Poles and zeros of a-P and a-p-y tracking

B. Ekstrand

filters

Abstract: Discrete transfer functions for the a-P and a-P-7 Kalman tracking filters are presented. Filters for the update, as well as the prediction, estimates are considered. Root loci for the poles and zeros are presented, parameterised in the manoeuvring index. Closed-form solutions for the a-/I-y gains are given as functions of the manoeuvring index.

1 Introduction

E-[{ and a-8-7 filters are basic discrete time filters for target tracking (of second- and third-order, respectively), which use the Kalman prediction-update structure based on simplified models for target motion. The filter gains are usually constants, determined by the dimensionless gain factors x, f l and y . These gain factors can bc designed by any method that will give the filters a suitable or required performance. We can then think of the filters as observers. However, the filters are often considered as Kalman filters, in which case the gain factors are obtained from the steady- state Kalman algorithm.

The 2 - j and 1-j-7 filters are discussed elsewhere [ 1, 21. For the second-order a-fl case, the steady-state Kalman gains [ 11 are based on a model that is often underlying the a-/j filter. The two gain factors can be expressed in closed form as functions of one parameter only, the so-called manoeuvring index, which includes the model parameters. The gain factors depend on the manoeuvring index, as demonstrated previously [3] , both for the second- and third-order case. The third-order case relations [3] have been used elsewhere [ 11 for numerical solution of the gains for this case.

The usual models underlying the a-b or x-P-y filter are somewhat different concerning the target manoeuvre model and the corresponding noise covariance. Conse- quently, different filters will be obtained for these models. As is well recognised, such models are not always adequate, since the behaviour of a manoeuvring target is difficult to catch in a simple model. The realism of the different models can be questioned. However, what counts in the end, of course, is the properties of the filter that are obtained for a certain model. The model can be seen as a vehicle to obtain a tracking filter, whose proper- ties have to be investigated anyway for the application in question. A suitable way to view the properties of a filter is by its transfer function.

By .-/I and a-fi-y filters we mean steady-state Kalman filters based on the models used elsewhere [l, 31. Transfer

C’ IEE, 2001 TEE PFoceedings online no. 20010578 DO/: 10.104Y/ip-cta:2001057X Paper first received 5th June 2001 The author is with Saab Dynamics AB, Sweden The author’s address is Ekelundsviigen 22, S-429 42 Saro, Sweden E-mail: bertil.ekstrand@telia.com

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functions are given for these filters, i.e. operators from the position measurement to different update and prediction estimates. Root loci, parameterised in the tracking index, are given for the poles and zeros of the transfer operators, from which the dynamic behaviour can be judged. In the third-order case, closed form solutions for a, p and y are given as functions of the tracking index; a partial solution for this case has been presented previously [4].

2 Filter equations

The a-/Y and x-fl-7 filters are based on the discrete time model

x(k + 1) &(k) + v(k) ( l a )

( l b ) y(k) = Hx(k) + w(k) For the second-order a-P case

and for thc third-order a-p-7 case

.[% 7 H = [ l 0 01 (3 )

T is the sampling interval. v(k) and w(k) are zero-mean white noise sequences with covariance

E[w(k)w(k)] = 0; (4)

and

T 2 / 4 T / 2 ( x - P case) ( 5 )

T 4 / 4 T 3 / 2 T 2 / 2

~[v(k)v’(k)l = ~ 3 1 2 ~2

( x - p - y case) (6)

In the state vectors x(k)= [x l (k ) x2(k)]’ or x(k) = [x, (k) xZ(k) xi@)]’, the components are position (x,), velocity (xz) and acceleration (x3), respectively. In the second-order case, eqn. la is a model for a system with

[ T 2 / 2 T lo’ IEE Piw.-Ctintrtil Theor}’ Appl., Vol, 148, No. 5, Septembw 2001

constant target acceleration during the sampling interval, where the acceleration value in different intervals is a random variable with standard deviation o,,. In the third- order case, it is the target acceleration increment that is assumed to be pieccwise constant. y(k) is the measurement of the position, where w(k) is the measurement noise.

Models of the same structure with different elements in the covariance matrices (eqns. 5 and 6) are also used, in which case the filters are also called a-b and a-8-y filters. However, we think of the a-/I and a-8-y filters as based on the models presented here, adopted from previous models [ I , 31.

The Kalman filter for eqn. 1 is given by

.?(k + 1 j k + 1) = i ( k + 1 I k ) + L[ y(k + 1) - H i ( k + 1 I k)]

(7)

where

i ( k + 1 I k ) = Fi(k I k ) (8)

and L is the gain. This is the filter for the update estimates. From this algorithm, we obtain the operator expression

i ( k I k ) = [ Z I - (I - LH)q- 'Lzy(k) (9)

where z is the forward shift operator.

prediction estimates From eqns. 7 and 8, we obtain the Kalman filter for the

i ( k + 1 I k ) = Fi(k I k - 1) + FL[ y (k) - Hi@ I k - 1 )I (10)

This gives the operator cxpression

i ( k I k - 1) = [zl - F(I - LH)]-'FLy(k) (1 1)

Let us introduce for our two model cases (of second- and third-order) the gains

= [ D;T]

and

respectively. SI, j and 1' are dimensionless constants, and the filters are named after these standard notations. All the parameters in the operator expressions (eqns. 9 and 11) are now defined.

a-b and a-B-y filters usually mean filters for the update estimates, but we also consider the prediction estimates, which are sometimes used as the Kalman filter as well.

If we use the gains that are obtained from the steady- state Kalman algorithm for the models given above, it turns out that a, p and y can actually be solved in closed form, and the solution can be expressed in one parameter, the so- called target manoeuvring index:

Thus, in steady-state the gain factors will be a function of this dimensionless parameter only, both for the second- and third-order cases. Note that the index I. is defined by the three parameters ov, o,+, and T that characterise the models.

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3 L Y - ~ filter

3. I Transfer functions From eqn. 9, for the second-order case we obtain

(1 5 ) P,(k I IC) - (az - a + B)Z

z2 + (a + p - 2)2 + 1 - a -

y(k)

These are the transfer functions from the filter input y(k) to the update estimates of position and velocity. From eqn. 11 we obtain the corresponding functions from y(k) to the prediction estimates:

(17) i ,(kIk- 1 ) - (a + P)2 - a

2 2 + (a + p - 2)z + 1 - M. -

y(k)

B -(z - I ) - T

z2 + (a + p - 212 + 1 - a (18) - i 2 ( k Ik - 1 )

y (k ) Except for the zero at z = 0, the same transfer function is obtained in eqns. 16 and 18 for the update and prediction velocity estimates. This is clear from the system matrix F, i.e. it is inherent in the basic model assumptions. Since ( z - 1)iTrepresents a derivative, the velocity estimates can be interpreted as obtained by low-pass filtering of thc derivative of y(k) .

Eqns. 15 and 17 have one zero each, which is not at the origin, and these zeros are different. Thus, from the transfer function point of view, the main difference between the update and the prediction position estimates is the location of a zero. The poles are, of course, the same for the different transfer functions.

By taking z= 1 in the transfer functions, we see that a step input J@) will be reproduced without error in the steady-state for both the update and the prediction position estimates. The velocity estimates will be zero in the steady state, i.e. for a step y(k), the velocity estimates tend to zero after an initial transient, as should be the case. Steady-state errors for other inputs (e.g. ramp or acceleration) can be calculated from the transfer functions.

3.2 Filter gains For the steady-state gains, the following relations can be derived [ 1, 31:

I 2

x = & q - - p

These equations can be solved for a and /I as functions of iL [ 11:

1* + S i - (1 + 4)4= 8

(21) G 11

U = -

l2 + 4a - a J F T Z 4 8=

These are the gain factors a and B corresponding to the steady-state Kalman gains for the second-order model (eqns. 1, 2 , 4, 5). We see that r , f i -+ 0 if 2 -+ 0. Fmther- more, we have ti -+ 1 and /I -+ 2 when 1 -+ 00. Between these limits a and /3 are monotonic increasing functions of 1 U].

371

3.3 Poles The poles are given by the solution to the characteristic equation

z 2 + ( u + p - 2 ) z + 1 - u = O (23)

which is

Using eqns. 21 and 22, the poles can be expressed in the manoeuvring index A. The root locus for the poles, para- meterised in 1, is given in Fig. 1.

For 1, = 0 we have a double pole at z = 1. With increasing I., the two branches of the root locus move out in the complex plane, cross the imaginary axis (where I = 4), meet at the negative real axis (i = 8), and finally end up at z=O andz=- l ( I=co) .

By comparison with the second-order root-locus diagram ofAstrom and Wittenmark [SI, we see that, for a certain range of values, the complex pole positions correspond (at least approximately) to the poles obtained when a contin- uous second-order system with damping factor 5 = 0.1 is sampled. Thus, loosely speaking, it could be said that the poles for the a-P filter have the damping factor 5 = 0.7.

The natural frequency corresponding to the poles is not immediately obtained from the pole positions, since the sampling interval T is also involved [ 5 ] . In our case, the root locus parameter I also depends on 7: However, taking T into account, the natural frequency ctn be deter- mined by comparison with the diagram ofAstrotn and Wittenmark [SI.

-1 .o -0.5 0 0.5 1 .o Fig. 1 Root locus for poles

From eqn. 19, we conclude that p ix -+ 0 when 2, + 0 and P / x --f 2 when I --f 00. Thus, for small I values, the zeros are close to z = 1. and they tend to z= -1 and z = 113, respectively, as iL + 00. The zeros are shown as functions of 2, in Fig. 2.

Between the limits, the zeros are monotonic decreasing functions of 1. The update zero is in the interval [-1, 11, and the prediction zero is in the interval [1/3, 11. The update zero is lcss than thc prediction zero for all I values.

4 a-B-y filter

4. I Transfer functions For the third-order case, we obtain the following transfer functions for the update estimates:

3.4 Zeros The dynamic properties of the filter depend not only on the poles, but also on the zeros. As we have already noted, the

18) corresponds to a differentiation. In the position transfer functions (eqns. 15 and 17), we have

position update zero

; l(k I k ) Y(k )

(MZ’ + ( - 2 ~ + p + Y / ~ ) z + M - P + ?/2)2 D(z)

~- - zero z= 1 in the velocity transfer functions (eqns. 16 and

(27)

(28)

I q k I k ) ,(Bz-P+l’)(z- 1)z

- -

x Y ( k ) D(Z)

P (25) z = 1 - -

position prediction zero z = a + 11 .

-1 .o 10“ 10-3 10-2 10-1 100 10’ 1 02 1 03

Fig. 2 - prcdiction - update

zeros in po.sitioii transfir functions

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where

The prediction transfer functions are

2,(/<lk - 1) (a + /?+ y/2)z2 - (2a + P - "/2)Z+ C i - - Y ( k ) D ( 4

(31)

(33)

These transfer functions can be obtained from eqns. 9 and 11, or more directly from the state equations (eqns. 10 and 8), which is perhaps easier in the third-order case.

We make some observations analogous to the .-/I case. For the update and prediction acceleration estimates (eqns. 29 and 33), we obtain the same transfer function (except for the zero z = O), which is to be expected from the model. This transfer function has a double zero at z = 1, corre- sponding to two derivatives; thus, the acceleration esti- mates can be interpretcd as obtained by low-pass filtering of two derivatives of the position.

Besides thc expccted zero at 2 = 1, each of the velocity transfer functions (eqns. 28 and 32) has one more zero (not at the origin), and these zeros are different for the two transfer functions. The position transfer hnctions (eqns. 27 and 31) havc two zcros each, which are different for the two cases. We note that differences, between update and prediction estimates for position and velocity, turn up as different locations of zeros in the transfer functions.

As for the x-p filtcr, by taking z= 1, we see that the position transfer functions will reproduce a step input without error. The velocity and acceleration estimates will he zero in the steady state for a step input.

4.2 Filter gains In the steady-state we havc the relations [3]

(34) 1 2

x=m--p

y = - P2 2a (35)

(36) Y2 12 - 1 - a

where 1. is the manoeuvring index (eqn. 14). Eqn. 34 is the same as eqn. 19 for the a-b filter. The gain factors U, f l and y can be solved from these equations in closed form as functions of 2. As demonstrated in Appendix 7, the solution is given by

CI = x(2 - x) (37)

p = 2x2 (38)

)' = (1 - x)% (39)

where x is obtained as follows. Introduce

(40) 1

36 Q = --I"(), - 18)

1 2 R = -1(1 - 27;- + 108) 216

1 1728

D = - - A2(1? - 432)

R 0 = arccos- m (43)

2.0-

1.0-

I I I

10-4 I 0-3 1 0-2 100 IO' 1 02 1 03

Fig. 3

-0

1, j. 7 us,fuiictions of i - a

_ _ ~ 11

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-7 .O -0.5 0 0.5 1 .o Fig. 4 Root locus for poles

The gain factors as functions of I are shown in Fig. 3 . The gain factors are monotonic increasing functions, starting at zero for i = O . M + 1, and

Note that the acceleration gain factor used by Kalata [3] is "J2, instead of y used in the present paper and previously [ 11. Furthermore, we note that numerical calculations of Bar-Shalom and Fortmann [ 11 of the gain factors are based on Kalata's relations [3]. Therefore, the 7-curve values in the gain factor diagram of Bar-Shalom and Fortmann [ l ] have to be divided by 2, to be consistent with the definition of the acceleration gain factor.

and y + 2 when ?,-t CO.

4.3 Poles The poles are the roots of the characteristic polynomial (eqn. 30). Since we know the gain factors M , p and 1' as functions of i, and by that the coefficients in eqn. 30, the poles can be also determined as functions of A. The characteristic equation is cubic, and so the general solution (eqns. 70 and 72) of the cubic equation in Appendix 7 can be used to obtain closed-form expressions for the poles.

In this way, we obtain the root locus for the poles (Fig. 4). For 1 = 0, the three poles are at z = I . For 2. > 0, one pole is in the interval [0,1] and ends up at z = 0 when

2, -+ 00. The other two branches first move out in the complex plane, then meet at the negative real axis and e n d u p a t z = O a n d z = - 1 .

By comparison with the diagram ofihtrom and Witten- mark [5] , we see that the complex poles have a damping factor of about 5 =0.5. However, the third real pole contributes to increased damping and the zeros also influ- ence the dynamic behaviour.

4.4 Velocity zeros In the velocity transfer functions (eqns. 28 and 32), we have the following 2-dependent zeros, respectively:

velocity update zero: z = 1 - - (46) B

B B + r (47) velocity prediction zero: z = __

These zeros are shown as functions of I in Fig. 5. The two zeros are within the unit circle. The values are monotonic decreasing, starting at z = 1. The prediction zero tends to 0.5, and the update zero tends to the value 0 as 3, + 00. The update zero is always less than the prediction zero.

4.5 Position zeros In the prediction transfer functions (eqn. 31) for the position estimate, we have two zeros given by the roots to the equation

From the values of the gain factors x, /I, y , it can be seen that these zeros start at z = 1 for I = 0; they are complex valued for i > O ; and they end up at 3/8f.2/?/8 as /1 i 00. The root locus is shown in Fig. 6.

The two zeros in the transfer function (eqn. 27) for the update position estimate are given by the roots to the equation

4c( - 2p - "4 2a - 2p + y - - 0 (49) = + 2c!

1- 2a

For these zeros, we obtain the root locus in Fig. 7. A similarity is evident here: the shape and position of the

locus in Fig. 6 appear to be the same as part of the locus in Fig. 7 . This is not only as it seems. In fact, by using eqns.

1.0-

0.5-

0 I h 1 0-4 1 0 4 10-2 lo-' 1 00 10' 1 02 103

Fig. 5 - prediction - update

374

Zeros in velocity transfer functions

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/ I -1 .o -0.5 0 0.5 1 .o

Fig. 6 Zeros in position prediction transfer junction

58-61, it can be proved that the locus for the prediction zeros (Fig. 6) is a subset of the locus for the update zeros (Fig. 7).

We also have a striking similarity between Fig. 7 and Fig. 1. Actually, it can be proved that these loci are the same; the locus for the position update zeros in the third- order case occupies the same set of the complex plane as the locus for the poles in the second-order case.

4.6 Pole placement We have considered the poles and zeros obtained when the steady-state Kalman gain is used in the filters. As we know, the poles can also bc given arbitrary positions by pole placement. Of course, the Kalman gain gives the optimal filter performance in the usual stochastic mean-square sense for the inodel in question. However, since the models are not always adequate for manoeuvring targets, it may be convenient in certain applications to use pole placement; for example, if a certain filter bandwidth is required. The zeros cannot be chosen independently of the poles, however. For certain poles, obtained by choices of E, p, y , we will necessarily obtain certain zeros according to the transfer functions, and those zeros are also deter- mined by 2, /I, 7 .

Re -

-0’5LlA -1 .o -1 .o -0.5 0 0.5 1 .o

Fig. 7 Zeros in pusition updute Irun.@ function

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Clearly, the transfer functions are useful in pole place- ment design. They can also be used for comparison of different filters with the a-P and a-fl-y Kalman filters.

5 Conclusions

Transfer functions for the a-8 and a-8-y Kalman tracking filters have been presented, i.e. transfer functions from position measurement to different update and prediction estimates. Poles and zeros have been discussed, and root loci for these quantities, paramcterised in the manoeuvring index, have been presented. In the cx-p-y case, the gain factors are solved in closed form as hnctions of the manoeuvring index.

The transfer functions contribute further information about the u-p and x - 8 - y filters and should be useful in the design process. The functions are also useful for pole placement design, and for comparison of filters with x-B and r-8-y Kalman filters.

6 References 1 BAR-SHALOM Y.. and FORTMANN. T.E.: ‘Tracking and data asso- -

ciation’ (Academic Press, 1988) 2 BLACKMAN, S.S.: ‘Multiple-target tracking with radar applications’

(Artech House, 1986) KALATA, ER.: ‘The tracking index: a generalized parameter for 1-b and E-{&)) target trackers’, IEEE Truns. Aerospnce Electron. Syst., 1984>

GRAY, J.E., and MURRAY, W.: ‘A derivation of an analytic expression for thc tracking index for the alpha-betl-gamma filter‘, IEEE Trans. .@space Electrotr. Sysf., 1993, AES-29, (3)

5 ASTROM, K.J., and WITTENMARK, E.: ‘Computer controlled systems’ (Prentice-Hall, 1997)

6 WEISSTEIN, E.W: ‘The CRC concise encyclopedia of mathcmatics‘ (CRC Press, 1999)

3

AES-20, (2) 4

7 Appendix

7.7 a-p-y gain factors For the third-order case, we have in the steady state the relations

1 2

E = & j - - P

02 P y = - 2%

(52) y2 - 1 2 1-Cc

where d is the manoeuvring index (eqn. 14). These relations are obtained by applying steady-state conditions to the covariance matrices in the Kalinan algorithm [3]. Eqn. 50 is an equivalent reformulation of the correspond- ing relation of Kalata [3]. Furthermore, we have

O < X < l (53)

0 < 8 < 2 (54)

o < y < 2 (55)

We know that the gains are positive U, I{, 7 > 0. Eqn. 53 then follows from eqn. 52 and eqns. 54 and 55 are to be expected from eqns. 50 and 51. Actually, eqns. 53-55 can be proved by using the fact that the error covariance matrices are positive semi-definite; the elements of these matrices are obtained in the derivation of eqns. 50-52.

It can also be shown that up > y . Furthermore, 0 < /z < CO, and from eqn. 50 we have

2

l - . = [ l - g ]

We want to solve eqns. 50-52 for U, /3, y as functions of 1,. For this purpose, we introduce the new variable

x = g (57)

We note that 0 i x < 1. From eqns. 56, 57 and 52, we obtain

c( = x(2 - x) (58)

p = 2x2 (59)

y = (1 - x ) i

From eqn. 51 we obtain

(61) 2x3 y=-

2 - x Combined with eqn. 60, this yields the following equation for x:

(62) 1 2

x3 - -(x - l)(x - 2) = 0

or

(63) i L 3A 2 2

x3 --x2 +-x - I L = 0

We have here a cubic equation for .x, whose appropriate real solution yields the gain factors via eqns. 58-61. By sketching the root locus for eqn. 62 with respect to i, it is seen that there is only one real solution 0 < x < 1 ; the other two roots are either complex-valued or have values x > 1 if they are real. Thus, we search for the least real solution to eqn. 63.

7.2 Cubic equation Consider a general cubic equation

23 + a2z2 + alz + a, = 0 (64) Introduce

(65) 1 9

Q = -(3a, - a;)

(66) 1

54 R = -(9ala2 - 20: - 27ao)

and

D = Q3 + R2 (67) One root is always real, since the equation is of order three. Whether the other two roots are real or complex is determined by the discriminant D:

D c 0 all real

D = 0

D > 0

all real, at least two coincide

one real, two complex

Introduce

S = (R + T = (R -

For D > 0 the solution to eqn. 64 is then given by

(70a)

00b)

( 7 0 ~ )

S and Tare real if D > 0; consequently, the real solution is then given by z1 (eqn. 70a). For the case D c 0, introduce

1 3

z2 = - -a2 - - (S + T ) + i- (S - T ) 3 2 2

z3 = - -a2 - - (S + T ) - i-(S - T ) 3 2 2

z1= --a2 + S + T

I 1 1/5

1 1 43

R U = arccos- m

The solution for this case is then given by

z , =

z2 = -

1 8 --a2 3 + 2 m c o s T

- - a 2 i 2 m c o s ( : i $ ) 1 (72h) 3

z3 = -ja2+2m~os(:+:) 1 (72c)

Of these three real roots, we search for the one of least value.

7.3 Solution In our case (eqn. 63), we have

(73) 3 I

a. = -i, al =-I,, 2 a2 = --I. 2

We then obtain 1

36 Q 1 ---(A - 18) (74)

1 R = - 1(A2 - 2 7 i + 108) (75) 216 ' 2 2 - n ( h -

1728 432) (76)

Thus, if i < J(432) the solution is given by eqn. 70a. If 2 > J(432), we have in eqn. 71 that 0 < 0 < n (as expectcd - 1 < R/J(I Q 13) < 1 in this case) and the real solution of least value is then given by eqn. 72h. Thus, we find that the required solution is given by two expressions, each valid in a certain region of 1 values. The two regions are separated by the point A = 2/(432), at which point the two expres- sions give the same value.

Summing up, the gain factors are obtained from eqns. 58-60, where

1 6

for i 5 2/432: x = -+ (R+ + ( R -

(77)

for

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