Mathematical Problems in Engineering

Volume 2013 (2013), Article ID 520390, 14 pages

http://dx.doi.org/10.1155/2013/520390

## Adaptive Stabilization for Nonholonomic Systems with Unknown Time Delays

^{1}College of Electric and Information Engineering, Zhengzhou University of Light Industry, Zhengzhou, Henan 450002, China^{2}Department of Mathematics, Southeast University, Nanjing, Jiangsu 210096, China^{3}Department of Mathematics and Information Science, Zhengzhou University of Light Industry, Zhengzhou, Henan 450002, China^{4}Research Institute of Automation, Qufu Normal University, Qufu, Shandong 273165, China

Received 19 June 2013; Revised 20 October 2013; Accepted 3 November 2013

Academic Editor: Oleg V. Gendelman

Copyright © 2013 Yuanyuan Wu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

This paper presents an adaptive control strategy for a class of nonholonomic systems in chained form with virtual control coefficients, nonlinear uncertainties, and unknown time delays. State scaling technique and backstepping recursive approach are applied to design a nonlinear state feedback controller, which can guarantee the stabilization of the closed-loop systems. The simulation results are provided to show the effectiveness of the proposed method.

#### 1. Introduction

In the last few decades, considerable efforts have been devoted to the research of nonholonomic system, which is a particular class of nonlinear systems and widespread in real world, such as mobile robots, car-like vehicle, underactuated satellites, and knife-edge. It is well known that the control of nonholonomic systems is extremely challenging, largely due to the impossibility of asymptotically stabilizing nonholonomic systems via smooth time-invariant state feedback, a well-recognized fact pointed out in [1, 2]. In order to overcome this obstruction, several approaches have been proposed, such as discontinuous feedback, time-varying feedback, and hybrid stabilization.

The discontinuous feedback stabilization was first proposed in [3], and then further discussion was made in [4–7]; especially an elegant discontinuous coordinate transformation approach was presented in [5] for the stabilization problem of nonholonomic systems. Meanwhile, the smooth time-varying feedback control strategies also have drawn much attention [8–11]. To date, there have been several controller design approaches for the asymptotic stabilization or exponential regulation of nonholonomic control systems [4–14].

As pointed out in [9], many nonlinear mechanical systems with nonholonomic constraints can be transformed, either locally or globally, into the nonholonomic systems in the so-called chained form. Therefore, a number of research literature resources [8–23] for such chained nonholonomic systems are provided. Recently, some new adaptive control strategies have been proposed to stabilize the nonholonomic systems. For instance, state feedback control is studied in [15–20] and output feedback control in [21–25].

From a practical point of view, when modeling a mechanical system, time delay should be taken into account, and there are a few literature resources [19, 20, 25] for the nonholonomic systems with time delay. In [19, 20, 25], the problem of stabilization is studied for delayed nonholonomic systems; however, the virtual control coefficients and unknown parameter vector are not considered.

In this paper, we introduce a new class of chained nonholonomic systems with unknown virtual control coefficients, uncertain nonlinearities, and unknown time delays and then study the problem of adaptive state feedback stabilization. Since the nonholonomic system considered in this paper contains the delayed terms, it cannot be handled by existing conventional methods. The proposed constructive design method is based on a combined application of the state scaling technique, the recursive backstepping approach, and the novel Lyapunov-Krasovskii functionals. The switching control strategy for the first subsystem is employed to achieve the asymptotic stabilization.

#### 2. Problem Formulation and Preliminaries

In this paper, we present an adaptive stabilization control design procedure for the following nonholonomic systems with nonlinear uncertainties and unknown time delays: where and are system states and control input, respectively. is an unknown bounded parameter vector. are disturbed virtual control coefficients, and the individual signs are known. denote the delayed terms, which contain output delays. are unknown constants; are vectors of smooth nonlinear functions and represent unmodeled dynamic and external disturbances.

*Assumption 1. *For nonlinear functions and , there exist (known) smooth nonnegative functions and such that
where .

*Assumption 2. *The nonlinear functions satisfy
in which are (known) smooth nonnegative nonlinear functions.

*Remark 3. *It is clear that system (1) covers a number of important classes of uncertain nonholonomic systems that have been investigated in some existing literature resources. For instance, when and , system (1) reduces the standard form of nonholonomic system which has been widely studied in the literature [15, 18–20]. Moreover, in Ge et al. [15], not only the virtual control coefficients and the dynamics satisfying are assumed but also the modeled dynamics do not exist. In Liu and Zhang [22], the virtual control coefficients and time delays have not been considered, and the expression is also required. While and and unknown parameters are not existent, system (1) degenerates to the one studied in Xi et al. [21]. When , together with , system (1) becomes the considered system in Ju et al. [23].

*Remark 4. *Note that here we only use the sign of without any knowledge of individual virtual control coefficient . Moreover, Assumptions 1 and 2 are imposed on the nonlinear functions and the delayed terms of system (1), respectively. It can be seen that some similar conditions are implied in [22].

Lemma 5. *For any real-valued continuous function , where and , there are smooth functions , , and such that
*

By Lemma 5 and Assumption 1, we know that there exist smooth functions , , , , , and such that

Denote that , then the above inequalities can be rewritten as follows:

#### 3. Adaptive Stabilization Control Design

In this section, we will design an adaptive stabilization controller for the case that , and the case that will be considered in the next section. Now, we use two separate stages to globally asymptotically stabilize the system (1). Firstly, the control should be designed for -subsystem; in the second stage, we design to guarantee all states of the rest in system (1) converge to zero.

##### 3.1. State Scaling

The following state scaling discontinuous transformation is introduced:

Under the new -coordinates, the system (1) is transformed into where

In order to obtain the estimation for the nonlinear functions and , the following lemmas are given.

Lemma 6. *For every , there exist smooth nonnegative functions and such that
**
where .*

*Proof. *By the above inequalities (6), it can be deduced that

Introduce the notation
It is clear that are smooth nonnegative functions, and inequality (11) holds.

Lemma 7. *For every , the following inequality holds:
**
where , , and are smooth nonnegative functions.*

*Proof. *According to Assumption 2, the nonlinear functions yield that

Let ; then the above inequality can be expressed as

It is seen that are smooth functions. Then using Lemma 5, there exist smooth functions and such that

##### 3.2. Control Design

In this section, we design the control inputs and subject to . The case that the initial condition will be treated in Section 4. The design of the control inputs here is based on the backstepping method for the transformed system (10). The recursive procedure stops once the true system inputs occur.

*Step 1. *For the -subsystem
define new variables , , and , where and are the estimates of and , respectively.

Consider the Lyapunov function candidate

Calculating the time derivative of along the system (18)

The controller can be chosen as where

With the controller in (21), the time derivative of satisfies

Choosing the following update laws and as we have where is a positive design parameter. Therefore, it implies that , , and are bound. By LaSalle’s Invariant Theorem, we can further achieve that , as .

*Remark 8. *The closed-loop dynamics of -subsystem is

It is seen that is bounded as , , , and are bounded. On the other hand, the solution of -system can be computed as

Obviously, for and , the solution exists and satisfies . That is, does not become zero at any time instant for . Therefore, the introduced state scaling above is effective.

Under the controller in (25), the -system can be rewritten as

*Step 2. *For -subsystem in (30)
let , and , , , , and , where , , , , and are the estimates of unknown parameters , , , , and , respectively. Introduce the coordinate transformations and , where is regarded as the virtual control input. Construct the following Lyapunov-Krasovskii functional:
where are scalars. Along (31), the time derivative of gives

By Lemmas 6 and 7, the following inequalities hold:
where and . Choose a virtual control function as follows:
where is a positive design parameter. With the choice of the update law
we can obtain
where

*Step i (). *Assume that, at step , a virtual control function and a Lyapunov functional have been designed for the -subsystem of (30) in such a way that

Now, we examine the -subsystem of (30). Define , and and , where and are the estimates of unknown parameters and , respectively. Introduce the coordinate transformations , where is regarded as a virtual control input, and construct the following Lyapunov-Krasovskii functional

Based on (39), the time derivative of along the solutions of (30) satisfies

By Lemmas 5 and 7 and Young inequality, the following inequality holds:

Using Lemma 6 and Young inequality, there are nonnegative smooth functions and such that
where and are known nonnegative functions.

Choose the following virtual control function :
where is a positive design parameter and , and are pending nonnegative functions to be specified in (47). Moreover, construct the following update law:

Substituting inequalities (42)–(45) into (41) yields
where

*Step n. *At the last step, we study the whole -subsystem (30), and the true input will be designed on the basis of the virtual control and the Lyapunov function introduced before. Here, let us consider a Lyapunov-Krasovskii function as follows:
Denote and , where is the estimate of unknown parameter . Recall that , with being a virtual control input, then

Differentiating along the solution of (30) gives

Similarly, by Lemmas 5–7 and Young inequality, we can easily obtain that there are scalars and smooth nonnegative functions and such that

Next, we can design the control input as follows:
where is a positive design parameter and , and are smooth nonnegative functions to be specified in (56). With the choice of the update law
it renders
where
Furthermore, employing the following update laws
eventually achieves

This together with (48) implies that , , , , and , are bounded. Since and are constant vector and constant, respectively, we know that , and , are also bounded. According to the definitions of the virtual control input in the above design procedure, are bounded as are bounded. It indicates that all signals of the closed loop system are bounded.

LaSalle Invariant Theorem further achieves that as . The boundedness of all signals and the choice of virtual control functions imply that converge to zero, which shows that also tend to zero. From the transformation , we can prove that , as .

The above analysis is summarized into the following theorem.

Theorem 9. *For the system (1), under Assumptions 1 and 2, if the control strategies (21) and (52) are applied with an appropriate choice of the design parameters , the global asymptotic stabilization of the closed loop system is achieved for .*

In the next section, we will deal with the stability analysis of the closed loop system with our control laws (21) and (52) as long as the initial condition is zero.

#### 4. Switching Controller

Several switching controllers have been proposed in some existing literature resources. As well known, the choice of a constant feedback for may lead to a finite escape. That is, the solution issued from the origin may blow up before the switch. Usually, the phenomenon occurs for systems with non-Lipschitz nonlinearities. In this paper, the term in -subsystem does not satisfy the Lipschitz conditions; then we should apply a novel switching control design. When the initial state , choose controller as where is a constant, is defined in (22), and update laws of the parameters and are chosen as in (25) and (26), respectively.

With controller in (59), the derivative of the Lyapunov function in (19) along -subsystem gives that

The above inequality indicates that , , and are bounded; then state cannot blow up during the time period .

For and Assumption 2, we have where . Obviously, is bounded.

The above inequality indicates that

It is clear that when , . Therefore, the state scaling coordinate transformation in (8) is effective, and we can use the following switching control strategy for .

During the time period , using the controller in (59), when and is a constant, the controller can be designed implying the simple nonlinear backstepping method. When , choose the controller as the iterative procedure in Section 3. Since at , we switch the control law and into (21) and (52), respectively.

Theorem 10. *For the system (1), under Assumptions 1 and 2, if the above switching control strategy is applied with an appropriate choice of the design parameters , then the closed-loop system is globally asymptotically regulated at the origin for .*

#### 5. Simulation Example

In this section, a numerical example will be given to illustrate that the proposed systematic control law design method is effective.

*Example 1. *Consider the following system:
where , , and are unknown virtual control coefficients, and , , and are unknown bounded parameters. Our purpose is to design controllers and such that the states of the closed-loop system tend to zero when .

To apply the proposed design method, we make the estimation of nonlinear functions in system (62) as follows:

Let ; then the above inequalities can be deduced as

Introduce the following coordinate transformation: then system (62) can be rewritten as

For -subsystem, design the following controller : with where and , and are design constants. Moreover, the adaptation laws of and are chosen as

Define and ; according to the design procedure in Section 3, the following controller can be given: with