# Global Finite-Time Stabilization for a Class of Uncertain High-Order Nonlinear Systems

Authors: Jian Wang; Jing Xie; Fangzheng Gao
Journal: Abstract and Applied Analysis
 (2014)

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## Abstract

This paper addresses the problem of global finite-time stabilization by state feedback for a class of high-order nonlinear systems under weaker condition. By using the methods of adding a power integrator, a continuous state feedback
controller is successfully constructed to guarantee the global finite-time stability of the resulting closed-loop system. A simulation example is provided to illustrate the effectiveness of the proposed approach.


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## Body

## 1. Introduction

In this paper, we consider the following high-order nonlinear systems:(1)x˙i=xi+1pi+fi(t,x,u),i=1,…,n-1,x˙n=upn+fn(t,x,u),
where x=(x1,…,xn)T∈Rn, u∈R are the system state and input, respectively; pi∈Rodd≥1:={p/q∣p and q are positive odd integers, and p≥q} are said to be the high orders of the system; fi:R+×Rn×R→R, i=1,…,n, are unknown continuous functions of all the states and the control input.The importance for studying such system is exemplified in [1], where state feedback controller was used to stabilize the underactuated, weakly coupled, unstable mechanical system. Since Jacobian linearization of system (1) at the origin is neither controllable nor feedback linearizable for the case of p>1, the traditional design tools including feedback linearization or backstepping are hardly applicable to the system (1). Mainly thanks to the adding a power integrator method, a series of stabilizing results have been achieved over the last decades; for example, one can see [2–10] and the references therein. 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 [11]. Hence it is more meaningful to investigate the finite-time stabilization problem than the classical asymptotical stabilization. In recent years, the finite-time stabilization of system (1) has been studied fairly extensively with various restrictions on the integrator powers and the system nonlinearities [12–21]. In particular, [22] solved the finite-time stabilization problem under the condition that fi satisfies
(2)|fi(t,x,u)|≤M∑j=1i‍|xj|(ri+ω)/rj
with constants M>0 and ω∈(-2/(2n+1)p1⋯pn-1,0). However, from both practical and theoretical points of view, it is somewhat restrictive to require system (1) to satisfy such restriction. Therefore, the following interesting problem is proposed:is it possible to relax the nonlinear growth condition? Under the weaker condition, can a finite-time stabilizing controller be designed?In this paper, by necessarily modifying the method of adding a power integrator and by successfully overcoming some essential difficulties such as the weaker assumption on the system growth, the appearance of the sign function, and the construction of a continuously differentiable, positive-definite, and proper Lyapunov function, we will focus on solving the above problem.Notations. Throughout this paper, the following notations are adopted. R+ denotes the set of all nonnegative real numbers and Rn denotes the real n-dimensional space. For any vector x=(x1,…,xn)T∈Rn denote x¯i=(x1,…,xi)T∈Ri, i=1,…,n, ∥x∥=(∑i=1n‍xi2)1/2. K denotes the set of all functions: R+→R+, which are continuous, strictly increasing, and vanishing at zero; K∞ denotes the set of all functions which are of class K and unbounded. A sign function sign⁡(x) is defined as follows: sign⁡(x)=1, if x>0; sign⁡(x)=0, if x=0; and sign⁡(x)=-1, if x<0. Besides, the arguments of the functions (or the functionals) will be omitted or simplified, whenever no confusion can arise from the context. For instance, we sometimes denote a function f(x(t)) by simply f(x), f(·), or f.


## 2. Problem Statement and Preliminaries

The objective of this paper is to develop a recursive design method for globally finite-time stabilizing system (1) via state feedback under the following assumption.Assumption 1.
Fori=1,…,n, there are smooth functions φi(x¯i)≥0 and constant ω∈(-1/∑l=1n‍p1⋯pl-1,0) such that
(3)|fi(t,x,u)|≤φi(x¯i)∑j=1i‍|xj|(ri+ω)/rj,
where ri’s are defined as
(4)r1=1,ri+1=ri+ωpi,i=1,…,n.Remark 2.
It is worth pointing out that Assumption1, which gives the nonlinear growth condition on the system drift terms, encompasses the assumptions in the closely related works [14, 21, 22]. To clearly show this, we would like to make the following comparisons to reveal the relationship between Assumption 1 and the counterparts in [14, 21, 22]; that is, Assumption 1 includes those as special cases.(i)
In [14, 21], the system nonlinearities fi’s are required to satisfy
(5)|fi(t,x,u)|≤γi(x¯i)(|x1|+⋯+|xi|),where γi(x¯i)≥0, i=1,…,n, are C1 functions. By (4), we get that ri=(1/p1⋯pi-1)+∑l=1i-1‍(ω/pl⋯pi-1). Furthermore, from ω∈(-1/∑l=1n‍p1⋯pl-1,0), we have (ri+ω)/rj<1, i=1,…,n. It means that
(6)|fi(t,x,u)|≤γi(x¯i)(|x1|+⋯+|xi|)≤γi(x¯i)(|x1|(ri+ω)/r1|x1|1-(ri+ω)/r1hhhhhhhh+⋯+|xi|(ri+ω)/ri|xi|1-(ri+ω)/ri).
Letting φi(x¯i) be smooth functions and satisfying φi(x¯i)≥max⁡{γi|x1|1-(ri+ω)/r1,…,γi|xi|1-(ri+ω)/ri}, we have
(7)|fi(t,x,u)|≤φi(x¯i)(|x1|(ri+ω)/r1+⋯+|xi|(ri+ω)/ri);
from this we can see that the main assumption (4) in [14, 21] is a special case of Assumption 1 above.(ii)
In [22], the system nonlinearities fi’s are required to satisfy
(8)|fi(t,x,u)|≤M∑j=1i‍|xj|(ri+ω)/rjwith constants M>0 and ω∈(-2/(2n+1)p1⋯pn-1,0). Obviously, when φi(x¯i)=M, inequality (4) degenerates to inequality (8). Moreover, the value range of ω in (4) is larger than that in (8). Thus, the main assumption (5) in [22] is a special case of Assumption 1 above.
In the remainder of this section, we present the following lemmas which play an important role in the design process.Lemma 3 (see [11]).
Consider the nonlinear system(9)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. 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 (9) is finite-time stable with
(10)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 radially unbounded (i.e., V(x)→+∞ as x→+∞), the origin of system (9) is globally finite-time stable.Lemma 4 (see [6]).
Forx,y∈R, and p≥1 being a constant, the following inequalities hold:
(11)|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
If p≥1 is odd, then
(12)|x-y|p≤2p-1|xp-yp||x1/p-y1/p|≤2(p-1)/p|x-y|1/p.Lemma 5 (see [10]).
Ifp=a/b∈Rodd≥1 with b≥1, then for any x,y∈R(13)|xp-yp|≤21-1/b|sgn⁡(x)|x|a-sgn⁡(y)|y|a|1/b.Lemma 6 (see [23]).
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/nhhhhhhhhhh×a(m+n)/nb-m/n|y|m+n,
where b>0 is any real number.


## 3. Finite-Time Control Design

In this section, we will construct a continuous state feedback controller by applying the method of adding a power integrator. For simplicity, we denotesgn⁡(x)|x|a≜[x]a for any a∈R+ and x∈R.Step 1.
Letξ1=[x1] and choose V1=W1=∫0x1‍s2-r2p1ds. Using (1) and Assumption 1, we have
(15)V˙1=x12-r2p1x2p1+x12-r2p1f1≤x12-r2p1x2p1+x12φ1.
Obviously, the first virtual controller
(16)x2*=-β1r2x1r2=-β1r2[ξ1]r2
with β1≥(n+φ1)1/r2p1 being smooth results in
(17)V˙1≤-n|ξ1|2+[ξ1]2-r2p1(x2p1-x2*p1).Step k
(
k
=
2
,
…
,
n
). Suppose at step k-1, there are a C1, proper and positive definite Lyapunov function Vk-1, and a set of virtual controllers x1*,…,xk* defined by
(18)x1*=0,x2*=-β1r2[ξ1]r2,⋮xk*=-βk-1rk[ξk-1]rk,ξ1=[x1]1/r1-[x1*]1/r1,ξ2=[x2]1/r2-[x2*]1/r2,⋮ξk=[xk]1/rk-[xk*]1/rk,
with β1>0,…,βk-1>0 being smooth, such that
(19)V˙k-1≤-(n-k+2)∑j=1k-1‍|ξj|2+[ξk-1]2-rkpk-1hhhhhhhhhhhhhhh×(xkpk-1-xk*pk-1).To complete the induction, at thekth step, we choose the following Lyapunov function:
(20)Vk(x¯k)=Vk-1(x¯k-1)+Wk(x¯k),
where
(21)Wk=∫xk*xk‍[[s]1/rk-[xk*]1/rk]2-rk+1pkds.Noting that2-rk+1pk≥1 and using a similar method as in [10], Vk can be shown to be C1, proper and positive definite. Moreover, we can obtain
(22)∂Wk∂xk=[ξk]2-rk+1pk∂Wk∂xj=-(2-rk+1pk)×∫xk*xk‍[[s]1/rk-[xk*]1/rk]1-rk+1pkds∂[xk*]1/rk∂xj,
where j=1,…,k-1.Using (20)–(22), it follows that
(23)V˙k≤-(n-k+2)∑j=1k-1‍|ξj|2+[ξk-1]2-rkpk-1(xkpk-1-xk*pk-1)+[ξk]2-rk+1pkxk+1pk+[ξk]2-rk+1pkfk+∑j=1k-1‍∂Wk∂xj(xj+1pj+fj).In order to proceed further, an appropriate bounding estimate should be given for the last three terms on the right-hand side of inequality (23). This is accomplished in the following propositions whose technical proofs are given in the appendix.Proposition 7.
There exists a positive constantlk1 such that
(24)[ξk-1]2-rkpk-1(xkpk-1-xk*pk-1)≤13|ξk-1|2+lk1|ξk|2.Proposition 8.
There exists a nonnegative smooth functionlk2 such that
(25)[ξk]2-rk+1pkfk≤13∑j=1k-1‍|ξj|2+lk2|ξk|2.Proposition 9.
There exists a nonnegative smooth functionlk3 such that
(26)∑j=1k-1‍∂Wk∂xj(xj+1pj+fj)≤13∑j=1k-1‍|ξj|2+lk3|ξk|2.
Substituting (24)–(26) into (23) yields
(27)V˙k≤-(n-k+1)∑j=1k-1‍|ξj|2+[ξk]2-rk+1pkxk+1pk+(lk1+lk2+lk3)|ξk|2.
Now, it easy to see that the virtual controller(28)xk+1*=-βkrk+1[ξk]rk+1,
where βk≥(n-k+1+lk1+lk2+lk3)1/(rk+1pk) is a smooth function, renders
(29)V˙k≤-(n-k+1)∑j=1k‍|ξj|2+[ξk]2-rk+1pk(xk+1pk-xk+1*pk).
This completes the inductive step.Using the inductive argument above, we can conclude that at thenth step, there exists a continuous state feedback controller of the form
(30)u=xn+1*=-βnrn+1[ξn]rn+1
such that
(31)V˙n≤-∑j=1n‍|ξj|2,
where
(32)Vn=∑k=1n‍Wk=∑k=1n‍∫xk*xk‍[[s]1/rk-[xk*]1/rk]2-rk+1pkds.


## 4. Stability Analysis

We state the main result in this paper.Theorem 10.
If Assumption1 holds for system (1), under the continuous state feedback controller (30), then the following holds:(i)
all the solutions of the closed-loop system are well defined on[0,+∞);(ii)
the equilibriumx=0 of the closed-loop system is globally finite-time stable.Proof.
(i) Considering (18), system (1) can be equivalently transformed into a ξ-system:
(33)ξ˙i(t)=φi(t,ξ,u),i=1,…,n.
By the existence and continuation of the solutions, statesξ(t) are defined on [0,tM], where the number tM>0 may be infinite or not. The following analysis focuses on [0,tM]. Noting that Vn is positive definite and radially unbounded, by (31), (32), and Lemma  4.3 in [24], it is not hard to obtain that there exist K∞ functions π1, π2, and π3 such that
(34)π1(∥ξ(t)∥)≤Vn(ξ(t))≤π2(∥ξ(t)∥)V˙n≤-π3(∥ξ(t)∥).
Sinceπ1 is a class K∞ function, then for any ɛ>0, one can always find a β=β(ɛ) with β>ɛ>0 such that π2(ɛ)≤π1(β), ∀t∈[0,tM], which means that ∥ξ(t)∥≤β, ∀t∈[0,tM]. Suppose that tM is finite; then lim⁡t→tM∥ξ(t)∥=+∞, which contradicts ∥ξ(t)∥≤β, ∀t∈[0,tM]. Hence, ξ(t) is well defined on [0,+∞), so is x(t).(ii)
Noticing thattM=+∞, by using (34), and Lyapunov stability theorem [24], we know that the equilibrium ξ=0 of the closed-loop ξ-systems (30) and (33) is globally asymptotically stable. According to the definition of finite-time stability [11], if the global finite-time attractivity of the closed-loop system can be guaranteed, then the global finite-time stabilization result will be obtained. To this end, let us prove the global finite-time attractivity. First of all, by using Lemma 5, it is easy to see that
(35)Wk=∫xk*xk‍[[s]1/rk-[xk*]1/rk]2-rk+1pkds≤|ξk|2-rk+1pk|xk-xk*|≤21-rk|ξk|2-ω. So we have the following estimate:
(36)Vn=∑k=1n‍Wk≤2∑k=1n‍|ξk|2-ω.
Letα=2/(2-ω). With (36) and (31) in mind, by Lemma 4, it is not difficult to obtain that
(37)V˙n≤-12Vnα.
Therefore, by Lemma3, we obtain that the equilibrium ξ=0 of the closed-loop ξ-systems (30) and (33) is globally finite-time stable. This together with the definitions of xi*’s directly concludes that the globally finite-time stability of the closed-loop systems (1), (18), and (30) at the equilibrium x=0.


## 5. Simulation Example

To illustrate the effectiveness of the proposed approach, we consider the following low-dimensional system:(38)x˙1=x25/3+x110/11,x˙2=u+x12sinx2,
where p1=5/3 and p2=1. It is worth pointing out that although system (38) is simple, it cannot be globally finite-time stabilized using the design method presented in [14, 21, 22] because of the presence of both low-order term x110/11 and high-order term x12sinx2. Choose ω=-1/11∈(-3/8,0]; then r1=1, r2=(r1+ω)/p1=6/11, and r3=(r2+ω)/p2=5/11. By Lemma 6, it is easy to get |f1|≤|x1|10/11 and |f2|≤(1+x12)|x1|5/11. Clearly, Assumption 1 holds with φ1=1 and φ2=1+x12; according to the design procedure proposed in Section 3, we can get
(39)x2*=-β16/11[x1]6/11,β1=311/10,u=-(1+l21+l22+l23)[ξ2]5/11,
where ξ2=[x2]11/6-[x2*]11/6, l21=(5/11)·(18/11)6/5·21/5, l22=(17/22)·(15/22)5/17·(1+x12)22/17, and l23=(6/11)·(30/11)5/6·(17/11)11/6·(3121/60+3121/30).In the simulation, by choosing the initial values asx1(0)=1 and x2(0)=-1, Figure 1 is obtained to demonstrate the effectiveness of the control scheme.The responses of the closed-loop system (38) and (39).
(a)
System states
(b)
Control input


## 6. Conclusion

In this paper, a continuous state feedback stabilizing controller is presented for a class of high-order nonlinear systems under weaker condition. The controller designed preserves the equilibrium at the origin and guarantees the global finite-time stability of the systems. It should be noted that the proposed controller can only work well when the whole state vector is measurable. Therefore, a natural and more interesting problem is how to design output feedback stabilizing controller for the systems studied in the paper if only partial state vector being measurable, which is now under our further investigation.



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