DYNAMIC POSITIONING CONTROL OF MARINE VEHICLE. Ting-Li Chien 1, Chia-Wei Lin 2, Chiou-Jye Huang 3 and Chung-Cheng Chen 4,

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1 International Journal o Innovative Computing, Inormation and Control ICIC International c 202 ISSN Volume 8, Number 5A), May 202 pp DYNAMIC POSITIONING CONTROL OF MARINE VEHICLE Ting-Li Chien, Chia-Wei Lin 2, Chiou-Jye Huang 3 and Chung-Cheng Chen 4, Department o Electronic Engineering Wueng Institute o Technology No 7, Sec 2, Chian-Kuo Rd, Ming-Hsiung, Chiayi 62, Taiwan 2 Institute o Computer Science and Inormation Engineering National Chiayi University No 300, Syueu Rd, Chiayi 60004, Taiwan 3 Green Energy and Environment Research Laboratories Industrial Technology Research Institute 95, Sec 4, Chung Hsing Rd, Chutung, Hsinchu 3040, Taiwan 4 Department o Electrical Engineering Hwa Hsia Institute o Technology No, Gong Jhuan Rd, Chung Ho, Taipei 235, Taiwan Corresponding author: ccc49827@ms25hinetnet Received January 20; revised May 20 Abstract The paper deals with a novel uzzy eedback linearization control o dynamic positioning marine vehicle with the almost disturbance decoupling perormance The main contribution o this study is to construct a controller such that the resulting dynamic positioning marine vehicle is valid or any initial and desired positions with the ollowing characteristics: input-to-state stability with respect to disturbance inputs and almost disturbance decoupling The dynamic positioning time based on our proposed novel nonlinear uzzy eedback linearization control is better than some existing approaches The proposed controller is constructed by eedback linearization controller and uzzy controller The eedback linearization controller guarantees the almost disturbance decoupling perormance and the uniorm ultimate bounded stability o the tracking error system Once the tracking errors are driven to touch the global inal attractor with the desired radius, the uzzy logic controller immediately is applied via a human expert s knowledge to improve the convergence rate Finally, the availability and easibility o the proposed approach are investigated with a numerical simulation Keywords: Marine vehicle, Dynamic positioning systems, Almost disturbance decoupling, Feedback linearization approach, Fuzzy logic control Introduction Oshore oilield development has orwarded to a deeper and severe environment or new oil sources and maintaining the position o an oshore platorm becomes a very challenging problem Thereore, dynamic position o marine vehicle using thrusters is oten used in many oshore oilield operations, such as drilling, pipe-laying, tanking between marine vehicles and diving support A dynamic positioning marine vehicle exhibits non-linear interaction in three degrees o reedom sway, surge and yaw) by means o main propellers at on the marine vehicle 8 Dynamic positioning marine vehicle has usually been designed by assuming that the kinematic equations o motions can be linearized about predeined constant yaw angles, meaning that linear optimal control approach and gain-scheduling techniques can be applied 4,36,40 It is obvious to 336

2 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 order to solve the linearization problem, 6,4,49 apply the Takagi-Sugenno uzzy model, which includes a set o IF-THEN rules, to investigating the non-linear dynamic positioning marine vehicle based on the parallel distributed compensation PDC) Its designing procedure is as ollows First, representing the non-linear system as the amous Takagi- Sugeno uzzy model oers an alternative to the conventional model The control design is carried out based on an aggregation o linear controllers constructed or each local linear element o the uzzy model via the parallel distributed compensation scheme 46 For the stability analysis o uzzy system, a lot o studies are reported See, eg, 27,4-43 and the reerences therein) The stability and controller design o uzzy system can be mainly discussed by Tanaka-Sugeno s theorem 42 However, it is diicult to ind the common positive deinite matrix P or linear matrix inequality LMI) problem even i P is a second order matrix 24 Moreover, the PDC will be subject to ailure i the uzzy model is decomposed by too many plant rules 0,45 This is because there are many linear-matrix-inequality constraints involved in optimization process and many decision variables that need to be carried out simultaneously Many approaches to stabilizing and tracking tasks have been proposed including eedback linearization, variable structure control sliding mode control), backstepping, regulation control, non-linear H control, internal model principle and H adaptive uzzy control 25 has proposed the use o variable structure control to deal with non-linear system However, chattering behaviour caused by discontinuous switching and imperect implementation that can drive the system into unstable regions is inevitable or variable structure control schemes Backstepping has proven to be a powerul tool or synthesizing controllers or non-linear systems 7,52 However, a disadvantage o this approach is an explosion in the complexity which is a result o repeated dierentiations o non-linear unctions 2,39,5 An alternative approach is to utilize output regulation control 2 in which the outputs are assumed to be excited by an exosystem However, the nonlinear regulation approach requires the solution o diicult partial-dierential algebraic equations Another diiculty is that the exosystem states need to be switched to describe changes in the output and this creates transient tracking errors 34 In general, non-linear H control requires the solution o the Hamilton-Jacobi equation, which is a diicult nonlinear partial-dierential equation 2,22,44 Only or some particular non-linear systems it is possible to derive a closed-orm solution 20 The control approach that is based on the internal model principle converts the tracking problem into a non-linear output regulation problem This approach depends on solving a irst-order partial-dierential equation o the center maniold 2 For some special non-linear systems and desired trajectories, the asymptotic solutions o this equation have been developed using ordinary dierential equations 5,8 Recently, H adaptive uzzy control has been proposed to systematically deal with non-linear systems 7 The drawback with H adaptive uzzy control is that the complex parameter update law makes this approach impractical in real-world situations During the past decade signiicant progress has been made in researching control approaches or non-linear systems based on the eedback linearization theory 9,25,33,38 Moreover, eedback linearization approach has been applied successully to many real control systems These include the control o an electromagnetic suspension system 23, pendulum system 9, spacecrat 37, electrohydraulic servosystem, car-pole system 3 and bank-to-turn missile system 28 The main contribution o this study is to solve the linearized and PDC shortcomings by using non-linear eedback linearization approach The almost disturbance decoupling problem, ie, the design o a controller that attenuates the eect o the disturbance on the output terminal to an arbitrary degree o accuracy,

3 DYNAMIC POSITIONING CONTROL OF MARINE VEHICLE 3363 was originally developed or linear and non-linear control systems by 30,48 respectively The problem has attracted considerable attention and many signiicant results have been developed or both linear and non-linear control systems 3,3,35,47 The almost disturbance decoupling problem o non-linear single-input single-output SISO) systems was investigated in 30 by using a state eedback approach and solved in terms o suicient conditions or systems with non-linearities that are not globally Lipschitz and disturbances bring linear but possibly actually bring multiples o non-linearities The resulting state eedback control is constructed ollowing a singular perturbation approach The suicient conditions in 30 require that the non-linearities multiplying the disturbances satisy structural triangular conditions 30 shows that or non-linear SISO systems the almost disturbance decoupling problem may not be solvable, as is case or ẋ t) = tan x 2 θt), θt) > π 2 ẋ 2 t) = u y = x where u, y denoted the input and output respectively and θt) was the disturbance o the system On the contrary, this example can be easily solved via the proposed approach in this paper To overcome the problems or linearized technique and uzzy-model approach, we will propose a new method to guarantee that the marine vehicle is stable and the almost disturbance decoupling perormance is achieved The designing structure is as ollows Firstly, based on the eedback linearization approach a tracking control is proposed in order to guarantee the almost disturbance decoupling property and the uniorm ultimate bounded stability o the tracking error system Once the tracking errors are driven to touch the global inal attractor, the conventional uzzy logic control is immediately applied via a human expert s knowledge to improve the convergence rate In order to exploit the signiicant applicability, this paper has also successully derived controller with almost disturbance decoupling or a marine vehicle According to the simulation results, it is easy to see that the dynamic positioning time based on our proposed novel non-linear uzzy eedback linearization control is shorter than the traditional uzzy-model approach 5 2 Feedback Linearization Controller Design The ollowing non-linear uncertain control system with disturbances is considered ẋ x, x 2,, x n ) ẋ 2 = 2 x, x 2,, x n ) ẋ n n x, x 2,, x n ) u x, x 2,, x n ) u 2 x, x 2,, x n ) g x, x 2,, x n ) g 2 x, x 2,, x n ) g m x, x 2,, x n ) qj θ jd x, x 2,, x n ) 2 x, x 2,, x n ) n x, x 2,, x n ) u m x, x 2,, x n ) a)

4 3364 T-L CHIEN, C-W LIN, C-J HUANG AND C-C CHEN that is y x, x 2,, x n ) y 2 x, x 2,, x n ) y m x, x 2,, x n ) = Ẋt) = Xt)) gxt))u yt) = hxt)) h x, x 2,, x n ) h 2 x, x 2,, x n ) x, x 2,, x n ) qj θ jd b) where Xt) x t) x 2 t) x n t) T R n is the state vector, u u u 2 u m T R m is the input vector, y y y 2 y m T R m is the output vector, θ d θ d t) θ 2d t) θ pd t) T is a bounded time-varying disturbances vector and 2 n R n is unknown non-linear unction representing uncertainty such as modelling error Let be described as = p q j θ uj, where θ u θ u X, t) θ u2 X, t) θ up X, t) T bounded time-varying vector and qj R n, j =, 2,, p, are coeicient vectors o disturbances The vectors R n and h R m are vectors o smooth real-valued unctions, 2,, n and h, h 2,,, respectively The matrix g R n m is a a matrix with columns being smooth vector ields g, g 2,, g m The nominal system is then deined as ollows: Ẋt) = Xt)) gxt))u 2a) yt) = hxt)) The nominal system o the orm 2) is assumed to have the vector relative degree {r, r 2,, r m } 6,9, ie, the ollowing conditions are satisied or all X R n : L gj L k h i X) = 0 3) or all i m, j m, k < r i, where the operator L is the Lie derivative 9 and r r 2 r m = r 2 The m m matrix L g L r h X) L gm L r h X) L g L r 2 h 2 X) L gm L r 2 h 2 X) A 4) L g L r m X) L gm L r m X) is non-singular The desired output trajectory yd i, i m and its irst r i derivatives are all uniormly bounded and yd, i yd i),, y ir i ) d B i d, i m 5) where Bd i is some positive constant The objective o the paper is to propose a control that includes a eedback linearization controller and a uzzy logic controller to achieve the almost disturbance decoupling and tracking perormances A Feedback linearization controller design Under the assumption o well-deined vector relative degree, it has been shown 9 that the mapping φ : R n R n 6) is a 2b)

5 deined as and satisying DYNAMIC POSITIONING CONTROL OF MARINE VEHICLE 3365 ξ i ξ i ξ i 2 ξ i r i φ i φ i 2 φ i r i L 0 h ix) L h ix) L r i h i X), i =, 2,, m 7) φ k Xt)) η k t), k = r, r 2,, n 8) L gj φ k Xt)) = 0, k = r, r 2,, n, j m 9) is a dieomorphism onto image, i the ollowing hold The distribution G span{g, g 2,, g m } 0) is involutive 2 The vector ields Y k j, j m, k r j ) are complete, where Y k j ) k ad k g j, j m, k r j 2) X) X) gx)a X)bX) 3) L r h X) L r 2 bx) h 2X) 4) X) L rm g g g 2 g m gx)a X) 5) ad k g, ad k g 6), g g X) gx) 7) For the sake o convenience, deine the trajectory error to be e i j ξ i j y ij ) d, i =, 2,, m, j =, 2,, r i 8) e i e i e i 2 e i r i T R r i 9) and the trajectory error to be multiplied with some adjustable positive constant ε and e i j εj e i j, i =, 2,, m, j =, 2,, r i 20) T e i e i e i 2 e i r i t) R r i 2) e e 2 e Rr 22) e m ξ ξ ξ 2 ξ r T R r, 23)

6 3366 T-L CHIEN, C-W LIN, C-J HUANG AND C-C CHEN ηt) η r t) η r2 t) η n t) T R n r 24) qξt), ηt)) L φ r t) L φ r2 t) L φ n t) T q r q r2 q n T 25) Deine a phase-variable canonical matrix A i c to be A i c α i α2 i α3 i αr i i r i r i, i m 26) where α, i α2, i, αr i i are any chosen parameters such that A i c is Hurwitz and the vector B i to be B i T r i, i m 27) Let P i be the positive deinite solution o the ollowing Lyapunov equation A i c) T P i P i A i c = I, i m 28) λ max P i ) the maximum eigenvalue o P i, i m 29) λ min P i ) the minimum eigenvalue o P i, i m 30) λ max min { λ max P ), λ max P 2 ),, λ max P m ) } 3) λ min min { λ min P ), λ min P 2 ),, λ min P m ) } 32) Assumption For all t 0, η R n r and ξ R r, there exists a positive constant M such that the ollowing inequality holds q 22 ξ, η, e) q 22 t, η, 0) M e ) 33) where q 22 t, η, ē) qξ, η) For the sake o stating precisely the investigated problem, deine and d ij L gj L r i h i X), i m, j m 34) c i L r i h ix), i m 35) e i α i e i α i 2e i 2 α i r i e i r i, i m 36) Deinition 2 25 Consider the system ẋ = t, x, θ), where : 0, ) R n R n R n is piecewise continuous in t and locally Lipschitz in x and θ This system is said to be input-to-state stable i there exists a class KL unction β, a class K unction γ and positive constants k and k 2 such that or any initial state x t 0 ) with x t 0 ) < k and any bounded input θt) with sup θ t) < k 2, the state exists and satisies t t 0 ) xt) β xt 0 ), t t 0 ) γ sup θτ) 37) t 0 τ t or all t t 0 0 Now we ormulate the tracking problem with almost disturbance decoupling as ollows: Deinition 22 3 The tracking problem with almost disturbance decoupling is said to be globally solvable by the state eedback controller u or the transormed-error system by a global dieomorphism 6), i the controller u enjoys the ollowing properties It is input-to-state stable with respect to disturbance inputs

7 DYNAMIC POSITIONING CONTROL OF MARINE VEHICLE For any initial value x e0 ēt 0 ) ηt 0 ) T, or any t t 0 and or any t 0 0 yt) y d t) β xt 0 ), t t 0 ) ) β 33 sup θτ) β22 and t t 0 yτ) y d τ) 2 dτ β 44 β 55 x e0 ) t t 0 β 33 t 0 τ t 38) θ τ) 2 ) dτ 39) where β 22, β 44 are some positive constants, β 33, β 55 are class K unctions and β is a class KL unction Theorem 2 Suppose that there exists a continuously dierentiable unction V 0 : R n r R such that the ollowing three inequalities hold or all η R n r : ω η 2 V η) ω 2 η 2, ω, ω 2 > 0 40a) t V η V ) T q 22 t, η, 0) 2α x V η), α x > 0 40b) η V ϖ 3 η, ϖ 3 > 0, 40c) then the tracking problem with almost disturbance decoupling is globally solvable by the controller deined by u eedback = A { b v} 4) b L r h L r 2 h 2 L r m T 42) v v v 2 v m T 43) v i y ir i ) d ε r i α i L 0 h i X) yd i ε r i α2 L i h i X) yd i) ε αr i i L r i h i X) y ir i ) d, i m 44) Moreover, the inluence o disturbances on the L 2 norm o the tracking error can be arbitrarily attenuated by increasing the ollowing adjustable parameter NN 2 > : H H Hε) 2 H 2 H 22 2α x 4ω2 3 φ η 2 w 3 M ω kε) 2w λ min w 3 M 8kε) φ ξ 2 P 2 8kε) φm ξ 2 P m 2 kε) 2w λ ελ min max ε 2 λ min P ) ε 2 λ min P m ) 45a) φ i ξε) α s ε) H H 22 H H 22 ) 2 4H2 2 /2 45b) 4 N 2α s ε) 45c) N m 2 sup θ d τ) θ u τ) ) 45d) 4 t 0 τ t { N 2 min ω, kε) } 2 λ min 45e) ε h iq ε h iqp, i m 45) ε r i h i q ε r i h i qq Lr i Lr i

8 3368 T-L CHIEN, C-W LIN, C-J HUANG AND C-C CHEN φ η ε) φ rq φ rqp φ nq φ nqq 45g) where H is positive deinite matrix and kε): R R is any continuous unction satisies ε lim kε) = 0 and lim ε 0 ε 0 kε) = 0 45h) Moreover, the output tracking error o system 2) is exponentially attracted into a N sphere B r, r = NN 2, with an exponential rate o convergence where NN2 N ) 2 max max r 2 2 α { max = max ω 2, k } 2 λ max 45i) 45j) Proo: Applying the coordinate transormation 6) yields ξ = φ dx dt = h g u qj θ jd = h h g u h g mu m = h h q j θ jd θ ju ) = ξ 2 h q j θ jd h q j θ jd θ ju ) h q j θ ju 46) ξ r = φ r h = Lr 2 dx dt = Lr 2 h Lr 2 h L r 2 h q j θ jd = Lr 2 h = L r h g u qj θ jd h g u Lr 2 L r 2 h q j θ ju L r 2 h q j θ jd θ ju ) L r 2 h q j θ jd θ ju ) g mu m 47)

9 ξ r = φ r = Lr DYNAMIC POSITIONING CONTROL OF MARINE VEHICLE 3369 dx dt = Lr h h Lr L r h h q j θ ju g u qj θ jd g u Lr h g mu m = L r h L g L r h u L gm L r h u m = c d u d m u m L r h q j θ jd θ ju ) L r h q j θ jd L r h q j θ jd θ ju ) 48) ξ m = φm dx dt = g u qj θ jd = g u g mu m = L q j θ jd θ ju ) = ξ m 2 q j θ jd q j θ jd θ ju ) q j θ ju 49) ξ m r m = φm r m = Lr m 2 dx dt = Lrm 2 Lrm 2 L rm 2 = L r m ξ m r m = φm r m = Lrm q j θ ju dx dt = Lrm g u qj θ jd g u Lrm 2 g mu m L r m 2 q j θ jd θ ju ) = ξr m m Lrm L r m q j θ ju g u qj θ jd g u Lrm g mu m L r m 2 q j θ jd L r m 2 q j θ jd θ ju ) L rm q j θ jd 50)

10 3370 T-L CHIEN, C-W LIN, C-J HUANG AND C-C CHEN = L r m L g L r m u L gm L r m u m L rm = c m d m u d mm u m q j θ jd θ ju ) η k t) = φ k dx dt = φ k g u qj θ jd Since = φ k φ k g u φ k g mu m = L φ k = q k φ k q j θ jd θ ju ) φ k q j θ jd θ ju ), L r m q j θ jd θ ju ) φ k q j θ jd k = r, r 2,, n φ k q j θ ju 5) 52) c i ξt), ηt)) L r i h ixt)), i m 53) d ij L gj L r i h i X), i m, j m 54) q k ξt), ηt)) = L φ k X), k = r, r 2,, n 55) the dynamic equations o system 2) in the new co-ordinates are as ollows: ξ i t) = ξit) Li h qj θ jd θ ju ), i =, 2,, r 56) ξ r t) = c ξt), ηt)) d ξt), ηt))u d m ξt), ηt))u m Lr h qj θ jd θ ju ) 57) ξ m i t) = ξ m it) Li q j θ jd θ ju ), i =, 2,, r m 58) ξ r m m t) = c m ξt), ηt)) d m ξt), ηt)) u d mm ξt), ηt))u m Lrm qj θ jd θ ju ) η k t) = q k ξt), ηt)) 59) φ kx)q j θ jd θ ju ), k = r,, n 60) y i t) = ξ i t), i m 6) According to Equations 8), 44), 53) and 54), the tracking controller can be rewritten as u eedback = A b v 62)

11 DYNAMIC POSITIONING CONTROL OF MARINE VEHICLE 337 Substituting Equation 62) into 57) and 59), the dynamic equations o system 2) can be shown as ollows: ξ i t) ξ i t) 0 ξ 2t) i ξ 2t) i = ξ r i i t) ξ i 0 r i t) 0 v i ξ r i i t) ξr i i t) h iqj θ jd θ ju ) 63) L h iqj θ jd θ ju ) Lr i h i qj θ jd θ ju ) η r t) η r2 t) η n t) η n t) = q r t) q r2 t) q n t) q n t) y i = r ξ i t) ξ i 2t) ξ i r i t) ξ i r i t) φ rqj θ jd θ ju ) φ r2qj θ jd θ ju ) φ n qj θ jd θ ju ) φ nqj θ jd θ ju ) r 64) = ξ i t), i m 65) Combining Equations 8), 20), 2), 26) and 44), it can be easily veriied that Equations 63)-65) can be transormed into the ollowing orm ηt) = qξt), ηt)) φ η θ d θ u ) q 22 t, ηt), e) φ η θ d θ u ) 66a) ε e i t) = A i ce i φ i ξ θ d θ u ), i m 66b) y i t) = ξ i t), i m 67) We consider L ē, η) deined by a weighted sum o V η) and W ē), ) ) L ē, η) V η) kε)w ē) V η) kε) W e W m e m ) where ) W ē) W e W m e m ) 69) ) as a composite Lyapunov unction o the subsystems 66a) and 66b) 26,29, where W e i satisies ) W i e i 2 eit P i e i 70) 68)

12 3372 T-L CHIEN, C-W LIN, C-J HUANG AND C-C CHEN In view o Equations 8), 33) and 40), the derivative o L along the trajectories o 66a) and 66b) is given by L = t V η V ) T η k ) ) e T ) 2 P e e P e ) ) e m P m e m e m ) T P m e m = t V η V ) T η k 2 ε A ce ) T ε φ ξθ d θ u ) P e ) T e P ε A ce ) ε φ ξθ d θ u ) ε Am c e m ) T ε φm ξ θ d θ u ) P m e m e m ) T P m ε Am c e m ) ε φm ξ θ d θ u ) = t V η V ) T q 22 t, ηt), ē) φ η θ d θ u )) { k ) T e 2ε A c ) T P P A c) e k 2ε em ) T A m c ) T P m P m A m c ) e m k θ d θ u ) T φ ξ) } T P e ε θ d θ u ) T φ mξ ) T P m e m t V η V ) T q 22 t, ηt), ē) η V ) T φ η θ d θ u ) k ) T e 2ε e e m ) T e m k θ d θ u ) φ ε ξ P e θ d θ u ) φ m ξ P m e m t V η V ) T q 22 t, ηt), 0) η V ) T q 22 t, ηt), e) q 22 t, ηt), 0) η V φ η θ d θ u ) k W ε λ max P ) W m λ max P m ) 4k2 φ ε 2 ξ 2 P 2 e 2 6 θ d θ u 2 4k2 ε 2 φm ξ 2 P m 2 e m 2 6 θ d θ u ) 2 t V η V ) T q 22 t, ηt), 0) η V q 22 t, ηt), ē) q 22 t, ηt), 0) η V φ η θ d θ u ) k W W m 4k2 φ ε λ max ε 2 ξ 2 P 2 e 6 θ d θ u ) 2 4k2 φ m ε 2 ξ 2 P m 2 e m 2 6 θ d θ u ) 2 2α x V ω 3 η M ē ω 3 η φ η θ d θ u ) k W 4k2 ε λ max ε 2 φ ξ 2 P 2 e 2 6 θ d θ u ) 2 4k2 ε 2 φm ξ 2 P m 2 e m 2 6 θ d θ u ) 2 2 2α x V ω 3 V M W ω3 V φη θ d θ u ) ω ω k ε λ max λ min W 4k2 ε 2 φ ξ 2 P 2 W /2λ min P ) 2

13 DYNAMIC POSITIONING CONTROL OF MARINE VEHICLE 3373 W m 4k2 ε 2 φm ξ 2 P m 2 /2λ min P m ) m 6 θ d θ u ) 2 ) ) 2 ω 3 M 2 ω3 ) 2 2α x V 2 2kλ V kw 4 φ η V min ω ω 6 θ d θ u ) 2 ε λ max km ) 2 4k 2 W m ε 2 φ ξ 2 P 2 W /2λ min P ) 4k2 ε 2 φm ξ 2 P m 2 /2λ min P m ) m 6 θ d θ u ) 2 ) ) 2α x 4ω2 ) 2 3 φ η 2 ω 3 M V V 2 kw ω 2ω kλ min 4k φ ξ 2 P 2 ελ max 2 ε2 λ min P ) 4k φm ξ 2 P m 2 ) ) 2 kw 2 ε2 λ min P m ) m θ d θ u ) 2 6 V = V kw H m θ d θ u ) 2 kw 6 ie, L λ min H)L m θ d θ u ) 2 72) 6 where λ min H) denotes the minimum eigenvalue o the matrix H Utilizing the act that λ min H) = 2α s, we obtain Deine Hence, L 2α s L m 6 θ d θ u ) 2 2α s V kw ) m 6 2α s ω η 2 k ) 2 λ min ē 2 m θ d θ u ) 2 6 NN 2 η 2 ē 2) m 6 e e e 2 e m L NN 2 η 2 Utilizing Equation 75) easily yields t t 0 e e θ d θ u ) 2 e rem 2 e rem θ d θ u ) 2 73), e rem R r 74) 2) m θ d θ u ) 2 75) 6 y τ) y τ)) 2 L t 0 ) d dτ m t NN 2 6NN 2 θ d τ) θ u τ)) 2 dτ 76) t 0 Similarly, it is easy to prove that t t 0 yi τ) ydτ) ) i 2 L t 0 ) dτ m t NN 2 6NN 2 θ d τ) θ u τ)) 2 dτ, 2 i m 77) t 0

14 3374 T-L CHIEN, C-W LIN, C-J HUANG AND C-C CHEN so that Equation 39) is satisied From Equation 73), we get where L NN 2 ytotal 2) m 6 y total 2 ē 2 η 2 θ d θ u ) 2 78a) 78b) By virtue o Theorem 52 25, Equation 78a) implies the input-to-state stability or the closed-loop system Furthermore, it is easy to see that that is min ē 2 η 2) L max ē 2 η 2) 79) min ytotal 2) L max ytotal 2) 80) where min min { ω, k 2 min} λ and max max { ω 2, k 2 max} λ From Equation 73) and Equation 80) yield that L NN 2 L m 2 sup θ d τ) θ u τ)) ) 8) max 6 t 0 τ t Hence, Lt) Lt 0 )e NN 2 max t t 0) maxm ) 6NN 2 which implies y t) y t) d Similarly, it is easy to prove that yi t) ydt) i 2Lt 0 ) 2Lt 0 ) e NN 2 kλ 2 max t t0) max m ) min 8kλ min NN 2 kλ min e NN 2 2 max t t 0) max m ) 8kλ min NN 2 2 sup θ d τ) θ u τ)) ), t t 0 82) t 0 τ t ) sup θ d τ) θ u τ)) t 0 τ t sup θ d τ) θ u τ)) t 0 τ t, 2 i m So that Equation 38) is proved and then the tracking problem with almost disturbance decoupling is globally solved Finally, we will prove that the sphere B r is a global attractor or the output tracking error o system 2) From Equation 78a) and Equation 45d), we get L NN 2 ytotal 2) N 85) For y total > r, we have L < 0 Hence, any closed ball deined by { } ē B r : ē 2 η 2 r η is a global inal attractor or the tracking error system o the non-linear control systems 2) Furthermore, it is easy routine to see that, or y / B r, we have L L NN 2 y total 2 N L NN 2 max N max r 2 α NN 2 y total 2 N max y total 2 NN 2 N max max y total 2 ) 83) 84) 86) 87)

15 DYNAMIC POSITIONING CONTROL OF MARINE VEHICLE 3375 ie, L α L According to the comparison theorem 32, we get Thereore, Consequently, we get Lt) Lt 0 ) exp α t t 0 ) min y total 2 Ly total t)) Ly total t 0 )) exp α t t 0 ) y total max y total t 0 ) 2 exp α t t 0 ) max y total t 0 ) exp min 2 α t t 0 ) ie, the convergence rate toward the sphere B r is equal to α /2 This completes our proo B Fuzzy controller design Ater the eedback linearization control is utilized as a guarantee o uniorm ultimate bounded stability, the multiple input/single output uzzy control design can be technically applied via human expert s knowledge to improve the convergence rate o tracking error The block diagram o the uzzy control is shown in Figure 88) Figure Fuzzy logic controller In general, the tracking error et) and its time derivative ėt) are utilized as the input uzzy variables o the IF-THEN control rules and the output is the control variable u uzzy For the sake o easy computation, the membership unctions o the linguistic terms or et), ėt) and u uzzy are all chosen to be the triangular shape unction We deine seven linguistic terms: PB Positive big), PM Positive medium), PS Positive small), ZE Zero), NS Negative small), NM Negative medium) and NB Negative big), or each uzzy variable, as shown in Figure 2 Fuzzy control rule table or u uzzy is shown in Table The rule base is heuristically built by the standard Macvicar-Whelan rule base 50 or usual servo control systems The Mamdani method is used or uzzy inerence The deuzziication o the output set membership value is obtained by the centroid method Thereore, we can combine the designs o eedback linearization control and uzzy control to construct the overall controller as ollows: u eu { b v}u s t) u uzzy u s t t ) u uzzy2 u s t t 2 ) u uzzym u s t t m ), i m 89)

16 3376 T-L CHIEN, C-W LIN, C-J HUANG AND C-C CHEN a) et) b) ėt) c) u uzzy Figure 2 Membership unctions or a) et), b) ėt) and c) u uzzy where u s t) denotes the unit step unction and t i, i m are the time variables that the tracking error o states touch the global inal attractor B r u uzzy ėt) Table Fuzzy control rule base et) NB NM NS ZE PS PM PB NB PB PB PB PB PM PS ZE NM PB PB PB PM PS ZE NS NS PB PB PM PS ZE NS NM ZE PB PM PS ZE NS NM NB PS PM PS ZE NS NM NB NB PM PS ZE NS NM NB NB NB PB ZE NS NM NB NB NB NB 3 Marine Vehicle in 6 Degrees o Freedom When investigating the behavior o marine vehicle in 6 degree o reedom, it is convenient to deine body-ixed coordinate rame and earth-ixed coordinate rame as indicated in Figure 3 Figure 3 Body-ixed and earth-ixed reerence rames or marine vehicle in 6 degrees o reedom

17 DYNAMIC POSITIONING CONTROL OF MARINE VEHICLE 3377 In general, the behavior o marine vehicle in 6 degree o reedom can be described by the ollowing vectors: η η T η 2 T T, η x y z T, η 2 φ θ ϕ T, ῡ ῡt ῡ2 T T, ῡ u v w T, ῡ 2 p q r T, τ τ T τ 2 T T, τ τ x τ y τ z T and τ 2 τ K τ M τ N T Here η and η 2 denote the position vector and orientation vector with coordinate in the earth-ixed rame, respectively, ῡ and ῡ 2 are used to describe the linear velocity vector and angular velocity vector with coordinates in the body-ixed rame, respectively, τ and τ 2 denote the orce vector and moment vector acting on the marine vehicle in the body-ixed rame, respectively Based on the derivation o, the kinematic equations relating the body-ixed reerence rame to the earth-ixed reerence rame and the 6 degree o reedom non-linear dynamic equations o motion can be expressed as η J η = 2 ) ῡ, 90) η J2 η 2 ) ῡ 2 ῡ = M Cῡ) ῡ M Dῡ) ῡ M g η) M τ, 9) cos ϕ cos θ sin ϕ cos φ cos ϕ sin θ sin φ sin ϕ sin φ cos ϕ cos φ sin θ J η 2 ) sin ϕ cos θ cos ϕ cos φ sin φ sin θ sin ϕ cos ϕ sin φ sin θ sin ϕ cos φ, sin θ cos θ sin φ cos θ cos φ 92) sin φ tan θ cos φ tan θ J 2 η 2 ) 0 cos φ sin φ, 93) 0 sin φ/cos θ cos φ/cos θ where M and Cῡ) denote the inertia matrix including added mass and the matrix o Coriolis and centripetal terms including added mass, respectively, Dῡ) denotes the damping matrix, τ denotes the vector o control inputs and g η) is used to describe the vector o gravitational orces and moments A dynamically positioned marine vehicle can be supplied with anchors i thruster-assisted mooring is to be investigated The mooring orces are described as spring orces, g η) = K η η 0 ), where η 0 is the equilibrium point o the mooring system For the sake o simplicity it is assumed that η 0 = 0 During station-keeping and low-speed application like dynamic positioning o marine vehicles, the linear velocities u, v and the angular velocity r are all small which contributes that a urther simpliication can be to neglect the term M Cῡ)ῡ Hence the equations o motion in surge, sway and yaw can be written as ollows according to η x y ϕ T, υ u v r T, τ τ x τ y τ N T : where η = Jη)υ, 94) υ = M Dυ) υ M K η M τ, 95) Jη) cos ϕ sin ϕ 0 sin ϕ cos ϕ 0 0 0, 96) is the rotation matrix in yaw, M is the inertia matrix including hydrodynamic added inertia, D is the damping matrix and τ are the control orces and moment provided by the thruster system Starboard-port symmetry o marine vehicles implies that the matrices M and D are constructed as the ollowing structure: M m m 22 m 23, 97) 0 m 32 m 33

18 3378 T-L CHIEN, C-W LIN, C-J HUANG AND C-C CHEN D d d 22 d 23, 98) 0 d 32 d 33 that is, there is no coupling item between the surge and sway-yaw subsystems Hence, the anchor orces and moment are related by the diagonal matrix K k k k 33 99) We consider a tanker with the numerical system matrices as ollows: Bis-scaled values ) M , 00) D , 0) K ) Ater arranging Equations 94)-96) in the state-space ormulation, it is easy to obtain the ollowing state equations: where ẋ = x 4 cos x 3 ) x 5 sin x 3 ), 03) ẋ 2 = x 4 sin x 3 ) x 5 cos x 3 ), 04) ẋ 3 = x 6, 05) ẋ 4 = 00358x 00797x u, 06) ẋ 5 = 00208x x x u 2 48u 3, 07) ẋ 6 = 00394x x x 6 48u u 3, 08) X x x 2 x 3 x 4 x 5 x 6 T x y ϕ u v r T, 09) u u u 2 u 3 T τx τ y τ z T 0) It is assumed that only the position and yaw angle measurements are available, that is, the output equations are given by y = x ) y 2 = x 2 2) y 3 = x 3 3) Now we will show how to explicitly construct a controller that tracks the desired signals yd = y2 d = y3 d = 0 and attenuates the disturbance s eect on the output terminal to an arbitrary degree o accuracy Let s arbitrarily choose α = α2 = 00, α 2 = α2 2 = 00, 0 α 3 = α2 3 = 0, A c = A 2 c = A 3 c = With the aid o Matlab, the solutions o Lyapunov equations are given by P = P 2 = P 3 = 4)

19 DYNAMIC POSITIONING CONTROL OF MARINE VEHICLE 3379 Hence, λ min = 005 and λ max = 005 From Equation 89), we obtain the desired tracking controllers u = A ) b v u s t) u uzzyu s t t ) u uzzy u s t t 2 ) 5) u uzzy u s t t 3 ) A a a 2 a 3 a 2 a 22 a 23 6) a 3 a 32 a 33 where u u u 2 u 3 T, b b b 2 b 3 T, v v v 2 v 3 T 7) a = 0925 cos x 3 8) a 2 = sin x 3 9) a 3 = 48 sin x 3 20) a 2 = 0925 sin x 3 2) a 22 = cos x 3 22) a 23 = 48 cos x 3 23) a 3 = 0 24) a 32 = 48 25) a 33 = ) b = x 6 ) x 4 sin x 3 x 5 cos x 3 ) 00358x 00797x 4 ) cos x 3 ) 00208x x x 6 ) sin x 3 ) 27) b 2 = x 6 ) x 4 cos x 3 x 5 sin x 3 ) 00358x 00797x 4 ) sin x 3 ) 00208x x x 6 ) cos x 3 ) 28) b 3 = 00394x x x 6 ) 29) v = 00 ε) 2 x 00 ε) x 4 cos x 3 ) x 5 sin x 3 )) 30) v 2 = 00 ε) 2 x 2 00 ε) x 4 sin x 3 ) x 5 cos x 3 )) 3) v 3 = 00 ε) 2 x 3 00 ε) x 6 ) 32) It can be veriied that the relative conditions o Theorem 2 are satisied with ε = 0, N = 995, N 2 = 079 and k = 00 ε Hence, the tracking controllers will steer the output tracking errors o the closed-loop system, starting rom any initial value, to be asymptotically attenuated to zero by virtue o Theorem 2 In the simulations, the ship is irst controlled by the dynamic positioning control law 5) using three control inputs and initially steered to the point x 0) x 2 0) x 3 0) x 4 0) x 5 0) x 6 0) T T The desired equilibrium point is x 0) x 2 0) x 3 0) x 4 0) x 5 0) x 6 0) T T The simulated results are shown in Figures 4-9 We see that the ship eventually converges to a neighborhood o the desired equilibrium point The dynamic positioning time based on our proposed novel non-linear uzzy eedback linearization control is approximately equal to be 06 On the contrary, the dynamic positioning time or the same ship control system using the traditional uzzy-model approach 5 is approximately equal to be 0 The simulation results show that our proposed control in this study is a marvelous approach or the control problem o non-linear dynamic positioning problem o ship

20 3380 T-L CHIEN, C-W LIN, C-J HUANG AND C-C CHEN Figure 4 Measured x-position x Figure 5 Measured y-position x 2 4 Comparative Example to Existing Approach 30 exploited the act that the almost disturbance decoupling problem could not be solvable or the ollowing system: ẋ t) tan = x 2 ) 0 u θt) 33) ẋ 2 t) 0 0 yt) = x t) := hxt)) 34) where u, y denoted the input and output respectively, θt) := 2 sin t u s t ) u s t 3) and u s t) denotes the unit step unction Assume that the desired tracking signal is equal to be sin t The almost disturbance decoupling problem can be easily solved via the proposed approach in this paper Following the same procedures shown in

21 DYNAMIC POSITIONING CONTROL OF MARINE VEHICLE 338 Figure 6 Measured yaw angle x 3 Figure 7 Surge velocity x 4 the demonstrated example, we can solve the tracking problem with almost disturbance decoupling problem by the controller u deined as u = x 2 2) sin t 42x sin t) 42tan x 2 cos t)u s t) u uzzy u s t t ) 35) The tracking error dynamics or 33) is depicted in Figure 0 It is worth noting that the suicient conditions given in 30 in particular the structural conditions on non-linearities multiplying disturbances) are not necessary in this study where a non-linear state eedback control is explicitly designed which solves the almost disturbance decoupling problem For instance, the almost disturbance decoupling problem is solvable or the system 33) by our proposed approach, while the suicient conditions

22 3382 T-L CHIEN, C-W LIN, C-J HUANG AND C-C CHEN Figure 8 Sway velocity x 5 Figure 9 Yaw velocity x 6 given in 30 ail when applied to the system 33) The design techniques in this study are also entirely dierent than those in 30 since the singular perturbation tools are not used 5 Conclusions A novel uzzy eedback control to globally solve the tracking problem with almost disturbance decoupling or dynamic positioning marine vehicle has been proposed A discussion and a practical application o eedback linearization o non-linear control systems using a parameterized coordinate transormation have been presented One comparative example is proposed to show the signiicant contribution o this paper with respect to existing approach A practical simulation example has been used to

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