An adaptive filter with periodic gain, and its application in autonomous satellite navigation

ControlEng. Practice, Vol. 4, No. 12, pp. 1727-1734, 1996 Copyright © 1996 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0967-0661/96 $15.00 + 0.00

Pergamon PII:S0967-0661(96)00190-6

AN ADAPTIVE FILTER WITH PERIODIC GAIN, AND ITS APPLICATION IN AUTONOMOUS SATELLITE NAVIGATION Li Jie, Wu Hongxin and Chen Yiqing Beijing Institute of Control Engineering, P.O. Box 2729, Beijing 100080, P.R. China

(Received March 1996; in final form April 1996)

Abstract: In the paper an Adaptive Filter with PEriodic Gain (AFPEG) is presented, which is applicable to Discrete-time Linear Periodic Systems (DLPS) and nonlinear systems with a periodic nominal trajectory. Its convergence properties subject to the DLPS are proved. Compared to conventional adaptive filtering algorithms, the AFPEG reduces on-line computation with limited storage requirements. The AFPEG is applied to an autonomous satellite navigation system. Simulation results demonstrate its effectiveness in improving the accuracy of navigation. Keywords: Adaptive filters, periodic motion, parameter identification, navigation systems, satellites.

1. INTRODUCTION In the past few years considerable interest has been devoted to periodically time-varying systems. Cyclic processes arise in nature and engineering often spontaneously or often intentionally, and thus application of periodic models may be found in a large spectrum of different fields ranging from economics and management, to biology, flight dynamics, control of multirate industrial plants, etc. (Bittanti and Guardabassi, 1986). A number of results have been obtained concerning the problem of optimal filtering and control of linear periodic systems and periodic solutions to the differential and difference periodic Riccati equation (Nishimura, 1972; Hewer, 1975; Bittanti, et al., 1988; De Nicolao, 1992a, 1992b, 1994). Some of them have been successfully applied in guidance, navigation and control systems of spacecraft. The technique of spectral factorization of periodically time-varying systems was utilized in a sensitivity study of orbit

determination problems of low-thrust spacecraft from an earth-based station (Nishimura, 1972). A quasi-steady periodic filter was applied in the attitude determination system of the Chinese recoverable satellite, and demonstrated good performance in flight tests (Chen, et aL, 1989). In the present paper the subject is extended to a periodic system with unknown parameters, and an Adaptive Filter with PEriodic Gain (AFPEG) is developed, which is applicable to a discrete-time linear periodic system (DLPS) and a nonlinear system with a periodic nominal trajectory. Its convergence properties subject to the DLPS are proved. The application of the AFPEG consists of two parts: off-line determination of gain vectors, and on-line state and parameter estimation. Through the off-line design two gain vector tables are determined, whose length is equal to the system period. During the on-line estimation, the gain vectors of the state prediction and parameter

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identification are picked out from the two gain vector tables. So the AFPEG reduces on-line computation, with limited storage requirements compared to conventional adaptive filtering algorithms. In the paper the AFPEG is applied to an autonomous satellite navigation system. The uncertainty of the atmospheric model is an important factor that degrades the accuracy of the Satellite Autonomous Navigation System by Starlight Atmospheric Refraction (SANSSAR). Based on the study of systematic variations of the atmospheric density, a simplified parameter model of the atmosphere is built up, and the adaptive filtering model of the SANSSAR is derived. Owing to the periodicity of the system, the navigation filter is developed based on the AFPEG, whose simple on-line computation accords with the requirement for real-time data processing. Simulation results demonstrate the effectiveness of the AFPEG in improving the accuracy of navigation. The paper is organized as follows. In Section 2 the AFPEG is developed, subject to the DLPS and the nonlinear system with a periodic nominal trajectory. Its convergence properties subject to the DLPS are proved. The application procedure of the AFPEG is also presented. In Section 3 the problem of the SANSSAR is described, and an adaptive filtering model is built up. A navigator is developed, based on the AFPEG, and simulation results are also provided. In Section 4 the main results of the paper are summarized.

the same period nT. Suppose that the parameter vector 0 is known, the Kalman predictor subject to DLPS (1) has the form ff(k+l)=Fr(k)ff(k )+DT(k)O+K(k)[z(k)-~,(k)]

(2)

2(k +l) =Hr(k+l)r X (k +1) +Er(k +l)rO where K(k) =

Fr(k)P(k)Hr(k) err(k) 2 + H r ( k ) r p ( k ) H r ( k )

(3) •

P(') corresponds to the symmetric positive semidefmite solution of the Difference Periodic Riccati Equation (DPRE) P ( k +1) = F ~ ( k ) P ( k ) F r ( k ) T + Q , ( k ) - Fr(k)P(k)Hr(k)Hr(k)rP(k)F~'(k)r err(k) 2 + Hr(k)r P ( k ) H r ( k )

(4)

The main results of periodic solutions of the DPRE (4) are stated in the following two lemmas. Lemma 1. (Bittanti, et al., 1988) There exists a unique symmetric periodic positive semidet'mite solution Pr (') of the DPRE (4) and the matrix

(s)

A ( k ) = Fr(k ) - K ( k ) H r ( k ) r is

asymptotically

stable is

if

and

( F~ (.),

QO.~( . ) )

stabilizable

(IT(),

cY-/(')Hr (") ) is detectable.

only

if and

2. ADAPTIVE FILTER WITH PERIODIC GAIN Lemma 2. (De Nicolao, 1992a) 2.1 Adaptive filtering of the DLPS

If

(IT(.),

(y-~'(.)H~(.))

is detectable, and

1-I0 > 0, then Consider the DLPS with unknown parameters lira {Pn (k ) - PT ( k ) } = 0 x ( k + l) = F T ( k ) X ( k ) + DT(k)O + U k )

z(k) = Hr(k)V X(k) + Er(k)rO +~(k)

(1)

(6)

where Pn (') is the solution of the DPRE (4) with initial condition Pn (0) = Flo.

where X ( . ) is the n-dimensional system state vector, 0 is the m-dimensional system parameter vector, and z(.) is the scalar measurement output. F~ (9, Dr ('), H~ (.), E~ (.) are periodic matrices or vectors with the period nr. (In the paper the subscript ' T' denotes a periodic matrix or vector.) ~('), ~(') are cyclostationary white noise sequences, whose covariance matrices QT(.) and e r r ( ) 2 have

Lemmas 1 and 2 guarantee the existence, uniqueness and attractiveness of the symmetric periodic positive semidefinite solution PT('), which can be determined with an iterative algorithm. It is well known that in the steady-state the gain vector of the Kalman predictor subject to a timeinvariant linear system converges to a constant gain

Adaptive Filter with Periodic Gain vector, and a suboptimal predictor can be built up with the constant gain (Gelb, 1974). Similarly, a predictor with periodic gain can be built up subject to the DLPS (1), which has the form as follows ( Suppose that 0 is known.)

d'(k + 1) = F~ (k)X(k)

+ D r (k )O + Kpr(k)[z(k ) - $(k)]

~,(k +l) = I t r ( k + l ) r X ( k + l ) + Er(k +1)~0

(7)

M(k+l)

1729 =Ar(k)M(k

N(k) r =Hr(k)rM(k)

) +BT(k ) +Er(k) r ,

(12)

From Lemma 1 it follows that Ar (') is a stable periodic matrix, so M ( . ) converges to a periodic matrix Mr (') for any initial conditions, and N ( . ) converges to a periodic vector Nr (') consequently. It is obvious that Mr (') and Nr (') satisfy equation

where K,, r (.) is the steady-state gain vector of the Kalman predictor (2), and

(11). [] In view of the above theorem, an iterative algorithm can be developed to determine the periodic matrix

G~(k) = ~(k),

+H~(k)~p~(k)g~(k).

(8)

Equation (7) can be rewritten in the equivalent form X ( k + 1) = A r ( k ) f ( ( k ) + B r (k)O + Kpr ( k ) z ( k ) ~(k + l) = H r ( k + l)r ) f ( k + l) + E r ( k +l)r0

(9)

where

Mr (') and the periodic vector Nr ('). Given the condition that the derivative of the predicted output with respect to the parameter vector is the periodic vector, a periodic sequential parameter-identification algorithm is derived, whose gain vector

(13)

K~r(k ) = N o' N r ( k )

A~(k) = G ( k ) - K . r ( k ) G ( k )

~

G(k) =G(k)-Gr(k)G(k)

~•

(lO)

where nr-I

(14)

No = Y, N r ( j ) N r ( j ) r j-~

Equation (9) has the form of a one-step-ahead predictor. An adaptive filter can be constructed by the application of the sequential prediction error algorithm to estimate the value of 0 that gives the best predictor of the given structure ( Goodwin and Sin, 1984). The steady-state periodicity of derivatives of predicted state and output of (9) with respect to the parameter vector is formulated by the following theorem.

Combining the periodic state predictor and the periodic parameter-identification algorithm together, an Adaptive Filter with PEriodic Gain (AFPEG) is obtained as follows (an objection facility is introduced in the recursive parameter identification algorithm to ensure stability): f((i* G + j + l) = 4 ( j ) f f ( i * G + j ) + B~ (j)O(i - 1) + Ker (j)z(i*n T + j ) 1 nt-]

OB(i) = O(i-1) + _ ~ { K g r ( j + l ) * [ z ( i * n r + j + l ) l 3=0

Theorem 1. Define M ( k ) = 3 f ( ( k ) / O 0

provided

N ( k ) r = O~(k)/O0

is

stabilizable

and

that

( Fr (.),

QOJ( . ) )

( Fr('),

6-rJ(')Hrr(') ) is detectable. Then M ( . )

periodic vector N~(.) respectively, which satisfy

Nr(k) r = Hr(k)r Mr(k) + Er(k) ~

{

Oa(i) f)(i-1)

-

if O~(i) sO, if O~(i) ~D--o

(15) where i 1,2 ..... 0 < j < n r 1, Do is a compact subset of an m-dimensional parameter space so that =

-

0 e Do, and let 0 (0) ~ D--0.

(11)

Consider the definition of M ( . ) and N ( . ) ; equation (9) leads to

Proof

f)(i) =

and

and N ( . ) converge to periodic matrix M r ( ' ) and

M~(k + O = aT(k)M~(k) + Br(k)

- H r ( j + l ) r X(i*nr + j + l ) - E ~ ( j + l ) ~ O ( i - 1 ) ] }

The convergence properties of the AFPEG (15) subject to the DLPS (1) are given in the following Theorems 2 and 3.

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Theorem 2. Consider the AFPEG (15) subject to the DLPS (1), provided that: (i) ( i T ( ' ) , stabilizable

and

( FT(. ),

Q~())

~ (.)H~ (.) )

is is

detectable; (ii) matrix No is nonsingular; (iii) ~(') and ~(') are periodically ergodic, or the following equations are satisfied with probability 1:

lim I £ ,-~ t i=l

t--~o t

•

i=1

t

(i*nr +J~)~ (i*nr +J2) = 0

[itno{aX a (k ) - aX e ( k ) } = O. (16)

lim E{SXp

Then limO(i) = 0 with probability 1

lim E {aX~ (k ) a X f (k )} = PT (k ) . k---~¢o

state prediction K,r(')

(20)

The derivation of equation (17) is straightforward. []

Q~,5( . ) ) is stabilizable and

( F r ( ' ) , c r - / ( ' ) H r ( ' ) ) is detectable, it follows from lemma 1 that the periodic gain vector of the is uniquely determined.

Provided that the matrix No is nonsingular; the periodic gain vector of the parameter identification

K~r(') is also uniquely determined. Therefore, the AFPEG (15) is uniquely determined. Then the AFPEG is rewritten in the form of arecursive identification algorithm. Finally, based on the ordinary differential equation approach to the analysis of recursive identification algorithms, the convergence property of the AFPEG is derived from Theorem 4.1 and its corollary of (Ljung and Soderstrom, 1983). A detailed derivation is given in (Li, 1993). Theorem 3. Consider the AFPEG (15) subject to the DLPS (1), and suppose that the conditions of Theorem 2 are satisfied. Then the predicted state error a X A(.) satisfies

lim E ( a X ~ (k )) = 0

k --+oo

Theorems 2 and 3 show that, when the number of the recursive computations tends to infinity, the estimated parameters of the AFPEG (15) subject to the DLPS (1) converge to their true values, and the statistic property of the AFPEG converges to the Kalman filter•

2.2 Adaptive filtering of the nonlinear system with a periodic nominal trajectory Consider the nonlinear system X(k+l) z(k)

= F(X(k), : h(X(k),

O) + ¢ ( k )

lim E [ a X A ( k ) a X ~ ( k ) ) = Pr ( k )

k ---~oo

where PT(') is the symmetric periodic positive semidefmite solution of the DPRE (4).

(21)

O) + U k )

which has a periodic nominal trajectory X~.~,(.) with the period nr, i.e.,

Xr.~r (k + nr) = Xr,~, ( k ) •

(17)

(19)

(k)} = 0

k --+oo

The proof of Theorem 2 consists of three parts. First,

is

It follows from Lemmas 1 and 2 that

where 0 _
provided that ( Fr ('),

(18)

probability 1, and the periodic matrix A t ( ' ) stable, so

(i*n r + j ) = 0

t ,=1 1

= A r ( j ) { a X A ( i *n r + j ) - S X e ( i *n r + j ) }

It follows from Theorem 2 that l/m®O(i) = 0 with

l i m l - ~ ; (i*nr +jl)~ (i*nr + j : ) = o~(j~)asj.s ~ '~

aX A (i *n r + j + 1) - aX e (i *n r + j + 1) + B r (j){O - 6 0 - 1)}

(i*n r + j ) = 0

l i m l £ ~ (i*nr +Jl)~ (i*nr +J2) r = Qr(Jl)as,,s2 '-~ t J=l liml~;

Proof. Let aXe (') denote the predicted state error of the predictor with periodic gain (7). From equations (1), (9) and (15), the following equation is derived

(22)

The linearized matrices and vectors are valued at the periodic nominal trajectory and reference parameters, i.e.,

Adaptive Filter with Periodic Gain

OF(X,O) XT,,.,,(k),O.,. = F(k) OX OF(X,O) XT:,,(k),O,,, = D(k) O0 Oh(X,O) x~,,,,(k),o,,, = H(k) r OX Oh(X,O) x~,,,(k),%,, 03

.=

E(k) r (23)

1731

The off-line determination of gain vectors consists of three steps. (1) determine periodic matrices and vectors Fr('), Dr('), Hr('), Er('), Qr('), err(') with linearization at the periodic nominal trajectory and the reference value of system parmneters; (2) compute Pr (') with the iterative algorithm. The periodic gain vector of the state prediction Kpr (') is determined with equation (8);

Obviously F(.), D(.), H(.), E(.) are periodic matrices and vectors, which are denoted as Fr ('), Dr ('), Hr ('), Er ('). Similar to the case of the DLPS, an adaptive filter is constructed by the application of the sequential prediction error parameter identification algorithm to the periodic state predictor of the nonlinear system (21). The periodic gain vectors Kpr (') and K~r (') can be determined similarly. The AFPEG subject to the nonlinear system (21) has the form as follows: X(i*n r + j + l ) = F[X(i*n r +j), 0(i-1)] + Ker(j){ z(i*n r +j) - h[f((i*n T +j), ()(i- 1)] }

08(i) = ~(i_ 1) + - l ~ 1{ K,,~(j + 1)* I j-o

[z(i*n r + j + l ) - h [ ) ( ( i * n r + j + l ) , 0(i-l)] ]}

o(i) =

{

On(i )

if On(i ) e D o if On(i ) ~D--o

6(i-1)

(24)

(3) compute Mr(') and Nr(') with the iterative algorithm. The periodic gain vector of the parameter identification KDr(') is determined with equations (13) and (14). Given the conditions of Theorem 2, two gain vector tables {Kpr(j)} and {KDr(j+I)} (O< j
2

tO

0

(if

-9o_
A practical problem in implementing the adaptive filtering algorithms is the conflict between the online computation and the real-time requirement of the autonomous navigation system. To solve the problem, the periodicity of the SANSSAR is utilized. It is well known that the orbit motion of the satellite is periodic. The position of the starlight refraction

0

5

10

15

20

25

30

time( number of orbits )

Fig. 3 Averaged navigation error of two navigators based on the Kalman filter and the AFPEG Figure 3 illustrates the averaged navigation error of 5 simulation tests of the two navigators, which implement the conventional Kalman filter and the

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AFPEG respectively. Since it estimates more variables on-line, the AFPEG converges more slowly in the first few orbits. Gradually, more and more information is obtained through onboard measurements, the parameters of the atmospheric model are identified, and the navigational accuracy of the AFPEG is improved. From Figure 3, the navigation error of the AFPEG is less than 150 meters, while the error of the Kalman filter remains in the range of 500-800 meters.

4. CONCLUSIONS In the paper an Adaptive Filter with PEriodic Gain (AFPEG) is developed, which is applicable to Discrete-time Linear Periodic Systems (DLPS) and nonlinear systems with a periodic nominal trajectory. Compared to conventional adaptive filtering algorithms, the AFPEG reduces on-line computation, and has limited storage requirements. It is proved that the estimated parameters of the AFPEG subject to the DLPS converge to their true values when the number of the recursive computation tends to infmity, and the statistic property of the AFPEG converges to the Kalman filter. The AFPEG has been applied to the Satellite Autonomous Navigation System by Starlight Atmospheric Refraction (SANSSAR). Simulation results show that the effect of the uncertainty of the atmospheric model is reduced and the accuracy of navigation is improved by the combined estimation of system states (satellite position and velocity vectors ) and parameters of the atmospheric model. Owing to the periodicity of the SANSSAR, the application of the AFPEG avoids the conflict between the complicated on-line computation of conventional adaptive filtering algorithms and the real-time requirements of the autonomous navigation system. Acknowledgment - - The authors wish to thank Professor Tu Shancheng for his expertise and suggestions.

REFERENCES Bittanti, S. and Guardabassi, G. (1986). Optimal periodic control and periodic systems analysis: an overview. Proc. 25th 1EEE Conference on Decision and Control, Dec. 1986, Athens, Greece.

Bittanti, S., Colaneri, P. and De Nicolao, G. (1988). The difference periodic Riccati equation for the periodic prediction problem. IEEE Trans. Automat. Contr., 33, 706-712. Chen, Y.Q., Chen Z.G., Sun, C.Q., Wang, X.D., Feng, X.Y. and Ding, G.C. (1989). Digital attitude determination and control system of Chinese recoverable satellite for scientific exploration and technical experiments. Advances in the Astronautical Sciences, Paper No. AAS 89-653. De Nicolao, G. (1992a). On the convergence to the strong solution of periodic Riccati equations. Int. J. Control, 56, 87-97. De Nicolao, G. (1992b). Cyclomonotonicity and stabilizability properties of solutions of the difference periodic Riccati equation. IEEE Trans. Automat. Contr., 37, 1405-1410. De Nicolao, G. (1994). Cyclomonotonicity, Riccati equations and periodic receding horizon control. Automatica, 30, 1375-1388. Gelb, A. (1974). Applied optimal estimation. The MIT Press, Cambridge, Massachusetts. Goodwin, G. C. and Sin, K. S. (1984). Adaptive filtering, prediction and control. Prentice-Hall, Inc., Englewood Cliffs, New Jersey. Gounley, R., White, R. and Gai, E. (1984). Autonomous satellite navigation by stellar refration. Journal of Guidance and Control, 7, 129-134. Hewer, G. A. (1975). Periodicity, detectability and the matrix Riccati equation. SIAM J. on Control, 13, 1235-1251. Lair, J.L., Duchon, P., Riant, P. and Muller, G. (1986). Satellite navigation by stellar refraction. Proc. 37th Congress of the International Astronautical Federation, Oct. 1986, Innsbruck, Austria. Li, J. (1993). Satellite attitude determination and autonomous navigation with star sensors. Ph.D. dissertation, Chinese Academy of Space Technology. Ljung, L. and Soderstrom, T. (1983). Theory and practice of recursive identification. The MIT Press, Cambridge, Massachusetts. Nishimura, T. (1972). Spectral factorization in periodically time-varying systems and application to navigation problems. J. of Spacecraft and Rockets, 9,540-546. White, R.L., Thurman, S.W. and Barnes, F.A. (1985). Autonomous satellite navigation using obvervations of starlight atmospheric refraction. Navigation, Journal of the Institute of Navigation, 32, 317-333.