Complexity

Volume 2019, Article ID 5823827, 7 pages

https://doi.org/10.1155/2019/5823827

## Concise Robust Control of Marine Engine Speed Based on Backstepping and Its Fuzzy Comprehension

^{1}School of Navigation, Guangdong Ocean University, Zhanjiang 524088, China^{2}Hubei Key Laboratory of Inland Shipping Technology, Wuhan 430063, China

Correspondence should be addressed to Sisi Wang; moc.anis@nil23sram

Received 14 March 2019; Accepted 21 April 2019; Published 2 May 2019

Academic Editor: Basil M. Al-Hadithi

Copyright © 2019 Lijun Wang and Sisi Wang. 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

In this paper, a concise robust control law based on Backstepping for marine engine speed regulation is presented with the uniform asymptotic stability of the closed-loop system proved by Lyapunov synthesis, and the control parameters have obvious physical meaning. Furthermore, parameter determination method is given by virtue of closed-loop gain shaping algorithm. To overcome the perturbation due to load or interference change, variable universe fuzzy inference is introduced to optimize the control system on-line. Compared with the existing research literature, the design method and performance of the controller are more in line with the ocean engineering practice. The results of the simulations of the proposed controller are presented and compared.

#### 1. Introduction

The speed regulation of marine main engine (MME) is related to its performance, service life, and ship safety. At present, the advanced MME speed governor is mostly based on PID digital controller [1–3]. There are typical nonlinearity, time-varying, uncertainty, and manual active intervention in the control process of ship main engine speed. Therefore, the traditional PID controller is difficult to obtain the optimal control performance [4, 5]. An improved PID tuning method was proposed for marine diesel engine governors to overcome load fluctuation due to weather and sea conditions [1]. In order to improve the robustness of ship engine speed control, active disturbance rejection controller was presented in [2, 6], and robust control method was recommended in [7, 8]. To optimize the control parameters, several intelligent algorithms have been used to achieve better control performance, such as fuzzy logic comprehension [3, 4], GA optimization [5], and Neural Network adaption [9].

Obviously, the current research mainly focuses on PID, robust, and intelligent algorithms; however, PID has poor self-adaptability, robust control is difficult to achieve in engineering, and intelligent algorithms often only have local optimal solutions. Inspired by previous studies, an optimal robust control system for marine engine speed regulation will be discussed in this paper. The main contributions of this work can be summarized as follows:(i)The uniform asymptotic stability proof of a concise robust control design for the marine engine speed is given, and the control parameters are of obvious physical significance and can be determined easily(ii)Variable universe fuzzy inference (VUFI) is recommended to solve the uncertainty caused by model perturbation and random disturbance for optimal control solution

The layout of the article is as follows: Section 2 presents a marine engine speed regulation model. Section 3 designs a concise robust controller and gives its stability proof. Section 4 provides online optimization of control parameters based on VUFI. Section 5 details the simulation results and discussion. Section 6 gives the conclusion.

#### 2. Marine Engine Speed Regulation Model

##### 2.1. Dynamic Model for Large Low Speed Marine Engines

This study is to design speed regulation controller for large low speed marine engines, such as MAN B&M S60M. The second order dynamic model can be described as follows [8, 10, 11].where is the time constant, is the amplification coefficient, is the dead time, and is the rate of revolution. And a nonlinear main engine model (NMEM) of transfer function form can be expressed aswhere is the fuel index position. The dead time is caused by the injection delay of the fuel system, which is estimated to be within the following range [1]. where is the rated speed and is the number of engine cylinders. For large low speed marine engine, the delay term can be replaced by the following first order inertial expression:Consequently, the propulsion plant dynamics are described with the following equation on the s-plane. Note that all poles in are real and stable.

##### 2.2. Actuating Mechanism

An electronic hydraulic actuator (EHA) controlled by a high-speed switch valve is used. The control signal is pulse width modulation, and the time delay of the link is not considered. The transfer function of the actuator can be described as follows:where is the time constant and is the damping coefficient. In the ship’s main engine operating system, the actuator has a displacement sensor, so it can be adjusted by feedback to make the output more stable.

#### 3. Control Design

##### 3.1. Concise Robust Control Design

In this section, a concise robust control law based on Backstepping for marine engine speed regulation is presented, and the control parameters have obvious physical meaning. Furthermore, parameter determination method is given.

Theorem 1. *Considering the marine engine speed regulation model (1), the proposed controller (7) based on the Backstepping can stabilize the speed motion and guaranteeing the uniformly asymptotic stability of the closed-loop speed regulation system.where , are the speed error and its derivative, and , are design parameters.*

*Proof. *Set , , , and , so we can obtain the state space model of (5) as follows:where , , and one defines the set revolution and the error variable :The first Lyapunov function is selected asDefine virtual control variable , and assumewhere design parameters , the function is negative definite, and one can get ; that is to say, the control of is realized.

Define another error variable and the second Lyapunov function :Substitute (8) and (12) in (14), so one can getTo guarantee the negative definiteness of , one can define a virtual control variable , and assumewhere design parameters , the function is negative definite, and one can get ; that is to say, the control of is realized by the control law (18), and all the variables in main engine speed control loop are uniformly asymptotic stable with equilibrium point .

This ends the proof of Theorem 1.

Substituting (16) into (18), the control law is transformed as follows:Substituting (8) into (20), the actual control law was deduced, if

*Remark 2. *The essence of the control law (21) is to compensate the system’s linearity or nonlinearity and to stabilize the control loop by a PD type controller . It is obvious that the Backstepping control design method is concise and effective. However, the design parameters , are of little engineering significance and can only be determined by trial and error, which is not conducive to the robustness and optimization of the control performance.

##### 3.2. Control Parameters Determination

In accordance with closed-loop gain shaping algorithm (CGSA) [12–17], a concise robust PID controller is presented as follows.where is the system transfer function model, stands for the system period, for the main engine speed control system, and is the design controller. Therefore, one can get a PID controller by substituting (24) into (23). (5) can be transformed into (26). Obviously, (26) has the standard form as (24), which is of strictly rational proper fraction function.On the basis of Theorem 1, the main engine speed steady-state error satisfies . As a result, the integral term is negligible. Define as the system period, and substitute the parameters of (26) into (25), so one can get

*Remark 3. *In accordance with ship main engine knowledge and the CGSA, the design parameters are of clear physical significance, which can be exactly determined. However, the control performance is not guaranteed to be adaptive to load changes and sea conditions.

#### 4. Parameter Online Tuning Based on VUFI

A concise robust PD controller based on Backstepping and CGSA (CRPD-BC) is brought up with definite parameter determination method. However, when the control system model has perturbation due to load or interference change, the fixed control parameter often means the control performance might get worse. Therefore, the on-line optimization of control parameters is very important for the optimal and stable control performance. Consequently, an adaptive PD controller based on VUFI (APD-VUFI) is presented, which can adjust the fuzzy domain and precision of control parameters according to input and output, just as shown in Figure 1. Furthermore, it has better adaptive ability than general fuzzy PID [18–20]. Specific parameter adjustment rules are as shown in (28).Fuzzy comprehensive reasoning takes the form of two inputs and two outputs. are the initial inputs and are the final outputs. According to the selection method of fuzzy scaling factor [19, 20], the domain scaling structure of fuzzy input , can be designed as follows:where , , , and are input regulation factors. At the same time, the domain scaling structure of fuzzy output is defined as follows.where* R* is a proportional constant and* P* is a constant vector.