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This paper is concerned with the global asymptotic stabilization control problem for a class of nonlinear systems with input-to-state stable (ISS) dynamic uncertainties and uncertain time-varying control coefficients. Unlike the existing works, the ISS dynamic uncertainty is characterized by the uncertain supply rates. By using the backstepping control approach, a systematic controller design procedure is developed. The designed control law can guarantee that the system states are asymptotically regulated to the origin from any initial conditions and the other signals are bounded in closed-loop systems. Moreover, it is shown that, under some additional conditions, a linear control law can be designed by the proposed methodology. The simulation example demonstrates its effectiveness.

The nonlinear control theory is an active research direction in the control field because of its widespread applications in the real world. During the past two decades, various novel methodologies have been generated for the nonlinear feedback control; see the recent survey [

It is noted that a unifying framework is presented in [

In this paper, we will further investigate this problem for a class of nonlinear systems with more general nonlinear uncertainties. Unlike the existing works such as in [

The rest of the paper is organized as follows. In Section

In this paper, we consider the following class of cascaded nonlinear systems with dynamic uncertainties:

The control objective in this paper is to find a smooth, dynamic, partial-state feedback law of the form

The

According to [

For the uncertain nonlinearities

There exist known positive constants

To deal with the unmeasured state

Consider the

If

For any

According to Lemma

In this section, we give the controller design procedure using the backstepping design method.

Starting with the

It is noted that, in (

Let

Let

In (

However, from (

In particular, when

After the above controller design procedure, we are now ready to state the main results.

Suppose the investigated system (

From the local conditions (

So far all the closed-loop system signals are bounded on

Again, according to (

It is noted that, under some stronger conditions, the designed control law can be a linear controller. In fact, we have the following statement.

Suppose that the conditions for Theorem

The uncertain functions

There exist known constants

Then, the proposed design method can result in a linear control law

Under the above hypotheses (i)–(iii), it is known that the constant

In this section, we provide a simulation example to illustrate the proposed method in the paper. Consider the following nonlinear systems:

Next, we use the proposed algorithm in Section

We consider the function

Define

Similar to (

To find the actual control law

As in Step

The Lyapunov function

The response of the closed-loop system.

The control input of the closed-loop system.

According to our results reported in Theorem

The state feedback stabilization problem is investigated for a class of nonlinear systems with dynamic uncertainties and uncertain control coefficients in this paper. The dynamic uncertainty is characterized by the uncertain ISS supply rates. A global asymptotic stabilization control scheme is proposed using the backstepping design scheme. The tuning function technique is applied in this procedure, which avoids the disadvantage of overparameterization. It is shown that, under some more restrictive conditions, a linear state feedback controller can be designed by the presented algorithm. The simulation example demonstrates the effectiveness of the proposed method.

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

This work is supported in part by the National Natural Science Foundation of China under Grant no. 61304008, the Shandong Provincial Natural Science Foundation of China under Grant no. ZR2013FQ033, and the Doctor Research Foundation of Shandong Jianzhu University under Grant no. XNBS1272.