Stabilization of a chaotic system described by Van der Pole equation

present input-output model (6) as input-state-output model

, ,                                              (7)

where  is state of model (7), , ,  и  are transformation matrices from input-output model (6) to input-state-output model (7). As polynomial  is Hurwitz and model (6) is strictly minimum phase, according to consequence 3 [4], it is possible to determine a symmetric positively determined matrix , satisfying two following matrix equations:

,  ,                                                                                                                             (8)

where , entries of matrix  depend on parameter  and do not depend on parameter .

Consider derivative of function of deviation

,                                                                                              (9)

where .

Let us present a theorem describing conditions for calculation of parameter  and function , ensuring accomplishing of purpose of control.

Theorem. There exist a parameter  and a function  such as all trajectories of system (7), (9) can be localized in any small area by increasing of parameter .

Proof. Consider a Lyapunov function of the form

,                                                                                                                                                     (10)

where

,                                                                                                                                                     (11)

.                                                                                                                                                          (12)

Differentiating (11) with respect to time subject to equation (7) we obtain

,                                                                                                                             (13)

Substituting equations (8) into (13) and considering the following equations

,

,

,

,

for derivative of Lyapunov function (11) we obtain

,                           (14)

where small value .

Differentiating (12) with respect to time subject to equation (7) we receive

                                                                          (15)

where in equation (15)  was substituted by summand

Then, considering inequalities

,

,

,

,

for derivative of function (12) we obtain

                         (16)

for derivative of Lyapunov function (10) we obtain

.                                                     (17)

Let us choose  such a way that the following inequality holds

,                                                                                                                        (18)

where  is a positively determined matrix.

Choose function , so the following inequality holds

,                                                                          (19)

where .

Then, according to restrictions on the nonlinearity for derivative of Lyapunov function (10) we obtain

                                                     (20)

Choosing number  as

,                                                                                                                                                     (21)

we obtain

.                                                                                                                         (22)

From inequality (22) as disturbance  is bounded, it follows that there exists such , that trajectories of system (7), (9) can be localized in any given area, which was to be proved.

Consider results of computer simulation of Van der Pole system. First let us assume that  and there appear stable oscillations in system (1) (system possesses a stable limit cycle). Then, with harmonic disturbance there arise chaotic phenomena in the system. And finally, on the last step of simulation of system (1) with control law (3), (4), we discover accomplishment of purpose of control.

Figures 1 and 2 show results of computer simulation of undisturbed system (1) for  , ,  (Figure 1) and disturbed system (1) for , , (Figure 2)  respectively. Results of computer simulations (Figures 1 and 2) show, that system (1) possesses stable limit cycle in absence of disturbance () and that in presence of harmonic disturbance () there appear chaotic processes in system (1).

 

Fig. 1. Phase portrait and transients of system (1) for , .

 

Fig. 2. Phase portrait and transients of system (1) for , .

To stabilize system (1) let us choose control algorithm (3). Choose polynomial , then

,                                                                                                        (23)

where function  is formed by estimation algorithm (4).

Function  is chosen so that inequality (19) holds, i.e.

.                                                                    (24)

Let us choose parameter , and simulate control system for different values of parameter . Transients of closed loop system for nonzero initial conditions () and values  and  are shown on Figure 3.

 

Fig. 3. Transients of system (1), (23), (24) for  and , .

4. Conclusion

An algorithm of output control of a nonlinear chaotic system, described by Van der Pole equality is proposed. The proposed control law ensures convergence of output trajectory of the nonlinear system to an area, and this area can be reduced by increasing of controller parameter.

5. References

[1] B.R. Andrievskii, A. L. Fradkov, “Control of Chaos: Methods and Applications. I. Methods”, Automation and Remote Control, vol. 64, no. 5, 2003, pp. 673-713.

[2] B.R. Andrievskii, A.L. Fradkov, “Control of Chaos: Methods and Applications. II. Applications”, Automation and Remote Control,  vol. 65, no. 4, 2004, pp. 505-533.

[3] T. Gilbert and R.V. Gammon, “Stable oscillations and devil’s staircase in the Van de Pole oscillator”, International Journal of bifurcation and chaos, vol. 20, no. 1, 2000, pp. 155-164.

[4] A.A. Bobtsov and N.A. Nikolaev, “Fradkov theorem-based design of the control of nonlinear systems with functional and parametric uncertainties”, Automation and remote control, vol. 66, no. 1, 2005, pp. 108–118.