Mathematical Problems in Engineering

Volume 2018, Article ID 3252653, 20 pages

https://doi.org/10.1155/2018/3252653

## Multiple Model Predictive Functional Control for Marine Diesel Engine

College of Power and Energy Engineering, Harbin Engineering University, Harbin 150001, China

Correspondence should be addressed to Xuemin Li; nc.ude.uebrh@mxl

Received 11 December 2017; Accepted 28 March 2018; Published 14 May 2018

Academic Editor: Xinkai Chen

Copyright © 2018 Runzhi Wang et al. 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 novel control scheme based on multiple model predictive functional control (MMPFC) is proposed to solve the cumbersome and time-consuming parameters tuning of the speed controller for a marine diesel engine. It combines the MMPFC with traditional PID algorithm. In each local linearization, a first-order plus time delay (FOPTD) model is adopted to be the approximate submodel. To overcome the model mismatches under the load disturbance conditions, we introduce a method to estimate the open-loop gain of the speed control model, by which the predictive multimodels are modified online. Thus, the adaptation and robustness of the proposed controller can be improved. A cycle-detailed hybrid nonlinear engine model rather than a common used mean value engine model (MVEM) is developed to evaluate the control performance. In such model, the marine engine is treated as a whole system, and the discreteness in torque generation, the working imbalance among different cylinders, and the cycle delays are considered. As a result, more reliable and practical validation can be achieved. Finally, numerical simulation of both steady and dynamic performances of the proposed controller is carried out based on the aforementioned engine model. A conventional well-tuned PID with integral windup scheme is adopted to make a comparison. The results emphasize that the proposed controller is with stable and adaptive ability but without needing complex and tough parameters regulation. Moreover, it has excellent disturbance rejection ability by modifying the predictive multimodels online.

#### 1. Introduction

Diesel engines are the most pervasive and favored prime movers or power sources in the domain of ship for their superior efficiency [1–3]. Speed control is the crucial task for the diesel engines, because the engine performance and service life of the engine and ship rely very much on the speed adjusting [4]. Especially for the diesel engines serving as the marine main engine for propulsion, it is necessary to govern the engine speed during the whole operating processes. Otherwise, the oscillation of engine speed leads to the engine being unable to operate normally [1]. Moreover, the overspeed caused by poor speed governing performance will bring irreversible damage in the marine main engine [5]. It should be pointed out that large speed fluctuation of the engine may cause heavy vibration and strong impact forces, inducing damaging vibration in the related structures leading to premature failure of the transmission system [6]. Hence, the crucial control targets should include tracking the speed set-point quickly and keeping the possible smallest deviations from the desired speed and recovering fast from the external disturbances. According to the practical needs, speed control strategies for the marine diesel engines have drawn considerable attention during the last decades.

However, the speed control algorithm design for the marine diesel engine remains to be a tough mission due to its inherent high nonlinearity. As for the engine speed, it is a function of various aspects, such as the ambient temperature and humidity, fuel injection timing, and compression and combustion processes. And some of these parameters also rely on the transient engine speed to a larger extent at the same time [7]. Furthermore, speed and load of the marine main engine are strongly affected by numerous other external aspects, such as the weather in the sea and the surface condition of the sea [2, 8]. Attempts have been made on the basis of variable control method, such as conventional PID method [9], sliding mode control (SMC) [2], control [10], active disturbance rejection control (ADRC) [11], fuzzy control [5], and model predictive control (MPC) [12].

Although remarkable progress has been acquired, there are still some drawbacks. Specifically, control and MPC techniques not only rely on exact mathematical model, but also require high-performance processor to execute the complex matrix computation. As shown in [12], the MPC method was carried out with the help of two 900-MHz IMB PPC processors, which actually cannot be accepted commercially. Despite the fact that there exists computationally efficient MPC algorithm with online linearization and quadratic optimization (e.g., [13–15]), where the online computational requirement is acceptable, such computational simplicity is generally realized by sacrificing the control performance as the linear approximation is invalid when the system state and input deviate far away from where they are linearized. Meanwhile, all these advanced methods still suffer from tiring tuning process [9].

As a result, most of the commercial controllers for marine main engine are based on PID or PI [3, 4, 9]. However, control parameter tuning is still intractable. Hence, the control gains in PID scheme need to be calibrated carefully locally, which is mostly achieved by complicated experiments via trial-and-error approach during various operating conditions, and such process is both costly and time-consuming. Unfortunately, when harsh uncertain external condition causes the operating point of the engine to deviate farther from its calibrated condition, the well-tuned PID controller still cannot guarantee good property [4].

Recently, modern control theory and intelligent control method are applied to improve the adaptive and robust performance of the traditional PID control scheme in marine engine speed control. For instance, a self-turning PID controller based on BP neural network was designed for a large-scale low-speed two-stroke marine diesel engine [16]; better control performance was achieved in simulation. Fuzzy control method was combined with PID to intelligently regulate the parameters of the speed controller online for a MAN B&M type diesel engine. The results showed the proposed fuzzy-PID controller possesses high anti-interference and strong robust abilities [17]. The control scheme was utilized as a tuning methodology for the PID speed regulator in [3] to guarantee the robustness against neglected dynamics in a propulsion marine engine and so on.

However, we still underline that the application of MPC method in engine control domain is an unstoppable tendency, various up-to-date relevant articles are giving clue (e.g., [18–21]), and just some issues as mentioned need to be solved furtherly. In many other industrial fields, some researchers started to pay attention to combine PID with MPC. In such applications, the tuning parameters of PID are obtained by MPC optimization law online to keep the advantages and alleviate the shortcomings in both methods. The control performance of these applications has been reported to be excellent. For example, PID was combined with predictive functional control (PFC) [22–25], synthesized with generalized predictive control (GPC) [26, 27], compounded with dynamic matric control (DMC) [28, 29], composited of some other predictive methods as shown in [30, 31].

Up to now, there is no related study published in the field of marine diesel engine yet. Hence, it is deserved to study such kind of application in marine engine speed control, expecting to get good performance and reduce parameters adjustment.

Apart from the above facts, for marine diesel engine speed control, temporally, it is reported that most of the control algorithms were verified via engine models. Some of these models were commonly used mean value engine models (MVEM) [2], and some were even simple low-order transfer function models [3, 32] or another kind of reduced complexity engine model [10]. In such situation, if a model could simulate the characteristics of the engine closer to the real one, it would help to make the verification of control algorithm more reliable and practical. Some innovative examples considered the discrete events in reciprocating engines, and hybrid models (the existence of discrete and continuous model) were studied to model the engine behaviors closer to the reality, such as [33, 34]. However, these models did not consider the whole engine system and only paid attention to intake manifold dynamic and the discrete events in torque generation, and they were merely limited at the idle mode. Some other important characteristics are ignored in such models, such as the working imbalance among different cylinders and the cycle delays, which have a significant effect in engine speed response and can influence the controller’s performance. Besides, cylinder-by-cylinder engine model (CCEM) also has been introduced into the engine control domain [35]. Although it can simulate engine more realistically, it requires in-cylinder pressure map, which has higher cost and is difficult to implement [36].

Motivated by the previous research and the challenges stated above, this paper develops a novel control scheme which combines the multiple model predictive functional control (MMPFC) with conventional PID for the marine diesel engine speed control, which markedly simplifies the parameter tuning work. The main contributions of this paper are listed as follows.

First, we adopt the multiple model strategy, which aims at solving the high nonlinearity and various working conditions of the marine diesel engine. With a set of locally linearized predictive models to approximate the nonlinear system, multimodel approach is able to deal with the nonlinear system with wide working range [37, 38]. Second, the PFC theory is introduced to design the optimizer, which guarantees good tracking performance and robustness with fewer requirements in model structure information. Meanwhile, the calculation amount within the PFC is acceptable, so that the processing speed is fast [39]. The proposed controller is developed via combining the PFC with a classical incremental PID. Third, an online identification algorithm is introduced to obtain the open-loop gain, by which system robustness towards mismatched model is enhanced. Fourth, considering the drawbacks in the existing hybrid model and the difficulty to obtain the CCEM, as a compromise, an accurate cycle-detailed hybrid nonlinear engine model is built in this study to evaluate the proposed controller. On the basis of such engine model, simulation results exhibit that the proposed controller with better performances compared with a well-tuned conventional PID dealt with integral windup scheme. And the online identification algorithm is effective in improving the system robustness under disturbance loads.

The rest of this paper is structured as follows. In Section 2, the combined MMPFC algorithm for engine speed control is described, and the modification method to update the model parameters online is discussed. In Section 3, a cycle-detailed hybrid nonlinear engine model is constructed for verifying the proposed controller, and the comparison between the proposed engine model and the classic MVEM is analyzed. In Section 4, the combined MMPFC method is illustrated in detail to apply in marine diesel engine speed control. And both steady and dynamic performances of the proposed controller are assessed by comparing with a well-tuned PID controller. In Section 5, a conclusion is given to sum up the whole work.

#### 2. Controller Design

##### 2.1. Description of the Predictive Multimodels for Marine Engine Speed Control

In order to design the control algorithm, the predictive multimodels must be obtained in a certain type. For diesel engine speed control, as shown in [40], the relationship between the control input (injection quality) and the system output (engine speed) is high nonlinear. It is difficult to get the accurate physical model, but fortunately there are simplified methods to obtain other types of model.

When engine runs around a certain speed and load condition, as shown in [41–44], the complex engine model can be regarded as simple low-order linear model. The explanation and proof can be found in [44], where the identified result indicates that it is reasonable to adopt the first-order autoregressive (AR) model to represent the speed control modeling in a diesel engine when it works around a specific speed range. In this study, the FOPTD model is adopted as the predictive model in each locally linearized zone, which holds the equivalent structure as the mentioned first-order AR model. Because it captures the process gain (indicates the size and direction of the process variable response to a control move), overall time constant (describes the speed of the response), and effective dead time (states the delay before the response begins) of the process [37]. Furthermore, this kind of model can be identified easily from both simulation and experimental data, generally, from step response data [42].

The working condition of the engine would vary largely as the engine speed and load change. Although the estimation of the load is difficult, it is easy to measure the engine speed. Hence, the zone of local linearization is divided according to the engine speed, and the predictive multimodels of the engine for designing the proposed controller can be written aswhere represents the different speed stage of the target engine under a certain load condition, and the speed range . are the transfer function model between output and the control input, the process gain, the overall time constant, and the effective dead time during the engine speed range , respectively.

*Remark 1. *To upgrade model accuracy, the speed stages above should be divided more and cover all the speed ranges. For example, the whole speed range of the target engine is from 800 rpm to 2000 rpm in this study, it can be divided every 100 rpm or 50 rpm during the whole speed range, and the local model can be identified by step speed response in each stage.

Note that the predictive multimodel in a certain speed stage is not unique, because different controller parameters will provide different result. In this study, the model is identified offline by using the data under a conventional PID controller. The parameters of such PID controller are obtained by simple - method. Hence, the engine speed controller can be designed on the basis of the identified models, which describe the basic dynamic for engine speed system in each speed range.

##### 2.2. Multimodel Switching Method

In multimodel predictive control (MMPC), the switching scheduler is necessary to shift the local model (or local controller) for keeping stable transition among different operating conditions [45]. And the design of the switching scheduler directly affects the stability of the close-loop system [38]. In this study, because the predictive multimodels are gained based on the different engine speed stages, a switching method by referring to the present engine speed is proposed.

As mentioned in (1), when the local model of the speed range is gained, assuming that this local model is the model at the middle speed point , then local models belong to speed points are obtained. Linear interpolation is applied to calculate the present model parameters based on the local models by referring to the present engine speed. This can be realized easily by look-up map method.

##### 2.3. Basic Combined MMPFC-PID Controller

For the predictive multimodels (1), ignoring the time delay, (1) can be discretized as follows:where is the output of the predictive model at sampling instant . and are process gain and time constant calculated by look-up map method from the predictive multimodels at the sampling instant . is the sample and control time.

The predictive output will be conducted based on the control input at sampling instant *,* namely, *.* First, assume that the control input will keep the same within future (predictive horizon) steps, that is, .

Hence, the output predicted 1 step ahead of the predictive model from data at the sampling instant which is

Then the output predicted 2 steps before the predictive model from data at the sampling instant :

Similarly, the output predicted steps ahead of the predictive model from data at the sampling instant :

By taking account of the ignored time delay, the corrected output of the predictive model is given as follows:where is the actual output of the plant at the sampling instant is the corrected output of the predictive model at step* k* is the time delay calculated by look-up map method from the predictive multimodels at the sampling instant .

The future reference trajectory is set aswhere represents the gentle factor of the reference; is the set-point at the sampling instant .

Here, cost function is chosen aswhere is the error between the corrected output and the output of the predictive model without time delay.

*Remark 2. *The cost function means getting the optimal solution within the predictive horizon rather than only at the predicted step as mentioned in [24].

By combining (5) with (8), can be written as

When , the optimal control input is

Conveniently, the incremental PID controller is proposed as follows:where presents the error between reference value and actual output of the plant at the sampling instant are PID parameters at the sampling instant *.*

The PID controller can be rewritten as

By solving (11) and (13) we can get as

Here, a variable is introduced, and thus the denominator will never become zero.

The optimized PID parameters can be calculated through (16), and they are

Here, the PID parameters are optimized by multimodel PFC method. This method can be called MMPFC-PID. And, about the practicability, robustness and stability of the proposed controller can be found similarly in [26]. The basic control structure diagram is illustrated in Figure 1. Provided with proper relative accurate predictive multimodels and basic parameters in the MMPFC-PID method, the proposed controller can optimize the PID parameters in each control step.