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 Sino-Atrial (SA) node is impulse generator and the Atrio-Ventricullar (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 Floret-Ponet and Lamnabhi-Lagarrigue (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 non-identical 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:

(1)

where, μ and ω₁ are the nonlinear damping coefficient and intrinsic frequency in the lack of nonlinear term, respectively. Suppose the output is:

(2)

The dynamical systems (1) and (2) in the state space representation are:

(3)

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:

(4)

where, is the observer state vector and f and g are appropriate functions. Note, in the observer equations, x₁ 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:

(5)

where, E = [e₁-e₂]^T is the error state vector. The error dynamical equations are:

(6)

The f and g functions are proposed as:

(7)

where, a is a positive scalar as observer gain. By substituting f and g in (6),

(8)

where, x₁(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:

(9)

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

(10)

The approximate equivalent is:

(11)

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:

(12)

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, ω₁ = 9 and the phase-margin 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 trade-off 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:

(13)

where, μ₁ and μ₂ are positive scalar damping coefficients, ω₁ and ω₂ are intrinsic frequencies of two oscillators and a is the positive coupling coefficient, X = [x₁ x₂ x₃ x₄]^T is the system state vector and y = [x₁ x₃]^T is the output vector. The observer system is proposed as follows:

(14)

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:

(15)

The error dynamical equations are:

(16)

(17)

Define f_1,2 and g_1,2 as follows:

(18)

where, a₁ and a₂ are positive scalar values called observer gains. According to f₁ and g₁ definitions in (18) and substitution in (16) and using Theorem 1, e₁ and e₂ tend to zero based on Theorem 1. By substituting f₂ and g₂ in (17), the achieved equations are similar to (8) except αe₁ term which enters as a disturbance. Since this system is robust against disturbance as discussed earlier and because the disturbance tends to zero, therefore e₃ and e₄ tend to zero and (17) is stable.

Alternative proof: Define g₂ in (17) as follows:

(19)

By substituting g₂ in (17) and rewriting (16) and (17), the error dynamical system is:

(20)

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 unidirectional-coupled Van der Pol systems.

Observer for two bidirectional coupled oscillators: The dynamical equations of bidirectional coupled oscillators in state space is:

(21)

where, α₁ and α₂ are the positive scalar coupling coefficients. Define observer equations as follow:

(22)

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:

(23)

By substituting the functions in (22), the error dynamical equations are obtained as:

(24)

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 bidirectional-coupled 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:

(25)

where, u(x₁, x₂, x₃, x₄) is the synchronizing signal. Define the synchronization error as,

(26)

The synchronization error dynamic is:

(27)

The control signal u is obtained from feedback linearization as follow:

(28)

where, k is a scalar number and used for error dynamic stabilization. By rewriting the error dynamical equation in matrix form:

(29)

The necessary and sufficient condition for stability is the satisfying the Hurwitz condition by A. The characteristic equation is:

(30)

where, λ is the eigenvalue of matrix A. All eigen values are negative if:

(31)

The control signal in (28) is dependent on x_i, (i = 1,2,3,4). There is access only to x₁ and x₃. To overcome this problem, the observer (14) with defining g as (19) is considered by revising it as follows:

(32)

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 unidirectional-coupled Van der Pol systems.

Simulation results
Example 1: Suppose 3 with ω₁ = 9, μ = 10 parameters. Figure 4 shows performance of the observer for a = 10. The initial conditions are x₀ = [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 ω₁ = 9, μ₁ = 10, ω₂ = 6, μ₂ = 15, a = 80. The observer system 14 performance with definition of 19 and a₁ = a₂=10 is shown in Fig. 5. The initial conditions are x₀ = [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 a₁ = a₂ = 10, a₁ = 30, a₂ = 80, ω₁ = 9, μ₁ = 10 and ω₂ = 6, μ₂ = 15. The initial conditions are selected as x₀ = [2 10 2 10]^T and . The observer error is shown in Fig. 6.

By increasing observer gain a₁ and a₂, 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 x1 and x3 state variables before and after synchronizer activation (synchronization error), (b) x3 and zstate variables and the error between them (the observer error)

Example 4: Assume parameters of two pacemakers as μ₁ = 10, ω₁ = 9, μ₁ = 15, ω₂ = 6, α = 20, observer gains as a₁ = a₂ = 10 and synchronization gain as k = μ₁+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.