Optimal control of Takagi–Sugeno fuzzy-model-based systems representing dynamic ship positioning systems

Applied Soft Computing 13 (2013) 3197–3210

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Applied Soft Computing journal homepage: www.elsevier.com/locate/asoc

Optimal control of Takagi–Sugeno fuzzy-model-based systems representing dynamic ship positioning systems Wen-Hsien Ho a , Shinn-Horng Chen b , Jyh-Horng Chou b,c,∗ a

Department of Healthcare Administration and Medical Informatics, Kaohsiung Medical University, 100 Shi-Chuan 1st Road, Kaohsiung 807, Taiwan, ROC Department of Mechanical Engineering, National Kaohsiung University of Applied Sciences, 415 Chien-Kung Road, Kaohsiung 807, Taiwan, ROC Institute of System Information and Control, National Kaohsiung First University of Science and Technology, 1 University Road, Yenchao, Kaohsiung 824, Taiwan, ROC b c

a r t i c l e

i n f o

Article history: Received 1 February 2012 Received in revised form 13 January 2013 Accepted 7 February 2013 Available online 19 March 2013 Keywords: Dynamic ship positioning systems Hybrid Taguchi-genetic algorithm Orthogonal functions Quadratic finite-horizon optimal control Takagi–Sugeno fuzzy model

a b s t r a c t Orthogonal function approach (OFA) and the hybrid Taguchi-genetic algorithm (HTGA) are used to solve quadratic finite-horizon optimal controller design problems in both a fuzzy parallel distributed compensation (PDC) controller and a non-PDC controller (linear state feedback controller) for Takagi–Sugeno (TS) fuzzy-model-based control systems for dynamic ship positioning systems (TS-DSPS). Based on the OFA, an algorithm requiring only algebraic computation is used to solve dynamic equations for TS-fuzzy-modelbased feedback and is then integrated with HTGA to design quadratic finite-horizon optimal controllers for TS-DSPS under the criterion of minimizing a quadratic finite-horizon integral performance index, which is also converted to algebraic form by the OFA. Integration of OFA and HTGA in the proposed approach enables use of simple algebraic computation and is well adapted to the computer implementation. Therefore, it facilitates design tasks of quadratic finite-horizon optimal controllers for the TS-DSPS. The applicability of the proposed approach is demonstrated in the example of a moored tanker designed using quadratic finite-horizon optimal controllers. © 2013 Elsevier B.V. All rights reserved.

1. Introduction A current issue in the literature on fuzzy-model-based control systems is the nonlinear control problem in dynamic ship positioning systems (DSPS) for controlling the thrusters and propellers of marine surface vessels when maneuvering at low speed or when maintaining a stationary position [1–6]. The DSPS maintains a floating vessel in the vicinity of a reference point (e.g., vertically above a well) and stabilizes the heading of a vessel. These maneuvers require controllers for multiple thrusters, including the main propellers aft of the ship, tunnel thrusters, and azimuth (rotatable) thrusters mounted under the hull. Since the mathematical model used for control in this study does not account for pitch and roll motion, Fig. 1 shows that only 3 degrees of freedom (DOF) are considered as discussed further in Section 2 [7]. That is, position and heading are controlled only for the horizontal plane. The practical implementation of this technique is in maintaining the stationary position of a ship by formulating the control of longitudinal

∗ Corresponding author at: Institute of System Information and Control, National Kaohsiung First University of Science and Technology, 1 University Road, Yenchao, Kaohsiung 824, Taiwan. Tel.: +886 7 6011000; fax: +886 7 6011066. E-mail addresses: [email protected] (W.-H. Ho), [email protected] (J.-H. Chou). 1568-4946/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.asoc.2013.02.019

movement (surge, earth-fixed position on X-axis), transverse movement (sway, earth-fixed position on Y-axis), and heading (yaw) as a 3 DOF problem in the horizontal plane. Based on 3 DOF errors resulting in offset of the vessel, the controller determines the thruster commands needed to return the vessel to its initial position. The thruster commands needed for correction are updated periodically [1]. The recent literature shows a rapidly growing interest in applying the Takagi–Sugeno (TS) fuzzy-model-based approach to nonlinear control systems [8–13]. The fuzzy model proposed by [14] is described by the fuzzy IF-THEN rules that represent local linear input-output relations of a nonlinear control system. The main feature of a TS fuzzy model is its use of a linear system model to express the local dynamics of each fuzzy rule. Once the TS fuzzy models are obtained, the linear control methodology can be used to design the local state feedback controllers for each linear model. However, recent applications of the TS fuzzy model in [4–6], in which nonlinear DSPS is represented using linear control methodology, have only designed the stabilizing controllers of the TS-fuzzy-model-based control system representing the DSPS (TS-DSPS), where the controllers belong to the type of parallel distributed compensation (PDC). Despite their success, it has become evident that many research issues for the TS-DSPS remain to be addressed. In control system design, the control objective of minimizing a quadratic finite-horizon integral performance criterion is often achieved by synthesizing a quadratic finite-horizon optimal

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Therefore, computational complexity is substantially reduced. Thus, the approach facilitates design tasks for quadratic finitehorizon optimal controllers for TS-DSPS. This paper is organized as follows. Section 2 describes the problem statement of the TS-DSPS. Section 3 presents a new approach for integrating OFA and HTGA in the designs for both a quadratic finite-horizon optimal fuzzy PDC controller and a quadratic finite-horizon optimal non-PDC controller for a TSDSPS while minimizing a defined quadratic finite-horizon integral performance index. Section 4 gives an illustrative example to demonstrate the applicability of the proposed approach. Finally, Section 5 concludes the study. 2. Problem statement of TS-DSPS

Fig. 1. Earth-fixed and vessel-fixed coordinate frames.

controller [15]. Thus, a key issue is the design of quadratic finitehorizon optimal controllers for the TS-DSPS. However, a literature review shows that this design issue has not been addressed. Recent studies such as [16–21], however, have applied orthogonal function approach (OFA) [22,23] to solve dynamic equations in TS-fuzzy-model-based systems. The OFA is characterized by the simplification of TS-fuzzy-model-based dynamic equations by converting them to straightforward algebraic equations that are easily solved by computer. Hence, the OFA has a relatively shorter and simpler solution procedure. The OFA has also shown highly satisfactory solution results. Therefore, the objectives of this study were to design both a quadratic finite-horizon optimal fuzzy PDC controller and a quadratic finite-horizon optimal non-PDC controller (quadratic finite-horizon optimal linear state feedback controller) for the TS-DSPS by integrating the OFA and the hybrid Taguchigenetic algorithm (HTGA) for direct minimization of a defined quadratic finite-horizon integral performance index representing the control objective. This study applies the HTGA introduced in [24,25] not only because it effectively finds optimal or close-tooptimal solutions, but also because its solution results are more robust compared to improved genetic algorithms reported in the literature. Therefore, to achieve the objective using the elegant operational properties of the OFA, this study first developed a computational algorithm for solving TS-DSPS feedback dynamic equations. The novel approach is the expression of state variables in terms of orthogonal functions. The proposed method simplifies the procedure for solving TS-DSPS feedback dynamic equations as successive solutions for a system of recursive formulae taking only the expansion coefficients. Based on the presented recursive formulae, only a straightforward algebraic computation is needed to perform the computation algorithm. The developed algorithm is then integrated with the HTGA in designs for both the quadratic finite-horizon optimal fuzzy PDC controller and the quadratic finite-horizon optimal non-PDC controller (quadratic finite-horizon optimal linear state feedback controller) of the TS-DSPS under the criterion of minimizing a quadratic finite-horizon integral performance index, where the OFA also converts the quadratic finite-horizon integral performance index to algebraic form. The proposed approach of integrating OFA and HTGA is non-differential, nonintegral, efficient, and well-suited for computer implementation.

In the normalized model of horizontal motion in a DSPS, the motion components are surge, sway and yaw [2]. This study evaluated the case of a DSPS controlled exclusively by thrusters. Anchors can also be included in the model of ship motion to analyze thruster-assisted mooring. Damping forces are assumed to be linear since the speed of the ship is always slow during dynamic positioning. The DSPS model also assumes a homogeneous mass distribution and xz-plane symmetry. The origin of the coordinates is set at the center line of the ship. Where the position (x, y) and heading of the vessel relative to an earth-fixed frame are expressed in vector form as  = [x, y, ]T , let the velocities be decomposed in a vessel-fixed reference frame as  = [u, v, r]T . These three modes are referred to as the surge, sway, and yaw modes of a ship. The origin of a vessel-fixed frame located at the vessel center line is expressed as the distance from the center of gravity. Since the mean roll and pitch angles are zero, DSPS applications typically assume that motions induced by rolling and pitching of the ship are negligible. Therefore, the rigid-body dynamic equations for motion of the vessel (surge, sway and yaw) obtain the vessel-fixed velocity vectors relative to the earth-fixed velocity (see Fig. 1, [7]) are (t) ˙ = J((t))(t),

(1)

and B(t) ˙ + D(t) + G(t) = (t),

(2)

(t)]T

denotes the earth-fixed orientawhere (t) = [x(t), y(t), tion vector describing the surge, sway and yaw modes, (t) = [u(t), v(t), r(t)]T denotes the body-fixed linear and angular velocity vector describing the surge, sway and yaw modes, (t) = [ 1 (t),  2 (t),  3 (t)]T denotes the input vector for the control forces and moment provided by the thruster system (main propellers aft of the ship  1 (t) tunnel thrusters  2 (t) and azimuth (rotatable) thrusters  3 (t)), and T means transpose of a matrix. In Eqs. (1) and (2),

⎡

− sin( (t))

cos( (t))

J((t)) = ⎣ sin( (t))

cos( (t))

0

0

0

⎤

0⎦

(3)

1

denotes the transformation matrix in yaw (assuming that J((t)) is non-singular), B denotes the inertia matrix including the hydrodynamic added inertia, D denotes the damping matrix, and G = diag {g11 , g22 , g33 } denotes the system state matrix of the surge, sway and yaw modes for the anchor forces and moment. Eqs. (1) and (2) are presented in detail in [2]. The starboard-port symmetry of ships implies that B and D have the following structure:

⎡ ⎢

b11

0

0

⎤ ⎥

B=⎣ 0

b22

b23 ⎦

0

b32

b33

(4a)

W.-H. Ho et al. / Applied Soft Computing 13 (2013) 3197–3210

3199

Fig. 2. Block diagram of a typical DSPS control system.

and

⎡ ⎢

d11

0

where

⎤

0

(t) = [1 (t), 2 (t), 3 (t), 4 (t), 5 (t), 6 (t)]

⎥

D=⎣ 0

d22

d23 ⎦ ,

0

d32

d33

(t) ˙ = A1 (t) + A2 (t) + A3 (t),

⎡

a111

a112

a113

A1 = −B−1 G = ⎣ a121

a122

a123

a131

a132

a133

a211

a212

a213

A2 = −B−1 D = ⎣ a221

a222

a223

a231

a232

a233

⎢

⎡ ⎢

and

⎡

a311

a312

a313

A3 = B−1 = ⎣ a321

a322

a323

a331

a332

a333

⎢

T

(t), u(t), v(t), r(t)] ,

(8)

and

that is, no coupling occurs between the surge and the sway–yaw subsystems. For this system, Eq. (2) can be re-written as

where

= [x(t), y(t),

(4b)

T

(5)

⎤ ⎥ ⎦,

(6a)

⎤ ⎥ ⎦,

(6b)

⎤ ⎥ ⎦.

(6c)

Combining Eqs. (1) and (5) in the space of state variables obtains the following state equations: ˙ 1 (t) = cos (3 (t))4 (t) − sin (3 (t))5 (t),

(7a)

˙ 2 (t) = sin(3 (t))4 (t) + cos(3 (t))5 (t),

(7b)

˙ 3 (t) = 6 (t),

(7c)

(t) = [1 (t), 2 (t), 3 (t)]T .

Fig. 2 shows an overall block diagram of a typical DSPS control system [26]. The set point for the control system is the ship position set by the operator. Comparison of the set point with the actual vessel position produces an error signal that the controller uses to derive a thrust demand for restoring the position of the ship. Thrust demands obtained for each of three axes, fore/aft (X), port/starboard (Y) and heading ( ) are combined and allocated to each thruster of the ship. The output of this algorithm, which is designated the thruster transform, is (t) demand to the thrusters. By responding to these commands, the thrusters (t) correct the position of the vessel, which then closes the control loop. Additionally, the proposed approach of using sector nonlinearity in the fuzzy model construction easily derives both the fuzzy sets of the premise part and the linear dynamic model of the consequent part of the TS fuzzy model from the physical model of the given nonlinear dynamic system [27]. This approach ensures an exact fuzzy model construction for a given nonlinear dynamic model [27]. The advantage of the sector nonlinearity approach is the elimination of approximation error between the original nonlinear system and its TS-fuzzy-model-based system [27]. Therefore, this study applies sector nonlinearity approach [27] so that the nonlinear equation for the DSPS motion can be exactly represented as a TS-fuzzy-model-based dynamic system. Assuming yaw angle  3 (t) = (t) varies between −/2 and /2, the exact TS-DSPS is obtained by applying the following rules: Rule 1 : IF z1 (t) is M11 and z1 (t) is M12 , ˙ ˜ 1 (t) + B˜ 1 (t), =A THEN (t) Rule 2 : IF z1 (t) is M21 and z2 (t) is M22 ,

˙ 4 (t) = a111 1 (t) + a112 2 (t) + a113 3 (t) + a211 4 (t) + a212 5 (t) + a213 6 (t) + a311 1 (t) + a312 2 (t) + a313 3 (t),

˙ ˜ 2 (t) + B˜ 2 (t), =A THEN (t) (7d)

Rule 3 : IF z1 (t) is M31 and z2 (t) is M32 , ˙ ˜ 3 (t) + B˜ 3 (t), =A THEN (t)

˙ 5 (t) = a121 1 (t) + a122 2 (t) + a123 3 (t) + a221 4 (t) + a222 5 (t) + a223 6 (t) + a321 1 (t) + a322 2 (t) + a323 3 (t),

Rule 4 : IF z1 (t) is M41 and z2 (t) is M42 , (7e)

and ˙ 6 (t) = a131 1 (t) + a132 2 (t) + a133 3 (t) + a231 4 (t) + a232 5 (t) + a233 6 (t) + a331 1 (t) + a332 2 (t) + a333 3 (t),

(7f)

(9)

˙ ˜ 4 (t) + B˜ 4 (t), =A THEN (t)

(10a)

(10b)

(10c)

(10d)

where z1 (t) = sin 3 (t) and z2 (t) = cos 3 (t) are the premise variables; (t) = [ 1 (t),  2 (t),  3 (t),  4 (t),  5 (t),  6 (t)]T denotes the state vector;  1 (t) and  4 (t) denote the earth-fixed position on the X-axis and the body-fixed velocity on the x-axis, respectively; 4 (t) = ˙ 1 (t),

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W.-H. Ho et al. / Applied Soft Computing 13 (2013) 3197–3210

 2 (t) and  5 (t) denote the earth-fixed position on the Y-axis and the body-fixed velocity on the y-axis, respectively; 5 (t) = ˙ 2 (t),  3 (t) and  6 (t) denote the yaw angle and yaw angular velocity, respectively; and 6 (t) = ˙ 3 (t), (t) = [ 1 (t),  2 (t),  3 (t)]T denotes the input vector with the control forces and the moment provided by the thruster system,  3 (t) ∈ [− /2, /2],

⎡

0

0

0

1

−1

0

⎤

⎡

0

0

is minimized to represent the control objective of maintaining the system state, (t) is minimized as close as possible to the desired state values of zero with a minimum expenditure of control effort, tf denotes a small time interval chosen for independent variable t, q is a positive integer specified by the designer, Q is a symmetric 0

0

−1

0

⎤

⎢ 0 ⎢ 0 0 0 1 1 0 ⎥ 0 0 1 0 0 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ 0 ⎥ ⎢ 0 0 0 0 1 0 0 0 0 0 1 ⎥ ⎢ ⎥ ⎢ ⎥ ˜1 = ⎢ A ⎥ , A˜ 2 = ⎢ a111 a112 a113 a211 a212 a213 ⎥ , ⎢ a111 a112 a113 a211 a212 a213 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ a121 a122 a123 a221 a222 a223 ⎦ ⎣ a121 a122 a123 a221 a222 a223 ⎦ a131

a132

a133

a231

a232

a233

0

0

0

1

1

0

⎡

⎤

a131

a132

a133

a231

a232

a233

0

0

0

0

1

0

⎡

⎤

⎢ 0 ⎢ 0 0 0 −1 1 0 ⎥ 0 0 −1 0 0 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ 0 ⎢ 0 0 0 0 0 1 ⎥ 0 0 0 0 1 ⎥ ⎢ ⎥ ⎢ ⎥ ˜3 = ⎢ A ⎥ , A˜ 4 = ⎢ a111 a112 a113 a211 a212 a213 ⎥ , ⎢ a111 a112 a113 a211 a212 a213 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ a121 a122 a123 a221 a222 a223 ⎦ ⎣ a121 a122 a123 a221 a222 a223 ⎦ a131

a132

a133

a231

⎡

a232

a131

a233

0

0 0

⎢ B˜ 1 = B˜ 2 = B˜ 3 = B˜ 4 = B˜ = ⎣ 0

a311

a321

a331

0 0

a312

a322

a332

0

0 0

a313

a323

a333

a132

⎤T

a231

a233

3. Quadratic finite-horizon optimal controllers design for the TS-DSPS

M11 (z1 (t)) = M21 (z1 (t)) =

z1 (t) + 1 , 2

(11a)

M31 (z1 (t)) = M41 (z1 (t)) =

1 − z1 (t) , 2

(11b) (11c)

and

For the TS-DSPS in Eq. (12), this section describes how both the OFA and the HTGA are used to find both the quadratic finite-horizon optimal fuzzy PDC controller and the quadratic finite-horizon optimal non-PDC controller for the TS-DSPS that minimize the quadratic finite-horizon integral performance index in Eq. (14). 3.1. Quadratic finite-horizon optimal fuzzy PDC controller

M22 (z2 (t)) = M42 (z2 (t)) = 1 − z2 (t).

(11d)

The resulting TS-DSPS inferred from Eq. (10) is represented as ˙ (t) =

4 

Consider the following fuzzy PDC controller for the TS-DSPS: 4 

˜ i (t) + B˜ i (t)), hi (z(t))(A

(12)

(t) = −

hi (z(t)) Fi (t),

(15)

i=1

i=1

where Fi (i = 1, 2, 3, 4) denote the 3 × 6 local feedback gain matrices. Substituting Eq. (15) into Eq. (12) obtains

in which z(t) = [z1 (t), z2 (t)] denotes the premise vector, h1 (z(t)) = M11 (z1 (t))M12 (z2 (t)),

(13a)

h2 (z(t)) = M21 (z1 (t))M22 (z2 (t)),

(13b)

h3 (z(t)) = M31 (z1 (t))M32 (z2 (t)),

˙ (t) =

4 4  

˜ i − B˜ i Fj )(t). hi (z(t))hj (z(t))(A

(16)

i=1 j=1

(13c)

The sufficient condition for quadratic stability is

and h4 (z(t)) = M41 (z1 (t))M42 (z2 (t)).

4

(13d)

Therefore, for all t, hi (z(t)) ≥ 0 and h (z(t)) = 1. i=1 i The considered problem is finding both the quadratic finitehorizon optimal fuzzy PDC controller and the quadratic finitehorizon optimal non-PDC controller (quadratic finite-horizon optimal linear state feedback controller) for the TS-DSPS in Eq. (12) such that the quadratic finite-horizon integral performance index





˜J =

qtf



T

T

q−1   k=0

˜ the equilibrium of a fuzzy Theorem 1. When B˜ 1 = B˜ 2 = B˜ 3 = B˜ 4 = B, control system (16) is globally and asymptotically stable if there exists a common positive definite matrix P such that T

˜ i − B˜ i Fi } P + P{A ˜ i − B˜ i Fi } < 0, {A

i = 1, 2, 3, 4.

(17)

Proof. The detailed procedure of this proof is shown in [27]. Now, assume that all elements of ␰(t) are absolutely integrable within ktf ≤ t ≤ (k + 1)tf , and define t = ktf + ε,

[ (t)Q(t) +  (t)R(t)]dt

0

=

a232

positive-semidefinite matrix, and R is a symmetric positive-definite matrix.

⎥ ⎦ ,

M12 (z2 (t)) = M32 (z2 (t)) = z2 (t),

a133

(18)

and (k+1) tf

ktf

[ T (t)Q(t) +  T (t)R(t)]dt

(14)

k = (ktf ), in which k = 0, 1, 2,. . ., q − 1, and 0 ≤ ␧ ≤ tf .

(19)

W.-H. Ho et al. / Applied Soft Computing 13 (2013) 3197–3210

Within ktf ≤ t ≤ (k + 1)tf , state vector ␰(t) can then be approximated as the truncated orthogonal function representation

(t) =

m−1 

(k) s Ts (t) = ˜ (k) T (t),

(20)

s=0

where m is the number of terms required for the orthogonal functions, T(t) = [T0 (t), T1 (t), . . ., Tm−1 (t)]T denotes the m × 1 orthogonal basis vector, Ti (t) (i = 0, 1,. . ., m − 1) denote the orthogonal (k) functions, s (s = 0, 1,. . ., m − 1) is the 6 × 1 coefficient vector, and (k) (k) (k) ˜ (k) = [0 , 1 , . . . , m−1 ] is the 6 × m coefficient matrix. Theoretically, the accuracy of approximate solutions should increase with the value of m. Based on the observations reported in [22] and [23], m∈ [4,8] generally enables the shifted Chebyshev function and the shifted Legendre function to obtain satisfactorily accurate results. After substituting Eq. (15) and the truncated orthogonal function representation of ␰(t) in Eq. (20) into the quadratic finite-horizon integral performance index in Eq. (14), the quadratic finite-horizon integral performance index ˜J becomes

˜J =

q−1 

⎡

⎛

trace ⎣W (˜ (k) )

T

⎝Q +

k=0

4 4  

⎞

⎤

hi (zk )hj (zk )FiT RFj ⎠ (˜ (k) )⎦ .

Since the degree of fulfillment of the antecedent must be computed before the consequent output can be inferred, and since tf is a small time interval, the value of hi (z(t)) within ktf ≤ t ≤ (k + 1)tf , is hi (z(k tf )) and is assumedly constant where hi (zk ) = hi (z(ktf )), and W denotes the product-integration matrix of two orthogonal basis vectors. The constant matrix W depends on the selected orthogonal function vector T(t). The constant matrices W for the shifted Chebyshev function were given in [17]. Integrating Eq. (12) from t = ktf to t = t within ktf ≤ t ≤ (k + 1)tf obtains 4 

  ˜i hi (zk ) A

i=1



t

t

(t)dt + B˜ i

(22)

ktf

where the value of hi (z(t)) is assumed to be hi (z(k tf )) within k tf ≤ t ≤ (k + 1) tf because the degree of fulfillment of the antecedent must be computed before the consequent output can be inferred and because tf is a small time interval. By applying the following integral property of the orthogonal functions



t

T (t)dt = HT (t),

(23)

ktf

and applying Eqs. (15), (19) and (20), Eq. (22) can be reformulated as

˜ (k) − [k , 0, 0, . . . , 0] =

4 4  



−

1 2

0

−1 2m(m − 2)

0 −

1 8

i=1 j=1

(24) where H is the operational matrix of integration for the orthogonal functions [17]. For example, the constant matrix H obtained by the shifted Chebyshev function is [17]

0

0

···

0

0

1 4

0

···

0

0

0

1 8

···

0

0

. . .

. . .

. . .

. . .

0

0

···

. . . 1 4(m − 3)

0

0

···

0

⎤

0

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ 0 ⎥ ⎥ ⎥. 0 ⎥ ⎥ ⎥ . ⎥ . ⎥ . ⎥ −1 ⎥ 4(m − 1) ⎥ ⎦ 0

0 1 4(m − 2)

0

Eq. (24) can be rewritten as ˜ (k) −

4 4  

˜ i − B˜ i Fj )˜ (k) H = Q˜ (k) , hi (zk )hj (zk )(A

(26)

i=1 j=1

where Q˜ (k) = [k , 0, 0, . . . , 0] is a 6 × m matrix. By applying the Kronecker product, Eq. (26) directly derives the explicit form for the coefficient matrix ˜ (k) as

⎡

ˆ (k) = ⎣I6m −

4 4  

⎤−1

˜ i − B˜ i Fj ))⎦ hi (zk )hj (zk )(H T ⊗ (A

Qˆ (k) ,

(27)

i=1 j=1



where I6m denotes the 6m × 6m identity matrix, ˆ (k) = (k)T

0

(k)T

, 1

(k)T

T

, . . . , m−1

,

T Qˆ (k) = [kT , 0T , 0T , . . . , 0T ] ,

⊗

and ˜ (k)

denotes the Kronecker product [28]. This implies that can be obtained from Eq. (27). If one set of local feedback gain matrices {F1 , F2 , F3 , F4 } is given, ˜ (k) (k = 0, 1,. . ., q − 1) can then be calculated from the following algorithm, which requires only algebraic computation.  Detailed Steps

Step 1: For a small time interval tf , a specified positive integer q, and an initial state vector ␰(0), set k = 0 Step 2: Calculate hi (z(ktf )) for i = 1, 2, 3, 4. Step 3: Calculate ˆ (k) from Eq. (27). Step 4: Compute  k + 1 by using k+1 = ((k + 1)tf ) = ˜ (k) T ((k + 1)tf ). Step 5: Set k = k + 1 If k > q − 1 then stop; otherwise, go to Step 2. The above algorithm clearly shows that ˜ (k) (k = 0, 1,. . ., q − 1) can be determined by specifying one set of local feedback gain matrices {F1 , F2 , F3 , F4 }; the value of the performance index in Eq. (21) corresponding to set {F1 , F2 , F3 , F4 } can then be calculated. Given another set of local feedback gain matrices {F1 , F2 , F3 , F4 } Eq. (21) obtains another performance index value. That is, the performance index value in Eq. (21) is actually dependent on the set of local feedback gain matrices {F1 , F2 , F3 , F4 }, i.e., ˜J = G(f111 , f112 , . . . , f436 ),



˜ i − B˜ i Fj ˜ (k) H, hi (zk )hj (zk ) A

···

(25)

Algebraic algorithm.

 (t)dt ,

ktf

1 2

⎢ ⎢ 1 ⎢ 8 ⎢ ⎢ 1 ⎢ − ⎢ 6 ⎢ ⎢ 1 H = tf ⎢ − 16 ⎢ ⎢ . ⎢ . ⎢ . ⎢ −1 ⎢ ⎢ 2(m − 1)(m − 3) ⎣

i=1 j=1

(21)

(t) − (ktf ) =

⎡

3201

(28)

where fijk (i = 1, 2, 3, 4, j = 1, 2, 3 and k = 1, 2, 3, 4, 5, 6) denote the elements of the local gain matrices Fi (i = 1, 2, 3, 4). Therefore, the design problem of the quadratic finite-horizon optimal fuzzy PDC controller for the TS-DSPS is to optimize fijk such that the performance index in Eq. (21) is minimized. The problem can be expressed as ˜ (f111 , f112 , . . . , f436 ) min ˜J = G

(29a)

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W.-H. Ho et al. / Applied Soft Computing 13 (2013) 3197–3210

subject to |fijk | ≤ Cijk ,

(29b)

for i = 1, 2, 3, 4 j = 1, 2, 3 and k = 1, 2, 3, 4, 5, 6, where Cijk are the positive real numbers given from the practical consideration. Eq. (29a) is a nonlinear function with continuous variables. To optimize nonlinear functions with the continuous variables, the HTGA discussed in [24,25,29–33] obtains results that are more accurate and robust compared to existing improved genetic algorithms reported in the literature. Therefore, this study used the HTGA to search for the optimal solution to the problem in Eq. (29). 3.2. Quadratic finite-horizon optimal non-PDC controller This subsection considers the following non-PDC controller (linear state feedback controller) for the TS-DSPS: (t) = −F(t),

(30)

where F denotes the 3 × 6 feedback gain matrix. Substituting Eq. (30) into Eq. (12) obtains ˙ (t) =

4 

˜ i − B˜ i F)(t). hi (z(t))(A

(31)

i=1

Besides, we have the following sufficient condition for the quadratic stability. Theorem 2. The equilibrium of a fuzzy control system (31) is globally and asymptotically stable if there exists a common positive definite matrix P such that T

˜ i − B˜ i F} P + P{A ˜ i − B˜ i F} < 0, {A

i = 1, 2, 3, 4.

(32)

Proof. The detailed procedure of this proof is the same as that for Theorem 1 presented in [27]. After applying the same procedures discussed in Section 3.1 above, performance index ˜J and the explicit form for the coefficient matrix ˜ (k) are, respectively, ˜J =

q−1 

T

trace[W (˜ (k) ) (Q + F T RF)(˜ (k) )],

(33)

k=0

and ˆ (k)



 =

I6m −

4 



T



˜ i − B˜ i F) hi (zk ) H ⊗ (A

−1 Qˆ (k) .

(34)

i=1

If one gain matrix F is given, then ˜ (k) (k = 0, 1,. . ., q − 1) can be calculated from the algebraic algorithm presented in Section 3.1, where Eq. (27) in Step 3 of the above algorithm is replaced with Eq. (34). According to the above algebraic algorithm, if one gain matrix F is specified, then ˜ (k) (k = 0, 1,. . ., q − 1) can be determined, which enables calculation of the performance index in Eq. (33) corresponding to this gain matrix F. Given a different gain matrix F, Eq. (33) obtains a different performance index value. That is, the value of the performance index in Eq. (33) is also dependent on the gain matrix F, i.e., ˜ 11 , f12 , . . . , f36 ), ˜J = G(f

(35)

where fij (i = 1, 2, 3 and j = 1, 2, 3, 4, 5, 6) denote the elements of gain matrix F. Hence, the design problem for the optimal non-PDC controller for the TS-DSPS is to optimize fij such that performance index in Eq. (33) is minimized. The equivalent problem is ˜ 11 , f12 , . . . , f36 ) min ˜J = G(f

(36a)

subject to |fij | ≤ Cij ,

(36b)

for i = 1, 2, 3 and j = 1, 2, 3, 4, 5, 6, where Cij are the positive real numbers obtained in the practical application.

The detailed steps of the HTGA described in [24,25,29–33] can then be performed to search for the optimal solution for the problem in Eq. (36), where Eq. (36a) is a nonlinear function with continuous variables.  Remark 1. Recent studies such as [27,34–36] have proposed LMI-based approaches to design quadratic-optimal controllers for TS-fuzzy-model-based control systems. Quadratic-optimal fuzzy PDC controller designs in Tanaka and Wang [27], Tong et al. [34], Zheng et al. [35] and Li [36] minimized the upper bound of a quadratic infinite-horizon integral performance index. However, if the design consideration is directly minimizing a quadratic infinite-horizon integral performance index, LMI-based approaches presented by Tanaka and Wang [27], Tong et al. [34], Zheng et al. [35] and Li [36] do not effectively solve the quadratic infinite-horizon optimal fuzzy PDC control problem in TS-fuzzymodel-based control systems. Practical applications require good solution performance in finite-horizon optimal control problems [37]. However, LMI-based approaches proposed by Tanaka and Wang [27], Tong et al. [34], Zheng et al. [35] and Li [36] have shown limited effectiveness for minimizing the finite-horizon performance index when solving the quadratic finite-horizon optimal PDC control problem of TS-fuzzy-model-based control systems. To solve optimal PDC control problems in TS-fuzzy-model-based control systems, other issues must be resolved, such as simplifying the computation needed to solve the quadratic finite-horizon optimal PDC control problem in TS-fuzzy-model-based control systems and ensuring that some characteristics of closed-loop systems are preserved [38]. Therefore, a key issue when developing TSfuzzy-model-based control systems is the computational methods for designing the quadratic-finite-horizon-optimal PDC controllers, where the performance index is directly minimized. In contrast, Wu and Lin used a local concept approach and the Pontryagin maximum principle to design the quadratic optimal fuzzy PDC controller in [39] and [40] but did not consider the effect of the coupled terms on the quadratic optimal fuzzy PDC controller design in [41]. That is, considering how the coupled terms of each submodel affect the quadratic optimal fuzzy PDC controller design, and in order to use PDC control as a function of the system state such that the TS-fuzzymodel-based system can be controlled by feedback, conventional optimal control methods are ineffective for finding the feedback gain matrices of the closed-loop quadratic finite-horizon optimal fuzzy PDC controller for a TS-fuzzy-model-based control system [41]. Therefore, the novel numerical optimization method proposed in this study integrates OFA and HTGA to find the quadratic finitehorizon optimal controller gains of both the closed-loop fuzzy PDC controller and the closed-loop non-PDC controller for the TS-fuzzy-model-based control system under the criterion of minimizing a quadratic finite-horizon integral performance index. The proposed approach is not only applicable for finding the feedback gain matrices of both the quadratic finite-horizon optimal fuzzy PDC controller and the quadratic finite-horizon optimal non-PDC controller for the TS-fuzzy-model-based dynamic system under minimization of a defined quadratic finite-horizon integral performance index, it also has practical applications in cases of constrained elements of feedback gain matrices. 4. Illustrative example Next, we consider the example of a moored tanker scaled according to the system matrices given for the bis-system. In this example of a mooring arrangement between a floating storage body moored in deep water and a shuttle tanker, a single point buoyant member is used for mooring the shuttle tanker in the offloading position relative to a floating production, storage and offloading

W.-H. Ho et al. / Applied Soft Computing 13 (2013) 3197–3210

vessel (FPSO) with a link between the floating storage body and the single point buoyant member. A DSPS installed on the mooring buoy is used to position the shuttle tanker according to environmental conditions or to move the tanker to a desired position during loading [2]:

⎡

B=⎣

⎡ D=⎣

and

⎡

G=⎣

1.0852

0

−0.4087 ⎦ ,

0

2.0575

0

−0.4087

0.0865

0

0

0.0762

0.1510 ⎦ ,

0

0.0151

0.0031

0.0389

0

0

0.0266

0

0

0

⎡

0

0

(38)

⎤

0⎦.

(39)

0

0

−0.0394

0

⎢ 0 0 ⎢ ⎢ 0 0 ˜2 = ⎢ A ⎢ −0.0358 0 ⎢ ⎣ 0 −0.0208 0

[ T (t)Q(t) +  T (t)R(t)]dt

=

−0.0394

0

1

−1

0

1

1

0

0

0

0

1

0

⎤

⎥ ⎥ ⎥ ⎥, ⎥ 0 −0.0797 0 0 ⎥ 0 0 −0.0818 −0.1224 ⎦ 0

0

0

−1

0

1

0

0

0

0

0

1

0

⎤

⎥ ⎥ ⎥ ⎥, ⎥ 0 −0.0797 0 0 ⎥ 0 0 −0.0818 −0.1224 ⎦ −0.2254 −0.2468 (41)

0



F1 =

0

0

⎢ 0 0 ⎢ ⎢ 0 0 ˜3 = ⎢ A ⎢ −0.0358 0 ⎢ ⎣ −0.0208 0 0

⎡

0

−0.0394 0

⎢ 0 0 ⎢ ⎢ 0 0 ˜4 = ⎢ A ⎢ −0.0358 0 ⎢ ⎣ 0 −0.0208 0 and

−0.0394

⎡

0

1

1

0

0

−1

1

0

0

−0.2254 −0.2468 (42)

0

0

0

1

0

−1

0

0

0

0

0

1

0

0

0

⎤

⎥ ⎥ ⎥ ⎥, ⎥ 0 −0.0797 0 0 ⎥ 0 0 −0.0818 −0.1224 ⎦

0 0

B˜ 1 = B˜ 2 = B˜ 3 = B˜ 4 = B˜ = ⎣ 0 0 0 0

−0.2254 −0.2468 (43)

0

⎤T

0

0.9215

0

0

0

0.7802

1.4811 ⎦ .

0

0

1.4811

7.4562 (44)

8.8828 5.5604

 F2 =

 F3 =

(45)

−7.6655

4.4991

8.5908

−9.3744 −10.0000

3.2725

6.9799

0.6383

−5.5728

−2.2292

−2.2743 −0.9285 3.2659

−6.0266

−7.6116

−1.5322

−4.3057

7.1810

−8.4534

1.2650

−0.7660

3.1762

−6.3887

−0.1642 −2.6643

−2.2489 −7.3854

0.4958

7.6066

5.2511

−8.6000

−1.2242

9.0753

−6.8434

8.7883

8.6406

2.5461 −8.1551

7.5967

−3.9296 −3.2861 8.8144

3.7853

−9.9527

−2.5230

7.8971



,

 ,



−4.2385

0.4735

−10.0000

−6.3025

−5.3147

,

and



⎤

⎥ ⎥ ⎥ 0 0 0 1 ⎥, ⎥ 0 −0.0797 0 0 ⎥ 0 0 −0.0818 −0.1224 ⎦

[ T (t)Q(t) +  T (t)R(t)]dt,

ktf

−3.3255

F4 =

⎡

(k+1)tf

in which q = 60, tf = 0.5, Q = diag{1, 1, 1, 1, 1, 1} and R = diag{1, 1, 1}. For the TS-DSPS in Eq. (10) with the system matrices in Eqs. (40)–(44), the proposed approach then integrates OFA and HTGA to find both the quadratic finite-horizon optimal fuzzy PDC controller and the quadratic finite-horizon optimal non-PDC controller such that the quadratic finite-horizon integral performance index in Eq. (45) is minimized. The orthogonal function considered in the OFA in this example is the shifted Chebyshev function. Based on the experience of the authors, the evolutionary environment for the HTGA was set to a population size of 100, a crossover rate of 0.8, a mutation rate of 0.1, and a generation number of 50 [16–21]. After first using the fuzzy PDC controller in Eq. (15), the proposed approach is applied where (0) = [− 10, − 10, 0, 0, 0, 0]T , m = 4 and |fijk |≤10 in which fijk (i = 1, 2, 3, 4, j = 1, 2, 3 and k = 1, 2, 3, 4, 5, 6) are the elements of the matrices Fi (i = 1, 2, 3, 4), and the orthogonal array L128 (2127 ) is applied in the HTGA. The resulting performance index is ˜J = 266.6111, and the optimal local feedback gain matrices are

−0.2254 −0.2468 (40)

0

0

q−1   k=0

⎤

0

30

0

0.2153

⎢ 0 0 ⎢ ⎢ 0 0 ˜1 = ⎢ A ⎢ −0.0358 0 ⎢ ⎣ 0 −0.0208 0



˜J =

(37)

In the DSPS system matrices, B = BT > 0 and B˙ = 0 [2]. Note that g33 = 0 (no mooring moment in yaw) is a good assumption for the moored tanker. The system matrices for the TS-DSPS then have the following forms:

⎡

The quadratic finite-horizon integral performance index is

⎤

0

3203

6.3635

−6.8542 −1.9182 −5.9585 −4.0295 −5.3087

3.1519

1.9577

8.0892

3.4169

3.4126

9.9811

−1.3292

−7.5521

6.6875

−0.2773

6.7095

0.4566

 .

Applying Theorem 1 then obtains a suitable common positive definite matrix P:

⎡

0.3971

⎢ −0.2923 ⎢ 0.0715 P=⎢ ⎢ 0.0581 ⎣

−0.2923 0.4926 −0.0162

0.0715

0.0581

−0.0162 −0.0853 0.0704

0.0340

0.1685 −0.0407 0.1896

−0.0853

0.0340

−0.0295 −0.0678

0.1685

−0.0407

0.1896

−0.0678 −0.0725

−0.0388

0.0076

−0.0470

0.0199

0.0249

−0.0388 0.0076

⎤

⎥ ⎥ × 10−11 . ⎥ ⎦

−0.0470 ⎥ 0.0199 0.0249 −0.0078

Therefore, the closed-loop fuzzy control system consisting of the fuzzy model and the quadratic finite-horizon optimal fuzzy PDC controller is globally and asymptotically stable according to the Lyapunov stability theorem. To confirm the effectiveness of the proposed method in terms of quantitative computation, Tables 1 and 2 compare the results obtained by the HTGA and by the traditional genetic algorithm (TGA) [42], respectively, in another five runs of the integrative approach. Five different sets of performance indices were obtained for five fuzzy PDC controllers in five independent runs. Tables 1 and 2 show that the minimal performance index ˜J is 264.4741 for the HTGA and 289.2708 for the TGA. Additional performance comparisons in Table 3 show that the mean performance

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W.-H. Ho et al. / Applied Soft Computing 13 (2013) 3197–3210

index and standard deviation are 265.9229 and 1.5233 in the HTGA and 317.0375 and 24.5504 in the TGA, respectively. In summary, the performance comparison results show that, compared to the TGA, the HTGA obtains (i) a smaller minimal performance index; (ii) a better mean performance index; (iii) a smaller standard deviation in its performance indices and, hence, a more stable solution quality. Thus, the comparison confirms that the HTGA is superior to the TGA for finding quadratic finite-horizon optimal fuzzy PDC controllers. Next, the non-PDC controller in Eq. (30) is used, and the proposed approach is applied where (0) = [−10, −10, 0, 0, 0, 0]T , m = 4, |fij |≤10 in which fij (i = 1, 2, 3 and j = 1, 2, 3, 4, 5, 6) are the elements of matrix F, and the orthogonal array L32 (231 ) is applied in the HTGA. The resulting performance index is ˜J = 258.9461, and the optimal feedback gain matrix is

 F=

6.6643 −2.7996 2.5744

−5.4166 −9.4646 6.1476 −7.1065 5.0050 4.7103 −5.7446 0.6073 8.5817 7.6703 −1.9355 7.6013 1.6451 −3.2030 9.0364

 .

Applying Theorem 2 then finds a suitable common positive definite matrix P:

⎡

⎢ P=⎢ ⎣

⎤

0.0025 −0.0013 0.0089 0.0021 −0.0020 0.0128 −0.0013 0.0007 −0.0022 −0.0012 0.0010 −0.0061 ⎥ 0.0089 −0.0022 0.3815 −0.0179 −0.0261 0.0647 ⎥ × 10−5 . 0.0021 −0.0012 −0.0179 0.0036 −0.0002 0.0094 ⎦ −0.0020 0.0010 −0.0261 −0.0002 0.0028 0.0249 0.0094 −0.0106 0.0691 0.0128 −0.0061 0.0647

That is, the closed-loop fuzzy control system consisting of the fuzzy model and the quadratic finite-horizon optimal non-PDC controller is globally and asymptotically stable according to the Lyapunov stability theorem. To confirm the effectiveness of the proposed method in terms of quantitative computation, Tables 4 and 5 compare the five sets of performance indices obtained by the HTGA and by the TGA [42], respectively, in another five independent runs of the algorithms for optimizing non-PDC controllers. The minimal performance index ˜J is 260.7422 for the HTGA and 275.4220 for the TGA. In Table 6, the performance comparison results show that the mean performance index and standard deviation obtained by the HTGA are 261.8071 and 0.8357 whereas those obtained by the TGA are 298.1852 and 19.8878, respectively. In summary, the results show that, compared to the TGA, the HTGA obtains (i) a smaller minimal performance index; (ii) a better mean performance index; and (iii) a smaller standard deviation in performance indices and, hence, a more stable solution quality. Therefore, compared to the TGA, the HTGA is more effective for finding quadratic finite-horizon optimal nonPDC controllers and obtains better solution quality. For the TS-DSPS with the designed quadratic finite-horizon optimal fuzzy PDC controller and the designed quadratic finitehorizon optimal non-PDC controller, Figs. 3–11, respectively, show the responses of the earth-fixed position on X-axis, the earth-fixed position  2 (t) on Y-axis, the yaw angle  1 (t) the body-fixed velocity  4 (t) on x-axis, the body-fixed velocity  5 (t) on y-axis, the yaw angular velocity  6 (t), and the control inputs  1 (t),  2 (t) and  3 (t). These results confirm the satisfactory results obtained by the proposed approach of integrating OFA and HTGA. To compare the performance of the integrated approach, the LMI-based approach presented in [27,34] was used to design the quadratic optimal fuzzy PDC controller in Eq. (15). Notably, local feedback gain matrices could not be obtained by Matlab LMI toolbox [43]. That is, in this example, LMI-based approach was inapplicable for finding quadratic optimal controllers. Therefore, the quantitative effectiveness of the proposed method was confirmed using the example given in Remark 4. The performance comparisons showed that the proposed integrative approach obtains a smaller performance index compared to the LMI-based approach.

Fig. 3. Earth-fixed position  1 (t) response on X-axis for the TS-DSPS with quadratic finite-horizon optimal fuzzy PDC controller (solid line) and the TS-DSPS with quadratic finite-horizon optimal non-PDC controller (dashed line).

That is, for finding quadratic optimal controllers, the proposed approach is again superior to the LMI-based approach [27,34]. Remark 2. Two-level orthogonal arrays L128 (2127 ) and L32 (231 ) are used in the HTGA by applying the Taguchi parameter design method. The Taguchi parameter design method is an engineering methodology for optimizing the product and process conditions that are minimally sensitive to the causes of variation and that produce high-quality products with low development and manufacturing costs. The Taguchi method is performed between the crossover and mutation operations of the HTGA. The systematic reasoning ability of the Taguchi method is then incorporated in the crossover operations for selecting better genes so that the crossover operations can be tailored to generate representative chromosomes of potential offspring. Therefore, using the Taguchi experimental design method to enhance genetic algorithms obtains an HTGA that is robust, statistically sound, and quickly convergent. The general symbol for two-level standard orthogonal arrays is Ln (2n − 1 ) where n = 2d is the number of experimental runs, d is a

Fig. 4. Earth-fixed position  2 (t) responses on Y-axis for the TS-DSPS with quadratic finite-horizon optimal fuzzy PDC controller (solid line) and the TS-DSPS with quadratic finite-horizon optimal non-PDC controller (dashed line).

Table 1 Five different sets of performance indices and fuzzy PDC controllers obtained by the HTGA in five independent runs. Run number

HTGA Performance index ˜J 264.4741

2

267.5606

3

267.5508

4

265.3303

5

264.6986









6.5688 1.5542 0.8265 3.1564 −4.0056 −3.1225 7.1090 5.9682 1.4466 3.9872 −1.3767 6.8631 −7.0673 6.9048 7.7627 4.9618 3.5959 9.5988 , F2 = −3.0723 2.0418 −5.8465 1.4839 −4.8859 −6.9238 , −7.8694 1.0507 0.4263 −8.9482 4.1994 −1.0160 1.7913 −9.6259 1.2576 4.0761 −8.0050  4.1792   5.3286 −7.3368 3.0563 3.9385 −2.9477 2.8186  8.0000 −9.8632 −3.7868 4.1705 −3.9890 −3.6452 F3 = −6.9994 10.0000 −0.5632 −6.0000 6.5346 6.5543 , F4 = −2.6425 6.8077 10.0000 −0.3421 2.5707 1.6353 . 4.0000 −9.8501 8.8128 6.9431 −7.8034 6.1517 −5.5778 −4.2867 −7.5915 −9.4324 6.7618 1.2841 F1 =

 −2.2642















−3.7749 −7.3960 −8.7650 3.4305 −6.8518 −3.6307 3.2417 −1.6167 5.1105 6.1295 , F2 = −3.0008 −1.6359 1.4915 1.0687 7.1484 1.0963 −1.4682 9.2480 4.4852  3.7786   −4.9720 4.3473 −10.0000 −7.0433 7.4738 0.3903 10.0000 F3 = −4.9144 3.9436 6.9738 0.0975 −2.8157 9.8460 5.2705 , F4 = 1.4141 6.2641 5.5880 −3.5912 −5.8548 5.4673 8.5602 F1 =

8.3412 −5.7273 7.7202 2.1114 −2.5756 7.7580 1.9842 −6.7323 6.5501 −0.6933 −1.9063 5.1211 10.0000 , F2 = 1.1297 5.4552 −6.3888 10.0000 −3.8093 0.4523 4.1052  10.0000   8.4755 7.5786 −9.5062 8.6853 2.3390 −8.1216 −2.4076 F3 = −8.7515 3.3919 8.3090 5.0164 4.6482 −4.0000 , F4 = −3.9192 −2.7683 1.1588 0.8616 4.1946 −9.2854 6.0002 9.7687 F1 =



0.2596 −6.2573  7.1217 9.1114 F3 = −4.8030 −8.3962 F1 =

 −3.4204

5.1977 4.2736 7.2670 −1.9364 −6.0722 −8.3582 −7.8384 −2.2152 4.0000 5.3522 −1.6531 −2.0816

−9.2025 8.7983 2.4721 10.0000 2.0040 −7.0355 −8.2048 0.7720 −2.3854 −0.6667 10.0000 −2.0723

−5.8367 −2.1376 −6.4928 −5.2611 −8.4532 −4.2132

5.6969 −6.6272 −7.5772 −9.1170 0.8209 −0.4448

−8.3553 6.4300 −3.6136 −1.6450 0.4682 −5.1229





−10.0000 8.1282 0.5962 0.8558 3.5019 −1.3681 , −3.7724 −1.3086 −3.5002  −4.7420 2.6103 −10.0000 . 5.4201 −1.8973 5.4467 −3.3709 −2.7507 −6.2504

1.5907 0.4958 5.5971 10.0000 −3.1784 −0.9782 −8.5644 3.5828 8.9649 −10.0000 , F2 = −8.1654 3.3373 6.5946 4.0837 7.8201 −7.7958 −6.8643 −2.3678 −0.0998 −2.2544 1.3040   7.7165 5.7108 −2.1356 −2.5481 −4.3811 7.8533 8.0603 −4.1074 −7.4589 −0.0914 −10.0000 , F4 = 9.9169 −2.4495 0.9609 2.6101 0.1126 −7.7245 −10.0000 −2.7763 −4.3378 4.3597





0.6628 −5.7873 −2.6426 −2.4122 , 0.2875 6.6251  8.5330 −5.8980 0.5784 8.8112 . 4.8476 −8.0753

0.0829 −3.1225 7.4921 9.2416 −6.9328 4.5380



−6.6415 −4.2837 , 5.7141  −4.3519 6.7411 . 8.7432

W.-H. Ho et al. / Applied Soft Computing 13 (2013) 3197–3210

1

Feedback gain matrices Fi



0.0905 −2.9026 6.0000 1.8310 −6.2422 −9.6203 −3.0349 −1.1247 8.0263 −4.7542 −6.4573 −7.9736 −0.1245 −2.9100 −9.5895 6.5438 −6.6312 , F2 = −3.6737 −8.6781 −1.9848 −3.9044 −8.0120 −9.1179 , 7.2991 7.3847 1.5094 2.7050 10.0000 9.4919 0.1110 5.3914 −7.7142 −4.2212  −2.2311  −6.7552  −3.0268 6.4733 0.6143 1.5834 1.0225 −6.3229 6.7372 9.1249 −8.0000 −6.6976 0.0344 −0.7801 F3 = −5.5563 −0.0712 −1.2778 0.7838 −0.3765 −5.8317 , F4 = −8.0229 −8.1783 −5.4108 −3.0745 −8.4117 5.2944 . −9.3690 6.5250 2.8540 −3.5841 −5.0621 −2.9630 −3.5564 −3.2165 3.7382 −2.4194 2.6206 9.6994 F1 =

3205

3206

Table 2 Five different sets of performance indices and fuzzy PDC controllers obtained by the TGA in five independent runs. Run number

TGA Performance index ˜J 314.5922



F1

F3

2

348.2154

F1

F3

3

334.7396

F1

F3

4

289.2708

F1

F3

5

298.3695

F1

F3







2.5906 2.3990 3.0006 5.0542 −4.4580 −1.1282 −3.7961 6.6327 3.8027 −5.2929 1.1213 −0.0306 = −0.2917 3.9229 −4.6109 −1.6359 −2.2101 6.3856 , F2 = −2.6161 0.0650 −2.3442 −9.5266 −6.5311 8.4066 , −1.4516 1.1237 −1.1440 −1.1831 4.8597 −3.3159 −3.6787 −2.0586 −0.2684 −4.3255 −5.2792  −5.2358   −0.4736  0.4492 −5.5934 −4.5645 2.5172 −2.6055 3.3185 −3.7396 2.6803 4.7925 −0.4455 −0.9075 = 3.1458 2.3760 −3.0862 −3.5014 1.1452 . 2.0135 −1.6149 0.3023 −3.1352 8.3888 −3.9401 , F4 = 1.9871 −0.1159 4.3270 2.8597 2.1818 −4.7751 4.3007 −0.4390 2.9551 −3.4804 −3.7490 3.9981 −6.3592   −2.3013   2.5821 5.6187 2.0811 −1.1391 −2.5654 4.4036 1.6385 −1.1526 4.9663 −0.4359 −3.1647 = 3.0538 −0.6427 4.1879 −4.6485 4.6595 −5.4664 , F2 = 6.1082 −0.5151 2.5830 2.6129 3.5534 4.8005 , 5.8510 −1.9163 −2.9874 4.3365 0.5437 0.1043 −3.3240 −1.4880 −3.2019 0.9812 −1.4041  0.3464    1.3366 5.0174 −0.8028 1.4881 −4.0778 1.0167 −0.5899 2.9030 −1.4097 −2.5243 −4.0332 1.5173 = −2.2818 2.8585 −2.3625 −0.3765 2.6827 −1.1250 , F4 = 0.3725 −0.7619 −1.1726 3.9713 −4.9350 7.5993 . 4.8255 −9.0955 0.1627 −2.7860 −4.8527 −2.0490 2.8257 −0.2728 1.6437 5.6112  −0.2073   1.8831  4.2206 3.4592 5.6434 6.4418 −2.2423 −2..0727 2.3551 1.3376 −2.8918 8.3573 −3.0399 −1.7681 = −1.1185 3.5680 6.9086 −4.2538 5.6228 , F2 = −4.8443 −5.6032 4.5550 −5.4003 0.6538 −6.4184 , 4.2320 3.8962 −3.3125 −3.5198 1.8918 −4.5199 −5.0134 −7.7664 1.1168 −0.4643 −6.0345 4.9498  −1.2281   3.2682  4.0545 −1.6466 −0.8220 7.4442 6.0999 −0.2098 7.6242 −1.1666 3.8582 2.8756 1.4471 = −5.0622 1.4659 4.7078 4.9981 3.5777 5.0053 , F4 = 4.6289 −1.1772 −5.3275 −1.4551 −2.4326 3.3423 . 7.3911 −4.7011 1.3189 4.9969 −8.1413 7.4821 4.6369 −1.4003 −0.4539 3.6139  −1.1023  −1.8024  8.7565 −8.4997 5.9194 −2.8888 7.9274 3.6174 3.9822 −4.6268 0.4047 3.0369 −1.8647 −6.0190 = −4.0998 3.9516 0.49966 −4.5674 5.6218 4.7990 , F2 = −0.1466 5.6784 −0.5654 9.2352 4.9054 −1.3157 , −3.9545 5.6807 −0.9772 6.3589 2.0904 0.2550 4.9668 −2.4793 4.0953 −0.0400 0.7765  2.6533    −3.7531 −1.6923 −6.0435 −5.5422 9.8608 −0.6384 0.1973 0.7316 3.4129 7.8966 −5.6271 6.8154 = 2.4700 6.9621 0.6403 −4.3315 6.4322 7.1884 , F4 = 6.1716 5.7719 5.2261 . 0.7747 −8.9634 1.5883 5.2722 −4.6501 −3.4519 1.2351 −6.3200 −6.5665 0.6742 3.5230 2.9108 5.5765 −7.6245 6.9941  −3.8461    4.0842 5.4368 6.7749 −7.4781 4.6057 −3.4903 1.6298 −3.3344 2.4596 −2.6349 6.3666 = −6.7082 4.9165 5.9162 0.4071 8.6612 2.2493 , F2 = 7.1126 7.1106 2.0385 −0.4759 , 2.6076 6.1507 −7.1340 0.3008 −2.2144 1.4054 3.9441 −2.4541 −4.9511 −6.9817 7.9660 8.0063 −4.8674  −1.9588   3.6337  0.8063 6.4003 −0.0086 0.1029 −1.7015 −1.3249 4.7061 −3.3851 4.4177 9.0683 −0.9135 = −1.4504 7.9157 0.1382 −4.3560 3.0435 −3.3250 , F4 = 4.9455 −2.6901 8.1535 6.9518 −9.6656 −2.2264 . 2.7775 1.6729 −3.2889 −8.4521 7.7309 7.0570 6.7310 −4.0813 −3.5903 5.0817 −9.5901 −3.3708

W.-H. Ho et al. / Applied Soft Computing 13 (2013) 3197–3210

1

Feedback gain matrices Fi

W.-H. Ho et al. / Applied Soft Computing 13 (2013) 3197–3210

3207

Table 3 Performance comparison of HTGA and TGA in finding the optimal fuzzy PDC controllers in five independent runs in the same evolutionary environment. Mean performance index (standard deviation)

Minimal performance index

HTGA

TGA

HTGA

TGA

265.9229 (1.5233)

317.0375 (24.5504)

264.4741

289.2708

positive integer greater than 1, 2 is the number of levels for each factor, and n − 1 is the number of columns in the orthogonal array. The letter ‘L’, which is the initial for ‘Latin’, represents the use of orthogonal arrays for experimental design by Latin square design from the outset. For example, the non-PDC controller design in Eq. (35) includes eighteen two-level factors. Since only eighteen columns are needed to allocate these factors, L32 (231 ) is sufficient because it has 31 columns. Remark 3. The regulator design problem considered here is to optimize time-response characteristics while maintaining the system outputs at zero. The proposed approach for integrating OFA and HTGA finds the steady-state optimal solution for this problem by minimizing the quadratic finite-horizon integral performance index Eq. (45), where Q and R are two weighting matrices, for both the fuzzy PDC controller Fi (i = 1, 2, 3, 4) and the non-PDC controller F. In the illustrative example of a moored tanker, a practicable time

Fig. 5. Yaw angle  3 (t) responses for the TS-DSPS with quadratic finite-horizon optimal fuzzy PDC controller (solid line) and the TS-DSPS with quadratic finite-horizon optimal non-PDC controller (dashed line).

Table 4 Five different sets of performance indices and non-PDC controllers obtained in five independent runs of the HTGA. Run number

HTGA Performance index ˜J

Feedback gain matrices F

1

261.3201

F=

2

262.9144

F

3

261.8087

F

4

262.2500

F

5

260.7422

F



4.4712 −3.5526  −1.5791 7.3416 = −7.3043  −6.6885 4.5782 = −6.0871  −0.2529 5.9973 = −2.0054  0.7528 2.6573 = −1.5724 −0.2596

−3.4183 4.6067 1.5108 −6.2325 7.9468 6.8466 −3.6203 7.1432 0.0476 −4.7586 3.3325 −0.3904 −2.2400 2.8509 0.8635



0.0218 6.8375 −9.0230 0.0311 2.0991 −1.8761 7.7246 2.1000 8.2232 −1.6160 −0.1092 8.1500  −3.9682 6.4819 −8.8824 3.4483 −0.3405 −2.0000 9.9668 −9.8009 9.9999 −4.8623 8.9696 6.3060  −3.3844 6.4302 −7.7628 −2.1430 −2.7814 −0.8587 9.8734 −4.9004 −8.7101 5.3531 −5.1411 −9.5403  −5.0288 6.9336 −8.7546 6.4527 0.9996 −0.7241 6.8547 1.4977 7.4842 −0.6836 0.2859 6.5830  −0.2754 5.0405 −7.7659 −3.6765 −1.5257 −1.1693 7.8889 0.6644 9.5632 1.3218 −4.4780 9.3798

Table 5 Five different sets of performance indices and non-PDC controllers obtained in five independent runs of the TGA. Run number

TGA Performance index ˜J

Feedback gain matrices F

 −4.4670

1

319.1676

F

2

275.4220

F

3

279.0650

F

4

304.1823

F

5

313.0892

F



7.0939 −2.2876 8.0460 4.6853 −5.2839 = 2.4317 −0.5480 −1.3541 6.3755 −3.5899 8.6770 8.2779 −8.0751 −1.8028 2.0660 −7.7429 6.1588  −3.0821  2.7862 −7.6997 −1.8257 2.4503 −9.4888 = −0.9547 2.1423 3.8273 −1.8437 7.1867 2.0955 0.1536 0.7651 −3.9056 7.2185 −4.3681 6.4405  5.9859  −4.9841 −1.4471 7.2493 −5.8481 −6.0704 = 0.9243 1.1039 3.8751 4.5704 0.6831 4.0170 −1.3759 3.7263 4.7000 −4.5855 6.0615  1.8949  −1.1606 3.9904 5.1967 9.8880 1.9147 −7.4379 = −7.9263 8.1201 −2.5767 −0.4471 3.3245 7.0639 3.6325 −7.4002 7.2982 7.7232 −8.4851 8.5102  5.3390  −5.6522 8.5257 8.1518 −9.0722 1.8614 = 7.1228 −6.4997 −3.0738 6.6141 0.9087 −3.3823 9.6207 5.4785 −3.4990 −5.0792 3.1284 8.6529

3208

W.-H. Ho et al. / Applied Soft Computing 13 (2013) 3197–3210

Table 6 Performance comparison of HTGA and TGA in finding non-PDC controllers in five independent runs in the same evolutionary environment. Mean performance index (standard deviation)

Minimal performance index

HTGA

TGA

HTGA

TGA

261.8071 (0.8357)

298.1852 (19.8878)

260.7422

275.4220

Fig. 8. Yaw angular velocity  6 (t) responses for the TS-DSPS with quadratic finitehorizon optimal fuzzy PDC controller (solid line) and the TS-DSPS with quadratic finite-horizon optimal non-PDC controller (dashed line).

Remark 4. Consider the nonlinear mass-spring-damper mechanical system discussed in Lee et al. [44]. The TS fuzzy model of the nonlinear mass-spring-damper mechanical system is Rule 1 : Fig. 6. Body-fixed position  4 (t) responses on x-axis for the TS-DSPS with quadratic finite-horizon optimal fuzzy PDC controller (solid line) and the TS-DSPS with quadratic finite-horizon optimal non-PDC controller (dashed line).

response is obtained by changing weighting matrices Q and R. In this illustrative example, stable control can be achieved within approximately 10 s. If the control objective is to design an energy optimal controller, the weighting matrix R must be enlarged. The time response of state vector (t) is stabilized over 10 s, and control input vector (t) is decreased. That is, the time response is optimized by adjusting weighting matrices Q and R according to the control objective.

˜ 1 (t) + B˜ 1 (t), IF z1 (t) is M11 , THEN (t) ˙ =A

˜ 2 (t) + B˜ 2 (t), Rule 2 : IF z1 (t) is M11 THEN (t) ˙ =A where (t) = [1 (t), 2

(46a) (46b)

(t)]T ,

1 (t) is the velocity, 2 (t) is the dis ˜ 1 = −1.0 −1.13 , placement, (t) is the force, z1 (t) = 13 (t), A 1.0 0











1.4387 0.5613 ˜ 2 = −1.0 −1.13 , , A B˜ 2 = 0 1.0 0 0 M11 = 0.5 + (z1 (t)/6.75), and M21 = 0.5 − (z1 (t)/6.75). The quadratic finite-horizon integral performance index is

B˜ 1 =





,

10

[T (t) Q(t) + T (t)R (t)]dt

˜J = 0

Fig. 7. Body-fixed position  5 (t) responses on y-axis for the TS-DSPS with quadratic finite-horizon optimal fuzzy PDC controller (solid line) and the TS-DSPS with quadratic finite-horizon optimal non-PDC controller (dashed line).

Fig. 9. Control input  1 (t) responses for the TS-DSPS with quadratic finite-horizon optimal fuzzy PDC controller (solid line) and the TS-DSPS with quadratic finitehorizon optimal non-PDC controller (dashed line).

W.-H. Ho et al. / Applied Soft Computing 13 (2013) 3197–3210

Fig. 10. Control input  2 (t) responses for the TS-DSPS with quadratic finite-horizon optimal fuzzy PDC controller (solid line) and the TS-DSPS with quadratic finitehorizon optimal non-PDC controller (dashed line).

=

q−1   k=0

(k+1)tf

[T (t) Q(t) + T (t)R (t)]dt,

(47)

ktf

in which q = 1000, tf = 0.01, Q = diag{1, 1}, and R = 1. In the TS-fuzzy-model-based control system in Eq. (46), the proposed approach, which integrates the OFA, the HTGA, and the presented LMI-based stabilizability condition, is applied in the design of both the quadratic finite-horizon optimal fuzzy PDC controller and the quadratic finite-horizon optimal non-PDC controller such that there exists a symmetric positive definite matrix P to make the LMIs in Eqs. (17) and (32) hold, respectively, and such that the quadratic integral performance index in Eq. (47) is minimized. Here, the conditions of the evolutionary environment of the HTGA are a population size of 100, a crossover rate of 0.8, a mutation rate of 0.1, and a generation number of 10. After using the proposed integrative approach and the LMI toolbox [44] with m = 4, |fijk |≤3.5 (i = 1, 2; j = 1 and k = 1, 2) and the

3209

orthogonal array L8 (27 ) for designing the fuzzy PDC controller, in which fijk are the elements of the local feedback gain matrices Fi (i = 1, 2), the results are performance index ˜J = 2.8124, local feedback gain matrices F1 = [0.4565 0.8698] and F2 = [0.9353 −0.1302],  10.8909 7.5414 and symmetric positive definite matrix P = . 7.5414 29.0336 Using the existing LMI-based approach [27,34] to design the fuzzy PDC controller then obtains performance index ˜J = 4.3202, local feedback gain matrices F1 = [−0.1432 0.0825] and F2 = [−0.5271 0.0767]   and symmetric positive definite matrix P = 0.0154 0.0044 . 0.0044 0.0159 Next, the proposed integrative approach and the LMI toolbox [44] with m = 4, |fij |≤3.5 (i = 1 and j = 1, 2) and the orthogonal array L4 (23 ) are used to design the non-PDC controller where fij are the elements of the feedback gain matrix F, we obtain performance index ˜J = 2.8125, feedback gain matrix  F = [0.6457 0.3823]  13.3892 8.5140 and symmetric positive definite matrix P = . 8.5140 34.6249 If the existing LMI-based approach [27,34] is then used to design the non-PDC controller, performance index ˜J = 3.7442, feedback gain matrix  F = [0.1653 −0.0511]  and symmetric positive definite matrix 0.0283 0.0089 P= . 0.0089 0.0231 The results of the performance comparisons between the proposed integrative approach and the LMI-based approach show that the proposed integrative approach obtains a smaller performance index compared to the LMI-based approach. Therefore, the proposed integrative approach is more effective for finding both the fuzzy PDC controller and non-PDC controller. Additionally, unlike the LMI-based approach, the proposed integrative approach can solve the problem of feedback gain matrix under constraints. 5. Conclusions This study combined OFA and HTGA to solve quadratic finitehorizon optimal controller design problems in both fuzzy PDC controllers and non-PDC controllers for the TS-DSPS. An OFA requiring only algebraic computation was used to solve TS-fuzzymodel-based feedback dynamic equations. The OFA was then integrated with HTGA to design quadratic finite-horizon optimal controllers for a TS-DSPS under the criterion of minimizing a quadratic finite-horizon integral performance index. The use of standard algebraic computation in the proposed approach of integrating OFA and HTGA enables easy computer implementation. Accordingly, the procedure for designing quadratic finite-horizon optimal controllers for the TS-DSPS is much shorter and simpler. Thus, the proposed approach facilitates the design tasks of quadratic finite-horizon optimal controllers for TS-DSPS. The illustrative example of a moored tanker confirmed that the proposed approach for designing quadratic finite-horizon optimal controllers for a TS-DSPS is more effective than the LMI-based approach presented by [27,34]. Acknowledgements This work was in part supported by the National Science Council, Taiwan, Republic of China, under Grant numbers NSC 98-2221-E037-006, NSC 99-2320-B-037-026-MY2, NSC 101-2221-E-151-031 and NSC 101-2320-B-037-022.

Fig. 11. Control input  3 (t) responses for the TS-DSPS with quadratic finite-horizon optimal fuzzy PDC controller (solid line) and the TS-DSPS with quadratic finitehorizon optimal non-PDC controller (dashed line).

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