INTRODUCTION
The Van der Pol oscillator was introduced in 1927 by Van der Pol and
Van der Mark (van der Pol and van der Mark, 1927). They succeeded to model
heartbeat by this oscillator. The modeling of cardiac pacemakers also
has been done with a little changes in the Van der Pol oscillator (Grudzinsky
and Zebrowski, 2004; Sato et al., 1994; Brando et al., 1998).
This oscillator has had many other applications in physics and biological
science. For instance, Fitzhugh (1961) and Nagumo et al. (1962)
extended it and introduced a model of neuron potential in biology.
Cardiac pacemakers modeling is one of the applications of Van der Pol
oscillator. There are two major coupled pacemakers in cardia that generate
and transfer the impulses which are needed for beating. The SinoAtrial
(SA) node is impulse generator and the AtrioVentricullar (AV) node is
a follower pacemaker that transfers impulses to ventricle. In the absence
of SA node impulses, the AV node initiates impulse generation. In the
normal mode, all impulses are generated in SA node and transferred to
ventricle with a small delay via AV node. When the coupling intensity
decreases, some impulses do not reach the ventricle and different types
of blocking arrhythmias occur such as wenkebach, mobitz and others (Guyton
and Hall, 2005). To prevent such arrhythmias, we propose a method to achieve
synchronization.
Knowing all the state variables of the system is necessary in the synchronization
method. Direct measurement of all states of system is impossible in practice,
or it is too cost consuming to use or even the states might be virtual.
Moreover, in some situations, the state estimation leads to better result
in comparison with direct measurement because of measurement noise interferences.
In linear systems, the Kalman filter has a good performance in the presence
of noise for state estimation (Kalman, 1960). The Kalman filter was extended
to nonlinear systems by Julier and Uhlmann (1997). In this method, first,
the Jacobean matrix of system is calculated then the Kalman filter is
used similar to linear system. Another method which has general usage
is State Dependent Ricatti Equation method which introduced by Pearson
(1962). Unfortunately, the two prior methods have stability issues.
There is not a general observer for nonlinear systems with guaranteed
stability. Most of nonlinear observers are designed for a certain class
of nonlinear systems or for a certain system. Hua et al. (2004)
introduced an adaptive observer for a class of nonlinear systems. In that
class, the system is divided into two parts: linear and nonlinear. The
linear part must be Strictly Positive Real (SPR) and the nonlinear part
must be globally Lipschitz. Many systems cannot satisfy these two conditions
such as van der Pol system. The major advantage of the Hua`s method is
its independence from Lipschitz constant which many observers depend on.
Another observer is introduced for a class of nonlinear systems by Liao
and Huang (1999) in which the nonlinear part should be globally Lipschitz
and the linear part should be observable. Two major drawbacks of this
method are its dependence on Lipschitz constant and the fact that the
output must be a scalar. A sliding mode state and parameter estimator
is presented by FloretPonet and LamnabhiLagarrigue (2001). For using
this method, the nonlinear part must be Lipschitz and the system must
be stable around the fixed point. The two aforementioned conditions are
very restricting because most of the stable oscillators are unstable around
the fixed point such as the van der Pol oscillator. The sliding mode is
used in Zhang et al. (1999), too in which the system structure
and parameters are unknown and only the output is known. It can estimate
all states of the system but the major disadvantages are chattering problem
and gain tuning which is difficult to achieve.
In this study, an observer is introduced for van der Pol oscillator and
its convergence is proved. In addition, the gain and phase margins are
obtained for a case study. The observer is extended to two van der Pol
oscillators with unidirectional and bidirectional couplings and its convergence
is proven in two ways. Finally, the introduced observer is used to design
a synchronization strategy for two nonidentical van der Pol oscillators
and it is proven that the synchronization error tends to zero exponentially.
DESIGN OF OBSERVER
The procedure of designing the observer is divided in three parts. At
first, an observer is designed for an insulated van der Pol oscillator.
Then, it is extended to two unidirectional coupled oscillators. Finally,
an observer is proposed for two bidirectional coupled oscillators.
Observer for an isolated van der pol: Consider the Van der Pol
dynamical equation as:
where, μ and ω_{1} are the nonlinear damping coefficient
and intrinsic frequency in the lack of nonlinear term, respectively. Suppose
the output is:
The dynamical systems (1) and (2) in the state space representation are:
where, X is the state vector. The goal of this research is to design
an observer that estimates all states of the system from the output.
The observer structure is proposed as:
where, is
the observer state vector and f and g are appropriate functions. Note,
in the observer equations, x_{1} and
are arranged particularly to guarantee the stability. We prove the observer
convergence in Theorem 1.
Theorem 1: There exist functions f and g such that the trajectories
of the observer (4) track those of the oscillator (3).
Proof: Define the observer errors as:
where, E = [e_{1}e_{2}]^{T} is the error state
vector. The error dynamical equations are:
The f and g functions are proposed as:
where, a is a positive scalar as observer gain. By substituting f and
g in (6),
where, x_{1}(t) is periodic and bounded. Equation set 8 represents
a linear time varying system. The block diagram of (8) can be shown in
Fig. 1 that the time varying and time invariant parts
are secluded from each other.
where transfer function G is:

Fig. 1: 
Block diagram of observer error dynamical system 
The approximate equivalent of time varying part is obtained using the
Describing Function (DF) method (Slotine and Li, 1991). Suppose
The approximate equivalent is:
Figure 2 shows the typical Nyquist diagram of G(jω)
and
According to the Nyquist criterion and Since G(p) is stable, if the
Nyquist diagram of G(jω) intersects the point
the system may have a limit cycle. If the diagram encircles the point,
the system will be unstable. It can easily be seen that for all bounded
X and φ in
space, the Nyquist diagram will be in the right hand of the critical point.
Therefore, the error dynamical system is stable.
It can be interpreted from Fig. 2 that the gain margin
is infinite. The phase margin is calculated as follows:
Finding a closed form for the phase margin for all states is complicated.
A numerical example is demonstrated in Fig. 3 in which
the parameters are μ = 10, ω_{1} = 9 and the phasemargin
is plotted versus the observer gain a.

Fig. 2: 
Nyquist diagram of a typical transfer function 

Fig. 3: 
Phase margin curve Vs. observer gain a 
It can be shown from Fig. 3 that the minimum phase
margin is 74°. The numerical simulation repeated for 0.1<ωSUB>1<15 (not shown here). The results show that
the phase margin is always greater than 65.5°. According to gain margin
and phase margin, it can be concluded that the observer is robust against
disturbances.
Remark 1: The reason why the frequencies of x and e are equal
is explained in Appendix A.
Remark 2: The parameter a controls the observer`s convergence
rate at the cost of noise sensitivity. A tradeoff must be made when selecting
its value.
Example 1 shown the observer performance for a unique Van der Pol system.
Observer for two unidirectional coupled van der Pol oscillators:
The two unidirectional coupled van der Pol equations in state space representation
are:
where, μ_{1} and μ_{2} are positive scalar
damping coefficients, ω_{1} and ω_{2} are intrinsic
frequencies of two oscillators and a is the positive coupling coefficient,
X = [x_{1 }x_{2} x_{3} x_{4}]^{T}
is the system state vector and y = [x_{1} x_{3}]^{T}
is the output vector. The observer system is proposed as follows:
where,
is the observer state vector and f_{1,2} and g_{1,2} are
appropriate functions. In Theorem 2, the convergence of observer (14)
is proven.
Theorem 2: There exist f_{1,2} and g_{1,2} functions
such that the error between trajectories of (13 and 14) tend to zero.
Proof: Define the observer errors as:
The error dynamical equations are:
Define f_{1,2} and g_{1,2} as follows:
where, a_{1} and a_{2} are positive scalar values called
observer gains. According to f_{1} and g_{1} definitions
in (18) and substitution in (16) and using Theorem 1, e_{1} and
e_{2} tend to zero based on Theorem 1. By substituting f_{2}
and g_{2} in (17), the achieved equations are similar to (8) except
αe_{1} term which enters as a disturbance. Since this system
is robust against disturbance as discussed earlier and because the disturbance
tends to zero, therefore e_{3} and e_{4} tend to zero
and (17) is stable.
Alternative proof: Define g_{2} in (17) as follows:
By substituting g_{2} in (17) and rewriting (16) and (17), the
error dynamical system is:
The Eq. 20 show that the system is composed of two
isolated subsystems that are stable as discussed in Theorem 1. Therefore
(20) is stable and the error tends to zero.
Example 2 shows the observer performance for two unidirectionalcoupled
Van der Pol systems.
Observer for two bidirectional coupled oscillators: The dynamical
equations of bidirectional coupled oscillators in state space is:
where, α_{1} and α_{2} are the positive scalar
coupling coefficients. Define observer equations as follow:
Theorem 3: For the observer definition (22), there exist f_{1,2
}and g_{1,2 }functions such that the error variables (15)
tend to zero.
Proof: The functions are proposed as:
By substituting the functions in (22), the error dynamical equations
are obtained as:
By looking carefully to (24), it is concluded that the error dynamical
system is composed of two isolated subsystem similar to (8) and it was
proved in Theorem 1 that such subsystems are stable. Since these subsystems
are isolated, therefore the whole system (24) is stable and error variables
tend to zero.
Example 3 shown the observer performance for two bidirectionalcoupled
van der Pol systems.
Notice: An alternative proof for the stability of (24) is given in Appendix
B.
DESIGN OF A SYNCHRONIZATION METHOD USING PROPOSED OBSERVER
As it is stated in Introduction, cardiac pacemakers modeling is one of
the usages of van der Pol equation. In normal mode, the two major coupled
pacemakers, SA and AV nodes are synchrony. When coupling intensity decreases,
the AV node could not follow the SA node and various blocking arrhythmias
arise such as Wenkebach, Mobitz and others. our goal is to synchronize
two coupled pacemakers using feedback linearization method. It is assumed
that only the SA and AV nodes action potential signals are available.
Therefore the other states must be estimated by an observer. We use our
proposed observer to do this task.
Since the effect of AV node on SA node is very little, it can be ignored
in synchronization problem. Therefore the unidirectional coupling is considered.
The equations are like (13) except:
where, u(x_{1}, x_{2}, x_{3}, x_{4})
is the synchronizing signal. Define the synchronization error as,
The synchronization error dynamic is:
The control signal u is obtained from feedback linearization as follow:
where, k is a scalar number and used for error dynamic stabilization.
By rewriting the error dynamical equation in matrix form:
The necessary and sufficient condition for stability is the satisfying
the Hurwitz condition by A. The characteristic equation is:
where, λ is the eigenvalue of matrix A. All eigen values are negative
if:
The control signal in (28) is dependent on x_{i}, (i = 1,2,3,4).
There is access only to x_{1} and x_{3}. To overcome this
problem, the observer (14) with defining g as (19) is considered by revising
it as follows:
By substituting
instead of u(x) in (25) and defining error as (15), the error dynamical
equation becomes similar to (20) that its stability was proven in Theorem
2. Therefore, the main system and observer trajectories converge together
and
can synchronize two pacemakers.
Example 4 shown the observer and synchronizer performances for two unidirectionalcoupled
Van der Pol systems.
Simulation results
Example 1: Suppose 3 with ω_{1} = 9, μ = 10
parameters. Figure 4 shows performance of the observer
for a = 10. The initial conditions are x_{0} = [2 10]^{T}
and
for the system and observer, respectively.
Figure 4 shown that the error tends to zero rapidly.
The convergence rate increases by increasing the observer gain a but the
undershoot and overshoot increase relatively.

Fig. 4: 
Proposed observer performance for a unique van der Pol 

Fig. 5: 
Observer performance for two unidirectional coupled
oscillators 
In addition, in the presence
of noise, the gain cannot be selected too large.
Example 2: Consider 13 with parameters ω_{1} = 9,
μ_{1} = 10, ω_{2} = 6, μ_{2} =
15, a = 80. The observer system 14 performance with definition of 19 and
a_{1} = a_{2}=10 is shown in Fig. 5.
The initial conditions are x_{0} = [2 10 2 10]^{T} and
It is observed from Fig. 5 that the error goes to zero
before 0.2 s. by comparing error peaks of two Fig. 4
and 5, it is concluded that the error peaks of those
are equal. The reason is that when (19) is used, the coupled system is
divided in two isolated subsystems like (8). Therefore it is expected
that their error peaks are equal.
Example 3: Consider (21) and (22) with parameters a_{1}
= a_{2} = 10, a_{1} = 30, a_{2} = 80, ω_{1}
= 9, μ_{1} = 10 and ω_{2} = 6, μ_{2}
= 15. The initial conditions are selected as x_{0} = [2 10 2 10]^{T}
and .
The observer error is shown in Fig. 6.
By increasing observer gain a_{1} and a_{2}, the speed
of convergence increases while the undershoot/overshoot increases.

Fig. 6: 
Observer performance for two bidirectional coupled oscillators 

Fig. 7: 
Synchronization and Observer performances. (a) error
between x _{1} and x _{3} state variables before and
after synchronizer activation (synchronization error), (b) x _{3}
and
zstate variables and the error between them (the observer error) 
Example 4: Assume parameters of two pacemakers as μ_{1}
= 10, ω_{1} = 9, μ_{1} = 15, ω_{2}
= 6, α = 20, observer gains as a_{1} = a_{2} = 10
and synchronization gain as k = μ_{1}+10. The parameters
are considered such that the impulse generation rate of SA node is 60
min^{1} and AV node is 40 min^{1}. The control signal
is applied at t = 2 sec. Figure 7a shows the synchronization
error and the control signal and Fig. 7b illustrates
the observer performance.
As the synchronization gain k increases, the convergence rate of error
increases too but the amplitude of the control signal increases.
CONCLUSION
In this research, we introduced a robust observer for van der Pol system.
We obtained the gain and phase margins for a range of parameters. The
results show that the gain margin is always infinity and the phase margin
is greater than 65°. The magnitudes of these criteria confirm the
observer robustness. We used describing function method to prove the introduced
observer stability. We extended the observer for two unidirectional and
bidirectional coupled van der Pol oscillators. To prove the stability
of new observer, we used the robustness properties. Since the coupled
Van der Pol oscillators are used as cardiac pacemakers` model, for illustrating
an application of the introduced observer, we designed a synchronization
method for two unidirectional coupled pacemakers based on introduced observer.
We used feedback linearization method to design synchronization system
and demonstrated analytically that the synchronization error tends to
zero exponentially.
APPENDIX
A: Consider the observer error dynamical equation for a unique
van der Pol oscillator as:
Assume the first harmonic frequencies of the variables of x and are
not equal. Therefore the frequencies of x and e are not equal:
By substituting (A2) in (A1) and considering first harmonic:
According to orthogonally of sinus and cosines functions and ω ≠
ω^{'} assumption, we must have.
And this is contradiction. Suppose ω = ώ:
It means that in steady state, in addition to frequency, the amplitude
of x and
is equal that is another confirmation to Theorem 1.
Notice: The (A5) cannot be achieved form (A3) directly. For that
case, it should be done from the (A1)
B: We rewrite (21) and (22) as follow (Liao and Huang, 1999):
where, L is observer gain vector and:
Since (A, C) is observable, there exist vector L such that (ALC) is
Hurwitz. According to (B1), the error dynamical equation is:
In addition, f(.,y) is globally Lipschitz, because:
where
x_{1} and x_{3 }are the van der Pol oscillator state variables.
Since van der Pol oscillator state variables are bounded therefore γ
exists.
Suppose as
error initial condition. The solution of (B3) is:
Since (ALC) is Hurwitz, therefore there exist α > 0, m >
1 such that:
And for e(t):
By multiplying (B7) to e^{αt} and using BellmanGrownwall
Lemma:
Therefore, if the condition α > mγ+δ,δ > 0
holds, the error dynamical system is exponentially stable. By choosing
a proper vector L, the condition holds for van der Pol oscillator.