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Proceedings of Engineering and Technology Innovation, vol. 10, 2018, pp. 13 - 18 

 

Robust Control of Delayed Fin Stabilizer Stochastic Systems of a Ship 

Cheung-Chieh Ku
*
, Chin-Yuan Hsu 

Department of Marine Engineering, National Taiwan Ocean University, Keelung, Taiwan, ROC 

Received 02 March 2018; received in revised form 24 June 2018; accepted 29 July 2018 
 

Abstract 

In this paper, the robust control problem of delayed fin stabilizer stochastic system of a ship with uncertainty is 

discussed and investigated. To describe the system, Linear Parameter Varying (LPV) modelling approach and 

multiplicative noise term are used to establish the corresponding polynomial model. For simulating the general 

operating environment, the delay effect is considered as time-varying case. Moreover, the gain-scheduled control 

scheme is employed to discuss the delay-dependent stabilization problem and to design the corresponding controller. 

Moreover, a novel Lyapunov-Kravoskii function is proposed by using parameter-dependent matrix and integral 

Lyapunov function to reduce the conservatism of the derived stability conditions. In order to apply the convex 

optimization algorithm, the derived conditions are converted into Linear Matrix Inequality (LMI) form. By solving 

the conditions, some feasible solutions can be obtained to establish the controller to guarantee robust stability of the 

delayed fin stabilizer stochastic system of a ship in the mean square. 

 

Keywords: LPV system, stochastic behavior, LMI, fin stabilizer system of a ship 

 

1. Introduction 

Generally, the comfort of passengers is deeply affected by sea waves according to violent rolling. Therefore, an important 

issue of reducing the roll motion of the ship is usually discussed for building a liner or a passenger ship. Besides, a large 

amplitude rolling motion may cause damage to the cargoes and vessels during  the shipping process. For the above reasons, an 

effectiveness of fin stabilizer was investigated in [1-7] to reduce the rolling motions. Generally, a pair of fins consists a ship fin 

stabilizer which locates approximately amidships on the bilge of the hull. Those fins can be used to control changes of ship roll 

angle and its rate. Through changing the angle [5-6], the hydrodynamic forces are induced on the fins to produce a moment that 

can reduce the wave induced roll motion. Furthermore, the state feedback control method [3-4] is usually used to deal with the 

fin stabilizer system. However, it is difficult for reaching control target of the fin stabilizer system because the uncertainties 

appear in calculating the fin lift from fin angle. For the reason, LPV system is applied to describe the dynamics of the fin 

stabilizer system in this paper and to discuss the stabilization problem. 

LPV system [8-13] has been built by several linear systems and a specific weighting function to represent uncertain 

systems or nonlinear systems.  According to the structure of LPV system, a general description for uncertain systems can be 

proposed to represent complex uncertainties. In [12-13], a Gain-Schedule (GS) control scheme has been applied to deal with 

the stabilization problem of LPV system. Because the structure of GS controller is similar to LPV system, the robustness of the 

described system can be increased. It means that the GS scheme is vary suitable control scheme for systems described by 

numerous sub-systems. Therefore, many robust stability criteria [10-13] have been proposed for LPV systems via applying the 

GS design scheme. Furthermore, many practical robust control applications [9, 13] have been achieved via LPV system and 

                                                           
*
 
Corresponding author. E-mail address: ccku@mail.ntou.edu.tw 

 
Tel.: +886-2-24622192; Fax: +886-2-24620724

 



Proceedings of Engineering and Technology Innovation, vol. 10, 2018, pp. 13 - 18 

Copyright ©  TAETI 

14 

GS control scheme. In this paper, the LPV system is applied to represent the uncertain fin stabilizer system and the GS control 

scheme is used to discuss the stabilization problem of the system. 

In addition to uncertainty, effect of time delay always exists in dynamic process such as propagation/transportation of 

material, information or energy. The similar effect of time delay appears in the fin stabilizer system due to signal transportation. 

Generally, delay-dependent criterion [11, 14-16] is discussed for general effect of time delay due to conservatism caused by 

constant delay. Besides, some relaxed technologies [15-16] for delay-dependent criteria are developed to reduce conservatism 

of stability conditions. Moreover, Lyapunov-Krasovskii function is often applied to derive sufficient conditions to analyze the 

stability of delayed system. Thus, time delay effecting on state is also concerned with the considered control problem of the 

uncertain fin stabilizer system. 

Moreover, stochastic behaviour [17-20] caused by high sea states often exists in operation during the shipping process. 

However, stochastic behaviour cannot be measured and predicted in the practical operation. Referring to [17], stochastic 

differential equation provides a powerful tool to describe the stochastic behaviours. In the stochastic differential equation, 

stochastic behaviour is modelled as multiplicative noise term which is consisted by system states and white noise. Hence, the 

stochastic difference equation has been applied in many control fields to extend design methods from deterministic systems to 

stochastic systems. Based on the stochastic differential equation, many stability criteria have been developed to deal with 

stability issue of stochastic systems. From [21-22], one can find that the LPV system with multiplicative noise term can be 

successfully employed to represent uncertain stochastic systems. Therefore, a stability criterion of delayed uncertain fin 

stabilizer stochastic system of a ship is discussed in this paper. 

To discuss the considered criterion, a delayed LPV stochastic system is built to describe practical fin stabilizer system of 

a ship with time delay, time-varying parameters and stochastic behaviors. Based on the modelling approach and stochastic 

differential equation, the linear systems with multiplicative noise term in LPV stochastic system can be modelled by setting 

varying range and number of time-varying parameters. In order to analyze stability of the system, some relaxed sufficient 

conditions are derived via Lyapunov-Krasovskii function and Jensen inequality [14]. In order to apply convex optimization 

algorithm, those stability conditions are converted into Linear Matrix Inequality (LMI) form [23]. Through solving the 

conditions, the feasible solutions can be obtained to build GS controller such that the considered fin stabilizer system of a ship 

is robust stable. Based on the simulation result, the fin stabilizer system of a ship can be stabilized with the added delay effect 

and stochastic behavior. 

2. System Description and Problem Formulation 

Referring to [4], the dynamics of fin stabilizer system of a ship can be represented as Fig. 1. Based on Fig. 1, the dynamic 

equation of fin stabilizer system can be described as follows: 

  3 51 2 1 3 5x x d sI I B B C C C M M               (1) 

where   denotes the ship’s rolling angle, 
x

I  and 
x

I  denote the mass inertia moment and affixing mass inertia moment 

relative to the vertical axes of ship, 
d

M  denotes the disturbance moment of sea waves, 
s

M denotes the stable moment of lift 

feedback fins. Besides, 
1

C , 
3

C , 
5

C , 
1

B , 
2

B  are constants and 
1

C Dh , where D  denotes the ship’s tonnage and h  denotes 

the height of steady center of ship’s rolling. In this paper, the parameters of a certain ship are assumed as 1457.26D t , 

1.15h m ,
6

3.4383 10
x x

I I   ,
6

3
2.097 10C   ,

6

5
4.814 10C   ,

6

1
0.636 10B   ,

6

2
0.79 10B   .Let 

t d s
M M M  , 

the nonlinear model of this ship’s rolling can be described as follows: 



Proceedings of Engineering and Technology Innovation, vol. 10, 2018, pp. 13 - 18 

Copyright ©  TAETI 

15 

3 5 7
0.185 0.23 0.4874 0.61 1.4 2.9084 10

t
M      


        (2) 

To consider Eq. (2) operated around 
0

0 , the nonlinear fin stabilizer system can be rewritten as a linear system. Moreover, the 

delay effect, uncertainties and stochastic behavior are added to describe the natural unstable source. Furthermore, the following 

delayed LPV stochastic systems is provided to represent (2) with the considered effects. 

                
2

1

i i ti i i

i

dx t t x t x t d t u t dt x t d t 


     A A B E  (3) 

where 𝐀𝟏 = [
0 1

−0.458156 −0.185
] , 𝐀𝟐 = [

0 1
−0.516644 −0.185

] , 𝐀𝒕𝟏 = [
0 1

−0.0195 0
] , 𝐀𝒕𝟐 = [

0 0
−0.0195 0

] , 𝐁𝟏 =

[
0

2.9084 × 10−7
] , 𝐁𝟐 = [

0
2.9084 × 10−7

] , 𝐄𝟏 = [
0 0
0 0.0698

] , 𝐄𝟐 = [
0 0
0 0.0698

] , 𝑎1(𝑡) = |sin⁡(𝑡)|  and 𝑎2(𝑡) = 1 −

|sin(𝑡)| ∙ ⁡0 ≤ 𝑑(𝑡) ≤ 𝑑𝑀, and 𝑑(𝑡) ≤ 𝜀 < 1. 𝑑𝛽(𝑡) is scalar continuous type Brownian motion satisfying the independent 

increment property [18]. 

 
Fig. 1 Dynamic of fin stabilizer system 

To deal with stabilization problem of Eq. (3), the following gain-scheduled controller is designed via state-feedback 

control concept. 

      
2

1

j j

j

u t t x t


  F  (4) 

Substituting Eq. (4) into Eq. (3), the following closed-loop system can be inferred: 

                  
2 2

1 1

i j i i j ti i

i j

dx t t t x t x t d t dt x t d t  
 

     A B F A E  

               
2 2

1 1

i j ij ti i

i j

t t x t x t d t dt x t d t  
 

    G A E  
(5) 

where 
ij i i j
 G A B F . 

2.1.   Lemma 1 

For any constant matrix 0M , scalars 1r  and 2r  satisfying 
2 1

r r , a vector function    : 0 xnx t ,   such that the 

integrals concerned are well defined, the following inequality holds. 



Proceedings of Engineering and Technology Innovation, vol. 10, 2018, pp. 13 - 18 

Copyright ©  TAETI 

16 

           
2 2 2

1 1 1

T

2 1

r r r

r r r
x s ds x s ds r r x s x s ds   M M   

In the following section, a stability criterion is proposed to deal with the stability issue of the close-loop system (5). Based 

on the proposed stability criterion, the robust stability of the closed-loop system (5) can be guaranteed. 

3. Delay-Dependent Stability Criterion 

In this section, the stability criterion subject to time-varying delay performances for closed-loop system Eq. (5) is 

developed. Some sufficient conditions are derived via applying Lyapunov-Krasovskii function and Itô’s formula in the 

following theorem. 

3.1.   Theorem 1 

For given scalars 
M

d  and  , the closed-loop system (5) is asymptotically stable if there exists T 0
i i
 P P , 0Q , and 

0R  such that the following conditions are satisfied. 

 

 

T T
0

1 0 0

1

i ij ij i i i i M i ti

M

d

*

* *
d





 
 

    
   
 

 
 

 

P G G P E P E Q R P A

Q

R

 for 1 2i, j ,  (6) 

In order to limitation of pages, the proof of this theorem is omitted. Although the condition in Theorem 1 belongs to 

bilinear matrix inequality, the similar conversion and algorithm in [21] can be used to convert Eq.(6) into LMI problem. In the 

following section, the numerical simulation is proposed. 

4. Simulation Results 

To apply the proposed design method, 10
M

d   and 0.7   are set. Moreover, the following feasible solutions can be 

obtained. 

 
 

Fig. 2 Responses of 𝜒1(𝑡) Fig. 3 Responses of 𝜒2(𝑡) 

𝐏𝟏 = [
0.9271 0.8592
0.8592 6.1864

] × 107, 𝐏𝟐 = [
0.3741 0.8592
0.8592 6.6161

] × 107, 𝐅1 = |−1.2315 3.1461| × 10
6, 𝐅2 = |−1.2315 3.1461| × 10

6, 

𝐐 = [
0.2910 0.1170
0.1170 7.5274

] × 106 , 𝐑 = [
0.0151 0.1212
0.1212 7.6452

] × 106 , and 𝛼 =⁡−0.1287  with the above feasible solutions, the GS 



Proceedings of Engineering and Technology Innovation, vol. 10, 2018, pp. 13 - 18 

Copyright ©  TAETI 

17 

controller (4) can be designed. Based on the designed GS controller, the responses of (3) are stated in Fig. 2-3 with initial 

condition 𝑥(0) = [
𝜋

18
0]

𝑇
. From Figs. (2)-(3), the state of Eq. (3) are converged to zero. According to simulation result, the 

proposed design method can be used to stabilize the uncertain fin stabilizer stochastic system with time delay. 

5. Conclusions 

In this paper, a GS controller design method was proposed to deal with stabilization problem of uncertain fin stabilizer 

stochastic system with time delay. Based on the modelling approaches, a LPV system with multiplicative noise was provided 

to describe the considered system. For the system, some conditions were derived via Lyapunov-Krasovskii function and Jensen 

inequality. By solving the derived conditions, the feasible solutions can be obtained to build GS controller such that the 

closed-loop system is robust stability. Finally, a simulation result has been provided to demonstrate the effectiveness and 

usefulness of the proposed design method. 

Acknowledgement 

This work was supported by the Ministry of Science and Technology, under Contract MOST 106-2221-E-019-006. 

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