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This paper discusses a novel technique and implementation to perform nonlinear control for two different forced model state oscillators and actuators. The paper starts by discussing the Duffing oscillator which features a second order non-linear differential equation describing complex motion whereas the second model is the Van der Pol oscillator with non-linear damping. A first order actuator is added to both models to expand on the chaotic behavior of the oscillators. In order to control the system without comprising linearization, Lyapunov non-linear control was used. A control Lyapunov function was tailored to the system. This led to improved maneuverability of the controller and the performance of the overall system. The controller was found to be highly efficient in system tracking and had swift response time. Simulations were performed on both the uncontrolled and controlled cases. Both simulation results ultimately confirmed the effectiveness of the proposed controller.

Duffing and Van der Pol Oscillators are typical examples of nonlinear dynamic systems and thus we will use them as a reference to test the implemented controllers. Both oscillators are good examples of periodically forced oscillators with non-linear elasticity. A Duffing oscillator can be represented by the mathematical model shown in Equation (1); on the other hand a Van der Pol Oscillator mathematical model is shown in Equation (2).

A forced or driven oscillator means that a driving function of

In our application, we apply an actuator for the oscillators and this yields the updated mathematical model that will be used in our calculations. Equation (5) represents a driven Duffing oscillator with an actuator. Equation (7) on the other hand, represents a driven Van der Pol Oscillator again with an actuator. Mathematical models of the forced Duffing and Van der Pol systems are shown respectively.

From Equation (5) the state space was deduced to be as shown below:

From Equation (7) the state space was deduced to be as shown below:

where

In a previous work [

There are numerous applications for Duffing and Van der Pol Oscillators. For example, the Duffing oscillator has been used to do chirp signal detection [

There has been some work done on the control of a Duffing Oscillator. Kuo et al. [

The authors in [

Equation (5) represents the forced Duffing oscillator alongside an actuator. Let the error

Let

Substituting

This means that

Rearranging Equation (5) will yield the following:

Now if we suppose that

The system performance was captured at the values below as shown in

The following figures show the results of the presented control system.

α | δ | β | k | ||
---|---|---|---|---|---|

0.0003 | 0.03 | 0.0002 | 1 | 0.7 | 190 |

The same principle and logic is applied again to the Van der Pol analysis. Equation (16) and Equation (17) represent the solution when

Using the same flow for showing the results for the Van der Pol system. Figures 7-12 shows the same series of plots as the Duffing system.

α | β | |||
---|---|---|---|---|

0.2 | 1.2 | 3 | 0.5 | 40 |

The potential and kinetic energies are studied in this section. The same analytical process is used as sections “a” and “b” earlier. The kinetic and the potential energy equations are shown in Equation (18) and (19) prospectively.

Using the same analysis as before the kinetic energy solution for both Duffing and Van der Pol oscillator, respectively, can be expressed as shown in Equations (20) and (21).

(21)

Using similar analogy, the potential energy for a Duffing and Van der Pol oscillator, respectively, can be expressed as shown in Equations (22) and (23).

The dissipation energy for both the Duffing and the Van der Pol oscillators can be expressed as shown in Equation (24).

This will yield dissipation energy solution as shown in Equations (25) and (26) representing the Van der Pol and Duffing respectively.

The system best performance was then recorded at the values below:

Using these solutions the results are shown in Figures 13-15.

α | Σ | b1 | b2 | g |
---|---|---|---|---|

1 | 0.1 | 1.5 | 1 | 1 |

s1 | s2 | g1 | g2 | b |
---|---|---|---|---|

1 | −0.1 | 1 | 1 | 0 |

s1 | a | b1 | b2 | g |
---|---|---|---|---|

0.2 | 1 | 1 | 0.5 | 0 |

a | s2 | s2 | g | b |
---|---|---|---|---|

1 | 0.1 | −0.1 | 1 | 0 |

the energy exchange rate in the unforced Duffing oscillator. This figure shows the kinetic, mechanical, and the potential energies as well as the dissipated power.

In this paper, we presented a novel nonlinear control method that was applied to forced Duffing and Van der Pol oscillators that were experiencing chaotic behavior to a prescribed performance. The oscillators had an actuator applied to them. We also presented the energy exchange in forced Duffing and Van der Pol oscillators. The paper illustrated the usefulness of the presented method in the unstable areas. The presented controllers achieved two objectives: we first stabilized both the Duffing oscillator and the Van der Pol oscillators. Secondly, we presented the transient performance of the system. Robustness can be added to the system as a future work. This can be achieved by incorporating states estimator, or parameters estimator or even both. These added estimators can be integrated into the design by introducing more virtual control constraints and changing the corresponding Lyapunov function. As an additional future work, we would like to incorporate the effort of this work into another work that we did earlier and more specifically to the photovoltaic system control with the presence of an electric vehicle and a home load as we showed in [

Alghassab, M., Mahmoud, A. and Zohdy, M.A. (2017) Nonlinear Control of Chaotic Forced Duffing and Van der Pol Oscillators. International Journal of Modern Nonlinear Theory and Application, 6, 26-37. https://doi.org/10.4236/ijmnta.2017.61003