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; 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:

(1)

where x₀ ∈ R and x = [x₁, …, x_n] ^T ∈ Rⁿ are the system states and u₀ ∈ R and u₁ ∈ R are the control inputs, respectively.

, (i = 1,2, …, n), and

; ω ∈ R^r is an r-dimensional standard Wiener process defined on the complete probability space (Ω, F, P) with Ω being a sample space, F being a filtration, and P being measure. The drift and diffusion terms f_i(·), φ_i(·) are assumed to be smooth, vanishing at the origin (x₁, x₂, …, x_i) = (0,0, …, 0); ∑(t) : R₊ → R^r×r is the Borel bounded measurable functions and is nonnegative definite for each t ≥ 0. d_i(t) are disturbed virtual control coefficients, where i = 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:

(2)

where x ∈ Rⁿ is the state of system (2), the Borel measurable functions: f : Rⁿ⁺¹ → Rⁿ and g : Rⁿ⁺¹ → R^n×r are assumed to be C¹ in their arguments, and ω ∈ R^r is an r-dimensional standard Wiener process defined on the complete probablity space (Ω, F, P).

Definition 1 (see [8].)Given any V(x, t) ∈ C^1,2, for stochastic nonlinear system (2), the differential operator L is defined as follows:

(3)

where C^1,2(Rⁿ × R₊; R₊) denotes all nonnegative functions V(x, t) on Rⁿ × R₊, which are C¹ in t and C² in x, and for simplicity, the smooth function f(·) is denoted by f.

Lemma 2 (see [8].)Let x and y be real variables. Then, for any positive integers m, n, and any real number ε > 0, the following inequality holds:

(4)

Lemma 3 (see [7].)Considering the stochastic nonlinear system (2), if there exist a C^1,2 function V(x, t), K_∞ class functionsand, constant, and a nonnegative functions W(x, t) such that

(5)

then for each x₀ ∈ Rⁿ. (1) For (2), there exists an almost surely unique solution on [0, ∞]. (2) When

, f(0, t) = 0, g(0, t) = 0, and W(x, t) = W(x) is continuous, the equilibrium x = 0 is globally stable in probability, and the solution x(t) satisfies P{lim _t→∞W(x(t) = 0} = 1. (3) For any given ε > 0, there exist a class KL function β_c(·, ·) and K function γ(·) such that P{|(x(t))| < β_c(|x₀|, t) + γ(c)} ≥ 1 − ε for any t ≥ 0, x₀ ∈ Rⁿ∖{0}.

Lemma 4 (see [20].)For any real-valued continuous function f(x, y), x ∈ R^m, y ∈ Rⁿ, there exist smooth scalar-value funcions a(x) ≥ 0, b(y) ≥ 0, c(x) > 1, and d(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 that x₀(t₀) ≠ 0 is firstly considered. Then, the case where the initial x₀(t₀) = 0 is dealt with later. The triangular structure of system (1) suggests that we should design the control inputs u₀ and u₁ in two separate stages.

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

Assumption 5. For 0 ≤ i ≤ n, there are some positive constants λ_i1 and λ_i2 that satisfy the inequality λ_i1 ≤ d_i(t) ≤ λ_i2.

Assumption 6. For f₀(t, x₀), there exists a nonnegative smooth function γ₀(t, x₀), such that | f₀(t, x₀)|≤|x₀ | γ₀(t, x₀).

For each, there exist nonnegative smooth functionsand, such that,.

3.1. Designing u₀ for x₀-Subsystem

For x₀-subsystem, the control u₀ can be chosen as

(6)

where λ₀ = (k₀ + γ₀)/λ₀₁ and k₀ is a positive design parameter.

Consider the Lyapunov function candidate

. From (6) and Assumptions 5 and 6, we have

(7)

So, we obtain the first result of this paper.

Theorem 7. The x₀-subsystem, under the control law (6) with an appropriate choice of the parameters k₀, λ₀₁, λ₀₂, is globally exponentially stable.

Proof. Clearly, from (7), LV₀ ≤ 0, which implies that. Therefore, x₀ is globally exponentially convergent. Consequently, x₀ can be zero only at t = t₀, when x(t₀) = 0 or t = ∞. It is concluded that x₀ does not cross zero for all t ∈ (t₀, ∞) provided that x(t₀) ≠ 0.

Remark 8. If x(t₀) ≠ 0, u₀ exists and does not cross zero for all t ∈ (t₀, ∞) independent of the x-subsystem from (6).

3.2. Backstepping Design for u₁

From the above analysis, the x₀-state in (1) can be globally exponentially regulated to zero as t → ∞, obviously. In this subsection, we consider the control law u₁ for the x-subsystem by using backstepping technique. To design a state-feedback controller, one first introduces the following discontinuous input-state-scaling transformation:

(8)

Under the new x-coordinates, x-subsystems is transformed into

(9)

where

(10)

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

Lemma 9. For i = 1,2 … n, there exist nonnegative smooth functions,, such that

(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

(13)

where

To design a state-feedback controller, one introduces the coordinate transformation

(14)

where α₂, …, α_n are smooth virtual control laws and will be designed later and α₁ = 0.

denotes the estimate of θ, where

(15)

Then using (9), (10), (14) and It

differentiation rule, one has

(16)

where η_n+1 = u,

, and

where i = 1,2 … n. Using Lemmas 2, 4, and 9 and (14), we easily obtain the following lemma.

Lemma 10. For 1 ≤ i ≤ n, there exist nonnegative smooth functions,, and, such that

(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 function. By (14), (15), and (16), we have

(18)

Using Lemma 10 and Lemma 4, we have

(19)

Substituting (19) into (18) and using (14), we have

(20)

where

. Substituting α₂ into (20), we have

(21)

where

Step i. (2 ≤ i ≤ n). Assume that at step i − 1, there exists a smooth state-feedback virtual control, such that

(22)

where

, and

where j = 1, …, n.

Then, define the ith Lyapunov candidate function. From (16) and (22), it follows that

(23)

Using Lemmas 9 and 4, there are always known nonnegative smooth functions

and constant ε_i > 0, ε_ij > 0, where i = 1, …, n and j = 1,2, 3,4.

Consider

(24)

where

(25)

where

(26)

where

(27)

where

, and

(28)

where

(29)

where c_i > 0 is a design parameter to be chosen.

With the aid of (24)–(29) and (14), (23) can be simplified as

(30)

Finally, when i = n, z_n+1 = u is the actual control. By choosing the actual control law and the adaptive law,

(31)

where c_n > 0 is a design parameter to be chosen and ψ_ni, i = 1, …4 are smooth functions; we get

(32)

where

. We have finished the controller design procedure for x₀(t₀) ≠ 0 and the parameter identification. Without loss of generality, we can assume that t₀ ≠ 0.

3.3. Switching Control and Main Result

In the preceding subsection, we have given controller design for x₀ ≠ 0. Now, we discuss how to choose the control laws u₀ and u₁ when x₀ = 0. We choose u₀ as. And choose the Lyapunov function. Its time derivative is given by, which leads to the bounds of x₀. During the time period [0, t_s), using, new control law u can be obtained by the control procedure described above to the original x-subsystem in (1). Then, we can conclude that the x-state of (1) cannot be blown up during the time period [0, t_s). Since at x(t_s) ≠ 0, we can switch the control inputs u₀ and u 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, the closed-loop system has an almost surely unique solution on [0, ∞), the solution process is bounded in probability, and.

Proof. According to the above analysis, it suffices to prove in the case x₀(0) ≠ 0. Since we have already proven that x₀ can be globally exponentially convergent to zero in probability in Section 3.1, we only need prove that x(t) is convergent to zero in probability also. In this case, we choose the Lyapunov function V = V_n, and c_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.

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