Research ArticleFault-Tolerant Supervisory Control forDynamic Positioning of Ships

Xiaogong Lin Heng Li Kun Liang Jun Nie and Juan Li

College of Automation Harbin Engineering University Harbin 150001 China

Correspondence should be addressed to Heng Li wsqessfoxmailcom

Received 6 August 2018 Accepted 17 December 2018 Published 3 January 2019

Academic Editor A M Bastos Pereira

Copyright copy 2019 Xiaogong Lin et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

A fault-tolerant supervisory control method for dynamic positioning of ships with actuator failures and sensor failures is presentedin this paper Unlike the traditional fault detection and control fault detection and fault-tolerant controller are designed as a unit inthis paper through a supervisor By introducing a nonlinear estimation error and virtual controller the sensor failures are separatedfrom the actuator failures in the supervisory control system It guarantees that the detectability property and matching property ofthe switched system are satisfied Firstly a new extended state observer is designed tomatch themodels of different actuator failuresSecondly by introducing a virtual controller the detectability property of the switched system is guaranteed Finally a nonlinearestimation error operator is used in the designing of switching logic to guarantee stability of the closed-loop system with sensorfailures When sensor failures and actuator failures occur we show that all the states of the closed-loop system are guaranteed to bebounded The effectiveness of the fault-tolerant control is verified by simulation experiments

1 Introduction

It is well known that failures can lead to a significantdegradation in the performance of plant and may even makethe system unstable These considerations provide a strongmotivation for the pursuit of systematic strategies for thedesigning of fault-tolerant controller (FTC) that ensure thestability of the system in case of failures and reduce the riskof safety hazards FTC has attracted much attention in controlcommunity in the last two decades and several methods canbe found in the literatures [1ndash5]

Dynamic positioning system (DPS) can only rely on itsown propulsion to counteract the interference of the exter-nal environment [6 7] Since the highly disturbed marineenvironment will lead to aging of the components whichcauses inevitable malfunctions in actuators measurementsensors and process equipment Therefore how to designa fault-tolerant controller for DPS is a critical problemThe challenges to be met when designing FTC for DPSare complex dynamic behavior due to strong nonlinearities

model uncertainty and environmental force which makesfault process difficult to monitor and control [8ndash10]

Over the last decade some research on FTC for DPShas been carried out Fu and Ning proposed a method foronline control reconfiguration of FTC to dynamic position-ing vessel using disturbance decouplingmethods [11] Andreaand Tor utilized an unknown input observer technique toproduce a fault detection and isolation (FDI) mechanismfor an overactuated marine vessel [5] Fault detection anddiagnosis mechanism based on two techniques the parityspace approach and the Luenberger observer was proposedto guarantee a fault-tolerant robust control for the dynamicpositioning of an overactuated offshore supply vessel [12] Duet al constructed an iterative learning observer to estimatethe fault signal and combined the pseudoinverse method togenerate a fault-tolerant controller for the dynamic position-ing system [13]Other remarkable results onFTC for dynamicpositioning vessels can be found in [14ndash16] So far most ofFTCmethods for DPS are based on the low-frequency modelof the ship and also generally assume that the upper boundof external disturbances is known which is often difficult

HindawiMathematical Problems in EngineeringVolume 2019 Article ID 9134952 11 pageshttpsdoiorg10115520199134952

2 Mathematical Problems in Engineering

to determine in practice Moreover the traditional FTC isbased on the FDImechanism which will reduce the real-timeperformance of the controller when the system is complexenough

Motivated by the above considerations a fault-tolerantsupervisory control method for dynamic positioning of shipswith actuator failures and sensor failures is presented in thispaperWe consider the possible faults as the uncertainty of thesystem and use the supervised switching mechanism insteadof the traditional fault detection and diagnosis mechanismto achieve fault-tolerant control in this paper When faultsoccur the dynamic characteristics of the controlled systemwill change abruptly so the controller can be quickly adaptedthrough the discontinuous changing switching signal Thenovelty of the approach with respect to existing resultsconsists of the following key-points(1) Unlike in [11 13 15] where FTC methods for DPS areonly based on the low-frequency model of the vessel whichlimits the range of applicability To this end in this paper FTCmethod for dynamic positioning vessel with wave-frequencymodel is introduced(2) In [13 14] the authors made restrictive assumptionson the external disturbance for example the upper bound ofdisturbances is known Instead in the proposed work suchrestrictive assumption is not needed(3)Unlike in [5] where FDI mechanism is needed in thispaper FDI mechanism is not needed due to the introductionof injected system and supervisor Besides the number offault actuators in [5] is affected by the structure of actuatormatrix while not in this paper

The remainder of this paper is organized as follows InSection 2 the model of ships is introduced and the structureof supervisory fault-tolerant control system is formulatedSection 31 presents a method to design multiestimator fordynamic positioning vessel based on wave-frequency modelA candidate controller for each estimator is proposed inSection 32 In Section 33 a nonlinear estimation erroroperator is proposed for sensor failures The fault-tolerantsupervisory control system is presented in Section 34 InSection 4 we illustrate the effectiveness of the proposedmethod via simulations on the DP vessel Finally someconcluding remarks are included in Section 5

2 Preliminaries

21 Modeling of Ships For the horizontal motion of a surfacevessel let the earth-fixed position (119909 119910) and the orientation120595 of the vessel relative to an earth-fixed frame 119883119864119884119864119885119864 beexpressed in vector form by 120578 = [119909 119910 120595]119879 and let thevelocities decomposed in a body-fixed reference frame 119883119884119885be represented by the state vector ] = [119906 V 119903]119879 These threemodes are referred to as the surge sway and yaw of a vessel[6 17] The origin of the body-fixed reference frame 119883119884119885is located at the vessel center line in a distance 119909119866 from thecenter of gravity as shown in Figure 1

The transformation between the body-fixed velocity andthe earth-fixed velocity vectors can be written as

120578 = 119869 (120578) ] (1)

XE

OE

(x y)

YEY

X

u

r

Figure 1 Vessel Reference Frames

where 119869(120578) is a rotation matrix In the remainder of the paperthe rotation matrix in yaw is defined as

119869 (120578) = 119869 (120595) = [[[

cos120595 minus sin120595 0sin120595 cos120595 00 0 1

]]]

(2)

In the mathematical modeling of ship dynamics it iscommon to separate the model into a low-frequency modeland wave-frequency model The wave-frequency motion ofthe ship is due to 1st-order wave loads The low-frequencymotion is driven by 2nd-order mean and slowly-varyingwave current wind and thrust forces and the restoring forcesfrom the mooring system [6 18] The total motion of theship is given as the sum of the low-frequency and the wave-frequency contributions

120585 = 119860119882120585 + 119861119882120596119908120578 = 119869 (120595) ]

= minus119860minus1119887 119887 + 119861119887120596119887119872] = 119879119870119879119901 + 119869119879 (120595) 119887 minus 119863] 119870 isin K

119910 = 120578 + 120578119908 + 120596119910

(3)

where 120585 isin R6 is the wave state vector and 119860119882 isin R6times6 119861119882 isinR6times3 are matrices related to the wave peak frequency 1205960119894 119894 =1 2 3 and relative damping ratio 120577119894 119894 = 1 2 3 120578119908 =119862119882120585 isin R3 is the wave frequency motion vector in the earth-fixed frame and 119862119882 isin R3times6 is wave frequency measurementmatrix 119887 isin R3 is a bias term representing slowly varyingenvironmental forces and moment and 119860119887 isin R3times3 119861119887 isinR3times3 are the diagonal matrix of positive bias time constantsand diagonal matrix scaling the amplitude of 120596119887 respectively120596119908 isin R3 120596119887 isin R3 120596119910 isin R3 are vectors of zero-meanGaussian white noise 119872 isin R3times3 is the inertia matrixincluding hydrodynamic added inertia and 119863 isin R3times3 is thedamping matrix 119879 isin R3times119898 119870 isin R119898times119898 and 119879119901 isin R119898 arethruster structure matrix coefficient matrix and thrust forcevector respectively K denotes all possible actuator failuresFor more details please refer to the literature [6 19]

22 Supervisory Control Supervisory control has attractedsignificant research efforts and various approaches have beendeveloped Supervisory control used for uncertain system is

Mathematical Problems in Engineering 3

Monitoring signalGenerator

pep

y

ypMultiminusEstimator

uSwitchingLogic

⨂

Figure 2 Supervisor Architecture

more advantageous than gain scheduling control in terms offlexibility andmodularity In supervisory control one builds abank of alternative candidate controllers and switches amongthem based onmeasurements collected onlineThe switchingis orchestrated by a specially designed logic that uses themeasurements to assess the performance of the candidatecontroller currently in use and also the potential performanceof alternative controllers The need for switching arises fromthe fact that no single candidate controller would be capableby itself of guaranteeing good performance when connectedwith the uncertain process [20ndash25]

In supervisory control system the controller selectionis carried out by a high-level supervisor The supervisor isa hybrid system with continuous states and discrete statesThe behavior of the process is compared with the behaviorof a several admissible process models by the supervisorto determine which model is more likely to describe theactual process Then the candidate controller that is moreadequate for the estimated model is placed in the loop[20 22 25] The supervisor comprises three subsystemsmultiestimator monitoring signal generator and switchinglogic (see Figure 2) where 119906 119910 are input and output of theprocess 119890119901(119901 isin 119875) is the estimation error 120587119901(119901 isin 119875) ismonitoring signal which is suitable function of the estimationerror and 120590 is the control switching signal There are fourimportant properties of supervisory control systems whichare matching property (MP) detectability property (DP)nondestabilization property (NP) and small error property(SP) [22 26]

(i) MPThemultiestimator should be designed such thatat least one of the estimation errors converges to zeroexponentially fast

(ii) DP For every fixed estimatorobserver the totalswitched system must be detectable with respect tothe corresponding estimation error 119890119901 when thecontrol switching signal120590 is frozen at120590 = 120594(119901) where120594 is controller selection function

(iii) SP The SP calls for a bound on 119890120588 in terms ofthe smallest of the signals 119890119901 119901 isin 119875 for a processswitching signal 120588 for which 120590 = 120594(120588)

(iv) NP The control switching signal 120590 is said to havethe NP if it preserves the detectability in a time-varying sense ie the switched system is detectablewith respect to the switched output 119890120588 for a processswitching signal 120588 for which 120590 = 120594(120588)

The stability of the supervisory control system can beguaranteed when the four properties above are satisfied[27]

23 Assumptions and Lemmas Throughout this paper thefollowing assumptions are made

Assumption 1 For low speed applications it can also beassumed that 119872119879 = 119872 gt 0 and 119863 = 119863119879 gt 0Remark 2 Assumption 1 is true if starboard and port sym-metries and low speed are assumed [28]

Assumption 3 In the Lyapunov stability analysis of theobserver some fair assumptions aremade120596119887 = 120596119910 = 120596119908 = 0and 119869(120595) = 119869(120595 + 120595119882)Remark 4 The bias and wave models are driven by zero-mean Gaussian white noise These terms are omitted in theobserver model and Lyapunov stability analysis since theestimator states are driven by the estimation error insteadBesides zero-mean Gaussian white measurement noise isnegligible compared to the wave disturbances 120578119908 [29]Assumption 5 The acceleration vector is available for mea-surement

Remark 6 Recently high performance inertial measurementunits (IMU) are becoming increasingly affordable and inte-grated navigation systems (INS) that integrate IMU and GPSreproduce not only positions but also accelerations with greataccuracy to a reasonable price So it is possible to adoptacceleration feedback in the control systems [30 31]

Before giving the main results we recalled some lemmaswhich will be utilized in the subsequent control developmentand analysis

Lemma 7 (Kalman-Yakubovich-Popov Lemma [32]) Let119866(119904) = 119862(119904119868 minus 119860)minus1119861 + 119863 be transfer function matrix where(119860 119861) is controllable and (119860 119862) is observable Then 119866(119904) isstrictly positive real (SPR) if and only if there exist matrices119875 = 119875119879 gt 0119876 = 119876119879 gt 0 such that

119875119860 + 119860119879119875 = minus119876119861119879119875 = 119862 (4)

Lemma 8 (certainty equivalence stabilization [33]) Forevery fixed 120590 = 120594(119901) the switched system is detectable withrespect to its equilibrium input-output pairs if (i) the processmodel is detectable about each of its equilibrium input-outputpairs and (ii) the injected system is input-to-state stable withrespect to its input

Lemma 9 (scale-independent hysteresis switching [34])Let 119873120590(119905 1199050) be the number of discontinuities of 120590 which isgenerated by hysteresis switching logic in the open interval(1199050 119905) Let 119875 be a finite set with 119898 elements For any 119901 isin 119875we have that

119873120590 (119905 1199050) le 1 + 119898+ 119898log (1 + ℎ) log[

120587119901 (119905)min119901isin119875120587119901 (1199050)]

(5)

int1199051199050

120590 (120591) 119889120591 le 119898[(1 + ℎ) 120587119901 (119905) minusmin119901isin119875

120587119901 (1199050)] (6)

4 Mathematical Problems in Engineering

3 Main Results

Based on the supervisory control system theory in thispaper we design a multiestimator that meets MP to coverall possible actuator failure modes a candidate controllerfor each estimator to make the fixed switched system satisfyDP and a nonlinear estimation error operator for differentsensor faults such that the switching signal satisfies SP andNP

31 Multiestimator for DPS with Actuator Failure By aug-menting a new state and utilizing acceleration feedbacktechnology the extended state observer proposed in thispaper based on (3) will improve disturbance attenuationperformance compared to the traditional one [9 35] Theobserver is designed as follows

120585 = 119860119908120585 + 1198701199101119910 + 1198701205881120588 + 119870]1119869 (120595) ]120578 = 119869 (120595) ] + 1198701199102119910 + 1198701205882120588 + 119870]2119869 (120595) ]119887 = minus119860minus1119887 + 1198701199103119910 + 1198701205883120588 + 119870]3119869 (120595) ]120588 = 1198701199104119910 + 1198701205884120588 + 119870]4119869 (120595) ]

119872 ] = 119879119870119879119901 + 119869119879 (120595) minus 119863] + 119869119879 (120595) 1198701199105119910+ 119869119879 (120595)1198701205885120588 + 119870]5

]

(7)

where 120585 isin R6 isin R3 120578 isin R3 ] isin R3 are the state vectorsof observer 120588 isin R3 is a new extended state 119910 = 119910 minus 119910 isthe estimation error of position ] = ] minus ] is accelerationestimation error and 119870119910119894 119870120588119894 and 119870]119894 119894 isin 1 5 areobserver gain matrices with an appropriate dimension Notethat for different thruster failures the value of the matrix 119870will be different

The acceleration feedback control law is designed as 119879119901 =(119879119870)+(119879119870119879119901 minus 119870]5]) where 119879119901 is called virtual control lawand 119870]5 is the designed matrix such that the new inertiamatrix 119872119870 = 119872 + 119870]5 is a positive definite diagonal matrixThen the velocity dynamics of ship and observer can berewritten as

119872119870] = 119879119870119879119901 + 119869119879119887 minus 119863]

119872119870 ] = 119879119870119879119901 + 119869119879 minus 119863] + 1198691198791198701199105119910 + 1198691198791198701205885120588(8)

Then acceleration estimation error equation can be rewrittenas

119872119870 ] = 119869119879 minus 119863] minus 1198691198791198701199105119910 minus 1198691198791198701205885120588 (9)

Now we can deduce the error dynamics equation accordingto Assumption 3

120585 = (119860119908 + 11987111119862119908) 120585 + 11987111120578 + 11987112120588 minus 11987113+ 119870V1119869119872minus1119870 119863]

120578 = 11987121119862119908120585 + 11987121120578 + 11987122120588 minus 11987123+ (119870V2119869119872minus1119870 119863 + 119869) ]

119887 = 11987131119862119908120585 + 11987131120578 + 11987132120588 minus (11987133 + 119860minus1119887 ) + 119870V3119869119872minus1119870 119863]

120588 = minus11987141119862119908120585 minus 11987141120578 minus 11987142120588 + 11987143 minus 119870V4119869119872minus1119870 119863]

(10)

where 119871 1198941 = 119870]119894119872minus1119870 1198701199105 minus119870119910119894 119871 1198942 = 119870]119894119872minus1119870 1198701205885 minus119870120588119894 119871 1198943 =119870]119894119872minus1119870 Since 119869(120595)2 = radic3 the parameters can alwaysbe selected to make 119870]2119869119872minus1119870 1198632 gtgt 119869(120595)2 Let 119909 =[120585119879 120578119879 119879 120588119879]119879 and then the error dynamics equation canbe written in compact form as follows

119909 = 119860119909 + 119861119869 (120595)119863] (11)

119872119870 ] = minus119863] minus 119869119879 (120595) 119862119909 (12)

where

119860 =[[[[[[

119860119908 + 11987111119862119908 11987111 minus11987113 1198711211987121119862119908 11987121 minus11987123 1198712211987131119862119908 11987131 minus (11987133 + 119860minus1119887 ) 11987132minus11987141119862119908 minus11987141 11987143 minus11987142

]]]]]]

119861 =[[[[[[[

119870V1119872minus1119870119870V2119872minus1119870119870V3119872minus1119870minus119870V4119872minus1119870

]]]]]]]

119862 = [1198701199105119862119908 1198701199105 minus119868 1198701205885]

(13)

Theorem 10 Considering the dynamic positioning vesseldefined by (3) under Assumptions 1 3 5 the proposed observergiven by (7) ensures the exponential convergence of the trackingerrors ie the multiestimator given by (7) satisfies MP

Proof By defining a new variable = 119862119909 the error dynamicscan be rewritten as

= 119860119909 + 119861119869 (120595)119863]

= 119862119909] = minus119872119870119863] minus 119872119870119869119879 (120595)

(14)

Since each element of system matrix 119860 119861 and 119862 containsa gain matrix to be designed it is always possible to designsystem (14) to satisfy Lemma 7 In addition as mentioned in[18] given a set of observer gains the existence of the systemto satisfy Lemma 7 can be checked numerically by using theFrequency Theorem

Mathematical Problems in Engineering 5

Consider the following Lyapunov function candidate

119881 = ]119879119863119879119872119870] + 119909119879119875119909 (15)

If we choose the acceleration feedback to make 119872119870 and119863 commutative then we have 119863119879119872119870 gt 0 according toAssumption 1

The time differentiation of 119881 along the trajectories of ]and 119909 according to Lemma 7 yields

= ]119879119863119879119872119870119863minus1119863] + (119863])119879119872119870119863minus1119863 ] + 2119909119879119875 119909= (minus119872minus1119870 119863] minus119872minus1119870 119869119879119862119909)119879119863119879119872119870119863minus1119863]

+ (119863])119879119872119870119863minus1119863(minus119872minus1119870 119863] minus119872minus1119870 119869119879119862119909)+ 119909119879 (119875119860 + 119860119879119875)119909 + 2119909119879119875119861119869119863]

= minus (119863])119879119872minus1119870 119863119872119870119863minus1119863]

minus (119863])119879 (119863minus1119872119870119863119872minus1119870 +119872119870119863minus1119863119872minus1119870 ) 119869119879119862119909minus (119863])119879119863] minus 119909119879119876119909 + 2 (119863])119879 119869119879119861119879119875119909

= minus2 (119863])119879119863] minus 2 (119863])119879 119869119879 (119862 minus 119861119879119875) 119909 minus 119909119879119876119909= minus2]119879119863119879119863] minus 119909119879119876119909 le minus12058211205822119881

(16)

where 1205821 ≜ min120582min(2119863119879119863) 120582min(119876) and 1205822 ≜max120582max(119863119879119872119870) 120582max(119875)

In conclusion we prove that the estimation error isexponentially convergent Then there must exists at least oneparticular [120585 120578 ]] to provide a good approximation of theoutput [120585 120578 119887 ]] for any actuator failure 119870119901 isin K

32 Candidate Virtual Controller In this subsection a can-didate controller is designed to make the injected systemconsisting of observer (7) and candidate controller itselfinput-to-state stable and ensure the switched system satisfiesDP

By defining 119910 = 119910minus 119890119910 119869(120595)] = 119869(120595) ]minus 119890] and designingan acceleration feedback control law 119879119901 = (119879119870)+(119879119870119879119901 minus119870]5]) where 119879119901 is virtual control law to be designed wecan transform the observer equation (7) into the followinginjected form

119864 = 119860119864 (120595) 119909119864 minus 119861119864 (120595) 119890 + 119861119906119879119901 (17)

where 119909119864 = [120585 120578 120588 ]]119879 119890 = [119890119910 119890]]119879

119860119864 =[[[[[[[[[

119860119908 0 0 1198701205881 00 0 0 1198701205882 1198690 0 minus119860minus1119887 1198701205883 00 0 0 1198701205884 00 0 119872minus1119870 J119879 119872minus1119870 1198691198791198701205885 minus119872minus1119870 119863

]]]]]]]]]

119861119864 =[[[[[[[[[

1198701199101 119870V1

1198701199102 119870V2

1198701199103 119870V3

1198701199104 119870V4

119872minus1119870 1198691198791198701199105 0

]]]]]]]]]

119861119906 =[[[[[[[[[

1198746times11198743times11198743times11198743times1

119872minus1119870 119879119870

]]]]]]]]]

(18)

Note that the inertia of vessel is very large and the accelerationcan be omitted hence it is reasonable to select accelerationas the input of the injected system Now we need to designcontrol law to make the injected system (17) input-to-statestable with respect to the input 119890 Note that the state 119909119864 isavailable for control which is reasonable because multiesti-mator is implemented by the controller designer Then wechoose the virtual control law as follows

119879119901 = (119872minus1119870 119879119870)+ (119880120585 diag (119869119879 119869119879) 120585 + 119880120578119869119879120578+ (119880119887 minus119872minus1119870 ) 119869119879 + (119880120588119869119879 minus119872minus1119870 1198691198791198701205885) 120588+ (119872minus1119870 119863 + 119880]) ] +119872minus1119870 1198691198791198701199105 (119910 minus 119910)

(19)

Theorem 11 Consider the injected system consisting ofobserver (7) and candidate controller 119879119901 = (119879119870)+(119879119870119879119901 minus119870]5]) where 119879119901 is chosen as (19) The switched system whichcontains injected system and plant will satisfy DP

Proof When the observer gain matrices are selected as thefollowing form 1198961199011 = diag(diag(119886 119886 119887) diag(119888 119888 119889)) 119870119901119894 =diag(119890 119890 119891)(119894 isin 2 3 4) one assumed that the bias timeconstants 119860119887 has the same form and wave peak frequency1205960119894 in surge and sway are the same so does relative dampingratio 120577119894 Then replace the virtual control law into the orginalinjected system (17) and we have

119864 = 119882119879 (120595)119860119882(120595) 119909119864 minus 119861 (120595) 119890 (20)

6 Mathematical Problems in Engineering

where119882(120595) = diag(119869119879(120595) 119869119879(120595) 1198683times3)

119860 =[[[[[[[[[

119860119908 0 0 1198701205881 00 0 0 1198701205882 1198680 0 minus119860minus1119887 1198701205883 00 0 0 1198701205884 0119880120585 119880120578 119880119887 119880120588 119880V

]]]]]]]]]

119861 =[[[[[[[[[

1198701199101 119870]1

1198701199102 119870]2

1198701199103 119870]3

1198701199104 119870]4

0 0

]]]]]]]]]

(21)

The control gains and observer gains which appear in 119860are chosen such that the matrix 119860 is asymptotically stableThen there exists a matrix 119875 = 119875119879 gt 0 such that 119875119860 + 119860119879119875 =minus119876 for a given matrix 119876 = 119876119879 gt 0 Besides it should benoted that the injected system (20) is input-to-state stable ifthe system = 119882119879(120595)119860119882(120595)119911 is asymptotically stable [27]Differentiate 119882(120595)119911 along the trajectories 119911 and we have

119889 (119882 (120595) 119911)119889119905 = 119878119882 (120595) 119911 + 119860119882(120595) 119911 (22)

where 119878 = diag(119878119879 119878119879 0)

119878 = [[[

0 minus1 01 0 00 0 0

]]]

(23)

Consider a Lyapunov function candiate

119881 (119911) = 119911119879119882119879 (120595) 119875119882(120595) 119911 (24)

Differentiate it119889119881119889119905 = [(119878 + 119860)119882 (120595) 119911]119879 119875119882(120595) 119911 + (119882 (120595) 119911)119879

sdot 119875 [(119878 + 119860)119882(120595) 119911]119879 = (119882 (120595) 119911)119879sdot [(119860119879 + 119878119879) 119875 + 119875 (119878 + 119860)]119882 (120595) 119911= (119882 (120595) 119911)119879 [minus119876 + 119903119878119879119875 + 119903119875119878]119882(120595) 119911le [minus120582min (119876) + 2119903max120582max (119875)] 1003817100381710038171003817119882 (120595) 11991110038171003817100381710038172

(25)

When 119903max is sufficiently small the system =119882119879(120595)119860119882(120595)119911 is asymptotically stable Then the injectedsystem (20) is input-to-state stable with respect to 119890

Obviously when there is no sensor failure the dynamicpositioning vessel satisfies detectability which means thatthe state of plant eventually becomes small if the inputsand outputs are small Then according to Lemma 8 we candeduce that the switched system satisfies the DP

33 NEEO for Different Sensor Failures Sensor failures willdamage the detectability of the vessel In order to illustrate theproblem the following possible sensor failures are consideredin this paper

119910 = 119892 ∘ ℎ (119909) (26)

where 119892 the sensor fault function satisfies 119892 ∘ ℎ(0) = 0ie the detectability of plant is not satisfied The detectabilityof switched system will not be guaranteed if we designsupervisor in traditional wayOne reason is that themeasuredoutput is no longer a true reflection of the actual value

In this paper a nonlinear estimation error operator(NEEO) is introduced to solve the above problems Firstlythe process model (3) is remodeled as follows

= 119891 (119909 119879119901)119910 =

119910119899 = ℎ (119901 119909) no sensor failure

119910119891 = 119892119895 ∘ ℎ (119901 119909) 119895119905ℎ sensor failure(27)

where 119909 = [120585119879 120578119879 119887119879 ]119879]119879 For convenience of analysis sup-pose that 119894 isin 119875 119895 isin 119870 which represents the different actuatorfailures and sensor failures respectively The multiestimatoris designed for process (27)

119909119864119894 = 119865119894 (119909119864119894 119879119901 119910)119910119894 = ℎ119894 (119909119864119894)

119864119895 = 119865119895 (119909119864119895 119879119901 119910) = 119865119894 (119909119864119895 119879119901 119892minus1119895 (119910))119910119895 = ℎ119894 (119909119864119895) 119895 = 119875 times 119896 + 119894 119894 isin 119875

(28)

where each 119865119894 119894 isin 119875 is the observer based on actuator failurejust like Section 31

Theorem 10 has proved that the estimator can track thetrue output 119892minus1119895 (119910) so we define the nonlinear estimationerror as 119890119901 = 119910119901 ⊟ 119910 where ⊟ is NEEO and the specific formis as follows

119890119901 = 119910119901 ⊟ 119910 = 119890119894 = 119910119894 minus 119910 119894 isin 119875119890119895 = 119910119895 minus 119892minus1119895 (119910) 119895 isin 119870 (29)

No matter if the sensor is fault or not there will always bea particular 119890119901lowast that is convergent which means the multi-estimator satisfies MP Note that although the nonlinearityof 119890119901 is caused by the existence of 119892 it does not affectthe design of the front candidate controller (19) Moreoversensor failures only change the controllerrsquos external inputwithout any changing in the internal structureTherefore thefollowing result is concluded

Theorem 12 Consider the dynamic positioning vessel de-scribed by (3) under Assumptions 1 3 5 the multiestimatorgiven by (28) satisfiesMP and the switched systemalso satisfiesDP

Remark 13 Obviously to solve the problem of detectabilityloss caused by sensor faults the fault information is neededwhich can be achieved through fault reconstruction [36]

Mathematical Problems in Engineering 7

y

y

y

y

y

SwitchingLogic

MonitoringSignal

p

epyp

xE

xE

Controller(1)

Controller(2)

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

Controller Set

Observer(1)

Observer(k)

Observer(k + 1)

Observer(k + (P))

Multi minus Estimator

u

u

ActuatorFailure

Actuator

b

y

Disturbances SensorFailure

Sensor

Supervisor

yp

Figure 3 Fault-Tolerant Supervisory Control with NEEO for DPS

Initialize (t0)

(1 + ℎ) p(t) le (t)

Yes

No

(t) = ＬＡＧＣＨ p

Figure 4 Scale-Independent Hysteresis Switching Logic

34 Supervisory Control for Fault Tolerant Taking intoaccount the contents of all three previous parts this sectionemploys the scale-independent hysteresis switching logic tocontrol the DPS with actuator failures and sensor failuresFigure 3 shows the structure of the proposed supervi-sory control system The multiestimator comprises ((119875) times(119870)) models and the controller set comprises (119875) can-didate controllers corresponding to the same (119875) actuatorfailures

The hysteresis switching logic will be used to slow downswitching based on the observed growth of the estima-tion errors [37] Figure 4 shows the architecture of scale-independent hysteresis switching logic where 120587119901 physicallyis a monitoring signal in terms of the error norm as follows[22]

120587119901 = 120598 + 119890minus1205821199051205980 + int1199050119890minus120582(119905minus120591)120574 (10038171003817100381710038171003817119890119901 (120591)10038171003817100381710038171003817) 119889120591 (30)

where 120598 1205980 are nonnegative constants of which at least oneof them is strictly positive and 120582 is a constant nonnegativeforgetting factor and 120574 is class119870 function

Theorem 14 Consider the dynamic positioning vessel (27) themultiestimator is designed as (28) the candidate accelerationfeedback control law is designed as 119879119901 = (119879119870)+(119879119870119879119901 minus119870]5])with a virtual control law 119879119901 (19) and the NEEO is chosenas (29) Then under the supervisor with the scale-independenthysteresis switching logic all states of the DPS are bounded nomatter if the failures occur or not

Proof The scaled version monitoring signal is defined as120587119901(119905) = 120587119901(119905)119890120582119905 1205980 = 0 Based on Theorem 12 multiesti-mator satisfies MP so there exists 1198881 gt 0 such that 119890lowast119901 lt 1198881Since the function 120574 is Lipschiz continuous then

120587lowast119901 (119905) = 120598119890120582119905 + int1199050119890120582119904120574 (10038171003817100381710038171003817119890lowast119901 (119904)10038171003817100381710038171003817) 119889119904

le 120598119890120582119905 + 120574 (1198881) 1120582 119890120582119904100381610038161003816100381610038161199050 le 119890120582119905 [120598 + 120574 (1198881)120582 ]le 1198901205821199051198882

(31)

Using Lemma 9 we have

119873120590 le 119898 + 119898ln (1 + ℎ) ln( 120587119902 (119905)

min120587119901 (1199050))

le 119898 + 119898ln (1 + ℎ) ln[11988821198901205821199051205981198901205821199050 ]

le 119898 + 119898ln (1 + ℎ) ln(1198882120598 ) + 119905 minus 1199050

ln (1 + ℎ) 120582119898

(32)

8 Mathematical Problems in Engineering

int1199050119890120582119904120574 (1003816100381610038161003816119890120590(119904) (119904)1003816100381610038161003816) 119889119904 + 120598119890120582119905 minus 120598le 119898 [(1 + ℎ) 120583119902 (119905) minusmin 120583119901 (1199050)]le 119898 (1 + ℎ) 1198882119890120582119905

(33)

Then we can choose ln(1 + ℎ)120582119898 gt ln120583(120582119900 minus 120582) such thatthe switching signal stasfities average dwell-time [38] Sincethe choice of the virtual control law ensures the input-to-state stability of the injected system then the state of switchedinjected system 119909119864 has the property (Theorem 31[25])

1198901205821199051205721 (1003817100381710038171003817119909119864 (119905)1003817100381710038171003817) le 1198881205722 (1003817100381710038171003817119909119864 (0)1003817100381710038171003817)+ 119888 int1199050119890120582119904120574 (1003817100381710038171003817119890120590 (119904)1003817100381710038171003817) 119889119904

(34)

Substitute (33) into (34) and then1003817100381710038171003817119909119864 (119905)1003817100381710038171003817 le 120572minus11 (1198881205722 (1003817100381710038171003817119909119864 (0)1003817100381710038171003817) + 119888119898 (1 + ℎ) 1198882) = 1198883 (35)

which means that the state of injected system is boundedThen the output of injected system119910119901 = ℎ119901(119909119864) (forall119901 isin 119875) andswitched thrust force 119879119901120590 are bounded too Since 119892minus1(119910) =119910119901lowast minus 119890119901lowast then exist119901lowast isin 119875 119892minus1(119910) is bounded too Then by thedetectability of surface vessel with respect to the 119892minus1(119910) ieℎ(0) = 0 we candeduce that the state of plant is bounded4 Simulation Research

A model of a DP vessel with 4 channel thrusters and 2 mainthrusters is used in the simulation to demonstrate the perfor-mance of the supervisory control in case of actuator failuresand sensor failures For simplicity we choose four types ofactuator stuck failures K ≜ 1198701198911 1198701198912 1198701198913 1198701198914 where1198701198911 1198701198912 denote the single actuator failure and 1198701198913 1198701198914denote the multiple actuator failure The simulation param-eters are given in Table 1

As shown in Table 2 only actuator failures are consideredin Case 1 and Case 2 and both actuator failures and sensorfailures are considered in Case 3 The expected position andheading vectors are taken as 120578119889 = [0119898 0119898 0119903119886119889]119879 andthe initial states are taken as 120578(1199050) = [20119898 20119898 1119903119886119889]119879Simulation results of the proposed fault-tolerant controlstrategy are depicted in Figures 5ndash9

It can be observed from Figure 5 that the accelerationestimation errors are very small enough to be ignored So itis reasonable to take the acceleration estimation errors as aninput of the injected system

To illustrate the performance of the proposed controlscheme comparison results are presented by applying theFTC based on FDI in [5] The initial states for both controlschemes are chosen to be the same to guarantee a faircomparison As shown in the detailed time responses ofFigure 7 both methods can stabilize the vesselrsquos state to thereference value with small bounded errors but the methodin [5] cannot deal with high frequency wave force very wellwhich is the weakness of the control based on low-frequency

Table 1 Main simulation parameters

Symbol Parameter valuesM 107 times [3640004352490249752095]D 106 times [322000322-2780-27881506]119860119887 diag[100010001000]119861119887 diag[5000056000500000]119860119908(21) diag[-081-081-081]119860119908(22) diag[-018-018-018]119860119908 [03times3 1198683times3 119860119908(21) 119860119908(22)]119861119908 [[03times3 diag[16 12 025]]1198701198911 diag[011111]1198701198912 diag[111101]1198701198913 diag[011010]1198701198914 diag[101011]1198921 1198921(119904 119905) = 2119904 + 10 sin(119905)119879 [000011111100504418-357-4545-45]

minus015minus01

minus0050

005

Acceleration estimation error du (Case1)

minus1minus05

005

Acceleration estimation error dv (Case1)

minus004minus002

0002

Acceleration estimation error dr (Case1)

minus101

minus101

150 250200

150 250200

024

0 50045040035030025020015010050time (s)

0 50045040035030025020015010050time (s)

0 50045040035030025020015010050time (s)

150 250200

times10minus3

times10minus3

times10minus5

(ms

)

(ms

)

(rad

s

)

Figure 5 Acceleration Estimation Error

modelTherefore the proposed fault-tolerant controlmethodis more effective for the dynamic positioning ships in thepresence of high frequency wave disturbance

Remark 15 The upper bound on the maximum number ofsimultaneously isolable actuator faults in [5] is determined bythe Uniform Sub-Rank of the matrix which is related to thecontrol input matrix 119861 and thruster matrix 119879 By calculationwe can deduce that themethod in [5] can only solve the singleactuator fault So only Case 1 is considered in the comparisonpart

Mathematical Problems in Engineering 9

Table 2 Condition of faults

Time(s) 0-100 100-250 250-300 300-500Case 1 119870 = 1198701198911 119870 = 1198701198912 119870 = 1198701198912 119870 = 1198701198911Case 2 119870 = 1198701198913 119870 = 1198701198914 119870 = 1198701198914 119870 = 1198701198913Case 3 119870 = 1198701198913 119870 = 1198701198914 119870 = 1198701198914 119892 = 1198921 119870 = 1198701198913 119892 = 1198921

minus20

0

20

(m)

1

15

2

sigm

a

minus20minus10

01020

(m)

1

15

2sig

ma

minus1minus05

005

115

(rad

)

1

15

2

sigm

a

0 50045040035030025020015010050time (s)

0 50045040035030025020015010050time (s)

0 50045040035030025020015010050time (s)

motion in the X direction(Case1)switching signal

motion in the Y direction(Case1)switching signal

rotation about the Z axis(Case1)switching signal

Figure 6 Position and Heading of DP Vessel (Case 1)

Control performances of Case 2 and Case 3 wheremultiple actuator failures occur are demonstrated in Figures8 and 9 respectively Firstly it is shown that the signal 120590 canswitch to the corresponding controller quickly and accuratelyin the presence of actuator failures Secondly when a sensorfailure occurs at 250119904 although there is a slight fluctuation inthe transition progress the signal 120590 can switch to the stablecontroller and positions of the vessel are stabilized near theequilibrium point eventually

5 Conclusion

This paper presented a fault-tolerant supervisory controlmethod for dynamic positioning ships with wave-frequencymodel The proposed fault-tolerant control method does notneed the FDI mechanism due to the introduction of injectedsystem and supervisor It has beenproved that by introducinga nonlinear estimation error and virtual controller thedetectability property and matching property of the switchedsystem are guaranteed in the presence of actuator failuresand sensor failures Simulation results and comparisons havedemonstrated the effectiveness of the proposed fault-tolerant

minus20

0

20

40

(m)

motion in the X direction(S2)(Case1)motion in the X direction(S1)(Case1)

minus20

0

20

40

(m)

motion in the Y direction(S2)(Case1)motion in the Y direction(S1)(Case1)

minus1

0

1

2

(rad

)

rotation about the Z axis(S2)(Case1)rotation about the Z axis(S1)(Case1)

minus100

1020

minus100

1020

0 2010

0 2010

0 2010

minus050

051

15

0 50045040035030025020015010050time (s)

0 50045040035030025020015010050time (s)

0 50045040035030025020015010050time (s)

Figure 7 The tracking performance comparison of the pro-posed method (S1)(Case 1) and fault-tolerant control allocation [5](S2)(Case 1)

control scheme In the future we will consider the unknownfaults which is more practical in realistic situation Furtherwe will improve the switching logic to make the switchingsignal more accurate and stable for more serious failure

Data Availability

All the data supporting the conclusions of the study have beenprovided in Simulation and readers can access these data in[27 29]

Conflicts of Interest

The authors declare that there are no conflicts of interestrelated to this paper

Acknowledgments

This research was partially supported by the National Sci-ence Technology Support Program of China (Project no51609046)

10 Mathematical Problems in Engineering

0 50045040035030025020015010050time (s)

0 50045040035030025020015010050time (s)

0 50045040035030025020015010050time (s)

minus20

0

20

(m)

1

15

2

sigm

a

motion in the X direction(Case2)switching signal

minus20minus10

01020

(m)

1

15

2

sigm

a

motion in the Y direction(Case2)switching signal

minus1minus05

005

115

(rad

)

1

15

2

sigm

a

rotation about the Z axis(Case2)switching signal

Figure 8 Position and Heading of DP Vessel (Case 2)

0 50045040035030025020015010050time (s)

0 50045040035030025020015010050time (s)

0 50045040035030025020015010050time (s)

minus20minus10

01020

(m)

1234

sigm

a

motion in the x direction(Case3)switching signal

0(m)

2

4

sigm

a

motion in the y direction(Case3)switching signal

minus1

0

1

(rad

)

1234

sigm

a

rotation about the z axis(Case3)switching signal

Figure 9 Position and Heading of DP Vessel (Case 3)

References

[1] H Azmi and M J Khosrowjerdi ldquoRobust adaptive faulttolerant control for a class of Lipschitz nonlinear systems withactuator failure and disturbancesrdquo Proceedings of the Institutionof Mechanical Engineers Part I Journal of Systems and ControlEngineering vol 230 no 1 pp 13ndash22 2016

[2] S Dai L and J Zhao ldquoReliable H8 controller design for a classof uncertain linear systems with actuator failuresrdquo InternationalJournal of Control Automation amp Systems vol 6 no 6 pp 954ndash959 2008

[3] Y L Hao and G H Yang ldquoRobust fault tolerant controlbased on sliding mode method for uncertain linear systemswith quantizationrdquo Proceedings of the Institution of MechanicalEngineers Part I Journal of Systems amp Control Engineering vol52 no 5 pp 692ndash703 2013

[4] Z Zuo D W C Ho and Y Wang ldquoFault tolerant controlfor singular systems with actuator saturation and nonlinearperturbationrdquo Automatica vol 46 no 3 pp 569ndash576 2010

[5] A Cristofaro and T A Johansen ldquoFault tolerant controlallocation using unknown input observersrdquoAutomatica vol 50no 7 pp 1891ndash1897 2014

[6] T I Fossen Handbook of Marine Craft Hydrodynamics andMotion Control 2011

[7] A J Soslashrensen ldquoA survey of dynamic positioning controlsystemsrdquo Annual Reviews in Control vol 35 no 1 pp 123ndash1362011

[8] J Liu R Allen andH Yi ldquoShipmotion stabilizing control usinga combination of model predictive control and an adaptiveinput disturbance predictorrdquo Proceedings of the Institution ofMechanical Engineers Part I Journal of Systems and ControlEngineering vol 225 no 5 pp 591ndash602 2011

[9] M Tomera ldquoNonlinear observers design for multivariable shipmotion controlrdquo Polish Maritime Research vol 19 pp 50ndash562012

[10] A Hajivand and S H Mousavizadegan ldquoThe effect of memoryin observer design for a dp systemrdquo in Proceedings of theASME 2010 International Mechanical Engineering Congress andExposition IMECE 2010 pp 53ndash59 Canada November 2010

[11] M Fu J Ning and Y Wei ldquoFault-tolerant control of dynamicpositioning vessel after thruster failures using disturbance

Mathematical Problems in Engineering 11

decoupling methodsrdquo in Proceedings of the 2011 IEEE Interna-tional Conference on Automation and Logistics ICAL 2011 pp433ndash437 China August 2011

[12] F Benetazzo G Ippoliti S Longhi and P Raspa ldquoAdvancedcontrol for fault-tolerant dynamic positioning of an offshoresupply vesselrdquoOcean Engineering vol 106 pp 472ndash484 2015

[13] Y Lin and J Du ldquoFault-tolerant control for dynamic positioningof ships based on an iterative learning observerrdquo in Proceedingsof the 35th Chinese Control Conference CCC 2016 pp 1116ndash1122China July 2016

[14] M Chen B Jiang and R Cui ldquoActuator fault-tolerant controlof ocean surface vessels with input saturationrdquo InternationalJournal of Robust amp Nonlinear Control vol 26 no 3 pp 542ndash564 2016

[15] Y Su C Zheng andPMercorelli ldquoNonlinear PDFault-TolerantControl for Dynamic Positioning of Ships With ActuatorConstraintsrdquo IEEEASME Transactions onMechatronics vol 22no 3 pp 1132ndash1142 2017

[16] W-Z Yu H-X Xu and H Feng ldquoRobust adaptive fault-tolerant control of dynamic positioning vessel with position ref-erence system faults using backstepping designrdquo InternationalJournal of Robust and Nonlinear Control vol 28 no 2 pp 403ndash415 2018

[17] N T Dong Design of hybrid marine control systems for dynamicpositioning [PhD thesis] 2006

[18] J P Strand and T I Fossen ldquoNonlinear passive observer designfor ships with adaptive wave filteringrdquo in New Directions inNonlinear Observer Design vol 244 of Lecture Notes in Controland Information Sciences pp 113ndash134 Springer London UK1999

[19] S Saelid J G Balchen and N A Jenssen ldquoDesign and analysisof a dynamic positioning system based on kalman filtering andoptimal controlrdquo IEEE Transactions on Automatic Control vol28 no 3 pp 331ndash339 1983

[20] G Battistelli J P Hespanha and P Tesi ldquoSupervisory control ofswitched nonlinear systemsrdquo International Journal of AdaptiveControl and Signal Processing vol 26 no 8 pp 723ndash738 2012

[21] I R Bertaska andK DVonEllenrieder ldquoSupervisory switchingcontrol of an unmanned surface vehiclerdquo in Proceedings of theMTSIEEE Washington OCEANS 2015 USA October 2015

[22] J P Hespanha ldquoTutorial on supervisory controlrdquo Overview2002

[23] J P Hespanha D Liberzon and A S Morse ldquoSupervision ofintegral-input-to-state stabilizing controllersrdquo Automatica vol38 no 8 pp 1327ndash1335 2002

[24] M Skogvold Supervisory-switched control for dynamic position-ing systems in arctic areas [Msc thesis] Department of MarineTechnology 2010

[25] L VuD Chatterjee andD Liberzon ldquoInput-to-state stability ofswitched systems and switching adaptive controlrdquo Automaticavol 43 no 4 pp 639ndash646 2007

[26] ED Sontag ldquoInput to state stability basic concepts and resultsrdquoinNonlinear andOptimal ControlTheory pp 163ndash220 SpringerHeidelberg Berlin Germany 2008

[27] T D Nguyen A J Soslashrensen and S T Quek ldquoDesign of hybridcontroller for dynamic positioning from calm to extreme seaconditionsrdquo Automatica vol 43 no 5 pp 768ndash785 2007

[28] T I Fossen ldquoMarine Control Systems Guidance Navigationand Control of Ships Rigs and Underwater Vehiclesrdquo 2002

[29] T I Fossen and J P Strand ldquoPassive nonlinear observer designfor ships using Lyapunov methods full-scale experiments witha supply vesselrdquo Automatica vol 35 no 1 pp 3ndash16 1999

[30] G Xia X Shao H Wang et al ldquoPassive nonlinear observerdesign for special structure vesselsrdquoOceans IEEE pp 1ndash8 2013

[31] K P Lindegaard Acceleration Feedback in Dynamic Position-ing Fakultet for Informasjonsteknologi Matematikk Og Elek-troteknikk 2012

[32] H K Khalil Nonlinear Systems Prentice-Hall Upper SaddleRiver NJ USA 3rd edition 2002

[33] J P Hespanha and A S Morse ldquoCertainty equivalence impliesdetectabilityrdquo Systems amp Control Letters vol 36 no 1 pp 1ndash131999

[34] J P Hespanha D Liberzon and A S Morse ldquoBounds onthe number of switchings with scale-independent hysteresisApplications to supervisory controlrdquo Proceedings of the IEEEConference on Decision and Control vol 4 pp 3622ndash3627 2000

[35] M H Kim and D J Inman ldquoDevelopment of a robust non-linear observer for dynamic positioning of shipsrdquo Proceedings ofthe Institution of Mechanical Engineers Part I Journal of Systemsamp Control Engineering vol 218 no 1 pp 1ndash11 2004

[36] H Alwi C Edwards and C P Tan Fault Detection and Fault-Tolerant Control Using Sliding Modes Advances in IndustrialControl Springer London UK 2011

[37] J P Hespanha D Liberzon and A S Morse ldquoHysteresis-based switching algorithms for supervisory control of uncertainsystemsrdquo Automatica vol 39 no 2 pp 263ndash272 2003

[38] J P Hespanha and A S Morse ldquoStability of switched systemswith average dwell-timerdquo in Proceedings of the 38th IEEEConference on Decision and Control (CDC rsquo99) pp 2655ndash2660Phoenix Ariz USA December 1999

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Mathematical Problems in Engineering 3

Monitoring signalGenerator

pep

y

ypMultiminusEstimator

uSwitchingLogic

⨂

Figure 2 Supervisor Architecture

more advantageous than gain scheduling control in terms offlexibility andmodularity In supervisory control one builds abank of alternative candidate controllers and switches amongthem based onmeasurements collected onlineThe switchingis orchestrated by a specially designed logic that uses themeasurements to assess the performance of the candidatecontroller currently in use and also the potential performanceof alternative controllers The need for switching arises fromthe fact that no single candidate controller would be capableby itself of guaranteeing good performance when connectedwith the uncertain process [20ndash25]

In supervisory control system the controller selectionis carried out by a high-level supervisor The supervisor isa hybrid system with continuous states and discrete statesThe behavior of the process is compared with the behaviorof a several admissible process models by the supervisorto determine which model is more likely to describe theactual process Then the candidate controller that is moreadequate for the estimated model is placed in the loop[20 22 25] The supervisor comprises three subsystemsmultiestimator monitoring signal generator and switchinglogic (see Figure 2) where 119906 119910 are input and output of theprocess 119890119901(119901 isin 119875) is the estimation error 120587119901(119901 isin 119875) ismonitoring signal which is suitable function of the estimationerror and 120590 is the control switching signal There are fourimportant properties of supervisory control systems whichare matching property (MP) detectability property (DP)nondestabilization property (NP) and small error property(SP) [22 26]

(i) MPThemultiestimator should be designed such thatat least one of the estimation errors converges to zeroexponentially fast

(ii) DP For every fixed estimatorobserver the totalswitched system must be detectable with respect tothe corresponding estimation error 119890119901 when thecontrol switching signal120590 is frozen at120590 = 120594(119901) where120594 is controller selection function

(iii) SP The SP calls for a bound on 119890120588 in terms ofthe smallest of the signals 119890119901 119901 isin 119875 for a processswitching signal 120588 for which 120590 = 120594(120588)

(iv) NP The control switching signal 120590 is said to havethe NP if it preserves the detectability in a time-varying sense ie the switched system is detectablewith respect to the switched output 119890120588 for a processswitching signal 120588 for which 120590 = 120594(120588)

The stability of the supervisory control system can beguaranteed when the four properties above are satisfied[27]

23 Assumptions and Lemmas Throughout this paper thefollowing assumptions are made

Assumption 1 For low speed applications it can also beassumed that 119872119879 = 119872 gt 0 and 119863 = 119863119879 gt 0Remark 2 Assumption 1 is true if starboard and port sym-metries and low speed are assumed [28]

Assumption 3 In the Lyapunov stability analysis of theobserver some fair assumptions aremade120596119887 = 120596119910 = 120596119908 = 0and 119869(120595) = 119869(120595 + 120595119882)Remark 4 The bias and wave models are driven by zero-mean Gaussian white noise These terms are omitted in theobserver model and Lyapunov stability analysis since theestimator states are driven by the estimation error insteadBesides zero-mean Gaussian white measurement noise isnegligible compared to the wave disturbances 120578119908 [29]Assumption 5 The acceleration vector is available for mea-surement

Remark 6 Recently high performance inertial measurementunits (IMU) are becoming increasingly affordable and inte-grated navigation systems (INS) that integrate IMU and GPSreproduce not only positions but also accelerations with greataccuracy to a reasonable price So it is possible to adoptacceleration feedback in the control systems [30 31]

Before giving the main results we recalled some lemmaswhich will be utilized in the subsequent control developmentand analysis

Lemma 7 (Kalman-Yakubovich-Popov Lemma [32]) Let119866(119904) = 119862(119904119868 minus 119860)minus1119861 + 119863 be transfer function matrix where(119860 119861) is controllable and (119860 119862) is observable Then 119866(119904) isstrictly positive real (SPR) if and only if there exist matrices119875 = 119875119879 gt 0119876 = 119876119879 gt 0 such that

119875119860 + 119860119879119875 = minus119876119861119879119875 = 119862 (4)

Lemma 8 (certainty equivalence stabilization [33]) Forevery fixed 120590 = 120594(119901) the switched system is detectable withrespect to its equilibrium input-output pairs if (i) the processmodel is detectable about each of its equilibrium input-outputpairs and (ii) the injected system is input-to-state stable withrespect to its input

Lemma 9 (scale-independent hysteresis switching [34])Let 119873120590(119905 1199050) be the number of discontinuities of 120590 which isgenerated by hysteresis switching logic in the open interval(1199050 119905) Let 119875 be a finite set with 119898 elements For any 119901 isin 119875we have that

119873120590 (119905 1199050) le 1 + 119898+ 119898log (1 + ℎ) log[

120587119901 (119905)min119901isin119875120587119901 (1199050)]

(5)

int1199051199050

120590 (120591) 119889120591 le 119898[(1 + ℎ) 120587119901 (119905) minusmin119901isin119875

120587119901 (1199050)] (6)

4 Mathematical Problems in Engineering

3 Main Results

Based on the supervisory control system theory in thispaper we design a multiestimator that meets MP to coverall possible actuator failure modes a candidate controllerfor each estimator to make the fixed switched system satisfyDP and a nonlinear estimation error operator for differentsensor faults such that the switching signal satisfies SP andNP

31 Multiestimator for DPS with Actuator Failure By aug-menting a new state and utilizing acceleration feedbacktechnology the extended state observer proposed in thispaper based on (3) will improve disturbance attenuationperformance compared to the traditional one [9 35] Theobserver is designed as follows

120585 = 119860119908120585 + 1198701199101119910 + 1198701205881120588 + 119870]1119869 (120595) ]120578 = 119869 (120595) ] + 1198701199102119910 + 1198701205882120588 + 119870]2119869 (120595) ]119887 = minus119860minus1119887 + 1198701199103119910 + 1198701205883120588 + 119870]3119869 (120595) ]120588 = 1198701199104119910 + 1198701205884120588 + 119870]4119869 (120595) ]

119872 ] = 119879119870119879119901 + 119869119879 (120595) minus 119863] + 119869119879 (120595) 1198701199105119910+ 119869119879 (120595)1198701205885120588 + 119870]5

]

(7)

where 120585 isin R6 isin R3 120578 isin R3 ] isin R3 are the state vectorsof observer 120588 isin R3 is a new extended state 119910 = 119910 minus 119910 isthe estimation error of position ] = ] minus ] is accelerationestimation error and 119870119910119894 119870120588119894 and 119870]119894 119894 isin 1 5 areobserver gain matrices with an appropriate dimension Notethat for different thruster failures the value of the matrix 119870will be different

The acceleration feedback control law is designed as 119879119901 =(119879119870)+(119879119870119879119901 minus 119870]5]) where 119879119901 is called virtual control lawand 119870]5 is the designed matrix such that the new inertiamatrix 119872119870 = 119872 + 119870]5 is a positive definite diagonal matrixThen the velocity dynamics of ship and observer can berewritten as

119872119870] = 119879119870119879119901 + 119869119879119887 minus 119863]

119872119870 ] = 119879119870119879119901 + 119869119879 minus 119863] + 1198691198791198701199105119910 + 1198691198791198701205885120588(8)

Then acceleration estimation error equation can be rewrittenas

119872119870 ] = 119869119879 minus 119863] minus 1198691198791198701199105119910 minus 1198691198791198701205885120588 (9)

Now we can deduce the error dynamics equation accordingto Assumption 3

120585 = (119860119908 + 11987111119862119908) 120585 + 11987111120578 + 11987112120588 minus 11987113+ 119870V1119869119872minus1119870 119863]

120578 = 11987121119862119908120585 + 11987121120578 + 11987122120588 minus 11987123+ (119870V2119869119872minus1119870 119863 + 119869) ]

119887 = 11987131119862119908120585 + 11987131120578 + 11987132120588 minus (11987133 + 119860minus1119887 ) + 119870V3119869119872minus1119870 119863]

120588 = minus11987141119862119908120585 minus 11987141120578 minus 11987142120588 + 11987143 minus 119870V4119869119872minus1119870 119863]

(10)

where 119871 1198941 = 119870]119894119872minus1119870 1198701199105 minus119870119910119894 119871 1198942 = 119870]119894119872minus1119870 1198701205885 minus119870120588119894 119871 1198943 =119870]119894119872minus1119870 Since 119869(120595)2 = radic3 the parameters can alwaysbe selected to make 119870]2119869119872minus1119870 1198632 gtgt 119869(120595)2 Let 119909 =[120585119879 120578119879 119879 120588119879]119879 and then the error dynamics equation canbe written in compact form as follows

119909 = 119860119909 + 119861119869 (120595)119863] (11)

119872119870 ] = minus119863] minus 119869119879 (120595) 119862119909 (12)

where

119860 =[[[[[[

119860119908 + 11987111119862119908 11987111 minus11987113 1198711211987121119862119908 11987121 minus11987123 1198712211987131119862119908 11987131 minus (11987133 + 119860minus1119887 ) 11987132minus11987141119862119908 minus11987141 11987143 minus11987142

]]]]]]

119861 =[[[[[[[

119870V1119872minus1119870119870V2119872minus1119870119870V3119872minus1119870minus119870V4119872minus1119870

]]]]]]]

119862 = [1198701199105119862119908 1198701199105 minus119868 1198701205885]

(13)

Theorem 10 Considering the dynamic positioning vesseldefined by (3) under Assumptions 1 3 5 the proposed observergiven by (7) ensures the exponential convergence of the trackingerrors ie the multiestimator given by (7) satisfies MP

Proof By defining a new variable = 119862119909 the error dynamicscan be rewritten as

= 119860119909 + 119861119869 (120595)119863]

= 119862119909] = minus119872119870119863] minus 119872119870119869119879 (120595)

(14)

Since each element of system matrix 119860 119861 and 119862 containsa gain matrix to be designed it is always possible to designsystem (14) to satisfy Lemma 7 In addition as mentioned in[18] given a set of observer gains the existence of the systemto satisfy Lemma 7 can be checked numerically by using theFrequency Theorem

Mathematical Problems in Engineering 5

Consider the following Lyapunov function candidate

119881 = ]119879119863119879119872119870] + 119909119879119875119909 (15)

If we choose the acceleration feedback to make 119872119870 and119863 commutative then we have 119863119879119872119870 gt 0 according toAssumption 1

The time differentiation of 119881 along the trajectories of ]and 119909 according to Lemma 7 yields

= ]119879119863119879119872119870119863minus1119863] + (119863])119879119872119870119863minus1119863 ] + 2119909119879119875 119909= (minus119872minus1119870 119863] minus119872minus1119870 119869119879119862119909)119879119863119879119872119870119863minus1119863]

+ (119863])119879119872119870119863minus1119863(minus119872minus1119870 119863] minus119872minus1119870 119869119879119862119909)+ 119909119879 (119875119860 + 119860119879119875)119909 + 2119909119879119875119861119869119863]

= minus (119863])119879119872minus1119870 119863119872119870119863minus1119863]

minus (119863])119879 (119863minus1119872119870119863119872minus1119870 +119872119870119863minus1119863119872minus1119870 ) 119869119879119862119909minus (119863])119879119863] minus 119909119879119876119909 + 2 (119863])119879 119869119879119861119879119875119909

= minus2 (119863])119879119863] minus 2 (119863])119879 119869119879 (119862 minus 119861119879119875) 119909 minus 119909119879119876119909= minus2]119879119863119879119863] minus 119909119879119876119909 le minus12058211205822119881

(16)

where 1205821 ≜ min120582min(2119863119879119863) 120582min(119876) and 1205822 ≜max120582max(119863119879119872119870) 120582max(119875)

In conclusion we prove that the estimation error isexponentially convergent Then there must exists at least oneparticular [120585 120578 ]] to provide a good approximation of theoutput [120585 120578 119887 ]] for any actuator failure 119870119901 isin K

32 Candidate Virtual Controller In this subsection a can-didate controller is designed to make the injected systemconsisting of observer (7) and candidate controller itselfinput-to-state stable and ensure the switched system satisfiesDP

By defining 119910 = 119910minus 119890119910 119869(120595)] = 119869(120595) ]minus 119890] and designingan acceleration feedback control law 119879119901 = (119879119870)+(119879119870119879119901 minus119870]5]) where 119879119901 is virtual control law to be designed wecan transform the observer equation (7) into the followinginjected form

119864 = 119860119864 (120595) 119909119864 minus 119861119864 (120595) 119890 + 119861119906119879119901 (17)

where 119909119864 = [120585 120578 120588 ]]119879 119890 = [119890119910 119890]]119879

119860119864 =[[[[[[[[[

119860119908 0 0 1198701205881 00 0 0 1198701205882 1198690 0 minus119860minus1119887 1198701205883 00 0 0 1198701205884 00 0 119872minus1119870 J119879 119872minus1119870 1198691198791198701205885 minus119872minus1119870 119863

]]]]]]]]]

119861119864 =[[[[[[[[[

1198701199101 119870V1

1198701199102 119870V2

1198701199103 119870V3

1198701199104 119870V4

119872minus1119870 1198691198791198701199105 0

]]]]]]]]]

119861119906 =[[[[[[[[[

1198746times11198743times11198743times11198743times1

119872minus1119870 119879119870

]]]]]]]]]

(18)

Note that the inertia of vessel is very large and the accelerationcan be omitted hence it is reasonable to select accelerationas the input of the injected system Now we need to designcontrol law to make the injected system (17) input-to-statestable with respect to the input 119890 Note that the state 119909119864 isavailable for control which is reasonable because multiesti-mator is implemented by the controller designer Then wechoose the virtual control law as follows

119879119901 = (119872minus1119870 119879119870)+ (119880120585 diag (119869119879 119869119879) 120585 + 119880120578119869119879120578+ (119880119887 minus119872minus1119870 ) 119869119879 + (119880120588119869119879 minus119872minus1119870 1198691198791198701205885) 120588+ (119872minus1119870 119863 + 119880]) ] +119872minus1119870 1198691198791198701199105 (119910 minus 119910)

(19)

Theorem 11 Consider the injected system consisting ofobserver (7) and candidate controller 119879119901 = (119879119870)+(119879119870119879119901 minus119870]5]) where 119879119901 is chosen as (19) The switched system whichcontains injected system and plant will satisfy DP

Proof When the observer gain matrices are selected as thefollowing form 1198961199011 = diag(diag(119886 119886 119887) diag(119888 119888 119889)) 119870119901119894 =diag(119890 119890 119891)(119894 isin 2 3 4) one assumed that the bias timeconstants 119860119887 has the same form and wave peak frequency1205960119894 in surge and sway are the same so does relative dampingratio 120577119894 Then replace the virtual control law into the orginalinjected system (17) and we have

119864 = 119882119879 (120595)119860119882(120595) 119909119864 minus 119861 (120595) 119890 (20)

6 Mathematical Problems in Engineering

where119882(120595) = diag(119869119879(120595) 119869119879(120595) 1198683times3)

119860 =[[[[[[[[[

119860119908 0 0 1198701205881 00 0 0 1198701205882 1198680 0 minus119860minus1119887 1198701205883 00 0 0 1198701205884 0119880120585 119880120578 119880119887 119880120588 119880V

]]]]]]]]]

119861 =[[[[[[[[[

1198701199101 119870]1

1198701199102 119870]2

1198701199103 119870]3

1198701199104 119870]4

0 0

]]]]]]]]]

(21)

The control gains and observer gains which appear in 119860are chosen such that the matrix 119860 is asymptotically stableThen there exists a matrix 119875 = 119875119879 gt 0 such that 119875119860 + 119860119879119875 =minus119876 for a given matrix 119876 = 119876119879 gt 0 Besides it should benoted that the injected system (20) is input-to-state stable ifthe system = 119882119879(120595)119860119882(120595)119911 is asymptotically stable [27]Differentiate 119882(120595)119911 along the trajectories 119911 and we have

119889 (119882 (120595) 119911)119889119905 = 119878119882 (120595) 119911 + 119860119882(120595) 119911 (22)

where 119878 = diag(119878119879 119878119879 0)

119878 = [[[

0 minus1 01 0 00 0 0

]]]

(23)

Consider a Lyapunov function candiate

119881 (119911) = 119911119879119882119879 (120595) 119875119882(120595) 119911 (24)

Differentiate it119889119881119889119905 = [(119878 + 119860)119882 (120595) 119911]119879 119875119882(120595) 119911 + (119882 (120595) 119911)119879

sdot 119875 [(119878 + 119860)119882(120595) 119911]119879 = (119882 (120595) 119911)119879sdot [(119860119879 + 119878119879) 119875 + 119875 (119878 + 119860)]119882 (120595) 119911= (119882 (120595) 119911)119879 [minus119876 + 119903119878119879119875 + 119903119875119878]119882(120595) 119911le [minus120582min (119876) + 2119903max120582max (119875)] 1003817100381710038171003817119882 (120595) 11991110038171003817100381710038172

(25)

When 119903max is sufficiently small the system =119882119879(120595)119860119882(120595)119911 is asymptotically stable Then the injectedsystem (20) is input-to-state stable with respect to 119890

Obviously when there is no sensor failure the dynamicpositioning vessel satisfies detectability which means thatthe state of plant eventually becomes small if the inputsand outputs are small Then according to Lemma 8 we candeduce that the switched system satisfies the DP

33 NEEO for Different Sensor Failures Sensor failures willdamage the detectability of the vessel In order to illustrate theproblem the following possible sensor failures are consideredin this paper

119910 = 119892 ∘ ℎ (119909) (26)

where 119892 the sensor fault function satisfies 119892 ∘ ℎ(0) = 0ie the detectability of plant is not satisfied The detectabilityof switched system will not be guaranteed if we designsupervisor in traditional wayOne reason is that themeasuredoutput is no longer a true reflection of the actual value

In this paper a nonlinear estimation error operator(NEEO) is introduced to solve the above problems Firstlythe process model (3) is remodeled as follows

= 119891 (119909 119879119901)119910 =

119910119899 = ℎ (119901 119909) no sensor failure

119910119891 = 119892119895 ∘ ℎ (119901 119909) 119895119905ℎ sensor failure(27)

where 119909 = [120585119879 120578119879 119887119879 ]119879]119879 For convenience of analysis sup-pose that 119894 isin 119875 119895 isin 119870 which represents the different actuatorfailures and sensor failures respectively The multiestimatoris designed for process (27)

119909119864119894 = 119865119894 (119909119864119894 119879119901 119910)119910119894 = ℎ119894 (119909119864119894)

119864119895 = 119865119895 (119909119864119895 119879119901 119910) = 119865119894 (119909119864119895 119879119901 119892minus1119895 (119910))119910119895 = ℎ119894 (119909119864119895) 119895 = 119875 times 119896 + 119894 119894 isin 119875

(28)

where each 119865119894 119894 isin 119875 is the observer based on actuator failurejust like Section 31

Theorem 10 has proved that the estimator can track thetrue output 119892minus1119895 (119910) so we define the nonlinear estimationerror as 119890119901 = 119910119901 ⊟ 119910 where ⊟ is NEEO and the specific formis as follows

119890119901 = 119910119901 ⊟ 119910 = 119890119894 = 119910119894 minus 119910 119894 isin 119875119890119895 = 119910119895 minus 119892minus1119895 (119910) 119895 isin 119870 (29)

No matter if the sensor is fault or not there will always bea particular 119890119901lowast that is convergent which means the multi-estimator satisfies MP Note that although the nonlinearityof 119890119901 is caused by the existence of 119892 it does not affectthe design of the front candidate controller (19) Moreoversensor failures only change the controllerrsquos external inputwithout any changing in the internal structureTherefore thefollowing result is concluded

Theorem 12 Consider the dynamic positioning vessel de-scribed by (3) under Assumptions 1 3 5 the multiestimatorgiven by (28) satisfiesMP and the switched systemalso satisfiesDP

Remark 13 Obviously to solve the problem of detectabilityloss caused by sensor faults the fault information is neededwhich can be achieved through fault reconstruction [36]

Mathematical Problems in Engineering 7

y

y

y

y

y

SwitchingLogic

MonitoringSignal

p

epyp

xE

xE

Controller(1)

Controller(2)

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

Controller Set

Observer(1)

Observer(k)

Observer(k + 1)

Observer(k + (P))

Multi minus Estimator

u

u

ActuatorFailure

Actuator

b

y

Disturbances SensorFailure

Sensor

Supervisor

yp

Figure 3 Fault-Tolerant Supervisory Control with NEEO for DPS

Initialize (t0)

(1 + ℎ) p(t) le (t)

Yes

No

(t) = ＬＡＧＣＨ p

Figure 4 Scale-Independent Hysteresis Switching Logic

34 Supervisory Control for Fault Tolerant Taking intoaccount the contents of all three previous parts this sectionemploys the scale-independent hysteresis switching logic tocontrol the DPS with actuator failures and sensor failuresFigure 3 shows the structure of the proposed supervi-sory control system The multiestimator comprises ((119875) times(119870)) models and the controller set comprises (119875) can-didate controllers corresponding to the same (119875) actuatorfailures

The hysteresis switching logic will be used to slow downswitching based on the observed growth of the estima-tion errors [37] Figure 4 shows the architecture of scale-independent hysteresis switching logic where 120587119901 physicallyis a monitoring signal in terms of the error norm as follows[22]

120587119901 = 120598 + 119890minus1205821199051205980 + int1199050119890minus120582(119905minus120591)120574 (10038171003817100381710038171003817119890119901 (120591)10038171003817100381710038171003817) 119889120591 (30)

where 120598 1205980 are nonnegative constants of which at least oneof them is strictly positive and 120582 is a constant nonnegativeforgetting factor and 120574 is class119870 function

Theorem 14 Consider the dynamic positioning vessel (27) themultiestimator is designed as (28) the candidate accelerationfeedback control law is designed as 119879119901 = (119879119870)+(119879119870119879119901 minus119870]5])with a virtual control law 119879119901 (19) and the NEEO is chosenas (29) Then under the supervisor with the scale-independenthysteresis switching logic all states of the DPS are bounded nomatter if the failures occur or not

Proof The scaled version monitoring signal is defined as120587119901(119905) = 120587119901(119905)119890120582119905 1205980 = 0 Based on Theorem 12 multiesti-mator satisfies MP so there exists 1198881 gt 0 such that 119890lowast119901 lt 1198881Since the function 120574 is Lipschiz continuous then

120587lowast119901 (119905) = 120598119890120582119905 + int1199050119890120582119904120574 (10038171003817100381710038171003817119890lowast119901 (119904)10038171003817100381710038171003817) 119889119904

le 120598119890120582119905 + 120574 (1198881) 1120582 119890120582119904100381610038161003816100381610038161199050 le 119890120582119905 [120598 + 120574 (1198881)120582 ]le 1198901205821199051198882

(31)

Using Lemma 9 we have

119873120590 le 119898 + 119898ln (1 + ℎ) ln( 120587119902 (119905)

min120587119901 (1199050))

le 119898 + 119898ln (1 + ℎ) ln[11988821198901205821199051205981198901205821199050 ]

le 119898 + 119898ln (1 + ℎ) ln(1198882120598 ) + 119905 minus 1199050

ln (1 + ℎ) 120582119898

(32)

8 Mathematical Problems in Engineering

int1199050119890120582119904120574 (1003816100381610038161003816119890120590(119904) (119904)1003816100381610038161003816) 119889119904 + 120598119890120582119905 minus 120598le 119898 [(1 + ℎ) 120583119902 (119905) minusmin 120583119901 (1199050)]le 119898 (1 + ℎ) 1198882119890120582119905

(33)

Then we can choose ln(1 + ℎ)120582119898 gt ln120583(120582119900 minus 120582) such thatthe switching signal stasfities average dwell-time [38] Sincethe choice of the virtual control law ensures the input-to-state stability of the injected system then the state of switchedinjected system 119909119864 has the property (Theorem 31[25])

1198901205821199051205721 (1003817100381710038171003817119909119864 (119905)1003817100381710038171003817) le 1198881205722 (1003817100381710038171003817119909119864 (0)1003817100381710038171003817)+ 119888 int1199050119890120582119904120574 (1003817100381710038171003817119890120590 (119904)1003817100381710038171003817) 119889119904

(34)

Substitute (33) into (34) and then1003817100381710038171003817119909119864 (119905)1003817100381710038171003817 le 120572minus11 (1198881205722 (1003817100381710038171003817119909119864 (0)1003817100381710038171003817) + 119888119898 (1 + ℎ) 1198882) = 1198883 (35)

which means that the state of injected system is boundedThen the output of injected system119910119901 = ℎ119901(119909119864) (forall119901 isin 119875) andswitched thrust force 119879119901120590 are bounded too Since 119892minus1(119910) =119910119901lowast minus 119890119901lowast then exist119901lowast isin 119875 119892minus1(119910) is bounded too Then by thedetectability of surface vessel with respect to the 119892minus1(119910) ieℎ(0) = 0 we candeduce that the state of plant is bounded4 Simulation Research

A model of a DP vessel with 4 channel thrusters and 2 mainthrusters is used in the simulation to demonstrate the perfor-mance of the supervisory control in case of actuator failuresand sensor failures For simplicity we choose four types ofactuator stuck failures K ≜ 1198701198911 1198701198912 1198701198913 1198701198914 where1198701198911 1198701198912 denote the single actuator failure and 1198701198913 1198701198914denote the multiple actuator failure The simulation param-eters are given in Table 1

As shown in Table 2 only actuator failures are consideredin Case 1 and Case 2 and both actuator failures and sensorfailures are considered in Case 3 The expected position andheading vectors are taken as 120578119889 = [0119898 0119898 0119903119886119889]119879 andthe initial states are taken as 120578(1199050) = [20119898 20119898 1119903119886119889]119879Simulation results of the proposed fault-tolerant controlstrategy are depicted in Figures 5ndash9

It can be observed from Figure 5 that the accelerationestimation errors are very small enough to be ignored So itis reasonable to take the acceleration estimation errors as aninput of the injected system

To illustrate the performance of the proposed controlscheme comparison results are presented by applying theFTC based on FDI in [5] The initial states for both controlschemes are chosen to be the same to guarantee a faircomparison As shown in the detailed time responses ofFigure 7 both methods can stabilize the vesselrsquos state to thereference value with small bounded errors but the methodin [5] cannot deal with high frequency wave force very wellwhich is the weakness of the control based on low-frequency

Table 1 Main simulation parameters

Symbol Parameter valuesM 107 times [3640004352490249752095]D 106 times [322000322-2780-27881506]119860119887 diag[100010001000]119861119887 diag[5000056000500000]119860119908(21) diag[-081-081-081]119860119908(22) diag[-018-018-018]119860119908 [03times3 1198683times3 119860119908(21) 119860119908(22)]119861119908 [[03times3 diag[16 12 025]]1198701198911 diag[011111]1198701198912 diag[111101]1198701198913 diag[011010]1198701198914 diag[101011]1198921 1198921(119904 119905) = 2119904 + 10 sin(119905)119879 [000011111100504418-357-4545-45]

minus015minus01

minus0050

005

Acceleration estimation error du (Case1)

minus1minus05

005

Acceleration estimation error dv (Case1)

minus004minus002

0002

Acceleration estimation error dr (Case1)

minus101

minus101

150 250200

150 250200

024

0 50045040035030025020015010050time (s)

0 50045040035030025020015010050time (s)

0 50045040035030025020015010050time (s)

150 250200

times10minus3

times10minus3

times10minus5

(ms

)

(ms

)

(rad

s

)

Figure 5 Acceleration Estimation Error

modelTherefore the proposed fault-tolerant controlmethodis more effective for the dynamic positioning ships in thepresence of high frequency wave disturbance

Remark 15 The upper bound on the maximum number ofsimultaneously isolable actuator faults in [5] is determined bythe Uniform Sub-Rank of the matrix which is related to thecontrol input matrix 119861 and thruster matrix 119879 By calculationwe can deduce that themethod in [5] can only solve the singleactuator fault So only Case 1 is considered in the comparisonpart

Mathematical Problems in Engineering 9

Table 2 Condition of faults

Time(s) 0-100 100-250 250-300 300-500Case 1 119870 = 1198701198911 119870 = 1198701198912 119870 = 1198701198912 119870 = 1198701198911Case 2 119870 = 1198701198913 119870 = 1198701198914 119870 = 1198701198914 119870 = 1198701198913Case 3 119870 = 1198701198913 119870 = 1198701198914 119870 = 1198701198914 119892 = 1198921 119870 = 1198701198913 119892 = 1198921

minus20

0

20

(m)

1

15

2

sigm

a

minus20minus10

01020

(m)

1

15

2sig

ma

minus1minus05

005

115

(rad

)

1

15

2

sigm

a

0 50045040035030025020015010050time (s)

0 50045040035030025020015010050time (s)

0 50045040035030025020015010050time (s)

motion in the X direction(Case1)switching signal

motion in the Y direction(Case1)switching signal

rotation about the Z axis(Case1)switching signal

Figure 6 Position and Heading of DP Vessel (Case 1)

Control performances of Case 2 and Case 3 wheremultiple actuator failures occur are demonstrated in Figures8 and 9 respectively Firstly it is shown that the signal 120590 canswitch to the corresponding controller quickly and accuratelyin the presence of actuator failures Secondly when a sensorfailure occurs at 250119904 although there is a slight fluctuation inthe transition progress the signal 120590 can switch to the stablecontroller and positions of the vessel are stabilized near theequilibrium point eventually

5 Conclusion

This paper presented a fault-tolerant supervisory controlmethod for dynamic positioning ships with wave-frequencymodel The proposed fault-tolerant control method does notneed the FDI mechanism due to the introduction of injectedsystem and supervisor It has beenproved that by introducinga nonlinear estimation error and virtual controller thedetectability property and matching property of the switchedsystem are guaranteed in the presence of actuator failuresand sensor failures Simulation results and comparisons havedemonstrated the effectiveness of the proposed fault-tolerant

minus20

0

20

40

(m)

motion in the X direction(S2)(Case1)motion in the X direction(S1)(Case1)

minus20

0

20

40

(m)

motion in the Y direction(S2)(Case1)motion in the Y direction(S1)(Case1)

minus1

0

1

2

(rad

)

rotation about the Z axis(S2)(Case1)rotation about the Z axis(S1)(Case1)

minus100

1020

minus100

1020

0 2010

0 2010

0 2010

minus050

051

15

0 50045040035030025020015010050time (s)

0 50045040035030025020015010050time (s)

0 50045040035030025020015010050time (s)

Figure 7 The tracking performance comparison of the pro-posed method (S1)(Case 1) and fault-tolerant control allocation [5](S2)(Case 1)

control scheme In the future we will consider the unknownfaults which is more practical in realistic situation Furtherwe will improve the switching logic to make the switchingsignal more accurate and stable for more serious failure

Data Availability

All the data supporting the conclusions of the study have beenprovided in Simulation and readers can access these data in[27 29]

Conflicts of Interest

The authors declare that there are no conflicts of interestrelated to this paper

Acknowledgments

This research was partially supported by the National Sci-ence Technology Support Program of China (Project no51609046)

10 Mathematical Problems in Engineering

0 50045040035030025020015010050time (s)

0 50045040035030025020015010050time (s)

0 50045040035030025020015010050time (s)

minus20

0

20

(m)

1

15

2

sigm

a

motion in the X direction(Case2)switching signal

minus20minus10

01020

(m)

1

15

2

sigm

a

motion in the Y direction(Case2)switching signal

minus1minus05

005

115

(rad

)

1

15

2

sigm

a

rotation about the Z axis(Case2)switching signal

Figure 8 Position and Heading of DP Vessel (Case 2)

0 50045040035030025020015010050time (s)

0 50045040035030025020015010050time (s)

0 50045040035030025020015010050time (s)

minus20minus10

01020

(m)

1234

sigm

a

motion in the x direction(Case3)switching signal

0(m)

2

4

sigm

a

motion in the y direction(Case3)switching signal

minus1

0

1

(rad

)

1234

sigm

a

rotation about the z axis(Case3)switching signal

Figure 9 Position and Heading of DP Vessel (Case 3)

References

[1] H Azmi and M J Khosrowjerdi ldquoRobust adaptive faulttolerant control for a class of Lipschitz nonlinear systems withactuator failure and disturbancesrdquo Proceedings of the Institutionof Mechanical Engineers Part I Journal of Systems and ControlEngineering vol 230 no 1 pp 13ndash22 2016

[2] S Dai L and J Zhao ldquoReliable H8 controller design for a classof uncertain linear systems with actuator failuresrdquo InternationalJournal of Control Automation amp Systems vol 6 no 6 pp 954ndash959 2008

[3] Y L Hao and G H Yang ldquoRobust fault tolerant controlbased on sliding mode method for uncertain linear systemswith quantizationrdquo Proceedings of the Institution of MechanicalEngineers Part I Journal of Systems amp Control Engineering vol52 no 5 pp 692ndash703 2013

[4] Z Zuo D W C Ho and Y Wang ldquoFault tolerant controlfor singular systems with actuator saturation and nonlinearperturbationrdquo Automatica vol 46 no 3 pp 569ndash576 2010

[5] A Cristofaro and T A Johansen ldquoFault tolerant controlallocation using unknown input observersrdquoAutomatica vol 50no 7 pp 1891ndash1897 2014

[6] T I Fossen Handbook of Marine Craft Hydrodynamics andMotion Control 2011

[7] A J Soslashrensen ldquoA survey of dynamic positioning controlsystemsrdquo Annual Reviews in Control vol 35 no 1 pp 123ndash1362011

[8] J Liu R Allen andH Yi ldquoShipmotion stabilizing control usinga combination of model predictive control and an adaptiveinput disturbance predictorrdquo Proceedings of the Institution ofMechanical Engineers Part I Journal of Systems and ControlEngineering vol 225 no 5 pp 591ndash602 2011

[9] M Tomera ldquoNonlinear observers design for multivariable shipmotion controlrdquo Polish Maritime Research vol 19 pp 50ndash562012

[10] A Hajivand and S H Mousavizadegan ldquoThe effect of memoryin observer design for a dp systemrdquo in Proceedings of theASME 2010 International Mechanical Engineering Congress andExposition IMECE 2010 pp 53ndash59 Canada November 2010

[11] M Fu J Ning and Y Wei ldquoFault-tolerant control of dynamicpositioning vessel after thruster failures using disturbance

Mathematical Problems in Engineering 11

decoupling methodsrdquo in Proceedings of the 2011 IEEE Interna-tional Conference on Automation and Logistics ICAL 2011 pp433ndash437 China August 2011

[12] F Benetazzo G Ippoliti S Longhi and P Raspa ldquoAdvancedcontrol for fault-tolerant dynamic positioning of an offshoresupply vesselrdquoOcean Engineering vol 106 pp 472ndash484 2015

[13] Y Lin and J Du ldquoFault-tolerant control for dynamic positioningof ships based on an iterative learning observerrdquo in Proceedingsof the 35th Chinese Control Conference CCC 2016 pp 1116ndash1122China July 2016

[14] M Chen B Jiang and R Cui ldquoActuator fault-tolerant controlof ocean surface vessels with input saturationrdquo InternationalJournal of Robust amp Nonlinear Control vol 26 no 3 pp 542ndash564 2016

[15] Y Su C Zheng andPMercorelli ldquoNonlinear PDFault-TolerantControl for Dynamic Positioning of Ships With ActuatorConstraintsrdquo IEEEASME Transactions onMechatronics vol 22no 3 pp 1132ndash1142 2017

[16] W-Z Yu H-X Xu and H Feng ldquoRobust adaptive fault-tolerant control of dynamic positioning vessel with position ref-erence system faults using backstepping designrdquo InternationalJournal of Robust and Nonlinear Control vol 28 no 2 pp 403ndash415 2018

[17] N T Dong Design of hybrid marine control systems for dynamicpositioning [PhD thesis] 2006

[18] J P Strand and T I Fossen ldquoNonlinear passive observer designfor ships with adaptive wave filteringrdquo in New Directions inNonlinear Observer Design vol 244 of Lecture Notes in Controland Information Sciences pp 113ndash134 Springer London UK1999

[19] S Saelid J G Balchen and N A Jenssen ldquoDesign and analysisof a dynamic positioning system based on kalman filtering andoptimal controlrdquo IEEE Transactions on Automatic Control vol28 no 3 pp 331ndash339 1983

[20] G Battistelli J P Hespanha and P Tesi ldquoSupervisory control ofswitched nonlinear systemsrdquo International Journal of AdaptiveControl and Signal Processing vol 26 no 8 pp 723ndash738 2012

[21] I R Bertaska andK DVonEllenrieder ldquoSupervisory switchingcontrol of an unmanned surface vehiclerdquo in Proceedings of theMTSIEEE Washington OCEANS 2015 USA October 2015

[22] J P Hespanha ldquoTutorial on supervisory controlrdquo Overview2002

[23] J P Hespanha D Liberzon and A S Morse ldquoSupervision ofintegral-input-to-state stabilizing controllersrdquo Automatica vol38 no 8 pp 1327ndash1335 2002

[24] M Skogvold Supervisory-switched control for dynamic position-ing systems in arctic areas [Msc thesis] Department of MarineTechnology 2010

[25] L VuD Chatterjee andD Liberzon ldquoInput-to-state stability ofswitched systems and switching adaptive controlrdquo Automaticavol 43 no 4 pp 639ndash646 2007

[26] ED Sontag ldquoInput to state stability basic concepts and resultsrdquoinNonlinear andOptimal ControlTheory pp 163ndash220 SpringerHeidelberg Berlin Germany 2008

[27] T D Nguyen A J Soslashrensen and S T Quek ldquoDesign of hybridcontroller for dynamic positioning from calm to extreme seaconditionsrdquo Automatica vol 43 no 5 pp 768ndash785 2007

[28] T I Fossen ldquoMarine Control Systems Guidance Navigationand Control of Ships Rigs and Underwater Vehiclesrdquo 2002

[29] T I Fossen and J P Strand ldquoPassive nonlinear observer designfor ships using Lyapunov methods full-scale experiments witha supply vesselrdquo Automatica vol 35 no 1 pp 3ndash16 1999

[30] G Xia X Shao H Wang et al ldquoPassive nonlinear observerdesign for special structure vesselsrdquoOceans IEEE pp 1ndash8 2013

[31] K P Lindegaard Acceleration Feedback in Dynamic Position-ing Fakultet for Informasjonsteknologi Matematikk Og Elek-troteknikk 2012

[32] H K Khalil Nonlinear Systems Prentice-Hall Upper SaddleRiver NJ USA 3rd edition 2002

[33] J P Hespanha and A S Morse ldquoCertainty equivalence impliesdetectabilityrdquo Systems amp Control Letters vol 36 no 1 pp 1ndash131999

[34] J P Hespanha D Liberzon and A S Morse ldquoBounds onthe number of switchings with scale-independent hysteresisApplications to supervisory controlrdquo Proceedings of the IEEEConference on Decision and Control vol 4 pp 3622ndash3627 2000

[35] M H Kim and D J Inman ldquoDevelopment of a robust non-linear observer for dynamic positioning of shipsrdquo Proceedings ofthe Institution of Mechanical Engineers Part I Journal of Systemsamp Control Engineering vol 218 no 1 pp 1ndash11 2004

[36] H Alwi C Edwards and C P Tan Fault Detection and Fault-Tolerant Control Using Sliding Modes Advances in IndustrialControl Springer London UK 2011

[37] J P Hespanha D Liberzon and A S Morse ldquoHysteresis-based switching algorithms for supervisory control of uncertainsystemsrdquo Automatica vol 39 no 2 pp 263ndash272 2003

[38] J P Hespanha and A S Morse ldquoStability of switched systemswith average dwell-timerdquo in Proceedings of the 38th IEEEConference on Decision and Control (CDC rsquo99) pp 2655ndash2660Phoenix Ariz USA December 1999

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4 Mathematical Problems in Engineering

3 Main Results

Based on the supervisory control system theory in thispaper we design a multiestimator that meets MP to coverall possible actuator failure modes a candidate controllerfor each estimator to make the fixed switched system satisfyDP and a nonlinear estimation error operator for differentsensor faults such that the switching signal satisfies SP andNP

31 Multiestimator for DPS with Actuator Failure By aug-menting a new state and utilizing acceleration feedbacktechnology the extended state observer proposed in thispaper based on (3) will improve disturbance attenuationperformance compared to the traditional one [9 35] Theobserver is designed as follows

120585 = 119860119908120585 + 1198701199101119910 + 1198701205881120588 + 119870]1119869 (120595) ]120578 = 119869 (120595) ] + 1198701199102119910 + 1198701205882120588 + 119870]2119869 (120595) ]119887 = minus119860minus1119887 + 1198701199103119910 + 1198701205883120588 + 119870]3119869 (120595) ]120588 = 1198701199104119910 + 1198701205884120588 + 119870]4119869 (120595) ]

119872 ] = 119879119870119879119901 + 119869119879 (120595) minus 119863] + 119869119879 (120595) 1198701199105119910+ 119869119879 (120595)1198701205885120588 + 119870]5

]

(7)

where 120585 isin R6 isin R3 120578 isin R3 ] isin R3 are the state vectorsof observer 120588 isin R3 is a new extended state 119910 = 119910 minus 119910 isthe estimation error of position ] = ] minus ] is accelerationestimation error and 119870119910119894 119870120588119894 and 119870]119894 119894 isin 1 5 areobserver gain matrices with an appropriate dimension Notethat for different thruster failures the value of the matrix 119870will be different

The acceleration feedback control law is designed as 119879119901 =(119879119870)+(119879119870119879119901 minus 119870]5]) where 119879119901 is called virtual control lawand 119870]5 is the designed matrix such that the new inertiamatrix 119872119870 = 119872 + 119870]5 is a positive definite diagonal matrixThen the velocity dynamics of ship and observer can berewritten as

119872119870] = 119879119870119879119901 + 119869119879119887 minus 119863]

119872119870 ] = 119879119870119879119901 + 119869119879 minus 119863] + 1198691198791198701199105119910 + 1198691198791198701205885120588(8)

Then acceleration estimation error equation can be rewrittenas

119872119870 ] = 119869119879 minus 119863] minus 1198691198791198701199105119910 minus 1198691198791198701205885120588 (9)

Now we can deduce the error dynamics equation accordingto Assumption 3

120585 = (119860119908 + 11987111119862119908) 120585 + 11987111120578 + 11987112120588 minus 11987113+ 119870V1119869119872minus1119870 119863]

120578 = 11987121119862119908120585 + 11987121120578 + 11987122120588 minus 11987123+ (119870V2119869119872minus1119870 119863 + 119869) ]

119887 = 11987131119862119908120585 + 11987131120578 + 11987132120588 minus (11987133 + 119860minus1119887 ) + 119870V3119869119872minus1119870 119863]

120588 = minus11987141119862119908120585 minus 11987141120578 minus 11987142120588 + 11987143 minus 119870V4119869119872minus1119870 119863]

(10)

where 119871 1198941 = 119870]119894119872minus1119870 1198701199105 minus119870119910119894 119871 1198942 = 119870]119894119872minus1119870 1198701205885 minus119870120588119894 119871 1198943 =119870]119894119872minus1119870 Since 119869(120595)2 = radic3 the parameters can alwaysbe selected to make 119870]2119869119872minus1119870 1198632 gtgt 119869(120595)2 Let 119909 =[120585119879 120578119879 119879 120588119879]119879 and then the error dynamics equation canbe written in compact form as follows

119909 = 119860119909 + 119861119869 (120595)119863] (11)

119872119870 ] = minus119863] minus 119869119879 (120595) 119862119909 (12)

where

119860 =[[[[[[

119860119908 + 11987111119862119908 11987111 minus11987113 1198711211987121119862119908 11987121 minus11987123 1198712211987131119862119908 11987131 minus (11987133 + 119860minus1119887 ) 11987132minus11987141119862119908 minus11987141 11987143 minus11987142

]]]]]]

119861 =[[[[[[[

119870V1119872minus1119870119870V2119872minus1119870119870V3119872minus1119870minus119870V4119872minus1119870

]]]]]]]

119862 = [1198701199105119862119908 1198701199105 minus119868 1198701205885]

(13)

Theorem 10 Considering the dynamic positioning vesseldefined by (3) under Assumptions 1 3 5 the proposed observergiven by (7) ensures the exponential convergence of the trackingerrors ie the multiestimator given by (7) satisfies MP

Proof By defining a new variable = 119862119909 the error dynamicscan be rewritten as

= 119860119909 + 119861119869 (120595)119863]

= 119862119909] = minus119872119870119863] minus 119872119870119869119879 (120595)

(14)

Since each element of system matrix 119860 119861 and 119862 containsa gain matrix to be designed it is always possible to designsystem (14) to satisfy Lemma 7 In addition as mentioned in[18] given a set of observer gains the existence of the systemto satisfy Lemma 7 can be checked numerically by using theFrequency Theorem

Mathematical Problems in Engineering 5

Consider the following Lyapunov function candidate

119881 = ]119879119863119879119872119870] + 119909119879119875119909 (15)

If we choose the acceleration feedback to make 119872119870 and119863 commutative then we have 119863119879119872119870 gt 0 according toAssumption 1

The time differentiation of 119881 along the trajectories of ]and 119909 according to Lemma 7 yields

= ]119879119863119879119872119870119863minus1119863] + (119863])119879119872119870119863minus1119863 ] + 2119909119879119875 119909= (minus119872minus1119870 119863] minus119872minus1119870 119869119879119862119909)119879119863119879119872119870119863minus1119863]

+ (119863])119879119872119870119863minus1119863(minus119872minus1119870 119863] minus119872minus1119870 119869119879119862119909)+ 119909119879 (119875119860 + 119860119879119875)119909 + 2119909119879119875119861119869119863]

= minus (119863])119879119872minus1119870 119863119872119870119863minus1119863]

minus (119863])119879 (119863minus1119872119870119863119872minus1119870 +119872119870119863minus1119863119872minus1119870 ) 119869119879119862119909minus (119863])119879119863] minus 119909119879119876119909 + 2 (119863])119879 119869119879119861119879119875119909

= minus2 (119863])119879119863] minus 2 (119863])119879 119869119879 (119862 minus 119861119879119875) 119909 minus 119909119879119876119909= minus2]119879119863119879119863] minus 119909119879119876119909 le minus12058211205822119881

(16)

where 1205821 ≜ min120582min(2119863119879119863) 120582min(119876) and 1205822 ≜max120582max(119863119879119872119870) 120582max(119875)

In conclusion we prove that the estimation error isexponentially convergent Then there must exists at least oneparticular [120585 120578 ]] to provide a good approximation of theoutput [120585 120578 119887 ]] for any actuator failure 119870119901 isin K

32 Candidate Virtual Controller In this subsection a can-didate controller is designed to make the injected systemconsisting of observer (7) and candidate controller itselfinput-to-state stable and ensure the switched system satisfiesDP

By defining 119910 = 119910minus 119890119910 119869(120595)] = 119869(120595) ]minus 119890] and designingan acceleration feedback control law 119879119901 = (119879119870)+(119879119870119879119901 minus119870]5]) where 119879119901 is virtual control law to be designed wecan transform the observer equation (7) into the followinginjected form

119864 = 119860119864 (120595) 119909119864 minus 119861119864 (120595) 119890 + 119861119906119879119901 (17)

where 119909119864 = [120585 120578 120588 ]]119879 119890 = [119890119910 119890]]119879

119860119864 =[[[[[[[[[

119860119908 0 0 1198701205881 00 0 0 1198701205882 1198690 0 minus119860minus1119887 1198701205883 00 0 0 1198701205884 00 0 119872minus1119870 J119879 119872minus1119870 1198691198791198701205885 minus119872minus1119870 119863

]]]]]]]]]

119861119864 =[[[[[[[[[

1198701199101 119870V1

1198701199102 119870V2

1198701199103 119870V3

1198701199104 119870V4

119872minus1119870 1198691198791198701199105 0

]]]]]]]]]

119861119906 =[[[[[[[[[

1198746times11198743times11198743times11198743times1

119872minus1119870 119879119870

]]]]]]]]]

(18)

Note that the inertia of vessel is very large and the accelerationcan be omitted hence it is reasonable to select accelerationas the input of the injected system Now we need to designcontrol law to make the injected system (17) input-to-statestable with respect to the input 119890 Note that the state 119909119864 isavailable for control which is reasonable because multiesti-mator is implemented by the controller designer Then wechoose the virtual control law as follows

119879119901 = (119872minus1119870 119879119870)+ (119880120585 diag (119869119879 119869119879) 120585 + 119880120578119869119879120578+ (119880119887 minus119872minus1119870 ) 119869119879 + (119880120588119869119879 minus119872minus1119870 1198691198791198701205885) 120588+ (119872minus1119870 119863 + 119880]) ] +119872minus1119870 1198691198791198701199105 (119910 minus 119910)

(19)

Theorem 11 Consider the injected system consisting ofobserver (7) and candidate controller 119879119901 = (119879119870)+(119879119870119879119901 minus119870]5]) where 119879119901 is chosen as (19) The switched system whichcontains injected system and plant will satisfy DP

Proof When the observer gain matrices are selected as thefollowing form 1198961199011 = diag(diag(119886 119886 119887) diag(119888 119888 119889)) 119870119901119894 =diag(119890 119890 119891)(119894 isin 2 3 4) one assumed that the bias timeconstants 119860119887 has the same form and wave peak frequency1205960119894 in surge and sway are the same so does relative dampingratio 120577119894 Then replace the virtual control law into the orginalinjected system (17) and we have

119864 = 119882119879 (120595)119860119882(120595) 119909119864 minus 119861 (120595) 119890 (20)

6 Mathematical Problems in Engineering

where119882(120595) = diag(119869119879(120595) 119869119879(120595) 1198683times3)

119860 =[[[[[[[[[

119860119908 0 0 1198701205881 00 0 0 1198701205882 1198680 0 minus119860minus1119887 1198701205883 00 0 0 1198701205884 0119880120585 119880120578 119880119887 119880120588 119880V

]]]]]]]]]

119861 =[[[[[[[[[

1198701199101 119870]1

1198701199102 119870]2

1198701199103 119870]3

1198701199104 119870]4

0 0

]]]]]]]]]

(21)

The control gains and observer gains which appear in 119860are chosen such that the matrix 119860 is asymptotically stableThen there exists a matrix 119875 = 119875119879 gt 0 such that 119875119860 + 119860119879119875 =minus119876 for a given matrix 119876 = 119876119879 gt 0 Besides it should benoted that the injected system (20) is input-to-state stable ifthe system = 119882119879(120595)119860119882(120595)119911 is asymptotically stable [27]Differentiate 119882(120595)119911 along the trajectories 119911 and we have

119889 (119882 (120595) 119911)119889119905 = 119878119882 (120595) 119911 + 119860119882(120595) 119911 (22)

where 119878 = diag(119878119879 119878119879 0)

119878 = [[[

0 minus1 01 0 00 0 0

]]]

(23)

Consider a Lyapunov function candiate

119881 (119911) = 119911119879119882119879 (120595) 119875119882(120595) 119911 (24)

Differentiate it119889119881119889119905 = [(119878 + 119860)119882 (120595) 119911]119879 119875119882(120595) 119911 + (119882 (120595) 119911)119879

sdot 119875 [(119878 + 119860)119882(120595) 119911]119879 = (119882 (120595) 119911)119879sdot [(119860119879 + 119878119879) 119875 + 119875 (119878 + 119860)]119882 (120595) 119911= (119882 (120595) 119911)119879 [minus119876 + 119903119878119879119875 + 119903119875119878]119882(120595) 119911le [minus120582min (119876) + 2119903max120582max (119875)] 1003817100381710038171003817119882 (120595) 11991110038171003817100381710038172

(25)

When 119903max is sufficiently small the system =119882119879(120595)119860119882(120595)119911 is asymptotically stable Then the injectedsystem (20) is input-to-state stable with respect to 119890

Obviously when there is no sensor failure the dynamicpositioning vessel satisfies detectability which means thatthe state of plant eventually becomes small if the inputsand outputs are small Then according to Lemma 8 we candeduce that the switched system satisfies the DP

33 NEEO for Different Sensor Failures Sensor failures willdamage the detectability of the vessel In order to illustrate theproblem the following possible sensor failures are consideredin this paper

119910 = 119892 ∘ ℎ (119909) (26)

where 119892 the sensor fault function satisfies 119892 ∘ ℎ(0) = 0ie the detectability of plant is not satisfied The detectabilityof switched system will not be guaranteed if we designsupervisor in traditional wayOne reason is that themeasuredoutput is no longer a true reflection of the actual value

In this paper a nonlinear estimation error operator(NEEO) is introduced to solve the above problems Firstlythe process model (3) is remodeled as follows

= 119891 (119909 119879119901)119910 =

119910119899 = ℎ (119901 119909) no sensor failure

119910119891 = 119892119895 ∘ ℎ (119901 119909) 119895119905ℎ sensor failure(27)

where 119909 = [120585119879 120578119879 119887119879 ]119879]119879 For convenience of analysis sup-pose that 119894 isin 119875 119895 isin 119870 which represents the different actuatorfailures and sensor failures respectively The multiestimatoris designed for process (27)

119909119864119894 = 119865119894 (119909119864119894 119879119901 119910)119910119894 = ℎ119894 (119909119864119894)

119864119895 = 119865119895 (119909119864119895 119879119901 119910) = 119865119894 (119909119864119895 119879119901 119892minus1119895 (119910))119910119895 = ℎ119894 (119909119864119895) 119895 = 119875 times 119896 + 119894 119894 isin 119875

(28)

where each 119865119894 119894 isin 119875 is the observer based on actuator failurejust like Section 31

Theorem 10 has proved that the estimator can track thetrue output 119892minus1119895 (119910) so we define the nonlinear estimationerror as 119890119901 = 119910119901 ⊟ 119910 where ⊟ is NEEO and the specific formis as follows

119890119901 = 119910119901 ⊟ 119910 = 119890119894 = 119910119894 minus 119910 119894 isin 119875119890119895 = 119910119895 minus 119892minus1119895 (119910) 119895 isin 119870 (29)

No matter if the sensor is fault or not there will always bea particular 119890119901lowast that is convergent which means the multi-estimator satisfies MP Note that although the nonlinearityof 119890119901 is caused by the existence of 119892 it does not affectthe design of the front candidate controller (19) Moreoversensor failures only change the controllerrsquos external inputwithout any changing in the internal structureTherefore thefollowing result is concluded

Theorem 12 Consider the dynamic positioning vessel de-scribed by (3) under Assumptions 1 3 5 the multiestimatorgiven by (28) satisfiesMP and the switched systemalso satisfiesDP

Remark 13 Obviously to solve the problem of detectabilityloss caused by sensor faults the fault information is neededwhich can be achieved through fault reconstruction [36]

Mathematical Problems in Engineering 7

y

y

y

y

y

SwitchingLogic

MonitoringSignal

p

epyp

xE

xE

Controller(1)

Controller(2)

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

Controller Set

Observer(1)

Observer(k)

Observer(k + 1)

Observer(k + (P))

Multi minus Estimator

u

u

ActuatorFailure

Actuator

b

y

Disturbances SensorFailure

Sensor

Supervisor

yp

Figure 3 Fault-Tolerant Supervisory Control with NEEO for DPS

Initialize (t0)

(1 + ℎ) p(t) le (t)

Yes

No

(t) = ＬＡＧＣＨ p

Figure 4 Scale-Independent Hysteresis Switching Logic

34 Supervisory Control for Fault Tolerant Taking intoaccount the contents of all three previous parts this sectionemploys the scale-independent hysteresis switching logic tocontrol the DPS with actuator failures and sensor failuresFigure 3 shows the structure of the proposed supervi-sory control system The multiestimator comprises ((119875) times(119870)) models and the controller set comprises (119875) can-didate controllers corresponding to the same (119875) actuatorfailures

The hysteresis switching logic will be used to slow downswitching based on the observed growth of the estima-tion errors [37] Figure 4 shows the architecture of scale-independent hysteresis switching logic where 120587119901 physicallyis a monitoring signal in terms of the error norm as follows[22]

120587119901 = 120598 + 119890minus1205821199051205980 + int1199050119890minus120582(119905minus120591)120574 (10038171003817100381710038171003817119890119901 (120591)10038171003817100381710038171003817) 119889120591 (30)

where 120598 1205980 are nonnegative constants of which at least oneof them is strictly positive and 120582 is a constant nonnegativeforgetting factor and 120574 is class119870 function

Theorem 14 Consider the dynamic positioning vessel (27) themultiestimator is designed as (28) the candidate accelerationfeedback control law is designed as 119879119901 = (119879119870)+(119879119870119879119901 minus119870]5])with a virtual control law 119879119901 (19) and the NEEO is chosenas (29) Then under the supervisor with the scale-independenthysteresis switching logic all states of the DPS are bounded nomatter if the failures occur or not

Proof The scaled version monitoring signal is defined as120587119901(119905) = 120587119901(119905)119890120582119905 1205980 = 0 Based on Theorem 12 multiesti-mator satisfies MP so there exists 1198881 gt 0 such that 119890lowast119901 lt 1198881Since the function 120574 is Lipschiz continuous then

120587lowast119901 (119905) = 120598119890120582119905 + int1199050119890120582119904120574 (10038171003817100381710038171003817119890lowast119901 (119904)10038171003817100381710038171003817) 119889119904

le 120598119890120582119905 + 120574 (1198881) 1120582 119890120582119904100381610038161003816100381610038161199050 le 119890120582119905 [120598 + 120574 (1198881)120582 ]le 1198901205821199051198882

(31)

Using Lemma 9 we have

119873120590 le 119898 + 119898ln (1 + ℎ) ln( 120587119902 (119905)

min120587119901 (1199050))

le 119898 + 119898ln (1 + ℎ) ln[11988821198901205821199051205981198901205821199050 ]

le 119898 + 119898ln (1 + ℎ) ln(1198882120598 ) + 119905 minus 1199050

ln (1 + ℎ) 120582119898

(32)

8 Mathematical Problems in Engineering

int1199050119890120582119904120574 (1003816100381610038161003816119890120590(119904) (119904)1003816100381610038161003816) 119889119904 + 120598119890120582119905 minus 120598le 119898 [(1 + ℎ) 120583119902 (119905) minusmin 120583119901 (1199050)]le 119898 (1 + ℎ) 1198882119890120582119905

(33)

Then we can choose ln(1 + ℎ)120582119898 gt ln120583(120582119900 minus 120582) such thatthe switching signal stasfities average dwell-time [38] Sincethe choice of the virtual control law ensures the input-to-state stability of the injected system then the state of switchedinjected system 119909119864 has the property (Theorem 31[25])

1198901205821199051205721 (1003817100381710038171003817119909119864 (119905)1003817100381710038171003817) le 1198881205722 (1003817100381710038171003817119909119864 (0)1003817100381710038171003817)+ 119888 int1199050119890120582119904120574 (1003817100381710038171003817119890120590 (119904)1003817100381710038171003817) 119889119904

(34)

Substitute (33) into (34) and then1003817100381710038171003817119909119864 (119905)1003817100381710038171003817 le 120572minus11 (1198881205722 (1003817100381710038171003817119909119864 (0)1003817100381710038171003817) + 119888119898 (1 + ℎ) 1198882) = 1198883 (35)

which means that the state of injected system is boundedThen the output of injected system119910119901 = ℎ119901(119909119864) (forall119901 isin 119875) andswitched thrust force 119879119901120590 are bounded too Since 119892minus1(119910) =119910119901lowast minus 119890119901lowast then exist119901lowast isin 119875 119892minus1(119910) is bounded too Then by thedetectability of surface vessel with respect to the 119892minus1(119910) ieℎ(0) = 0 we candeduce that the state of plant is bounded4 Simulation Research

A model of a DP vessel with 4 channel thrusters and 2 mainthrusters is used in the simulation to demonstrate the perfor-mance of the supervisory control in case of actuator failuresand sensor failures For simplicity we choose four types ofactuator stuck failures K ≜ 1198701198911 1198701198912 1198701198913 1198701198914 where1198701198911 1198701198912 denote the single actuator failure and 1198701198913 1198701198914denote the multiple actuator failure The simulation param-eters are given in Table 1

As shown in Table 2 only actuator failures are consideredin Case 1 and Case 2 and both actuator failures and sensorfailures are considered in Case 3 The expected position andheading vectors are taken as 120578119889 = [0119898 0119898 0119903119886119889]119879 andthe initial states are taken as 120578(1199050) = [20119898 20119898 1119903119886119889]119879Simulation results of the proposed fault-tolerant controlstrategy are depicted in Figures 5ndash9

It can be observed from Figure 5 that the accelerationestimation errors are very small enough to be ignored So itis reasonable to take the acceleration estimation errors as aninput of the injected system

To illustrate the performance of the proposed controlscheme comparison results are presented by applying theFTC based on FDI in [5] The initial states for both controlschemes are chosen to be the same to guarantee a faircomparison As shown in the detailed time responses ofFigure 7 both methods can stabilize the vesselrsquos state to thereference value with small bounded errors but the methodin [5] cannot deal with high frequency wave force very wellwhich is the weakness of the control based on low-frequency

Table 1 Main simulation parameters

Symbol Parameter valuesM 107 times [3640004352490249752095]D 106 times [322000322-2780-27881506]119860119887 diag[100010001000]119861119887 diag[5000056000500000]119860119908(21) diag[-081-081-081]119860119908(22) diag[-018-018-018]119860119908 [03times3 1198683times3 119860119908(21) 119860119908(22)]119861119908 [[03times3 diag[16 12 025]]1198701198911 diag[011111]1198701198912 diag[111101]1198701198913 diag[011010]1198701198914 diag[101011]1198921 1198921(119904 119905) = 2119904 + 10 sin(119905)119879 [000011111100504418-357-4545-45]

minus015minus01

minus0050

005

Acceleration estimation error du (Case1)

minus1minus05

005

Acceleration estimation error dv (Case1)

minus004minus002

0002

Acceleration estimation error dr (Case1)

minus101

minus101

150 250200

150 250200

024

0 50045040035030025020015010050time (s)

0 50045040035030025020015010050time (s)

0 50045040035030025020015010050time (s)

150 250200

times10minus3

times10minus3

times10minus5

(ms

)

(ms

)

(rad

s

)

Figure 5 Acceleration Estimation Error

modelTherefore the proposed fault-tolerant controlmethodis more effective for the dynamic positioning ships in thepresence of high frequency wave disturbance

Remark 15 The upper bound on the maximum number ofsimultaneously isolable actuator faults in [5] is determined bythe Uniform Sub-Rank of the matrix which is related to thecontrol input matrix 119861 and thruster matrix 119879 By calculationwe can deduce that themethod in [5] can only solve the singleactuator fault So only Case 1 is considered in the comparisonpart

Mathematical Problems in Engineering 9

Table 2 Condition of faults

Time(s) 0-100 100-250 250-300 300-500Case 1 119870 = 1198701198911 119870 = 1198701198912 119870 = 1198701198912 119870 = 1198701198911Case 2 119870 = 1198701198913 119870 = 1198701198914 119870 = 1198701198914 119870 = 1198701198913Case 3 119870 = 1198701198913 119870 = 1198701198914 119870 = 1198701198914 119892 = 1198921 119870 = 1198701198913 119892 = 1198921

minus20

0

20

(m)

1

15

2

sigm

a

minus20minus10

01020

(m)

1

15

2sig

ma

minus1minus05

005

115

(rad

)

1

15

2

sigm

a

0 50045040035030025020015010050time (s)

0 50045040035030025020015010050time (s)

0 50045040035030025020015010050time (s)

motion in the X direction(Case1)switching signal

motion in the Y direction(Case1)switching signal

rotation about the Z axis(Case1)switching signal

Figure 6 Position and Heading of DP Vessel (Case 1)

Control performances of Case 2 and Case 3 wheremultiple actuator failures occur are demonstrated in Figures8 and 9 respectively Firstly it is shown that the signal 120590 canswitch to the corresponding controller quickly and accuratelyin the presence of actuator failures Secondly when a sensorfailure occurs at 250119904 although there is a slight fluctuation inthe transition progress the signal 120590 can switch to the stablecontroller and positions of the vessel are stabilized near theequilibrium point eventually

5 Conclusion

This paper presented a fault-tolerant supervisory controlmethod for dynamic positioning ships with wave-frequencymodel The proposed fault-tolerant control method does notneed the FDI mechanism due to the introduction of injectedsystem and supervisor It has beenproved that by introducinga nonlinear estimation error and virtual controller thedetectability property and matching property of the switchedsystem are guaranteed in the presence of actuator failuresand sensor failures Simulation results and comparisons havedemonstrated the effectiveness of the proposed fault-tolerant

minus20

0

20

40

(m)

motion in the X direction(S2)(Case1)motion in the X direction(S1)(Case1)

minus20

0

20

40

(m)

motion in the Y direction(S2)(Case1)motion in the Y direction(S1)(Case1)

minus1

0

1

2

(rad

)

rotation about the Z axis(S2)(Case1)rotation about the Z axis(S1)(Case1)

minus100

1020

minus100

1020

0 2010

0 2010

0 2010

minus050

051

15

0 50045040035030025020015010050time (s)

0 50045040035030025020015010050time (s)

0 50045040035030025020015010050time (s)

Figure 7 The tracking performance comparison of the pro-posed method (S1)(Case 1) and fault-tolerant control allocation [5](S2)(Case 1)

control scheme In the future we will consider the unknownfaults which is more practical in realistic situation Furtherwe will improve the switching logic to make the switchingsignal more accurate and stable for more serious failure

Data Availability

All the data supporting the conclusions of the study have beenprovided in Simulation and readers can access these data in[27 29]

Conflicts of Interest

The authors declare that there are no conflicts of interestrelated to this paper

Acknowledgments

This research was partially supported by the National Sci-ence Technology Support Program of China (Project no51609046)

10 Mathematical Problems in Engineering

0 50045040035030025020015010050time (s)

0 50045040035030025020015010050time (s)

0 50045040035030025020015010050time (s)

minus20

0

20

(m)

1

15

2

sigm

a

motion in the X direction(Case2)switching signal

minus20minus10

01020

(m)

1

15

2

sigm

a

motion in the Y direction(Case2)switching signal

minus1minus05

005

115

(rad

)

1

15

2

sigm

a

rotation about the Z axis(Case2)switching signal

Figure 8 Position and Heading of DP Vessel (Case 2)

0 50045040035030025020015010050time (s)

0 50045040035030025020015010050time (s)

0 50045040035030025020015010050time (s)

minus20minus10

01020

(m)

1234

sigm

a

motion in the x direction(Case3)switching signal

0(m)

2

4

sigm

a

motion in the y direction(Case3)switching signal

minus1

0

1

(rad

)

1234

sigm

a

rotation about the z axis(Case3)switching signal

Figure 9 Position and Heading of DP Vessel (Case 3)

References

[1] H Azmi and M J Khosrowjerdi ldquoRobust adaptive faulttolerant control for a class of Lipschitz nonlinear systems withactuator failure and disturbancesrdquo Proceedings of the Institutionof Mechanical Engineers Part I Journal of Systems and ControlEngineering vol 230 no 1 pp 13ndash22 2016

[2] S Dai L and J Zhao ldquoReliable H8 controller design for a classof uncertain linear systems with actuator failuresrdquo InternationalJournal of Control Automation amp Systems vol 6 no 6 pp 954ndash959 2008

[3] Y L Hao and G H Yang ldquoRobust fault tolerant controlbased on sliding mode method for uncertain linear systemswith quantizationrdquo Proceedings of the Institution of MechanicalEngineers Part I Journal of Systems amp Control Engineering vol52 no 5 pp 692ndash703 2013

[4] Z Zuo D W C Ho and Y Wang ldquoFault tolerant controlfor singular systems with actuator saturation and nonlinearperturbationrdquo Automatica vol 46 no 3 pp 569ndash576 2010

[5] A Cristofaro and T A Johansen ldquoFault tolerant controlallocation using unknown input observersrdquoAutomatica vol 50no 7 pp 1891ndash1897 2014

[6] T I Fossen Handbook of Marine Craft Hydrodynamics andMotion Control 2011

[7] A J Soslashrensen ldquoA survey of dynamic positioning controlsystemsrdquo Annual Reviews in Control vol 35 no 1 pp 123ndash1362011

[8] J Liu R Allen andH Yi ldquoShipmotion stabilizing control usinga combination of model predictive control and an adaptiveinput disturbance predictorrdquo Proceedings of the Institution ofMechanical Engineers Part I Journal of Systems and ControlEngineering vol 225 no 5 pp 591ndash602 2011

[9] M Tomera ldquoNonlinear observers design for multivariable shipmotion controlrdquo Polish Maritime Research vol 19 pp 50ndash562012

[10] A Hajivand and S H Mousavizadegan ldquoThe effect of memoryin observer design for a dp systemrdquo in Proceedings of theASME 2010 International Mechanical Engineering Congress andExposition IMECE 2010 pp 53ndash59 Canada November 2010

[11] M Fu J Ning and Y Wei ldquoFault-tolerant control of dynamicpositioning vessel after thruster failures using disturbance

Mathematical Problems in Engineering 11

decoupling methodsrdquo in Proceedings of the 2011 IEEE Interna-tional Conference on Automation and Logistics ICAL 2011 pp433ndash437 China August 2011

[12] F Benetazzo G Ippoliti S Longhi and P Raspa ldquoAdvancedcontrol for fault-tolerant dynamic positioning of an offshoresupply vesselrdquoOcean Engineering vol 106 pp 472ndash484 2015

[13] Y Lin and J Du ldquoFault-tolerant control for dynamic positioningof ships based on an iterative learning observerrdquo in Proceedingsof the 35th Chinese Control Conference CCC 2016 pp 1116ndash1122China July 2016

[14] M Chen B Jiang and R Cui ldquoActuator fault-tolerant controlof ocean surface vessels with input saturationrdquo InternationalJournal of Robust amp Nonlinear Control vol 26 no 3 pp 542ndash564 2016

[15] Y Su C Zheng andPMercorelli ldquoNonlinear PDFault-TolerantControl for Dynamic Positioning of Ships With ActuatorConstraintsrdquo IEEEASME Transactions onMechatronics vol 22no 3 pp 1132ndash1142 2017

[16] W-Z Yu H-X Xu and H Feng ldquoRobust adaptive fault-tolerant control of dynamic positioning vessel with position ref-erence system faults using backstepping designrdquo InternationalJournal of Robust and Nonlinear Control vol 28 no 2 pp 403ndash415 2018

[17] N T Dong Design of hybrid marine control systems for dynamicpositioning [PhD thesis] 2006

[18] J P Strand and T I Fossen ldquoNonlinear passive observer designfor ships with adaptive wave filteringrdquo in New Directions inNonlinear Observer Design vol 244 of Lecture Notes in Controland Information Sciences pp 113ndash134 Springer London UK1999

[19] S Saelid J G Balchen and N A Jenssen ldquoDesign and analysisof a dynamic positioning system based on kalman filtering andoptimal controlrdquo IEEE Transactions on Automatic Control vol28 no 3 pp 331ndash339 1983

[20] G Battistelli J P Hespanha and P Tesi ldquoSupervisory control ofswitched nonlinear systemsrdquo International Journal of AdaptiveControl and Signal Processing vol 26 no 8 pp 723ndash738 2012

[21] I R Bertaska andK DVonEllenrieder ldquoSupervisory switchingcontrol of an unmanned surface vehiclerdquo in Proceedings of theMTSIEEE Washington OCEANS 2015 USA October 2015

[22] J P Hespanha ldquoTutorial on supervisory controlrdquo Overview2002

[23] J P Hespanha D Liberzon and A S Morse ldquoSupervision ofintegral-input-to-state stabilizing controllersrdquo Automatica vol38 no 8 pp 1327ndash1335 2002

[24] M Skogvold Supervisory-switched control for dynamic position-ing systems in arctic areas [Msc thesis] Department of MarineTechnology 2010

[25] L VuD Chatterjee andD Liberzon ldquoInput-to-state stability ofswitched systems and switching adaptive controlrdquo Automaticavol 43 no 4 pp 639ndash646 2007

[26] ED Sontag ldquoInput to state stability basic concepts and resultsrdquoinNonlinear andOptimal ControlTheory pp 163ndash220 SpringerHeidelberg Berlin Germany 2008

[27] T D Nguyen A J Soslashrensen and S T Quek ldquoDesign of hybridcontroller for dynamic positioning from calm to extreme seaconditionsrdquo Automatica vol 43 no 5 pp 768ndash785 2007

[28] T I Fossen ldquoMarine Control Systems Guidance Navigationand Control of Ships Rigs and Underwater Vehiclesrdquo 2002

[29] T I Fossen and J P Strand ldquoPassive nonlinear observer designfor ships using Lyapunov methods full-scale experiments witha supply vesselrdquo Automatica vol 35 no 1 pp 3ndash16 1999

[30] G Xia X Shao H Wang et al ldquoPassive nonlinear observerdesign for special structure vesselsrdquoOceans IEEE pp 1ndash8 2013

[31] K P Lindegaard Acceleration Feedback in Dynamic Position-ing Fakultet for Informasjonsteknologi Matematikk Og Elek-troteknikk 2012

[32] H K Khalil Nonlinear Systems Prentice-Hall Upper SaddleRiver NJ USA 3rd edition 2002

[33] J P Hespanha and A S Morse ldquoCertainty equivalence impliesdetectabilityrdquo Systems amp Control Letters vol 36 no 1 pp 1ndash131999

[34] J P Hespanha D Liberzon and A S Morse ldquoBounds onthe number of switchings with scale-independent hysteresisApplications to supervisory controlrdquo Proceedings of the IEEEConference on Decision and Control vol 4 pp 3622ndash3627 2000

[35] M H Kim and D J Inman ldquoDevelopment of a robust non-linear observer for dynamic positioning of shipsrdquo Proceedings ofthe Institution of Mechanical Engineers Part I Journal of Systemsamp Control Engineering vol 218 no 1 pp 1ndash11 2004

[36] H Alwi C Edwards and C P Tan Fault Detection and Fault-Tolerant Control Using Sliding Modes Advances in IndustrialControl Springer London UK 2011

[37] J P Hespanha D Liberzon and A S Morse ldquoHysteresis-based switching algorithms for supervisory control of uncertainsystemsrdquo Automatica vol 39 no 2 pp 263ndash272 2003

[38] J P Hespanha and A S Morse ldquoStability of switched systemswith average dwell-timerdquo in Proceedings of the 38th IEEEConference on Decision and Control (CDC rsquo99) pp 2655ndash2660Phoenix Ariz USA December 1999

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Submit your manuscripts atwwwhindawicom

Mathematical Problems in Engineering 5

Consider the following Lyapunov function candidate

119881 = ]119879119863119879119872119870] + 119909119879119875119909 (15)

If we choose the acceleration feedback to make 119872119870 and119863 commutative then we have 119863119879119872119870 gt 0 according toAssumption 1

The time differentiation of 119881 along the trajectories of ]and 119909 according to Lemma 7 yields

= ]119879119863119879119872119870119863minus1119863] + (119863])119879119872119870119863minus1119863 ] + 2119909119879119875 119909= (minus119872minus1119870 119863] minus119872minus1119870 119869119879119862119909)119879119863119879119872119870119863minus1119863]

+ (119863])119879119872119870119863minus1119863(minus119872minus1119870 119863] minus119872minus1119870 119869119879119862119909)+ 119909119879 (119875119860 + 119860119879119875)119909 + 2119909119879119875119861119869119863]

= minus (119863])119879119872minus1119870 119863119872119870119863minus1119863]

minus (119863])119879 (119863minus1119872119870119863119872minus1119870 +119872119870119863minus1119863119872minus1119870 ) 119869119879119862119909minus (119863])119879119863] minus 119909119879119876119909 + 2 (119863])119879 119869119879119861119879119875119909

= minus2 (119863])119879119863] minus 2 (119863])119879 119869119879 (119862 minus 119861119879119875) 119909 minus 119909119879119876119909= minus2]119879119863119879119863] minus 119909119879119876119909 le minus12058211205822119881

(16)

where 1205821 ≜ min120582min(2119863119879119863) 120582min(119876) and 1205822 ≜max120582max(119863119879119872119870) 120582max(119875)

In conclusion we prove that the estimation error isexponentially convergent Then there must exists at least oneparticular [120585 120578 ]] to provide a good approximation of theoutput [120585 120578 119887 ]] for any actuator failure 119870119901 isin K

32 Candidate Virtual Controller In this subsection a can-didate controller is designed to make the injected systemconsisting of observer (7) and candidate controller itselfinput-to-state stable and ensure the switched system satisfiesDP

By defining 119910 = 119910minus 119890119910 119869(120595)] = 119869(120595) ]minus 119890] and designingan acceleration feedback control law 119879119901 = (119879119870)+(119879119870119879119901 minus119870]5]) where 119879119901 is virtual control law to be designed wecan transform the observer equation (7) into the followinginjected form

119864 = 119860119864 (120595) 119909119864 minus 119861119864 (120595) 119890 + 119861119906119879119901 (17)

where 119909119864 = [120585 120578 120588 ]]119879 119890 = [119890119910 119890]]119879

119860119864 =[[[[[[[[[

119860119908 0 0 1198701205881 00 0 0 1198701205882 1198690 0 minus119860minus1119887 1198701205883 00 0 0 1198701205884 00 0 119872minus1119870 J119879 119872minus1119870 1198691198791198701205885 minus119872minus1119870 119863

]]]]]]]]]

119861119864 =[[[[[[[[[

1198701199101 119870V1

1198701199102 119870V2

1198701199103 119870V3

1198701199104 119870V4

119872minus1119870 1198691198791198701199105 0

]]]]]]]]]

119861119906 =[[[[[[[[[

1198746times11198743times11198743times11198743times1

119872minus1119870 119879119870

]]]]]]]]]

(18)

Note that the inertia of vessel is very large and the accelerationcan be omitted hence it is reasonable to select accelerationas the input of the injected system Now we need to designcontrol law to make the injected system (17) input-to-statestable with respect to the input 119890 Note that the state 119909119864 isavailable for control which is reasonable because multiesti-mator is implemented by the controller designer Then wechoose the virtual control law as follows

119879119901 = (119872minus1119870 119879119870)+ (119880120585 diag (119869119879 119869119879) 120585 + 119880120578119869119879120578+ (119880119887 minus119872minus1119870 ) 119869119879 + (119880120588119869119879 minus119872minus1119870 1198691198791198701205885) 120588+ (119872minus1119870 119863 + 119880]) ] +119872minus1119870 1198691198791198701199105 (119910 minus 119910)

(19)

Theorem 11 Consider the injected system consisting ofobserver (7) and candidate controller 119879119901 = (119879119870)+(119879119870119879119901 minus119870]5]) where 119879119901 is chosen as (19) The switched system whichcontains injected system and plant will satisfy DP

Proof When the observer gain matrices are selected as thefollowing form 1198961199011 = diag(diag(119886 119886 119887) diag(119888 119888 119889)) 119870119901119894 =diag(119890 119890 119891)(119894 isin 2 3 4) one assumed that the bias timeconstants 119860119887 has the same form and wave peak frequency1205960119894 in surge and sway are the same so does relative dampingratio 120577119894 Then replace the virtual control law into the orginalinjected system (17) and we have

119864 = 119882119879 (120595)119860119882(120595) 119909119864 minus 119861 (120595) 119890 (20)

6 Mathematical Problems in Engineering

where119882(120595) = diag(119869119879(120595) 119869119879(120595) 1198683times3)

119860 =[[[[[[[[[

119860119908 0 0 1198701205881 00 0 0 1198701205882 1198680 0 minus119860minus1119887 1198701205883 00 0 0 1198701205884 0119880120585 119880120578 119880119887 119880120588 119880V

]]]]]]]]]

119861 =[[[[[[[[[

1198701199101 119870]1

1198701199102 119870]2

1198701199103 119870]3

1198701199104 119870]4

0 0

]]]]]]]]]

(21)

The control gains and observer gains which appear in 119860are chosen such that the matrix 119860 is asymptotically stableThen there exists a matrix 119875 = 119875119879 gt 0 such that 119875119860 + 119860119879119875 =minus119876 for a given matrix 119876 = 119876119879 gt 0 Besides it should benoted that the injected system (20) is input-to-state stable ifthe system = 119882119879(120595)119860119882(120595)119911 is asymptotically stable [27]Differentiate 119882(120595)119911 along the trajectories 119911 and we have

119889 (119882 (120595) 119911)119889119905 = 119878119882 (120595) 119911 + 119860119882(120595) 119911 (22)

where 119878 = diag(119878119879 119878119879 0)

119878 = [[[

0 minus1 01 0 00 0 0

]]]

(23)

Consider a Lyapunov function candiate

119881 (119911) = 119911119879119882119879 (120595) 119875119882(120595) 119911 (24)

Differentiate it119889119881119889119905 = [(119878 + 119860)119882 (120595) 119911]119879 119875119882(120595) 119911 + (119882 (120595) 119911)119879

sdot 119875 [(119878 + 119860)119882(120595) 119911]119879 = (119882 (120595) 119911)119879sdot [(119860119879 + 119878119879) 119875 + 119875 (119878 + 119860)]119882 (120595) 119911= (119882 (120595) 119911)119879 [minus119876 + 119903119878119879119875 + 119903119875119878]119882(120595) 119911le [minus120582min (119876) + 2119903max120582max (119875)] 1003817100381710038171003817119882 (120595) 11991110038171003817100381710038172

(25)

When 119903max is sufficiently small the system =119882119879(120595)119860119882(120595)119911 is asymptotically stable Then the injectedsystem (20) is input-to-state stable with respect to 119890

Obviously when there is no sensor failure the dynamicpositioning vessel satisfies detectability which means thatthe state of plant eventually becomes small if the inputsand outputs are small Then according to Lemma 8 we candeduce that the switched system satisfies the DP

33 NEEO for Different Sensor Failures Sensor failures willdamage the detectability of the vessel In order to illustrate theproblem the following possible sensor failures are consideredin this paper

119910 = 119892 ∘ ℎ (119909) (26)

where 119892 the sensor fault function satisfies 119892 ∘ ℎ(0) = 0ie the detectability of plant is not satisfied The detectabilityof switched system will not be guaranteed if we designsupervisor in traditional wayOne reason is that themeasuredoutput is no longer a true reflection of the actual value

In this paper a nonlinear estimation error operator(NEEO) is introduced to solve the above problems Firstlythe process model (3) is remodeled as follows

= 119891 (119909 119879119901)119910 =

119910119899 = ℎ (119901 119909) no sensor failure

119910119891 = 119892119895 ∘ ℎ (119901 119909) 119895119905ℎ sensor failure(27)

where 119909 = [120585119879 120578119879 119887119879 ]119879]119879 For convenience of analysis sup-pose that 119894 isin 119875 119895 isin 119870 which represents the different actuatorfailures and sensor failures respectively The multiestimatoris designed for process (27)

119909119864119894 = 119865119894 (119909119864119894 119879119901 119910)119910119894 = ℎ119894 (119909119864119894)

119864119895 = 119865119895 (119909119864119895 119879119901 119910) = 119865119894 (119909119864119895 119879119901 119892minus1119895 (119910))119910119895 = ℎ119894 (119909119864119895) 119895 = 119875 times 119896 + 119894 119894 isin 119875

(28)

where each 119865119894 119894 isin 119875 is the observer based on actuator failurejust like Section 31

Theorem 10 has proved that the estimator can track thetrue output 119892minus1119895 (119910) so we define the nonlinear estimationerror as 119890119901 = 119910119901 ⊟ 119910 where ⊟ is NEEO and the specific formis as follows

119890119901 = 119910119901 ⊟ 119910 = 119890119894 = 119910119894 minus 119910 119894 isin 119875119890119895 = 119910119895 minus 119892minus1119895 (119910) 119895 isin 119870 (29)

No matter if the sensor is fault or not there will always bea particular 119890119901lowast that is convergent which means the multi-estimator satisfies MP Note that although the nonlinearityof 119890119901 is caused by the existence of 119892 it does not affectthe design of the front candidate controller (19) Moreoversensor failures only change the controllerrsquos external inputwithout any changing in the internal structureTherefore thefollowing result is concluded

Theorem 12 Consider the dynamic positioning vessel de-scribed by (3) under Assumptions 1 3 5 the multiestimatorgiven by (28) satisfiesMP and the switched systemalso satisfiesDP

Remark 13 Obviously to solve the problem of detectabilityloss caused by sensor faults the fault information is neededwhich can be achieved through fault reconstruction [36]

Mathematical Problems in Engineering 7

y

y

y

y

y

SwitchingLogic

MonitoringSignal

p

epyp

xE

xE

Controller(1)

Controller(2)

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

Controller Set

Observer(1)

Observer(k)

Observer(k + 1)

Observer(k + (P))

Multi minus Estimator

u

u

ActuatorFailure

Actuator

b

y

Disturbances SensorFailure

Sensor

Supervisor

yp

Figure 3 Fault-Tolerant Supervisory Control with NEEO for DPS

Initialize (t0)

(1 + ℎ) p(t) le (t)

Yes

No

(t) = ＬＡＧＣＨ p

Figure 4 Scale-Independent Hysteresis Switching Logic

34 Supervisory Control for Fault Tolerant Taking intoaccount the contents of all three previous parts this sectionemploys the scale-independent hysteresis switching logic tocontrol the DPS with actuator failures and sensor failuresFigure 3 shows the structure of the proposed supervi-sory control system The multiestimator comprises ((119875) times(119870)) models and the controller set comprises (119875) can-didate controllers corresponding to the same (119875) actuatorfailures

The hysteresis switching logic will be used to slow downswitching based on the observed growth of the estima-tion errors [37] Figure 4 shows the architecture of scale-independent hysteresis switching logic where 120587119901 physicallyis a monitoring signal in terms of the error norm as follows[22]

120587119901 = 120598 + 119890minus1205821199051205980 + int1199050119890minus120582(119905minus120591)120574 (10038171003817100381710038171003817119890119901 (120591)10038171003817100381710038171003817) 119889120591 (30)

where 120598 1205980 are nonnegative constants of which at least oneof them is strictly positive and 120582 is a constant nonnegativeforgetting factor and 120574 is class119870 function

Theorem 14 Consider the dynamic positioning vessel (27) themultiestimator is designed as (28) the candidate accelerationfeedback control law is designed as 119879119901 = (119879119870)+(119879119870119879119901 minus119870]5])with a virtual control law 119879119901 (19) and the NEEO is chosenas (29) Then under the supervisor with the scale-independenthysteresis switching logic all states of the DPS are bounded nomatter if the failures occur or not

Proof The scaled version monitoring signal is defined as120587119901(119905) = 120587119901(119905)119890120582119905 1205980 = 0 Based on Theorem 12 multiesti-mator satisfies MP so there exists 1198881 gt 0 such that 119890lowast119901 lt 1198881Since the function 120574 is Lipschiz continuous then

120587lowast119901 (119905) = 120598119890120582119905 + int1199050119890120582119904120574 (10038171003817100381710038171003817119890lowast119901 (119904)10038171003817100381710038171003817) 119889119904

le 120598119890120582119905 + 120574 (1198881) 1120582 119890120582119904100381610038161003816100381610038161199050 le 119890120582119905 [120598 + 120574 (1198881)120582 ]le 1198901205821199051198882

(31)

Using Lemma 9 we have

119873120590 le 119898 + 119898ln (1 + ℎ) ln( 120587119902 (119905)

min120587119901 (1199050))

le 119898 + 119898ln (1 + ℎ) ln[11988821198901205821199051205981198901205821199050 ]

le 119898 + 119898ln (1 + ℎ) ln(1198882120598 ) + 119905 minus 1199050

ln (1 + ℎ) 120582119898

(32)

8 Mathematical Problems in Engineering

int1199050119890120582119904120574 (1003816100381610038161003816119890120590(119904) (119904)1003816100381610038161003816) 119889119904 + 120598119890120582119905 minus 120598le 119898 [(1 + ℎ) 120583119902 (119905) minusmin 120583119901 (1199050)]le 119898 (1 + ℎ) 1198882119890120582119905

(33)

Then we can choose ln(1 + ℎ)120582119898 gt ln120583(120582119900 minus 120582) such thatthe switching signal stasfities average dwell-time [38] Sincethe choice of the virtual control law ensures the input-to-state stability of the injected system then the state of switchedinjected system 119909119864 has the property (Theorem 31[25])

1198901205821199051205721 (1003817100381710038171003817119909119864 (119905)1003817100381710038171003817) le 1198881205722 (1003817100381710038171003817119909119864 (0)1003817100381710038171003817)+ 119888 int1199050119890120582119904120574 (1003817100381710038171003817119890120590 (119904)1003817100381710038171003817) 119889119904

(34)

Substitute (33) into (34) and then1003817100381710038171003817119909119864 (119905)1003817100381710038171003817 le 120572minus11 (1198881205722 (1003817100381710038171003817119909119864 (0)1003817100381710038171003817) + 119888119898 (1 + ℎ) 1198882) = 1198883 (35)

which means that the state of injected system is boundedThen the output of injected system119910119901 = ℎ119901(119909119864) (forall119901 isin 119875) andswitched thrust force 119879119901120590 are bounded too Since 119892minus1(119910) =119910119901lowast minus 119890119901lowast then exist119901lowast isin 119875 119892minus1(119910) is bounded too Then by thedetectability of surface vessel with respect to the 119892minus1(119910) ieℎ(0) = 0 we candeduce that the state of plant is bounded4 Simulation Research

A model of a DP vessel with 4 channel thrusters and 2 mainthrusters is used in the simulation to demonstrate the perfor-mance of the supervisory control in case of actuator failuresand sensor failures For simplicity we choose four types ofactuator stuck failures K ≜ 1198701198911 1198701198912 1198701198913 1198701198914 where1198701198911 1198701198912 denote the single actuator failure and 1198701198913 1198701198914denote the multiple actuator failure The simulation param-eters are given in Table 1

As shown in Table 2 only actuator failures are consideredin Case 1 and Case 2 and both actuator failures and sensorfailures are considered in Case 3 The expected position andheading vectors are taken as 120578119889 = [0119898 0119898 0119903119886119889]119879 andthe initial states are taken as 120578(1199050) = [20119898 20119898 1119903119886119889]119879Simulation results of the proposed fault-tolerant controlstrategy are depicted in Figures 5ndash9

It can be observed from Figure 5 that the accelerationestimation errors are very small enough to be ignored So itis reasonable to take the acceleration estimation errors as aninput of the injected system

To illustrate the performance of the proposed controlscheme comparison results are presented by applying theFTC based on FDI in [5] The initial states for both controlschemes are chosen to be the same to guarantee a faircomparison As shown in the detailed time responses ofFigure 7 both methods can stabilize the vesselrsquos state to thereference value with small bounded errors but the methodin [5] cannot deal with high frequency wave force very wellwhich is the weakness of the control based on low-frequency

Table 1 Main simulation parameters

Symbol Parameter valuesM 107 times [3640004352490249752095]D 106 times [322000322-2780-27881506]119860119887 diag[100010001000]119861119887 diag[5000056000500000]119860119908(21) diag[-081-081-081]119860119908(22) diag[-018-018-018]119860119908 [03times3 1198683times3 119860119908(21) 119860119908(22)]119861119908 [[03times3 diag[16 12 025]]1198701198911 diag[011111]1198701198912 diag[111101]1198701198913 diag[011010]1198701198914 diag[101011]1198921 1198921(119904 119905) = 2119904 + 10 sin(119905)119879 [000011111100504418-357-4545-45]

minus015minus01

minus0050

005

Acceleration estimation error du (Case1)

minus1minus05

005

Acceleration estimation error dv (Case1)

minus004minus002

0002

Acceleration estimation error dr (Case1)

minus101

minus101

150 250200

150 250200

024

0 50045040035030025020015010050time (s)

0 50045040035030025020015010050time (s)

0 50045040035030025020015010050time (s)

150 250200

times10minus3

times10minus3

times10minus5

(ms

)

(ms

)

(rad

s

)

Figure 5 Acceleration Estimation Error

modelTherefore the proposed fault-tolerant controlmethodis more effective for the dynamic positioning ships in thepresence of high frequency wave disturbance

Remark 15 The upper bound on the maximum number ofsimultaneously isolable actuator faults in [5] is determined bythe Uniform Sub-Rank of the matrix which is related to thecontrol input matrix 119861 and thruster matrix 119879 By calculationwe can deduce that themethod in [5] can only solve the singleactuator fault So only Case 1 is considered in the comparisonpart

Mathematical Problems in Engineering 9

Table 2 Condition of faults

Time(s) 0-100 100-250 250-300 300-500Case 1 119870 = 1198701198911 119870 = 1198701198912 119870 = 1198701198912 119870 = 1198701198911Case 2 119870 = 1198701198913 119870 = 1198701198914 119870 = 1198701198914 119870 = 1198701198913Case 3 119870 = 1198701198913 119870 = 1198701198914 119870 = 1198701198914 119892 = 1198921 119870 = 1198701198913 119892 = 1198921

minus20

0

20

(m)

1

15

2

sigm

a

minus20minus10

01020

(m)

1

15

2sig

ma

minus1minus05

005

115

(rad

)

1

15

2

sigm

a

0 50045040035030025020015010050time (s)

0 50045040035030025020015010050time (s)

0 50045040035030025020015010050time (s)

motion in the X direction(Case1)switching signal

motion in the Y direction(Case1)switching signal

rotation about the Z axis(Case1)switching signal

Figure 6 Position and Heading of DP Vessel (Case 1)

Control performances of Case 2 and Case 3 wheremultiple actuator failures occur are demonstrated in Figures8 and 9 respectively Firstly it is shown that the signal 120590 canswitch to the corresponding controller quickly and accuratelyin the presence of actuator failures Secondly when a sensorfailure occurs at 250119904 although there is a slight fluctuation inthe transition progress the signal 120590 can switch to the stablecontroller and positions of the vessel are stabilized near theequilibrium point eventually

5 Conclusion

This paper presented a fault-tolerant supervisory controlmethod for dynamic positioning ships with wave-frequencymodel The proposed fault-tolerant control method does notneed the FDI mechanism due to the introduction of injectedsystem and supervisor It has beenproved that by introducinga nonlinear estimation error and virtual controller thedetectability property and matching property of the switchedsystem are guaranteed in the presence of actuator failuresand sensor failures Simulation results and comparisons havedemonstrated the effectiveness of the proposed fault-tolerant

minus20

0

20

40

(m)

motion in the X direction(S2)(Case1)motion in the X direction(S1)(Case1)

minus20

0

20

40

(m)

motion in the Y direction(S2)(Case1)motion in the Y direction(S1)(Case1)

minus1

0

1

2

(rad

)

rotation about the Z axis(S2)(Case1)rotation about the Z axis(S1)(Case1)

minus100

1020

minus100

1020

0 2010

0 2010

0 2010

minus050

051

15

0 50045040035030025020015010050time (s)

0 50045040035030025020015010050time (s)

0 50045040035030025020015010050time (s)

Figure 7 The tracking performance comparison of the pro-posed method (S1)(Case 1) and fault-tolerant control allocation [5](S2)(Case 1)

control scheme In the future we will consider the unknownfaults which is more practical in realistic situation Furtherwe will improve the switching logic to make the switchingsignal more accurate and stable for more serious failure

Data Availability

All the data supporting the conclusions of the study have beenprovided in Simulation and readers can access these data in[27 29]

Conflicts of Interest

The authors declare that there are no conflicts of interestrelated to this paper

Acknowledgments

This research was partially supported by the National Sci-ence Technology Support Program of China (Project no51609046)

10 Mathematical Problems in Engineering

0 50045040035030025020015010050time (s)

0 50045040035030025020015010050time (s)

0 50045040035030025020015010050time (s)

minus20

0

20

(m)

1

15

2

sigm

a

motion in the X direction(Case2)switching signal

minus20minus10

01020

(m)

1

15

2

sigm

a

motion in the Y direction(Case2)switching signal

minus1minus05

005

115

(rad

)

1

15

2

sigm

a

rotation about the Z axis(Case2)switching signal

Figure 8 Position and Heading of DP Vessel (Case 2)

0 50045040035030025020015010050time (s)

0 50045040035030025020015010050time (s)

0 50045040035030025020015010050time (s)

minus20minus10

01020

(m)

1234

sigm

a

motion in the x direction(Case3)switching signal

0(m)

2

4

sigm

a

motion in the y direction(Case3)switching signal

minus1

0

1

(rad

)

1234

sigm

a

rotation about the z axis(Case3)switching signal

Figure 9 Position and Heading of DP Vessel (Case 3)

References

[1] H Azmi and M J Khosrowjerdi ldquoRobust adaptive faulttolerant control for a class of Lipschitz nonlinear systems withactuator failure and disturbancesrdquo Proceedings of the Institutionof Mechanical Engineers Part I Journal of Systems and ControlEngineering vol 230 no 1 pp 13ndash22 2016

[2] S Dai L and J Zhao ldquoReliable H8 controller design for a classof uncertain linear systems with actuator failuresrdquo InternationalJournal of Control Automation amp Systems vol 6 no 6 pp 954ndash959 2008

[3] Y L Hao and G H Yang ldquoRobust fault tolerant controlbased on sliding mode method for uncertain linear systemswith quantizationrdquo Proceedings of the Institution of MechanicalEngineers Part I Journal of Systems amp Control Engineering vol52 no 5 pp 692ndash703 2013

[4] Z Zuo D W C Ho and Y Wang ldquoFault tolerant controlfor singular systems with actuator saturation and nonlinearperturbationrdquo Automatica vol 46 no 3 pp 569ndash576 2010

[5] A Cristofaro and T A Johansen ldquoFault tolerant controlallocation using unknown input observersrdquoAutomatica vol 50no 7 pp 1891ndash1897 2014

[6] T I Fossen Handbook of Marine Craft Hydrodynamics andMotion Control 2011

[7] A J Soslashrensen ldquoA survey of dynamic positioning controlsystemsrdquo Annual Reviews in Control vol 35 no 1 pp 123ndash1362011

[8] J Liu R Allen andH Yi ldquoShipmotion stabilizing control usinga combination of model predictive control and an adaptiveinput disturbance predictorrdquo Proceedings of the Institution ofMechanical Engineers Part I Journal of Systems and ControlEngineering vol 225 no 5 pp 591ndash602 2011

[9] M Tomera ldquoNonlinear observers design for multivariable shipmotion controlrdquo Polish Maritime Research vol 19 pp 50ndash562012

[10] A Hajivand and S H Mousavizadegan ldquoThe effect of memoryin observer design for a dp systemrdquo in Proceedings of theASME 2010 International Mechanical Engineering Congress andExposition IMECE 2010 pp 53ndash59 Canada November 2010

[11] M Fu J Ning and Y Wei ldquoFault-tolerant control of dynamicpositioning vessel after thruster failures using disturbance

Mathematical Problems in Engineering 11

decoupling methodsrdquo in Proceedings of the 2011 IEEE Interna-tional Conference on Automation and Logistics ICAL 2011 pp433ndash437 China August 2011

[12] F Benetazzo G Ippoliti S Longhi and P Raspa ldquoAdvancedcontrol for fault-tolerant dynamic positioning of an offshoresupply vesselrdquoOcean Engineering vol 106 pp 472ndash484 2015

[13] Y Lin and J Du ldquoFault-tolerant control for dynamic positioningof ships based on an iterative learning observerrdquo in Proceedingsof the 35th Chinese Control Conference CCC 2016 pp 1116ndash1122China July 2016

[14] M Chen B Jiang and R Cui ldquoActuator fault-tolerant controlof ocean surface vessels with input saturationrdquo InternationalJournal of Robust amp Nonlinear Control vol 26 no 3 pp 542ndash564 2016

[15] Y Su C Zheng andPMercorelli ldquoNonlinear PDFault-TolerantControl for Dynamic Positioning of Ships With ActuatorConstraintsrdquo IEEEASME Transactions onMechatronics vol 22no 3 pp 1132ndash1142 2017

[16] W-Z Yu H-X Xu and H Feng ldquoRobust adaptive fault-tolerant control of dynamic positioning vessel with position ref-erence system faults using backstepping designrdquo InternationalJournal of Robust and Nonlinear Control vol 28 no 2 pp 403ndash415 2018

[17] N T Dong Design of hybrid marine control systems for dynamicpositioning [PhD thesis] 2006

[18] J P Strand and T I Fossen ldquoNonlinear passive observer designfor ships with adaptive wave filteringrdquo in New Directions inNonlinear Observer Design vol 244 of Lecture Notes in Controland Information Sciences pp 113ndash134 Springer London UK1999

[19] S Saelid J G Balchen and N A Jenssen ldquoDesign and analysisof a dynamic positioning system based on kalman filtering andoptimal controlrdquo IEEE Transactions on Automatic Control vol28 no 3 pp 331ndash339 1983

[20] G Battistelli J P Hespanha and P Tesi ldquoSupervisory control ofswitched nonlinear systemsrdquo International Journal of AdaptiveControl and Signal Processing vol 26 no 8 pp 723ndash738 2012

[21] I R Bertaska andK DVonEllenrieder ldquoSupervisory switchingcontrol of an unmanned surface vehiclerdquo in Proceedings of theMTSIEEE Washington OCEANS 2015 USA October 2015

[22] J P Hespanha ldquoTutorial on supervisory controlrdquo Overview2002

[23] J P Hespanha D Liberzon and A S Morse ldquoSupervision ofintegral-input-to-state stabilizing controllersrdquo Automatica vol38 no 8 pp 1327ndash1335 2002

[24] M Skogvold Supervisory-switched control for dynamic position-ing systems in arctic areas [Msc thesis] Department of MarineTechnology 2010

[25] L VuD Chatterjee andD Liberzon ldquoInput-to-state stability ofswitched systems and switching adaptive controlrdquo Automaticavol 43 no 4 pp 639ndash646 2007

[26] ED Sontag ldquoInput to state stability basic concepts and resultsrdquoinNonlinear andOptimal ControlTheory pp 163ndash220 SpringerHeidelberg Berlin Germany 2008

[27] T D Nguyen A J Soslashrensen and S T Quek ldquoDesign of hybridcontroller for dynamic positioning from calm to extreme seaconditionsrdquo Automatica vol 43 no 5 pp 768ndash785 2007

[28] T I Fossen ldquoMarine Control Systems Guidance Navigationand Control of Ships Rigs and Underwater Vehiclesrdquo 2002

[29] T I Fossen and J P Strand ldquoPassive nonlinear observer designfor ships using Lyapunov methods full-scale experiments witha supply vesselrdquo Automatica vol 35 no 1 pp 3ndash16 1999

[30] G Xia X Shao H Wang et al ldquoPassive nonlinear observerdesign for special structure vesselsrdquoOceans IEEE pp 1ndash8 2013

[31] K P Lindegaard Acceleration Feedback in Dynamic Position-ing Fakultet for Informasjonsteknologi Matematikk Og Elek-troteknikk 2012

[32] H K Khalil Nonlinear Systems Prentice-Hall Upper SaddleRiver NJ USA 3rd edition 2002

[33] J P Hespanha and A S Morse ldquoCertainty equivalence impliesdetectabilityrdquo Systems amp Control Letters vol 36 no 1 pp 1ndash131999

[34] J P Hespanha D Liberzon and A S Morse ldquoBounds onthe number of switchings with scale-independent hysteresisApplications to supervisory controlrdquo Proceedings of the IEEEConference on Decision and Control vol 4 pp 3622ndash3627 2000

[35] M H Kim and D J Inman ldquoDevelopment of a robust non-linear observer for dynamic positioning of shipsrdquo Proceedings ofthe Institution of Mechanical Engineers Part I Journal of Systemsamp Control Engineering vol 218 no 1 pp 1ndash11 2004

[36] H Alwi C Edwards and C P Tan Fault Detection and Fault-Tolerant Control Using Sliding Modes Advances in IndustrialControl Springer London UK 2011

[37] J P Hespanha D Liberzon and A S Morse ldquoHysteresis-based switching algorithms for supervisory control of uncertainsystemsrdquo Automatica vol 39 no 2 pp 263ndash272 2003

[38] J P Hespanha and A S Morse ldquoStability of switched systemswith average dwell-timerdquo in Proceedings of the 38th IEEEConference on Decision and Control (CDC rsquo99) pp 2655ndash2660Phoenix Ariz USA December 1999

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6 Mathematical Problems in Engineering

where119882(120595) = diag(119869119879(120595) 119869119879(120595) 1198683times3)

119860 =[[[[[[[[[

119860119908 0 0 1198701205881 00 0 0 1198701205882 1198680 0 minus119860minus1119887 1198701205883 00 0 0 1198701205884 0119880120585 119880120578 119880119887 119880120588 119880V

]]]]]]]]]

119861 =[[[[[[[[[

1198701199101 119870]1

1198701199102 119870]2

1198701199103 119870]3

1198701199104 119870]4

0 0

]]]]]]]]]

(21)

The control gains and observer gains which appear in 119860are chosen such that the matrix 119860 is asymptotically stableThen there exists a matrix 119875 = 119875119879 gt 0 such that 119875119860 + 119860119879119875 =minus119876 for a given matrix 119876 = 119876119879 gt 0 Besides it should benoted that the injected system (20) is input-to-state stable ifthe system = 119882119879(120595)119860119882(120595)119911 is asymptotically stable [27]Differentiate 119882(120595)119911 along the trajectories 119911 and we have

119889 (119882 (120595) 119911)119889119905 = 119878119882 (120595) 119911 + 119860119882(120595) 119911 (22)

where 119878 = diag(119878119879 119878119879 0)

119878 = [[[

0 minus1 01 0 00 0 0

]]]

(23)

Consider a Lyapunov function candiate

119881 (119911) = 119911119879119882119879 (120595) 119875119882(120595) 119911 (24)

Differentiate it119889119881119889119905 = [(119878 + 119860)119882 (120595) 119911]119879 119875119882(120595) 119911 + (119882 (120595) 119911)119879

sdot 119875 [(119878 + 119860)119882(120595) 119911]119879 = (119882 (120595) 119911)119879sdot [(119860119879 + 119878119879) 119875 + 119875 (119878 + 119860)]119882 (120595) 119911= (119882 (120595) 119911)119879 [minus119876 + 119903119878119879119875 + 119903119875119878]119882(120595) 119911le [minus120582min (119876) + 2119903max120582max (119875)] 1003817100381710038171003817119882 (120595) 11991110038171003817100381710038172

(25)

When 119903max is sufficiently small the system =119882119879(120595)119860119882(120595)119911 is asymptotically stable Then the injectedsystem (20) is input-to-state stable with respect to 119890

Obviously when there is no sensor failure the dynamicpositioning vessel satisfies detectability which means thatthe state of plant eventually becomes small if the inputsand outputs are small Then according to Lemma 8 we candeduce that the switched system satisfies the DP

33 NEEO for Different Sensor Failures Sensor failures willdamage the detectability of the vessel In order to illustrate theproblem the following possible sensor failures are consideredin this paper

119910 = 119892 ∘ ℎ (119909) (26)

where 119892 the sensor fault function satisfies 119892 ∘ ℎ(0) = 0ie the detectability of plant is not satisfied The detectabilityof switched system will not be guaranteed if we designsupervisor in traditional wayOne reason is that themeasuredoutput is no longer a true reflection of the actual value

In this paper a nonlinear estimation error operator(NEEO) is introduced to solve the above problems Firstlythe process model (3) is remodeled as follows

= 119891 (119909 119879119901)119910 =

119910119899 = ℎ (119901 119909) no sensor failure

119910119891 = 119892119895 ∘ ℎ (119901 119909) 119895119905ℎ sensor failure(27)

where 119909 = [120585119879 120578119879 119887119879 ]119879]119879 For convenience of analysis sup-pose that 119894 isin 119875 119895 isin 119870 which represents the different actuatorfailures and sensor failures respectively The multiestimatoris designed for process (27)

119909119864119894 = 119865119894 (119909119864119894 119879119901 119910)119910119894 = ℎ119894 (119909119864119894)

119864119895 = 119865119895 (119909119864119895 119879119901 119910) = 119865119894 (119909119864119895 119879119901 119892minus1119895 (119910))119910119895 = ℎ119894 (119909119864119895) 119895 = 119875 times 119896 + 119894 119894 isin 119875

(28)

where each 119865119894 119894 isin 119875 is the observer based on actuator failurejust like Section 31

Theorem 10 has proved that the estimator can track thetrue output 119892minus1119895 (119910) so we define the nonlinear estimationerror as 119890119901 = 119910119901 ⊟ 119910 where ⊟ is NEEO and the specific formis as follows

119890119901 = 119910119901 ⊟ 119910 = 119890119894 = 119910119894 minus 119910 119894 isin 119875119890119895 = 119910119895 minus 119892minus1119895 (119910) 119895 isin 119870 (29)

No matter if the sensor is fault or not there will always bea particular 119890119901lowast that is convergent which means the multi-estimator satisfies MP Note that although the nonlinearityof 119890119901 is caused by the existence of 119892 it does not affectthe design of the front candidate controller (19) Moreoversensor failures only change the controllerrsquos external inputwithout any changing in the internal structureTherefore thefollowing result is concluded

Theorem 12 Consider the dynamic positioning vessel de-scribed by (3) under Assumptions 1 3 5 the multiestimatorgiven by (28) satisfiesMP and the switched systemalso satisfiesDP

Remark 13 Obviously to solve the problem of detectabilityloss caused by sensor faults the fault information is neededwhich can be achieved through fault reconstruction [36]

Mathematical Problems in Engineering 7

y

y

y

y

y

SwitchingLogic

MonitoringSignal

p

epyp

xE

xE

Controller(1)

Controller(2)

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

Controller Set

Observer(1)

Observer(k)

Observer(k + 1)

Observer(k + (P))

Multi minus Estimator

u

u

ActuatorFailure

Actuator

b

y

Disturbances SensorFailure

Sensor

Supervisor

yp

Figure 3 Fault-Tolerant Supervisory Control with NEEO for DPS

Initialize (t0)

(1 + ℎ) p(t) le (t)

Yes

No

(t) = ＬＡＧＣＨ p

Figure 4 Scale-Independent Hysteresis Switching Logic

34 Supervisory Control for Fault Tolerant Taking intoaccount the contents of all three previous parts this sectionemploys the scale-independent hysteresis switching logic tocontrol the DPS with actuator failures and sensor failuresFigure 3 shows the structure of the proposed supervi-sory control system The multiestimator comprises ((119875) times(119870)) models and the controller set comprises (119875) can-didate controllers corresponding to the same (119875) actuatorfailures

The hysteresis switching logic will be used to slow downswitching based on the observed growth of the estima-tion errors [37] Figure 4 shows the architecture of scale-independent hysteresis switching logic where 120587119901 physicallyis a monitoring signal in terms of the error norm as follows[22]

120587119901 = 120598 + 119890minus1205821199051205980 + int1199050119890minus120582(119905minus120591)120574 (10038171003817100381710038171003817119890119901 (120591)10038171003817100381710038171003817) 119889120591 (30)

where 120598 1205980 are nonnegative constants of which at least oneof them is strictly positive and 120582 is a constant nonnegativeforgetting factor and 120574 is class119870 function

Theorem 14 Consider the dynamic positioning vessel (27) themultiestimator is designed as (28) the candidate accelerationfeedback control law is designed as 119879119901 = (119879119870)+(119879119870119879119901 minus119870]5])with a virtual control law 119879119901 (19) and the NEEO is chosenas (29) Then under the supervisor with the scale-independenthysteresis switching logic all states of the DPS are bounded nomatter if the failures occur or not

Proof The scaled version monitoring signal is defined as120587119901(119905) = 120587119901(119905)119890120582119905 1205980 = 0 Based on Theorem 12 multiesti-mator satisfies MP so there exists 1198881 gt 0 such that 119890lowast119901 lt 1198881Since the function 120574 is Lipschiz continuous then

120587lowast119901 (119905) = 120598119890120582119905 + int1199050119890120582119904120574 (10038171003817100381710038171003817119890lowast119901 (119904)10038171003817100381710038171003817) 119889119904

le 120598119890120582119905 + 120574 (1198881) 1120582 119890120582119904100381610038161003816100381610038161199050 le 119890120582119905 [120598 + 120574 (1198881)120582 ]le 1198901205821199051198882

(31)

Using Lemma 9 we have

119873120590 le 119898 + 119898ln (1 + ℎ) ln( 120587119902 (119905)

min120587119901 (1199050))

le 119898 + 119898ln (1 + ℎ) ln[11988821198901205821199051205981198901205821199050 ]

le 119898 + 119898ln (1 + ℎ) ln(1198882120598 ) + 119905 minus 1199050

ln (1 + ℎ) 120582119898

(32)

8 Mathematical Problems in Engineering

int1199050119890120582119904120574 (1003816100381610038161003816119890120590(119904) (119904)1003816100381610038161003816) 119889119904 + 120598119890120582119905 minus 120598le 119898 [(1 + ℎ) 120583119902 (119905) minusmin 120583119901 (1199050)]le 119898 (1 + ℎ) 1198882119890120582119905

(33)

Then we can choose ln(1 + ℎ)120582119898 gt ln120583(120582119900 minus 120582) such thatthe switching signal stasfities average dwell-time [38] Sincethe choice of the virtual control law ensures the input-to-state stability of the injected system then the state of switchedinjected system 119909119864 has the property (Theorem 31[25])

1198901205821199051205721 (1003817100381710038171003817119909119864 (119905)1003817100381710038171003817) le 1198881205722 (1003817100381710038171003817119909119864 (0)1003817100381710038171003817)+ 119888 int1199050119890120582119904120574 (1003817100381710038171003817119890120590 (119904)1003817100381710038171003817) 119889119904

(34)

Substitute (33) into (34) and then1003817100381710038171003817119909119864 (119905)1003817100381710038171003817 le 120572minus11 (1198881205722 (1003817100381710038171003817119909119864 (0)1003817100381710038171003817) + 119888119898 (1 + ℎ) 1198882) = 1198883 (35)

which means that the state of injected system is boundedThen the output of injected system119910119901 = ℎ119901(119909119864) (forall119901 isin 119875) andswitched thrust force 119879119901120590 are bounded too Since 119892minus1(119910) =119910119901lowast minus 119890119901lowast then exist119901lowast isin 119875 119892minus1(119910) is bounded too Then by thedetectability of surface vessel with respect to the 119892minus1(119910) ieℎ(0) = 0 we candeduce that the state of plant is bounded4 Simulation Research

A model of a DP vessel with 4 channel thrusters and 2 mainthrusters is used in the simulation to demonstrate the perfor-mance of the supervisory control in case of actuator failuresand sensor failures For simplicity we choose four types ofactuator stuck failures K ≜ 1198701198911 1198701198912 1198701198913 1198701198914 where1198701198911 1198701198912 denote the single actuator failure and 1198701198913 1198701198914denote the multiple actuator failure The simulation param-eters are given in Table 1

As shown in Table 2 only actuator failures are consideredin Case 1 and Case 2 and both actuator failures and sensorfailures are considered in Case 3 The expected position andheading vectors are taken as 120578119889 = [0119898 0119898 0119903119886119889]119879 andthe initial states are taken as 120578(1199050) = [20119898 20119898 1119903119886119889]119879Simulation results of the proposed fault-tolerant controlstrategy are depicted in Figures 5ndash9

It can be observed from Figure 5 that the accelerationestimation errors are very small enough to be ignored So itis reasonable to take the acceleration estimation errors as aninput of the injected system

To illustrate the performance of the proposed controlscheme comparison results are presented by applying theFTC based on FDI in [5] The initial states for both controlschemes are chosen to be the same to guarantee a faircomparison As shown in the detailed time responses ofFigure 7 both methods can stabilize the vesselrsquos state to thereference value with small bounded errors but the methodin [5] cannot deal with high frequency wave force very wellwhich is the weakness of the control based on low-frequency

Table 1 Main simulation parameters

Symbol Parameter valuesM 107 times [3640004352490249752095]D 106 times [322000322-2780-27881506]119860119887 diag[100010001000]119861119887 diag[5000056000500000]119860119908(21) diag[-081-081-081]119860119908(22) diag[-018-018-018]119860119908 [03times3 1198683times3 119860119908(21) 119860119908(22)]119861119908 [[03times3 diag[16 12 025]]1198701198911 diag[011111]1198701198912 diag[111101]1198701198913 diag[011010]1198701198914 diag[101011]1198921 1198921(119904 119905) = 2119904 + 10 sin(119905)119879 [000011111100504418-357-4545-45]

minus015minus01

minus0050

005

Acceleration estimation error du (Case1)

minus1minus05

005

Acceleration estimation error dv (Case1)

minus004minus002

0002

Acceleration estimation error dr (Case1)

minus101

minus101

150 250200

150 250200

024

0 50045040035030025020015010050time (s)

0 50045040035030025020015010050time (s)

0 50045040035030025020015010050time (s)

150 250200

times10minus3

times10minus3

times10minus5

(ms

)

(ms

)

(rad

s

)

Figure 5 Acceleration Estimation Error

modelTherefore the proposed fault-tolerant controlmethodis more effective for the dynamic positioning ships in thepresence of high frequency wave disturbance

Remark 15 The upper bound on the maximum number ofsimultaneously isolable actuator faults in [5] is determined bythe Uniform Sub-Rank of the matrix which is related to thecontrol input matrix 119861 and thruster matrix 119879 By calculationwe can deduce that themethod in [5] can only solve the singleactuator fault So only Case 1 is considered in the comparisonpart

Mathematical Problems in Engineering 9

Table 2 Condition of faults

Time(s) 0-100 100-250 250-300 300-500Case 1 119870 = 1198701198911 119870 = 1198701198912 119870 = 1198701198912 119870 = 1198701198911Case 2 119870 = 1198701198913 119870 = 1198701198914 119870 = 1198701198914 119870 = 1198701198913Case 3 119870 = 1198701198913 119870 = 1198701198914 119870 = 1198701198914 119892 = 1198921 119870 = 1198701198913 119892 = 1198921

minus20

0

20

(m)

1

15

2

sigm

a

minus20minus10

01020

(m)

1

15

2sig

ma

minus1minus05

005

115

(rad

)

1

15

2

sigm

a

0 50045040035030025020015010050time (s)

0 50045040035030025020015010050time (s)

0 50045040035030025020015010050time (s)

motion in the X direction(Case1)switching signal

motion in the Y direction(Case1)switching signal

rotation about the Z axis(Case1)switching signal

Figure 6 Position and Heading of DP Vessel (Case 1)

Control performances of Case 2 and Case 3 wheremultiple actuator failures occur are demonstrated in Figures8 and 9 respectively Firstly it is shown that the signal 120590 canswitch to the corresponding controller quickly and accuratelyin the presence of actuator failures Secondly when a sensorfailure occurs at 250119904 although there is a slight fluctuation inthe transition progress the signal 120590 can switch to the stablecontroller and positions of the vessel are stabilized near theequilibrium point eventually

5 Conclusion

This paper presented a fault-tolerant supervisory controlmethod for dynamic positioning ships with wave-frequencymodel The proposed fault-tolerant control method does notneed the FDI mechanism due to the introduction of injectedsystem and supervisor It has beenproved that by introducinga nonlinear estimation error and virtual controller thedetectability property and matching property of the switchedsystem are guaranteed in the presence of actuator failuresand sensor failures Simulation results and comparisons havedemonstrated the effectiveness of the proposed fault-tolerant

minus20

0

20

40

(m)

motion in the X direction(S2)(Case1)motion in the X direction(S1)(Case1)

minus20

0

20

40

(m)

motion in the Y direction(S2)(Case1)motion in the Y direction(S1)(Case1)

minus1

0

1

2

(rad

)

rotation about the Z axis(S2)(Case1)rotation about the Z axis(S1)(Case1)

minus100

1020

minus100

1020

0 2010

0 2010

0 2010

minus050

051

15

0 50045040035030025020015010050time (s)

0 50045040035030025020015010050time (s)

0 50045040035030025020015010050time (s)

Figure 7 The tracking performance comparison of the pro-posed method (S1)(Case 1) and fault-tolerant control allocation [5](S2)(Case 1)

control scheme In the future we will consider the unknownfaults which is more practical in realistic situation Furtherwe will improve the switching logic to make the switchingsignal more accurate and stable for more serious failure

Data Availability

All the data supporting the conclusions of the study have beenprovided in Simulation and readers can access these data in[27 29]

Conflicts of Interest

The authors declare that there are no conflicts of interestrelated to this paper

Acknowledgments

This research was partially supported by the National Sci-ence Technology Support Program of China (Project no51609046)

10 Mathematical Problems in Engineering

0 50045040035030025020015010050time (s)

0 50045040035030025020015010050time (s)

0 50045040035030025020015010050time (s)

minus20

0

20

(m)

1

15

2

sigm

a

motion in the X direction(Case2)switching signal

minus20minus10

01020

(m)

1

15

2

sigm

a

motion in the Y direction(Case2)switching signal

minus1minus05

005

115

(rad

)

1

15

2

sigm

a

rotation about the Z axis(Case2)switching signal

Figure 8 Position and Heading of DP Vessel (Case 2)

0 50045040035030025020015010050time (s)

0 50045040035030025020015010050time (s)

0 50045040035030025020015010050time (s)

minus20minus10

01020

(m)

1234

sigm

a

motion in the x direction(Case3)switching signal

0(m)

2

4

sigm

a

motion in the y direction(Case3)switching signal

minus1

0

1

(rad

)

1234

sigm

a

rotation about the z axis(Case3)switching signal

Figure 9 Position and Heading of DP Vessel (Case 3)

References

[1] H Azmi and M J Khosrowjerdi ldquoRobust adaptive faulttolerant control for a class of Lipschitz nonlinear systems withactuator failure and disturbancesrdquo Proceedings of the Institutionof Mechanical Engineers Part I Journal of Systems and ControlEngineering vol 230 no 1 pp 13ndash22 2016

[2] S Dai L and J Zhao ldquoReliable H8 controller design for a classof uncertain linear systems with actuator failuresrdquo InternationalJournal of Control Automation amp Systems vol 6 no 6 pp 954ndash959 2008

[3] Y L Hao and G H Yang ldquoRobust fault tolerant controlbased on sliding mode method for uncertain linear systemswith quantizationrdquo Proceedings of the Institution of MechanicalEngineers Part I Journal of Systems amp Control Engineering vol52 no 5 pp 692ndash703 2013

[4] Z Zuo D W C Ho and Y Wang ldquoFault tolerant controlfor singular systems with actuator saturation and nonlinearperturbationrdquo Automatica vol 46 no 3 pp 569ndash576 2010

[5] A Cristofaro and T A Johansen ldquoFault tolerant controlallocation using unknown input observersrdquoAutomatica vol 50no 7 pp 1891ndash1897 2014

[6] T I Fossen Handbook of Marine Craft Hydrodynamics andMotion Control 2011

[7] A J Soslashrensen ldquoA survey of dynamic positioning controlsystemsrdquo Annual Reviews in Control vol 35 no 1 pp 123ndash1362011

[8] J Liu R Allen andH Yi ldquoShipmotion stabilizing control usinga combination of model predictive control and an adaptiveinput disturbance predictorrdquo Proceedings of the Institution ofMechanical Engineers Part I Journal of Systems and ControlEngineering vol 225 no 5 pp 591ndash602 2011

[9] M Tomera ldquoNonlinear observers design for multivariable shipmotion controlrdquo Polish Maritime Research vol 19 pp 50ndash562012

[10] A Hajivand and S H Mousavizadegan ldquoThe effect of memoryin observer design for a dp systemrdquo in Proceedings of theASME 2010 International Mechanical Engineering Congress andExposition IMECE 2010 pp 53ndash59 Canada November 2010

[11] M Fu J Ning and Y Wei ldquoFault-tolerant control of dynamicpositioning vessel after thruster failures using disturbance

Mathematical Problems in Engineering 11

decoupling methodsrdquo in Proceedings of the 2011 IEEE Interna-tional Conference on Automation and Logistics ICAL 2011 pp433ndash437 China August 2011

[12] F Benetazzo G Ippoliti S Longhi and P Raspa ldquoAdvancedcontrol for fault-tolerant dynamic positioning of an offshoresupply vesselrdquoOcean Engineering vol 106 pp 472ndash484 2015

[13] Y Lin and J Du ldquoFault-tolerant control for dynamic positioningof ships based on an iterative learning observerrdquo in Proceedingsof the 35th Chinese Control Conference CCC 2016 pp 1116ndash1122China July 2016

[14] M Chen B Jiang and R Cui ldquoActuator fault-tolerant controlof ocean surface vessels with input saturationrdquo InternationalJournal of Robust amp Nonlinear Control vol 26 no 3 pp 542ndash564 2016

[15] Y Su C Zheng andPMercorelli ldquoNonlinear PDFault-TolerantControl for Dynamic Positioning of Ships With ActuatorConstraintsrdquo IEEEASME Transactions onMechatronics vol 22no 3 pp 1132ndash1142 2017

[16] W-Z Yu H-X Xu and H Feng ldquoRobust adaptive fault-tolerant control of dynamic positioning vessel with position ref-erence system faults using backstepping designrdquo InternationalJournal of Robust and Nonlinear Control vol 28 no 2 pp 403ndash415 2018

[17] N T Dong Design of hybrid marine control systems for dynamicpositioning [PhD thesis] 2006

[18] J P Strand and T I Fossen ldquoNonlinear passive observer designfor ships with adaptive wave filteringrdquo in New Directions inNonlinear Observer Design vol 244 of Lecture Notes in Controland Information Sciences pp 113ndash134 Springer London UK1999

[19] S Saelid J G Balchen and N A Jenssen ldquoDesign and analysisof a dynamic positioning system based on kalman filtering andoptimal controlrdquo IEEE Transactions on Automatic Control vol28 no 3 pp 331ndash339 1983

[20] G Battistelli J P Hespanha and P Tesi ldquoSupervisory control ofswitched nonlinear systemsrdquo International Journal of AdaptiveControl and Signal Processing vol 26 no 8 pp 723ndash738 2012

[21] I R Bertaska andK DVonEllenrieder ldquoSupervisory switchingcontrol of an unmanned surface vehiclerdquo in Proceedings of theMTSIEEE Washington OCEANS 2015 USA October 2015

[22] J P Hespanha ldquoTutorial on supervisory controlrdquo Overview2002

[23] J P Hespanha D Liberzon and A S Morse ldquoSupervision ofintegral-input-to-state stabilizing controllersrdquo Automatica vol38 no 8 pp 1327ndash1335 2002

[24] M Skogvold Supervisory-switched control for dynamic position-ing systems in arctic areas [Msc thesis] Department of MarineTechnology 2010

[25] L VuD Chatterjee andD Liberzon ldquoInput-to-state stability ofswitched systems and switching adaptive controlrdquo Automaticavol 43 no 4 pp 639ndash646 2007

[26] ED Sontag ldquoInput to state stability basic concepts and resultsrdquoinNonlinear andOptimal ControlTheory pp 163ndash220 SpringerHeidelberg Berlin Germany 2008

[27] T D Nguyen A J Soslashrensen and S T Quek ldquoDesign of hybridcontroller for dynamic positioning from calm to extreme seaconditionsrdquo Automatica vol 43 no 5 pp 768ndash785 2007

[28] T I Fossen ldquoMarine Control Systems Guidance Navigationand Control of Ships Rigs and Underwater Vehiclesrdquo 2002

[29] T I Fossen and J P Strand ldquoPassive nonlinear observer designfor ships using Lyapunov methods full-scale experiments witha supply vesselrdquo Automatica vol 35 no 1 pp 3ndash16 1999

[30] G Xia X Shao H Wang et al ldquoPassive nonlinear observerdesign for special structure vesselsrdquoOceans IEEE pp 1ndash8 2013

[31] K P Lindegaard Acceleration Feedback in Dynamic Position-ing Fakultet for Informasjonsteknologi Matematikk Og Elek-troteknikk 2012

[32] H K Khalil Nonlinear Systems Prentice-Hall Upper SaddleRiver NJ USA 3rd edition 2002

[33] J P Hespanha and A S Morse ldquoCertainty equivalence impliesdetectabilityrdquo Systems amp Control Letters vol 36 no 1 pp 1ndash131999

[34] J P Hespanha D Liberzon and A S Morse ldquoBounds onthe number of switchings with scale-independent hysteresisApplications to supervisory controlrdquo Proceedings of the IEEEConference on Decision and Control vol 4 pp 3622ndash3627 2000

[35] M H Kim and D J Inman ldquoDevelopment of a robust non-linear observer for dynamic positioning of shipsrdquo Proceedings ofthe Institution of Mechanical Engineers Part I Journal of Systemsamp Control Engineering vol 218 no 1 pp 1ndash11 2004

[36] H Alwi C Edwards and C P Tan Fault Detection and Fault-Tolerant Control Using Sliding Modes Advances in IndustrialControl Springer London UK 2011

[37] J P Hespanha D Liberzon and A S Morse ldquoHysteresis-based switching algorithms for supervisory control of uncertainsystemsrdquo Automatica vol 39 no 2 pp 263ndash272 2003

[38] J P Hespanha and A S Morse ldquoStability of switched systemswith average dwell-timerdquo in Proceedings of the 38th IEEEConference on Decision and Control (CDC rsquo99) pp 2655ndash2660Phoenix Ariz USA December 1999

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Mathematical Problems in Engineering 7

y

y

y

y

y

SwitchingLogic

MonitoringSignal

p

epyp

xE

xE

Controller(1)

Controller(2)

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

middotmiddotmiddot

Controller Set

Observer(1)

Observer(k)

Observer(k + 1)

Observer(k + (P))

Multi minus Estimator

u

u

ActuatorFailure

Actuator

b

y

Disturbances SensorFailure

Sensor

Supervisor

yp

Figure 3 Fault-Tolerant Supervisory Control with NEEO for DPS

Initialize (t0)

(1 + ℎ) p(t) le (t)

Yes

No

(t) = ＬＡＧＣＨ p

Figure 4 Scale-Independent Hysteresis Switching Logic

34 Supervisory Control for Fault Tolerant Taking intoaccount the contents of all three previous parts this sectionemploys the scale-independent hysteresis switching logic tocontrol the DPS with actuator failures and sensor failuresFigure 3 shows the structure of the proposed supervi-sory control system The multiestimator comprises ((119875) times(119870)) models and the controller set comprises (119875) can-didate controllers corresponding to the same (119875) actuatorfailures

The hysteresis switching logic will be used to slow downswitching based on the observed growth of the estima-tion errors [37] Figure 4 shows the architecture of scale-independent hysteresis switching logic where 120587119901 physicallyis a monitoring signal in terms of the error norm as follows[22]

120587119901 = 120598 + 119890minus1205821199051205980 + int1199050119890minus120582(119905minus120591)120574 (10038171003817100381710038171003817119890119901 (120591)10038171003817100381710038171003817) 119889120591 (30)

where 120598 1205980 are nonnegative constants of which at least oneof them is strictly positive and 120582 is a constant nonnegativeforgetting factor and 120574 is class119870 function

Theorem 14 Consider the dynamic positioning vessel (27) themultiestimator is designed as (28) the candidate accelerationfeedback control law is designed as 119879119901 = (119879119870)+(119879119870119879119901 minus119870]5])with a virtual control law 119879119901 (19) and the NEEO is chosenas (29) Then under the supervisor with the scale-independenthysteresis switching logic all states of the DPS are bounded nomatter if the failures occur or not

Proof The scaled version monitoring signal is defined as120587119901(119905) = 120587119901(119905)119890120582119905 1205980 = 0 Based on Theorem 12 multiesti-mator satisfies MP so there exists 1198881 gt 0 such that 119890lowast119901 lt 1198881Since the function 120574 is Lipschiz continuous then

120587lowast119901 (119905) = 120598119890120582119905 + int1199050119890120582119904120574 (10038171003817100381710038171003817119890lowast119901 (119904)10038171003817100381710038171003817) 119889119904

le 120598119890120582119905 + 120574 (1198881) 1120582 119890120582119904100381610038161003816100381610038161199050 le 119890120582119905 [120598 + 120574 (1198881)120582 ]le 1198901205821199051198882

(31)

Using Lemma 9 we have

119873120590 le 119898 + 119898ln (1 + ℎ) ln( 120587119902 (119905)

min120587119901 (1199050))

le 119898 + 119898ln (1 + ℎ) ln[11988821198901205821199051205981198901205821199050 ]

le 119898 + 119898ln (1 + ℎ) ln(1198882120598 ) + 119905 minus 1199050

ln (1 + ℎ) 120582119898

(32)

8 Mathematical Problems in Engineering

int1199050119890120582119904120574 (1003816100381610038161003816119890120590(119904) (119904)1003816100381610038161003816) 119889119904 + 120598119890120582119905 minus 120598le 119898 [(1 + ℎ) 120583119902 (119905) minusmin 120583119901 (1199050)]le 119898 (1 + ℎ) 1198882119890120582119905

(33)

Then we can choose ln(1 + ℎ)120582119898 gt ln120583(120582119900 minus 120582) such thatthe switching signal stasfities average dwell-time [38] Sincethe choice of the virtual control law ensures the input-to-state stability of the injected system then the state of switchedinjected system 119909119864 has the property (Theorem 31[25])

1198901205821199051205721 (1003817100381710038171003817119909119864 (119905)1003817100381710038171003817) le 1198881205722 (1003817100381710038171003817119909119864 (0)1003817100381710038171003817)+ 119888 int1199050119890120582119904120574 (1003817100381710038171003817119890120590 (119904)1003817100381710038171003817) 119889119904

(34)

Substitute (33) into (34) and then1003817100381710038171003817119909119864 (119905)1003817100381710038171003817 le 120572minus11 (1198881205722 (1003817100381710038171003817119909119864 (0)1003817100381710038171003817) + 119888119898 (1 + ℎ) 1198882) = 1198883 (35)

which means that the state of injected system is boundedThen the output of injected system119910119901 = ℎ119901(119909119864) (forall119901 isin 119875) andswitched thrust force 119879119901120590 are bounded too Since 119892minus1(119910) =119910119901lowast minus 119890119901lowast then exist119901lowast isin 119875 119892minus1(119910) is bounded too Then by thedetectability of surface vessel with respect to the 119892minus1(119910) ieℎ(0) = 0 we candeduce that the state of plant is bounded4 Simulation Research

A model of a DP vessel with 4 channel thrusters and 2 mainthrusters is used in the simulation to demonstrate the perfor-mance of the supervisory control in case of actuator failuresand sensor failures For simplicity we choose four types ofactuator stuck failures K ≜ 1198701198911 1198701198912 1198701198913 1198701198914 where1198701198911 1198701198912 denote the single actuator failure and 1198701198913 1198701198914denote the multiple actuator failure The simulation param-eters are given in Table 1

As shown in Table 2 only actuator failures are consideredin Case 1 and Case 2 and both actuator failures and sensorfailures are considered in Case 3 The expected position andheading vectors are taken as 120578119889 = [0119898 0119898 0119903119886119889]119879 andthe initial states are taken as 120578(1199050) = [20119898 20119898 1119903119886119889]119879Simulation results of the proposed fault-tolerant controlstrategy are depicted in Figures 5ndash9

It can be observed from Figure 5 that the accelerationestimation errors are very small enough to be ignored So itis reasonable to take the acceleration estimation errors as aninput of the injected system

To illustrate the performance of the proposed controlscheme comparison results are presented by applying theFTC based on FDI in [5] The initial states for both controlschemes are chosen to be the same to guarantee a faircomparison As shown in the detailed time responses ofFigure 7 both methods can stabilize the vesselrsquos state to thereference value with small bounded errors but the methodin [5] cannot deal with high frequency wave force very wellwhich is the weakness of the control based on low-frequency

Table 1 Main simulation parameters

Symbol Parameter valuesM 107 times [3640004352490249752095]D 106 times [322000322-2780-27881506]119860119887 diag[100010001000]119861119887 diag[5000056000500000]119860119908(21) diag[-081-081-081]119860119908(22) diag[-018-018-018]119860119908 [03times3 1198683times3 119860119908(21) 119860119908(22)]119861119908 [[03times3 diag[16 12 025]]1198701198911 diag[011111]1198701198912 diag[111101]1198701198913 diag[011010]1198701198914 diag[101011]1198921 1198921(119904 119905) = 2119904 + 10 sin(119905)119879 [000011111100504418-357-4545-45]

minus015minus01

minus0050

005

Acceleration estimation error du (Case1)

minus1minus05

005

Acceleration estimation error dv (Case1)

minus004minus002

0002

Acceleration estimation error dr (Case1)

minus101

minus101

150 250200

150 250200

024

0 50045040035030025020015010050time (s)

0 50045040035030025020015010050time (s)

0 50045040035030025020015010050time (s)

150 250200

times10minus3

times10minus3

times10minus5

(ms

)

(ms

)

(rad

s

)

Figure 5 Acceleration Estimation Error

modelTherefore the proposed fault-tolerant controlmethodis more effective for the dynamic positioning ships in thepresence of high frequency wave disturbance

Remark 15 The upper bound on the maximum number ofsimultaneously isolable actuator faults in [5] is determined bythe Uniform Sub-Rank of the matrix which is related to thecontrol input matrix 119861 and thruster matrix 119879 By calculationwe can deduce that themethod in [5] can only solve the singleactuator fault So only Case 1 is considered in the comparisonpart

Mathematical Problems in Engineering 9

Table 2 Condition of faults

Time(s) 0-100 100-250 250-300 300-500Case 1 119870 = 1198701198911 119870 = 1198701198912 119870 = 1198701198912 119870 = 1198701198911Case 2 119870 = 1198701198913 119870 = 1198701198914 119870 = 1198701198914 119870 = 1198701198913Case 3 119870 = 1198701198913 119870 = 1198701198914 119870 = 1198701198914 119892 = 1198921 119870 = 1198701198913 119892 = 1198921

minus20

0

20

(m)

1

15

2

sigm

a

minus20minus10

01020

(m)

1

15

2sig

ma

minus1minus05

005

115

(rad

)

1

15

2

sigm

a

0 50045040035030025020015010050time (s)

0 50045040035030025020015010050time (s)

0 50045040035030025020015010050time (s)

motion in the X direction(Case1)switching signal

motion in the Y direction(Case1)switching signal

rotation about the Z axis(Case1)switching signal

Figure 6 Position and Heading of DP Vessel (Case 1)

Control performances of Case 2 and Case 3 wheremultiple actuator failures occur are demonstrated in Figures8 and 9 respectively Firstly it is shown that the signal 120590 canswitch to the corresponding controller quickly and accuratelyin the presence of actuator failures Secondly when a sensorfailure occurs at 250119904 although there is a slight fluctuation inthe transition progress the signal 120590 can switch to the stablecontroller and positions of the vessel are stabilized near theequilibrium point eventually

5 Conclusion

This paper presented a fault-tolerant supervisory controlmethod for dynamic positioning ships with wave-frequencymodel The proposed fault-tolerant control method does notneed the FDI mechanism due to the introduction of injectedsystem and supervisor It has beenproved that by introducinga nonlinear estimation error and virtual controller thedetectability property and matching property of the switchedsystem are guaranteed in the presence of actuator failuresand sensor failures Simulation results and comparisons havedemonstrated the effectiveness of the proposed fault-tolerant

minus20

0

20

40

(m)

motion in the X direction(S2)(Case1)motion in the X direction(S1)(Case1)

minus20

0

20

40

(m)

motion in the Y direction(S2)(Case1)motion in the Y direction(S1)(Case1)

minus1

0

1

2

(rad

)

rotation about the Z axis(S2)(Case1)rotation about the Z axis(S1)(Case1)

minus100

1020

minus100

1020

0 2010

0 2010

0 2010

minus050

051

15

0 50045040035030025020015010050time (s)

0 50045040035030025020015010050time (s)

0 50045040035030025020015010050time (s)

Figure 7 The tracking performance comparison of the pro-posed method (S1)(Case 1) and fault-tolerant control allocation [5](S2)(Case 1)

control scheme In the future we will consider the unknownfaults which is more practical in realistic situation Furtherwe will improve the switching logic to make the switchingsignal more accurate and stable for more serious failure

Data Availability

All the data supporting the conclusions of the study have beenprovided in Simulation and readers can access these data in[27 29]

Conflicts of Interest

The authors declare that there are no conflicts of interestrelated to this paper

Acknowledgments

This research was partially supported by the National Sci-ence Technology Support Program of China (Project no51609046)

10 Mathematical Problems in Engineering

0 50045040035030025020015010050time (s)

0 50045040035030025020015010050time (s)

0 50045040035030025020015010050time (s)

minus20

0

20

(m)

1

15

2

sigm

a

motion in the X direction(Case2)switching signal

minus20minus10

01020

(m)

1

15

2

sigm

a

motion in the Y direction(Case2)switching signal

minus1minus05

005

115

(rad

)

1

15

2

sigm

a

rotation about the Z axis(Case2)switching signal

Figure 8 Position and Heading of DP Vessel (Case 2)

0 50045040035030025020015010050time (s)

0 50045040035030025020015010050time (s)

0 50045040035030025020015010050time (s)

minus20minus10

01020

(m)

1234

sigm

a

motion in the x direction(Case3)switching signal

0(m)

2

4

sigm

a

motion in the y direction(Case3)switching signal

minus1

0

1

(rad

)

1234

sigm

a

rotation about the z axis(Case3)switching signal

Figure 9 Position and Heading of DP Vessel (Case 3)

References

[1] H Azmi and M J Khosrowjerdi ldquoRobust adaptive faulttolerant control for a class of Lipschitz nonlinear systems withactuator failure and disturbancesrdquo Proceedings of the Institutionof Mechanical Engineers Part I Journal of Systems and ControlEngineering vol 230 no 1 pp 13ndash22 2016

[2] S Dai L and J Zhao ldquoReliable H8 controller design for a classof uncertain linear systems with actuator failuresrdquo InternationalJournal of Control Automation amp Systems vol 6 no 6 pp 954ndash959 2008

[3] Y L Hao and G H Yang ldquoRobust fault tolerant controlbased on sliding mode method for uncertain linear systemswith quantizationrdquo Proceedings of the Institution of MechanicalEngineers Part I Journal of Systems amp Control Engineering vol52 no 5 pp 692ndash703 2013

[4] Z Zuo D W C Ho and Y Wang ldquoFault tolerant controlfor singular systems with actuator saturation and nonlinearperturbationrdquo Automatica vol 46 no 3 pp 569ndash576 2010

[5] A Cristofaro and T A Johansen ldquoFault tolerant controlallocation using unknown input observersrdquoAutomatica vol 50no 7 pp 1891ndash1897 2014

[6] T I Fossen Handbook of Marine Craft Hydrodynamics andMotion Control 2011

[7] A J Soslashrensen ldquoA survey of dynamic positioning controlsystemsrdquo Annual Reviews in Control vol 35 no 1 pp 123ndash1362011

[8] J Liu R Allen andH Yi ldquoShipmotion stabilizing control usinga combination of model predictive control and an adaptiveinput disturbance predictorrdquo Proceedings of the Institution ofMechanical Engineers Part I Journal of Systems and ControlEngineering vol 225 no 5 pp 591ndash602 2011

[9] M Tomera ldquoNonlinear observers design for multivariable shipmotion controlrdquo Polish Maritime Research vol 19 pp 50ndash562012

[10] A Hajivand and S H Mousavizadegan ldquoThe effect of memoryin observer design for a dp systemrdquo in Proceedings of theASME 2010 International Mechanical Engineering Congress andExposition IMECE 2010 pp 53ndash59 Canada November 2010

[11] M Fu J Ning and Y Wei ldquoFault-tolerant control of dynamicpositioning vessel after thruster failures using disturbance

Mathematical Problems in Engineering 11

decoupling methodsrdquo in Proceedings of the 2011 IEEE Interna-tional Conference on Automation and Logistics ICAL 2011 pp433ndash437 China August 2011

[12] F Benetazzo G Ippoliti S Longhi and P Raspa ldquoAdvancedcontrol for fault-tolerant dynamic positioning of an offshoresupply vesselrdquoOcean Engineering vol 106 pp 472ndash484 2015

[13] Y Lin and J Du ldquoFault-tolerant control for dynamic positioningof ships based on an iterative learning observerrdquo in Proceedingsof the 35th Chinese Control Conference CCC 2016 pp 1116ndash1122China July 2016

[14] M Chen B Jiang and R Cui ldquoActuator fault-tolerant controlof ocean surface vessels with input saturationrdquo InternationalJournal of Robust amp Nonlinear Control vol 26 no 3 pp 542ndash564 2016

[15] Y Su C Zheng andPMercorelli ldquoNonlinear PDFault-TolerantControl for Dynamic Positioning of Ships With ActuatorConstraintsrdquo IEEEASME Transactions onMechatronics vol 22no 3 pp 1132ndash1142 2017

[16] W-Z Yu H-X Xu and H Feng ldquoRobust adaptive fault-tolerant control of dynamic positioning vessel with position ref-erence system faults using backstepping designrdquo InternationalJournal of Robust and Nonlinear Control vol 28 no 2 pp 403ndash415 2018

[17] N T Dong Design of hybrid marine control systems for dynamicpositioning [PhD thesis] 2006

[18] J P Strand and T I Fossen ldquoNonlinear passive observer designfor ships with adaptive wave filteringrdquo in New Directions inNonlinear Observer Design vol 244 of Lecture Notes in Controland Information Sciences pp 113ndash134 Springer London UK1999

[19] S Saelid J G Balchen and N A Jenssen ldquoDesign and analysisof a dynamic positioning system based on kalman filtering andoptimal controlrdquo IEEE Transactions on Automatic Control vol28 no 3 pp 331ndash339 1983

[20] G Battistelli J P Hespanha and P Tesi ldquoSupervisory control ofswitched nonlinear systemsrdquo International Journal of AdaptiveControl and Signal Processing vol 26 no 8 pp 723ndash738 2012

[21] I R Bertaska andK DVonEllenrieder ldquoSupervisory switchingcontrol of an unmanned surface vehiclerdquo in Proceedings of theMTSIEEE Washington OCEANS 2015 USA October 2015

[22] J P Hespanha ldquoTutorial on supervisory controlrdquo Overview2002

[23] J P Hespanha D Liberzon and A S Morse ldquoSupervision ofintegral-input-to-state stabilizing controllersrdquo Automatica vol38 no 8 pp 1327ndash1335 2002

[24] M Skogvold Supervisory-switched control for dynamic position-ing systems in arctic areas [Msc thesis] Department of MarineTechnology 2010

[25] L VuD Chatterjee andD Liberzon ldquoInput-to-state stability ofswitched systems and switching adaptive controlrdquo Automaticavol 43 no 4 pp 639ndash646 2007

[26] ED Sontag ldquoInput to state stability basic concepts and resultsrdquoinNonlinear andOptimal ControlTheory pp 163ndash220 SpringerHeidelberg Berlin Germany 2008

[27] T D Nguyen A J Soslashrensen and S T Quek ldquoDesign of hybridcontroller for dynamic positioning from calm to extreme seaconditionsrdquo Automatica vol 43 no 5 pp 768ndash785 2007

[28] T I Fossen ldquoMarine Control Systems Guidance Navigationand Control of Ships Rigs and Underwater Vehiclesrdquo 2002

[29] T I Fossen and J P Strand ldquoPassive nonlinear observer designfor ships using Lyapunov methods full-scale experiments witha supply vesselrdquo Automatica vol 35 no 1 pp 3ndash16 1999

[30] G Xia X Shao H Wang et al ldquoPassive nonlinear observerdesign for special structure vesselsrdquoOceans IEEE pp 1ndash8 2013

[31] K P Lindegaard Acceleration Feedback in Dynamic Position-ing Fakultet for Informasjonsteknologi Matematikk Og Elek-troteknikk 2012

[32] H K Khalil Nonlinear Systems Prentice-Hall Upper SaddleRiver NJ USA 3rd edition 2002

[33] J P Hespanha and A S Morse ldquoCertainty equivalence impliesdetectabilityrdquo Systems amp Control Letters vol 36 no 1 pp 1ndash131999

[34] J P Hespanha D Liberzon and A S Morse ldquoBounds onthe number of switchings with scale-independent hysteresisApplications to supervisory controlrdquo Proceedings of the IEEEConference on Decision and Control vol 4 pp 3622ndash3627 2000

[35] M H Kim and D J Inman ldquoDevelopment of a robust non-linear observer for dynamic positioning of shipsrdquo Proceedings ofthe Institution of Mechanical Engineers Part I Journal of Systemsamp Control Engineering vol 218 no 1 pp 1ndash11 2004

[36] H Alwi C Edwards and C P Tan Fault Detection and Fault-Tolerant Control Using Sliding Modes Advances in IndustrialControl Springer London UK 2011

[37] J P Hespanha D Liberzon and A S Morse ldquoHysteresis-based switching algorithms for supervisory control of uncertainsystemsrdquo Automatica vol 39 no 2 pp 263ndash272 2003

[38] J P Hespanha and A S Morse ldquoStability of switched systemswith average dwell-timerdquo in Proceedings of the 38th IEEEConference on Decision and Control (CDC rsquo99) pp 2655ndash2660Phoenix Ariz USA December 1999

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Mathematical Problems in Engineering

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8 Mathematical Problems in Engineering

int1199050119890120582119904120574 (1003816100381610038161003816119890120590(119904) (119904)1003816100381610038161003816) 119889119904 + 120598119890120582119905 minus 120598le 119898 [(1 + ℎ) 120583119902 (119905) minusmin 120583119901 (1199050)]le 119898 (1 + ℎ) 1198882119890120582119905

(33)

Then we can choose ln(1 + ℎ)120582119898 gt ln120583(120582119900 minus 120582) such thatthe switching signal stasfities average dwell-time [38] Sincethe choice of the virtual control law ensures the input-to-state stability of the injected system then the state of switchedinjected system 119909119864 has the property (Theorem 31[25])

1198901205821199051205721 (1003817100381710038171003817119909119864 (119905)1003817100381710038171003817) le 1198881205722 (1003817100381710038171003817119909119864 (0)1003817100381710038171003817)+ 119888 int1199050119890120582119904120574 (1003817100381710038171003817119890120590 (119904)1003817100381710038171003817) 119889119904

(34)

Substitute (33) into (34) and then1003817100381710038171003817119909119864 (119905)1003817100381710038171003817 le 120572minus11 (1198881205722 (1003817100381710038171003817119909119864 (0)1003817100381710038171003817) + 119888119898 (1 + ℎ) 1198882) = 1198883 (35)

which means that the state of injected system is boundedThen the output of injected system119910119901 = ℎ119901(119909119864) (forall119901 isin 119875) andswitched thrust force 119879119901120590 are bounded too Since 119892minus1(119910) =119910119901lowast minus 119890119901lowast then exist119901lowast isin 119875 119892minus1(119910) is bounded too Then by thedetectability of surface vessel with respect to the 119892minus1(119910) ieℎ(0) = 0 we candeduce that the state of plant is bounded4 Simulation Research

A model of a DP vessel with 4 channel thrusters and 2 mainthrusters is used in the simulation to demonstrate the perfor-mance of the supervisory control in case of actuator failuresand sensor failures For simplicity we choose four types ofactuator stuck failures K ≜ 1198701198911 1198701198912 1198701198913 1198701198914 where1198701198911 1198701198912 denote the single actuator failure and 1198701198913 1198701198914denote the multiple actuator failure The simulation param-eters are given in Table 1

As shown in Table 2 only actuator failures are consideredin Case 1 and Case 2 and both actuator failures and sensorfailures are considered in Case 3 The expected position andheading vectors are taken as 120578119889 = [0119898 0119898 0119903119886119889]119879 andthe initial states are taken as 120578(1199050) = [20119898 20119898 1119903119886119889]119879Simulation results of the proposed fault-tolerant controlstrategy are depicted in Figures 5ndash9

It can be observed from Figure 5 that the accelerationestimation errors are very small enough to be ignored So itis reasonable to take the acceleration estimation errors as aninput of the injected system

To illustrate the performance of the proposed controlscheme comparison results are presented by applying theFTC based on FDI in [5] The initial states for both controlschemes are chosen to be the same to guarantee a faircomparison As shown in the detailed time responses ofFigure 7 both methods can stabilize the vesselrsquos state to thereference value with small bounded errors but the methodin [5] cannot deal with high frequency wave force very wellwhich is the weakness of the control based on low-frequency

Table 1 Main simulation parameters

Symbol Parameter valuesM 107 times [3640004352490249752095]D 106 times [322000322-2780-27881506]119860119887 diag[100010001000]119861119887 diag[5000056000500000]119860119908(21) diag[-081-081-081]119860119908(22) diag[-018-018-018]119860119908 [03times3 1198683times3 119860119908(21) 119860119908(22)]119861119908 [[03times3 diag[16 12 025]]1198701198911 diag[011111]1198701198912 diag[111101]1198701198913 diag[011010]1198701198914 diag[101011]1198921 1198921(119904 119905) = 2119904 + 10 sin(119905)119879 [000011111100504418-357-4545-45]

minus015minus01

minus0050

005

Acceleration estimation error du (Case1)

minus1minus05

005

Acceleration estimation error dv (Case1)

minus004minus002

0002

Acceleration estimation error dr (Case1)

minus101

minus101

150 250200

150 250200

024

0 50045040035030025020015010050time (s)

0 50045040035030025020015010050time (s)

0 50045040035030025020015010050time (s)

150 250200

times10minus3

times10minus3

times10minus5

(ms

)

(ms

)

(rad

s

)

Figure 5 Acceleration Estimation Error

modelTherefore the proposed fault-tolerant controlmethodis more effective for the dynamic positioning ships in thepresence of high frequency wave disturbance

Remark 15 The upper bound on the maximum number ofsimultaneously isolable actuator faults in [5] is determined bythe Uniform Sub-Rank of the matrix which is related to thecontrol input matrix 119861 and thruster matrix 119879 By calculationwe can deduce that themethod in [5] can only solve the singleactuator fault So only Case 1 is considered in the comparisonpart

Mathematical Problems in Engineering 9

Table 2 Condition of faults

Time(s) 0-100 100-250 250-300 300-500Case 1 119870 = 1198701198911 119870 = 1198701198912 119870 = 1198701198912 119870 = 1198701198911Case 2 119870 = 1198701198913 119870 = 1198701198914 119870 = 1198701198914 119870 = 1198701198913Case 3 119870 = 1198701198913 119870 = 1198701198914 119870 = 1198701198914 119892 = 1198921 119870 = 1198701198913 119892 = 1198921

minus20

0

20

(m)

1

15

2

sigm

a

minus20minus10

01020

(m)

1

15

2sig

ma

minus1minus05

005

115

(rad

)

1

15

2

sigm

a

0 50045040035030025020015010050time (s)

0 50045040035030025020015010050time (s)

0 50045040035030025020015010050time (s)

motion in the X direction(Case1)switching signal

motion in the Y direction(Case1)switching signal

rotation about the Z axis(Case1)switching signal

Figure 6 Position and Heading of DP Vessel (Case 1)

Control performances of Case 2 and Case 3 wheremultiple actuator failures occur are demonstrated in Figures8 and 9 respectively Firstly it is shown that the signal 120590 canswitch to the corresponding controller quickly and accuratelyin the presence of actuator failures Secondly when a sensorfailure occurs at 250119904 although there is a slight fluctuation inthe transition progress the signal 120590 can switch to the stablecontroller and positions of the vessel are stabilized near theequilibrium point eventually

5 Conclusion

This paper presented a fault-tolerant supervisory controlmethod for dynamic positioning ships with wave-frequencymodel The proposed fault-tolerant control method does notneed the FDI mechanism due to the introduction of injectedsystem and supervisor It has beenproved that by introducinga nonlinear estimation error and virtual controller thedetectability property and matching property of the switchedsystem are guaranteed in the presence of actuator failuresand sensor failures Simulation results and comparisons havedemonstrated the effectiveness of the proposed fault-tolerant

minus20

0

20

40

(m)

motion in the X direction(S2)(Case1)motion in the X direction(S1)(Case1)

minus20

0

20

40

(m)

motion in the Y direction(S2)(Case1)motion in the Y direction(S1)(Case1)

minus1

0

1

2

(rad

)

rotation about the Z axis(S2)(Case1)rotation about the Z axis(S1)(Case1)

minus100

1020

minus100

1020

0 2010

0 2010

0 2010

minus050

051

15

0 50045040035030025020015010050time (s)

0 50045040035030025020015010050time (s)

0 50045040035030025020015010050time (s)

Figure 7 The tracking performance comparison of the pro-posed method (S1)(Case 1) and fault-tolerant control allocation [5](S2)(Case 1)

control scheme In the future we will consider the unknownfaults which is more practical in realistic situation Furtherwe will improve the switching logic to make the switchingsignal more accurate and stable for more serious failure

Data Availability

All the data supporting the conclusions of the study have beenprovided in Simulation and readers can access these data in[27 29]

Conflicts of Interest

The authors declare that there are no conflicts of interestrelated to this paper

Acknowledgments

This research was partially supported by the National Sci-ence Technology Support Program of China (Project no51609046)

10 Mathematical Problems in Engineering

0 50045040035030025020015010050time (s)

0 50045040035030025020015010050time (s)

0 50045040035030025020015010050time (s)

minus20

0

20

(m)

1

15

2

sigm

a

motion in the X direction(Case2)switching signal

minus20minus10

01020

(m)

1

15

2

sigm

a

motion in the Y direction(Case2)switching signal

minus1minus05

005

115

(rad

)

1

15

2

sigm

a

rotation about the Z axis(Case2)switching signal

Figure 8 Position and Heading of DP Vessel (Case 2)

0 50045040035030025020015010050time (s)

0 50045040035030025020015010050time (s)

0 50045040035030025020015010050time (s)

minus20minus10

01020

(m)

1234

sigm

a

motion in the x direction(Case3)switching signal

0(m)

2

4

sigm

a

motion in the y direction(Case3)switching signal

minus1

0

1

(rad

)

1234

sigm

a

rotation about the z axis(Case3)switching signal

Figure 9 Position and Heading of DP Vessel (Case 3)

References

[1] H Azmi and M J Khosrowjerdi ldquoRobust adaptive faulttolerant control for a class of Lipschitz nonlinear systems withactuator failure and disturbancesrdquo Proceedings of the Institutionof Mechanical Engineers Part I Journal of Systems and ControlEngineering vol 230 no 1 pp 13ndash22 2016

[2] S Dai L and J Zhao ldquoReliable H8 controller design for a classof uncertain linear systems with actuator failuresrdquo InternationalJournal of Control Automation amp Systems vol 6 no 6 pp 954ndash959 2008

[3] Y L Hao and G H Yang ldquoRobust fault tolerant controlbased on sliding mode method for uncertain linear systemswith quantizationrdquo Proceedings of the Institution of MechanicalEngineers Part I Journal of Systems amp Control Engineering vol52 no 5 pp 692ndash703 2013

[4] Z Zuo D W C Ho and Y Wang ldquoFault tolerant controlfor singular systems with actuator saturation and nonlinearperturbationrdquo Automatica vol 46 no 3 pp 569ndash576 2010

[5] A Cristofaro and T A Johansen ldquoFault tolerant controlallocation using unknown input observersrdquoAutomatica vol 50no 7 pp 1891ndash1897 2014

[6] T I Fossen Handbook of Marine Craft Hydrodynamics andMotion Control 2011

[7] A J Soslashrensen ldquoA survey of dynamic positioning controlsystemsrdquo Annual Reviews in Control vol 35 no 1 pp 123ndash1362011

[8] J Liu R Allen andH Yi ldquoShipmotion stabilizing control usinga combination of model predictive control and an adaptiveinput disturbance predictorrdquo Proceedings of the Institution ofMechanical Engineers Part I Journal of Systems and ControlEngineering vol 225 no 5 pp 591ndash602 2011

[9] M Tomera ldquoNonlinear observers design for multivariable shipmotion controlrdquo Polish Maritime Research vol 19 pp 50ndash562012

[10] A Hajivand and S H Mousavizadegan ldquoThe effect of memoryin observer design for a dp systemrdquo in Proceedings of theASME 2010 International Mechanical Engineering Congress andExposition IMECE 2010 pp 53ndash59 Canada November 2010

[11] M Fu J Ning and Y Wei ldquoFault-tolerant control of dynamicpositioning vessel after thruster failures using disturbance

Mathematical Problems in Engineering 11

decoupling methodsrdquo in Proceedings of the 2011 IEEE Interna-tional Conference on Automation and Logistics ICAL 2011 pp433ndash437 China August 2011

[12] F Benetazzo G Ippoliti S Longhi and P Raspa ldquoAdvancedcontrol for fault-tolerant dynamic positioning of an offshoresupply vesselrdquoOcean Engineering vol 106 pp 472ndash484 2015

[13] Y Lin and J Du ldquoFault-tolerant control for dynamic positioningof ships based on an iterative learning observerrdquo in Proceedingsof the 35th Chinese Control Conference CCC 2016 pp 1116ndash1122China July 2016

[14] M Chen B Jiang and R Cui ldquoActuator fault-tolerant controlof ocean surface vessels with input saturationrdquo InternationalJournal of Robust amp Nonlinear Control vol 26 no 3 pp 542ndash564 2016

[15] Y Su C Zheng andPMercorelli ldquoNonlinear PDFault-TolerantControl for Dynamic Positioning of Ships With ActuatorConstraintsrdquo IEEEASME Transactions onMechatronics vol 22no 3 pp 1132ndash1142 2017

[16] W-Z Yu H-X Xu and H Feng ldquoRobust adaptive fault-tolerant control of dynamic positioning vessel with position ref-erence system faults using backstepping designrdquo InternationalJournal of Robust and Nonlinear Control vol 28 no 2 pp 403ndash415 2018

[17] N T Dong Design of hybrid marine control systems for dynamicpositioning [PhD thesis] 2006

[18] J P Strand and T I Fossen ldquoNonlinear passive observer designfor ships with adaptive wave filteringrdquo in New Directions inNonlinear Observer Design vol 244 of Lecture Notes in Controland Information Sciences pp 113ndash134 Springer London UK1999

[19] S Saelid J G Balchen and N A Jenssen ldquoDesign and analysisof a dynamic positioning system based on kalman filtering andoptimal controlrdquo IEEE Transactions on Automatic Control vol28 no 3 pp 331ndash339 1983

[20] G Battistelli J P Hespanha and P Tesi ldquoSupervisory control ofswitched nonlinear systemsrdquo International Journal of AdaptiveControl and Signal Processing vol 26 no 8 pp 723ndash738 2012

[21] I R Bertaska andK DVonEllenrieder ldquoSupervisory switchingcontrol of an unmanned surface vehiclerdquo in Proceedings of theMTSIEEE Washington OCEANS 2015 USA October 2015

[22] J P Hespanha ldquoTutorial on supervisory controlrdquo Overview2002

[23] J P Hespanha D Liberzon and A S Morse ldquoSupervision ofintegral-input-to-state stabilizing controllersrdquo Automatica vol38 no 8 pp 1327ndash1335 2002

[24] M Skogvold Supervisory-switched control for dynamic position-ing systems in arctic areas [Msc thesis] Department of MarineTechnology 2010

[25] L VuD Chatterjee andD Liberzon ldquoInput-to-state stability ofswitched systems and switching adaptive controlrdquo Automaticavol 43 no 4 pp 639ndash646 2007

[26] ED Sontag ldquoInput to state stability basic concepts and resultsrdquoinNonlinear andOptimal ControlTheory pp 163ndash220 SpringerHeidelberg Berlin Germany 2008

[27] T D Nguyen A J Soslashrensen and S T Quek ldquoDesign of hybridcontroller for dynamic positioning from calm to extreme seaconditionsrdquo Automatica vol 43 no 5 pp 768ndash785 2007

[28] T I Fossen ldquoMarine Control Systems Guidance Navigationand Control of Ships Rigs and Underwater Vehiclesrdquo 2002

[29] T I Fossen and J P Strand ldquoPassive nonlinear observer designfor ships using Lyapunov methods full-scale experiments witha supply vesselrdquo Automatica vol 35 no 1 pp 3ndash16 1999

[30] G Xia X Shao H Wang et al ldquoPassive nonlinear observerdesign for special structure vesselsrdquoOceans IEEE pp 1ndash8 2013

[31] K P Lindegaard Acceleration Feedback in Dynamic Position-ing Fakultet for Informasjonsteknologi Matematikk Og Elek-troteknikk 2012

[32] H K Khalil Nonlinear Systems Prentice-Hall Upper SaddleRiver NJ USA 3rd edition 2002

[33] J P Hespanha and A S Morse ldquoCertainty equivalence impliesdetectabilityrdquo Systems amp Control Letters vol 36 no 1 pp 1ndash131999

[34] J P Hespanha D Liberzon and A S Morse ldquoBounds onthe number of switchings with scale-independent hysteresisApplications to supervisory controlrdquo Proceedings of the IEEEConference on Decision and Control vol 4 pp 3622ndash3627 2000

[35] M H Kim and D J Inman ldquoDevelopment of a robust non-linear observer for dynamic positioning of shipsrdquo Proceedings ofthe Institution of Mechanical Engineers Part I Journal of Systemsamp Control Engineering vol 218 no 1 pp 1ndash11 2004

[36] H Alwi C Edwards and C P Tan Fault Detection and Fault-Tolerant Control Using Sliding Modes Advances in IndustrialControl Springer London UK 2011

[37] J P Hespanha D Liberzon and A S Morse ldquoHysteresis-based switching algorithms for supervisory control of uncertainsystemsrdquo Automatica vol 39 no 2 pp 263ndash272 2003

[38] J P Hespanha and A S Morse ldquoStability of switched systemswith average dwell-timerdquo in Proceedings of the 38th IEEEConference on Decision and Control (CDC rsquo99) pp 2655ndash2660Phoenix Ariz USA December 1999

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Mathematical Problems in Engineering 9

Table 2 Condition of faults

Time(s) 0-100 100-250 250-300 300-500Case 1 119870 = 1198701198911 119870 = 1198701198912 119870 = 1198701198912 119870 = 1198701198911Case 2 119870 = 1198701198913 119870 = 1198701198914 119870 = 1198701198914 119870 = 1198701198913Case 3 119870 = 1198701198913 119870 = 1198701198914 119870 = 1198701198914 119892 = 1198921 119870 = 1198701198913 119892 = 1198921

minus20

0

20

(m)

1

15

2

sigm

a

minus20minus10

01020

(m)

1

15

2sig

ma

minus1minus05

005

115

(rad

)

1

15

2

sigm

a

0 50045040035030025020015010050time (s)

0 50045040035030025020015010050time (s)

0 50045040035030025020015010050time (s)

motion in the X direction(Case1)switching signal

motion in the Y direction(Case1)switching signal

rotation about the Z axis(Case1)switching signal

Figure 6 Position and Heading of DP Vessel (Case 1)

Control performances of Case 2 and Case 3 wheremultiple actuator failures occur are demonstrated in Figures8 and 9 respectively Firstly it is shown that the signal 120590 canswitch to the corresponding controller quickly and accuratelyin the presence of actuator failures Secondly when a sensorfailure occurs at 250119904 although there is a slight fluctuation inthe transition progress the signal 120590 can switch to the stablecontroller and positions of the vessel are stabilized near theequilibrium point eventually

5 Conclusion

This paper presented a fault-tolerant supervisory controlmethod for dynamic positioning ships with wave-frequencymodel The proposed fault-tolerant control method does notneed the FDI mechanism due to the introduction of injectedsystem and supervisor It has beenproved that by introducinga nonlinear estimation error and virtual controller thedetectability property and matching property of the switchedsystem are guaranteed in the presence of actuator failuresand sensor failures Simulation results and comparisons havedemonstrated the effectiveness of the proposed fault-tolerant

minus20

0

20

40

(m)

motion in the X direction(S2)(Case1)motion in the X direction(S1)(Case1)

minus20

0

20

40

(m)

motion in the Y direction(S2)(Case1)motion in the Y direction(S1)(Case1)

minus1

0

1

2

(rad

)

rotation about the Z axis(S2)(Case1)rotation about the Z axis(S1)(Case1)

minus100

1020

minus100

1020

0 2010

0 2010

0 2010

minus050

051

15

0 50045040035030025020015010050time (s)

0 50045040035030025020015010050time (s)

0 50045040035030025020015010050time (s)

Figure 7 The tracking performance comparison of the pro-posed method (S1)(Case 1) and fault-tolerant control allocation [5](S2)(Case 1)

control scheme In the future we will consider the unknownfaults which is more practical in realistic situation Furtherwe will improve the switching logic to make the switchingsignal more accurate and stable for more serious failure

Data Availability

All the data supporting the conclusions of the study have beenprovided in Simulation and readers can access these data in[27 29]

Conflicts of Interest

The authors declare that there are no conflicts of interestrelated to this paper

Acknowledgments

This research was partially supported by the National Sci-ence Technology Support Program of China (Project no51609046)

10 Mathematical Problems in Engineering

0 50045040035030025020015010050time (s)

0 50045040035030025020015010050time (s)

0 50045040035030025020015010050time (s)

minus20

0

20

(m)

1

15

2

sigm

a

motion in the X direction(Case2)switching signal

minus20minus10

01020

(m)

1

15

2

sigm

a

motion in the Y direction(Case2)switching signal

minus1minus05

005

115

(rad

)

1

15

2

sigm

a

rotation about the Z axis(Case2)switching signal

Figure 8 Position and Heading of DP Vessel (Case 2)

0 50045040035030025020015010050time (s)

0 50045040035030025020015010050time (s)

0 50045040035030025020015010050time (s)

minus20minus10

01020

(m)

1234

sigm

a

motion in the x direction(Case3)switching signal

0(m)

2

4

sigm

a

motion in the y direction(Case3)switching signal

minus1

0

1

(rad

)

1234

sigm

a

rotation about the z axis(Case3)switching signal

Figure 9 Position and Heading of DP Vessel (Case 3)

References

[1] H Azmi and M J Khosrowjerdi ldquoRobust adaptive faulttolerant control for a class of Lipschitz nonlinear systems withactuator failure and disturbancesrdquo Proceedings of the Institutionof Mechanical Engineers Part I Journal of Systems and ControlEngineering vol 230 no 1 pp 13ndash22 2016

[2] S Dai L and J Zhao ldquoReliable H8 controller design for a classof uncertain linear systems with actuator failuresrdquo InternationalJournal of Control Automation amp Systems vol 6 no 6 pp 954ndash959 2008

[3] Y L Hao and G H Yang ldquoRobust fault tolerant controlbased on sliding mode method for uncertain linear systemswith quantizationrdquo Proceedings of the Institution of MechanicalEngineers Part I Journal of Systems amp Control Engineering vol52 no 5 pp 692ndash703 2013

[4] Z Zuo D W C Ho and Y Wang ldquoFault tolerant controlfor singular systems with actuator saturation and nonlinearperturbationrdquo Automatica vol 46 no 3 pp 569ndash576 2010

[5] A Cristofaro and T A Johansen ldquoFault tolerant controlallocation using unknown input observersrdquoAutomatica vol 50no 7 pp 1891ndash1897 2014

[6] T I Fossen Handbook of Marine Craft Hydrodynamics andMotion Control 2011

[7] A J Soslashrensen ldquoA survey of dynamic positioning controlsystemsrdquo Annual Reviews in Control vol 35 no 1 pp 123ndash1362011

[8] J Liu R Allen andH Yi ldquoShipmotion stabilizing control usinga combination of model predictive control and an adaptiveinput disturbance predictorrdquo Proceedings of the Institution ofMechanical Engineers Part I Journal of Systems and ControlEngineering vol 225 no 5 pp 591ndash602 2011

[9] M Tomera ldquoNonlinear observers design for multivariable shipmotion controlrdquo Polish Maritime Research vol 19 pp 50ndash562012

[10] A Hajivand and S H Mousavizadegan ldquoThe effect of memoryin observer design for a dp systemrdquo in Proceedings of theASME 2010 International Mechanical Engineering Congress andExposition IMECE 2010 pp 53ndash59 Canada November 2010

[11] M Fu J Ning and Y Wei ldquoFault-tolerant control of dynamicpositioning vessel after thruster failures using disturbance

Mathematical Problems in Engineering 11

decoupling methodsrdquo in Proceedings of the 2011 IEEE Interna-tional Conference on Automation and Logistics ICAL 2011 pp433ndash437 China August 2011

[12] F Benetazzo G Ippoliti S Longhi and P Raspa ldquoAdvancedcontrol for fault-tolerant dynamic positioning of an offshoresupply vesselrdquoOcean Engineering vol 106 pp 472ndash484 2015

[13] Y Lin and J Du ldquoFault-tolerant control for dynamic positioningof ships based on an iterative learning observerrdquo in Proceedingsof the 35th Chinese Control Conference CCC 2016 pp 1116ndash1122China July 2016

[14] M Chen B Jiang and R Cui ldquoActuator fault-tolerant controlof ocean surface vessels with input saturationrdquo InternationalJournal of Robust amp Nonlinear Control vol 26 no 3 pp 542ndash564 2016

[15] Y Su C Zheng andPMercorelli ldquoNonlinear PDFault-TolerantControl for Dynamic Positioning of Ships With ActuatorConstraintsrdquo IEEEASME Transactions onMechatronics vol 22no 3 pp 1132ndash1142 2017

[16] W-Z Yu H-X Xu and H Feng ldquoRobust adaptive fault-tolerant control of dynamic positioning vessel with position ref-erence system faults using backstepping designrdquo InternationalJournal of Robust and Nonlinear Control vol 28 no 2 pp 403ndash415 2018

[17] N T Dong Design of hybrid marine control systems for dynamicpositioning [PhD thesis] 2006

[18] J P Strand and T I Fossen ldquoNonlinear passive observer designfor ships with adaptive wave filteringrdquo in New Directions inNonlinear Observer Design vol 244 of Lecture Notes in Controland Information Sciences pp 113ndash134 Springer London UK1999

[19] S Saelid J G Balchen and N A Jenssen ldquoDesign and analysisof a dynamic positioning system based on kalman filtering andoptimal controlrdquo IEEE Transactions on Automatic Control vol28 no 3 pp 331ndash339 1983

[20] G Battistelli J P Hespanha and P Tesi ldquoSupervisory control ofswitched nonlinear systemsrdquo International Journal of AdaptiveControl and Signal Processing vol 26 no 8 pp 723ndash738 2012

[21] I R Bertaska andK DVonEllenrieder ldquoSupervisory switchingcontrol of an unmanned surface vehiclerdquo in Proceedings of theMTSIEEE Washington OCEANS 2015 USA October 2015

[22] J P Hespanha ldquoTutorial on supervisory controlrdquo Overview2002

[23] J P Hespanha D Liberzon and A S Morse ldquoSupervision ofintegral-input-to-state stabilizing controllersrdquo Automatica vol38 no 8 pp 1327ndash1335 2002

[24] M Skogvold Supervisory-switched control for dynamic position-ing systems in arctic areas [Msc thesis] Department of MarineTechnology 2010

[25] L VuD Chatterjee andD Liberzon ldquoInput-to-state stability ofswitched systems and switching adaptive controlrdquo Automaticavol 43 no 4 pp 639ndash646 2007

[26] ED Sontag ldquoInput to state stability basic concepts and resultsrdquoinNonlinear andOptimal ControlTheory pp 163ndash220 SpringerHeidelberg Berlin Germany 2008

[27] T D Nguyen A J Soslashrensen and S T Quek ldquoDesign of hybridcontroller for dynamic positioning from calm to extreme seaconditionsrdquo Automatica vol 43 no 5 pp 768ndash785 2007

[28] T I Fossen ldquoMarine Control Systems Guidance Navigationand Control of Ships Rigs and Underwater Vehiclesrdquo 2002

[29] T I Fossen and J P Strand ldquoPassive nonlinear observer designfor ships using Lyapunov methods full-scale experiments witha supply vesselrdquo Automatica vol 35 no 1 pp 3ndash16 1999

[30] G Xia X Shao H Wang et al ldquoPassive nonlinear observerdesign for special structure vesselsrdquoOceans IEEE pp 1ndash8 2013

[31] K P Lindegaard Acceleration Feedback in Dynamic Position-ing Fakultet for Informasjonsteknologi Matematikk Og Elek-troteknikk 2012

[32] H K Khalil Nonlinear Systems Prentice-Hall Upper SaddleRiver NJ USA 3rd edition 2002

[33] J P Hespanha and A S Morse ldquoCertainty equivalence impliesdetectabilityrdquo Systems amp Control Letters vol 36 no 1 pp 1ndash131999

[34] J P Hespanha D Liberzon and A S Morse ldquoBounds onthe number of switchings with scale-independent hysteresisApplications to supervisory controlrdquo Proceedings of the IEEEConference on Decision and Control vol 4 pp 3622ndash3627 2000

[35] M H Kim and D J Inman ldquoDevelopment of a robust non-linear observer for dynamic positioning of shipsrdquo Proceedings ofthe Institution of Mechanical Engineers Part I Journal of Systemsamp Control Engineering vol 218 no 1 pp 1ndash11 2004

[36] H Alwi C Edwards and C P Tan Fault Detection and Fault-Tolerant Control Using Sliding Modes Advances in IndustrialControl Springer London UK 2011

[37] J P Hespanha D Liberzon and A S Morse ldquoHysteresis-based switching algorithms for supervisory control of uncertainsystemsrdquo Automatica vol 39 no 2 pp 263ndash272 2003

[38] J P Hespanha and A S Morse ldquoStability of switched systemswith average dwell-timerdquo in Proceedings of the 38th IEEEConference on Decision and Control (CDC rsquo99) pp 2655ndash2660Phoenix Ariz USA December 1999

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

10 Mathematical Problems in Engineering

0 50045040035030025020015010050time (s)

0 50045040035030025020015010050time (s)

0 50045040035030025020015010050time (s)

minus20

0

20

(m)

1

15

2

sigm

a

motion in the X direction(Case2)switching signal

minus20minus10

01020

(m)

1

15

2

sigm

a

motion in the Y direction(Case2)switching signal

minus1minus05

005

115

(rad

)

1

15

2

sigm

a

rotation about the Z axis(Case2)switching signal

Figure 8 Position and Heading of DP Vessel (Case 2)

0 50045040035030025020015010050time (s)

0 50045040035030025020015010050time (s)

0 50045040035030025020015010050time (s)

minus20minus10

01020

(m)

1234

sigm

a

motion in the x direction(Case3)switching signal

0(m)

2

4

sigm

a

motion in the y direction(Case3)switching signal

minus1

0

1

(rad

)

1234

sigm

a

rotation about the z axis(Case3)switching signal

Figure 9 Position and Heading of DP Vessel (Case 3)

References

[1] H Azmi and M J Khosrowjerdi ldquoRobust adaptive faulttolerant control for a class of Lipschitz nonlinear systems withactuator failure and disturbancesrdquo Proceedings of the Institutionof Mechanical Engineers Part I Journal of Systems and ControlEngineering vol 230 no 1 pp 13ndash22 2016

[2] S Dai L and J Zhao ldquoReliable H8 controller design for a classof uncertain linear systems with actuator failuresrdquo InternationalJournal of Control Automation amp Systems vol 6 no 6 pp 954ndash959 2008

[3] Y L Hao and G H Yang ldquoRobust fault tolerant controlbased on sliding mode method for uncertain linear systemswith quantizationrdquo Proceedings of the Institution of MechanicalEngineers Part I Journal of Systems amp Control Engineering vol52 no 5 pp 692ndash703 2013

[4] Z Zuo D W C Ho and Y Wang ldquoFault tolerant controlfor singular systems with actuator saturation and nonlinearperturbationrdquo Automatica vol 46 no 3 pp 569ndash576 2010

[5] A Cristofaro and T A Johansen ldquoFault tolerant controlallocation using unknown input observersrdquoAutomatica vol 50no 7 pp 1891ndash1897 2014

[6] T I Fossen Handbook of Marine Craft Hydrodynamics andMotion Control 2011

[7] A J Soslashrensen ldquoA survey of dynamic positioning controlsystemsrdquo Annual Reviews in Control vol 35 no 1 pp 123ndash1362011

[8] J Liu R Allen andH Yi ldquoShipmotion stabilizing control usinga combination of model predictive control and an adaptiveinput disturbance predictorrdquo Proceedings of the Institution ofMechanical Engineers Part I Journal of Systems and ControlEngineering vol 225 no 5 pp 591ndash602 2011

[9] M Tomera ldquoNonlinear observers design for multivariable shipmotion controlrdquo Polish Maritime Research vol 19 pp 50ndash562012

[10] A Hajivand and S H Mousavizadegan ldquoThe effect of memoryin observer design for a dp systemrdquo in Proceedings of theASME 2010 International Mechanical Engineering Congress andExposition IMECE 2010 pp 53ndash59 Canada November 2010

[11] M Fu J Ning and Y Wei ldquoFault-tolerant control of dynamicpositioning vessel after thruster failures using disturbance

Mathematical Problems in Engineering 11

decoupling methodsrdquo in Proceedings of the 2011 IEEE Interna-tional Conference on Automation and Logistics ICAL 2011 pp433ndash437 China August 2011

[12] F Benetazzo G Ippoliti S Longhi and P Raspa ldquoAdvancedcontrol for fault-tolerant dynamic positioning of an offshoresupply vesselrdquoOcean Engineering vol 106 pp 472ndash484 2015

[13] Y Lin and J Du ldquoFault-tolerant control for dynamic positioningof ships based on an iterative learning observerrdquo in Proceedingsof the 35th Chinese Control Conference CCC 2016 pp 1116ndash1122China July 2016

[14] M Chen B Jiang and R Cui ldquoActuator fault-tolerant controlof ocean surface vessels with input saturationrdquo InternationalJournal of Robust amp Nonlinear Control vol 26 no 3 pp 542ndash564 2016

[15] Y Su C Zheng andPMercorelli ldquoNonlinear PDFault-TolerantControl for Dynamic Positioning of Ships With ActuatorConstraintsrdquo IEEEASME Transactions onMechatronics vol 22no 3 pp 1132ndash1142 2017

[16] W-Z Yu H-X Xu and H Feng ldquoRobust adaptive fault-tolerant control of dynamic positioning vessel with position ref-erence system faults using backstepping designrdquo InternationalJournal of Robust and Nonlinear Control vol 28 no 2 pp 403ndash415 2018

[17] N T Dong Design of hybrid marine control systems for dynamicpositioning [PhD thesis] 2006

[18] J P Strand and T I Fossen ldquoNonlinear passive observer designfor ships with adaptive wave filteringrdquo in New Directions inNonlinear Observer Design vol 244 of Lecture Notes in Controland Information Sciences pp 113ndash134 Springer London UK1999

[19] S Saelid J G Balchen and N A Jenssen ldquoDesign and analysisof a dynamic positioning system based on kalman filtering andoptimal controlrdquo IEEE Transactions on Automatic Control vol28 no 3 pp 331ndash339 1983

[20] G Battistelli J P Hespanha and P Tesi ldquoSupervisory control ofswitched nonlinear systemsrdquo International Journal of AdaptiveControl and Signal Processing vol 26 no 8 pp 723ndash738 2012

[21] I R Bertaska andK DVonEllenrieder ldquoSupervisory switchingcontrol of an unmanned surface vehiclerdquo in Proceedings of theMTSIEEE Washington OCEANS 2015 USA October 2015

[22] J P Hespanha ldquoTutorial on supervisory controlrdquo Overview2002

[23] J P Hespanha D Liberzon and A S Morse ldquoSupervision ofintegral-input-to-state stabilizing controllersrdquo Automatica vol38 no 8 pp 1327ndash1335 2002

[24] M Skogvold Supervisory-switched control for dynamic position-ing systems in arctic areas [Msc thesis] Department of MarineTechnology 2010

[25] L VuD Chatterjee andD Liberzon ldquoInput-to-state stability ofswitched systems and switching adaptive controlrdquo Automaticavol 43 no 4 pp 639ndash646 2007

[26] ED Sontag ldquoInput to state stability basic concepts and resultsrdquoinNonlinear andOptimal ControlTheory pp 163ndash220 SpringerHeidelberg Berlin Germany 2008

[27] T D Nguyen A J Soslashrensen and S T Quek ldquoDesign of hybridcontroller for dynamic positioning from calm to extreme seaconditionsrdquo Automatica vol 43 no 5 pp 768ndash785 2007

[28] T I Fossen ldquoMarine Control Systems Guidance Navigationand Control of Ships Rigs and Underwater Vehiclesrdquo 2002

[29] T I Fossen and J P Strand ldquoPassive nonlinear observer designfor ships using Lyapunov methods full-scale experiments witha supply vesselrdquo Automatica vol 35 no 1 pp 3ndash16 1999

[30] G Xia X Shao H Wang et al ldquoPassive nonlinear observerdesign for special structure vesselsrdquoOceans IEEE pp 1ndash8 2013

[31] K P Lindegaard Acceleration Feedback in Dynamic Position-ing Fakultet for Informasjonsteknologi Matematikk Og Elek-troteknikk 2012

[32] H K Khalil Nonlinear Systems Prentice-Hall Upper SaddleRiver NJ USA 3rd edition 2002

[33] J P Hespanha and A S Morse ldquoCertainty equivalence impliesdetectabilityrdquo Systems amp Control Letters vol 36 no 1 pp 1ndash131999

[34] J P Hespanha D Liberzon and A S Morse ldquoBounds onthe number of switchings with scale-independent hysteresisApplications to supervisory controlrdquo Proceedings of the IEEEConference on Decision and Control vol 4 pp 3622ndash3627 2000

[35] M H Kim and D J Inman ldquoDevelopment of a robust non-linear observer for dynamic positioning of shipsrdquo Proceedings ofthe Institution of Mechanical Engineers Part I Journal of Systemsamp Control Engineering vol 218 no 1 pp 1ndash11 2004

[36] H Alwi C Edwards and C P Tan Fault Detection and Fault-Tolerant Control Using Sliding Modes Advances in IndustrialControl Springer London UK 2011

[37] J P Hespanha D Liberzon and A S Morse ldquoHysteresis-based switching algorithms for supervisory control of uncertainsystemsrdquo Automatica vol 39 no 2 pp 263ndash272 2003

[38] J P Hespanha and A S Morse ldquoStability of switched systemswith average dwell-timerdquo in Proceedings of the 38th IEEEConference on Decision and Control (CDC rsquo99) pp 2655ndash2660Phoenix Ariz USA December 1999

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Mathematical Problems in Engineering 11

decoupling methodsrdquo in Proceedings of the 2011 IEEE Interna-tional Conference on Automation and Logistics ICAL 2011 pp433ndash437 China August 2011

[12] F Benetazzo G Ippoliti S Longhi and P Raspa ldquoAdvancedcontrol for fault-tolerant dynamic positioning of an offshoresupply vesselrdquoOcean Engineering vol 106 pp 472ndash484 2015

[13] Y Lin and J Du ldquoFault-tolerant control for dynamic positioningof ships based on an iterative learning observerrdquo in Proceedingsof the 35th Chinese Control Conference CCC 2016 pp 1116ndash1122China July 2016

[14] M Chen B Jiang and R Cui ldquoActuator fault-tolerant controlof ocean surface vessels with input saturationrdquo InternationalJournal of Robust amp Nonlinear Control vol 26 no 3 pp 542ndash564 2016

[15] Y Su C Zheng andPMercorelli ldquoNonlinear PDFault-TolerantControl for Dynamic Positioning of Ships With ActuatorConstraintsrdquo IEEEASME Transactions onMechatronics vol 22no 3 pp 1132ndash1142 2017

[16] W-Z Yu H-X Xu and H Feng ldquoRobust adaptive fault-tolerant control of dynamic positioning vessel with position ref-erence system faults using backstepping designrdquo InternationalJournal of Robust and Nonlinear Control vol 28 no 2 pp 403ndash415 2018

[17] N T Dong Design of hybrid marine control systems for dynamicpositioning [PhD thesis] 2006

[18] J P Strand and T I Fossen ldquoNonlinear passive observer designfor ships with adaptive wave filteringrdquo in New Directions inNonlinear Observer Design vol 244 of Lecture Notes in Controland Information Sciences pp 113ndash134 Springer London UK1999

[19] S Saelid J G Balchen and N A Jenssen ldquoDesign and analysisof a dynamic positioning system based on kalman filtering andoptimal controlrdquo IEEE Transactions on Automatic Control vol28 no 3 pp 331ndash339 1983

[20] G Battistelli J P Hespanha and P Tesi ldquoSupervisory control ofswitched nonlinear systemsrdquo International Journal of AdaptiveControl and Signal Processing vol 26 no 8 pp 723ndash738 2012

[21] I R Bertaska andK DVonEllenrieder ldquoSupervisory switchingcontrol of an unmanned surface vehiclerdquo in Proceedings of theMTSIEEE Washington OCEANS 2015 USA October 2015

[22] J P Hespanha ldquoTutorial on supervisory controlrdquo Overview2002

[23] J P Hespanha D Liberzon and A S Morse ldquoSupervision ofintegral-input-to-state stabilizing controllersrdquo Automatica vol38 no 8 pp 1327ndash1335 2002

[24] M Skogvold Supervisory-switched control for dynamic position-ing systems in arctic areas [Msc thesis] Department of MarineTechnology 2010

[25] L VuD Chatterjee andD Liberzon ldquoInput-to-state stability ofswitched systems and switching adaptive controlrdquo Automaticavol 43 no 4 pp 639ndash646 2007

[26] ED Sontag ldquoInput to state stability basic concepts and resultsrdquoinNonlinear andOptimal ControlTheory pp 163ndash220 SpringerHeidelberg Berlin Germany 2008

[27] T D Nguyen A J Soslashrensen and S T Quek ldquoDesign of hybridcontroller for dynamic positioning from calm to extreme seaconditionsrdquo Automatica vol 43 no 5 pp 768ndash785 2007

[28] T I Fossen ldquoMarine Control Systems Guidance Navigationand Control of Ships Rigs and Underwater Vehiclesrdquo 2002

[29] T I Fossen and J P Strand ldquoPassive nonlinear observer designfor ships using Lyapunov methods full-scale experiments witha supply vesselrdquo Automatica vol 35 no 1 pp 3ndash16 1999

[30] G Xia X Shao H Wang et al ldquoPassive nonlinear observerdesign for special structure vesselsrdquoOceans IEEE pp 1ndash8 2013

[31] K P Lindegaard Acceleration Feedback in Dynamic Position-ing Fakultet for Informasjonsteknologi Matematikk Og Elek-troteknikk 2012

[32] H K Khalil Nonlinear Systems Prentice-Hall Upper SaddleRiver NJ USA 3rd edition 2002

[33] J P Hespanha and A S Morse ldquoCertainty equivalence impliesdetectabilityrdquo Systems amp Control Letters vol 36 no 1 pp 1ndash131999

[34] J P Hespanha D Liberzon and A S Morse ldquoBounds onthe number of switchings with scale-independent hysteresisApplications to supervisory controlrdquo Proceedings of the IEEEConference on Decision and Control vol 4 pp 3622ndash3627 2000

[35] M H Kim and D J Inman ldquoDevelopment of a robust non-linear observer for dynamic positioning of shipsrdquo Proceedings ofthe Institution of Mechanical Engineers Part I Journal of Systemsamp Control Engineering vol 218 no 1 pp 1ndash11 2004

[36] H Alwi C Edwards and C P Tan Fault Detection and Fault-Tolerant Control Using Sliding Modes Advances in IndustrialControl Springer London UK 2011

[37] J P Hespanha D Liberzon and A S Morse ldquoHysteresis-based switching algorithms for supervisory control of uncertainsystemsrdquo Automatica vol 39 no 2 pp 263ndash272 2003

[38] J P Hespanha and A S Morse ldquoStability of switched systemswith average dwell-timerdquo in Proceedings of the 38th IEEEConference on Decision and Control (CDC rsquo99) pp 2655ndash2660Phoenix Ariz USA December 1999

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom