Factorized extended Kalman filter: case study results

James L. Fisher; David P. Casasent; Charles P. Neuman

doi:10.1364/AO.27.001877

Applied Optics
Vol. 27,
Issue 9,
pp. 1877-1885
(1988)
•https://doi.org/10.1364/AO.27.001877

Factorized extended Kalman filter: case study results

James L. Fisher, David P. Casasent, and Charles P. Neuman

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James L. Fisher, David P. Casasent, and Charles P. Neuman, "Factorized extended Kalman filter: case study results," Appl. Opt. 27, 1877-1885 (1988)

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Abstract

Simulation results for an optical realization of a factorized extended Kalman filter algorithm are presented, minimum word lengths required for accurate tracking are empirically determined, and computation times for an optical realization are quantified.

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$h [x_{k}] = {[\dot{r}, r, θ, ϕ]}^{T}$
where	$\dot{r} = (x \dot{x} + y \dot{y} + z \dot{z}) / \sqrt{x^{2} + y^{2} + z^{2}}$
	$r = \sqrt{x^{2} + y^{2} + z^{2}}$
	$θ = {\begin{array}{l} \begin{matrix} arctan (y / x) & if x > 0 \end{matrix} \\ \begin{matrix} π + arctan (y / x) & if x < 0 \end{matrix} \end{array}$
	$ϕ = arctan (- z / \sqrt{x^{2} + y^{2}})$

β	0.1sec⁻¹	continuous-time system poles
T	0.1 sec	sampling period
A_max	49.05 meters/sec² = 5 g’s	maximum acceleration
P₀	0.5	probability of no acceleration
P_max	0.05	probability of maximum acceleration
R = Diag[9, 16, 10⁻⁶, 10⁻⁶]
$M (k = 0) = Diag [\begin{matrix} 400, 400, 400, & 10, 10, 10, & 10, 10, 10 \end{matrix}]$

Linear System Model	$x_{k = 1} = Φ x_{k} + Γ u_{k} + w_{k}$ (11)
Nonlinear Measurement Model	$z_{k} = h [x_{k}] + v_{k}$ (12)
Error Covariance Measurement-Update	${P_{k}}^{- 1} = {M_{k}}^{- 1} + H^{T} [{\bar{x}}_{k}] {R_{k}}^{- 1} H [{\bar{x}}_{k}]$ (13)
Gain Matrix	$K_{k} = P_{k} H^{T} [{\bar{x}}_{k}] {R_{k}}^{- 1}$ (14)
State Estimate Measurement-Update	${\hat{x}}_{k} = {\bar{x}}_{k} + K_{k} {z_{k} - h [{\bar{x}}_{k}]}$ (15)
Error Covariance Time-Update	$M_{k + 1} = Φ P_{k} Φ^{T} + Q_{k}$ (16)
State Estimate Time-Update	${\bar{x}}_{k + 1} = Φ {\hat{x}}_{k} + Γ u_{k}$ (17)

Symbol	Dimension	Description
E{.}		Expectation operator
Γ	(n × m)	Input distribution matrix
$h [{\bar{x}}_{k}]$	(r × 1)	Nonlinear coordinate transformation vector which transforms the Cartesian coordinates of ${\bar{x}}_{k}$ , to the polar coordinate frame of the measurement vector z_k
$H [{\bar{x}}_{k}]$	(r × n)	Linearized measurement matrix: $H [{\bar{x}}_{k}] = \partial h [x_{k}] / \partial x_{k}$ evaluated at $x_{k} = {\bar{x}}_{k}$
k		Discrete-time index
K_k	(n × r)	Kalman gain matrix
M_k	(n × n)	A priori error covariance matrix
m		Number of inputs
n		Number of states
P_k	(n × n)	A posteriori error covariance matrix
Φ	(n × n)	System matrix
Q_k	(n × n)	System driving noise covariance matrix of w_k
r		Number of measurements
R_k	(r × r)	Measurement noise covariance matrix of v_k
T		Data sampling period
u_k	(m × 1)	Input (control) vector
v_k	(r × 1)	Measurement noise vector
w_k	(n × 1)	System noise vector
x_k	(n × 1)	System state vector
${\bar{x}}_{k}$	(n × 1)	A priori state vector estimate
${\hat{x}}_{k}$	(n × 1)	A posteriori state vector estimate
z_k	(r × 1)	Measurement vector

Operation	Dimension	No. of VIPs	No. of Mults	No. of Adds	Misc.
VIP	(1 × n) × (n × 1)	1	$\frac{1}{2} n (n + 1)$	$\frac{1}{2} n (n - 1)$
Scaled VOP	(n × 1) × (1 × n)	$\frac{1}{2} n (n + 3)$	$\frac{1}{2} n (n + 3)$	0
M-V mult	lower(n × n) ×(n × 1)	n	$\frac{1}{2} n (n + 1)$	$\frac{1}{2} n (n - 1)$	non-unit diagonal
″	″	″	$\frac{1}{2} n (n - 1)$	$\frac{1}{2} n (n - 1)$	unit diagonal
″	full(n × r) × (r × 1)	r	nr	n(r − 1)
M-M mult	lower(n × n) × lower(n × n)	$\frac{1}{2} n (n + 1)$	$\frac{n^{3}}{6} + \frac{n^{2}}{2} + \frac{n}{3}$	$\frac{1}{6} n (n^{2} - 1)$	non-unit diags
″	″	″	$\frac{1}{6} n (n^{2} - 1)$	$\frac{1}{6} n (n^{2} - 1)$	1 mat unit diag
″	″	″	$\frac{n^{3}}{6} - \frac{n^{2}}{2} + \frac{n}{3}$	$\frac{1}{6} n (n^{2} - 1)$	2 mat unit diag
Cholesky factorization	lower(n × n), diag(n × n)	n(n − 1)	$\frac{n^{3}}{6} + n^{2} - \frac{n}{6}$	$\frac{n^{3}}{6} + \frac{n^{2}}{2} - \frac{2 n}{3}$

Operation	No. of VIPs	No. of Mults	No. of Adds
1. i ← 1.
2. L_G ← L_M and D_G ← D_M.
3. $Tmp 1 \leftarrow L_{G}^{T} h_{i}$ .	n	$\frac{1}{2} n (n - 1)$	$\frac{1}{2} n (n - 1)$
4. $d_{i} \leftarrow D_{G} Tmp 1 = D L_{G}^{T} h_{i}$ .	*or n	n	0
5. Tmp1 ← (Tmp1)^Td_i.	1	n	n−1
6. α ← −1/(Tmp1 + R_i).	*	1	1
7. $Tmp 1 \leftarrow α d_{i} d_{i}^{T}$ .	$\frac{1}{2} n (n + 3)$	$\frac{1}{2} n (n + 3)$	0
8. S ← D_G + Tmp1.	*	0	n
9. Compute the Cholesky factors L_S and D_S of S.	n(n − 1)	$\frac{n^{3}}{6} + n^{2} - \frac{n}{6}$	$\frac{n^{3}}{6} + \frac{n^{2}}{2} - \frac{2 n}{3}$
10. D_G ← D_S.
11. L_G ← L_GL_S.	$\frac{1}{2} n (n + 1)$	$\frac{n^{3}}{6} - \frac{n^{2}}{2} + \frac{n}{3}$	$\frac{1}{6} n (n^{2} - 1)$
12. If i=r, then L_P ← L_G and D_P ← D_G; Else i ← i+1 and go to Step 3.
13. Stop.

Operation	No. of VIPs	No. of Mults	No. of Adds
1. Tmp1 ← ΦL_P	$\frac{1}{2} n (n + 1)$	$\frac{1}{6} n (n^{2} - 1)$	$\frac{1}{6} n (n^{2} - 1)$
2. L_G ← Tmp1 D.	$\frac{1}{2} n (n + 1)$	$\frac{1}{2} n (n - 1)$	0
3. D_G ← (D⁻¹)²D_P	*	n	0
4. i ← 1.
5. Back substitute L_Gd_i = b_i for d_i.	*	$\frac{1}{2} n (n - 1)$	$\frac{1}{2} n (n - 1)$
6. $Tmp 1 \leftarrow c d_{i} d_{i}^{T}$ .	$\frac{1}{2} n (n + 1)$	$\frac{1}{2} n (n + 3)$	0
7. S ← D_G + Tmp1.	*	0	n
8. Compute the Cholesky factors L_S and D_S of S.	n(n − 1)	$\frac{n^{3}}{6} + n^{2} - \frac{n}{6}$	$\frac{n^{3}}{6} + \frac{n^{2}}{2} - \frac{2 n}{3}$
9. D_G ← D_S.
10. L_G ← L_GL_S.
11. If i=n, then L_M ← L_G and D_M ← D_G; Else i ← i+1 and go to Step 3.
12. Stop.

Equation Calculated	Calculation Time Required (inT₂units)	Numerical Values for the Case ofn=9 andr=4 (inT₂units)
Eq. (13): Error Covariance Measurement-Update	r(2n²+5n+1)T₂	832 T₂
Eq. (14): Gain Matrix and
Eq. (15): State Estimate Measurement-Update	(3n+2r)T₂	35 T₂
Eq. (16): Error Covariance Time-Update	(2n³+n²+2n)T₂	1557 T₂
Eq. (17): State Estimate Time-Update	nT₂	9 T₂

Axis\Time	20 < k < 60	60 < k < 100
x-axis	+5.0g’s	−5.0g’s
y-axis	−5.0g’s	+5.0g’s
z-axis	+3.0g’s	−3.5g’s

Abstract

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Equations (21)

Applied Optics