Robust Fuzzy Controller Design for Dynamic ... - Springer Link

International Journal of Control, Automation, and Systems (2015) 13(5):1294-1305 DOI 10.1007/s12555-014-0239-5

ISSN:1598-6446 eISSN:2005-4092 http://www.springer.com/12555

Robust Fuzzy Controller Design for Dynamic Positioning System of Ships Werneld Egno Ngongi*, Jialu Du, and Rui Wang Abstract: This paper presents a robust fuzzy controller design approach for dynamic positioning (DP) system of ships using optimal H∞ control techniques. The H∞ control technique is used to exterminate the effects of environmental disturbances. Firstly, a Takagi-Sugeno (TS) fuzzy model is applied to approximate the nonlinear DP system. Next, linear matrix inequality (LMI) and general eigenvalue problem (GEVP) methods are employed to find a positive definite matrix and controller gains. The stability of the controller is proven by using Lyapunov stability theorems. A positive definite matrix is determined by solving LMI equations using robust control toolbox available in MATLAB. The obtained positive definite matrix proves that the designed fuzzy controller is stable. Finally, a uniformly ultimately bound (UUB) and control performance for the dynamic position system is guaranteed. Simulation is carried out, and results are presented to validate the effectiveness and performance of the proposed control system. Keywords: Dynamic positioning, eigenvalue problem, linear matrix inequality, robust fuzzy controller, Takagi-Sugeno fuzzy model, uniformly ultimately bound.

1. INTRODUCTION Offshore exploration and exploitation of hydrocarbons have stimulated the popularity of dynamically positioned (DP) vessels. Recently, there are 2000 and more DP vessels of various types operating world-wide [1]. Thus, it implies that the significant technological advances in the DP control system design have taken place for more than thirty years of research and improvement. However, more advanced researches need to be addressed in the DP control systems so as to improve more and more. By definition, dynamic positioning (DP) is a computer controlled system to automatically maintain the position and heading of the ship by using of its own propellers and thrusters. Information to the computer pertaining to the ship's position, magnitude and direction of the environmental forces affecting its position is provided by position reference sensors, together with wind sensors, motion sensors and gyro-compasses. The computer software contains a mathematical model of the ship that includes information relating to the wind and ocean current drag of the vessel and the location of the thrusters. Thus, dynamic positioning systems are essential when operating in the deep sea where anchors are not __________ Manuscript received June 16, 2014; revised October 4, 2014 and November 5, 2014; accepted November 25, 2014. Recommended by Associate Editor Do Wan Kim under the direction of Editor Euntai Kim. This work was supported in part by the Natural Science Foundation of China (51079013), in part by Applied Basic Research Program of Ministry of Transport of P. R. China (2012329-225-070). Werneld Egno Ngongi, Jialu Du, and Rui Wang are with the School of Information Science and Technology, Dalian Maritime University, Dalian, Liaoning Province 116026, China (e-mails: [email protected], [email protected], [email protected]). * Corresponding author. © ICROS, KIEE and Springer 2015

appropriate. DP system can be employed to oil platforms, supply ships, cable and pipeline laying ships, drilling, and many more. In such type of applications, the performances of DP systems are important because they determine whether the operation can be successfully. The first DP system was designed using proportional integrator derivative controllers cascaded with low pass or notch filters in the early 1960’s. In the mid of 1970’s, many various control methodologies based on optimal control theory of linear Kalman filter were introduced [2,3]. The main drawbacks of these approaches were that the kinetic equations of motions must be linearized under certain conditions. For every linearized equation, optimal Kalman filter and feedback-gain control was computed, and the gains were modified online through gain scheduling method. A passive nonlinear observer was introduced to overcome the problem [4]. In 2000, a nonlinear globally asymptotically stable output feedback controller for dynamic positioning of ships using separation principle was introduced [5]. The objective was to solve the DP problem and not to vocalize new separation principle for systems of nonlinear. Another work to study the effects of extreme sea on the DP control system design has been given by [6]. The study in [6] suggested enhanced control strategies for operation of DP in extreme seas condition. The study of DP continued, and more researches conducted whereas, in 2005, a nonlinear optimal observer and suboptimal observer using state dependent and Riccati equation technique were given by [7]. In this work, a new observer design method was introduced, and suboptimal observer was developed to compare to an existing observer. The main advantage of the method in [7] is the flexibility offered by the observer because it was based on general mathematical framework which allowed the use of dissimilar models of ships. However, the shortcomings of the approach were that, optimality

Robust Fuzzy Controller Design for Dynamic Positioning System of Ships

could not be guaranteed and the loss of global asymptotic stability. In 2007, a global robust and adaptive output feedback controller for DP system of surface ships in the presence of environmental disturbances was given by [8]. A gorgeous feature of the approach was that, the adaptive observer can act at the same time as an estimator and a filter to estimate unavailable state vectors and filter out high frequency noise. A study on stabilization and tracking control for dynamically positioned vessels using nonlinear sampled data control theory based on Euler approximation introduced in [9]. Unfortunately in [9], the designed state feedback tracking laws did not theoretically guarantee the convergence of the tracking errors. In 2010, a type-two fuzzy logic controller for dynamic positioning systems of ships was presented in [10]. The objective of [10] was to design a controller such that, the requirement for linearization of the kinematic equations of motion when designing controllers for the DP systems is removed using a combination of fuzzy logic controllers and nonlinear observers. A discrete time variable controller for the dynamic positioning system of ships was presented by [11]. The method proposed in [11] guarantees the robustness with admiration to the environmental disturbances. A recursive vectorial backstepping controller was introduced for keeping a fixed position and heading of the ship in the presence of wave disturbances [12]. In this work, the observer parameters were optimized off-line by the application of the genetic algorithm (GA). Also, a neural network controller based on vectorial backstepping approach for DP with uncertainties and unknown disturbances was developed in [13]. The benefit of [13] is that there is no need to know about the ship dynamics and environmental disturbances. A fuzzy control for dynamic positioning of ship has been given in [25]. The proposed controller in [25] employs Takagi-Sugeno fuzzy model and linear feedback control techniques to design nonlinear fuzzy controller for DP system. However, the proposed controller [25] considers dynamic positioning with mooring and does not include environmental disturbances. A dynamic ship positioning systems with multiplicative noises has been proposed in [28]. In [28], the sufficient conditions formulated to ensure the closedloop system stability. The stability conditions established in this paper can be synchronized with the LMI constraints. Thus, by solving these LMIs stability conditions, a PDC (Parallel Distributed Compensation) based fuzzy controller was obtained for stabilizing the TS fuzzy models with multiplicative noises. Again, a PD controller based on high-gain observer for dynamic position systems of ships was proposed by [14]. The application of high-gain observer for the proposed controller in [14] offers a high ability to reject environmental disturbance. A number of studies (researches) on control systems design for dynamic positioning of ships have been successfully conducted and given in [2-14,25,28]. Therefore, it has been witnessed that dynamic positioning systems have continued to improve from time to time in an attempt to

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reduce the DP control problems. These improvements are proportional with developments in technology. It is expected that these improvements will continue and that many new researches are needed for DP systems. In recent years, there has been a fast growth of interest in fuzzy control design of nonlinear systems and there have been successful results. For instance, a robust fuzzy controller for the discrete perturbed time delay affine T-S fuzzy model has been developed [29]. The purpose of the study given in [29] was to achieve the H∞ performance constraints and cope with the nastiest case effects of the disturbances. More works related to robust fuzzy have been given in [27-35,37-38]. Some of these literatures have no direct comparison to the proposed work but show the richness of fuzzy control designs. The performances of some literatures proposed in [28-34] are discussed in section four. Thus, this paper proposes a linear fuzzy controller design based on the nonlinear H∞ control scheme for the control of dynamic positioning of the system of ships. A Takagi-Sugeno (T-S) linear fuzzy model introduced in [17] is used to approximate a nonlinear DP model. Then, a crossbreed fuzzy controller is employed to stabilize the nonlinear DP model while eliminating the effects of environmental disturbance below a prescribed level; as a result the desired H∞ control performance is guaranteed. Hence, this method combines fuzzy linear model and the H∞ performance so as to obtain a simple and practical algorithm for robust performance control design for DP system. Finally, the optimal H∞ of fuzzy control system is formulated into eigenvalue problem to minimize the eigenvalue of a matrix depending on a variable subject to linear matrix inequality (LMI) constraints. For design simplicity, in this paper, it has been assumed that all state variables are available for measurements and that the premise variables are dependent of state variables. The contribution of this study is: 1) the robust linear fuzzy controller design for DP system via optimal Hinfinity disturbance attenuation is achieved, 2), the stability of the fuzzy controller and H∞ performance on the effect of the environment disturbance on the control performance is ensured. A simulation of ship model is presented to demonstrate the performance of the proposed approach. The rest of the work in this paper organized as follows: section two presents mathematical model of a DP ship and its T-S fuzzy model. Controllers design and its stability analysis are given in section three while simulation and analysis presented in section four. Lastly, section five gives the conclusion. 2. MATHEMATICAL MODEL For the description for the motion of the ship at sea, it is important to define the two coordinate systems [15]. The moving coordinate system [X, Y, Z] is fixed to the ship is called the body-fixed frame. The origin of the body-fixed frame is usually picked to coincide with the centre of the gravity of the ship. Then the motion of the body-fixed frame is described relative to the earth-fixed frame, namely [XE, YE, ZE]. The position (x, y) and

Werneld Egno Ngongi, Jialu Du, and Rui Wang

1296

heading ψ of the ship relative to the earth-fixed frame can be expressed in form of vector by η = [ x, y, ψ ]T . The velocity of the body-fixed frame is represented in vector form as υ = [u , ν , r ]T , where u is surge velocity, v is the sway velocity and r is the yaw velocity. Thus the relation between the body-fixed frame and earth-fixed frame can be expressed by the following transformation η = J (η )υ ,

(1)

where J (η ) is the state dependent transformation matrix given in the form ⎡ cosψ J (η ) = J (ψ ) = ⎢⎢sinψ ⎢⎣ 0

− sinψ

0⎤ 0 ⎥⎥ , 1 ⎥⎦

cosψ 0

0 − Nν

let us define the following variables x(t ) = [η , υ ]T = [ x1 , x2 , x3 , x4 , x5 , x6 ]T ,

(5) (6)

b(t ) = [b1 , b2 , b3 ] .

0

⎤ ⎥ mxG − Yr ⎥ , I z − N r ⎥⎦

where m is the body (marine vessel) mass, Iz is the moment of inertia about the body-fixed z – axis and xG is the distance between the centre of gravity and the origin of the body-fixed frame. X u , Yν , Yr , Nν and N r are added mass and inertia terms. Lastly, the linear damping matrix D ∈ » 3×3 is defined as − Yν

J (η ) ⎤ ⎡η ⎤ ⎡ 0 ⎤ 0 ⎡ ⎤ ⎡η ⎤ ⎡0 τ + ⎢ −1 T ⎥ b, (4) + ⎢υ ⎥ = ⎢ −1 ⎥ ⎢υ ⎥ ⎢ −1 ⎥ ⎣ ⎦ ⎣0 − M D ⎦ ⎣ ⎦ ⎣ M ⎦ ⎣ M J (η ) ⎦

T

where τ ∈ »3 is a vector of control forces and moments and b ∈ »3 is a vector describing all unmodeled forces and moments caused by wind, ocean currents and waves. The inertia matrix M ∈ » 3×3 is assumed to take the following form

⎡− X u ⎢ D=⎢ 0 ⎢⎣ 0

where τ = [τ1 , τ 2 , τ 3 ]T is the vector of input control forces and moment. Rewriting (3) in matrix form, yields the following

uc (t ) = τ = [uc1 , uc 2 , uc 3 ] ,

(2)

0 ⎡ m − X u ⎢ M =⎢ 0 m − Yν ⎢⎣ 0 mxG − Nν

(3)

υ = − M −1 Dυ + M −1τ + M −1 J T (η )b,

T

where J (ψ ) is non-singular for all ψ and that J −1 (ψ ) = J T (ψ ). At low speed the motion of ship can be described by the following mathematical model Mν + Dυ = τ + J T (η )b,

η = J (η )υ ,

0

⎤ ⎥ ⎥, mxG u0 − N r ⎥⎦ mu0 − Yr

where the ship’s speed u0 = 0 for DP and u0 > 0 when the ship is moving forward. Normally, the damping forces will be nonlinear; nevertheless for DP ship cruising at low constant speed linear damping is a good assumption. Fig. 2 in [27] shows overall block diagram of a dynamic positioning (DP) control system. The reference (set) point for the control system is the ship position. The comparison between the set point and real ship positions yields an error signal. The produced error signal is used by controller to derive thrust demand for restoring the ship’s position. Additionally, thrust demand obtained from each of the three axes namely fore/aft (X), port/starboard (Y) and heading (ψ) are combined and located to each thruster of the ship. Next we combine (1) and (2) together to obtain the following

(7)

Next, for simplicity of computation and design, we calculate the parameters of the products of matrices M −1 D, M −1 and M −1 J T (η ) as follows: ⎡ d11 0 ⎢ M −1 D = ⎢ 0 a22 ⎢⎣ 0 a32

0 ⎤ ⎥ a23 ⎥ , a33 ⎥⎦

⎡1 0 ⎢ M −1 = ⎢ 0 m01 ⎢⎣ 0 m20 sin( x3 (t ))

0 ⎤ ⎥ m02 ⎥ , m10 ⎥⎦

0 ⎤ ⎡ cos( x3 (t )) ⎢ ⎥ M J (η ) = ⎢ −m33 sin( x3 (t )) m33 cos( x3 (t )) − m32 ⎥ . ⎢⎣ −m23 sin( x3 (t )) − m23 cos( x3 (t )) m22 ⎥⎦ −1 T

Then, substituting (5)-(7) into (4) gives the following result ⎡ x1 ⎤ ⎡ 0 0 0 cos x3 − sin x3 0 ⎤ ⎡ x1 ⎤ ⎡0 ⎢ x ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎢ 2 ⎥ ⎢0 0 0 sin x3 cos x3 0 ⎥ ⎢ x2 ⎥ ⎢0 ⎢ x3 ⎥ ⎢0 0 0 0 0 1 ⎥ ⎢ x3 ⎥ ⎢0 ⎢ ⎥=⎢ ⎥⎢ ⎥+⎢ 0 ⎥ ⎢ x4 ⎥ ⎢1 ⎢ x4 ⎥ ⎢0 0 0 − d11 0 ⎢ x ⎥ ⎢0 0 0 0 − a − a ⎥ ⎢ x ⎥ ⎢0 22 23 ⎢ 5⎥ ⎢ ⎥⎢ 5⎥ ⎢ ⎢⎣ x6 ⎥⎦ ⎢⎣0 0 0 0 − a32 − a33 ⎥⎦ ⎢⎣ x6 ⎥⎦ ⎢⎣0 ⎡uc1 ⎤ ⎡b ⎤ 0 ⎤⎢ 1 ⎥ ⎢ ⎥ ⎡ × ⎢ u c 2 ⎥ + ⎢ −1 T ⎥ b2 , M J (η ) ⎦ ⎢ ⎥ ⎢⎣uc 3 ⎥⎦ ⎣ ⎢⎣b3 ⎥⎦

0 0 ⎤ 0 0 ⎥⎥ 0 0 ⎥ ⎥ 0 0 ⎥ m01 m02 ⎥ ⎥ m20 m10 ⎥⎦

(8)

which can be re-written in more compact form as x = Ax(t ) + Buc (t ) + Eb(t ),

(9)

where J (η ) ⎤ ⎡0 Ax(t ) = ⎢ , −1 ⎥ ⎣0 − M D ⎦

T

B = ⎡⎣ 0 M −1 ⎤⎦ ,

T

E = ⎡⎣ 0 M −1 J T (η ) ⎤⎦ .

3. T-S LINEAR FUZZY MODELLING OF THE NONLINEAR DP MODEL Note that, in order to apply fuzzy linear control design method [17-22], the fuzzy model which represents the


nonlinear ship DP model must be obtained. Then, the ship (DP) model in (9) is represented by Takagi-Sugeno fuzzy model. The model in (9) is approximated by using three rule fuzzy model. Note that the system in (9) can be represented as (1) in [16]. In this paper we put an assumption that the disturbance is unknown but with known upper bound such that d Eub ≥ d E (t ) , where d E ∈ »3 is the unknown disturbances. Definition 1 [15]: The solutions of a dynamic system are said to be uniformly ultimately bound (UUB) if there exist positive constants α and φ, and for every δ ∈ (0, ϕ ) there is positive constant Γ = Γ(δ ), such that x(t0 ) < δ ⇒ x(t ) ≤ α , for t ≥ t0 + Γ. Note that, the above definition (Definition 1) and (2) will be recalled later in the following sections. Hence it follows that

1297

Note that assumptions stated in (15)-(16) will be recalled later in the following sections. Therefore, from (1) [16], we have the following x (t ) = F ( x(t )) + H ( x(t ))uc (t ) + d E (t ) n

= ∑ µi ( z (t ))[ Ai x(t ) + Bi uc (t )] i =1

n ⎡⎛ ⎞ + ⎢⎜ F ( x) − ∑ µi ( z (t ))Ai x(t ) ⎟ ⎢⎣⎝ i =1 ⎠ n ⎤ ⎛ ⎞ + ⎜ H ( x) − ∑ µi ( z (t ))Bi ⎟ uc (t ) ⎥ + d E (t ), i =1 ⎝ ⎠ ⎦⎥

where

Plant Rule i:

n ⎡⎛ ⎞ ⎢⎜ F ( x) − ∑ µi ( z (t ))Ai x(t ) ⎟ ⎢⎣⎝ i =1 ⎠

IF z1 (t ) is f i1 and and zn (t ) is f in THEN x (t ) = Ai x(t ) + Bi uc (t ) + d E (t )

n ⎤ ⎛ ⎞ + ⎜ H ( x) − ∑ µi ( z (t ))Bi ⎟ uc (t ) ⎥ i =1 ⎝ ⎠ ⎦⎥

(10)

for i = 1, 2, , n where fij is the fuzzy set, Ai ∈ » n×n and Bi ∈ » n×m are state and input matrices respectively, n is the number of IF-THEN rules, and z1 (t ), z2 (t ), , z n (t ) are premise variables. Then the overall fuzzy system is represented as follows [17-21]: n

x (t ) =

∑ ωi ( z (t ))[ Ai x(t ) + Bi uc (t )] i =1

n

∑ ωi ( z(t ))

is defined as an approximation error between the nonlinear system and the fuzzy model in (11). Suppose the following fuzzy controller is applied to control the system in (17); hence, we have: Control rule j: IF z1 (t ) is F j1 and and zn (t ) is F jn THEN uc (t ) = L j x(t ), for j = 1, 2, ..., r

+ d E (t )

(11)

i =1

n

= ∑ μi ( z (t ))[ Ai x(t ) + Bi uc (t )] + d E (t ),

n

u (t ) =

where

∑ ω j ( z (t ))[ L j x(t )] j =1

n

∑ ω j ( z (t ))

q

j =1

μi ( z (t )) =

ωi ( z (t ))

n

∑ ωi ( z (t ))

,

(12) (13)

(14)

∑ ωi ( z (t )) > 0,

i =1 j =1

n ⎛ ⎞ + ⎜ F ( x) − ∑ µi ( z (t ))Ai x(t ) ⎟ i =1 ⎝ ⎠ n

i =1

i =1

+ d E (t ).

(15)

and n

i =1

n

+ ∑ µi ( z (t ))∑ µ j ( z (t ))[ H ( x) − Bi ]L j x(t )

∀t ,

therefore

∑ µi ( z (t )) = 1.

n

x (t ) = ∑∑ µi ( z (t )) µ j ( z (t ))[ Ai + Bi L j ] x(t )

i =1

µi ( z (t )) ≥ 0, for i = 1, 2, ..., n

(16)

(19)

j =1

where µ j ( z (t )) are defined as in (15) and (16). Next, substituting (19) into (17) gives the following result

and z (t ) = f ij ( z j (t)) is the grade membership of z j (t ) in fij. Assuming that ωi ( z (t )) ≥ 0, for i = 1, 2, ..., n and n

n

= ∑ μ j ( z (t ))L j x(t ),

j =1

n

i =1

z (t ) = [ z1 (t ), z2 (t ), ..., zn (t )],

(18)

where Lj are the control gains. Thus, this gives the following overall fuzzy controller

i =1

ωi ( z (t )) = ∏ fij ( z j (t )),

(17)

Defining and applying n ⎛ ⎞ ΔF = ⎜ F ( x) − ∑ µi ( z (t ))Ai x(t ) ⎟ i =1 ⎝ ⎠

and

(20)


1298 n

n

i =1

i =1

n ⎛ n ⎞ ⋅ ⎜ ∑ μi ( z (t ))∑ μ j ( z (t ))[ H ( x) − Bi ]L j x(t ) ⎟ i =1 ⎝ i =1 ⎠

ΔH = ∑ µi ( z (t ))∑ µ j ( z (t ))[ H ( x) − Bi ]L j x(t ),

T

consequently, equation (20) reduces to n

n ⎛ n ⎞ = ⎜ ∑ μi ( z (t ))∑ μ j ( z (t )) ρi Bo L j x(t ) ⎟ i =1 ⎝ i =1 ⎠

n

x (t ) = ∑∑ µi ( z (t )) µ j ( z (t ))[ Ai + Bi L j ] x(t )

(21)

i =1 j =1

n ⎛ n ⎞ ⋅ ⎜ ∑ μi ( z (t ))∑ μ j ( z (t )) ρi Bo L j x(t ) ⎟ . i =1 ⎝ i =1 ⎠

+ ΔF + ΔH + d E (t ).

Now, assuming that there exist two bounding matrices, ΔAi and ΔBi such as ΔF ≤

n

∑ µi ( z (t ))ΔAi x(t )

(22)

i =1

and ΔH ≤

n

n

i =1

j =1

∑ µi ( z (t ))∑ µ j ( z (t ))ΔBi L j x(t ) .

[ΔAi ΔBi ]T = [δ i Ao ρi Bo ]T ,

(24)

ρi ≤ 1, for i = 1, 2, ..., n.

n

∑ µi ( z (t ))ΔAi x(t ) ( ≥

ΔF

i =1

)

n

n

i =1

i =1

⎛ n ⎞ × ⎜ ∑ µ j ( z (t )) Bo L j x(t ) ⎟ . ⎜ j =1 ⎟ ⎝ ⎠

tF

ΔH

∫

)

(26)

Remember, in this paper we put an assumption that the disturbance dE(t) is unknown and bounded. Nevertheless, the presence of dE(t) will decline the control performance of the fuzzy controller. Thus, it is important to eradicate the effect of dE(t) so that we can guarantee the control performance of the controller. In this paper we will employ H∞ control so as to effectually abolish the effects of dE(t) on the fuzzy controller. Based on [22-24], we design H∞ as follows:

∫0

and

∑ µi ( z (t ))∑ µ j ( z (t ))ΔBi L j x(t ) ( ≥

T

⎛ n ⎞ (ΔH )T (ΔH ) ≤ ⎜ ∑ µ j ( z (t )) Bo L j x(t ) ⎟ ⎝ i =1 ⎠

(23)

For all solution of x(t) and bounding matrices ΔAi and ΔBi can be described by the following

where δ i ≤ 1 and Then

Noting that ρi ≤ 1 for i = 1, 2, ..., n and using condition (16) we get the following result

tF

xT (t )Qx(t )dt

d ET (t )d E (t )dt 0

0). The implication of (29) is that, the effect of dE(t) on x(t) must be attenuated below the anticipated level σ from lookout of energy regardless how high the disturbance dE(t) is. This means the gain L2 from dE(t) to x(t) must be equal or less than the set value of σ2. In addition, σ is chosen to be small positive value less than 1 for attenuation of the disturbance dE(t). If we place initial conditions, (28) becomes tF

∫0

xT (t )Qx(t )dt < ∫

tF 0

xT (0) Px(0)dt

+σ 2 ∫

tF 0

d ET (t )d E (t )dt ,

(29)

where P is a symmetry positive definite matrix (i.e., P = PT > 0). The objective of the approach proposed here is to specify a linear fuzzy controller in (19) such that the stability of fuzzy linear control system and the H∞ control performance in (19) with attenuation σ are guaranteed. The purpose of robust optimization is to attain the minimum σ2 in (29) so as to acquire maximum


elimination of the effect of disturbance dE(t). For (10), the design problem is how to stipulate a stabilizable fuzzy controller in (19) so as to minimize σ2 subject to the constraint in (29).

+ n

1

1299

xT (t ) PPx(t ) + σ 2 d ET (t )d E (t )

σ2 n

≤ ∑∑ µi ( z (t ))µ j ( z (t )){xT (t )[( AiT P + PAi i =1 j =1

4. CONTROLLER DESIGN

+ PBi L j + LTj BiT P + AoT Ao + ( Bo L j )T ( Bo L j )

In this section, the H∞ control design based on fuzzy linear control techniques will be presented. The aim is to specify the fuzzy linear control law presented in (19) for the system in given in (21) with the guaranteed H∞ control performance in (29). Hence, we proceed as follows: let a Lyapunov candidate function be chosen as T

V ( x(t )) = x (t ) Px (t ),

(30)

+2 PP] x(t )} +

σ2 +σ 2 d ET (t )d E (t ).

V ( x(t )) = xT (t ) Px(t ) + xT (t ) Px (t ) n

n

i =1

j =1

≤ ∑ µi ( z (t ))∑ µ j ( z (t )){xT (t )[ P ( Ai + Bi L j )

+ xT (t ) Pd E (t ) + d ET (t ) Px(t ).

(31)

Substituting (25) and (26) into (31) we have n

n

V ( x(t )) ≤ ∑∑ µi ( z (t ))µ j ( z (t )){xT (t )[( AiT P + PAi i =1 j =1

+ PBi L j + LTj BiT P + AoT Ao + ( Bo L j )T ( Bo L j ) +2 PP] x(t )} + xT (t ) Pd E (t ) + d ET (t ) Px(t ). (32)

Theorem 1: If the fuzzy controller in (19) is applied to the nonlinear DP model given in (9), and there exist a positive definite matrix P ( P = PT > 0) such that the matrix inequalities AiT P + PAi + PBi L j + LTj BiT P + AoT Ao

(33)

1 ⎞ ⎛ + ( Bo L j ) ( Bo L j ) + ⎜ 2 + 2 ⎟ PP + Q < 0 σ ⎠ ⎝ T

for i, j = 1, 2, ..., n. Then the nonlinear DP system given in (21) is said to be uniformly ultimately bound (UUB) and H∞ control performance of (29) is guaranteed for a given value of σ2. Proof: From (32), we have n

n

V ( x(t )) ≤ ∑∑ µi ( z (t ))µ j ( z (t )){x

T

i =1 j =1

+ PBi L j +

LTj BiT P +

(

AoT

(t )[( AiT P +

PAi

T

Ao + ( Bo L j ) ( Bo L j )

+2 PP] x(t )} + xT (t ) Pd E (t ) + d ET (t ) Px(t ) −σ 2 d ET (t )d E (t ) −

1

⎞ xT (t ) PPx (t ) ⎟ σ ⎠ 2

(34)

AiT P + PAi + PBi L j + LTj BiT P + AoT Ao + ( Bo L j )T ( Bo L j ) + 2 PP < −

1 σ

2

(35)

PP − Q

for i, j = 1, 2, ..., n. Substituting (34) into (35) we obtain n n 1 ⎞ V ( x (t )) < ∑∑ μi ( z (t ))μ j ( z (t ))xT (t ) ( −Q − 2 PP ⎟ σ ⎠ i =1 j =1

⋅x(t ) +

+( Ai + Bi L j )T P]x(t ) + (ΔF )T (ΔF ) + xT (t ) ⋅PPx(t ) + (ΔH )T (ΔH ) + xT (t ) PPx(t )}

xT (t ) PPx(t )

Applying the condition given in (33), we obtain

T

where P = P > 0. Differentiating (30) with respect to time and substituting (21) into it, we have the following

1

1 σ2

xT (t ) PPx(t ) + σ 2 d ET (t )d E (t ). (36)

Applying the condition of (15) and (16) to (36), it implies that 1 V ( x (t )) < − xT (t )Qx(t ) − 2 xT PPx(t ) σ 1 T + 2 x (t ) PPx(t ) + σ 2 d ET (t )d E (t ) σ = − xT (t )Qx(t ) + σ 2 d ET (t )d E (t ).

(37)

But d E (t ) ≤ d Eub , then (37) reduces to 2 V ( x (t )) < − xT (t )Qx(t ) + σ 2 d Eub

(38)

2 ≤ −a1 xT (t ) x(t ) + σ 2 d Eub ,

where a1 = λmin (Q) is the minimum eigenvalue of Q. Therefore, we can observe that, whenever x(t ) >

σ d Eub a1

V ( x(t )) < 0.

,

Hence, this shows that the trajectories of (21) are uniformly ultimately bound (UUB). Now taking time integral of (37) yields the following result V (t F ) − V (0) < −∫

tF 0

xT (t )Qx(t )dt + σ 2 ∫

tF 0

d T (t )d E (t )dt.

(39) Hence, using (30) we have tF

∫0

xT (t )Qx (t )dt < xT (0)Qx(0) + σ 2 ∫

tF 0

d ET (t )d E (t )dt .

(40) Looking at equation (40) we can notice that it is the same as equation (29) which implies that the H∞ control performance is attained with a prescribed value of σ2. Usually it is difficult to determine a common solution


1300

of P = PT > 0 analytically, for inequality (33). Thus, we employ a linear matrix inequality (LMI) method to compute the value of P [25,26]. LMI problem can be solved efficiently using a technique so called the interior point. This can be achieved by firstly converting the matrix inequalities of (33) into the equivalent LMI’s using the following procedures: multiplying by P-1on the left and right sides of equation (33) and then defining X = P-1 we have the following result

for i, j = 1, 2, ..., n. Next we define Y j = L j X substitute into (41) yields

(41)

and

XAiT + XAi + BiY j + Y jT BiT + XAoT Ao X 1 ⎞ ⎛ + ( BoY j )T ( BoY j ) + ⎜ 2 + 2 ⎟ I + XQX < 0 σ ⎠ ⎝

(42)

for i, j = 1, 2, ..., n. Using Schur complements described in [20], we can re-write (31) in the following LMIs as ⎡ ⎧ XAiT + XAi + BiY j + Y jT BiT ⎫ ⎤ ⎢⎪ ⎥ ⎪ T X ⎬ ( BoY j ) ⎢⎨ ⎥ 1 ⎞ ⎛ +⎜2+ 2 ⎟I ⎪ ⎢⎪ ⎥ σ ⎠ ⎝ ⎭ ⎢⎩ ⎥ ⎢ ⎥ 0

⎡ ⎧ XAiT + XAi + BiY j + Y jT BiT ⎫ ⎤ ⎢⎪ ⎥ ⎪ T X ⎬ ( BoY j ) ⎢⎨ ⎥ 1 ⎞ ⎛ +⎜2+ 2 ⎟I ⎪ ⎢⎪ ⎥ σ ⎠ ⎝ ⎭ ⎢⎩ ⎥ ⎢ ⎥ 0). Hence, given below is the value matrix Q.

1302


0 0 0 0 0 ⎤ ⎡0.0100 ⎢ 0 0.0100 0 0 0 0 ⎥⎥ ⎢ ⎢ 0 0 0.0100 0 0 0 ⎥ Q=⎢ ⎥ 0 0 0.0100 0 0 ⎥ ⎢ 0 ⎢ 0 0 0 0 0.0100 0 ⎥ ⎢ ⎥ 0 0 0 0 0.0100 ⎥⎦ ⎢⎣ 0

14 12 10 ] 8 [m1 x 6

4 2

Note that, because the LMI equations contains matrix Q, then, Q must be defined. The simulation of LMIs gives the following results P= ⎡ 0.9433 − 0.0000 0.0000 0.1507 − 0.1719 0.0000⎤ ⎢−0.0000 0.9433 − 0.0000 0.1719 0.1506 − 0.0000 ⎥ ⎢ ⎥ ⎢ 0.0000 − 0.0000 0.0010 − 0.0000 0.0000 0.0005 ⎥ ⎢ ⎥, ⎢ 0.1507 0.1719 − 0.0000 0.0702 − 0.0000 − 0.0000 ⎥ ⎢−0.1719 0.1506 0.0000 − 0.0000 0.0702 0.0000 ⎥ ⎢ ⎥ ⎢⎣ 0.0000 − 0.0000 0.0005 − 0.0000 0.0000 0.0002 ⎥⎦ L1 =

⎡ 5.0157 5.7244 − 0.0001 2.3377 − 0.0001 − 0.0000 ⎤ ⎢ −2.08851.8297 0.0039 − 0.0000 0.8529 0.0020 ⎥ , ⎢ ⎥ ⎢⎣ −1.2774 1.1192 0.0210 − 0.0001 0.5227 0.0107 ⎥⎦ L2 =

⎡ 4.5718 5.2178 − 0.0001 2.1308 − 0.0001 − 0.0000 ⎤ ⎢ −1.9937 1.7467 0.0029 − 0.0001 0.8142 0.0015⎥ , ⎢ ⎥ ⎣⎢ −0.9259 0.8111 0.0165 − 0.0001 0.3789 0.0084 ⎦⎥ L3 = ⎡ 5.0149 5.7235 − 0.0001 2.3374 − 0.0001 − 0.0000 ⎤ ⎢ −2.0870 1.8284 0.0039 − 0.0001 0.8523 0.0020 ⎥ , ⎢ ⎥ ⎢⎣ −1.2777 1.1194 0.0210 − 0.0001 0.5228 0.0107 ⎥⎦

and

Position response in surge [x]

16

0 -2

0

0.2

0.4

0.6

0.8

1 time(s)

1.2

1.4

1.6

1.8

2

Fig. 1. Simulation curves of controller response for the position in surge direction [x=x1]. Position response in sway [y]

2 0 -2

] m [2 x

-4 -6 -8 -10 -12 0

0.1

0.2

0.3

0.4

0.5 time(s)

0.6

0.7

0.8

0.9

1

Fig. 2. Simulation curves of controller response for the position in sway direction [y=x2]. response in yaw [ ψ]

20 10 0 -10 ] -20 g e [d3 x -30

σ 2 = 1.6634 × 10−2 .

Note that the values of feedback gains L1, L2, and L3 are in multiple of 106. The above fuzzy controller satisfies Theorem 1. Thus, we conclude that the nonlinear DP system given in (21) is uniformly upper bound (UUB) and H∞ control performance in (19) is guaranteed for a given value of σ2. Then, based on controller gains computed (i.e., L1, L2 and L3) and nonlinear ship model given in (3), we can construct and simulate the responses of earth-fixed position ( x, y ) and heading ψ as well as body-fixed velocities v = [u, v, r ]T of the ship. The following initial conditions were set for simulation purpose: T

x(0) = ⎡⎣15m, 0.5m, 0 , 0.16ms −1 , 3ms −1 , 0.5rads −1 ⎤⎦ .

Hence, the simulation results of the fuzzy controller for the dynamic position (DP) system are shown in Fig. 1 to Fig. 8.

-40 -50 -60 -70

0

0.1

0.2

0.3

0.4

0.5 time(s)

0.6

0.7

0.8

0.9

1

Fig. 3. Simulation curves of controller response for the yaw angle [ψ=x3]. Figs. 1, 2 and 3 show the ship position [x, y] and heading ψ control response of the proposed controller respectively. It is clear that all figures (i.e., Fig. 1 to Fig. 3) the responses of variables [x, y, ψ] converge to zero at a finite time. This indicates that the proposed fuzzy controller uniformly upper-bound and H∞ control performance is guaranteed and hence the ship can maintain the desired position. Similarly Figs. 4, 5 and 6 show velocity response [u, v, r] of the controller. Also,

Robust Fuzzy Controller Design for Dynamic Positioning System of Ships response in surge velocity [u]

0.35

control force in surge

0

0.3

1303

-20

] N k [

τ1

-40

0.25 -60

0.2 ] /s [m4 x 0.15

10

20

30

40

50 60 time(s) control force in sway

70

80

90

100

0

10

20

30

40

70

80

90

100

0

10

20

30

40

70

80

90

100

10

] N k [

τ2

0.1 0.05 0

0

5

0

-5

0

0.5

1

time(s)

1.5

2

0

2.5

-0.5 -1

]

Fig. 4. Simulation curves of controller response for the surge velocity [u=x4].

m N k [

τ3 -1.5

-2 -2.5

response in sway velocity [v]

3

50 60 time(s) control moment in yaw

50 time(s)

60

Fig. 7. Control signal [uc1, uc2, uc1]. 2.5 2

2

1.5

1.5

1

1

0.5

0.5

0

0

Disturbance Signals b1(N)

] /s m [5 x

-0.5 0

0.5

1

time(s)

1.5

2

2.5

Fig. 5. Simulation curves of controller response for the sway velocity [v=x5]. response vwkocity in yaw [r]

0.5

b3(Nm)

-0.5 -1 -1.5 -2

0.45

b2(N)

0

10

20

30

40

50 time(s)

60

70

80

90

100

Fig. 8. Disturbance signal [b1, b2, b3].

0.4 0.35

simulation results, the proposed controller performs better compared to that given in [25,27] and [35], by having small response time of less than 0.5 seconds. Fig. 7 and Fig. 8 shows simulation results for control and disturbance signals respectively.

0.3

] /s d 0.25 a [r 6 x

0.2

0.15 0.1

6. CONCLUSION

0.05 0

0

0.5

1

time(s)

1.5

2

2.5

Fig. 6. Simulation curves of controller response for the yaw velocity [r=x6]. we can observe from Fig. 5 to Fig. 6 that the velocity responses converge to zero as time approaches to a given time. This is another proof that the objective of this paper is achieved. Additionally, it can be observed that the proposed controller is more effective since the response time is as small as below 0.2 seconds for position and heading, and 0.5 seconds for velocities. Based on the

In this paper, a fuzzy linear control method based on H∞ control technique for dynamic positioning (DP) system of the ships in the presence of bound disturbances has been designed. All state variables were assumed to be available for measurements. Takagi-Sugeno (TS) fuzzy model was employed to approximate the nonlinear DP system. With the application of H∞ control method, the effects of environmental disturbance can be eliminated to its minimum prescribed level. The stability of the controller was proven using Lyapunov stability theorems. Finally, the linear matrix inequality tool and eigenvalue problem method were utilized to find a

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positive definite matrix and controller gains. Simulation results were given to exemplify the applicability of the designed fuzzy controller. However, this work did not include adaptive laws for model and disturbances uncertainties. Therefore, for future studies the model and disturbance uncertainties should be taken into account. REFERENCES A. J. Sørensen, “A survey of dynamic positioning control systems,” Annual Reviews in Control, vol. 35, no. 1, pp. 123-136, 2011. [2] A. Jensen, J. G. Balchen, and S. Sælid, “Dynamic positioning of floating vessels based on Kalman filtering and optimal control,” Proc. of the 19th IEEE Conf. on Decision and Control, New York, pp. 852864, 1980. [3] A. Jensen, E. Mathisen, J. G. Balchen, and S. Sælid, “A dynamic positioning system based on Kalman filtering and optimal control,” Modeling, Identification and Control, vol. 1, no. 3, pp. 135-163, 1980. [4] T. I. Fossen and J. P. Strand, “Passive nonlinear observer design for ships using Lyapunov methods: full-scale experiments with a supply vessel,” Automatica, vol. 35, no. 1, pp. 3-16, 1999. [5] A. Loria, T. I. Fossen, and E. Panteley, “A separation principle for dynamic positioning of ships: theoretical and experimental results,” IEEE Trans. on Control Systems Technology, vol. 8, no. 2, pp. 332-343, 2000. [6] A. J. Smensenl, J. P. Strand, and H. Nybergl, “Dynamic positioning of ships and floaters in extreme seas,” Oceans MTS/IEEE, vol. 3, pp. 1849-1854, 2002. [7] J. G. Snijders and J. W. van der Woude, “Nonlinear observer design for dynamic positioning,” Proc. of Dynamic Positioning Conf., 2005. [8] K. D. Do, “Global robust and adaptive output feedback dynamic positioning of surface ships,” Proc. of IEEE International Conf. on Robotics and Automation, Roma, Italy, pp. 4271-4276, 2007. [9] H. Kaji and H. Katayama, “Digital control problems for dynamically positioned ships,” Proc. of 18th IEEE International Conf. on Control Applications, Part of IEEE Multi-conf. on Systems and Control, Saint Petersburg, Russia, pp. 1288-1293, 2009. [10] X. T. Chen and W. W. Tan, “A type-2 fuzzy logic controller for dynamic positioning systems,” Proc. of 8th IEEE International Conf. on Control and Automation, Xiamen, China, pp. 1013-101, 2010. [11] F. Benetazzo, G. Ippoliti, S. Longhi, P. Raspa, and A. J. Sørensen, “dynamic positioning of a marine vessel using DTVSC and robust control allocation,” Proc. of 20th Mediterranean Conf. on Control & Automation (MED), Barcelona, Spain, pp. 12111216, 2012. [12] A. Witkowska, “Dynamic positioning system with vectorial backstepping controller,” Proc. of 18th International Conf. on Methods and Models in Automation and Robotics (MMAR), IEEE Conf. Publi[1]

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Werneld Egno Ngongi was born on April, 1976 in Tanzania. Received the M.Sc. degree in Electronics Engineering (2008) and the B.Sc. degree in Electrical Engineering (2002) at the University of Dar es Salaam, Tanzania. He is working as assistant lecturer at Dar es Salaam Maritime Institute, Tanzania. Presently, he is a Ph.D. candidate in Control Theory and Control Engineering at the School of Information Science and Technology, Dalian Maritime University, Dalian, China. His research interests include intelligent control design, nonlinear control design, ship position and motion control.

Jialu Du was born in Liaoning Province, P. R. China, in 1966. She received her Ph.D. degree from Dalian Maritime University, China, in 2005. She was a visiting scholar in Norwegian University of Science and Technology from 2006 to 2007 and in University of California, San Diego from 2012 to 2013, respectively. Currently she is a professor at the School of Information Science and Technology, Dalian Maritime University, China. Her current research interests include nonlinear control theory, intelligent control, and ship motion control.

Rui Wang was born in Shandong Province, P. R. China, in 1990. He received his B.Sc. degree from Shandong University of Technology, China in 2012. Now he is working as a master student at the Department of Information Science and Technology, Dalian Maritime University, China. His current research interests include model prediction control, ship motion control.