# Global Finite-Time Output Feedback Stabilization for a Class of Uncertain Nonholonomic Systems

Authors: Baojian Du; Fangzheng Gao; Fushun Yuan
Journal: Abstract and Applied Analysis
 (2013)

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## 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.


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## Body

## 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:(1)x˙0=u0,x˙1=x1u0,x˙2=u1,
where x0 and x1 are measurable and x2 is not available for feedback.In discontinuous approach, as seen, for example, in [26–28], assuming that x0(t0)≠0 one might design the control u0 as follows:
(2)u0(x0)=-k0x0α0,0<α0=α1α2<1,
where k0 is a positive design parameter and αi, i=1,2, are positive odd numbers. It is easy to verify that the u0 in (2) renders x0 globally converging to zero in a finite time T0.Next, we need to stabilize thex-subsystem
(3)x˙1=-k0x0α0x2,x˙2=u1
within a settling time T1 satisfying T1<T0. By introducing the input-state-scaling transformation x1=x1/u0 and z2=x2, the system (3) can be rewritten as
(4)z˙1=z2+u˙0u0z1,z˙2=u1.However, the system (3) possesses the time-varying coefficient -k0x0α0 (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 x-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., theσ-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. Section2 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:(5)x˙0=d0u0+ϕ0(t,x0),x˙i=dixi+1u0+ϕi(t,x0,x),i=1,…,n-1,x˙n=dnu1+ϕn(t,x0,x),y=(x0,x1)T,
where (x0,x)T=(x0,x1,…,xn)T∈Rn+1, u=(u0,u1)T∈R2, y∈R2 are the system state, control input, and system output, respectively, di’s are disturbed virtual control coefficients, and ϕi’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(6)x^˙=ϑ(x^,y),u0=u0(x0),u1=u1(x^,y),
such that the finite-time regulation of the states is achieved; that is, limt→T(|x0(t)|+|x(t)|)=0 and (x0(t),x(t))=(0,0) for any t≥T, where T is a finite settling time.To this end, the following assumptions regarding system (5) are imposed.Assumption 1.
Fori=0,1,…,n, there are positive constants ci1 and ci2 such that
(7)ci1≤di≤ci2.Assumption 2.
Forϕ0, there is a positive constant a such that
(8)|ϕ0(t,x0)|≤a|x0|.Assumption 3.
Fori=1,…,n, there are constants b>0 and τ∈(-1/n,0) such that
(9)|ϕi(t,x0,x)|≤b(|x1|(ri+τ)/r1+⋯+|xi|(ri+τ)/ri),
where ri=1+(i-1)τ.For simplicity, in this paper we assumeτ=-p/q with p being any even integer and q being any odd integer. Based on this, we know that ri∈(0,1) is a ratio of two positive odd integers.Remark 4.
Assumptions1-2 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 sinx and ln(1+x2) that can be bounded by a function |x|m for any constant m∈(0,1) 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(10)x˙=f(x)withf(0)=0,x∈Rn,
where f:U0→Rn is continuous with respect to x on an open neighborhood U0 of the origin x=0. The equilibrium x=0 of the system is (locally) finite-time stable if it is Lyapunov stable and finite-time convergent in a neighborhood U∈U0 of the origin. By “finite-time convergence,” we mean that if, for any initial condition, x(0)∈U, there is a settling time T>0, such that every solution x(t) with x(0) as its initial condition of (10) is well defined with x(0)∈U∖{0} for t∈[0,T) and satisfies limt→Tx(t)=0 and x(t)=0 for any t≥T. If U=U0=Rn, 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 C1 function V(x) defined in a neighborhood U^∈Rn of the origin, real numbers c>0, and 0<α<1, such that(i)
V(x) is positive definite on U^;(ii)
V˙(x)+cVα(x)≤0, ∀x∈U^.
Then, the origin of system (10) is locally finite-time stable with
(11)T≤V1-α(x(0))c(1-α)
for initial condition x(0) in some open neighborhood U∈U^ of the origin. If U=Rn and V(x) is also radially unbounded (i.e., V(x)→+∞ as x→+∞), the origin of system (10) is globally finite-time stable.Definition 7 (see [29]).
Weighted homogeneity: for fixed coordinates(x1,…,xn)∈Rn and real numbers ri>0, i=1,…,n, one has the following.(i)
The dilationΔε(x) is defined by Δε(x)=(εr1x1,…,εrnxn) for any ε>0, where ri is called the weights of the coordinates. For simplicity, we define dilation weight Δ=(r1,…,rn).(ii)
A functionV∈(Rn,R) is said to be homogeneous of degree τ if there is a real number τ∈R such that V(Δε(x))=ετV(x1,…,xn) for any x∈Rn∖{0}, ε>0.(iii)
A vector fieldf∈(Rn,Rn) is said to be homogeneous of degree τ if there is a real number τ∈R such that fi(Δε(x))=ετ+rifi(x), for any x∈Rn∖{0}, ε>0, i=1,…,n.(iv)
A homogeneousp-norm is defined as ∥x∥Δ,p=(∑i=1n‍|xi|p/ri)1/p for all x∈Rn, for a constant p≥1. For simplicity, in this paper, one chooses p=2 and writes ∥x∥Δ for ∥x∥Δ,2.Lemma 8 (see [30]).
Suppose thatV:Rn→R is a homogeneous function of degree τ with respect to the dilation weight Δ. Then the following hold.(i)
∂V/∂xi is homogeneous of degree τ-ri with ri being the homogeneous weight of xi.(ii)
There is a constantc such that V(x)≤c∥x∥Δτ. Moreover, if V(x) is positive definite, then c_∥x∥Δτ≤V(x), where c_ is a constant.Lemma 9 (see [31]).
Forx∈R, y∈R, and p≥1 which is a constant, the following inequalities hold:
(12)|x+y|p≤2p-1|xp+yp|,(|x|+|y|)1/p≤|x|1/p+|y|1/p≤2(p-1)/p(|x|+|y|)1/p.
Ifp≥1 is odd, then
(13)|x-y|p≤2p-1|xp-yp|,|x1/p-y1/p|≤2(p-1)/p(|x-y|)1/p.Lemma 10 (see [32]).
Letx, y be real variables; then for any positive real numbers a, m, and n, one has
(14)a|x|m|y|n≤b|x|m+n+nm+n(m+nm)-m/na(m+n)/nb-m/n|y|m+n,
where b>0 is any real number.Lemma 11 (see [33]).
Forx,y∈R and positive real number p, the following inequality holds:
(15)|xp-yp|≤p|x-y||xp-1+yp-1|≤c|x-y||(x-y)p-1+yp-1|,
where c=p for 1<p≤2 and c=p2p-1 for p>2.


## 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 thex-subsystem in a finite time by output feedback.(ii)
Then we design a controller such that thex0-subsystem is finite-time stable.


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

For thex0-subsystem, we choose the control u0 as
(16)u0≡u0*,
where u0* is a positive constant. In this case, the x0-subsystem becomes
(17)x˙0=d0u0*+ϕ0(t,x0).Noting thatϕ0(t,x0) satisfies the linear growth condition, it is easy to obtain that the solution of x0-subsystem is bounded, for any given finite time ts>0. Hence, x0 is well defined on [0,ts]. Under the control law (16), the x-subsystem can be written as
(18)x˙i=diu0*xi+1+ϕi(t,x0,x),i=1,…,n-1,x˙n=dnu1+ϕn(t,x0,x).Next we consider the finite-time output feedback stabilizer for system (18). For convenience, we define the following change of coordinates:
(19)ζ1=x1,ζi=d1⋯di-1u0*i-1xi,i=2,…,n
under which system (18) is transformed into
(20)ζ˙i=ζi+1+fi(t,x0,ζ),i=1,…,n-1,ζ˙n=du1+fn(t,x0,ζ),
where d=d1⋯dnu0*n-1, fi=d1⋯di-1u0*i-1ϕi, and the state ζ1=x1 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 fi, i=1,…,n, satisfy
(21)|fi(t,x0,ζ)|≤b(|ζ1|(ri+τ)/r1+⋯+|ζi|(ri+τ)/ri)
with a new growth rate b.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 fi’s. Then, we utilize a scaling gain in the controller to dominate the uncertain nonlinearities fi’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:(22)z˙i=zi+1,i=1,…,n-1,z˙n=dv.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 DesignStep 1.
Choose the Lyapunov functionV1=x12/2. Clearly, the first virtual controller
(23)z2*=-nz1r2∶=-β1ξ1r2
with ξ1=z1 and β1=n renders
(24)V˙1≤-nξ12+τ+ξ1(z2-z2*).Step i (2≤i≤n).
In this step, we can obtain the following property.Proposition 13.
For theith Lyapunov function defined by
(25)Vi=Vi-1+∫zi*zi(s1/ri-zi*1/ri)2-rids
under the coordinate transformation
(26)zk*=-βk-1ξk-1rk,ξk=zk1/rk-zk*1/rk,k=2,…,i
there exists the C0 virtual controller zi+1*=-βiξi1/ri+1 such that
(27)V˙i≤-(n-i+1)(ξ12+τ+⋯+ξi2+τ)+ξi2-ri(zi+1-zi+1*),
where βj>0, j=1,…,i are constants.Proof.
The detailed proof can be found in [20] and hence is omitted here.From the inductive steps, we can design(28)zn+1*=-βnξnrn+τ=-βn(+β21/r3(z21/r2+β11/r2z1))zn1/rn+βn-11/rn×(zn-11/rn-1+⋯+β21/r3(z21/r2+β11/r2z1)))rn+τ=-βn(β-nzn1/rn+β-n-1zn-11/rn-1+⋯+β-1z1)rn+τ,
where
(29)β-i={βn-11/rn⋯βi1/ri+1,i=1,…,n-11,i=n
such that
(30)V˙n≤-(ξ12+τ+⋯+ξn2+τ)+dξn2-rn(v-zn+1*).(B) Output Feedback Controller Design. Since z2,…,zn are unmeasurable, we construct a homogeneous observer
(31)η˙i=-li-1z^i,z^i=(ηi+li-1z^i-1)ri/ri-1,i=2,…,n,
where z^1=z1 and li>0; i=1,…,n-1 are the gains to be determined. By the certainty equivalence principle, we can replace zi with z^i in (28) and obtain an output feedback controller
(32)v(z^)=-βn(β-nz^n1/rn+β-n-1z^n-11/rn-1+⋯+β-1z1)rn+τ,
where z^=(z1,z^2,…,z^n).Considering(33)Wi=∫γi(2-ri-1)/ri-1zi(2-ri-1)/ri(sri-1/(2-ri-1)-γi)ds,
where γi=ηi+li-1zi-1, and setting ei=(zi-z^i)1/ri, for i=2,…,n, from (22), (31), and (33), it follows that
(34)W˙i=2-ri-1rizi(2-ri-1-ri)/ri(ziri-1/ri-γi)zi+1-li-1eiri(zi(2-ri-1)/ri-z^i(2-ri-1)/ri)-li-1eiri(z^i(2-ri-1)/ri-γi(2-ri-1)/ri-1),
where zn+1=v(z^).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 constantλi such that
(35)-li-1eiri(zi(2-ri-1)/ri-z^i(2-ri-1)/ri)≤-li-1λiei2+τ.Proposition 15.
Fori=2,…,n-1,
(36)2-ri-1rizi(2-ri-1-ri)/ri(ziri-1/ri-γi)zi+1≤112∑j=i-1i+1ξj2+τ+miei2+τ+gi(li-1)ei-12+τ,
where gi is a continuous function of li-1, mi>0 is a constant, and g2=0.Proposition 16.
For the controllerv(z^), one obtains
(37)2-rn-1rnzn(2-rn-1-rn)/rn(znrn-1/rn-γn)v(z^)≤18∑j=1n‍ξj2+τ+c∑i=2n‍ei2+τ+gn(ln-1)en-12+τ,
where gn is a continuous function of ln-1 and c>0 is a constant.Proposition 17.
Fori=3,…,n,
(38)-li-1eiri(z^i(2-ri-1)/ri-γi(2-ri-1)/ri-1)≤116(ξi-12+τ+ξi2+τ)+ei2+τ+hi(li-1)ei-12+τ,
where hi is a continuous function of li-1.ChoosingW=∑i=2n‍Wi, by Propositions 14–17, we get
(39)W˙=12∑i=1nξi2+τ+(-l1λ2+m2+c+g3(l2)+h3(l2))e22+τ+∑i=3n-1‍(-li-1λi+mi+1+c+gi+1(li)+hi+1(li))ei2+τ+(-ln-1λn+1+c)en2+τ.By (28), (32), and Assumption 1, we can estimate dξi2-rn(v-zn+1*) in (29) by the following proposition, whose proof is given in the Appendix.Proposition 18.
There exists a positive constantμ such that
(40)dξi2-rn(v-zn+1*)≤14∑i=1n‍ξi2+τ+μ∑i=2n‍ei2+τ,
where gn is a continuous function of ln-1.With the help of Proposition18, defining U=Vn+W, combining (29) and (39), and recursively choosing
(41)ln-1≥λn-1(14+1+c+μ),li-1≥λi-1(14+mi+1+c+μ+gi+1(li)+hi+1(li)),hhhhhhhhhhhhhhhhhhhhhhhhhhhhi=n-1,…,3,l1≥λ2-1(14+m2+c+μ+g3(l2)+h3(l2))
we obtain
(42)U˙=-14∑i=1n‍ξi2+τ-14∑i=2n‍ei2+τ.SinceU is positive definite and proper with respect to Z=(z1,…,zn,η2…,ηn)T, (42) implies that the closed-loop system can be rewritten as the following compact form:
(43)Z˙=F(Z)=(z2,…,zn,dv,η˙2…,η˙n)T
which is homogeneous with the dilation weight
(44)Δ=(r1,…,rn︸forz1,…,zn,r1,…,rn-1︸forη2…,ηn)=(1,…,1+(n-1)τ︸forz1,…,zn,1,…,1+(n-2)τ︸forη2…,ηn).It can be shown that (43) is homogeneous of degree τ. In addition, U is homogeneous of degree 2. By Lemma 8, there is a constant c-1, such that
(45)U≤c-1∥Z∥Δ2,
where c-1>0 and ∥Z∥Δ=(∑i=12n-1‍|Zi|2/ri). Similarly, since the right-hand side of (42) is homogeneous of degree 2+τ, by Lemma 8 there is a constant c-2 such that
(46)∂U∂ZF(Z)≤-14∑i=1n‍ξi2+τ-14∑i=2n‍ei2+τ≤-c-2∥Z∥Δ2+τ.Combining (45) and (46), it can be deduced from (42) that
(47)U˙≤-kU(2+τ)/2
for a constant k>0. By Lemma 6 with α=(2+τ)/2<1, 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 rn+τ, 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
(48)zi=ηiLi-1,i=2,…,n,v=u1Ln,
where L≥1 is a constant to be determined later. Under (48), system (20) can be rewritten as
(49)z˙i=Lzi+1+fi(·)Li-1,i=1,…,n-1,z˙n=Ldv++fn(·)Ln-1.Now we construct an observer with a gainL as follows:
(50)η˙i=-Lliz^i,z^i=(ηi+li-1z^i-1)ri/ri-1,i=2,…,n.In addition, we designu1 using the same construction of (32), specifically,
(51)u1=-Lnβn(β-nz^n1/rn+β-n-1z^n-11/rn-1+⋯+β-1z1)rn+τ.Now, the closed-loop system (49)–(51) can be written as
(52)Z˙=LF(Z)+(f1(·),f2(·)L,…,fn(·)Ln-1,0,…,0)T.Hence, it can be concluded from (46) that
(53)U˙≤-Lc-2∥Z∥Δ2+τ+∂U∂Z(f1(·),f2(·)L,…,fn(·)Ln-1,0,…,0)T.From (21), (48), and L≥1, we can find constants δi>0 and αi<1 such that
(54)|fi(·)Li-1|≤b∑j=1i‍L(j-1)(ri+τ)/rjLi-1|zj|(ri+τ)/rj≤δiLαi∥Z∥Δri+τ.Noting that, fori=1,…,n, ∂U/∂Zi is homogeneous of degree 2-ri, we know that
(55)∂U∂Zi(|z1|(ri+τ)/r1+⋯+|zi|(ri+τ)/ri)
is homogeneous of degree 2+τ.With (54) and (55) in mind, we can find a positive constant ci such that
(56)|∂U∂Zi||fi(·)Li-1|≤ciLαi∥Z∥Δ2+τ.Substituting (56) into (53) yields
(57)U˙≤-L(c-2-∑i=1n‍ciLαi-1)∥Z∥Δ2+τ≤-L(c-2-∑i=1n‍ciLαmax-1)∥Z∥Δ2+τ,
where αmax=max1≤i≤n{αi}<1. Apparently, by choosing a large enough L, the right-hand side of (57) is negative definite.Furthermore, it can be deduced from (57) that there is a constant c-3 such that
(58)U˙≤-c-3U(2+τ)/2.By Lemma6 (U=V, c=c-3, and α=(2+τ)/2<1), (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 T1 satisfies
(59)T1≤-2U(-τ)/2(0)c-3τ.


### 3.2. Finite-Time Output Feedback Stabilization of thex0-Subsystem

From Section3.1, we know that x(t)≡0 when t≥T1. Therefore, we just need to stabilize the x0-subsystem in a finite time. When t≥T1, for the x0-subsystem, we can take the following control law:
(60)u0(x0)=g0(x0)x0α0,0<α0=α1α2<1,g0(x0)=-1c01(k0+ϕ0(x0)),
where k0 is a positive design constant, αi, i=1,2 are positive odd numbers, and ϕ0(x0)≥a|x0|1-α0≥0 is a smooth function. For instance, we can simply choose ϕ0(x0)=a(1+x02).Taking the Lyapunov functionV0=x02/2, a simple computation gives
(61)V˙0≤-k0x01+α0≤-k0V0(1+α0)/2.Thus, by Lemma6, x0 tends to 0 within a settling time denoted by T2 and
(62)T2≤2V0(1-α0)/2(0)k0(1-α0).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 Assumptions1–3, 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 (x0,x)∈Rn+1, the closed-loop system is globally finite-time regulated at origin.


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

For thex0-subsystem, we choose the control u0 as
(16)u0≡u0*,
where u0* is a positive constant. In this case, the x0-subsystem becomes
(17)x˙0=d0u0*+ϕ0(t,x0).Noting thatϕ0(t,x0) satisfies the linear growth condition, it is easy to obtain that the solution of x0-subsystem is bounded, for any given finite time ts>0. Hence, x0 is well defined on [0,ts]. Under the control law (16), the x-subsystem can be written as
(18)x˙i=diu0*xi+1+ϕi(t,x0,x),i=1,…,n-1,x˙n=dnu1+ϕn(t,x0,x).Next we consider the finite-time output feedback stabilizer for system (18). For convenience, we define the following change of coordinates:
(19)ζ1=x1,ζi=d1⋯di-1u0*i-1xi,i=2,…,n
under which system (18) is transformed into
(20)ζ˙i=ζi+1+fi(t,x0,ζ),i=1,…,n-1,ζ˙n=du1+fn(t,x0,ζ),
where d=d1⋯dnu0*n-1, fi=d1⋯di-1u0*i-1ϕi, and the state ζ1=x1 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 fi, i=1,…,n, satisfy
(21)|fi(t,x0,ζ)|≤b(|ζ1|(ri+τ)/r1+⋯+|ζi|(ri+τ)/ri)
with a new growth rate b.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 fi’s. Then, we utilize a scaling gain in the controller to dominate the uncertain nonlinearities fi’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:(22)z˙i=zi+1,i=1,…,n-1,z˙n=dv.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 DesignStep 1.
Choose the Lyapunov functionV1=x12/2. Clearly, the first virtual controller
(23)z2*=-nz1r2∶=-β1ξ1r2
with ξ1=z1 and β1=n renders
(24)V˙1≤-nξ12+τ+ξ1(z2-z2*).Step i (2≤i≤n).
In this step, we can obtain the following property.Proposition 13.
For theith Lyapunov function defined by
(25)Vi=Vi-1+∫zi*zi(s1/ri-zi*1/ri)2-rids
under the coordinate transformation
(26)zk*=-βk-1ξk-1rk,ξk=zk1/rk-zk*1/rk,k=2,…,i
there exists the C0 virtual controller zi+1*=-βiξi1/ri+1 such that
(27)V˙i≤-(n-i+1)(ξ12+τ+⋯+ξi2+τ)+ξi2-ri(zi+1-zi+1*),
where βj>0, j=1,…,i are constants.Proof.
The detailed proof can be found in [20] and hence is omitted here.From the inductive steps, we can design(28)zn+1*=-βnξnrn+τ=-βn(+β21/r3(z21/r2+β11/r2z1))zn1/rn+βn-11/rn×(zn-11/rn-1+⋯+β21/r3(z21/r2+β11/r2z1)))rn+τ=-βn(β-nzn1/rn+β-n-1zn-11/rn-1+⋯+β-1z1)rn+τ,
where
(29)β-i={βn-11/rn⋯βi1/ri+1,i=1,…,n-11,i=n
such that
(30)V˙n≤-(ξ12+τ+⋯+ξn2+τ)+dξn2-rn(v-zn+1*).(B) Output Feedback Controller Design. Since z2,…,zn are unmeasurable, we construct a homogeneous observer
(31)η˙i=-li-1z^i,z^i=(ηi+li-1z^i-1)ri/ri-1,i=2,…,n,
where z^1=z1 and li>0; i=1,…,n-1 are the gains to be determined. By the certainty equivalence principle, we can replace zi with z^i in (28) and obtain an output feedback controller
(32)v(z^)=-βn(β-nz^n1/rn+β-n-1z^n-11/rn-1+⋯+β-1z1)rn+τ,
where z^=(z1,z^2,…,z^n).Considering(33)Wi=∫γi(2-ri-1)/ri-1zi(2-ri-1)/ri(sri-1/(2-ri-1)-γi)ds,
where γi=ηi+li-1zi-1, and setting ei=(zi-z^i)1/ri, for i=2,…,n, from (22), (31), and (33), it follows that
(34)W˙i=2-ri-1rizi(2-ri-1-ri)/ri(ziri-1/ri-γi)zi+1-li-1eiri(zi(2-ri-1)/ri-z^i(2-ri-1)/ri)-li-1eiri(z^i(2-ri-1)/ri-γi(2-ri-1)/ri-1),
where zn+1=v(z^).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 constantλi such that
(35)-li-1eiri(zi(2-ri-1)/ri-z^i(2-ri-1)/ri)≤-li-1λiei2+τ.Proposition 15.
Fori=2,…,n-1,
(36)2-ri-1rizi(2-ri-1-ri)/ri(ziri-1/ri-γi)zi+1≤112∑j=i-1i+1ξj2+τ+miei2+τ+gi(li-1)ei-12+τ,
where gi is a continuous function of li-1, mi>0 is a constant, and g2=0.Proposition 16.
For the controllerv(z^), one obtains
(37)2-rn-1rnzn(2-rn-1-rn)/rn(znrn-1/rn-γn)v(z^)≤18∑j=1n‍ξj2+τ+c∑i=2n‍ei2+τ+gn(ln-1)en-12+τ,
where gn is a continuous function of ln-1 and c>0 is a constant.Proposition 17.
Fori=3,…,n,
(38)-li-1eiri(z^i(2-ri-1)/ri-γi(2-ri-1)/ri-1)≤116(ξi-12+τ+ξi2+τ)+ei2+τ+hi(li-1)ei-12+τ,
where hi is a continuous function of li-1.ChoosingW=∑i=2n‍Wi, by Propositions 14–17, we get
(39)W˙=12∑i=1nξi2+τ+(-l1λ2+m2+c+g3(l2)+h3(l2))e22+τ+∑i=3n-1‍(-li-1λi+mi+1+c+gi+1(li)+hi+1(li))ei2+τ+(-ln-1λn+1+c)en2+τ.By (28), (32), and Assumption 1, we can estimate dξi2-rn(v-zn+1*) in (29) by the following proposition, whose proof is given in the Appendix.Proposition 18.
There exists a positive constantμ such that
(40)dξi2-rn(v-zn+1*)≤14∑i=1n‍ξi2+τ+μ∑i=2n‍ei2+τ,
where gn is a continuous function of ln-1.With the help of Proposition18, defining U=Vn+W, combining (29) and (39), and recursively choosing
(41)ln-1≥λn-1(14+1+c+μ),li-1≥λi-1(14+mi+1+c+μ+gi+1(li)+hi+1(li)),hhhhhhhhhhhhhhhhhhhhhhhhhhhhi=n-1,…,3,l1≥λ2-1(14+m2+c+μ+g3(l2)+h3(l2))
we obtain
(42)U˙=-14∑i=1n‍ξi2+τ-14∑i=2n‍ei2+τ.SinceU is positive definite and proper with respect to Z=(z1,…,zn,η2…,ηn)T, (42) implies that the closed-loop system can be rewritten as the following compact form:
(43)Z˙=F(Z)=(z2,…,zn,dv,η˙2…,η˙n)T
which is homogeneous with the dilation weight
(44)Δ=(r1,…,rn︸forz1,…,zn,r1,…,rn-1︸forη2…,ηn)=(1,…,1+(n-1)τ︸forz1,…,zn,1,…,1+(n-2)τ︸forη2…,ηn).It can be shown that (43) is homogeneous of degree τ. In addition, U is homogeneous of degree 2. By Lemma 8, there is a constant c-1, such that
(45)U≤c-1∥Z∥Δ2,
where c-1>0 and ∥Z∥Δ=(∑i=12n-1‍|Zi|2/ri). Similarly, since the right-hand side of (42) is homogeneous of degree 2+τ, by Lemma 8 there is a constant c-2 such that
(46)∂U∂ZF(Z)≤-14∑i=1n‍ξi2+τ-14∑i=2n‍ei2+τ≤-c-2∥Z∥Δ2+τ.Combining (45) and (46), it can be deduced from (42) that
(47)U˙≤-kU(2+τ)/2
for a constant k>0. By Lemma 6 with α=(2+τ)/2<1, 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 rn+τ, 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
(48)zi=ηiLi-1,i=2,…,n,v=u1Ln,
where L≥1 is a constant to be determined later. Under (48), system (20) can be rewritten as
(49)z˙i=Lzi+1+fi(·)Li-1,i=1,…,n-1,z˙n=Ldv++fn(·)Ln-1.Now we construct an observer with a gainL as follows:
(50)η˙i=-Lliz^i,z^i=(ηi+li-1z^i-1)ri/ri-1,i=2,…,n.In addition, we designu1 using the same construction of (32), specifically,
(51)u1=-Lnβn(β-nz^n1/rn+β-n-1z^n-11/rn-1+⋯+β-1z1)rn+τ.Now, the closed-loop system (49)–(51) can be written as
(52)Z˙=LF(Z)+(f1(·),f2(·)L,…,fn(·)Ln-1,0,…,0)T.Hence, it can be concluded from (46) that
(53)U˙≤-Lc-2∥Z∥Δ2+τ+∂U∂Z(f1(·),f2(·)L,…,fn(·)Ln-1,0,…,0)T.From (21), (48), and L≥1, we can find constants δi>0 and αi<1 such that
(54)|fi(·)Li-1|≤b∑j=1i‍L(j-1)(ri+τ)/rjLi-1|zj|(ri+τ)/rj≤δiLαi∥Z∥Δri+τ.Noting that, fori=1,…,n, ∂U/∂Zi is homogeneous of degree 2-ri, we know that
(55)∂U∂Zi(|z1|(ri+τ)/r1+⋯+|zi|(ri+τ)/ri)
is homogeneous of degree 2+τ.With (54) and (55) in mind, we can find a positive constant ci such that
(56)|∂U∂Zi||fi(·)Li-1|≤ciLαi∥Z∥Δ2+τ.Substituting (56) into (53) yields
(57)U˙≤-L(c-2-∑i=1n‍ciLαi-1)∥Z∥Δ2+τ≤-L(c-2-∑i=1n‍ciLαmax-1)∥Z∥Δ2+τ,
where αmax=max1≤i≤n{αi}<1. Apparently, by choosing a large enough L, the right-hand side of (57) is negative definite.Furthermore, it can be deduced from (57) that there is a constant c-3 such that
(58)U˙≤-c-3U(2+τ)/2.By Lemma6 (U=V, c=c-3, and α=(2+τ)/2<1), (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 T1 satisfies
(59)T1≤-2U(-τ)/2(0)c-3τ.


## 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:(22)z˙i=zi+1,i=1,…,n-1,z˙n=dv.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 DesignStep 1.
Choose the Lyapunov functionV1=x12/2. Clearly, the first virtual controller
(23)z2*=-nz1r2∶=-β1ξ1r2
with ξ1=z1 and β1=n renders
(24)V˙1≤-nξ12+τ+ξ1(z2-z2*).Step i (2≤i≤n).
In this step, we can obtain the following property.Proposition 13.
For theith Lyapunov function defined by
(25)Vi=Vi-1+∫zi*zi(s1/ri-zi*1/ri)2-rids
under the coordinate transformation
(26)zk*=-βk-1ξk-1rk,ξk=zk1/rk-zk*1/rk,k=2,…,i
there exists the C0 virtual controller zi+1*=-βiξi1/ri+1 such that
(27)V˙i≤-(n-i+1)(ξ12+τ+⋯+ξi2+τ)+ξi2-ri(zi+1-zi+1*),
where βj>0, j=1,…,i are constants.Proof.
The detailed proof can be found in [20] and hence is omitted here.From the inductive steps, we can design(28)zn+1*=-βnξnrn+τ=-βn(+β21/r3(z21/r2+β11/r2z1))zn1/rn+βn-11/rn×(zn-11/rn-1+⋯+β21/r3(z21/r2+β11/r2z1)))rn+τ=-βn(β-nzn1/rn+β-n-1zn-11/rn-1+⋯+β-1z1)rn+τ,
where
(29)β-i={βn-11/rn⋯βi1/ri+1,i=1,…,n-11,i=n
such that
(30)V˙n≤-(ξ12+τ+⋯+ξn2+τ)+dξn2-rn(v-zn+1*).(B) Output Feedback Controller Design. Since z2,…,zn are unmeasurable, we construct a homogeneous observer
(31)η˙i=-li-1z^i,z^i=(ηi+li-1z^i-1)ri/ri-1,i=2,…,n,
where z^1=z1 and li>0; i=1,…,n-1 are the gains to be determined. By the certainty equivalence principle, we can replace zi with z^i in (28) and obtain an output feedback controller
(32)v(z^)=-βn(β-nz^n1/rn+β-n-1z^n-11/rn-1+⋯+β-1z1)rn+τ,
where z^=(z1,z^2,…,z^n).Considering(33)Wi=∫γi(2-ri-1)/ri-1zi(2-ri-1)/ri(sri-1/(2-ri-1)-γi)ds,
where γi=ηi+li-1zi-1, and setting ei=(zi-z^i)1/ri, for i=2,…,n, from (22), (31), and (33), it follows that
(34)W˙i=2-ri-1rizi(2-ri-1-ri)/ri(ziri-1/ri-γi)zi+1-li-1eiri(zi(2-ri-1)/ri-z^i(2-ri-1)/ri)-li-1eiri(z^i(2-ri-1)/ri-γi(2-ri-1)/ri-1),
where zn+1=v(z^).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 constantλi such that
(35)-li-1eiri(zi(2-ri-1)/ri-z^i(2-ri-1)/ri)≤-li-1λiei2+τ.Proposition 15.
Fori=2,…,n-1,
(36)2-ri-1rizi(2-ri-1-ri)/ri(ziri-1/ri-γi)zi+1≤112∑j=i-1i+1ξj2+τ+miei2+τ+gi(li-1)ei-12+τ,
where gi is a continuous function of li-1, mi>0 is a constant, and g2=0.Proposition 16.
For the controllerv(z^), one obtains
(37)2-rn-1rnzn(2-rn-1-rn)/rn(znrn-1/rn-γn)v(z^)≤18∑j=1n‍ξj2+τ+c∑i=2n‍ei2+τ+gn(ln-1)en-12+τ,
where gn is a continuous function of ln-1 and c>0 is a constant.Proposition 17.
Fori=3,…,n,
(38)-li-1eiri(z^i(2-ri-1)/ri-γi(2-ri-1)/ri-1)≤116(ξi-12+τ+ξi2+τ)+ei2+τ+hi(li-1)ei-12+τ,
where hi is a continuous function of li-1.ChoosingW=∑i=2n‍Wi, by Propositions 14–17, we get
(39)W˙=12∑i=1nξi2+τ+(-l1λ2+m2+c+g3(l2)+h3(l2))e22+τ+∑i=3n-1‍(-li-1λi+mi+1+c+gi+1(li)+hi+1(li))ei2+τ+(-ln-1λn+1+c)en2+τ.By (28), (32), and Assumption 1, we can estimate dξi2-rn(v-zn+1*) in (29) by the following proposition, whose proof is given in the Appendix.Proposition 18.
There exists a positive constantμ such that
(40)dξi2-rn(v-zn+1*)≤14∑i=1n‍ξi2+τ+μ∑i=2n‍ei2+τ,
where gn is a continuous function of ln-1.With the help of Proposition18, defining U=Vn+W, combining (29) and (39), and recursively choosing
(41)ln-1≥λn-1(14+1+c+μ),li-1≥λi-1(14+mi+1+c+μ+gi+1(li)+hi+1(li)),hhhhhhhhhhhhhhhhhhhhhhhhhhhhi=n-1,…,3,l1≥λ2-1(14+m2+c+μ+g3(l2)+h3(l2))
we obtain
(42)U˙=-14∑i=1n‍ξi2+τ-14∑i=2n‍ei2+τ.SinceU is positive definite and proper with respect to Z=(z1,…,zn,η2…,ηn)T, (42) implies that the closed-loop system can be rewritten as the following compact form:
(43)Z˙=F(Z)=(z2,…,zn,dv,η˙2…,η˙n)T
which is homogeneous with the dilation weight
(44)Δ=(r1,…,rn︸forz1,…,zn,r1,…,rn-1︸forη2…,ηn)=(1,…,1+(n-1)τ︸forz1,…,zn,1,…,1+(n-2)τ︸forη2…,ηn).It can be shown that (43) is homogeneous of degree τ. In addition, U is homogeneous of degree 2. By Lemma 8, there is a constant c-1, such that
(45)U≤c-1∥Z∥Δ2,
where c-1>0 and ∥Z∥Δ=(∑i=12n-1‍|Zi|2/ri). Similarly, since the right-hand side of (42) is homogeneous of degree 2+τ, by Lemma 8 there is a constant c-2 such that
(46)∂U∂ZF(Z)≤-14∑i=1n‍ξi2+τ-14∑i=2n‍ei2+τ≤-c-2∥Z∥Δ2+τ.Combining (45) and (46), it can be deduced from (42) that
(47)U˙≤-kU(2+τ)/2
for a constant k>0. By Lemma 6 with α=(2+τ)/2<1, 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 rn+τ, 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
(48)zi=ηiLi-1,i=2,…,n,v=u1Ln,
where L≥1 is a constant to be determined later. Under (48), system (20) can be rewritten as
(49)z˙i=Lzi+1+fi(·)Li-1,i=1,…,n-1,z˙n=Ldv++fn(·)Ln-1.Now we construct an observer with a gainL as follows:
(50)η˙i=-Lliz^i,z^i=(ηi+li-1z^i-1)ri/ri-1,i=2,…,n.In addition, we designu1 using the same construction of (32), specifically,
(51)u1=-Lnβn(β-nz^n1/rn+β-n-1z^n-11/rn-1+⋯+β-1z1)rn+τ.Now, the closed-loop system (49)–(51) can be written as
(52)Z˙=LF(Z)+(f1(·),f2(·)L,…,fn(·)Ln-1,0,…,0)T.Hence, it can be concluded from (46) that
(53)U˙≤-Lc-2∥Z∥Δ2+τ+∂U∂Z(f1(·),f2(·)L,…,fn(·)Ln-1,0,…,0)T.From (21), (48), and L≥1, we can find constants δi>0 and αi<1 such that
(54)|fi(·)Li-1|≤b∑j=1i‍L(j-1)(ri+τ)/rjLi-1|zj|(ri+τ)/rj≤δiLαi∥Z∥Δri+τ.Noting that, fori=1,…,n, ∂U/∂Zi is homogeneous of degree 2-ri, we know that
(55)∂U∂Zi(|z1|(ri+τ)/r1+⋯+|zi|(ri+τ)/ri)
is homogeneous of degree 2+τ.With (54) and (55) in mind, we can find a positive constant ci such that
(56)|∂U∂Zi||fi(·)Li-1|≤ciLαi∥Z∥Δ2+τ.Substituting (56) into (53) yields
(57)U˙≤-L(c-2-∑i=1n‍ciLαi-1)∥Z∥Δ2+τ≤-L(c-2-∑i=1n‍ciLαmax-1)∥Z∥Δ2+τ,
where αmax=max1≤i≤n{αi}<1. Apparently, by choosing a large enough L, the right-hand side of (57) is negative definite.Furthermore, it can be deduced from (57) that there is a constant c-3 such that
(58)U˙≤-c-3U(2+τ)/2.By Lemma6 (U=V, c=c-3, and α=(2+τ)/2<1), (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 T1 satisfies
(59)T1≤-2U(-τ)/2(0)c-3τ.


## 3.2. Finite-Time Output Feedback Stabilization of thex0-Subsystem

From Section3.1, we know that x(t)≡0 when t≥T1. Therefore, we just need to stabilize the x0-subsystem in a finite time. When t≥T1, for the x0-subsystem, we can take the following control law:
(60)u0(x0)=g0(x0)x0α0,0<α0=α1α2<1,g0(x0)=-1c01(k0+ϕ0(x0)),
where k0 is a positive design constant, αi, i=1,2 are positive odd numbers, and ϕ0(x0)≥a|x0|1-α0≥0 is a smooth function. For instance, we can simply choose ϕ0(x0)=a(1+x02).Taking the Lyapunov functionV0=x02/2, a simple computation gives
(61)V˙0≤-k0x01+α0≤-k0V0(1+α0)/2.Thus, by Lemma6, x0 tends to 0 within a settling time denoted by T2 and
(62)T2≤2V0(1-α0)/2(0)k0(1-α0).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 Assumptions1–3, 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 (x0,x)∈Rn+1, 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:(63)x˙0=d0u0+θ0(t)x0,x˙1=d1x2u0+θ1(t)ln(1+x12),x˙2=d2u1+θ2sinx1,
where di, i=0,1,2 are unknown constants and θi(t), i=0,1,2 are unknown functions.It should be mentioned that, whend1=d2=1 and θ0(t)=θ1(t)=θ2(t)=0, 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 thatdi∈[0.5,1] and |θi(t)|≤1, i=1,2,3. From this and Remark 4, it is not difficult to verify that Assumptions 1–3 hold. Firstly, we define the control law u0=1 and introduce the change of coordinates
(64)ζ1=x1,ζ2=d1x2
under which the x-subsystem of (63) is transformed into
(65)ζ˙1=ζ2+θ1(t)ln(1+ζ12),ζ˙2=d1d2u1+d1θ2(t)sinζ1.If we pickτ=-2/5, the dilation is defined as r1=1 and r2=3/5. 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
(66)η˙2=-Ll1z^2,z^2=(η2+l1z1)3/5,u1=-L2β2(z^25/3+β15/3z1)1/5
with appropriate positive constants l1, β1, β2, and a large enough gain L such that output feedback controller (66) renders the system (65) (i.e., the x-subsystem of (63)) globally finite-time stable with a settling time T1.Then, whent≥T1, for the x0-subsystem, we switch the control input u0 to
(67)u0(x0)=-2k0x01/3,
where k0 is a positive design constant.In the simulation, we assumed0=d1=d2=1 and θ0(t)=θ1(t)=θ2(t)=sint. When (x0(0),x1(0), x2(0),η2(0))=(0,1,-1,0), by choosing the gains for the output laws as L=2, β1=2, β2=9, l1=20, and k0=1, the simulation shown in Figure 1 demonstrates the global finite-time stability property of the closed-loop system (63)–(67).The responses of the closed-loop system (63)–(67).
(a)
x
0
(b)
x
1
(c)
x
2
(d)
η
2
(e)
u
0
(f)
u
1Remark 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.



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*Source: 926971-2013-11-24.xml*