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This paper investigates the stabilization problem for a class of nonlinear systems, whose control coefficient is uncertain and varies continuously in value and sign. The study emphasizes the development of a robust control that consists of a modified Nussbaum function to tackle the uncertain varying control coefficient. By such a method, the finite-time escape phenomenon has been prevented when the control coefficient is crossing zero and varying its sign. The proposed control guarantees the asymptotic stabilization of the system and boundedness of all closed-loop signals. The control performance is illustrated by a numerical simulation.

The control design for nonlinear uncertain systems has been the research focus in the community for decades [

However, most previous results only investigate a relatively simple case that the sign of the uncertain control coefficient is fixed, that is, either positive or negative. It is because of the fact that the conventional Nussbaum function requires the control coefficient to be sign-fixed or “bounded away from zero.” Unfortunately, the general case of sign-varying uncertain control coefficient has received much less attention and has not been fully solved yet. Undoubtedly, this problem is technically more challenging and cannot be directly handled by the conventional Nussbaum function. In particular, to design a successful controller, two critical issues have to be taken into full consideration. First, the sign may vary very rapidly and be difficult to track. Second, any control will lose its power when the control coefficient is crossing zero; that is, singular points of control exist and improper controllers probably result in finite-time escape phenomenon [

The first attempt in addressing the problem of sign-varying uncertain control coefficient for a scalar nonlinear system was reported in [

In this paper, we propose a new Nussbaum function, which does not require such assumptions of [

In order to highlight the development of the proposed control approach, we will only consider the following scalar uncertain nonlinear system in this paper:

However, in this paper

Assume that there always exists an arbitrarily small positive constant

In addition, since

Assume that the derivative of

Though

Though

As stated before, the focus of this research is to tackle the continuously varying “sign” function

Let us define the Lyapunov function

Now, design the control

Rewrite (

Now, we will design the Nussbaum function

Conventionally, the Nussbaum gain often adopts the form

Suppose that

The detailed proof of Lemma

Before presenting the main theorem of asymptotic convergence, the following lemma is introduced first.

If

The proof of Lemma

Now, the main result is stated below.

The proposed controller in (

As shown in Lemma

Since

Next, we will deduce a result for

Then, we will estimate the increment of

Again, note that the sign of

In summary,

Since

The proof of the Theorem is organized as follows. First, the finite increment of

Note that according to Lemma

An example is used to illustrate the performance of the proposed control. Consider system (

The profiles of

The profiles of

In this paper, the control problem is studied for a class of nonlinear uncertain systems with the uncertain control coefficient, which is allowed to vary continuously between positive and negative. A new Nussbaum gain is designed and integrated with robust controller to tackle this problem. By following the Lyapunov-fashion controller design procedure, the potential finite-time escape phenomenon is avoided. It is proven that the proposed control approach yields asymptotic stability and guarantees the boundedness of the closed-loop signals.

For the concise of the proof, we will only consider the case of

Because

Because

Similarly, when

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

This work was supported by the 863 Program under Grant 2012AA041709 and by the NSFC under Grant no. 61004057.