Alpha-Beta-Gamma Filters This filter works in two steps: prediction and correction. The smoothed parameters (often called innovations) are computed for position, velocity and acceleration. Using these values, the prediction for position and velocity are made. The alpha-beta-gamma equationsError: Reference source not found are given below: 2
x p ( k +1 ) =x s ( k ) +T v s ( k )+
T as( k ) 2
v p ( k +1 )=v s ( k )+T a s ( k ) x s ( k ) =x p ( k ) +α ( x 0 ( k )− x p ( k ) )
v s ( k )=v p ( k )+
β ( x ( k )−x p ( k ) ) T 0
a s ( k ) =a p ( k ) +
γ x ( k ) −x p ( k ) ) 2( 0 T
x, v, and a are position, velocity, and acceleration respectively. x 0 is the measurement. The filter tuning parameters are α, β, & γ. The parameters are manipulated to deal with stability, noise, and tracking performance. The above equations may be put in a state space format for observer design. The filter parameters may be considered as an observer gain vector and computed by pole placement. A filter approach has been given for a 2-state version. The 1 st four equations out the five given above are for the 2-state version. A 3-state development follows from the five equations given above. The state space equations are:
[ ][
][ ]
x p ( k +1 ) 1 dt dt 2 /2 x p ( k ) v p( k + 1) = 0 1 dt v p ( k ) + 0 0 1 a p ( k +1 ) ap(k)
[ ] α + β+ β+ γ dt γ dt 2
γ 2
[ x 0 ( k )− x p ( k ) ]
The above is the identical construction of the 3 state observer system given above which is repeated below.
x k +1= A x k + L ( y k −C x k )
is
the
basic
observer
equation,
where
] []
[
L1 1 dt dt∗dt /2 A= 0 1 , L= L2 , C=[ 1 0 0 ] dt . Thus, we can now solve for the alpha0 0 1 L3 beta-gamma filter parameters in terms of desired poles. We have the following.
[ ] α+ β+
[]
L1 L2 = L3
β +γ dt γ dt 2
γ 2
[][
γ L 1− β− α 2 → β = dt L 2−γ γ dt 2 L 3
]
, by solving starting with the bottom equation.
With this approach we have a solution for the alpha-beta-gamma filter parameters in terms of classical filter performance characteristics in addition to those normally used to solve for the alpha-beta-gamma filter parameters. The following plots show deviation detections for the different state observers outlined above. The test signal consists of a small ramp appended to a straight line with a large amount of noise added. Indicators using the different states are compared for each case. The overall performance of the detections depends on the observer poles, therefore they should be adjusted according to the application. The direction change begins at time = 100 seconds. Position Detection at 144, Velocity Detection at 130 signal & filtered Y
1.5 1 0.5 0 -0.5
0
50
100
150
200
0
50
100
150
200
Velocity indicator
0.06 0.04 0.02 0 -0.02
(Variable dt) Position Detection at 144, Velocity Detection at 138 signal & filtered Y
1.5 1 0.5 0 -0.5
0
50
100
150
200
0
50
100
150
200
Velocity indicator
0.15 0.1 0.05 0 -0.05
The above figures using a velocity indicator show improvement over a position indicator for time of indication. Note the direction change begins at 100 seconds.
signal & filtered Y
Tracking Suppress = 1, Position Detection at 135, Velocity Detection at 122 1.5 1 0.5 0 -0.5
0
50
100
150
200
0
50
100
150
200
Velocity indicator
0.06 0.04 0.02 0 -0.02
signal & filtered Y
(Variable dt) Tracking Suppress = 1, Position Detection at 135, Velocity Detection at 125 1.5 1 0.5 0 -0.5
0
50
100
150
200
0
50
100
150
200
Velocity indicator
0.08 0.06 0.04 0.02 0 -0.02
The above two figures show at full tracking the velocity indicator is better than the position indicator. Position Detection at 147, Acceleration Detection at 125 signal & filtered Y
1.5 1 0.5 0
Acceleration indicator
-0.5
6
0 x 10
50
100
150
200
50
100
150
200
-3
4 2 0 -2
0
(Variable dt) Position Detection at 144, Acceleration Detection at 132 signal & filtered Y
1.5 1 0.5 0
Acceleration indicator
-0.5
15
0 x 10
50
100
150
200
50
100
150
200
-3
10 5 0 -5
0
The above two figures indicate that the acceleration indicator is better than either the position or velocity indicator for this application. Position Detection at 147, Velocity Detection at 127 signal & filtered Y
1.5 1 0.5 0 -0.5
0
50
100
150
200
0
50
100
150
200
Velocity indicator
0.06 0.04 0.02 0 -0.02
Repeats = 2, Position Detection at 147, Velocity Detection at 125 signal & filtered Y
1.5 1 0.5 0 -0.5
0
50
100
150
200
0
50
100
150
200
Velocity indicator
0.06 0.04 0.02 0 -0.02
The above two figures give results for a two iteration estimator with two states. The results for the velocity indictor are similar to the acceleration indicator results. The alpha-beta-gamma filter may also be derived using the Kalman filter approach 6. The alpha-beta filter will be shown 1 st and may be considered as a steady-state Kalman filter that is applied to a two-state system with a position measurement, as shown below.
[]
T2 x k = 1 T x k −1+ 2 w 'k −1 0 1 T
[ ]
y k =[ 1 0 ] x k + v k , where T is the sample time, w’k and vk are uncorrelated white noise processes. The Kalman filter is shown below. +¿ F T + Q −¿=F P¿ P¿
−¿ H T + R ¿ HP ¿ ¿ −¿ H T ¿ K =P ¿ +¿ −¿=F ̂x ¿k−1 ¿ ̂x k
−¿ y k −H k ̂x ¿k −¿+ K ¿ +¿= ̂x ¿k ̂x¿k −¿ +¿=( I − KH ) P ¿ P¿
[
[ ]
P P 12 F = 1 T , H =[ 1 0 ] , P= 11 0 1 P 12 P 22
]
After much algebra the unknowns are given below. K1=
−1 2 [ λ +8λ−( λ+ 4 ) √ λ2 +8λ ] 8
K 2=
1 2 [ λ + 4λ−λ √ λ 2+8λ ] 4T
K 1 σ 2w 1−K 1 P ¿11
−¿=
K 2 σ 2w 1−K 1 ¿ P 12
−¿=
−¿ K K −¿= 1 + 2 P¿12 T 2 ¿ P 22
(
)
σ 2w T 2 λ= R The variable λ is called the target maneuvering index or target tracking index. It gives the ratio of the motion uncertainty to the measurement uncertainty. It may also be used to help tune the tracker. The similar development for the alpha-beta-gamma filter will be given next6. Consider the three-state system below with position measurement.
[ ] [] 1 T
xk =
0 0
1 0
T2 2 x + T k −1 1
T2 2 w' k−1 T 1
y k =[ 1 0 0 ] x k + v k The process equation is given next.
[
]
1 T T 2 /2 xk = 0 1 T x k−1+ w k−1 0 0 1 w k ( 0, Q )
[]
T2 2 2 ' 'T T Q= E [wk wk ] 2 T 1
[
T
][
The alpha-beta-gamma gains are given next. ∝=1−s
2
β=2 ( 1−s2 ) γ =2λs λ b= −3 2 λ c= +3 2
]
T 4 /4 T 3 /2 T 2 /2 2 1 = T 3 /2 T 2 T σw T 2 /2 T 1
2
p=c−
b 3
3
q=
2 b bc − −1 27 3
[
−q + √ q 2+ 4 p 3 /27 z= 2 s=z−
p b − 3z 3
1/ 3
]