A model predictive vertical motion control of a passenger ship
Introduction
Seakeeping ability of a ship is strongly associated with the improvements of undesired motions in rough seas. Seasickness level of passengers on board should be reduced sufficiently for passenger comfort and crew operability. The first experimental studies that reveal the influence of ship vertical accelerations on the human body were conducted in 1974 (O'Hanlon and McCauley, 1974). This experimental work was carried out with a large group of voluntary male students. Subjects were undergone different amplitude of acceleration at different oscillation frequencies in order to observe vomiting tolerance. According to the results of this work, number of graphs were plotted describing the seasickness regions by means of the frequency and amplitude of the oscillation. Additionally, it is understood that the most dangerous frequency to the human body is in the vicinity of 1 rad/s and seasickness incident is proportional to the exposure time. Since most wave frequencies at sea are observed in the vicinity of 1 rad/s, reducing the pitch motion of the ship by a control system is required for safer operations in a seaway.
The techniques used to reduce the pitch motion of ships have been primarily passive. However, using passive methods has significantly increased the resistance of the ship (Lewis, 1955), (Abkowitz, 1959), (Avis, 1991). Hence, it is understood that when the forward speed of the ship is higher than knots, active actuators such as fins yielded more effective solutions (Perez, 2005), (Huang et al., 2018a).
It is well known that controller design plays a significant role in ship motion control. Therefore, a large variety of control approaches for reducing the vertical motions of ships have been presented in the literature. A proportional, integral, derivative (PID) based controller with the aim of reducing pitch and heave acceleration for a high-speed ferry form using flap and T-foil is proposed in (Esteban et al., 2004). An experimental campaign was carried out in (Giron-Sierra et al., 2001) to demonstrate the effectiveness of active control to damp the vertical motion of a scaled down replica of a fast ferry. In this study, vertical accelerations which are the main cause of seasickness has been reduced with a fuzzy control system based on the fuzzy model of a ship is proposed in (López et al., 2002). A new model-free control approach to reduce the vertical motions induced by random waves on a highspeed ferry was also introduced at (Ticherfatine and Quidan, 2018a), (Ticherfatine and Quidan, 2018b). In these studies, a comparison study between Proportional-Derivative (PD) and so-called i-PD (intelligent PD) was performed and the results showed that i-PD exhibited a better ability to handle the varying system parameters and operating velocity. On the other hand, Zhang et al. proposed a output feedback control method using Ricatti equations to reduce the longitudinal motion and the sickness incidence of the wave piercing catamaran. They used instantaneous heave and pitch velocities as feedback signals (Zhang et al., 2014). (Huang et al., 2018a) applied a numerical and experimental study for pitch stabilization in head waves. In this work, a short term predictor has been proposed to predict hydrodynamic forces. Then these predicted motions are used in a force estimator to forecast the ship's hydrodynamics. In a similar way (Huang et al., 2018b), studied on pitch-roll stabilization problem by active fins. However, although the proposed control strategy considers magnitude bound on the angle-of-attack, it does not consider any rate constraint on the control signal which makes it difficult to realise on a full-sized ship having huge control fins with very large time constants. Similarly, a Linear Matrix Inequality (LMI) based robust static output feedback controller design was developed in (Cakici et al., 2018) to mitigate vertical acceleration of a motor yacht form. Among these aforementioned works, it is apparently seen that to the best of our knowledge, no study exists in literature utilizing the MPC strategy for reducing vertical ship motion by the use of anti-pitching. Moreover, actuator amplitude and rate saturation problems have never been jointly taken into consideration in controller design as provided in this paper. These investigations motivate us to develop a practically applicable model predictive controller for ships having actuator amplitude and fin velocity saturation. It is noted that the optimal controller is also applied in this paper in order to have comparative results.
In this work, a disturbance attenuation type discrete-time MPC under different wave conditions and ship forward speeds is proposed which considers actuator amplitude and rate saturation. The popularity of MPC arises from the fact that the resulting control action respects all the system and problem information, in conjunction with interactions and constraints of the system parameters, which would be very difficult to achieve by any other controller (Borrelli et al., 2017). The bounds on the rate of the actuator force (torque, thrust, etc.) have also been identified as a source of severe performance degradation or instability in control applications and might yield critical limitations on the system (Kose and Jabbari, 2001). Different from the literature, the aim of this study is to develop a more realistic, practically implementable optimal controller which attenuates vertical motions of a passenger ship subject to irregular wave disturbances by respecting actuator constraints.
This paper is organized in a way to develop from modelling of a ship and wave disturbance to the MPC design for the active foil system having actuator amplitude and rate saturation. Therefore, Section 2 describes the mathematical model of the ship and the wave disturbance. Disturbance rejection type MPC strategy is developed in Section 3. Section 4 considers the design of a discrete-time state feedback controller for systems having magnitude and rate based actuator limitations. Section 5 provides extensive simulations on the system under different sea conditions and ship speeds. Finally, Section 6 concludes the paper with some final remarks on possible research directions. The paper is further structured into subsections for better readability.
Notation. Throughout the paper, a fairly standard notation is used. The symbol denotes the set of real numbers, stands for matrices having real entries. Column vectors having n entries are represented by . The symbol ⌃ is used to represent estimated signals. stands for the quadratic term . denotes a rectangular null matrix having a size whereas represents a square identity matrix. Lower case italic letters are generally used to represent vectors whereas capital italic letters are used for matrices. An ellipsoidal set having a weighting matrix and centred at the origin is defined as
For matrices and vectors, indicates the transpose operator. For symmetrical matrix elements, denotes the transposed symmetric element induced by the symmetry. indicates that is a negative semi-definite (positive semi-definite) matrix. stands for the diagonal matrix having elements X and Y on its main dioganal. Finally, stands for the supremum (smallest upper bound) of a set X.
Section snippets
Mathematical model of a passenger ship
During the derivation of the mathematical model, Cummins’ equation is used to represent the vertical ship motions of the considered passenger ship subject to irregular waves. First, the frequency domain coefficients are calculated using an in house code based on strip theory. Then, the convolution integrals used in the equation of motions are approximated by a well-known time-domain identification method. Finally, irregular head wave scenario is realized with wave-based excitation signals by
Disturbance rejection based MPC design
It is well known that input saturation, which can restrict the capability of actuators might cause remarkable performance degradation on the control system performance, sometimes even destabilization (Sadeghi Reineh et al., 2018–11). In many electro-mechanical applications including the active foil systems considered in this study, actuator can be saturated both in terms of the magnitude and the rate of change of the signal that is applied to the system. Therefore, the controllers that were
Discrete-time state-feedback control for systems having magnitude and rate-saturated actuators
In this section, to demonstrate the efficiency of the proposed MPC strategy and to provide a fair comparison with MPC approach, derivation of a novel optimal state-feedback controller for discrete-time systems having amplitude and rate limited actuators is considered.
Assume that the vertical motions of a ship are governed by the difference equation (30). Also, assume that the system is subject to control constraints defined by (31) and (32). Note that the control action can be modelled in
Simulation study
In this section, the results of the simulation studies which were performed for mitigating the vertical accelerations of the passenger ship in irregular head waves having Froude number and 0.50 are presented with the help of several tables and graphs. The optimisation problem (51) is solved in real time for a suitable MPC using the system matrices given in (54). All computations are accomplished using MATLAB along with quadprog solver by taking all initial conditions equal to
Conclusion
In this study, it was aimed to reduce the vertical accelerations of a passenger ship which is operating at two different advance speeds and in the different level magnitude of head waves. For this reason, Cummins’ equation was solved with the time domain identification of fluid memory effects. Firstly, radiation terms were calculated in the frequency domain. By the aid of the information in the frequency domain, all parameters in the Cummins equation were set for the solution in the time
Acknowledgements
The second author was supported by ASELSAN Graduate Scholarship for Turkish Academicians.
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