THE EXTENDED KALMAN FILTERThe Kalman filtering problem considered up to this point has addressed the estimation of as state vectorin a linear model of a dynamical system. If, however, the model is nonlinear, we may extend the use ofKalman filteringthrough a linearizationprocedure. The resulting filter is naturally referred to as extendedKalman filter (EFK). Such an extension is feasible by virtue of fact that Kalman filters described in termsof differential equations (in the case of continuous–time systems) or difference equations (in the case ofdiscrete-time systems). This is in contrast to Wiener filter that is limited to linearsystems,since thenotion of an impulse response (on which the Wiener filter is based ) is meaningful only in the context oflinear systems . Hereis another important advantage of Kalman filter over the Wiener filter.To set the stage for development of the extended Kalman filter in the discrete-time domain, considerfirst the standard linear state –space model that we studied in the earlier part of this chapter [Eqs. (7.17)and (7.19)], reproduced here for convenience of presentation:Where v1(n) and v2(n) are uncorrelated zero-mean white-noise processes with correlation matricesQ1(n) and Q2(n), respectively, as defined in equations(7.18),(7.20), and (7.21). thecorresponding Kalmanfilter equations are summarized in Table. In this section, however we will rewrite these equations in aslightly modified form that is more convenient for our present discussion. Specifically,the update of thesate estimate is performed in two steps . The first step updates this updateequation is simply (7.59).The second step updates and is obtained by substituting Eq.(7.45)into Eq. (7.60),and by defining a new gain matrix:We may thus writeWe next make the following observation. Suppose thatinstead of the state equations (7.99) and (7.100),weare given the alternative state vector model
Whered(n) is a known (i.e., nonrandom)vector . In this case, it is easily verified that the same Kalmanequations (7.103) through (7.107) apply except for a modification in the first equation (7.102)), whichnow reads as followsThis modification arises in the derivation of extended Kalmanfilter, as discussed in the sequel. As mentioned previously, the extended Kalman filter is an approximate solution that allows us toextend the Kalman filtering idea to anonlinear state space models (Jazwimski, 1970 ;Maybeck,1982;Lung and Soderstorm, 1983). In particular, the non-linear model considered here has the followingform:Where,as before,v1 (n) and v2 (n) are uncorrelated zero-mean white–noise processes with correlationmatrices Q1(n) and Q2 (n) respectively. Here, however, the functionalF(n,x(n))denotes a nonlineartransition matrix function that is not possibly time-variant. In the linear case, we simply haveBut in a general nonlinear setting, the entriesof the state vector x(n) may combined nonlinearly by theaction of the functional; F(n.x (n) ).Moreover, this nonlinear operation may vary with time. Likewise, thefunctional C(n, x(n))) denotes a nonlinear measurement matrix that may be time – variant too.As an example, consider the following two-dimensional nonlinear state – space model:In this example, we have
and The basic idea of the extended Kalman filter is to linearize the state –space model of Eqs. (7.11) and(7.112) at each time instant around the most recent state estimate , which is taken to be either ,depending on which particular functional is being considered. Once a linear model obtained, thestandard Kalman filter equations are applied.More explicitly, the approximation proceeds in two stages.Stage1. The following two matrices are constructedAndThat is, the ijth entry of F(n+1,n) is equal to the partial derivative of the ith component of F(n,x) withrespect to theijth component of x. Likewise, the ijth entry of C(n) is equal to the partial derivative of theith component of C(n,x). With respect to jth component of x.In former case, the derivatives areevaluated at while in the latter case the derivatives are evaluated at The entries ofthe matrices F(n+1,n) and C(n) are all known (i.e. , computable), since andare made available as described later .Applyingthe definitions ofEqs.(7.113) and (7.114) to the previous example, we getWhich leads toAnd
Stage 2.Once the matrices F(n+1,n) and C(n) are evaluated , they are then employed in a first- orderTaylor approximation of the nonlinear functionalF(n,x(n))and C(n,x(n)) aroundand respectively. Specifically, F(n,x(n)) and C(n,x(n)) are approximated as follows,respectively :With the above approximation expressions at hand, we may now proceed to approximate the nonlinearstate-equations (7.111) and (7.112) as shown by, respectively,Where we have introduced two new quantities:AndThe entries in the term are known at time n, and, therefore, can be regarded as anobservationvector at timen. Likewise, the entries in the term d(n) are all known at time n. Figure7.6. One-step predictor for the extended Kalman filter.
The approximate state-space model of Eqs.( 7.117) and (7.118) is a linear model of the samemathematic al form as that described in Eqs. (7.108) and (7.109); indeed, it is with this objective in mindthat earlier on we formulated the state–space model of Eqs. (7.108)standard Kalman equations (7.13)through (7.109) and (7.110) to the above linear model. This leads to the following set of equations:On the basis of Eqs.(7.121) and (7.122), we may formulate the signal flow graph of Fig 7.6 for updatingthe one-step prediction in the extended Kalman filter.In table 7.5 we present a summary of the extended Kalman filtering algorithm, where the linearizedmatricesF(n+1,n) and C(n) are computed from their respective nonlinear counterparts using Eqs.7.113and (7.114). Given a nonlinear state- space model of the form described in Eqs. (7.111)and(7.112), wemay thus use this algorithm to compute state estimate recursively. Comparing the equations of theextended Kalman filter summarized here in with those of the standard Kalman filter given Eqs.7.102
Table7.5 Summary of extended Kalman filterthrough (7.107) we see that the only difference between them arise in the computations of theinnovcatiopns vector nad the updated estimate Specifically, the linear terms in the standard Kalman filter are replaced by the approximate terms respectively , in the extended Kalman filter . These differences also inthe standard Kalman gfilter with that of Fig7.6 for one-step prediction in the extended Kalman filter.