#### Abstract

This paper investigates the state-feedback stabilization of stochastic nonholonomic systems with an unknown time-varying delay. Without imposing any assumptions on the time-varying delay, a state-feedback controller is skillfully designed by using input-state-scaling technique and backstepping control approach. The switching strategy is proposed to eliminate the phenomenon of uncontrollability and to guarantee that the closed-loop system has an almost surely unique solution for any initial state and the equilibrium of interest is globally asymptotically stable in probability. A simulation example demonstrates the effectiveness of the proposed scheme.

#### 1. Introduction

Nonholonomic systems represent an important class of control systems, which arises in many mechanical systems such as mobile robots, car-like vehicles, underactuated satellites, and the knife-edge. It is well known that the control of nonholonomic systems is extremely challenging, largely due to the limitation imposed by Brockett’s stability condition [1]. This class of nonlinear systems cannot be stabilized to a point by smooth or even continuous pure state feedback, which makes the well-developed smooth and continuous control methods cannot be directly applied to these systems. This motivates researchers to seek for effective control strategies such as discontinuous feedback, time-varying feedback, and hybrid control laws.

Since the existences of stochastic disturbances in many practical systems may change deterministic systems into stochastic systems, the control and design of stochastic systems has become an important role in the field of engineering. In [2], the backstepping design was firstly introduced, which makes the stochastic nonlinear control to obtain a breakthrough. Then, based on the backstepping technique and quartic Lyapunov functions, many results have been obtained for stochastic nonlinear systems with different structures [3–8]. Specially, by using the stochastic Lyapunov-like theorem and backstepping design technique, the state or output-feedback stabilization for stochastic nonholonomic systems was obtained in [9–14]. Zhang et al. [11] studied the problem of adaptive stabilization of stochastic nonholonomic systems with nonhomogeneous uncertainties. Zhao et al. [12] designed a state-feedback controller to stabilize a class of more general high-order stochastic nonholonomic systems. Du et al. [13, 14] studied the design of controllers for a class of stochastic high-order nonholonomic systems, which were considered to cancel the power order restriction in [12]. However, the aforementioned contributions have not taken into account the effect of time delay on the systems.

The research of time delay systems has been received widespread attentions [15–25]. One reason is that the time-delay phenomenon frequently arises and is inevitable in many practical systems, such as chemical engineering systems, communication systems, and mechanical systems. Another reason is that time delay is a primary source of instability and performance and makes practical control systems hard to control. Naturally, the research of this kind of systems plays an important role in control theory and practical applications. Compared with the previous results without time delay, the control problem of a time-delay system is more challenging. One main obstacle that exists in the stabilization procedure is the restrictive growth conditions imposed on delay-dependent nonlinear functions, for example, Liu et al. [16], Chen et al. [17], and Liu and Wu [18]. Wu and Wu [19] proposed a robust state-feedback switching controller to stabilize time-delay nonholonomic systems with strongly nonlinear uncertainties by using discontinuous transformation and dynamic feedback approach. Subsequently, Wu and Liu [20] addressed the output-feedback stabilization problem for time-delay nonholonomic systems whose time-delay exists in polynomial nonlinear growing conditions, and they considered the output-feedback control problem for high-order nonholonomic time-delay systems [21]. However, all the above references considered the time-delay nonholonomic systems in the deterministic case, and up to now, fewer researchers studied this kind of time-delay systems with stochastic disturbance. Compared with the deterministic systems, obviously, stochastic time-delay systems contain more uncertainties in drift and diffusion terms than those in the related papers. Qin and Min [22] studied the adaptive stabilization problem for a class of stochastic nonholonomic systems with time delays. This work gave the adaptive state-feedback control at the expense of delay-dependent nonlinear diffusion terms and time-delay assumption, i.e., for a known positive constant Wu et al. [23] studied state-feedback stabilization of stochastic nonholonomic systems under arbitrary switchings with time-varying delays. But the time-delay terms of stochastic nonholonomic systems is required to satisfy . Hence, it is necessary to introduce stochastic nonholonomic systems with an unknown time-varying delay and explore how to remove the restrictions assumed on time-varying delay and to weaken the growth assumptions imposed on and

Recently, Min et al. [24] have studied the problem of globally adaptive control for stochastic nonlinear time-delay systems with perturbations and its application. Without imposing any assumptions on the time-varying delay, an adaptive state-feedback controller was skillfully designed by using adaptive backstepping control technique, and it was proven that the constructed controller can guarantee the closed-loop system to be globally asymptotically stable in probability. So, how to extend this method to stochastic nonholonomic systems with an unknown time-varying delay is very interesting and significant.

In the following, we will make efforts to solve this issue. The main contributions are summarized as follows:(i)Stochastic nonholonomic systems with an unknown time-varying delay are globally stabilized for the first time. To compensate for the unknown time-varying delay, the Lyapunov–Razumikhin lemma is employed.(ii)We use subtly input-state-scaling technique and Itô formula to change the stochastic nonholonomic systems into stochastic nonlinear systems that is a more complicated system in the study of Min et al. [24], which is very important. Since Itô stochastic differentiation involves not only the gradient but also the Hessian term in the Lyapunov design procedure of stochastic systems, it will produce much more nonlinear terms than those in the stochastic nonlinear systems in the study of Min et al. [24]. How to deal with them skillfully is one of the main technical obstacles in our paper.(iii)Compared with the related references, a distinctive feature is that the restrictions assumed on time-varying delay are removed and the growth assumptions imposed on and are somewhat weakened. Then, a state-feedback controller is skillfully designed which renders the closed-loop system globally asymptotically stable (GAS) in probability.

The paper is organized as follows: Section 2 begins with the preliminary results. Section 3 shows the input-state-scaling technique and the backstepping design procedure, while Section 4 provides the switching control strategy and Section 5 provides the main result. Section 6 gives a simulation example to illustrate the theoretical finding of this paper. Finally, Section 7 concludes the paper.

#### 2. Preliminary Results

Consider the following stochastic nonlinear time-delay system:where is the state, is a time-varying delay which is the Borel measurable function, the initial value is defined as , and is an *r*-dimensional independent standard Wiener process defined on the complete probability space The Borel measurable functions are locally the Lipschitz functions with and

For any given , the differential operator along (1) is defined aswhere is said to be the Hessian term of .

The following definitions and lemmas will be used throughout the paper:

*Definition 1 [2]. *The equilibrium of system (1) with and is(i) Globally stable in probability if for any , there exists a class function such that (ii)Globally asymptotically stable in probability if it is globally stable in probability and

Lemma 1 [2]. *Consider the stochastic system (1); if there exist a function , class functions and , constants , and a nonnegative function such thatthen*(1)*For (1), there exists an almost surely unique solution on for each belongs to *(2)*When , and is continuous, then the equilibrium is globally stable in probability and *

Lemma 2 [24]. *For system (1), let be positive constants and . Assume that there exists a continuous, positive definite Lyapunov function and such that , for all , and moreover, provided , satisfying . Then, for all initial values the solution of system (1) is GAS in probability.*

Lemma 3 [26]. *Let be real numbers, then*

Lemma 4 [27]. *Let be real variables. For any positive real numbers and the following inequality holds:*

#### 3. State-Feedback Controller Design

Let us consider the following stochastic nonholonomic systems with an unknown time-varying delay:where and are the system states, and are the control inputs; , and ; and is an unknown time-varying delay. is an *r*-dimensional standard Wiener process defined on a probability space , with being a sample space, being a filtration, and being a probability measure; the drift terms and the diffusion terms , are unknown locally Lipschitz functions with , , .

The following assumptions are made on systems (7a) and (7b). For simplicity, sometimes we write as and as

*Assumption 1. *For smooth function , there is a known constant vector such that

*Assumption 2. *For the drift terms and diffusion terms and in system (7a) and (7b), there exist smooth nonnegative functions and such that

*Remark 1. *There are some differences between the assumptions here and the existing ones. When , systems (7a) and (7b) become the time-delay nonholonomic systems in the deterministic case, which has been considered in the literature [19–21]. In [20], the constant time-delay was studied for the deterministic system and the time-delay terms are required to satisfy , which were studied in [18, 21]. Compared with the deterministic systems, obviously, systems (7a) and (7b) contain more uncertainties in drift and diffusion terms than those in the related papers. In detail, of Assumption 2 was assumed being positive constants in [18]. Compared with the related references, a distinctive feature is that the restrictions assumed on time-varying delay are removed and the growth assumptions imposed on and are somewhat weakened by allowing and being functions related to system states. These differences show the assumptions of this paper are more general.

#### 4. State-Feedback Control Design

In this section, we present the design procedure. For clarity, the case that is considered first. Then the case that the initial is dealt later. The inherent structure of system (7a) suggests that we should design the control inputs and in two separate stages.

Let us consider the subsystem (7a) in stochastic nonholonomic time-delay systems (7a) and (7b). In order to guarantee that converges to zero, one can take as follows:where is a positive constant.

If we take a Lyapunov function of the formfrom (2), Assumption 1, one can obtain

Theorem 1. *If Assumption 1 holds for the stochastic nonholonomic time-delay subsystem (7a), then under the smooth controller (10), one has the following:*(1)*The closed-loop system has an almost surely unique solution on for *(2)*The equilibrium of the closed-loop system is globally asymptotically stable in probability*

*Remark 2. *Proof of Theorem 1 is similar to Theorem 1 in [13]. Due to the space limitation, we omit the proof details.

Consider the second subsystem of stochastic nonholonomic time-delay system (7b). In order to design a state-feedback controller, the following state-input scaling transformation is needed:Under the new coordinates, with choice as in (10), the subsystem is transformed intowhereFor using backstepping technique, we define the error variable:where are the virtual control laws to be designed.

Then, by ’s rule in [28], (14)–(16), one haswhere

Lemma 5. *For , there exist positive smooth functions such that*

*Remark 3. *Proof of the above inequalities is based on (13), (15), and Assumption 2. Due to the space limitation, we omit the proof details.

By the following the standard procedures, a state-feedback controller can be obtained:

*Step 1. *Consider the Lyapunov functionfrom Lemma 5, there exist nonnegative smooth functions such thatThus, we havewhere are positive design constants:Constructing , . Then, adding and subtracting the term on the right-hand side of (22), one getswhere is a parameter to be designed, .

Inductive step : assume that, at step , there are , proper and positive definite Lyapunov function , and the virtual controllers defined bywith being smooth such thatwhere , are the design parameters. Then, for the Lyapunov functionthere exists a virtual control lawsuch thatwhere is the known continuous function and and for are the design constants.

*Proof. *Firstly, according to Definition 1, (26) and (27), one obtainsTo proceed further, we estimate each term on the right-hand side of (30). Using Assumption 2 and Lemma 4, we havewhere are the arbitrary position constants; , are the nonnegative continuous functions; and