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

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.


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][3][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][20][21].By using statescaling 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][12][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 statefeedback 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][12][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 closedloop 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.

Preliminaries and Problem Formulation
2.1.Preliminaries.Consider the following stochastic nonlinear system: where  ∈ R  is the state, the Borel measurable functions  : R  → R  and  : R  → R × are locally Lipschitz in , and  ∈ R  is an -dimensional independent standard Wiener process defined on the complete probability space (Ω, F, ).
The following definitions and lemmas will be used in the paper.
Definition 1 (see [16]).For any given () ∈ C 2 , associated with stochastic system (1), the differential operator L is defined as follows: Definition 2 (see [24]).The equilibrium  = 0 of system ( 1) is (i) globally stable in probability if for ∀ > 0, there exists a class K function (⋅) such that (ii) globally asymptotically stable in probability if it is globally stable in probability and Definition 3 (see [25]).A stochastic process () is said to be bounded in probability if the random variable |()| is bounded in probability uniformly in ; that is, lim Lemma 4 (see [24]).Considering the stochastic system (1) (ii) when  2 = 0, (0) = 0, (0) = 0, and () is continuous, then the equilibrium  = 0 is globally stable in probability and Lemma 5 (see [26]).Let  and  be real variables.Then, for any positive integers ,  and any real number  > 0, the following inequality holds: 2.2.Problem Formulation.Hespanha et al. introduced the mobile robot with parametric uncertainties [11], which were further discussed in [12,13] as follows: where V and  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  are subject to some stochastic disturbances.Based on the similar methods in [27, Page 1-2], velocity V and the angular velocity  with stochastic disturbances can be expressed as follows: where Ḃ () is the derivative of a Brownian motion ().Remark 6.The second equality of (10) is the same as that of Remark 2 in [19].Moreover, (10) means that () can be divided into two parts, with the second parts being stochastic disturbances and the same for V(  ,   , ).Substituting (10) into (9), the system (9) can be transformed into where  * 1 is unknown parameter taking values in a known interval [ min ,  max ] with 0 <  min <  max < ∞;  * 2 is unknown positive parameter.
For system (11), we introduce the following state and input transformation: and it is easy to see that 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.

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  2 (), there exists a known positive constant  1 , such that Assumption 10.For smooth function V 2 (  ,   , ) and any positive constant , there exists a known nonnegative constant  2 , such that Remark 11.For the adaptive controllers' design in the following, if we let  =  1 , this assumption will change to , where  2 is defined in (25) and  1 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  0 ( 0 ) ̸ = 0.The case of  0 ( 0 ) = 0 will be discussed in Section 3.

The First State Stabilization.
Let us consider the subsystem (13a) of stochastic nonholonomic nonlinear systems (13a) and (13b): In order to guarantee that  0 converges to zero, one can take  0 as follows: where  > 0 is a design parameter.
If we employ a Lyapunov function of the form: From (13a), ( 17), ( 18), and Assumption 9, one can obtain (ii) the equilibrium  0 = 0 of the closed-loop subsystem composed by (13a) and ( 17) is globally asymptotically stable in probability.
Remark 13.From Theorem 12, one has the state  0 bounded in probability; that is, there exists a positive constant  3 , such that lim Substituting (17) into the subsystem (13a), one gets Proposition 14.For initial state  0 ( 0 ) ̸ = 0, the solution of (21) will never reach the zero, which avoids the uncontrollability of the subsystem (13b).
In the following Section 3.2, the other states will be regulated to the origin in probability by the design of the control input .

Other States Stabilization.
In order to design a smooth adaptive state-feedback controller, the following state-input scaling discontinuous transformation is needed: Remark 15.For the initial state  0 ( 0 ) ̸ = 0, from Proposition 14, one can obtain that transformation (23) is meaningful.
Under the new -coordinate, the subsystem (13b) is transformed into To invoke the backstepping method, the error variables  1 and  2 are given by Step 1. Define the first Lyapunov candidate function: By ( 24)-( 26) and Definition 1, one has The virtual control can be chosen as where  1 is a positive constant, which will be chosen later.From ( 27), Lemma 5, and simple operation, we have the following inequalities: where  > 0 is a design parameter.Substituting these above inequalities into (27), it is easy to see that where  is a design parameter and 0 <  < 1.If we select parameters  0 and  1 to satisfy one has (32) Step 2. By ( 24), ( 25), (32), and Itô formula (Theorem 6.2, [27, Page 32]), one gets To deal with the uncertain parameter  * 2 , define parameter and Θ = Θ− Θ being the parameter estimation error, Θ being the estimate of Θ. Define the second Lyapunov candidate function: From ( 33), (35), and Definition 1, one can obtain By ( 34), (36), and Lemma 5, we have the following inequalities: Substituting the above inequalities into (36) and adding and subtracting the term  2  4  2 on the right-hand side of (36), we have where One can choose the actual control law  and the adaptive laws Θ as follows: Substituting (40) into (38), one gets Choosing the Lyapunov function as together with (19) and (41), we have Theorem 16.If Assumptions 9 and 10 hold, one can choose positive constants ,  1 ,  2 ,  3 , and  max , with  > 0 and 0 <  < 1 satisfying  1 > 0 and (31); for positive constant  2 , one has the following.

Switching Control Stability
In Section 2, the case of  0 ( 0 ) ̸ = 0 is discussed.We design controllers  0 and  for systems (13a) and (13b) as in (17) and (40), respectively.Now we turn to the case of  0 ( 0 ) = 0.When the initial  0 ( 0 ) = 0, one can choose an open loop control  0 = − * 0 ̸ = 0 to drive the state  0 away from zero in a limited time.
In fact, when we choose an open loop control  0 = − * 0 ̸ = 0, system (13a) will be in the following form: For a given constant  > 0, define a stopping time   = inf{ :  ≥  0 , | 0 ()| ≥ }.With the similar analysis in Section V in [22], we have (  −  0 ≥ ) ≤ /T * 0 , which means that (  = ∞) = 0 for any  > 0. (i) when the initial state belongs to (ii) when the initial state belongs to One designs control inputs  0 and  in form Then, for any initial condition in the state space, the states of system (11) are asymptotically regulated to zero in probability.
From Figure 1, it is easy to see that the states ,   , and   are asymptotically regulated to zero in probability in spite of the stochastic disturbances.As shown in Figure 2, the control inputs  0 and  are convergent to a small neighborhood of zero asymptotically.Figure 3 indicates that the estimated parameter Θ is bounded.

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.

Figure 1 :
Figure 1: The responses of states ,   , and   with respect to time.

Figure 2 :
Figure 2: The responses of controllers  0 and  with respect to time.

Figure 3 :
Figure 3: The response of estimate parameter Θ with respect to time.