What do filter coefficient relationships . , mean? John E. Gray, A. Sunshine Smith-Carroll, and William J. Murray. Naval Surfare Warfare Center Dahlgren Division.
What do filter coefficient relationships mean? .,
John E. Gray, A. Sunshine Smith-Carroll, and William J. Murray Naval Surfare Warfare Center Dahlgren Division Systems Research and Technology Department Code B32 Dahlgren, VA 22448-5150 Phone: 540-653-1259 F a : 540-653-7775
A b s t w c t - There are three commonly used relationships between alpha and beta that are reported in the literature: Kalata, Benedict-Bordner, and Continuous White Noise. The Kalata relation is obtained from steady state Kalman filter theory assuming zero mean white noise in the position and velocity state equations. The Benedict-Bordner relation is derived based on good noise reduction and good tracking through maneuvers. Both the Kalata and BenedictBordner relationships can be derived without any reference to a Kalman filter. T h e question, given the variety of filter coefficient relationships, is which relationship should be chosen as part of a filter design and why? What does it mean to choose a particular filter coefficient relationship? What is the difference between filter coefficient relationship and a criteria to maximize performance?
1. Introduction 2. Cost Functions 3. Cost Function hIodels 4. Which Filter Relationship Should One Choose? 5. Conclusions 6. References
p=-
K ~ y i t ~ m ~ d sfilter coefficient relationship, alpha-beta filter, cost function,
I. ISTRODUCTIOS The idea of a filter coefficient relationship can be traced back to one of the pioneering papers in target tracking by [12]. In that paper he suggested that for a a-P tracking filter, wit,h predictmionequations given by
z p ( k )= xs(k - 1)
+ @,(k
u p ( k )= US(k- 1)
- 1)T
tracking. Thus. filter coefficient relationships p = p ( a ) were born. One, the Benedict-Bordner relation 121, can be said to predate the Kalman filter, since its derivation is based on =-transforms and classical analog filter design principles. The Benedict-Bordner relation is derived based on a cost function that consists of two terms, one of which has a term which represents good noise reduction while the other term represents good tracking through maneuvers. A somewhat convoluted (to modern eyes) argument based on defining an idealized two state optimized (in terms of pole structure) tracker which is compared to the pole structure of the a-P filter. This leads to the first “optimal coefficient relationship” , the Benedict-Bordner relation:
(1)
a2
2-a!.
(5)
Two subsequent papers continued in this tradition [ll]and [101Subsequent to these three papers, filter coefficients were cast in terms of steady state solutions to the Kalman filter as is discussed in [l].The Kalata relation, which is obtained from steady state Kalman filter theory assuming zero mean white noise in the position and velocity state equations [6], is given by: p = 2(2 - a!) - 4 d E (6) While another is the- Continuous White Noise (CTW’N) relation: [I] a!
= J?p
+ P2 - 2. P
(7)
While there are three named relationships between a! and ,B that are in common usage in the literature. A number of different filter relationships including the Kalata zs@) = z p ( k ) a!(x,rl(k)- z p ( k ) ) , (3) and Benedict-Bordner relationships can be derived without any reference to a Kalman filter. The question then and occurs, given the variety of filter coefficient relationships, P us@) = u p @ ) +n(k) - zp(W), (4) is which relationship should we choose as part of a filter design and why? What does it mean to choose a particua relat,ionship between a! and p t.hat represented critical lar filter coefficient relationship? What is the relationship damping of the poles of the filt.er would be useful for target between filter coefficient relationship and a criteria to maximize performance? We now try to answer some of these U .S. Govcriiinriit. work not prot.oc.tod by U .S. copyriglit. and smoot.hing equat,ions given by
+
+
an explicit geometric interpretation of filter coefficient relationship as a tangent curve which holds the lines B = const and A = constfor all points that satisfy Eq. (12). Since the functions A and B are used to represent separate costs associated with filter performance, the filter coefficient relationship is the optimum relationship which jointly mininiizes these joint costs or restraints.
questions. without reference to the Kalman filter, in order to provide a straightforward interpretation of the answer to these questions.
11. COST
FUSCTIOSS
Cost functions can used to determine a in terms of knon-n system and design parameters (This idea dates back at least to Sklansky, though it was not widely pursued in the open literature.) This gives one an alternative to the Kalata tracking index. This cost function approach can also be used to determine filter coefficient relationships. Once the relevant equations are solved. a single functional relationship is P = @ ( a )achieved that satisfies the minimization criteria. This is exactly what we mean when n-e say there is a filter coefficient relationship @ = p(a). The Jacobian of variables 2 and y with respect to wriables U and z‘ is defined as IS]
,A= 1
I
Figure 1.
2
- y plane n-ith the contours A(a. y) = c o m t and B(x.y) = const
If the independent variables are discernible from the context, then the Jacobian is simply written as J(x.y). Jacobians are quite useful when one is considering functions that are joint functions of two variables. The particular coordinate system one is working in is unimportant in many cases. it is the mutual property of two variables rather than the joint property of three variables ( J a p e s lias made this observation in some unpublished notes on tliermodynamics). This follows from the observatioii that a integral over a region in one coordinate system can be espressed in terms of another using the Jacobian of the transformation. Another method for looking at this is to draw the x - y plane with the contours A(z.y) = const and B(z,y) = coizst 51s shown in the Figure 1. With the z-axis pointing out of the plane. the Jacobian can be written as a cross product J(A,B;x,y) = =
(2)y (VAXVB),
-
111. COSTFUXCTIOS MODELS A cost function for a tracking filter consists of two com-
(z)y (9)
ponents. a noise reduction component and a componet due to un-modeled behavior. There are three niodels one could take that contribute un-modeled behavior in a tracking filter to the squared error output of the filter. One possibility is uncertainty in position alone due t o a transient acceleration in position only n-hich we refer to as Model 1, if the output is a lag. we refer to the niodel as Model 2. and uncertainty in acceleration that contributes to both position and velocity which we refer to as Model 3. The scalar cost functions will be the noise reduction ratios of the position noise reduction term l i p ( a p) . or the velocity noise reduction term K2.(a, ,B) and an un-modeled response term that consists of transient terms D p A f o d e l 1 2 , 3 ( a . p) or D v ~ f ~ d 1e. 2i , 3 ( a . @)* (The context for the models is found in the 151.) The position error function Model 1 is
which is equal to the area of the parallelogram with sides VA and VB. If we explicitly take the functions as
2 2 + 2p - 3a9 ( 2 - a)(1- ay
C$)(a,p)= (
D
D
)
(13)
m-hile the velocity error function Model 1 is
g,
272 -a)
The Jacobian for functions of this form leads to the equation
Cp’(aJ) =
D This equation produces an explicit filter coefficient relationship. At the point where the Jacobian vanishes, any infinitesimal change to A also holds B constant. This gives
F~~
37
lielocitJ.
+ 2p(1 -a)
PD
= a(4 - 2a - p). function, f = 2,@, so fa = 0,
)
partials are
(16)
While we have derived three different coefficient relationships, a number of others are probably possible depending on what choice one makes for the second scalar funct,ion g(z,y) in Eq. (14). For most applications one wants to (18) consider noise reduction as t.he first componnet contributing to the determination of P = P ( a ! ) , but the "acceleration t.erm" could be replaced with a variety of other possibilities. The natural question with a variety of different relationships is which one is preferable?
f o = 4P, while for h = a ( 4 - 2a! - ,B), so t.he partials are
(17)
h,=4-4a-p
and for
a y 2 - a ) + 2P(l - a!)
9=
Ph
Iv.
t,he part,ials are
((4a- 1) - 2 P ) 12. - ( 4 2 - a!) gQ =
+ 2,8(1-
FILTER RELATIOSSHIP SHOULD CHOOSE?
OSE
a)) ha,
One can define t,he velocity lag
8122
g.9 =
IvHICH
T'
as
2(1 - a)hp h2 .
If we substitute these into
The acceleration bias of tshesmoothed velocity response of the filter is defined as
and simplify: we get where we define the velocity lag coefficient as
The same things happens for CF)(a,P). so our formalism leads to the Benedict-Bordner relationship for both cost functions. For Model 2, a cost function that consists of the velocity lag and velocity noise reduction ratio has the cost function (25)
Xote the velocity lag coefficient does not depend on which relationship one chooses betn7een a and 0. The acceleration bias of the smoothed position response filter is
L, = CIOT'I,.
(33)
n-here the position lag coefficient is defined as l , = - - -1- - a - 1
Our formalism leads to zero. which implies that there is no optimal relationship between the coefficients for the velocity cost function. position lag plus noise reduction ratio cost function can be defined as
1
13 - P
(34)
Note both p and a can ah-ajs be n-ritten in terms of lag coefficients as 2 P = 2 (Is 7) 1 ' (35) 2a2 - $(sa!- 2) 1 - a ~;)(a. J) = 44-2a-b) ' B ' (2G) and . (27 1) a!= (36) Appl) ing the previous formalism to this cost function gives 2(L+T)+l' a ielationship betm-een cr and 9, while the noise reduction ratios expressed in the lag form
+ + +
-)
(
3=2-2
d3,
(27)
can be expressed as
P,
which appears to be unknown in the literature. The position noise reduction ratio along n-ith the new position transient error response derived from Model 3, used to form the follon-ing function
cp(a.p)=
(
2a2
-
2P2 T2n(4- 2 a - 0)
2
T2(2Z,
+ ~ ) ( 2 +7 1)'
and
+;2
-3
4 (1 - a)* '
-)
2aB
.
(28)
Applying the previous formalism to this cost function gives a relationship between cu and 3!, is
p
=
= 2(2 - a) - 4
G
.
(29) 38
P,
= -
- P(3a - 2 ) 4 4 - 2a - p)
2a2
=
+ 21, + + 7)(2T + 1)
(272
(21, T2
--Pt,(2T2 2
7)
+ 21, + T )
(37)
Now if we look at each one of the filter coefficient relationships in terms of the velocity lag coefficient 7 . we get some curious results. For all the filter coefficient re1ationships.n.e can express the filter coefficients in terms of the velocity lag coefficient form of the filter coefficients (e.g. the form that depends just on T and not I,) which gives the following Table 1.
(I
I
3
i
4
5
6
1
li
9
IO
9
10
lil"
Figure 3. Position Xoise Reduction Ratio 7
Name
~
Kalata
~
B-B
~
1.8
CWN
16 I 4
-
1.2
$
1
'$ 0.8 0.6 0.f
0.2
U
I
2
3
4
5
h
7
8
19"
Table 1. Filter coefficient relatioiiships Figure 4. Velocity Yoise Reduction Ratio Once all the relevant expressions for the noise reduction ratios and the lags have been espressed in a common form; a coinparison can be made between them. 11
0
I
5
15
10 1
Figure 2.
V. COSCLUSIOS
Isolated Comparison Betn-een cy and
T
The cost function approach to track filtering is as complete as the Kalman filtering approach. One can implement this entire approach to filtering as a table of filter coefficients provided the measurement noise is determined either by using formulas based on knon-n angle and range covariances of the sensors. or can estimate the noise directly from measurements. Alternatively, one can estimate the noise covariance of ones measurements and updatc the filter coefficients by solving for the optimized value of r from the chosen cost function with the same degree of fidelity as the ICalman filter. Noise reduction relations can be used to produce the filters covariances based on the noise reduction ratios. IT-hile we have done all of this n-oil
