Mathematical Problems in Engineering

Volume 2019, Article ID 4868473, 17 pages

https://doi.org/10.1155/2019/4868473

## H2/H∞ Antivertical Controller Based on Particle Swarm Optimization (PSO) Using Active T-Foils and Trim Tabs for a Fast Catamaran

College of Automation, Harbin Engineering University, Harbin 150001, China

Correspondence should be addressed to Yu Ma; nc.ude.uebrh@ym_yrolg

Received 30 May 2019; Revised 2 September 2019; Accepted 21 September 2019; Published 13 October 2019

Academic Editor: Łukasz Jankowski

Copyright © 2019 Qidan Zhu and Yu Ma. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

A wave-piercing catamaran (WPC) is a high-performance ship that has been developed in recent years. Compared to other common ships, the WPC has higher lateral stability, larger deck area, lower oil consumption, and higher speed. However, under rough seas and at high speeds, the coupled heave/pitch motions of the WPC can easily produce coupled oscillations that seriously affect its seaworthiness. To solve these problems, a ride control system was designed for the WPC in this study. This system comprises two T-foils, two trim tabs, and a catamaran motion controller. The H2/H∞ controller was designed based on memory-based particle swarm optimization to alleviate the coupled oscillation resulting from heave/pitch motions. Numerical simulations were conducted to validate the proposed method, and the results showed that the proposed motion controller could obviously improve the sea-keeping performance of a WPC.

#### 1. Introduction

A wave-piercing catamaran (WPC) is a fast, high-performance ship that has been developed in recent years [1]. Its main advantage is that, in rough seas, it can pierce a wave rather than ride on it, thereby significantly reducing the wave-added resistance and speed loss. Therefore, a WPC normally has a maximum speed exceeding 40 knots [2]. Further, because a catamaran is much wider than a single-hull vessel, it has relatively much higher lateral stability. Finally, under adverse sea conditions, it shows ∼15% lower roll angle and can withstand large waves; these are useful characteristics in practice [3].

However, a WPC, owing to its higher speed, is more susceptible to disturbances caused by wind, wave, current, and other relative elements. In particular, the influence of heave and heave motions should be considered. Coupled heave/pitch motions have direct and negative effects on the seaworthiness of a ship. For example, harmonic shake can threaten the crew, cargo, and equipment on a ship. Therefore, studies have investigated the longitudinal fin performance and developed vertical stabilizers for high-speed catamarans. Further, T-foil and flaps have been used as subsidiary mechanisms in WPCs to improve their dynamic performance [4].

Active fin stabilizers are commonly used for ship roll motion control, and their control methods, including LQ control algorithm [5], model predictive control (MPC) [6], neural network and fuzzy logic algorithms [7], and robust control algorithm [8], have been studied extensively.

However, few studies have focused on coupled heave/pitch control. Since 2001, the Giron-Sierra team at the University of Madrid has been studying the hydrodynamic characteristics of the T-foil and trim tab for high-speed passenger ferries ferry through extensive pool experiments [9, 10]. A classic proportional-integral-derivative (PID) controller was used for the actuators [11, 12]; however, the joint control algorithm did not fully address coupling problems. Although the control effect was stable, the control system performance should be improved further using modern artificial intelligence control theory and corresponding strategies. A ride control system (RCS) is considered feasible for improving vertical motion. Further, studies have already designed an RCS [13] and a longitudinal motion controller [14] for a wave-piercing boat; therefore, an RCS can also be considered feasible for improving longitudinal motion. We also design a ride control system for WPC by NSGA-II [15] and use *μ* synthesis theory and improved GA for its controller design [16].

However, with rapid developments in ocean engineering, military science, and modern industry, ship RCSs are becoming more complex and system performance requirements are increasing. An important factor hindering the acquisition of high-performance systems is the uncertainty of systems and external disturbances. Uncertainty can cause deterioration of control quality and even system instability. Therefore, uncertainties and their impact on accuracy must be considered when designing control systems. In this light, robust control theory is used to solve ride control problems.

In robust control, the H2 and H∞ norms are the most widely studied measures of robust performance indicators. Hybrid H2/H∞ control considering a system’s H2 and H∞ performance indicators is a type of multiobjective control problem. However, owing to the nonconvexity of the conventional hybrid norm control problem, traditional mathematical methods cannot effectively solve this problem. Further, a multiobjective optimization problem does not have only one solution; instead, its solution is the set of all noninferior solutions or Pareto optimal solution.

In recent years, the linear matrix inequality (LMI) method has been used most commonly to solve the multiobjective H2/H∞ control problem. The LMI method requires the set of parameters satisfying constrained LMIs to be convex. Therefore, some additional constraints are inevitably introduced to solve the control problem, making the optimization results conservative. If the H2/H∞ hybrid multiobjective problem satisfies the time-domain dynamic performance index and has stable poles, the system can simultaneously have satisfactory robustness, steady-state performance, and dynamic performance. Therefore, the study of this problem has important application value; nonetheless, it remains a very difficult multiobjective control problem. Because this type of multiobjective control problem has both time-domain characteristics and frequency-domain characteristics of the optimal design of the multiobjective robust control system, it is difficult to solve using the LMI method. However, the robust control system design using a memory-based particle swarm optimization algorithm (MBPSO) can solve a multiobjective optimal control problem with both time- and frequency-domain characteristics, and because its constrained parameter set does not need to be a convex set, the conservativeness of the system is reduced. Therefore, we apply the multiobjective particle swarm optimization (MOPSO) algorithm to the design of a robust control system.

In this study, we establish an uncertain mathematical model for a WPC and then propose an MBPSO-based H2/H∞ controller for the RCS with a suitable T-type hydrofoil (T-foil) and trim tab. Finally, we conduct a numerical simulation to validate the proposed controller. The study results should have academic and industrial implications for improving the RCS of current catamarans.

#### 2. Model of Coupled Heave/Pitch Motions of a Catamaran

Assuming a homogenous fluid medium in quiet seas, for small disturbances in the vertical plane, the catamaran’s frequency domain formulation can be expressed as [17]where , respectively, represent the displacement, velocity, and acceleration of the vessel; the indices *k* and *j* indicate the *j*-mode oscillatory motion caused by the k-direction force; *M* is the rigid body mass matrix; *A* and *B* are the added mass and damping matrix, and their values are based on the shape of the ship, forward speed, and water depth; *C* is the restoring matrix; and *F* is the excitation force.

For heave and pitch, the coupled two-degree-of-freedom (2-DOF) equations arewhere the numerals 3 and 5 denote the third and fifth DOFs that, respectively, indicate the heave and pitch movements. Correspondingly, *F*_{3} and *F*_{5}, respectively, denote the heave force and pitch force of the target WPC. The hydrodynamic coefficients in these equations are mainly related to three factors: the static buoyancy effect of the ship, potential flow motion of the ship with a free water surface caused by a disturbance, and the effect of viscous flow on the naked hull.

The hydrodynamic coefficients related to the static buoyancy of ships can be obtained easily, whereas those caused by the potential flow are mainly calculated using the 2.5D potential flow theory [18, 19]. The calculations of the hydrodynamic coefficients related to the viscosity and the actuators are based on experimental and semiempirical data. Finally, the total hydrodynamic coefficient is obtained by adding each individual hydrodynamic coefficient described above. In the above equations, only *C* is a constant; *A* and *B* depend on the encounter frequency of the waves.

Figure 1 shows the catamaran model investigated in this study. Further, Table 1 lists the main characteristics of the ship. The hydrodynamic coefficients can be obtained using the 2.5D strip method. Figure 2 and Table 2, respectively, show the added mass and the damping and restoring coefficients of the WPC when its speed is 40 knots.