Concise Robust Control of Marine Engine Speed Based on ...downloads.hindawi.com/journals/complexity/2019/5823827.pdf · ResearchArticle Concise Robust Control of Marine Engine Speed

Research ArticleConcise Robust Control of Marine Engine Speed Based onBackstepping and Its Fuzzy Comprehension

Lijun Wang12 and Sisi Wang 2

1School of Navigation Guangdong Ocean University Zhanjiang 524088 China2Hubei Key Laboratory of Inland Shipping Technology Wuhan 430063 China

Correspondence should be addressed to Sisi Wang mars32linsinacom

Received 14 March 2019 Accepted 21 April 2019 Published 2 May 2019

Academic Editor Basil M Al-Hadithi

Copyright copy 2019 Lijun Wang and Sisi Wang This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

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

1 Introduction

Thespeed regulation ofmarinemain engine (MME) is relatedto its performance service life and ship safety At presentthe advanced MME speed governor is mostly based on PIDdigital controller [1ndash3] There are typical nonlinearity time-varying uncertainty and manual active intervention in thecontrol process of ship main engine speed Therefore thetraditional PID controller is difficult to obtain the optimalcontrol performance [4 5] An improved PID tuning methodwas proposed formarine diesel engine governors to overcomeload fluctuation due to weather and sea conditions [1] Inorder to improve the robustness of ship engine speed controlactive disturbance rejection controller was presented in [26] and robust control method was recommended in [7 8]To optimize the control parameters several intelligent algo-rithms have been used to achieve better control performancesuch as fuzzy logic comprehension [3 4] GA optimization[5] and Neural Network adaption [9]

Obviously the current research mainly focuses on PIDrobust and intelligent algorithms however PID has poorself-adaptability robust control is difficult to achieve inengineering and intelligent algorithms often only have localoptimal solutions Inspired by previous studies an optimal

robust control system formarine engine speed regulation willbe discussed in this paper The main contributions of thiswork can be summarized as follows

(i) The uniform asymptotic stability proof of a conciserobust control design for the marine engine speedis given and the control parameters are of obviousphysical significance and can be determined easily

(ii) Variable universe fuzzy inference (VUFI) is recom-mended to solve the uncertainty caused by modelperturbation and random disturbance for optimalcontrol solution

The layout of the article is as follows Section 2 presentsa marine engine speed regulation model Section 3 designsa concise robust controller and gives its stability proofSection 4 provides online optimization of control parametersbased on VUFI Section 5 details the simulation results anddiscussion Section 6 gives the conclusion

2 Marine Engine Speed Regulation Model

21 DynamicModel for Large Low SpeedMarine Engines Thisstudy is to design speed regulation controller for large low

HindawiComplexityVolume 2019 Article ID 5823827 7 pageshttpsdoiorg10115520195823827

2 Complexity

speedmarine engines such asMANBampMS60MThe secondorder dynamic model can be described as follows [8 10 11]

1198961198791 119899119891 (119905) + 119896119899119891 (119905) = 119904 (119905 minus 120591) (1)

where 1198791 is the time constant 119896 is the amplification coeffi-cient 120591 is the dead time and 119899119891 is the rate of revolution Anda nonlinear main engine model (NMEM)of transfer functionform can be expressed as

119866119873 (119904) =119899119891 (119904)119901 (119904) = 119890minus120591119904

1198961198791119904 + 119896 (2)

where 119901 is the fuel index position The dead time is caused bythe injection delay of the fuel system which is estimated tobe within the following range [1]

15119873119891 lt 120591 lt 15

119873119891 + 60119873119891 sdot 119885 (3)

where 119873119891 is the rated speed and 119885 is the number of enginecylinders For large low speed marine engine the delayterm can be replaced by the following first order inertialexpression

119890minus120591119904 asymp 1120591119904 + 1 (4)

Consequently the propulsion plant dynamics are describedwith the following equation on the s-plane Note that all polesin 119866119873(119904) are real and stable

119866119873 (119904) = 1(120591119904 + 1) (1198961198791119904 + 119896)

= 11198961205911198791 (119904 + 1120591) (119904 + 11198791)

(5)

22 Actuating Mechanism An electronic hydraulic actuator(EHA) controlled by a high-speed switch valve is used Thecontrol signal is pulse width modulation and the time delayof the link is not considered The transfer function of theactuator can be described as follows

119866119860 (119904) = 111987922 1199042 + 21205851198792119904 + 1 (6)

where1198792 is the time constant and 120585 is the damping coefficientIn the shiprsquos main engine operating system the actuator hasa displacement sensor so it can be adjusted by feedback tomake the output more stable

3 Control Design

31 Concise Robust Control Design In this section a conciserobust control law based on Backstepping for marine enginespeed regulation is presented and the control parametershave obvious physical meaning Furthermore parameterdetermination method is given

Theorem 1 Considering the marine engine speed regulationmodel (1) the proposed controller (7) based on the Backsteppingcan stabilize the speed motion and guaranteeing the uniformlyasymptotic stability of the closed-loop speed regulation system

119906 = 1119887 [minus119891 (1199092) + 119910119904119890119905]

minus 1119887 [(1 + 11989611198962) 119890 + (1198961 + 1198962) 119890119888]

(7)

where 119890 119890119888 are the speed error and its derivative and 1198961 1198962 aredesign parameters

Proof Set 1199091 = 119899119891 1199092 = 119899119891 119906 = 119901 and 119910 = 1199091 so we canobtain the state space model of (5) as follows

1 = 11990922 = 119891 (1199092) + 119887119906119910 = 1199091

(8)

where 119891(1199092) = minus((120591 + 1198791)1205911198791)1199092 minus (11205911198791) int 1199092119889119905 119887 =11198961205911198791 and one defines the set revolution 119910119904119890119905 and the errorvariable 1205781

1205781 = 119910 minus 119910119904119890119905 = 1199091 minus 119910119904119890119905 (9)

The first Lyapunov function is selected as

V1 = 1212057821 (10)

V1 = 1205781 1205781 = 1205781 ( 1199091 minus 119910119903) = 1205781 (1199092 minus 119910119903) (11)

Define virtual control variable 1199092119889 = 1 and assume

1199092119889 = minus11989611205781 + 119910119904119890119905 (12)

V1 = minus119896112057821 lt 0 (13)

where design parameters 1198961 gt 0 the function V1 is negativedefinite and one can get lim119905997888rarrinfin(1199091 minus119910119904119890119905) = 0 that is to saythe control of 1199091 is realized

Define another error variable 1205782 and the second Lyapunovfunction V2

1205782 = 1199092 minus 1199092119889 (14)

V2 = 1212057821 + 1

212057822 (15)

Substitute (8) and (12) in (14) so one can get

1205782 = 1 minus 119910119904119890119905 + 11989611205781 = 1205781 + 11989611205781 (16)

V2 = minus119896112057821 + 1205782 (1205781 + 2 minus 1199092119889)= minus119896112057821 + 1205782 [(1 minus 11989621) 1205781 + 11989611205782 + 1199092 minus 119904119890119905]

(17)

To guarantee the negative definiteness of V2 one can define avirtual control variable 1199093119889 = 2 and assume

1199093119889 = minus (1 minus 11989621) 1205781 minus (1198961 + 1198962) 1205782 + 119910119904119890119905 (18)

V2 = minus119896112057821 minus 119896212057822 lt 0 forall1198961 = 0 1198962 = 0 (19)

Complexity 3

where design parameters 1198962 gt 0 the function V2 is negativedefinite and one can get lim119905997888rarrinfin(1199092 minus 1199092119889) = 0 that isto say the control of 1199092119889 is realized by the control law (18)and all the variables in main engine speed control loop areuniformly asymptotic stablewith equilibriumpoint [1199091 1199092] =[119910119904119890119905 119910119904119890119905]

This ends the proof of Theorem 1

Substituting (16) into (18) the control law is transformedas follows

1199093119889 = minus (1 + 11989611198962) 1205781 minus (1198961 + 1198962) 1205781 + 119910119904119890119905 (20)

Substituting (8) into (20) the actual control law was deducedif

119906 = 1119887 [minus119891 (1199092) + 119910119904119890119905]

minus 1119887 [(1 + 11989611198962) 1205781 + (1198961 + 1198962) 1205781]

= 1119887 [minus119891 (1199092) + 119910119904119890119905]

minus 1119887 [(1 + 11989611198962) (1199091 minus 119910119904119890119905) + (1198961 + 1198962) (1199092 minus 119910119904119890119905)]

= 1119887 [minus119891 (1199092) + 119910119904119890119905]

minus 1119887 [(1 + 11989611198962) 119890 + (1198961 + 1198962) 119890119888]

= minus1119887 [119891 (1199092) minus 119910119904119890119905 + 119906119875119863]

(21)

Remark 2 The essence of the control law (21) is to compen-sate the systemrsquos linearity or nonlinearity and to stabilize thecontrol loop by a PD type controller 119906119875119863 It is obvious that theBackstepping control design method is concise and effectiveHowever the design parameters1198961 1198962 are of little engineeringsignificance and can only be determined by trial and errorwhich is not conducive to the robustness and optimization ofthe control performance

32 Control Parameters Determination In accordance withclosed-loop gain shaping algorithm (CGSA) [12ndash17] a con-cise robust PID controller is presented as follows

1119879119904 + 1 = 119866119862

1 + 119866119862 (22)

119862 = 1119866119879119904 (23)

119866 (119904) = 12059012057521199042 + 1205751119904 + 1205750 (24)

where 119866 is the system transfer function model 119879 stands forthe system period for the main engine speed control systemand 119862 is the design controller Therefore one can get a PIDcontroller by substituting (24) into (23)

119862 = 1205751120590119879 + 1205752119904

120590119879 + 1205750120590119879119904 = 119870119901 + 119870119889119904 + 119870119894 1119904 (25)

(5) can be transformed into (26) Obviously (26) has thestandard form as (24) which is of strictly rational properfraction function

119866119873 (119904) = 111989612059111987911199042 + (120591 + 1198791) 119896119904 + 119896 (26)

On the basis ofTheorem 1 themain engine speed steady-stateerror satisfies lim119905997888rarrinfin(119899119891 minus 119910119904119890119905) = 0 As a result the integralterm 119870119894119904 is negligible Define 119879119872 as the system period andsubstitute the parameters of (26) into (25) so one can get

119880119875119863 = 119870119901 + 119870119889119904 = 120591 + 1198791119879119872 + 1198961205911198791

119879119872 119904 (27)

Remark 3 In accordance with ship main engine knowledgeand the CGSA the design parameters are of clear physicalsignificance which can be exactly determined However thecontrol performance is not guaranteed to be adaptive to loadchanges and sea conditions

4 Parameter Online Tuning Based on VUFI

A concise robust PD controller based on Backstepping andCGSA (CRPD-BC) is brought up with definite parameterdetermination method However when the control systemmodel has perturbation due to load or interference changethe fixed control parameter often means the control perfor-mance might get worse Therefore the on-line optimizationof control parameters is very important for the optimaland stable control performance Consequently an adaptivePD controller based on VUFI (APD-VUFI) is presentedwhich can adjust the fuzzy domain and precision of controlparameters according to input and output just as shownin Figure 1 Furthermore it has better adaptive ability thangeneral fuzzy PID [18ndash20] Specific parameter adjustmentrules are as shown in (28)

1198701015840119901 = 119870119901 + Δ119870119901 = 119870119901 (1 + 119874119901)1198701015840119889 = 119870119889 + Δ119870119889 = 119870119889 (1 + 119874119889)

(28)

Fuzzy comprehensive reasoning takes the form of two inputsand two outputs 119890 119890119888 are the initial inputs and Δ119870119875 Δ119870119863 arethe final outputs According to the selection method of fuzzyscaling factor [19 20] the domain scaling structure of fuzzyinput 119868119890 119868119890119888 can be designed as follows

119868119890 = 119890120572 (119890) = 119890 ( |119890|119864 )1205821

(29)

119868119890119888 = 119890119888120573 (119890119888) = 119890119888 ( |119890119888|119864119862)1205822

(30)

where 119890 isin [minus119864 119864] 119890119888 isin [minus119864119862 119864119862] 1205821 1205822 isin (0 1) and 120572 120573are input regulation factors At the same time the domainscaling structure of fuzzy output 119874119901 119874119889 is defined as follows

120574 (119905) = 119877 [119890 119890119888] 119875 = 119877 [119890 119890119888] [1199011 1199012]T (31)

where R is a proportional constant and P is a constant vector

4 Complexity

VariableUniverse

Adaptation

FuzzyInference

CRPD-BC NMEMSetSpeed

Output

ddt

ddt

e

EHA+

- -

+

ec

Ie

Iec

Op

Od

Kp

Kd

ΔKPΔ

KD

Figure 1 The control flow chart of robust fuzzy PD control of marine engine speed regulation

minus1

0

1

minus1

0

1

minus05

0

05

Ie

Iec

OP

(a)minus1

0

1

minus1

0

1

minus05

0

05

IeIec

OD

(b)

Figure 2 The surface views of fuzzy inference rules on Δ119870119875 (a) and Δ119870119863 (b)

The input and output of fuzzy reasoning are dividedby triangular membership function with fuzzy linguisticvariables NB NM NS ZO PS PM and PB According to thespecific physical meaning and function of the control param-eters combined with field debugging and expert operationexperience the fuzzy control rules are shown in Figure 2

5 Simulation

51 Simulation Configuration In this section the effective-ness of the presented control scheme in marine engineer-ing practice is illustrated by several simulation examplesThe simulation model described in Section 2 employs theparameters of MAN BampM S60M which is a widely usedlarge low speed diesel engine The time constant 1198791 = 121119904amplification coefficient 119896 = 938 pure delay time 120591 =0037119904 when the model is perturbed the above parametersare multiplied by 50 In the actuating mechanism 1198792 =00307 120585 = 0704 The speed of the shiprsquos main engine isusually divided into four grades dead slow slow half andfull respectively corresponding to 20 40 60 and 80rpm inthe simulation tests It is assumed that the main engine speedis equal to the propeller speed Furthermore two kinds ofinterference are considered the first is the sudden increase ordecrease of load with an equivalent speed change of 10rpm

and the second is a sinusoidal wave interference shown asfollows

119889119904 = lim119899997888rarrinfin

120579119899 (119905) =

0 119905 lt 3010rpm 30 le 119905 lt 80minus10rpm 119905 ge 80

(32)

119889V = 119860 sin (120596119905) 119860 = 10 119908 = 01120587 (33)

The fuzzy domain scaling factors are 1205821 = 1205822 = 06 119877 = 2and [1199011 1199012] = [2 2]

52 Simulation Results and Discussion The control perfor-mances of APD-VUFI under different settings of speeds anddisturbances are shown in Figures 3ndash5 which indicate thatthe proposed control scheme can achieve good performancesof speed regulation even in case of sudden load changingsinusoidal wave interference and model perturbations

The speed regulation performances and fuel index posi-tions at different set speeds are shown in Figure 3 It is clearthat the performances of different rotating speeds meet therequirements Moreover it has typical quasi-linear transientperformance without overshoot and good steady state perfor-mance From the point of control input big throttle is given

Complexity 5

80 rmin60 rmin

40 rmin20 rmin

0

20

40

60

80

spee

d (r

min

)

20 40 60 80 1000time (s)

(a)

0 20 40 60 80 100

10

time (s)

0

2

4

6

8

posit

ion

(mm

)

80 rmin60 rmin

40 rmin20 rmin

(b)

Figure 3 Speed regulation performances (a) and fuel index positions (b) at different set speeds

20 40 60 80 1000time (s)

0

20

40

60

80

spee

d (r

min

)

(a)

0

2

4

6

8

10po

sitio

n (m

m)

20 40 60 80 1000time (s)

(b)

20 40 60 80 1000time (s)

0

10

20

30

40

KP

(c)

35

40

45

50

55

60

KD

20 40 60 80 1000time (s)

(d)

Figure 4The speed regulation performances (a) fuel index positions (b) and adaptive variation of control parameters ((c)amp(d)) at differentset speeds with sudden step disturbances

firstly to speed up the start and then the throttle is reducedat the right time to stabilize the speed at the set value whichis of typical engineering operation significance

Figure 4 shows the control performance control inputand control parameters adaptation in case of the set speedchanging and even sudden load increases or decreasesFigure 4(a) indicates that the control system is well qualifiedfor the above tasks The control input in Figure 4(b) isreasonable but there are some overshoots in the face of steptype interference At 30 seconds the control input can beappropriately reduced when the load is suddenly reducedso that the marine engine speed can be stabilized at theset value Furthermore it can be proved that the control

system can also cope with sudden increase of load at 80seconds Figures 4(c) and 4(d) illustrate that the optimalcontrol performance can be obtained by adjusting the controlparameters under different set speed and different load condi-tions

As shown in Figure 5(a) when faced with 50 modelperturbation APD-VUFI performs better than CRPD-BCand traditional PD (TPD) CRPD-BC has the longest adjust-ment time and TPD has certain overshoot When there issinusoidal wave interference (gt50s) APD-VUFI and CRPD-BC can eliminate interference greatly while TPC cannot Interms of control rules the control inputs given by APD-VUFIand CRPD-BC are similar but APD-VUFI can provide larger

6 Complexity

APD-VUFICRPD-BCTPD

20 40 60 80 1000time (s)

0

20

40

60

80

100sp

eed

(rm

in)

(a)

APD-VUFICRPD-BCTPD

0

2

4

6

8

10

posit

ion

(mm

)

20 40 60 80 1000time (s)

(b)

Figure 5 The comparison of speed regulation performances (a) and fuel index positions (b) by different controllers with a 50 modelperturbation and sinusoidal wave disturbances after 50 seconds

input for quick startup while TPD has smaller input and acertain phase delay in face of wave interference

6 Conclusion

In this paper one focuses on the optimal speed regulation forlarge low speed marine engines in field of engineering prac-tice A concise robust PID control law based on Backsteppingis presented with the uniform asymptotic stability and thecontrol parameters have obvious physical meaning Further-more parameter determination method is given by virtue ofclosed-loop gain shaping algorithm To overcome the pertur-bation due to load or interference change variable universefuzzy inference is used to optimize the control systemon-lineCompared with the existing research literature the designmethod and performances are more in line with the oceanengineering practice Simulation results have illustrated theperformances and effectiveness of the proposed system

Abbreviations

e abbreviations and symbols adopted throughout the paperare listed

MME Marine main engineGA Generic algorithmVUFI Variable universe fuzzy inferenceCGSA Closed-loop gain shaping algorithmEHA Electronic hydraulic actuatorNMEM Nonlinear main engine modelRPM Revolutions per minuteTPD Traditional PDAPD-VUFI Adaptive PD controller based on VUFICRPD-BC Concise robust PD controller based on

Backstepping and CGSAPID Proportional integral derivative controllerPD Proportional derivative controller119879 System period

119879119872 System period of marine engine speedsystem

1198791 Time constant of marine engine speedsystem

119896 Amplification coefficient of marine enginespeed system

119899119891 Shaft speed of marine engine speed system119901 Fuel index position120591 Dead time of marine engine speed system119885 Number of engine cylinders119866 System transfer function model119866119873 Transfer function of marine engine speed

system119866119860 Transfer function of electronic hydraulic

actuator119862 Designed controller1198792 Time constant of electronic hydraulic

actuator120585 Damping constant of electronic hydraulic

actuator119890 Speed error119890119888 Speed error derivative119906 The proposed controller119880119875119863 PD type controller1198961 1198962 Design parameters of the controller119910119904119890119905 Set revolution1205781 1205782 Set revolution errors1199091 1199092 State variablesV1 V2 Lyapunov functions1199092119889 1199093119889 Virtual control variables1205750 1205751 1205752 Denominator parameters of a second

order strictly rational proper system120590 Numerator parameters of a second order

strictly rational proper system119896119901 119896119889 119896119894 Proportion integration and

differentiation parameters119896lowast119901 119896lowast119889 Adaptive proportion and differentiation

parameters

Complexity 7

119874119901 119874119889 Fuzzy adjusting output factors ofproportion and differentiation parameters

Δ119870119875 Δ119870119863 Fuzzy outputs of the of proportion anddifferentiation parameters

119868119890 119868119890119888 Fuzzy inputs119877 1205821 1205822 Fuzzy domain scaling factors119875 Constant vector120572 120573 Fuzzy input regulation factors120574 Fuzzy output regulation factor[minus119864 119864] Fuzzy domain of speed error[minus119864119862 119864119862] Fuzzy domain of speed error derivative119889119904 Disturbance due to sudden increase or

decrease of load with an equivalent speedchange

119889V Sinusoidal wave interference119860 Amplitude of sinusoidal wave interference120596 Frequency of sinusoidal wave interference

Data Availability

The authors obtain data from the third parties and thereforedo not have the right to make that dataset publicly availableBut data can be available upon request through the ChinaClassification Society

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

This work was supported by Fund of Hubei Key Laboratoryof Inland Shipping Technology (Grant NHHY2018003) Sci-entific Research Start-up Funds of Guangdong Ocean Uni-versity (Grants E15031 and R17012) Characteristic InnovationProjects of Guangdong Province (Grants 2017KTSCX088and 2017KTSCX092) and College Student Innovation andEntrepreneurship Training Program of Guangdong Province(Grant 530002001095)

References

[1] N I Xiros ldquoPID marine engine speed regulation under fullload conditions for sensitivity Hinfin-norm specifications againstpropeller disturbancerdquo Journal of Marine Engineering and Tech-nology vol 3 no 2 pp 3ndash11 2004

[2] W Pan H Xiao Y Han et al ldquoNonlinear active disturbancerejection controller research of main engine for shiprdquo inIntelligent Control amp Automation IEEE 2010

[3] X Shen and Y Su ldquoMarine diesel engine speed control systembased on fuzzy-PIDrdquo AppliedMechanics andMaterials vol 152-154 pp 1589ndash1594 2012

[4] T A Tran ldquoThe optimization of marine diesel engine rotationalspeed control process by fuzzy logic control based on Particleswarm optimization algorithmrdquo Future Internet vol 10 no 10pp 1ndash18 2018

[5] V Raval ldquoThe rotation speed control of marine diesel enginerdquoJournal of Automation and Automobile Engineering vol 2 no 1pp 38ndash41 2017

[6] H-D Hua N Ma J Ma and X-Y Zhu ldquoRobust intelligentcontrol design for marine diesel enginerdquo Journal of ShanghaiJiaotong University (Science) vol 18 no 6 pp 660ndash666 2013

[7] G Papalambrou and N P Kyrtatos ldquoHinfin robust control ofmarine diesel engine equipped with power-take-in systemrdquoIFAC Proceedings Volumes vol 39 no 12 pp 591ndash596 2006

[8] M Tuffaha and J T Gravdahl ldquoModeling and control of amarine diesel engine driving a synchronous machine and apropellerrdquo in Proceedings of the 2014 IEEE Conference on ControlApplications CCA 2014 pp 897ndash904 France October 2014

[9] W Meng and C Guo ldquoResearch on speed intelligent controlbased on neural networks for large marine main diesel enginerdquoin Proceedings of the 2010 8th World Congress on IntelligentControl and Automation WCICA 2010 pp 4667ndash4670 July2010

[10] R P Sinha and R Balaji ldquoA mathematical model of marinediesel engine speed control systemrdquo Journal of e Institutionof Engineers vol 99 no 2 pp 1ndash8 2017

[11] H M Nahim R Younes C Nohra and M Ouladsine ldquoCom-pletemodeling for systems of a marine diesel enginerdquo Journal ofMarine Science and Application vol 14 no 1 pp 93ndash104 2015

[12] L Wang S Wang J Liu and J Han ldquoRobust pid control ofcourse-keeping with RRS and its MCO based on NSGA-IIrdquoICIC Express Letters vol 9 no 11 pp 3113ndash3119 2015

[13] X Zhang G Yang Q Zhang G Zhang and Y ZhangldquoImproved concise backstepping control of course keeping forships using nonlinear feedback techniquerdquo Journal of Naviga-tion vol 70 no 6 pp 1ndash14 2017

[14] X-K Zhang Q Zhang H-X Ren and G-P Yang ldquoLin-ear reduction of backstepping algorithm based on nonlineardecoration for ship course-keeping control systemrdquo OceanEngineering vol 147 pp 1ndash8 2018

[15] G Zhang Y Deng W Zhang and C Huang ldquoNovel DVSguidance and path-following control for underactuated shipsin presence of multiple static and moving obstaclesrdquo OceanEngineering vol 170 pp 100ndash110 2018

[16] G Zhang B Tian W Zhang and X Zhang ldquoOptimized robustcontrol for industrial unstable process via the mirror-mappingmethodrdquo ISA Transactions vol 86 pp 9ndash17 2019

[17] S Wang L Wang Z Qiao and F Li ldquoOptimal robust controlof path following and rudder roll reduction for a container shipin heavy wavesrdquo Applied Sciences vol 8 no 9 p 1631 2018

[18] PHui L Fan andX Zeren ldquoVariable universe fuzzy control forvehicle semi-active suspension system with MR damper com-bining fuzzy neural network and particle swarm optimizationrdquoNeurocomputing vol 306 pp 130ndash140 2018

[19] N Arun and B Mohan ldquoModeling stability analysis andcomputational aspects of some simplest nonlinear fuzzyrdquo ISATransactions vol 70 pp 16ndash29 2017

[20] S Askari ldquoA novel and fast MIMO fuzzy inferencerdquo ExpertSystems with Applications vol 84 pp 301ndash322 2017

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2 Complexity

speedmarine engines such asMANBampMS60MThe secondorder dynamic model can be described as follows [8 10 11]

1198961198791 119899119891 (119905) + 119896119899119891 (119905) = 119904 (119905 minus 120591) (1)

where 1198791 is the time constant 119896 is the amplification coeffi-cient 120591 is the dead time and 119899119891 is the rate of revolution Anda nonlinear main engine model (NMEM)of transfer functionform can be expressed as

119866119873 (119904) =119899119891 (119904)119901 (119904) = 119890minus120591119904

1198961198791119904 + 119896 (2)

where 119901 is the fuel index position The dead time is caused bythe injection delay of the fuel system which is estimated tobe within the following range [1]

15119873119891 lt 120591 lt 15

119873119891 + 60119873119891 sdot 119885 (3)

where 119873119891 is the rated speed and 119885 is the number of enginecylinders For large low speed marine engine the delayterm can be replaced by the following first order inertialexpression

119890minus120591119904 asymp 1120591119904 + 1 (4)

Consequently the propulsion plant dynamics are describedwith the following equation on the s-plane Note that all polesin 119866119873(119904) are real and stable

119866119873 (119904) = 1(120591119904 + 1) (1198961198791119904 + 119896)

= 11198961205911198791 (119904 + 1120591) (119904 + 11198791)

(5)

22 Actuating Mechanism An electronic hydraulic actuator(EHA) controlled by a high-speed switch valve is used Thecontrol signal is pulse width modulation and the time delayof the link is not considered The transfer function of theactuator can be described as follows

119866119860 (119904) = 111987922 1199042 + 21205851198792119904 + 1 (6)

where1198792 is the time constant and 120585 is the damping coefficientIn the shiprsquos main engine operating system the actuator hasa displacement sensor so it can be adjusted by feedback tomake the output more stable

3 Control Design

31 Concise Robust Control Design In this section a conciserobust control law based on Backstepping for marine enginespeed regulation is presented and the control parametershave obvious physical meaning Furthermore parameterdetermination method is given

Theorem 1 Considering the marine engine speed regulationmodel (1) the proposed controller (7) based on the Backsteppingcan stabilize the speed motion and guaranteeing the uniformlyasymptotic stability of the closed-loop speed regulation system

119906 = 1119887 [minus119891 (1199092) + 119910119904119890119905]

minus 1119887 [(1 + 11989611198962) 119890 + (1198961 + 1198962) 119890119888]

(7)

where 119890 119890119888 are the speed error and its derivative and 1198961 1198962 aredesign parameters

Proof Set 1199091 = 119899119891 1199092 = 119899119891 119906 = 119901 and 119910 = 1199091 so we canobtain the state space model of (5) as follows

1 = 11990922 = 119891 (1199092) + 119887119906119910 = 1199091

(8)

where 119891(1199092) = minus((120591 + 1198791)1205911198791)1199092 minus (11205911198791) int 1199092119889119905 119887 =11198961205911198791 and one defines the set revolution 119910119904119890119905 and the errorvariable 1205781

1205781 = 119910 minus 119910119904119890119905 = 1199091 minus 119910119904119890119905 (9)

The first Lyapunov function is selected as

V1 = 1212057821 (10)

V1 = 1205781 1205781 = 1205781 ( 1199091 minus 119910119903) = 1205781 (1199092 minus 119910119903) (11)

Define virtual control variable 1199092119889 = 1 and assume

1199092119889 = minus11989611205781 + 119910119904119890119905 (12)

V1 = minus119896112057821 lt 0 (13)

where design parameters 1198961 gt 0 the function V1 is negativedefinite and one can get lim119905997888rarrinfin(1199091 minus119910119904119890119905) = 0 that is to saythe control of 1199091 is realized

Define another error variable 1205782 and the second Lyapunovfunction V2

1205782 = 1199092 minus 1199092119889 (14)

V2 = 1212057821 + 1

212057822 (15)

Substitute (8) and (12) in (14) so one can get

1205782 = 1 minus 119910119904119890119905 + 11989611205781 = 1205781 + 11989611205781 (16)

V2 = minus119896112057821 + 1205782 (1205781 + 2 minus 1199092119889)= minus119896112057821 + 1205782 [(1 minus 11989621) 1205781 + 11989611205782 + 1199092 minus 119904119890119905]

(17)

To guarantee the negative definiteness of V2 one can define avirtual control variable 1199093119889 = 2 and assume

1199093119889 = minus (1 minus 11989621) 1205781 minus (1198961 + 1198962) 1205782 + 119910119904119890119905 (18)

V2 = minus119896112057821 minus 119896212057822 lt 0 forall1198961 = 0 1198962 = 0 (19)

Complexity 3




1199093119889 = minus (1 + 11989611198962) 1205781 minus (1198961 + 1198962) 1205781 + 119910119904119890119905 (20)


119906 = 1119887 [minus119891 (1199092) + 119910119904119890119905]

minus 1119887 [(1 + 11989611198962) 1205781 + (1198961 + 1198962) 1205781]

= 1119887 [minus119891 (1199092) + 119910119904119890119905]


= 1119887 [minus119891 (1199092) + 119910119904119890119905]

minus 1119887 [(1 + 11989611198962) 119890 + (1198961 + 1198962) 119890119888]

= minus1119887 [119891 (1199092) minus 119910119904119890119905 + 119906119875119863]

(21)



1119879119904 + 1 = 119866119862

1 + 119866119862 (22)

119862 = 1119866119879119904 (23)

119866 (119904) = 12059012057521199042 + 1205751119904 + 1205750 (24)


119862 = 1205751120590119879 + 1205752119904

120590119879 + 1205750120590119879119904 = 119870119901 + 119870119889119904 + 119870119894 1119904 (25)


119866119873 (119904) = 111989612059111987911199042 + (120591 + 1198791) 119896119904 + 119896 (26)


119880119875119863 = 119870119901 + 119870119889119904 = 120591 + 1198791119879119872 + 1198961205911198791

119879119872 119904 (27)




1198701015840119901 = 119870119901 + Δ119870119901 = 119870119901 (1 + 119874119901)1198701015840119889 = 119870119889 + Δ119870119889 = 119870119889 (1 + 119874119889)

(28)


119868119890 = 119890120572 (119890) = 119890 ( |119890|119864 )1205821

(29)

119868119890119888 = 119890119888120573 (119890119888) = 119890119888 ( |119890119888|119864119862)1205822

(30)


120574 (119905) = 119877 [119890 119890119888] 119875 = 119877 [119890 119890119888] [1199011 1199012]T (31)


4 Complexity

VariableUniverse

Adaptation

FuzzyInference


Output

ddt

ddt

e

EHA+

- -

+

ec

Ie

Iec

Op

Od

Kp

Kd

ΔKPΔ

KD


minus1

0

1

minus1

0

1

minus05

0

05

Ie

Iec

OP

(a)minus1

0

1

minus1

0

1

minus05

0

05

IeIec

OD

(b)



5 Simulation



119889119904 = lim119899997888rarrinfin

120579119899 (119905) =


(32)

119889V = 119860 sin (120596119905) 119860 = 10 119908 = 01120587 (33)




Complexity 5

80 rmin60 rmin

40 rmin20 rmin

0

20

40

60

80

spee

d (r

min

)

20 40 60 80 1000time (s)

(a)

0 20 40 60 80 100

10

time (s)

0

2

4

6

8

posit

ion

(mm

)

80 rmin60 rmin

40 rmin20 rmin

(b)


20 40 60 80 1000time (s)

0

20

40

60

80

spee

d (r

min

)

(a)

0

2

4

6

8

10po

sitio

n (m

m)

20 40 60 80 1000time (s)

(b)

20 40 60 80 1000time (s)

0

10

20

30

40

KP

(c)

35

40

45

50

55

60

KD

20 40 60 80 1000time (s)

(d)






6 Complexity

APD-VUFICRPD-BCTPD

20 40 60 80 1000time (s)

0

20

40

60

80

100sp

eed

(rm

in)

(a)

APD-VUFICRPD-BCTPD

0

2

4

6

8

10

posit

ion

(mm

)

20 40 60 80 1000time (s)

(b)



6 Conclusion


Abbreviations















parameters

Complexity 7






Data Availability




Acknowledgments


References




























Journal of












Journal of







Volume 2018






Volume 2018








Complexity 3




1199093119889 = minus (1 + 11989611198962) 1205781 minus (1198961 + 1198962) 1205781 + 119910119904119890119905 (20)


119906 = 1119887 [minus119891 (1199092) + 119910119904119890119905]

minus 1119887 [(1 + 11989611198962) 1205781 + (1198961 + 1198962) 1205781]

= 1119887 [minus119891 (1199092) + 119910119904119890119905]


= 1119887 [minus119891 (1199092) + 119910119904119890119905]

minus 1119887 [(1 + 11989611198962) 119890 + (1198961 + 1198962) 119890119888]

= minus1119887 [119891 (1199092) minus 119910119904119890119905 + 119906119875119863]

(21)



1119879119904 + 1 = 119866119862

1 + 119866119862 (22)

119862 = 1119866119879119904 (23)

119866 (119904) = 12059012057521199042 + 1205751119904 + 1205750 (24)


119862 = 1205751120590119879 + 1205752119904

120590119879 + 1205750120590119879119904 = 119870119901 + 119870119889119904 + 119870119894 1119904 (25)


119866119873 (119904) = 111989612059111987911199042 + (120591 + 1198791) 119896119904 + 119896 (26)


119880119875119863 = 119870119901 + 119870119889119904 = 120591 + 1198791119879119872 + 1198961205911198791

119879119872 119904 (27)




1198701015840119901 = 119870119901 + Δ119870119901 = 119870119901 (1 + 119874119901)1198701015840119889 = 119870119889 + Δ119870119889 = 119870119889 (1 + 119874119889)

(28)


119868119890 = 119890120572 (119890) = 119890 ( |119890|119864 )1205821

(29)

119868119890119888 = 119890119888120573 (119890119888) = 119890119888 ( |119890119888|119864119862)1205822

(30)


120574 (119905) = 119877 [119890 119890119888] 119875 = 119877 [119890 119890119888] [1199011 1199012]T (31)


4 Complexity

VariableUniverse

Adaptation

FuzzyInference


Output

ddt

ddt

e

EHA+

- -

+

ec

Ie

Iec

Op

Od

Kp

Kd

ΔKPΔ

KD


minus1

0

1

minus1

0

1

minus05

0

05

Ie

Iec

OP

(a)minus1

0

1

minus1

0

1

minus05

0

05

IeIec

OD

(b)



5 Simulation



119889119904 = lim119899997888rarrinfin

120579119899 (119905) =


(32)

119889V = 119860 sin (120596119905) 119860 = 10 119908 = 01120587 (33)




Complexity 5

80 rmin60 rmin

40 rmin20 rmin

0

20

40

60

80

spee

d (r

min

)

20 40 60 80 1000time (s)

(a)

0 20 40 60 80 100

10

time (s)

0

2

4

6

8

posit

ion

(mm

)

80 rmin60 rmin

40 rmin20 rmin

(b)


20 40 60 80 1000time (s)

0

20

40

60

80

spee

d (r

min

)

(a)

0

2

4

6

8

10po

sitio

n (m

m)

20 40 60 80 1000time (s)

(b)

20 40 60 80 1000time (s)

0

10

20

30

40

KP

(c)

35

40

45

50

55

60

KD

20 40 60 80 1000time (s)

(d)






6 Complexity

APD-VUFICRPD-BCTPD

20 40 60 80 1000time (s)

0

20

40

60

80

100sp

eed

(rm

in)

(a)

APD-VUFICRPD-BCTPD

0

2

4

6

8

10

posit

ion

(mm

)

20 40 60 80 1000time (s)

(b)



6 Conclusion


Abbreviations















parameters

Complexity 7






Data Availability




Acknowledgments


References




























Journal of












Journal of







Volume 2018






Volume 2018








4 Complexity

VariableUniverse

Adaptation

FuzzyInference


Output

ddt

ddt

e

EHA+

- -

+

ec

Ie

Iec

Op

Od

Kp

Kd

ΔKPΔ

KD


minus1

0

1

minus1

0

1

minus05

0

05

Ie

Iec

OP

(a)minus1

0

1

minus1

0

1

minus05

0

05

IeIec

OD

(b)



5 Simulation



119889119904 = lim119899997888rarrinfin

120579119899 (119905) =


(32)

119889V = 119860 sin (120596119905) 119860 = 10 119908 = 01120587 (33)




Complexity 5

80 rmin60 rmin

40 rmin20 rmin

0

20

40

60

80

spee

d (r

min

)

20 40 60 80 1000time (s)

(a)

0 20 40 60 80 100

10

time (s)

0

2

4

6

8

posit

ion

(mm

)

80 rmin60 rmin

40 rmin20 rmin

(b)


20 40 60 80 1000time (s)

0

20

40

60

80

spee

d (r

min

)

(a)

0

2

4

6

8

10po

sitio

n (m

m)

20 40 60 80 1000time (s)

(b)

20 40 60 80 1000time (s)

0

10

20

30

40

KP

(c)

35

40

45

50

55

60

KD

20 40 60 80 1000time (s)

(d)






6 Complexity

APD-VUFICRPD-BCTPD

20 40 60 80 1000time (s)

0

20

40

60

80

100sp

eed

(rm

in)

(a)

APD-VUFICRPD-BCTPD

0

2

4

6

8

10

posit

ion

(mm

)

20 40 60 80 1000time (s)

(b)



6 Conclusion


Abbreviations















parameters

Complexity 7






Data Availability




Acknowledgments


References




























Journal of












Journal of







Volume 2018






Volume 2018








Complexity 5

80 rmin60 rmin

40 rmin20 rmin

0

20

40

60

80

spee

d (r

min

)

20 40 60 80 1000time (s)

(a)

0 20 40 60 80 100

10

time (s)

0

2

4

6

8

posit

ion

(mm

)

80 rmin60 rmin

40 rmin20 rmin

(b)


20 40 60 80 1000time (s)

0

20

40

60

80

spee

d (r

min

)

(a)

0

2

4

6

8

10po

sitio

n (m

m)

20 40 60 80 1000time (s)

(b)

20 40 60 80 1000time (s)

0

10

20

30

40

KP

(c)

35

40

45

50

55

60

KD

20 40 60 80 1000time (s)

(d)






6 Complexity

APD-VUFICRPD-BCTPD

20 40 60 80 1000time (s)

0

20

40

60

80

100sp

eed

(rm

in)

(a)

APD-VUFICRPD-BCTPD

0

2

4

6

8

10

posit

ion

(mm

)

20 40 60 80 1000time (s)

(b)



6 Conclusion


Abbreviations















parameters

Complexity 7






Data Availability




Acknowledgments


References




























Journal of












Journal of







Volume 2018






Volume 2018








6 Complexity

APD-VUFICRPD-BCTPD

20 40 60 80 1000time (s)

0

20

40

60

80

100sp

eed

(rm

in)

(a)

APD-VUFICRPD-BCTPD

0

2

4

6

8

10

posit

ion

(mm

)

20 40 60 80 1000time (s)

(b)



6 Conclusion


Abbreviations















parameters

Complexity 7






Data Availability




Acknowledgments


References




























Journal of












Journal of







Volume 2018






Volume 2018








Complexity 7






Data Availability




Acknowledgments


References




























Journal of












Journal of







Volume 2018






Volume 2018















Journal of












Journal of







Volume 2018






Volume 2018








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Concise Robust Control of Marine Engine Speed Based on ...downloads.hindawi.com/journals/complexity/2019/5823827.pdf · ResearchArticle Concise Robust Control of Marine Engine Speed