W.W. Zhou and. M. Blanke. Servolaboratory, build. 326. S6ren T. Lyngs~ A'S ..... 400.0. 150.0. 200.0. 250.0. 300.0. 350.0. NUMBER OF SAMPLES (-05 SEC) ?-
Proceedings of 25th Conference on Declsion and Control Athens, 0December 1988
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FA7 9%
Identification of a Class of Nonlinear State-Space Models using RPE Techniques W.W. Zhou
and
M. Blanke S6ren T.L y n g s ~A'S Lyngs6 Alle 2970 Horsholm, Denmark
Servolaboratory, build. 326 Technical Universityof Denmark 2800 Lyngby on leave from Jimei Yavigation Institute Xiamen, China
Summary The recursive prediction error methods in state-space as p a r a m e t e r form have been efficiently used identifiers for linear systems, and especially Ljung's innovations filter using a Newton search direction has nonlinear the filtering theory [Jazwinski, R P E method From proved to be quite ideal. In this paper, the in state-space form is developed to the nonlinear case 1970][Mayback, 19821 it is known that an attractive and a nedx t e n d e d to i n c l u dteheex a cf ot r m of a appliable nonlinear filter is the first-order filter with bias nonlinearity, thus enabling structure preservation for correction term (FOFBC), which is based on using first-order certain classes of nonlinear systems. Both the discrete covariance and gain computations, but with the second-order and the continuous-discrete versions of the algorithm terms in state expectation and prediction error equations. In this studywe use the FOFBC method for identification of the are investigated,anda i na ni n n o v a t i o n m s odel shows a quite nonlinear model (1-a, b). When a fixed value 0 is given, the nonlinear simulation example convincingperformance of t h e filter as combined predictor correspondingto (1-a, b)will be parameter and state estimator.
I j ; ( t + 1 , 8 ) = f ( 8 , u ; t , i ( t , 8 ) ) + B ~ t ) + K ! t i [ v i t ) - h ( 8 ; t , ~ ( t , 8 ) j - B y ( t )(2-a) l 1
1. Introduction where the
second order term Bx(t) the is
Inthispaper we present two parameteridentifiers for nonlinear discrete and continuous-discrete state-space models.Thesealgorithmsareinvestigated by usingthe linear recursive prediction error (RPE) method, Ljung and Soderstrom [1983], in combination with nonlinear secondorderfilteringtheoryJazwinski [ 19701, Mayback [ 19821, Zhou [ 19851.
(2-b)
;ctle)=h ce;t,at,e))
n,-vector with K t h
component
B,(t)=
2
and By(t) is the ny-vector with Kth component 2. Model and algorithm in discrete version We assumeanonlineardiscretestate-space following form,
model of the
by One finds thatuse of the recursive prediction error method LjungandSoderstrom,[1983],directlyonthenonlinear predictor mndel (2-a, b) is hardly feasible, due to computationalcomplexity. If alinearmeasurementequationis chosen instcad, however, complexity of the algorithm is rewhere f() and h() are nonlinear functionsof the state, v(t) is ducedsignificantly.Thenthepredictorhasthefollowing white process noise, and e(t) is uncorrelated measurement form noise with statistics i(t+I,e)=f(e,u;t,~(t,e))+Bsit)+K[tj[v!t)-~Iie)ict,O)l (3-3)
I
jYt/e)=H (e)E(t,e)
(3-b)
The assumption of a linear measurement is valid in a wide class of practical applications. Then the recursive prediction error method using a Newton search directionfor parameter updating can be applied to the model (3-a, b). The algorithm will consist of the following set of recursive equations: The initial valueof the statex(o) has the properties 1637
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where
-
Ro(t), Dt aredefinedin(5-c,d,f)respectively,and E , ( t ) is definedin (2-c). It is noted that in version (7-a-h) one has to use equations (4-e) and (4-0 in order to obtain the covariance matrix P(t) in B,(t). If the measurement vector x, and the matrixH y( t ) has the same dimension as the state is ;in identity matrix then the covariance matrix is vd,(t,H),
P(t)= E{cxr~l-iil))cx(tj-t(t)jT] =E{c(t)cT(t)/
where
Since y(t)= H, .?it) =i(t). Consequently, the matrix P(t) can be replaced by A(t) in this case, and P(t) need no longer be ralculated. 3. Model and algorithm in continuous-discrete version In most applicationsinvolvingtheidentification of parameters of a physical continuous time system, it is generally is the derivativeof x(t,6) in the right-hand side of (3-a) with preferable to use continuous-discrete a algorithm. The reasons are primarily structure preservationof known parts respect to8. Further of thesystemandthepossibilitytoinclmebounds on parameterestimates of physicalparameters whoseconstraints areknown. The latter is a practical way overcome to part of thedifficultieswith possible local minimawhen identifyingparameters of nonlinearsystems. As inthe is the derivative of the parameter matrices in the bracket presentation in section 2, the discrete measurement equation with respect to 8, and will be chosen in its linear version, and an innovations model isemployed. We hence assume the nonlinear continuousdiscrete state-spacemodel of the form:
(5-f) BJt) isdelined in (2 -c)
(5-9)
This versionof the filter(4-a-&includes a calculation of the where f( ) is the nonlinear function of state. v(t(ti) is white Kalman gains in (I-d,e,O andK t is calculated from (4-d,e,f). process noise, e(ti) is uncorrelated measurement noise with As per the suggestion given by Ljung (1979), the parameter statistics, identifier can assume an innovations model of the form:
where &(t)is the innovation dueto measurement t, and K(8) is a setof (as yet undetermined) steady state Kalman gains, which is parameterized and willbe identified directly along The second order predictor using an innovations model will with the system parameters. This gives less complex be compatations, and the algorithm corresponding to (6-a,b)will then be as follows:
where E( ti+ 1 ) is the innovation due to measurement t i + 1 , and K( ti+ 1 ,e) comprise parameterized steady state Kalman fains. The algorithm corresponding to (9-a,b,c) will be as ollows:
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4. Example ( 1 0-b) ( 1 0-c)
After integration of (lO-a,b,c),:(ti available, and
- I - ) , P(ti+ I-), W(tiA 1-1 are
Theability of thenonlinearRPEmethodtoestimate of practical parameters and states of a nonlinear system importance is demonstrated in this example. Thecontinuousdiscreteversion of thenonlinearfilter derived above iscomparedwiththecorrespondinglinear algorithm by Gavel and Azevedo [1982]. The results of bias correctionof the demonstrate the advantages in terms nonlinear filter. The nonlinear system consideredis an equivalentto the ship speed equation. The parameters identified will, for the real ship,meanhullresistanceand efficiency inutilizingthe prime moverof the vesselfor forward thrust. Both values are of major technical importance and as they change over time, theyhavevastimpact on theships'sfuel economy and efficiency. The criteria for maintenence of the ship's hull, propeller, and prime mover system can be directly derived from these parameters, and it is hence of prime importance that they are estimated without bias.
The second order nonlinearity typeof system is furthermore technically important when identifying propulsion losses of ships at sea aiming at autopilot and steering gear performance evaluation, Blanke [1981], Blanke and Sorensen [1984],Blanke [1986]. The responses and parameter estimatesbelow were obtained usingasquarewavepertubation to theinputu(t).The amplitude of the pertubation is10 percent of its steady state value. The practical equivalentto this experimentwould be a stepwise increaseidecrease in propeller thrust.
w,*,
Re, and N in the algorithm (10-a-1) The matrices B,, of x(t)ti,8) in the right-hand side of (9-a) with corresponding to the examplewill be is the derivative respect to8. B, (tlk) = a p(ti) = a A(ti) Wx*(tit;) = 2aii(ti)w(tIti) -
Me (ti) = [(kz(ti) +P(ti)),u(ti),oI = [(?2(ti)
(12)
+ A(ti)),u(ti),~l
is the derivativeof the parameter matrices in the right-hand N (ti)= [o,o,c(ti)l side of (9-a) with respect to8. Further the following notation is used Figure 1 shows results of identifying the parameters a and b in the nonlinear equation using the nonlinear filter. The 2 illustrate the performance of a curves plotted in figure linear RPE filter applied to the same nonlinear equation. Although the driving signal's pertubation is only 10 percent of its average, the bias of the linear estimator is apparent, andthesuperiorperformance of thenonlinearfilteris obvious.
The same treatment will be usedwhen Hi is an identity matrix and has the same dimension as the statevector x. In this case the P(ti) matrix will not be calculated any longer and is replaced by h(to. 1639 Authorized licensed use limited to: Danmarks Tekniske Informationscenter. Downloaded on July 01,2010 at 08:51:08 UTC from IEEE Xplore. Restrictions apply.
Ljung, L.: Analysis of Recursive Stochastic Algorithms. IEEE Trans. Automatic Control, Vol. 22, No. 4,1977
5 . Conclusions
Thispaperhaspresented two algorithms for identifying parameters of a nonlinear discrete state-space system model and a nonlinearcontinuous-discretestate-spacesystem model. Bothversionsaretreatedusingalineardiscrete measurement equation. These algorithms were investigated with reference to the theory of linear RPE methods and the theory of nonlinear filtering. The innovations model formulationwasfound to be attractive, and the algorithms were implemented and tested against computer simulations showing excellent convergence, and bias properties that by far exceed those of a linear continuouddiscrete filter. The of thenonlinear analysis of theconvergenceproperties estimator and further tests of applications of these algorithms shouldbe persued ina further study.
Ljung, L.: Asymptotic Behavior ofthe Extended Kalman Filter as a Parameter Estimatorfor Linear Systems. Maybeck, P.S.: Stochastic Models, Estimation, and Control. Volume 1,Academic Press, 1979 Volume 2, Academic Press, 1982 McGarty, T.P.: Stochastic Systems& State Estimation. John Wiley & Sons, 1974 Young, P.: Recursive Estimation and Time-Series Analysis. Springer-Verlag, 1984 Zhou, W.-W.: Filtering and Recursive Identification. Servolaboraty,TechnicalUniversity of Denmark, October 1985.
6. References Astrom, K.J.: Introductionto Stochastic Control Theory. Academic Press, 1970 Blanke, M.: Propulsion losses Related to Automatic Steering andPrime Mover Control.PhDthesis.Technical Univ. of Denmark, Dec. 1981. Blanke, M.: Cross-Bispectrum Technique Identificationof Nonlinear Ship Speed Dynamics, International Journal of Computers and Control, Vol. 2, 1986 pp. 54-62. Blanke, M. & I,.S.Sflrensen: The Ljung Innovations Filter Used for Identification of NonlinearShipSpeed Dynamics.7thShipControlSystemsSymposium, Bath, IJ.K., 1984 Eykhoff, P.: System Identification- Parameter and State Estimation. John Wiley& Sons, 1977 Gavel, D.T. & S.G. Azevedo: Identification of Continuous Time Systems - An Application of Ljung's Corrected Extended Kalman Filter Sixth. IFAC Symposiumon IdentificationandSystemParameterEstimation, USA, June, 1982 Gelb, A.:Applied Optimal Estimation,MIT Press, 1974 Goodwin, G.C. & R.L. Payne: Dynamic System Identification: ExperimentDesignandDataAnalysis.Academic Press, 1977 Jazwinski, A.H.: Stochastic Processes and Filtering Theory. Academic Press, 1970 Ljung, L.: System Identification - theory for the user. Lecture Notes, Universityof Linkoping, Sweden 1984 of Recursive Ljung, L.& T. Soderstrom: Theory and Practice Identification. TheMIT Press, 1983 Ljung, L.: Analysis of a General Recursive Prediction Error Identification Algorithm. Automatica, Vol. 17, No. 1, pp 89-99,1981 Ljung, L.: Identification Methods,Model Validation Recursive Identification Methods for Off-line Identification Problems. Sixth IFAC Symposium on IdentificationandSystemParameterEstimation, USA, June, 1982 1640 Authorized licensed use limited to: Danmarks Tekniske Informationscenter. Downloaded on July 01,2010 at 08:51:08 UTC from IEEE Xplore. Restrictions apply.
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