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 pitchroll 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 pitchroll 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 pitchroll 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 antiroll tanks, gyrostabilizers, fin stabilizers, rudderroll stabilization, and so forth (Wang et al., 2017). However, it is seldom reported to the stabilization of coupled pitchroll motion (Huang et al., 2018).
This paper has employed a quasisliding mode control synthesis to suppress the coupled pitchroll 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 tradeoff between the control performance and its cost.
The sliding mode control (SMC) and higherorder 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 nonmotorized barge (Lee et al., 2020a). The key contribution of this work is to achieve the ship stabilization of coupled pitchroll motions governed by a twodegreeoffreedom (2DOF) motions and chattering reduction of the actuators using a quasiSMC method.
The rest of this paper is represented as follows. Section 2 accounts for the mathematical formulation of pitchroll ship dynamics. Then, quasiSMC is implemented to control the pitchroll motions, and the Lyapunov theory is shown to prove the closedloop 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 pitchroll system
The coupled pitchroll modes in regular wave governed by 2DOF (twodegreeoffreedom) can be addressed as follows (Lee and You, 2018a):
where M and K describe the pitch and roll motions, respectively; I_{θ} and I_{ϕ} are the inertial moments for the pitch and roll directions; ${M}_{\ddot{\theta}}$ and ${K}_{\ddot{\varphi}}$ denote the hydrodynamic added masses for the pitch and roll directions; ${M}_{\dot{\theta}}$ and ${K}_{\dot{\varphi}}$ show the hydrodynamic damping terms; M_{θ} and K _{ϕ} describe the hydrostatic linear coefficients for the 1^{st} order restoring responses; M_{θ θ} and K_{ϕ θ} represent the 2^{nd} order restoring terms; M_{s} (t) and K_{s} (t) indicate the wave excitation forces on the pitch and roll modes, respectively. One can expect the autoparametric resonance phenomena according to the coupled pitch and roll motions. Some analytical approaches are useful to solve the above pitchroll modes using the perturbation theory. Otherwise, a nondimensional process is helpful using a small amount of perturbation parameter. The coupled pitchroll vessel system under regular waves can be expressed as the following nonlinear dynamical equations through transformation step (Kamel, 2007;Zhou and Chen, 2008):
with
where ϵ is the small amount of perturbation parameter; μ_{1} and μ_{2} are the hydrodynamic damping terms for the pitch and roll modes, respectively, Ω_{n}_{1} and Ω_{n}_{2} describe the natural angular frequencies; k_{1} and k_{2} indicate the nonlinear restoring terms; ω_{e}_{1} and ω_{e}_{2} represent the wave encountering frequencies; F_{1} and F_{2} describe the wave excitation amplitudes divided by the inertia terms.
One of the important dynamical features in the pitchroll system is the internal resonance case (Ω_{n}_{1} ≅ 2Ω_{n}_{2} ) leading to the undesirable large roll motions in the angle and velocity. Kamel (2007) investigated the resonance cases such as the combination type (ω_{n}_{1} = ω_{n}_{2} ≅ Ω_{n}_{1} + Ω_{n}_{2} ) and internal type (Ω_{n}_{1} ≅ 2Ω_{n}_{2} ). Since the pitchroll 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 pitchroll 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 antirolling devices, rudderroll stabilizing system, gyrostabilizing system, canted rudders, and active fin stabilizers. The controlled pitchroll system can be expressed as the forced pitchroll system by considering the timevarying two disturbances (d = [d_{1}, d_{2} ]T ) and the input control vector (u = [u_{1}, u_{2} ]T ),
where d denotes all the quantities of disturbances acting on the longitudinal and transversal directions such as the unexpected impact on the pitchroll system, some environmental factors (wind, waves, currents), and shiptoship hydrodynamic interaction forces nearby (Lee, 2017). A proposed controller should be designed to meet the goal of suppression of pitchroll responses within finite times. By considering the vector of state variables as $x=\left[\theta ,\dot{\theta},\dot{\varphi},\dot{\varphi}\right]=\left[{x}_{1},{x}_{2},{x}_{3},{x}_{4}\right]$, the controlled pitchroll system will be expressed by the statespace representative as follows:
where the terms are given as
As seen in Fig. 1, the quasisliding mode controller will be implemented to suppress the coupled pitchroll system exposed to two regular disturbances. For the first step, the sliding surfaces should be generated using the tracking error e = [e_{1}, e_{2} ]^{T} = [x_{1}  x_{d}_{1}, x_{3}  x_{3d} ]^{T} as follows:
where p_{i} (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
where
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,
where $\frac{{T}_{1}}{{p}_{1}}$ and $\frac{{T}_{2}}{{p}_{3}}$ indicate the equivalent part; $\frac{1}{{p}_{1}}{u}_{sw1}$, $\frac{1}{{p}_{3}}{u}_{sw2}$ denote the switching control part. Therefore, the equation (6) can be rearranged as
When implementing the SMC scheme on the pitchroll system, one of the main drawbacks is the chattering phenomenon. The quasiSMC 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 $\left{d}_{i}\right\le {\sigma}_{i}\left(i=1,2\right)$ the switching control scheme can be realized as
where ρ_{i} (i = 1, 2 ) denotes a positive constant in large number; ξ_{i} (i = 1, 2 ) describes an arbitrary parameter designated by the system designer.
Theorem 1. The state variable vector (x) of the coupled pitchroll system in equation (4) is regulated by tracking the reference responses x_{d} , 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 closedloop stability,
The derivative of the Lyapunov function is shown as follows:
Substituting (8) into (11) leads to
From (9) and (12), it can be further given as
The parameters ρ_{i}, σ_{i} (i = 1, 2) and σ_{j} (j = 1, 2, 3) are positive constants (R^{+} ); $\left{d}_{i}\right\le {\sigma}_{i}\hspace{0.17em}\left(i=1,2\right)$. Indeed, V > 0 (positive definite) and $\dot{V}\le 0$ (negative semidefinite) are realized by using the proposed control scheme. Therefore the convergence of tracking error $\underset{t\to \infty}{\text{lim}}\Vert e\left(t\right)\Vert =0$ 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 pitchroll 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:

μ_{1} = 0.2 , μ_{2} = 0.2 , Ω_{n}_{2} = 1.0, Ω_{n}_{1} = 2Ω_{n}_{2} ,

k_{1} = 1.5, k_{2} = 1.2 , ϵ = 0.01 ,

F_{1} = 0.7, F_{2} = 0.02 , ω_{e}_{1} = 1.03, ω_{e}_{2} = 1.03 ,

p_{1} = 0.05, p_{2} = 0.04 , p_{3} = 0.05, p_{4} = 0.04 ,

ρ_{1} = 0.1, ρ_{2} = 0.1 .
where μ_{1} and μ_{2} mean the hydrodynamic damping coefficients for the pitch and roll modes in a marine vessel; Ω_{n}_{1} and Ω_{n}_{2} denote the natural angular frequencies; k_{1} and k_{2} indicate the nonlinear restoring terms; F_{1} and F_{2} describe the amplitude of wave excitation in the longitudinal and transversal directions; ω_{e}_{1} and ω_{e}_{2} 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 pitchroll models under sea disturbances, focused on the internal resonance case (Ω_{n}_{1} ≅ 2Ω_{n}_{2} ). 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; F_{1} = 0.7 ) and side directions (small secondary wave; F_{2} = 0.02 ) of the marine vessel. In addition, the values of restoring forces are set to k_{1} = 1.5 and k_{2} = 1.2 , respectively. However, the wave encountering frequencies (ω_{e}_{1} = 1.03 , ω_{e}_{2} = 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 pitchroll 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 (e_{1} ) for the pitch angle after control action. As illustrated in the magnified zoom (Fig. 4), all the errors are not eliminated by using the quasiSMC due to its strong disturbance (F_{1} ).
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 (e_{2} ) of roll angle is illustrated in Fig. 6. Still, some errors exist when implementing quasiSMC method.
Fig. 7 indicates the time histories of control inputs for both pitch and roll motion (u_{1} = pitch control; u_{2} = 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 integerorder SMC presents high control costs at the control action. When increasing the values of control parameters (ρ_{1}, ρ_{1} ) 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 (s_{1} ) and roll motion (s_{2} ). 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 pitchroll motion can be successfully stabilized with the help of the proposed control strategy under two regular disturbances.
4. Conclusions
The paper investigated the pitchroll stabilization of marine vessels subjected to the external disturbances in the longitudinal and transversal directions. The quasiSMC 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 pitchroll 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 integerorder type. However, it can be concluded that the complex pitchroll 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.