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

Authors: Lijun Wang; Sisi Wang
Journal: Complexity
 (2019)

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## Abstract

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


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## Body

## 1. Introduction

The speed regulation of marine main engine (MME) is related to its performance, service life, and ship safety. At present, the advanced MME speed governor is mostly based on PID digital controller [1–3]. There are typical nonlinearity, time-varying, uncertainty, and manual active intervention in the control process of ship main engine speed. Therefore, the traditional PID controller is difficult to obtain the optimal control performance [4, 5]. An improved PID tuning method was proposed for marine diesel engine governors to overcome load fluctuation due to weather and sea conditions [1]. In order to improve the robustness of ship engine speed control, active disturbance rejection controller was presented in [2, 6], and robust control method was recommended in [7, 8]. To optimize the control parameters, several intelligent algorithms have been used to achieve better control performance, such as fuzzy logic comprehension [3, 4], GA optimization [5], and Neural Network adaption [9].Obviously, the current research mainly focuses on PID, robust, and intelligent algorithms; however, PID has poor self-adaptability, robust control is difficult to achieve in engineering, and intelligent algorithms often only have local optimal solutions. Inspired by previous studies, an optimal robust control system for marine engine speed regulation will be discussed in this paper. The main contributions of this work can be summarized as follows:(i)
The uniform asymptotic stability proof of a concise robust control design for the marine engine speed is given, and the control parameters are of obvious physical significance and can be determined easily(ii)
Variable universe fuzzy inference (VUFI) is recommended to solve the uncertainty caused by model perturbation and random disturbance for optimal control solutionThe layout of the article is as follows: Section2 presents a marine engine speed regulation model. Section 3 designs a concise robust controller and gives its stability proof. Section 4 provides online optimization of control parameters based on VUFI. Section 5 details the simulation results and discussion. Section 6 gives the conclusion.


## 2. Marine Engine Speed Regulation Model

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

This study is to design speed regulation controller for large low speed marine engines, such as MAN B&M S60M. The second order dynamic model can be described as follows [8, 10, 11].(1)kT1n˙ft+knft=st-τwhere T1 is the time constant, k is the amplification coefficient, τ is the dead time, and nf is the rate of revolution. And a nonlinear main engine model (NMEM) of transfer function form can be expressed as(2)GNs=nfsps=e-τskT1s+kwhere p is the fuel index position. The dead time is caused by the injection delay of the fuel system, which is estimated to be within the following range [1]. (3)15Nf<τ<15Nf+60Nf·Zwhere Nf is the rated speed and Z is the number of engine cylinders. For large low speed marine engine, the delay term can be replaced by the following first order inertial expression:(4)e-τs≈1τs+1Consequently, the propulsion plant dynamics are described with the following equation on the s-plane. Note that all poles in GNs are real and stable.(5)GNs=1τs+1kT1s+k=1kτT1s+1/τs+1/T1


### 2.2. Actuating Mechanism

An electronic hydraulic actuator (EHA) controlled by a high-speed switch valve is used. The control signal is pulse width modulation, and the time delay of the link is not considered. The transfer function of the actuator can be described as follows:(6)GAs=1T22s2+2ξT2s+1where T2 is the time constant and ξ is the damping coefficient. In the ship’s main engine operating system, the actuator has a displacement sensor, so it can be adjusted by feedback to make the output more stable.


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

This study is to design speed regulation controller for large low speed marine engines, such as MAN B&M S60M. The second order dynamic model can be described as follows [8, 10, 11].(1)kT1n˙ft+knft=st-τwhere T1 is the time constant, k is the amplification coefficient, τ is the dead time, and nf is the rate of revolution. And a nonlinear main engine model (NMEM) of transfer function form can be expressed as(2)GNs=nfsps=e-τskT1s+kwhere p is the fuel index position. The dead time is caused by the injection delay of the fuel system, which is estimated to be within the following range [1]. (3)15Nf<τ<15Nf+60Nf·Zwhere Nf is the rated speed and Z is the number of engine cylinders. For large low speed marine engine, the delay term can be replaced by the following first order inertial expression:(4)e-τs≈1τs+1Consequently, the propulsion plant dynamics are described with the following equation on the s-plane. Note that all poles in GNs are real and stable.(5)GNs=1τs+1kT1s+k=1kτT1s+1/τs+1/T1


## 2.2. Actuating Mechanism

An electronic hydraulic actuator (EHA) controlled by a high-speed switch valve is used. The control signal is pulse width modulation, and the time delay of the link is not considered. The transfer function of the actuator can be described as follows:(6)GAs=1T22s2+2ξT2s+1where T2 is the time constant and ξ is the damping coefficient. In the ship’s main engine operating system, the actuator has a displacement sensor, so it can be adjusted by feedback to make the output more stable.


## 3. Control Design

### 3.1. Concise Robust Control Design

In this section, a concise robust control law based on Backstepping for marine engine speed regulation is presented, and the control parameters have obvious physical meaning. Furthermore, parameter determination method is given.Theorem 1.
Considering the marine engine speed regulation model (1), the proposed controller (7) based on the Backstepping can stabilize the speed motion and guaranteeing the uniformly asymptotic stability of the closed-loop speed regulation system.(7)u=1b-fx2+y¨set-1b1+k1k2e+k1+k2ecwhere e, ec are the speed error and its derivative, and k1, k2 are design parameters.Proof.
Setx1=nf, x2=n˙f, u=p, and y=x1, so we can obtain the state space model of (5) as follows:(8)x˙1=x2x˙2=fx2+buy=x1where fx2=-τ+T1/τT1x2-1/τT1∫x2dt, b=1/kτT1, and one defines the set revolution yset and the error variable η1:(9)η1=y-yset=x1-ysetThe first Lyapunov function is selected as(10)v1=12η12(11)v˙1=η1η˙1=η1x˙1-yr=η1x2-yrDefine virtual control variable x2d=x˙1, and assume(12)x2d=-k1η1+yset,(13)v˙1=-k1η12<0where design parameters k1>0, the function v1 is negative definite, and one can get limt→∞⁡x1-yset=0; that is to say, the control of x1 is realized.
Define another error variableη2 and the second Lyapunov function v2:(14)η2=x2-x2d(15)v2=12η12+12η22Substitute (8) and (12) in (14), so one can get(16)η2=x˙1-y˙set+k1η1=η˙1+k1η1,(17)v˙2=-k1η12+η2η1+x˙2-x˙2d=-k1η12+η21-k12η1+k1η2+x˙2-y¨setTo guarantee the negative definiteness of v2, one can define a virtual control variable x3d=x˙2, and assume(18)x3d=-1-k12η1-k1+k2η2+y¨set,(19)v˙2=-k1η12-k2η22<0,∀k1≠0,k2≠0where design parameters k2>0, the function v2 is negative definite, and one can get limt→∞⁡x2-x2d=0; that is to say, the control of x2d is realized by the control law (18), and all the variables in main engine speed control loop are uniformly asymptotic stable with equilibrium point x1,x2=yset,y˙set.
This ends the proof of Theorem1.Substituting (16) into (18), the control law is transformed as follows:(20)x3d=-1+k1k2η1-k1+k2η˙1+y¨setSubstituting (8) into (20), the actual control law was deduced, if (21)u=1b-fx2+y¨set-1b1+k1k2η1+k1+k2η˙1=1b-fx2+y¨set-1b1+k1k2x1-yset+k1+k2x2-y˙set=1b-fx2+y¨set-1b1+k1k2e+k1+k2ec=-1bfx2-y¨set+uPDRemark 2.
The essence of the control law (21) is to compensate the system’s linearity or nonlinearity and to stabilize the control loop by a PD type controller uPD. It is obvious that the Backstepping control design method is concise and effective. However, the design parameters k1, k2 are of little engineering significance and can only be determined by trial and error, which is not conducive to the robustness and optimization of the control performance.


### 3.2. Control Parameters Determination

In accordance with closed-loop gain shaping algorithm (CGSA) [12–17], a concise robust PID controller is presented as follows.(22)1Ts+1=GC1+GC(23)C=1GTs(24)Gs=σδ2s2+δ1s+δ0where G is the system transfer function model, T stands for the system period, for the main engine speed control system, and C is the design controller. Therefore, one can get a PID controller by substituting (24) into (23).(25)C=δ1σT+δ2sσT+δ0σTs=Kp+Kds+Ki1s (5) can be transformed into (26). Obviously, (26) has the standard form as (24), which is of strictly rational proper fraction function.(26)GNs=1kτT1s2+τ+T1ks+kOn the basis of Theorem 1, the main engine speed steady-state error satisfies limt→∞⁡nf-yset=0. As a result, the integral term Ki/s is negligible. Define TM as the system period, and substitute the parameters of (26) into (25), so one can get(27)UPD=Kp+Kds=τ+T1TM+kτT1TMsRemark 3.
In accordance with ship main engine knowledge and the CGSA, the design parameters are of clear physical significance, which can be exactly determined. However, the control performance is not guaranteed to be adaptive to load changes and sea conditions.


## 3.1. Concise Robust Control Design

In this section, a concise robust control law based on Backstepping for marine engine speed regulation is presented, and the control parameters have obvious physical meaning. Furthermore, parameter determination method is given.Theorem 1.
Considering the marine engine speed regulation model (1), the proposed controller (7) based on the Backstepping can stabilize the speed motion and guaranteeing the uniformly asymptotic stability of the closed-loop speed regulation system.(7)u=1b-fx2+y¨set-1b1+k1k2e+k1+k2ecwhere e, ec are the speed error and its derivative, and k1, k2 are design parameters.Proof.
Setx1=nf, x2=n˙f, u=p, and y=x1, so we can obtain the state space model of (5) as follows:(8)x˙1=x2x˙2=fx2+buy=x1where fx2=-τ+T1/τT1x2-1/τT1∫x2dt, b=1/kτT1, and one defines the set revolution yset and the error variable η1:(9)η1=y-yset=x1-ysetThe first Lyapunov function is selected as(10)v1=12η12(11)v˙1=η1η˙1=η1x˙1-yr=η1x2-yrDefine virtual control variable x2d=x˙1, and assume(12)x2d=-k1η1+yset,(13)v˙1=-k1η12<0where design parameters k1>0, the function v1 is negative definite, and one can get limt→∞⁡x1-yset=0; that is to say, the control of x1 is realized.
Define another error variableη2 and the second Lyapunov function v2:(14)η2=x2-x2d(15)v2=12η12+12η22Substitute (8) and (12) in (14), so one can get(16)η2=x˙1-y˙set+k1η1=η˙1+k1η1,(17)v˙2=-k1η12+η2η1+x˙2-x˙2d=-k1η12+η21-k12η1+k1η2+x˙2-y¨setTo guarantee the negative definiteness of v2, one can define a virtual control variable x3d=x˙2, and assume(18)x3d=-1-k12η1-k1+k2η2+y¨set,(19)v˙2=-k1η12-k2η22<0,∀k1≠0,k2≠0where design parameters k2>0, the function v2 is negative definite, and one can get limt→∞⁡x2-x2d=0; that is to say, the control of x2d is realized by the control law (18), and all the variables in main engine speed control loop are uniformly asymptotic stable with equilibrium point x1,x2=yset,y˙set.
This ends the proof of Theorem1.Substituting (16) into (18), the control law is transformed as follows:(20)x3d=-1+k1k2η1-k1+k2η˙1+y¨setSubstituting (8) into (20), the actual control law was deduced, if (21)u=1b-fx2+y¨set-1b1+k1k2η1+k1+k2η˙1=1b-fx2+y¨set-1b1+k1k2x1-yset+k1+k2x2-y˙set=1b-fx2+y¨set-1b1+k1k2e+k1+k2ec=-1bfx2-y¨set+uPDRemark 2.
The essence of the control law (21) is to compensate the system’s linearity or nonlinearity and to stabilize the control loop by a PD type controller uPD. It is obvious that the Backstepping control design method is concise and effective. However, the design parameters k1, k2 are of little engineering significance and can only be determined by trial and error, which is not conducive to the robustness and optimization of the control performance.


## 3.2. Control Parameters Determination

In accordance with closed-loop gain shaping algorithm (CGSA) [12–17], a concise robust PID controller is presented as follows.(22)1Ts+1=GC1+GC(23)C=1GTs(24)Gs=σδ2s2+δ1s+δ0where G is the system transfer function model, T stands for the system period, for the main engine speed control system, and C is the design controller. Therefore, one can get a PID controller by substituting (24) into (23).(25)C=δ1σT+δ2sσT+δ0σTs=Kp+Kds+Ki1s (5) can be transformed into (26). Obviously, (26) has the standard form as (24), which is of strictly rational proper fraction function.(26)GNs=1kτT1s2+τ+T1ks+kOn the basis of Theorem 1, the main engine speed steady-state error satisfies limt→∞⁡nf-yset=0. As a result, the integral term Ki/s is negligible. Define TM as the system period, and substitute the parameters of (26) into (25), so one can get(27)UPD=Kp+Kds=τ+T1TM+kτT1TMsRemark 3.
In accordance with ship main engine knowledge and the CGSA, the design parameters are of clear physical significance, which can be exactly determined. However, the control performance is not guaranteed to be adaptive to load changes and sea conditions.


## 4. Parameter Online Tuning Based on VUFI

A concise robust PD controller based on Backstepping and CGSA (CRPD-BC) is brought up with definite parameter determination method. However, when the control system model has perturbation due to load or interference change, the fixed control parameter often means the control performance might get worse. Therefore, the on-line optimization of control parameters is very important for the optimal and stable control performance. Consequently, an adaptive PD controller based on VUFI (APD-VUFI) is presented, which can adjust the fuzzy domain and precision of control parameters according to input and output, just as shown in Figure1. Furthermore, it has better adaptive ability than general fuzzy PID [18–20]. Specific parameter adjustment rules are as shown in (28).(28)Kp′=Kp+ΔKp=Kp1+OpKd′=Kd+ΔKd=Kd1+OdFuzzy comprehensive reasoning takes the form of two inputs and two outputs. e,ec are the initial inputs and ΔKP,ΔKD are the final outputs. According to the selection method of fuzzy scaling factor [19, 20], the domain scaling structure of fuzzy input Ie, Iec can be designed as follows:(29)Ie=e.αe=e.eEλ1(30)Iec=ec.βec=ec.ecECλ2where e∈-E,E, ec∈-EC,EC, λ1,λ2∈0,1, and α,β are input regulation factors. At the same time, the domain scaling structure of fuzzy output Op,Od is defined as follows.(31)γt=Re,ecP=Re,ecp1,p2TwhereR is a proportional constant andP is a constant vector.Figure 1
The control flow chart of robust fuzzy PD control of marine engine speed regulation.The input and output of fuzzy reasoning are divided by triangular membership function with fuzzy linguistic variables NB, NM, NS, ZO, PS, PM, and PB. According to the specific physical meaning and function of the control parameters, combined with field debugging and expert operation experience, the fuzzy control rules are shown in Figure2.Figure 2
The surface views of fuzzy inference rules onΔKP (a) and ΔKD (b).
(a) (b)


## 5. Simulation

### 5.1. Simulation Configuration

In this section, the effectiveness of the presented control scheme in marine engineering practice is illustrated by several simulation examples. The simulation model described in Section2 employs the parameters of MAN B&M S60M, which is a widely used large low speed diesel engine. The time constant T1=12.1s, amplification coefficient k=93.8, pure delay time τ=0.037s, when the model is perturbed, the above parameters are multiplied by 50%. In the actuating mechanism, T2=0.0307, ξ=0.704. The speed of the ship’s main engine is usually divided into four grades: dead slow, slow, half, and full, respectively, corresponding to 20, 40, 60, and 80rpm in the simulation tests. It is assumed that the main engine speed is equal to the propeller speed. Furthermore, two kinds of interference are considered: the first is the sudden increase or decrease of load with an equivalent speed change of 10rpm, and the second is a sinusoidal wave interference, shown as follows:(32)ds=limn→∞⁡θnt=0,t<3010rpm,30≤t<80-10rpm,t≥80(33)dv=Asin⁡ωt,A=10,w=0.1πThe fuzzy domain scaling factors are λ1=λ2=0.6, R=2, and p1,p2=2,2.


### 5.2. Simulation Results and Discussion

The control performances of APD-VUFI under different settings of speeds and disturbances are shown in Figures3–5, which indicate that the proposed control scheme can achieve good performances of speed regulation even in case of sudden load changing, sinusoidal wave interference, and model perturbations.Figure 3
Speed regulation performances (a) and fuel index positions (b) at different set speeds.
(a) (b)Figure 4
The speed regulation performances (a), fuel index positions (b), and adaptive variation of control parameters ((c)&(d)) at different set speeds with sudden step disturbances.
(a) (b) (c) (d)Figure 5
The comparison of speed regulation performances (a) and fuel index positions (b) by different controllers with a 50% model perturbation and sinusoidal wave disturbances after 50 seconds.
(a) (b)The speed regulation performances and fuel index positions at different set speeds are shown in Figure3. It is clear that the performances of different rotating speeds meet the requirements. Moreover, it has typical quasi-linear transient performance without overshoot and good steady state performance. From the point of control input, big throttle is given firstly to speed up the start, and then the throttle is reduced at the right time to stabilize the speed at the set value, which is of typical engineering operation significance.Figure4 shows the control performance, control input, and control parameters adaptation in case of the set speed changing and even sudden load increases or decreases. Figure 4(a) indicates that the control system is well qualified for the above tasks. The control input in Figure 4(b) is reasonable, but there are some overshoots in the face of step type interference. At 30 seconds, the control input can be appropriately reduced when the load is suddenly reduced, so that the marine engine speed can be stabilized at the set value. Furthermore, it can be proved that the control system can also cope with sudden increase of load at 80 seconds. Figures 4(c) and 4(d) illustrate that the optimal control performance can be obtained by adjusting the control parameters under different set speed and different load conditions.As shown in Figure5(a), when faced with 50% model perturbation, APD-VUFI performs better than CRPD-BC and traditional PD (TPD), CRPD-BC has the longest adjustment time, and TPD has certain overshoot. When there is sinusoidal wave interference (>50s), APD-VUFI and CRPD-BC can eliminate interference greatly, while TPC cannot. In terms of control rules, the control inputs given by APD-VUFI and CRPD-BC are similar, but APD-VUFI can provide larger input for quick startup, while TPD has smaller input and a certain phase delay in face of wave interference.


## 5.1. Simulation Configuration

In this section, the effectiveness of the presented control scheme in marine engineering practice is illustrated by several simulation examples. The simulation model described in Section2 employs the parameters of MAN B&M S60M, which is a widely used large low speed diesel engine. The time constant T1=12.1s, amplification coefficient k=93.8, pure delay time τ=0.037s, when the model is perturbed, the above parameters are multiplied by 50%. In the actuating mechanism, T2=0.0307, ξ=0.704. The speed of the ship’s main engine is usually divided into four grades: dead slow, slow, half, and full, respectively, corresponding to 20, 40, 60, and 80rpm in the simulation tests. It is assumed that the main engine speed is equal to the propeller speed. Furthermore, two kinds of interference are considered: the first is the sudden increase or decrease of load with an equivalent speed change of 10rpm, and the second is a sinusoidal wave interference, shown as follows:(32)ds=limn→∞⁡θnt=0,t<3010rpm,30≤t<80-10rpm,t≥80(33)dv=Asin⁡ωt,A=10,w=0.1πThe fuzzy domain scaling factors are λ1=λ2=0.6, R=2, and p1,p2=2,2.


## 5.2. Simulation Results and Discussion

The control performances of APD-VUFI under different settings of speeds and disturbances are shown in Figures3–5, which indicate that the proposed control scheme can achieve good performances of speed regulation even in case of sudden load changing, sinusoidal wave interference, and model perturbations.Figure 3
Speed regulation performances (a) and fuel index positions (b) at different set speeds.
(a) (b)Figure 4
The speed regulation performances (a), fuel index positions (b), and adaptive variation of control parameters ((c)&(d)) at different set speeds with sudden step disturbances.
(a) (b) (c) (d)Figure 5
The comparison of speed regulation performances (a) and fuel index positions (b) by different controllers with a 50% model perturbation and sinusoidal wave disturbances after 50 seconds.
(a) (b)The speed regulation performances and fuel index positions at different set speeds are shown in Figure3. It is clear that the performances of different rotating speeds meet the requirements. Moreover, it has typical quasi-linear transient performance without overshoot and good steady state performance. From the point of control input, big throttle is given firstly to speed up the start, and then the throttle is reduced at the right time to stabilize the speed at the set value, which is of typical engineering operation significance.Figure4 shows the control performance, control input, and control parameters adaptation in case of the set speed changing and even sudden load increases or decreases. Figure 4(a) indicates that the control system is well qualified for the above tasks. The control input in Figure 4(b) is reasonable, but there are some overshoots in the face of step type interference. At 30 seconds, the control input can be appropriately reduced when the load is suddenly reduced, so that the marine engine speed can be stabilized at the set value. Furthermore, it can be proved that the control system can also cope with sudden increase of load at 80 seconds. Figures 4(c) and 4(d) illustrate that the optimal control performance can be obtained by adjusting the control parameters under different set speed and different load conditions.As shown in Figure5(a), when faced with 50% model perturbation, APD-VUFI performs better than CRPD-BC and traditional PD (TPD), CRPD-BC has the longest adjustment time, and TPD has certain overshoot. When there is sinusoidal wave interference (>50s), APD-VUFI and CRPD-BC can eliminate interference greatly, while TPC cannot. In terms of control rules, the control inputs given by APD-VUFI and CRPD-BC are similar, but APD-VUFI can provide larger input for quick startup, while TPD has smaller input and a certain phase delay in face of wave interference.


## 6. Conclusion

In this paper, one focuses on the optimal speed regulation for large low speed marine engines in field of engineering practice. A concise robust PID control law based on Backstepping is presented with the uniform asymptotic stability, and the control parameters have obvious physical meaning. Furthermore, parameter determination method is given by virtue of closed-loop gain shaping algorithm. To overcome the perturbation due to load or interference change, variable universe fuzzy inference is used to optimize the control system on-line. Compared with the existing research literature, the design method and performances are more in line with the ocean engineering practice. Simulation results have illustrated the performances and effectiveness of the proposed system.



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*Source: 5823827-2019-05-02.xml*