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This paper investigates the global stabilization problem for a class of nonholonomic systems in chained form with input delay. A particular transformation is introduced to convert the original time-delay system into a delay-free form. Then, by using input-state-scaling technique and the method of sliding mode control, a constructive design procedure for state feedback control is given, which can guarantee that all the system states globally asymptotically converge to the origin. An illustrative example is also provided to demonstrate the effectiveness of the proposed scheme.

Nonholonomic systems, which can model many classes of mechanical systems such as mobile robots and wheeled vehicles, have attracted intensive attention over the past decades. However, due to the limitation imposed by Brockett’s condition [

On the other hand, sliding mode control (SMC), also known as variable structure control (VSC), in essence, is a special nonlinear control, and its nonlinearity is reflected in the noncontinuity of control. Since the properties and parameters of the SMC just depend on the design of the switching hyperplane and have nothing with the external interferences, the SMC has many advantages such as simple algorithm, fast response, and robustness to external noise and parameter perturbation. During the past few years, the SMC strategy has been also applied to the nonholonomic system control [

In this paper, we introduce a new class of nonholonomic chained systems with input delay and then study the problem of robust state feedback stabilization for the concerned systems. Since the nonholonomic system considered in this paper contains input delay, therefore it cannot be handled by general existing methods. By composing linear transformation and input-state-scaling techniques with the SMC strategy, a state feedback controller is constructed to guarantee that the states of the closed-loop systems are asymptotically regulated to the origin.

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

Since many mechanical systems with nonholonomic constraints, such as wheeled mobile robot, can be transformed to a kind of nonholonomic systems in the so-called chained form [

The control objective is to find a state feedback controller which makes the closed-loop system be globally asymptotically regulated at origin.

Before the analysis of system (

Consider the following linear system with input delay:

For system (

Let

If there exists a state feedback controller in the form

From the linear transformation (

In this section, we present a systematic controller design procedure for the system (

Consider the control input

Under the control law (

Under the control law (

Since the time-delay nonlinear system (

The inherent structure of system (

For

Obviously, the control input

As a result, the following lemma can be easily established by direct calculation.

For any initial

Substituting (

Furthermore, from (

Hence, we can see that the

The design in Section

Under the new

Obviously,

In terms of Lemma

To fulfill the controller design of

The above proposed control scheme will drive the state to approach the sliding mode surface

Under the control law (

From (

Substituting (

In the following lemma, the sufficient condition for the asymptotic stability of the system (

If there exists a positive definite matrix

It is straightforward and thus is omitted here.

Based on the input-state-scaling transformation and Lemmas

If there exists a positive definite matrix

Considering the robustness of SMC, the following result is a slight extension of Theorem

For the uncertain chained system with input delay

Consider a car-like mobile robot as shown in Figure

Schematic of the mobile robot.

Using the same modeling method in [

Considering that the delay in the input is often unavoidable due to sensors, calculation, information processing, or transport, here we assume that the forward velocity

By taking the following state and input transformation:

Assume that

Transient responses of the closed system.

System states

Control inputs

In this paper, we consider the global stabilization problem for a class of nonholonomic systems in chained form with input delay via state feedback. First, a particular linear transformation is introduced to convert the original time-delay system into a delay-free form. Then, by using input-state-scaling technique and the SMC approach to design control laws, global asymptotic regulation of the closed-loop system is guaranteed. Simulation results demonstrate the effectiveness of the proposed scheme.

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

This work has been supported in part by the National Nature Science Foundation of China under Grant no. 61073065 and the Key Program of Science Technology Research of Education Department of Henan Province under Grant nos. 13A120016 and 14A520003. The authors thank the editor and the anonymous reviewers for their constructive comments and suggestions for improving the quality of the paper.