## Nonlinear Systems in Function Spaces and Applications in Biomedical Sciences, Control Theory, and Engineering

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Hua Chen, Wen Chen, Chaoli Wang, Dongkai Zhang, Binwu Zhang, "Finite-Time Stabilization of Dynamic Nonholonomic Wheeled Mobile Robots with Parameter Uncertainties", *Journal of Function Spaces*, vol. 2013, Article ID 595849, 7 pages, 2013. https://doi.org/10.1155/2013/595849

# Finite-Time Stabilization of Dynamic Nonholonomic Wheeled Mobile Robots with Parameter Uncertainties

**Academic Editor:**Jin Liang

#### Abstract

The finite-time stabilization problem of dynamic nonholonomic wheeled mobile robots with parameter uncertainties is considered for the first time. By the equivalent coordinate transformation of states, an uncertain 5-order chained form system can be obtained, based on which a discontinuous switching controller is proposed such that all the states of the robots can be stabilized to the origin equilibrium point within any given settling time. The systematic strategy combines the theory of finite-time stability with a new switching control design method. Finally, the simulation result illustrates the effectiveness of the proposed controller.

#### 1. Introduction

Stabilization problem of nonholonomic systems is theoretically challenging and practically interesting. As pointed out in [1], although every nonholonomic system is controllable, it cannot be stabilized to a point with pure smooth (or even continuous) state feedback law. In order to overcome the difficulty of Brocket's condition [1], a variety of sophisticated feedback stabilization methods have been proposed which mainly include continuous time-varying feedback control laws [2–4], discontinuous feedback control laws [5–8], and hybrid feedback control laws [9].

A common characteristic of these designs of controllers above is based on kinematic model, where only a kinematic model is considered and the velocities are taken as control inputs. But in fact, for some mechanical systems with nonholonomic constraints, it is more realistic to formulate the control problems at dynamic levels, where the torque and force are chosen as new inputs. Some results can be found in recent papers, for example, the dynamic tracking control of wheeled mobile robots in the presence of both actuator saturations and external disturbances is considered in [10], where a computationally tractable moving horizon tracking scheme is presented. In [11, 12], the saturated stabilization and tracking control are discussed for simple dynamic nonholonomic mobile robot. For uncertain dynamic nonholonomic systems, Ma and Tso [13] have given a robust control law for the exponential regulation of an uncertain dynamic nonholonomic wheeled mobile robot, in which the authors improved the convergence speed of regulating the state to a desired set point for the first time.

In order to drive a system to the equilibrium point with a fast convergence rate, finite-time stability theory has become a studying focus recently, for example, finite-time stabilization problems have been studied mostly in the contexts of optimality, controllability, and deadbeat control for several decades [14–16]. Compared to the asymptotic stabilization, the finite-time stabilization, which renders the trajectories of the closed-loop systems convergent to the origin in a finite time, has many advantages such as fast response, high tracking precision, and disturbance rejection properties.

For the nonholonomic systems, a few researchers have got some excellent results in finite-time control field. In [17], the relay switching technique and the terminal sliding mode control scheme with finite-time convergence are used for the design of the controller to address the tracking control of the nonholonomic systems with extended chained form. For a class of uncertain nonholonomic chained form systems, Hong et al. [18] have designed a nonsmooth state feedback law such that the controlled chained form system is both Lyapunov stable and finite-time convergent within any given settling time. And the finite-time tracking control for single mobile robots or multiple nonholonomic mobile robots is considered in [19–21]. The previous developed controller for nonholonomic systems can be divided into two categories: one is for the finite-time stabilization problem of chained form systems and the other is for the tracking control problem of mobile robots. However, to the best of our knowledge, there exist no results to deal with the robust finite-time stabilization of uncertain dynamic nonholonomic mobile robots.

This paper considers the stabilization problem of dynamic nonholonomic mobile robots with uncertain parameters in a finite time. The main results and contributions can be summarized as the following two respects.(a)An uncertain 5-order chained form system can be obtained under the equivalent coordinate transformation of states, which means the finite-time stabilization of the chained form system is equivalent to the finite-time stabilization of the original dynamic robot system.(b)Applying the theory of finite-time stability and the switching control method, we design a discontinuous robust controller to make the states of the chained form system converge to the equilibrium point in a finite time.

The structure of the paper is as follows: Section 2 gives a formalization of the problem considered in the paper. Section 3 states our main results. Section 4 provides an illustrative numerical example and the corresponding simulation results of the proposed methodology. Finally, a conclusion is shown in Section 5.

#### 2. Problem Statement

A class of nonholonomic wheeled mobile robots are shown in Figure 1, the two fixed rear wheels of the robot are controlled independently by motors, and a front castor wheel prevents the robot from tipping over as it moves on a plane. Assuming that the geometric center point and the mass center point of the robot are the same and that the radiuses are identical for all the rear wheels, is the length of the fixed two rear wheels, where and are known positive constants. Its kinematic and dynamics model can be described by the following differential equations [13]: where , are the position of the mass center of the robot moving in the plane and , are the mass and inertia of mobile robots; respectively, is the forward velocity, is the steering velocity and denotes its heading angle from the horizontal axis, and , are driving torques on the right and left rear wheels.

The geometric and inertia parameters are all assumed to be unknown positive constants, but are bounded by some known positive constants , that is, One of the equilibrium states of systems (1) is .

The control objective is to design a discontinuous state feedback law , such that the state trajectory of dynamic nonholonomic mobile robot system (1) starting from an arbitrary initial state converges to the origin equilibrium point in a finite time with the unknown parameters satisfying (2).

As pointed out in [11], take an orthogonal coordinate transformation Then system (1) can be converted to the following equation: where , are new control inputs and , are new unknown parameters with their bounds derived from (2) as follows: Because the coordinate transformation (3) is globally invertible and does not change the origin, it is obvious that the equilibrium point is finite-time stable for its closed-loop system of (4) it is means that is also a finite time stable equilibrium point for the corresponding closed-loop system of (1).

Hence, the control task is to design a discontinuous finite-time stabilizing controller for system (4) with the unknown parameters (5). Here, it should be noted that Hong et al. have designed a switching control strategy to discuss the finite-time stabilization of uncertain chained form systems in [18], however, it is invalid to control the dynamic chained system (4); thus, a new improved discontinuous design method is required.

The following definition and lemmas are needed for our controller design later.

*Definition 1 (see [14–16]). *Consider a time-invariant system in the form of
where is continuous on an open neighborhood of the origin. The equilibrium of the system is (locally) finite-time stable if (i) it is asymptotically stable, in , an open neighborhood of the origin, with ; (ii) it is finite-time convergent in , that is, for any initial condition , there is a settling time such that every solution of system (6) is defined with for and satisfies , and if . Moreover, if , the origin is globally finite-time stable.

Lemma 2 (see [15, 18]). *Suppose there exist a positive-definite and proper function , real numbers , , such that is negative semidefinite. Then, the origin is a globally finite-time stable equilibrium point of system (6).*

Lemma 3 (see [18]). *For the uncertain time-varying chained form system:
**
where , are the state and control input, respectively; are uncertain parameters located in known intervals, that is,
** is an uncertain function satisfying
**
and satisfies .**Let , , and (with and odd integers) be real numbers satisfying
**
Then, finite-time stabilizing control law of (7) can be constructed in the form of
**
where is defined as follows:
**
where denotes the sign function and , are suitable constants.*

*Proof. *See [18] for details.

#### 3. Main Results

In this section, the main results will be presented. Firstly, we will state the basic idea to design a finite-time switching controller for system (4).

Note that system (4) can be decoupled into two subsystems, one of which is -subsystem and the other describes the rest of (4), that is, -subsystem By designing , the state of (13) can be driven to any predetermined point in a finite time, based on which of (14) can be stabilized to by designing the finite-time controller , and the last step is to redesign such that can be driven to in a finite time.

The following are the switching controller design and the corresponding proofs of Theorems 4 and 6.

Theorem 4. *Given , for system (4), take the following switching control law.**Step 1*. Let , , where , . Until , then go to Step 2.*Step 2*. Let
*
where
**
Unitll , , then go to Step 3.**Step 3*. Let ,
*
where
**
Until , stop.**Then system (4) can be stabilized to the origin equilibrium point in a finite time by the switching controller Step 1–Step 3.*

*Proof. *In the first step, let , , then
which means that ; according to the conclusion of Lemma 2, there exists a finite time such that for all , that is, after this time.

In Step 2, for the subsystem (14)
Let
One has
Because , where
the finite-time stabilization problems of (20) and (22) are equivalent.

Set , , , , and from Lemma 3, system (20) can be stabilized to zero by in Step 2 after some finite time ; that is, for all .

Similarly, consider subsystem (13) again in the last step
Set , , , , , by using Lemma 3, it is clear that there exists a finite time such that system (13) can also be stabilized to zero after .

This completes the proof of the theorem.

*Remark 5. * Then control objective can be completed in each step within a finite time, and thus system (4) can be stabilized to zero in a finite time.

On the other hand, from (3), we have the following:
Therefore, the finite-time switching controller for the original robot system (1) can be stated as follows.

Theorem 6. *Given , for system (1), take the following switching control law.** Step 1′*. Let
*
where , . Until , then go to Step 2′.** Step 2′*. Let
*
where
**
Unitl , , then go to Step 3′.** Step 3′*. Let
*
where
**
Until , stop.**Then system (1) can be stabilized to the origin equilibrium point in a finite time by the switching controller Step 1′–Step 3′.*

*Proof. *According to (25), we have the following:
where
Because the determinant , then it is obvious to see that is equivalent to .

On the other hand, the switching controller in Theorem 4 can be used to stabilize the states of system (4) in a finite time; hence, the control task is changed to find the relation between the original controller of system (1) and the controller of system (4).

Comparing system (4) with system (1), we have the following:
from which, and by using the switching controller of Step 1–Step 3 in Theorem 4, we can solve the corresponding and thus the conclusion can be obtained in Theorem 6.

This completes the proof of the theorem.

#### 4. Simulations

In this section, the discontinuous switching controller proposed in theorems above is used to show how to stabilize the state of (4) and the state of (1) to the zero equilibrium point in a finite time. We will demonstrate the effectiveness of our methods by a numerical example.

In the following simulation, we assume that , , , , , , , . The initial condition of system (1) is . From (3) and (5), the initial value of system (4) is , and , , , . Given , the design parameters are taken as follows: , , , , , , , , , , , . The settling time in every step is given in advance; , , .

Figures 2–5 show some simulation results with Matlab. From Figures 2 and 3, it can be seen that all the state variables of the closed system (4) are driven to the origin equilibrium point in a given settling time . Observing Figure 2, in time interval 0*~*5 s, the first step control task is completed, that is, as . Next, from Figure 3, it is clear that can be stabilized to zero by the controller in Step 2 as and remain unchanged. Finally, the controller in Step 3 drives to zero in the settling time .

Figures 4 and 5 show the finite-time convergence of the state variables of the original robot system (1) exactly as the state of the transformed system.

#### 5. Conclusion

The robust finite-time stabilization problem is discussed in this paper for a class of uncertain dynamic nonholonomic wheeled mobile robot. The contributions of this paper include having applied finite-time control technique and a new switching design method such that all the states can be stabilized to the zero point by the proposed discontinuous controller. And we will work on extending the results to consider the corresponding trajectory tracking control problem in the coming time.

#### Acknowledgments

This paper was supported by the Natural Science Foundation of China (61304004, 61374040, and 11372097), China Postdoctoral Science Foundation (2013M531263), the Scientific Innovation Program of Shanghai Education Committee (13ZZ115), the National Basic Research Program of China (973 Project, 2010CB832702), the National Science Funds for Distinguished Young Scholars (11125208), the 111 Project (B12032), the R&D Special Fund for Public Welfare Industry (Hydrodynamics Project, 201101014), and the the Natural Science Foundation of Hebei Province (A2014106035).

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#### Copyright

Copyright © 2013 Hua Chen et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.