Adaptive Exponential Stabilization for a Class of Stochastic Nonholonomic Systems

and Applied Analysis 3 x(t 0 ) = 0 or t = ∞. It is concluded that x 0 does not cross zero for all t ∈ (t 0 ,∞) provided that x(t 0 ) ̸ = 0. Remark 8. If x(t 0 ) ̸ = 0, u 0 exists and does not cross zero for all t ∈ (t 0 ,∞) independent of the x-subsystem from (6). 3.2. Backstepping Design for u 1 . From the above analysis, the x 0 -state in (1) can be globally exponentially regulated to zero as t → ∞, obviously. In this subsection, we consider the control law u 1 for the x-subsystem by using backstepping technique. To design a state-feedback controller, one first introduces the following discontinuous input-state-scaling transformation:


Introduction
The nonholonomic systems cannot be stabilized by stationary continuous state feedback, although it is controllable, due to Brockett's theorem [1].So the well-developed smooth nonlinear control theory and the method cannot be directly used in these systems.Many researchers have studied the control and stabilization of nonholonomic systems in the nonlinear control field and obtained some success [2][3][4][5][6].It should be mentioned that many literatures consider the asymptotic stabilization of nonholonomic systems; the exponential convergence is also an important topic theme, which is demanded in many practical applications.However, the exponential regulation problem, particularly the systems with parameterization, has received less attention.Recently, [3] firstly introduced a class of nonholonomic systems with strong nonlinear uncertainties and obtained global exponential regulation.References [4,5] studied a class of nonholonomic systems with output feedback control.Reference [6] combined the idea of combined input-statescaling and backstepping technology, achieving the asymptotic stabilization for nonholonomic systems with nonlinear parameterization.
It is well known that when the backstepping designs were firstly introduced, the stochastic nonlinear control had obtained a breakthrough [7].Based on quartic Lyapunov functions, the asymptotical stabilization control in the large of the open-loop system was discussed in [8].Further research was developed by the recent work [9][10][11][12][13][14][15][16].[17][18][19] studied a class of nonholonomic systems with stochastic unknown covariance disturbance.Since stochastic signals are very prevalent in practical engineering, the study of nonholonomic systems with stochastic disturbances is very significant.So, there exists a natural problem that is how to design an adaptive exponential stabilization for a class of nonholonomic systems with stochastic drift and diffusion terms.Inspired by these papers, we will study the exponential regulation problem with nonlinear parameterization for a class of stochastic nonholonomic systems.We use the inputstate-scaling, the backstepping technique, and the switching scheme to design a dynamic state-feedback controller with ∑  ∑ ̸ = ; the closed-loop system is globally exponentially regulated to zero in probability.
This paper is organized as follows.In Section 2, we give the mathematical preliminaries.In Section 3, we construct the new controller and offer the main result.In the last section, we present the conclusions.

Problem Statement and Preliminaries
In this paper, we consider a class of stochastic nonholonomic systems as follows: where  0 ∈  and  = [ 1 , . . .,   ]  ∈   are the system states and  0 ∈  and  1 ∈  are the control inputs, respectively.Consider the following stochastic nonlinear system: where  ∈   is the state of system (2), the Borel measurable functions:  :  +1 →   and  :  +1 →  × are assumed to be  1 in their arguments, and  ∈   is an -dimensional standard Wiener process defined on the complete probablity space (Ω, , ).

Controller Design and Analysis
The purpose of this paper is to construct a smooth statefeedback control law such that the solution process of system (1) is bounded in probability.For clarity, the case that  0 ( 0 ) ̸ = 0 is firstly considered.Then, the case where the initial  0 ( 0 ) = 0 is dealt with later.The triangular structure of system (1) suggests that we should design the control inputs  0 and  1 in two separate stages.
To design the controller for system (1), the following assumptions are needed.
Theorem 7. The  0 -subsystem, under the control law (6) with an appropriate choice of the parameters  0 ,  01 ,  02 , is globally exponentially stable.

Backstepping
Design for  1 .From the above analysis, the  0 -state in (1) can be globally exponentially regulated to zero as  → ∞, obviously.In this subsection, we consider the control law  1 for the -subsystem by using backstepping technique.To design a state-feedback controller, one first introduces the following discontinuous input-state-scaling transformation: Under the new -coordinates, -subsystems is transformed into where In order to obtain the estimations for the nonlinear functions   and   , the following Lemma can be derived by Assumption 6. Lemma 9.For  = 1, 2 . . ., there exist nonnegative smooth functions   (⋅),   (⋅), such that Proof.We only prove (11).The proof of ( 12) is similar to that of (11).In view of ( 6), ( 8), (10) where To design a state-feedback controller, one introduces the coordinate transformation where  2 , . . .,   are smooth virtual control laws and will be designed later and  1 = 0. θ denotes the estimate of , where Then using ( 9), ( 10), (14) and It ô differentiation rule, one has where The proof of Lemma 10 is similar to that of Lemma 9, so we omitted it.
We now give the design process of the controller.

Switching Control and Main
Result.In the preceding subsection, we have given controller design for  0 ̸ = 0. Now, we discuss how to choose the control laws  0 and  1 when  0 = 0. We choose  0 as  0 = − 0  0 + * 0 ,  * 0 > 0. And choose the Lyapunov function  0 = (1/2) 2 0 .Its time derivative is given by  0 = − 0  2 0 +  * 0 , which leads to the bounds of  0 .During the time period [0,   ), using  0 = − 0  0 +  * 0 , new control law  can be obtained by the control procedure described above to the original -subsystem in (1).Then, we can conclude that the -state of (1) cannot be blown up during the time period [0,   ).Since at (  ) ̸ = 0, we can switch the control inputs  0 and  to ( 6) and (31), respectively.Now, we state the main results as follows.
Theorem 11.Under Assumption 5, if the proposed adaptive controller (31) together with the above switching control strategy is used in (1), then for any initial contidion ( 0 , , θ) ∈   , the closed-loop system has an almost surely unique solution on [0, ∞), the solution process is bounded in probability, and {lim  → ∞ θ() exists and is finite} = 1.
Proof.According to the above analysis, it suffices to prove in the case  0 (0) ̸ = 0. Since we have already proven that  0 can be globally exponentially convergent to zero in probability in Section 3.1, we only need prove that () is convergent to zero in probability also.In this case, we choose the Lyapunov function  =   , and   >   +   ; from (32) and Lemma 3, we know that the closed-loop system has an almost surely unique solution on [0, ∞), and the solution process is bounded in probability.

Conclusions
This paper investigates the globally exponential stabilization problem for a class of stochastic nonholonomic systems in chained form.To deal with the nonlinear parametrization problem, a parameter separation technique is introduced.With the help of backstepping technique, a smooth adaptive controller is constructed which ensures that the closed-loop system is globally asymptotically stable in probability.A further work is how to design the output-feedback tracking control for more high-order stochastic nonholonomic systems.