Factorized extended Kalman filter for optical processing

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

doi:10.1364/AO.25.001615

Applied Optics
Vol. 25,
Issue 10,
pp. 1615-1621
(1986)
•https://doi.org/10.1364/AO.25.001615

Factorized extended Kalman filter for optical processing

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 for optical processing," Appl. Opt. 25, 1615-1621 (1986)

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Abstract

Kalman filtering represents formidable linear algebra computational requirements for each new input measurement vector. An architecture-motivated implementation of a discrete-time extended Kalman filter algorithm is presented. This particular formulation takes advantage of the following features of the optical processor architecture: the ability to perform matrix–vector operations, floating-point capabilities, and specially designed matrix–vector L U decomposition operations. A factorized L D L^T algorithm is used to propagate the covariance matrices between sample times. The air-to-air missile guidance problem is used as a case study wherein an extended Kalman filter is required due to the nonlinear nature of the measurement equations.

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Time	Multiplier	AO channel contents

		a₃	a₂	a₁
1T₂	3.5	1	1	1
2T₂	1.0	1	0	0
3T₂	2.5	1	0	1

Point modulator	Multiplicand mantissa	Delay Δ_i	Input time slots

			5T₁	4T₁	3T₁	2T₁	1T₁
Top	2.0	2	1	0	0	0_d	0_d
Middle	1.5	0	0_P	0_p	1	1	0
Bottom	1.0	1	0_P	1	0	0	0_d

Symbol	Dimension	Description
E{·}		Expectation operator
Γ	(n × m)	Input distribution matrix
$h [{\overset{̅}{x}}_{k}]$	(r × 1)	Nonlinear coordinate transformation vector which transforms the Cartesian coordinates of ${\overset{̅}{x}}_{k}$ to the polar coordinate frame of the measurement vector z_k
H	(r × n)	Linear coordinate transformation matrix (in KF)
$H [{\overset{̅}{x}}_{k}]$	(r × n)	Linearized measurement matrix (in EKF): $H [{\overset{̅}{x}}_{k}] = \partial h [x_{h}] / \partial x_{k}$ evaluated at $x_{k} = {\overset{̅}{x}}_{k}$
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
${\overset{̅}{x}}_{k}$	(n × 1)	A priori state vector estimate
${\hat{x}}_{k}$	(n × 1)	A posteriori state vector estimate
Z_k	(r × 1)	Measurement vector

Linear system model	(8) $x_{k + 1} = Φ x_{k} + Γ u_{k} + w_{k}$
Linear measurement model	(9) $z_{k} = {Hx}_{k} + v_{k}$
Error covariance measurement update	(10) ${P_{k}}^{- 1} = {M_{k}}^{- 1} + H^{T} {R_{k}}^{- 1} H$
Gain matrix	(11) $K_{k} = P_{k} H^{T} {R_{k}}^{- 1}$
State estimate measurement update	(12) ${\hat{x}}_{k} = {\overset{̅}{x}}_{k} + K_{k} {z_{k} - H {\overset{̅}{x}}_{k}}$
Error covariance time update	(13) $M_{k + 1} = Φ P_{k} Φ^{T} + Q_{k}$
State estimate time update	(14) ${\overset{̅}{x}}_{k + 1} = Φ {\hat{x}}_{k} + Γ u_{k}$

Linear system model	(8) $x_{k + 1} = Φ x_{k} + Γ u_{k} + w_{k}$
Nonlinear measurement model	(15) $z_{k} = h [x_{k}] + v_{k}$
Error covariance measurement update	(16) ${P_{k}}^{- 1} = {M_{k}}^{- 1} + H^{T} [{\overset{̅}{x}}_{k}] {R_{k}}^{- 1} H [{\overset{̅}{x}}_{k}]$
Gain matrix	(17) $K_{k} = P_{k} H^{T} [{\overset{̅}{x}}_{k}] {R_{k}}^{- 1}$
State estimate measurement update	(18) ${\hat{x}}_{k} = {\overset{̅}{x}}_{k} + K_{k} {z_{k} - h [{\overset{̅}{x}}_{k}]}$
Error covariance time update	(13) $M_{k + 1} = Φ P_{k} Φ^{T} + Q_{k}$
State estimate time update	(14) ${\overset{̅}{x}}_{k + 1} = Φ {\hat{x}}_{k} + Γ u_{k}$

Development
(13) $M_{k + 1} = Φ P_{k} Φ^{T} + Q = Φ P_{k} Φ^{T} + L_{Q} D_{Q} L_{Q}^{T} .$
Column-wise partition L_Q as L_Q = [b₁ … b_n], where b_i is an n-vector.
Write D_Q as D_Q = diag[q₁… q_n], where each q_i is a scalar.
$M_{k + 1} = Φ P_{k} Φ^{T} + q_{1} b_{1} b_{1}^{T} + \dots + q_{n} b_{n} b_{n}^{T} .$
Let $G = Φ P_{k} Φ^{T} = (Φ L_{P} D) (D^{- 1} D_{P} D^{- T}) {(Φ L_{P} D)}^{T} = L_{G_{1}} D_{G_{1}} L_{G_{1}}^{T}$ , where $D$ is a diagonal matrix whose nonzero elements are the reciprocals of the corresponding diagonal elements of Φ.
In recursive form, $G_{i + 1} = G_{i} + q_{i} b_{i} b_{i}^{T} = L_{G_{i}} S_{i} L_{G_{i}}^{T}$ . We first form $S_{i} = D_{G_{i}} + q_{i} d_{i} d_{i}^{T}$ (where L_Gid_i=b_i), and then decompose S_i as $S_{i} = L_{S_{i}} D_{S_{i}} L_{S_{i + 1}}^{T}$ .
Thus L_{G_i+1} = L_{G_i}L_{S_i} and D_{G_i+1} = D_{S_i}.
The following are unit diagonal lower triangular matrices: L_Q, $(Φ D)$ (Φ by itself is lower triangular but not unit diagonal), Lp, Tmp1 (in step 1), L_s, and L_M.
The following are diagonal matrices: D_Q, $D$ , D_G, Dp, Ds, and D_M.
The following are full matrices: S, Tmp1 (in steps 6 and 7).
Computation
1.	Tmp1 ← ΦL_P.
2.	$L_{G} \leftarrow Tmp 1 D$ .
3.	$D_{G} \leftarrow {(D^{- 1})}^{2} D_{P}$ .
4.	i ← 1.
5.	Back substitute L_Gd_i= b_i for d_i.
6.	$Tmp 1 \leftarrow c d_{i} d_{i}^{T} .$
7.	S ← D_G+ Tmp1.
8.	Compute the Cholesky factors L_s and D_s of S.
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 5.
12.	Stop.

Development
Column-wise partition $H^{T} [{\overset{̅}{x}}_{k}]$ as [h₁…h_r], where h_i is an n vector.
Let $R_{k}^{- 1} = diag [ℛ_{1}^{- 1} \dots ℛ_{r}^{- 1}] .$
Rewrite Eq. (16) as $P_{k}^{- 1} = M_{k}^{- 1} + ℛ_{1}^{- 1} h_{1} h_{1}^{T} + \dots + ℛ_{1}^{- 1} h_{r} h_{r}^{T} .$
We will process one vector outer product at a time. Therefore, we use the notation
(19) $G_{i + 1}^{- 1} = G_{i}^{- 1} + ℛ_{i}^{- 1} h_{i} h_{i}^{T},$
where $G_{1}^{- 1} = M_{k}^{- 1}$ and $P_{k}^{- 1} = G_{r + 1}^{- 1}$ . (Subscripts k and i are time iteration indices, respectively.)
Take the inverse of both sides of Eq. (19) and apply the matrix inversion lemma:
(20) ${(A + {XCY}^{T})}^{- 1} = A^{- 1} - A^{- 1} X {(C^{- 1} + Y^{T} A^{- 1} X)}^{- 1} Y^{T} A^{- 1}$
with the substitutions $A = G_{i}^{- 1}$ , X = Y = h_i, and $C = R_{i}^{- 1}$ to obtain
(21) $G_{i + 1} = G_{i} + α [G_{i} h_{i}] {[G_{i} h_{i}]}^{T},$
where the scalar α satisfies $1 / α = - (ℛ_{i} + h_{i}^{T} G_{i} h_{i})$ .
Let $G_{i} = L_{G} D_{G} L_{G}^{T}$ and $d_{i} = D_{G} L_{G}^{T} h_{i}$ . Then Eq. (21) becomes
(22) $L_{G_{i + 1}} D_{G_{i + 1}} L_{G_{i + 1}}^{T} = L_{G_{i}} [D_{G_{i}} + α d_{i} d_{i}^{T}] L_{G}^{T} = L_{G_{i}} S_{i} L_{G_{i}}^{T} .$
If $S_{i} = L_{S_{i}} D_{S_{i}} L_{S_{i}}^{T}$ is the factored form of the matrix $S_{i} = D_{G_{i}} = α d_{i} d_{i}^{T}$ , then
(23) $L_{G_{i + 1}} = L_{G_{i}} L_{S_{i}} and D_{G_{i + 1}} = D_{S_{i}} .$
Computation
1.	i ← 1.
2.	L_G←L_M and D_G←D_M.
3.	$Tmp 1 \leftarrow L_{G}^{T} h_{i}$ .
4.	d_i ← D_GTmp1 = ${DL}_{G}^{T} h_{i}$ .
5.	Tmp1 ← (Tmp1)^Td_i.
6.	α ← −1/(Tmp1) + ℛ_i).
7.	Tmp1 ← $α d_{i} d_{i}^{T}$ .
8.	S ← D_G+ Tmp1.
9.	Compute the Cholesky factors L_S and D_S of S.
10.	D_G ← D_S[D_G is the appropriate component of the left side of Eq. (21) at step i + 1].
11.	L_G ← L_GL_S.
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.

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Equations (38)

Applied Optics