IEEE TRANSACTIONS ON INTELLIGENT VEHICLES 1 Fast and ... · control of an autonomous surface vehicle (ASV) with com-plex unknowns including unmodeled dynamics, uncertainties and/or

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IEEE TRANSACTIONS ON INTELLIGENT VEHICLES 1

Fast and Accurate Trajectory Tracking Control ofan Autonomous Surface Vehicle with Unmodelled

Dynamics and DisturbancesNing Wang,Senior Member, IEEE,Shuailin Lv, Meng Joo Er,Senior Member, IEEE

and Wen-Hua Chen,Senior Member, IEEE

Abstract—In this paper, fast and accurate trajectory trackingcontrol of an autonomous surface vehicle (ASV) with com-plex unknowns including unmodeled dynamics, uncertaintiesand/or unknown disturbances is addressed within a proposedhomogeneity-based finite-time control (HFC) framework. Majorcontributions are as follows: (1) In the absence of externaldisturbances, a nominal HFC framework is established to achieveexact trajectory tracking control of an ASV, whereby global finite-time stability is ensured by combining homogeneous analysis andLyapunov approach; (2) Within the HFC scheme, a finite-timedisturbance observer (FDO) is further nested to rapidly andaccurately reject complex disturbances, and thereby contributingto an FDO-based HFC (FDO-HFC) scheme which can realizeexactness of trajectory tracking and disturbance observation; (3)Aiming to exactly deal with complicated unknowns includingunmodeled dynamics and/or disturbances, a finite-time unknownobserver (FUO) is deployed as a patch for the nominal HFCframework, and eventually results in an FUO-based HFC (FUO-HFC) scheme which guarantees that accurate trajectory trackingcan be achieved for an ASV under harsh environments. Simu-lation studies and comprehensive comparisons conducted on abenchmark ship demonstrate the effectiveness and superiority ofthe proposed HFC schemes.

Index Terms—Global finite-time stability, accurate trajectorytracking, finite-time disturbance observer (FDO), finite-time un-known observer (FUO), autonomous surface vehicle (ASV).

I. I NTRODUCTION

I N RECENT YEARS, autonomous surface vehicles (ASVs)have been widely deployed for various missions related

with observations, military tasks, coastal and inland watersmonitoring,etc., [1]. Generally, tracking control of an ASVto a prescribed trajectory/path with an acceptable accuracyplays a key role within the entire autopilot system and hasthus attracted great attention from both marine engineeringand control communities [2]. It is much more demanding toachieve accurate tracking of a pre-determined trajectory in

Manuscript received October 17, 2016; revised January 20, 2017. This workis supported by the National Natural Science Foundation of P. R. China (underGrants 51009017 and 51379002), the Fund for Dalian Distinguished YoungScholars (under Grant 2016RJ10), the Innovation Support Plan for DalianHigh-level Talents (under Grant 2015R065), and the Fundamental ResearchFunds for the Central Universities (under Grant 3132016314).

N. Wang and S. Lv are with the Marine Engineering College,Dalian Maritime University, Dalian 116026, China. (email:{n.wang.dmu.cn,lvshuailin}@gmail.com).

M. Er is with the School of Electrical and Electronic Engineering, NanyangTechnological University, Singapore. (e-mail: [email protected]).

W.-H. Chen is with the Department of Aeronautical and AutomotiveEngineering, Loughborough University, Loughborough, LE11 3TU, U.K. (e-mail: [email protected]).

some applications; for example, marine surveying and map-ping on the sea in the presence of complicated uncertaintiesand variations including system uncertainties and externaldisturbances due to ocean winds, waves and currents [3]–[7].In this context, it becomes extremely challenging to achieveaccurate trajectory tracking of an ASV sailing in such harshenvironments.

The sliding mode control (SMC) technique has been in-vestigated as a promising approach to achieve high accuratetrajectory tracking control of an ASV [8]. However, the SMC-based approaches have to incur high-frequency chattering withconservatively large magnitude around the sliding surface todominate unknowns and achieve the robustness. Furthermore,the SMC technique can only handle matched unknowns. Byvirtue of the (vectorial) backstepping technique [9], a trajec-tory tracking controller has been designed for an ASV in thepresence of (mismatched) unknowns including time-varyingdisturbances and system uncertainties. It should be noted thattracking errors can only be made globally uniformly ultimatelybounded. Applying the integrator backstepping [10] techniqueto the design of trajectory tracking control law for an underac-tuated ASV contributes to semi-globally exponentially stabletracking errors. Unfortunately, uncertainties and disturbanceshave not been addressed. In combination with neural networks(NNs) and adaptive robust control techniques [11], a saturatedtracking controller that renders tracking errors semi-globallyuniformly ultimately bounded has been proposed to preservethe robustness against time-varying disturbances induced bywaves and ocean currents. In addition to NNs [12]–[14], a lotof efforts on adaptive approximation based tracking controlhave also been made via fuzzy systems (FS) [15]–[19], andfuzzy neural networks (FNN) [20]–[25],etc., and can rough-ly compensate unknown dynamics. Recently, a significantprogress has been made by an innovative approximator termedself-constructing fuzzy neural network (SCFNN) [26]–[29]towards the dynamic-structure-approximation based adaptivecontrol approaches [20] with much higher accuracy of bothreconstruction and trajectory tracking. It should be highlightedthat accurate tracking control can still hardly be achieved bythe foregoing SCFNN-based control approaches since therestill exist unexpected approximation residuals. Nevertheless,the convergence rate of tracking errors is usually somewhatslow since only asymptotic or exponential closed-loop stabilitycan be derived from previous tracking control approaches.

In order to further pursue better tracking performance,



2 IEEE TRANSACTIONS ON INTELLIGENT VEHICLES

finite-time control approaches have been implemented in theliterature [30], [31]. In [30], the SMC technique has beenemployed to realize attitude tracking of a rigid spacecraftand a modified differentiator has been incorporated to com-pensate disturbances and inertia uncertainties, whereby finite-time convergence of tracking errors can be obtained. Byvirtue of the homogeneous method in [31], finite-time stabilityof the closed-loop control system with negative degree ofhomogeneous can be ensured if tracking error dynamics canbe proved to be asymptotically stable. It should be noted that,compared with traditional asymptotic convergent methods, theaforementioned finite-time control approaches can achieve notonly faster convergence rate within the vicinity of the originbut also stronger disturbance rejection. Meaningfully, finite-time control approaches ensure that tracking errors can reachto zero within a finite time. Note that, unlike SMC-basedapproaches, the homogeneity-based method [31] contributes toa straightforward solution without any couplings of trackingerrors. Besides, external disturbances can even excite un-modeled dynamics of the ASV, and thus require to be wellestimated. Otherwise, complex disturbances pertaining to anASV cannot be exactly observed within a short time, and maketrajectory tracking inaccurate.

Recently, disturbance observer based control (DOBC) tech-nique has also been proposed by Chen [32] to not only improvesystem robustness but also enhance the entire performancewithout sacrificing the nominal one [33]–[35]. In this context,the DOBC schemes have been extensively studied and widelyapplied to various industrial sectors, including mechatronicssystems [36], aerospace systems [37], and process controlsystems [38]. Clearly, it is innovative within the DOBC frame-work that all disturbances and/or unknowns are addressed asa lumped nonlinearity estimated by a nonlinear disturbanceobserver (NDO) which is usually involved.

In this paper, an ambitious goal of achieving fast and accu-rate trajectory control of an ASV in the presence of unknownsincluding disturbances and unmodelled dynamics is pursued.To be specific, a homogeneity-based finite-time control (HFC)scheme is developed to achieve accurate trajectory tracking ofan ASV in the absence of external disturbances. In conjunctionwith a finite-time disturbance observer (FDO) which can befurther devised to exactly estimate external disturbances withina short time, a FDO-based HFC (FDO-HFC) scheme is thusimplemented to exactly track an ASV suffering from complexdisturbances. In order to further address unmodelled dynamicsincluding uncertainties and/or excited dynamics due to dis-turbances, a high-order sliding mode estimator is designedto realize a finite-time unknown observer (FUO) which canexactly capture unknown dynamics. Incorporating the FUOinto the HFC scheme contributes to the FUO-based HFC(FUO-HFC) scheme which can achieve fast and accuratetrajectory tracking with complex unknowns including unmod-elled dynamics and/or uncertainties in addition to disturbances.

The rest of this paper is organized as follows. In SectionII, preliminaries together with the trajectory tracking problemassociated with an ASV are addressed. The HFC, FDO-HFCand FUO-HFC schemes together with theoretical analysis onfinite-time stability are presented in Section III. Simulation

studies and discussions are conducted in Section IV. Conclu-sions are drawn in Section V.

II. PRELIMINARIES AND PROBLEM STATEMENT

A. Preliminaries

For the convenience of readers, we collect the key defini-tions and lemmas frequently used in this paper in the sequel.

Consider an autonomous nonlinear system as follows:

xxx(t) = fff(xxx(t)), x(0) = 0, fff(0) = 0, xxx ∈ U0 ⊂ Rn (1)

where xxx = [x1, · · · , xn]T and nonlinear functionfff(·) is

continuous on a open neighborhoodU0 of the origin.Definition 1 (Globally Asymptotic Stability [39]):The equi-

librium xxxe = 0 of system (1) is globally asymptotically stableif there exits a functionV (xxx) satisfying

(a) V (0) = 0;(b) V (xxx) > 0, ∀ xxx 6= 0, andV (xxx) is radically unbounded;(c) V (xxx) ≤ 0;(d) V (xxx) does not vanish identically along any trajectory in

Rn, other than the null solutionxxx = 0. �

Definition 2 (Homogeneity [31]):DenoteV (xxx) : Rn →

R be a continuous scalar function.V (xxx) is said to be ahomogeneous function of degreeσ with respect to weights(r1, · · · , rn) ∈ R

n with ri > 0, i = 1, 2, · · · , n, if, for anygiven ε > 0,

V (εr1x1, · · · , εxnxn) = εσV (xxx), i = 1, · · · , n, ∀xxx ∈ R

n.

Denotefff(xxx) = [f1(xxx), · · · , fn(xxx)]T be a continuous vector

field. fff(xxx) is homogeneous of degreek ∈ R with respect toweights(r1, · · · , rn), if, for any givenε > 0,

fi(εr1x1, · · · , ε

rnxn) = εk+rifi(xxx), i = 1, · · · , n, ∀xxx ∈ Rn.

And, system (1) is said to be homogeneous iffff(xxx) is homo-geneous. �

In combination with Definitions 1 and 2, a fundamentalresult on global finite-time stability can be obtained as follows:

Lemma 1 (Global Finite-Time Stability [31]):System (1) isglobally finite-time stable if system (1) is globally asymptoti-cally stable and is homogeneous of a negative degree.�

By virtue of Lemma 1, we can derive a cornerstone result,whose proof is presented in details in Appendix A, for finite-time observer design and analysis in this paper.

Lemma 2:The following system:

z1 = z2 − l1sigα2(z1)

z2 = z3 − l2sigα3(z1)

...

zn = −lnsigαn+1(z1) (2)

whereli > 0, i = 1, 2, · · · , n are appropriate constants, and

sigαi(z1) := |z1|αisgn(z1) (3)

with αi+1 = αi+τ, i = 1, 2, · · · , n andα1 = 1 for anyτ < 0,is globally finite-time stable. �



WANG et al.: FAST AND ACCURATE TRAJECTORY TRACKING CONTROL OF AN AUTONOMOUS SURFACE VEHICLE 3

O Yo

Xo

ψ

xg

G

Ax

y

u

V

r

v

X

Y

Fig. 1: Earth-fixedOXoYo and body-fixedAXY coordinateframes of an ASV

B. Problem Formulation

Let ηηη = [x, y, ψ]T denote the 3-DOF position(x, y) andheading angle(ψ) of the ASV in the earth-fixed inertialframe as shown in Fig. 1, and letννν = [u, v, r]T denote thecorresponding linear velocities(u, v), i.e., surge and swayvelocities, and angular rate(r), i.e., yaw, in the body-fixedframe. An ASV sailing in a planar space can be modeled asfollows [3]:

ηηη = R(ψ)ννν

Mννν = fff(ηηη,ννν) + τττ + τττd (4)

with dynamicsfff(ηηη,ννν) usually modeled by

fff(ηηη,ννν) = −C(ννν)ννν −D(ννν)ννν − ggg(ηηη,ννν) (5)

where τττ = [τ1, τ2, τ3]T and τττd := MRT (ψ)ddd(t) with

ddd(t) = [d1(t), d2(t), d3(t)]T are control input and mixed

disturbances, respectively, andggg denote the restoring forcesand moments due to gravitation/buoyancy. The termR(ψ) isa rotation matrix given by

R(ψ) =

cosψ − sinψ 0sinψ cosψ 00 0 1

(6)

with the following properties:

RT (ψ)R(ψ) =I, and ‖R(ψ)‖ = 1, ∀ ψ ∈ [0, 2π] (7a)

R(ψ) = R(ψ)S(r) (7b)

RT (ψ)S(r)R(ψ) = R(ψ)S(r)RT (ψ) = S(r) (7c)

where S(r) =

0 −r 0r 0 00 0 0

, the inertia matrixM =

MT > 0, the skew-symmetric matrixC(ννν) = −C(ννν)T and

the damping matrixD(ννν) are given by

M =

m11 0 00 m22 m23

0 m32 m33

(8a)

C(ννν) =

0 0 c13(ννν)0 0 c23(ννν)

−c13(ννν) −c23(ννν) 0

(8b)

D(ννν) =

d11(ννν) 0 0

0 d22(ννν) d23(ννν)0 d32(ννν) d33(ννν)

(8c)

wherem11 = m − Xu, m22 = m − Yv, m23 = mxg −Yr, m32 = mxg − Nv, m33 = Iz − Nr; c13(ννν) =−m11v − m23r, c23(ννν) = m11u; d11(ννν) = −Xu −X|u|u|u| − Xuuuu

2, d22(ννν) = −Yv − Y|v|v|v|, d23(ννν) =−Yr−Y|v|r|v|−Y|r|r|r|, d32(ννν) = −Nv−N|v|v|v|−N|r|v|r|and d33(ννν) = −Nr − N|v|r|v| − N|r|r|r|. Here,m is themass of the vessel,Iz is the moment of inertia about the yawrotation,Yr = Nv, andX∗, Y∗ andN∗ denote correspondinghydrodynamic derivatives which are actually difficult to beaccurately obtained.

Consider the following desired trajectory:

ηηηd = R (ψd)νννd

Mνννd = fff0(ηηηd, νννd) (9)

where fff0(·) is the nominal dynamics,ηηηd = [xd, yd, ψd]T

andνννd = [ud, vd, rd]T are the desired position and velocity

vectors, respectively.In this context, the control objective is to design a controller

τττ such that the actual position and velocity vectors (i.e.,ηηη andννν) of the ASV in (4) can track exactly the desired trajectory(i.e., ηηηd andνννd) generated by (9) in a finite time.

Remark 1:Clearly, in addition to unknown disturbancesτττd,if the dynamicsfff of the ASV in (4) cannot be sufficientlymodeled due to parametric unknowns includingC, D andggg, and/or structural unmodeled dynamics, accurate trajectorytracking control of an ASV under harsh environments wouldbecome extremely challenging.

III. H OMOGENEITY-BASED FINITE-TIME TRACKING

CONTROL SCHEME

A. Nominal Homogeneity-Based Finite-Time Contol

In order to facilitate the controller design and analysis, weintroduce an auxiliary velocity vector as follows:

www = Rννν (10a)

wwwd = R(ψd)νννd (10b)

wherewww = [w1, w2, w3]T , wwwd = [wd,1, wd,2, wd,3]

T , R =R(ψ) andRd = R(ψd).

Together with (4) and (10a), using properties in (7), we have

ηηη = www

www = RM−1τττ + hhh(ηηη,www) + ddd(t) (11)

where

hhh(ηηη,www) = S(w3)www +RM−1fff(ηηη,RTwww) (12)




Similarly, using (9) and (10b), we have

ηηηd = wwwd

wwwd = S(wd,3)wwwd +RdM−1fff0

(ηηηd,R

Tdwwwd

)(13)

Combining with (11) and (13), we have

ηηηe = wwwe

wwwe = RM−1τττ + hhhe(ηηη,www,ηηηd,wwwd) + ddd(t) (14)

whereηηηe = ηηη − ηηηd := [ηe,1, ηe,2, ηe,3]T , wwwe = www − wwwd :=

[we,1, we,2, we,3]T , and

hhhe(ηηη,www,ηηηd,wwwd) = Swww − Sdwwwd +RM−1fff(ηηη,RTwww)

−RdM−1fff0(ηηηd,R

Tdwwwd) (15)

with S = S(w3) andSd = S(wd,3).Starting from the tracking error dynamics (14), we set out to

design a nominal homogeneity-based finite-time control (HFC)scheme which is expected to ensure that the tracking errorsηηηeandwwwe converge to zero in a finite time.

To this end, a nominal HFC scheme for the ASV in (4)without disturbances (i.e.,τττd = 0 orddd(t) = 0) is developed byemploying the homogeneous theory. Moreover, we show thatfinite-time stability of the entire closed-loop tracking systemcan be ensured by using the Lyapunov synthesis.

Design the nominal HFC lawτττHFC as follows:

τττHFC =−MR−1(K1sig

β1(ηηη − ηηηd) +K2sigβ2(Rννν −Rdνννd)

)

−MSννν +MR−1SdRdνννd

− fff(ηηη,ννν) +MR−1RdM−1fff0(ηηηd, νννd) (16)

where sigβi(xxx) =[sigβi(x1), · · · , sig

βi(xn)]T, i = 1, 2,

K1 > 0, K2 > 0, 0 < β1 < 1 andβ2 = 2β1/(1 + β1).It is essential that the proposed HFC scheme can make the

ASV in (4) track exactly the desired trajectory generated by(9) in a finite time. The key result ensuring the closed-loopfinite-time stability is now stated.

Theorem 1 (HFC):Using the HFC scheme governed by(16), the ASV in (4) can exactly track the desired trajectorygenerated by (9) within a finite time0 < T <∞, i.e.,ηηη(t) ≡ηηηd(t), ννν(t) ≡ νννd(t), ∀ t ≥ T .

Proof: Substituting the HFC law (16) into the trackingerror system (14) without considering disturbances yields theclosed-loop tracking error dynamics as follows:

ηe,j = we,j

we,j = −K1sigβ1(ηe,j)−K2sig

β2(we,j) (17)

for j = 1, 2, 3.In light of Lemma 1, global asymptotic stability and nega-

tive homogeneity of system (17) are expected to be guaranteedrespectively in the sequel.

1) Global Asymptotic Stability: Consider the followingLyapunov function:

V (ηηηe,wwwe) =

3∑

j=1

(K1

∫ ηe,j

0

sigβ1(µ)dµ+1

2w2

e,j

)(18)

DifferentiatingV (ηηηe,wwwe) along the tracking error dynamics(17), we have

V (ηηηe,wwwe) = K1

3∑

j=1

sigβ1(ηe,j)we,j

−

3∑

j=1

we,j

(K1sig

β1(ηe,j) +K2sigβ2(we,j)

)

= −K2

3∑

j=1

we,jsigβ2(we,j)

= −K2

3∑

j=1

|we,j |1+β2 (19)

which yieldsV (t) is bounded as timet tends to infinity, i.e.,

1

2‖wwwe(t)‖

2≤ V (t) <∞ (20)

Using ‖wwwe(t)‖1+β2 ≤ ‖wwwe(t)‖

2+ 1, we further have

∥∥∥www(1+β2)/2e (t)

∥∥∥ ≤√2V (t) + 1 <∞ (21)

Note, from (19), that∫ t

0

∥∥∥www(1+β2)/2e (τ)

∥∥∥2

dτ =V (0)− V (t)

K2<∞ (22)

Combining with (21) and (22) and using Barbalat’s lemma[40] yields

limt→∞

wwwe(t) = 0 (23)

In what follows, we expect to prove thatηηηe(t) also con-verges to zero as timet tends to infinity. To this end, a proofby contradiction is employed by assuming thatηηηe(t) convergesto a nonzero constantηηη0 6= 0.

Together with (17) and (23), we have, ast→ ∞,

ηe,j = 0

we,j = −K1sigβ1(ηe,j) (24)

which, combining with the hypothesis, implies thatηe,j con-verges to a nonzero constantη0,j 6= 0, and therebywe,j =−K1sig

β1(ηe,j) 6= 0. In this context,we,j deviates from theorigin and makesηe,j 6= 0, and thereby resulting in a newconvergent constantη0,j different from the assumed one, i.e.,η0,j 6= η0,j . This leads to a contradiction and thus yields

limt→∞

ηηηe(t) = 0 (25)

It follows from (23) and (25) that using the HFC law in (16),system (14) without disturbances (i.e.,ddd(t) = 0) is globallyasymptotically stable.

2) Negative Homogeneity: For system (17), selecting adilation as follows:

(r1, r2) = (1,1 + β1

2) (26)

for any givenε > 0, yields

f1(εr1ηe,j , ε

r2we,j) = εσ+r1f1(ηe,j , we,j)

f2(εr1ηe,j , ε

r2we,j) = εσ+r2f2(ηe,j , we,j) (27)




with f1(·) = we,j , f2 = −K1sigβ1(ηe,j)−K2sig

β2(we,j) anda negative degree of homogeneous with respect to the dilationin (26), i.e.,σ = (β1 − 1)/2 < 0.

By Lemma 1, we can conclude that the tracking error system(14) without disturbances controlled by the HFC scheme (16)is globally finite-time stable, i.e., there exists a finite time0 < T <∞ such that

ηηηe(t) ≡ 0, wwwe(t) ≡ 0, ∀ t ≥ T (28)

Together with (10), we further have

ηηηe(t) ≡ 0, νννe(t) ≡ 0, ∀ t ≥ T (29)

This concludes the proof.Remark 2:If the powersβ1 andβ2 of the HFC scheme in

(16) are set asβ1 = β2 = 1, the closed-loop system composedof the tracking error system (14) without disturbances and theHFC law (16) degrades to be globally asymptotically stable,i.e.,

ηe,j = we,j (30)

we,j = −K1ηe,j −K2we,j (31)

which can be derived easily from the conventional backstep-ping technique. �

Remark 3:Note that the external disturbanceddd(t) in (14)is not addressed in the nominal HFC scheme. In this context,within the HFC framework, the disturbance observer is ex-pected to be developed for enhancing the robustness and evenachieving exact disturbance rejection. �

B. Finite-Time Disturbance Observer Based HFC

In this subsection, the finite-time disturbance observer basedHFC (FDO-HFC) scheme is proposed. To this end, a genericassumption on the disturbanceddd(t) is required as follows:

Assumption 1:The external time-varying disturbanceddd(t)in (11) satisfies

ddd(n)(t) =

n−1∑

i=0

Hn−iddd(i)(t), i = 0, 1, · · · , n− 1 (32)

wheren is a positive integer andHi = diag(hi,1, hi,2, hi,3)with any constantshi,j ∈ R, j = 1, 2, 3. �

The key result pertaining to the FDO-HFC scheme is nowsummarized as follows:

Theorem 2 (FDO-HFC):Consider the ASV in (11) withunknown external disturbancesddd(t) satisfying Assumption 1,an FDO-HFC scheme designed as follows:

τττFDO =−MR−1(K1sig

β1(ηηη − ηηηd) +K2sigβ2(Rννν −Rdνννd)

)

−MSννν +MR−1SdRdνννd

− fff(ηηη,ννν) +MR−1RdM−1fff0(ηηηd, νννd)−MR−1ddd

(33)

with the FDO governed by

ddd = ppp1 +H1ppp0 (34)

whereppp1 andppp0 are derived by

˙ppp0 = ppp1 +H1ppp0 + uuu+ L0sigα1(ppp0 − ppp0)

...˙pppn−1 = pppn +Hnppp0 −Hn−1uuu+ Ln−1sig

αn(ppp0 − ppp0)

˙pppn = −Hnuuu+ Lnsigαn+1(ppp0 − ppp0) (35)

withppp0 = wwwe, uuu = RM−1τττ + hhhe (36)

andLi = diag(li,1, li,2, li,3), i = 0, 1, · · · , n, αi = 1+iϑ with−1/(n+ 1) < ϑ < 0, andϑ = −q1/q2 with q1 andq2 beingpositive even and odd integers, can render the ASV in (4)exactly track the desired trajectory generated by (9) within ashort time0 < T <∞, i.e.,ηηη(t) ≡ ηηηd(t), ννν(t) ≡ νννd(t), ∀ t ≥T .

Proof: In order to examine finite-time stability of theclosed-loop system (14) and (33) including an FDO (35), weneed to obtain the disturbance observation error dynamics. Tothis end, we define auxiliary variables as follows:

εεε0 = wwwe, εεε1 = ddd(t), εεε2 = ddd(t), · · · , εεεn = ddd(n−1)(t) (37)

Together with (14) and (32), we thus have

εεε0 = εεε1 + uuu

εεεi = εεεi+1, i = 1, 2, · · · , n− 1

εεεn = Hnεεε1 +Hn−1εεε2 + · · ·+H1εεεn (38)

Consider a coordinate transformation governed by

ppp0 = εεε0

ppp1 = εεε1 −H1εεε0...

pppn = εεεn −H1εεεn−1 − · · · −Hnεεε0 (39)

we further have

ppp0 = ppp1 +H1ppp0 + uuu

...

pppn−1 = pppn +Hnppp0 −Hn−1uuu

pppn = −Hnuuu (40)

Together with (35) and (40), the disturbance observationerror dynamics can be derived as follows:

pppe,0 = pppe,1 − L0sigα1(pppe,0)

...

pppe,n−1 = pppe,n − Ln−1sigαn(pppe,0)

pppe,n = −Lnsigαn+1(pppe,0) (41)

wherepppe,i = pppi − pppi := [p1e,i, p2e,i, p

3e,i]

T , i = 0, 1, · · · , n, and

pje,0 = pje,1 − l0,jsigα1(pje,0)

...

pje,n−1 = pje,n − ln−1,jsigαn(pje,0)

pje,n = −ln,jsigαn+1(pje,0), j = 1, 2, 3 (42)




Applying Lemma 2 to system (42), we can conclude thatthe disturbance observation errorspppe,i, i = 0, 1, · · · , n areglobally finite-time stable, i.e., the FDO in (35) can exactlyobserve the dynamics in (40) within a finite time0 < T <∞.Together with (34), (37) and (39), we can immediately obtainthat ppp0 and ppp1 can exactly estimatewwwe and ddd − H1wwwe,respectively, andddd governed by (34) can thus exactly observethe disturbanceddd in a finite time. Actually, the derivativesddd(i)(t), i = 1, 2, · · · , n−1 can also be exactly observed withina finite time bypppi+1 +H1pppi + · · ·+Hi+1ppp0.

In this context, we eventually have

ddd(i)(t) ≡ ddd(i)(t), ∀ t > T, i = 0, 1, · · · , n− 1 (43)

with ddd(i)

= pppi+1 +H1pppi + · · ·+Hi+1ppp0.Substituting (33) together with (34) and (43) into (14) yields

ηe,j = we,j

we,j = −K1sigβ1(ηe,j)−K2sig

β2(we,j) + dj (44)

wheredj = dj − dj .Together with (43) and (44), we further have

ηe,j(t) = we,j(t)

we,j(t) = −K1sigβ1(ηe,j(t))−K2sig

β2(we,j(t)) (45)

for any t > T with a finite time0 < T <∞.In what follows, similar to the proof of Theorem 1, global

finite-time stability of the closed-loop system (45) can beensured. As a consequence, in the presence of complex dis-turbances, using the FDO-HFC in (33), the ASV in (4) canexactly track the desired trajectory(ηηηd, νννd) generated by (9)in a finite time. This concludes the proof.

Remark 4: In addition to disturbancesτττd, the dynamicshhhe given by (15) might be at least partially unknown due tononlinearitiesfff(·) andfff0(·), and will make the foregoing HFCin (16) and FDO-HFC in (33) unavailable. In this context, theobserver for accurate estimate on mixed unknowns includingnot only unmodeled dynamics but also external disturbancesis required to be a patch within the HFC framework.

C. Finite-Time Unknown Observer Based HFC

Rewrite the tracking error dynamics in (14) as follows:

ηηηe = wwwe

wwwe = RM−1τττ + Swww − Sdwwwd + fffu(ηηη,www,ηηηd,wwwd, t) (46)

with the lumped unknownsfffu including unmodeled dynamicsfff , desired dynamicsfff0 and disturbancesddd, i.e.,

fffu(ηηη,www,ηηηd,wwwd, t) = RM−1fff(ηηη,RTwww)

−RdM−1fff0(ηηηd,R

Tdwwwd) + ddd(t) (47)

From (47), it reasonably requires to assume that dynamicsfff , fff0 andddd are twice differentiable, and thereby contributingto the following hypothesis:

‖fffu‖ ≤ Lu (48)

for a bounded constantLu <∞.

In this context, a finite-time unknown observer based HFC(FUO-HFC) scheme will be proposed to accurately track theASV in (4) with complex unknowns including both unmod-elded dynamics and disturbances to a desired trajectory withcompletely unknown dynamics. This challenging problem willbe solved by the proposed FUO-HFC scheme with globalfinite-time stability presented as follows.

Theorem 3 (FUO-HFC):Consider the ASV in (4) withunmodeled dynamicsfff and unknown external disturbancesddd(t), an FUO-HFC scheme designed as follows:

τττFUO = −MR−1(K1sig

β1(ηηη − ηηηd)

+K2sigβ2(Rννν −Rdνννd) + zzz1

)(49)

with zzz1 estimated by the following FUO:

zzz0 = ζζζ0 +RM−1τττ + Swww − Sdwwwd

ζζζ0 = −λ1LLL1/3sig2/3(zzz0 −wwwe) + zzz1

zzz1 = ζζζ1

ζζζ1 = −λ2LLL1/2sig1/2(zzz1 − ζζζ0) + zzz2

zzz2 = −λ3LLLsgn(zzz2 − ζζζ1) (50)

where zzzj := [zj,1, zj,2, zj,3]T , j = 0, 1, 2, ζζζk :=

[ζk,1, ζk,2, ζk,3]T , k = 0, 1, λi > 0, i = 1, 2, 3 and LLL =

diag(ℓ1, ℓ2, ℓ3), can render the ASV exactly track the desiredtrajectory generated by (9) with completely unknown dynam-ics fff0 in a short time0 < T < ∞, i.e.,ηηη(t) ≡ ηηηd(t), ννν(t) ≡νννd(t), ∀ t ≥ T .

Proof: Define the FUO observation errors as follows:

eee1 = zzz0 −wwwe, eee2 = zzz1 − fffu, eee3 = zzz2 − fffu (51)

Combining with (46) and (50), we have the FUO observa-tion error dynamics as follows:

eee1 = −λ1LLL1/3sig2/3(eee1) + eee2

eee2 = −λ2LLL1/2sig1/2(eee2 − eee1) + eee3

eee3 = −λ3LLLsgn(eee3 − eee2)− fffu (52)

i.e.,

e1,j = −λ1ℓ1/3j sig2/3(e1,j) + e2,j

e2,j = −λ2ℓ1/2j sig1/2(e2,j − e1,j) + e3,j

e3,j ∈ −λ3ℓjsgn(e3,j − e2,j) + [−Lu, Lu] (53)

where “∈” denotes the differential inclusion understood in theFilippov sense [41].

According to [42, Lemma 2] together with its detailed proof,we can immediately conclude that the tracking error dynamicsin (53) is globally finite-time stable, i.e., there exists a finitetime 0 < T <∞ such that

zzz0(t) ≡ wwwe(t), zzz1(t) ≡ fffu(t), zzz2(t) ≡ fffu(t), ∀ t > T (54)

Substituting (49) into (46) and using (54) yields

ηηηe(t) = wwwe(t)

wwwe(t) = −K1sigβ1(ηηηe(t))−K2sig

β2(wwwe(t)), ∀ t > T (55)




TABLE I: Main parameters of CyberShip II

m 23.8000 Yv −0.8612 Xu −2.0

Iz 1.7600 Y|v|v −36.2823 Yv −10.0

xg 0.0460 Yr 0.1079 Yr −0.0

Xu −0.7225 Nv 0.1052 Nv −0.0

X|u|u −1.3274 N|v|v 5.0437 Nr −1.0

Xuuu −5.8664

6 8 10 12 14 16

9

10

11

12

13

14

15

16

17

y/m

x/m

DesiredHFC (β1 = 1)

HFC (β1 = 1/3)

Fig. 2: Desired and actual trajectories in thexy plane

which has been proven to be globally finite-time stable inTheorem 1. In this context, the proposed FUO-HFC in (49)renders the ASV in (4) with lumped unmodeled dynamics anddisturbances can exactly track the desired trajectory(ηηηd, νννd)generated by (9) with completely unknown dynamics in a finitetime. This concludes the proof.

Remark 5:It should be noted that in addition to unmodeleddynamics and external disturbances pertaining to the ASV,the desired dynamics of the trajectory to be tracked are alsonot necessarily known for the FUO-HFC scheme. It impliesthat the proposed FUO-HFC approach is completely inde-pendent on dynamics of the ASV and the desired trajectory,and thereby contributing to a both task-free and model-freemethodology.

IV. SIMULATION STUDIES AND DISCUSSIONS

In order to demonstrate the effectiveness and superiority ofthe proposed control schemes for trajectory tracking control ofan ASV, simulation studies and comprehensive comparisonsare conducted on a well-known surface vehicle CyberShip II[43] of which the main parameters are listed in Table I.

Our objective is to track exactly the desired trajectory(ηηηd, νννd) governed by (9) with assumed dynamicsfff0(ηηηd, νννd) =−C(νννd)νννd − D(νννd)νννd − g(ηηηd, νννd) + τττ0, where τττ0 =[6, 3 cos2(0.2πt), sin2(0.2πt)]T , and the initial conditions areset asηηηd(0) = [15.6, 6.8, π/4]T , νννd(0) = [1, 0, 0]T , ηηη(0) =[15.5, 5.5, π/2]T andννν(0) = [0, 0, 0]T .

In what follows, 3 cases will be deployed to evaluate theperformance of the proposed HFC, FDO-HFC and FUO-HFCapproaches, respectively.

0 5 10 15 20 25 30

10

15

t/s

x/m

Desired

HFC (β1 = 1/3)

0 5 10 15 20 25 3068

101214

t/s

y/m

Desired

HFC (β1 = 1/3)

0 5 10 15 20 25 30

246

t/s

ψ/rad

Desired

HFC (β1 = 1/3)

Fig. 3: Desired and actual statesx, y, andψ

0 5 10 15 20 25 300

0.51

t/s

u/m

·s−

1

Desired

HFC (β1 = 1/3)

0 5 10 15 20 25 30−0.6−0.4−0.2

0

t/s

v/m

·s−

1

Desired

HFC (β1 = 1/3)

0 5 10 15 20 25 30

00.20.4

t/s

r/rad·s−

1

Desired

HFC (β1 = 1/3)

Fig. 4: Desired and actual statesu, v, andr

0 5 10 15 20 25 30−0.8−0.6−0.4−0.20

0.2

t/s

xe/m

HFC (β1 = 1)

HFC (β1 = 1/3)

0 5 10 15 20 25 30−1.5

−1−0.5

00.5

t/s

y e/m

HFC (β1 = 1)

HFC (β1 = 1/3)

0 5 10 15 20 25 30−0.2

00.20.40.6

t/s

ψe/rad

HFC (β1 = 1)

HFC (β1 = 1/3)

Fig. 5: Tracking errorsxe, ye, andψe

A. Performance Evaluation on the HFC

In this subsection, a nominal case where the ASV is suffi-ciently modeled (i.e.,fff is known) and external disturbancesare not considered (i.e.,ddd = 0) is employed to demonstratethe effectiveness and superiority of the proposed HFC scheme.




0 5 10 15 20 25 30−1

−0.50

0.5

t/s

ue/m

·s−

1

HFC (β1 = 1)

HFC (β1 = 1/3)

0 5 10 15 20 25 30−0.4−0.2

00.2

t/s

v e/m

·s−

1

HFC (β1 = 1)

HFC (β1 = 1/3)

0 5 10 15 20 25 30−0.3−0.2−0.1

00.1

t/s

r e/rad·s−

1

HFC (β1 = 1)

HFC (β1 = 1/3)

Fig. 6: Tracking errorsue, ve, andre

0 5 10 15 20 25 3005

10

t/s

τ 1/N

0 5 10 15 20 25 30−20−10

0

t/s

τ 2/N

0 5 10 15 20 25 30−1

012

t/s

τ 3/N

·m

HFC (β1 = 1/3)

Fig. 7: Control forcesτ1, τ2, and torqueτ3

User-defined parameters are chosen as follows:K1 = 0.3,K2 = 0.3, β1 = 1/3 andβ2 = 1/2.

Simulation results are shown in Figs. 2–7. Comparing withthe traditional asymptotic approach, i.e.,β1 = β2 = 1 withinthe HFC scheme, we can see from Fig. 2 that the HFC withβ1 = 1/3 can achieve much faster convergence. In addition,as shown in Figs. 3–6, the ASV can exactly track the desiredtrajectory within a finite time by virtue of the HFC laws shownin Fig. 7. In comparison with the asymptotic approach (i.e.,β1 = 1), the HFC scheme withβ1 = 1/3 is able to rendertracking errors converge to the origin in a very short time.

B. Performance Evaluation on the FDO-HFC

In this subsection, a much more practical case with unknowndisturbances is deployed to demonstrate the performance e-valuation and comparisons. In order to facilitate simulationstudies, the external disturbances are assumed to be governedby

ddd(t) =

10 cos(0.1πt− π/5)8 cos(0.3πt+ π/6)6 cos(0.2πt+ π/3)

(56)

6 8 10 12 14 16

9

10

11

12

13

14

15

16

17

18

y/m

x/m

DesiredHFC (β1 = 1)

HFC (β1 = 1/3)

FDO-HFC (β1 = 1/3)


0 5 10 15 20 25 30

10

15

t/s

x/m

Desired

FDO-HFC (β1 = 1/3)

0 5 10 15 20 25 3068

101214

t/s

y/m

Desired

FDO-HFC (β1 = 1/3)

0 5 10 15 20 25 30

246

t/s

ψ/rad

Desired

FDO-HFC (β1 = 1/3)


0 5 10 15 20 25 300

1

2

t/s

u/m

·s−

1

Desired

FDO-HFC (β1 = 1/3)

0 5 10 15 20 25 30−1.5

−1−0.5

0

t/s

v/m

·s−

1

Desired

FDO-HFC (β1 = 1/3)

0 5 10 15 20 25 30−0.5

0

0.5

t/s

r/rad·s−

1

Desired

FDO-HFC (β1 = 1/3)


with H1 = diag(0, 0, 0) and H2 =diag(−0.01π2,−0.09π2,−0.04π2).

Accordingly, user-defined parameters of the HFC and FDO-HFC schemes are commonly chosen asK1 = 2.6, K2 =2.6, β1 = 1/3 and β2 = 1/2. The other parameters of theFDO are selected as follows:L0 = diag(10, 10, 10), L1 =




0 5 10 15 20 25 30−0.4−0.2

00.20.4

t/s

xe/m

HFC (β1 = 1) HFC (β1 = 1/3) FDO-HFC (β1 = 1/3)

0 5 10 15 20 25 30

−1−0.5

0

t/s

y e/m

0 5 10 15 20 25 30−1

0

1

t/s

ψe/rad


0 5 10 15 20 25 30−1

0

1

t/s

ue/m

·s−

1

HFC (β1 = 1) HFC (β1 = 1/3) FDO-HFC (β1 = 1/3)

0 5 10 15 20 25 30

−101

t/s

v e/m

·s−

1

0 5 10 15 20 25 30−1

−0.50

0.5

t/s

r e/rad·s−

1


0 5 10 15 20 25 30

−505

t/s

d1

d1

d1

0 5 10 15 20 25 30

−505

t/s

d2

d2

d2

0 5 10 15 20 25 30−5

0

5

t/s

d3

d3

d3

Fig. 13: Disturbances and their finite-time observation

diag(32, 32, 32), L2 = diag(20, 20, 20), α1 = 7/9, α2 = 5/9andα3 = 1/3.

The actual and desired trajectories in the planar space areshown in Fig. 8–12, from which we can see that, in comparisonwith the conventional asymptotic control scheme (i.e.,β1 = 1),the proposed HFC scheme (i.e.,β1 = 1/3) achieves faster

6 8 10 12 14 16

9

10

11

12

13

14

15

16

17

y/m

x/m

Desired TrajectoryFUO−HFC


0 5 10 15 20 25 30

10

15

t/s

x/m

DesiredFUO-HFC

0 5 10 15 20 25 3068

101214

t/s

y/m

DesiredFUO-HFC

0 5 10 15 20 25 30

246

t/s

ψ/rad

DesiredFUO-HFC


0 5 10 15 20 25 300

1

2

t/s

u/m

·s−

1

DesiredFUO-HFC

0 5 10 15 20 25 30−1.5

−1−0.5

00.5

t/s

v/m

·s−

1

DesiredFUO-HFC

0 5 10 15 20 25 30

−0.50

0.5

t/s

r/rad·s−

1

DesiredFUO-HFC


convergence and stronger disturbance rejection simultaneous-ly, and thereby resulting in higher tracking accuracy, and theFDO-HFC approach is able to realize exact trajectory trackingwithin a short time since unknown disturbances can be finite-time observed exactly as shown in Fig. 13.




0 5 10 15 20 25 30−0.1

00.10.2

t/s

xe/m

FUO-HFC

0 5 10 15 20 25 30

−1−0.5

0

t/s

y e/m

0 5 10 15 20 25 300

0.20.40.6

t/s

ψe/rad


0 5 10 15 20 25 30−1

0

1

t/s

ue/m

·s−

1

FUO-HFC

0 5 10 15 20 25 30−1.5

−1−0.5

0

t/s

v e/m

·s−

1

0 5 10 15 20 25 30−0.8−0.6−0.4−0.2

0

t/s

r e/rad·s−

1


0 5 10 15 20 25 30

−505

10

t/s

f u,1

fu,1z1,1

0 5 10 15 20 25 30

−505

t/s

f u,2

fu,2z1,2

0 5 10 15 20 25 30−5

0

5

t/s

f u,3

fu,3z1,3

Fig. 19: Lumped unknowns and their finite-time observation

C. Performance Evaluation on the FUO-HFC

In order to demonstrate the superiority of the proposedFUO-HFC scheme, we employ a much more complex casewhere both unmodeled dynamics of the ASV and desiredtrajectory and unknown disturbances are included. The control

parameters are selected as follows:K1 = 2.6, K2 = 2.6,β1 = 1/3, β2 = 1/2, λ1 = 2, λ2 = 1.5, λ3 = 1.1andLLL = diag(30, 30, 30). Corresponding simulation resultsand comparisons are shown in Figs. 14–19. The actual andreference trajectories in the planar space are shown in Fig.14, which indicates that the actual trajectory can exactlytrack the desired one in a very short time although the ASVsuffers from unmodeled dynamics and unknown disturbances.Actually, the previous HFC and FDO-HFC approaches becomeunavailable due to the unexpected unmodeled dynamics of theASV and the desired trajectory. From the tracking performanceon position and velocity shown in Figs. 15–18, we can see thattrajectory tracking errors converge to zero in a very short timein spite of complex unknowns including unmodeled dynamicsand unknown disturbances. In essence, the remarkable perfor-mance of the proposed FUO-HFC scheme on exact trajectorytracking relies on the accurate observation on the lumpedunknowns via a FUO, whereby the finite-time observationresults are shown in Fig. 19. In this context, the FUO-HFCmethodology can achieve fast and exact trajectory trackingtogether with accurate reconstruction on complex unknownsincluding unmodeled dynamics, uncertainties and unknowndisturbances.

V. CONCLUSIONS

In this paper, in order to achieve trajectory fast and accuratetracking control of an autonomous surface vehicle (ASV)subject to unmodeled dynamics and unknown disturbances, ahomogeneity-based finite-time control (HFC) framework hasbeen innovatively proposed. For exactly dealing with externaldisturbances, a finite-time disturbance observer (FDO) hasbeen developed and has been incorporated into the HFCframework, thereby contributing the FDO-based HFC (termedFDO-HFC) scheme which can realize exact trajectory trackingcontrol of an ASV in the presence of complex disturbances.To further accurately handle complicated unknowns includingboth unmodeled dynamics and unknown disturbances, a finite-time unknown observer based HFC (FUO-HFC) scheme hasbeen proposed to enhance the entire performance includingboth trajectory tracking and unknowns identification, wherebyhigh accuracy and fast convergence can be ensured simulta-neously. Simulation studies and comprehensive comparisonshave been conducted on a benchmark ship, i.e., CyberShipII, and have demonstrated the effectiveness and superiority ofthe proposed HFC schemes in term of exact trajectory trackingand unknowns rejection.

ACKNOWLEDGMENTS

The authors would like to thank the Editor-in-Chief, As-sociate Editor and anonymous referees for their invaluablecomments and suggestions.

APPENDIX APROOF OFLEMMA 2

Proof: In the light of Lemma 1, the entire proof can bedivided into 2 phases, i.e., proves of global asymptotic stabilityand negative homogeneity.




Phase I: Global Asymptotic StabilityIn order to facilitate an inductive proof, applying a set of

coordinate transformations as follows:

zi = li−1xi, i = 1, 2, · · · , n (A.1)

with l0 = 1, to system (2), we have

x1 = c1 (x2 − xα2

1 )

x2 = c2 (x3 − xα3

1 )

...

xn = −cnxαn+1

1 (A.2)

wherexαi1 := sigαi(x1) andci = li/li−1, i = 1, 2, · · · , n.

Using (A.2), a backward recursive procedure will be estab-lished in the sequel.

Initial step: We first consider the following system:

xn = −cnxαn+1/αnn (A.3)

Choosing a Lyapunov function as follows:

Vn(xn) =αn

2|xn|

2/αn (A.4)

we have

Vn(xn)|(A.3) ≤ −kn|xn|(2+τ)/αn (A.5)

with kn = cn.Inductive step:Assume there exists a Lyapunov function as

follows:

Vi+1(xi+1, xi+2, · · · , xn)

=

n∑

j=i+1

∫ xj

xαj/αj+1

j+1

(s(2−αj)/αj − x

(2−αj)/αj+1

j+1

)ds

(A.6)

such thatVi+1 along the following system:

xi+1 = ci+1

(xi+2 − x

αi+2/αi+1

i+1

)

xi+2 = ci+2

(xi+3 − x

αi+3/αi+1

i+1

)

...

xn = −cnxαn+1/αi+1

i+1 (A.7)

satisfies

Vi+1|(A.7) ≤ −ki+1

n∑

j=i+1

∣∣xαj+1/αj

j − xj+1

∣∣(2+τ)/αj+1

(A.8)

with ki+1 > 0. It is easily verified that (A.8) holds for system(A.3) at the initial step, i.e.,i = n− 1.

In this context, we are expected to prove that for the systemas follows:

xi = ci(xi+1 − x

αi+1/αi

i

)

xi+1 = ci+1

(xi+2 − x

αi+2/αi

i

)

...

xn = −cnxαn+1/αi

i (A.9)

there exists the following Lyapunov function:

Vi(xi,xi+1, · · · , xn) = Vi+1(xi+1, xi+2, · · · , xn)

+

∫ xi

xαi/αi+1

i+1

(s(2−αi)/αi − x

(2−αi)/αi+1

i+1

)ds (A.10)

such thatVi along (A.9) satisfies the form like (A.8).To this end, together with (A.8), we have

Vi|(A.9) = Vi+1|(A.7) −

n∑

j=1+1

cj∂Vi+1

∂xj

(xαj+1/αi

i − xαj+1/αi+1

i+1

)

+d

dt

∫ xi

xαi/αi+1

i+1

(s(2−αi)/αi − x

(2−αi)/αi+1

i+1

)ds∣∣∣(A.9)

≤ −ki+1

n∑

j=i+1

∣∣xαj+1/αj

j − xj+1

∣∣(2+τ)/αj+1

− ci(x(2−αi)/αi

i − x(2−αi)/αi+1

i+1

)(xαi+1/αi

i − xi+1

)

−ci+1(2− αi)

αi+1x(2−αi−αi+1)/αi+1

i+1

×(xi+2 − x

αi+2/αi

i

)(xi − x

αi/αi+1

i+1

)

−n∑

j=1+1

cj∂Vi+1

∂xj

(xαj+1/αi

i − xαj+1/αi+1

i+1

)(A.11)

Note that

− ci(x(2−αi)/αi

i − x(2−αi)/αi+1

i+1

)(xαi+1/αi

i − xi+1

)

≤ −ciki+1

∣∣xαi+1/αi

i − xi+1

∣∣(2+τ)/αi+1 (A.12)

−ci+1(2− αi)

αi+1x(2−αi−αi+1)/αi+1

i+1

×(xi+2 − x

αi+2/αi

i

)(xi − x

αi/αi+1

i+1

)

≤ µ1

n∑

j=i+1

∣∣xαj+1/αj

j − xj+1

∣∣(2+τ)/αj+1

+ ρ1(µ1)∣∣xαi+1/αi

i − xi+1

∣∣(2+τ)/αi+1 (A.13)

−

n∑

j=1+1

cj∂Vi+1

∂xj

(xαj+1/αi

i − xαj+1/αi+1

i+1

)

≤ µ2

n∑

j=i+1

∣∣xαj+1/αj

j − xj+1

∣∣(2+τ)/αj+1

+ ρ2(µ2)∣∣xαi+1/αi

i − xi+1

∣∣(2+τ)/αi+1 (A.14)

with positive continuous functionsρ1(·) andρ2(·) with respectto any positive constantsµ1 andµ2, respectively.

Substituting (A.12)–(A.14) into (A.11) yields

Vi|(A.9) ≤ −(ki+1 − µ1 − µ2)n∑

j=i+1

∣∣xαj+1/αj

j − xj+1

∣∣(2+τ)/αj+1

− (ciki+1 − ρ1 − ρ2)∣∣xαi+1/αi

i − xi+1

∣∣(2+τ)/αi+1

(A.15)

Selecting parametersµ1, µ2, ki+1 andci such that

ci ≥ki + ρ1(µ1) + ρ2(µ2)

ki+1, ki ≤ ki+1 − µ1 − µ2 (A.16)




yields

Vi|(A.9) ≤ −ki

n∑

j=i

∣∣xαj+1/αj

j − xj+1

∣∣(2+τ)/αj+1 (A.17)

which implies that (A.8) also holds for theith inductive step.Recursively, we can eventually construct a Lyapunov func-

tion as follows:

V1(x1, x2, · · · , xn)

=

n∑

j=1

∫ xj

xαj/αj+1

j+1

(s(2−αj)/αj − x

(2−αj)/αj+1

j+1

)ds

(A.18)

with xn+1 = 0 andα1 = 1, such that

V1|(A.2) ≤ −k1

n∑

j=1

∣∣xαj+1/αj

j − xj+1

∣∣(2+τ)/αj+1 (A.19)

with k1 > 0 recursively determined by (A.16).It follows from (A.19) that system (A.2) is globally asymp-

totically stable. Together with the global diffeomorphism(A.1), we have system (2) is globally asymptotically stable.

Phase II: Negative HomogeneityApplying a dilation as follows:

(r1, r2, · · · , rn) = (α1, α2, · · · , αn) (A.20)

with α1 = 1, to system (2) yields a negative homogeneous ofdegree, i.e.,τ = αi+1 − αi < 0, i = 1, 2, · · · , n.

In this context, combining with Phases I and II, and usingLemma 1, we immediately have system (2) is globally asymp-totically stable. This concludes the proof.

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Ning Wang (S’08-M’12-SM’15) received his B.Eng. degree in Marine Engineering and the Ph.D.degree in control theory and engineering from theDalian Maritime University (DMU), Dalian, Chinain 2004 and 2009, respectively. From September2008 to September 2009, he was financially support-ed by China Scholarship Council (CSC) to work asa joint-training PhD student at the Nanyang Tech-nological University (NTU), Singapore. In the lightof his significant research at NTU, he received theExcellent Government-funded Scholars and Students

Award in 2009. From August 2014 to August 2015, he worked as a VisitingScholar at the University of Texas at San Antonio. He is currently a FullProfessor with the Marine Engineering College, DMU, Dalian 116026, China.

Dr. Wang received the Nomination Award of Liaoning Province ExcellentDoctoral Dissertation, the DMU Excellent Doctoral Dissertation Award andthe DMU Outstanding PhD Student Award in 2010, respectively. He also wonthe Liaoning Province Award for Technological Invention and the honourof Liaoning BaiQianWan Talents, Liaoning Excellent Talents, Science andTechnology Talents the Ministry of Transport of the P. R. China, YouthScience and Technology Award of China Institute of Navigation, and DalianLeading Talents. His research interests include fuzzy neural systems, deeplearning, nonlinear control, self-organizing fuzzy neural modeling and control,unmanned vehicles and autonomous control. He currently serves as AssociateEditors of theNeurocomputingand theInternational Journal of Fuzzy Systems.

Shuailin Lv received his B. Eng. degree in MarineElectronic and Electrical Engineering and B. E.degree in International Economics and Trade fromthe Dalian Maritime University (DMU), China in2015. He is currently pursuing his Master degree atthe same university. His research interests includeelectric drive and control, sliding mode control,marine control, and finite-time control.

Meng Joo Er (S’82-M’87-SM’07) is currently aFull Professor in Electrical and Electronic Engineer-ing, Nanyang Technological University, Singapore.He has authored 5 books, 16 book chapters and morethan 500 refereed journal and conference papers inhis research areas of interest. His areas of researchinterests are intelligent control theory and applica-tions, computational intelligence, robotics and au-tomation, sensor networks, biomedical engineeringand cognitive science.

In recognition of the significant and impactfulcontributions to Singapores development by his research project entitledDevelopment of Intelligent Techniques for Modelling, Controlling and Op-timizing Complex Manufacturing Systems, Professor Er won the Institutionof Engineers, Singapore (IES) Prestigious Engineering Achievement Award2011. He is also the only dual winner in Singapore IES Prestigious PublicationAward in Application (1996) and IES Prestigious Publication Award in Theory(2001). He received the Teacher of the Year Award for the School of EEE in1999, School of EEE Year 2 Teaching Excellence Award in 2008 and the MostZealous Professor of the Year Award 2009. He also received the Best SessionPresentation Award at the World Congress on Computational Intelligence in2006 and the Best Presentation Award at the International Symposium onExtreme Learning Machine 2012. Under his leadership as Chairman of theIEEE CIS Singapore Chapter from 2009 to 2011, the Singapore Chapter wonthe CIS Outstanding Chapter Award 2012. In recognition of his outstandingcontributions to professional bodies, he was bestowed the IEEE OutstandingVolunteer Award (Singapore Section) and the IES Silver Medal in 2011. Ontop of this, he has more than 40 awards at international and local competitions.Currently, Professor Er serves as the Editor-in-Chief of the Transaction onMachine Learning and Artificial Intelligence, an Area Editor of InternationalJournal of Intelligent Systems Science, an Associate Editor of eleven refereedinternational journals including the IEEE Transaction on Fuzzy Systems andIEEE Transactions on Cybernetics, and an editorial board member of the EETimes. He has been invited to deliver more than 60 keynote speeches andinvited talks overseas.

Wen-Hua Chen (SM’06) received the M.Sc. andPh.D. degrees from Northeast University, Shenyang,China, in 1989 and 1991, respectively.

From 1991 to 1996, he was a Lecturer and thenAssociate Professor with the Department of Auto-matic Control, Nanjing University of Aeronauticsand Astronautics, Nanjing, China. From 1997 to2000, he held a research position and then was aLecturer of control engineering with the Centre forSystems and Control, University of Glasgow, Glas-gow, U.K. In 2000, he moved to the Department of

Aeronautical and Automotive Engineering, Loughborough University, Lough-borough, U.K., as a Lecturer, where he was appointed as a Professor in 2012.His research interests include the development of advanced control strategies(nonlinear model predictive control, disturbance-observer-based control, etc.)and their applications in aerospace and automotive engineering. Currently,much of his work has also been involved with the development of unmannedautonomous intelligent systems.

Dr. Chen is a Fellow of The Institution of Engineering and Technology,U.K., and a Fellow of The Institution of Mechanical Engineers, U.K.

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