_{1}

^{*}

In this paper, a super twisting controller (STC) is designed to control the chaotic behavior of the Duffing-Holmes system in stabilization and tracking cases. Due to lack of availability of the performance evaluation of STC in controlling Duffing-Holmes system, this paper aims to test the performance of STC in controlling Duffing-Holmes system. In order to achieve this control design, a modification of the conventional super twisting algorithm is adapted. Numerical simulations showed that the modified STC had high performance and ability to ensure robustness with respect to bounded external disturbances.

Chaos’s behavior is very sensitive to initial conditions. It has been utilized in many modern applications such as oscillators [

The sliding mode control is a well-known control technique and its fundamentals are available in many books and manuscripts [

The super-twisting controller (STC) is the most known second order sliding mode controller. It provides finite time and exact convergence in the presence of bounded perturbations. Recently, a strict Lyapunov functions for the STC, to analyze its robustness for a wide class of perturbations, make the possibility to obtain an explicit relation for the controller design parameters has been introduced [

In this paper, the super twisting controller (STC) is used to control the chaotic behavior of the Duffing-Holmes system (DHS). The goal is to test the performance of STC in controlling DHS. Based in our best knowledge, there is a lack of studies that evaluate or test the STC performance in controlling DHSs.

The organization of rest of this article is as follows. Section 2 describes the DHS. Section 3 reviews the second order sliding mode method and describes the STC. In Section 4, simulation results are provided to show the effectiveness of the proposed method in controlling the DHS. Section 5 concludes the paper.

A nonlinear oscillator with a cubic stiffness term to describe the hardening spring effect observed in many mechanical problems was introduced by Duffing. Duffing’s equation has been modified in different manners afterwards such as Holmes. In this paper we consider a modified Duffing equation named Duffing-Holmes described as [

where x is the oscillation displacement, p_{0} is the damping constant, p_{1} is the linear stiffness constant, p_{2} is the cubic stiffness constant, q is the excitation amplitude, and ω is excitation frequency. By defining the states of Equation (1)

Extreme sensitivity to initial conditions is the fundamental characteristic of a chaotic system thus; small differences in the initial conditions can lead to differences in the system states response. To show this behavior, parameters of chaotic DHS is selected as

The dynamic behavior of a second order nonlinear system can be written as follows:

where x(t), is the state and u(t) is the control input vectors, respectively. f(x, t) and b(x, t) are unknown nonlinear functions of time and states. The functions f(x, t) and b(x, t) are not exactly known with upper bounded uncertainties. The control problem let the state x track a specified time dependent state x_{r}.

_{m}, K_{M}, and C the following inequalities hold globally.

2-SM controllers may be considered as controllers for the following differential inclusion [

2-SM controllers allow to solve the problem of finite-time stabilization, the only information needed from the system is the output. The control u(t) can be given as a sum of two components: The first one is defined by means of its discontinuous time derivative, while the another is a continuous function of the available sliding variable.

where k_{1} and k_{2} positive tuning parameters. Recently a modified super-twisting controller was proposed in [

where k_{3} a positive tuning parameter. The signum function is defined as given below:

The 2-SMC method reduces a suitably-defined sliding variable to zero by the use of a discontinuous control action. The vanishing of the sliding variable guarantees the achievement of the control objective. The sliding variable is a linear combination between the tracking error and its first derivatives. In 1-SMC, the discontinuous control operates on the first time-derivative of the sliding variable. In 2-SM the discontinuous control affects the second derivative of the sliding variable. In this work the sliding variable σ in Equation (7) is designed as follows:

The error is defined as:

The constant c value can be selected to make the sliding variable converges to zero in a very short time. The first derivative of the error

where τ is a time constant. A small value for τ in the noise-free case, leads to an accurate estimation.

The Saturation block imposes upper and lower limits on the control signal. Output the signal, but only up to some limited magnitude, then caps the output to a value of T. The saturation function is an odd function. The saturation function is given by:

The procedure described above can be represented by flowchart in

In this section, the super twisting controller is used to control the DHS. This dynamic behavior will be controlled under the two schemes namely a set point and tracking tasks. To test the performance of the controller, a plant uncertainty representing the unmodeled dynamics or structural variation of the system,

Then, the uncertainty,

The external disturbance δ(t) is given as:

In general, the uncertainty and the disturbance are assumed to be bounded and the corresponding upper bounds can be obtained as follows:

Set point or stabilizing problem is to find a control u(t) for stabilizing the state of the system at one of the unstable equilibrium points. This can be considered as a special case of the general tracking control with a constant reference signal. To demonstrate the performance of the STC, we present the results of the numerical simulations that have been obtained using MatLab/Simulink. The STC is used to set states

The time-step used in the simulation is equal to 0.001 second.

behavior of the controlled DHS. The states are initiates from the initial condition x_{0} = [1, 2] and by applying the STC, the system states converge to the steady state equilibrium point i.e. [0, 0]. The time needed for the first state x_{1} to reach zero error is approximately 5 seconds.

The control objective is to solve the tracking problem stated below:

For any bounded reference trajectory x_{r} whose derivative _{r} asymptotically as t goes to ∞ for any initial conditions

To test the performance of the STC in tracking task, the DHS is controlled to follow the trajectory given as:

The parameters of the DHS and the STC are similar to the previous section. _{r}. Note that the control signal of the STC does not include chattering. A good view of tracking the DHS, the reference trajectory, is shown in phase plane plot depicted in

In this paper, the super-twisting controller has been applied to control a Duffing chaotic

system. An appropriate sliding variable has been selected to solve the stabilization, tracking cases. The simulation results obtained clearly show the good performance of the controller in controlling a chaotic system with uncertainty to any arbitrarily desired trajectory with high accuracy. The steady state error was reduced to zero. The STC controller can be also applied to synchronization of chaos, since the problem can be changed into the nth order tracking problem of state. It has been observed that a proper selection of the control parameters influences the control effort and the error, so that a method for tuning the parameters is required.

Khadra, F.A. (2016) Super-Twisting Control of the Duffing-Hol- mes Chaotic System. International Journal of Modern Nonlinear Theory and Application, 5, 160-170. http://dx.doi.org/10.4236/ijmnta.2016.54016