TDOF PID Controller for Enhanced Disturbance Rejection with MS-Constraints for Speed Control of Marine Diesel Engine

Abstract

This study proposes a two-degree-of-freedom PID controller design and tuning method based on a simple pole placement approach to enhance servo and regulatory performance while ensuring the stability of diesel engines. In the modeling of the control target, the actuator is analytically modeled. In contrast, the type of model for the diesel engine is derived analytically, and its parameter estimation uses operational data from naval ships. The proposed controller consists of a PID controller to improve regulatory performance and a set-point filter to enhance servo response. PID controller parameters consist of the parameters of the controlled plant model and a single tuning variable. At the same time, the set-point filter comprises the controller parameters and a single weighting factor. To ensure the robust stability of the controller, the controller parameters are tuned based on the maximum sensitivity. To verify the effectiveness of this study, simulations for the speed control of a diesel engine with inherent nonlinearity were conducted under three scenarios. Performance was quantitatively analyzed using the integral of time-weighted absolute error, 2% settling time, 2% recovery time, specified maximum sensitivity, and maximum peak response value, and was compared with Skogestad’s IMC and Lee’s IMC. Based on evaluation indices, the proposed controller demonstrated superior performance in both servo and regulatory responses compared to the two existing control techniques while ensuring stability.

1. Introduction

Diesel engines (DE) have long served as the core of propulsion systems in most maritime vessels [1,2]. Due to their excellent reliability, efficiency, and robustness, these engines are preferred and utilized across various ships, including cargo vessels, passenger liners, and naval ships. Among the different types of marine diesel engines, medium-speed diesel engines hold a prominent position due to their balanced performance characteristics that suit a wide range of operational requirements. These engines typically operate at speeds between 250 and 900 revolutions per minute, compromising the high efficiency of low-speed engines and the compact size and flexibility of high-speed engines.

The speed control of the engine is a critical aspect of marine diesel engine operation. Precise speed control is essential for optimal performance, fuel efficiency [3], and compliance with safety and environmental regulations [4]. One of the main challenges in the speed control of marine diesel engines is handling the various load disturbance conditions during vessel operation. These load disturbances can affect the responsiveness and stability of the engine. Therefore, a robust control system that maintains desired speed levels despite these variations is necessary. The performance of a diesel engine must exhibit excellent tracking response to changes in rotational speed and be robust against disturbance inputs and parameter variations. To achieve this, it is essential to have a stable and robust speed controller that can continuously supply fuel oil and air to the combustion chamber by receiving feedback from sensors on the engine’s rotational speed, temperature, etc.

However, controlling the speed of diesel engines remains challenging due to the inherently high nonlinearity [5,6] of diesel engines, time delays, and the combustion process, as well as the parameter variations of the engine under different operating conditions and unpredictable load disturbances. Load disturbances, in particular, are strongly influenced by various external factors such as marine weather and sea conditions, with the type and magnitude of these disturbances being highly variable. Various strategies have been applied to speed control of marine main engines so far, including model predictive control (MPC) algorithms [7,8], sliding mode control (SMC) algorithms [9,10], H-infinity control algorithms [11,12], fuzzy control (FC) algorithms [13,14], neural network (NN) control algorithms [15,16], and active disturbance rejection control (ADRC) algorithms [17,18].

Shu et al. [7] conducted a study applying the MPC algorithm to the step control of marine diesel engine speed, considering model mismatch and external disturbances. The algorithm was simplified by converting the nonlinear model to a linear model. A disturbance observer-based nonlinear MPC and a linear multiple MPC were proposed. The experimental results showed that both controllers performed better than the PID controller.

Li and Yurkovich [9] applied a linear transformation to convert a time delay system into a delay-free system. They used a SMC technique to control the idle speed of an internal combustion engine. Using this method, they demonstrated the asymptotic stability of the speed and the disturbance rejection capability. A robust nonlinear controller using SMC algorithms was proposed by Li et al. [10] to maintain the desired speed of marine diesel engines under harsh sea conditions. The simulation results showed that this controller outperformed classical PID controllers, enhancing speed performance in both transient and steady-states. H-infinity control involves very complex controller design, requiring extensive mathematical knowledge, and may be difficult to apply in real-time systems due to computational time constraints [11,12].

Tran [13] proposed a PSO-based fuzzy logic controller by combining fuzzy logic control and particle swarm optimization (PSO) for marine main diesel engine speed control. The membership functions of the fuzzy logic controller were optimized using PSO. The validity of the proposed method was demonstrated through digital simulation by comparing its performance with traditional controllers. Li and Zheng [14] studied the optimal operating speed to minimize the fuel consumption rate of diesel engines and designed a fuzzy adaptive PID control algorithm for optimization. They verified improved dynamic response and stability through MATLAB (2021a) simulations.

Lynch et al. [15] proposed an embedded real-time type-2 neuro-fuzzy controller for marine diesel engines, which reduces computation time to enhance real-time performance and provides more accurate and robust speed control compared to current commercial engine controllers. Li et al. [16] proposed a speed control method combining a backpropagation neural network and PID control, using the sigmoid function as the activation function of the neural network. By adjusting the proportional gain and integral gain in real-time through the backpropagation neural network, they achieved ideal speed tracking and maintained a stable speed under variable working conditions.

Han [17] anticipated that ADRC, as a new digital control technology, would address the shortcomings of PID and become a practical alternative for various engineering applications. Wang et al. [18] addressed the modeling and pressure control of the high-pressure common rail injection system in diesel engines. They compared the performance of the ESO-based PI controller with that of conventional PID and ADRC controllers.

Despite the advantages of the aforementioned control algorithms, they also have the following drawbacks: MPC has complex control system structures and hardware, which are expensive. Sliding mode control offers fast response speeds and strong robustness against external noise interference, overcoming the drawbacks of system uncertainty. However, it relies on an accurate mathematical model of the control target, making its application range limited compared to PID control technology, especially for complex nonlinear systems like diesel engines [19]. Fuzzy control can become difficult to determine control rules as the number of control variables increases because the rules can grow exponentially, and it heavily depends on the actual experience of the designer. NNs have the disadvantages of high computational load and low convergence speed, making it difficult to meet the real-time requirements of online learning. ADRC offers high reliability and robustness against uncertain disturbances or parameter changes, but it requires the design of an extended state observer and often demands advanced optimization theories like particle swarm optimization or genetic algorithms to obtain the observer’s gains [20].

Unlike these, the internal model control (IMC) method [21,22] and the direct synthesis (DS) method [23,24] have simpler control system structures and straightforward tuning parameters for PID controller settings. A representative IMC method is the SIMC method proposed by Skogestad [21] to control plants with time delays, which has been used as a benchmark for verifying controller performance by many control engineers. Lee [22] applied the IMC method for speed control of medium-speed diesel engines for propulsion, demonstrating their effectiveness through simulations. Anil and Sree [23] designed DS-based PID controllers for various integrating models with time delays. In So’s study [24], DS-based PID controllers were applied to pure integrating models with time delays.

From the literature review, various methods exist for designing and tuning controllers for the speed control of diesel engines. In general, the most desirable control system can tune the controller parameters with a minimal number of adjustment variables in a simple structure while also achieving excellent performance and robustness. Therefore, this study focuses on the structure of a simple control system and proposes a TDOF controller that is robust against parameter uncertainties and provides excellent servo and regulatory responses in the speed control of a marine medium-speed diesel engine. The TDOF controller consists of a PID controller, which emphasizes regulatory response, and a set-point filter designed to mitigate overshoot in the servo response. The modeling of the control target is divided into two parts: the actuator and the diesel engine. The actuator part, including the electro-hydraulic governor, is analytically modeled. On the other hand, since the diesel engine part is a complex and highly nonlinear device, it is difficult to establish a complete mathematical engine model analytically. Therefore, while the form of the mathematical model for the engine is derived analytically, its parameter estimation uses operational data from naval ships.

The PID controller is implemented using a simple pole placement method, and the set-point filter comprises the PID controller parameters and a weighting factor. A specified maximum sensitivity MS is used for a stability measure of the controller parameters.

The proposed method is applied to the speed control of a medium-speed diesel engine, and its effectiveness is verified by comparing it with conventional control methods.

The core contributions of the proposed method, which consists of a parallel PID controller and a set-point filter, are as follows:

(1)

The diesel engine model is obtained from sea trial data of a naval ship.

(2)

The control system structure is simple, and the PID parameters are expressed with model parameters and a single tuning variable.

(3)

The set-point filter consists of controller parameters and a single weighting factor.

(4)

The set-point filter can be used to improve servo response.

(5)

Since there is only one tuning variable for setting PID parameters, parameter tuning is very easy.

(6)

MS-based PID parameter settings can adjust a balance between stability and response performance.

The organization and contents of this study are as follows: In Section 2, the diesel engine, along with the actuator, is modeled as a third-order plant with time delay, which is then approximated to a first-order model with time delay for controller design. In Section 3, the core of this study, a parallel structure PID controller, is designed using a simple pole placement method, and a set-point filter is constructed. Additionally, considering the stability of the control system, the specified maximum sensitivity is used to tune the controller parameters. In Section 4, performance evaluation indices are explained to validate the proposed method, and simulations are performed under the nominal conditions of the model and parameter uncertainty conditions. Finally, Section 5 summarizes the conclusions.

2. Modeling of a Marine Diesel Engine System

The Pielstick 16PC2.5 STC diesel engine(MAN Energy Solutions France, Saint-Nazaire, France), which is subject to speed control, is a V-type 16-cylinder, 4-stroke cycle engine. Its key specifications include a rated speed of 500 rpm, a rated power output of approximately 7.7 MW, a cylinder bore of 400 mm, a piston stroke of 460 mm, and a fuel consumption rate of 200 g/kWh.

Figure 1 shows a block diagram of a control system composed of a controller, an electro-hydraulic governor, a fuel injection system, and an engine assembly. The torque motor embedded in the electro-hydraulic governor changes the position of the hydraulic pilot spool according to the voltage value received from the local control panel (LCP). The spool displacement operates the fuel injection pump rack, thereby controlling the fuel flow rate supplied to the fuel injector. Additionally, the fuel index transmitter provides feedback on the angular displacement of the torque motor, and the speed sensor measures the engine’s rotational speed and sends feedback signals to the controller.

Mathematical modeling is conducted by dividing it into the actuator, including the governor and the diesel engine assembly, as shown in Figure 2. The actuator is modeled as a second-order system, and the diesel engine (DE) is represented as a first-order plus time delay (FOPTD) system using actual operating data. Therefore, the overall diesel engine plant is represented as a third-order plus time delay (TOPTD) system. For convenience, transfer functions and signals are represented using uppercase and lowercase letters, respectively.

In Figure 2, $(s),$ $z\left(s\right)$, and $y(s)$ represent the controller output, actuator output, and diesel engine rotational speed, respectively. The $G_A\left(s\right) \mathrm{a}\mathrm{n}\mathrm{d} G_E\left(s\right)$ represent the transfer functions of the actuator and the DE, respectively.

2.1. Actuator Modeling

The actuator is simplified and modeled as a torque motor connected to a ball screw. First, the electrical circuit related to the torque motor can be represented as shown in Figure 3, where the input voltage results in angular velocity as the output.

The Kirchhoff voltage equation for Figure 3 is given by Equation (1).

$$L_a{\displaystyle \frac{d{i}_a(t)}{dt}}=v\left(t\right)-R_a{i}_a\left(t\right)-e_b(t)$$

(1)

where $v\left(t\right)$, $e_b(t)$, $L_a$, $R_a$, and $i_a\left(t\right)$ represent the input voltage, back electromotive force (EMF), inductance, armature resistance, and armature current, respectively.

When the field is a permanent magnet, the motor driving torque ${\tau }_m(t)$ is proportional to the current, and the back electromotive force $e_b(t)$ is proportional to the angular velocity, which can be expressed by Equations (2) and (3), respectively.

$${\tau }_m(t)=K_t{i}_a(t)$$

(2)

$$e_b\left(t\right)=K_b{\omega }_m\left(t\right),$$

(3)

where $K_t$, $K_b$, and ${\omega }_m\left(t\right)$ represent the torque constant, EMF constant, and the angular velocity of the motor rotor, respectively.

Since the inductance is very small, it can be neglected. Substituting Equations (2) and (3) into Equation (1) and simplifying them, it can be expressed as Equation (4).

$$i_a\left(t\right)=-{\displaystyle \frac{K_b}{R_a}}{\omega }_m\left(t\right)+{\displaystyle \frac{1}{R_a}}v(t)$$

(4)

Considering the motor driving torque and friction torque, the equation of the motion for the motor can be expressed as Equation (5).

$$J_m{\displaystyle \frac{d{\omega }_m(t)}{dt}}={\tau }_m(t)-{\tau }_{mf}({\omega }_m)$$

(5)

where $J_m$ represents the moment of inertia, including the torque motor and the linkage mechanism, and ${\tau }_{mf}({\omega }_m)$ is the nonlinear friction torque associated with ${\omega }_m.$

Neglecting friction, the transfer function of the motor’s angular velocity with respect to the input voltage is expressed as Equation (6)

$${\displaystyle \frac{{\omega }_m(s)}{v(s)}}={\displaystyle \frac{K_m}{1+T_{m}s}}$$

(6)

where $T_m=J_m{R}_a/K_b{K}_t$, $K_m=1/K_b$.

The relationship between the angular velocity of the torque motor and the position of the fuel rack is expressed as

$${\displaystyle \frac{dz(t)}{d(t)}}=K_v{\omega }_m(t)$$

(7)

$$z(s)={\displaystyle \frac{K_v}{s}}{\omega }_m\left(s\right)$$

(8)

where $K_v$ is the gain between them and $z(s)$ represents the position of the fuel rack.

Figure 4 shows the block diagram of the actuator, and

Table 1. Actuator parameters.

Parameters	Descriptions	Values
$R_a$	Armature winding resistance [Ω]	0.25
$L_a$	Armature inductance [H]	-
${\omega }_m$	Angular velocity [rad/s] of a torque motor	-
${\theta }_m$	Rotor angle [rad] of a torque motor	-
$K_b$	Back EMF constant [V·rad/s]	0.4202
$K_t$	Torque constant [N·m/A]	0.4202
$K_{tg}$	Gain [V·s/rad] of a tacho-generator	0.03184
$J_m$	Inertial moment of actuator with motor [N·m·s2]	0.01107
$K_a$	Gain of amplifier	10
$K_n$	Proportional gain	8
$K_v$	Ball screw constant [cm/rad]	0.0796

summarizes its parameters.

The actuator can be represented as follows:

$$G_A\left(s\right)={\displaystyle \frac{z\left(s\right)}{u\left(s\right)}}={\displaystyle \frac{{K_{a}K}_n{K}_t{K}_v}{J_m{R}_a{s}^2+(K_b{K}_t+K_n{K}_t{K}_{tg})s+{K_{a}K}_n{K}_t{K}_v}}$$

(9)

2.2. Diesel Engine Modeling

The equation of motion for the engine rotational speed can be expressed as

$$J_e{\displaystyle \frac{d\omega (t)}{dt}}={\tau }_e\left(t\right)-{\tau }_l\left(t\right)-{\tau }_f\left(t\right)$$

(10)

where $J_e$ and $\omega (t)$ represent the moment of inertia and angular velocity of the crankshaft, respectively; ${\tau }_e\left(t\right)$ is the engine driving torque due to fuel combustion; ${\tau }_l\left(t\right)$ is the load torque; and ${\tau }_f\left(t\right)$ represents various friction torques.

The right side of Equation (10) represents a highly complex nonlinearity to the position of the fuel rack, which determines the fuel flow rate and the engine’s rotational speed. Therefore, it is treated as a nonlinear function of these variables. By converting the engine’s angular velocity $\omega (t)$ to the engine’s revolutions per minute $y(t)$, it can be expressed as

$$R{\displaystyle \frac{dy(t)}{dt}}=f(y,z,t)$$

(11)

where $R={\displaystyle \frac{\pi }{30}}{J}_e,$ $y={\displaystyle \frac{30}{\pi }}\omega .$

Let $y_0$ and $z_0$ be the operating points for the engine’s RPM and the fuel rack position, respectively, and let Δ$y$ and Δ$z$ be their respective small variations. Then, $y=y_0$ + Δ$y$ and $z=z_0$ + Δ$z$. Expanding Equation (11) in a Taylor series around $y_0$ and $z_0$ yields a first-order linear differential equation as

$$R{\displaystyle \frac{\mathsf{\Delta }y(t)}{dt}}=-\alpha \mathsf{\Delta }y\left(t\right)+\beta \mathsf{\Delta }z\left(t\right)$$

(12)

where α = ${-\left.{\displaystyle \frac{\partial f(y,z,t)}{\partial y}}\right|}_{\begin{array}{c}y=y_0\\ z=z_0\end{array}}$ and β = ${\left.{\displaystyle \frac{\partial f(y,z,t)}{\partial z}}\right|}_{\begin{array}{c}y=y_0\\ z=z_0\end{array}}$.

Applying the Laplace transform to Equation (12), the transfer function given by Equation (13) can be obtained.

$$G_E\left(s\right)={\displaystyle \frac{y\left(s\right)}{z\left(s\right)}}={\displaystyle \frac{K_e}{T_{e}s+1}}$$

(13)

where $K_e=\beta /\alpha$ is the steady-state gain and $T_e=R/\alpha$ is time constant for the DE.

Considering the time delay ${\theta }_E$ as the sum of the time from the fuel rack adjustment to the increase or decrease in the injected fuel and the time from the injected fuel to ignition, the final transfer function of the DE is derived as shown in Equation (14).

$$G_E\left(s\right)={\displaystyle \frac{y\left(s\right)}{z\left(s\right)}}={\displaystyle \frac{K_e}{T_{e}s+1}}{e}^{-{\theta }_{e}s}$$

(14)

By combining (9) and (14), the entire DE system can be represented as (15).

$$G_P\left(s\right)={\displaystyle \frac{y\left(s\right)}{u\left(s\right)}}={\displaystyle \frac{b_0}{{a_{3}s}^3+a_2{s}^2+a_{1}s+a_0}}{e}^{-{\theta }_{e}s}$$

(15)

where, $a_3$, $a_2$, $a_1$, $a_0$, and $b_0$ are as follows:

$$\begin{array}{l}{a}_3=J_m{R}_a{T}_e,a_2=J_m{R}_a{+K_b{K}_{t}T}_e{+K_n{K_{t}K}_{tg}T}_e,\\ a_1=K_b{K}_t+K_n{K_{t}K}_{tg}+{K_{a}K}_n{K}_t{K}_v{T}_e,a_0={K_{a}K}_n{K}_t{K}_v,b_0={K_{a}K}_n{K}_t{K}_v{K}_e.\end{array}$$

2.3. Parameter Estimation of Models

The gain $K_e$ and time constant $T_e$ of the Pielstic 16PC DE were estimated using sea trial and operational data from naval vessels through the Piecewise Cubic Hermite Interpolating Polynomial [25].

Figure 5, Figure 6, Figure 7 and Figure 8 show the relationships between fuel flow rate, engine speed, rack position, and engine time constant obtained from the engine operating data. From these relationships, the engine time constant $T_e$ = 0.565 and steady-state gain $K_e$ = 11.251 of the DE at the operating point of 330 rpm are determined.

The time delay of the DE is approximately determined from the equation. In a two-stroke cycle engine, the camshaft completes one rotation for each crankshaft rotation. Assuming the effective stroke of the fuel injection pump is one-fourth of the cycle, the time delay ${\theta }_2$ of the two-stroke cycle engine can be expressed as the sum of the fuel injection pump’s adequate stroke time and the interval between the cylinder’s ignition times, as shown in Equation (16).

$${\theta }_2={\displaystyle \frac{15}{N}}+{\displaystyle \frac{60}{N·n}},$$

(16)

where N and n represent the engine’s revolutions per minute and the number of cylinders, respectively. It is also assumed that the fuel is not adjusted during the effective stroke of the fuel injection pump. In a four-stroke cycle engine, the camshaft makes one rotation for every two rotations of the crankshaft. Therefore, the time delay ${\theta }_e$ for the four-stroke cycle engine is twice that of ${\theta }_2$, as given by Equation (17).

$${\theta }_e={\theta }_4=2{\theta }_2.$$

(17)

The Pielstick 16PC diesel engine is a 16-cylinder, 4-stroke cycle engine, so from Equation (17), $\mathrm{t}\mathrm{i}\mathrm{m}\mathrm{e}\mathrm{d}\mathrm{e}\mathrm{l}\mathrm{a}\mathrm{y} {\theta }_e$ is approximately determined to be 0.114. Therefore, the transfer function of the DE at the operating point of 330 rpm is given by Equation (18).

$$G_E\left(s\right)={\displaystyle \frac{11.251}{0.565s+1}}{e}^{-0.114s}$$

(18)

The DE parameters estimated through Figure 5, Figure 6, Figure 7 and Figure 8 and Equation (17) are summarized in

Table 2. DE parameters at the operating point of 330 rpm.

Parameters	Values
Ke	11.251
Te	0.565
θe	0.114

.

Although the DE parameters at the operating point of 330 rpm have been determined, it should be noted that these parameters can also be obtained for various operating points within the operating range using Figure 5, Figure 6, Figure 7 and Figure 8 and Equation (17).

Furthermore, by substituting the values from Table 1 into Equation (9) and simplifying the transfer function, the actuator is determined as Equation (19).

$$G_A\left(s\right)={\displaystyle \frac{z\left(s\right)}{u\left(s\right)}}={\displaystyle \frac{2.6772}{0.0028s^2+0.2838s+2.6772}}$$

(19)

2.4. Approximation of Model

In the previous section, the DE was modeled as a first-order plus time delay (FOPTD) system, and the actuator was modeled as a second-order system without time delay, resulting in the overall DE plant being a third-order plus time delay (TOPTD) plant. This TOPTD system is approximated as a FOPTD model for model-based controller design.

$$G_M(s)={\displaystyle \frac{K}{Hs+1}}{e}^{-\theta s}$$

(20)

where K is the steady-state gain, H is the time constant, and θ is the time delay of the FOPTD model. The process reaction curve (PRC) method is used to identify the parameters for the FOPTD model [26]. The parameters of the FOPTD model are shown in

Table 3. Parameters of FOPTD model.

Parameters	Values
K	11.251
H	0.577
θ	0.216

.

Figure 9 illustrates the responses of the TOPTD plant and the FOPTD model to a unit step input, along with the error between these responses. Figure 9 reveals that the FOPTD model aligns closely with the TOPTD plant, exhibiting a maximum error of approximately 0.6 rpm. This indicates that the FOPTD model is appropriate for use in controller design.

3. Design and Tuning of TDOF PID Controller

The proposed simple pole placement-based TDOF PID (SPP-TDOF PID) controller consists of a conventional PID controller and a set-point filter, as shown in Figure 10.

Here, r(s), d(s), r_f(s), u(s), y(s), and e(s) represent the set-point, load disturbance, set-point filter output, controller output, model output, and error, respectively.

3.1. Simple Pole Placement-Based TDOF PID Controller

The proposed SPP method determines parameters for the controller by aligning the desired characteristic equation with that of the closed-loop system, including the controller. To facilitate the design and tuning of the controller, the roots of the desired characteristic equation are configured as multiple roots.

The FOPTD model for controller design is Equation (20). The PID controller is as follows:

$$G_C(\mathrm{s})=K_p(1+{\displaystyle \frac{1}{T_{i}s}}+T_{d}s)$$

(21)

where K_p, T_i, and T_d represent the controller’s proportional gain, integral time, and derivative time, respectively.

In Figure 10, the characteristic equation of the closed-loop transfer function for the control system is as follows:

$$1+G_C(s)G_M(s)=1+K_p(1++{\displaystyle \frac{1}{T_{i}s}}+T_{d}s){\displaystyle \frac{K}{Hs+1}}{e}^{-\theta s}=0$$

(22)

The time delay $e^{-\theta s}$ is approximated as ${\displaystyle \frac{1-0.5\theta s}{1+0.5\theta s}}$. Equation (22) is rearranged as Equation (23).

$$c_3{s}^3+c_2{s}^2+c_{1}s+1=0,$$

(23)

where $c_3$, $c_2$, and $c_1$ are as follows:

$$c_3={\displaystyle \frac{0.5H{T}_i\theta }{K{K}_p}}-0.5T_d{T}_i\theta ,c_2=T_i(T_d-0.5\theta )+{\displaystyle \frac{T_i(H+0.5\theta )}{K{K}_p}},c_1={\displaystyle \frac{T_i(K{K}_p+1)}{K{K}_p}}-0.5\theta .$$

For the purpose of controller design, the desired characteristic Equation (23) is selected.

$${(\mathsf{\lambda }s+1)}^3=0$$

(24)

Using characteristic Equations (23) and (24), the PID parameters can be derived as follows:

$$K_p={\displaystyle \frac{{\theta }^3+2\left(3\lambda +2H\right){\theta }^2-12\left({\lambda }^2-2H\lambda \right)\theta -8{\lambda }^3}{K{(\theta +2\lambda )}^3}}$$

(25)

$$T_i={\displaystyle \frac{{\theta }^3+2\left(3\lambda +2H\right){\theta }^2-12\left({\lambda }^2-2H\lambda \right)\theta -8{\lambda }^3}{4\theta (\theta +2H)}}$$

(26)

$$T_d={\displaystyle \frac{H{\theta }^3+6H\lambda {\theta }^2-4\left(2\lambda -3H\right){\theta \lambda }^2-8H{\lambda }^3}{{\theta }^3+2\left(3\lambda +2H\right){\theta }^2-12\left({\lambda }^2-2H\lambda \right)\theta -8{\lambda }^3}}$$

(27)

Since K, H, and θ are given from the FOPTD model, the only variable that needs to be adjusted is the time constant λ.

3.2. Set-Point Filter

To reduce overshoot in the servo response, the controller output of Equation (28) is applied.

$$u\left(t\right)=K_p\left(wr\left(t\right)-y\left(t\right)\right)+{\displaystyle \frac{1}{T_i}}{\int }_0^t(r\left(\tau \right)-y\left(\tau \right))d\tau +T_d({\displaystyle \frac{dr\left(t\right)}{dt}}-{\displaystyle \frac{dy\left(t\right)}{dt}})$$

(28)

where the weighting coefficient is $w$ = [0 1].

Applying the Laplace transform to Equation (28), the set-point filter is as follows:

$$F_r\left(s\right)={\displaystyle \frac{T_d{T}_i{s}^2+w{T}_{i}s+1}{T_d{T}_i{s}^2+T_{i}s+1}}$$

(29)

3.3. Controller Parameter Tuning

The only tuning variable required for the PID controller in the proposed method is the desired time constant λ. This tuning variable is adjusted based on the maximum sensitivity function MS, as in Equation (30). Its physical meaning is the reciprocal of the shortest distance from the point (−1, 0j) to the Nyquist plot. Therefore, the PID tuning in this study allows a suitable compromise between stability and response performance.

$$\mathrm{M}\mathrm{S}=\underset{\omega }{\max }\left|{\displaystyle \frac{1}{1+G_C(j\omega )G_M(j\omega )}}\right|$$

(30)

A higher MS value indicates an excellent control response but reduced robustness, and vice versa. Typically, for FOPTD models, controllers are designed with an MS value ranging from 1.4 to 2.0 [23]. In this study, the designed MS_d value is set at 1.6703 for the FOPTD model and the PID controller system to balance system stability and performance. Accordingly, the time constant λ is adjusted to 0.2317 to achieve MS_d = 1.6703. The findings are presented in

Table 4. Controller parameter setting.

Tuning Methods		Parameters			Remarks
	Kp	Ti	Td	λ	MSa	MSd
Proposed	0.1805	0.5382	0.0700	0.2317	1.4769	1.6703
Lee-IMC	0.1807	0.6850	0.0910	0.2290	1.4043	1.7213
SIMC	0.1187	0.5770	-	0.2160	1.5113	1.5904

, alongside the parameters of the Lee-IMC controller and Skogestad’s SIMC controller for comparison. Due to modeling discrepancies between the TOPTD plant and the FOPTD model, the actual MS_a is determined using the TOPTD DE plant $G_P(j\omega )$ rather than the FOPTD model $G_M(j\omega )$.

4. Simulation and Discussion

Figure 11 shows the overall control system consisting of a set-point filter, controller, actuator, saturator, and DE. A saturator with a limit set at 40 mm was incorporated, taking into account the maximum displacement of the fuel rack.

Simulations were conducted under nominal and parameter uncertainty conditions to verify the performance of the proposed method. Nominal conditions refer to the case where the parameters of the DE do not change, while parameter uncertainty conditions refer to the case where the parameters of the DE change. Parameter uncertainty conditions are further divided into cases caused by the aging of the DE and those caused by modeling errors and changes in the operating point. The proposed method’s servo and regulatory response performance were compared with those of Skogestad’s SIMC method [21] and Lee-IMC method [22]. For the three scenarios, the weight of the set-point filter in the proposed method was fixed at 0.66.

In each simulation, performance evaluation is quantitatively compared considering the integral of time-weighted absolute error (ITAE) defined by Equation (31), a 2% settling time (T_s), a 2% recovery time (T_rcy), MS, and a maximum response value (M_peak). M_peak represents the value exceeding the steady-state value.

$$ITAE={\int }_0^t\tau \left|e\left(\tau \right)\right|d\tau$$

(31)

Furthermore, the robust stability of the system is evaluated through MS. The smaller each performance index, the better the performance of the controller is considered.

4.1. Nominal Conditions

Figure 12 shows the response under nominal conditions when a set-point of 330 rpm is applied as a step input at t = 0 s while the DE operates at a rotational speed of 290 rpm, and a load disturbance input is introduced as a step input at t = 5 s. The load disturbance assumes that wind is applied from the stern to the naval ship’s bow, reducing the fuel rack’s position by about 5.9 mm. When operating at 330 rpm, the position of the fuel rack is about 29.3 mm, corresponding to a fuel flow rate of about 900 kg/h as derived from Figure 8. Due to the wind disturbance, the fuel flow rate must be reduced by approximately 210 kg/h.

Table 5. Performance comparisons against nominal conditions.

Tuning Methods	Servo Response			Regulatory Response			MSa
	Ts	Mpeak	ITAEs	Trcy	Mpeak	ITAEd
Proposed	0.912	0.289	5.060	1.994	23.055	12.182	1.477
Lee-IMC	1.704	1.667	5.830	2.978	22.255	20.346	1.404
SIMC	1.410	1.822	5.883	2.892	29.095	25.723	1.511

provides the response performance metrics.

In the servo response analysis, the proposed method exhibited the lowest values across all performance indices compared to the three tuning methods. The Lee-IMC approach demonstrated a settling time T_s about 1.9 times greater than that of the proposed method, while the SIMC approach exhibited a maximum response value M_peak approximately 6.3 times higher than that of the proposed method.

Regarding regulatory response, the proposed method achieved very low performance index values, except for M_peak. Although the Lee-IMC method had a smaller M_peak than the proposed method, its T_rcy and ITAE were about 1.5 times and 1.7 times higher, respectively. The SIMC method recorded about 1.5 times T_rcy and 2.1 times ITAE greater than the proposed method.

The Nyquist plots in Figure 13 reveal that the proposed controller exhibits a MS_a value comparable to that of the SIMC controller, whereas the Lee-IMC controller demonstrates the lowest MS_a value. Evaluating the performance indices indicates that the proposed method outperforms the others.

4.2. Robustness Against Aging

Its parameters were modified to verify the controller’s robustness against the aging of the DE. Generally, as the DE ages, the steady-state gain of the engine tends to decrease, while the time delay and time constant tend to increase. Therefore, considering the effects of aging, the steady-state gain of the DE is reduced by 20%, and both the time delay and time constant are increased by 20% from the nominal condition. Except for the changes in the DE parameters due to aging, the simulation scenario is the same as in the nominal condition.

As illustrated in Figure 14 and detailed in

Table 6. Performance comparisons against aging.

Tuning Methods	Tracking Performance			Disturbance Performance			MSa
	Ts	Mpeak	ITAEs	Trcy	Mpeak	ITAEd
Proposed	2.195	2.192	10.824	2.652	19.500	15.389	1.401
Lee-IMC	2.380	2.345	10.856	2.933	19.506	22.238	1.314
SIMC	2.404	2.008	12.316	2.712	23.982	29.136	1.411

, the servo and regulatory responses of all three tuning methods were similar; however, the proposed method demonstrated the lowest values across all of the performance indices. Notably, for the ITAE_d metric, the Lee-IMC method recorded a value approximately 1.4 times higher than the proposed method, while the SIMC method’s value was approximately 1.9 times greater.

Figure 15’s Nyquist plots and MS circle demonstrate that the Lee-IMC method possesses the lowest MS_a value, while the proposed and SIMC methods exhibit similar MS_a values.

Consequently, the simulation results indicate that the proposed method is superior to the others.

4.3. Robustness Against Parameter Change

To assess the controller’s robust stability, a simulation was performed under the assumption that the steady-state gain and time delay of the DE is increased by 20% and that there is a 20% relative decrease in the time constant to the nominal DE. Except for parameter changes due to operating condition variations or modeling errors, the simulation scenario is the same as in the nominal condition.

As illustrated in Figure 16 and detailed in

Table 7. Performance comparisons against 20% parameter uncertainty.

Tuning Methods	Tracking Performance			Disturbance Performance			MSa
	Ts	Mpeak	ITAEs	Trcy	Mpeak	ITAEd
Proposed	1.401	0.990	4.767	2.008	30.999	11.410	1.996
Lee-IMC	0.868	5.052	3.955	2.825	29.784	19.099	1.864
SIMC	1.769	6.685	7.569	2.599	38.082	23.672	1.935

, the proposed method had a very small M_peak in the servo response, but T_s and ITAE_s were at an intermediate level. In the regulatory response, the proposed method had the smallest T_rcy and ITAE_d values, with particularly excellent performance in ITAE_d. The M_peak value was comparable to that of the Lee-IMC method. On the other hand, the SIMC method had very large M_peak and ITAE_d values.

Figure 17’s Nyquist plots and MS circle illustrate that the Lee-IMC method achieves the lowest MS_a value, whereas the proposed and SIMC methods exhibit comparable MS_a values. All three methods fall within the ensuring robustness range of 1.4 to 2.0 [23].

Consequently, the simulation results indicate that the proposed method is superior to the others.

5. Conclusions

This study introduced a TDOF PID speed controller to control medium-speed diesel engines, focusing on a straightforward control system architecture. The TDOF controller consists of a PID controller and a set-point filter. The PID controller focuses on improving regulatory response, while the set-point filter focuses on improving servo response.

The modeling of the diesel engine plant was conducted in two parts. The actuator part was modeled as a second-order system using an analytical method. For the diesel engine part, the form of the model was derived as a first-order system with time delay using the analytical method. Still, its parameters were estimated using operational data from naval ships. A third-order system with time delay was approximated to a first-order model with time delay for controller design, and a PID controller was designed by applying a simple pole placement method based on the model. The PID parameters comprise known parameters from the model and a single unknown tuning variable, whereas the set-point filter includes the controller parameters and a fixed weighting value. This configuration facilitates the straightforward tuning of the controller parameters using only one adjustment variable. The controller parameters were adjusted based on maximum sensitivity to balance stability and response performance.

To evaluate the response performance of the proposed controller, simulations were conducted for the speed control of a diesel engine with inherent nonlinearity under three scenarios. Based on various evaluation indices, the proposed controller was compared with two existing control techniques, demonstrating superior performance in servo and regulatory responses while ensuring stability. Future research plans include designing controllers for various operating points and exploring the fuzzy combinations or adaptive control of these controllers.