# Speed Control for a Marine Diesel Engine Based on the Combined Linear-Nonlinear Active Disturbance Rejection Control

Authors: Runzhi Wang; Xuemin Li; Jiguang Zhang; Jian Zhang; Wenhui Li; Yufei Liu; Wenjie Fu; Xiuzhen Ma
Journal: Mathematical Problems in Engineering
 (2018)

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## Abstract

In this paper, a compound control scheme with linear active disturbance rejection control (LADRC) and nonlinear active disturbance rejection control (NLADRC) is designed to stabilize the speed control system of the marine engine. To deal with the high nonlinearity and the complex disturbance and noise conditions in marine engines, the advantages of both LADRC and NLADRC are employed. As the extended state observer (ESO) is affected severely by the inherent characteristics (cyclic speed fluctuation, cylinder-to-cylinder deviations, etc.) of the reciprocating engines, a cycle-detailed hybrid nonlinear engine model is adopted to analyze the impact of such characteristics. Hence, the controller can be evaluated based on the modified engine model to achieve more reliable performance. Considering the mentioned natural properties in reciprocating engines, the parameters of linear ESO (LESO), nonlinear ESO (NLESO), and the switching strategy between LADRC and NLADRC are designed. Finally, various comparative simulations are carried out to show the effectiveness of the proposed control scheme and the superiority of switching strategy. The simulation results demonstrate that the proposed control scheme has prominent control effects both under the speed tracking mode and the condition with different types and levels of load disturbance. This study also reveals that when ADRC related approaches are employed to the reciprocating engine, the impact of the inherent characteristics of such engine on the ESO should be considered well.


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## Body

## 1. Introduction

Compared with the aviation, rail, and automobile transport, shipping is known as the most energy efficient and environmental friendly classical mode of transport [1]. The energy generated by marine diesel engines is widely used in the domain of ship propulsion [2–5]. In such application, speed control for the marine main engine becomes a crucial task.On the one hand, the engine speed should be regulated effectively over all working points of the engine. Otherwise, the oscillation of engine speed would lead to abnormal operating conditions [4], which decreases the service life of engines and even results in premature failure of the transmission system in case of severe speed fluctuation [6]. Moreover, sustained overspeed will cause irreversible damage to the marine main engine [7]. On the other, an excellent speed controller can help the engine keep good power performance under its complex operation conditions, thus reducing the fuel consumption and emission [8, 9] and releasing partial burden of the engine control unit (ECU) from the increasingly strict diesel emission regulations.The main task of marine engine speed control comprises tracking the target speed fast and maintaining the engine speed steady in the presence of the intrinsic instabilities and disturbances coupled with the fast and dynamic changes of external environment, load, and operation conditions [7].Historically, various different strategies have been adopted on the speed regulation for marine main engines, such as traditional PID [10], sliding mode control (SMC) [3], H∞ control [11], fuzzy control [7], and model predictive control (MPC) [12]. Although these control strategies have been attempted to provide feasible ways to deal with the marine engine speed control issue, most of them have corresponding drawbacks. For instance, the parameters in classical PID need to be readjusted when the engine operation deviates further from the calibrated situation caused by the changing of external condition; the chattering phenomenon in SMC is hard to be eliminated; the control effect of H∞ and fuzzy methods is limited because the operation condition of the marine engine varies widely; and the implementation of MPC is complex and with considerable high cost.So far, speed regulation for diesel engines remains to be a challengeable mission due to the fact that diesel engines are inherently high nonlinear, and their load disturbance is harsh and unpredictable. In fact, the load disturbance not only varies with the operation condition but also is affected strongly by numerous other external aspects, such as the marine weather and the sea surface condition [3, 13]. In the moderate sea conditions, the load disturbance is mainly caused by ocean current. However, when a ship navigates in extreme seas, partial or whole, the propeller disk would emerge from the water. Hence, different levels of change in propeller torque might exist and would result in a large fluctuation in marine engine speed [14].Focusing on the mentioned sophisticated load disturbance, active disturbance rejection control (ADRC) has been applied to the marine engine speed control because of its considerable control effect in dealing with the system with uncertain disturbances. It has been proved by enormous practical applications in extensive industrial domains, such as in [15–18], that ADRC has strong robustness towards parameters variations, disturbances, and noises. In the field of marine main engine, for instance, in [19], a nonlinear active disturbance rejection controller was designed for a MAN B&W large low-speed diesel engine via a simplified transfer function engine model. In [20], a combined controller based on cerebellar model articulation controller (CMAC) and ADRC was presented to control the engine speed on a simplified empirical diesel engine model coupled with the model of propeller and hull dynamics.However, in most of the previous articles concerning marine engine speed control, the controllers were only evaluated by using simple engine models. The impact of the intrinsical characteristic of the engine speed on the control effect of these controllers has been ignored. In terms of the reciprocating engine, engine speed is naturally cyclic fluctuation due to the existence of the in-cylinder discrete torque generation [21, 22]. The periodic instant speed signal in the crank-angle (CA) domain is cyclic but aperiodic in the time domain as the engine speed varies [23]. The inherent speed fluctuation would be more serious when affected by cylinder-to-cylinder and cycle-to-cycle differences in torque production [21, 24]. Furthermore, the deviations in engine speed caused by imbalance working in cylinders are characterized as periodic disturbances in the CA domain [21] rather than the general time domain, which has been proved as a difficulty for asymptotic tracking and disturbance rejection [23]. In this study, it is found that this phenomenon has a significant impact on the performance of the ESO.In general, it is reported that the mean value engine model (MVEM) is sufficient for controller design [25, 26]. But for some control algorithms it can be summed up from previous articles that it is hard to guarantee the objectivity when designing a speed controller for the reciprocating engine, because of the existence of these ignored characteristics in engine speed. For example, as for SMC, there are papers researched the application by testing MVEM or else simplified engine model, and the results are found to be satisfactory [3, 27]. Nevertheless, when the similar methods were tested on a more complex engine model or a real engine, the results turned out to be less ideal. In [28, 29], the oscillation in speed caused by the coupling of chattering phenomenon in SMC and the inherent speed fluctuation in reciprocating engine cannot be alleviated easily, and the oscillation of control input is also apparent [27]. Likewise, this is also an inevitable impact for the controller based on ADRC as the control performance of ADRC is affected by noise and disturbance [17, 30].It should be pointed out that the mentioned issue has not drawn any attention when the ADRC method is applied on the engine speed regulation, whereas its impact can be observed in related articles. In [8], a linear ADRC (LADRC) framework was adopted to compensate the total disturbance for idle speed control in a diesel engine. Although the control performance for sudden load disturbances has been improved effectively, the steady-state speed variation is obviously larger than it in the commercial controller where PID method is employed. As far as the authors know, there is no analysis about the reason therein.As mentioned above, the noise and disturbance are extremely complex and uncertain in marine engines, and the type and extent of the disturbance are various. Moreover, the performance of ESO is also affected by the natural speed fluctuation. Single LADRC or nonlinear ADRC (NLADRC) may not guarantee control performance over the whole work conditions. In [31], a systemic analysis of the characteristics of LADRC and NLADRC is provided. Normally, NLADRC outperforms LADRC, but when the level of the total noise and disturbance turns to a certain degree, the performance of NLADRC will decrease sharply [31]. It is summarized that by combining LADRC and NLADRC, it leads to a better method called LADRC/NLADRC switching control (SADRC), which keeps the merits of both methods. Better control performance can be obtained when the total disturbance and noise are complicated, and their amplitudes change widely. This kind of control strategy has not been tested in marine engine speed control domain which suffered from the mentioned sophisticated load disturbance, noise, and the inherent speed fluctuation.Motivated by the previous research and the challenges stated above, a speed controller that employs the SADRC scheme is designed to deal with the complex noise and disturbance in the marine engine. Considering the analysis above about the inadequacy in the validation of the previous ADRC based controllers for the diesel engine speed control via MVEMs or simpler engine models, a more detailed engine model is adopted, which can exhibit the inherent speed fluctuation. Aiming at the special features of the engine speed, the parameters for the proposed controller are adjusted via analyzing the impact of the inherent speed fluctuation on the performance of ESO. Finally, in order to show the superiority of the proposed scheme, comparisons between the proposed control scheme and other controllers are carried out under speed tracking mode and various external load disturbance conditions.The rest of this paper is organized as follows. A cycle-detailed hybrid nonlinear engine model is presented in Section2, and the detailed differences between the proposed engine model and the classic MVEM are compared. In Section 3, the SADRC algorithm for engine speed control is described. And the impact of the inherent speed fluctuation on the performance of ESO is studied, based on which the parameters in the proposed controller are designed. Section 4 exhibits the control performance of the proposed controller by comparing with LADRC, NLADRC, and PID controller. In Section 5, the conclusion is summarized about the whole work in this study, and a research direction for further work is discussed.


## 2. The Cycle-Detailed Hybrid Nonlinear Engine Model

In this study, the idea in [22, 32] is employed to modify the common MVEM. By doing such, the advantages in the MVEM are kept; meanwhile, the inherent speed fluctuation caused by the discrete torque generation and cyclic deviations among cylinders can be simulated without making the engine model more complex and harder to be executed in computation. Figure 1 demonstrates the thermodynamic volumes for the marine engine. The engine model is composed of five parts which are the intake and exhaust manifold, cylinders, intercooler, and turbocharger. In this paper, only the discrete torque generation process and the final crankshaft dynamic are provided. Else contents of the engine model, such as engine cycle delays, intake and exhaust manifold, intercooler, turbocharger, can refer to the authors’ previous work [5].Figure 1
Schematic diagram of the marine diesel engine for propulsion.


### 2.1. The Discrete Torque Generation Process

The CA signalφ is calculated by(1)φ=mod⁡∫6nedt,720,where ne is engine speed and operator “mod” represents modulus.Such CA signal is used to decide the timing sequence for the in-cylinder process. For the individual cylinder, the discrete torque generation mechanism and the in-cylinder evolution process can be shown in the form of finite-state machines (FSMs) [22, 32]. Figure 2 shows that the indicated torque is produced only within the phases “C” to “E”. Assuming the indicated torque is sustained unchanged during such phase, the mean indicated torque value in the whole working cycle can be replaced by the mean indicated torque during the process from “C” to “E”.Figure 2
In-cylinder evolution process (where “I” represents intake process, “C” compression process, “E” expansion process, and “H” exhaust process).Moreover, to simulate the imbalance working ability among cylinders, a factorξi is defined, which denotes the cylinder-by-cylinder variation of the cylinder i. As a result, the gross indicated torque Migi for the cylinder i is shown as follows(2)Migi=Pdurtion30·Wfi·qHV·ηigiπ·ne720Nc·φFξi,(3)Pdurtion=1i-1360Nc<φ<i-1360Nc+φf0otherwise,(4)ηigi=fne,λi,where Pdurtion is pulse function, Wfi is the fuel mass flow rate of the cylinderi, qHV is the low calorific value of diesel, ηigi represents the indicated efficiency of the cylinderi, which can be defined as a function between the engine speed and air to fuel ratio (AFR) (defined as λi for the cylinderi), φf means the lasting angle of the expansion phase, and Nc is the number of cylinders.The mass fuel flow of the cylinderi,Wfi, can be calculated by(5)Wfi=ne·Nc·mf120·10-6,where mf is the control input (fuel injection quality per stroke per cylinder).The total gross indicated torqueMig is(6)Mig=∑1NcMigi.


### 2.2. Engine Rotational Dynamic

Combing the indicated torque from all cylinders, the engine rotational dynamic equation can be described by(7)Jednedt=30πMig-Mp+Mf+Mload+Mwave+Mnoise,where Je is the total rotary inertia, Mp is the pumping torque, Mf is the friction torque, Mload means the load torque, Mwave represents the load disturbance from wave, and Mnoise is the total bounded disturbance.For more information about the engine model and the mentioned symbols, refer to [5].


### 2.3. The Comparison between the Cycle-Detailed Hybrid Nonlinear Engine Model and the MVEM

Under the same control parameters (a PID controller), noise condition, and tested process, the individual cylinder indicted torque waveform of the hybrid engine model can be demonstrated in Figure3(a). The comparisons of the total indicated torque and the engine speed for both engine models are shown in Figures 3(b) and 3(c), respectively. The right side of Figure 3 denotes the enlarged plots for the corresponding compared variables within around one engine working cycle.Figure 3
The comparisons of both engine models. (a) Individual cylinder indicated torque in the proposed engine model. (b) Total indicated torque of both engine models. (c) The speed responses of both engine models.Figure3(a) shows the pulse indicated torque in individual cylinder. It can clearly illustrate the discrete torque generation process. Meanwhile, it can be known from Figure 3(b) that the mean effect of the total indicated torque in the proposed engine model is almost the same in MVEM. It can be observed from Figure 3(c) that the inherent speed fluctuation can be modeled in the proposed engine model. The engine speed fluctuation (instant and average speed) in the engine model is significantly larger than that in the MVEM. And the control effect in the proposed engine model is inferior to that in the MVEM. Note that the MVEM has been verified in authors’ previous research [3]. It can be noticed from Figure 3(c) that the speed responses in the two models are the same in overall, which can be believed as the verification of the proposed model in this study.


## 2.1. The Discrete Torque Generation Process

The CA signalφ is calculated by(1)φ=mod⁡∫6nedt,720,where ne is engine speed and operator “mod” represents modulus.Such CA signal is used to decide the timing sequence for the in-cylinder process. For the individual cylinder, the discrete torque generation mechanism and the in-cylinder evolution process can be shown in the form of finite-state machines (FSMs) [22, 32]. Figure 2 shows that the indicated torque is produced only within the phases “C” to “E”. Assuming the indicated torque is sustained unchanged during such phase, the mean indicated torque value in the whole working cycle can be replaced by the mean indicated torque during the process from “C” to “E”.Figure 2
In-cylinder evolution process (where “I” represents intake process, “C” compression process, “E” expansion process, and “H” exhaust process).Moreover, to simulate the imbalance working ability among cylinders, a factorξi is defined, which denotes the cylinder-by-cylinder variation of the cylinder i. As a result, the gross indicated torque Migi for the cylinder i is shown as follows(2)Migi=Pdurtion30·Wfi·qHV·ηigiπ·ne720Nc·φFξi,(3)Pdurtion=1i-1360Nc<φ<i-1360Nc+φf0otherwise,(4)ηigi=fne,λi,where Pdurtion is pulse function, Wfi is the fuel mass flow rate of the cylinderi, qHV is the low calorific value of diesel, ηigi represents the indicated efficiency of the cylinderi, which can be defined as a function between the engine speed and air to fuel ratio (AFR) (defined as λi for the cylinderi), φf means the lasting angle of the expansion phase, and Nc is the number of cylinders.The mass fuel flow of the cylinderi,Wfi, can be calculated by(5)Wfi=ne·Nc·mf120·10-6,where mf is the control input (fuel injection quality per stroke per cylinder).The total gross indicated torqueMig is(6)Mig=∑1NcMigi.


## 2.2. Engine Rotational Dynamic

Combing the indicated torque from all cylinders, the engine rotational dynamic equation can be described by(7)Jednedt=30πMig-Mp+Mf+Mload+Mwave+Mnoise,where Je is the total rotary inertia, Mp is the pumping torque, Mf is the friction torque, Mload means the load torque, Mwave represents the load disturbance from wave, and Mnoise is the total bounded disturbance.For more information about the engine model and the mentioned symbols, refer to [5].


## 2.3. The Comparison between the Cycle-Detailed Hybrid Nonlinear Engine Model and the MVEM

Under the same control parameters (a PID controller), noise condition, and tested process, the individual cylinder indicted torque waveform of the hybrid engine model can be demonstrated in Figure3(a). The comparisons of the total indicated torque and the engine speed for both engine models are shown in Figures 3(b) and 3(c), respectively. The right side of Figure 3 denotes the enlarged plots for the corresponding compared variables within around one engine working cycle.Figure 3
The comparisons of both engine models. (a) Individual cylinder indicated torque in the proposed engine model. (b) Total indicated torque of both engine models. (c) The speed responses of both engine models.Figure3(a) shows the pulse indicated torque in individual cylinder. It can clearly illustrate the discrete torque generation process. Meanwhile, it can be known from Figure 3(b) that the mean effect of the total indicated torque in the proposed engine model is almost the same in MVEM. It can be observed from Figure 3(c) that the inherent speed fluctuation can be modeled in the proposed engine model. The engine speed fluctuation (instant and average speed) in the engine model is significantly larger than that in the MVEM. And the control effect in the proposed engine model is inferior to that in the MVEM. Note that the MVEM has been verified in authors’ previous research [3]. It can be noticed from Figure 3(c) that the speed responses in the two models are the same in overall, which can be believed as the verification of the proposed model in this study.


## 3. Controller Design

### 3.1. Basic Description of the ADRC

It was proved in previous papers, such as [8, 19, 20], that second-order or even first-order ADRC is suitable to the engine speed control by some corresponding simplifications. To improve the speed control accuracy, we decide to adopt second-order ADRC, although the engine rotational dynamic model Equation (7) is first-order. On one hand, having the engine system with inevitable delay (such as turbocharge lag and cyclic combustion delay), the first-order ADRC would not guarantee control performance in some working conditions. On the other, the second-order ADRC has better adaptability and disturbance prediction ability [33], which will improve the control performance in the case of sophisticated working conditions for marine diesel engines. In order to show clearly the advantage of the using second-order ADRC, Figure 4 gives the control performance comparison between them. It is obvious that the overshoot, settling time, and the steady-state speed fluctuation in the former are larger than those in the latter.Figure 4
The control performance comparison between first-order and second-order ADRC.Taking the derivative of both sides of (7), we can get(8)n¨e=30πJeM˙ig-fne,n˙e,λi,ωtwhere n˙e is the derivate of engine speed ne, ωt means the unknown disturbance and unmodeled dynamics, fne,n˙e,λi,ωt=30M˙p+M˙f+M˙load+M˙noise+M˙wave/πJe, is an unknown and time-varying function, and M˙ig can be written as(9)M˙ig=∑1NcM˙igi=K·gne,n˙e,λi·mf˙where K=(180qHVξi/πφFJe)10-6, gne,ne˙,λi·mf˙=∑1Nc(dPdurtionne·ηigine,λi·mf/dt) is a nonlinear time-varying function and mf˙ is an intermediate variable without practical significance.Combining (8) and (9), a new second-order dynamic model for engine speed can be constructed as follows(10)n¨e=K·gne,n˙e,λi·mf˙-fne,n˙e,λi,ωt=bt·mf-fne,n˙e,λi,ωt,where bt=(30K·gne,n˙e,λi·mf˙)/πJemf.Definingx1=ne, x2=n˙e, x3=fne,n˙e,λi,ωt, u=mf, (10) can be rewritten in the form of state-space as follows(11)x˙1=x2x˙2=btu-x3x˙3=f˙ne,n˙e,λi,ωty=x1where x3 represents the nonlinear dynamics of the engine, including the total lumped disturbance.Hence, focusing on the state-space Equation (11), we study the second-order ADRC for diesel engine speed control in this study. According to [34], a general third-order ESO for the second-order system Equation (11) can be given as (12)ek=z1k-yz1k+1=z1k+hz2k-β01φ1ekz2k+1=z2k+hb0uk+z3k-β02φ2ekz3k+1=z3k-hβ03φ3ekwhere y is the system output, h is step size, u(k) is the control input at instantk, b0 is the control gain, β0i (i=1,2,3) are the gains of observer, z1k and z2k are the estimation of the system states at the instantk, z3k is the estimation of the total disturbance at the instantk,and φi(ek) (i=1,2,3) are nonlinear functions, which can be defined as(13)φiek=falek,αi,δ=ekδ1-αiek≤δekαisgn⁡ekek>δwhere αi and δ can be decided in advance; some recommended values are summed up in Han’s research [35]. When αi<1, this function can represent the characteristic of “small error, big gain; big error, small gain”. When αi=1, it becomes a linear function.To show clearly the effect of “small error, big gain; big error, small gain”, according to [31], function falek,αi,δ can be rewritten as(14)falek,αi,δ=falek,αi,δekek=λiek,αi,δekFor instance,α=0.25, and taking δ as 0.05, 0.1, and 0.15, respectively, a set of curves for λiek,αi,δ can be obtained in Figure 5.Figure 5
Demonstration of the functionλiek,αi,δ when α=0.25.Hence, state error feedback control law can be defined as(15)e1=v1k-z1ke2=v2k-z2ku0=β1fale1,αi,δ+β2fale2,αi′,δ′where v1k and v2k are the set-points and their differential at the sampling instant k, respectively, and β1 and β2 are the control gains, which can be considered as nonlinear or linear control law based on the value of αi′.In this study, to reduce both the amplification effect of noise and the burden of parameters tuning, we adopt linear control law. Hence, the state error feedback control law can be written as(16)u0=β1e1+β2e2Considering the disturbance compensation, the control input can be given as(17)uk=u0-z3kb0


### 3.2. Scheme of SADRC for Marine Engine Speed Control

As the state error feedback control law is chosen to be linear, the switching scheme for SADRC refers to the shift between the linear ESO (LESO) and the nonlinear ESO (NLESO).Motivated by [31], the switch process depends on the value of ek. We do not consider the transition time as another shift condition as in [31]. Hence, only a specific value em should be decided, thus the shift between LESO and NLESO can be completed. To be specific, in the case of |ek|<em, NLESO works; otherwise, LESO takes the place of it. The method for selecting em proposed in [31] cannot be used directly in this study. The reason will be explained after introducing the parameters tuning of ESO (see Remark 2).Given the input saturation of the actuator, the input signal for ESO diverges from the original control input, which causes the inaccurate estimation of the total disturbance in ESO. As shown in [36] and the references therein, a simple solution is to replace the control input for ESO by the saturated value of actuator.To sum up, we can get the SADRC controller for marine engine speed control as illustrated in Figure6.Figure 6
Basic control structure diagram of the SADRC method for marine engine speed control.


### 3.3. Parameters Tuning of the SADRC for Marine Engine Speed Control

The parameters tuning in ADRC are sophisticated as there are multiple parameters that need to be adjusted, especially for NLADRC. Gao proposed a tuning method based on observer bandwidth for LADRC [30]. The parameters of LESO can be chosen as(18)
β
01
=
3
ω
o
,
β
02
=
3
ω
o
2
,
β
03
=
ω
o
3
,
(19)
ω
o
=
5
~
10
ω
c
,where ωo denotes the bandwidth of observer, which has a relationship with the control bandwidth ωc.The determination ofωc can be found in [37], and it can be calculated by(20)ωc=10ts∗where ts∗ denotes the desired setting time, which can be obtained by practical demand.Initially, in this study, for the target engine, according to its response time (refering to Figure4, the settling time for the acceleration and deceleration processes is within the range from 1s to 2s), the desired setting time is set to be 2s, i.e., ts∗≈2s. The choice of ts∗ should consider the limitation of the engine, including the mechanical safety and emission; for example, to obtain small ts∗, the engine needs to run with quick speedup procedure, which would cause incomplete combustion or even damage to engine. Then we can get ωc=5 by (20). Choosing ωo=8ωc=40, the parameters in LESO can be obtained with the method of (18) Thus, we have β01=120, β02=4800, β03=64000. Note that the variations among cylinders have not been considered firstly, i.e., the factor ξi=1.On the basis of the initial parameters of LESO, the proper control parameters of linear error state feedback (LESF) are decided via trial-and-error approach in the proposed engine model. During such process, we follow some fundamentals: the delay in disturbance estimation depends onβ03, the bigger β03, the less delay, but oversize β03 leads to oscillation; meanwhile, the control performance can be improved to a certain degree by adjusting β01 and β02 coordinately. Finally, we get the proper parameters for LESO: β01=200,β02=10000,β03=60000, and parameters for LESF: β1=4.5, β2=0.15, b0=500. The control effect will be given in next section.As for NLADRC, in [31], empirical formulas are summarized for the parameters design in NLESO. They can be decided as follows(21)β01′=3ωo,β02′=3ωo25,β03′=ωo39.Remark 1.
It is not possible to get feasible parameters to keep stability by (21) in this study. To get suitable parameters for NLESO, the impact of the inherent characteristics of the engine (such as speed fluctuation) on the performance of the ESO needs to be analyzed. But firstly, some parameters can be decided as α1=1, α2=0.5, α3=0.25 by common previous experience [35].Meanwhile, to illustrate the necessity of taking into account the inherent speed fluctuation while designing ADRC related controller for engine speed control, the estimation performances of the ESO between the proposed engine model and the MVEM are compared in detail. We find it easier to get the parameters of LESO in MVEM. The values calculated directly from (18) are adequate, i.e., β01=120, β02=4800, β03=64000. Moreover, the parameters for LESO in the MVEM have a wider range than those in the proposed engine model.The tracking curves of LESO statez1and z2 for both engine models are shown in Figure 7. The tracking performance for z3 is not given for the total disturbance is unknown. Note that the noise condition is the same in both engine models. Moreover, the sampling and control time are designed to be 0.01s (h=0.01) in both engine models.Figure 7
The tracking condition of the LESO state under different engine models. (a1) The tracking curves ofz1 and y in the proposed engine model. (b1) The error between z1 and y in the proposed engine model. (c1) The tracking curves of z2 and dy/dt in the proposed engine model. (d1) The error between z2 and dy/dt in the proposed engine model. (a2), (b2), (c2), and (d2) represent the corresponding variables in the MVEM, respectively.It can be observed from Figure7 that z1 and z2 can appropriately track the system states. However, significant differences can be seen between the tracking curves of the two engine models. In the proposed engine model, the bound of the distribution of the tracking error ek is -1.5,1.5, and most of ek distributes in the range -1.0,1.0, whereas there is an order of magnitude difference for the corresponding values in the MVEM. Also, the changing frequency of these variables is apparently different in the two engine models. It is higher in the MVEM. The main reason is that the speed or torque changing is cycle based in the proposed engine model, while it is time based in the MVEM. From Figure 7 (c1), (c2), we know that the estimated differential z2 is more smooth than the time-derivative of y(dy/dt). This means that the LESO can effectively filter the disturbed time-derivative of y then lessen the noise effect on the control performance in the MVEM. On the contrary, this effect is not obvious or weak in the proposed engine model.Since the tracking performance can hardly be improved by tuning the parameters of LESO in this study, it can be concluded that the performance of ESO (including LESO and NLESO) is limited by the inherent property in speed fluctuation, and the distribution ofek mentioned above is a kind of inherent property in reciprocating engines. This is different with other references, such as in [30, 31, 38, 39], where the tracking error ek is with a very small distribution range under steady-state. This property cannot be eliminated in the reciprocating engine. Furthermore, in a real engine, when ek is affected by the coupled effect of the inherent speed fluctuation and other noise and disturbance, the actual ek would far exceed the bound -1.0,1.0 inevitably. As a result, the estimating capacity of NLESO would deteriorate sharply, due to the fact shown in Figure 5 that the equivalent gain λiek,αi,δ<1 when ek>1.Remark 2.
Under the background mentioned above, an enlightenment is got for designing the parametersβ01′,β02′,β03′,δ′ in NLESO and em in SADRC. The basic rule is to avoid excessively amplifying the inherent property in speed fluctuation when ek<1, which requires that the nonlinear gains maintain with relatively small value while ek<1. Therefore, δ′ is chosen to be 0.1 to limit the maximum in gains. Then β01′,β02′,β03′ are regulated by reducing the corresponding value in obtained β01,β02,β03 (for LESO). Eventually, the proper parameters for NLESO can be determined as β01′=150,β02′=1200,β03′=5800. The switching condition is chosen as em=1, it means that, on the one hand, the advantage in NLESO is kept when ek<1, and, on the other, LESO can avoid the performance degrading in NLESO when ek>1. The choice of em is based on the characteristics in engine rather than the recommended value in [31]. The rationality to choose em=1 as the switching condition also was mentioned in [40].


## 3.1. Basic Description of the ADRC

It was proved in previous papers, such as [8, 19, 20], that second-order or even first-order ADRC is suitable to the engine speed control by some corresponding simplifications. To improve the speed control accuracy, we decide to adopt second-order ADRC, although the engine rotational dynamic model Equation (7) is first-order. On one hand, having the engine system with inevitable delay (such as turbocharge lag and cyclic combustion delay), the first-order ADRC would not guarantee control performance in some working conditions. On the other, the second-order ADRC has better adaptability and disturbance prediction ability [33], which will improve the control performance in the case of sophisticated working conditions for marine diesel engines. In order to show clearly the advantage of the using second-order ADRC, Figure 4 gives the control performance comparison between them. It is obvious that the overshoot, settling time, and the steady-state speed fluctuation in the former are larger than those in the latter.Figure 4
The control performance comparison between first-order and second-order ADRC.Taking the derivative of both sides of (7), we can get(8)n¨e=30πJeM˙ig-fne,n˙e,λi,ωtwhere n˙e is the derivate of engine speed ne, ωt means the unknown disturbance and unmodeled dynamics, fne,n˙e,λi,ωt=30M˙p+M˙f+M˙load+M˙noise+M˙wave/πJe, is an unknown and time-varying function, and M˙ig can be written as(9)M˙ig=∑1NcM˙igi=K·gne,n˙e,λi·mf˙where K=(180qHVξi/πφFJe)10-6, gne,ne˙,λi·mf˙=∑1Nc(dPdurtionne·ηigine,λi·mf/dt) is a nonlinear time-varying function and mf˙ is an intermediate variable without practical significance.Combining (8) and (9), a new second-order dynamic model for engine speed can be constructed as follows(10)n¨e=K·gne,n˙e,λi·mf˙-fne,n˙e,λi,ωt=bt·mf-fne,n˙e,λi,ωt,where bt=(30K·gne,n˙e,λi·mf˙)/πJemf.Definingx1=ne, x2=n˙e, x3=fne,n˙e,λi,ωt, u=mf, (10) can be rewritten in the form of state-space as follows(11)x˙1=x2x˙2=btu-x3x˙3=f˙ne,n˙e,λi,ωty=x1where x3 represents the nonlinear dynamics of the engine, including the total lumped disturbance.Hence, focusing on the state-space Equation (11), we study the second-order ADRC for diesel engine speed control in this study. According to [34], a general third-order ESO for the second-order system Equation (11) can be given as (12)ek=z1k-yz1k+1=z1k+hz2k-β01φ1ekz2k+1=z2k+hb0uk+z3k-β02φ2ekz3k+1=z3k-hβ03φ3ekwhere y is the system output, h is step size, u(k) is the control input at instantk, b0 is the control gain, β0i (i=1,2,3) are the gains of observer, z1k and z2k are the estimation of the system states at the instantk, z3k is the estimation of the total disturbance at the instantk,and φi(ek) (i=1,2,3) are nonlinear functions, which can be defined as(13)φiek=falek,αi,δ=ekδ1-αiek≤δekαisgn⁡ekek>δwhere αi and δ can be decided in advance; some recommended values are summed up in Han’s research [35]. When αi<1, this function can represent the characteristic of “small error, big gain; big error, small gain”. When αi=1, it becomes a linear function.To show clearly the effect of “small error, big gain; big error, small gain”, according to [31], function falek,αi,δ can be rewritten as(14)falek,αi,δ=falek,αi,δekek=λiek,αi,δekFor instance,α=0.25, and taking δ as 0.05, 0.1, and 0.15, respectively, a set of curves for λiek,αi,δ can be obtained in Figure 5.Figure 5
Demonstration of the functionλiek,αi,δ when α=0.25.Hence, state error feedback control law can be defined as(15)e1=v1k-z1ke2=v2k-z2ku0=β1fale1,αi,δ+β2fale2,αi′,δ′where v1k and v2k are the set-points and their differential at the sampling instant k, respectively, and β1 and β2 are the control gains, which can be considered as nonlinear or linear control law based on the value of αi′.In this study, to reduce both the amplification effect of noise and the burden of parameters tuning, we adopt linear control law. Hence, the state error feedback control law can be written as(16)u0=β1e1+β2e2Considering the disturbance compensation, the control input can be given as(17)uk=u0-z3kb0


## 3.2. Scheme of SADRC for Marine Engine Speed Control

As the state error feedback control law is chosen to be linear, the switching scheme for SADRC refers to the shift between the linear ESO (LESO) and the nonlinear ESO (NLESO).Motivated by [31], the switch process depends on the value of ek. We do not consider the transition time as another shift condition as in [31]. Hence, only a specific value em should be decided, thus the shift between LESO and NLESO can be completed. To be specific, in the case of |ek|<em, NLESO works; otherwise, LESO takes the place of it. The method for selecting em proposed in [31] cannot be used directly in this study. The reason will be explained after introducing the parameters tuning of ESO (see Remark 2).Given the input saturation of the actuator, the input signal for ESO diverges from the original control input, which causes the inaccurate estimation of the total disturbance in ESO. As shown in [36] and the references therein, a simple solution is to replace the control input for ESO by the saturated value of actuator.To sum up, we can get the SADRC controller for marine engine speed control as illustrated in Figure6.Figure 6
Basic control structure diagram of the SADRC method for marine engine speed control.


## 3.3. Parameters Tuning of the SADRC for Marine Engine Speed Control

The parameters tuning in ADRC are sophisticated as there are multiple parameters that need to be adjusted, especially for NLADRC. Gao proposed a tuning method based on observer bandwidth for LADRC [30]. The parameters of LESO can be chosen as(18)
β
01
=
3
ω
o
,
β
02
=
3
ω
o
2
,
β
03
=
ω
o
3
,
(19)
ω
o
=
5
~
10
ω
c
,where ωo denotes the bandwidth of observer, which has a relationship with the control bandwidth ωc.The determination ofωc can be found in [37], and it can be calculated by(20)ωc=10ts∗where ts∗ denotes the desired setting time, which can be obtained by practical demand.Initially, in this study, for the target engine, according to its response time (refering to Figure4, the settling time for the acceleration and deceleration processes is within the range from 1s to 2s), the desired setting time is set to be 2s, i.e., ts∗≈2s. The choice of ts∗ should consider the limitation of the engine, including the mechanical safety and emission; for example, to obtain small ts∗, the engine needs to run with quick speedup procedure, which would cause incomplete combustion or even damage to engine. Then we can get ωc=5 by (20). Choosing ωo=8ωc=40, the parameters in LESO can be obtained with the method of (18) Thus, we have β01=120, β02=4800, β03=64000. Note that the variations among cylinders have not been considered firstly, i.e., the factor ξi=1.On the basis of the initial parameters of LESO, the proper control parameters of linear error state feedback (LESF) are decided via trial-and-error approach in the proposed engine model. During such process, we follow some fundamentals: the delay in disturbance estimation depends onβ03, the bigger β03, the less delay, but oversize β03 leads to oscillation; meanwhile, the control performance can be improved to a certain degree by adjusting β01 and β02 coordinately. Finally, we get the proper parameters for LESO: β01=200,β02=10000,β03=60000, and parameters for LESF: β1=4.5, β2=0.15, b0=500. The control effect will be given in next section.As for NLADRC, in [31], empirical formulas are summarized for the parameters design in NLESO. They can be decided as follows(21)β01′=3ωo,β02′=3ωo25,β03′=ωo39.Remark 1.
It is not possible to get feasible parameters to keep stability by (21) in this study. To get suitable parameters for NLESO, the impact of the inherent characteristics of the engine (such as speed fluctuation) on the performance of the ESO needs to be analyzed. But firstly, some parameters can be decided as α1=1, α2=0.5, α3=0.25 by common previous experience [35].Meanwhile, to illustrate the necessity of taking into account the inherent speed fluctuation while designing ADRC related controller for engine speed control, the estimation performances of the ESO between the proposed engine model and the MVEM are compared in detail. We find it easier to get the parameters of LESO in MVEM. The values calculated directly from (18) are adequate, i.e., β01=120, β02=4800, β03=64000. Moreover, the parameters for LESO in the MVEM have a wider range than those in the proposed engine model.The tracking curves of LESO statez1and z2 for both engine models are shown in Figure 7. The tracking performance for z3 is not given for the total disturbance is unknown. Note that the noise condition is the same in both engine models. Moreover, the sampling and control time are designed to be 0.01s (h=0.01) in both engine models.Figure 7
The tracking condition of the LESO state under different engine models. (a1) The tracking curves ofz1 and y in the proposed engine model. (b1) The error between z1 and y in the proposed engine model. (c1) The tracking curves of z2 and dy/dt in the proposed engine model. (d1) The error between z2 and dy/dt in the proposed engine model. (a2), (b2), (c2), and (d2) represent the corresponding variables in the MVEM, respectively.It can be observed from Figure7 that z1 and z2 can appropriately track the system states. However, significant differences can be seen between the tracking curves of the two engine models. In the proposed engine model, the bound of the distribution of the tracking error ek is -1.5,1.5, and most of ek distributes in the range -1.0,1.0, whereas there is an order of magnitude difference for the corresponding values in the MVEM. Also, the changing frequency of these variables is apparently different in the two engine models. It is higher in the MVEM. The main reason is that the speed or torque changing is cycle based in the proposed engine model, while it is time based in the MVEM. From Figure 7 (c1), (c2), we know that the estimated differential z2 is more smooth than the time-derivative of y(dy/dt). This means that the LESO can effectively filter the disturbed time-derivative of y then lessen the noise effect on the control performance in the MVEM. On the contrary, this effect is not obvious or weak in the proposed engine model.Since the tracking performance can hardly be improved by tuning the parameters of LESO in this study, it can be concluded that the performance of ESO (including LESO and NLESO) is limited by the inherent property in speed fluctuation, and the distribution ofek mentioned above is a kind of inherent property in reciprocating engines. This is different with other references, such as in [30, 31, 38, 39], where the tracking error ek is with a very small distribution range under steady-state. This property cannot be eliminated in the reciprocating engine. Furthermore, in a real engine, when ek is affected by the coupled effect of the inherent speed fluctuation and other noise and disturbance, the actual ek would far exceed the bound -1.0,1.0 inevitably. As a result, the estimating capacity of NLESO would deteriorate sharply, due to the fact shown in Figure 5 that the equivalent gain λiek,αi,δ<1 when ek>1.Remark 2.
Under the background mentioned above, an enlightenment is got for designing the parametersβ01′,β02′,β03′,δ′ in NLESO and em in SADRC. The basic rule is to avoid excessively amplifying the inherent property in speed fluctuation when ek<1, which requires that the nonlinear gains maintain with relatively small value while ek<1. Therefore, δ′ is chosen to be 0.1 to limit the maximum in gains. Then β01′,β02′,β03′ are regulated by reducing the corresponding value in obtained β01,β02,β03 (for LESO). Eventually, the proper parameters for NLESO can be determined as β01′=150,β02′=1200,β03′=5800. The switching condition is chosen as em=1, it means that, on the one hand, the advantage in NLESO is kept when ek<1, and, on the other, LESO can avoid the performance degrading in NLESO when ek>1. The choice of em is based on the characteristics in engine rather than the recommended value in [31]. The rationality to choose em=1 as the switching condition also was mentioned in [40].


## 4. Simulation and Analysis

### 4.1. Engine Speed Tracking Performance

As it is exhibited in Figure8, to simulate the real acceleration and deceleration processes in a marine diesel engine for propulsion, ramp references are adopted. And four controllers (LADRC, NLADRC, SADRC, and PID) are compared to prove the superiority of the proposed method. The control parameters of the classical PID are well tuned by trail-and-error scheme with the consideration of wind-up scheme. Note that the load torque is normalized during these processes, and noise load is set as banded white noise with a basic constant 200 N·m. Besides, the cylinder-by-cylinder variation degree is designed as [ξ1,ξ2,ξ3,ξ4,ξ5,ξ6]T=[1.0,0.95,1.0,0.95,0.95,1.0]T. Although the control parameters in the comparative controllers are obtained under the condition that there is no variation between cylinders, it is reasonable and necessary to assess the control effect under the imbalance cylinder condition. Because the imbalance cylinder working ability would occur and be different with its original condition when the controller was designed due to varying dynamics and ageing of components of the fuel injection system [41].Figure 8
The comparisons of speed tracking performance.From Figure8: Zoom1 and Zoom3, it can be observed that in both the acceleration and deceleration processes, LADRC, NLADRC, and SADRC are better than PID in terms of the control performance in overshoot and settling time. Moreover, LADRC and SADRC present slight advantage in overshoot when compared with NLADRC. As shown in Figure 8: Zoom2, the steady-state speed fluctuation in NLADRC and SARDC is obviously smaller than that in LADRC, followed by PID. When synthetically considering the control effects in overshoot and steady-state speed fluctuation, it is apparent that SADRC is the best one during the speed tracking process.


### 4.2. Anti-Interference Ability under Mutation Load Disturbances

As mentioned in the section above, when the ship voyages in the sea, the load torque is affected by wave. The load conditions become complex for marine engine. Especially, when sea waves make the partial or total propeller out of water surface then drop into water again, the load of engine would change violently. Under the same simulation condition as mentioned above, we design three different mutation loads to validate the proposed control scheme.As displayed in Figure9(a), when the mutation load is large (100% full load), the speed variations and settling time in LADRC and SADRC are obviously smaller than those in NLADRC and PID. Unexpectedly, the settling time in NLADRC is the longest; it is even far inferior to that in the classical PID. When the mutation load becomes to medium degree (60% full load), as shown in Figure 9(b), the speed variations in LADRC, NLADRC, and SADRC are similar. Although the settling time in NLADRC catches up with that in PID, it is still not as good as that in the LADRC and SADRC. When the mutation load is small (20% full load), seen in Figure 9(c), both the speed variations and settling time in LADRC, NLADRC, and SADRC are almost the same. To understand well the impact of the extents of the load changing on the control performance of the comparative controllers, the estimate error of ESOs is compared in Figure 10. When the extents of load changing are larger (100% and 60% full load), the tracking error ek in ESOs is far beyond the bound [-1,1] (see Figure 10, (a1, a2, a3, b1, b2, b3)), which results in performance deterioration for NLESO. This is the reason why the NLADRC shows bad control performance in such conditions. On the contrary, when the load change is smaller (20% full load), the tracking error ek in ESOs is similar and within the bound [-1,1] (see Figure 10, (c1, c2, c3)); as a result, the speed deviation and settling time during such processes for the ADRC related controllers are almost the same.Figure 9
The speed responses under different levels of mutation load. (a) 100% full load. (b) 60% full load. (c) 20% full load.Figure 10
The tracking errorek in ESOs under different levels of mutation load. (a1, a2, a3) 100% full load. (b1, b2, b3) 60% full load. (c1, c2, c3) 20% full load.Besides, the control effects of the steady states after unloading different loads (during the time from 1.5s to 2.5s) are enlarged in the corresponding subplot in Figure9. It can be observed that only SADRC can consistently maintain the smallest speed fluctuation. To distinguish more clearly the control effect of the controllers under steady-state, the criteria in integral absolute error (IAE) of the system output and the total variation (TV) of the control signal are calculated. TheTV index represents the manipulated input usage, which can be computed by (22), and detailed information can be found in [42–44].(22)TV=∑i=1∞ui+1-ui,where [u1,u2,…,ui,…] represents the discretized control input u.Hence, a more detailed comparison in the criteria ofIAE andTV for the three mentioned steady states is manifested in Table 1. It can be seen that the value ofIAE in SADRC is significantly smaller than that in another three controllers, which means SADRC has better adaptability under steady-state after different levels of sudden load changing. As for theTV values, PID has less values in all the compared cases, which means that the control signal in PID is more smooth (as shown in Figure 11). From Figure 11, it can be observed that the control input in LADRC and SADRC adapts more quickly. On the one hand, it is similar to the explanation in [8] that faster antidisturbance ability is achieved by regulating the control input (fuel injection quality) quickly. On the other hand, it is also a drawback if high frequency oscillation with large amplitude occurs in control input. The existence of the inherent speed fluctuation and the imbalance working ability among cylinders are the main reason of such oscillation. From Table 1, we also know that, compared with LADRC, after unloading a larger load (100% and 60% full load), theTV values in SADRC have a reduction of about 10% to 6%. When after unloading a smaller load (20% full load), such reduction is unapparent. The reason is that the estimate error ek starts to exceed the bound [-1,1] (as shown in Figure 10 (c1, c2, c3), during the time from 1.5s to 2.5s), resulting in the performance deterioration in NLESO. This is also the reason why theIAE value in NLADRC gets larger than that in LADRC and SADRC after unloading 20% full load.Table 1
The criteria ofIAE and TV for different controllers after different mutation load processes.
After unloading 100 % full load After unloading 60 % full load After unloading 20 % full load LADRC NLADRC SADRC PID LADRC NLADRC SADRC PID LADRC NLADRC SADRC PID IAE 0.193 0.148 0.136 0.241 0.241 0. 330 0.222 0.392 0.350 0.546 0.314 0.808 TV 65.3 59.5 58.8 12.1 96.7 88.2 90.9 21.7 141.5 140.6 138.2 31.1Figure 11
The comparison of control signal for the four controllers under mutation load (change 100% full load) condition.In overall, SADRC maintains the same speed deviation as LADRC under load changing process, but smaller speed fluctuation and less oscillation in control signal under the steady-state after unloading process. Compared with NLADRC, SADRC has less settling time when the level of sudden load change is large and less speed fluctuation under steady-state after unloading. It can be concluded that SADRC has evident superiority to deal with the different degrees of sudden load change.


### 4.3. Anti-Interference Ability under Wave Load Disturbances

The wave load disturbance is another typical inevitable perturbation for marine engine. Considering the value of normalized load torque at 1800 rpm, three sine waves with different amplitudes are designed to represent different intensities of the wave load; the amplitudes are 100 N·m, 200 N·m, and 300 N·m, respectively. And the frequency is set to be 0.2 Hz.Figure12 gives the speed responses of the four controllers under the three different wave load disturbances. Intuitively, the speed fluctuation in PID is the largest among them. To show clearly the control performance in these controllers, the comparison in the indexes ofIAE andTV are given in Table 2. When wave load is smaller (amplitude is 100 N·m or 200 N·m), the values ofIAE for SADRC are the smallest among the four controllers. Inversely, theIAE values of the LADRC show significant disadvantage when wave load is larger than 100 N·m. Note that when wave amplitude is 300 N·m, the wave load disturbance reaches to a limiting case compared with the normalized load torque at 1800 rpm; hence, it can be concluded that the SADRC has the best performance in the index ofIAE under different levels of possible wave disturbance. As for the index inTV, compared with the ADRC related controllers, the PID controller gains the smallest values. One reason is as mentioned above that the ADRC related controllers can react faster to the load disturbance, leading to the oscillation in control signal. By combing the NLADRC and LADRC, theTVvalues in SADRC have been reduced by 13%, 8%, and 10% under the three cases, respectively. From Figure 13, we know that, under different levels of wave load, the tracking error ek in ESOs exceeds the bound [-1,1], especially when wave load is larger. With the switching scheme, the SADRC can combine the use of NLADRC and LADRC to obtain better control performance when the tracking error ek in ESOs changes with the disturbance load. It can be regarded as an adaptation ability for SADRC.Table 2
The indexes ofIAE and TV for different controllers during different wave load conditions.
Wave amplitude: 100 N ⋅ m Wave amplitude: 200 N ⋅ m Wave amplitude: 300 N ⋅ m LADRC NLADRC SADRC PID LADRC NLADRC SADRC PID LADRC NLADRC SADRC PID IAE 6.2 6.4 5.9 11.3 9.3 8.0 7.9 20.1 12.8 10.3 10.4 28.1 TV 1546.2 1425.9 1349.1 379.5 1540.5 1431.2 1416.4 391.8 1612.5 1464.9 1444.4 469.9Figure 12
The speed responses under different levels of wave load. (a) Wave amplitude is 100 N⋅m. (b) Wave amplitude is 200 N⋅m. (c) Wave amplitude is 300 N⋅m.Figure 13
The tracking errorek in ESOs under different levels of wave load. (a1, a2, a3) Wave amplitude: 100 N⋅m. (b1, b2, b3) Wave amplitude: 200 N⋅m. (c1, c2, c3) Wave amplitude: 300 N⋅m.


## 4.1. Engine Speed Tracking Performance

As it is exhibited in Figure8, to simulate the real acceleration and deceleration processes in a marine diesel engine for propulsion, ramp references are adopted. And four controllers (LADRC, NLADRC, SADRC, and PID) are compared to prove the superiority of the proposed method. The control parameters of the classical PID are well tuned by trail-and-error scheme with the consideration of wind-up scheme. Note that the load torque is normalized during these processes, and noise load is set as banded white noise with a basic constant 200 N·m. Besides, the cylinder-by-cylinder variation degree is designed as [ξ1,ξ2,ξ3,ξ4,ξ5,ξ6]T=[1.0,0.95,1.0,0.95,0.95,1.0]T. Although the control parameters in the comparative controllers are obtained under the condition that there is no variation between cylinders, it is reasonable and necessary to assess the control effect under the imbalance cylinder condition. Because the imbalance cylinder working ability would occur and be different with its original condition when the controller was designed due to varying dynamics and ageing of components of the fuel injection system [41].Figure 8
The comparisons of speed tracking performance.From Figure8: Zoom1 and Zoom3, it can be observed that in both the acceleration and deceleration processes, LADRC, NLADRC, and SADRC are better than PID in terms of the control performance in overshoot and settling time. Moreover, LADRC and SADRC present slight advantage in overshoot when compared with NLADRC. As shown in Figure 8: Zoom2, the steady-state speed fluctuation in NLADRC and SARDC is obviously smaller than that in LADRC, followed by PID. When synthetically considering the control effects in overshoot and steady-state speed fluctuation, it is apparent that SADRC is the best one during the speed tracking process.


## 4.2. Anti-Interference Ability under Mutation Load Disturbances

As mentioned in the section above, when the ship voyages in the sea, the load torque is affected by wave. The load conditions become complex for marine engine. Especially, when sea waves make the partial or total propeller out of water surface then drop into water again, the load of engine would change violently. Under the same simulation condition as mentioned above, we design three different mutation loads to validate the proposed control scheme.As displayed in Figure9(a), when the mutation load is large (100% full load), the speed variations and settling time in LADRC and SADRC are obviously smaller than those in NLADRC and PID. Unexpectedly, the settling time in NLADRC is the longest; it is even far inferior to that in the classical PID. When the mutation load becomes to medium degree (60% full load), as shown in Figure 9(b), the speed variations in LADRC, NLADRC, and SADRC are similar. Although the settling time in NLADRC catches up with that in PID, it is still not as good as that in the LADRC and SADRC. When the mutation load is small (20% full load), seen in Figure 9(c), both the speed variations and settling time in LADRC, NLADRC, and SADRC are almost the same. To understand well the impact of the extents of the load changing on the control performance of the comparative controllers, the estimate error of ESOs is compared in Figure 10. When the extents of load changing are larger (100% and 60% full load), the tracking error ek in ESOs is far beyond the bound [-1,1] (see Figure 10, (a1, a2, a3, b1, b2, b3)), which results in performance deterioration for NLESO. This is the reason why the NLADRC shows bad control performance in such conditions. On the contrary, when the load change is smaller (20% full load), the tracking error ek in ESOs is similar and within the bound [-1,1] (see Figure 10, (c1, c2, c3)); as a result, the speed deviation and settling time during such processes for the ADRC related controllers are almost the same.Figure 9
The speed responses under different levels of mutation load. (a) 100% full load. (b) 60% full load. (c) 20% full load.Figure 10
The tracking errorek in ESOs under different levels of mutation load. (a1, a2, a3) 100% full load. (b1, b2, b3) 60% full load. (c1, c2, c3) 20% full load.Besides, the control effects of the steady states after unloading different loads (during the time from 1.5s to 2.5s) are enlarged in the corresponding subplot in Figure9. It can be observed that only SADRC can consistently maintain the smallest speed fluctuation. To distinguish more clearly the control effect of the controllers under steady-state, the criteria in integral absolute error (IAE) of the system output and the total variation (TV) of the control signal are calculated. TheTV index represents the manipulated input usage, which can be computed by (22), and detailed information can be found in [42–44].(22)TV=∑i=1∞ui+1-ui,where [u1,u2,…,ui,…] represents the discretized control input u.Hence, a more detailed comparison in the criteria ofIAE andTV for the three mentioned steady states is manifested in Table 1. It can be seen that the value ofIAE in SADRC is significantly smaller than that in another three controllers, which means SADRC has better adaptability under steady-state after different levels of sudden load changing. As for theTV values, PID has less values in all the compared cases, which means that the control signal in PID is more smooth (as shown in Figure 11). From Figure 11, it can be observed that the control input in LADRC and SADRC adapts more quickly. On the one hand, it is similar to the explanation in [8] that faster antidisturbance ability is achieved by regulating the control input (fuel injection quality) quickly. On the other hand, it is also a drawback if high frequency oscillation with large amplitude occurs in control input. The existence of the inherent speed fluctuation and the imbalance working ability among cylinders are the main reason of such oscillation. From Table 1, we also know that, compared with LADRC, after unloading a larger load (100% and 60% full load), theTV values in SADRC have a reduction of about 10% to 6%. When after unloading a smaller load (20% full load), such reduction is unapparent. The reason is that the estimate error ek starts to exceed the bound [-1,1] (as shown in Figure 10 (c1, c2, c3), during the time from 1.5s to 2.5s), resulting in the performance deterioration in NLESO. This is also the reason why theIAE value in NLADRC gets larger than that in LADRC and SADRC after unloading 20% full load.Table 1
The criteria ofIAE and TV for different controllers after different mutation load processes.
After unloading 100 % full load After unloading 60 % full load After unloading 20 % full load LADRC NLADRC SADRC PID LADRC NLADRC SADRC PID LADRC NLADRC SADRC PID IAE 0.193 0.148 0.136 0.241 0.241 0. 330 0.222 0.392 0.350 0.546 0.314 0.808 TV 65.3 59.5 58.8 12.1 96.7 88.2 90.9 21.7 141.5 140.6 138.2 31.1Figure 11
The comparison of control signal for the four controllers under mutation load (change 100% full load) condition.In overall, SADRC maintains the same speed deviation as LADRC under load changing process, but smaller speed fluctuation and less oscillation in control signal under the steady-state after unloading process. Compared with NLADRC, SADRC has less settling time when the level of sudden load change is large and less speed fluctuation under steady-state after unloading. It can be concluded that SADRC has evident superiority to deal with the different degrees of sudden load change.


## 4.3. Anti-Interference Ability under Wave Load Disturbances

The wave load disturbance is another typical inevitable perturbation for marine engine. Considering the value of normalized load torque at 1800 rpm, three sine waves with different amplitudes are designed to represent different intensities of the wave load; the amplitudes are 100 N·m, 200 N·m, and 300 N·m, respectively. And the frequency is set to be 0.2 Hz.Figure12 gives the speed responses of the four controllers under the three different wave load disturbances. Intuitively, the speed fluctuation in PID is the largest among them. To show clearly the control performance in these controllers, the comparison in the indexes ofIAE andTV are given in Table 2. When wave load is smaller (amplitude is 100 N·m or 200 N·m), the values ofIAE for SADRC are the smallest among the four controllers. Inversely, theIAE values of the LADRC show significant disadvantage when wave load is larger than 100 N·m. Note that when wave amplitude is 300 N·m, the wave load disturbance reaches to a limiting case compared with the normalized load torque at 1800 rpm; hence, it can be concluded that the SADRC has the best performance in the index ofIAE under different levels of possible wave disturbance. As for the index inTV, compared with the ADRC related controllers, the PID controller gains the smallest values. One reason is as mentioned above that the ADRC related controllers can react faster to the load disturbance, leading to the oscillation in control signal. By combing the NLADRC and LADRC, theTVvalues in SADRC have been reduced by 13%, 8%, and 10% under the three cases, respectively. From Figure 13, we know that, under different levels of wave load, the tracking error ek in ESOs exceeds the bound [-1,1], especially when wave load is larger. With the switching scheme, the SADRC can combine the use of NLADRC and LADRC to obtain better control performance when the tracking error ek in ESOs changes with the disturbance load. It can be regarded as an adaptation ability for SADRC.Table 2
The indexes ofIAE and TV for different controllers during different wave load conditions.
Wave amplitude: 100 N ⋅ m Wave amplitude: 200 N ⋅ m Wave amplitude: 300 N ⋅ m LADRC NLADRC SADRC PID LADRC NLADRC SADRC PID LADRC NLADRC SADRC PID IAE 6.2 6.4 5.9 11.3 9.3 8.0 7.9 20.1 12.8 10.3 10.4 28.1 TV 1546.2 1425.9 1349.1 379.5 1540.5 1431.2 1416.4 391.8 1612.5 1464.9 1444.4 469.9Figure 12
The speed responses under different levels of wave load. (a) Wave amplitude is 100 N⋅m. (b) Wave amplitude is 200 N⋅m. (c) Wave amplitude is 300 N⋅m.Figure 13
The tracking errorek in ESOs under different levels of wave load. (a1, a2, a3) Wave amplitude: 100 N⋅m. (b1, b2, b3) Wave amplitude: 200 N⋅m. (c1, c2, c3) Wave amplitude: 300 N⋅m.


## 5. Conclusions and Future Work

Among the previous ADRC related articles on the marine engine speed control, the impact of the intrinsical characteristics of the reciprocating engine on the control effect of ADRC is ignored. One important intention of this paper is to design an ADRC based controller for marine diesel engine with the consideration of the mentioned characteristics. To this end, a cycle-detailed hybrid nonlinear engine model which can simulate the inherent speed fluctuation is employed to evaluate the ADRC based controller.Single LADRC or NLADRC is not enough to keep good control effect due to the strong nonlinearity, complex disturbance, and the extra inherent speed fluctuation in marine engines. The compound of LADRC and NLADRC is introduced to keep the merits of both methods. On the basis of the proposed engine model, the impact of the inherent speed fluctuation on the performance of the ESO is analyzed. Then the control parameters are adjusted with the consideration of such features for the proposed controller by modifying the previous approaches in the references. The proposed scheme is compared with LADRC, NLADRC, and well-tuned conventional PID via numerical simulations. The results indicate that the SADRC controller gains the advantages of both LADRC and NLADRC. It provides better control effect in speed tracking and also has preponderance in keeping better control performance under different levels of both mutation disturbance and wave disturbance. However, we also find that the control input oscillation in ADRC related controllers is stronger than that in PID under the proposed engine model.As it is found that, for ADRC related approaches, the existence of inherent speed fluctuation and cylinder variations in reciprocating engines would affect its control performance, making the parameters tuning difficult and causing the oscillation in control input, our future work will be focused on studying the method to improve the control performance of ADRC for engine speed control and alleviate such oscillation. One possible way would be making use of filter method [45, 46]. The proposed method needs to be further verified on the real engine bench as well.



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*Source: 7641862-2018-12-13.xml*