Bulletin of Computer Center, Hosei University

最小自乗の原理

非線形方程式

ニュートン・ラプソン法

non-linear Kalman filter

minimum quadratic loss (least square method)

non-linear equations

Newton-Raphson method

Non-linear discrete-time Kalman filter is also derived on the principle of minimum quadratic loss in the same process as that of linear discrete-time Kalman filter. A non-linear discrete-time dynamic system is: {x(j+1)=Φ(j+1;j)x(j)+v(j) {y(j)=f(x(j))+w(j) for which, Kalman filter equations are: {Δ(k)={P(k|k-1)J'_k+Z(k|k-1)}[J'_kP(k|k-1)J'_k+R(k)]^-1 {x^ﾟ(k|k)=x^ﾟ(k|k-1)+Δ(k){y(k)-f(x^ﾟ(k|k-1))} {x^ﾟ(k+1|k)=Φ(k+1;k)x^ﾟ(k|k) {P(k|k)=P(k|k-1)-Δ(k){J_kP(k|k+1)+Z'(k|k-1)} {={I-Δ(k)J_k}P(k|k-1)-Δ(k)Z'(k|k-1) {P(k+1|k)=Φ(k+1;k)P(k|k)Φ'(k+1;k)+Q(k) where I is unit matrix, J_k is Jacobian (functional determinant), J'_k is transposed matrix and, covariance matrices are: Q(k)=E{v(k)v'(k)], R(k)=E[{r(k|k-1)+w(k)}{r(k|k-1)+w(k)}'], Z(k|k-1)=E[x~(k|k-1)r'(k|k-1)], x~(k|k-1)=x(k)-x^ﾟ(k|k-1), r(k|k-1) is the residual of the first order Taylor approximation i.e. f(x(k))=f(x^ﾟ(k|k-1))+J_kx~(k|k-1)+r(k|k-1). For the time-invariant case, it is shown that the filter is applicable to solve the non-linear simultaneous equations f(x)=0. {Δ(k)={P(k|k-1)J'_k+Z(k|k-1)}[J_kP(k|k-1)J'k+R(k)]^-1 {x^ﾟ(k+1|k)=x^ﾟ(k|k-1)-Δ(k)f(x^ﾟ(k|k-1)) {P(k+1|k)={I-Δ(k)J_k}P(k|k-1)-Δ(k)Z'(k|k-1)+Q(k). And its algorithm is something like "damped" Newton-Raphson scheme. But it has much more flexibility than Newton-Raphson method.

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