Adaptive State-Feedback Stabilization for Stochastic Nonholonomic Mobile Robots with Unknown Parameters

Abstract

The stabilizing problem of stochastic nonholonomic mobile robots with uncertain parameters is addressed in this paper. The nonholonomic mobile robots with kinematic unknown parameters are extended to the stochastic case. Based on backstepping technique, adaptive state-feedback stabilizing controllers are designed for nonholonomic mobile robots with kinematic unknown parameters whose linear velocity and angular velocity are subject to some stochastic disturbances simultaneously. A switching control strategy for the original system is presented. The proposed controllers that guarantee the states of closed-loop system are asymptotically stabilized at the zero equilibrium point in probability.

1. Introduction

In the past decades, the control of nonholonomic systems has been widely pursued. By the results of Brockett [1], the nonholonomic system cannot be stabilized at a single equilibrium point by any static smooth pure state-feedback controller. To solve this problem, lots of novel approaches have been considered: discontinuous feedback control [2–4], smooth time-varying feedback controller [5], and the method of LMI [6]. The control of nonholonomic mobile robots plays an important role in that of nonholonomic systems because they are a benchmark for these systems. There is much attention devoted to the control of nonholonomic mobile robots. The nonholonomic mobile robots were classified into four types, which were characterized by generic structures of the model equations [7]. Based on the backstepping technique, the control for nonholonomic mobile robots was discussed: tracking problems [8] and stabilizing problems [9, 10]. Hespanha et al. introduced the mobile robot with parametric uncertainties [11], which were further discussed [12, 13]. But all the above articles discussed the nonholonomic systems in the deterministic case, which was not considered a stochastic disturbance.

In recent years, stochastic nonlinear systems have received much attention [14, 15], especially for stochastic control when backstepping designs were firstly introduced [16, 17]. For stochastic nonholonomic systems, there were a few papers. The almost global adaptive asymptotical controllers of stochastic nonholonomic chained form systems were discussed by using discontinuous control [18]. The adaptive stabilization problem of stochastic nonholonomic systems with nonlinear drifts was considered [19–21]. By using state-scaling method, backstepping controllers were proposed to deal with exponential stabilization for nonholonomic mobile robots with stochastic disturbance [22, 23]. But the above two papers did not consider unknown parameters. To our knowledge, the problem of adaptive state-feedback stabilization for nonholonomic mobile robots with kinematic unknown parameters, whose linear velocity and angular velocity are subject to some stochastic disturbances simultaneously, has not been reported. So, there exists a natural problem which is how to extend the models in [11–13] to the stochastic case and design an adaptive state-feedback stabilizing controller for stochastic nonholonomic mobile robots with uncertain parameters.

The purpose of this paper is to design adaptive state-feedback stabilizing controllers for stochastic nonholonomic mobile robots with unknown parameters. The main idea of this paper is highlighted as follows.(i)We extend the models of nonholonomic mobile robots with unknown parameters in [11–13] to the stochastic case. The stabilizing controllers are designed for stochastic nonholonomic mobile robots with unknown parameters by adaptive state-feedback backstepping technique.(ii)A switching control strategy for the original system is presented. It guarantees that the states of closed-loop system are asymptotically stabilized at the zero equilibrium point in probability.

The paper is organized as follows. Section 1 begins with the mathematical preliminaries. In Section 2, the adaptive state-feedback backstepping controller is designed. In Section 3, a switching control strategy for the original system is discussed. Finally, a simulation example is given to show the effectiveness of the controller in Section 4.

2. Preliminaries and Problem Formulation

2.1. Preliminaries

Consider the following stochastic nonlinear system: d𝑥=𝑓(𝑥)d𝑡+𝑔(𝑥)d𝐵,𝑥𝑡₀∈ℝ^𝑛, (1) where 𝑥∈ℝ^𝑛 is the state, the Borel measurable functions 𝑓∶ℝ^𝑛→ℝ^𝑛 and 𝑔∶ℝ^𝑛→ℝ^𝑛×𝑟 are locally Lipschitz in 𝑥, and 𝐵∈ℝ^𝑟 is an 𝑟-dimensional independent standard Wiener process defined on the complete probability space (Ω,ℱ,𝑃).

The following definitions and lemmas will be used in the paper.

Definition 1 (see [16]). For any given 𝑉(𝑥)∈𝒞², associated with stochastic system (1), the differential operator ℒ is defined as follows: ℒ𝑉(𝑥)=𝜕𝑉𝑓(𝑥)+1𝜕𝑥2Tr𝑔^𝑇(𝑥)𝜕²𝑉𝜕𝑥²𝑔(𝑥). (2)

Definition 2 (see [24]). The equilibrium 𝑥=0 of system (1) is(i)globally stable in probability if for ∀𝜀>0, there exists a class 𝒦 function 𝛾(⋅) such that 𝑃|𝑥(𝑡)|<𝛾𝑥𝑡₀≥1−𝜀,∀𝑡≥0,𝑥𝑡₀∈ℝ^𝑛⧵{0}; (3)(ii)globally asymptotically stable in probability if it is globally stable in probability and 𝑃lim_𝑡→∞|𝑥(𝑡)|=0=1,∀𝑥𝑡₀∈ℝ^𝑛. (4)

Definition 3 (see [25]). A stochastic process 𝑥(𝑡) is said to be bounded in probability if the random variable |𝑥(𝑡)| is bounded in probability uniformly in 𝑡; that is, lim_𝑅→∞sup_𝑡>𝑡0𝑃{|𝑥(𝑡)|>𝑅}=0. (5)

Lemma 4 (see [24]). Considering the stochastic system (1), if there exist a 𝒞² function 𝑉(𝑥), class 𝒦_∞ functions 𝛼₁(⋅) and 𝛼₂(⋅), constants 𝑐₁>0, 𝑐₂≥0, and a nonnegative function 𝑊(𝑥) such that 𝛼₁(|𝑥|)≤𝑉(𝑥)≤𝛼₂𝑥,(𝑥)=(||)ℒ𝑉𝜕𝑉1𝜕𝑥𝑓+2Tr𝑔^𝑇𝜕²𝑉𝜕𝑥²𝑔≤−𝑐₁𝑊(𝑥)+𝑐₂, (6) then(i)for (1), there exists an almost surely unique solution on [𝑡₀,∞) for each 𝑥(𝑡₀)∈ℝ^𝑛;(ii)when 𝑐₂=0, 𝑓(0)=0, 𝑔(0)=0, and 𝑊(𝑥) is continuous, then the equilibrium 𝑥=0 is globally stable in probability and 𝑃lim_𝑡→∞𝑊(𝑥(𝑡))=0=1,for∀𝑥𝑡₀∈ℝ^𝑛. (7)

Lemma 5 (see [26]). Let 𝑥 and 𝑦 be real variables. Then, for any positive integers 𝑚, 𝑛 and any real number 𝜀>0, the following inequality holds: |𝑥|^𝑚𝑦^𝑛≤𝑚𝜀|𝑥|𝑚+𝑛^𝑚+𝑛+𝑛𝜀𝑚+𝑛^−𝑚/𝑛𝑦^𝑚+𝑛. (8)

2.2. Problem Formulation

Hespanha et al. introduced the mobile robot with parametric uncertainties [11], which were further discussed in [12, 13] as follows: 𝜃̇=𝑝^∗₁𝜔,𝑥̇_𝑐=𝑝^∗₂𝑦̇𝑣cos𝜃,_𝑐=𝑝^∗₂𝑣sin𝜃, (9) where 𝑣 and 𝜔 are two control inputs to denote the forward velocity and angular velocity, respectively.

Here we assume that the forward velocity 𝑣 and the angular velocity 𝜔 are subject to some stochastic disturbances. Based on the similar methods in [27, Page 1-2], velocity 𝑣 and the angular velocity 𝜔 with stochastic disturbances can be expressed as follows: 𝜔(𝜃)=𝜔₁(𝜃)+𝜔₂(𝜃)𝐵̇(𝑡),𝑣𝑥_𝑐,𝑦_𝑐,𝜃=𝑣₁𝑥_𝑐,𝑦_𝑐,𝜃+𝑣₂𝑥_𝑐,𝑦_𝑐,𝜃𝐵̇(𝑡), (10) where 𝐵̇(𝑡) is the derivative of a Brownian motion 𝐵(𝑡).

Remark 6. The second equality of (10) is the same as that of Remark 2 in [19]. Moreover, (10) means that 𝜔(𝜃) can be divided into two parts, with the second parts being stochastic disturbances and the same for  𝑣(𝑥_𝑐,𝑦_𝑐,𝜃).

Substituting (10) into (9), the system (9) can be transformed into d𝜃=𝑝^∗₁𝜔₁d𝑡+𝑝^∗₁𝜔₂d𝐵,d𝑥_𝑐=𝑝^∗₂𝑣₁cos𝜃d𝑡+𝑝^∗₂𝑣₂cos𝜃d𝐵,d𝑦_𝑐=𝑝^∗₂𝑣₁sin𝜃d𝑡+𝑝^∗₂𝑣₂sin𝜃d𝐵, (11) where 𝑝^∗₁ is unknown parameter taking values in a known interval [𝑝_min,𝑝_max] with 0<𝑝_min<𝑝_max<∞; 𝑝^∗₂ is unknown positive parameter.

For system (11), we introduce the following state and input transformation: 𝑥₀=𝜃,𝑢₀=𝜔₁,𝑢=𝑣₁,𝑥₁=𝑥_𝑐sin𝜃−𝑦_𝑐𝑥cos𝜃,₂=𝑥_𝑐cos𝜃+𝑦_𝑐sin𝜃, (12) and it is easy to see thatd𝑥₀=𝑝^∗₁𝑢₀d𝑡+𝑝^∗₁𝜔₂𝑥₀d𝐵,d𝑥₁=𝑝^∗₂𝑥₂𝑢₀1d𝑡−2𝑝^∗₁²𝑥₁𝜔²₂d𝑡+𝑝^∗₁𝑝^∗₂𝑣₂𝜔₂d𝑡+𝑝^∗₁𝑥₂𝜔₂d𝐵,d𝑥₂=𝑝^∗₂𝑢d𝑡−𝑝^∗₁𝑥₁𝑢₀+12𝑝^∗₁²𝑥₂𝜔²₂+𝑝d𝑡_∗2𝑣₂−𝑝^∗₁𝑥₁𝜔₂d𝐵. (13a)(13b)

Remark 7. The main difference between this paper and [22] is that the unknown parameter exists in this paper. The controller design of systems (13a) and (13b) will be more difficult.

Remark 8. For system (13a) and (13b), the variable 𝑥₂ appears in the term 𝑝^∗₁𝑥₂𝜔₂d𝐵 in the first equation of (13b); this is different from the traditional stochastic backstepping technique in [16, 17, 24].

3. Adaptive Controller Design

In this section, we will design state-feedback controllers such that all the signals in closed-loop system are regulated to the origin in probability. The following assumptions are needed.

Assumption 9. For the smooth function 𝜔₂(𝜃), there exists a known positive constant 𝑚₁, such that 𝜔₂(𝜃)=𝑚₁𝜃. (14)

Assumption 10. For smooth function 𝑣₂(𝑥_𝑐,𝑦_𝑐,𝜃) and any positive constant 𝐶, there exists a known nonnegative constant 𝑚₂, such that 𝑣₂𝑥_𝑐,𝑦_𝑐,𝜃≤𝑚₂𝑥₂𝑥−𝐶¹𝑢₀. (15)

Remark 11. For the adaptive controllers’ design in the following, if we let 𝐶=𝑐₁, this assumption will change to |𝑣₂(𝑥_𝑐,𝑦_𝑐,𝜃)|≤𝑚₂|𝜀₂|, where 𝜀₂ is defined in (25) and 𝑐₁ is the same as that in (28) in the following Section 3.2.

Firstly, we will consider the problem of stabilization for systems (13a) and (13b) under the condition of 𝑥₀(𝑡₀)≠0. The case of 𝑥₀(𝑡₀)=0 will be discussed in Section 3.

3.1. The First State Stabilization

Let us consider the subsystem (13a) of stochastic nonholonomic nonlinear systems (13a) and (13b): d𝑥₀=𝑝^∗₁𝑢₀d𝑡+𝑝^∗₁𝜔₂𝑥₀d𝐵. (16)

In order to guarantee that 𝑥₀ converges to zero, one can take 𝑢₀ as follows: 𝑢₀=−𝜂₀𝑥₀,𝜂₀3=𝜆+2𝑚²₁𝑝_max, (17) where 𝜆>0 is a design parameter.

If we employ a Lyapunov function of the form:𝑉₀𝑥₀=14𝑝^∗₁𝑥⁴₀. (18) From (13a), (17), (18), and Assumption 9, one can obtain ℒ𝑉₀≤𝑥³₀𝑢₀+32𝑝^∗₁𝑥²₀𝑝^∗₁²𝜔²₂≤−𝜆𝑥⁴₀. (19)

Theorem 12. If Assumption 9 holds, one can choose positive constants 𝑚₁, 𝜆, and 𝑝_max  and the controller 𝑢₀ as (17), respectively, then(i)the closed-loop subsystem composed by (13a) and (17) has an almost surely unique solution on [𝑡₀,∞) for ∀𝑥₀(𝑡₀);(ii)the equilibrium 𝑥₀=0 of the closed-loop subsystem composed by (13a) and (17) is globally asymptotically stable in probability.

Proof. Choosing Lyapunov function as (18), by (19), 𝜆>0, and Lemma 4, (i) holds and the equilibrium 𝑥₀=0 of the closed-loop subsystem which contained (13a) and (17) is globally stable in probability and for ∀𝑥₀(𝑡₀)≠0, 𝑃{lim_𝑡→∞(|𝑥₀(𝑡)|=0)}=1. From Definition 2, (ii) holds.

Remark 13. From Theorem 12, one has the state 𝑥₀ bounded in probability; that is, there exists a positive constant 𝑚₃, such that lim_𝑚3→∞sup_𝑡>𝑡0𝑃𝑥₀(𝑡)>𝑚₃=0. (20)
Substituting (17) into the subsystem (13a), one gets d𝑥₀=−𝑝^∗₁𝜂₀𝑥₀d𝑡+𝑝^∗₁𝑚₁𝑥₀d𝐵. (21)

Proposition 14. For initial state 𝑥₀(𝑡₀)≠0, the solution of (21) will never reach the zero, which avoids the uncontrollability of the subsystem (13b).

Proof. From Lemma 2.3 ([27, Page 93]), the following equality will be the solution of (21): 𝑥₀(𝑡)=𝑥₀𝑡₀exp^𝑡_𝑡0−𝑝^∗₁𝜂₀−12𝑝^∗₁𝑚₁²+d𝑠_𝑡𝑡0𝑝^∗₁𝑚₁d𝜔. (22) From the above expression of 𝑥₀(𝑡) and 𝑥₀(𝑡₀)≠0, it is easy to see that 𝑥₀(𝑡) will never cross the origin at the time interval 𝑡∈(𝑡₀,+∞).

In the following Section 3.2, the other states will be regulated to the origin in probability by the design of the control input 𝑢.

3.2. Other States Stabilization

In order to design a smooth adaptive state-feedback controller, the following state-input scaling discontinuous transformation is needed: 𝑧₁=𝑥₁𝑢₀,𝑧₂=𝑥₂. (23)

Remark 15. For the initial state 𝑥₀(𝑡₀)≠0, from Proposition 14, one can obtain that transformation (23) is meaningful.

Under the new 𝑧-coordinate, the subsystem (13b) is transformed into d𝑧₁=𝑝^∗₁−𝜂₀z₂−0.5𝑝^∗₁𝑧₁𝜔²₂−𝑝^∗₁𝑧₂𝜔²₂𝑥²₀+𝜂₀𝑧₁+𝑝^∗₂𝑣₂𝜔₂𝑥₀+𝑝^∗₁𝑧₁𝜔²₂𝑥²₀d𝑡+𝑝^∗₁𝑧₂𝜔₂𝑥₀−𝑧₁𝜔₂𝑥₀d𝐵,d𝑧₂=𝑝^∗₂𝑢d𝑡+𝑝^∗₁𝜂₀𝑧₁𝑥²₀−0.5𝑝^∗₁²𝑧₂𝜔²₂+𝑝d𝑡_∗2𝑣₂−𝑝^∗₁𝑧₁𝑥₀𝜔₂d𝐵. (24)

To invoke the backstepping method, the error variables 𝜀₁ and 𝜀₂ are given by 𝜀₁=𝑧₁,𝜀₂=𝑧₂−𝛼₁𝜀₁. (25)

Step 1. Define the first Lyapunov candidate function: 𝑉₁=14𝑝^∗₁𝜀⁴₁. (26)
By (24)–(26) and Definition 1, one has ℒ𝑉₁≤𝜀³₁−𝜂₀𝑧₂−12𝑝^∗₁𝑧₁𝜔²₂−𝑝^∗₁𝑧₂𝜔²₂𝑥²₀+𝜂₀𝑧₁+𝑝^∗₂𝑣₂𝜔₂𝑥₀+𝑝^∗₁𝑧₁𝜔²₂𝑥²₀+3𝜀²₁𝑝^∗₁²𝑧²₂𝜔²₂𝑥²₀+𝑝^∗₁²𝑧²₁𝜔²₂𝑥²₀. (27)
The virtual control can be chosen as 𝛼₁𝜀₁=𝑐₁𝜀₁, (28) where 𝑐₁ is a positive constant, which will be chosen later. From (27), Lemma 5, and simple operation, we have the following inequalities: −𝜂₀𝑧₂𝜀³₁≤𝜂₀3𝑑4𝜀⁴₁+34𝑑³𝜀⁴₂−𝜂₀𝑐₁𝜀⁴₁,−12𝜀³₁𝑝^∗₁𝑧₁𝜔²₂≤12𝑝_max𝑚²₁𝑚²₃𝜀⁴₁,−𝑝^∗₁𝜀³₁𝑧₂𝜔²₂𝑥²₀≤𝑝_max𝑚²₁34+𝑐₁𝜀⁴₁+14𝑝_max𝑚²₁𝜀⁴₂,𝑝^∗₂𝑣₂𝜔₂𝑥₀≤14𝜀⁴₁+34𝑚₁𝑚₂^4/3𝑝^∗₂^4/3𝜀⁴₂,𝑝^∗₁𝑧₁𝜔²₂𝑥²₀≤𝑝_max𝑚²₁𝜀⁴₁,3𝑝^∗₁²𝜀³₁𝑧²₂𝜔²₂𝑥²₀≤3𝑝²_max𝑚²₁2𝑐²₁+1𝜀⁴₁+3𝑝²_max𝑚²₁𝜀⁴₂,3𝑝^∗₁²𝜀³₁𝑧²₁𝜔²₂𝑥²₀≤3𝑝²_max𝑚²₁𝜀⁴₁, (29) where 𝑑>0 is a design parameter. Substituting these above inequalities into (27), it is easy to see thatℒ𝑉₁≤−𝜂₀𝑐₁𝑒+1+3𝑑4𝜂₀−𝑐₁𝜂₀()+𝑝1−𝑒_max𝑚²₁2𝑚²₃+4𝑐₁+24𝑝_max𝑐²₁4+𝑝_max𝑚²₁36𝑝_max+7+14𝜀⁴₁+𝜂₀4𝑑³+𝑝_max𝑚²₁4+3𝑝²_max𝑚²₁𝜀⁴₂+34𝑚₁𝑚₂^4/3𝑝^∗₂^4/3𝜀⁴₂, (30) where 𝑒 is a design parameter and 0<𝑒<1. If we select parameters 𝜂₀ and 𝑐₁ to satisfy 𝑐₁≥4+3𝑑,𝜂4𝑒₀≥𝑝_max𝑚²₁2𝑚²₃+4𝑐₁+24𝑝_max𝑐²₁+36𝑝_max+7+12𝑐₁(),1−𝑒 (31) one has ℒ𝑉₁𝑐≤−¹𝜂₀(1−𝑒)2𝜀⁴₁+34𝑚₁𝑚₂^4/3𝑝^∗₂^4/3𝜀⁴₂+𝜂₀4𝑑³+𝑝_max𝑚²₁4+3𝑝²_max𝑚²₁𝜀⁴₂. (32)

Step 2. By (24), (25), (32), and It𝑜̂ formula (Theorem 6.2, [27, Page 32]), one gets d𝜀₂=𝑝^∗₂𝑢d𝑡+𝑝^∗₁𝜂₀𝑧₁𝑥²₀−12𝑝^∗₁²𝑧₂𝜔²₂+𝑝^∗₁𝑐₁𝜂₀𝑧₂+12𝑝^∗₁²𝑐₁𝑧₁𝜔²₂−𝑝^∗₁𝑐₁𝜂₀𝑧₁+𝑝^∗₁²𝑐₁𝑧₂𝜔²₂𝑥²₀−𝑝^∗₁𝑝^∗₂𝑐₁𝑣₂𝜔₂𝑥₀−𝑝^∗₁²𝑐₁𝑧₁𝜔²₂𝑥²₀+𝑝d𝑡_∗2𝑣₂−𝑝^∗₁𝑧₁𝑥₀𝜔₂−𝑝^∗₁𝑐₁𝑧₂𝜔₂𝑥₀+𝑝^∗₁𝑐₁𝑧₁𝜔₂𝑥₀d𝐵. (33)
To deal with the uncertain parameter 𝑝^∗₂, define parameter Θ=max𝑝^∗₂,𝑝^∗₂^4/3,1𝑝^∗₂,1𝑝^∗₂^4/3,1𝑝^∗₂², (34) and Θ=Θ−Θ being the parameter estimation error, Θ being the estimate of Θ. Define the second Lyapunov candidate function: 𝑉₂=𝑉₁+14𝑝^∗₂𝜀⁴₂+12Θ². (35)
From (33), (35), and Definition 1, one can obtain ℒ𝑉₂𝑐≤−¹𝜂₀(1−𝑒)2𝜀⁴₁+34𝑚₁𝑚₂^4/3𝑝^∗₂^4/3𝜀⁴₂+𝜂₀4𝑑³+𝑝_max𝑚²₁4+3𝑝²_max𝑚²₁𝜀⁴₂+𝜀³₂𝑝𝑢+^∗1𝑝^∗₂𝜂₀𝑧₁𝑥²₀−12𝑝^∗₁²𝑝^∗₂𝑧₂𝜔²₂+𝑝^∗₁𝑝^∗₂𝑐₁𝜂₀𝑧₂+12𝑝^∗₁²𝑝^∗₂𝑐₁𝑧₁𝜔²₂−𝑝^∗₁𝑝^∗₂𝑐₁𝜂₀𝑧₁+𝑝^∗₁²𝑝^∗₂𝑐₁𝑧₂𝜔²₂𝑥²₀−𝑝^∗₁𝑐₁𝑣₂𝜔₂𝑥₀−𝑝^∗₁²𝑝^∗₂𝑐₁𝑧₁𝜔²₂𝑥²₀d𝑡+6𝜀²₂𝑝^∗₂𝑣²₂+𝑝^∗₁²𝑝^∗₂𝑧²₁𝑥²₀𝜔²₂+𝑝^∗₁²𝑝^∗₂𝑐²₁𝑧²₂𝜔²₂𝑥²₀+𝑝^∗₁²𝑝^∗₂𝑐²₁𝑧²₁𝜔²₂𝑥²₀−ΘΘ. (36)
By (34), (36), and Lemma 5, we have the following inequalities: 𝑝^∗₁𝑝^∗₂𝜂₀𝑧₁𝑥²₀𝜀³₂3≤Θ4𝑝_max𝜂₀𝑚²₃^4/3𝜀⁴₂+14𝜀⁴₁,−12𝑝^∗₁²𝑝^∗₂𝑧₂𝜔²₂𝜀⁴₂1≤Θ2𝑝²_max𝑚²₃𝑚²₁+38𝑐₁𝑝²_max𝑚²₃𝑚²₁^4/3𝜀⁴₂+18𝜀⁴₁,𝑝^∗₁𝑝^∗₂𝑐₁𝜂₀𝑧₂𝜀³₂≤Θ𝑐₁𝑝_max𝜂₀+34𝑐₁^8/3𝑝^4/3_max𝜂₀^4/3𝜀⁴₂+14𝜀⁴₁,12𝑝^∗₁²𝑝^∗₂𝑐₁𝑧₁𝜔²₂𝜀³₂3≤Θ8𝑐₁𝑝²_max𝑚²₃𝑚²₁^4/3𝜀⁴₂+18𝜀⁴₁,−𝑝^∗₁𝑝^∗₂𝑐₁𝜂₀𝑧₁𝜀³₂3≤Θ4𝑐₁𝜂₀𝑝_max^4/3𝜀⁴₂+14𝜀⁴₁,𝑝^∗₁²𝑝^∗₂𝑐₁𝑧₂𝜔²₂𝑥²₀𝜀³₂≤Θ𝑐₁𝑝²_max𝑚²₁𝜀⁴₂3+Θ4𝑐²₁𝑚²₁𝑝²_max^4/3𝜀⁴₂+14𝜀⁴₁,−𝑝^∗₁𝑐₁𝑣₂𝜔₂𝑥₀𝜀³₂≤𝑝_max𝑚₁𝑚₂𝑐₁𝜀⁴₂,−𝑝^∗₁²𝑝^∗₂𝑐₁𝑧₁𝜔²₂𝑥²₀𝜀³₂≤14𝜀⁴₁3+Θ4𝑝²_max𝑐₁𝑚²₁^4/3𝜀⁴₂,6𝜀²₂𝑝^∗₂𝑣²₂𝑧²₁𝑥²₀+𝑝^∗₁²𝑝^∗₂𝜔²₂+𝑝^∗₁²𝑝^∗₂𝑐²₁𝑧²₂𝜔²₂𝑥²₀+𝑝^∗₁²𝑝^∗₂𝑐²₁𝑧²₁𝜔²₂𝑥²₀≤Θ6𝑚²₂+3𝑝⁴_max𝑚⁸₃𝑚⁴₁+12𝑝²_max𝑚²₁+6𝑝⁴_max𝑚⁴₁𝑐⁴₁+3𝑝⁴_max𝑚⁴₁𝑐⁴₁𝜀⁴₂+12𝜀⁴₁. (37)
Substituting the above inequalities into (36) and adding and subtracting the term 𝑐₂𝜀⁴₂ on the right-hand side of (36), we have ℒ𝑉₂𝑐≤−¹𝜂₀(1−𝑒)2−13.5𝜀⁴₁−𝑐₂𝜀⁴₂+𝐻₂₁𝜀⁴₂+𝜀³₂𝑢+Θ𝐻₂₂𝜀⁴₂−ΘΘ̇≤−𝑐₁𝜀⁴₁−𝑐₂𝜀⁴₂+𝜀³₂𝑢+Θ𝐻₂₂𝜀⁴₂−Θ̇+𝐻₂₁+1+Θ²𝐻₂₂𝜀⁴₂, (38) where 𝐻₂₁=𝜂₀4𝑑³+𝑝_max𝑚²₁4+3𝑝²_max𝑚²₁+𝑝_max𝑚₁𝑚₂𝑐₁,𝑐₁=𝑐₁𝜂₀(1−𝑒)2𝐻−13.5,₂₂=𝑐₂+34𝑝_max𝜂₀𝑚²₃^4/3+12𝑝²_max𝑚²₃𝑚²₁+𝑐₁𝑝_max𝜂₀+38𝑐₁𝑝²_max𝑚²₃𝑚²₁^4/3+34𝑐₁^8/3𝑝^4/3_max𝜂₀^4/3+38𝑐₁𝑝²_max𝑚²₃𝑚²₁^4/3+34𝑐₁𝜂₀𝑝_max^4/3+𝑐₁𝑝²_max𝑚²₁+34𝑐²₁𝑚²₁𝑝²_max^4/3+6𝑚²₂+12𝑝²_max𝑚²₁+3𝑝⁴_max𝑚⁸₃𝑚⁴₁+6𝑝⁴_max𝑚⁴₁𝑐⁴₁+3𝑝⁴_max𝑚⁴₁𝑐⁴₁+34𝑝³_max𝑐₁𝑚²₁^4/3+34𝑚₁𝑚₂^4/3. (39)
One can choose the actual control law 𝑢 and the adaptive laws Θ as follows: 𝑢=−𝐻₂₁+1+Θ²𝐻₂₂𝜀₂,Θ̇=𝐻₂₂𝜀⁴₂. (40)
Substituting (40) into (38), one gets ℒ𝑉₂≤−𝑐₁𝜀⁴₁−𝑐₂𝜀⁴₂. (41)
Choosing the Lyapunov function as 𝑉=𝑉₀+𝑉₂, (42) together with (19) and (41), we have ℒ𝑉≤−𝜆𝑥²₀−𝑐₁𝜀⁴₁−𝑐₂𝜀⁴₂. (43)

Theorem 16. If Assumptions 9 and 10 hold, one can choose positive constants 𝜆, 𝑚₁, 𝑚₂, 𝑚₃, and  𝑝_max, with  𝑑>0 and 0<𝑒<1 satisfying 𝑐₁>0 and (31); for positive constant 𝑐₂, one has the following.(i)The closed-loop system composed by (13a), (17), (24), and (40) has an almost surely unique solution on [𝑡₀,∞) for ∀𝑥₀(𝑡₀),𝑧(𝑡₀) and Θ(𝑡₀).(ii)The equilibrium (𝑥₀,𝑧,Θ)=(0,0,0) of the closed-loop system is globally stable in probability.(iii)For initial condition ∀𝑥₀(𝑡₀), 𝑧(𝑡₀), and Θ(𝑡₀), 𝑃{lim_𝑡→∞(|𝑥₀(𝑡)|+|𝑧(𝑡)|)=0}=1, 𝑃{lim_𝑡→∞Θ(𝑡)exists𝑎𝑛𝑑𝑖𝑠𝑓𝑖𝑛𝑖𝑡𝑒}=1, where Θ=Θ−Θ and 𝑧(𝑡)=(𝑧₁(𝑡),𝑧₂(𝑡)).

Proof. From conditions in Theorem 16, it is easy to see that constants 𝜆>0, 𝑐₁>0, and 𝑐₂>0. So, ℒ𝑉 in (43) becomes the same form as (3.19) in [21]. Using (43) and Lemma 4, Theorem 16 can be proved.

4. Switching Control Stability

In Section 2, the case of 𝑥₀(𝑡₀)≠0 is discussed. We design controllers 𝑢₀ and 𝑢 for systems (13a) and (13b) as in (17) and (40), respectively. Now we turn to the case of 𝑥₀(𝑡₀)=0. When the initial 𝑥₀(𝑡₀)=0, one can choose an open loop control 𝑢₀=−𝑢^∗₀≠0 to drive the state 𝑥₀ away from zero in a limited time.

In fact, when we choose an open loop control 𝑢₀=−𝑢^∗₀≠0, system (13a) will be in the following form: d𝑥₀=−𝑢^∗₀d𝑡+𝑝^∗₁𝜔₂𝑥₀d𝐵. (44) For a given constant 𝑙>0, define a stopping time 𝜏_𝑙=inf{𝑡∶𝑡≥𝑡₀,|𝑥₀(𝑡)|≥𝑙}. With the similar analysis in Section V in [22], we have 𝑃(𝜏_𝑙−𝑡₀≥𝑇)≤𝑙/T𝑢^∗₀, which means that 𝑃(𝜏_𝑙=∞)=0 for any 𝑙>0. Letting 𝑡^∗_𝑠=𝜏_𝑙, it is easy to see that 𝑥₀𝑡^∗_𝑠=𝑥₀𝜏_𝑙=𝑙≠0. (45) So, there exists 𝑡^∗_𝑠>0, such that 𝑥₀(𝑡^∗_𝑠)≠0. After that, at the time 𝑡=𝑡^∗_𝑠, we switch the control inputs 𝑢₀ and 𝑢 to (17) and (40) in 𝑡∈[𝑡^∗_𝑠,+∞), respectively.

Theorem 17. If Assumptions 9 and 10 hold, one can apply the following switching control procedure to system (11): (i)when the initial state belongs to 𝜃𝑡₀,𝑥_𝑐𝑡₀,𝑦_𝑐𝑡₀∈ℝ³∣𝜃𝑡₀≠0, (46)(ii)when the initial state belongs to 𝜃𝑡₀,𝑥_𝑐𝑡₀,𝑦_𝑐𝑡₀∈ℝ³∣𝜃𝑡₀=0. (47)One designs control inputs 𝑢₀ and 𝑢 in form (17) and (40), respectively; for 𝑡∈[𝑡₀,𝑡^∗_𝑠), one can choose the control law 𝑢₀=−𝑢^∗₀≠0 and 𝑢=𝑢^∗; for 𝑡∈[𝑡^∗_𝑠,+∞), at the time 𝑡=𝑡^∗_𝑠, one switches the control inputs 𝑢₀ and 𝑢 to (17) and (40), respectively.
Then, for any initial condition in the state space, the states of system (11) are asymptotically regulated to zero in probability.

Proof. Firstly, we consider the case that the initial state belongs to 𝜃𝑡₀,𝑥_𝑐𝑡₀,𝑦_𝑐𝑡₀∈ℝ³∣𝜃𝑡₀≠0. (48) From Theorems 12 and 16, for the closed-loop system composed by (13a), (17), (24), and (40), states 𝑥₀ and 𝑧(𝑡) are regulated to zero in probability, Θ(𝑡) is bounded in probability, and 𝑃{lim_𝑡→∞(|𝑥₀(𝑡)|+|𝑧(𝑡)|)=0}=1. This implies that states 𝑥₀ and 𝑧(𝑡) are globally asymptotically regulated to zero in probability and bounded in probability. As a result of (23), one gets that the states 𝑥₀, 𝑥₁, and 𝑥₂ of closed-loop system composed by (11), (17), and (40) asymptotically converge to zero in probability and all bounded in probability. By orthogonal transformation (12), one can obtain that the states 𝜃, 𝑥_𝑐, and 𝑦_𝑐 of closed-loop system composed by (11), (17), and (40) are asymptotically stabilized in probability.
Secondly, when the initial state belongs to 𝜃𝑡₀,𝑥_𝑐𝑡₀,𝑦_𝑐𝑡₀∈ℝ³∣𝜃𝑡₀=0, (49) we use the constant control 𝑢₀=−𝑢^∗₀≠0 in order to drive 𝑥₀ far away from the origin, which guarantees that all the signals are bounded in probability during [𝑡₀,𝑡^∗_𝑠). Then, in view of 𝑥₀(𝑡^∗_𝑠)≠0, the switching control strategy is applied to system (11) at the time instant 𝑡^∗_𝑠>0. This completes the proof.

5. A Simulation Example

Consider the system (11) with 𝜔₂=0.5𝜃 and 𝑣₂=𝑥₂−(3.5𝑥₁/𝑢₀). In simulation, one can choose 𝑝^∗₁=0.2, 𝑝^∗₂=0.1, 𝑒=0.5, 𝑝_max=0.25, 𝑑=1, 𝑐₁=5.5, 𝑐₂=0.2, 𝜂₀=10.5, 𝑚₁=0.5, 𝑚₂=1, and 𝑚₃=3 and the initial values 𝜃(0)=2.8, 𝑥_𝑐(0)=−0.678, 𝑦_𝑐(0)=0.528, and Θ(0)=0.38. Figures 1, 2, and 3 give the responses of the closed-loop system consisting of (11), (17), and (40).

Figure 1 

The responses of states 𝜃, 𝑥𝑐, and 𝑦𝑐 with respect to time.

Figure 2 

The responses of controllers 𝑢0 and 𝑢 with respect to time.

Figure 3 

The response of estimate parameter Θ with respect to time.

From Figure 1, it is easy to see that the states 𝜃, 𝑥_𝑐, and 𝑦_𝑐 are asymptotically regulated to zero in probability in spite of the stochastic disturbances. As shown in Figure 2, the control inputs 𝑢₀ and 𝑢 are convergent to a small neighborhood of zero asymptotically. Figure 3 indicates that the estimated parameter Θ is bounded.

6. Conclusions

In this paper, we extend the nonholonomic mobile robots with unknown parameters to the stochastic case. Based on backstepping technique, adaptive state-feedback stabilizing controllers are designed for stochastic nonholonomic mobile robots with unknown parameters. A switching control strategy for the original system is given, which guarantees that the states of closed-loop system are asymptotically stabilized at the zero equilibrium point in probability.

There exist some problems to be discussed, for example, how to design the controller for the dynamic stochastic nonholonomic systems with unknown parameters.

Acknowledgments

The authors would like to express sincere gratitude to the editor and reviewers’ hard work. This paper is supported by the National Science Foundation (no. 61304004) and Natural Science Foundation of Hebei Province of China (no. A2014106035), Doctoral Natural Science Foundation of Shijiazhaung University.