Kalman filter with unknown inputs and robust two

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Kalman filter with unknown inputs and robust twostage filter a

a

J.Y. KELLER , M. DAROUACH & L. CARAMELLE

a

a

C.R.A.N.-E.A.R.A.L.Univscrsité Henri Poincaré , Nancy I, 186 rue de Lorraine, Cosncs et Romain, 54400, France Published online: 16 May 2007.

To cite this article: J.Y. KELLER , M. DAROUACH & L. CARAMELLE (1998) Kalman filter with unknown inputs and robust twostage filter, International Journal of Systems Science, 29:1, 41-47, DOI: 10.1080/00207729808929494 To link to this article: http://dx.doi.org/10.1080/00207729808929494

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International Journal of Systems Science, 1998, volume 29, number I, pages 41-47

Kalman filter with unknown inputs and robust two-stage filter 1 Y.

KELLERt,

M,

DAROUACHt

and L

CARAMELLEt

Downloaded by [McMaster University] at 07:25 26 February 2015

A nell' approach for state filtering in linear discrete-time stochastic systems with unknown inputs is presented, The obtained estimator, optimal in the unbiased minimum variance sense, is usedfor robust decentralized state and constant bias filtering,

I.

Introduction

The problem of state estimation in the presence of unknown inputs has received considerable attention in the last two decades. Many papers (Kudva et al. 1980, Yang and Richard 1988, Darouach et al. 1994) have been published in which the problem of designing the unknown input observer has been studied. Wiinnenberg and Frank (1987) and Patton et al. (1989) have used the unknown input observer approach for robust fault diagnosis. However, for stochastic systems only a few appraochcs have been developed. The most common approach is to treat the unknown inputs as a stochastic process with known wide-sense description. Generally, the unknown inputs are not constrained to evolve in accordance with a dynamic state equation. Kitanidis (1987) has developed a Kalman filter with unknown inputs by minimizing the trace of the state error covariance matrix under an algebraic contraint. Darouch et al. (1992) have proposed an approach for the design of unknown input decoupled optimal observer by transforming a standard system with unknown inputs into a singular system without unknown inputs. Chen and Patton (1996) have proposed another observer, where the remaining design freedom, after unknown inputs decoupling, is used to ensure that the state estimation has a minimal variance. Under the condition that the covariance matrix of plant noises is strictly positive defined, Keller et al. (1996) have shown that the Kalman filter with unknown input can be derived from the maximum likelihood descriptor Kalman filter developed by Nikhoukhah et al. (1992). Recently Darouach and Zasadzinski (1997) have extended the results of the approach developed by Kitanidis (1987). Received 3 February t997. Revised 19 June 1997. Accepted 30 June 1997. tC.R.A.N.-E.A.R.A.L.. Univsersite Henri Poincare, Nancy 1,186. rue de Lorraine. 54400 Cosncs ct Romain, France.

The gain matrix of the filter is obtained by parameterizing the solution of the algebraic constraint given by Kitanidis (1987) with an arbitrary matrix designed to lead to a minimum-variance estimator. In the first part of this paper we propose a new approach to derive an unbiased minimum-variance linear filter. As in the work of Darouach and Zasadzinski (1997), we propose to parametrize the solution of the algebraic constraint given by Kitanidis (1987) in relation to the standard Kalman filter's gain. The arbitrary matrix designed to obtain a minimumvariance estimator is then equal to the covariance matrix of the standard Kalman filter's innovation sequence. This new approach leads to a structure of the estimator very simular to the predictor-corrector structure of the standard Kalman filter. For state and bias filtering, the basic approach is to augment the system state to include the bias vector as part of the state, and then to implement the augmented state Kalman filter (ASKF). With this approach the problem is that the implementation of this ASKF can be computationally expensive. In addition, numerical problems may arise during the implementation, mainly for ill-conditioned systems. To avoid the use of the ASKF, Friedland (1969) has shown that the optimum estimate can be expressed as the output of the bias-free filter, based on the non-existence of the bias, corrected with the output of the bias filter estimating the bias vector. Mendel and Washburn (1978) and Ignani (1981) have proposed different alternative derivations of Friedland's method. Keller et al. (1996) have presented a state and bias filtering technique for systems with unknown inputs. Unfortunately, the obtained results are restricted to the case of systems with a covariance matrix of plant noises strictly positive defined. To remove this restrictive assumption, this paper presents a new two-stage estimator with unknown inputs. The only difference to Friedland's estimator concerns the gain matrices of

0020-7721/98 S12.00!) 1995 Taylor & Francis Ltd.

42

J. Y. Keller et al.

bias-free and bias filters determined in regard to the results of the approach developed in the first part of this paper.

(6a) with

(6b) T

2. Kalman filter with unknown inputs Consider the discrete-time stochastic systems described by


Yk = CXk +

(I b)

I'k>

where Xk E ~I/ is the state vector, Yk E ~II/ is the observation vector, Uk E~' is the known input vector, and dk E ~q is the unknown input vector. Matrices A, B, C and M have the appropriate dimensions. We assume that rank (M) = rank (CM) = q and q < m. The noise II'~ and I'k arc zero mean uncorrelated random sequences with

K k+1 = Pk+ 1/kC (CPk+I/kCT

Theorem I: If (Xk+1 = Ck~I' where CHI = CPk+I/kCT + V is the covariance matrix of the Kalman filter's innovation. the unbiased minimum variance filter can be expressed as

+ BUk + (K k+ 1 + 1Jk+lllk+Il1'HI,

xHI/HI = AXk/k

where 111'\';::: 0, V > 0 and {jkj is the Kronecker delta. The initial statc xo, a gaussian random variable with and

E{(xo - .\"o)(xo - .\"o)T} =

+ 1Jk+lllk+ICk+ lnJ+I1JJ+I,

1Jk+1 = (M - Kk+ 1CM), llk+l

(3 a)

Vk+1/ k+1

= [(CM)TCk~ICMrl(CM)TCk~I'

(8 a) x nX CTC-I K k+l = rk+l/k k+l,

= Xk+l/k + Lk+I(Yk+1

- CXk+l/k),

= (1- LH1C)Pk+l/k(l- Lk+ICl

(7d)

where

Consider the linear filler

Xk+l/k+l

(7 c)

(3 b)

P"k/k = E[ek/kek/k]'

(7 b)

with

satisfies

= 0,

(7a)

Pk+l/k+! = (I - Kk+ 1C)Pk+ l/k

Po,

is uncorrclated with thc white noise processes w~ and Vk' From measurement up to timc k, assume that we know the optimal statc estimate of Xk noted Xk/k' Let ek/k = Xk/k - Xk be the estimation error vector that

E(ek/k)

(6 c)

where the free parameter O

(17 e)

" = rk/k nb ST b-I K k+1 k+IG k+l, T GZ+ I = CPk+l/kC + SHI "z/kSk+1 + V.

(17d)


(1311) The above filter is of dimension 11 + p. When p is comparable to 11, the dimension of Xk / k becomes substantially larger than the original system state. The computation of the ASKF with unknown input may become excessive. The problem is now to find two parallel reduced-order filters that optimally implement the augmented state filter. For systems with constant bias but without unknown input, Friedland (1969) has shown that the optimum estimate Xk/k of the state Xk can be expressed as

Assuming that PiJ is positive definite, the initial parameters of the two-stage estimator are given by '\"0/0 = Xo - Vob o and p~/o = P~ - voPiJ vl, where

Vo = Po"(PiJ)-I.

To take into account the unknown inputs, the gain matrices K k+1 and Kt+1 of the two-stage estimator must satisfy Kk+ICM = M,

-"

KHICM = O. where .\"k/k is the bias-free estimate computed with the standard Kalman filter designed as if no bias was presented to the system, bk / k is the optimal estimate of bias h DX'" h . . all( I II k = I'\''' k/k ,>"-1 k/k' were r k/k IS t e eovananee matnx of Xk/k and bk/k and "z/k is the covariance matrix of bk / k . Friedland's algorithm is described by

(17 e)

(18a)

(18 b)

Proceeding as in §2, the solutions of (18a) and (18b) that minimize the trace of the estimation error covariance matrix are given by (19a)

(19 b) with

XHI/HI

= '\"HI/HI +

k+l/k+1 = I'>\'k+l/k+1

I>\'

+

VHlbk+I/HI,

(14a)

T Vk+1 rk+l/k+1 nb Vk+l,

(14 b)

where the bias-free filter, coupling equations and bias filter arc defined as follows. Bias free filler:

1]k+1 = (M - K k+1CM),

(20 a)

ITk+1 = [(CM)TGk:;::CMrl(CM)TGk:;::'

(20b) (20 e)

" b CM, 1/k+1 = K k+1

112+1 = [(CM)T G2:;:1cMrl (CM)TGZ:;:\.

(20d)

Replacing (15a) and (15b) by Sk+l/k+1 = A.\"k/k + BUk

+ (Kk+1 + 'Yk+1 x K- k+1

= 'YHI =

C(A'\"kfk +

oX CTGx-1 rk+l/k k+l,

-"

-x·

T

I'HI/k = APk/kA + W,

T Gk+1 = CPk+l/kC + V.

Bud,

(15 e)

( 15d )

f>k+I/HI

= (I -

1]k+1 llk+1 )'Yk+l,

Kk+1C)f>k+l/k

+ 1/k+1 llk+1 Gk+ 1llk:11/k":I,

bk+l / k+1 = bk/k + (Kt+1 + 1/2+1 112+1 h%+"

(15f)

"z+I/k+1 = (I - K%+ISk+I)"z/k b b G" llbT bT +1]H IllHI HI k+11/k+l,

(16a) ( 16b) (16 e)

(21 b)

and (17 a) and ( 17 b) by

(15e)

Coupling equations:

(21 a)

(22 a)

(22 b)

the robust two-stage Kalman filter is finally obtained by substituting the expression of the gain matrix Kk+1 into the coupling equation (16 e), which gives Vk+1 = (I - Kk+IC)Uk+1 - Kk+IG -1/k+lllk+ISk+I'

(23)

Kalman filter with unknown inputs

45

22,-------,--,---------,--,--------,-------,,.------.------r----.-----, , , 21

20 19 18

17 16


15 14

13

10

20

30

,

,

40

50

, 60

70

80

90

Time (k) Figure t. 4.

Bias estimate (ASKF with unknown inputs).

Illustrative example

To illustrate the equivalence between the ASKF with unknown inputs and the proposed two-stage filter, we consider a process of the form (9a), (9h) and (9c) descri bed by

the covariance matrix Pk / k of the ASK F, and Fig. 4 the trace of

obtained from the robust two-stage estimator. Figures 1-4 show that the robust two-stage filter optimally implements the ASKF with unknown inputs described by (12) and (13). Under the condition W' > 0, other simulations not presented here show that the robust two-stage Kalman filter gives the same results as those obtained by Keller et al. (1996).

5.

For this numerical example, the unknown input is taken to be dk = 10 sin (0.006 k) + Zb where Zk is a correlated noise sequence. Figs I and 2 show the results of bias estimation obtained with the robust ASKF and the two-stage filter, respectively. Fig. 3 shows the trace of

Conclusion

We have developed a new state observer for linear stochastic systems subject to unknown inputs or disturbances. Compared with the work of Darouach Zasadzinski (1997), we have proposed to parametrize the solution of the algebraic constraint given by Kitanidis (1987) in relation to the standard Kalman filter gain. The structure of the obtained unbiased minimum-variance estimator is very similar to the predictor-eorrector structure of the standard Kalman filter, and this fact has simplified the development of a new two-stage Kalman filter with unknown inputs.

46

J. Y. Keller et al. ,

22 21 20 19

~

18 17

16


15 14 13 12 0

,

10

.

,

20

30

40

50

60

70

80

90

100

Time (k) Figure 2.

Bias estimate (robust two-stage filter).

, , 300,------,.------,----,---.-----,----.---,-----,-----.----,

250

200

.

150

100

50

10

20

30

40

50

60

70

80

Time (k) Figure 3.

Trace of the covariance matrix (ASKF with unknown inputs).

90

100

47

Kalman filter with unknown inputs

, , , , 300 r----.---r----.------,r----,-----,----.------r---r-----,

250

200

150


100

-

50

. 10

20

30

50

40

60

70

80

90

100

Time (k)

Figure 4.

Trace of

p./.

(robust two-stage filter).

References

KELLER,J. Y., SUMMERER, L., and DAROUACH, M., 1996, Extension of

AWUANI, A. T., XIA, P., RICE, T. R., and BLAIR, W. D., 1993,

Friedland's bias filtering technique to discrete-time systems with unknown inputs. International Journal of Systems Sciences; 27, 1219-1229. KITANIDIS, P. K.; 1987. Unbiased minimum-variance linear state estimation. Automatica; 23, 775-778. KUDVA, P., VISWANADHAM, N., and RAMAKRISHNA, A" 1980, Observer for linear system with unknown inputs. IEEE Transactions Oil Automatic Control; 25, 125-133. MENDEL, J. M., and WASHBURN, H. D., 1978, Multistage estimation of bias states in linear systems. International Journal of Control, 28, 411-524. NIKOUKHAH, R., WILLSKY, A. S.. and LEVY, B. C., 1992, Kalman filtering and Riccati equations for descriptor systems. IE£I:.- Transactions on Automatic Control, 37, 1325-1342. PATTON, R. J., FRANK, P. M., and CLARK, R. N" 1989, Foul, Diagnosis in Dynamic Systems: Theory and Application (Prentice-Hall). WUNNENBERG, J., and FRANK, P. M.; 1987, Sensor Fault Using Detection via Robust Observers: System Fault Diagnostics. Reliabilityand Related Knowledge-Based Approaches (Reidel Press), pp. 147-160. YANG, F., and RICHARD, W, 1988, Observers for linear systems with unknown inputs. IEEE Transactions on Automatic Control, 33; 677-681.

On the optimality of two-stage state estimation in the presence of random bias. IEEE Transactions on Automatic Control, 38, 1279-1282. CHEN, J .. and PATTON, R. J., 1996, Optimal filtering and robust fault diagnosis of stochastic systems with unknown disturbances. lEE Proceedings, Control Theory and Applications, I, 31-36. DAROUACH, M., ZASAI)ZINSKI, M., and Xu, S. J., 1994, Full-order observers for linear systems with unknown inputs. IEEE Transactions all Automatic Control, 39, 606-609. DARouAcH. M., and ZASADZINSKI, M., 1997, Unbiased minimum variance estimation for systems with unknown exogenous inputs. Autontatica, 33, 717-719. DAROUACH, M., ZASADZINSKI, M., and KELLER, J. Y., 1992, Stale estimation for discrete systems with unknown inputs using state estimation of singular systems. Proceedings of the American Control Conference. Chicago, pp. 3014-3015. FRIEDLAND, B., 1969, Treatment of bias in recursive fillering. IEEE Transactions on Automatic Control, 14,359-367. IGNANI. M., 1981, An alternate derivation and extension of Friedland's two-stage Kalman estimator. IEEE Transactions on Automatic Control, 26, 746-750.