I. INTRODUCTION

Optimal FIR Filters for Time-Varying State-Space Models

WOOK HYUN KWON

KYU SEUNG LEE Seoul National University

0. K. KWON Inha University Korea

An optimal finite-impulse response (FIR) filter and an optimal

FIR smoother are introduced for tinie-varying state-space models.

The suggested filter not only possesses an FIR structure but also

utilizes a finite observation It is shown that the impulse response

of the optimal FIR filter can be obtained by a simple Riccati-type

matrix differential equation. Especially for time-invariant system,

this FIR filter is reduced to the previously known simple forms.

For implementation, a recursive form of the optimal FIR filter and

smoother is derived by using adjoint variables, and coniputational

algorithns are suggested. It is also shown by sensitivity analysis

that the proposed optimal FIR filter alleviates potential divergence

characteristics of the standard Kalman filter.

Manuscript received December 20, 1989; revised March 23, 1990.

IEEE Log No. 39257.

Authors’ addresses: W. H. Kwon and K. S . Lee, Department of Control and Instrumentation Engineering and Automation and Systems Research Institute, Seoul National university, San 56-1 Silim-dong Kwanak-gu, Seoul 151-742, Korea; 0. H. Kwon, Dep’t. of Electrical Engineering, Inha University, Inchon 402-751, Korea.

Kalman filters have been extensively used in such applications as the tracking of missiles or planes and the orbit determination of spacecraft. But in designing a Kalman filter, two important problems arise. One of the problems is that very often a precise knowledge of the a priori statistics of the noise models as well as system models is unknown. This makes Kalman filters possess poor performance or divergence phenomenon 131. Another is the computation problem such as roundoff errors or coefficient quantization errors which may cause the Kalman filter to diverge when it is implemented using a digital computer with finite word length [4].

of the standard Kalman filters, Jazwinski [l] and Schweppe [2] introduces the limited memory filters for discrete-time state-space models. The limited memory filters in [l, 21 are obtained with the finite observation data by the maximum likelihood criterion. Bruckstein and Kailath [5] derived a sliding window filter for both the continuous-time and discrete-time state-space models on similar conditions but with the minimum variance criterion. But all these filters are obtained with assumption of finite observation and have recursive forms. They only guessed that the limited memory filter will prevent divergence due to finite observation when there exist modeling errors. But this claim has not been rigorously analyzed.

To overcome the numerical problems, alternate recursive algorithms of Kalman filters such as square root and factorization algorithms have been developed. These filters are known to exhibit improved numerical characteristics, particularly in ill-conditioned problems associated with the solution of the Riccati equation [6]. But even if Kalman gains are precomputed on a high-precision computer so that the gains are accurate, the finite-precision implementation degrades the filter performance [4].

optimal FIR filter for the time-invariant state-space models. This optimal FIR filter also uses finite observation and this prevents divergence. In addition, due to FIR structure it is believed that this optimal FIR filter has the built-in BIB0 stability and is effective on the numerical problems such as coefficient quantization errors and roundoff errors [7]. But the optimal FIR filter given in [14] is applied to time-invariant systems and no analysis has been done on the divergence problem. Since time-varying state-space models are quite often used in practical applications in the aerospace, the optimal FIR filter for time-varying systems is derived here. Also, we analyze the effects of the modeling errors on the performance and show that the suggested optimal FIR filters alleviate potential divergence characteristics of the standard Kalman filter.

In order to overcome the divergence characteristics

For another approach, Kwons [14] introduced the

IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 26, NO. 6 NOVEMBER 1990 101 1

For general time-varying systems, the finite impulse implies that the impulse response H(t , ,; T ) satisfies response H(t , r ;T ) of the optimal FIR filter has the property H(t,r;Tj = o for G < t - T . n u s a linear H(t , s ;T ) = P(t,s)Cl- H ( t , r ; T ) C T P ( ~ , s ) C ; d r , filter 2( t ;T) with the FIR structure in the state-space model can be represented by

(1) where

It is noted that the FIR filter (1) utilizes finite observation data z[t - T,t] . Here, the optimal FIR filter is obtained with minimum variance criterion while satisfying the above FIR constraint (1) and is called the optimal FIR filter.

In Section 11, the optimal FIR filter for the general linear time-varying systems is discussed and a Riccati-type matrix differential equation is obtained for the computation of the filter impulse response function. It is shown that the optimal FIR filter is reduced to a simple form for time-invariant systems. In Section 111, the optimal FIR smoother is introduced. In Section IV, the recursive forms of the optimal FIR filter and smoother are derived for implementational purpose. In Section V, sensitivity analysis is evaluated. The simulation result is given in Section VI and the conclusion in Section VII.

I I . OPTIMAL FIR FILTER

Let us consider the continuous time-varying state-space model

(2) X(t ) = A,x(t) + B,w(t)

z ( t ) = Ctx(t> + V(t) (3)

where the state x(.) and the observation z( . ) are zero-mean stochastic processes. Here the initial time to is zero, the initial state x ( 0 ) is a zero-mean random variable, the system noise w(.) and the observation noise v(.) are zero-mean white, and

E[x(O)x’(O)] = Po;

E[v(t)v’(s)] = Z6(t - s )

E [ W ( t ) V ’ ( S ) ] = E[w(t)x’(O)] = E[v(t)x’(O)] = 0

E[w(c)w’(s)] = Qt6(t - s);

for all t and s. Now we determine the optimal FIR filter P(.;T) for the state x ( . )

t - T < s < t ( 5 )

P(t,s) = E[x(t)x’(s)]

with a(.,.) being the state transition matrix of A, and

-P(t , t ) d = A,P( t , t ) + P(t , t )A: d t

+ &QtB:, P(0,O) = PO. (7)

For fixed t and T , (5) is a Fredholm integral equation of the 2nd kind and there are direct methods for this equation [9], whose computational burden is very large. These direct methods require operations of the order N 3 at each time t , where N is the number of nodal points in its discretized computation. For the Fredholm integral equations with certain kernel, Schumitzky [lo], Baggeroer [l l] , and Kailath [12] proposed easier methods in which they can be solved by the Riccati-type differential equations. We show that due to the special FIR structure of (4), the solution of (5) can be given by a simpler form than those of the above methods.

THEOREM 1 For the state-space model (2), (3) the impulse response H(t, .; T ) of the optimal FIR filter equation (4) is givne by H( t , s ;T ) = H ( ~ , s ; o ) ~ ~ - T , where

a - H ( f , S ; U ) a U = [Ar -T+o - R(t,U)C;-~+ucr-T+o]

x H(t ,s;u) , 0 5 T - t + s < U 5 T

(sa)

H(t ,s;T - t + s) = R(t,T - t + s)C:

a aU -R(t,u) = Ar-T+oR(t ,U) + R ( ~ , ~ ) A : - T + ~

B ~ - T + ~ Q ~ - T + ~ B ~ - T + ~ - R(t,u)C/-T+o

x C r - ~ + o R ( t , ~ ) , 0 < U i T (sa)

R(t,O) = P(t - T , t - T ) . (9b)

PROOF. interval T - t + s < o 5 T as follows:

Let us define H(t , s; .) and R(t , .) on the i ( t ; T ) = 11, H ( t , T ; T ) z ( T ) d T (4)

with the minimum variance criterion

J = E [ x ( t ) - 2(t;T)]’[x(t) - .?(t;T)]. H ( ~ , s ; ( T ) = P( t - T + CT,S)C~

The orthogonal property of the optimal filter i ( . ;T ) , i.e., [ ~ ( t ) - 2(t;T)] I z(s) for t - T 5 s 5 t

1012 IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 26, NO. 6 NOVEMBER 1990

R(t,g) = P(t - T + U,C - T + U )

Then we have H(t , s ;T) = H(t , s ;a )J ,=~ and the boundary conditions (8b) and (9b). Differentiating (10) with respect to U , we can obtain (8a), i.e.,

a - H ( t , s ; ~ ) = [ A t - ~ + o - H ( t , t - T + ( T ; u ) C ~ - T + ~ ] a0

x P ( t - T + G , s ) c ;

a I - T + a --H(t,7; o)C,P(r,s)C,' dT -L, ao

= [At-T+o - R(t,u)C:_T+uC,-T+o]-H(t,s;u)

where the second equality comes from the well-known property of Fredholm integral equations. Differentiating (11) and using (sa), we can obtain (sa). Hence the Proof is completed.

The computation algorithms for the impulse response H(t , .; T ) corresponding to Theorem 1 are as follows.

1) At the same point t , compute R(t ,a) on 0 5 U 5 T from (9).

2) For each s, t - T 5 s 5 t , compute H(t , s ;T) from (8) according to the procedure as shown in Fig. 1.

If the impulse response H(t , . ;T) is derived by Schumitzky's method [lo], its computation will be more complicated than that of Theorem 1. If N is the number of nodal points in discretized computation, the required number of operations in the above algorithm is on the order of N 2 for each t and T . This requires less computation compared with the order N 3 of the direct method in [9] for Fredholm integral equations.

Note that R(t,T) in (11) is the estimation error covariance at time t , as shown by the calculation

E [ x ( t ) - a ( t ; T ) ] [ ~ ( t ) - f ( t ;T ) ] '

= E [ x ( t ) - J' H(t , r ;T)z (r )dr]x ' ( t ) I - T

r , = P(t,t) - H(t,T;T)C,P(T,t)dT

L T

= R(t,T).

The optimal FIR filter i ( t ; T ) in Theorem 1 discards all the measurement information before time t - T but retains the previous statistical information. It would, however, be interesting to assume that we also discard the statistical information before time t - T . In this case the state x ( t - T ) , which is the initial state on the interval [t - T, t ] , can be assumed to be completely unknown and then P(t - T,t - T ) in (9b) is very large. T h i s special case of P(t - T, t - T ) = coI is

--

Fig. 1. Computation procedure for H ( t , s ; T ) .

given in the next theorem, whose result turns out to be practically useful.

THEOREM 2 Suppose that P(t - T,t - T ) = col, the pair [A,,C,] is uniformly completely observable, and B, and Qt are uniformly bounded. Then the impulse response H(t, .;T) of the optimal FIR filter (4) is given bY

W ( t , s ; T ) = S- ' ( t ,T)L(t ,s;T) , T 2 60 (12)

a - a,L(t ,s;o) = - [A:-*+o + S ( I , U ) B , - T + o

x Q, - T + ~ B:-T+~ ]L(t , S ; a) ( 1 3 )

L ( t , s ; T - t + s) = C:, O<T - t + s < U 5 T (13b) a

a U -S( t ,a ) = -S(t ,a)A,-T+u - A:-T+J(t ,a)

+ C , ' - T + ~ C ~ - T + ~ - S ( t , a ) B r - ~ + u Q r - ~ + o

x B;-T+,S(t,a), 0 < U 5 T (14)

S( t ,O) = 0

where 60 is the uniform complete observability index. This result is derived from the relations S(t ,o) :=

R-l(t ,o) and L(t ,s;a) := S(t,a)H(t,s;cr) The uniform complete observability of [A,, Ct ] and the uniform boundedness of B, and Q, guarantee the existence of the positive definite matrix S( t ,a ) for 60 5 U 5 T, which is the dual property of the moving horizon controller [13]. The Proof is omitted for brevity.

It is noted that the filter in Theorem 2 can be considered as suboptimal to that in Theorem 1 from the viewpoint of the criterion given in Theorem 1. However, the algorithm of Theorem 2 is more efficient in computation than that of Theorem 1. Especially for time-invariant systems, the former becomes a very simple form as is shown later.

For general linear time-varying systems, the impulse response H ( t , .; T ) should be recomputed on [t - T,t] at each time t and hence its computation burden is large. For time-invariant systems, however, the algorithms in Theorem 1 and Theorem 2 can be

KWON, ET AL.: OPTIMAL FIR FILTERS FOR TIME-VARYING STATE-SPACE MODELS 1013

reduced to very simple forms and the computation on only one interval [O,T] is only required once. Although time-varying systems are often used in many areas such as the tracking and guidance in the aerospace industry, time-invariant systems are also used due to simplicity.

Let us consider the time-invariant state-space model (2), (3) with constant matrices of A,, Bt , C, and Q,. It is well known that the stable time-invariant system becomes stationary, i.e., P(t ,s) = P(t - s ) in the steady state case (to = -CO). Theorem 1 is written in very simple form for this stationary time-invariant system. The optimal FIR filter (4) can be shown to be time invariant as follows:

= iT H ( T ; T)z( t - T ) d r (15)

It is noted that in (17), S(a) > 0 for any a > 0 and Q 2 0 with the complete observable pair [A,C] and the filter impulse response does not vanish even for the zero input noise (Q = 0). The above results given in equations (15)-(17) coincide with those of [14].

Ill. OPTIMAL FIR SMOOTHER

The optimal FIR smoother for the time-varying system (2), (3) is also derived from the similar procedure to that of the optimal FIR filter. Therefore we give a brief summary of the results without proof.

We define an optimal FIR smoother on the interval [t - T, t ] by

i , ( t - T ; T ) = H,(t,r;T)Z(r)dT (18) L T

with the minimum variance criterion

where the impulse response H(.;T) is obtained by H(t ;T ) = H(t;(T)I,=T and

J, = E [ x ( t - T ) - 2,(t - T;T)]’[x(t - T ) - 2,(t - T ; T ) ] .

The orthogonal property implies that H,(t, .; T ) satisfies a the integral equation aU --H(t;a) = [A - R(a)C’C]H(t,a),

H,(t,s;T) = P( t - T,s)Ci - H,(t,r;T) 0 5 T - t < cr 5 T (16a) 1 : T

H(t,T - t ) = R(T - t )@ x C, P ( r , s ) C ~ d r , t - T 5 s 5 t . d

da -R(a) = AR(a) + R(a)A’ + BQB’

- R(a)C’CR(a), 0 < (T 5 T

(16b) R(0) = P(0).

In (16b), the covariance P(0) = E[x(t)x’(t)] satisfies the algebraic Lyapunov equation AP(0) + P(0)A’ + BQB’ = 0.

1 holds only for the stationary system. But we can show that, even for the nonstationary system, the optimal FIR filter is time invariant for the special case of P(t - T,t - T ) = col. Therefore the assumption of steady state, i.e., to = -a, can be avoided in this case. That is, from Theorem 2 we have the time-invariant optimal FIR filter (15) whose impulse response H(.; T ) is given by

Note that the time-invariant filter (15) by Theorem

H(t;T) = S-’(T)L(t; T ) (17a) a G L ( t ; a ) = -[A’ + S(a)BQB’]L(t;c7),

0 5 T - t < c7 5 T (17b) L(t;T - t ) = C‘

d du -S(U) = -S(a)A - A’S(a) + C’C - S(U)BQB’S(U),

(19)

Then the impulse response H,(t,.;T) of the optimal FIR smoother is obtained as follows:

Hs(l,s;T) = K ( t r S ; g ) J u = T (20) a all - - N s ( t , s ; o ) = -W(~,~)C:-~+oC,-T+u~(f,S;ff),

os T - r + S < (T 5 T (21a)

H,(t ,s;T-t + s ) = W ( t , T - r +S)C,’ (21b) a

--W(t,(T) = W ( t , u ) [ A , - ~ + u - R(f,ff)C:-~+~Cr--~+ol’, aa o < ~ < T (22a)

W(t,O) = P(t - T , t - T )

where H(t , s; a ) and R(t ,a) are computed by Theorem 1. This result comes from the definitions

(22b)

r-T+a

H, (2, r; g) 1 - T

H,(t,s;a) := P(t - T,s)Ci -

x C, P ( r , s)C: d r , t - T 5 s 5 t - T + a

(23)

and

W(t,a) := P(t - T,t - T + a ) -

0 < U 5 T (17c) x C, P(r,t - T + a ) d r , 0 5 a 5 T

S(0) = 0. (24)

1014 IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 26, NO. 6 NOVEMBER 1990

It is noted that as the optimal FIR filter is to the standard Kalman filter, so the optimal FIR smoother is to the well-known fixed-lag smoother. The optimal FIR smoother (18) has the BIB0 stability due to the FIR structure, though the stability is not generally guaranteed in recursive fixed-lag smoother [SI.

response H,(.,.;T) of the optimal FIR smoother (18) is given under the same conditions as those of the optimal FIR filter as follows:

In the case of P(t - T,t - T ) = col, the impulse

H,(t, s;T) = n(t,T)S-'(t,T)D(t, s;T),

t - T < ~ < t , TL6o (Z) a -D(t,s;a) = -[A,-T+o - S-'(t,fl)C:-T+a

BU

x c~-T+u]'D(t, s; fl)

- C:-T+UCI-T+US-*(~~~)L(~,S;(J), 0 5 T - t + s < fl 5 T (26a)

D(t,s;T - t + s) = C: (26b)

a - , , f l ( t ,~) = - f l ( t , u ) [ A - ~ + ~ + & - T + ~ Q , - T + ~

B:-T+uS(t, (27a)

f l ( t ,O) = z (27b)

where 60 is the observability index and L(t , s; a) and S(t,n) are computed by Theorem 2. This result directly comes from the definition R(t,a) := W(t,a)S( t ,a) and D(t,s;a) := W-'(t,cr)H,(t,s;a) using the relations (S), (9) and (12)-(14).

IV. RECURSIVE FORMS OF OPTIMAL FIR FILTER

One of the disadvantages of the optimal FIR filter and smoother given in the previous sections is the inefficiency in computation since they are of nonrecursive forms. In order to overcome this problem, we introduce recursive forms of these estimators.

THEOREM 3 For the rime-varying state-space model (2), (3) the optimal FIR filter can be represented by the following recursion for t > T,

1 5 [ f(t;T) ] = [ M ( t ; T ) f f ( t , t -T ;T)CI -T

dt n,(t - T ; T ) -H,(t,r;T)C, M2(r;T)

H ( t , t ; T ) -H(t,t - T ; T ) ] [ z ( t ) ] + [ H,(t,t;T) -H,(t,t - T ; T ) z(t - T )

(28)

PROOF. Differentiating (4) for t > T , we have

+ H(t , t ;T)z( t ) - H ( t , t - T;T)z ( t - T ) .

(29)

We can see from ( 5 ) a at -H( t , s ;T) = [A, - H(t , t ;T)C,]H(t ,s;T)

+ H(t,t - T;T)Ct-TH,(t,s;T). (30)

Combining (29) and (30) yields the upper half of (28). Differentiating (18) gives

f a d -i?,(t - T ; T ) = -Hs( t ,T ;T)z (T)dT dt r - T a t

+ H,(t,t;T)z(t)- H,(t,t - T;T)z( t - T ) .

(31)

Similarly as in (30) H,(t, .;T) satisfies a %H,(t,s;T) = [A,-T + H , ( f , t - T;T)C,-r

Br-&-TB:-TP-l(f - T,t - T)]f&(r,S;T)

- H, (t , f ; T)C, H ( t , s; T ) . (32)

Substituting (32) into (31), we have the lower half of (28). Hence the Proof is completed.

The initial conditions for (28) are given by

i ( T ; T ) = H(T,r;T)z(T)dT (33a) 1' iT i,(O;T) = H,(T,r;T)z(r)dr (33b)

where H(T, . ;T) is computed by Theorem 1 and H,(T, .; T ) by Theorem 3.2.

THEOREM 4 The boundary values of H( t , s ;T ) and H,(t,s;T) at s = t and s = t - T in Theorem 3 are obtained as follows:

H( t , t ;T ) = R(t,T)Ci (34a)

(34b)

H,(t,t;T) = W(t,T)C: (34c)

H(t , - T ; T ) = W'(t,T)C:-T

H,( t , t -T ;T) = - W(t,fl)Ci-~+~Cr-~+aW'(t,fl)d6 1' x C:-T + P ( t - T , t - T)C;-'. (34d)

KWON, ET AL.: OPTIMAL FIR FILTERS FOR TIME-VARYING STATE-SPACE MODELS 1015

PROOF. obtained from (Sb) and (21b), respectively. From (22a) we have

Equations (34a) and (34c) are directly

where $J(.,.) is the transition matrix of [ A t - ~ + o -

gives R(t ,U)Ci_~+~ct-T+g] for fixed t and T . Now (sa)

H(t,t - T ; u ) = $(O,O)H(t,t - T;O)

= W'(t,u)c;-, (35)

which implies (34b). From (24a) and (22b),

which is equivalent to (34) . This completes the Proof.

Gain matrices for the time-varying systems require repeated computation at each time point. However, these gain matrices can also be computed recursively as shown in the next Theorem.

THEOREM 5 H,(t,s;T) at s = t and s = t - T in (28) can be obtained as follows:

The boundary value of H(t ,s;T) and

H( t , t ;T) = R(t,T)C: (37a)

(37b)

H,(t,t;T) = W(t,T)C: (37c)

(37d)

H(t,t - T ; T ) = Wf(t,T)Ci-T

Hs(t,t - T ; T ) = F(t,T)Ci-,

where R(t,T), W( t ,T ) and F(t ,T) sati f i the following relations at t > T:

x P-'(t - T,t - T)]W(t ,T)

- w(t,T)C,'CtR(t,T) + F(t,T)C,LTC,-TW(r,T)

(39)

The above results in Theorem 5 are derived from the following definitions:

R(t,T) := P( t , t ) - H(f,T;T)C,P(T,f)dT (41a) 6,

F ( t , T ) := P ( t - T,t - T )

The initial values of R(t ,T) , W( t ,T ) and F(t ,T) at t = T are computed by (41) using H(T,s ;T) and H,(T,s;T) for 0 5 s 5 T . The Proof is omitted for brevity.

The usefulness of Theorem 3 and Theorem 4 can be demonstrated by the corresponding results for time-invariant systems which also appear in practical applications. For the stationary time-invariant systems, the recursive form of the optimal FIR filter P(.;T) can be obtained directly from Theorems 3 and 4 as follows:

I A - H(0 T)C H(T;T)C

-&(O;T)C A + H,(T;T)C + BQB'P-'(0)

where H(O;T) = R(T)C', H ( T ; T ) = W'(T)C', H,(O;T) = W(T)C' ,

H,(T;T) = - W(u)C'CW(a) doc' + P(O)C', I' (43)

and R(.) is computed from equation (16b) and W ( . ) satisfies

d du -W(u) = W ( a ) [ A - R(a)C'C]', W(0) = P(0).

(44) It is noted that the gain matrices in (42) are

constant and determined on the only one interval [O,T]. It is also noted that similar results can be obtained for the special case of P(t - T,t - T ) = wZ.

V. SENSITIVITY ANALYSIS

It has been known that the standard Kalman filter possesses divergence phenomenon due to modeling errors. Instead, the limited memory filter is believed to overcome divergence characteristics, which has not been rigorously proved to the authors' knowledge.

1016

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We show that the optimal FIR filter introduced here has better characteristics in divergence problems. To evaluate the effects of modeling errors, an expression for the covariance matrix of the estimation error due to the suboptimal gain H ( t , .; T ) which is caused by modeling errors is obtained.

from H(t, .; T ) , i.e., Let the resulting state estimate %(t;T) be obtained

%(t;T) = l :TH( t , r ;T ) z ( r )dr (45)

where H(t, .; T ) corresponds to the incorrect model with &, Bf, C t , Qf and initial state covariance Po and can be written by

H(t ,r;T) = H(t , r ;T ) + o H ( t , r ; T ) ,

t - T l r S t . (46)

Then the estimation difference DL(t; T ) between the optimal filter and the suboptimal filter is represented bY

VR( t ;T ) := %(t;T) - i ( t ; T ) r t

and the real estimation error covariance defined by

R,(t,T) := E[x( t ) - %(t;T)][x( t ) - %(t;T)]’ (48)

can be given by Lemma 1.

LEMMA 1 If the suboptimal filter gain H(t, .; T ) is used, the real estimation error covariance R,(t,T) is given by

R,(t,T) = R(t ,T) + U(t ,T ) (49)

where

PROOF. By definition, R,(t,T) becomes

R,(t,T) = E[x( t ) - i ( t ; T ) - o i ( t ; T ) ]

x [x( t ) - i ( t ; T ) - Vi( t ;T)] ’

= R(t ,T) - EVa( t ;T)[x( t ) - a(t;T)]’

- E[x( t ) - i ( t ; T ) ] o i ’ ( t ; T )

+ EVP(t; T)Vi ’ ( t ; T ) . (52)

Substituting (5) into (52), we can obtain (49)-(51). This completes the Proof.

The suboptimal gain of (45) is computed by

dH(t, 3; u ) / ~ u = [ A r - ~ + u - R(t, g ) c ; - ~ + ~ C t -T+U I

x H(r,s;o) 05 T - t + s < U < T (53a)

(53b) H ( f , s ; T - f + s ) = R(f ,T - t +S)C:

aR(t,u)/aU = Ar-T+uR(l,b) + R(t,a)A:-T+u

+ B, - T + a - + B:-

- R(t,U)C:_T+~Cr-T+oR(f ,U),

O < U < T (54a)

R(f , 0) = P(t - T , t - T ) (54b)

@a)

P(0,O) = Po. (55b)

d P ( t , t ) / d f = & P ( t , t ) + P(t,t)Ai + BfQ,B:

In Theorem 6 we obtain the estimation error covariance for the given modeling errors.

THEOREM 6 If the incorrect model is represented with At, Bt, C f , Q f , and Po, the real estimation error covariance R,(t,T) can be obtained as follows:

R,(t,T) = R(t ,T) + U(t ,T) (56)

where a ~ u ( t , g ) = [ A t - ~ + o - K ( t l ~ ) C , - ~ + o l u ( f , o )

+ u( t ,o) [Ar-T+u - K(t,~‘)Ct-~+nl’

+ VK ( f , a )VK ’ (f , U)

+ [VA,-T+u - K ( t , a ) v c , - , + u l ~ ’ ( ~ , ~ )

+ T(t,n)[VAf-T+a - K(t,U)

x VC,-T+,]’, o < U T (57a)

U(t,O) = 0 (57b)

a %T(‘,Q) = [ A , - T + ~ K(t ,o )C , -~+u lT( t , g )

+ T(t,o)A:-T+o + V K ( ~ , o ) C f - T + o R ( ~ P )

+ [ v A , - ~ + o - K(r,o)VCr-~+,l

x [P(r - T + o,T - I +U) - R(r,o)]

O < U L T (58a)

T(t,O) = 0 (58b)

and

K( t ,o) = R(w)C:-*+a;

K(t , 0) = R(t , c)C: - T+,, ;

V K ( t , o ) = K ( t , o ) - K ( t , o ) .

R(t,a), R(t ,a) sari@ (9) and (54), respective@.

PROOF. Let us define U(t ,a) on the interval 0 5 (T 5

KWON, ET AL.: OPTIMAL FIR FTLTERS FOR TIME-VARYING STATE-SPACE MODELS 1017

T as the solution of the equation t-T+a I -T+a

V H ( t , 71; a)Rz (71,r2) 1 - T 1 - T

U( t ,u ) :=

x VH’(t , r2; a ) dr1 dr2. (59) Then it is clear that U(t ,T) = U(t,a)l ,=~. Differentiating (59) with respect to a , we obtain

a a a aU dU

l-T (ma) - ,,U(t,a) = -U1(t,a) + -Ui(t,a)

f - T + a VH( t , t - T + a;a)

a -U1(t,(T) = aU

x R,(t - T + a , r ) V H ’ ( t , r ; ~ ) d ~

By substituting (61a) into (64) and using the definition fo R(t,a), (5%) can be obtained. The initial conditions (57b), (58b) come from the definitions of U(t ,u) and T ( t , a). This completes the Proof.

From Theorem 6, the error divergence condition can be derived. When the filter diverges, some of the elements of U(t ,T ) increase without limit while the solution R(t, T ) has a bounded value for the observable system. Now we discuss two separate cases, depending on whether V A I - T + ~ - K ( ~ , u ) V C ~ - T + , is non-zero.

Case 1. V A , - T + ~ = V C I - ~ + a = 0. (No modeling errors exist in system dynamics).

Since VAr-T+a - K(t,a)VCr-T+o is zero, U(t ,T ) is independent of the real system state covariance P(t, t) from (57b). Therefore, (57a) becomes

From (8) and (53), we have + u( t ,a )[At -~+, - K ( ~ , ~ ) C I - T + U ] ’

a -VH(t,s; a ) a0

+ VK(t ,u)VK’(t ,a) , 0 < a 5 T .

(65)

-- Os T - t + s < a 5 T (61a) VH( t , t - T + ~ ; a ) = VK( t ,a ) . (61b)

Substituting (61a) and (61b) into (60b) gives

1 2

+ -VK(t,u)VK’(t,a)

+ [VAr-T+a - K(t,a)~CI--T+uJ ,-I-T+u

x VH’(t,7.;a)dr. (62) By letting

f -T+a

T ( t , 4 = l-T VH(~,T;CJ)CI-T+,P(T,~ - T a)dT

(63) (57a) is obtained. Differentiating (63) with respect to U , we obtain a -gp,rJ) = V H ( t , t - T + f f ; f f )C, -T+oP( t - T + f f , t - T + f f )

a aff

t -T+o

VH(t , r ;o)C,~~+,- -P(s , t - T + f f ) d r .

Rr(t,O) = P( t - T, t - T ) .

Since the above error equation is defined on the finite interval, Rr(t , T ) has the finite value regardless of any errors present in BI , Ql, and Po. In other words, when there exist no modeling errors in system dynamics, the optimal FIR filter will not represent the divergence phenomenon. This is better characteristics than that of Kalman filter since Kalman filter may diverge for the errors of system noises.

Case 2. If there exist modeling errors in A, and C,,

VAr-T+a - K(t,a)VCI-~+,, is not generally zero. Thus, the behavior of R,(t,T) depends on that of T( t ,a) from (57a). It can be seen from (58a) and (7) that if AI is a stable matrix, then T ( t , T ) is stable and cannot cause R,(t,T) to diverge. If A, is unstable, T( t , T ) diverges, which causes R, ( t , T ) to diverge.

Thus we conclude that the divergence in the optimal FIR filter occurs only when the original system AI is unstable and V A I or VC, is not zero. This divergence condition is much stronger than that of the standard Kalman filter [3].

For the stationary process the optimal FIR filter becomes time invariant. Then the real estimation error

Modeling errors exist in system dynamics.

1018 IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 26, NO. 6 NOVEMBER 1990

covariance R,(T) is constant and obtained by

R,(T) = R(T) + U ( T ) (67) d

-U(o) = [A - K(a)C]U(o) + U ( a ) [ A - K(a)C]’ d o

+ [VA - K(o)VC]T’(o) + T ( a )

x [VA - K(c)VC]’ + VK(C)VK’(C),

d da -T(c) = [A - K(cT)C]T(O) + T(a)A’

+ VK(o)CR(a)

+ [VA - K(o)VC][P(O) - R(o)],

0 < a 5 T (69a)

and K(a) = R(a)C’; K(a) = R(a)C’;

(70) VK(t ,a) = K ( t , g ) - K(t,g).

VI. SIMULATION

The rectilinear orbit determination problem [l] is chosen for the simulation. Its dynamics is

h ( t ) = x2( t )

R2(t) = -p /x f ( t ) , x(0) = [8.0 2.01’ (71)

where x l ( t ) is position, x2(t) velocity, and p the gravitational constant times Earth mass, 19.9094165 er3/h2 (h = hour, er = Earth radii).

represented by The observation is assumed to be continuous and is

z ( t ) = X , ( t ) + V ( l ) (72)

with E[v(r)] = 0, E[v(t)v(s)] = r6(t - s), and r = 1 x 10-1°er2.

optimal FIR filter are compared. The initial state estimation of both filter is assumed as follows:

The continuous extended Kalman filter and the

7.997

a(o) = [1.9921

The filter length T of the optimal FIR filter is chosen as T = 0.1 h. On the first interval, the extended Kalman filter is used for the optimal FIR filter since it is not defined on the first interval. Because the optimal FIR filter is applied to linear models, we derive the linearized model of (71) at each nominal trajectory.

problem, the optimal FIR filter (4) i s modified as Since the state x(.) is not zero-mean process in this

H ( ~ , T ; T ) [ z ( T ) - C m , ( ~ ) ] d r

(73)

-

KWON, ET AL.: OPTIMAL FIR FILTERS FOR TIME-VARYING STATE-SPACE MODELS 1019

Fig. 2. Position estimation error

where 2 ( t ; T ) is the state estimate and m,(t) is the mean value of state x(.) at t which is obtained from (71) with the initial condition mx(0) = [7.997 The optimal FIR filter of Theorem 2 is used in this simulation and the impulse response of the optimal FIR filter is computed from (12)-(14).

Since the system (71) has no system noise, the extended Kalman filter is expected to diverge. As shown in Fig. 2, the position estimation error of the extended Kalman filter for this problem diverges rapidly. On the other hand, the optimal FIR filter does not diverge.

1.9921’.

VII. CONCLUSION

In this paper we introduced the optimal FIR filter and smoother for time-varying state-space signal models which has both the finite observation and the FIR structure. The optimal FIR filter is different from the limited memory filters [ l , 21 and the sliding window filter [5) in that the former has FIR structure but the latter filters have the infinite-impulse response (IIR) (or recursive) structure. Though they are derived from different conditions, the optimal FIR filter appears to have the simplest form among them.

The optimal FIR filter for linear time-varying systems is of a simpler form than that given in [lo]. The optimal FIR filter can be reduced to very simple form in an important case of time-invariant systems in which case the computation of the filter impulse response is required on only one interval [O,T]. Using the optimal FIR smoother as an adjoint variable, we suggested a recursive form of the optimal FIR filter in Theorem 3. Computation algorithms for filter gains are suggested in Theorem 4 and Theorem 5 for time-varying systems.

To consider the robustness of the optimal FIR filter, the error covariance of the FIR filter which is obtained under incorrect system models and noise information is derived. It is shown that the FIR filter never diverges if there exists no modeling error in

system dynamics even though system and measurement noises exist. The divergence condition is much stronger than that of the standard Kalman filter. In addition, due to the FIR structure, it automatically possesses robustness to filter coefficient quantization error and roundoff error. It is noted that the recursive forms of the optimal FIR filter may lose the above numerical merit since it is of IIR type rather than FIR type. Therefore when the computation power exists, the FIR structure is recommended rather than the recursive form.

Disadvantages of the optimal FIR filter compared with the standard Kalman filter are its computation burden and memory requirement, which are inherent in the filters of this kind [l, 2, 51. But for the large dimensional systems whose states to be estimated are small, by calculating the optimal FIR filter by off-line, the computation burden can be greatly reduced since FIR filters can be allocated to each state independent of other states. We believe therefore that there exist many applications in which the optimal FIR filter can be better used instead of the standard Kalman filter. See [lq for example.

REFERENCES

[1] Jaminski, A. H. (1968) Limited memory optimal filtering. IEEE Transactions on Automatic Control, AC-13 (Oct. 1W), 558-563.

Uncertain Dynamic Systems. Englewood Cliffs, N J Prentice-Hall, 1973.

Divergence of the Kalman filter. IEEE Transactions on Automatic Control, AC-16 (Dec.

[2] Schweppe, E C. (1973)

[3] Fitzgerald, R. J. (1971)

1971), 736-747. [4] Daum, E D., and Fitzgerald, R. J. (1983)

Decoupled Kalman filters for phased array radar tracking IEEE Transactions on Automatic Control, AC-28 (Mar. 1983), 269-283.

Recursive limited memory filtering and scattering theory. IEEE Transactions on Information nieory, IT-31 (May 1985), 440-443.

[5] Bruckstein, A. M., and Kailath, T. (1985)

Bierman, G. J. (1977) Factorization Methods for Discrete Sequential Estimation. New York: Academic, 1977.

Rabiner, L. R., and Gold, B. (1975) Theory and Application of Digital Signal Processing. Englewood Cliffs, NJ: Prentice-Hall, 1975, 75-84.

On the stability of fiued-lag smoothing algorithms. Journal of the Franklin Institute, 291 (Apr. 1971), 271-281.

A Survey of Numerical Methods for the Solution of Fredholm Integral Equations of the Second Kind. Philadelphia, PA: Society for Industrial and Applied Mathematics, 1976.

Schumitzky, A. (1968) On the equivalence between matrix Riccati equations and Fredholm resobents. Journal of Computer Science, 2 (June 1%8), 76437.

Baggeroer, A. B. (1969) A state-variable approach to the solution of Fredholm integral equations. IEEE Pansactions on Information Theory, IT-15 (Sept.

Kelly, C. N., and Anderson, B. D. 0. (1971)

Atkinson, K. E. (1976)

1%9), 557-570. Kailath, T. (1969)

Fredholm resolvent, Wiener-Hopf equation, and Riccati differential equations. IEEE Transactions Information Theory, IT-15 (Nov. 1969), 665472.

A modified quadratic cost problem and feedback stabilization of a linear system. IEEE Transactions on Automatic Control, AC-22 (Oct. 1977), 838-842.

FIR filters and recursive forms for continuous time-invariant state-space models. IEEE Transactions on Automatic Control, AC-32 (Apr. 1987), 352-356.

FIR filters and recursive form for discrete-time state-space models. Automatica, 25 (Sept. 1989), 715-728.

Optimal linear phase FIR filter for 2-D state-space models with application to image processing. In Proceedings of the ISL Winter Workshop, Depi. of Control and Instrumentation Engineering, Seoul National University, Seoul, Korea, 3 (Feb. 1990), 106-129.

Kwon, W. H., and Pearson, A. E. (1977)

Kwon, W. H., and Kwon, 0. K. (1987)

Kwon, 0. K., Kwon, W. H., and Lee, K. S. (1989)

Kang, J. Y., and Lee, K. S. (1990)

1020 IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 26, NO. 6 NOVEMBER 1990

Wook Hyun Kwon was born on January 19, 1943. He received the B.S. and M.S. degrees in electrical engineering from Seoul National University, Seoul, Korea, in 1966 and 1972, and the Ph.D. degree in electrical engineering from Brown University, Providence, RI, in 1975.

1976 to 1977 he was an Assistant Professor at the University of Iowa, Ames. Since 1977 he has been with the Department of Control and Instrumentation Engineering, Seoul National University, Seoul, Korea, where he is currently a Professor. From 1981 to 1982 he was a Visiting Assistant Professor at the Information Systems Laboratory at Stanford University, Stanford, CT. His fields of interest are in multivariable control and systems theory, statistical signal processing, digital system design for real time control and industrial automation.

From 1975 to 1976 he was a Research Associate at Brown University. From

Oh Kyu Kwon was born in Seoul, Korea, in 1952. He received the B.S. and M.S. degrees in electrical engineering and a Ph.D. degree in control and instrumentation engineering from Seoul National University, Seoul, Korea, in 1978 and 198.5, respectively.

He served as an Instructor from 1982 to 1983, and as an Assistant Professor from 1984 to 1987, at the Department of Electrical Engineering, Inha University, Inchon, Korea. He was also a Visiting Professor at the Department of Electrical Engineering and Computer Science, The University of Newcastle, Newcastle, Australia, during his academic year from August 1988 to August 1989. Since 1988 he has been an Associate Professor at the Department of Electrical Engineering, Inha University. His current research interests include estimation, signal processing, robust fault detection and digital control, with special applications in spacecraft guidance and attitude control.

Kyu Seung Lee was born in Jaechun, Korea, in 1963. He received the B.S. and M.S. degrees in control and instrumentation engineering from Seoul National University, Seoul, Korea, in 1986 and 1988, respectively. He is presently a Ph.D. student in the Department of Control and Instrumentation Engineering at Seoul National University.

Mr. Lee is a Research Assistant at Information Systems Laboratory, Seoul National University. His main research interests are in filtering theory, signal processing, optimal control and system theory.

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