Adaptive Exponential Stabilization for a Class of Stochastic Nonholonomic Systems

Abstract

This paper investigates the adaptive stabilization problem for a class of stochastic nonholonomic systems with strong drifts. By using input-state-scaling technique, backstepping recursive approach, and a parameter separation technique, we design an adaptive state feedback controller. Based on the switching strategy to eliminate the phenomenon of uncontrollability, the proposed controller can guarantee that the states of closed-loop system are global bounded in probability.

1. 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 [2–6]. 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-state-scaling 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 [9–16]. [17–19] 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 input-state-scaling, the backstepping technique, and the switching scheme to design a dynamic state-feedback controller with∑^𝑇∑≠𝐼; 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.

2. Problem Statement and Preliminaries

In this paper, we consider a class of stochastic nonholonomic systems as follows: 𝑑𝑥₀=𝑑₀(𝑡)𝑢₀𝑑𝑡+𝑓₀𝑡,𝑥₀𝑑𝑡𝑑𝑥_𝑖=𝑑_𝑖(𝑡)𝑥_𝑖+1𝑢₀𝑑𝑡+𝑓_𝑖𝑡,𝑥₀,𝑥_𝑖𝑑𝑡+𝜑_𝑖𝑥_𝑖(𝑡)𝑑𝜔,𝑖=1,…,𝑛−1,𝑑𝑥_𝑛=𝑑_𝑛(𝑡)𝑢₁𝑑𝑡+𝑓_𝑛𝑡,𝑥₀,𝑥𝑑𝑡+𝜑_𝑛(𝑥)(𝑡)𝑑𝜔, (1) where𝑥₀∈𝑅and𝑥=[𝑥₁,…,𝑥_𝑛]^𝑇∈𝑅^𝑛are the system states and𝑢₀∈𝑅and𝑢₁∈𝑅are the control inputs, respectively.𝑥_𝑖=[𝑥₁,𝑥₂,…,𝑥_𝑖]^𝑇∈𝑅^𝑖,(𝑖=1,2,…,𝑛), and𝑥_𝑛=𝑥;𝜔∈𝑅^𝑟is an𝑟-dimensional standard Wiener process defined on the complete probability space(Ω,𝐹,𝑃)withΩbeing a sample space,𝐹being a filtration, and𝑃being measure. The drift and diffusion terms𝑓_𝑖(⋅),𝜑_𝑖(⋅)are assumed to be smooth, vanishing at the origin(𝑥₁,𝑥₂,…,𝑥_𝑖)=(0,0,…,0);∑(𝑡)∶𝑅₊→𝑅^𝑟×𝑟is the Borel bounded measurable functions and is nonnegative definite for each𝑡≥0.𝑑_𝑖(𝑡)are disturbed virtual control coefficients, where𝑖=0,1…𝑛.

Next we introduce several technical lemmas which will play an important role in our later control design.

Consider the following stochastic nonlinear system: 𝑑𝑥=𝑓(𝑥,𝑡)𝑑𝑡+𝑔(𝑥,𝑡)𝑑𝜔,𝑥(0)=𝑥₀∈𝑅^𝑛, (2) where𝑥∈𝑅^𝑛is the state of system (2), the Borel measurable functions:𝑓∶𝑅^𝑛+1→𝑅^𝑛and𝑔∶𝑅^𝑛+1→𝑅^𝑛×𝑟are assumed to be𝐶¹in their arguments, and𝜔∈𝑅^𝑟is an𝑟-dimensional standard Wiener process defined on the complete probablity space(Ω,𝐹,𝑃).

Definition 1 (see [8]). Given any𝑉(𝑥,𝑡)∈𝐶^1,2, for stochastic nonlinear system (2), the differential operator𝐿is defined as follows: 𝐿𝑉(𝑥,𝑡)=𝜕𝑉+𝜕𝑡𝜕𝑉1𝜕𝑥𝑓+2tr𝑔^𝑇𝜕²𝑉𝜕𝑥²𝑔, (3) where𝐶^1,2(𝑅^𝑛×𝑅₊;𝑅₊)denotes all nonnegative functions𝑉(𝑥,𝑡)on𝑅^𝑛×𝑅₊, which are𝐶¹in𝑡and𝐶²in𝑥, and for simplicity, the smooth function𝑓(⋅)is denoted by𝑓.

Lemma 2 (see [8]). Let𝑥and𝑦be real variables. Then, for any positive integers𝑚,𝑛, and any real number𝜀>0, the following inequality holds: 𝛼(⋅)𝑥^𝑚𝑦^𝑛≤𝜀|𝑥|^𝑚+𝑛+𝑛𝑚+𝑛𝑚+𝑛𝑚^−𝑚/𝑛×𝛼(⋅)^{(𝑚+𝑛)/𝑛}𝜀^−𝑚/𝑛𝑦^𝑚+𝑛. (4)

Lemma 3 (see [7]). Considering the stochastic nonlinear system (2), if there exist a𝐶^1,2function𝑉(𝑥,𝑡),𝐾_∞class functions𝛼and𝛼, constant𝑐, and a nonnegative functions𝑊(𝑥,𝑡)such that 𝛼|(𝑥)|≤𝑉(𝑥)≤𝛼|(𝑥)|,𝐿𝑉(𝑥)≤−𝑊(𝑥,𝑡)+𝑐, (5) then for each𝑥₀∈𝑅^𝑛. (1) For (2), there exists an almost surely unique solution on[0,∞]. (2) When𝑐=0,𝑓(0,𝑡)=0,𝑔(0,𝑡)=0, and𝑊(𝑥,𝑡)=𝑊(𝑥)is continuous, the equilibrium𝑥=0is globally stable in probability, and the solution𝑥(𝑡)satisfies𝑃{lim_𝑡→∞𝑊(𝑥(𝑡)=0}=1.(3) For any given𝜀>0, there exist a class𝐾𝐿function𝛽_𝑐(⋅,⋅)and𝐾function𝛾(⋅)such that𝑃{|(𝑥(𝑡))|<𝛽_𝑐(|𝑥₀|,𝑡)+𝛾(𝑐)}≥1−𝜀for any𝑡≥0,𝑥₀∈𝑅^𝑛⧵{0}.

Lemma 4 (see [20]). For any real-valued continuous function𝑓(𝑥,𝑦),𝑥∈𝑅^𝑚,𝑦∈𝑅^𝑛, there exist smooth scalar-value funcions𝑎(𝑥)≥0,𝑏(𝑦)≥0,𝑐(𝑥)>1,and𝑑(𝑦)≥1, such that|𝑓(𝑥,𝑦)|≤𝑎(𝑥)+𝑏(𝑦), and|𝑓(𝑥,𝑦)|≤𝑐(𝑥)𝑑(𝑦).

3. Controller Design and Analysis

The purpose of this paper is to construct a smooth state-feedback control law such that the solution process of system (1) is bounded in probability. For clarity, the case that𝑥₀(𝑡₀)≠0is firstly considered. Then, the case where the initial𝑥₀(𝑡₀)=0is dealt with later. The triangular structure of system (1) suggests that we should design the control inputs𝑢₀and𝑢₁in two separate stages.

To design the controller for system (1), the following assumptions are needed.

Assumption 5. For0≤𝑖≤𝑛, there are some positive constants𝜆_𝑖1and𝜆_𝑖2that satisfy the inequality𝜆_𝑖1≤𝑑_𝑖(𝑡)≤𝜆_𝑖2.

Assumption 6. For𝑓₀(𝑡,𝑥₀), there exists a nonnegative smooth function𝛾₀(𝑡,𝑥₀), such that|𝑓₀(𝑡,𝑥₀)|≤|𝑥₀|𝛾₀(𝑡,𝑥₀).
For each𝑓_𝑖(𝑡,𝑥₀,𝑥_𝑖),𝜑_𝑖(𝑥_𝑖), there exist nonnegative smooth functions𝛾_𝑖(𝑡,𝑥₀,𝑥_𝑖)and𝜌_𝑖(𝑥_𝑖), such that|𝑓_𝑖(𝑡,𝑥₀,𝑥_𝑖)|≤(∑^𝑖_𝑘=1|𝑥_𝑘|)𝛾_𝑖(𝑡,𝑥₀,𝑥_𝑖),|𝜑_𝑖(𝑥_𝑖)|≤(∑^𝑖_𝑘=1|𝑥_𝑘|)𝜌_𝑖(𝑥_𝑖).

3.1. Designing𝑢₀for𝑥₀-Subsystem

For𝑥₀-subsystem, the control𝑢₀can be chosen as 𝑢₀=−𝜆₀𝑥₀, (6) where𝜆₀=(𝑘₀+𝛾₀)/𝜆₀₁and𝑘₀is a positive design parameter.

Consider the Lyapunov function candidate𝑉₀=𝑥²₀/2. From (6) and Assumptions 5 and 6, we have 𝐿𝑉₀=𝑥₀𝑑₀𝑢₀+𝑓₀𝑡,𝑥₀≤𝑑₀𝑢₀𝑥₀+𝑥²₀𝛾₀≤−𝑘₀𝑥²₀=−2𝑘₀𝑉₀. (7) So, we obtain the first result of this paper.

Theorem 7. The𝑥₀-subsystem, under the control law (6) with an appropriate choice of the parameters𝑘₀,𝜆₀₁,𝜆₀₂, is globally exponentially stable.

Proof. Clearly, from (7),𝐿𝑉₀≤0, which implies that|𝑥₀(𝑡)|≤|𝑥₀(𝑡₀)|𝑒^{−𝑘0(𝑡−𝑡0)}. Therefore,𝑥₀is globally exponentially convergent. Consequently,𝑥₀can be zero only at𝑡=𝑡₀, when𝑥(𝑡₀)=0or𝑡=∞. It is concluded that𝑥₀does not cross zero for all𝑡∈(𝑡₀,∞)provided that𝑥(𝑡₀)≠0.

Remark 8. If𝑥(𝑡₀)≠0,𝑢₀exists and does not cross zero for all𝑡∈(𝑡₀,∞)independent of the𝑥-subsystem from (6).

3.2. Backstepping Design for𝑢₁

From the above analysis, the𝑥₀-state in (1) can be globally exponentially regulated to zero as𝑡→∞, obviously. In this subsection, we consider the control law𝑢₁for the𝑥-subsystem by using backstepping technique. To design a state-feedback controller, one first introduces the following discontinuous input-state-scaling transformation: 𝜂_𝑖=𝑒^𝛼𝑡𝑥_𝑖𝑢₀^𝑛−𝑖,𝑖=1…,𝑛,𝑢=𝑒^𝛼𝑡𝑢₁. (8) Under the new𝑥-coordinates,𝑥-subsystems is transformed into 𝑑𝜂_𝑖=𝑑_𝑖𝜂_𝑖+1𝑑𝑡+𝑓_𝑖𝑑𝑡+𝜙^𝑇_𝑖^𝑇(𝑡)𝑑𝜔,𝑖=1,…,𝑛−1,𝑑𝜂_𝑛=𝑑_𝑛𝑢𝑑𝑡+𝑓_𝑛𝑑𝑡+𝜙^𝑇_𝑛^𝑇(𝑡)𝑑𝜔, (9) where 𝑓_𝑖=𝛼𝜂_𝑖+𝑒^𝛼𝑡𝑓_𝑖𝑢₀^𝑛−𝑖−(𝑛−𝑖)𝜂_𝑖𝑢₀𝜕𝑢₀𝜕𝑥₀𝑑₀𝑢₀+𝑓₀,𝜙_𝑖=𝑒^𝛼𝑡𝜑_𝑖𝑢₀^𝑛−𝑖. (10)

In order to obtain the estimations for the nonlinear functions𝑓_𝑖and𝜙_𝑖, the following Lemma can be derived by Assumption 6.

Lemma 9. For𝑖=1,2…𝑛,there exist nonnegative smooth functions𝛾_𝑖(⋅),𝜌_𝑖(⋅), such that 𝑓_𝑖≤^𝑖_𝑘=1𝜂_𝑘𝛾_𝑖𝑥₀,𝑥_𝑖,𝜙_𝑖≤^𝑖_𝑘=1𝜂_𝑘𝜌_𝑖𝑥_𝑖. (11)(12)

Proof. We only prove (11). The proof of (12) is similar to that of (11). In view of (6), (8), (10) and Assumption 6, one obtains 𝑓_𝑖=𝛼𝜂_𝑖+𝑒^𝛼𝑡𝑓_𝑖𝑢₀^𝑛−𝑖−(𝑛−𝑖)𝜂_𝑖𝑢₀𝜕𝑢₀𝜕𝑥₀𝑑₀𝑥₀+𝑓₀≤𝛼𝜂_𝑖+^𝑖_𝑘=1𝑒^𝛼𝑡𝑥_𝑘𝑢₀^𝑛−𝑘𝑢₀^𝑖−𝑘𝛾_𝑖+(𝑛−𝑖)𝜆₀𝜆₀₂+𝛾₀𝜂_𝑖≤|𝛼|𝜂_𝑖+^𝑖_𝑘=1𝜂_𝑘𝜆₀^𝑖−𝑘𝑥₀^𝑖−𝑘𝛾_𝑖+(𝑛−𝑖)𝜆₀𝜆₀₂+𝛾₀𝜂_𝑖≤^𝑖_𝑘=1𝜂_𝑘|𝛼|+𝜆₀^𝑖−𝑘𝑥₀^𝑖−𝑘𝛾_𝑖+(𝑛−𝑖)𝜆₀𝜆₀₂+𝛾₀≤^𝑖_𝑘=1𝜂_𝑘𝛾_𝑖𝑥₀,𝑥_𝑖, (13) where𝛾_𝑖(𝑥₀,𝑥_𝑖)≥|𝛼|+|𝜆₀^𝑖−𝑘|𝑥₀^𝑖−𝑘|𝛾_𝑖+(𝑛−𝑖)(𝜆₀𝜆₀₂+𝛾₀).
To design a state-feedback controller, one introduces the coordinate transformation 𝑧₁=𝜂₁,𝑧_𝑖=𝜂_𝑖−𝛼_𝑖𝑧_𝑖−1,𝑖=1,2…,𝑛, (14) where𝛼₂,…,𝛼_𝑛are smooth virtual control laws and will be designed later and𝛼₁=0.𝜃denotes the estimate of𝜃, where 𝜃=sup_𝑡≥0max(𝑡)^𝑇(𝑡)²,(𝑡)^𝑇(𝑡)^4/3,(𝑡)^𝑇(𝑡). (15) Then using (9), (10), (14) and It𝑜̂differentiation rule, one has 𝑑𝑧_𝑖=𝑑𝜂_𝑖−𝛼_𝑖=𝑑_𝑖𝜂_𝑖+1+𝐹_𝑖𝑧_𝑖,𝑥₀−𝜕𝛼_𝑖𝜕𝜃̇𝜃𝑑𝑡+𝐺^𝑇_𝑖𝑧_𝑖^𝑇(𝑡)−1𝑑𝜔2^𝑖−1_𝑘,𝑚=1𝜕²𝛼_𝑖𝜕𝑧_𝑘𝜕𝑧_𝑚𝜙^𝑇_𝑘𝑧_𝑘^𝑇(𝑡)(𝑡)𝜙_𝑚𝑧_𝑚𝑑𝑡,𝑖=1,2…𝑛, (16) where𝜂_𝑛+1=𝑢,𝐹_𝑖(𝑧_𝑖,𝑥₀)=𝑓_𝑖+∑^𝑖−1_𝑘=1(𝜕𝛼_𝑖/𝜕𝑧_𝑘)(𝑑_𝑘𝜂_𝑘+1+𝑓_𝑘), and𝐺_𝑖(𝑧_𝑖,𝑥₀)=𝜙_𝑖+∑^𝑖−1_𝑘=1(𝜕𝛼_𝑖/𝜕𝑧_𝑘)𝜙_𝑘,where𝑖=1,2…𝑛.Using Lemmas 2, 4, and 9 and (14), we easily obtain the following lemma.

Lemma 10. For1≤𝑖≤𝑛,there exist nonnegative smooth functions𝛾_𝑖1(𝑧_𝑖,𝑥₀),𝑝_𝑖1(𝑧_𝑖), and𝑝_𝑖(𝑧_𝑖), such that 𝐹_𝑖≤^𝑖_𝑘=1𝑧_𝑘𝛾_𝑖1𝑧_𝑖,𝑥₀,𝐺_𝑖≤^𝑖_𝑘=1𝑧_𝑘𝑝_𝑖1𝑧_𝑖,Φ_𝑖≤^𝑖_𝑘=1𝑧_𝑘𝑝_𝑖𝑧_𝑖. (17)

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.

Step 1. Consider the first Lyapunov function𝑉₁(𝑧₁,𝜃)=(1/4)𝑧⁴₁+(1/2)(𝜃−𝜃)². By (14), (15), and (16), we have 𝐿𝑉₁=𝑧³₁𝑑₁𝜂₂+𝐹₁+32𝑧²₁Tr𝐺^𝑇₁^𝑇(𝑡)(𝑡)𝐺₁+̇𝜃−𝜃𝜃. (18)
Using Lemma 10 and Lemma 4, we have 𝑧³₁𝐹₁≤𝑧⁴₁𝛾₁₁𝑧₁,𝑥₀32𝑧²₁Tr𝐺^𝑇₁^𝑇(𝑡)(𝑡)𝐺₁≤𝑧⁴₁𝑝²₁₁𝑧₁,𝑥₀^𝑇(𝑡)(𝑡)≤𝑧⁴₁𝑝²₁₁𝑧₁,𝑥₀𝜃. (19) Substituting (19) into (18) and using (14), we have 𝐿𝑉₁≤𝑑₁𝑧³₁𝜂₂−𝛼₂+𝑑₁𝑧³₁𝛼₂+𝑧⁴₁𝑝²₁₁𝑧₁,𝑥₀𝜃+𝑧⁴₁𝛾₁₁𝑧₁,𝑥₀+̇𝜃−𝜃𝜃≤𝑑₁𝑧³₁𝑧₂+𝑑₁𝑧³₁𝛼₂+𝑧⁴₁𝑝²₁₁𝑧₁,𝑥₀𝜃+𝑧⁴₁𝛾₁₁𝑧₁,𝑥₀+̇𝜃−𝜃𝜃, (20) where𝛼₂=−𝑧₁𝛽₁=−𝑧₁((𝑐₁+𝛾₁₁+𝑝²₁₁𝜃)/𝜆₁₁). Substituting𝛼₂into (20), we have 𝐿𝑉₁≤𝑑₁𝑧³₁𝑧₂−𝑐₁𝑧⁴₁+̇𝜃−𝜃𝜃−𝜏₁, (21) where𝜏₁=𝑧⁴₁𝑝²₁₁.

Step i. (2≤𝑖≤𝑛). Assume that at step𝑖−1, there exists a smooth state-feedback virtual control𝛼_𝑖=−𝑧_𝑖−1𝛽_𝑖−1(𝑧_𝑖−1,𝜃)=−𝑧_𝑖−1((𝑐_𝑖−1+𝜃1+(𝜓_𝑖−12+𝜓_𝑖−13)²+𝑏_𝑖−1+𝜓_𝑖−11+𝜓_𝑖−14)/𝜆_𝑖−11), such that 𝐿𝑉_𝑖−1≤−^𝑖−2_𝑗=1𝑐_𝑗−𝜀_𝑗−𝑒_𝑗𝑧⁴_𝑗−𝑐_𝑖−1𝑧⁴_𝑖−1+𝑑_𝑖−1𝑧³_𝑖−1𝑧_𝑖+𝜃−𝜃−∑^𝑖−1_𝑘=2𝑧³_𝑘𝜕𝛼_𝑘𝜕𝜃̇𝜃−𝜏_𝑖−1, (22) where𝑉_𝑖−1=∑^𝑖−1_𝑗=1(1/4)𝑧⁴_𝑗+(1/2)(𝜃−𝜃)²,𝜏_𝑖−1=𝜏₁+∑^𝑖−1_𝑘=2𝑧⁴_𝑘(𝜓_𝑖−12+𝜓_𝑖−13), and𝜀_𝑗=∑^𝑗_𝑘=1(𝜀_𝑘1+𝜀_𝑘2+𝜀_𝑘3+𝜀_𝑘4),where𝑗=1,…,𝑛.
Then, define the𝑖th Lyapunov candidate function𝑉_𝑖(𝑧_𝑖,𝜃)=𝑉_𝑖−1+(1/4)𝑧⁴_𝑖. From (16) and (22), it follows that 𝐿𝑉_𝑖≤−^𝑖−2_𝑗=1𝑐_𝑗−𝜀_𝑗−𝑒_𝑗𝑧⁴_𝑗−𝑐_𝑖−1𝑧⁴_𝑖−1+𝑑_𝑖−1𝑧³_𝑖−1𝑧_𝑖+𝑧³_𝑖𝑑_𝑖𝜂_𝑖+1+𝐹_𝑖𝑧_𝑖,𝑥₀−𝜕𝛼_𝑖𝜕𝜃̇𝜃−12^𝑖−1_𝑘,𝑚=1𝜕²𝛼_𝑖𝜕𝑧_𝑘𝜕𝑧_𝑚𝜙^𝑇_𝑘𝑧_𝑘^𝑇(𝑡)(𝑡)𝜙_𝑚𝑧_𝑚+32𝑧²_𝑖Tr𝐺^𝑇_𝑖𝑧_𝑖^𝑇(𝑡)(𝑡)𝐺_𝑖𝑧_𝑖+𝜃−𝜃−^𝑖−1_𝑘=2𝑧³_𝑘𝜕𝛼_𝑘𝜕𝜃̇𝜃−𝜏_𝑖−1. (23) Using Lemmas 9 and 4, there are always known nonnegative smooth functions𝜓_𝑖1(𝑧_𝑖),𝜓_𝑖2(𝑧_𝑖),𝜓_𝑖3(𝑧_𝑖),𝜓_𝑖4(𝑧_𝑖)and constant𝜀_𝑖>0,  𝜀_𝑖𝑗>0, where  𝑖=1,…,𝑛and  𝑗=1,2,3,4.
Consider 𝑧³_𝑖𝐹_𝑖≤𝑧³_𝑖^𝑖−1_𝑘=1𝑧_𝑘𝛾_𝑖1𝑧_𝑖,𝑥₀≤𝛾_𝑖1𝑧⁴_𝑖+^𝑖−1_𝑘=1𝜀_𝑘1𝑧⁴_𝑘+344𝜀_𝑘1^−1/3𝛾^4/3_𝑖1𝑧⁴_𝑖≤^𝑖−1_𝑘=1𝜀_𝑘1𝑧⁴_𝑘+𝜓_𝑖1𝑧⁴_𝑖, (24) where𝜓_𝑖1≥𝛾_𝑖1+∑^𝑖−1_𝑘=1(3/4)(4𝜀_𝑘1)^−1/3𝛾^4/3_𝑖1.−12𝑧³_𝑖^𝑖−1_𝑘,𝑚=1𝜕²𝛼_𝑖𝜕𝑧_𝑘𝜕𝑧_𝑚𝜙^𝑇_𝑘^𝑇(𝑡)(𝑡)𝜙_𝑚≤12𝑧³_𝑖^𝑖−1_𝑘,𝑚=1𝜕²𝛼_𝑖𝜕𝑧_𝑘𝜕𝑧_𝑚^𝑘_𝑗=1𝑧_𝑗𝑝_𝑘𝑧_𝑘×^𝑚_𝑗=1𝑧_𝑗𝑝_𝑚𝑧_𝑚^𝑇(𝑡)(𝑡)≤𝑧³_𝑖^𝑖−1_𝑘=1𝑧²_𝑘𝑝_𝑖𝑧_𝑖^𝑇(𝑡)(𝑡)≤𝑧⁴_𝑖𝜓_𝑖2𝑧_𝑖𝜃+^𝑖−1_𝑘=1𝜀_𝑘2𝑧⁴_𝑘, (25) where𝜓_𝑖2≥∑^𝑖−1_𝑘=1(3/4)(4𝜀_𝑘2)^−1/3(𝑝_𝑖(𝑧_𝑖))^3/4. 32𝑧²_𝑖Tr𝐺^𝑇_𝑖^𝑇(𝑡)(𝑡)𝐺_𝑖≤32𝑧²_𝑖𝑝²_𝑖2𝑧_𝑖^𝑖_𝑘=1𝑧_𝑘^2𝑇(𝑡)(𝑡≤3)2𝑧²_𝑖𝑖𝑝²_𝑖2𝑧_𝑖^𝑖_𝑘=1𝑧²_𝑘^𝑇(𝑡)(𝑡≤3)2𝑧⁴_𝑖𝑖𝑝²_𝑖2𝑧_𝑖𝜃+^𝑖−1_𝑘=1𝜀_𝑘3𝑧⁴_𝑘+^𝑖−1_𝑘=114𝜀_𝑘332𝑖𝑝²_𝑖2²𝑧⁴_𝑖𝜃≤^𝑖−1_𝑘=1𝜀_𝑘3𝑧⁴_𝑘+𝜓_𝑖3𝑧⁴_𝑖𝜃, (26) where𝜓_𝑖3≥(3/2)𝑖𝑝²_𝑖2+∑^𝑖−1_𝑘=1(1/4𝜀_𝑘3)((3/2)𝑖𝑝²_𝑖2)². 𝑑_𝑖−1𝑧³_𝑖−1𝑧_𝑖≤𝜆_𝑖−12𝑧³_𝑖−1𝑧_𝑖≤𝑒_𝑖−1𝑧⁴_𝑖−1+1443𝑒_𝑖−1⁻³𝑧⁴_𝑖𝜆⁴_𝑖2≤𝑒_𝑖−1𝑧⁴_𝑖−1+𝑏_𝑖𝑧⁴_𝑖, (27) where𝑏_𝑖≥(1/4)((4/3)𝑒_𝑖−1)⁻³𝜆⁴_𝑖2,𝜏_𝑖−1=𝑧⁴₁𝑝²₁₁+∑^𝑖−1_𝑘=2𝑧⁴_𝑘(𝜓_𝑘2+𝜓_𝑘3), and𝜏_𝑖=𝜏_𝑖−1+(𝜓_𝑖2+𝜓_𝑖3)𝑧⁴_𝑖. −𝑧³_𝑖𝜕𝛼_𝑖𝜕𝜃𝜏_𝑖≤𝑧³_𝑖𝜕𝛼_𝑖𝜕𝜃𝜏_𝑖−1+𝑧⁴_𝑖𝜓_𝑖2+𝜓_𝑖3≤𝑧⁴_𝑖1+𝑧³_𝑖𝜕𝛼_𝑖𝜕𝜃²𝜓_𝑖2+𝜓_𝑖3+𝑧³_𝑖𝜕𝛼_𝑖𝜕𝜃𝑧⁴₁𝑝²₁₁+^𝑖−1_𝑘=2𝑧⁴_𝑘𝜓_𝑘2+𝜓_𝑘3+34(4)^−1/3𝜕𝛼₂𝜕𝜃𝑧³₁𝑝²₁₁^4/3𝜀_𝑖4^−1/3𝑧⁴_𝑖≤𝜀_𝑖4𝑧⁴₁+34(4)^−1/31+𝜕𝛼_𝑖𝜕𝜃𝑧³₁𝑃²₁₁^24/3𝜀^−1/3_𝑖4𝑧⁴_𝑖+^𝑖−1_𝑘=2𝜀_𝑘4𝑧⁴_𝑘+^𝑖−1_𝑘=234(4)^−1/31+𝜕𝛼_𝑖𝜕𝜃𝑧³_𝑘𝜓_𝑘2+𝜓_𝑘3^24/3𝜀^−1/3_𝑘4𝑧⁴_𝑖≤^𝑖−1_𝑘=1𝜀_𝑘4𝑧⁴_𝑘+𝜓_𝑖4𝑧⁴_𝑖, (28) where𝜓_𝑖4≥(3/4)(4𝜀_𝑖4)^−1/31+((𝜕𝛼_𝑖/𝜕𝜃)𝑧³₁𝑃²₁₁)^24/3+∑^𝑖−1_𝑘=2(3/4)(4𝜀_𝑘4)^−1/31+((𝜕𝛼_𝑖/𝜕𝜃)𝑧³_𝑘(𝜓_𝑘2+𝜓_𝑘3))^24/3. 𝛼_𝑖+1𝑧_𝑖,𝜃=−𝑧_𝑖𝛽_𝑖𝑧_𝑖,𝜃,𝛽_𝑖𝑧_𝑖,𝜃=𝑐_𝑖+𝜓_𝑖1+𝜓_𝑖4+𝑏_𝑖+1+𝜓_𝑖2+𝜓_𝑖3²𝜃𝜆_𝑖1, (29) where𝑐_𝑖>0is a design parameter to be chosen.
With the aid of (24)–(29) and (14), (23) can be simplified as 𝐿𝑉_𝑖≤−^𝑖−1_𝑗=1𝑐_𝑗−𝜀_𝑗−𝑒_𝑗𝑧⁴_𝑗−𝑐_𝑖𝑧⁴_𝑖+𝑑_𝑖𝑧³_𝑖𝑧_𝑖+1+𝜃−𝜃−^𝑖_𝑘=2𝜕𝛼_𝑘𝜕𝜃𝑧³_𝑘̇𝜃−𝜏_𝑖. (30)
Finally, when𝑖=𝑛,  𝑧_𝑛+1=𝑢is the actual control. By choosing the actual control law and the adaptive law, 𝑢𝑧_𝑛,𝜃=−𝑧_𝑛𝛽_𝑛𝑧_𝑛,𝜃,̇𝜃=𝜏_𝑛=𝑧⁴₁𝑝²₁₁+^𝑛_𝑘=2𝑧⁴_𝑘𝜓_𝑘2+𝜓_𝑘3,𝛽_𝑛𝑧_𝑛,𝜃=𝑐_𝑛+𝑏_𝑛+𝜓_𝑛1+𝜓_𝑛4+1+𝜓_𝑛2+𝜓_𝑛3²𝜃𝜆_𝑛1,𝑢₁=𝑒^−𝛼𝑡𝑢, (31) where𝑐_𝑛>0is a design parameter to be chosen and𝜓_𝑛𝑖,𝑖=1,…4are smooth functions; we get 𝐿𝑉_𝑛≤−^𝑛_𝑗=1𝑐_𝑗−𝜀_𝑗−𝑒_𝑗𝑧⁴_𝑗, (32) where𝑉_𝑛(𝑧,𝜃)=∑^𝑛_𝑘=1(1/4)𝑧⁴_𝑘+(1/2)(𝜃−𝜃)²,𝑧=(𝑧₁,…𝑧_𝑛). We have finished the controller design procedure for𝑥₀(𝑡₀)≠0and the parameter identification. Without loss of generality, we can assume that𝑡₀≠0.

3.3. Switching Control and Main Result

In the preceding subsection, we have given controller design for𝑥₀≠0.Now, we discuss how to choose the control laws𝑢₀and𝑢₁when𝑥₀=0.We choose𝑢₀as𝑢₀=−𝜆₀𝑥₀+𝑢^∗₀,𝑢^∗₀>0. And choose the Lyapunov function𝑉₀=(1/2)𝑥²₀. Its time derivative is given by𝐿𝑉₀=−𝜆₀𝑥²₀+𝑢^∗₀, which leads to the bounds of𝑥₀. During the time period[0,𝑡_𝑠), using𝑢₀=−𝜆₀𝑥₀+𝑢^∗₀, new control law𝑢can be obtained by the control procedure described above to the original𝑥-subsystem in (1). Then, we can conclude that the𝑥-state of (1) cannot be blown up during the time period[0,𝑡_𝑠). Since at𝑥(𝑡_𝑠)≠0,we can switch the control inputs𝑢₀and𝑢to (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(𝑥₀,𝑥,𝜃)∈𝑅^𝑛, the closed-loop system has an almost surely unique solution on[0,∞), the solution process is bounded in probability, and𝑃{lim_𝑡→∞𝜃(𝑡)𝑒𝑥𝑖𝑠𝑡𝑠𝑎𝑛𝑑𝑖𝑠𝑓𝑖𝑛𝑖𝑡𝑒}=1.

Proof. According to the above analysis, it suffices to prove in the case𝑥₀(0)≠0.Since we have already proven that𝑥₀can be globally exponentially convergent to zero in probability in Section 3.1, we only need prove that𝑥(𝑡)is convergent to zero in probability also. In this case, we choose the Lyapunov function𝑉=𝑉_𝑛, and𝑐_𝑖>𝜀_𝑖+𝑒_𝑖; 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.

4. 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.

Acknowledgments

This work was supported by the university research projects of Department of Education in Shandong Province, China (J13LI03). The author would like to thank the reviewers for their helpful comments.