Research Article Recursive Subspace Identification of AUV...

Research ArticleRecursive Subspace Identification of AUV Dynamic Model underGeneral Noise Assumption

Zheping Yan1 Di Wu1 Jiajia Zhou1 and Lichao Hao2

1 College of Automation Harbin Engineering University Harbin Heilongjiang 150001 China2 China Institute of Marine Technology and Economy Beijing 100081 China

Correspondence should be addressed to Di Wu roaddywugmailcom

Received 5 October 2013 Accepted 10 December 2013 Published 16 January 2014

Academic Editor Xiaojie Su

Copyright copy 2014 Zheping Yan 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 recursive subspace identification algorithm for autonomous underwater vehicles (AUVs) is proposed in this paper Due tothe advantages at handling nonlinearities and couplings the AUV model investigated here is for the first time constructed as aHammersteinmodelwith nonlinear feedback in the linear part To better take the environment and sensor noises into considerationthe identification problem is concerned as an errors-in-variables (EIV) one which means that the identification procedure isunder general noise assumption In order to make the algorithm recursively propagator method (PM) based subspace approachis extended into EIV framework to form the recursive identification method called PM-EIV algorithm With several identificationexperiments carried out by the AUV simulation platform the proposed algorithm demonstrates its effectiveness and feasibility

1 Introduction

In recent years autonomous underwater vehicles (AUVs)have attracted increasing attentions due to their remarkablefeatures such as high agility excellent convenience and lowcost in applications of underwater explorations and develop-ments However contradictions lay between more and morecomplicated missions for AUV and the control and naviga-tion systems that are not accurate enough System identifi-cation methods have provided an alternative way other thantraditional expensive instruments dependent approaches toimprove the abilities of autonomous systems in variousaspects [1ndash5] As a result a variety of researches have been putforward to identify the ordinary differential model of AUVsfor model based control and navigation Rentschler andcoworkers [1] have demonstrated an iterative procedure torevise the model and controller of Odyssey III AUV toobtain better flight performances Nonlinear observers basedidentification algorithmwith slidingmode observer and EKFis also proposed for designation of nonlinear controller in [2]For a more robust navigation system in case of sensor faultHegrenaes and Hallingstad [3] have used least squares algo-rithm to estimate both sea current disturbances and model

parameters of the HUGIN 4500 AUV However due to thecomplexity of AUV system many nonlinearities and coupledterms exist in ordinary differential equations that make theidentification of whole pack of hydrodynamic coefficientsquite complicated and time consuming As a consequenceAUVdifferential model usually need to be simplified througheliminations of nonlinear and coupled terms before identifi-cation process For example in the research of Tiano et al [6]a set of decoupled AUV models concerning different degreeof freedom were set up and the yaw dynamics of the Ham-merhead AUV was identified according to observer Kalmanfilter identification (OKID) method However nonlinearitiesand couplings are two significant features being researched inthe area of AUV so as it is said in [6] construction of MIMOcoupled AUV model is a rather important and challengingmodeling issue

In this paper compared with the traditional differentialequation used in AUV modeling a Hammerstein modelwhich consists of a static nonlinear part and a dynamic linearone is adopted in order to deal with nonlinear and linearproperty of AUV system separately Due to the particularcharacteristics of AUV system the Hammerstein model hasto be modified with a static nonlinear feedback part added

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2014 Article ID 547539 10 pageshttpdxdoiorg1011552014547539

2 Mathematical Problems in Engineering

on the linear part As illustrated in Figure 1 one remarkablebenefit of Hammerstein system is that it possesses advantagesof linear MIMO system and can approximate nonlinear andcoupled terms of AUV to a large extent at the same timeTo the best of the authorsrsquo acknowledge this is the firsttime that Hammerstein model is applied in the area of AUVmodeling Because of the conveniences brought by Hammer-stein system extensive attentions has been paid to obtainsystem parameters from inputoutput data Cai and Bai [7]proposed a method to make the identification of parametricHammerstein system a linear problem through regardingthe average squared error cost function as the inner prod-uct between the true but unknown parameter vector andits estimations But this method was discussed under theassumption that Hammerstein system only consists of singleinput and single output As to MIMO Hammerstein systemidentification a nonparametric algorithmbased on stochasticapproximation approach is proposed in [8] However nonlin-ear MIMO Hammerstein identification problem concernedin this paper is still challenging

One group of widely studiedMIMO system identificationalgorithms for Hammerstein system is subspace identifi-cation methods [9] which mainly include three differentbranches numerical algorithms for state-space subspace sys-tem identification (N4SID) MIMO output-error state-spacemodel identification (MOESP) and canonical variate analy-sis (CVA) Compared with other identification algorithmssubspace identification methods are more attractive due toseveral advantages [10] For example subspace identificationmethods can circumvent the complicated parameterizationprocedure forMIMOsystemof prediction errormethods [11]What ismore subspace identificationmethods do not requirenonlinear searches in the parameter space based on com-putational tools such as QR factorization and singular valuedecomposition (SVD) [12] Since the Hammerstein AUVmodel is a parametric onewithmultiinputs andmultioutputsMIMO MOESP algorithm is adopted as the theoretical basisfor further investigation in this paper

In addition instead of regarding the identification proce-dure in ideal situations practical engineering circumstancesneed to be taken into consideration A widely studied oneis that the general noise assumption has to be made whichmeans ldquothe measured input is corrupted by a white measure-ment noise while the measured output is corrupted by thesum of a white measurement noise and a term due to a whiteprocess noiserdquo [13] Another practical situation which need tobe considered is that different oceanic environment may leadto different hydrodynamic coefficients in which case off-lineidentification results of Hammerstein AUV model will bringerrors to control and navigation systems [14] Besides itis often the case that the structure of AUV usually hasto be modified mildly in order to fulfill various tasks inpractice Therefore recursive identification methods whichcan adjustmodel parameters online become rather significantand attractive in applications of such as adaptive controlmodel-aided navigation and so on

As a result in order to fulfill the situations concernedabove the identification of Hammerstein AUV model isregarded as an errors-in-variables (EIV) problem in this

Nonlinear part Linear part

Nonlinear feedback

u(k)

(k)υ u(k)

d(k) ω

y(k) y(k)

(k)

Figure 1 Hammerstein AUVModel

paper and we are going to make the MOESP method recur-sively under EIV framework so as to carry out recursive iden-tification algorithm for the Hammerstein AUV model Onemajor obstacle in recursive subspace identification is theincreasing computational complexity of SVD [15 16] Inprevious studies projection approximation subspace tracking(PAST) algorithm designed by Yang [17] was widely used toupdate the SVD However approximation in the algorithmwill bring slight difference between identified model and theoriginal one In addition IV-PAST algorithm [18] and gradi-ent type subspace tracking method [19] are also developed toestimate the signal subspace In [20] an instrumental variablepropagator method (IVPM) based recursive identificationalgorithmwas proposedNevertheless the paper only studiedthe method within past input (PI)past output (PO) MOESPframework So in this paper combining with HammersteinAUV identification problem described above IVPMmethodis extended into EIV framework and the PM-EIV algorithmfor recursive subspace identification of errors-in-variablesproblem is derived Compared with previous algorithmsmentioned the PM-EIV algorithm is more suitable to handlethe MIMO AUV Hammerstein model identification prob-lems under errors-in-variables framework recursively As amatter of fact since the Hammerstein model constructed canbe viewed as a generalized one for mechanical systems theproposed PM-EIV can be extended to identification of othersystems as well

The remainder of this paper is organized as follows InSection 2 a Hammerstein AUV model with nonlinear feed-back is formed with proper transformation from an ordinarydifferential one Then the linearization process of nonlinearpart is presented In Section 3 the MOESP identificationmethod is described with no consideration about the systemnoise After that a recursive subspacemethod called PM-EIVis derived under the general noise assumption Identificationand validation of theHammersteinAUVmodel are presentedin Section 4 At last conclusions are made in Section 5

Some notations used in this paper are followed Thesuperscript (sdot)119879 denotes the transposition operator 119864(sdot) is theexpectation operator R119898times119899 represents the set of 119898 times 119899 realmatrices119872perp is the orthogonal complement of119872

2 AUV Modeling

In this section a Hammerstein AUV model is formed afterintroducing the ordinary differential one Then in order tomake the Hammerstein AUV model suitable for subspace

Mathematical Problems in Engineering 3

identification method linearization of the nonlinear part isbrought out

21 Hammerstein AUV Model According to [21] a couplednonlinear ordinary differential model for AUV based onNewtonian mechanics can be described as the followingequation

119872] + 119862 (]) ] + 119863 (]) ] = 120591 (1)

where ] isin R6 represents the state vector of AUV119872 isin R6 times 6consists of inertia matrix and add mass matrix119863(]) isin R6 times 6is the linear and quadratic damping matrix 119862(]) isin R6 times 6is the coriolis and centripetal matrix 120591 isin R6 represents theforces and moments acted on the vehicle

The coupled and nonlinear terms inmatrix119863(]) and119862(])make the identification process quite complicated and timeconsuming To obtain Hammerstein model of AUV simpli-fications and transformations have to be made First somecoupled terms with little influences are eliminated Secondthe nonlinear terms are separated from the system andthe system is divided into static nonlinear input part andnonlinear feedback part Then remaining nonlinear andcoupled terms constitute the nonlinear feedback part At lastthe remaining part of the model can be described as a linearMIMO state-space model After those steps a HammersteinAUV model with nonlinear feedback part can be formed asin Figure 1 119889(119896) 120592(119896) and 120596(119896) are process noise input mea-surement noise and output measurement noise respectively

According to the system structure above define thenonlinear input function as 119891(sdot) nonlinear feedback as119892(sdot) so discrete time MIMO state-space equations can beconstructed as below

119909 (119896 + 1) = 119860119909 (119896) + 1198611119891 (119906 (119896)) + 1198612119892 (119910 (119896)) + 119889 (119896)

119910 (119896) = 119862119909 (119896) + 1198631119891 (119906 (119896)) + 1198632119892 (119910 (119896))

119910 (119896) = 119910 (119896) + 120596 (119896)

(119896) = 119906 (119896) + 120592 (119896)

(2)

where 119909(119896) isin 1198776 represent the system states 119906(119896) isin 1198773 are thesystem inputs including thruster revolution rudder deflec-tion and elevator deflection 119910(119896) isin 1198776 is output vector con-sists of 119906 V 119908 119901 119902 119903 that represents the surge velocity swayvelocity heave velocity roll rate pitch rate yaw rate respec-tively 119860 119861

1 1198612 119862 119863

1 and 119863

2are the corresponding linear

subsystem matrices with appropriate dimensions (119896) 119910(119896)are corrupted system inputs and outputs

Many identification methods have focused on handlingoutput measurement noise and process noise In fact inputmeasurement noise is inevitable in any engineering processesincluding practical identification experiments So in thispaper 119889(119896) 120592(119896) and 120596(119896) are considered under the generalnoise assumption

22 Linearization of AUV Model According to theSection 21 AUV model can be described as Hammersteinone with nonlinear feedback In order to identify the

parameters of corresponding matrices nonlinearity inthe equations needs to be linearly parameterized so thatrecursive subspace identification methods can be appliedOne traditional approach to approximate nonlinearity ofthe system is linear combination of basic functions InLovera [12] Tchebiceff polynomials are chosen to linearizenonlinearities However as the system studied in this papercontains nonlinear feedback part Tchebiceff polynomialsapproximation of 119892(sdot) will influence the identifiabilityof system matrix So in this paper truncated Fourierseries described in Luo and Leonessa [22] also known astrigonometric polynomials are adopted to linearize thenonlinear functions 119891(sdot) and 119892(sdot) Define the basic functionas follows

120593119896 (119909) = 1 119896 = 0

120593119896 (119909) = [cos(

119896120587 (119909 minus 119909119898)

119909119889

) sin(119896120587 (119909 minus 119909

119898)

119909119889

)]

119879

119896 ge 1

(3)

Then nonlinear function can be approximated by follow-ing equation

119865 (119909) = 119908119900+

119873

sum119894=1

119908119894120593119894 (119909) (4)

where 119909119898

= (119909max + 119909min)2 119909119889 = (119909max minus 119909min)2 and119908119894= [119908119894 cos 119908

119894 sin] Since the Hammerstein AUV model isMIMO define Φ(119909) = [120593119879

0(119909) 120593119879

1(119909) sdot sdot sdot 120593119879

119873(119909)]119879

119882 =

[1199080 1199081 sdot sdot sdot 119908119873] then 119891(sdot) 119892(sdot) can be expressed as

119891 (119906) = [1198821198791199061

1198821198791199062

sdot sdot sdot 119882119879119906119903]119879

sdot [Φ119879 (1199061) Φ119879 (1199062) sdot sdot sdot Φ

119879 (119906119898)]119879

119892 (119910) = [119882119879

11991011198821198791199102

sdot sdot sdot 119882119879119910119904]119879

sdot [Φ119879 (1199101) Φ119879 (119910

2) sdot sdot sdot Φ119879 (119910

119897)]119879

(5)

where 119898 = 3 119897 = 6 and corresponding coefficient vector119882119906119894

isin 1198771 times 3(2119873+1) and 119882119910119894

isin 1198771 times 6(2119873+1) A simplifiedexpression for nonlinear part can be presented as follows

119891 (119906) = 119870120585 (119906)

119892 (119910) = 119875120577 (119910) (6)

with definitions that120585(119906)

Δ

= [Φ119879(1199061) Φ119879(119906

2) sdot sdot sdot Φ119879(119906

119898)]119879

119870Δ

=

[1198821198791199061

1198821198791199062

sdot sdot sdot 119882119879119906119903]119879

The equation for 119892(119910) can alsobe formed in a similar principle Then a new state-spacemodel with no nonlinearity can be described as follow

119909 (119896 + 1) = 119860119909 (119896) +1198721120585 (119906 (119896)) + 1198722120577 (119910 (119896)) + 119889 (119896)

119910 (119896) = 119862119909 (119896) + 1198731120585 (119906 (119896)) + 1198732120577 (119910 (119896))

(7)


where 1198721

= 1198611119870 isin 119877119899times3(2119873+1) 119872

2= 119861

2119875 isin

119877119899times6(2119873+1) 1198731= 1198631119870 isin 119877119897times3(2119873+1) and 119873

2= 1198632119875 isin

119877119897times6(2119873+1) represent new coefficients matrices 120585(119906(119896)) isin

1198773(2119873+1)times1 120577(119910(119896)) isin 1198776(2119873+1)times1 respectively represent theinput vector and the feedback vector Further define 119872 =

[1198721 1198722] 119873 = [1198731 1198732] 119890(119896) = [120585119879(119906(119896)) 120577119879(119910(119896))]119879

asthe system matrix and input vector and the HammersteinAUV model can be presented in classical form

119909 (119896 + 1) = 119860119909 (119896) + 119872119890 (119896) + 119889 (119896)

119910 (119896) = 119862119909 (119896) + 119873119890 (119896) (8)

By now a Hammerstein AUV model is established andhas been linearized based on trigonometric polynomials witha description as in (8) being obtained In the followingsection a recursive subspace method to identify the systemunder EIV frame will be derived

3 Recursive Subspace Identification Method

In this section the PM-EIV subspace identification methodfor the Hammerstein AUVmodel will be proposed To betterillustrate themethod theMOESP algorithm have to be intro-duced first to lay the foundation so that the PM-EIV methodcan be derived Before that several assumptions have to bemade

Assumption 1 The difference between the nonlinear func-tions 119891(sdot) 119892(sdot) and their linear approximations respectivelycan be neglected

Assumption 2 Persistent exciting condition is satisfiedaccording to the definition from Ljung [23]

Assumption 3 The system noise sequences 119889(119896) 120592(119896) and120596(119896) are white noises independent from each other

31 MOESP Identification Method The fundamental featureof MOESP method is to estimate the extended observermatrix of the system based on inputoutput dataThen systemmatrices can be obtained through applying least-squaresalgorithms To give a brief introduction of MOESP methodsystem noises are assumed to be zero in this section that is119889(119896) equiv 0 120592(119896) equiv 0 and 120596(119896) equiv 0 According to Verhaegen[24] The following equations need to be formed firstly

119884119895119894119873

= Γ119894119883119895119873

+ 119867119894119864119895119894119873 (9)

with definitions of 119884119895119894119873

119883119895119873

and 119864119895119894119873

as follows

119884119895119894119873

=[[

[

119910119895

sdot sdot sdot 119910119895+119873minus1

d

119910119895+119894minus1

sdot sdot sdot 119910119895+119873+119894minus2

]]

]

119864119895119894119873

=[[

[

119890119895


d

119890119895+119894minus1


]]

]

119883119895119873

= [119909119895 119909119895+1 sdot sdot sdot 119909119895+119873minus1]

(10)

Therefore extended observermatrix Γ119894and low triangular

block Toeplitz matrix119867119894have the following structures

Γ119894= [119862119879 (119862119860)

119879sdot sdot sdot (119862119860119894minus1)

119879]119879

119867119894=

[[[[

[

119873 0 sdot sdot sdot 0

119862119872 119873 sdot sdot sdot 0

d

119862119860119894minus2119872 119862119860119894minus3119872 sdot sdot sdot 119873

]]]]

]

(11)

Inputoutput data matrix is formed and factorized by RQfactorization The following equation can be acquired

[119864119895119894119873

119884119895119894119873

] = [11987711

0

11987721

11987722

] [1198761

1198762

] (12)

Combining (12) with (9) we can obtain that

Γ119894119883119895119873

= [11987721 minus 11986711989411987711 11987722] [1198761

1198762

] (13)

It can be derived that the column space of Γ119894and the

column space of 11987722are equal After carrying out SVD of 119877

22

as in (14) columns of 119880119899can be regarded as a basis for Γ

119894

11987722= [119880119899 119880

perp

119899] [1198781

0

119900 1198782

] [119881119879119899

(119881perp119899)119879] (14)

Based on the definition of extended observer matrix Γ119894

transformed system matrices 119860119879 119862119879can be easily calculated

with 119880119899

119880(1)

119899119860119879= 119880(2)

119899

119862119879= 119880119899 (1 119897 )

(15)

And119872119879119873 can be acquired by least-squares solutions for

(119880perp

119899)119879

119867119894minus (119880perp

119899)119879

11987721119877minus1

11= 0 (16)

32 Errors-in-Variables Problem Because the identificationproblem of the Hammerstein AUV model is an EIV onea more practical subspace identification method handlingEIV problem is introduced in this section which takesdisturbances 119889(119896) 120592(119896) 120596(119896) into consideration based on thebasic MOESP algorithm above In this case (9) needs to bemodified as

119895119894119873

= Γ119894119883119895119873

+ 119867119894119864119895119894119873

minus 1198671198941198811015840

119895119894119873+ 119866119894119875119895119894119873

+119882119895119894119873

(17)


where 119875119895119894119873

119882119895119894119873

are block Hankel matrices of noises 119889(119896)and 120596(119896) 119866

119894is defined as

119866119894=

[[[[

[



d


]]]]

]

(18)

Notice that 1198811015840119895119894119873

is not only made of 120592(119896) output mea-surement noise 120596(119896) is also involved due to the linearizationof nonlinear feedback partThis brings the closed-loop prob-lem into the identification process One significant obsta-cle resulting from closed-loop situation is the interferencebetween input signals and output measurement noise whichviolates the following equation

119864 [119906119896119908119879

119895] = 0 for (119895 lt 119896) (19)

It means that input signals are no longer unrelated to thepast noises However problems studied in this paper onlyneed the condition that future noise is unrelated to past inputThat is

119864 [119906119896119908119879

119895] = 0 for (119895 ge 119896) (20)

So in order to eliminate the influence from the noisesinstrumental variables can still be formed as past input andoutput signals Then following relation can be obtained

119894+1119894119873

[1198641198791119894119873

1198791119894119873

]

= Γ119894119883119894+1119873

[1198641198791119894119873

1198791119894119873

]

+ 119867119894119864119894+1119894119873

[1198641198791119894119873

1198791119894119873

]

(21)

[119864119894+1119894119873

1198641198791119894119873

119864119894+1119894119873

1198791119894119873

119894+1119894119873

1198641198791119894119873

119894+1119894119873

1198791119894119873

]

= [11987711 (119905) 0

11987721 (119905) 119877

22 (119905)] [

1198761 (119905)

1198762 (119905)

]

(22)

According to Theorem 3 of Chou and Verhaegen [13]the column space of Γ

119894can be consistently estimated from

11987722 119860119879 119872119879 119862119879 and 119873 can be acquired based on OE PIV

algorithm To save the space more details can be found in[13]

33 PM-EIV Subspace Identification In the above sectionssubspace identification method for AUV Hammerstein sys-tem has been developed under general noise assumption Asmentioned in Section 1 recursive method for AUV identifi-cation can be much more suitable due to the properties ofoceanic environment and tasks So in this section identifi-cation method will be modified recursively One of the keyproblems handled in recursive identification is the recursiveupdate of SVD process in order to reduce the computationalburden which comes with increasing inputoutput data In[20] propagator method used in array signal processingarea is introduced for recursive subspace identification under

PIPOMOESP schemes Considering about the EIV problemin the identification of AUV model propagator method willbe extended into EIV framework in this paper for the firsttime and the resulting algorithm will be called PM-EIVsubspace identification method An important step in PMsubspacemethod is the calculation of observer vector definedas

119911119894 (119905 + 1) = 119910

119894 (119905 + 1) minus 119867119894 (119905 + 1) 119890119894 (119905 + 1) (23)

where 119905 = 119895 + 119873 minus 1 119910119894(119905 + 1) 119890

119894(119905 + 1) are new updated

outputinput data vectors respectively

331 Update of Observer Vector 119911119894in EIV Framework In this

section RQ factorization method is adopted for updating ofobserve vector Based on (22) in Section 32 update of thedata matrix can be expressed as follows

[119864119894+1119894119873+1

1198641198791119894119873+1

119864119894+1119894119873+1

1198791119894119873+1

119894+1119894119873+1

1198641198791119894119873+1

119894+1119894119873+1

1198791119894119873+1

]

= [11987711 (119905 + 1) 0

11987721 (119905 + 1) 119877

22 (119905 + 1)] [

1198761 (119905 + 1)

1198762 (119905 + 1)

]

(24)

where 119864119894+1119894119873+1

= [119864119894+1119894119873

119890119894(119905 + 1)] 1198641119894119873+1 and 119894+1119894119873+1

1119894119873+1

are formed in similar manner A transformation of(24) is as follows

[119864119894+1119894119873

119890119894 (119905 + 1)

119894+1119894119873

119910119894 (119905 + 1)

] [1198641198791119894119873

1198791119894119873

119890119894(119873 + 1)

119879119910119894(119873 + 1)

119879]

= [11987711 (119905) 0 119890

119894 (119905 + 1) 120601 (119873 + 1)

11987721 (119905) 119877

22 (119905) 119910119894 (119905 + 1) 120601 (119873 + 1)

][

[

1198761 (119905)

1198762 (119905)

119868

]

]

(25)

where 120601(119873 + 1) = [119890119894(119873 + 1)

119879119910119894(119873 + 1)

119879] denotes theupdate instrumental variables

Then given rotation can be implemented to eliminate the119890119894(119905 + 1)120601(119873 + 1) in the above equation in order to make a

lower triangle form

[11987711 (119905) 0 119890

119894 (119905 + 1) 120601 (119873 + 1)

11987721 (119905) 119877

22 (119905) 119910119894 (119905 + 1) 120601 (119873 + 1)

]Giv (119905 + 1)

= [11987711 (119905 + 1) 0 0

11987721 (119905 + 1) 119877

22 (119905) 119894 (119905 + 1)

]

(26)

So the following relation can be obtained

11987722 (119905 + 1) 11987722(119905 + 1)

119879= 11987722 (119905) 11987722(119905)

119879+ 119894 (119905 + 1)

119879

119894(119905 + 1)

(27)

In addition the following equation holds

11987722 (119905 + 1)1198762 (119905 + 1)

= (119894+1119894119873+1

minus 119894119864119894+1119894119873+1

) [1198641198791119894119873+1

1198791119894119873+1

]

= 11987722 (119905) 1198762 (119905) + 119911119894 (119905 + 1) [119890119894(119873 + 1)

119879119910119894(119873 + 1)

119879]

(28)


Combining (27) and (28) relation between 119911119894(119905 + 1) and

119894(119905 + 1) can be found

119894 (119905 + 1) 119894(119905 + 1)

119879

= 11987722 (119905) 1198762 (119905) 120601(119873 + 1)

119879119911119894(119905 + 1)

119879

+ 119911119894 (119905 + 1) 120601 (119873 + 1) (11987722 (119905) 1198762 (119905))

119879

+ 119911119894 (119905 + 1) 120601 (119873 + 1) 120601(119873 + 1)

119879119911119894(119905 + 1)

119879

(29)

Assuming there is a matrix119870(119905+1) satisfies the followingequation

119870 (119905 + 1)119870(119905 + 1)119879= 119894 (119905 + 1) 119894(119905 + 1)

119879+ 11987722 (119905) 11987722(119905)

119879

(30)

Then observer vector 119911119894can be updated according to (31)

119911119894 (119905 + 1) [119890

119879

119894(119873 + 1) 119910119879

119894(119873 + 1)]

= (119870 (119905 + 1) minus 11987722 (119905) 1198762 (119905))

119911119894 (119905 + 1) = 119870

minus1

120601(119870 (119905 + 1) minus 11987722 (119905) 1198762 (119905)) 120601

119879

119873

(31)

where 119870120601= 120601119873120601119879119873is the coefficient related to instrumental

variables andnotice that119870(119905+1) is not necessary to be square

332 Estimation of Observer Matrix Γ119894 Since the observer

vector 119911119894can be obtained in EIV scheme Then extended

observer matrix Γ119894can be found through propagator method

Observer matrix Γ119894can be expressed in the following form

Γ119894= [

Γ1198941

Γ1198942

] = [119868

119875119879119898

] Γ1198941= 119876119904Γ1198941 (32)

Since the matrix Γ1198941isin R119899times119899 has full rank column space of

Γ119894equals that of 119876

119904 Combining with (23) observer vector 119911

119894

can be divided as

119911119894 (119905 + 1) = 119876

119904Γ1198941119909 (119905 + 1) + 119887119894 (119905 + 1) = [

1199111198941 (119905 + 1)

1199111198942 (119905 + 1)

] (33)

Without consideration of noise term 119887119894(119905 + 1) it can

be easily established that 1199111198942

= 1198751198791198981199111198941 Then the 119875119879

119898can

be solved with least-squares methods However existence ofnoise termwill lead to a biased estimation of119875119879

119898 So the IVPM

algorithm proposed by Mercere [20] is adopted to estimate119875119879119898 A suitable variable 120574 isin 119877119899times1 needs to be found with no

correlation with system noises According to Section 32 pastsystem input date may satisfy the condition

Begin

Initialization

Deploy missions(SIC)

Fault diagnosis

AUV (MC)Mission end

Manage missions(MiC)

(MiC)

(MiC)Path plans

Yes

NoFault

No

Yes

End

Control outputs(MoC)

(MC)(MC)Control surface Thruster

States monitor(SIC)

Visual reality(3DC)

Figure 2 Information flow of AUV simulation platform

Finally recursive estimation of 119875119879119898can be acquired in the

following RLS form

119870119901 (119905 + 1)

= 120574119879(119905 + 1) 119877 (119905) (1 + 120574

119879(119905 + 1) 119877 (119905) 1199111198941 (119905 + 1))

minus1

119875119879

119898(119905 + 1)

= 119875119879

119898(119905) + [1199111198942 (119905 + 1) minus 119875

119879

119898(119905) 1199111198941 (119905 + 1)]119870119901 (119905 + 1)

119877 (119905 + 1) = 119877 (119905) minus 119877 (119905) 1199111198941 (119905 + 1)119870119901 (119905 + 1)

(34)

where 119877(119905) = 119864[1199111198941(119905)120574119879(119905)]

minus1With the estimation of observer matrix Γ

119894 algorithms

mentioned in Section 32 can be applied here to estimate thesystem matrices 119860

119879 119872119879 119862119879 and 119873 Therefore propagator

based subspace identification method in errors-in-variablesscheme (PM-EIV) is derived Based on the IVPM methodthe algorithm is more suitable for the identification ofHammerstein AUV model proposed in Section 2 In the fol-lowing the identification algorithm will be verified throughexperiments carried out by the AUV simulation platform

4 Simulations and Results

In this section simulation experiments are carried out toevaluate the performance of the PM-EIV algorithm proposedin this paper through identifying the Hammerstein AUVmodel based on data from the AUV simulation platformshown as in Figures 2-3 After a brief introduction of theAUVsimulation platform two typical identification cases areinvestigated One is the identification of AUV model basedon the MOESP method without consideration of noise Theother one is the verification of PM-EIV algorithm undergeneral noise assumption Finally to be more practical


Surface interface computer Motion control computer Mission control computer

Ethernet

AUV model Control surface model Thruster model

3D simulation computerModel computer

Figure 3 AUV simulation platform

model identification based on data from a closed-loop pathfollowing simulation experiment is performed

41 AUV Simulation Platform The basic structure of AUVsimulation platform is depicted as in Figure 3The whole sys-tem is connected through Ethernet and responsibilities of fivemain components are introduced here

(C1) Surface Interface Computer (SIC) Surface interface soft-ware is running on SIC which is in charge of deploying mis-sions for AUV and monitoring the states of the system Thesoftware also allows for manual intervention in case ofemergency

(C2) Mission Control Computer (MiC) Mission Managementsoftware developed in QNX real-time system is the core ofMiC MiC is mainly responsible for path plan according tomissions from SIC and fault diagnosis of the system

(C3) Motion Control Computer (MoC) Motion control soft-ware running on MoC aims at controlling the states such asheading speed and depth of AUV based on the preplannedpaths from MiC In field experiments MoC is also in chargeof navigation of the system

(C4) Model Computer (MC) MC is the host for mathematicmodels of AUV thruster and control surfaceThemathematicAUV model is a validated model with full coefficientsobtained from water-tank experiments

(C5) 3D Simulation Computer (3DC) 3DC provides anapproach for visual simulation relied on vir-tual reality tech-nology Vega Prime is used to construct the virtual oceanicenvironment and AUV model is developed in MultigenCreator

A more detailed process flow is described in Figure 2

42 Case 1 Identification without Consideration of Noise Inan ideal situation process noise and measurement noises canbe ignored so that the ordinary MOESP algorithm can beapplied to identify theHammersteinAUVmodel described asin Figure 1This case aims at testifying the reasonability of the

0 200 400 600 800 1000minus1

minus05

0

05

1

15

2

25

3

35

Time (s)

u (m

s)

System outputPrediction output

Prediction error

Figure 4 Prediction of surge speed

Hammersteinmodel for anAUVsystemAll system input sig-nals are chosen as sinusoid curves with different amplitudesand periods and the value of 119873 in (4) is set to be 4 Systemidentification results based on O-MOESP algorithm can beobtained Figures 4 5 and 6 shows the prediction errorsbetween the outputs of the identified model and the originaloutputsThree main system outputs surge velocity pitch rateand yaw rate that play important roles in control and naviga-tion of underactuated AUV which are considered here FromFigures 4ndash6 it can be concluded that even though iden-tification errors exist Hammerstein model constructed inSection 2 can act as a suitable structure for AUV dynamics

43 Case 2 Identification under EIV Framework In this casegeneral noises are added on the model computer in the AUVsimulation platform So the identification problem becomesan EIV one which can be solved recursively by the PM-EIValgorithm proposed in this paper System inputs of the simu-lation platform are still sinusoid curves and the value of119873 ischosen to be 6 Since the PM-EIValgorithm is a recursive one


minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1Ya

w ra

te (r

ads

)

0 200 400 600 800 1000Time (s)


Prediction error

Figure 5 Prediction of yaw rate

minus015

minus01

minus005

0

005

01

Pitc

h ra

te (r

ads

)

0 200 400 600 800 1000Time (s)


Prediction error

Figure 6 Prediction of pitch rate

a relative small amount of data is used to acquire an initialvalue of the model at the beginning Identification results areshown in Figures 7 8 and 9 Based on the Figures 7ndash9 it isreliable to conclude that the PM-EIV algorithm is effectiveand feasible in identifying the Hammerstein AUV modelunder general noise assumption

Remark Through the above two different identification casesunder different situations it has been approved that Ham-mersteinmodel proposed in Section 2 is capable of represent-ing the AUVdynamic system and the PM-EIVmethod is ableto identify theHammersteinmodel recursively under generalnoise assumption It is also interesting to notice that steadystate prediction errors in Case 2 are decreased comparisonwith those in Case 1 it is also interesting to notice that steadystate prediction errors in Case 2 are decreased comparingwith those in Case 1 due to the increase of119873 adopted

minus1

minus05

0

05

1

15

2

25

u (m

s)

0 100 200 300 400 500 600 700 800Time (s)


Prediction error


0 100 200 300 400 500 600 700 800minus025

minus02

minus015

minus01

minus005

0

005

01

015

Time (s)

Pitc

h ra

te (r

ads

)


Prediction error

Figure 8 Prediction of system pitch rate

minus06

minus04

minus02

0

02

04

06

08

Yaw

rate

(rad

s)

0 100 200 300 400 500 600 700 800Time (s)


Prediction error

Figure 9 Prediction of system yaw rate


0 200 400 600 800 10000

100

200

300

400

500

600

700

800

900

North (m)

East

(m)

Start point

AUV trajectoryPreplanned path

(a)

0 200400

600800

0200

400600

800

02468

1012

North (m)

Start point

East (m)

Dep

th (m

)

AUV diving trajectoryPreplanned path

(b)

Figure 10 Simulation results (a) horizontal projection of the trajectory (b) 3D trajectory of AUV

0 100200300400500600700800900

02004006008001000

02468

1012

Cali point

North (m)

Start Point

East (m)

Dep

th (m

)

AUV trajectoryModel trajectory

Figure 11 Identification result

44 Identification from Closed-Loop Simulation In the abovesimulations AUV model is identified using data from openloop control of surge speed heading and yaw However inpractical applications data for model identification is usuallycollected from field experiments with close loop control forspecific preplanned paths So in this section AUV identifica-tion process is carried out based on data from a closed-loopsimulation experiment To be more pellucid the feasibilityof PM-EIV algorithm is not illustrated by predicting thesurge speed pitch rate and yaw rate but shown by predictingthe trajectory of AUV in the test Figures 10 and 11 are thesimulation results from the AUV simulation platform Thepreplanned path is a circle with an origin at (500 500)mand radius of 300mThe depth command is 10m Simulationresults have shown that AUV can follow the preplanned circlevery well and the diving process is stable and fast

Then identification of the Hammerstein AUV model isbased on the inputsoutputs of this experiment Noises arealso considered Figure 11 has shown the identification result

in a different point of view It can be seen that predictionerror between the identified model of AUV and the actualtrajectory of AUV is large at the beginning because therecursive identification procedure needs time to convergeTherefore a position calibration operation is carried outwhen the identification results have converged at simulationtime 350 sTheCali-Point in Figure 11 is where the calibrationis implementedThe green line indicates the distance betweenCali-Point and expected point After the calibration thetrajectory of the identified Hammerstein AUV model canfollow the trajectory of AUV with satisfying accuracy

5 Conclusions

In this paper a recursive subspace identification algorithmPM-EIV is derived under general noise assumption subspaceidentification algorithm based on propagator method isextended into EIV framework is extended into EIV frame-work In order to implement the method on identificationof AUV model with consideration about nonlinearities andcouplings at the same time a Hammerstein AUV model isconstructed for the first time Three simulation experimentsunder different conditions are carried out to verify thefeasibility of the model and the effectiveness of the proposedalgorithm

In the future study system noises will not be restricted towhite noise and noise models related with oceanic environ-ment and sensors will be introduced to make the algorithmmore practical

Conflict of Interests

The authors Zheping Yan Di Wu Jiajia Zhou and LichaoHao declare that there is no conflict of interests regarding thepublication of this paper


Acknowledgments

This work was supported by the National Natural ScienceFoundation of China (Grant no 5179038) Program for NewCentury Excellent Talents in University of China (Grant noNCET-10-0053) and Fundamental Research Funds for theCentral Universities (Grant no HEUCF041318)

References

[1] M E Rentschler F S Hover and C Chryssostomidis ldquoSystemidentification of open-loop maneuvers leads to improved AUVflight performancerdquo IEEE Journal of Oceanic Engineering vol31 no 1 pp 200ndash208 2006

[2] J Kim K KimH S ChoiW Seong and K-Y Lee ldquoEstimationof hydrodynamic coefficients for an AUV using nonlinearobserversrdquo IEEE Journal of Oceanic Engineering vol 27 no 4pp 830ndash840 2002

[3] Oslash Hegrenaes and O Hallingstad ldquoModel-aided INS with seacurrent estimation for robust underwater navigationrdquo IEEEJournal of Oceanic Engineering vol 36 no 2 pp 316ndash337 2011

[4] R Qi L Zhu and B Jiang ldquoFault tolerant reconfigurablecontrol for MIMO systems using online fuzzy idenficationrdquoInternational Journal of Innovative Computing Information andControl vol 10 no 10 pp 3915ndash3928 2013

[5] S A Hisseini S A Hadei M B Menhaj et al ldquoFast euclideandirection search algorithm in adaptive noise cancellation andsystem identificationrdquo International Journal of Innovative Com-puting Information and Control vol 9 no 1 pp 191ndash206 2013

[6] A Tiano R Sutton A Lozowicki and W Naeem ldquoObserverKalman filter identification of an autonomous underwatervehiclerdquo Control Engineering Practice vol 15 no 6 pp 727ndash7392007

[7] Z Cai and E-W Bai ldquoMaking parametric Hammerstein systemidentification a linear problemrdquo Automatica vol 47 no 8 pp1806ndash1812 2011

[8] X-M Chen and H-F Chen ldquoRecursive identification forMIMOHammerstein systemsrdquo IEEETransactions onAutomaticControl vol 56 no 4 pp 895ndash902 2011

[9] J P Noel S Marchesiello and G Kerschen ldquoSubspace-basedidentification of a nonlinear spacecraft in the time and fre-quency domainsrdquo Mechanical Systems and Signal Processingvol 43 no 1-2 pp 217ndash236 2013

[10] S J Qin ldquoAn overview of subspace identificationrdquoComputers ampChemical Engineering vol 30 no 10ndash12 pp 1502ndash1513 2006

[11] W Lin S J Qin and L Ljung ldquoOn consistency of closed-loop subspace identification with innovation estimationrdquo inProceedings of the 43rd IEEEConference onDecision and Control(CDC rsquo04) vol 2 pp 2195ndash2200 Nassau Bahamas December2004

[12] M Lovera T Gustafsson and M Verhaegen ldquoRecursive sub-space identification of linear and non-linearWiener state-spacemodelsrdquo Automatica vol 36 no 11 pp 1639ndash1650 2000

[13] C T Chou and M Verhaegen ldquoSubspace algorithms forthe identification of multivariable dynamic errors-in-variablesmodelsrdquo Automatica vol 33 no 10 pp 1857ndash1869 1997

[14] Z Yan DWu J Zhou andW Zhang ldquoRecursive identificationof autonomous underwater vehicle for emergency navigationrdquoin Proceedings of the Oceans pp 1ndash6 IEEE Hampton Roads VaUSA 2012

[15] A Alenany and H Shang ldquoRecursive subspace identificationwith prior information using the constrained least squaresapproachrdquo Computers amp Chemical Engineering vol 54 pp 174ndash180 2013

[16] A Akhenak Deviella L Bako et al ldquoOnline fault diagnosisusing recursive subspace identification application to a dam-gallery open channel systemrdquo Control Engineering Practice vol21 no 6 pp 791ndash806 2013

[17] B Yang ldquoProjection approximation subspace trackingrdquo IEEETransactions on Signal Processing vol 43 no 1 pp 95ndash107 1995

[18] T Gustafsson ldquoInstrumental variable subspace tracking usingprojection approximationrdquo IEEETransactions on Signal Process-ing vol 46 no 3 pp 669ndash681 1998

[19] H Oku and H Kimura ldquoRecursive 4SID algorithms usinggradient type subspace trackingrdquo Automatica vol 38 no 6 pp1035ndash1043 2002

[20] G Mercere L Bako and S Lecœuche ldquoPropagator-basedmethods for recursive subspace model identificationrdquo SignalProcessing vol 88 no 3 pp 468ndash491 2008

[21] T I FossenGuidance and Control of Ocean Vehicles JohnWileyamp Sons New York NY USA 1994

[22] D Luo andA Leonessa ldquoIdentification ofMIMOHammersteinsystems with non-linear feedbackrdquo IMA Journal of Mathemati-cal Control and Information vol 25 no 3 pp 367ndash392 2008

[23] L Ljung System Identification Theory for Users Prentice HallUpper Saddle River NJ USA 1987

[24] MVerhaegen andPDewilde ldquoSubspacemodel identificationmdashpart 1The output-error state-spacemodel identification class ofalgorithmsrdquo International Journal of Control vol 56 no 5 pp1187ndash1210 1992

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


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Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


on the linear part As illustrated in Figure 1 one remarkablebenefit of Hammerstein system is that it possesses advantagesof linear MIMO system and can approximate nonlinear andcoupled terms of AUV to a large extent at the same timeTo the best of the authorsrsquo acknowledge this is the firsttime that Hammerstein model is applied in the area of AUVmodeling Because of the conveniences brought by Hammer-stein system extensive attentions has been paid to obtainsystem parameters from inputoutput data Cai and Bai [7]proposed a method to make the identification of parametricHammerstein system a linear problem through regardingthe average squared error cost function as the inner prod-uct between the true but unknown parameter vector andits estimations But this method was discussed under theassumption that Hammerstein system only consists of singleinput and single output As to MIMO Hammerstein systemidentification a nonparametric algorithmbased on stochasticapproximation approach is proposed in [8] However nonlin-ear MIMO Hammerstein identification problem concernedin this paper is still challenging

One group of widely studiedMIMO system identificationalgorithms for Hammerstein system is subspace identifi-cation methods [9] which mainly include three differentbranches numerical algorithms for state-space subspace sys-tem identification (N4SID) MIMO output-error state-spacemodel identification (MOESP) and canonical variate analy-sis (CVA) Compared with other identification algorithmssubspace identification methods are more attractive due toseveral advantages [10] For example subspace identificationmethods can circumvent the complicated parameterizationprocedure forMIMOsystemof prediction errormethods [11]What ismore subspace identificationmethods do not requirenonlinear searches in the parameter space based on com-putational tools such as QR factorization and singular valuedecomposition (SVD) [12] Since the Hammerstein AUVmodel is a parametric onewithmultiinputs andmultioutputsMIMO MOESP algorithm is adopted as the theoretical basisfor further investigation in this paper

In addition instead of regarding the identification proce-dure in ideal situations practical engineering circumstancesneed to be taken into consideration A widely studied oneis that the general noise assumption has to be made whichmeans ldquothe measured input is corrupted by a white measure-ment noise while the measured output is corrupted by thesum of a white measurement noise and a term due to a whiteprocess noiserdquo [13] Another practical situation which need tobe considered is that different oceanic environment may leadto different hydrodynamic coefficients in which case off-lineidentification results of Hammerstein AUV model will bringerrors to control and navigation systems [14] Besides itis often the case that the structure of AUV usually hasto be modified mildly in order to fulfill various tasks inpractice Therefore recursive identification methods whichcan adjustmodel parameters online become rather significantand attractive in applications of such as adaptive controlmodel-aided navigation and so on

As a result in order to fulfill the situations concernedabove the identification of Hammerstein AUV model isregarded as an errors-in-variables (EIV) problem in this

Nonlinear part Linear part

Nonlinear feedback

u(k)

(k)υ u(k)

d(k) ω

y(k) y(k)

(k)

Figure 1 Hammerstein AUVModel

paper and we are going to make the MOESP method recur-sively under EIV framework so as to carry out recursive iden-tification algorithm for the Hammerstein AUV model Onemajor obstacle in recursive subspace identification is theincreasing computational complexity of SVD [15 16] Inprevious studies projection approximation subspace tracking(PAST) algorithm designed by Yang [17] was widely used toupdate the SVD However approximation in the algorithmwill bring slight difference between identified model and theoriginal one In addition IV-PAST algorithm [18] and gradi-ent type subspace tracking method [19] are also developed toestimate the signal subspace In [20] an instrumental variablepropagator method (IVPM) based recursive identificationalgorithmwas proposedNevertheless the paper only studiedthe method within past input (PI)past output (PO) MOESPframework So in this paper combining with HammersteinAUV identification problem described above IVPMmethodis extended into EIV framework and the PM-EIV algorithmfor recursive subspace identification of errors-in-variablesproblem is derived Compared with previous algorithmsmentioned the PM-EIV algorithm is more suitable to handlethe MIMO AUV Hammerstein model identification prob-lems under errors-in-variables framework recursively As amatter of fact since the Hammerstein model constructed canbe viewed as a generalized one for mechanical systems theproposed PM-EIV can be extended to identification of othersystems as well

The remainder of this paper is organized as follows InSection 2 a Hammerstein AUV model with nonlinear feed-back is formed with proper transformation from an ordinarydifferential one Then the linearization process of nonlinearpart is presented In Section 3 the MOESP identificationmethod is described with no consideration about the systemnoise After that a recursive subspacemethod called PM-EIVis derived under the general noise assumption Identificationand validation of theHammersteinAUVmodel are presentedin Section 4 At last conclusions are made in Section 5

Some notations used in this paper are followed Thesuperscript (sdot)119879 denotes the transposition operator 119864(sdot) is theexpectation operator R119898times119899 represents the set of 119898 times 119899 realmatrices119872perp is the orthogonal complement of119872

2 AUV Modeling

In this section a Hammerstein AUV model is formed afterintroducing the ordinary differential one Then in order tomake the Hammerstein AUV model suitable for subspace




119872] + 119862 (]) ] + 119863 (]) ] = 120591 (1)




119909 (119896 + 1) = 119860119909 (119896) + 1198611119891 (119906 (119896)) + 1198612119892 (119910 (119896)) + 119889 (119896)

119910 (119896) = 119862119909 (119896) + 1198631119891 (119906 (119896)) + 1198632119892 (119910 (119896))

119910 (119896) = 119910 (119896) + 120596 (119896)

(119896) = 119906 (119896) + 120592 (119896)

(2)


1 1198612 119862 119863

1 and 119863






120593119896 (119909) = 1 119896 = 0

120593119896 (119909) = [cos(

119896120587 (119909 minus 119909119898)

119909119889

) sin(119896120587 (119909 minus 119909

119898)

119909119889

)]

119879

119896 ge 1

(3)


119865 (119909) = 119908119900+

119873

sum119894=1

119908119894120593119894 (119909) (4)

where 119909119898



0(119909) 120593119879

1(119909) sdot sdot sdot 120593119879

119873(119909)]119879

119882 =


119891 (119906) = [1198821198791199061

1198821198791199062

sdot sdot sdot 119882119879119906119903]119879


119879 (119906119898)]119879

119892 (119910) = [119882119879

11991011198821198791199102

sdot sdot sdot 119882119879119910119904]119879

sdot [Φ119879 (1199101) Φ119879 (119910

2) sdot sdot sdot Φ119879 (119910

119897)]119879

(5)


isin 1198771 times 3(2119873+1) and 119882119910119894


119891 (119906) = 119870120585 (119906)

119892 (119910) = 119875120577 (119910) (6)


Δ

= [Φ119879(1199061) Φ119879(119906


119898)]119879

119870Δ

=

[1198821198791199061

1198821198791199062

sdot sdot sdot 119882119879119906119903]119879


119909 (119896 + 1) = 119860119909 (119896) +1198721120585 (119906 (119896)) + 1198722120577 (119910 (119896)) + 119889 (119896)

119910 (119896) = 119862119909 (119896) + 1198731120585 (119906 (119896)) + 1198732120577 (119910 (119896))

(7)


where 1198721

= 1198611119870 isin 119877119899times3(2119873+1) 119872

2= 119861

2119875 isin


2= 1198632119875 isin



[1198721 1198722] 119873 = [1198731 1198732] 119890(119896) = [120585119879(119906(119896)) 120577119879(119910(119896))]119879


119909 (119896 + 1) = 119860119909 (119896) + 119872119890 (119896) + 119889 (119896)

119910 (119896) = 119862119909 (119896) + 119873119890 (119896) (8)








119884119895119894119873

= Γ119894119883119895119873

+ 119867119894119864119895119894119873 (9)


119883119895119873

and 119864119895119894119873

as follows

119884119895119894119873

=[[

[

119910119895


d

119910119895+119894minus1


]]

]

119864119895119894119873

=[[

[

119890119895


d

119890119895+119894minus1


]]

]

119883119895119873


(10)



Γ119894= [119862119879 (119862119860)


119879]119879

119867119894=

[[[[

[


119862119872 119873 sdot sdot sdot 0

d


]]]]

]

(11)


[119864119895119894119873

119884119895119894119873

] = [11987711

0

11987721

11987722

] [1198761

1198762

] (12)


Γ119894119883119895119873

= [11987721 minus 11986711989411987711 11987722] [1198761

1198762

] (13)



22


119894

11987722= [119880119899 119880

perp

119899] [1198781

0

119900 1198782

] [119881119879119899

(119881perp119899)119879] (14)



with 119880119899

119880(1)

119899119860119879= 119880(2)

119899

119862119879= 119880119899 (1 119897 )

(15)


(119880perp

119899)119879

119867119894minus (119880perp

119899)119879

11987721119877minus1

11= 0 (16)


119895119894119873

= Γ119894119883119895119873

+ 119867119894119864119895119894119873

minus 1198671198941198811015840

119895119894119873+ 119866119894119875119895119894119873

+119882119895119894119873

(17)


where 119875119895119894119873

119882119895119894119873


119894is defined as

119866119894=

[[[[

[



d


]]]]

]

(18)

Notice that 1198811015840119895119894119873


119864 [119906119896119908119879

119895] = 0 for (119895 lt 119896) (19)


119864 [119906119896119908119879

119895] = 0 for (119895 ge 119896) (20)


119894+1119894119873

[1198641198791119894119873

1198791119894119873

]

= Γ119894119883119894+1119873

[1198641198791119894119873

1198791119894119873

]

+ 119867119894119864119894+1119894119873

[1198641198791119894119873

1198791119894119873

]

(21)

[119864119894+1119894119873

1198641198791119894119873

119864119894+1119894119873

1198791119894119873

119894+1119894119873

1198641198791119894119873

119894+1119894119873

1198791119894119873

]

= [11987711 (119905) 0

11987721 (119905) 119877

22 (119905)] [

1198761 (119905)

1198762 (119905)

]

(22)







119911119894 (119905 + 1) = 119910

119894 (119905 + 1) minus 119867119894 (119905 + 1) 119890119894 (119905 + 1) (23)

where 119905 = 119895 + 119873 minus 1 119910119894(119905 + 1) 119890

119894(119905 + 1) are new updated




[119864119894+1119894119873+1

1198641198791119894119873+1

119864119894+1119894119873+1

1198791119894119873+1

119894+1119894119873+1

1198641198791119894119873+1

119894+1119894119873+1

1198791119894119873+1

]

= [11987711 (119905 + 1) 0

11987721 (119905 + 1) 119877

22 (119905 + 1)] [

1198761 (119905 + 1)

1198762 (119905 + 1)

]

(24)

where 119864119894+1119894119873+1

= [119864119894+1119894119873

119890119894(119905 + 1)] 1198641119894119873+1 and 119894+1119894119873+1

1119894119873+1


[119864119894+1119894119873

119890119894 (119905 + 1)

119894+1119894119873

119910119894 (119905 + 1)

] [1198641198791119894119873

1198791119894119873

119890119894(119873 + 1)

119879119910119894(119873 + 1)

119879]

= [11987711 (119905) 0 119890

119894 (119905 + 1) 120601 (119873 + 1)

11987721 (119905) 119877

22 (119905) 119910119894 (119905 + 1) 120601 (119873 + 1)

][

[

1198761 (119905)

1198762 (119905)

119868

]

]

(25)

where 120601(119873 + 1) = [119890119894(119873 + 1)

119879119910119894(119873 + 1)



lower triangle form

[11987711 (119905) 0 119890

119894 (119905 + 1) 120601 (119873 + 1)

11987721 (119905) 119877

22 (119905) 119910119894 (119905 + 1) 120601 (119873 + 1)

]Giv (119905 + 1)

= [11987711 (119905 + 1) 0 0

11987721 (119905 + 1) 119877

22 (119905) 119894 (119905 + 1)

]

(26)


11987722 (119905 + 1) 11987722(119905 + 1)

119879= 11987722 (119905) 11987722(119905)

119879+ 119894 (119905 + 1)

119879

119894(119905 + 1)

(27)


11987722 (119905 + 1)1198762 (119905 + 1)

= (119894+1119894119873+1

minus 119894119864119894+1119894119873+1

) [1198641198791119894119873+1

1198791119894119873+1

]

= 11987722 (119905) 1198762 (119905) + 119911119894 (119905 + 1) [119890119894(119873 + 1)

119879119910119894(119873 + 1)

119879]

(28)



119894(119905 + 1) can be found

119894 (119905 + 1) 119894(119905 + 1)

119879

= 11987722 (119905) 1198762 (119905) 120601(119873 + 1)

119879119911119894(119905 + 1)

119879

+ 119911119894 (119905 + 1) 120601 (119873 + 1) (11987722 (119905) 1198762 (119905))

119879

+ 119911119894 (119905 + 1) 120601 (119873 + 1) 120601(119873 + 1)

119879119911119894(119905 + 1)

119879

(29)


119870 (119905 + 1)119870(119905 + 1)119879= 119894 (119905 + 1) 119894(119905 + 1)

119879+ 11987722 (119905) 11987722(119905)

119879

(30)


119911119894 (119905 + 1) [119890

119879

119894(119873 + 1) 119910119879

119894(119873 + 1)]

= (119870 (119905 + 1) minus 11987722 (119905) 1198762 (119905))

119911119894 (119905 + 1) = 119870

minus1

120601(119870 (119905 + 1) minus 11987722 (119905) 1198762 (119905)) 120601

119879

119873

(31)







Γ119894= [

Γ1198941

Γ1198942

] = [119868

119875119879119898

] Γ1198941= 119876119904Γ1198941 (32)




119894

can be divided as

119911119894 (119905 + 1) = 119876

119904Γ1198941119909 (119905 + 1) + 119887119894 (119905 + 1) = [

1199111198941 (119905 + 1)

1199111198942 (119905 + 1)

] (33)



= 1198751198791198981199111198941 Then the 119875119879

119898can


119898 So the IVPM



Begin

Initialization


Fault diagnosis

AUV (MC)Mission end


(MiC)

(MiC)Path plans

Yes

NoFault

No

Yes

End



States monitor(SIC)

Visual reality(3DC)



following RLS form

119870119901 (119905 + 1)

= 120574119879(119905 + 1) 119877 (119905) (1 + 120574

119879(119905 + 1) 119877 (119905) 1199111198941 (119905 + 1))

minus1

119875119879

119898(119905 + 1)

= 119875119879

119898(119905) + [1199111198942 (119905 + 1) minus 119875

119879

119898(119905) 1199111198941 (119905 + 1)]119870119901 (119905 + 1)

119877 (119905 + 1) = 119877 (119905) minus 119877 (119905) 1199111198941 (119905 + 1)119870119901 (119905 + 1)

(34)

where 119877(119905) = 119864[1199111198941(119905)120574119879(119905)]


119894 algorithms








Ethernet













0 200 400 600 800 1000minus1

minus05

0

05

1

15

2

25

3

35

Time (s)

u (m

s)


Prediction error





minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1Ya

w ra

te (r

ads

)

0 200 400 600 800 1000Time (s)


Prediction error


minus015

minus01

minus005

0

005

01

Pitc

h ra

te (r

ads

)

0 200 400 600 800 1000Time (s)


Prediction error




minus1

minus05

0

05

1

15

2

25

u (m

s)

0 100 200 300 400 500 600 700 800Time (s)


Prediction error


0 100 200 300 400 500 600 700 800minus025

minus02

minus015

minus01

minus005

0

005

01

015

Time (s)

Pitc

h ra

te (r

ads

)


Prediction error


minus06

minus04

minus02

0

02

04

06

08

Yaw

rate

(rad

s)

0 100 200 300 400 500 600 700 800Time (s)


Prediction error



0 200 400 600 800 10000

100

200

300

400

500

600

700

800

900

North (m)

East

(m)

Start point


(a)

0 200400

600800

0200

400600

800

02468

1012

North (m)

Start point

East (m)

Dep

th (m

)


(b)


0 100200300400500600700800900

02004006008001000

02468

1012

Cali point

North (m)

Start Point

East (m)

Dep

th (m

)






5 Conclusions






Acknowledgments


References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












119872] + 119862 (]) ] + 119863 (]) ] = 120591 (1)




119909 (119896 + 1) = 119860119909 (119896) + 1198611119891 (119906 (119896)) + 1198612119892 (119910 (119896)) + 119889 (119896)

119910 (119896) = 119862119909 (119896) + 1198631119891 (119906 (119896)) + 1198632119892 (119910 (119896))

119910 (119896) = 119910 (119896) + 120596 (119896)

(119896) = 119906 (119896) + 120592 (119896)

(2)


1 1198612 119862 119863

1 and 119863






120593119896 (119909) = 1 119896 = 0

120593119896 (119909) = [cos(

119896120587 (119909 minus 119909119898)

119909119889

) sin(119896120587 (119909 minus 119909

119898)

119909119889

)]

119879

119896 ge 1

(3)


119865 (119909) = 119908119900+

119873

sum119894=1

119908119894120593119894 (119909) (4)

where 119909119898



0(119909) 120593119879

1(119909) sdot sdot sdot 120593119879

119873(119909)]119879

119882 =


119891 (119906) = [1198821198791199061

1198821198791199062

sdot sdot sdot 119882119879119906119903]119879


119879 (119906119898)]119879

119892 (119910) = [119882119879

11991011198821198791199102

sdot sdot sdot 119882119879119910119904]119879

sdot [Φ119879 (1199101) Φ119879 (119910

2) sdot sdot sdot Φ119879 (119910

119897)]119879

(5)


isin 1198771 times 3(2119873+1) and 119882119910119894


119891 (119906) = 119870120585 (119906)

119892 (119910) = 119875120577 (119910) (6)


Δ

= [Φ119879(1199061) Φ119879(119906


119898)]119879

119870Δ

=

[1198821198791199061

1198821198791199062

sdot sdot sdot 119882119879119906119903]119879


119909 (119896 + 1) = 119860119909 (119896) +1198721120585 (119906 (119896)) + 1198722120577 (119910 (119896)) + 119889 (119896)

119910 (119896) = 119862119909 (119896) + 1198731120585 (119906 (119896)) + 1198732120577 (119910 (119896))

(7)


where 1198721

= 1198611119870 isin 119877119899times3(2119873+1) 119872

2= 119861

2119875 isin


2= 1198632119875 isin



[1198721 1198722] 119873 = [1198731 1198732] 119890(119896) = [120585119879(119906(119896)) 120577119879(119910(119896))]119879


119909 (119896 + 1) = 119860119909 (119896) + 119872119890 (119896) + 119889 (119896)

119910 (119896) = 119862119909 (119896) + 119873119890 (119896) (8)








119884119895119894119873

= Γ119894119883119895119873

+ 119867119894119864119895119894119873 (9)


119883119895119873

and 119864119895119894119873

as follows

119884119895119894119873

=[[

[

119910119895


d

119910119895+119894minus1


]]

]

119864119895119894119873

=[[

[

119890119895


d

119890119895+119894minus1


]]

]

119883119895119873


(10)



Γ119894= [119862119879 (119862119860)


119879]119879

119867119894=

[[[[

[


119862119872 119873 sdot sdot sdot 0

d


]]]]

]

(11)


[119864119895119894119873

119884119895119894119873

] = [11987711

0

11987721

11987722

] [1198761

1198762

] (12)


Γ119894119883119895119873

= [11987721 minus 11986711989411987711 11987722] [1198761

1198762

] (13)



22


119894

11987722= [119880119899 119880

perp

119899] [1198781

0

119900 1198782

] [119881119879119899

(119881perp119899)119879] (14)



with 119880119899

119880(1)

119899119860119879= 119880(2)

119899

119862119879= 119880119899 (1 119897 )

(15)


(119880perp

119899)119879

119867119894minus (119880perp

119899)119879

11987721119877minus1

11= 0 (16)


119895119894119873

= Γ119894119883119895119873

+ 119867119894119864119895119894119873

minus 1198671198941198811015840

119895119894119873+ 119866119894119875119895119894119873

+119882119895119894119873

(17)


where 119875119895119894119873

119882119895119894119873


119894is defined as

119866119894=

[[[[

[



d


]]]]

]

(18)

Notice that 1198811015840119895119894119873


119864 [119906119896119908119879

119895] = 0 for (119895 lt 119896) (19)


119864 [119906119896119908119879

119895] = 0 for (119895 ge 119896) (20)


119894+1119894119873

[1198641198791119894119873

1198791119894119873

]

= Γ119894119883119894+1119873

[1198641198791119894119873

1198791119894119873

]

+ 119867119894119864119894+1119894119873

[1198641198791119894119873

1198791119894119873

]

(21)

[119864119894+1119894119873

1198641198791119894119873

119864119894+1119894119873

1198791119894119873

119894+1119894119873

1198641198791119894119873

119894+1119894119873

1198791119894119873

]

= [11987711 (119905) 0

11987721 (119905) 119877

22 (119905)] [

1198761 (119905)

1198762 (119905)

]

(22)







119911119894 (119905 + 1) = 119910

119894 (119905 + 1) minus 119867119894 (119905 + 1) 119890119894 (119905 + 1) (23)

where 119905 = 119895 + 119873 minus 1 119910119894(119905 + 1) 119890

119894(119905 + 1) are new updated




[119864119894+1119894119873+1

1198641198791119894119873+1

119864119894+1119894119873+1

1198791119894119873+1

119894+1119894119873+1

1198641198791119894119873+1

119894+1119894119873+1

1198791119894119873+1

]

= [11987711 (119905 + 1) 0

11987721 (119905 + 1) 119877

22 (119905 + 1)] [

1198761 (119905 + 1)

1198762 (119905 + 1)

]

(24)

where 119864119894+1119894119873+1

= [119864119894+1119894119873

119890119894(119905 + 1)] 1198641119894119873+1 and 119894+1119894119873+1

1119894119873+1


[119864119894+1119894119873

119890119894 (119905 + 1)

119894+1119894119873

119910119894 (119905 + 1)

] [1198641198791119894119873

1198791119894119873

119890119894(119873 + 1)

119879119910119894(119873 + 1)

119879]

= [11987711 (119905) 0 119890

119894 (119905 + 1) 120601 (119873 + 1)

11987721 (119905) 119877

22 (119905) 119910119894 (119905 + 1) 120601 (119873 + 1)

][

[

1198761 (119905)

1198762 (119905)

119868

]

]

(25)

where 120601(119873 + 1) = [119890119894(119873 + 1)

119879119910119894(119873 + 1)



lower triangle form

[11987711 (119905) 0 119890

119894 (119905 + 1) 120601 (119873 + 1)

11987721 (119905) 119877

22 (119905) 119910119894 (119905 + 1) 120601 (119873 + 1)

]Giv (119905 + 1)

= [11987711 (119905 + 1) 0 0

11987721 (119905 + 1) 119877

22 (119905) 119894 (119905 + 1)

]

(26)


11987722 (119905 + 1) 11987722(119905 + 1)

119879= 11987722 (119905) 11987722(119905)

119879+ 119894 (119905 + 1)

119879

119894(119905 + 1)

(27)


11987722 (119905 + 1)1198762 (119905 + 1)

= (119894+1119894119873+1

minus 119894119864119894+1119894119873+1

) [1198641198791119894119873+1

1198791119894119873+1

]

= 11987722 (119905) 1198762 (119905) + 119911119894 (119905 + 1) [119890119894(119873 + 1)

119879119910119894(119873 + 1)

119879]

(28)



119894(119905 + 1) can be found

119894 (119905 + 1) 119894(119905 + 1)

119879

= 11987722 (119905) 1198762 (119905) 120601(119873 + 1)

119879119911119894(119905 + 1)

119879

+ 119911119894 (119905 + 1) 120601 (119873 + 1) (11987722 (119905) 1198762 (119905))

119879

+ 119911119894 (119905 + 1) 120601 (119873 + 1) 120601(119873 + 1)

119879119911119894(119905 + 1)

119879

(29)


119870 (119905 + 1)119870(119905 + 1)119879= 119894 (119905 + 1) 119894(119905 + 1)

119879+ 11987722 (119905) 11987722(119905)

119879

(30)


119911119894 (119905 + 1) [119890

119879

119894(119873 + 1) 119910119879

119894(119873 + 1)]

= (119870 (119905 + 1) minus 11987722 (119905) 1198762 (119905))

119911119894 (119905 + 1) = 119870

minus1

120601(119870 (119905 + 1) minus 11987722 (119905) 1198762 (119905)) 120601

119879

119873

(31)







Γ119894= [

Γ1198941

Γ1198942

] = [119868

119875119879119898

] Γ1198941= 119876119904Γ1198941 (32)




119894

can be divided as

119911119894 (119905 + 1) = 119876

119904Γ1198941119909 (119905 + 1) + 119887119894 (119905 + 1) = [

1199111198941 (119905 + 1)

1199111198942 (119905 + 1)

] (33)



= 1198751198791198981199111198941 Then the 119875119879

119898can


119898 So the IVPM



Begin

Initialization


Fault diagnosis

AUV (MC)Mission end


(MiC)

(MiC)Path plans

Yes

NoFault

No

Yes

End



States monitor(SIC)

Visual reality(3DC)



following RLS form

119870119901 (119905 + 1)

= 120574119879(119905 + 1) 119877 (119905) (1 + 120574

119879(119905 + 1) 119877 (119905) 1199111198941 (119905 + 1))

minus1

119875119879

119898(119905 + 1)

= 119875119879

119898(119905) + [1199111198942 (119905 + 1) minus 119875

119879

119898(119905) 1199111198941 (119905 + 1)]119870119901 (119905 + 1)

119877 (119905 + 1) = 119877 (119905) minus 119877 (119905) 1199111198941 (119905 + 1)119870119901 (119905 + 1)

(34)

where 119877(119905) = 119864[1199111198941(119905)120574119879(119905)]


119894 algorithms








Ethernet













0 200 400 600 800 1000minus1

minus05

0

05

1

15

2

25

3

35

Time (s)

u (m

s)


Prediction error





minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1Ya

w ra

te (r

ads

)

0 200 400 600 800 1000Time (s)


Prediction error


minus015

minus01

minus005

0

005

01

Pitc

h ra

te (r

ads

)

0 200 400 600 800 1000Time (s)


Prediction error




minus1

minus05

0

05

1

15

2

25

u (m

s)

0 100 200 300 400 500 600 700 800Time (s)


Prediction error


0 100 200 300 400 500 600 700 800minus025

minus02

minus015

minus01

minus005

0

005

01

015

Time (s)

Pitc

h ra

te (r

ads

)


Prediction error


minus06

minus04

minus02

0

02

04

06

08

Yaw

rate

(rad

s)

0 100 200 300 400 500 600 700 800Time (s)


Prediction error



0 200 400 600 800 10000

100

200

300

400

500

600

700

800

900

North (m)

East

(m)

Start point


(a)

0 200400

600800

0200

400600

800

02468

1012

North (m)

Start point

East (m)

Dep

th (m

)


(b)


0 100200300400500600700800900

02004006008001000

02468

1012

Cali point

North (m)

Start Point

East (m)

Dep

th (m

)






5 Conclusions






Acknowledgments


References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










where 1198721

= 1198611119870 isin 119877119899times3(2119873+1) 119872

2= 119861

2119875 isin


2= 1198632119875 isin



[1198721 1198722] 119873 = [1198731 1198732] 119890(119896) = [120585119879(119906(119896)) 120577119879(119910(119896))]119879


119909 (119896 + 1) = 119860119909 (119896) + 119872119890 (119896) + 119889 (119896)

119910 (119896) = 119862119909 (119896) + 119873119890 (119896) (8)








119884119895119894119873

= Γ119894119883119895119873

+ 119867119894119864119895119894119873 (9)


119883119895119873

and 119864119895119894119873

as follows

119884119895119894119873

=[[

[

119910119895


d

119910119895+119894minus1


]]

]

119864119895119894119873

=[[

[

119890119895


d

119890119895+119894minus1


]]

]

119883119895119873


(10)



Γ119894= [119862119879 (119862119860)


119879]119879

119867119894=

[[[[

[


119862119872 119873 sdot sdot sdot 0

d


]]]]

]

(11)


[119864119895119894119873

119884119895119894119873

] = [11987711

0

11987721

11987722

] [1198761

1198762

] (12)


Γ119894119883119895119873

= [11987721 minus 11986711989411987711 11987722] [1198761

1198762

] (13)



22


119894

11987722= [119880119899 119880

perp

119899] [1198781

0

119900 1198782

] [119881119879119899

(119881perp119899)119879] (14)



with 119880119899

119880(1)

119899119860119879= 119880(2)

119899

119862119879= 119880119899 (1 119897 )

(15)


(119880perp

119899)119879

119867119894minus (119880perp

119899)119879

11987721119877minus1

11= 0 (16)


119895119894119873

= Γ119894119883119895119873

+ 119867119894119864119895119894119873

minus 1198671198941198811015840

119895119894119873+ 119866119894119875119895119894119873

+119882119895119894119873

(17)


where 119875119895119894119873

119882119895119894119873


119894is defined as

119866119894=

[[[[

[



d


]]]]

]

(18)

Notice that 1198811015840119895119894119873


119864 [119906119896119908119879

119895] = 0 for (119895 lt 119896) (19)


119864 [119906119896119908119879

119895] = 0 for (119895 ge 119896) (20)


119894+1119894119873

[1198641198791119894119873

1198791119894119873

]

= Γ119894119883119894+1119873

[1198641198791119894119873

1198791119894119873

]

+ 119867119894119864119894+1119894119873

[1198641198791119894119873

1198791119894119873

]

(21)

[119864119894+1119894119873

1198641198791119894119873

119864119894+1119894119873

1198791119894119873

119894+1119894119873

1198641198791119894119873

119894+1119894119873

1198791119894119873

]

= [11987711 (119905) 0

11987721 (119905) 119877

22 (119905)] [

1198761 (119905)

1198762 (119905)

]

(22)







119911119894 (119905 + 1) = 119910

119894 (119905 + 1) minus 119867119894 (119905 + 1) 119890119894 (119905 + 1) (23)

where 119905 = 119895 + 119873 minus 1 119910119894(119905 + 1) 119890

119894(119905 + 1) are new updated




[119864119894+1119894119873+1

1198641198791119894119873+1

119864119894+1119894119873+1

1198791119894119873+1

119894+1119894119873+1

1198641198791119894119873+1

119894+1119894119873+1

1198791119894119873+1

]

= [11987711 (119905 + 1) 0

11987721 (119905 + 1) 119877

22 (119905 + 1)] [

1198761 (119905 + 1)

1198762 (119905 + 1)

]

(24)

where 119864119894+1119894119873+1

= [119864119894+1119894119873

119890119894(119905 + 1)] 1198641119894119873+1 and 119894+1119894119873+1

1119894119873+1


[119864119894+1119894119873

119890119894 (119905 + 1)

119894+1119894119873

119910119894 (119905 + 1)

] [1198641198791119894119873

1198791119894119873

119890119894(119873 + 1)

119879119910119894(119873 + 1)

119879]

= [11987711 (119905) 0 119890

119894 (119905 + 1) 120601 (119873 + 1)

11987721 (119905) 119877

22 (119905) 119910119894 (119905 + 1) 120601 (119873 + 1)

][

[

1198761 (119905)

1198762 (119905)

119868

]

]

(25)

where 120601(119873 + 1) = [119890119894(119873 + 1)

119879119910119894(119873 + 1)



lower triangle form

[11987711 (119905) 0 119890

119894 (119905 + 1) 120601 (119873 + 1)

11987721 (119905) 119877

22 (119905) 119910119894 (119905 + 1) 120601 (119873 + 1)

]Giv (119905 + 1)

= [11987711 (119905 + 1) 0 0

11987721 (119905 + 1) 119877

22 (119905) 119894 (119905 + 1)

]

(26)


11987722 (119905 + 1) 11987722(119905 + 1)

119879= 11987722 (119905) 11987722(119905)

119879+ 119894 (119905 + 1)

119879

119894(119905 + 1)

(27)


11987722 (119905 + 1)1198762 (119905 + 1)

= (119894+1119894119873+1

minus 119894119864119894+1119894119873+1

) [1198641198791119894119873+1

1198791119894119873+1

]

= 11987722 (119905) 1198762 (119905) + 119911119894 (119905 + 1) [119890119894(119873 + 1)

119879119910119894(119873 + 1)

119879]

(28)



119894(119905 + 1) can be found

119894 (119905 + 1) 119894(119905 + 1)

119879

= 11987722 (119905) 1198762 (119905) 120601(119873 + 1)

119879119911119894(119905 + 1)

119879

+ 119911119894 (119905 + 1) 120601 (119873 + 1) (11987722 (119905) 1198762 (119905))

119879

+ 119911119894 (119905 + 1) 120601 (119873 + 1) 120601(119873 + 1)

119879119911119894(119905 + 1)

119879

(29)


119870 (119905 + 1)119870(119905 + 1)119879= 119894 (119905 + 1) 119894(119905 + 1)

119879+ 11987722 (119905) 11987722(119905)

119879

(30)


119911119894 (119905 + 1) [119890

119879

119894(119873 + 1) 119910119879

119894(119873 + 1)]

= (119870 (119905 + 1) minus 11987722 (119905) 1198762 (119905))

119911119894 (119905 + 1) = 119870

minus1

120601(119870 (119905 + 1) minus 11987722 (119905) 1198762 (119905)) 120601

119879

119873

(31)







Γ119894= [

Γ1198941

Γ1198942

] = [119868

119875119879119898

] Γ1198941= 119876119904Γ1198941 (32)




119894

can be divided as

119911119894 (119905 + 1) = 119876

119904Γ1198941119909 (119905 + 1) + 119887119894 (119905 + 1) = [

1199111198941 (119905 + 1)

1199111198942 (119905 + 1)

] (33)



= 1198751198791198981199111198941 Then the 119875119879

119898can


119898 So the IVPM



Begin

Initialization


Fault diagnosis

AUV (MC)Mission end


(MiC)

(MiC)Path plans

Yes

NoFault

No

Yes

End



States monitor(SIC)

Visual reality(3DC)



following RLS form

119870119901 (119905 + 1)

= 120574119879(119905 + 1) 119877 (119905) (1 + 120574

119879(119905 + 1) 119877 (119905) 1199111198941 (119905 + 1))

minus1

119875119879

119898(119905 + 1)

= 119875119879

119898(119905) + [1199111198942 (119905 + 1) minus 119875

119879

119898(119905) 1199111198941 (119905 + 1)]119870119901 (119905 + 1)

119877 (119905 + 1) = 119877 (119905) minus 119877 (119905) 1199111198941 (119905 + 1)119870119901 (119905 + 1)

(34)

where 119877(119905) = 119864[1199111198941(119905)120574119879(119905)]


119894 algorithms








Ethernet













0 200 400 600 800 1000minus1

minus05

0

05

1

15

2

25

3

35

Time (s)

u (m

s)


Prediction error





minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1Ya

w ra

te (r

ads

)

0 200 400 600 800 1000Time (s)


Prediction error


minus015

minus01

minus005

0

005

01

Pitc

h ra

te (r

ads

)

0 200 400 600 800 1000Time (s)


Prediction error




minus1

minus05

0

05

1

15

2

25

u (m

s)

0 100 200 300 400 500 600 700 800Time (s)


Prediction error


0 100 200 300 400 500 600 700 800minus025

minus02

minus015

minus01

minus005

0

005

01

015

Time (s)

Pitc

h ra

te (r

ads

)


Prediction error


minus06

minus04

minus02

0

02

04

06

08

Yaw

rate

(rad

s)

0 100 200 300 400 500 600 700 800Time (s)


Prediction error



0 200 400 600 800 10000

100

200

300

400

500

600

700

800

900

North (m)

East

(m)

Start point


(a)

0 200400

600800

0200

400600

800

02468

1012

North (m)

Start point

East (m)

Dep

th (m

)


(b)


0 100200300400500600700800900

02004006008001000

02468

1012

Cali point

North (m)

Start Point

East (m)

Dep

th (m

)






5 Conclusions






Acknowledgments


References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










where 119875119895119894119873

119882119895119894119873


119894is defined as

119866119894=

[[[[

[



d


]]]]

]

(18)

Notice that 1198811015840119895119894119873


119864 [119906119896119908119879

119895] = 0 for (119895 lt 119896) (19)


119864 [119906119896119908119879

119895] = 0 for (119895 ge 119896) (20)


119894+1119894119873

[1198641198791119894119873

1198791119894119873

]

= Γ119894119883119894+1119873

[1198641198791119894119873

1198791119894119873

]

+ 119867119894119864119894+1119894119873

[1198641198791119894119873

1198791119894119873

]

(21)

[119864119894+1119894119873

1198641198791119894119873

119864119894+1119894119873

1198791119894119873

119894+1119894119873

1198641198791119894119873

119894+1119894119873

1198791119894119873

]

= [11987711 (119905) 0

11987721 (119905) 119877

22 (119905)] [

1198761 (119905)

1198762 (119905)

]

(22)







119911119894 (119905 + 1) = 119910

119894 (119905 + 1) minus 119867119894 (119905 + 1) 119890119894 (119905 + 1) (23)

where 119905 = 119895 + 119873 minus 1 119910119894(119905 + 1) 119890

119894(119905 + 1) are new updated




[119864119894+1119894119873+1

1198641198791119894119873+1

119864119894+1119894119873+1

1198791119894119873+1

119894+1119894119873+1

1198641198791119894119873+1

119894+1119894119873+1

1198791119894119873+1

]

= [11987711 (119905 + 1) 0

11987721 (119905 + 1) 119877

22 (119905 + 1)] [

1198761 (119905 + 1)

1198762 (119905 + 1)

]

(24)

where 119864119894+1119894119873+1

= [119864119894+1119894119873

119890119894(119905 + 1)] 1198641119894119873+1 and 119894+1119894119873+1

1119894119873+1


[119864119894+1119894119873

119890119894 (119905 + 1)

119894+1119894119873

119910119894 (119905 + 1)

] [1198641198791119894119873

1198791119894119873

119890119894(119873 + 1)

119879119910119894(119873 + 1)

119879]

= [11987711 (119905) 0 119890

119894 (119905 + 1) 120601 (119873 + 1)

11987721 (119905) 119877

22 (119905) 119910119894 (119905 + 1) 120601 (119873 + 1)

][

[

1198761 (119905)

1198762 (119905)

119868

]

]

(25)

where 120601(119873 + 1) = [119890119894(119873 + 1)

119879119910119894(119873 + 1)



lower triangle form

[11987711 (119905) 0 119890

119894 (119905 + 1) 120601 (119873 + 1)

11987721 (119905) 119877

22 (119905) 119910119894 (119905 + 1) 120601 (119873 + 1)

]Giv (119905 + 1)

= [11987711 (119905 + 1) 0 0

11987721 (119905 + 1) 119877

22 (119905) 119894 (119905 + 1)

]

(26)


11987722 (119905 + 1) 11987722(119905 + 1)

119879= 11987722 (119905) 11987722(119905)

119879+ 119894 (119905 + 1)

119879

119894(119905 + 1)

(27)


11987722 (119905 + 1)1198762 (119905 + 1)

= (119894+1119894119873+1

minus 119894119864119894+1119894119873+1

) [1198641198791119894119873+1

1198791119894119873+1

]

= 11987722 (119905) 1198762 (119905) + 119911119894 (119905 + 1) [119890119894(119873 + 1)

119879119910119894(119873 + 1)

119879]

(28)



119894(119905 + 1) can be found

119894 (119905 + 1) 119894(119905 + 1)

119879

= 11987722 (119905) 1198762 (119905) 120601(119873 + 1)

119879119911119894(119905 + 1)

119879

+ 119911119894 (119905 + 1) 120601 (119873 + 1) (11987722 (119905) 1198762 (119905))

119879

+ 119911119894 (119905 + 1) 120601 (119873 + 1) 120601(119873 + 1)

119879119911119894(119905 + 1)

119879

(29)


119870 (119905 + 1)119870(119905 + 1)119879= 119894 (119905 + 1) 119894(119905 + 1)

119879+ 11987722 (119905) 11987722(119905)

119879

(30)


119911119894 (119905 + 1) [119890

119879

119894(119873 + 1) 119910119879

119894(119873 + 1)]

= (119870 (119905 + 1) minus 11987722 (119905) 1198762 (119905))

119911119894 (119905 + 1) = 119870

minus1

120601(119870 (119905 + 1) minus 11987722 (119905) 1198762 (119905)) 120601

119879

119873

(31)







Γ119894= [

Γ1198941

Γ1198942

] = [119868

119875119879119898

] Γ1198941= 119876119904Γ1198941 (32)




119894

can be divided as

119911119894 (119905 + 1) = 119876

119904Γ1198941119909 (119905 + 1) + 119887119894 (119905 + 1) = [

1199111198941 (119905 + 1)

1199111198942 (119905 + 1)

] (33)



= 1198751198791198981199111198941 Then the 119875119879

119898can


119898 So the IVPM



Begin

Initialization


Fault diagnosis

AUV (MC)Mission end


(MiC)

(MiC)Path plans

Yes

NoFault

No

Yes

End



States monitor(SIC)

Visual reality(3DC)



following RLS form

119870119901 (119905 + 1)

= 120574119879(119905 + 1) 119877 (119905) (1 + 120574

119879(119905 + 1) 119877 (119905) 1199111198941 (119905 + 1))

minus1

119875119879

119898(119905 + 1)

= 119875119879

119898(119905) + [1199111198942 (119905 + 1) minus 119875

119879

119898(119905) 1199111198941 (119905 + 1)]119870119901 (119905 + 1)

119877 (119905 + 1) = 119877 (119905) minus 119877 (119905) 1199111198941 (119905 + 1)119870119901 (119905 + 1)

(34)

where 119877(119905) = 119864[1199111198941(119905)120574119879(119905)]


119894 algorithms








Ethernet













0 200 400 600 800 1000minus1

minus05

0

05

1

15

2

25

3

35

Time (s)

u (m

s)


Prediction error





minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1Ya

w ra

te (r

ads

)

0 200 400 600 800 1000Time (s)


Prediction error


minus015

minus01

minus005

0

005

01

Pitc

h ra

te (r

ads

)

0 200 400 600 800 1000Time (s)


Prediction error




minus1

minus05

0

05

1

15

2

25

u (m

s)

0 100 200 300 400 500 600 700 800Time (s)


Prediction error


0 100 200 300 400 500 600 700 800minus025

minus02

minus015

minus01

minus005

0

005

01

015

Time (s)

Pitc

h ra

te (r

ads

)


Prediction error


minus06

minus04

minus02

0

02

04

06

08

Yaw

rate

(rad

s)

0 100 200 300 400 500 600 700 800Time (s)


Prediction error



0 200 400 600 800 10000

100

200

300

400

500

600

700

800

900

North (m)

East

(m)

Start point


(a)

0 200400

600800

0200

400600

800

02468

1012

North (m)

Start point

East (m)

Dep

th (m

)


(b)


0 100200300400500600700800900

02004006008001000

02468

1012

Cali point

North (m)

Start Point

East (m)

Dep

th (m

)






5 Conclusions






Acknowledgments


References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119894(119905 + 1) can be found

119894 (119905 + 1) 119894(119905 + 1)

119879

= 11987722 (119905) 1198762 (119905) 120601(119873 + 1)

119879119911119894(119905 + 1)

119879

+ 119911119894 (119905 + 1) 120601 (119873 + 1) (11987722 (119905) 1198762 (119905))

119879

+ 119911119894 (119905 + 1) 120601 (119873 + 1) 120601(119873 + 1)

119879119911119894(119905 + 1)

119879

(29)


119870 (119905 + 1)119870(119905 + 1)119879= 119894 (119905 + 1) 119894(119905 + 1)

119879+ 11987722 (119905) 11987722(119905)

119879

(30)


119911119894 (119905 + 1) [119890

119879

119894(119873 + 1) 119910119879

119894(119873 + 1)]

= (119870 (119905 + 1) minus 11987722 (119905) 1198762 (119905))

119911119894 (119905 + 1) = 119870

minus1

120601(119870 (119905 + 1) minus 11987722 (119905) 1198762 (119905)) 120601

119879

119873

(31)







Γ119894= [

Γ1198941

Γ1198942

] = [119868

119875119879119898

] Γ1198941= 119876119904Γ1198941 (32)




119894

can be divided as

119911119894 (119905 + 1) = 119876

119904Γ1198941119909 (119905 + 1) + 119887119894 (119905 + 1) = [

1199111198941 (119905 + 1)

1199111198942 (119905 + 1)

] (33)



= 1198751198791198981199111198941 Then the 119875119879

119898can


119898 So the IVPM



Begin

Initialization


Fault diagnosis

AUV (MC)Mission end


(MiC)

(MiC)Path plans

Yes

NoFault

No

Yes

End



States monitor(SIC)

Visual reality(3DC)



following RLS form

119870119901 (119905 + 1)

= 120574119879(119905 + 1) 119877 (119905) (1 + 120574

119879(119905 + 1) 119877 (119905) 1199111198941 (119905 + 1))

minus1

119875119879

119898(119905 + 1)

= 119875119879

119898(119905) + [1199111198942 (119905 + 1) minus 119875

119879

119898(119905) 1199111198941 (119905 + 1)]119870119901 (119905 + 1)

119877 (119905 + 1) = 119877 (119905) minus 119877 (119905) 1199111198941 (119905 + 1)119870119901 (119905 + 1)

(34)

where 119877(119905) = 119864[1199111198941(119905)120574119879(119905)]


119894 algorithms








Ethernet













0 200 400 600 800 1000minus1

minus05

0

05

1

15

2

25

3

35

Time (s)

u (m

s)


Prediction error





minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1Ya

w ra

te (r

ads

)

0 200 400 600 800 1000Time (s)


Prediction error


minus015

minus01

minus005

0

005

01

Pitc

h ra

te (r

ads

)

0 200 400 600 800 1000Time (s)


Prediction error




minus1

minus05

0

05

1

15

2

25

u (m

s)

0 100 200 300 400 500 600 700 800Time (s)


Prediction error


0 100 200 300 400 500 600 700 800minus025

minus02

minus015

minus01

minus005

0

005

01

015

Time (s)

Pitc

h ra

te (r

ads

)


Prediction error


minus06

minus04

minus02

0

02

04

06

08

Yaw

rate

(rad

s)

0 100 200 300 400 500 600 700 800Time (s)


Prediction error



0 200 400 600 800 10000

100

200

300

400

500

600

700

800

900

North (m)

East

(m)

Start point


(a)

0 200400

600800

0200

400600

800

02468

1012

North (m)

Start point

East (m)

Dep

th (m

)


(b)


0 100200300400500600700800900

02004006008001000

02468

1012

Cali point

North (m)

Start Point

East (m)

Dep

th (m

)






5 Conclusions






Acknowledgments


References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











Ethernet













0 200 400 600 800 1000minus1

minus05

0

05

1

15

2

25

3

35

Time (s)

u (m

s)


Prediction error





minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1Ya

w ra

te (r

ads

)

0 200 400 600 800 1000Time (s)


Prediction error


minus015

minus01

minus005

0

005

01

Pitc

h ra

te (r

ads

)

0 200 400 600 800 1000Time (s)


Prediction error




minus1

minus05

0

05

1

15

2

25

u (m

s)

0 100 200 300 400 500 600 700 800Time (s)


Prediction error


0 100 200 300 400 500 600 700 800minus025

minus02

minus015

minus01

minus005

0

005

01

015

Time (s)

Pitc

h ra

te (r

ads

)


Prediction error


minus06

minus04

minus02

0

02

04

06

08

Yaw

rate

(rad

s)

0 100 200 300 400 500 600 700 800Time (s)


Prediction error



0 200 400 600 800 10000

100

200

300

400

500

600

700

800

900

North (m)

East

(m)

Start point


(a)

0 200400

600800

0200

400600

800

02468

1012

North (m)

Start point

East (m)

Dep

th (m

)


(b)


0 100200300400500600700800900

02004006008001000

02468

1012

Cali point

North (m)

Start Point

East (m)

Dep

th (m

)






5 Conclusions






Acknowledgments


References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1Ya

w ra

te (r

ads

)

0 200 400 600 800 1000Time (s)


Prediction error


minus015

minus01

minus005

0

005

01

Pitc

h ra

te (r

ads

)

0 200 400 600 800 1000Time (s)


Prediction error




minus1

minus05

0

05

1

15

2

25

u (m

s)

0 100 200 300 400 500 600 700 800Time (s)


Prediction error


0 100 200 300 400 500 600 700 800minus025

minus02

minus015

minus01

minus005

0

005

01

015

Time (s)

Pitc

h ra

te (r

ads

)


Prediction error


minus06

minus04

minus02

0

02

04

06

08

Yaw

rate

(rad

s)

0 100 200 300 400 500 600 700 800Time (s)


Prediction error



0 200 400 600 800 10000

100

200

300

400

500

600

700

800

900

North (m)

East

(m)

Start point


(a)

0 200400

600800

0200

400600

800

02468

1012

North (m)

Start point

East (m)

Dep

th (m

)


(b)


0 100200300400500600700800900

02004006008001000

02468

1012

Cali point

North (m)

Start Point

East (m)

Dep

th (m

)






5 Conclusions






Acknowledgments


References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 200 400 600 800 10000

100

200

300

400

500

600

700

800

900

North (m)

East

(m)

Start point


(a)

0 200400

600800

0200

400600

800

02468

1012

North (m)

Start point

East (m)

Dep

th (m

)


(b)


0 100200300400500600700800900

02004006008001000

02468

1012

Cali point

North (m)

Start Point

East (m)

Dep

th (m

)






5 Conclusions






Acknowledgments


References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Acknowledgments


References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Research Article Recursive Subspace Identification of AUV...

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