An extended nonsingular terminal sliding surface is proposed for second-order nonlinear systems. It is shown that the proposed surface is a superset of a conventional nonsingular terminal sliding surface which guarantees that the system state gets to zero in finite time. The conventional nonsingular sliding surfaces have been designed using a power function whose exponent is a rational number with positive odd numerator and denominator. The proposed nonsingular terminal sliding surface overcomes the restriction on the exponent of a power function; that is, the exponent can be a positive real number. Simulation results are provided to show the validity of the main result.

According to the progress of control schemes, a variety of nonlinear control systems have been proposed: H-infinity optimal control, fuzzy control, neural network control, controller using genetic algorithms, and so on [

Although most of control systems proposed so far have been designed such that the closed-loop system is asymptotic, finite time stabilization is very important to many actual applications, such as motor, power, robot, and aerospace systems because, in the actual applications, the main objective of a control system is to make system’s state to a desired one in a finite time interval which is determined a priori. Thus, many recent studies have focused on the finite time stabilization [

Finite time control systems based on sliding mode control schemes have been called terminal sliding mode control systems since their sliding surfaces have been designed as terminal attractors [

Thus, in this paper, a novel nonsingular terminal sliding surface is proposed for second-order nonlinear systems. It is shown that the proposed scheme guarantees that the system state gets to zero in finite time and it does not suffer from singularity problems. Furthermore, the proposed nonsingular terminal sliding surface overcomes the restriction on the exponent of a power function by being a superset of a conventional nonsingular terminal sliding surface. That is, we extend the range of an exponent of a power function in the nonsingular terminal sliding surface from a rational number with an odd numerator and an odd denominator to a real number.

Simulation results and experimental results are given to show the validity of the main result.

Consider a second-order nonlinear system of the following form:

The uncertainty

In previous works on terminal sliding mode control systems, the conventional terminal sliding surfaces have been designed as

Recently, a nonsingular terminal sliding surface was proposed to overcome the singularity problem [

However, this surface still has the restrictions on the exponent of the power function; that is,

Figure

Example of a nonsingular terminal sliding surface.

It is clear that the proposed nonsingular terminal sliding surface is stable because the sliding surface is in the second and fourth quadrants in the phase space with the axes of

For the proposed nonsingular terminal sliding surface, we derived the following theorem for the finite time convergence.

The proposed nonsingular terminal sliding surface (

If the system is in the sliding mode and

From the above equation, if

If

This completes the proof.

In the above theorem, we proved that the system state gets to zero in finite time if it is in the sliding mode. Thus, in the following theorem, we propose a controller that guarantees that the sliding mode existence condition holds such that the overall system will be in the sliding mode.

For system (

Let the Lyapunov function candidate be

If

Therefore, we can conclude that

Since the proposed controller (

To show the validity of the proposed method, the simulation results for the following system are given:

Figures

Sliding variable (

Phase portrait (

Output (

Control input (

It is clear that the sliding mode existence condition,

Figure

The control input signal can be seen in Figure

To show the nonsingular property, we simulated the system with the initial condition near the vertical axis in the phase space because, from (

Phase portrait (

Control input (

In addition, we also applied the proposed method to the actual DC motor system. For the control unit in the experimental system, a TMS320F2812 DSP processor was used. The sampling time was set to 1 msec. The following model was used for the DC motor system:

Figure

Output angle (

The control input signal is given in Figure

Control input (

Figure

Phase portrait (

In this paper, the extended nonsingular terminal sliding surface for second-order nonlinear systems has been proposed. It has been shown that the proposed nonsingular terminal sliding surface guarantees finite time convergence and is singularity-free. Furthermore, the exponent of the power function in the proposed sliding surface can be a real number in contrast to conventional nonsingular terminal sliding surfaces where the exponent should be a rational number with an odd numerator and odd denominator. Simulation and experimental results have shown the validity of the main result.

The authors declare that there is no conflict of interests regarding the publication of this paper.

This work was supported by a Korea University Grant.