#### Abstract

This paper investigates the problem of global finite-time stabilization by output feedback for a class of nonholonomic systems in chained form with uncertainties. By using backstepping recursive technique and the homogeneous domination approach, a constructive design procedure for output feedback control is given. Together with a novel switching control strategy, the designed controller renders that the states of closed-loop system are regulated to zero in a finite time. A simulation example is provided to illustrate the effectiveness of the proposed approach.

#### 1. Introduction

Over the past decade, nonholonomic systems have attracted much attention because they can be used to model many real systems, such as mobile robots, car-like vehicle, and under-actuated satellites. An important feature of a nonholonomic system is that the number of its inputs is less than the number of its degree of freedom, which makes the control problems of a nonholonomic system challenging. As pointed out by Brockett in [1], there does not exist a pure-state feedback control law for a nonholonomic system such that its state converges to its equilibrium. To overcome this difficulty, with the effort of many researchers a number of intelligent approaches have been proposed, which can be classified into discontinuous control laws [2, 3], time-varying control laws [4–6], and hybrid control laws [7, 8]; see the survey paper [9] for more details and references therein. Considering the difficulty of measuring full states and the inevitability of uncertainties in engineering practice, the output feedback issue of nonholonomic systems with drift uncertainties has recently been studied [10–16]. However, it should be mentioned that the aforementioned works only consider the feedback stabilizer that makes the trajectories of the systems converge to the equilibrium as the time goes to infinity.

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 [17]. Hence, it is more meaningful to investigate the finite-time stabilization problem than the classical asymptotical stability. In recent years, the problem of finite-time stabilization for nonlinear systems has been studied and some interesting results have been obtained [18–25]. However, the finite-time stabilization of nonholonomic systems is a relatively new problem. In fact, even in the case of finite-time stabilization using state feedback, there are very few results in the literature [26–28]. In the case when parts of the states are not measurable, to stabilize a nonholonomic system in a finite time only using limited measurable states becomes challenging.

To illustrate the difficulties in finite-time control of nonholonomic systems via output feedback, let us consider a problem of finite time stabilizing the following simple system at the origin: where and are measurable and is not available for feedback.

In discontinuous approach, as seen, for example, in [26–28], assuming that one might design the control as follows: where is a positive design parameter and , , are positive odd numbers. It is easy to verify that the in (2) renders globally converging to zero in a finite time .

Next, we need to stabilize the -subsystem within a settling time satisfying . By introducing the input-state-scaling transformation and , the system (3) can be rewritten as

However, the system (3) possesses the time-varying coefficient (or the system (4) dissatisfies the low-order growth condition), which renders the existing finite-time control methods highly difficult to the control problem of the -subsystem or even inapplicable. To the best of the authors’ knowledge, there is no result referred to the finite-time stabilization of nonholonomic systems by output feedback.

Motivated by the aforementioned discussion, in this paper we aim to tackle this challenging question and provide a solution to the problem of global finite-time output feedback stabilization for nonholonomic systems with uncertainties by applying the homogeneous domination approach. The main contribution of this paper is twofold. (i) Compared to the existing output feedback stabilization results for nonholonomic systems, the finite-time stabilizer proposed in this paper leads to faster convergence rate. (ii) As the common assumption to guarantee the existence of global finite-time output feedback stabilizer for a nonlinear system, the low-order growth (the order less than one) of system nonlinearities renders the discontinuous change of coordinates (i.e., the -process) inapplicable to the finite-time output feedback control problem of the nonholonomic systems, even the ideal chained systems, and how to deal with this constitutes one of the main contributions of this paper.

The rest of this paper is organized as follows. Section 2 provides the problem formation and preliminary knowledge. Section 3 presents the control design procedure and the main result, while Section 4 gives a simulation example to illustrate the theoretical finding of this paper. Finally, concluding remarks are proposed in Section 5.

#### 2. Problem Formulation and Preliminaries

In this paper, we consider the following uncertain nonholonomic systems: where , , are the system state, control input, and system output, respectively, ’s are disturbed virtual control coefficients, and ’s denote the input and states driven uncertainties, which are called the nonlinear drifts of the system (5).

The objective of this paper is to design an output feedback controller in the form such that the finite-time regulation of the states is achieved; that is, and for any , where is a finite settling time.

To this end, the following assumptions regarding system (5) are imposed.

*Assumption 1. *For , there are positive constants and such that

*Assumption 2. *For , there is a positive constant such that

*Assumption 3. *For , there are constants and such that
where .

For simplicity, in this paper we assume with being any even integer and being any odd integer. Based on this, we know that is a ratio of two positive odd integers.

*Remark 4. *Assumptions 1-2 are common and similar to the one usually imposed on the nonlinear systems [10]. Relatively speaking, Assumption 3 seems to be quite restrictive; however, it plays an essential role in ensuring the existence of finite-time output feedback stabilizer for nonholonomic system (5). Furthermore, it is worth pointing out that there are a number of nonlinear functions such as and that can be bounded by a function for any constant actually satisfying this assumption.

The following definitions and lemmas will serve as the basis of the coming control design and performance analysis.

*Definition 5 (see [17]). *Consider a system
where is continuous with respect to on an open neighborhood of the origin . The equilibrium of the system is (locally) finite-time stable if it is Lyapunov stable and finite-time convergent in a neighborhood of the origin. By “finite-time convergence,” we mean that if, for any initial condition, , there is a settling time , such that every solution with as its initial condition of (10) is well defined with for and satisfies and for any . If , the origin is a globally finite-time stable equilibrium.

Lemma 6 (see [17]). *Consider the nonlinear system described in (10). Suppose that there is a function defined in a neighborhood of the origin, real numbers , and , such that*(i)* is positive definite on ;*(ii)*, .**
Then, the origin of system (10) is locally finite-time stable with
**
for initial condition in some open neighborhood of the origin. If and is also radially unbounded (i.e., as ), the origin of system (10) is globally finite-time stable.*

*Definition 7 (see [29]). *Weighted homogeneity: for fixed coordinates and real numbers , , one has the following.(i)The dilation is defined by for any , where is called the weights of the coordinates. For simplicity, we define dilation weight .(ii)A function is said to be homogeneous of degree if there is a real number such that for any , .(iii)A vector field is said to be homogeneous of degree if there is a real number such that , for any , , .(iv)A homogeneous -norm is defined as for all , for a constant . For simplicity, in this paper, one chooses and writes for .

Lemma 8 (see [30]). *Suppose that is a homogeneous function of degree with respect to the dilation weight . Then the following hold.*(i)* is homogeneous of degree with being the homogeneous weight of .*(ii)*There is a constant such that . Moreover, if is positive definite, then , where is a constant.*

Lemma 9 (see [31]). *For , , and which is a constant, the following inequalities hold:
**
If is odd, then
*

Lemma 10 (see [32]). *Let , be real variables; then for any positive real numbers , , and , one has
**
where is any real number.*

Lemma 11 (see [33]). *For and positive real number , the following inequality holds:
**
where for and for .*

#### 3. Finite-Time Output Feedback Controller Design

In this section, we give a constructive procedure for the finite-time stabilizer of system (5) by output feedback. The design of finite-time output feedback controller is divided into the following two steps.(i)We first stabilize the -subsystem in a finite time by output feedback.(ii)Then we design a controller such that the -subsystem is finite-time stable.

##### 3.1. Finite-Time Output Feedback Stabilization of the -Subsystem

For the -subsystem, we choose the control as where is a positive constant. In this case, the -subsystem becomes

Noting that satisfies the linear growth condition, it is easy to obtain that the solution of -subsystem is bounded, for any given finite time . Hence, is well defined on . Under the control law (16), the -subsystem can be written as

Next we consider the finite-time output feedback stabilizer for system (18). For convenience, we define the following change of coordinates: under which system (18) is transformed into where , , and the state is measurable.

*Remark 12. *It is worth pointing out that, in terms of the transformation (19), the stabilizing control design of system (18) is equivalent to that of system (20). Thus, in what follows, we turn to designing the output feedback stabilizing controller for system (20) rather than (18). Moreover, with the help of Assumptions 1 and 3, it can be verified that , , satisfy
with a new growth rate .

To construct a global output feedback controller for system (20), we will employ the homogeneous domination approach introduced in [34]. We will first construct specifically a homogeneous output feedback controller for the nominal system without considering perturbing terms ’s. Then, we utilize a scaling gain in the controller to dominate the uncertain nonlinearities ’s.

###### 3.1.1. Homogeneous Output Feedback Control of the Nominal System

In this subsection, we will construct an output feedback stabilizer for the following nominal system:

The design of output feedback controller is divided into two steps. In Step A, we suppose that all the states are measurable and develop a recursive design method to explicitly construct a state feedback control law for system (22). Then in Step B, by constructing a nonsmooth reduced-order observer, we design an output feedback controller.

*(A) State Feedback Controller Design*

*Step 1. *Choose the Lyapunov function . Clearly, the first virtual controller
with and renders

*Step i (). *In this step, we can obtain the following property.

Proposition 13. *For the th Lyapunov function defined by
**
under the coordinate transformation
**
there exists the virtual controller such that
**
where , are constants.*

*Proof. *The detailed proof can be found in [20] and hence is omitted here.

From the inductive steps, we can design where such that

*(B) Output Feedback Controller Design.* Since are unmeasurable, we construct a homogeneous observer
where and ; are the gains to be determined. By the certainty equivalence principle, we can replace with in (28) and obtain an output feedback controller
where .

Considering where , and setting , for , from (22), (31), and (33), it follows that where .

Each term on the right-hand side of (34) can be estimated by the following propositions whose proofs are given in the Appendix.

Proposition 14. *There exists a positive constant such that
*

Proposition 15. *For ,
**
where is a continuous function of , is a constant, and .*

Proposition 16. *For the controller , one obtains
**
where is a continuous function of and is a constant.*

Proposition 17. *For ,
**
where is a continuous function of .*

Choosing , by Propositions 14–17, we get

By (28), (32), and Assumption 1, we can estimate in (29) by the following proposition, whose proof is given in the Appendix.

Proposition 18. *There exists a positive constant such that
**
where is a continuous function of .*

With the help of Proposition 18, defining , combining (29) and (39), and recursively choosing we obtain

Since is positive definite and proper with respect to , (42) implies that the closed-loop system can be rewritten as the following compact form: which is homogeneous with the dilation weight

It can be shown that (43) is homogeneous of degree . In addition, is homogeneous of degree 2. By Lemma 8, there is a constant , such that where and . Similarly, since the right-hand side of (42) is homogeneous of degree , by Lemma 8 there is a constant such that

Combining (45) and (46), it can be deduced from (42) that for a constant . By Lemma 6 with , the closed-loop system is globally finite-time stable.

*Remark 19. *It should be pointed out that the output feedback controller (32) is only continuous (rather than continuously differentiable) due to the presence of the powers , which is less than one. As a consequence, the closed-loop system (22) and (32) is not locally Lipschitz. Therefore, the uniqueness of the solution of system (22) and (32) is not guaranteed. Fortunately, as shown in the work [35], the existence of the solution can still be guaranteed for a continuous system without Lipschitz condition.

###### 3.1.2. Homogeneous Output Feedback Control of the System (20)

Together with the homogeneous controller and observer established previously, in this subsection we are ready to use the homogeneous domination approach to globally stabilize (20) via output feedback under (21). First, we introduce the change of coordinates where is a constant to be determined later. Under (48), system (20) can be rewritten as

Now we construct an observer with a gain as follows:

In addition, we design using the same construction of (32), specifically,

Now, the closed-loop system (49)–(51) can be written as

Hence, it can be concluded from (46) that

From (21), (48), and , we can find constants and such that

Noting that, for , is homogeneous of degree , we know that is homogeneous of degree .

With (54) and (55) in mind, we can find a positive constant such that

Substituting (56) into (53) yields where . Apparently, by choosing a large enough , the right-hand side of (57) is negative definite.

Furthermore, it can be deduced from (57) that there is a constant such that

By Lemma 6 (, , and ), (58) leads to the conclusion that the closed-loop system (20), (50), and (51) is globally finite-time stable, which yields that system (18) can be globally finite-time stabilized by the output feedback. In addition, the settling time satisfies

##### 3.2. Finite-Time Output Feedback Stabilization of the -Subsystem

From Section 3.1, we know that when . Therefore, we just need to stabilize the -subsystem in a finite time. When , for the -subsystem, we can take the following control law: where is a positive design constant, , are positive odd numbers, and is a smooth function. For instance, we can simply choose .

Taking the Lyapunov function , a simple computation gives

Thus, by Lemma 6, tends to within a settling time denoted by and

Up to now, we have finished the finite-time output feedback stabilizing controller design of the system (5). Consequently, the following theorem can be obtained to summarize the main result of the paper.

Theorem 20. *Under Assumptions 1–3, if the proposed control design procedure together with the above switching control strategy is applied to system (5), then, for any initial conditions in the state space , the closed-loop system is globally finite-time regulated at origin.*

#### 4. Simulation Example

To verify our proposed controller, we consider the following low-dimensional system: where , are unknown constants and , are unknown functions.

It should be mentioned that, when and , the system (63) collapses into a third-order chained form system which can be viewed as the bilinear model of a mobile robot with small angle measurement error (see [10, 36] for more details). This means that the system (63) is a simple one; however, it comes from real world.

For simplicity, it is assumed that and , . From this and Remark 4, it is not difficult to verify that Assumptions 1–3 hold. Firstly, we define the control law and introduce the change of coordinates under which the -subsystem of (63) is transformed into

If we pick , the dilation is defined as and . Then, according to the design procedure shown in Section 3, we can explicitly construct an output feedback controller for system (65). We can choose specifically with appropriate positive constants , , , and a large enough gain such that output feedback controller (66) renders the system (65) (i.e., the -subsystem of (63)) globally finite-time stable with a settling time .

Then, when , for the -subsystem, we switch the control input to where is a positive design constant.

In the simulation, we assume and . When , , by choosing the gains for the output laws as , , , , and , the simulation shown in Figure 1 demonstrates the global finite-time stability property of the closed-loop system (63)–(67).

(a) |

(b) |

(c) |

(d) |

(e) |

(f) |

*Remark 21. *Although system (63) was asymptotically stabilized by the existing output feedback controller in [10, 13], the system (66)-(67) is the first output feedback controller which globally finite-time stabilizes system (63). Compared to the existing asymptotical stabilization results, the proposed controller demonstrates more advantages such as faster convergence rates, higher accuracies, and better disturbance rejection properties [17].

#### 5. Conclusion

This paper has solved the problem of global finite-time output feedback stabilization for a class of nonholonomic systems in chained form with uncertainties. With the help of backstepping recursive technique and the homogeneous domination approach, a constructive design procedure for output feedback control is given. It is shown that the designed control laws can guarantee that the closed-loop system states are globally finite-time regulated to zero. In this direction, there are still remaining problems to be investigated. For example, an interesting research problem is how to design a finite-time output feedback stabilizing controller for nonholonomic systems in stochastic setting.

#### Appendix

*Proof of Proposition 14. *By Lemma 9, one obtains
where is a constant.

*Proof of Proposition 15. *Using , (26), (31), and Lemmas 9–11, it follows that
where , are constants and is a continuous function of . By , one has .

*Proof of Proposition 16. *By (26), (32), , and the definition of the homogeneous norm, one gets
where , , and are positive constants.

Similar to (A.2), with the use of Lemmas 9–11 and (A.3), (37) holds immediately.

*Proof of Proposition 17. *From , (26), and Lemma 9, there is a positive constant such that

According to , (31), (A.4), and Lemmas 9–11, one obtains
where is a continuous function of .

*Proof of Proposition 18. *By (26), (31), and Lemmas 9–11, it follows that
where , , and are positive constants.

#### Acknowledgments

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