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    ADAPTIVE FILTE APPLICATIONS TO HEAVE COMPENSATIOND.G. Lainiotis, K. Plataniotis, and C. ChardamgoasFlorida Institute of Technology, Melbourne,FL and

    University of Patras, Patras, Greece.

    Abstract - The problem of heave motion compensation isaddressed in this paper. A significant class of the Eainiotispartitioning approach is applied and comparisons are madewith the Kalman filter based approach, with respect to theircomputational complexity and performance. It is shown thatthe linearLainiotis filter is well suited for on-line implemen-tation, since orders of magnitude reduction of processing timeis achieved, while in the case where model parameter uncer-tainty exists, the adaptive Lainiotis filter has excellentperformance.

    1. INTRODUCTIONIn many sea-related problem, as seismic experiments for oilexploration, a deep towed signal source and a sensor areemployed. The motion in the vertical axis of the source and thesensor (heave) affect the reflection records. The heave effectscan be partially removed by using the estimates of the verticalsource motions from hydrostatic pressure and motion sensorsto delay or advance the pulse firing instants relative to a clockpulse transfer [11.It is preferable that filtering can be applied in a real-time modeduring acquisition of the reflection responses. Sensors and ma-nipulation process are subject to random noise which isdependenton the sea state, the current and the relative positionof the ship with respect to the waves. The appropriate choiceof the covariances matrices of the the noise is an important is-sue that is crucial to the success of the heave compensationstrategy. The need for the design of fast, efficient and practi-cally implementable optimal filters that provide the requiredestimates is apparent.A lot of studies have been reported for the solution to thisproblem, most of them utilizing Kalman filter-based approach113, [21. This approach has two main drawbacks :1.Due to the fact that the design of the Kalman filter isbasedon the assumption of complete knowledge of the modelwhich describes the heave dynamics, there is a degradation

    in the estimate quality in the case of a mismatch between

    the model used to desig,n he filter and the actualmodel,2. When the model is peric&c in time, somethiing very reasin-

    able dor sea-related motion dynamics, the Kalman filter,due to its computational complexity, is inappropriate foron-line applications,and

    In this paper, the Lainiotir; multimodel partitioning approach[3]-[5] is proposed. The linear Lainiotis filter is used in thecase where the model is (completely known, but periodicintime, reducing signficantly the processing time. The adaptiveLainiotis filter is used in order to provide adaptability in achanging environment and its performance is evaluated withrespect to that of the Kalman filter.Specifically, the paper isorganizedas follows: n section 2 themodel that describes the heave motion dynamics is given. Insection 3 the linear Kalmaun and Lainiotis filters are given anddiscussed for the case of time-invariant and periodic models,when completely model knowledge is assumed. In section 4the adaptive Lainiotis filter is presented and discussed for thecase of partially unknown linear systems.In section 5 the sim-ulation resultsarepresented, and finally, conclusions are givenin section6.

    2. PROBLEM FORMULATIONThe mathematical model nsed, that describes the heave mo-tion dynamics appeared in [ ], is given by :

    x(t) =F x(t) t G w(t)z(t)=H x(t)+ v(t)

    where x(t) and z(t) is the 2x1 and 1x1 state and measurementprocesses, respectively; (vv(t)) and (v(t)) are the 1x1 and 1x1plant and measurement nciiserandom processes, respectively,which are independent, zero mean white Gaussian processeswith covariancesQ(t) and R(t), respectively. F is the 2x2statetransition matrix, G is the 2x1 noise matrix, and H is the 1x1observation matrix. The initial stae vector x(0)t is independent

    2770-7803-0838-7/92 $3.00 0 1992 IEEE

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    The matricesF,G, H are given by :r

    H =- -1l o

    where wo s the natural frequency of the system, and Q, is aquality factor.The discretized motion equations are :

    where T is the discretization interval@(k+l,k) = exp(IT) ( 5 )r(k) exp(IT)r G (6 )

    More details about the model and the values of the parameters,can be found in [l].The objective is to obtain the optimal, in themean square sense(mmse) estimate;'x(k/k)of x(k), using the noisy measurementsz(k)=(z(l),.. (k)). In the next sections such estimation algo-rithms will be presented and discussed.

    3. LINEAR FlLTERINGAssuming that all the parameters of the model are known inadvanced, the most common approachused is the design of theKalman filter in order to obtain the required estimates. Themmse state estimates ^x(k/k)and the corresponding error co-variance P(k/k) are given by [lo] :

    P(k+l/k+l) = [I-k(k+l) H(k+l)] P(k+l/k) (8)

    K(k+l) = P(k+l/k) HT (k+l) P -l(k+l/k) (12)TP (k+l/k) = H(k+l) P(k+l/k) H (k+1) + R(k+l) (13)

    It can be noticed from theabove equations that even in the caseof time invariant models, the Kalman filter is time varying.Due to its computational complexity it is inappropriate foron-line applications.In a radicallydifferent approach taken by Lainiotis [3]-[51 theinitial state vector is partitioned into the sum of two indepen-dent gaussian vectors, the nominal vector x and the unknownand random vector x . In other words, th% partitioning ap-proach decomposes the original estimation problem into asimpler one, namely the one with partially known initial con-dition, and a parameter estimation problem pertaining to theunknown part x of the initial state. The resulting filter is thelinear Lainiotisrper step partitioning filter (LLPSPF), and isgiven by :"x+l/k+l) =2 (k+l/k+l) + @ (k+lJc) P(k/k+l)

    2 (k+ /k+ I) = K (k+1) z(k+l)n

    0,&+1) = aT(k+l,k) HT(k+l) A(k+l) H(k) @(k+l,k) (18)P (k+l/k+l) = [I - K (k+1) H(k+l) ] Q(k) (19)0 k+l,k) = [I - K (k+l) H(k+l)] w+ l ,k) (20)K,(k+l) = Q(k) HT(k+l) A(k+l)

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    Comments : timal mmse estimate?(I&) f x(k) and the com:sponding er-ror covarianceP(k/k) are given by [31-[51:- Both the above filters, have the same performance for linear,

    Gaussian models, since they are different realizations of thesame optimal mmse estimator. They only differ on the com-putational requirements, and thus the amount of processingtime required for their implementation.

    - In the case where the model is time invariant, the Kalmanfilter is time varying, while the LLPSPF equations can begreatly simplified, namely the quantities 0 , ,@ ,K ,Kbecome time invariant and can be computdonfy o k e ? thebeginningof the filtering session[7].

    - Even in the case where the model is periodic in time, thequantities of the LLPSPF referred above, need only be com-puted for the first period, stored and then used as needed. Inthis situation the LLPSPF requires remarkable less compu-tations than the Kalman filter. This results to processing timesavings which is very useful in on-line applications.

    Fig. 1shows the computational requirements (in terms of nor-malized operations) of these filters when the number ofsensors is increasing from 1 to 100, for both time invariant andperiodic models. It can be easily seen the superiority of theLLPSPF in processing time savings, especially when the num-ber of sensors is very large.

    4.ADAPTIVE FILTERINGIt is very unrealistic to expect the filter designer to know inadvance which model describes the heave motion dynamics abany time. Thus, adaptive estimation techniques must be usedin order to improve the estimation results.The adaptive estimation problem considered is specified bythe following equations :

    where al l quantities are as described previously, and 0 is anunknown finite dimensional parameter vector, which ifknown, would completely specify the model. Moreover, 0 isconsidered to be a random variable with known or assumeda-priori density p(Q/O)=p(0).The processes [w(k)) and (v(k))are still uncorrelated when conditioned on 0, with covariancesQ(k,O)andR(k,O), respectively.Given the measurement set Z(k) = [z(l), z(2), ...,z(k)), the op-

    where ?@&;e) and P(k/k;Bi) are the 0-conditional mse stateestimate and the corresponding B-conditional error covariancematrix. They are obtained from the corresponding linear filtermatched to the model with parameter value 0 and initializedwith initial conditions x(O/Ol;0) and P(O/O;0). !2 is the samplespace of 0 and p(0/k) is the a-posteriori pdf given by the fol-lowing recursive Baps rule formula :

    where L(k/k;0) is the likelihiood ratio given by :

    where Pz(k/k-1;0) is the 0-conditional measurement error co-variance matrix and z(k/k-1;8) s the $-conditional inn ovationsequence.Comments :- The above equations pertain to the case that tlhe pdf associ-ated with 0 isa continuous function of 0. When th is is the case,one i s faced with the need for a nondenumemhle infinity oflinear filters for the exact realization of the opti~nal stimator.The usual approximation performed to overcolme this diffi-culty is to approximate 0:s.pdf by a finite sum, i.e., to dis-cretize the sample space Cb. There exist, of come, cases inwhich the sample space is in itself naturally discrete. In thiscase, the integrals in (27)-(30) are replaced b~ summationsrunning over all possible values of parameter 0.- It is comforting to know that when the true pameter valuelies inside the sample space that the adaptive estimator as-sumes, the estimator converges to this value. TWhen the trueparameter value is outside the assumed sample-space, the es-timator converges to that value in the sample space that is

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    "closer" to the true value, in the sense of Kullback's informa-tion measure minimization [9]. mci= 1

    2MSE = [ x(k) - &/k) 3- It is well known that the adaptive estimation problem consti-tutes a class of nonlinear estimator problems. Lainiotis'partitioning approach decomposes this nonlinear problem intoa linear nonadaptivepart, cosisting of a bank of linear filters,each filter matched to an admissible value of 6, and a nonlin-earpart., consisting of the a-posteriori pdf's p(6/k), that incor-porates the adaptive, learning, or system identifying nature ofthe adaptive estimator.- An important feature of the partitioning realization of theoptimal estimator is its natural decoupled structure. Indeed allthe filters needed to implement the adaptive estimatorcanbeindependently realized. This fact has the following great ad-vantages [SI :

    - these filters can be implemented using parallel processing- the overall realization is robust with respect to failure ofmachines, saving enormous computational timeany of the parallel processors

    The simulation results are shown in figs. 2-9. From thesegraphs the following conclusions can be made :- Mismodeled dynamics play an important role in the overall

    filter performance.The performance of the Kalman filtercorresponding to the matched cases is better than hat corre-sponding to the mismatched ones, while the performance ofthe ALF is bounded above by that of the matched Kalmanfilter and below by that of the mismatched Kalman filter.

    - Both in the time invariant and time varying situation, theALF perfoms better and correctly identifies the model in alimited number of steps.

    - The decisiondirected ALF identifies the correct model fasterthan the regular ALF.

    4.CONCLUSIONS5. SIMULATIONRESULTS

    Since the plant noise (w(k)) is dependent on the sea state andthe current, it is reasonable to consider its covariance matrixQ(k) as unknown.During the f is t simulation experimentQ(k)was assumed un-known but constant while in the second experimentQ(k) wasassumed unknown and periodic in time.The Kalman filter was designed and tested for the matched andthe unmatched case.The Adaptive Lainiotis Filter (ALF) was designed with twolinear filters, each one matched to a specific value of Q . Twodifferent realizations of the ALF were tested. The first was thisdescribed by (27)-(30) in the previous section. The second re-alization, instead of averaging the 6-conditional states esti-mates with respect to the a-posteriori probabilities p(O/k),simply selects the 0-conditionalstateestimate with the highesta-posteriori probability. This is reffered to as the decision-directed ALF (ALF-DD).In order to asses the performance of the above filters, the meansquare error, averaged over 50 Monte Carlo runs was used :

    The problem of heave motion estimation was considered inthis paper. The need for on-line implementation of the filteringalgorithm lead us to propose the LLPSPF that has great ad-vantages in processing time for both the time invariant modeland periodic case.Since the pameters of the model are not all known in ad-vance, the ALF was used, and its performance was comparedwith that of the Kalman fiter. It is shown via simulation, thatthe ALF identifies the correct model in limited number of stepsand its performance is far better than that of the Kalman filter.

    REFERENCES[11 Ferial El-Hawary, "Compensation for source heave by useof a Kalman filter", IEEE J. of Oceanic Eng., OE-7, n. 2,

    1982.[2] Ferial El-Hawary, "Pattern recognition for marine seismicexplorations", in Automated pattern analysis in petroleum

    explorations, eds. I . Palaz, S.K. Sengupta, Springer-Verlag, 1991.[3] D.G. Lainiotis, "Optimal adaptive estimation : Structure

    and parameter adaptation", IEEE Trans. on AC, v.16,10-170, 1971.

    [4] D.G. Lainiotis, "Partitioned estimation algorithms, I :Non-linear estimation",J. Information Sciences, 7, 203-255,

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    1974.[51 D.G. Lainiotis, "Partitioning :A unifying frameworks for

    adaptive systems , :Estimation",Roc. of the IEEE,@ , 8 ,1976.[6 ]D.G. Lainiotis et al., "Real time ship motion estimation us-

    ing Lainiotis filters", IFAC Workshop on Expert systemsand signal processing in marine automation, Denmark,1989.[7] D.G. Lainiotis, S.K. Katsikas, Linearand nonlinearLain-iotis filters :A survey and comparative evaluation",IFACWorkshop on Expert systems and signal processing in ma-rine automation, Denmark, 1989.[8] S.K. Katsikas, S.D. Likothanassis, D.G. Lainiotis, "On the

    parallel implementationsof the linear Kalman and Laini-otis filters and their efficiency", Signal Processing, 25,[91 R.M. Hawkes, R J . Moore, "Performanceof Bayesian pa-

    rameter estimators for linear signal models", IEEE Trans.on Automatic Control, AC-21,523-527,1976.[lo]R.E. Kalman, "A new approach to linear filtering and pre-diction problems", Trans. ASME J Bas. Engng., Ser D,

    289-305, 1991.

    35-45,1960.

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    : ALF-OD----------: Mismatched KF- _ - _-+-+-+--+-+ : Matched Kf

    stepsGg . 9 Motcned Model p!am noiae covariance Q I al/kco7$l!)l

    Mismotcisd Model piant noiae ccvoiionce 92=aZ(k, jsln,bZ(i./3j)

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