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ISSN : 1229-3431(Print)
ISSN : 2287-3341(Online)
Journal of the Korean Society of Marine Environment and Safety Vol.27 No.2 pp.211-218
DOI : https://doi.org/10.7837/kosomes.2021.27.2.211

# Suppression of Coupled Pitch-Roll Motions using Quasi-Sliding Mode Control

Sang-Do Lee*, Truong Ngoc Cuong**, Xiao Xu**, Sam-Sang You***
*Professor, Division of Navigation & Information System, Mokpo National Maritime University, Mokpo, Republic of Korea
**Doctor candidate, Department of Logistics Engineering, Korea Maritime and Ocean University, Busan, Republic of Korea
***Professor, Division of Mechanical Engineering, Korea Maritime and Ocean University, Busan, Republic of Korea

* First Author : oksangdo@mmu.ac.kr, 061-240-7257

Corresponding Author : ssyou@kmou.ac.kr, 051-410-4366
January 28, 2021 March 15, 2021 April 27, 2021

## Abstract

This paper addressed the problems of controlling the coupled pitch-roll motions in a marine vessel exposed to the regular waves in the longitudinal and transversal directions. Stabilization of the pitch and roll motions can be regarded as the essential task to ensure the safety of a ship’s navigation. One of the important features in the pitch-roll motions is the resonance phenomena, which result in unexpected large responses in terms of pitch and roll modes in some specific conditions. Besides, owing to its inherent characteristics of coupled combination and nonlinearity of restoring terms, the vessel shows various dynamical behaviors according to the system parameters, especially in the pitch responses. Above all, it can be seen that suppression of pitch rate remains the most significant challenge to overcome for ship maneuvering safety studies. To secure the stable upright condition, a quasi-sliding mode control scheme is employed to reduce the undesirable pitch and roll responses as well as chattering elimination. The Lyapunov theory is adopted to guarantee the closed stability of the pitch-roll system. Numerical simulations demonstrate the effectiveness of the control scheme. Finally, the control goals of state convergences and chattering reduction are effectively realized through the proposed control synthesis.

# 준 슬라이딩 모드 제어를 이용한 선박의 종동요 및 횡동요 억제

이 상도*, 트롱 엥곡 쿠옹**, 서 효**, 유 삼상***
*목포해양대학교 항해정보시스템학부 교수
**한국해양대학교 대학원 박사과정
***한국해양대학교 기계공학부 교수

## 초록

본 연구는 종방향과 횡방향에서 규칙파가 외란으로 입사하는 선박의 종동요 및 횡동요를 억제하는 문제를 다룬다. 선박의 안 전 운항을 위해서 파랑 중 선박의 종동요 및 횡동요를 안정화시키는 것은 필수적인 과제로 여겨진다. 종동요 및 횡동요 거동에서 중요한 특징 중에 하나는 공진인데, 이 현상은 특정한 조건에서 예상치 못한 큰 응답을 초래한다. 종동요와 횡동요는 두 운동이 상호 결합되어 있고 복원항이 강한 비선형을 갖고 있음으로 계의 파라미터에 따라 다양한 동적 거동을 보이는 것이 중요한 특징인데, 특히 종동요에서 이 특성이 두드러지게 나타난다. 무엇보다, 선박의 조종성능 및 안전을 위해 선박의 종동요 가속도 응답을 완전히 억제하는 것은 상당히 도전적인 과제이다. 이 연구에서는 준 슬라이딩 모드 제어 기법을 이용하여 종동요와 횡동요를 줄임으로 파랑 중의 선박을 안정적인 직 립 상태로 유지시키고, 아울러 채터링을 감소시키는 목적을 달성하였다. 리아프노프 이론으로 준 슬라이딩 모드 제어의 안정성을 판명하 였고, 수치 시뮬레이션으로 제어 방식의 유효성을 증명하였다. 제시한 기법으로 종동요 및 횡동요 응답의 수렴 및 채터링 감소라는 두 가 지 목표가 효과적으로 달성되었다.

## 1. Introduction

The problem of stabilization for the dynamical behaviors in a ship influenced by the unexpected internal impacts or experienced environmental factors can be regarded as one of the important issues for the safety of a ship’s navigation. Also, controlling the undesirable ship’s motions will play an important role in the maritime autonomous surface ship shortly. Despite the accident measures, it seems a difficult challenge to overcome the capsizing problem of uncontrolled vessels such as fishing boats, passenger ferries. Especially, fishing vessels have a high risk of capsize due to the roll resonance mode. Moreover, it is a crucial issue for a containership to prevent the parametric roll resonance eventually leading to cargo damage and loss of containers. Therefore, this paper intends to suppress the dynamical pitch-roll resonance modes exposed to the extreme sea disturbances in the longitudinal and transversal directions.

Typically, the marine vessel system shows two dangerous behaviors such as chaos roll (Lee et al., 2020a) and parametric roll resonance (Lee et al., 2021). The reason why chaos is dangerous that the phenomenon may appear at the end of stable responses even in regular waves (Lee and You, 2018a;Lee and You, 2018b). Due to its strong nonlinearity in the restoring term, the vessel system is inherently vulnerable to the nonlinear large responses. On the contrary, in case of decreasing the restoring forces, marine vessels may navigate a kind of tender ship. Similarly, the pitch-roll system exhibits the huge jump of roll response under resonance mode as well as nonlinear pitch responses. However, it is considered as a challenging task to suppress pitch rate mode perfectly.

There have been reports of several researches on pitch-roll ship dynamics and its responses (Nayfeh et al., 1973;Kamel, 2007;Zhou and Chen, 2008). Several mechanisms have been reported to suppress the roll dynamics such as anti-roll tanks, gyro-stabilizers, fin stabilizers, rudder-roll stabilization, and so forth (Wang et al., 2017). However, it is seldom reported to the stabilization of coupled pitch-roll motion (Huang et al., 2018).

This paper has employed a quasi-sliding mode control synthesis to suppress the coupled pitch-roll motions. When implementing the sliding mode type controller, a system designer agonizes how to solve the chattering problem as well as the performance of state convergence simultaneously. Sometimes two control goals are mutually incompatible or competing. Thus several approaches have been proposed to balance a trade-off between the control performance and its cost.

The sliding mode control (SMC) and higher-order SMC strategies are successfully realized in controlling the marine vessel system such as a moored ship (Lee and You, 2018c;Lee et al., 2020b), a patrol boat (Lee et al., 2019), and a non-motorized barge (Lee et al., 2020a). The key contribution of this work is to achieve the ship stabilization of coupled pitch-roll motions governed by a two-degree-of-freedom (2DOF) motions and chattering reduction of the actuators using a quasi-SMC method.

The rest of this paper is represented as follows. Section 2 accounts for the mathematical formulation of pitch-roll ship dynamics. Then, quasi-SMC is implemented to control the pitch-roll motions, and the Lyapunov theory is shown to prove the closed-loop stability of ship dynamics. Section 3 shows the effectiveness of the proposed control synthesis by numerical simulations. In conclusion, some remarks will be addressed in section 4.

## 2. Mathematical Formulation

### 2.1 Coupled pitch-roll system

The coupled pitch-roll modes in regular wave governed by 2DOF (two-degree-of-freedom) can be addressed as follows (Lee and You, 2018a):

$( I θ + M θ ¨ ) θ ¨ + M θ ˙ θ ˙ + M θ θ + M θ θ ϕ 2 = M s ( t ) , ( I ϕ + K ϕ ¨ ) ϕ ¨ + K ϕ ˙ ϕ ˙ + K ϕ ϕ + K ϕ θ ϕ θ = K s ( t )$
(1)

where M and K describe the pitch and roll motions, respectively; Iθ and Iϕ are the inertial moments for the pitch and roll directions; $M θ ¨$ and $K ϕ ¨$ denote the hydrodynamic added masses for the pitch and roll directions; $M θ ˙$ and $K ϕ ˙$ show the hydrodynamic damping terms; Mθ and K ϕ describe the hydrostatic linear coefficients for the 1st order restoring responses; Mθ θ and Kϕ θ represent the 2nd order restoring terms; Ms (t) and Ks (t) indicate the wave excitation forces on the pitch and roll modes, respectively. One can expect the auto-parametric resonance phenomena according to the coupled pitch and roll motions. Some analytical approaches are useful to solve the above pitch-roll modes using the perturbation theory. Otherwise, a non-dimensional process is helpful using a small amount of perturbation parameter. The coupled pitch-roll vessel system under regular waves can be expressed as the following nonlinear dynamical equations through transformation step (Kamel, 2007;Zhou and Chen, 2008):

$θ ¨ + 2 ∈ μ 1 θ ˙ + Ω n 1 2 θ + ∈ k 1 ϕ 2 = F 1 sin ( ω e 1 t ) , ϕ ¨ + 2 ∈ μ 2 ϕ ˙ + Ω n 2 2 ϕ + ∈ k 2 ϕ θ = F 2 sin ( ω e 2 t )$
(2)

with

$μ 1 = M θ ˙ / ( I θ + M θ ¨ ) , μ 2 = K ϕ ˙ / ( I ϕ + K ϕ ¨ ) , Ω 1 = 0.5 M θ / ( I θ + M θ ¨ ) , Ω 2 = 0.5 K ϕ / ( I ϕ + K ϕ ¨ ) , k 1 = M ϕ ϕ / ( I θ + M θ ¨ ) , k 2 = K ϕ θ / ( I ϕ + K ϕ ¨ ) , F 1 = M s ( t ) / ( I θ + M θ ¨ ) , F 2 = K s ( t ) / ( I ϕ + K ϕ ¨ ) ,$

where ϵ is the small amount of perturbation parameter; μ1 and μ2 are the hydrodynamic damping terms for the pitch and roll modes, respectively, Ωn1 and Ωn2 describe the natural angular frequencies; k1 and k2 indicate the nonlinear restoring terms; ωe1 and ωe2 represent the wave encountering frequencies; F1 and F2 describe the wave excitation amplitudes divided by the inertia terms.

One of the important dynamical features in the pitch-roll system is the internal resonance case (Ωn1 ≅ 2Ωn2 ) leading to the undesirable large roll motions in the angle and velocity. Kamel (2007) investigated the resonance cases such as the combination type (ωn1 = ωn2Ωn1 + Ωn2 ) and internal type (Ωn1 ≅ 2Ωn2 ). Since the pitch-roll system has many influencing parameters on the pitch and roll resonance phenomena, this paper deals with the ship dynamical issues restricted to the internal resonance. Surely, extensive complex dynamical behaviors under the internal resonance can be observed depending on considering other major parameters such as wave encountering frequencies and wave amplitudes, initial conditions, and so forth.

### 2.2 Control synthesis

The pitch-roll vessel system under regular waves represents the undesirable complex nonlinear behaviors in both pitch and roll responses. It should be properly suppressed to secure the safety of ship navigation despite unwanted severe influences. Generally, there exist many mechanisms to reduce the pitch and roll motions such as anti-rolling devices, rudder-roll stabilizing system, gyro-stabilizing system, canted rudders, and active fin stabilizers. The controlled pitch-roll system can be expressed as the forced pitch-roll system by considering the time-varying two disturbances (d = [d1, d2 ]T ) and the input control vector (u = [u1, u2 ]T ),

$θ ¨ + 2 ∈ μ 1 θ ˙ + Ω n 1 2 θ + ∈ k 1 ϕ 2 = u 1 + d 1 , ϕ ¨ + 2 ∈ μ 2 ϕ ˙ + Ω n 2 2 ϕ + ∈ k 2 ϕ θ = u 2 + d 2$
(3)

where d denotes all the quantities of disturbances acting on the longitudinal and transversal directions such as the unexpected impact on the pitch-roll system, some environmental factors (wind, waves, currents), and ship-to-ship hydrodynamic interaction forces nearby (Lee, 2017). A proposed controller should be designed to meet the goal of suppression of pitch-roll responses within finite times. By considering the vector of state variables as $x = [ θ , θ ˙ , ϕ ˙ , ϕ ˙ ] = [ x 1 , x 2 , x 3 , x 4 ]$, the controlled pitch-roll system will be expressed by the state-space representative as follows:

$x ˙ = ν 1 x + γ ( x , t ) + ν 2 u + ν 3 d$
(4)

where the terms are given as

$x = [ x 1 x 2 x 3 x 4 ] , ν 1 = [ 0 1 0 0 − Ω n 1 2 − 2 ∈ μ 1 0 0 0 0 0 1 0 0 − Ω n 2 2 − 2 ∈ μ 2 ] , ν 1 = ν 2 = [ 0 0 1 0 0 0 0 1 ] , γ ( x , t ) = [ 0 − ∈ k 1 x 1 2 0 − ∈ k 2 x 1 x 3 ]$

As seen in Fig. 1, the quasi-sliding mode controller will be implemented to suppress the coupled pitch-roll system exposed to two regular disturbances. For the first step, the sliding surfaces should be generated using the tracking error e = [e1, e2 ]T = [x1 - xd1, x3 - x3d ]T as follows:

$s = [ s 1 s 2 ] = [ p 1 e ˙ + p 2 e p 3 e ˙ + p 4 e ]$
(5)

where pi (i = 1,2,3, 4 ) is the positive constant (R+ ), which is related to the accuracy and rate of convergence for four state variables (pitch angle, pitch rate, roll angle, roll rate). Then, the sliding mode dynamics can be written as

$s ˙ = [ s ˙ 1 s ˙ 2 ] = [ T 1 + p 1 ( d 1 + u 1 ) T 2 + p 3 ( d 2 + u 2 ) ]$
(6)

where

$[ T 1 T 2 ] = [ p 1 ( − Ω n 1 2 x 1 − 2 ∈ μ 1 x 2 − ∈ k 1 x 2 2 − x ¨ d 1 + p 2 ( x 2 − x ˙ d 1 ) p 3 ( − Ω n 2 2 x 3 − 2 ∈ μ 2 x 4 − ∈ k 2 x 1 x 3 − x ¨ d 3 + p 2 ( x 4 − x ˙ d 3 ) ] .$

The control goal is to achieve the robust suppression of the above dynamics asymptotically under two regular disturbances. The active control synthesis can be realized with the equivalent and switching parts,

$u = [ u 1 u 2 ] = [ − T 1 p 1 + 1 p 1 u s w 1 − T 2 p 3 + 1 p 3 u s w 2 ]$
(7)

where $− T 1 p 1$ and $− T 2 p 3$ indicate the equivalent part; $1 p 1 u s w 1$, $1 p 3 u s w 2$ denote the switching control part. Therefore, the equation (6) can be rearranged as

$s ˙ = [ s ˙ 1 s ˙ 2 ] = [ p 1 d 1 + u s w 1 p 3 d 2 + u s w 2 ]$
(8)

When implementing the SMC scheme on the pitch-roll system, one of the main drawbacks is the chattering phenomenon. The quasi-SMC approach is helpful to attenuate high control activities by replacing a sigmoid function (Cuong et al., 2020;Lee et al, 2020b). By assuming that the external disturbances is bounded as $| d i | ≤ σ i ( i = 1 , 2 )$ the switching control scheme can be realized as

$u s w = [ u s w 1 u s w 2 ] = [ − ρ 1 s 1 | s 1 | + ξ 1 − p 1 σ 1 − ρ 2 s 2 | s 2 | + ξ 2 − p 3 σ 2 ]$
(9)

where ρi (i = 1, 2 ) denotes a positive constant in large number; ξi (i = 1, 2 ) describes an arbitrary parameter designated by the system designer.

Theorem 1. The state variable vector (x) of the coupled pitch-roll system in equation (4) is regulated by tracking the reference responses xd , asymptotically as t → ∞ for all initial conditions in the sliding surface (5) by using the proposed control schemes in equations (7) and (9).

Proof: The Lyapunov function candidate with positive definite is chosen to prove the closed-loop stability,

$V ( s 1 , s 2 ) = 1 2 ( s 1 2 + s 2 2 )$
(10)

The derivative of the Lyapunov function is shown as follows:

$V ˙ = s 1 s ˙ 1 + s 2 s ˙ 2$
(11)

Substituting (8) into (11) leads to

$V ˙ = s 1 ( λ 1 d 1 + u s w 1 ) + s 2 ( λ 3 d 2 + u s w 2 )$
(12)

From (9) and (12), it can be further given as

$V ˙ = s 1 ( p 1 d 1 − ρ 1 s 1 | s 1 | + ξ 1 − p 1 σ 1 ) + s 2 ( p 3 d 2 − ρ 2 s 2 | s 2 | + ξ 2 − p 3 σ 2 ) = p 1 d 1 s 1 − ρ 1 s 1 2 | s 1 | + ξ 1 − p 1 σ 1 s 1 + p 3 d 2 s 2 − ρ 2 s 2 2 | s 2 | + ξ 2 − p 3 σ 2 s 2 ) ≤ λ 1 | s 1 | ( | d 1 | − σ 1 ) − ρ 1 s 1 2 | s 1 | + ξ 1 + λ 3 | s 2 | ( | d 2 | − σ 2 ) − ρ 2 s 2 2 | s 2 | + ξ 2$
(13)

The parameters ρi, σi (i = 1, 2) and σj (j = 1, 2, 3) are positive constants (R+ ); $| d i | ≤ σ i ( i = 1 , 2 )$. Indeed, V > 0 (positive definite) and $V ˙ ≤ 0$ (negative semi-definite) are realized by using the proposed control scheme. Therefore the convergence of tracking error $lim t → ∞ ‖ e ( t ) ‖ = 0$ is guaranteed by the Lyapunov stability theory.

## 3. Simulation tests

In this section, numerical simulations are illustrated to demonstrate the effectiveness of the proposed control method. As discussed in the previous section, the pitch-roll system shows a series of complex dynamical behaviors depending on the system parameters such as amplitudes of wave excitation forces, wave encountering frequencies, and natural frequencies for pitch and roll motions. The main parameters for the simulation tests are selected as follows:

• μ1 = 0.2 , μ2 = 0.2 , Ωn2 = 1.0, Ωn1 = 2Ωn2 ,

• k1 = 1.5, k2 = 1.2 , ϵ = 0.01 ,

• F1 = 0.7, F2 = 0.02 , ωe1 = 1.03, ωe2 = 1.03 ,

• p1 = 0.05, p2 = 0.04 , p3 = 0.05, p4 = 0.04 ,

• ρ1 = 0.1, ρ2 = 0.1 .

where μ1 and μ2 mean the hydrodynamic damping coefficients for the pitch and roll modes in a marine vessel; Ωn1 and Ωn2 denote the natural angular frequencies; k1 and k2 indicate the nonlinear restoring terms; F1 and F2 describe the amplitude of wave excitation in the longitudinal and transversal directions; ωe1 and ωe2 are the wave encountering frequencies acting on the pitch and roll motions, respectively. The above values are selected to achieve the control goals for the practical pitch-roll models under sea disturbances, focused on the internal resonance case (Ωn1 ≅ 2Ωn2 ). Some control method shows good performance in case of restricted low circumstances, such as low disturbances and low restoring values. In this paper, high values of parameters are selected to make extreme responses for pitch angle/rate and roll angle/rate, such as two disturbances from the front (big primary wave; F1 = 0.7 ) and side directions (small secondary wave; F2 = 0.02 ) of the marine vessel. In addition, the values of restoring forces are set to k1 = 1.5 and k2 = 1.2 , respectively. However, the wave encountering frequencies (ωe1 = 1.03 , ωe2 = 1.03 ) are chosen as the near pitch frequency, which makes sense of practical sea situations and shows torsional pictures. When the vessel situate on the many resonance conditions, it results in capsizing phenomena. Since the natural pitch frequency is twice the natural roll frequency, it is expected to resulting in a large roll motion. To avoid the extreme resonance phenomena, the values of wave encounter frequencies are not matched with the natural frequencies. The initial starting points of pitch and roll angle are designated as 0.05(rad) equally. Control action activates at 330 seconds through all the simulation tests.

Figs. 2 and 3 show the comparison of the uncontrolled cases (upper figure) and the controlled cases (lower figure) of pitch angle and pitch rate, respectively. Since the pitch-roll system is highly sensitive to the parameters in equation (2), various nonlinear motions are demonstrated, especially in pitch motion (Lee et al., 2021). In the upper figures of Figs. 2 and 3, one can see the torsional oscillations in some bounded areas. Torsional behaviors of pitch rate are more clear before control action. Therefore, controlling the pitch rate can be regarded as one of the challenging tasks. Thus, control performance is usually recognized by the pitch rate stabilization for marine vessels. Fig. 4 indicates the tracking error (e1 ) for the pitch angle after control action. As illustrated in the magnified zoom (Fig. 4), all the errors are not eliminated by using the quasi-SMC due to its strong disturbance (F1 ).

Fig. 5 describes the simulation results for the uncontrolled case (upper figure) and controlled case (lower figure) of roll angle. Since the patterns of roll angle and roll rate are almost identical, a mere roll is illustrated. As depicted in the upper figure of Fig. 5, the increasing and decreasing oscillations are repeated in the roll motion. It may result in a capsizing accident owing to the resonance oscillations. Fishing vessels are more prone to capsize due to the frequencies of encountering disturbances and dynamical behaviors as well as the magnitudes of excitation forces. The tracking errors (e2 ) of roll angle is illustrated in Fig. 6. Still, some errors exist when implementing quasi-SMC method.

Fig. 7 indicates the time histories of control inputs for both pitch and roll motion (u1 = pitch control; u2 = roll control). There is no chattering phenomenon during control action until 400 seconds. However, it is observed that the peak jump at the activating point for pitch control. It means that the control burden of pitch motion is higher than the roll control. As pointed out in the previous report (Lee et al., 2021), a type of integer-order SMC presents high control costs at the control action. When increasing the values of control parameters (ρ1, ρ1 ) for high performance, the peak jump will be higher. Also, in case of high disturbances, the values of control parameters should be increased to stabilize the ship’s motions. At that case, it is useful to solve the problem by adding some kinds of adaptation algorithm (Xu et al., 2020).

Fig. 8 represents the sliding surfaces of pitch motion (s1 ) and roll motion (s2 ). Both values are recorded in a smooth line. Even though the sums of errors for the angle and angle rate fluctuate, it shows almost zero degrees of both sliding surfaces in the finite times. Finally, it concludes that the coupled pitch-roll motion can be successfully stabilized with the help of the proposed control strategy under two regular disturbances.

## 4. Conclusions

The paper investigated the pitch-roll stabilization of marine vessels subjected to the external disturbances in the longitudinal and transversal directions. The quasi-SMC method has been implemented to achieve two control goals of state convergences and chattering reduction. Numerical simulations have been presented to demonstrate the effectiveness of the active control scheme. The coupled pitch-roll system presents complex dynamical behaviors according to various system parameters. Especially, it is hard to eliminate all amounts of pitch rate errors. Even though the SMC has the merits of robust performance in the state convergences, it needs high control burden owing to the sliding surface based on the integer-order type. However, it can be concluded that the complex pitch-roll motions can be suppressed effectively by using the proposed control method. Future work will be progressed on mitigating control burden of vessel actuators by considering a novel fractional order sliding surface.

## Acknowledgements

This research was supported by Mokpo National Maritime University Research Grant in 2020.

## Figure

Block diagram of pitch-roll stabilization using active control synthesis.

Uncontrolled/controlled pitch angles when control action starts at t = 330 sec.

Uncontrolled/controlled pitch rates when control action starts at t = 330 sec.

Pitch angle error (e1 ) during control action.

Uncontrolled/controlled roll angle when control action starts at t = 330 sec.

Roll angle error (e2 ) during control action.

Control inputs (u1 = pitch control; u2 = roll control).

Sliding surface variables in the equation (5) (s1 = pitch motion; s2 = roll motion).

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