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GLI I

N r N

r.!ju-q , t4.N

Fig. 2. New configuration with a feedback to "real world" system.

hand, it is suggested that the system can be fed back with weighted measurements for better estimation. The ideas presented here lead to model error compensation of hear filters which is dealt with elsewhere 131.

b R E N C E 3

[I] J. S. Lo* Wptimal linear filter and its relation to Singular optimal control" in Pra.

[2] A. GeIb, Ed., Applied optimal Esfimation. Cambridge, MA: MJ.T. Re$ 1974. [3] S. Vathsal, 'The modelling m r compensation of digital filters for the ali-ent of

[4] C T. Leondes and J. 0. Pearson, "A minimax filter for system with b e plant

I970 Joint Aufomal. Confr. C o d , session paper 14-4 pp. 313-319.

inertial platformrs" DFVLR-FB 79-19. 1979.

rmccrtain~" IEEE Trans. Automat. bnf r . , vol. AG17. pp. 266-268, Apr. 1972.

Asymptotic Convergence Properties of the Extended Balman Filter Using Filtered State Estimates

BJBRN URSIN

A b s ~ - I n a repent paper, Ljnng has given a convergence anal@ of theeKten&dKalmnnfilter@KF)asaparameterestimatorforlinear s y s t e m a ' I h e ~ i s d o n e f o r a v e r s i o n o f t h e ~ u s i n g p r e c t i e t e d values of the state vector. In this note a similar convergence analysis is done for the EKF using f i i e d values of the state vector. 'zbe amver- geore properties of the two algorithms are similar, but not identical. The reealculatioo of a simple exampk given by Ljung indicate that using the filtered estimate of the state vector gives improved rwwvergence properties of the algorithm.

I. INTRODUCTION

In a recent paper Ljung [ l ] gives a convergence analysis of the extended Kalman filter (EKF) as a parameter estimator for linear discrete-time systems. However, only the version of the EKF using predicted values of the state vector was considered. In practice, one often applies the EKF using filtered values of the state vector, thus utilizing an improved estimate of the state vector. In this note a similar convergence analysis for the filtering procedure will be given.

11. COM'ERGENCE h U Y S I S

In the following a similar notation as in Ljung [ 11 is used, The model is given by

x ( t + l ) = A x ( t ) + B u ( t ) + a ( t )

y ( t ) = C x ( t ) + e ( r ) (1)

where

Manuscript received December 19,1979; revised August 11,1980. The author is witb the Division of Petroleum Technology, SINTEF, Trondheim-NTH,

Nomy.

The matrices A, B, C, and JI, depend on a finitedimensional parameter vector 8.

The EKF approach to determine the unknown parameter vector 0 is obtained by extending the state vector x with the parameter vector 8=8(r). This gives

with the nonlinear model

where

We have to solve a nonlinear filtering problem. The EKF technique [2] gives the following equations:

E ( t ) = y ( t ) - h ( z ^ ( t , t - l ) )

i ( t , t ) = i ( t , t - 1 ) + i v 1 ( t ) e ( t )

i ( t + l , t ) = f ( z ^ ( t , t ) , u ( t ) ) + & s - ' ( t ) e ( t ) (6)

i(0, - I)=;, = [ lo] where z^(t, t - 1) denotes the predicted value of z ( t ) and z^(t, t ) denotes the filtered value. From (6) with p defined below, it is seen that

The filter gain and the covariance matrices are

ivI(t)=P(t)HT(r)s-'(r) (8)

p ( t + ~ ) = ~ ( r ) ~ ( t ) ~ ~ ( t ) + @ - ~ ( t ) . ~ ( t ) i v ~ ( t ) (9)

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1208

with

First, we note that we can write

which gives

With the block partitions

Note that (D), with the aid of (21) and (24), can be rewritten as

EM3 TRANSACTIONS ON AUTOMATIC CONTROG VOL. AC-3, NO. 6, DECEMBER 1980

P2(f+l)=[A(t)-A(I)G(t)C(t)-QcS-1(t)C(t)]P2(I)

(11) - [ A ( t ) G ( f ) D ( t ) . + Q ' S - ' ( r ) D ( r ) l P , ( r ) + M ( t ) P ~ ( t + l ) . (25)

Ljung's algorithm [l, eq. (3.14)-(3.20)] and the one used here (15)-(25) are very similar, but not identical. Ljung's (3.14) can be written

i(t+l,t)=A(I)i(t,f-1)+B(f)u(I)+K(t)c(t) (26)

with

K ( f ) = [ A ( t ) P l ( t ) C r ( ( t ) + ~ ( I ) P ~ ( f ) C T ( t )

( 12) +A(f)P2(t)DT(t)+M(f)P3(f)DT(t)+Qc]S-1(f) (27)

while our (16) and (18) can be written

i(t+l,f)=A(t)i(t,t-l)+B(f)~(t)+KI(f)c(t) (28)

(13) with

~l(r)=[A(t)Pl(r)CT(r)+A(t)P2(~)Dr(t)+Q~]S-1(f). (29)

Using (21), (27) and (29) give

K ( t ) = K , ( t ) + M ( t ) L ( t ) . (30)

We also note that M(r) in (1 1) is computed using i ( t , t ) , in contrast to Ljung's algorithm where i ( r , f - 1) is used

In the convergence analysis I? and P3 are kept constant. In [l] it is (14) shown that, when t+m, P l ( f ) , S ( t ) and K ( t ) will tend, respectively, to

the solutions of the equations

P , = A P , A ~ + Q ~ - K S K ~

(15) S= CPICT+ Qe (3 1)

K = [ A P l C T + Q c ] S - '

where all the variables except Q' and Qe are functions of I?. In a similar (16)

(I7) way it can be shown that G ( t ) tends to

G=PICTSs-' . (32)

From (25) we can now define the new F(I) process (the dependence (18)

on I? is not denoted explicitly) as

F(f+I)=(A-AGC-QcS-'C)F(f)+M(t)-(AG+QCS-')D(t). (20)

(21) (33)

Using (32) and (31) this can be written

w(r+l)-(A-KC)F(r)+M(f)-KD(t) (34)

which is (4.4) in [l] except that M ( t ) is defined slightly differently. All proofs and all results can now be reproduced with this slightly different process F(t). This constitutes a convergence analysis of the EKF based

(22) on the filtering procedure.

111. A S w m EXAMPLE

In [l] a global converge analysis is done for a very simple example consisting of a fiirst-order system with one unknown parameter (see [l, (5.4)-(5.11)]. We shall not repeat all the equations, but only comment on the differences-which arise by using the filtering approach. In (5.7) the

(23) equations for <( I ; a ) and q r ; a ) have to be replaced by

2( I+1 , t+ l ;a )= (a -E(a ) )2 ( r , f ;a )+C(a )y ( I+1 ) -

(24) ~(f+l;a)=(a--K(a))W(t;a)+i(t,l;a) (35)

~(t,a)=[l--q-'(a-K(a)~l-2~(o)q-'y(t) (36)

where ~ ? ( u ) = ~ ( u ) / a . We obtain

TRANSACTIONS ON AUTOMATIC CONTROL, VOL AC-25, NO. 6, D-BR 1980 1209

o. ml

C L C

-0. rml I -0.1

AO=O LAMBOA=I no a1

Fig. 1. Thcfunctionsj,(a)and~f(a)for~,~O,X-1,andh-10.

which gives

where

and

Assuming that la,l<l and If l<l , the unit circle may be used as an integration path in (37).

Using the predicted value of the state vector gives

The functions &(a) and fF(a) given in (40) and (37) are plotted in Figs. 1-5 for a, -0, 0.2, 0.4, 0.6, and 0.8 and for A = 1 and A = 10. The function $,(a) is plotted with a broken line, and for a, -40.2 and 0.4 it is also plotted separately with a different scaling to display its behavior for small functional values. From the figures we can draw the following conclusions about the asymptotic behavior of the two algorithms (for this particular example and 0 6 a. < 1).

1) The filtering algorithm always converges to a value io close to the correct value.

2) The prediction algorithm converges to a value io close to the correct value only if the initial est i t~te io(0) is positive. For &(O) < 0 it may converge to a value io with the wrong sign and incorrect absolute value. This algorithm may even converge to a value of io, which corresponds to an unstable system (io < - 1).

3) The filtering algorithm has a much faster rate of convergence than the prediction algorithm since IfF(a)l>l/,(a)l and Idf,(a)/da)> Idf,(a)/daI close to convergence. For smal l values of a,, the prediction algorithm is expected to converge very slowly (see Figs. 1 and 2).

4) For an incorrect value of the noise covariance (A= 10) and for Ci,(O)>O the prediction algorithm converges to a value io, which is closer to the correct value than the value to which the filtering algorithm converges. (That is, the prediction algorithm converges to a less biased estimate than the filtering algorithm, provided io@) > 0.)

Similar results have been derived by Westerlund and Tyssei [3]. Their comment does not contain a convergence analysis of the filtering algo- rithm. Their result for the functionfF(a) has to be divided by 1 -c(u) (1 - 2 ( u ) in their notation) in order to obtain the correct result.

IV. CQNCLUSION

A convergence analysis of the extended Kalman filter using filtered values of the state vector has been given. For a simple example it has been shown that the use of filtered values instead of predicted values of the state vector yields an algorithm with improved convergence proper- ties

ACKNOWLEDGMENT

The author wishes to thank H. Mordt, Division of Technical Cybernetics, The Norwegian Institute of Technology, for writing the computer programs and for producing the figures.

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IBEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. AC-25, NO. 6, DECEMBER 1980

a m I I I U S U t -a 2 no a2

AO=O. 2 LAMBOA=iO

Fig. Z The functionsf,(u) andfr(u) for uo -0.2, X== 1, and X- 10.

1 I

a a - n d no a4 AO=O. 4 LAMBDA=lO

0 . 2

ao

-0.2

-0. 4

-0.6

I -a 1

hO=O. 4 LAMBOh=IO 0 . 0 a4

Fig. 3. The Functionsfp(u) andfdu) forao -0.4, A= 1, and X- 10.

no

4.0

ao

-4. 0

-a o

Fig. 4. The functionsfp(u) andfF(u) for uo -0.6, X- 1, and X- 10.

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linear timeinvariant multivariable system has been solved in [l] and [2] using geometric concepts. This paper attempts to interpret the geometric solvability conditions of [I] and [2] in transfer function terms since the original problem specification is often in these terms.

The system considered is

i ( t ) = A x ( t ) + B u ( t ) + D E ( t ) (1)

y ( t ) = C ' x ( t ) t > O x(0)=xo (2)

where x E R n , y E R m , uER', and .$€Ktr are the state, output, control, and disturbance, respectively. Considering the controls

u=Fux, Fa EWrXn ( 3 4

u=Fbx+G[, Fb ER"", G€RrX' (3b)

and the corresponding disturbance transfer functions

H,(s )L C'(sZ-A-BFJ'D

L C ~ ~ Z - A -BF,)-'(D+BG)

the results of [l] and [2] state that 1) there exists F, such that H,(s)=O if and only if

REFERENCES

2) there exists &, G such that Hb(s)dJ if and only if

DCB+V*. (B)

1970. T- Wcsteriund and A. T W , "RemarLs on 'Asymptotic behavim of the extended

Norwegian Inst TecbnoL. Trondheim, Norway, 1979.

To interpret (A) and (B) in frequency domain terms, define the ~alman ~ t c r as a parameter estimator for timar systems: 9. &. open-loop control and disturbance transfer functions, respectively,

Frequency Domain Conditions for Disturbance Rejection

S. P. BHA'ITACHARWA

I. INTRODUC~ON

Tbe problem of zeroing the disturbance transfer function using state feedback or state feedback and disturbance feedfornard control in a

G(s) C'(sZ-A)-'B

H ( s ) C'(sz-A)-'D.

In the sequel, conditions on G(s) and H(s) are given that are necessarily satisfied when (A) and (B) hold. These neceSSary conditions are shown to be sufficient for (A) and (B) to hold in singleoutput systems and in multivariate systems satisfying a mild restriction- Further, it is shown that when r > m, these necessary conditions are generically sufficient for (A) and (B) to hold, and that when r<m, (A) and (B) fail to hold generically.

n. SPIGLE-OUTPUT SYSTEMS

In this section, we treat single-output systems with m= 1 and C = c ' a row vector. The proofs of the results amount to simple calculations and are omitted.

Lemma I : Let m = l , C'=c' in (2). If G(s)=O, (4)

then ( A ) or (3) holds ifund on& 9 H( s) EO.

Suppose now that G(s)zO. Let

BLminj:c'AjB#o, j€[O,l,*.-,n-l]. (64

0018-9286/80/1200-1211%00.75 01980 IEEE