Nonlinear Kalman filtering via ultradiscretization procedure

Abstract

A new filtering method for a nonlinear system is proposed. The original nonlinear system is reduced to a piecewise linear system through the procedure of ultradiscretization. Then the discrete-time Kalman filter is readily applied to the obtained system by imposing some conditions on system variables and parameters. Some numerical experiments are given to show the efficiency of the method.

Appendix

We derive the three periodic solution (26). If we put $\alpha =-1$ and $\beta =1$, (17b) and (17a) reduce to

$$\begin{aligned}&\max \left[ S(\xi _{n+1}\xi _n) + X_{n+1} + X_n, S(\eta _n) + A + Y_n \right] \nonumber \\&\quad = \max \left[ S(-\xi _{n+1}\xi _n) + X_{n+1} + X_n, S(-\eta _n) + A + Y_n, B, 2Y_n \right] , \end{aligned}$$

(38a)

$$\begin{aligned}&\max \left[ S(\eta _{n+1}\eta _n) + Y_{n+1} + Y_n, S(\xi _{n+1}) + A + X_{n+1} \right] \nonumber \\&\quad = \max \left[ S(-\eta _{n+1}\eta _n) + Y_{n+1} + Y_n, S(-\xi _{n+1}) + A + X_{n+1}, B, 2X_{n+1} \right] , \end{aligned}$$

(38b)

respectively. Substituting the initial values $(\xi _0, X_0) = (-1,C)$ and $(\eta _0, Y_0) = (1,A)$ into (38a) with $n=0$, we have

$$\begin{aligned} \max \left[ S(-\xi _{1}) + X_{1} + C, 2A \right] = \max \left[ S(\xi _{1}) + X_{1} + C, B, 2A \right] . \end{aligned}$$

(39)

Noting $B=A+\tilde{B}<2A$, we find that the equation is solved by

$$\begin{aligned} \xi _1=1 \text{ or } -1,\quad X_1 \le 2A -C. \end{aligned}$$

(40)

Remembering we choose $(\xi _{n+1},X_{n+1})=(-\xi _n, X_n)$ if $(\xi _{n+1},X_{n+1})$ becomes indeterminate, we take $(\xi _1,X_1)=(1, C)$, which actually satisfies (40). Next, substituting $(\xi _1,X_1)$ and $(\eta _0, Y_0)$ into (38b) with $n=0$, we have

$$\begin{aligned} \max \left[ S(\eta _{1}) + Y_{1} + A, A + C \right] = \max \left[ S(-\eta _{1}) + Y_{1} + A, B, 2C \right] . \end{aligned}$$

(41)

This equation is uniquely solved by $(\eta _1, Y_1) = (-1, C)$. Similarly, we obtain $(\xi _2, X_2)=(1,A)$ (unique), $(\eta _2, Y_2)=(1,C)$ (indeterminate) and further $(\xi _3,X_3)=(-1,C)=(\xi _0,X_0)$ (indeterminate), $(\eta _3, Y_3)=(1,A)=(\eta _0,Y_0)$ (unique).