Time Varying or Stationary?
一般而言 Kalman filter 是 time varying 的 filter. 多數 Kalman filter 的應用也是 time varying system, 例如 navigation, gesture recognition, etc.
另一類的應用: 如果 A, B, Bw, C, (參考 Berkeley article) matrices and (A, C) is observable or detectable; (A, Bw) is controllable or stabilizable.
同時 covariance matrices Q, R 是 stationary and semi-positive definite. Kalman filter 有 steady state 的 solution.
Steady state 的參數包含:
- state mean, x(k) -> x, or m(k) -> m (only one)
- state covariance 有兩個: P+ 是 prediction step 之後的 covariance; P- 是 update step 之後的 covariance.
P+ > P-; P+ = A' P- A + Q
or P- = A'.inv (P+ - Q) A.inv
- Kalman gain (only one)
Method A: Brute force iterative simulation
Method B: P = P
Method C: discrete algebraic Ricartti Equation (DARE) steady state equation 如下:
(可以參考 const_v.py code for simulation example)
ARE
RE: Riccarti Equation ARE: (Algebraic Riccarti Equation)
Stanford EE363 note
Another Berkeley article
Ms 是 prediction step 之後的 covariance.
Fs 是 steady state 時的 Kalman Gain!!!
Example1: DC offset estimation: A=C=1 ; Bw=1
P+ = P- +Q
P+ = P+ + Q - P+ ( P+ + R)^-1 P+
P+ + R = P+^2 / Q
P+ = 0.5 * (Q + sqrt(Q^2 + 4QR)) as expected
Example 2: Constant Velocity Model: A = [1, 1; 0, 1] , C =[1, 0], Bw=1, V=R, W=[qo, q1; q1, q2]
case 1: W = [q0, 0; 0, 0] process noise only on position, reduce to example 1?? Q = q0
case 2: W = ?? process noise only on velocity ->?
