Performance analysis of interacting multiple model algorithm

PERFORMANCE ANALYSIS OF INTERACTING MULTIPLE MODEL...

14th World Congress of IFAC

Copyright © 1999 IFAC 14th Triennial World Congress, Beijing, P.R. China

H-3a-05-5

PERFORM:ANCE ANALYSIS OF INTERACTING MmLTIWLE MODEL ALGOIDTHM Pan Quan, Liang Van, Liu Gang, Zhang Hongcai and Dai Guanzhong Department ofAutomatic Control, Northwestern Po{vtechnical (FniversifJJ, Xi : an. P. R. China, 7 J0072~ Email: [email protected]

Abstract: A new non-simulation method for perfonnance analysis of Interacting Multiple Model Algorithtn is proposed. Firstly~ how input-interaction effects model-conditional estimation is analyzed. Four conclusions are made qualitatively. Besides this, the compression ratio of model-conditional error is defined. So the parameters and modeling can be chosen quantitatively. Secondly, ho\v input-interaction effects model probability is analyzed. Input-interaction is found not only to decide the upper and lower limits of model probability but also to lessen the difference among model probabilities, then in the sense of model probability!' decaying-memory filtering~ damping coefficient and regulating-time are defined. All these \vorks may be useful to choose optimal parameters and design new adaptive filters. Copyright @ 1999 IFAC Keywords: Adaptive filtering, Kahnan filter, Manoeuvering target, Performance analysis, and Markov models

1.

INTRODUCTJON 1

Hybrid estimation has been successfully applied in the systems with uncertain structure or randomvarying parameters such as tracking maneuvering targets systems (Bar-shalom, 1995~ Li, 1993a and 1996~ Pan, 1997), on-line supervising failure/repair systems (B assevilIe, 1993), and etc. It regards decision in discrete space S as a special state estimation, \vhich means that estimation is

accomplished in the space Rn X S \V·here Rn is the continuous state space with n dimensions.

1. This paper is partly supported by the National

Now in hybrid estimation, the Multiple Model Algorithms are the mainstream. These algoritluns can be divided into tlrree steps. First, the panuneter space is mapped to a model set. Then:- modelconditional filters run in parallel. Finally the overall state estimation is made as a combination of modelconditional estimations. Compared with Single Model Adaptive Filters, Multiple Model Algorithms have the following advantages. (1) The parameter space can be described more accurately_ (2) They are all optimal estimators in the minimum mean-square error sense under the assumptions. (3) They have distinct parallel structures, which often means that their implementations can be real-time and effective. Among Multiple Model AIgoritluns, Interacting Multiple Model Algorithm (11\.11\1) is more popular

Nature Science Foundation of China.

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ISBN: 0 08 043248 4

PERFORMANCE ANALYSIS OF INTERACTING MULTIPLE MODEL...

14th World Congress ofIFAC

(Mazor!, 1998). It is tllOre adaptive than first-order Generalized Pseudo Bayesian algorithm (GPBl)~ and as accurate as second-order Generalized Pseudo Bayesian algorithm (GPB2) while only

1

~

m

u

sir

(k) A P{ms(k) j m/(k + l),Zk} -

1

= C· K ts · us(k)

of the

m

computation of GPB2 is needed ( m is model number). l1\.flvf is also better than Full-Hypothesis Tree Algorithm (FlIT) \vhose computation increases

C t == ""' ~ Jr ts . u s (k) s::;l

Assumption two: The model-conditionaJ estimation is unbiased when it matches the real movement mode. That is

exponentially. As a result of input-interaction, tlle coupling exists between model-conditional estimations and modelconditional calculating covariance. So up to now, the petformance evaluation of Th11vf is still limited in Monte Carlo simulation (Li, 1993b). It is difficult to choose optimal paralncters and design ne",' adaptive filters.

E[1]s(k)lms(k+l)~Zk]=O

(4)

17s(k)~Xs(k'k)-X(k)

(5)

where Then we can get

~o(k' k) ~ E{[XtO(ktk) -X(k)]

-[X~ (k Ik) INTERACTING MULTIPLE MODEL ALGORITHM

2.

(3)

t

X(k)]T IZk ,mt(k + I)}

m

=

2:{ [Xy(kl k)-~(klk)] [Xs(k\k)-k;(k )k)f s::=l

(6)

Consider the state equation and measurement equation based on the i-th model. Xi(k+ 1) = F;(k, Xi (k)) + W;(k~Xj(k))· qi(k)

{

m

ui(k);=

H;(k:1 ~(k)) + ri(k~ Xi (k)) . r,(k) where Xi(k + 1) and ZJ(k) are the state and measurement vectors, respectively~ q,. (k) and

Step 2

r (k) are mutually uncorrelated white Gaussian noise vectors ,vith zero mean respectively~ F; C·) )

the

r

are assumed to be known. The initial state is assumed to be Gaussian and uncorrelated with the process and measurement noises. i

state

estimation

X

t

(k + 1 Ik + 1) ,

the

calculating

covariallce

~

(k + 1 Ik + 1),

the

innovation

Vt(k + I)?

Step 3

innovation

Alodel Probability Updating

=

The model probability updating equation is

I ut(k+l) ==-~At(k+l)·u~(k)

Assumption 1 can leads to (1).

C

m :=

1

(9)

C == LAJ(k+l).u~(k)

(1)

$=1

t

XtO(k Ik) ~ E[X{k) l me (k + 1), Zk] rn

s-=-l

the

The likelihood function based on the t-th model is At (k + 1) AT[Y; (k = 1) : O,St (k + 1)] (8) \vhere AT*l is density function with Guassian distribution.

Assumption one: The model space covers the moveluent mode space of the system. That is, the model set is of perfectness. And the matched model is of exclusiveness.

= LU

and

covariance St (k + 1) .

Input-Interaction

LP{ms(k)}

(7)

Filtering

(·)

IM1v1 can be divided into 4 steps as follo\vs.

Step 1

-us(k)

Based on the t-th model, the filtering can obtain

j

HlC·), and

ts

s=1

~.(k) ==

~(.),

LK

sIt

(k)~

X

(k Ik)

Step 4

Output-Interaction

(2)

S

Xs(k (k) ~ E[X(k) ImsCk),Zk]

3933

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PERFORMANCE ANALYSIS OF INTERACTING MULTIPLE MODEL...

P(k + 1) k

+ 1) == L~(k + 11 k + 1)· ut(k + 1)

The compression ratio of input-interaction error based on the mismatched model is denoted by

t

+

14th World Congress of IFAC

L 1]t(k + 1) "1}t(k + l)T · ut(k + 1) t

(11)

YS(j) 3~

EFFECT OF INPUT-INTERACTION ON MODEL-CONDITIONAL ESTIMATION

11~(kjk~1 IIMJ(kjk~1

Then 1t .. 1~

>-

1

m

or :Jr.. > H

Here the i-th model is supposed as the matched one in [le.. }, k]. Because model probability is calculated not only based on the current information but also based on the history information, the model \vith the maximum probability may not be the matched one. If there is no priori information, the following can be assumed.

={

. _

J~ 1-1, .. ~~m

(12)

l=j

2-u.(k) > 1 (abandoned) 2-m-ui (k) J

So

1

u. (k)

YS(i)=

1+

1

I-nu

Conclusion One:

Jr 11

rn, j 7:- i

(15)

Model match is defined as tIle consistency between the model and the valid mode of the system. And the lnatched luodel is one of the most consistent with the valid mode of the system. It means that the estimation based on the matched model is more accurate.

/;t=j

~

3. 1 Effect of Input-Interaction on the matched model-conditional estimation If

Obviously, estimation accuracy is not only dependent on modeling, but also related to inputinteraction. It is important to analyze the effect of input-interaction on model-conditional estimation. It is expected to find some rules or conditions to improve estimation accuracy by input-interaction.

l- P mp-l

1~ j

<1 •

'~lhen

Cm· 'IT...

1)

-

11

1t _. n

I > -m

~

the previous

matched model-conditional estimation after input... interaction will be still better than the previous overall estimation. From the compression ratio of input-interaction error based on the matched model, The follo~rjng is obvious. 1) YS(i) will decrease \,,'ith

model number.

2) YS( i) becomes smaller as

Jr i i

approaches 1.

where m is the model number.

3) The model probability indicates the relatively matched extent of the model. Thus in order to improve the matched model estimation by inputinteraction) ",re should choose the models "With large differences.

Because the Inodeling information of the matched model is the most accurate and effective, the estimation based on the matched model is the optimal estimation under these kno\vn modeling information. So the model-conditional estimations and the overall estimation with the matched modelconditional estimation can be compared as follows.

~(klk) = ~(klk) - ~.(kJk)

3~

(13)

2 Effect of Input-Interaction on tnismatched model-conditional estimation in IM1v1 \vith 2

models

The overall estilnating error is

M(klk)=X(klk)-~(klk)

(14)

3.2. 1

The compression ratio of input-interact.ion error based on the matched model may be denoted by

.

IIM~(k'k)11

Improvement of mismatched Inodel.. conditional estimation

If

YS(l)== 11.....

AX(klk)11

then

3934

Copyright 1999 IFAC

ISBN: 0 08 043248 4

PERFORMANCE ANALYSIS OF INTERACTING MULTIPLE MODEL...

1

YS(j)=

1-1r i i

1+---

n

ii

ui(k)

11~(l(lk~1~ IIAXlkjk~1

<1

Then from

uj(k)

uj(k) where history

u;(k -1)

u~(k -1)

before

and

Ilill/(k\k)11

. u1(k)

1- trjj u/k)

u.(k)

4·Jr ... uj(k)

u/(k)

I-tr jj

_+

IIM](k Ik)11

JJ

u,(k) (21)

11 a + b II~II a I1 + conditions are

From

I! b 11,

the

IIAX/k Ik)1I < 1 + u;(k) IIM](k Ik)11

Ai (k ) Aj(k)

sufficient

(22)

u,(k)

and

IIAX] (k Ik)11 < 1 + u, (k) IIAXj(k Ik)11 u/k)

represents the relative extent of match at time k. Thus the matched model should have great difference from mismatched models. This is consistent with Conclusion One. 3.3.2 3.2. 2

uj(k)

~-------":":-<1+-1

u;o(k -1) . A;(k) u~(k -1) i\.j(k)·

k,

<1+ uj(k) + 4·trjj

(20)

represents the effect of the

information

the necessary

and

the condition to improve the previous matched Jllodel-conditional estimation after inputinteraction. Thus the Markovian transmission matrix \\~in be a tradeoff result. =

11 a + b II~ III a 11-11 b Ill,

IIMj(klk)11 IIAX,(k Ik)1I

In order Lo minimize YS(j)., the parameters should be chosen as follows 1) To minimum tr i i. Bnt it is incompatible with

u;(k)

(19)

conditions are

Conclusion Two: in !MM with 2 models., the estimation based on the previous mismatched model will be improved after input-interaction.

2) To maximum

14th World Congress ofIFAC

Effect of Input-Interaction on the previous lnisruatched model-conditional estimation

(23)

To make the previous mismatched modelconditional estimations worse

If

If

IIAX~(k Ik)/I < IIAk(k I k)11

then

~ii

(17)

(24)

llAiJ (kl k ~I > IIM] (kj k ~I

(25)

and

1

< 0.5 = m

Conclusion Three: in !MM with 2 models, inputinteraction can cause one model-conditional estimation better than the overall one \vhile the other is improved but still inferior to the overall one.

Then from "a + b conditions are

II Ak .(klk)1I

II

1

IIs/l a " + 11 b )J,

the necessary

u(k)

ll>l+~

IltiX] (klk~1

3. 3 Effect of Input-Interaction on mismatched model-conditional estimation in INnvf with 3 models

3. 3. 1

1I~(klk~1>"AX/klk~1

(26)

u/ (k)

and

]AX/(klk)~ > 1+ u,.(k)

IIM/k/k)/I

ImproveUlent of nlismatched modelconditional estimation

(27)

u/k)

Thus (26) and (27) can lead to (28).

>1

If

_~_....--:ul:..........c(_k)_,_u""""",,--j(_k_)_~_ (18)

[ul(k) + uj(k)]· [uj(k) + uj(k)]

and

Obviously, (28)

~11

(28)

never come true.

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PERFORMANCE ANALYSIS OF INTERACTING MOLTIPLE MODEL...

Conclusion Four: in llv1M: with 3 models, it is impossible for input-interaction to make all previous mismatched lllodel-conditional estimations worse.

4.

14th World Congress ofIFAC

Obviously, input-interaction can adaptively forget some old probability information if 7r ii

1 m

~ -.

To further indicate how input-interaction lessens the difference among model probabilities, modelconditional likelihood functions are supposed as

EFFECT OF INPUT-INTERACTION ON MODEL PROBABILITY

fo110\\l8.

The model probability) as the weight of modelconditional estimation in the overall estimation, plays an important role in the overall estimation.

(38)

Then

(k) _ u~(k -1) uj(k) - u~(k-l)

1-7[" ~r • [l-uj(k)]

uNk) = ff i ( uj(k) +

IJ j

m~

(39)

(29)

JI .. >

If

lZ

Theorem 1: if (38) can be satisfied,. model probability can be expressed as follo\vs

1

- ~ then m u~(k) <

ff i(

ui(k) +

7r

it·

] ] 1 u?(k) =[l-m·1t F}..] k -r L ·[u-(O)-1 1 m +m

[1- uj(k)] (30)

u?(k) >

If

Jr

iiS

u? (k) ~

1 m

i

(31)

12

m-I

o .) 1] 1 u i (0) == [1- m · 7r iJ· [u,(O - m + m

(uf(k) +

u?(k)

s:

7T

if'

[1-

lli

(k)]

(32)

(33)

1] 1 u?(k+l) =[l-m-;rr .. ] k+2 ·[u(O)-m +m

1-7[ ..

I

n

m-I

1

j

(42)

l-ff

i

[min-{ m-I ., tr i i}7

i

In Multiple Model Algorithms, the history information is contained not only in modelconditional state estimations but also in model probabilities.

j

max{ m-I

1r

I;}] (34)

Obviously, input-interaction can limit the most and the least effect of the history information on model probability

The probability ofTh11vf based on the i-th model can be expressed as

?(k) _ (m·

1

7r ii -

1) . u i ( k )

+ 1-

K

If u·1 (k) 2:] u -(k) and n ii ~

1 -,

m

(36)

the reciprocal of model number. So too many models often make the matched model probability too small, v-'hich leads the overall estimation drops in precision. This is why the desirable model number is often not too large and 7r:i j (i +- j) is

then

uiO(k) < ui(k) u~(k) - uj(k)

often very small in Monte Carlo simulation. From above, I1vfl\.1 can be looked as decaying-memory filter in the sense of model probabilities.

1

If u.(k) < u.(k) and ;rH ~ -~ then ' J

u;(k) > ui(k) uJ(k) - uj(k)

Obviously,

(35)

j j

uJ(k) - (m-;r ji-l)ouj(k)+l-n u

dk) ·~(k-l)·A,(k).

input-interaction can lessen the difference of model probability. And this effect is greater as model nwnber increases or 7r j j (t:;t: j) approaches to

Besides this, (7) can leads to (35). U

J})

It is still compatible to (40). So (40) exists.

Thus the model probability interval after inputinteraction is as follows.

1-1Z"

(41)

It is compatible to (40). Suppose (40) is correct. Then (39) and (40) can leads to (42)

then

-:0

ff

Proof Obviously, (39) can leads to (41)

1-1l ..

1

(40)

m

The probability ofFHT based on the i-th model can

(37)

3936

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PERFORMANCE ANALYSIS OF INTERACTING MULTIPLE MODEL...

1 be expressed as - - . u.(O)· C(k)

of model probability but also to lessen the difference among model probabilities, In the sense of model probability, decaying-memory filtering, damping coefficient and regulating-time are defined. The

IT A.( i) . From it~ we

I.'

k

j~l

J

14th World Congress of IFAC

~

can find that the likelihood functions at any time in history makes the same effect on the current model probabi1ity~ and thus FlIT is full-memory filter.

conclusions and conditions do not depend on tracking scenarios~ All these works may be useful to choose optimal parameters and design new adaptive filters.

Denoted Damping Coefficient by

1 &A-=

m· lr Jj

REFERENCES

1 If 1r .. ~7rr' is positive~ then 0 < Jr ij< - - i -:F- ,/ . l)· I m-I

1

m , then 8

If 0 < 7! ' .< l)

Bar-..shalom, Y. and X.R. Li (1995). Estimation and Tracking: Principle and Techniques, MA: Artech House~ Boston. Basseville, M~ and I. Nikiforov (1993), Detection of Abrupt Changes: Theory and Application. Nj: Prenticc Hall, Engle"'ood Cliffs. Li, X.R. and Y. Bar--shalom (1993a). Design of an Interacting Multiple Model Algorithm fOT Air Traffic Control Tracking. IEEE Trans. on Control Systems Technology, 1, pp186-194.

>1 which means

system is in over-damping. If Jr

1

i

j~ - ,

then G

m = 1 which means system is in critical damping. If

1 < m

-

7r .. < 11

1 - - ,then 0 < /; < 1 which means

m-I

.

system is in nnder..damping.

Li, X.R. and Y. Bar..-shalom (1993b). Performance

If

"i'

E

J

1[1- m·

I

111 (0,< - ) V ( - , - - ) i m m m-I n"

Thus

if

-:t:-

Electronic System, 29, pp755-771. Li, X.R (1996). Hybrid Estimation Techniques. Control and Dynamic Systems (Lecondes, C.T. (ED.)), Academic Press Inc, ppl-75.

j, then

~~

'ft . [uj(O) -

:::; [l-m-7t

Prediction of the Interacting Multiple Models Algoritlun~ IEEE Trans. on Aerospace

C x 100%

Supposed model probability comes into error band after the time of k.

Mazof, E.

(1998).

Interacting Multiple Model

Methods in Target Tracking: A Survey. IEEE

1 I m-C 1 l k -[1- m1=

Trans. on Aerospace Electronic System, 34, ppl03-122. Q.~ Y. Liang H.C. Zhang and G.Z. Dai (1997). Interacting multiple model algorithm based on alpha--betaJalpha--beta--gamma filter. Proceeding of American Control Conference, pp3698-3701

Pan,

k== InC-ln[m-l] Inll- m· 1r ijl

The regulating period can be denoted as

In C - In[ m-I] . t Injl-m'1r j j l

(41) s

where t s is the sampling period and

1 m

10%. If Jr ij is equal to -

~

C is 5% or

there is no regulating

period.

5.

SUMMARY

A new non-simulation method for perfonnance analysis of Interacting Multiple Model Algorithm is proposed. Four conclusions are made qualitatively. The compression ratio of model-conditional error is defined. So the parameters and modeling can be chosen quantitatively. Besides this, Input-interaction is found not only to decide the upper and lo\ver liLnits

3937

Copyright 1999 IFAC

ISBN: 0 08 043248 4