Further Result on Finite-Time Stabilization of Stochastic Nonholonomic Systems

Abstract

This paper further investigates the problem of finite-time state feedback stabilization for a class of stochastic nonholonomic systems in chained form. Compared with the existing literature, the stochastic nonholonomic systems under investigation have more uncertainties, such as the 𝑥₀-subsystem contains stochastic disturbance. This renders the existing finite-time control methods highly difficult to the control problem of the systems or even inapplicable. In this paper, by extending adding a power integrator design method to a stochastic system and by skillfully constructing 𝐶² Lyapunov function, a novel switching control strategy is proposed, which renders that the states of closed-loop system are almost surely regulated to zero in a finite time. A simulation example is provided to demonstrate the effectiveness of the theoretical results.

1. Introduction

The nonholonomic systems, which can be used to model many frequently met mechanical systems, such as wheeled mobile robot, knife edge and rolling disk, have been an active research field over the past decades. From Brockett’s necessary condition [1], it is well known that the nonholonomic systems cannot be stabilized to the origin by any static continuous state feedback, so the classical smooth control theory cannot be applied directly. In order to overcome this obstruction, several novel approaches have been developed for the problem, such as discontinuous time-invariant stabilization [2, 3], smooth time-varying stabilization [4–6], and hybrid stabilization [7]. Using these valid approaches, many fruitful results have been developed [8–15]. Particularly, the stochastic nonholonomic systems, which can be viewed as the extension of the classical nonholonomic systems, have been recently achieved investigations [16–18]. However, it should be noted that those aforementioned papers consider the feedback stabilizer that makes the trajectories of the systems converge to the equilibrium as the time goes to infinity.

Compared to asymptotic stabilization, the closed-loop system with finite-time convergence usually demonstrates faster convergence rates, higher accuracies, better disturbance rejection properties, and robustness against uncertainties [19]. Hence, it is more meaningful to investigate the finite-time stabilization problem than the classical asymptotical stability. For the deterministic case, in [20], a novel switching finite-time control strategy was proposed to nonholonomic systems in a chained form with uncertain parameters and perturbed terms by the use of time rescaling and Lyapunov based method. Later this result was essentially extended under weaker constraints on drift terms in [21]. As a natural extension, the finite-time stabilization for stochastic nonholonomic systems is an interesting and challenging subject of intensive study. However, it cannot be solved by simply extending the methods for deterministic systems because of the presence of stochastic disturbance. As pointed out by Yin et al. [22], “the existence of a unique solution and the non-satisfaction of local Lipschitz condition are two prerequisites of discussing the finite-time stability for a stochastic nonlinear system. That is, as a common assumption condition to guarantee the existence of a unique solution for stochastic nonlinear system, local Lipschitz condition cannot be applied to study stochastic finite-time stability.” Therefore, the finite-time controller design for stochastic nonholonomic systems should solve the following questions: under what conditions, the stochastic nonholonomic systems exist possibly finite-time stabilizer? Under these conditions, how can one design a finite-time state feedback stabilizing controller?

To the best of the authors’ knowledge, only the authors in [23] considered the aforementioned problems when the𝑥₀-subsystem is ordinary differential equation. This paper continues the investigation in [23] and addresses the finite-time stabilizing control design for stochastic nonholonomic systems with more uncertainties than those in [16–18, 23], which cannot be handled by general existing methods. A constructive method in designing finite-time stabilizing controller for such uncertain system is proposed. The contribution of this paper is highlighted as follows. First, inspired by the deterministic works in [24, 25], we generalize adding a power integrator design method [26] to a stochastic system, and a novel 𝐶² Lyapunov function is constructed to deal with stochastic disturbances. Second, based on a new switching method, a systematic control design procedure is proposed to solve the finite-time stabilization problem for all plants in the considered class.

The remainder of this paper is organized as follows. Section 2 presents some necessary notations, definitions, and preliminary results. Section 3 describes the systems to be studied and formulates the control problem. Section 4 presents the design scheme to the controller and the main result. Section 5 gives a simulation example to illustrate the theoretical finding of this paper. Finally, concluding remarks are proposed in Section 6.

2. Notations and Preliminary Results

The following notations, definitions, and lemmas are to be used throughout the paper. 𝑅⁺ denotes the set of all nonnegative real numbers, and 𝑅^𝑛 denotes the real 𝑛-dimensional space. (𝑎,𝑏)_odddenotes the set{𝑥∣𝑥=𝑝/𝑞and𝑥∈(𝑎,𝑏), where𝑝 and 𝑞 are odd integers}. (𝑎,𝑏)_even denotes the set {𝑥∣𝑥=𝑝/𝑞 and𝑥∈(𝑎,𝑏), where 𝑝 is an even integer and 𝑞 is an odd integer}. For a given vector or matrix 𝑋, 𝑋^𝑇 denotes its transpose, Tr{𝑋} denotes its trace when 𝑋 is square, and |𝑋| is the Euclidean norm of a vector 𝑋. 𝐶^𝑖 denotes the set of all functions with continuous 𝑖th partial derivatives. 𝐾 denotes the set of all functions:𝑅⁺→𝑅⁺, which are continuous, strictly increasing, and vanishing at zero; 𝐾_∞ denotes the set of all functions which are of class 𝐾 and unbounded. The arguments of the functions (or the functionals) will be omitted or simplified, whenever no confusion can arise from the context. For instance, we sometimes denote a function 𝑓(𝑥(𝑡)) by simply 𝑓(𝑥) or 𝑓.

We begin with some basic concepts and terminologies related to the notion of stochastic finite-time stability. The reader is referred to [22, 27, 28] as well as the references therein for additional details.

Consider the stochastic nonlinear system 𝑑𝑥=𝑓(𝑡,𝑥)𝑑𝑡+𝑔(𝑡,𝑥)𝑑𝜔, (1) where 𝑥∈𝑅^𝑛 is the system state with the initial condition 𝑥(0)=𝑥₀; 𝜔 is an m-dimensional independent standard Wiener process defined on a complete probability space (Ω,ℱ,{ℱ_𝑡}_𝑡≥0,𝑃) withΩbeing a sample space, ℱ being a 𝜎-field,{ℱ_𝑡}_𝑡≥0 being a filtration, and 𝑃 being a probability measure. The functions: 𝑓∶𝑅⁺×𝑅^𝑛→𝑅^𝑛 and 𝑔∶𝑅⁺×𝑅^𝑛→𝑅^𝑛×𝑚 are piecewise continuous and continuous with respect to the first and second arguments, respectively, and satisfy 𝑓(𝑡,0)≡0 and 𝑔(𝑡,0)≡0. Moreover, system (1) is assumed to have a pathwise unique strong solution, denoted by 𝑥(𝑡,𝑥₀).

Definition 1. For system (1), define 𝑇₀(𝑥₀,𝑤)=inf{𝑇≥0∶𝑥(𝑡,𝑥₀)=0, for all  𝑡≥𝑇}, which is called the stochastic settling time function. Especially,𝑇₀(𝑥₀,𝑤)∶=+∞if𝑥(𝑡,𝑥₀)≠0, for all 𝑡≥0.

Definition 2. The equilibrium 𝑥≡0 of the system (1) is said to be a stochastic finite-time stable equilibrium if(i)it is stable in probability: for every pair of 𝜀∈(0,1) and𝑟>0, there exists𝛿>0such that 𝑃{|𝑥(𝑡,𝑥₀)|<𝑟,  for all 𝑡≥0}≥1−𝜀, whenever|𝑥|<𝛿;(ii)its stochastic settling-time function𝑇₀(𝑥₀,𝑤)exists finitely with probability and𝐸[𝑇₀(𝑥₀,𝑤)]<∞.

Next, we give two lemmas where the first one provides sufficient conditions to ensure the existence of pathwise unique strong solution to system (1), and the other one has been used to determine the finite-time stability of stochastic nonlinear systems.

Lemma 3 (see [23, 29]). Assume that 𝑓(𝑡,𝑥) and 𝑔(𝑡,𝑥) are continuous in 𝑥. Further, for any 0<𝛿<1, each 𝑁=1,2,…, and each 0≤𝑇<∞, if the following conditions hold: 𝑓((𝑡|𝑥,𝑔(𝑡,𝑥)≤𝑐)(1+|)𝑡,𝑥)²≤𝑐(𝑡)1+|𝑥|²,𝑓𝑡,𝑥₁−𝑓𝑡,𝑥₂≤𝑐^𝑁_𝑇(𝑡)𝑥₁−𝑥₂,𝑔𝑡,𝑥₁−𝑔𝑡,𝑥₂≤𝑐^𝑁_𝑇(𝑡)𝑥₁−𝑥₂, (2) as 0<𝛿≤|𝑥_𝑖|≤𝑁,𝑖=1,2,𝑡∈[0,𝑇], where 𝑐(𝑡) and𝑐^𝑁_𝑇(𝑡)are nonnegative functions such that ∫^𝑇₀𝑐(𝑡)𝑑𝑡<∞ and ∫^𝑇₀𝑐^𝑁_𝑇(𝑡)𝑑𝑡<∞. Then for any given𝑥₀∈𝑅^𝑛, system (1) has a pathwise unique strong solution.

Lemma 4 (see [30]). Consider the stochastic nonlinear system described in (1). Suppose that there exist a 𝐶² function 𝑉(𝑥), class 𝐾_∞   functions 𝜇₁ and 𝜇₂, real numbers𝑐>0and0<𝛼<1, such that 𝜇₁(|𝑥|)≤𝑉(𝑥)≤𝜇₂𝑥,(𝑥)=(||)ℒ𝑉𝜕𝑉1𝜕𝑥𝑓+2Tr𝑔^𝑇𝜕²𝑉𝜕𝑥²𝑔≤−𝑐𝑉^𝛼(𝑥). (3) Then it is globally finite-time stable in probability, and the stochastic settling time function 𝑇₀(𝑥₀,𝑤) satisfies 𝐸𝑇₀𝑥₀≤𝑉,𝑤^1−𝛼𝑥₀𝑐().1−𝛼 (4)

Then we list two lemmas that serve as the basis of the key tools for the adding a power integrator technique.

Lemma 5 (see [31]). For 𝑥∈𝑅, 𝑦∈𝑅, and 𝑝≥1 being a constant, the following inequalities hold: 𝑥+𝑦^𝑝≤2^𝑝−1𝑥^𝑝+𝑦^𝑝,|𝑥|+𝑦^1/𝑝≤|𝑥|^1/𝑝+𝑦^1/𝑝≤2^{(𝑝−1)/𝑝}|𝑥|+𝑦^1/𝑝. (5) If𝑝≥1is odd, then 𝑥−𝑦^𝑝≤2^𝑝−1𝑥^𝑝−𝑦^𝑝,𝑥^1/𝑝−𝑦^1/𝑝≤2^{(𝑝−1)/𝑝}𝑥−𝑦^1/𝑝. (6)

Lemma 6 (see [32]). Let𝑥,𝑦be real variables; then for any positive real numbers𝑎,𝑚, and𝑛, one has 𝑎|𝑥|^𝑚𝑦^𝑛≤𝑏|𝑥|^𝑚+𝑛+𝑛𝑚+𝑛𝑚+𝑛𝑚^−𝑚/𝑛×𝑎^{(𝑚+𝑛)/𝑛}𝑏^−𝑚/𝑛𝑦^𝑚+𝑛, (7) where𝑏>0is any real number.

3. Problem Formulation

Motivated by the statement in [18, 23] that many nonholonomic mechanical systems, such as wheeled mobile robot, subject to stochastic disturbances, can be transformed to a kind of stochastic nonholonomic systems in the so-called chained form, this paper considers the following class of stochastic nonholonomic systems in chained form: 𝑑𝑥₀=𝑑₀(𝑡)𝑢₀𝑑𝑡+𝑓₀𝑡,𝑥₀𝑑𝑡+𝑔^𝑇₀𝑡,𝑥₀𝑑𝑤,𝑑𝑥_𝑖=𝑑_𝑖(𝑡)𝑥_𝑖+1𝑢₀𝑑𝑡+𝑓_𝑖𝑡,𝑥₀,𝑥𝑑𝑡+𝑔^𝑇_𝑖𝑡,𝑥₀,𝑥𝑑𝑤,𝑖=1,…,𝑛−1,𝑑𝑥_𝑛=𝑑_𝑛(𝑡)𝑢₁𝑑𝑡+𝑓_𝑛𝑡,𝑥₀,𝑥𝑑𝑡+𝑔^𝑇_𝑛𝑡,𝑥₀,𝑥𝑑𝑤, (8) where𝑥₀∈𝑅and 𝑥=(𝑥₁,…,𝑥_𝑛)^𝑇∈𝑅^𝑛 are system states and 𝑢₀∈𝑅 and 𝑢₁∈𝑅 are control inputs, respectively; 𝑑_𝑖∶𝑅⁺→𝑅, 𝑖=0,…,𝑛 are uncertain and continuous, called the control coefficients. The nonlinear functions 𝑓₀∶𝑅⁺×𝑅→𝑅 and 𝑔₀∶𝑅⁺×𝑅→𝑅^𝑚, 𝑓_𝑖∶𝑅⁺×𝑅^𝑖+1→𝑅 and 𝑔_𝑖∶𝑅⁺×𝑅^𝑖+1→𝑅^𝑚, 𝑖=1,…,𝑛 are assumed to be 𝐶¹ with their arguments with 𝑓₀(𝑡,0)=0, 𝑔₀(𝑡,0)=0, 𝑓_𝑖(𝑡,𝑥₀,0)=0 and 𝑔_𝑖(𝑡,𝑥₀,0)=0.  and 𝜔 is an m-dimensional independent standard Wiener process defined on a complete probability space (Ω,𝐹,𝑃) with Ω being a sample space, 𝐹 being a filtration, and 𝑃 being a probability measure.

Remark 7. Obviously, the stochastic nonholonomic system (8) contains more uncertainties in drift and diffusion terms than those in the closely related papers [16–18, 23]. This, especially the existence of stochastic disturbance in the𝑥₀-subsystem, renders the procedure of control design in [23] inapplicable to the finite-time control problem of the system. Up to now, how to design a stabilizing controller to achieve the finite-time stabilization of the system (8) is unsolved. It is precisely our intention of this paper.
The following assumptions regarding system (8) are imposed throughout the paper.

Assumption 8. For 𝑖=0,…,𝑛, there are known positive constants 𝑐_𝑖1 and 𝑐_𝑖2 such that 𝑐_𝑖1≤𝑑_𝑖(𝑡)≤𝑐_𝑖2. (9)

Assumption 9. For 𝑓₀ and 𝑔₀, there are constants𝑎>0and 𝑟∈(0,1)_odd such that 𝑓₀𝑡,𝑥₀−𝑓₀𝑡,𝑥₀≤𝑎𝑥₀−𝑥₀^𝑟,𝑔₀𝑡,𝑥₀−𝑔₀𝑡,𝑥₀≤𝑎𝑥₀−𝑥₀^(𝑟+1)/2. (10)

Assumption 10. For 𝑖=1,…,𝑛, there is a constant𝑏>0such that for any𝜏∈(−2/(4𝑛+1),0), 𝑓_𝑖𝑡,𝑥₀,𝑥−𝑓_𝑖𝑡,𝑥₀,𝑥≤𝑏𝑥₁−𝑥₁^{(𝑚𝑖+𝜏)/𝑚1}+⋯+𝑥_𝑖−𝑥_𝑖^{(𝑚𝑖+𝜏)/𝑚𝑖},𝑔_𝑖𝑡,𝑥₀,𝑥−𝑔_𝑖𝑡,𝑥₀,𝑥≤𝑏𝑥₁−𝑥₁^{(2𝑚𝑖+𝜏)/2𝑚1}+⋯+𝑥_𝑖−𝑥_𝑖^{(2𝑚𝑖+𝜏)/2𝑚𝑖}, (11) where 𝑚_𝑖=1+(𝑖−1)𝜏.
For simplicity, in this paper we assume that 𝜏=−𝑝/𝑞 with 𝑝 being any even integer and 𝑞 being any odd integer, under which and the definition of 𝑚_𝑖 in Assumption 10, we know that 𝑚_𝑖 is an odd number.

Remark 11. Noting that 𝑓₀(𝑡,0)=0, 𝑔₀(𝑡,0)=0, 𝑓_𝑖(𝑡,𝑥₀,0)=0 and 𝑔_𝑖(𝑡,𝑥₀,0)=0 are assumed, Assumptions 9 and 10 imply that 𝑓₀𝑡,𝑥₀≤𝑎𝑥₀^𝑟,𝑔₀𝑡,𝑥₀≤𝑎𝑥₀^(𝑟+1)/2,𝑓_𝑖𝑡,𝑥₀,𝑥≤𝑏𝑥₁^{(𝑚𝑖+𝜏)/𝑚1}+⋯+𝑥_𝑖^{(𝑚𝑖+𝜏)/𝑚𝑖},𝑔_𝑖𝑡,𝑥₀,𝑥≤𝑏𝑥₁^{(2𝑚𝑖+𝜏)/2𝑚1}+⋯+𝑥_𝑖^{(2𝑚𝑖+𝜏)/2𝑚𝑖}. (12) It is noteworthy that Assumption 10 is a generalization of the homogeneous growth condition introduced in [33] where 𝑥=0 and 𝜏≥0. Assumptions 9 and 10 are necessary, which play an essential role in ensuring the existence of finite-time stabilizer for stochastic nonholonomic system (8).

4. Finite-Time Control Design

In this section, we give a constructive procedure for the finite-time stabilizer of system (8). The design of finite-time state feedback controller is divided into two steps.

Step 𝐴. We first stabilize the 𝑥-subsystem in a finite time almost surely.

For the 𝑥₀-subsystem, we choose the control 𝑢₀ as 𝑢₀≡𝑢^∗₀, (13) where 𝑢^∗₀ is a positive constant. In this case, the 𝑥₀-subsystem becomes 𝑑𝑥₀=𝑑₀𝑢^∗₀+𝑓₀𝑑𝑡+𝑔^𝑇₀𝑑𝑤. (14)

With the help of Assumptions 8 and 9 and Lemma 3, it easy to obtain that system (14) has a unique solution on [0,𝑇], for any given finite time 𝑇>0. Hence, 𝑥₀ is well defined on [0,𝑇]. Next, we stabilize the 𝑥-subsystem 𝑑𝑥_𝑖=𝑑_𝑖(𝑡)𝑢^∗₀𝑥_𝑖+1𝑑𝑡+𝑓_𝑖𝑡,𝑥₀,𝑥𝑑𝑡+𝑔^𝑇_𝑖𝑡,𝑥₀,𝑥𝑑𝑤,𝑖=1,…,𝑛−1,𝑑𝑥_𝑛=𝑑_𝑛(𝑡)𝑢₁𝑑𝑡+𝑓_𝑛𝑡,𝑥₀,𝑥𝑑𝑡+𝑔^𝑇_𝑛𝑡,𝑥₀,𝑥𝑑𝑤, (15) within a finite time. The control law 𝑢₁ can be recursively constructed by applying the method of adding a power integrator.

Step 1. Let 𝜉₁=𝑥^𝜎₁, where 𝜎∈(2,∞)_odd is a constant. Choosing the Lyapunov function 𝑉₁=𝑥₁^{(4𝑙𝜎−𝜏)}/(4𝑙𝜎−𝜏), where 𝑙 is a constant satisfying 4𝑙∈(0,∞)_even and (4𝑙−2)𝜎≥1. With the help of Remark 11 and Lemma 4, it can be verified that ℒ𝑉₁≤𝑑₁𝑢^∗₀𝑥₁^{4𝑙𝜎−𝜏−1}𝑥₂−𝑥^∗₂+𝑑₁𝑢^∗₀𝑥₁^{4𝑙𝜎−𝜏−1}𝑥^∗₂+𝑏𝑥₁^4𝑙𝜎+12(4𝜎−𝜏−1)𝑏²𝑥₁^4𝑙𝜎. (16)

Obviously, the first virtual controller 𝑥^∗₂1=−𝑐₁₁𝑢^∗₀1𝑀+𝑛−1+𝑏+2(4𝜎−𝜏−1)𝑏²𝑥₁^1+𝜏∶=−𝛽^𝑚2₁^/𝜎𝜉^𝑚2₁^/𝜎, (17) with design constant 𝑀>0, results in ℒ𝑉₁≤−(𝑀+𝑛−1)𝜉₁^4𝑙+𝑑₁𝑢^∗₀𝜉₁^{(4𝑙𝜎−𝜏−1)/𝜎}𝑥₂−𝑥^∗₂. (18)

Inductive Step. Suppose at step 𝑘−1, there are a 𝐶², proper and positive definite Lyapunov function 𝑉_𝑘−1, and a set of virtual controllers 𝑥^∗₁,…,𝑥^∗_𝑘 defined by 𝑥^∗₁=0,𝜉₁=𝑥^𝜎/𝑚1₁−𝑥^{∗𝜎/𝑚1}₁,𝑥^∗₂=−𝛽^𝑚2₁^/𝜎𝜉^𝑚2₁^/𝜎,𝜉₂=𝑥^𝜎/𝑚2₂−𝑥^{∗𝜎/𝑚2}₂,𝑥⋮⋮_∗𝑘=−𝛽^{𝑚𝑘/𝜎}_𝑘−1𝜉^{𝑚𝑘/𝜎}_𝑘−1,𝜉_𝑘=𝑥^{𝜎/𝑚𝑘}_𝑘−𝑥^{∗𝜎/𝑚𝑘}_𝑘, (19) with constants 𝛽₁>0,…,𝛽_𝑘−1>0, such that ℒ𝑉_𝑘−1≤−(𝑀+𝑛−𝑘+1)𝜉₁^4𝑙+⋯+𝜉^4𝑙_𝑘−1+𝑑_𝑘−1𝑢^∗₀𝜉^{(4𝑙𝜎−𝜏−𝑚𝑘−1)/𝜎}_𝑘−1𝑥_𝑘−𝑥^∗_𝑘. (20) To complete the induction, at the 𝑘th step, we choose the following Lyapunov function: 𝑉_𝑘𝑥₁,…,𝑥_𝑘=𝑉_𝑘−1𝑥₁,…,𝑥_𝑘−1+𝑊_𝑘𝑥₁,…,𝑥_𝑘, (21) where 𝑊_𝑘𝑥₁,…,𝑥_𝑘=^𝑥𝑘_𝑥∗𝑘(𝑠^{𝜎/𝑚𝑘}−𝑥^{∗𝜎/𝑚𝑘}_𝑘)^{(4𝑙𝜎−𝜏−𝑚𝑘)/𝜎}𝑑𝑠. (22)

Noting that 𝑥^{∗𝜎/𝑚𝑘}_𝑘=−𝛽_𝑘−1𝜉_𝑘−1,𝜉_𝑘=𝑥^{𝜎/𝑚𝑘}_𝑘−𝛽_𝑘−1𝑥^{𝜎/𝑚𝑘−1}_𝑘−1+⋯+(−1)^𝑘−1×𝛽_𝑘−1⋯𝛽₁𝑥^𝜎/𝑚1₁, (23) with the help of the fact 𝜎/𝑚_𝑖>2, the function 𝑉_𝑘 can be shown to be 𝐶², proper and positive definite using a similar method as in [34]. Moreover, we can obtain 𝜕𝑊_𝑘𝜕𝑥_𝑘=𝜉^{(4𝑙𝜎−𝜏−𝑚𝑘}_𝑘^)/𝜎,𝜕²𝑊_𝑘𝜕𝑥²_𝑘=4𝑙𝜎−𝜏−𝑚_𝑘𝑚_𝑘𝜉^{((4𝑙−1)𝜎−𝜏−𝑚𝑘}_𝑘^)/𝜎𝑥^{(𝜎−𝑚𝑘)/𝑚𝑘}_𝑘,𝜕²𝑊_𝑘𝜕𝑥_𝑘𝜕𝑥_𝑖=𝜕²𝑊_𝑘𝜕𝑥_𝑖𝜕𝑥_𝑘=4𝑙𝜎−𝜏−𝑚_𝑘𝜎×𝜉^{((4𝑙−1)𝜎−𝜏−𝑚𝑘}_𝑘^)/𝜎𝜕𝑥^{∗𝜎/𝑚𝑘}_𝑘𝜕𝑥_𝑖,𝜕𝑊_𝑘𝜕𝑥_𝑖=−4𝑙𝜎−𝜏−𝑚_𝑘𝜎𝜕𝑥^{∗𝜎/𝑚𝑘}_𝑘𝜕𝑥_𝑖×^𝑥𝑘_𝑥∗𝑘𝑠^{𝜎/𝑚𝑘}−𝑥^{∗𝜎/𝑚𝑘}_𝑘^{((4𝑙−1)𝜎−𝜏−𝑚𝑘)/𝜎}𝜕𝑑𝑠,₂𝑊_𝑘𝜕𝑥²_𝑖=4𝑙𝜎−𝜏−𝑚_𝑘𝜎(4𝑙−1)𝜎−𝜏−𝑚_𝑘𝜎𝜕𝑥^{∗𝜎/𝑚𝑘}_𝑘𝜕𝑥_𝑖²×^𝑥𝑘_𝑥∗𝑘𝑠^{𝜎/𝑚𝑘}−𝑥^{∗𝜎/𝑚𝑘}_𝑘^{((4𝑙−2)𝜎−𝜏−𝑚𝑘)/𝜎}−𝑑𝑠4𝜎−𝜏−𝑚_𝑘𝜎𝜕²𝑥^{∗𝜎/𝑚𝑘}_𝑘𝜕𝑥²_𝑖×^𝑥𝑘_𝑥∗𝑘(𝑠^{𝜎/𝑚𝑘}−𝑥^{∗𝜎/𝑚𝑘}_𝑘)^{((4𝑙−1)𝜎−𝜏−𝑚𝑘)/𝜎}𝜕𝑑𝑠,₂𝑊_𝑘𝜕𝑥_𝑖𝜕𝑥_𝑗=4𝑙𝜎−𝜏−𝑚_𝑘𝜎⋅(4𝑙−1)𝜎−𝜏−𝑚_𝑘𝜎𝜕𝑥^{∗𝜎/𝑚𝑘}_𝑘𝜕𝑥_𝑖𝜕𝑥^{∗𝜎/𝑚𝑘}_𝑘𝜕𝑥_𝑗×^𝑥𝑘_𝑥∗𝑘𝑠^{𝜎/𝑚𝑘}−𝑥^{∗𝜎/𝑚𝑘}_𝑘^{((4𝑙−2)𝜎−𝜏−𝑚𝑘)/𝜎}𝑑𝑠, (24) where 𝑖,𝑗=1,…,𝑘−1,   𝑖≠𝑗.

Using (20)–(22) and (24), it follows that ℒ𝑉_𝑘≤−(𝑀+𝑛−𝑘+1)𝜉₁^4𝑙+⋯+𝜉^4𝑙_𝑘−1+𝑑_𝑘𝑢^∗₀𝜉^{(4𝑙𝜎−𝜏−𝑚𝑘}_𝑘^)/𝜎𝑥_𝑘+1+𝑑_𝑘−1𝑢^∗₀𝜉^{(4𝑙𝜎−𝜏−𝑚𝑘−1)/𝜎}_𝑘−1𝑥_𝑘−𝑥^∗_𝑘+𝜉^{(4𝑙𝜎−𝜏−𝑚𝑘}_𝑘^)/𝜎𝑓_𝑘+^𝑘−1_𝑖=1𝜕𝑊_𝑘𝜕𝑥_𝑖𝑑_𝑖𝑢^∗₀𝑥_𝑖+1+𝑓_𝑖+12^𝑘−1_{𝑖,𝑗=1,𝑖≠𝑗}𝜕²𝑊_𝑘𝜕𝑥_𝑖𝜕𝑥_𝑗𝑔^𝑇_𝑖𝑔^𝑇_𝑗+12^𝑘−1_𝑖=1𝜕²𝑊_𝑘𝜕𝑥²_𝑖𝑔^𝑇_𝑖²+12^𝑘−1_𝑖=1𝜕²𝑊_𝑘𝜕𝑥_𝑘𝜕𝑥_𝑖𝑔^𝑇_𝑘𝑔^𝑇_𝑖+12𝜕²𝑊_𝑘𝜕𝑥²_𝑘𝑔^𝑇_𝑘². (25)

Based on Lemmas 5 and 6, the following proposition is given to estimate the last seven terms on the right-hand side of (25).

Proposition 12. There exists a positive constant 𝑙_𝑘 such that 𝑑_𝑘−1𝑢^∗₀𝜉^{(4𝑙𝜎−𝜏−𝑚𝑘)/𝜎}_𝑘−1𝑥_𝑘−𝑥^∗_𝑘+𝜉^{(4𝑙𝜎−𝜏−𝑚𝑘}_𝑘^)/𝜎𝑓_𝑘+^𝑘−1_𝑖=1𝜕𝑊_𝑘𝜕𝑥_𝑖𝑑_𝑖𝑢^∗₀𝑥_𝑖+1+𝑓_𝑖+12^𝑘−1_{𝑖,𝑗=1,𝑖≠𝑗}𝜕²𝑊_𝑘𝜕𝑥_𝑖𝜕𝑥_𝑗𝑔^𝑇_𝑖𝑔^𝑇_𝑗+12^𝑘−1_𝑖=1𝜕²𝑊_𝑘𝜕𝑥²_𝑖𝑔^𝑇_𝑖²+12^𝑘−1_𝑖=1𝜕²𝑊_𝑘𝜕𝑥_𝑘𝜕𝑥_𝑖𝑔^𝑇_𝑘𝑔^𝑇_𝑖+12𝜕²𝑊_𝑘𝜕𝑥²_𝑘𝑔^𝑇_𝑘²≤𝜉₁^4𝑙+⋯+𝜉^4𝑙_𝑘−1+𝑙_𝑘𝜉_𝑘^4𝑙. (26)

Proof. The similar proof can be found in [23] and thus it is omitted here.
Substituting (26) into (25), we obtain ℒ𝑉_𝑘≤−(𝑀+𝑛−𝑘)𝜉₁^4𝑙+⋯+𝜉^4𝑙_𝑘−1+𝑑_𝑘𝑢^∗₀𝜉^{(4𝑙𝜎−𝜏−𝑚𝑘}_𝑘^)/𝜎𝑥_𝑘+1+𝑙_𝑘𝜉_𝑘^4𝑙. (27)
Clearly, the 𝐶⁰ virtual controller 𝑥^∗_𝑘+11=−𝑐_𝑘1𝑢^∗₀𝜉^(𝑚𝑘_𝑘^+𝜏)/𝜎𝑀+𝑛−𝑘+𝑙_𝑘∶=−𝛽^𝑚𝑘+1_𝑘^/𝜎𝜉^𝑚𝑘+1_𝑘^/𝜎, (28) with 𝛽_𝑘>0 being constant, results in ℒ𝑉_𝑘≤−(𝑀+𝑛−𝑘)𝜉₁^4𝑙+⋯+𝜉_𝑘^4𝑙+𝑑_𝑘𝑢^∗₀𝜉^{(4𝑙𝜎−𝜏−𝑚𝑘}_𝑘^)/𝜎𝑥_𝑘+1−𝑥^∗_𝑘+1. (29) This completes the proof of the inductive step.

Using the previous inductive argument, one concludes that at the 𝑛th step, there exists a non-Lipschitz continuous state feedback control law of the form 𝑢=𝑥^∗_𝑛+1=−𝛽^𝑚𝑛+1_𝑛^/𝜎𝜉^𝑚𝑛+1_𝑛^/𝜎, (30) with 𝛽_𝑛>0 being constant and a 𝐶² positive definite and proper Lyapunov function 𝑉_𝑛 of the form (21)-(22), such that ℒ𝑉_𝑛≤−𝑀𝜉₁^4𝑙+⋯+𝜉_𝑛^4𝑙. (31)

So the following result is obtained.

Lemma 13. If Assumptions 8 and 10 hold for the stochastic nonlinear system (15), then the solution of the closed-loop system consisting of (15) and (30) is finite-time stable in probability.

Proof. From (23) and (30), it is not hard to verify that all conditions in Lemma 3 are satisfied, which means that the closed-loop system admits a unique solution. Next we prove that the closed-loop system is globally finite-time stable in probability.
First of all, by using Lemma 5, it is easy to see that 𝑊_𝑘=^𝑥𝑘_𝑥∗𝑘𝑠^{𝜎/𝑚𝑘}−𝑥^{∗𝜎/𝑚𝑘}_𝑘^{(4𝑙𝜎−𝜏−𝑚𝑘)/𝜎}≤𝜉𝑑𝑠_{𝑘(4𝑙𝜎−𝜏−𝑚𝑘)/𝜎}𝑥_𝑘−𝑥^∗_𝑘≤2^{(𝜎−𝑚𝑘)/𝜎}𝜉_𝑘^{(4𝑙𝜎−𝜏)/𝜎}. (32)
So we have the following estimate 𝑉_𝑛=1()𝑥4𝑙𝜎−𝜏^{14𝑙𝜎−𝜏}+^𝑛_𝑘=2𝑊_𝑘≤1()𝜉4𝑙𝜎−𝜏^{1(4𝑙𝜎−𝜏)/𝜎}+^𝑛_𝑘=22^{(𝜎−𝑚𝑘)/𝜎}𝜉_𝑘^{(4𝑙𝜎−𝜏)/𝜎}≤2^𝑛_𝑘=1𝜉_𝑘^{(4𝑙𝜎−𝜏)/𝜎}. (33)
Let 𝛼=4𝑙𝜎/(4𝑙𝜎−𝜏). With (33) and (31) in mind, by Lemma 5, it is not difficult to obtain that ℒ𝑉_𝑛≤−𝑀𝑉^𝛼_𝑛2^𝛼. (34)
Therefore, by Lemma 4, the uncertain system (15) with the controller (30) is globally finite-time stable in probability and its stochastic settling time 𝑇₁(𝑥(0),𝑤) satisfies 𝐸𝑇₁(𝑥(0)≤2,𝑤)^𝛼𝑉_𝑛^1−𝛼(𝑥(0))𝑀().1−𝛼 (35)

Step 𝐵. Then we design a state feedback controller such that the 𝑥₀-subsystem is finite-time stable in probability.

From Step 𝐴, we know that 𝑥(𝑡)≡0 when 𝑡≥𝑇₁(𝑥(0),𝑤). Therefore, we just need to stabilize the 𝑥₀-subsystem in a finite time almost surely. When 𝑡≥𝑇₁(𝑥(0),𝑤), for the 𝑥₀-subsystem, we can take the following control law: 𝑢₀𝑥₀1=−𝑐₀₁𝑘₀3+𝑎+2𝑎²𝑥^𝑟₀, (36) where 𝑘₀ is a positive constant.

Taking the Lyapunov function 𝑉₀=𝑥⁴₀/4, a simple computation gives ℒ𝑉₀≤−𝑘₀𝑥₀^3+𝑟. (37)

Thus by Lemma 4, 𝑥₀ stochastically tends to 0 within a settling time denoted by 𝑇₂(𝑥₀(0),𝑤) and 𝐸𝑇₂𝑥₀(0),𝑤≤2𝑉₀^(1−𝑟)/4𝑥₀(0)𝑘₀().1−𝑟 (38)

Up to now, we have finished the finite-time stabilizing controller design of the system (8). Consequently, the following theorem can be obtained to summarize the main result of the paper.

Theorem 14. Under Assumptions 8–10, if the proposed control design procedure together with the previous switching control strategy is applied to system (8), then, for any initial conditions in the state space (𝑥₀,𝑥)∈𝑅^𝑛+1, the closed-loop system is globally finite time regulated at origin in probability.

Remark 15. The settling time of the closed-loop system depends on design parameters 𝑘₀, 𝑀 and initial conditions, that is, when the initial conditions are known, the settling-time can be arbitrarily adjusted by the choice of appropriate design parameters 𝑘₀ and 𝑀.

5. Simulation Example

To verify our proposed controller, we consider the following low-dimensional system: 𝑑𝑥₀=(1.5+0.5cos𝑡)𝑢₀1𝑑𝑡+4𝑥₀^4/5𝑑𝑤,𝑑𝑥₁=𝑥₂1𝑑𝑡+4𝑥₁^9/111𝑑𝑡+8sin𝑥₁𝑑𝑤,𝑑𝑥₂1=𝑢𝑑𝑡+4𝑥₁^7/111𝑑𝑡+8cos2𝑥₂𝑑𝑤. (39)

We choose 𝜏=−2/11, which together with 𝑚₁=1 implies that 𝑚₂=9/11. By Lemma 5, it is verified that Assumptions 9-10 are satisfied with 𝑎=𝑏=1/2. Assumption 8 holds with 1≤1.5+0.5cos𝑡≤2; hence the controller proposed in this paper is applicable.

When (𝑥₀(0),𝑥₁(0),𝑥₂(0))=(1,−1,1), by choosing 𝜎=3 and 𝑙=2/3, according to the design procedure proposed in Section 4, a state feedback stabilizing controller can be explicitly given. Using MATLAB, Figures 1 and 2 are obtained to exhibit the trajectories of the closed-loop system. From these figures, we can see that the states of the closed-loop system are finite time regulated to zero almost surely.

(a) 𝑥0

(b) 𝑥1

(c) 𝑥2

Figure 1 

States of the closed-loop system.

(a) 𝑢0

(b) 𝑢1

Figure 2 

Control inputs of the closed-loop system.

6. Conclusion

In this paper, the finite-time state feedback stabilization problem has been investigated for a class of nonholonomic systems with more general stochastic disturbances. With the help of adding a power integrator technique, a systematic control design procedure is developed in the stochastic setting. It should be noted that the proposed controller can only work well when the whole state vector is measurable. Therefore, a natural and more interesting problem is how to design output feedback stabilizing controller for the system studied in the paper if only partial state vector is measurable, which is now under our further investigation.

Acknowledgments

This work has been supported in part by National Nature Science Foundation of China under Grants 61073065 and 61273091 and the Key Program of Science Technology Research of Education Department of Henan Province under Grant 13A120016. The authors thank the editor and the anonymous reviewers for their constructive comments and suggestions for improving the quality of the paper.