0
0
0
1
4"
For the linear oscillator to be marginally stable and the have
a periodic solution, the eigenvalues of the state matrix must lie
on the imaginary axis and they should be a multiple of some
fundamental frequency., i. e. let the spectra of A be (A) =
f 1 ; 2 ; : : : ; 14 g then
3
7
7
7
7
5
Re ( i ) = 0 and
b
=
0
0
0
0
0
v
0 g1 (x) g2 (x) g3 (x)
0 g4 (x) g5 (x) g6 (x) g7 (x)
=
=
=
=
=
=
=
= jm! 0
8i = 1; : : : ; 14
(12)
2
=
R14 ;
b=
Kx ;
0
0 ::: 1
K 2 R1
T
14
The b used above is such that the pair (A; b) must be
controllable [7]. Then K must be such that the matrix A =
A bK has its eigenvalues on the vertical axis of the complex
plane and these satisfy multiplicity described in (12), see [7]
and [9]. The effect of this state feedback is exempli ed in Fig.
T
where
g1 (x)
g2 (x)
g3 (x)
g4 (x)
g5 (x)
g6 (x)
g7 (x)
i
To make this we use a control signal such that
d
x = Ax + bv
dt
where
However, as we saw in the previous section, (11) is a
truncated linearization, where the higher order terms have been
deleted, in fact exists a G(x) of the form
G(x)
4
"x41 x2
2"x31 x22
3"x21 x32
"x51 x2
2"x41 x22
3"x31 x32
4"x21 x42
1
Then the equation that describes the real dynamics of x
is
d
x
dt
~ + G(x)
= Ax
Now, we can calculate i;k , for which we use least square
method for each gi (x) above, the following example shows
a particular case of this linearization for the van der Pol
Oscillator.
From the van der Pol Oscillator equation in (9) when ! =
" = 1, we calculate the value of i;k using the least squares
method as described above. As example g1 (x) is approximate
by
g1 (x)
Fig. 1. Eigenvalue assignment in order to satisfy the
constraint in (12).
1x
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2008 5th International Conference on Electrical Engineering, Computing Science and Automatic Control (CCE 2008)
5
Example 7: The simple projection of the eigenvalues to
the imaginary axis does not guarantee that the eigenvalues
be multiple of the lowest one, then we may interpret as a
nonlinear projection as show in gure 1. Hence K must be
such that one of the eigenvalues for the matrix A must be the
corresponding to the natural frequency of (9) on the vertical
axis and the rest, multiples of this one. This fact makes that
the linearized van der Pol systems oscillates. Then
(A)
0
= f ij
= !0 j
i
=
n
0;
n 2 N;
i = 1; :::; 7g
A way to obtain ! 0 when " is large is in [8], otherwise we
can the less precise formula in (9).
Fig. 3. Symmetric coupled linearized van der
Pol oscillators.
In this way the system
d
x = Ax
(13)
dt
is an stable, but not asymptotically stable, nite order
linearization of (9). The Fig. 2 shows the waveforms of the
original and the linearized van der Pol oscillators. Notice
how well approximates the linear system to the van der Pol
oscillator.
Then we add a control signal in (13) for both oscillators
such that.
d
x
dt i
yi
yi0
(14)
= Ai xi + Bui
= Cxi
= xi;1
B 2 R14 1 ; C 2 R1 14 ;
ui =
k(yi yj ); i 6= j;
B =
. .
.
01..010..0100..01000
=
. .
.
01..010..0100..01000
C
Fig. 2. First state of the original van der Pol oscillator
compared with the rst component of the 14th order
linear approximation.
T
ui 2 R
k 2 R;
k>0
Thus
VI. D IFFUSED COUPLED LINEARIZED VAN DER P OL
d
x
dt i
O SCILLATORS
1)z 1 + z1 = 0
z1 (0) = 2; z 1 (0) = 0
z 2 + (z22
1)z 2 + (1:01) z2 = 0
2
z2 (0) = 2; z 2 (0) = 0
kBxj
In this case the both pairs (Ai ; B) i = 1; 2; are controllable
and both pairs (C; Ai ) are observable respectively.
i. e., ! 1 = "1 = "2 = 1 and ! 2 = 1:01; we can use A1
from example above and obtain A2 using the method described
in the previous section.
In this way the behavior of x1 and x2 depends directly
from the value of k, in fact both systems become unstable for
any value of k as show in the gure 4
For coupling the two oscillators, we use following diagram,
see [12], [14] and [11]:
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kBC)xi
these values for C and B correspond to the velocity z in
the original van der Pol oscillator . It is routine verify that the
pairs (Ai ; B) and (C; Ai ), for i = 1; 2; are controllable and
observable respectively.
In (13) we have a stable nite 14th order linearization of
the van der Pol Oscillator for ! = " = 1. Now apply our
procedure for two original van der Pol oscillators with slightly
different frequencies as
z 1 + (z12
= (Ai
17
2008 5th International Conference on Electrical Engineering, Computing Science and Automatic Control (CCE 2008)
Fig. 4 The coupled oscillators cannot synchronize
Fig. 1.
The fact that the amplitude increases is due to during
the state feedback through ui = k(yi yj ) modi es the
eigenvalues of the matrix (Ai kBC) to the left and to the
right side on the complex plane. The Fig. 5 shows the roots
locus of the eigenvalues of the matrix (A1 kBC).
6
Two nonlinear van der Pol oscillators synchronized
VII. C ONCLUSIONS
In the rst part of the paper we propose a Modi ed version
of the Truncated Carleman Linearization method, then we
applied this novel procedure to the van der Pol oscillator and
used to try to synchronize to linearized van der Pol oscillators
with coupling proportional to velocity. We conclude that it is
not possible to synchronize them as they do in the original
systems using the same coupling.
R EFERENCES
[1] Bellman, R. Perturbation Techniques in Mathematics, Physics, and
Engineering. Holt, Rinehart and Winston, Inc., pp. 69-74, 1966.
[2] Bellman, R. & Richardson, J.M., “On some questions arising in the
approximative solution of nonlinear differential equations”, Quart. Appl.
Math, Vol. 20, pp. 333-339, 1963.
[3] A. Björck, "Numerical Methods for Least Squares Problems", SIAM,
Philadelphia, 1996.
[4] Brewer, J. "Kronecker products and matrix calculus in system theory".
IEEE Trans. on Circuits and Systems, Volume CAS-25, Issue 9, pp. 772
- 781, Sep 1978.
[5] Brockett, R. "Lie Algebras and Lie Groups in Control Theory". In
Geometric Methods In Systems Theory. D. Mayne and R. Brockett
(Eds.), Reidel Dordrecht, 1973.
[6] Carleman T. "Application de la Théorie des Équations Intégrales
Linéaires aux Systémes D 'Équations Différentialles Non Linéaires",
Acta. Math., Vol. 59, pp. 63-87, 1932.
[7] Chen, C. T. Linear System Theory and Design, Third Edition, Oxford
University Press, pp. 235-239, 1999.
[8] Dorodnicyn, A. A. "Asymptotic Solution of van der Pol´s Equation",
Engl. transl. in Amer. Math. Soc. Transl., Series 1, 88, pp. 1-24, 1953.
[9] Kailath, T., Linear Systems, Prentice-Hall, Inc., Englewood Cliffs, N. J.
07632, 1980.
[10] H. K. Khalil. "Nonlinear Systems", 3rd. Ed. Upper Saddle
River:Prentice-Hall, USA, 2002.
[11] Kuramoto Y., "Chemical Oscillations,Waves, and Turbulence", New
York: Springer, 1984.
[12] Mimila P. O., "Sincronización de Osciladores", Tesis de maestría,
Automatic Control Dept. Cinvestav-IPN, Sep. 2006.
[13] Nayfeh, A. H. Nonlinear Oscillations, John Wiley & Sons Inc., pp. 3-6,
1979.
[14] Pikovsky, A., M. Rosenblum, and J. Kurths. "Synchronization: A
universal concept in nonlinear sciences". Cambridge University Press,
Cambridge, 2001.
[15] Qiu, L. and E. J. Davison. "The Stability Robustness Determination
of State Space Models with Real Unstructured Uncertainty". Math. of
Contr., Signals and Systems, Vol. 4, pp. 247-267, 1991.
[16] Rugh, W. J. Non Linear System Theory The Volterra/Wiener Approach.
The Johns HopkinsUniversity Press, pp. 103-116, 1981.
Fig. 5. Root Locus of coupled Linearized 14th order van
der Pol oscillators.
This show that the eigenvalues turn on the right side at
the same moment when k is different of zero, thus we can
say it is no possible to synchronize the linear oscillators with
diffusively symmetric coupling. Notice this behavior is not for
the coupling proposed, but for any linear coupling, because
to synchronize linear oscillator is required that eigenvalues
coincide on the imaginary axis, that keeps the multiplicity
property and that these conditions holds for a range of values
of k greater that a threshold [14].
The Fig. 6 shows how two original nonlinear van der Pol
oscillators can be synchronized with diffusively symmetric
coupling, the natural frecuencies are the same that the example
with linear oscillators above
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2008 5th International Conference on Electrical Engineering, Computing Science and Automatic Control (CCE 2008)
[17] Stewart, G. W. Introduction to Matrix Computations, Academic Press,
pp. 217-221, 1973.
[18] Stewart, G. W. Matrix Algorithms: Basic Decompositions, SIAM,
Philadelphia, 1998.
[19] L. N. Trefethen and D. Bau, III, "Numerical Linear Algebra", SIAM,
Philadelphia, 1997.
[20] van der Pol, B. L. . "The nonlinear theory of electric oscillators", Proc
of the IRE, Vol. 22, Num. 9, pp. 1051-1056, September 1934.
[21] Charles F. Van Loan, "The ubiquitous Kronecker product". Journal of
Computational and Applied Mathematics, 123 (2000) 85–100.
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