dynamic positioning control of marine vehicledynamic positioning control of marine vehicle ting-li...

International Journal of InnovativeComputing, Information and Control ICIC International c©2012 ISSN 1349-4198Volume 8, Number 5(A), May 2012 pp. 3361–3385

DYNAMIC POSITIONING CONTROL OF MARINE VEHICLE

Ting-Li Chien1, Chia-Wei Lin2, Chiou-Jye Huang3

and Chung-Cheng Chen4,∗

1Department of Electronic EngineeringWufeng Institute of Technology

No. 117, Sec. 2, Chian-Kuo Rd., Ming-Hsiung, Chiayi 621, Taiwan

2Institute of Computer Science and Information EngineeringNational Chiayi University

No. 300, Syuefu Rd., Chiayi 60004, Taiwan

3Green Energy and Environment Research LaboratoriesIndustrial Technology Research Institute

195, Sec. 4, Chung Hsing Rd., Chutung, Hsinchu 31040, Taiwan

4Department of Electrical EngineeringHwa Hsia Institute of Technology

No. 111, Gong Jhuan Rd., Chung Ho, Taipei 235, Taiwan∗Corresponding author: [email protected]

Received January 2011; revised May 2011

Abstract. The paper deals with a novel fuzzy feedback linearization control of dynamicpositioning marine vehicle with the almost disturbance decoupling performance. The maincontribution of this study is to construct a controller such that the resulting dynamicpositioning marine vehicle is valid for any initial and desired positions with the follow-ing characteristics: input-to-state stability with respect to disturbance inputs and almostdisturbance decoupling. The dynamic positioning time based on our proposed novel non-linear fuzzy feedback linearization control is better than some existing approaches. Theproposed controller is constructed by feedback linearization controller and fuzzy controller.The feedback linearization controller guarantees the almost disturbance decoupling perfor-mance and the uniform ultimate bounded stability of the tracking error system. Once thetracking errors are driven to touch the global final attractor with the desired radius, thefuzzy logic controller immediately is applied via a human expert’s knowledge to improvethe convergence rate. Finally, the availability and feasibility of the proposed approach areinvestigated with a numerical simulation.Keywords: Marine vehicle, Dynamic positioning systems, Almost disturbance decou-pling, Feedback linearization approach, Fuzzy logic control

1. Introduction. Offshore oilfield development has forwarded to a deeper and severeenvironment for new oil sources and maintaining the position of an offshore platform be-comes a very challenging problem. Therefore, dynamic position of marine vehicle usingthrusters is often used in many offshore oilfield operations, such as drilling, pipe-laying,tanking between marine vehicles and diving support. A dynamic positioning marine ve-hicle exhibits non-linear interaction in three degrees of freedom (sway, surge and yaw)by means of main propellers aft on the marine vehicle [8]. Dynamic positioning marinevehicle has usually been designed by assuming that the kinematic equations of motionscan be linearized about predefined constant yaw angles, meaning that linear optimal con-trol approach and gain-scheduling techniques can be applied [4,36,40]. It is obvious to

3361

3362 T.-L. CHIEN, C.-W. LIN, C.-J. HUANG AND C.-C. CHEN

see that some modeling errors do exist during the overall linearization process. In or-der to solve the linearization problem, [6,14,49] apply the Takagi-Sugenno fuzzy model,which includes a set of IF-THEN rules, to investigating the non-linear dynamic position-ing marine vehicle based on the parallel distributed compensation (PDC). Its designingprocedure is as follows. First, representing the non-linear system as the famous Takagi-Sugeno fuzzy model offers an alternative to the conventional model. The control design iscarried out based on an aggregation of linear controllers constructed for each local linearelement of the fuzzy model via the parallel distributed compensation scheme [46]. Forthe stability analysis of fuzzy system, a lot of studies are reported (See, e.g., [27,41-43]and the references therein). The stability and controller design of fuzzy system can bemainly discussed by Tanaka-Sugeno’s theorem [42]. However, it is difficult to find thecommon positive definite matrix P for linear matrix inequality (LMI) problem even if Pis a second order matrix [24]. Moreover, the PDC will be subject to failure if the fuzzymodel is decomposed by too many plant rules [10,45]. This is because there are manylinear-matrix-inequality constraints involved in optimization process and many decisionvariables that need to be carried out simultaneously.Many approaches to stabilizing and tracking tasks have been proposed including feed-

back linearization, variable structure control (sliding mode control), backstepping, regu-lation control, non-linear H∞ control, internal model principle and H∞ adaptive fuzzycontrol. [25] has proposed the use of variable structure control to deal with non-linearsystem. However, chattering behaviour caused by discontinuous switching and imperfectimplementation that can drive the system into unstable regions is inevitable for variablestructure control schemes. Backstepping has proven to be a powerful tool for synthesizingcontrollers for non-linear systems [17,52]. However, a disadvantage of this approach isan explosion in the complexity which is a result of repeated differentiations of non-linearfunctions [12,39,51]. An alternative approach is to utilize output regulation control [21]in which the outputs are assumed to be excited by an exosystem. However, the non-linear regulation approach requires the solution of difficult partial-differential algebraicequations. Another difficulty is that the exosystem states need to be switched to describechanges in the output and this creates transient tracking errors [34]. In general, non-linearH∞ control requires the solution of the Hamilton-Jacobi equation, which is a difficult non-linear partial-differential equation [2,22,44]. Only for some particular non-linear systemsit is possible to derive a closed-form solution [20]. The control approach that is basedon the internal model principle converts the tracking problem into a non-linear outputregulation problem. This approach depends on solving a first-order partial-differentialequation of the center manifold [21]. For some special non-linear systems and desired tra-jectories, the asymptotic solutions of this equation have been developed using ordinarydifferential equations [15,18]. Recently, H∞ adaptive fuzzy control has been proposed tosystematically deal with non-linear systems [7]. The drawback with H∞ adaptive fuzzycontrol is that the complex parameter update law makes this approach impractical inreal-world situations. During the past decade significant progress has been made in re-searching control approaches for non-linear systems based on the feedback linearizationtheory [19,25,33,38]. Moreover, feedback linearization approach has been applied suc-cessfully to many real control systems. These include the control of an electromagneticsuspension system [23], pendulum system [9], spacecraft [37], electrohydraulic servosys-tem [1], car-pole system [3] and bank-to-turn missile system [28]. The main contributionof this study is to solve the linearized and PDC shortcomings by using non-linear feedbacklinearization approach.The almost disturbance decoupling problem, i.e., the design of a controller that attenu-

ates the effect of the disturbance on the output terminal to an arbitrary degree of accuracy,

DYNAMIC POSITIONING CONTROL OF MARINE VEHICLE 3363

was originally developed for linear and non-linear control systems by [30,48] respectively.The problem has attracted considerable attention and many significant results have beendeveloped for both linear and non-linear control systems [13,31,35,47]. The almost dis-turbance decoupling problem of non-linear single-input single-output (SISO) systems wasinvestigated in [30] by using a state feedback approach and solved in terms of sufficientconditions for systems with non-linearities that are not globally Lipschitz and distur-bances bring linear but possibly actually bring multiples of non-linearities. The resultingstate feedback control is constructed following a singular perturbation approach. Thesufficient conditions in [30] require that the non-linearities multiplying the disturbancessatisfy structural triangular conditions. [30] shows that for non-linear SISO systems thealmost disturbance decoupling problem may not be solvable, as is case for

x1(t) = tan−1 x2 + θ(t), |θ(t)| > π

2x2(t) = u

y = x1

where u, y denoted the input and output respectively and θ(t) was the disturbance of thesystem. On the contrary, this example can be easily solved via the proposed approach inthis paper.

To overcome the problems for linearized technique and fuzzy-model approach, we willpropose a new method to guarantee that the marine vehicle is stable and the almostdisturbance decoupling performance is achieved. The designing structure is as follows.Firstly, based on the feedback linearization approach a tracking control is proposed inorder to guarantee the almost disturbance decoupling property and the uniform ultimatebounded stability of the tracking error system. Once the tracking errors are driven totouch the global final attractor, the conventional fuzzy logic control is immediately appliedvia a human expert’s knowledge to improve the convergence rate. In order to exploit thesignificant applicability, this paper has also successfully derived controller with almostdisturbance decoupling for a marine vehicle. According to the simulation results, it iseasy to see that the dynamic positioning time based on our proposed novel non-linearfuzzy feedback linearization control is shorter than the traditional fuzzy-model approach[5].

2. Feedback Linearization Controller Design. The following non-linear uncertaincontrol system with disturbances is considered.

x1

x2...xn

=

f1(x1, x2, . . . , xn)f2(x1, x2, . . . , xn)

...fn(x1, x2, . . . , xn)

+ [g1(x1, x2, . . . , xn) g2(x1, x2, . . . , xn) · · · gm(x1, x2, . . . , xn)]

u1(x1, x2, . . . , xn)u2(x1, x2, . . . , xn)

...um(x1, x2, . . . , xn)

+

p∑j=1

q∗j θjd +

∆f1(x1, x2, . . . , xn)∆f2(x1, x2, . . . , xn)

...∆fn(x1, x2, . . . , xn)

(1a)

3364 T.-L. CHIEN, C.-W. LIN, C.-J. HUANG AND C.-C. CHENy1(x1, x2, . . . , xn)y2(x1, x2, . . . , xn)

...ym(x1, x2, . . . , xn)

=

h1(x1, x2, . . . , xn)h2(x1, x2, . . . , xn)

...hm(x1, x2, . . . , xn)

(1b)

that is

X(t) = f(X(t)) + g(X(t))u+

p∑j=1

q∗j θjd +∆f

y(t) = h(X(t))

where X(t) ≡ [x1(t) x2(t) · · · xn(t)]T ∈ <n is the state vector, u ≡ [u1 u2 · · · um]

T ∈ <m

is the input vector, y ≡ [y1 y2 · · · ym]T ∈ <m is the output vector, θd ≡ [θ1d(t) θ2d(t)

· · · θpd(t)]T is a bounded time-varying disturbances vector and ∆f ≡ [∆f1 ∆f2 · · · ∆fn] ∈<n is unknown non-linear function representing uncertainty such as modelling error. Let

∆f be described as ∆f =p∑

j=1

q∗j θuj, where θu ≡ [θu1(X, t) θu2(X, t) · · · θup(X, t)]T is a

bounded time-varying vector and q∗j ∈ <n, j = 1, 2, . . . , p, are coefficient vectors of dis-turbances. The vectors f ∈ <n and h ∈ <m are vectors of smooth real-valued functionsf1, f2, · · · , fn and h1, h2, · · · , hm, respectively. The matrix g ∈ <n×m is a a matrix withcolumns being smooth vector fields g1, g2, · · · , gm. The nominal system is then defined asfollows:

X(t) = f(X(t)) + g(X(t))u (2a)

y(t) = h(X(t)) (2b)

The nominal system of the form (2) is assumed to have the vector relative degree{r1, r2, · · · , rm} [16,19], i.e., the following conditions are satisfied for all X ∈ <n:〈1〉

LgjLkfhi(X) = 0 (3)

for all 1 ≤ i ≤ m, 1 ≤ j ≤ m, k < ri − 1, where the operator L is the Lie derivative [19]and r1 + r2 + · · ·+ rm = r.〈2〉 The m×m matrix

A ≡

Lg1L

r1−1f h1(X) · · · LgmL

r1−1f h1(X)

Lg1Lr2−1f h2(X) · · · LgmL

r2−1f h2(X)

......

Lg1Lrm−1f hm(X) · · · LgmL

rm−1f hm(X)

(4)

is non-singular.The desired output trajectory yid, 1 ≤ i ≤ m and its first ri derivatives are all uniformly

bounded and ∥∥∥[yid, yi(1)d , · · · , yi(ri)d

]∥∥∥ ≤ Bid, 1 ≤ i ≤ m (5)

where Bid is some positive constant.

The objective of the paper is to propose a control that includes a feedback linearizationcontroller and a fuzzy logic controller to achieve the almost disturbance decoupling andtracking performances.A. Feedback linearization controller design.Under the assumption of well-defined vector relative degree, it has been shown [19] that

the mapping

φ : <n → <n (6)


defined as

ξi ≡

ξi1ξi2...ξiri

≡

φi1

φi2...φiri

≡

L0fhi(X)

L1fhi(X)

...Lri−1f hi(X)

, i = 1, 2, . . . ,m (7)

φk(X(t)) ≡ ηk(t), k = r + 1, r + 2, · · · , n (8)

and satisfying

Lgjφk(X(t)) = 0, k = r + 1, r + 2, · · · , n, 1 ≤ j ≤ m (9)

is a diffeomorphism onto image, if the following hold〈1〉 The distribution

G ≡ span{g1, g2, · · · , gm} (10)

is involutive.〈2〉 The vector fields

Y kj , 1 ≤ j ≤ m, 1 ≤ k ≤ rj (11)

are complete, where

Y kj ≡ (−1)k−1 adk−1

fgj, 1 ≤ j ≤ m, 1 ≤ k ≤ rj (12)

f(X) ≡ f(X)− g(X)A−1(X)b(X) (13)

b(X) ≡

Lr1f h1(X)

Lr2f h2(X)

...Lrmf hm(X)

(14)

g ≡ [g1 g2 · · · gm] ≡ g(X)A−1(X) (15)

adkfg ≡⌊f, adk−1

f g⌋

(16)

[f, g] ≡ ∂g

∂Xf(X)− ∂f

∂Xg(X) (17)

For the sake of convenience, define the trajectory error to be

eij ≡ ξij − yi(j−1)d , i = 1, 2, · · · ,m, j = 1, 2, · · · , ri (18)

ei ≡[ei1 ei2 · · · eiri

]T ∈ <ri (19)

and the trajectory error to be multiplied with some adjustable positive constant ε

eij ≡ εj−1eij, i = 1, 2, · · · ,m, j = 1, 2, · · · , ri (20)

ei ≡[ei1 ei2 · · · eiri(t)

]T∈ <r

i (21)

e ≡

e1

e2...em

∈ <r (22)

and

ξ ≡

ξ1ξ2...ξr

T

∈ <r, (23)


η(t) ≡ [ηr+1(t) ηr+2(t) · · · ηn(t)]T ∈ <n−r (24)

q(ξ(t), η(t)) ≡ [Lfφr+1(t) Lfφr+2(t) · · · Lfφn(t)]T ≡ [qr+1 qr+2 · · · qn]

T (25)

Define a phase-variable canonical matrix Aic to be

Aic ≡

0 1 0 · · · 00 0 1 · · · 0

......

0 0 0 · · · 1−αi

1 −αi2 −αi

3 · · · −αiri

ri×ri

, 1 ≤ i ≤ m (26)

where αi1, α

i2, · · · , αi

riare any chosen parameters such that Ai

c is Hurwitz and the vectorBi to be

Bi ≡ [0 0 · · · 0 1]Tri×1 , 1 ≤ i ≤ m (27)

Let P i be the positive definite solution of the following Lyapunov equation

(Aic)

TP i + P iAic = −I, 1 ≤ i ≤ m (28)

λmax(Pi) ≡ the maximum eigenvalue of P i, 1 ≤ i ≤ m (29)

λmin(Pi) ≡ the minimum eigenvalue of P i, 1 ≤ i ≤ m (30)

λ∗max ≡ min

{λmax(P

1), λmax(P2), · · · , λmax(P

m)}

(31)

λ∗min ≡ min

{λmin(P

1), λmin(P2), · · · , λmin(P

m)}

(32)

Assumption 1. For all t ≥ 0, η ∈ <n−r and ξ ∈ <r, there exists a positive constant Msuch that the following inequality holds

‖q22(ξ, η, e)− q22(t, η, 0)‖ ≤ M (‖e‖) (33)

where q22(t, η, e) ≡ q(ξ, η).For the sake of stating precisely the investigated problem, define

dij ≡ LgjLri−1f hi(X), 1 ≤ i ≤ m, 1 ≤ j ≤ m (34)

ci ≡ Lrif hi(X), 1 ≤ i ≤ m (35)

and

ei ≡ αi1e

i1 + αi

2ei2 + · · ·+ αi

rieiri , 1 ≤ i ≤ m (36)

Definition 2.1. [25] Consider the system x = f(t, x, θ), where f : [0,∞)×<n×<n → <n

is piecewise continuous in t and locally Lipschitz in x and θ. This system is said to beinput-to-state stable if there exists a class KL function β, a class K function γ andpositive constants k1 and k2 such that for any initial state x (t0) with ‖x (t0)‖ < k1 andany bounded input θ(t) with sup

t≥t0

‖θ (t)‖ < k2, the state exists and satisfies

‖x(t)‖ ≤ β (‖x(t0)‖ , t− t0) + γ

(sup

t0≤τ≤t‖θ(τ)‖

)(37)

for all t ≥ t0 ≥ 0.

Now we formulate the tracking problem with almost disturbance decoupling as follows:

Definition 2.2. [31] The tracking problem with almost disturbance decoupling is said tobe globally solvable by the state feedback controller u for the transformed-error system bya global diffeomorphism (6), if the controller u enjoys the following properties.〈1〉 It is input-to-state stable with respect to disturbance inputs.


〈2〉 For any initial value xe0 ≡ [e(t0) η(t0)]T , for any t ≥ t0 and for any t0 ≥ 0

|y(t)− yd(t)| ≤ β11 (‖x(t0)‖, t− t0) +1√β22

β33

(sup

t0≤τ≤t‖θ(τ)‖

)(38)

andt∫

t0

[y(τ)− yd(τ)]2 dτ ≤ 1

β44

β55 (‖xe0‖) +t∫

t0

β33

(‖θ (τ)‖2

)dτ

(39)

where β22, β44 are some positive constants, β33, β55 are class K functions and β11 is aclass KL function.

Theorem 2.1. Suppose that there exists a continuously differentiable function V0: <n−r →<+ such that the following three inequalities hold for all η ∈ <n−r:

ω1‖η‖2 ≤ V (η) ≤ ω2‖η‖2, ω1, ω2 > 0 (40a)

∇tV + (∇ηV )T q22(t, η, 0) ≤ −2αxV (η), αx > 0 (40b)

‖∇ηV ‖ ≤ $3‖η‖, $3 > 0, (40c)

then the tracking problem with almost disturbance decoupling is globally solvable by thecontroller defined by

ufeedback = A−1{−b+ v} (41)

b ≡[Lr1f h1 Lr2

f h2 · · · Lrmf hm

]T(42)

v ≡ [v1 v2 · · · vm]T (43)

vi ≡ yi(ri)

d − ε−riαi1

[L0fhi(X)− yid

]− ε1−riαi

2

[L1fhi(X)− yi

(1)

d

]− · · · − ε−1αi

ri

[Lri−1f hi(X)− yi

(ri−1)

d

], 1 ≤ i ≤ m (44)

Moreover, the influence of disturbances on the L2 norm of the tracking error can bearbitrarily attenuated by increasing the following adjustable parameter NN2 > 1:

H(ε) ≡[H11 H12

H12 H22

]

≡

2αx −

4ω23

ω1

‖φη‖2 − 1√k(ε)

[w3M√2w1λ∗

min

]

− 1√k(ε)

[w3M√2w1λ∗

min

]1

ελ∗max

−8k(ε)‖φ1

ξ‖2‖P 1‖2

ε2λmin(P 1)− · · · −

8k(ε)‖φmξ ‖2‖Pm‖2

ε2λmin(Pm)

(45a)

αs(ε) ≡H11 +H22 − [(H11 −H22)

2 + 4H212]

1/2

4(45b)

N ≡ 2αs(ε) (45c)

N1 ≡m+ 1

4

(sup

t0≤τ≤t‖θd (τ) + θu (τ)‖

)2

(45d)

N2 ≡ min

{ω1,

k(ε)

2λ∗min

}(45e)

φiξ(ε) ≡

ε ∂∂X

hiq∗1 · · · ε ∂

∂Xhiq

∗p

......

εri ∂∂X

Lri−1f hiq

∗1 · · · εri ∂

∂XLri−1f hiq

∗q

, 1 ≤ i ≤ m (45f)


φη(ε) ≡

∂∂X

φr+1q∗1 · · · ∂

∂Xφr+1q

∗p

......

∂∂X

φnq∗1 · · · ∂

∂Xφnq

∗q

(45g)

where H is positive definite matrix and k(ε): <+ → <+ is any continuous function satisfies

limε→0

k(ε) = 0 and limε→0

ε

k(ε)= 0 (45h)

Moreover, the output tracking error of system (2.1) is exponentially attracted into a

sphere Br, r =√

N1

NN2, with an exponential rate of convergence

1

2

(NN2

∆max

− N1

∆maxr2

)≡ 1

2α∗ (45i)

where

∆max = max

{ω2,

k

2λ∗max

}(45j)

Proof: Applying the coordinate transformation (6) yields

ξ11 =∂φ1

1

∂X

dX

dt=

∂h1

∂X

[f + g · u+

p∑j=1

q∗j θjd +∆f

]

=∂h1

∂Xf +

∂h1

∂Xg1u1 + · · ·+ ∂h1

∂Xgmum +

p∑j=1

∂h1

∂Xq∗j θjd +

p∑j=1

∂h1

∂Xq∗j θju

=∂h1

∂Xf +

p∑j=1

∂h1

∂Xq∗j (θjd + θju) = ξ12 +

p∑j=1

∂h1

∂Xq∗j (θjd + θju)

(46)

...

ξ1r1−1 =∂φ1

r1−1

∂X

dX

dt=

∂Lr1−2f h1

∂X

[f + g · u+

p∑j=1

q∗j θjd +∆f

]

=∂Lr1−2

f h1

∂Xf +

∂Lr1−2f h1

∂Xg1u1 + · · ·+

∂Lr1−2f h1

∂Xgmum

+

p∑j=1

∂Lr1−2f h1

∂Xq∗j θjd +

p∑j=1

∂Lr1−2f h1

∂Xq∗j θju

=∂Lr1−2

f h1

∂Xf +

p∑j=1

∂Lr1−2f h1


= Lr1−1f h1 +

p∑j=1

∂Lr1−2f h1


(47)


ξ1r1 =∂φ1

r1

∂X

dX

dt=

∂Lr1−1f h1

∂X

[f + g · u+

p∑j=1

q∗j θjd +∆f

]

=∂Lr1−1

f h1

∂Xf +

∂Lr1−1f h1

∂Xg1u1 + · · ·+

∂Lr1−1f h1

∂Xgmum +

p∑j=1

∂Lr1−1f h1

∂Xq∗j θjd

+

p∑j=1

∂Lr1−1f h1

∂Xq∗j θju

= Lr1f h1 + Lg1L

r1−1f h1u1 + · · ·+ LgmL

r1−1f h1um +

p∑j=1

∂Lr1−1f h1


= c1 + d11u1 + · · ·+ d1mum +

p∑j=1

∂Lr1−1f h1


(48)

...

ξm1 =∂φm

1

∂X

dX

dt=

∂hm

∂X

[f + g · u+

p∑j=1

q∗j θjd +∆f

]

=∂hm

∂Xf +

∂hm

∂Xg1u1 + · · ·+ ∂hm

∂Xgmum +

p∑j=1

∂hm

∂Xq∗j θjd +

p∑j=1

∂hm

∂Xq∗j θju

= L1fhm +

p∑j=1

∂hm

∂Xq∗j (θjd + θju) = ξm2 +

p∑j=1

∂hm


(49)

...

ξmrm−1 =∂φm

rm−1

∂X

dX

dt=

∂Lrm−2f hm

∂X

[f + g · u+

p∑j=1

q∗j θjd +∆f

]

=∂Lrm−2

f hm

∂Xf +

∂Lrm−2f hm

∂Xg1u1 + · · ·+

∂Lrm−2f hm

∂Xgmum +

p∑j=1

∂Lrm−2f hm

∂Xq∗j θjd

+

p∑j=1

∂Lrm−2f hm

∂Xq∗j θju

= Lrm−1f hm +

p∑j=1

∂Lrm−2f hm

∂Xq∗j (θjd + θju) = ξmrm +

p∑j=1

∂Lrm−2f hm


(50)

ξmrm =∂φm

rm

∂X

dX

dt=

∂Lrm−1f hm

∂X

[f + g · u+

p∑j=1

q∗j θjd +∆f

]

=∂Lrm−1

f hm

∂Xf +

∂Lrm−1f hm

∂Xg1u1 + · · ·+

∂Lrm−1f hm

∂Xgmum +

p∑j=1

∂Lrm−1f hm

∂Xq∗j θjd

+

p∑j=1

∂Lrm−1f hm

∂Xq∗j θju


= Lrmf hm + Lg1L

rm−1f hmu1 + · · ·+ LgmL

rm−1f hmum +

p∑j=1

∂Lrm−1f hm


= cm + dm1u1 + · · ·+ dmmum +

p∑j=1

∂Lrm−1f hm


(51)

ηk(t) =∂φk

∂X

dX

dt=

∂φk

∂X

[f + g · u+

p∑j=1

q∗j θjd +∆f

]

=∂φk

∂Xf +

∂φk

∂Xg1u1 + · · ·+ ∂φk

∂Xgmum +

p∑j=1

∂φk

∂Xq∗j θjd +

p∑j=1

∂φk

∂Xq∗j θju

= Lfφk +

p∑j=1

∂φk


= qk +

p∑j=1

∂φk

∂Xq∗j (θjd + θju) , k = r + 1, r + 2, · · · , n

(52)

Since

ci(ξ(t), η(t)) ≡ Lrif hi(X(t)), 1 ≤ i ≤ m (53)

dij ≡ LgjLri−1f hi(X), 1 ≤ i ≤ m, 1 ≤ j ≤ m (54)

qk(ξ(t), η(t)) = Lfφk(X), k = r + 1, r + 2, · · · , n (55)

the dynamic equations of system (2.1) in the new co-ordinates are as follows:

ξ1i (t) = ξ1i+1(t) +

p∑j=1

∂

∂XLi−1f h1q

∗j (θjd + θju), i = 1, 2, · · · , r1 − 1 (56)

ξ1r1(t) = c1(ξ(t), η(t)) + d11(ξ(t), η(t))u1 + · · ·+ d1m(ξ(t), η(t))um

+

p∑j=1

∂

∂XLr1−1f h1q

∗j (θjd + θju)

(57)

...

ξmi (t) = ξmi+1(t) +

p∑j=1

∂

∂XLi−1f hmq

∗j (θjd + θju), i = 1, 2, · · · , rm − 1 (58)

ξmrm(t) = cm(ξ(t), η(t)) + dm1 (ξ(t), η(t))u1 + · · ·+ dmm(ξ(t), η(t))um

+

p∑j=1

∂

∂XLrm−1

f hmq∗j (θjd + θju)

(59)

ηk(t) = qk(ξ(t), η(t)) +

p∑j=1

∂

∂Xφk(X)q∗j (θjd + θju) , k = r + 1, · · · , n (60)

yi(t) = ξi1(t), 1 ≤ i ≤ m (61)

According to Equations (18), (44), (53) and (54), the tracking controller can be rewrit-ten as

ufeedback = A−1 [−b+ v] (62)


Substituting Equation (62) into (57) and (59), the dynamic equations of system (2.1)can be shown as follows:

ξi1(t)

ξi2(t)...

ξiri−1(t)

ξiri(t)

=

0 1 0 · · · 00 0 1 · · · 0

......

0 0 0 · · · 10 0 0 · · · 0

ξi1(t)ξi2(t)...

ξiri−1(t)ξiri(t)

+

00...01

vi

+

p∑j=1

∂∂X

hiq∗j (θjd + θju)

p∑j=1

∂∂X

L1fhiq

∗j (θjd + θju)

...p∑

j=1

∂∂X

Lri−1f hiq

∗j (θjd + θju)

(63)

ηr+1(t)ηr+2(t)

...ηn−1(t)ηn(t)

=

qr+1(t)qr+2(t)

...qn−1(t)qn(t)

+

p∑j=1

∂∂X

φr+1q∗j (θjd + θju)

p∑j=1

∂∂X

φr+2q∗j (θjd + θju)

...p∑

j=1

∂∂X

φn−1q∗j (θjd + θju)

p∑j=1

∂∂X

φnq∗j (θjd + θju)

(64)

yi = [1 0 · · · 0 0]r×1

ξi1(t)ξi2(t)...

ξiri−1(t)ξiri(t)

r×1

= ξi1(t), 1 ≤ i ≤ m (65)

Combining Equations (18), (20), (21), (26) and (44), it can be easily verified thatEquations (63)-(65) can be transformed into the following form.

η(t) = q(ξ(t), η(t)) + φη(θd + θu) ≡ q22(t, η(t), e) + φη(θd + θu) (66a)

ε.

ei(t) = Aice

i + φiξ (θd + θu) , 1 ≤ i ≤ m (66b)

yi(t) = ξi1(t), 1 ≤ i ≤ m (67)

We consider L (e, η) defined by a weighted sum of V (η) and W (e),

L (e, η) ≡ V (η) + k(ε)W (e) ≡ V (η) + k(ε)(W 1

(e1)+ · · ·+Wm (em)

)(68)

where

W (e) ≡ W 1(e1)+ · · ·+Wm (em) (69)

as a composite Lyapunov function of the subsystems (66a) and (66b) [26,29], whereW(ei)

satisfies

W i(ei)≡ 1

2ei

TP iei (70)


In view of Equations (18), (33) and (40), the derivative of L along the trajectories of(66a) and (66b) is given by

L =[∇tV + (∇ηV )T η

]+

k

2

[(e1)P 1e1 +

(e1)T

P 1(e1)

+ · · ·+(

˙em)Pmem + (em)

TPm

(˙em)]

=[∇tV + (∇ηV )T η

]+

k

2

[(1

εA1

ce1 +

1

εφ1ξ(θd + θu)

)T

P 1e1

+(e1)T

P 1

(1

εA1

ce1 +

1

εφ1ξ(θd + θu)

)+ · · ·+

(1

εAm

c em +

1

εφmξ (θd + θu)

)T

Pmem

+(em)TPm

(1

εAm

c em +

1

εφmξ (θd + θu)

)]=[∇tV + (∇ηV )T (q22 (t, η(t), e) + φη(θd + θu))

]+

{k

2ε

(e1)T [

(A1c)

TP 1 + P 1(A1c)]e1

+ · · ·+ k

2ε(em)

T [(Am

c )TPm + Pm(Am

c )]em

+k

ε

[(θd + θu)

T (φ1ξ)

TP 1e1 + · · ·+ (θd + θu)T (φm

ξ )TPmem

]}≤[∇tV + (∇ηV )T q22 (t, η(t), e) + (∇ηV )T φη(θd + θu)

]− k

2ε

[(e1)T

e1 + · · ·

+ (em)Tem]+

k

ε

[‖(θd + θu)‖

∥∥φ1ξ

∥∥∥∥P 1∥∥∥∥∥e1∥∥∥+ · · ·+ ‖(θd + θu)‖‖φm

ξ ‖‖Pm‖‖em‖]

≤[∇tV + (∇ηV )T q22 (t, η(t), 0)

]+ (∇ηV )T [q22 (t, η(t), e)− q22 (t, η(t), 0)]

+ ‖∇ηV ‖‖φη‖‖(θd + θu)‖ −k

ε

[W 1

λmax(P 1)+ · · ·+ Wm

λmax(Pm)

]+

4k2

ε2∥∥φ1

ξ

∥∥2 ∥∥P 1∥∥2 ∥∥∥e1∥∥∥2 + 1

16‖θd + θu‖2 + · · ·

+4k2

ε2‖φm

ξ ‖2‖Pm‖2‖em‖2 + 1

16‖(θd + θu)‖2

≤[∇tV + (∇ηV )T q22(t, η(t), 0)

]+ ‖∇ηV ‖‖q22 (t, η(t), e)− q22 (t, η(t), 0) ‖

+ ‖∇ηV ‖‖φη‖‖(θd + θu)‖ −k

ε

1

λ∗max

[W 1 + · · ·+Wm

]+

4k2

ε2∥∥φ1

ξ

∥∥2 ∥∥P 1∥∥2 ∥∥∥e1∥∥∥2

+1

16‖(θd + θu)‖2 + · · ·+ 4k2

ε2∥∥φm

ξ

∥∥2 ‖Pm‖2 ‖em‖2 + 1

16‖(θd + θu)‖2

≤ − 2αxV + ω3 ‖η‖M ‖e‖+ ω3‖η‖‖φη‖‖(θd + θu)‖

− k

ε

1

λ∗max

W +4k2

ε2‖φ1

ξ‖2∥∥P 1

∥∥2 ∥∥∥e1∥∥∥2+

1

16‖(θd + θu)‖2 + · · ·+ 4k2

ε2‖φm

ξ ‖2‖Pm‖2‖em‖2 + 1

16‖(θd + θu)‖2

≤ − 2αxV + ω31

√ω1

√VM

√2

λ∗min

√W + ω3

1√ω1

√V ‖φη‖‖ (θd + θu) ‖

− k

ε

1

λ∗max

W +4k2

ε2∥∥φ1

ξ

∥∥2 ∥∥P 1∥∥2 W 1

1/2λmin(P 1)


+ · · ·+ 4k2

ε2‖φm

ξ ‖2‖Pm‖2 Wm

1/2λmin(Pm)+

m

16‖(θd + θu)‖2

≤ − 2αx

(√V)2

+ 2ω3M√2kλ∗

minω1

√V√kW + 4

(ω3√ω1

‖φη‖)2 (√

V)2

+1

16‖(θd + θu)‖2 −

1

ε

1

λ∗max

(√kM

)2+

4k2

ε2‖φ1

ξ‖2‖P 1‖2 W 1

1/2λmin(P 1)

+ · · ·+ 4k2

ε2‖φm

ξ ‖2‖Pm‖2 Wm

1/2λmin(Pm)+

m

16‖(θd + θu)‖2

≤ −(2αx −

4ω23

ω1

‖φη‖2)(√

V)2

+ 2

(ω3M√2ω1kλ∗

min

)√V√kW

−(

1

ελ∗max

−4k‖φ1

ξ‖2‖P 1‖212ε2λmin(P 1)

− · · · −4k‖φm

ξ ‖2‖Pm‖212ε2λmin(Pm)

)(√kW

)2+

m+ 1

16‖(θd + θu)‖2

= −[√

V√kW

]H

[ √V√kW

]+

m+ 1

16‖(θd + θu)‖2

i.e.,

L ≤ −λmin(H)L+m+ 1

16‖(θd + θu)‖2 (72)

where λmin(H) denotes the minimum eigenvalue of the matrix H. Utilizing the fact thatλmin(H) = 2αs, we obtain

L ≤ − 2αsL+m+ 1

16‖(θd + θu)‖2 ≤ −2αs (V + kW ) +

m+ 1

16‖(θd + θu)‖2

≤ − 2αs

(ω1 ‖η‖2 +

k

2λ∗min ‖e‖

2

)+

m+ 1

16‖(θd + θu)‖2

≤ −NN2

(‖η‖2 + ‖e‖2

)+

m+ 1

16‖(θd + θu)‖2

(73)

Define

e ≡

e1

e2...em

≡[

e11e1rem

], e1rem ∈ <r−1 (74)

Hence,

L ≤ −NN2

(‖η‖2 +

∥∥∥e11∥∥∥2 + ∥∥∥e1rem∥∥∥2)+m+ 1

16‖(θd + θu)‖2 (75)

Utilizing Equation (75) easily yields

t∫t0

(y1(τ)− y1

d(τ))2dτ ≤ L (t0)

NN2

+m+ 1

16NN2

t∫t0

‖(θd(τ) + θu(τ))‖2dτ (76)

Similarly, it is easy to prove that

t∫t0

(yi(τ)− yid(τ)

)2dτ ≤ L (t0)

NN2

+m+ 1

16NN2

t∫t0

‖(θd(τ) + θu(τ))‖2dτ, 2 ≤ i ≤ m (77)


so that Equation (39) is satisfied. From Equation (73), we get

L ≤ −NN2

(‖ytotal‖2

)+

m+ 1

16‖(θd + θu)‖2 (78a)

where

‖ytotal‖2 ≡ ‖e‖2 + ‖η‖2 . (78b)

By virtue of Theorem 5.2 [25], Equation (78a) implies the input-to-state stability forthe closed-loop system. Furthermore, it is easy to see that

∆min

(‖e‖2 + ‖η‖2

)≤ L ≤ ∆max

(‖e‖2 + ‖η‖2

)(79)

that is

∆min

(‖ytotal‖2

)≤ L ≤ ∆max

(‖ytotal‖2

)(80)

where ∆min ≡ min{ω1,

k2λ∗min

}and ∆max ≡ max

{ω2,

k2λ∗max

}. From Equation (73) and

Equation (80) yield that

L ≤ −NN2

∆max

L+m+ 1

16

(sup

t0≤τ≤t‖(θd(τ) + θu(τ))‖

)2

(81)

Hence,

L(t) ≤ L(t0)e− NN2

∆max(t−t0) +

∆max(m+ 1)

16NN2

(sup

t0≤τ≤t‖(θd(τ) + θu(τ))‖

)2

, t ≥ t0 (82)

which implies∣∣y1(t)− y1d(t)∣∣ ≤√2L(t0)

kλ∗min

e−NN2

2∆max(t−t0) +

√∆max(m+ 1)

8kλ∗minNN2

(sup

t0≤τ≤t‖(θd(τ) + θu(τ))‖

)(83)

Similarly, it is easy to prove that∣∣yi(t)− yid(t)∣∣ ≤√2L(t0)

kλ∗min

e−NN2

2∆max(t−t0)

+

√∆max(m+ 1)

8kλ∗minNN2

(sup

t0≤τ≤t‖(θd(τ) + θu(τ))‖

), 2 ≤ i ≤ m

(84)

So that Equation (38) is proved and then the tracking problem with almost disturbancedecoupling is globally solved. Finally, we will prove that the sphere Br is a global attractorfor the output tracking error of system (2.1). From Equation (78a) and Equation (45d),we get

L ≤ −NN2

(‖ytotal‖2

)+N1 (85)

For ‖ytotal‖ > r, we have L < 0. Hence, any closed ball defined by

Br ≡{[

eη

]: ‖e‖2 + ‖η‖2 ≤ r

}(86)

is a global final attractor for the tracking error system of the non-linear control systems(2.1). Furthermore, it is easy routine to see that, for y /∈ Br, we have

L

L≤ −NN2 ‖ytotal‖2 +N1

L≤ −NN2 ‖ytotal‖2 +N1

∆max ‖ytotal‖2≤ −NN2

∆max

+N1

∆max ‖ytotal‖2

≤ −NN2

∆max

+N1

∆maxr2≡ −α∗

(87)


i.e.,L ≤ −α∗L

According to the comparison theorem [32], we get

L(t) ≤ L(t0) exp [−α∗(t− t0)]

Therefore,

∆min ‖ytotal‖2 ≤ L(ytotal(t)) ≤ L(ytotal(t0)) exp [−α∗(t− t0)]

≤ ∆max ‖ytotal(t0)‖2 exp [−α∗(t− t0)](88)

Consequently, we get

‖ytotal‖ ≤√

∆max

∆min

‖ytotal(t0)‖ exp[−1

2α∗(t− t0)

]i.e., the convergence rate toward the sphere Br is equal to α

∗/2. This completes our proof.B. Fuzzy controller design.

After the feedback linearization control is utilized as a guarantee of uniform ultimatebounded stability, the multiple input/single output fuzzy control design can be technicallyapplied via human expert’s knowledge to improve the convergence rate of tracking error.The block diagram of the fuzzy control is shown in Figure 1.

Figure 1. Fuzzy logic controller

In general, the tracking error e(t) and its time derivative e(t) are utilized as the inputfuzzy variables of the IF-THEN control rules and the output is the control variable ufuzzy.For the sake of easy computation, the membership functions of the linguistic terms fore(t), e(t) and ufuzzy are all chosen to be the triangular shape function. We define sevenlinguistic terms: PB (Positive big), PM (Positive medium), PS (Positive small), ZE (Zero),NS (Negative small), NM (Negative medium) and NB (Negative big), for each fuzzyvariable, as shown in Figure 2.

Fuzzy control rule table for ufuzzy is shown in Table 1. The rule base is heuristicallybuilt by the standard Macvicar-Whelan rule base [50] for usual servo control systems.The Mamdani method is used for fuzzy inference. The defuzzification of the outputset membership value is obtained by the centroid method. Therefore, we can combinethe designs of feedback linearization control and fuzzy control to construct the overallcontroller as follows:

ufe+fu ≡ {−b+ v}us(t) +

ufuzzy1us(t− t1)ufuzzy2us(t− t2)

...ufuzzymus(t− tm)

, 1 ≤ i ≤ m (89)


(a) e(t) (b) e(t) (c) ufuzzy

Figure 2. Membership functions for (a) e(t), (b) e(t) and (c) ufuzzy

where us(t) denotes the unit step function and ti, 1 ≤ i ≤ m are the time variables thatthe tracking error of states touch the global final attractor Br.

Table 1. Fuzzy control rule base

ufuzzye(t)

NB NM NS ZE PS PM PB

e(t)

NB PB PB PB PB PM PS ZENM PB PB PB PM PS ZE NSNS PB PB PM PS ZE NS NMZE PB PM PS ZE NS NM NBPS PM PS ZE NS NM NB NBPM PS ZE NS NM NB NB NBPB ZE NS NM NB NB NB NB

3. Marine Vehicle in 6 Degrees of Freedom. When investigating the behavior ofmarine vehicle in 6 degree of freedom, it is convenient to define body-fixed coordinateframe and earth-fixed coordinate frame as indicated in Figure 3.

Figure 3. Body-fixed and earth-fixed reference frames for marine vehiclein 6 degrees of freedom


In general, the behavior of marine vehicle in 6 degree of freedom can be described

by the following vectors: η ≡[ηT1 ηT2

]T, η1 ≡ [x y z]T , η2 ≡ [φ θ ϕ]T , υ ≡[

υT1 υT

2

]T, υ1 ≡ [u v w]T , υ2 ≡ [p q r]T , τ ≡

[τT1 τT2

]T, τ1 ≡ [τx τy τz]

T

and τ2 ≡ [τK τM τN ]T . Here η1 and η2 denote the position vector and orientation vec-

tor with coordinate in the earth-fixed frame, respectively, υ1 and υ2 are used to describethe linear velocity vector and angular velocity vector with coordinates in the body-fixedframe, respectively, τ1 and τ2 denote the force vector and moment vector acting on themarine vehicle in the body-fixed frame, respectively. Based on the derivation of [11],the kinematic equations relating the body-fixed reference frame to the earth-fixed refer-ence frame and the 6 degree of freedom non-linear dynamic equations of motion can beexpressed as [

˙η1˙η2

]=

[J1(η2) 03×3

03×3 J2(η2)

] [υ1υ2

], (90)

˙υ =[−M−1C(υ)

]υ +

[−M−1D(υ)

]υ +

[−M−1g(η)

]+[M−1τ

], (91)

J1(η2) ≡

cosϕ cos θ − sinϕ cosφ+ cosϕ sin θ sinφ sinϕ sinφ+ cosϕ cosφ sin θsinϕ cos θ cosϕ cosφ+ sinφ sin θ sinϕ − cosϕ sinφ+ sin θ sinϕ cosφ− sin θ cos θ sinφ cos θ cosφ

,

(92)

J2(η2) ≡

1 sinφ tan θ cosφ tan θ0 cosφ − sinφ0 sinφ/cos θ cosφ/cos θ

, (93)

where M and C(υ) denote the inertia matrix including added mass and the matrix of Cori-olis and centripetal terms including added mass, respectively, D(υ) denotes the dampingmatrix, τ denotes the vector of control inputs and g(η) is used to describe the vectorof gravitational forces and moments. A dynamically positioned marine vehicle can besupplied with anchors if thruster-assisted mooring is to be investigated. The mooringforces are described as spring forces, g(η) = K (η − η0), where η0 is the equilibrium pointof the mooring system. For the sake of simplicity it is assumed that η0 = 0. Duringstation-keeping and low-speed application like dynamic positioning of marine vehicles,the linear velocities u, v and the angular velocity r are all small which contributes thata further simplification can be to neglect the term −M−1C(υ)υ. Hence the equations of

motion in surge, sway and yaw can be written as follows according to η ≡ [x y ϕ]T ,

υ ≡ [u v r]T , τ ≡ [τx τy τN ]T :

˙η = J(η)υ, (94)

υ =[−M−1D(υ)

]υ +

[−M−1K

]η +

[M−1

]τ, (95)

where

J(η) ≡

cosϕ − sinϕ 0sinϕ cosϕ 00 0 1

, (96)

is the rotation matrix in yaw, M is the inertia matrix including hydrodynamic addedinertia, D is the damping matrix and τ are the control forces and moment providedby the thruster system. Starboard-port symmetry of marine vehicles implies that thematrices M and D are constructed as the following structure:

M ≡

m11 0 00 m22 m23

0 m32 m33

, (97)


D ≡

d11 0 00 d22 d230 d32 d33

, (98)

that is, there is no coupling item between the surge and sway-yaw subsystems. Hence,the anchor forces and moment are related by the diagonal matrix.

K ≡

k11 0 00 k22 00 0 k33

. (99)

We consider a tanker with the numerical system matrices as follows: (Bis-scaled values[11])

M ≡

1.0852 0 00 2.0575 −0.40870 −0.4087 0.2153

, (100)

D ≡

0.0865 0 00 0.0762 0.15100 0.0151 0.0031

, (101)

K ≡

0.0389 0 00 0.0266 00 0 0

. (102)

After arranging Equations (94)-(96) in the state-space formulation, it is easy to obtainthe following state equations:

x1 = x4 (cosx3)− x5 (sinx3) , (103)

x2 = x4 (sinx3)− x5 (cosx3) , (104)

x3 = x6, (105)

x4 = −0.0358x1 − 0.0797x4 + 0.9215u1, (106)

x5 = −0.0208x2 − 0.0818x5 − 0.1224x6 + 0.7802u2 + 1.4811u3, (107)

x6 = −0.0394x2 − 0.2254x5 − 0.2468x6 + 1.4811u2 + 7.4562u3, (108)

whereX ≡

[x1 x2 x3 x4 x5 x6

]T ≡[x y ϕ u v r

]T, (109)

u ≡[u1 u2 u3

]T ≡[τx τy τz

]T. (110)

It is assumed that only the position and yaw angle measurements are available, that is,the output equations are given by

y1 = x1 (111)

y2 = x2 (112)

y3 = x3 (113)

Now we will show how to explicitly construct a controller that tracks the desired signalsy1d = y2d = y3d = 0 and attenuates the disturbance’s effect on the output terminal to anarbitrary degree of accuracy. Let’s arbitrarily choose α1

1 = α12 = 100, α2

1 = α22 = 100,

α31 = α3

2 = 10, A1c = A2

c = A3c =

[0 1100 100

]. With the aid of Matlab, the solutions of

Lyapunov equations are given by

P 1 = P 2 = P 3 =

[1.005 0.0050.005 0.005

](114)


Hence, λ∗min = 0.05 and λ∗

max = 1.005. From Equation (89), we obtain the desired trackingcontrollers

u = A−1(−

⇀b +

⇀v)us(t) +

ufuzzyus(t− t1)ufuzzyus(t− t2)ufuzzyus(t− t3)

(115)

A ≡

a11 a12 a13a21 a22 a23a31 a32 a33

(116)

u ≡[u1 u2 u3

]T, b ≡

[b1 b2 b3

]T, v ≡

[v1 v2 v3

]T(117)

wherea11 = 0.9215 cos x3 (118)

a12 = −0.7802 sinx3 (119)

a13 = −1.4811 sinx3 (120)

a21 = 0.9215 sin x3 (121)

a22 = −0.7802 cosx3 (122)

a23 = −1.4811 cosx3 (123)

a31 = 0 (124)

a32 = 1.4811 (125)

a33 = 7.4562 (126)

b1 = (x6) (−x4 sin x3 − x5 cos x3) + (−0.0358x1 − 0.0797x4) (cos x3)

+ (−0.0208x2 − 0.0818x5 − 0.1224x6) (− sin x3)(127)

b2 = (x6) (x4 cosx3 + x5 sin x3) + (−0.0358x1 − 0.0797x4) (sin x3)

+ (−0.0208x2 − 0.0818x5 − 0.1224x6) (− cosx3)(128)

b3 = (−0.0394x2 − 0.2254x5 − 0.2468x6)× 1 (129)

v1 = −100 (ε)−2 x1 − 100 (ε)−1 (x4 (cosx3)− x5 (sinx3)) (130)

v2 = −100 (ε)−2 x2 − 100 (ε)−1 (x4 (sinx3)− x5 (cosx3)) (131)

v3 = −100 (ε)−2 x3 − 100 (ε)−1 (x6) (132)

It can be verified that the relative conditions of Theorem 2.1 are satisfied with ε = 0.1,N = 9.95, N2 = 0.79 and k = 100

√ε. Hence, the tracking controllers will steer the

output tracking errors of the closed-loop system, starting from any initial value, to beasymptotically attenuated to zero by virtue of Theorem 2.1.

In the simulations, the ship is first controlled by the dynamic positioning control law(115) using three control inputs and initially steered to the point[

x1(0) x2(0) x3(0) x4(0) x5(0) x6(0)]T ≡

[−10 −10 0 0 0 0

]TThe desired equilibrium point is[

x1(0) x2(0) x3(0) x4(0) x5(0) x6(0)]T ≡

[0 0 0 0 0 0

]TThe simulated results are shown in Figures 4-9. We see that the ship eventually con-

verges to a neighborhood of the desired equilibrium point. The dynamic positioning timebased on our proposed novel non-linear fuzzy feedback linearization control is approxi-mately equal to be 0.6. On the contrary, the dynamic positioning time for the same shipcontrol system using the traditional fuzzy-model approach [5] is approximately equal tobe 10. The simulation results show that our proposed control in this study is a marvelousapproach for the control problem of non-linear dynamic positioning problem of ship.


Figure 4. Measured x-position x1

Figure 5. Measured y-position x2

4. Comparative Example to Existing Approach. [30] exploited the fact that thealmost disturbance decoupling problem could not be solvable for the following system:[

x1(t)x2(t)

]=

[tan−1 (x2)

0

]+

[01

]u+

[10

]θ(t) (133)

y(t) = x1(t) := h(X(t)) (134)

where u, y denoted the input and output respectively, θ(t) := 2 sin t [us(t− 1.1)− us(t−1.31)] and us(t) denotes the unit step function. Assume that the desired trackingsignal is equal to be sin t. The almost disturbance decoupling problem can be easilysolved via the proposed approach in this paper. Following the same procedures shown in


Figure 6. Measured yaw angle x3

Figure 7. Surge velocity x4

the demonstrated example, we can solve the tracking problem with almost disturbancedecoupling problem by the controller u defined as

u =(1 + x2

2

)[− sin t− 42(x1 − sin t)− 42(tan−1 x2 − cos t)]us(t) + ufuzzyus(t− t1) (135)

The tracking error dynamics for (133) is depicted in Figure 10.It is worth noting that the sufficient conditions given in [30] (in particular the structural

conditions on non-linearities multiplying disturbances) are not necessary in this studywhere a non-linear state feedback control is explicitly designed which solves the almostdisturbance decoupling problem. For instance, the almost disturbance decoupling problemis solvable for the system (133) by our proposed approach, while the sufficient conditions


Figure 8. Sway velocity x5

Figure 9. Yaw velocity x6

given in [30] fail when applied to the system (133). The design techniques in this studyare also entirely different than those in [30] since the singular perturbation tools are notused.

5. Conclusions. A novel fuzzy feedback control to globally solve the tracking problemwith almost disturbance decoupling for dynamic positioning marine vehicle has been pro-posed. A discussion and a practical application of feedback linearization of non-linearcontrol systems using a parameterized coordinate transformation have been presented.One comparative example is proposed to show the significant contribution of this pa-per with respect to existing approach. A practical simulation example has been used to


Figure 10. The tracking error for (133)

demonstrate the applicability of the proposed fuzzy feedback linearization approach andthe composite Lyapunov approach. Simulation results have been presented to show thatthe proposed methodology can be successfully applied to feedback linearization problemand is able to achieve the desired tracking and almost disturbance decoupling perfor-mances of the controlled system. Moreover, it is easy to see that the dynamic positioningtime based on our proposed novel non-linear fuzzy feedback linearization control is shorterthan the traditional fuzzy-model approach.

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