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Abstract and Applied Analysis

Volume 2014 (2014), Article ID 650835, 17 pages

http://dx.doi.org/10.1155/2014/650835

## Output Feedback Adaptive Stabilization of Uncertain Nonholonomic Systems

^{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}Research Institute of Automation, Qufu Normal University, Qufu, Shandong 273165, China^{4}College of Mathematics and Information Science, Zhengzhou University of Light Industry, Zhengzhou, Henan 450002, China

Received 23 January 2014; Accepted 15 April 2014; Published 12 May 2014

Academic Editor: Hao Shen

Copyright © 2014 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 investigates the problem of output feedback adaptive stabilization control design for a class of nonholonomic chained systems with uncertainties, involving virtual control coefficients, unknown nonlinear parameters, and unknown time delays. The objective is to design a robust nonlinear output-feedback switching controller, which can guarantee the stabilization of the closed loop systems. An observer and an estimator are employed for states and parameters estimates, respectively. A constructive controller design procedure is proposed by applying input-state scaling transformation, parameter separation technique, and backstepping recursive approach. Simulation results are provided to show the effectiveness of the proposed method.

#### 1. Introduction

The control and feedback stabilization problems of nonholonomic systems have been widely studied by many researchers. It is well known that 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, a number of approaches have been proposed for the problem, which mainly include discontinuous feedback, time-varying feedback, and hybrid stabilization. The discontinuous feedback stabilization was first proposed by [3], and then further discussion was made in [4–7]; especially an elegant discontinuous coordinate transformation approach is proposed in [5] for the stabilization problem of nonholonomic systems. Meanwhile, the smooth time-varying feedback control strategies also have drawn much attention [8–11].

As pointed out in [9], many nonlinear mechanical systems with nonholonomic constraints can be transformed, either locally or globally, to the nonholonomic systems in the so-called chained form. So far, there have been a number of controller design approaches [8–25] for such chained nonholonomic systems. Recently, adaptive control strategies have been proposed to stabilize the nonholonomic systems. For instance, the problem of adaptive state-feedback control is studied in [15–19], while output feedback controller design in [20–24]. Considering the actual modeling perspective, time delay should be taken into account. The problem of state feedback stabilization is studied for the delayed nonholonomic systems in [25, 26]. However, the virtual control coefficients and unknown parameter vector are not considered in its system models. Here, an iterative controller design method will be proposed for the output feedback adaptive stabilization of the concerned delayed nonholomic systems.

In this paper, we study a class of chained nonholonomic systems with strong nonlinear drifts, and the problem of adaptive output-feedback stabilization for the concerned nonholonomic systems is investigated. The constructive design method proposed in this note is based on a combined application of the input scaling technique, the backstepping recursive approach, and the novel Lyapunov-Krasovskii functionals. The switching control strategy for the first subsystem is employed to achieve the asymptotic stabilization.

The rest of this paper is organized as follows. In Section 2, the problem formulation and some preliminary knowledge are given. Section 3 presents the controller design procedure and stability analysis. Section 4 gives the switching control strategy. In Section 5, numerical simulations testify to the effectiveness of the proposed method, and Section 6 summarizes the paper.

#### 2. Problem Formulation and Preliminaries

In this paper, we deal with a class of nonholonomic systems described by where , , and are system states, control input, and measurable output, respectively; is an unknown parameter vector; (known) and (unknown) denote the possible modeling error and neglected dynamics; are known modeled dynamics, which contain output delays; are unknown constants, and referred to the respective virtual control coefficients.

In this paper, we make the following assumptions on the virtual control directions and nonlinear functions in system (1).

*Assumption 1. * is a known constant and the sign of is known, where .

*Assumption 2. *There exist known smooth nonnegative functions and such that
for all .

*Assumption 3. *For every , the nonlinear function satisfies inequality
in which and are known smooth nonnegative nonlinear functions.

*Remark 4. *Compared with some existing literatures in recent years, the structure of our concerned system (1) is more general. For instance, in [15], it is assumed that not only the virtual control directions and the dynamics satisfy , but also the modeled dynamics do not exist. In [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 [21]. When , together with , system (1) becomes the considered system in [23].

*Remark 5. *Note that here we only use the sign of without any knowledge of individual virtual control direction . Moreover, Assumptions 2 and 3 are imposed on the nonlinear functions and , respectively. In fact, if the modeled dynamics do not involve time delays, inequality (3) is reduced into
It can be seen that the above inequality condition is used in some existing literatures, such as [20, 21], and so on.

Our object of this paper is to design adaptive output feedback control laws under Assumptions 1–3, such that the system states converge to zero, while other signals of the closed-loop system are bounded. The designed control laws can be expressed in the following form: Next, we list some lemmas which will be applied in the coming controller design.

Lemma 6 (see [27]). *For any real-valued continuous function , where , there are smooth functions such that
*

*Lemma 7 (see [19]). For any continuous function there exist two strictly positive real numerates and such that the unique solution of the following matrix differential equation:
satisfies .*

*By Lemma 6 and Assumption 1, we know that there exist smooth functions , and such that
Furthermore, we denote ; then it yields
*

*3. Output Feedback Adaptive Stabilization Control Design*

*In this paper, we design control laws and separately to globally asymptotically stabilize the system (1). According to the structure of system (1), we can see that when converges to zero, will be uncontrollable. A widely used method to design control law is to introduce a discontinuous input scaling transformation (12). On the other hand, the control directions are unknown; then we should employ another coordinate transformation to overcome the obstacle.*

*3.1. State Coordinate Transformation*

*Firstly, we design the coordinate transformation as follows:
where and . Then, the system (1) can be transformed into
*

*Next, the following input-state scaling discontinuous transformation is introduced:
*

*Under the new -coordinates, the -subsystem (10) is changed into
*

*Next, we can design the control laws and to asymptotically stabilize the states and , respectively. Rewrite system (13) in the compact form
where
with
*

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

*Lemma 8. For every , there exists smooth nonnegative function such that
*

*Lemma 9. For every , there exist smooth nonnegative functions such that
*

*Remark 10. *By lemmas and assumptions before, Lemmas 8 and 9 can be derived easily, and then the proof is omitted.

*3.2. Observer Design*

*3.2. Observer Design**Define the following filter/estimator:
where , , , and are design parameters to be determined later. Let ; then, the estimation error and the newly defined parameter satisfy the dynamical equations
*

*3.3. Control Design*

*3.3. Control Design**In this section, the intergrator backstepping approach will be used to design the control laws and subject to . The case that the initial condition will be treated in Section 4.*

*Step 0.* At this step, control law will be designed, which is essential to guarantee the effectiveness of the subsequent steps. For the -subsystem, choose the control as follows:
where is a constant satisfying . Introduce the Lyapunov function candidate , and the time derivative of satisfies
where . This indicates that converges to zero exponentially.

*Since is a smooth function, then there exist a constant , such that for . Therefore, the following inequality is true with :
which implies that when , the state converges to zero with a rate less than a certain constant . It is which does not become zero in any time instant. Therefore, the adopted input-state scaling discontinuous transformation in (12) is effective.*

*According to the design of control law in (23), it can be computed that
where and − + .*

*Remark 11. *From (26), we know that is a constant and is a function with respect to . Moreover, we can conclude that is smooth because is a nonnegative smooth function.

Denote ; we can choose appropriate design parameters such that is Hurwitz. Then there exists a positive definite matrix satisfying , and is a positive constant.

*Step **1*. For -subsystem in (13),
let , and . Introduce the following Lyapunov functional:
where
with being positive constants to be designed; , where is an unknown parameter vector to be specified later, and is an estimate of .

*Associated with (22) and (27), the time derivatives of and can be calculated, respectively, that
*

*For some terms on the right-hand side of (30), the following estimations (32)–(34) should be conducted. Firstly, by Lemma 8 and Young’s inequality, we can obtain that there exist positive constants to make the following inequalities hold:
where . Next, employ Lemma 9 and Young’s inequality, and we have
where , and is a positive constant.*

*By completing the square, the following estimations are also true:
*

*Substitute (31)–(34) into , it yields
where , and with
*

*Choose the virtual control function and the adaptation law of as follows:
*

*Notice that , then it follows from (35)–(38) that
*

* Step 2.* Introduce the new variable , where is regarded as the virtual control input, and take the Lyapunov functional as
where , is an unknown parameter vector to be defined later, and is an estimate of . Then, combined with (20), (37), and (39), we have

*Using Lemmas 8 and 9 and Young’s inequality, the following inequalities hold:
*

*By the above inequalities, we get
where and . By taking the adaptation law and the virtual control function as
we can obtain
*

* Step 3*. Define that , where is the virtual control input, and consider the following Lyapunov functional:

*The time derivative of along the estimator system (20) satisfies
*

*By similar conduction method in (42), we have
where is a scalar. Based on (48), it yields
where and . Choose the tuning function , and the virtual control function as follows:
*

*Under the virtual control function and the tuning function defined above, the derivative of becomes that
*

*Step i (). *Assume that, at Step* i−*1, a virtual control function , a tuning function , and a Lyapunov functional have been designed in such a way that

*Let , where is regarded as the virtual control input, and choose Lyapunov functional as
Based on (52), the time derivative of satisfies
*

*Next, we estimate the following terms in the right-hand side of (53) by Lemmas 8 and 9 and Young’s inequality as follows:
*

*Choosing the virtual control function as
*