# Finite-Time Speed Control of Marine Diesel Engine Based on ADRC

Authors: Yuanqing Wang; Guichen Zhang; Zhubing Shi; Qi Wang; Juan Su; Hongyu Qiao
Journal: Mathematical Problems in Engineering
 (2020)

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## Abstract

In this paper, in order to handle the nonlinear system and the sophisticated disturbance in the marine engine, a finite-time convergence control method is proposed for the diesel engine rotating speed control. First, the mean value model is established for the diesel engine, which can represent response of engine fuel injection to engine speed. Then, in order to deal with parameter perturbation and load disturbance of the marine diesel engine, a finite-time convergence active disturbance rejection control (ADRC) is proposed. At the last, simulation experiments are conducted to verify the effectiveness of the proposed controller under the different load disturbances for the 7RT-Flex60C marine diesel engine. The simulation results demonstrate that the proposed control scheme has better control effect and stronger anti-interference ability than the linear ADRC.


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## Body

## 1. Introduction

For the energy efficient and environmental friendly, diesel engines are widely used in ship transportation [1, 2]. For ship propulsion systems, the primary task of the control system of the diesel engine is to keep the speed steady in the inherently unstable and unpredictable external environment [3]. With the development of diesel engine technology, electronically controlled diesel engines have replaced mechanically controlled, and many speed control methods have been developed [4]. These methods fall into two main categories: model-based control method and black box method [5]. The control methods based on the model mainly include adaptive control, system identification, robust control, variable structure control, stochastic system theory, and optimal control [6–8]. These above methods depend on an accurate mathematical model, but it is difficult to achieve in practical engineering applications. Different from the first method, the black box method does not need precise mathematical model and usually includes the classic PID control [9], fuzzy control [10], and active disturbance rejection control (ADRC) [11] as mentioned previously.The speed control effect of the diesel engine is affected strongly by the inherently unstable and unpredictable external load disturbances; therefore, the active disturbance rejection control has attracted much attention due to its excellent disturbance resistance [12–14]. In the theory of ADRC, both inherently unstable and unpredictable external load disturbances are treated as a “total disturbance,” which is expanded to a new state of the system and then estimated and compensated by the extended state observer (ESO) [15–17]. Due to its excellent antidisturbance ability, ADRC has been widely used in industrial applications. Castillo et al., Zheng et al., and Gao proposed a linear active disturbance rejection control (LADRC), which reduced its adjustment parameters to two, greatly promoting its industrial application [18–20]. Compared with the wide application and theoretical research of LADRC, it is noted that the nonlinear ESO proposed in this manuscript is rarely studied [2, 21]. Therefore, it is essential to further study the nonlinear extended observer.According to the theory of the active disturbance rejection control, the ADRC nonlinear state error feedback control law is similar to the PD control and converges to the equilibrium point when time tends to be infinite, and the convergence time is not controllable. Finite-time control has higher accuracies, faster convergence rates, and better disturbance rejection properties. Meanwhile, the convergence time of the finite-time control method is controllable. In recent years, finite-time convergence theory has attracted wide attention [22, 23]. However, we know that finite-time convergence theory is rarely used in ADRC.In this paper, a new ADRC method, which can converge in finite time for uncertain nonlinear systems, is designed. According to the theory of ADRC, the possibly unknown diesel system dynamics, external load disturbance and parameter perturbation, are treated as an extended state of the system, which are estimated and compensated in real time by a new ESO. This new type of ESO, which is established by inverse hyperbolic sine function, has fewer adjustment parameters and simple theoretical analysis. On the basis of feedback linearization of the ESO, the sliding mode error feedback control law is designed based on the theory of finite-time convergence [24]. According to Lyapunov stability theory, the control law can converge in finite time.The content of this article is as follows. In the “Mathematic model” section, the mathematic model of the diesel engine is established. In the “ADRC design” section, inverse hyperbolic sine function is used to design nonlinear ESO, and the new nonlinear ESO method, which has few adjustment parameters, is used to observe and compensate various load disturbances of the diesel engine. And then, we design a finite-time sliding mode controller for ADRC, and it is proved that the controller can converge in finite time. In the “Simulation experiments” section, several experiments are carried out with load disturbance and speed fluctuation, respectively, and the performance of the finite-time active disturbance rejection control (FT-SADRC) is compared with the linear ADRC. In the “Conclusion” section, the last paragraph will summarize the research work of this paper.


## 2. Mathematical Model

In this paper, the research object is the 7RT-Flex60C diesel engine. The maximum sustained power is 16520 kW, and the rated speed is 114 rpm. This paragraph established the mean value model of the diesel engine [25, 26]. The details are as follows:(1)dnedt=−103Vdk1ne+k2ne2NstI30π2−KQρne2D5I30π+30π2103ηiHum˙fneI,where net is the diesel engine speed, Vd is the volume of air cylinder per cycle, k1, k2 is the coefficient of fitting function, Nst is the number of strokes, KQ is the torque coefficient, ρ is the seawater density, D is the diameter of the propeller, ηi is the thermal efficiency, I is the total inertia of the diesel engine, shaft system, and propeller, Hu is the low heat value of the fuel, and m˙f is the average mass of fuel flowing into the cylinder per cycle. Suppose u=m˙f, fnet=−103Vdk1net+k2net2/NstI30/π2−KQρnet2D5/I30/π, gnet=30/π2103ηiHu/netI, then formula (1) can be expressed asdne/dt=fne+gneu.We definex1t=∫netdt and x2t=net. Then, the state-space model can be obtained:(2)x˙1t=x2t,x˙2t=fx1t,x2t+gx1t,x2tu−b0ut+b0ut,y=x2t,where b0>0 is the input channel process gain, treated as a (locally) constant parameter; y is the system output; and u is the control input for the system.The load perturbation of the diesel engine is the disturbance caused by complex sea conditions. The ADRC system with load disturbance characteristics is shown in Figure1.Figure 1
The propulsion control system with load disturbances.According to the working principle of Figure1, the control system with load disturbance can be expressed as follows:(3)x˙1t=x2t,x˙2t=fx1t,x2t+gx1t,x2tut−b0ut+b0ut+lt,y=x2t,where lt is the system load disturbance, lt<L, and L is a constant. Here,(4)fx1t,x2t+gx1t,x2tut−b0ut+ltis the new lumped disturbance and has bounded. We define that(5)x3t=fx1t,x2t+gx1t,x2tut−b0ut+ltas the extended state, and suppose x˙3=ot. Equation (3) is extended in the following equation:(6)x˙1t=x2t,x˙2t=x3t+b0ut,x˙3t=ot,yt=x2t.


## 3. Controller Design

The ADRC approach consists of three modules: tracking differentiator (TD), extended state observer (ESO), and nonlinear state error feedback control law (NLSEF) [27]. TD generates a transient trajectory to avoid the setpoint jump. The system’s total disturbance is estimated by the ESO. The NLSEF is a nonlinear PD controller.The modules’ design is independent. Each module can be designed separately according to the controlled object and combined together to acquire the ADRC controller. For system equation (6), the TD in the continuous form is as follows:(7)v˙1t=v2t,v˙2t=R2−a1e1arshe1−a2e2arshe2/R,where arsh is the inverse hyperbolic sine function [28] (arshx=lnx+x2+1); e1t=∫e2tdt, e2t=v2t−rt; rt is the system reference signal; R>0, a1>0, a2>0. For any bounded integrable function rt, the tracking value of rt is v2t which satisfies the following formula:(8)limR⟶∞∫0Tv2t−rtdt=0,where T > 0.The ESO is selected as follows:(9)e3t=z1t−∫0tytdt,z˙1t=z2t−β1e3t,z˙2t=z3t−β2⋅arshe3t+b0ut,z˙3t=−β3⋅arshe3t,where β1>0, β2>0, and β3>0. z1t and z2t track the output variable x1t and x2t, respectively. x3t is the extended state and is used to estimate the total disturbance. For the ESO, the method of parameter tuning is used and described as follows [29]:(10)β1=9veso,β2=27veso2,β3=27veso3,where veso is the observer observation speed. Because of β1β2−β3=243veso3−27veso3 and ωeso>0, the stability condition of the observer can be satisfied [30]: β1β2−β3>0.In order to improve the control performance, we apply the sliding mode error feedback control law to the ADRC. We definee6t=∫e7tdt and e7t=yt−r=x2t−r. Here, r is the system reference signal. And hence, the error system corresponding to equation (6) is(11)e˙6t=e7t,e˙7t=fx1t,x2t+gx1t,x2tut−b0ut+b0ut+lt.In order to apply the ADRC, a following governing action is proposed:ut=u0t−z3t/b0 [31], and system (11) can be rewritten as(12)e˙6t=e7t,e˙7t=fx1t,x2t+gx1t,x2tut−b0ut+lt+u0t−z3t,where u0t is the new control input for the system, which can be designed even as formula (21).Due to the expansion of observer (9) convergence, that is, z1t⟶x1t, z2t⟶x2t, z3t⟶x3t=fx1t,x2t+gx1t,x2tut−b0ut+lt. System (12) can be rewritten as(13)e˙6t=e7t,e˙7t=u0t.And then, we introduce the theory of finite-time convergence.For the nonlinear systems [32],(14)x˙=gx,x⊆Rn,g0=0,where f:D⟶Rn satisfies the locally Lipschitz continuous condition. x=x1,x2,…,xnT is the state.Lemma 1 (see [31]).
For system (14), if there exist any real numbers and 0<β<1, β∈C1, radially unbounded Lyapunov function V(x), the system such that V˙x≤−kVβx, k > 0, then system (14) is global finite-time stable. The settling time x0 satisfies Txx0≤V1−αx0/c1−α.
In the following, finite-time stability theory is used to analyze the convergence of the sliding mode error feedback control.
And then, we design a finite-time convergence error feedback control rate for ADRC.Definition 1.
For system (6), we design the system control input as equation (17).(15)ut=1b0−z3t−k1γe6tγ−1e7t−αRsgnssη,where 0 < γ < 1, k1 > 0, α > 0, and η > 0. Then, the state error e6t and e7t converges to zero in finite time, and the diesel engine speed x2t converges to r. Convergence time will be satisfied in equation (23).(16)TH≤R0s1−η0α1−η+S′1−γ0k11−γ.Proof.
According to sliding mode control theory, we choose the sliding surface as follows [33]:(17)S=e7t+k1e6tγsgne6t,where 0<γ<1 and k1>0.
Because the sliding surface satisfiesSS˙≤0, we define the reaching law as the following equation:(18)S˙=−αsgnSRSη,α>0,η>0,where R=e7, 0<R<R0, and R0=r.
From (17) and (18), we have(19)SS˙=−αRSη+1<0.
From equation (19), system (13) can come to the sliding mode surface.
We take the derivative of formula (17), and then connect formulas (13) and (18):(20)ut=1b0−fx1t,x2t−gx1t,x2tut+b0ut−lt−k1γe6tγ−1e7t−αRsgnssη=1b0−z3t−k1γe6tγ−1e7t−αRsgnssη.
We selected the nonlinear control law as(21)u0=−k1γe6tγ−1e7t−αRsgnssη.
Therefore, control input can be obtained as(22)ut=1b0−z3t+u0t.
And then, we prove that the system converges in finite time. According to sliding mode control theory, the system moves in two steps: (i) the approaching stage; (ii) the sliding state.
The Lyapunov function of the approaching stage can be defined as(23)V=S2.
If the derivative of equation (23) is taken, we can get(24)V˙=2SS˙.
From equations (13) and (24), we have(25)V˙=−2αRSη+1<0.
From formulas (23) and (25), we have(26)V˙=−2αRV1/2η+1.
Consider the operation of the control systemR=e7t, 0<R<R0, R0=r.
Therefore,(27)V˙≤−2αR0V1/2η+1,∀t>0.
According to Lemma1, the first stage can converge for a finite time. Convergence time T1 satisfaction is(28)T1≤R0V1/21−η0α1−η=R0S1−η0α1−η.
And then, the system will move along the surface until it converges to zero. So, the state meets the following requirements:(29)S=e7t+k1e6tγsgne6t=0.
Consider that(30)e˙6t=e7t.
And hence, from formulas (29) and (30), we have(31)e˙6t=k1e6tγsgne6t.
We define that(32)e6=S′.
Formula (31) can be written as(33)S˙′=k1S′γsgnS′.
Choose the Lyapunov function for formula (33) as follows:(34)V′=S′2.
And then, we have the time derivative ofV′(35)V˙′=2S′S˙′.
From formulas (31) and (33), we have(36)V˙′=−2k1S′γ+1≤0.
According to Lemma1, the convergence time of system (33) satisfies(37)T2≤V′1/21−γ0k11−γ=S′1−γ0k11−γ.
Therefore, the whole convergence time of the control system can be given as(38)TH≤T1+T2≤R0s1−η0α1−η+S′1−γ0k11−γ.
The proof of Definition1 is complete.Remark 1.
In this paper, we use arsh(S) instead of the sign function sgn(S). Because it is a smooth function, the control process is stable, and the control precision is improved.


## 4. Simulation Study

The research object of this paper is the RT-Flex60C low-speed marine diesel engine, whose main parameters are shown in Table1.Table 1
RT-Flex60C engine parameters.
Parameters
Value
Amount of cylinders
7
Stroke
2250 mm
Crank link ratio
0.489
Compression ratio
18.4
Stroke number
2
Cylinder diameter
600 mm
Rated speed
114 r/min
Rated power
16520 kW
Rated torque
1390 kN·m
Specific fuel consumption
177 g/kWhIn order to verify the control effect of the proposed algorithm, the simulation experiment of the proposed method (FT-SADRC) is compared with the linear ADRC in different circumstances.


### 4.1. Analysis of the Diesel Engine Running in the Bad Sea Condition

When ships encounter such bad sea conditions as typhoons and tangled propellers, the simulation analysis of the ship propulsion control system under load torque variation is carried out. Here, as an unknown external disturbance,l(t) is 25×103sint⋅20π N·m, the simulation time is set to 100 s, and the initial speed of the diesel engine is rated speed 114 r/min. In the process, the load torque of the propeller is 1390 kN·m. At 50 s, the load torque was suddenly reduced to 900 kN·m to simulate the propeller load change for the wind and waves.Ge et al. proposed a tuning method based on observer bandwidth for LADRC [29]. In this test, the bandwidth of the observer ω0=37 and the bandwidth of the controller ωc=3.5.FT-SADRC parameter is selected as follows:R = 10, b0 = 1000 veso=2, k1 = 0.6, γ = 0.8, and α = 0.4.The comparison of simulation experiments under load disturbance is shown in Figures2–8. We can find the following:(1)
It can be seen from Figures2–4 that the new ESO can better track the system state quantity, and because it has few adjusting parameters and simple theoretical analysis, it is conducive to practical engineering application.(2)
Although two kinds of control methods can achieve good control effect, the rotation speed error of the FT-SADRC is obviously smaller than that of LADRC, and the speed error of the FT-SADRC and the LADRC are 0.559 and 1.785 r/min, respectively.(3)
In Figures7 and 8, we find that control input and torque are smaller for the FT-SADRC, which can save energy and reduce actuator wear.Figure 2
x1 and z1 curve.Figure 3
x2 and z2 curve.Figure 4
x3 and z3 curve.Figure 5
Diesel engine speed y curve.Figure 6
Speed error curve.Figure 7
Diesel engine control input u curve.Figure 8
Diesel engine torque curve.


### 4.2. Analysis of Maneuvering Operation

In this section, the simulation analysis is carried out on the working conditions of ship maneuvering such as entering and leaving port, overtaking and avoiding, without considering the influence of wind waves, ship inertia, and other factors. Here, as an unknown external disturbance,l(t) is 25×103sint⋅20π N·m, the simulation time is set to 100 s, and the initial speed of the diesel engine is rated speed 114r/min. In the process, the load torque of the propeller is 1390 kN·m, and then, the given rotation speed of the bridge is set to 90.5 r/min in 30 s and to 103.6 r/min in 60 s according to the maneuvering navigation conditions.For LADRC [29], the bandwidth of the observer ω0=43, and the bandwidth of the control ωc=4.1. FT-SADRC parameter is selected as follows: R = 10, b0 = 1000, veso=2.7, k1 = 0.4, γ = 0.9, and α = 0.5.From Figure9, the maximum speed of the proposed FT-SADRC method is less than that of the LADRC method, and the time required to restore normal speed is also less than that of the LADRC method.Figure 9
Diesel engine speed curve.From Figure10, it can be seen that the control input of FT-SADRC requires only about 3 s from sudden dumping load to restore stability while the LADRC requires more than 5 s. Therefore, the adjustment speed of the control input of the proposed method is faster than that of LADRC.Figure 10
Speed error curve.Besides, from Figures11 and 12, when ship maneuvering, the FT-SADRC method has small fluctuation, which is beneficial to reduce the vibration of the equipment and extend the service life of the equipment.Figure 11
Diesel engine control input curve.Figure 12
Diesel engine torque curve.


## 4.1. Analysis of the Diesel Engine Running in the Bad Sea Condition

When ships encounter such bad sea conditions as typhoons and tangled propellers, the simulation analysis of the ship propulsion control system under load torque variation is carried out. Here, as an unknown external disturbance,l(t) is 25×103sint⋅20π N·m, the simulation time is set to 100 s, and the initial speed of the diesel engine is rated speed 114 r/min. In the process, the load torque of the propeller is 1390 kN·m. At 50 s, the load torque was suddenly reduced to 900 kN·m to simulate the propeller load change for the wind and waves.Ge et al. proposed a tuning method based on observer bandwidth for LADRC [29]. In this test, the bandwidth of the observer ω0=37 and the bandwidth of the controller ωc=3.5.FT-SADRC parameter is selected as follows:R = 10, b0 = 1000 veso=2, k1 = 0.6, γ = 0.8, and α = 0.4.The comparison of simulation experiments under load disturbance is shown in Figures2–8. We can find the following:(1)
It can be seen from Figures2–4 that the new ESO can better track the system state quantity, and because it has few adjusting parameters and simple theoretical analysis, it is conducive to practical engineering application.(2)
Although two kinds of control methods can achieve good control effect, the rotation speed error of the FT-SADRC is obviously smaller than that of LADRC, and the speed error of the FT-SADRC and the LADRC are 0.559 and 1.785 r/min, respectively.(3)
In Figures7 and 8, we find that control input and torque are smaller for the FT-SADRC, which can save energy and reduce actuator wear.Figure 2
x1 and z1 curve.Figure 3
x2 and z2 curve.Figure 4
x3 and z3 curve.Figure 5
Diesel engine speed y curve.Figure 6
Speed error curve.Figure 7
Diesel engine control input u curve.Figure 8
Diesel engine torque curve.


## 4.2. Analysis of Maneuvering Operation

In this section, the simulation analysis is carried out on the working conditions of ship maneuvering such as entering and leaving port, overtaking and avoiding, without considering the influence of wind waves, ship inertia, and other factors. Here, as an unknown external disturbance,l(t) is 25×103sint⋅20π N·m, the simulation time is set to 100 s, and the initial speed of the diesel engine is rated speed 114r/min. In the process, the load torque of the propeller is 1390 kN·m, and then, the given rotation speed of the bridge is set to 90.5 r/min in 30 s and to 103.6 r/min in 60 s according to the maneuvering navigation conditions.For LADRC [29], the bandwidth of the observer ω0=43, and the bandwidth of the control ωc=4.1. FT-SADRC parameter is selected as follows: R = 10, b0 = 1000, veso=2.7, k1 = 0.4, γ = 0.9, and α = 0.5.From Figure9, the maximum speed of the proposed FT-SADRC method is less than that of the LADRC method, and the time required to restore normal speed is also less than that of the LADRC method.Figure 9
Diesel engine speed curve.From Figure10, it can be seen that the control input of FT-SADRC requires only about 3 s from sudden dumping load to restore stability while the LADRC requires more than 5 s. Therefore, the adjustment speed of the control input of the proposed method is faster than that of LADRC.Figure 10
Speed error curve.Besides, from Figures11 and 12, when ship maneuvering, the FT-SADRC method has small fluctuation, which is beneficial to reduce the vibration of the equipment and extend the service life of the equipment.Figure 11
Diesel engine control input curve.Figure 12
Diesel engine torque curve.


## 5. Conclusions

In view of the marine diesel engine speed control problem under the random load disturbances and unknown uncertainties, the sliding-mode control method is put forward by combining a new ESO (based on the inverse hyperbolic sine function). Although the new observer is nonlinear, the adjustment parameters are the same as linear ESO, which reduce the difficulty of application of the nonlinear observer. The control law is constructed by the sliding-mode control for ADRC, and the method can realize finite-time convergence and improve the convergence speed and control precision. From the simulation experiment using the model of a large low-speed two-stroke marine diesel engine RT-Flex60c, the proposed method has more advantages than the LADRC method, such as control accuracy and disturbance rejection (both random disturbance and sudden dumping load disturbance).Then, the main contributions of this article are summarized as follows: (1) in theory, a new ESO and finite-time convergence control law is proposed, which makes full use of the known nonlinear term in the diesel engine model and further improves the adaptability of the controller. The proposed method can be applied to control issues of other systems which have known nonlinear terms. (2) In practical, the proposed method can maintain quick regulation and fewer tuning parameters of diesel engine speed under different external and system disturbances and has more practical significance compared with LADRC. Actually, the fuel injection signal of a marine diesel engine is usually maintained at several fixed positions for safety and economy. Therefore, the proposed method under a certain output constraint could be considered further.



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