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.
In the past decades, the control of nonholonomic systems has been widely pursued. By the results of Brockett [
In recent years, stochastic nonlinear systems have received much attention [
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. We extend the models of nonholonomic mobile robots with unknown parameters in [ 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
Consider the following stochastic nonlinear system:
The following definitions and lemmas will be used in the paper.
For any given
The equilibrium globally stable in probability if for globally asymptotically stable in probability if it is globally stable in probability and
A stochastic process
Considering the stochastic system ( for ( when
Let
Hespanha et al. introduced the mobile robot with parametric uncertainties [
Here we assume that the forward velocity
The second equality of (
Substituting (
For system (
The main difference between this paper and [
For system (
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.
For the smooth function
For smooth function
For the adaptive controllers’ design in the following, if we let
Firstly, we will consider the problem of stabilization for systems (
Let us consider the subsystem (
In order to guarantee that
If we employ a Lyapunov function of the form:
If Assumption the closed-loop subsystem composed by ( the equilibrium
Choosing Lyapunov function as (
From Theorem
Substituting (
For initial state
From Lemma 2.3 ([
In the following Section
In order to design a smooth adaptive state-feedback controller, the following state-input scaling discontinuous transformation is needed:
For the initial state
Under the new
To invoke the backstepping method, the error variables
Define the first Lyapunov candidate function:
By (
The virtual control can be chosen as
By (
To deal with the uncertain parameter
From (
By (
Substituting the above inequalities into (
One can choose the actual control law
Substituting (
Choosing the Lyapunov function as
If Assumptions The closed-loop system composed by ( The equilibrium For initial condition
From conditions in Theorem
In Section
In fact, when we choose an open loop control
If Assumptions when the initial state belongs to
when the initial state belongs to
One designs control inputs
Then, for any initial condition in the state space, the states of system (
Firstly, we consider the case that the initial state belongs to
Secondly, when the initial state belongs to
Consider the system (
The responses of states
The responses of controllers
The response of estimate parameter
From Figure
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.
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.