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Output feedback adaptive stabilization of uncertain nonholonomic systems.

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

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (1)

where [[[x.sub.0](t), x(t)].sup.T] = [[[x.sub.0](t), [x.sub.i](t),..., [x.sub.n](t)].sup.T] [member of] [R.sup.n+i], u(t) = [[[u.sub.0](t), [u.sub.i](t)].sup.T] [member of] [R.sup.2], and y(t) [member of] [R.sup.2] are system states, control input, and measurable output, respectively; [theta] [member of] [R.sup.m] is an unknown parameter vector; [[phi].sub.0] (known) and (1 [less than or equal to] i < [less than or equal to] n) (unknown) denote the possible modeling error and neglected dynamics; [[phi].sub.i](1 [less than or equal to] i [less than or equal to] n) are known modeled dynamics, which contain output delays; [[tau].sub.i](1 [less than or equal to] i [less than or equal to] n) are unknown constants, and [d.sub.i] (0 [less than or equal to] i [less than or equal to] n) referred to the respective virtual control coefficients.

In this paper, we make the following assumptions on the virtual control directions [d.sub.i] and nonlinear functions [[phi].sub.i], [[phi].sub.i] in system (1).

Assumption 1. [d.sub.0] is a known constant and the sign of [[bar.d].sub.n] is known, where [[bar.d].sub.n] = [d.sub.1] [d.sub.2] ... [d.sub.n].

Assumption 2. There exist known smooth nonnegative functions [[bar.[phi]].sub.o] and [[bar.[phi]].sub.i] (1 [less than or equal to] i [less than or equal to] n) such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (2)

Assumption 3. For every 1 [less than or equal to] i [less than or equal to] n, the nonlinear function [[phi].sub.i]; satisfies inequality

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (3)

in which [[bar.[phi]].sub.i] and [[psi].sub.i] 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 [d.sub.i] = 1 and the dynamics [[phi].sub.i] satisfy [[phi].sub.i] = [[??].sup.T.sub.i][theta] but also the modeled dynamics [[phi].sub.i] do not exist. In [22], the virtual control coefficients and time delays have not been considered, and the expression = [[phi].sub.i] = [[??].sup.T.sub.i][theta] is also required. While [d.sub.i] = 1 and [[phi].sub.i] and unknown parameters [theta] are not existent, system (1) degenerates to the one studied in [21]. When [[phi].sub.i] = 0, together with [[phi].sub.i] = [[??].sup.T.sub.i][theta], system 1) becomes the considered system in [23].

Remark 5. Note that here we only use the sign of [[bar.d].sub.n] = [d.sub.1][d.sub.2] ... [d.sub.n] without any knowledge of individual virtual control direction [d.sub.i] (1 [less than or equal to] i [less than or equal to] n). Moreover, Assumptions 2 and 3 are imposed on the nonlinear functions and [[phi].sub.i] and [[phi].sub.i], respectively. In fact, if the modeled dynamics [[phi].sub.i] do not involve time delays, inequality (3) is reduced into

[absolute value of [[phi].sub.i]([u.sub.0](t), y(t))] [less than or equal to] [absolute value of [x.sub.i](t)][[psi].sub.i]([u.sub.0](t), y(t)). (4)

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 ([x.sub.0](t), x(t)) converge to zero, while other signals of the closed-loop system are bounded. The designed control laws can be expressed in the following form:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (5)

Next, we list some lemmas which will be applied in the coming controller design.

Lemma 6 (see [27]). For any real-valued continuous function f(x, y), where x [member of] [R.sup.n], y [member of] [R.sup.m], there are smooth functions a(x) [greater than or equal to] 0, b(y) [greater than or equal to] 0, c(x) [greater than or equal to] 1, d(y) [greater than or equal to] 1 such that

[absolute value of f(x, y)] [less than or equal to] a(x) + b(y), [absolute value of f(x, y)] [less than or equal to] c(x)d(y).. (6)

Lemma 7 (see [19]). For any continuous function [[mu].sub.0](t) there exist two strictly positive real numerates [p.sub.min] and [p.sub.max] such that the unique solution P(t) of the following matrix differential equation:

[??] = P[(A - [[mu].sub.0](t)L).sup.T] + (A - [[mu].sub.0](t) L)P - [PC.sup.T]CP + I, P(0) = [P.sub.0] > 0, (7)

satisfies [p.sub.min] I [less than or equal to] P(t) [less than or equal to] [P.sub.max] I, t [greater than or equal to] 0.

By Lemma 6 and Assumption 1, we know that there exist smooth functions [[omega].sub.i] [greater than or equal to] 1, and [[zeta].sub.i] such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (8)

Furthermore, we denote [??] = [[summation].sup.n.sub.i=1][[zeta].sub.i]([theta]); then it yields

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (9)

3. Output Feedback Adaptive Stabilization Control Design

In this paper, we design control laws [u.sub.0](t) and [u.sub.1] (t) separately to globally asymptotically stabilize the system (1). According to the structure of system (1), we can see that when [x.sub.0](t) converges to zero, [x.sub.i]) (1 [less than or equal to] i [less than or equal to] n) will be uncontrollable. A widely used method to design control law [u.sub.1](t) is to introduce a discontinuous input scaling transformation (12). On the other hand, the control directions [d.sub.i] 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:

[[bar.x].sub.i](t) = [[bar.d].sub.i-1][x.sub.i](t), 1 [less than or equal to] i [less than or equal to] n, (10)

where [[bar.d].sub.0] = 1 and [[bar.d].sub.i-1] = [d.sub.1][d.sub.2] ... [d.sub.i-1] (1 [less than or equal to] i [less than or equal to] n + 1). Then, the system (1) can be transformed into

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (11)

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

[z.sub.i](t)= [[bar.x].sub.i](t)/[u.sub.n-i.sub.0], 1 [less than or equal to] i [less than or equal to] n. (12)

Under the new z(i)-coordinates, the [bar.x](t)-subsystem (10) is changed into

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (13)

Next, we can design the control laws [u.sub.0](t) and [u.sub.1](t) to asymptotically stabilize the states [x.sub.0] (t) and z(t), respectively. Rewrite system (13) in the compact form

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (14)

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (15)

with

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (16)

In order to obtain the estimation for the nonlinear functions [[PSI].sub.i] and [[PHI].sub.i], the following lemmas are given.

Lemma 8. For every 1 [less than or equal to] i [less than or equal to] n, there exists smooth nonnegative function [[??].sub.i]([u.sub.0](t), [x.sub.0](t), [z.sub.1] (t)) such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (17)

Lemma 9. For every 1 [less than or equal to] i [less than or equal to] n, there exist smooth nonnegative functions [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (18)

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. Define the following filter/estimator:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (19)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (20)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (21)

where y(t) = [z.sub.i](t), [e.sub.n] = [[0,..., 1].sup.T], [[xi].sub.0] = [[[xi].sub.01],..., [[xi].sub.0n].sup.T], v = [[[v.sub.1],..., [v.sub.n]].sup.T], [A.sub.0] = A - KC, C = [1, 0,..., 0], K = [[[k.sub.i],..., [k.sub.n]].sup.T], and [k.sub.i] (1 [less than or equal to] i [less than or equal to] n) are design parameters to be determined later. Let [??](t) = [[xi].sub.0](t) + [[bar.d].sub.n]v, [sigma](t) = z(t) - [bar.d].sub.n]v(t); then, the estimation error [epsilon](t) = z(t) - [??](t) and the newly defined parameter [sigma](t) satisfy the dynamical equations

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (22)

3.3. Control Design. In this section, the intergrator back-stepping approach will be used to design the control laws [u.sub.0](t) and [u.sub.i](t) subject to [x.sub.0]([t.sub.0]) [not equal to] 0. The case that the initial condition [x.sub.0] ([t.sub.0]) = 0 will be treated in Section 4.

Step 0. At this step, control law [u.sub.0](t) will be designed, which is essential to guarantee the effectiveness of the subsequent steps. For the [x.sub.0](t)-subsystem, choose the control [u.sub.0](t) as follows:

[u.sub.0](t) = -[[lambda].sub.0][x.sub.0](t) - [[lambda].sub.0][x.sub.0](t)[[bar.[phi]].sub.0]([x.sub.0](t)), (23)

where [[lambda].sub.0] is a constant satisfying [[lambda].sub.0][d.sub.0] > 1. Introduce the Lyapunov function candidate [V.sub.0] = (1/2)[x.sup.2.sub.0](t), and the time derivative of [V.sub.0] satisfies

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (24)

where [c.sub.0] = [[lambda].sub.0][d.sub.0] > 1. This indicates that [x.sub.0](t) converges to zero exponentially.

Since [[bar.[phi]].sub.0]([x.sub.0](t)) is a smooth function, then there exist a constant [M.sub.0] > 1, such that [absolute value of [[bar.[phi]].sub.0] ([x.sub.0](t))] [less than or equal to] [M.sub.0] for [absolute value of [x.sub.0](t)] [less than or equal to] 1. Therefore, the following inequality is true with [absolute value of [x.sub.0](t)] [less than or equal to] 1:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (25)

which implies that when [absolute value of [x.sub.0](t)] [less than or equal to] 1, the state [x.sub.0](t) converges to zero with a rate less than a certain constant [rho]. It is [x.sub.0](t) 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 [u.sub.0](t) in (23), it can be computed that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (26)

where [beta] = -[[lambda].sub.0][d.sub.0] and [[??].sub.0] = - ([[lambda].sub.0][d.sub.0] - 1)[[bar.[phi]].sub.0]([x.sub.0](t)) - [[lambda].sub.0][d.sub.0][x.sub.0])(t)([partial derivative] [[bar.[phi]].sub.0]([x.sub.0](t))/[partial derivative][x.sub.0](t)) + ([x.sub.0](t)[[bar.[phi]].sub.0]([x.sub.0](t))/(1 + [[bar.[phi]].sub.0]([x.sub.0](t))))([partial derivative] [[bar.[phi]].sub.0]([x.sub.0](t))/[partial derivative][x.sub.0](t)).

Remark 11. From (26), we know that [beta] is a constant and [[??].sub.0]([x.sub.0](t)) is a function with respect to [x.sub.0](t). Moreover, we can conclude that [[??].sub.0]([x.sub.0](t)) is smooth because [[bar.[phi]].sub.0] ([x.sub.0](t)) is a nonnegative smooth function.

Denote [A.sub.1] = [A.sub.0] - KC - L[beta]; we can choose appropriate design parameters [k.sub.i](1 [less than or equal to] i [less than or equal to] n) such that [A.sub.1] is Hurwitz. Then there exists a positive definite matrix Q satisfying Q[A.sub.1] + [A.sup.T.sub.1]Q = -[mu]I, and [mu] is a positive constant.

Step 1. For [z.sub.1](t)-subsystem in (13),

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (27)

let [[eta].sub.1](t) = [z.sub.1](t), and [[eta].sub.2](t) = [v.sub.2](t) - [[alpha].sub.1]. Introduce the following Lyapunov functional:

[V.sub.1] = [[bar.V].sub.1] + [[??].sub.1], (28)

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (29)

with [l.sub.1], [[delta].sub.2] being positive constants to be designed; [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], where [[THETA].sub.1] is an unknown parameter vector to be specified later, and [[??].sub.1] is an estimate of [[THETA].sub.1].

Associated with (22) and 27), the time derivatives of [[bar.V].sub.1] and [[??].sub.1] can be calculated, respectively, that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (30)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (31)

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 [l.sub.1], [[delta].sub.1] to make the following inequalities hold:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (32)

where [[??].sub.1] = [[??].sup.2] + [[summation].sup.n- 1.sub.j=1][[bar.d].sup.2.sub.j][[??].sup.2]. Next, employ Lemma 9 and Young's inequality, and we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (33)

where d = 1 + [[summation].sup.n-1.sub.j=1][[bar.d].sup.2.sub.j] + [[summation].sup.n-1.sub.j=1][[bar.d].sup.4.sub.j], and [[delta].sub.2] is a positive constant.

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

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (34)

Substitute (31)-(34) into [[??].sub.2], it yields

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (35)

where [bar.[mu]] = [mu] - 1/[[delta].sub.1] - 1/[[delta].sub.2] - 3, [[bar.c].sub.1] = [c.sub.1] - 3 - [K.sup.T][Q.sup.T]QK - 4[l.sub.1][K.sup.T]K - [l.sub.1][P.sup.2.sub.max] + (n - 1)[beta], [[THETA].sup.T.sub.1] = (1/[[bar.d].sub.n])[1, d, [[??].sub.1]], and [Y.sub.1] = [[[Y.sub.11], [Y.sub.12], [Y.sub.13].sup.T] with

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (36)

Choose the virtual control function [[alpha].sub.1] and the adaptation law of [[??].sub.1] as follows:

[[alpha].sub.1] = -[[??].sup.T.sub.1][Y.sub.1], (37)

[[??].sub.1] = sign([[bar.d].sub.n])[Y.sub.1][[eta].sub.1](t). (38)

Notice that [[bar.d].sub.n][[eta].sub.1](t)[[eta].sub.2](t) [less than or equal to] [[eta].sup.2.sub.1](t) + ([[bar.d].sup.2.sub.n]/4)[[eta].sup.2.sub.2](t), then it follows from (35)-(38) that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (39)

Step 2. Introduce the new variable [[eta].sub.3](t) = [v.sub.3](t)- [[alpha].sub.2], where [[alpha].sub.2] is regarded as the virtual control input, and take the Lyapunov functional as

[V.sub.2] = [V.sub.1] + 1/2[[eta].sup.2.sub.2](t) + 1/2[[??].sup.T.sub.2] [[??].sub.2], (40)

where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is an unknown parameter vector to be defined later, and [[??].sub.2] is an estimate of [[THETA].sub.2]. Then, combined with (20), (37), and (39), we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (41)

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

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (42)

By the above inequalities, we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (43)

where [[THETA].sup.T.sub.2] = [[[??].sup.2], [[bar.d].sup.2.sub.n], [[bar.d].sub.n]] and [Y.sub.2] = [[(1/2)[([partial derivative][[alpha].sub.1]/[partial derivative][z.sub.1].sup.2] [[??].sup.2.sub.1][[eta].sub.2](t), [[eta].sub.2](t)/4, - [partial derivative][[alpha].sub.1]/[partial derivative][z.sub.1])[v.sub.2](t)].sup.T]. By taking the adaptation law [[??].sub.2] = [Y.sub.2][[eta].sub.2](t) and the virtual control function [[alpha].sub.2] as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (44)

we can obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (45)

Step 3. Define that [[eta].sub.4](t) = [v.sub.4](t) - [[alpha].sub.3], where [[alpha].sub.3] is the virtual control input, and consider the following Lyapunov functional:

[V.sub.3] = [V.sub.2] + 1/2 [[eta].sup.2.sub.3](t) + 1/2 [[??].sup.T.sub.3][[??].sub.3]. (46)

The time derivative of [V.sub.3] along the estimator system (20) satisfies

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (47)

By similar conduction method in 42), we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (48)

where [l.sub.3] > 0 is a scalar. Based on (48), it yields

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (49)

where [[THETA].sup.T.sub.3] = [[[??].sup.2], [[bar.d].sub.n]] and [Y.sub.3] = [[(1/2)[([partial derivative][a.sub.2]/[partial derivative][z.sub.i]).sup.2][[??].sup.2.sub.1][[eta].sub.3](t), -([partial derivative][[alpha].sub.2]/[partial derivative][z.sub.1]) [v.sub.2](t)].sup.T]. Choose the tuning function [[pi].sub.3][Y.sub.3][[eta].sub.3](t), and the virtual control function [[alpha].sub.3] as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (50)

Under the virtual control function [[alpha.sub.3] and the tuning function [[pi].sub.3] defined above, the derivative of [V.sub.3] becomes that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (51)

Step i (4 [less than or equal to] i [less than or equal to] n). Assume that, at Step i-1, a virtual control function [[alpha].sub.i-1], a tuning function [[pi].sub.i-1], and a Lyapunov functional [V.sub.i-1] have been designed in such a way that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (52)

Let [[eta].sub.i+1](t) = [v.sub.i+1](t)-[[alpha].sub.i], where [[alpha].sub.i] is regarded as the virtual control input, and choose Lyapunov functional as

[V.sub.i] = [V.sub.i-1] + 1/2 [[eta].sup.2.sub.i](t). (53)

Based on (52), the time derivative of [V.sub.i] satisfies

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (54)

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

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (55)

Choosing the virtual control function [[alpha].sub.i] as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (56)

and the tuning function [[pi].sub.i] = [[pi].sub.i-1] + [Y.sub.i][[eta].sub.i](t) with [Y.sub.i] = [[(1/2)[([partial derivative][[alpha].sub.i-1]/[partial derivative][z.sub.1]).sup.2][[??].sup.2.sub.1][[eta].sub.i](t), -([partial derivative][[alpha].sub.i-1]/[partial derivative][z.sub.1])[v.sub.2](t)].sup.T]. Then, we can show that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (57)

At the last step (i = n), the true input [u.sub.1](t) will be designed on the basis of the virtual control [[alpha]'.sub.i]s and the Lyapunov function [V.sub.n-1] introduced before.

The actual control input [u.sub.1](t) can be designed as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (58)

and the update law [[??].sub.3] = [[pi].sub.n] with [[pi].sub.n] = [[pi].sub.n-1] + [Y.sub.n][[eta].sub.n](t) and [Y.sub.n] = [[(1/2)[([partial derivative][[alpha].sub.n-1]/[partial derivative][z.sub.1]).sup.2][[??].sup.2.sub.1][[eta].sub.n](t), -([partial derivative][[alpha].sub.n-1]/[partial derivative][z.sub.1])[v.sub.2](t)].sup.T]. Eventually, it can be achieved that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (59)

3.4. Stability Analysis. Notice that [??]([x.sub.0](t)) tends to zero as [x.sub.0](t) converges to origin, and [[delta].sub.1], [[delta].sub.2], [l.sub.i], [c.sub.i] (1 [less than or equal to] i [less than or equal to] n) in (59) are positive design parameters. Therefore, by an appropriate parameter choice, there exist positive constants [[lambda].sub.i] > 0(1 [less than or equal to] i [less than or equal to] n + 2) such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (60)

It can be seen that [[eta].sub.i](t), [epsilon](t), [sigma](t), [[??].sub.1], [[??].sub.2], [[??].sub.3] are bounded. Since [theta] and [d.sub.i] are unknown bounded parameters, [[??].sub.1], [[??].sub.2], [[??].sub.1] are bounded. According to estimator equations (19)-(21), it can be deduced that the boundedness of [z.sub.1](t) = [[eta].sub.1](t) guarantees the boundedness of [[xi].sub.0](t), and then [v.sub.1](t) = (1/[[bar.d].sub.n])([z.sub.1](t) - [[sigma].sub.1](t)) and [[alpha].sub.1] are also bounded. By similar analysis, we can conclude that all signals of the closed loop system are bounded.

By LaSalle invariant Theorem, it further achieves that [[eta].sub.i](t), [epsilon](t), [sigma](t), [[??].sub.1], [[??].sub.2], [[??].sub.3] [right arrow] 0 as t [right arrow] [infinity]. By the controller design procedure, we get that [[xi].sub.0](t), v(t), [[alpha].sub.i], [u.sub.i](t) asymptotically tend to zero. Then, the definitions [??](t) = [[xi].sub.0](t) + [[bar.d].sub.n]v(t) and z(t) = [epsilon](t) + [??](t) show the asymptotical convergence of [??](t) and z(t). Finally, from the transformations (10) and (12), we know [x.sub.i](t) = (1/[[bar.d].sub.n])[u.sup.n- i].sub.0](t)[z.sub.i](t), which indicates that the states [x.sub.i](t) asymptotically converge to zero with the initial condition [x.sub.0]([t.sub.0]) [not equal to] 0.

For purposes of analysis, we can rewrite the system (14) as follows:

[??](t) = ([A.sub.1] - L[[bar.[phi]].sub.0]([x.sub.0](t))) z(t) + K[z.sub.i](t) + B[u.sub.1](t) + [PSI] + [PHI]. (61)

To solve the above differential equation, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (62)

Notice that [A.sub.i] = A - KC - L[beta] is Hurwitz, and [[phi].sub.0]([x.sub.0](t)) tends to zero as [x.sub.0](t) [right arrow] 0, then by Lemmas 8 and 9, there exist constants [[??].sub.1] > 0, [[??].sub.2] > 0 such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (63)

where [[??].sub.1] is a nonnegative smooth function of [d.sub.i], [u.sub.0](s), [u.sub.0](s - [[tau].sub.i]), y(s), y(s - [[tau].sub.i]), and [[??].sub.2] is a nonnegative smooth function of [d.sub.1], [u.sub.0](s), [x.sub.0](s), [z.sub.1](s), [??].

Since [x.sub.0](t), [x.sub.i](t), [u.sub.0](t) and the system parameters are all bounded, then [[??].sub.1], [[??].sub.2] in (63) are also bounded. Employing the convergence of [x.sub.0](t), [z.sub.1](t), [u.sub.1](t), we can get that z(t)-system is globally asymptotically convergent. From the introduced transformations before, it can be deduced that system (1) is also asymptotically convergent. Now, we can express the following theorem.

Theorem 12. For system (1), under Assumptions 1-3, if the control strategies (23) and (58) are applied with an appropriate choice of the design parameters, the global asymptotic stabilization of the closed loop system is achieved for [x.sub.0]([t.sub.0]) [not equal to] 0.

In the next section, we will deal with the stability analysis of the closed loop as long as the initial condition [x.sub.0] ([t.sub.0]) is zero.

4. Switching Controller

Several switching controllers have been proposed in some existing literatures. As well known, the choice of a constant feedback for [u.sub.0](t) may lead to a finite escape. In this note, the following switching category can be designed for the stabilization of system (1) with the initial sate [x.sub.0]([t.sub.0]) = 0. Choosing controller [u.sub.0](t) as

[u.sub.0](t) = sign ([d.sub.0])[u.sub.*.sub.0], when [absolute value of [x.sub.0](t)] [less than or equal to] [[??].sub.3] < [x.sup.*.sub.0], (64)

where [u.sup.*.sub.0] > 0 and [[??].sub.3] > 0 are constants.

Since [x.sub.0]([t.sub.0]) = 0, then [[??].sub.0]([t.sub.0]) with [u.sub.0](t) can be deduced

[[??].sub.0] ([t.sub.0]) = [absolute value of [d.sub.0]][u.sup.*.sub.0] + [phi](t, [x.sub.0]([t.sub.0])) = [absolute value of [d.sub.0]][u.sup.*.sub.0] > 0, (65)

then during the initial small time period, [x.sub.0](t) is increasing and satisfies [absolute value of [x.sub.0](t)] + [absolute value of [x.sub.0](t)][[bar.[phi]].sub.0]([x.sub.0](t)) < [absolute value of [[d.sub.0]][u.sup.*.sub.0].

Choose [x.sup.*.sub.0] that satisfy

[absolute value of [x.sup.*.sub.0]] + [absolute value of [x.sup.*.sub.0]] [[bar.[phi]].sub.0]([x.sup.*.sub.0]) = [absolute value of [d.sub.0]][u.sup.*.sub.0]. (66)

Obviously, [x.sub.0](t) is increasing when [x.sub.0](t) [less than or equal to] [x.sup.*.sub.0]. When [absolute value of [x.sub.0](t)] [less than or equal to] [??] < [x.sup.*.sub.0], choose the controller [u.sub.0](t) = sign([d.sub.0])[u.sup.*.sub.0], and the controller [u.sub.1](t) can be designed according to the simple nonlinear backstepping iterative approach. Since [absolute value of [x.sub.0](t)] > [[??].sub.3], at [t.sub.s], we switch the control laws [u.sub.0](t) and [u.sub.1](t) into (23) and (58), respectively.

Theorem 13. For system (1), under Assumptions 1-3, if above switching control strategy is applied with an appropriate choice of the design parameters, then the closed-loop system is globally asymptotic regulated at the origin for [x.sub.0]([t.sub.0]) = 0.

5. Simulation Example

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

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (67)

where [d.sub.0], [d.sub.1], [d.sub.2] are virtual control directions with [d.sub.1], [d.sub.2] unknown and [d.sub.0] known, and the sign of [[bar.d].sub.2] = [d.sub.1][d.sub.2] is also known. [[theta].sub.1], [[theta].sub.2] are unknown bounded parameters. Next, we consider to design the controller [u.sub.0](t) and [u.sub.1](t) to asymptotically stabilize system (67) by the measurable output. We assume that [x.sub.0]([t.sub.0]) [not equal to] 0 and make the following estimation for some nonlinear terms in system (67):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (68)

where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Firstly, we introduce the following transformation:

[[bar.x].sub.1](t) = [x.sub.i](t), [[bar.x].sub.2](t) = [d.sub.1][x.sub.2](t), (69)

and then the system 67) can be rewritten as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (70)

where [[bar.d].sub.2] = [d.sub.1][d.sub.2], and assume that the sign of [[bar.d].sub.2] is known.

Next, make the following input scaling transformation for [bar.x](t)-system:

[z.sub.1](t) = [[bar.x].sub.1](t)/[u.sub.0](t), [z.sub.2](t) = [[bar.x].sub.2](t), (71)

and then the transformed system is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (72)

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (73)

Design the following controller [u.sub.0](t):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (74)

and then [[??].sub.0](t)/[u.sub.0](t) can be calculated as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (75)

For system (72), constructing the following estimator:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (76)

where y(t) = [z.sub.1](t), [e.sub.n] = [[0, 1].sup.T], [[xi].sub.0] = [[[xi].sub.01], [[xi].sub.02].sup.T], v = [[v.sub.1], [v.sub.2]].sup.T], [A.sub.0] = A - KC, C = [1, 0], and K = [[[k.sub.1], [k.sub.2]].sup.T]. The design of [k.sub.1], [k.sub.2] can guarantee that [A.sub.1] = [A.sub.0] - KC - L[beta] is Hurwitz. It is further achieved that there exists plosive definite matrix Q satisfying Q[A.sub.1] + [A.sup.T.sub.1]Q = -[mu]I, in which [mu] > 0 is a constant. Denote [??](t) = [[xi].sub.0](t) + [[bar.d].sub.n]v, [sigma](t) = z(t) - [[bar.d].sub.n]v(t) and [epsilon](t) = z(t) - [??](t), and then the observation error [epsilon](t) and parameter invariable [sigma](t) satisfy

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (77)

Define the invariable that [[eta].sub.1](t) = [z.sub.1](t), [[eta].sub.2](t) = [v.sub.2](t) - [[alpha].sub.1]. According to the iterative procedure in Section 3, we can design the virtual control function and controller [u.sub.1](t) as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (78)

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (79)

The adoption laws of the parameter invariable in controller [u.sub.i](t) are chosen as

[[??].sub.1] = sign ([[bar.d].sub.2])[Y.sub.1][[eta].sub.1](t), [[??].sub.2] = [Y.sub.2][[eta].sub.2](t). (80)

For simulation use, we pick the unknown parameters [d.sub.1] = 1.5, [d.sub.2] = 2.5, [[theta].sub.1] = [[theta].sub.2] = 0.5. In addition, we take the other controller design parameters as [c.sub.0] = 1, [c.sub.1] = 130, [c.sub.2] = 2, [k.sub.1] = 4, [k.sub.2] = 1, [l.sub.1] = 2, [l.sub.2] = 3,[[delta].sub.1] = [[delta].sub.2] = 4. Moreover, The initial state condition is [[0.2,0, -0.1].sup.T]. Simulation results are shown in Figures 1, 2, 3, and 4. It is obvious that the states [x.sub.0](t), [x.sub.1](t), [x.sub.2](t) and control input [u.sub.0](t), [u.sub.1](t) converge to zero, and the parameters estimation invariable tend to constants.

6. Conclusion

The output-feedback adaptive stabilization was investigated for a class of nonholonomic systems with unknown virtual control coefficients, nonlinear uncertainties, and unknown time delays. In order to overcome the difficulties, we introduce suitable transformation and novel Lyapunov-Krasovskii functionals, and then a recursive technique is given to design the adaptive controller. To make the input-state scaling transformation effective, the switching control strategy is employed to achieve the asymptotic stabilization.

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

Conflict of Interests

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

Acknowledgments

This work is partially supported by National Natural Science Foundation (U1304620,61273091,61374079), Basic and Frontier Technologies Research Program of Henan Province (122300410279), and Doctoral Fund of Zhengzhou University of Light Industry (201BSJJ006).

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Yuanyuan Wu, (1, 2) Zicheng Wang, (1) Yuqiang Wu, (3) and Qingbo Li (4)

(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

Correspondence should be addressed to Qingbo Li; qbliyy@163.com

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

Academic Editor: Hao Shen
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