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An adaptive second-order sliding mode controller is proposed for a class of nonlinear systems with unknown input. The proposed controller continuously drives the sliding variable and its time derivative to zero in the presence of disturbances with

Sliding mode control (SMC) has gained much attention due to its attractive characteristics of finite time convergence and robustness against uncertainties [

There are several ways to avoid chattering problem when using sliding mode control. The conventional SMC uses a control law with large control gains yielding the undesired chattering while the control system is in the sliding mode. To eliminate the chattering, the discontinuous control function is replaced by “saturation” or continuous “sigmoid” functions in [

In this paper, an adaptive controller gain super-twisting algorithm (ASTW) is proposed for nonlinear system with unknown input. Based on the Lyapunov theory, the proposed control law continuously drives the sliding variable and its time derivative to zero in the presence of bounded unknown input but without knowing the boundary. The stability and the robustness of the control system are proven, and the tracking performance is ensured.

The super-twisting control law (STW) is effective to remove the chattering when the relative degree equals one. It generates the continuous control function that drives the sliding variable and its time derivative to zero in finite time in the presence of bounded unknown disturbance. The main disadvantage of STW algorithm is that it requires the bounded value of

Consider a class of

Let the desired state vector be

The relative degree of system (

Under Assumption

The first-order time derivative of the uncertain function

The uncertain function

With Assumptions

The control objective is to drive the sliding variable

In this note, an adaptive-gain approach is used to solve this problem with STW algorithm. STW controller generates the continuous control function to remove the chattering effect while adaptive-gain approach allows controller gain nonoverestimation.

The STW algorithm can be described by differential inclusion as follows [

Substituting (

The term

In this paper, the gains

Now, the control objective is reduced to driving

Consider system (

A new state vector is introduced in order to present system (

Then, the following Lyapunov function candidate is introduced for system (

Taking the derivative of (

Since

In order to show that the

Substituting (

The positive term in (

Now, we can conclude that after some time

In view of practical implementation, the condition

Consider the following nonlinear system [

The desired state trajectory is supposed to be

According to (

The time derivative of

The adaptive-gain STW control laws (

Figures

The performance of state

The performance of state

The performance of state

Sliding surface function of (

Control input

Time varying function

In order to avoid high frequency control activity, the boundary layer technique is employed during the simulation. Thus, the saturation function

Adaptive second-order sliding mode control design is studied for a class of nonlinear systems with unknown inputs. In real applications, the upper boundary of uncertainty is difficult to obtain, which is required for calculating control gains of super-twisting sliding mode control. To solve this problem, a simple control design method is proposed based on Lyapunov function. The idea is very simple; the gains are increased according to a dynamic law until the sliding mode is attained and stops increasing thereafter. The proposed approach has two advantages:

Only one parameter

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

This work was supported by Zhejiang Provincial Natural Science Foundation of China (no. LQ14F010001), Ningbo City Natural Science Foundation of China (no. 2013A610114, no. 2014A610088, and no. 2013A610150), Key Scientific and Technological Projects of Ningbo (no. 2014B92001), and the Fundamental Research Funds for the Central Universities (Grant no. HIT. NSRIF201625).

_{∞}control strategy for switching converters in sliding-mode current control