Robust Finite-Time Stability Control of a Class of High-Order Uncertain Nonlinear Systems

ROBUST FINITE-TIME STABILITY CONTROL OF A CLASS OFHIGH-ORDER UNCERTAIN NONLINEAR SYSTEMS

Haitao Liu, Xiaozhen Wang, and Tie Zhang

ABSTRACT

In this study, a novel robust finite-time stability controller is proposed for a class of high-order uncertain nonlinear systems. It usesthe dynamic surface control (DSC) approach to simplify the traditional backstepping design for high-order nonlinear systems, thus avoidingthe “explosion of terms”. The finite-time stability of the closed-loop system is guaranteed to have high performance, such as fast transientand strong robustness to dynamic uncertainties, and the tracking error is made arbitrarily small. Simulation results of two examples indicatethat the proposed controller is effective.

Key Words: Nonlinear system, uncertainty, finite-time stability, dynamic surface control.

I. INTRODUCTION

Unlike asymptotic stability, finite-time stability [1] meansthat system states can be stabilized to equilibrium in finite timeand then remain at equilibrium. As finite-time stability may giverise to fast transient and high precision, in recent years, finite-time stability control of nonlinear systems has been paidincreasing attention. Bhat et al. presented continuous finite-time-stabilizing feedback control laws of translational and rota-tional double integrators [2]. Feng et al. proposed a non-singularterminal sliding mode controller of robot systems [3]. Honget al. were the first to propose a finite-time controller with statefeedback and dynamic output feedback control [4]. Su presenteda global finite-time tracking controller by replacing linear errorsin the linear proportional-derivative plus scheme with non-smooth but continuous exponential errors [5], and implementedthe global finite-time tracking of robotic manipulators by replac-ing linear errors with nonlinear exponential-like ones [6].

However, all the above-mentioned approaches did not takeinto consideration the uncertainties of systems. In order to solvethis problem, Huang et al. applied the Hölder continuous statefeedback to the global finite-time stabilization of a class ofuncertain nonlinear systems dominated by a lower-triangularsystem [7]. Hong and Jiang investigated the non-smooth finite-time stabilization of nonlinear systems with parametric anddynamic uncertainties [8]. Hong, Wang, and Cheng studied thefinite-time stabilization of high-order lower-triangular nonlinearsystems [9] . Zhao et al. designed a robust finite-time stability

controller based on the backstepping approach [10], but a singu-larity problem consequently appears.

In order to avoid the “explosion of terms” caused by thebackstepping approach with repeated differentiations of certainnonlinear functions, Swaroop et al. proposed a dynamic surfacecontrol (DSC) to eliminate the problem by introducing a first-order filtering of the synthetic input in each step of the traditionalbackstepping approach [11]. In addition, Wang and Huang inves-tigated the adaptive dynamic surface control of a class ofuncertain nonlinear systems based on the neural network [12].Yoo et al. developed a self-recurrent wavelet neural network-based DSC method for the robust control of flexible-joint (FJ)robots with model uncertainties [13] and further applied it toparametric strict-feedback nonlinear systems with unknown timedelay and linear parametric uncertainties [14]. Recently, Zhaoet al. proposed a novel DSC algorithm for a class of uncertainnonlinear systems in a completely non-affine pure-feedbackform [15], and Zhang et al. proposed an adaptive DSC schemefor a class of nonlinear systems with unknown backlash-likehysteresis to reduce the impact of the hysteresis and guaranteethe L∞ performance of the system’s tracking error [16].

In this paper, we further consider the finite-time stabiliza-tion of a class of high-order nonlinear systems with uncertainties,and propose a robust finite-time stability controller which over-comes the “explosion of terms” by introducing the dynamicsurface control and guarantees the finite-time stability of closed-loop systems. As compared with the finite-time controllers basedon the traditional backstepping approach, the proposed controlleris much simpler. In addition, we perform an analysis to reveal thefinite-time stability of the closed-loop system and to verify theeffectiveness of the controller in making the steady-sate trackingerror arbitrarily small.

The presentation of this paper is organized as follows:Section II briefly reviews the Lyapunov theory for finite-timestability of continuous systems and the relative lemmas; SectionIII gives the main results of this paper, namely the robust finite-time stability controller based on the dynamic surface control

Manuscript received March 12, 2013; revised February 28, 2014; accepted March19, 2014.

Haitao Liu (corresponding author, e-mail: [email protected]) is with EngineeringCollege, Guangdong Ocean University, Zhanjiang 524088, China.

Xiaozhen Wang is with Guangdong Communication Polytechnic, Guangzhou510650, China (e-mail: [email protected]).

Tie Zhang is with School of Mechanical and Automotive Engineering, South ChinaUniversity of Technology, Guangzhou 510640, China (e-mail: [email protected]).

This work was supported by the Project of DEGP (2013KJCX0100) and the Projecton the IIERGP (2012B010300023).

Asian Journal of Control, Vol. 17, No. 1, pp. 1–7, January 2015Published online in Wiley Online Library (wileyonlinelibrary.com) DOI: 10.1002/asjc.916

–Brief Paper–

© 2014 Chinese Automatic Control Society and Wiley Publishing Asia Pty Ltd

approach and the stability analysis of the closed-loop system;and, in Section IV, two illustrative examples are presented toverify the proposed controller. Finally, some conclusions aredrawn in Section V.

II. PRELIMINARIES

The concepts related to the finite-time stability controlare given in [1]. Here, we recall the corresponding Lyapunovstability theorem for the finite-time stability of autonomoussystems [17], which will be used in Section III.

Definition 1. Considering a time-invariant nonlinear system inthe form of

�x f x f x Rn= = ∈( ), ( ) ,0 0 (1)

where f U Rn: 0ˆ → is continuous in an open neighborhood U0 of

the origin, the equilibrium x = 0 of the system is (locally)finite-time stable if:

(i) it is asymptotically stable in U, an open neighborhood of

the origin, with ˆ ˆU U⊆ 0;(ii) it is finite-time convergent in U, that is, for any initial

condition x U0 0∈ ˆ \ { }, there is a settling time T > 0 suchthat each solution x(t,x0) of system (1) is defined withx t x U( , ) \ {0}0 ∈ ˆ for t ∈ [0, T) and satisfies

lim ( , )t T

x t x→

=0 0 (2)

and, if t ≥ T, x(t, x0) = 0. Moreover, if U Rn= , the origin x = 0 isglobally finite-time stable.

Definition 2. Consider a nonlinear system in the form of

�x f x u f x R u Rn m= = ∈ ∈( , ), ( , ) , ,0 0 0 (3)

where f : Rn × Rm → Rn is globally defined. The origin of system(3) is finite-time stable if there is a state feedback control law inthe form of u = ϕ(x) with ϕ(0) = 0, such that the origin of theclosed-loop system �x f x x= ( , ( ))φ is in a finite-time stableequilibrium state.

Lemma 1 [18]. Consider the nonlinear system (1). Suppose thatthere is C1 function V(x) defined in neighborhoodU Rn⊂ of the origin, and that real numbers c > 0 and 0 < α < 1,such that:

(i) V(x) is positive definite in U ; and(ii) �V x cV x x U( ) ( ) 0, \ {0}+ ≤ ∀ ∈α ˆ .

Then, the origin of system (1) is locally finite-time stable. Thesettling time, which depends on the initial state x(0) = x0,satisfies

T xV x

cx ( )

( )

( )0

01

1≤

−

−α

α(4)

for all x0 in some open neighborhoods of the origin. If U Rn=and V(x) are also radially unbounded (i.e. V(x) → +∞ as||x|| → +∞), the origin of system (3) is globally finite-time stable.

Lemma 2 [19]. For any real number li, i = 1, . . . , n, when0 < ζ < 1 and 0 < λ < 2, the following inequalities hold:

l l l ln n1 1+ +( ) ≤ + +� �ζ ζ ζ (5)

l l l ln n12 2

1

2+ +( ) ≤ + +( )� �

λ λ λ (6)

For the convenience of control design and analysis, wedefine vector Sig(·)α ∈ Rn as follows:

Sig sig sig nT

n n

( ) [ ( ) , , ( ) ]

sgn( ), , sgn( )

ξ ξ ξ

ξ ξ ξ ξ

α α α

α α

=

= ⎡⎣ ⎤⎦

1

1 1

�

�TT (7)

where ξ = [ξ1, . . . , ξn]T ∈ Rn, 0 < α < 1, and sgn(·) is the signfunction defined as

sgn( )

,

,

,

ξξξξ

=>=

− <

⎧⎨⎪

⎩⎪

1 0

0 0

1 0

(8)

III. DESIGN OF ROBUST FINITE-TIMESTABILITY CONTROLLER

Here, we consider a class of uncertain nonlinear systems inthe form of

��

x x f x

x x f x x

x u f x xn n n

1 2 1 1

2 3 2 1 2

1

= += +

= +

( )

( , )

( , , )

(9)

where x = (x1, . . . , xn)T ∈ Rn and u ∈ Rm are the system state andinput, respectively, and fn(x1, . . . , xn) is a C1 uncertain functionwith fn(0, . . . , 0) = 0.

In practice, it is difficult to determine fi(x1, . . . , xi) exactly.Assuming that the following expression holds:

f x x f x x f x xi i i i i i( , , ) ( , , ) ( , , )10

1 1� � �= + Δ (10)

where f x xi i0

1( , , )� is the nominal part of fi(x1, . . . , xi), andΔfi(x1, . . . , xi) is the unknown part of fi (x1, . . . , xi), thus, we have

��

x x f x f x

x x f x x f x x

x u fn

1 2 10

1 1 1

2 3 20

1 2 2 1 2

= + += + +

= +

( ) ( )

( , ) ( , )

ΔΔ

nn n n nx x f x x01 1( , , ) ( , , )� �+ Δ

(11)

The following assumptions are given:

2 Asian Journal of Control, Vol. 17, No. 1, pp. 1–7, January 2015


1. Δfi(x1, . . . , xi)| ≤ ρi(x1, . . . , xi), where ρi(x1, . . . , xi) is a C1

function in its arguments.2. f x xi i

01( , , )� is a smooth function in its arguments, and

fi0 0 0 0( , , )� = .

The objective of the controller is to make x1(t) track thereference input x1d(t) with a prescribed small error in finite time,where x1d(t) is a sufficiently smooth function of t and x x xd d d1 1 1, ,� ��are all bounded for t ≥ 0.

3.1 Robust dynamic surface control approach

In this section, we introduce the DSC approach in thedesign of the finite-time stability controller for the nth-ordersystem (11). Similar to the traditional backstepping approach,the recursive design procedure contains n steps.

Step 1. Letting

s x x d1 1 1= − (12)

s1 is called the error surface with the desired trajectory x1d, oneobtains

� � � �s x x x f x f x xd d1 1 1 2 10

1 1 1 1= − = + + −( ) ( )Δ (13)

Then, choosing a virtual control x2 as follows:

x f x K Sig s s x d2 10

1 1 1 1 1 1= − − − +( ) ( )α γ � (14)

where K1 > 0, 0 < α < 1, γ ρε112

2

3

20= + > and ε > 0.

If x2 tracks x2 asymptotically, s1 will converge to aneighborhood around 0. Introducing a new state variable x2d andletting x2 pass through a first-order filter with time constant τ2 toobtain x2d, there is

τ 2 2 2 2 2 20 0�x x x x xd d d+ = =, ( ) ( ) (15)

Step 2. Defining the 2nd error surface as

s x x d2 2 2= − (16)

one can obtain

� �s x f x x f x x x d2 3 20

1 2 2 1 2 2= + + −( , ) ( , )Δ (17)

The following virtual control is chosen:

x f x x K Sig s s x d3 20

1 2 2 2 2 2 2= − − − +( , ) ( )α γ � (18)

where γ ρε222

2

3

2= + . Introducing a state variable x3d and letting x3

pass through a first-order filter with time constant τ3 to obtain x3d

τ3 3 3 3 3 30 0�x x x x xd d d+ = =, ( ) ( ) (19)

Step i. Defining the ith error surface

s x xi i id= − (20)

one obtains

� � � �s x f x x f x x xi i i i i i id= + + −+10

1 1( , , ) ( , , )Δ (21)

The following virtual control is chosen:

x f x x K Sig s s xi i i i i i i id+ = − − − +10

1( , , ) ( )� �α γ (22)

where γ ρεii= +2

2

3

2. Similarly, letting xi+1 pass through a

first-order filter with time constant τi+1 to obtain xi+1d

τ i i d i d i i d ix x x x x+ + + + + ++ = =1 1 1 1 1 10 0� , ( ) ( ) (23)

Step n. Finally, defining the nth error surface

s x xn n nd= − (24)

one obtains

� � � �s u f x x f x x xn n n n n nd= + + −01 1( , , ) ( , , )Δ (25)

Choosing the virtual control

x f x x K Sig s s xn n n n n n n n d= − − − +− − − − − − −10

1 1 1 1 1 1 1( , , ) ( )� �α γ (26)

where γ ρεn

n−

−= +11

2

2

3

2, and letting xnd be the filtered version of xn

with time constant τn,

τ n nd nd n nd nx x x x x� + = =, ( ) ( )0 0 (27)

such that the actual control law can be chosen as

u K Sig s f x x s xn n n n n n n nd= − − − +( ) ( , , )α γ01 � � (28)

It is notable that this control law does not involve modeldifferentiation, so that it helps to avoid the “explosion of terms”.

Remark 1. The differentiation of xnd in (28) is obtained by

�xx x

ndn nd

n

= −τ

(29)

3.2 Finite-time stability analysis

Theorem 1. For the nonlinear dynamic system (9), if we giveany positive number p, for all initial conditions satisfyings s y y pn n1

2 222 2

02+ + + + ≤� � and s s pn12 2 2+ + ≤� (p ≤ p0),

there exist a set of surface gains K1, . . . , Kn, filter time constantsτ1, . . . , τn and λ > 0, such that the dynamic surface controller isdesigned as (28). Thus, the closed-loop system is finite-timestable, and the steady-state tracking error is smaller than theprescribed error bound.

Defining the following boundary layer error equations:

3H. Liu et al.: Robust Finite-Time Stability Control of a Class of High-Order Uncertain Nonlinear Systems


y x x x f x K Sig s s x

y x x

d d d

i i d i

2 2 2 2 10

1 1 1 1 1 1

1 1 1

= − = + + + −

= −+ + +

( ) ( )α γ ��

== + + + −+x f x x K Sig s s xi d i i i i i i id10

1( , , ) ( )� �α γ(30)

The closed-loop system is the new coordinate si, which can beexpressed as

��

�

s s y K Sig s s f x

s s y K Sig si i i i i

1 2 2 1 1 1 1 1 1

1 1

= + − − +

= + −+ +

( ) ( )

( )

α

α

γ Δ

−− +

= − − +

γ

γα

i i i i

n n n n n n n

s f x x

s K Sig s s f x x

Δ

Δ

( , , )

( ) ( , , )

1

1

��

� �

(31)

and

� � � � �yy

xy f x

xx K s s s2

2

22

2

2

10

1

11 1 1 1 1

12

2

3

2= − − = − + ∂

∂+ + +⎛

⎝τ τα ρ

εα( )

⎜⎜⎞⎠⎟

+ ∂∂

−s

xx x d

1 1 1

11 1

ρε

ρ � ��(32)

Clearly,

� � ��yy

s s y K x x xd d d22

22 1 2 2 1 1 1 1+ ≤

τη ( , , , , , , ) (33)

Similarly, we have

� � � �yy f

xx K s s si

i

i

i

jj

i

i i i ii

++

+= − + ∂

∂+ + +⎛

⎝⎜⎞⎠⎟

+

∑11

1

0

1

2

2

3

2τα ρ

εα

ss

xx x

ys s y y

i i i

jj

i

id

i

ii i i

ρε

ρ

τη

∂∂

−

= − +

∑+

++ +

� ��

� �

1

1

11 1 1 2( , , , , , , xx x xid id id, , )� ��

(34)

where

η αi i i id id idi

jj

i

i is s y y x x xf

xx K s+ + = ∂

∂+∑1 1 1 2

0

1

( , , , , , , , , )� � � �� αα

ρε

ρε

ρ

�

� � ��

s

ss

xx x

i

ii i i i

jj

i

id+ +⎛⎝⎜

⎞⎠⎟+ ∂

∂−∑

2

12

3

2

(35)

Since any x1d and its time derivatives up to the secondorder are supposed to be continuous and bounded, thereexists a positive number R0 such that Ω1 1 1 1= {( , , ) :x x xd d d� ��

12

12

12

03+ + ≤ ∈}x x x R Rd d d� �� . Defining the compact set

Ωi n n ji

j ji

jns y s y p R i n= ∑ + ∑ ≤ ∈ == ={( , ) : } , , ,1

22

20

22 1 � , andnoting that Ω1 × Ωi is also compact in R2n+3, therefore, |ηi+1| has amaximum η on Ω1 × Ωi.

Proof.Step 1. Here, we consider the following Lyapunovfunction:

V s yi

i

n

i

i

n

= +=

+=

−

∑ ∑1

2

1

22

11

2

1

1

(36)

Differentiating V with respect to time and then combiningit with (31) and (34), by using Young’s inequality,namely s f s si i i i i iΔ ≤ ≤ +ρ ρ ε ε2 2 2 2, ε > 0 andy yi i+ +≤ +1 1

2 2 2 2η η ω ω , ω > 0, we can obtain

� � �

�

V s s y y

s s s s y K Sig s

i i

i

n

i i

i

n

n n i i i i i

= +

≤ + + −

=+ +

=

−

+ +

∑ ∑1

1 11

1

1 1( ( )) ( , , ))α γ

τη

− +

+ − +⎛⎝⎜

⎞⎠⎟

=

−

+

++ +

=

∑ i i i i

i

n

i

ii i

i

s f x x

yy

Δ 11

1

12

11 1

1

�

nn

n n n i i i i i

i

n

is K Sig s s s y K Sig s s

−

+ +=

−

∑

∑≤ − + + − −

1

1 11

12

1

3

2( ) ( ( ) )α α

nn

i

ii i

i

n

i i i

i

n

ny

y

s K Sig s

∑

∑+ + − +⎛⎝⎜

⎞⎠⎟

≤ −

+

++ +

=

−

=

ετ

η

α

21

2

11 1

1

1

1

( )∑∑ ∑ ∑+ + + − +

+ −( ) − −

+ +

=

−

+

2

2

3

2 2

12

1

2

21

21

2

1

12

1

1

2

s s ys n

n

i i i

i

n

i

n

i

ε

ωτ

ηωω

ε ωτ

ηω

⎛⎝⎜

⎞⎠⎟

≤ + − − − −⎛⎝⎜

⎞⎠⎟

+=

−

++

∑ y

n n y

i

i

n

ii

i

12

1

1

1

2

12

21

2

1

2

1

2( )

==

−

+=

−

∑

∑≤ −

1

1

12

1

1

n

i

i

n

l yσ

(37)

where σ is chosen properly such that ∫ < ∞∞0 σdt , and

σ ε ω= + −n n2

12

( ) (38)

li

= − −⎛⎝⎜

⎞⎠⎟>

+min

1

2

1

20

1

2

τηω

(39)

Integrating both sides of (37) from t = 0 to T, one obtains

V T V l y dt dti

i

nT T( ) ( )− ≤ − ++

=

−

∑∫ ∫0 12

1

1

0 0σ (40)

Since V(T) ≥ 0 and inequality ∫ < ∞∞0 σdt holds, we can obtain

lim sup ( ) limT

i

i

nT T

TTy dt

lV dt

T→∞+

=

−

→∞∑∫ ∫≤ +( ) =1 10

101

2

1

1

0 0σ (41)

from which we directly obtain yi → 0 as t → ∞, i.e., yi isuniformly and ultimately bounded. Moreover, by reducing thevalue of τi, yi may be arbitrarily small.



Step 2. The analysis of finite-time stability.For the Lapunov function

V ss i

i

n

==∑1

22

1

(42)

its time derivative is

� �

� �

V s s

s s s s y K Sig s s f x

s i i

i

n

n n i i i i i i i i

=

≤ + + − − +

=

+ +

∑1

1 1 1( ( ) ( ,α γ Δ ,, ))

( ) ( ( ) )

x

s K Sig s s s y K Sig s

i

i

n

n n n i i i i i

i

n

=

−

+ +=

−

∑

≤ − + + −

1

1

1 11

1α α∑∑ ∑

∑

− +

≤ − − − + +=

+=

3

2 2

1

2 2

1

2

2

1

112 2

12

s n

s K Sig s s s n y

i

n

i i i

i

n

n i

i

ε

εα( )11

1

1

n

i i i

i

n

s K Sig s

−

=

∑

∑≤ − +( )α δ (43)

where

δ ε= + +=

−

∑n yi

i

n

2

1

21

2

1

1

(44)

Defining μ = (1 + α)/2,1/2 < μ < 1, Kmin = min{Ki} and�K K= 2μ

min , the following inequality can be established:

�

�

V K s

K s

KV

s i i

i

n

i

i

n

s

≤ − +

≤ − ⎛⎝⎜

⎞⎠⎟

+

≤ − +

+

=

=

+

∑

∑

1

1

2

1

1 2

α

α

μ

δ

δ

δ

min

( ) (45)

On the level set defined by Vs(s1, . . . , sn) = p, letting�K p> +λ δ μ , there is �V Vs s≤ −λ μ on Vs = p. Thus, Vs ≤ p is an

invariant set, i.e., if Vs(0) ≤ p, Vs(t) ≤ p for all t ≥ 0. Furthermore,according to Lemma 1, the closed-loop system is finite-timestable. In addition, the setting time is estimated as

T sK

V ss( )( )

( )≤−

−1

10

1

� μμ (46)

�V Vs s≤ − ≤λ μ 0 means that Vs(t) is eventually bounded. Thus, s1,. . . , sn are also uniformly and ultimately bounded. Moreover, byincreasing the value of λ, the tracking error may be arbitrarilysmall. This completes the proof.

Remark 2. From the two above-mentioned steps of proof, thefollowing filter time constant should be selected:

τ ωη ωi+ <

+1 2

2(47)

Remark 3. In Theorem 1, we consider the system whose initialconditions all satisfy s s s s pT

nT

n1 1 2+ + ≤� . From the physicalviewpoint, this is a restriction on the initial values of all variablesof the system, i.e., all related variables must be in a ball of radius

2 p . As p can be arbitrarily large, the condition is not reallyrestrictive; and, a set of initial conditions in the desired ball wereprovided in [11], which can be used for our case.

IV. ILLUSTRATIVE EXAMPLES

In this section, two examples are presented.

Example 4.1. Here, we consider the following example [11]:

��

x x f x

x x

x u

y x

1 2 1 1

2 3

3

1

= +===

⎧

⎨⎪⎪

⎩⎪⎪

Δ ( )

(48)

The control objective is to synthesize a state feedback law for usuch that the output y(t) can track the reference signal xr(t)effectively. The uncertainty Δf x x x1 1 1

21( ) sin( )= is assumed to be

bounded by x12, the reference signal xr(t) = sin(2πt), and the

initial states x1 = x2 = x3 = 0.2. Moreover, the controllerparameters are set as K1 = 20, K2 = 30, K3 = 30, γ1 = 5, γ2 = 5 andα = 0.7. Then, the controller is compared with the dynamicsurface controller presented in [11], with the simulationparameters being set as K1 = 40, K2 = 40 and K3 = 60.

Figs 1 and 2 show the results when filter time constantsτ2 = τ3 = 0.01, and Figs 3 and 4 show the results whenτ2 = τ3 = 0.005. It can be seen that the proposed controllerprovides good finite-time stability performance, such as fasttransient and strong robustness to dynamic uncertainties.Moreover, it is obvious that, compared with the traditional DSCapproach, the proposed finite-time dynamic surface control

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-0.05

0

0.05

0.1

0.15

0.2

Time (s)

Tra

ckin

g er

ror

FTDSC

DSC

Fig. 1. Tracking error (τ2 = τ3 = 0.01).



(FTDSC) approach helps to achieve smaller tracking error andless control input.

Example 4.2. Here, we consider a chain of integrators in thecase of n = 2, i.e.,

� �x x x u1 2 2= =, (49)

and the Hölder continuous finite-time stability (FTS) controllerin [7], that is,

u x x= − +1 5 0 1525 3

11 5. ( . ) (50)

The FTDSC parameters we set here are, respectively, K1 = 1,K2 = 10, γ1 = 1, γ2 = 50, τ2 = 0.01 and α = 0.8.

Figs 5 and 6, respectively, show the simulation results ofthe proposed finite-time controller and the backstepping-basedFTS controller proposed by Huang and Lin [7]. From Fig. 5, itcan be seen that the proposed controller is effective due to itstransient response to the closed-loop system. The two controllersboth possess high performances, and it is more important that,unlike the exponent qk in [7], qk = (2n + 3 − 2k)/(2n + 1), k = 1, 2,. . . , n, the exponent α ∈ (0 1) can be set arbitrarily.

Therefore, the dynamic surface control approach isfeasible in designing the finite-time stability controller ofhigh-order uncertain nonlinear systems and is effective insolving the “explosion of terms” or the singularity problem.

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-1.5

-1

-0.5

0

0.5

1

1.5x 10

4

Time (s)

Con

trol

sig

nal

FTDSC

DSC

Fig. 2. Control signal (τ2 = τ3 = 0.01).

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-0.05

0

0.05

0.1

0.15

0.2

Time (s)

Tra

ckin

g er

ror

FTDSC

DSC

Fig. 3. Tracking error (τ2 = τ3 = 0.005).

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2x 10

4

Time (s)

Con

trol

sig

nal

FTDSC

DSC

Fig. 4. Control signal (τ2 = τ3 = 0.005).

0 5 10 15-25

-20

-15

-10

-5

0

5

Time (s)

FTDSC

x1

x2

Fig. 5. Trajectories of the closed-loop system (FTDCS).



V. CONCLUSION

In this paper, the finite-time stability controller of a class ofhigh-order uncertain nonlinear systems is proposed, which usesthe dynamic surface control approach to avoid the “explosion ofterms” caused by the traditional backstepping approach. Withthis controller, all signals in the closed-loop system are guaran-teed to be uniformly and ultimately bounded, and the trackingerror may be arbitrarily small if the control parameters are suit-ably adjusted. Simulation results indicate that the proposed con-troller is easier to implement than the controllers based on thetraditional backstepping approach.

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0 5 10 15-3

-2

-1

0

1

2

3

4

5

Time (s)

FTS

x1

x2

Fig. 6. Trajectories of the closed-loop system (FTS).



Robust Finite-Time Stability Control of a Class of High-Order Uncertain Nonlinear Systems

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