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Research Article
Concise Robust Control of Marine Engine Speed Based on
Backstepping and Its Fuzzy Comprehension
Lijun Wang1,2 and Sisi Wang 2
1School of Navigation, Guangdong Ocean University, Zhanjiang 524088, China
2Hubei Key Laboratory of Inland Shipping Technology, Wuhan 430063, China
Correspondence should be addressed to Sisi Wang; marslin@sina.com
Received 14 March 2019; Accepted 21 April 2019; Published 2 May 2019
Academic Editor: Basil M. Al-Hadithi
Copyright © Lijun Wang and Sisi Wang. is is an open access article distributed under the Creative Commons Attribution
License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly
cited.
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. e results of the simulations of the proposed controller are presented and compared.
1. Introduction
e 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 [–]. ere are typical nonlinearity, time-
varying, uncertainty, and manual active intervention in the
control process of ship main engine speed. erefore, the
traditional PID controller is dicult to obtain the optimal
control performance [, ]. An improved PID tuning method
was proposed for marine diesel engine governors to overcome
load uctuation due to weather and sea conditions []. In
order to improve the robustness of ship engine speed control,
active disturbance rejection controller was presented in [,
], and robust control method was recommended in [, ].
To optimize the control parameters, several intelligent algo-
rithms have been used to achieve better control performance,
such as fuzzy logic comprehension [, ], GA optimization
[], and Neural Network adaption [].
Obviously, the current research mainly focuses on PID,
robust, and intelligent algorithms; however, PID has poor
self-adaptability, robust control is dicult to achieve in
engineering, and intelligent algorithms oen 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. e main contributions of this
work can be summarized as follows:
(i) e 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 signicance and can be determined easily
(ii) Variable universe fuzzy inference (VUFI) is recom-
mended to solve the uncertainty caused by model
perturbation and random disturbance for optimal
control solution
e layout of the article is as follows: Section presents
a marine engine speed regulation model. Section designs
a concise robust controller and gives its stability proof.
Section provides online optimization of control parameters
based on VUFI. Section details the simulation results and
discussion. Section gives the conclusion.
2. Marine Engine Speed Regulation Model
2.1. Dynamic Model for Large Low Speed Marine Engines. is
study is to design speed regulation controller for large low
Hindawi
Complexity
Volume 2019, Article ID 5823827, 7 pages
https://doi.org/10.1155/2019/5823827
Complexity
speed marine engines, such as MAN B&M SM. e second
order dynamic model can be described as follows [, , ].
1
𝑓()+𝑓()=(−)()
where 1is the time constant, is the amplication coe-
cient, is the dead time, and 𝑓is the rate of revolution. And
a nonlinear main engine model (NMEM) of transfer function
form can be expressed as
𝑁()=𝑓()
()=−𝜏𝑠
1+ ()
where is the fuel index position. e dead time is caused by
the injection delay of the fuel system, which is estimated to
be within the following range [].
15
𝑓<< 15
𝑓+60
𝑓⋅ ()
where 𝑓is the rated speed and is the number of engine
cylinders. For large low speed marine engine, the delay
term can be replaced by the following rst order inertial
expression:
−𝜏𝑠 ≈1
+ 1 ()
Consequently, the propulsion plant dynamics are described
with the following equation on the s-plane. Note that all poles
in 𝑁()are real and stable.
𝑁()=1
(+ 1)1+
=1
1(+1/)+ 1/1
()
2.2. Actuating Mechanism. An electronic hydraulic actuator
(EHA) controlled by a high-speed switch valve is used. e
control signal is pulse width modulation, and the time delay
of the link is not considered. e transfer function of the
actuator can be described as follows:
𝐴()=1
2
22+2
2+1 ()
where 2is the time constant and is the damping coecient.
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.
eorem 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.
=1
−2+
𝑠𝑒𝑡
−1
1+ 12+1+2 ()
where ,are the speed error and its derivative, and 1,2are
design parameters.
Proof. Set 1=
𝑓,2=
𝑓,=,and=
1,sowecan
obtain the state space model of () as follows:
1=
2
2=
2+
=
1
()
where (2) = −(( + 1)/1)2−(1/
1)∫2,=
1/1, and one denes the set revolution 𝑠𝑒𝑡 and the error
variable 1:
1=−
𝑠𝑒𝑡 =
1−
𝑠𝑒𝑡 ()
e rst Lyapunov function is selected as
V1=1
22
1()
V1=
1
1=
1
1−
𝑟=
12−
𝑟()
Dene virtual control variable 2𝑑 =
1, and assume
2𝑑 =−
11+
𝑠𝑒𝑡,()
V1=−
12
1<0 ()
where design parameters 1>0,thefunctionV1is negative
denite, and one can get lim𝑡→∞(1−𝑠𝑒𝑡)=0;thatistosay,
the control of 1is realized.
Dene another error variable 2and the second Lyapunov
function V2:
2=
2−2𝑑 ()
V2=1
22
1+1
22
2()
Substitute () and () in (), so one can get
2=
1−
𝑠𝑒𝑡 +11=
1+11,()
V2=−
12
1+
21+
2−
2𝑑
=−
12
1+
21− 2
11+12+
2−
𝑠𝑒𝑡()
To guarantee the negative deniteness of V2, one can dene a
virtual control variable 3𝑑 =
2, and assume
3𝑑 =−1−
2
11−1+22+
𝑠𝑒𝑡,()
V2=−
12
1−22
2<0, ∀
1=0,
2=0 ()
Complexity
where design parameters 2>0,thefunctionV2is negative
denite, and one can get lim𝑡→∞(2−
2𝑑)=0;thatis
to say, the control of 2𝑑 is realized by the control law (),
and all the variables in main engine speed control loop are
uniformly asymptotic stable with equilibrium point [1,2]=
[𝑠𝑒𝑡,
𝑠𝑒𝑡].
is ends the proof of eorem .
Substituting () into (), the control law is transformed
as follows:
3𝑑 =−1+
121−1+2
1+
𝑠𝑒𝑡 ()
Substituting () into (), the actual control law was deduced,
if
=1
−2+
𝑠𝑒𝑡
−1
1+ 121+1+2
1
=1
−2+
𝑠𝑒𝑡
−1
1+ 121−
𝑠𝑒𝑡+1+22−
𝑠𝑒𝑡
=1
−2+
𝑠𝑒𝑡
−1
1+ 12+1+2
=−1
2−
𝑠𝑒𝑡 +𝑃𝐷
()
Remark 2. e essence of the control law () is to compen-
sate the system’s linearity or nonlinearity and to stabilize the
control loop by a PD type controller 𝑃𝐷. It is obvious that the
Backstepping control design method is concise and eective.
However, the design parameters 1,2areoflittleengineering
signicance 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) [–], a con-
cise robust PID controller is presented as follows.
1
+ 1 =
1+ ()
= 1
()
()=
22+
1+
0()
where is the system transfer function model, stands for
the system period, for the main engine speed control system,
and is the design controller. erefore, one can get a PID
controller by substituting () into ().
= 1
+2
+0
=
𝑝+
𝑑+
𝑖1
()
() can be transformed into (). Obviously, () has the
standard form as (), which is of strictly rational proper
fraction function.
𝑁()=1
12++
1+ ()
On the basis of eorem , the main engine speed steady-state
error satises lim𝑡→∞(𝑓−𝑠𝑒𝑡)=0.Asaresult,theintegral
term 𝑖/ is negligible. Dene 𝑀as the system period, and
substitute the parameters of () into (), so one can get
𝑃𝐷 =
𝑝+
𝑑=+
1
𝑀+1
𝑀()
Remark 3. In accordance with ship main engine knowledge
and the CGSA, the design parameters are of clear physical
signicance, 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 denite parameter
determination method. However, when the control system
model has perturbation due to load or interference change,
the xed control parameter oen means the control perfor-
mance might get worse. erefore, 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 Figure . Furthermore, it has better adaptive ability than
general fuzzy PID [–]. Specic parameter adjustment
rules are as shown in ().
𝑝=
𝑝+𝐾𝑝 =
𝑝1+𝑝
𝑑=
𝑑+𝐾𝑑 =
𝑑1+ 𝑑()
Fuzzy comprehensive reasoning takes the form of two inputs
and two outputs. , are the initial inputs and 𝐾𝑃,𝐾𝐷 are
the nal outputs. According to the selection method of fuzzy
scaling factor [, ], the domain scaling structure of fuzzy
input 𝑒,𝑒𝑐 can be designed as follows:
𝑒=.()=.||
𝜆1 ()
𝑒𝑐 =.()=.||
𝜆2 ()
where ∈ [−,], ∈ [−,],1,2∈(0,1),and,
are input regulation factors. At the same time, the domain
scaling structure of fuzzy output 𝑝,𝑑is dened as follows.
()=[,]=[,]1,2T()
where Ris a proportional constant and Pis a constant vector.
Complexity
Var i a b l e
Univers e
Adaptation
Fuzzy
Inference
CRPD-BC NMEM
Set
Speed
Output
d/dt
d/dt
e
EHA
+
--
+
ec
Ie
Iec
Op
Od
Kp
Kd
ΔKP
ΔKD
F : e control ow chart of robust fuzzy PD control of marine engine speed regulation.
−1
0
1
−1
0
1
−0.5
0
0.5
Ie
Iec
OP
(a)
−1
0
1
−1
0
1
−0.5
0
0.5
Ie
Iec
OD
(b)
F : e surface views of fuzzy inference rules on 𝐾𝑃 (a) and 𝐾𝐷 (b).
e input and output of fuzzy reasoning are divided
by triangular membership function with fuzzy linguistic
variablesNB,NM,NS,ZO,PS,PM,andPB.Accordingtothe
specic physical meaning and function of the control param-
eters, combined with eld debugging and expert operation
experience, the fuzzy control rules are shown in Figure .
5. Simulation
5.1. Simulation Configuration. In this section, the eective-
ness of the presented control scheme in marine engineer-
ing practice is illustrated by several simulation examples.
e simulation model described in Section employs the
parameters of MAN B&M SM, which is a widely used
large low speed diesel engine. e time constant 1= 12.1,
amplication coecient = 93.8,puredelaytime=
0.037, when the model is perturbed, the above parameters
are multiplied by %. In the actuating mechanism, 2=
0.0307, = 0.704.espeedoftheship’smainengineis
usually divided into four grades: dead slow, slow, half, and
full, respectively, corresponding to , , , and rpm 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 rst is the sudden increase or
decrease of load with an equivalent speed change of rpm,
and the second is a sinusoidal wave interference, shown as
follows:
𝑠=lim
𝑛→∞ 𝑛()=
0, < 30
10rpm,30≤<80
−10rpm,≥80
()
V=sin (), = 10, = 0.1 ()
e fuzzy domain scaling factors are 1 = 2 = 0.6,=2,
and [1,2]=[2,2].
5.2. Simulation Results and Discussion. e control perfor-
mances of APD-VUFI under dierent settings of speeds and
disturbances are shown in Figures –, 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.
e speed regulation performances and fuel index posi-
tions at dierent set speeds are shown in Figure . It is clear
that the performances of dierent rotating speeds meet the
requirements. Moreover, it has typical quasi-linear transient
performance without overshoot and good steady state perfor-
mance. From the point of control input, big throttle is given
Complexity
80 r/min
60 r/min
40 r/min
20 r/min
0
20
40
60
80
speed (r/min)
20 40 60 80 1000
time (s)
(a)
0 20 40 60 80 100
10
time (s)
0
2
4
6
8
position (mm)
80 r/min
60 r/min
40 r/min
20 r/min
(b)
F : Speed regulation performances (a) and fuel index positions (b) at dierent set speeds.
20 40 60 80 1000
time (s)
0
20
40
60
80
speed (r/min)
(a)
0
2
4
6
8
10
position (mm)
20 40 60 80 1000
time (s)
(b)
20 40 60 80 1000
time (s)
0
10
20
30
40
KP
(c)
35
40
45
50
55
60
KD
20 40 60 80 1000
time (s)
(d)
F : e speed regulation performances (a), fuel index positions (b), and adaptive variation of control parameters ((c)&(d)) at dierent
set speeds with sudden step disturbances.
rstly 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 signicance.
Figure 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 (a) indicates that the control system is well qualied
for the above tasks. e control input in Figure (b) is
reasonable, but there are some overshoots in the face of step
type interference. At 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
seconds. Figures (c) and (d) illustrate that the optimal
control performance can be obtained by adjusting the control
parameters under dierent set speed and dierent load condi-
tions.
AsshowninFigure(a),whenfacedwith%model
perturbation, APD-VUFI performs better than CRPD-BC
and traditional PD (TPD), CRPD-BC has the longest adjust-
ment time, and TPD has certain overshoot. When there is
sinusoidal wave interference (>s), 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
Complexity
APD-VUFI
CRPD-BC
TPD
20 40 60 80 1000
time (s)
0
20
40
60
80
100
speed (r/min)
(a)
APD-VUFI
CRPD-BC
TPD
0
2
4
6
8
10
position (mm)
20 40 60 80 1000
time (s)
(b)
F : e comparison of speed regulation performances (a) and fuel index positions (b) by dierent controllers with a % model
perturbation and sinusoidal wave disturbances aer seconds.
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 eld of engineering prac-
tice. A concise robust PID control law based on Backstepping
is presented with the uniform asymptotic stability, and the
control parameters have obvious physical meaning. Further-
more, parameter determination method is given by virtue of
closed-loop gain shaping algorithm. To overcome the pertur-
bation 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 eectiveness of the proposed system.
Abbreviations
e abbreviations and symbols adopted throughout the paper
are listed:
MME: Marine main engine
GA: Generic algorithm
VUFI: Variable universe fuzzy inference
CGSA: Closed-loop gain shaping algorithm
EHA: Electronic hydraulic actuator
NMEM: Nonlinear main engine model
RPM: Revolutions per minute
TPD: Traditional PD
APD-VUFI: Adaptive PD controller based on VUFI
CRPD-BC: Concise robust PD controller based on
Backstepping and CGSA
PID: Proportional integral derivative controller
PD: Proportional derivative controller
:Systemperiod
𝑀: System period of marine engine speed
system
1: Time constant of marine engine speed
system
: Amplication coecient of marine engine
speed system
𝑓: Sha speed of marine engine speed system
:Fuelindexposition
: Dead time of marine engine speed system
: Number of engine cylinders
:Systemtransferfunctionmodel
𝑁: Transfer function of marine engine speed
system
𝐴: Transfer function of electronic hydraulic
actuator
: Designed controller
2: Time constant of electronic hydraulic
actuator
: Damping constant of electronic hydraulic
actuator
: Speed error
: Speed error derivative
: e proposed controller
𝑃𝐷: PD type controller
1,2: Design parameters of the controller
𝑠𝑒𝑡:Setrevolution
1,2: Set revolution errors
1,2: State variables
V1,V2:Lyapunovfunctions
2𝑑,3𝑑: Virtual control variables
0,1,2: Denominator parameters of a second
order strictly rational proper system
: Numerator parameters of a second order
strictly rational proper system
𝑝,𝑑,𝑖: Proportion, integration, and
dierentiation parameters
∗
𝑝,∗
𝑑: Adaptive proportion and dierentiation
parameters
Complexity
𝑝,𝑑: Fuzzy adjusting output factors of
proportion and dierentiation parameters
𝐾𝑃,𝐾𝐷: Fuzzy outputs of the of proportion and
dierentiation parameters
𝑒,𝑒𝑐: Fuzzy inputs
,1,2: Fuzzy domain scaling factors
:Constantvector
,: Fuzzy input regulation factors
: Fuzzy output regulation factor
[−,]: Fuzzy domain of speed error
[−,]: Fuzzy domain of speed error derivative
𝑠: Disturbance due to sudden increase or
decrease of load with an equivalent speed
change
V:Sinusoidalwaveinterference
: Amplitude of sinusoidal wave interference
: Frequency of sinusoidal wave interference.
Data Availability
e authors obtain data from the third parties and therefore
do not have the right to make that dataset publicly available.
But data can be available upon request through the China
Classication Society.
Conflicts of Interest
e authors declare that there are no conicts of interest
regarding the publication of this paper.
Acknowledgments
is work was supported by Fund of Hubei Key Laboratory
of Inland Shipping Technology (Grant NHHY); Sci-
entic Research Start-up Funds of Guangdong Ocean Uni-
versity (Grants E and R); Characteristic Innovation
Projects of Guangdong Province (Grants KTSCX
and KTSCX); and College Student Innovation and
Entrepreneurship Training Program of Guangdong Province
(Grant ).
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