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Unbiased minimum-variance input and state estimation for systems with unknown inputs: A system reformation approach
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Technical Communique
Unbiased minimum-variance input and state estimation for systems with unknown inputs: A system reformation approach✩ Chien-Shu Hsieh 1 Department of Electrical and Electronic Engineering, Ta Hwa University of Science and Technology, 1, Dahua Road, Qionglin, Hsinchu 30740, Taiwan, ROC
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Article history: Received 27 April 2016 Received in revised form 22 April 2017 Accepted 14 May 2017 Available online xxxx Keywords: Unbiased minimum-variance estimation Unknown input filtering Optimal input and state estimation System reformation
a b s t r a c t This paper presents a system reformation based unbiased minimum-variance input and state estimation for systems with unknown inputs which can be reconstructed with a one-step delay. It is shown that, within this new filtering approach the optimal unknown input and state estimation can be simultaneously achieved through the filter developed by Gillijns and De Moor. An illustrative example is given to show the effectiveness of the proposed results. Moreover, under some additional assumptions the proposed system reformation can be easily extended to consider a multi-step delayed input estimation. © 2017 Elsevier Ltd. All rights reserved.
wk ∈ Rn and the measurement noise vk ∈ Rp are assumed to
1. Introduction Simultaneous input and state estimation (SISE) has received much research attention due to its vast applications in different research areas, e.g., input reconstruction (Chavan, Fitch, & Palanthandalam-Madapusi, 2014), input estimation (De Nicolao, Sparacino, & Cobelli, 1997; Yong, Zhu, & Frazzoli, 2016), sensor fault diagnosis (Gao & Ho, 2006), robust state estimation (Gillijns & De Moor, 2007b; Hsieh, 2000), and descriptor state estimation (Hsieh, 2013). In the above SISE problem, there may not be any knowledge concerning the model of the unknown inputs. A common approach to solve the SISE problem is by making use of unbiased minimum-variance estimation, which yields globally optimal unbiased minimum-variance filter for systems without direct feedthrough (Gillijns & De Moor, 2007a; Hsieh, 2000) and for systems with direct feedthrough (Cheng, Ye, Wang, & Zhou, 2009; Gillijns & De Moor, 2007b; Hsieh, 2009, 2013; Yong et al., 2016). The system under consideration is given as follows: xk+1 = Ak xk + Gk dk + wk ,
(1)
yk = Ck xk + Hk dk + vk , n
(2) m
where xk ∈ R is the state vector, dk ∈ R is an unknown input vector, and yk ∈ Rp is the measurement vector. The process noise ✩ This work was supported by the Ministry of Science and Technology, R.O.C. under Grant MOST 104-2221-E-233-004 & 105-2221-E-233-004. The material in this paper was not presented at any conference. This paper was recommended for publication in revised form by Associate Editor Henrik Sandberg under the direction of Editor André L. Tits. E-mail address: [email protected]. 1 Fax: +886 35927085.
be mutually uncorrelated, zero-mean, white random signals with known covariance matrices, Qk = E [wk wkT ] ≥ 0 and Rk = E [vk vkT ] > 0, respectively. It is well-known that, for system without direct feedthrough, i.e. Hk = 0, the filter developed by Hsieh (2000) gives the globally optimal SISE (OSISE) (Gillijns & De Moor, 2007a). On the other hand, under the full-rank direct feedthrough matrix, i.e. rank[Hk ] = m, the filter RTSF developed by Gillijns and De Moor (2007b) gives the globally OSISE (Hsieh, 2010). However, for rank deficient direct feedthrough matrix, no corresponding results are addressed in the above works. In solving the aforementioned rank deficient direct feedthrough case, the previously proposed result in Hsieh (2009) serves as a convenient refinement of Gillijns and De Moor’s filter to optimally estimate the system state and the unknown inputs. It is shown that, subject to a specific rank condition, i.e. the first equality of (6), the globally optimal state estimator can be obtained; however, the inherently obtained unknown input estimates are biased due to that the delayed input estimation is not considered. Some other approaches can also be used to solve the SISE problem, e.g., singular value decomposition (SVD) based output transformation method (Cheng et al., 2009; Yong et al., 2016) and descriptor Kalman filtering (Hsieh, 2013). Note that, in the above works only the result of Yong et al. (2016) explicitly addresses the optimal one-step delayed input estimation, which has been extended to multi-step delayed input estimation (Yong, Zhu, & Frazzoli, 2015). The main aim of this technical communique is to extend the previous works (Gillijns & De Moor, 2007a,2007b; Hsieh, 2000, 2009) for the following more general rank deficient conditions: 0 < rank[Hk ] < m,
(3)
http://dx.doi.org/10.1016/j.automatica.2017.06.037 0005-1098/© 2017 Elsevier Ltd. All rights reserved.
Please cite this article in press as: Hsieh, C.-S., Unbiased minimum-variance input and state estimation for systems with unknown inputs: A system reformation approach. Automatica (2017), http://dx.doi.org/10.1016/j.automatica.2017.06.037.
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C.-S. Hsieh / Automatica (
in achieving the OSISE. This can be solved by using a novel system reformation, based on the estimable input generating model (EIGM) used in Hsieh (2009), and the filter in Gillijns and De Moor (2007b). The results of this research can be extremely useful to gain the insight of optimal delayed unknown input estimation, which can be seen as a byproduct of the globally optimal state estimation. Note that, the assumptions of this new method are more restrictive than those in Yong et al. (2015); however, it may yield a less complex OSISE design.
)
which is alternative to (4) by using (5). Thus, subject to the input reconstruction condition (4), and using (12) and (13), we can obtain the reformed system as:
ˇ k dˇ k + wk , x¯ k+1 = Ak x¯ k + G
(15)
ˇ k dˇ k + vk , yk = Ck x¯ k + H
(16)
where
Πk∗−1 ≡ Πk−1 (I − Ωk+ Ωk ) = 0,
(4)
where M + denotes the Moore–Penrose pseudo-inverse of M,
Πk = Im − Hk+ Hk ,
Ωk = Ck Gk−1 Πk−1 .
(5)
(2) For the filter design, we assume that an unbiased estimate x˘ 0 of the initial state x0 is available with covariance matrix P0x ; furthermore, the following filter existence rank condition holds:
[
rank Hk
] Ωk = rank[Hk ] + rank[Gk−1 Πk−1 ] = m.
(6)
The main idea to solve the OSISE problem is to reconstruct the onestep delayed unknown input dk−1 by using a linear combination of the delay of the online estimable input d¯ k−1 and the one-step-delay estimable input d˜ k−1 , which is represented as follows: dk−1 = d¯ k−1 + Πk−1 d˜ k−1 .
(7)
,
d˜ Tk−1
ˇ k = Gk G
Ak Gk−1 Πk−1 ,
[
Assumptions. (1) For the system dynamics, we assume that (Ak , Ck ) is observable and that x0 is independent of vk and wk for all k; furthermore, the following one-step delayed unknown input reconstruction condition holds:
]T
dˇ k = d¯ Tk
[
2. The EIGM based system reformation
–
(17)
ˇ k = Hk H
]
[
Ωk .
]
(18)
Remark 1. The implication of the above system reformation is to transform the original system into one which can reflect the delay estimable input on the measurement equation in order to achieve the complete input estimation within the existing OSISE design, which also holds for multi-step delayed input estimation under some additional assumptions (see Section 5 for details). This reformed system serves as a unified framework to solve the existing OSISE problems. If matrix Hk is of full-column rank, the reformed system remains the same as the original one. This is the case considered in Gillijns and De Moor (2007b), where dk can be reconstructed without a delay, i.e. dk = d¯ k . On the other hand, for the case Hk = 0 one has Ωk = Ck Gk−1 . Thus, the unknown input vector dk can only be reconstructed through one-step delayed estimation, i.e. dk−1 = d˜ k−1 . This is the situation considered in Hsieh (2000). 3. Optimal input and state estimation design
the one-step delay of the EIGM of dk in (2), which is expressed as follows:
Based on the proposed system reformation in Section 2, the OSISE design problem remains to find optimal estimators of x¯ k and dˇ k such that the optimal estimation of dk−1 and xk can be simultaneously achieved.
d¯ k = (Hk+ Hk )dk ,
3.1. Optimal one-step delayed input estimator design
From Hsieh (2009), one can easily choose the above input d¯ k−1 as
(8)
and illustrates that both d¯ k and dk will yield the same optimal input estimate. Thus, the input reconstruction problem (7) remains to determine the input d˜ k−1 . Using (7) and (8), system (1), (2) can be rewritten as follows: xk+1 = Ak xk + Gk d¯ k + Gk Πk d˜ k + wk , yk = Ck xk + Hk d¯ k + vk .
(9) (10)
Next, we define a reformed state of xk that does not contain the delayed input d˜ k−1 as follows: x¯ k = xk − Gk−1 Πk−1 d˜ k−1
= Ak−1 xk−1 + Gk−1 d¯ k−1 + wk−1 ,
(11)
where dck = S˜k dˇ k ∈ Rm is a linear combination of dˇ k that has the minimum order. Using (19), the system dynamic (15) can be expressed as follows:
ˇ k S˜ + dck + Gˇ k (I − S˜ + S˜k )dˇ k + wk . x¯ k+1 = Ak x¯ k + G k k
ˇ + Hˇ k dˇ k = S˜ + S˜k dˇ k = S˜ + dck , dˇ k = H k k k (13)
(20)
(14)
(21)
by using which in (20) yields
¯ k dck + wk , x¯ k+1 = Ak x¯ k + G
where d˜ k−1 can be chosen as the EIGM of dk−1 as: d˜ k−1 = (Ωk+ Ωk )dk−1 .
(19)
(12)
and the measurement equation (10) is reformulated as: yk = Ck x¯ k + Hk d¯ k + Ωk d˜ k−1 + vk ,
yk = Ck x¯ k + S¯k dck + vk ,
From (16) and using the EIGM concept, we have the following relationships:
by which the system dynamic of x¯ k is obtained as follows: x¯ k+1 = Ak x¯ k + Gk d¯ k + Ak Gk−1 Πk−1 d˜ k−1 + wk ,
In order to solve the reformed system (15), (16) for the estimates of dˇ k using the result of Gillijns and De Moor (2007b), we ˇ k as Hˇ k = S¯k S˜k , where S¯k first define the full-rank factorization of H is of full-column rank and S˜k of full-row rank. Thus, output (16) can be rewritten as:
¯ k = Gˇ k S˜ + . G k
(22)
Next, applying the filter developed in Gillijns and De Moor (2007b) to system (19), (22), one obtains the unbiased minimumvariance estimate of dck as follows:
Finally, with the inputs d¯ k−1 and d˜ k−1 defined by (8) and (14), respectively, we can obtain the condition that achieves the reconstruction (7) as follows:
dˆ ck|k = Mkc (yk − Ck xˆ¯ k|k−1 ),
I = Hk+−1 Hk−1 + Πk−1 Ωk+ Ωk ,
1 ¯ −1 Pkd|k = (S¯kT R¯ − , k Sk )
c
c
1 Mkc = Pkd|k S¯kT R¯ − k ,
(23)
R¯ k = Ck Pkx¯|k−1 CkT + Rk .
(24)
Please cite this article in press as: Hsieh, C.-S., Unbiased minimum-variance input and state estimation for systems with unknown inputs: A system reformation approach. Automatica (2017), http://dx.doi.org/10.1016/j.automatica.2017.06.037.
C.-S. Hsieh / Automatica (
Finally, assuming that the conditions in (4)–(6) hold and using the following relationships due to (17) and (21): dˆ¯ k|k = T¯k dˆ ck|k ,
c
¯ Pkd|k = T¯k Pkd|k T¯kT ,
ˆ d˜ k−1|k = T˜k dˆ ck|k ,
(25)
c Tk Pkd|k TkT
˜ Pkd−1|k
˜ ,
=˜
3
R˜ k = Ck Pkx|k−1 CkT + Rk ,
1 Kk = Pkx|k−1 CkT R˜ − k ,
ˇ k∗ = (Hˇ kT R˜ −1 Hˇ k )+ Hˇ kT R˜ −1 , H k k 0 Sk ,
T¯k = I
[
˜+
]
I Sk ,
T˜k = 0
[
]
˜+
(27)
and the reconstruction model (7), we have the optimal one-step delayed unknown input estimator as follows:
[
dˆ k−1|k = Vkd (dˆ ck−1|k−1 )T
[ =
Vkd
c Pkd−1|k−1 T
(dˆ ck|k )T
{•} c
Pkd|k
{•}
]T
,
(28)
,
(29)
] (Vkd )T
L k = Kk +
([
ˇ k Hk , Gk−1 Πk−1 − Kk H
0
= (I −
Lk Ck )Pkx|k−1 (I
ˇ∗
[ Pkx+1|k = Ak
Gk
]
(41) (42)
T
− Lk Ck ) +
Lk Rk LTk
,
xˆ k+1|k = Ak xˆ k|k + Gk dˆ k|k ,
[
(39) (40)
)
]
xˆ k|k = xˆ k|k−1 + Lk (yk − Ck xˆ k|k−1 ), Pkx|k
where
(43) (44)
Pkx|k (Pkxd|k )T
Pkxd|k Pkd|k
][
ATk GTk
]
+ Qk ,
(45)
where
Πk−1 T˜k , c {•} = E [(dk−1 − dˆ ck−1|k−1 )(dck − dˆ ck|k )T ] = −E [(dc − dˆ c )(x¯ k − xˆ¯ k|k−1 )T ]C T (M c )T Vkd
–
Theorem 1. Let the filter existence condition (6) hold. The filter given by (37)–(38) is globally optimal in the sense that it is equivalent to the following ERTSF (Hsieh, 2009), which is globally optimal (Hsieh, 2010):
(26)
where
Pkd−1|k
)
[
]
= T¯k−1
k−1
k−1|k−1
k
(30)
dˆ k|k = Mk∗ (yk − Ck xˆ k|k−1 ), Pkxd|k
k
ˇ k∗ , 0 H
Mk∗ = I
[
]
= (Lk − Kk )R˜ k (Mk ) , ∗ T
(47)
Pkd|k = Mk∗ R˜ k (Mk∗ )T .
c c = −(Ak−1 Pkx¯−d 1|k−1 + G¯ k−1 Pkd−1|k−1 )T CkT (Mkc )T .
(46)
(48)
Proof. We assume that the following identities hold: Remark 2. The above optimal one-step delayed input estimator dˆ k−1|k can be shown to be globally optimal among all possible solutions. Although this can be verified by following a similar approach given in Gillijns and De Moor (2007a) to show the global optimality of a dedicated input estimator, an intuitive reasoning can also be conducted as below. Using (7), (15)–(17), (21), (25)– (28), and the fact that the filter dˆ ck|k is a globally optimal estimator of the reformed system (19), (22), one can deduce that dˆ k−1|k is a globally optimal estimator.
xˆ¯ k|k−1 = xˆ k|k−1 ,
Pkx¯|k−1 = Pkx|k−1 ,
(49)
which will be verified by inductive reasoning later. Then, we have ˇ ∗ = S˜ + M c and (27) Kk = K¯ k and R˜ k = R¯ k . Using the relationship H k k k in (41) yields Lk − Kk = −Kk S¯k Mkc + Gk−1 Πk−1 T˜k Mkc ,
(50)
by which and using (23), (32) and (33), we can rewrite (42) and (43), respectively, as follows: xˆ k|k = xˆ k|k−1 + Kk (yk − Ck xˆ k|k−1 ) + (Lk − Kk )
3.2. Globally optimal state estimator design First, the optimal reformed state filter of (15)–(16) is obtained by using the RTSF in (19) and (22) as follows: 1 K¯ k = Pkx¯|k−1 CkT R¯ − k ,
c
c
Pkx¯|dk = −K¯ k S¯k Pkd|k ,
[ ¯k G
= Ak
]
=
Pkx¯|k
(51) T
Kk Rk KkT
+ (Lk − Kk )R˜ k (Lk − Kk ) − ˜ c c + Gk−1 Πk−1 T˜k (Pkx¯|dk )T + Pkx¯|dk T˜kT ΠkT−1 GTk−1
(33)
(35)
Pkx¯|k c (Pkx¯|dk )T
c Pkx¯|dk c Pkd|k
][
ATk GTk
¯
]
+ Qk .
+ Gk−1 Πk−1 T˜k Pkd|k T˜kT ΠkT−1 GTk−1 .
(32)
(34)
¯ k dˆ ck|k , xˆ¯ k+1|k = Ak xˆ¯ k|k + G
[
=
Pkx|k−1
c
Pkx¯|k = Pkx¯|k−1 − K¯ k R¯ k K¯ kT + K¯ k S¯k Pkd|k S¯kT K¯ kT ,
Pkx¯+1|k
Pkx|k
(31)
xˆ¯ k|k = xˆ¯ k|k−1 + K¯ k (yk − Ck xˆ¯ k|k−1 ) − K¯ k S¯k dˆ ck|k ,
c
× (yk − Ck xˆ k|k−1 ) = xˆ¯ k|k + Gk−1 Πk−1 T˜k dˆ ck|k ,
(36)
Second, using the reformed state x¯ k defined in (11), we can reconstruct the original state estimator xˆ k|k as:
(52)
Using (51) and (52), we obtain that (42) and (43) are equivalent to (37) and (38), respectively. Now, the proof remains to verify that (44) and (45) are equivalent to (35) and (36), respectively. This is achieved by using (51), (52), and the following relationships: dˆ k|k = T¯k dˆ ck|k ,
(53)
¯ k = Gk T¯k + Ak Gk−1 Πk−1 T˜k , G Pkxd|k
=
c (Pkx¯|dk
+
(54)
c Gk−1 Πk−1 Tk Pkd|k )TkT
˜
¯ ,
(55)
c
Pkd|k = T¯k Pkd|k T¯kT .
(56)
(37)
Thus, by inductive reasoning the identities in (49) hold for all k, and hence the theorem is proved.
(38)
Remark 3. From (26) and (51), it is clear that the state estimator in Hsieh (2009) is a specific Kalman filter which implicitly determines
where Vkx = I Gk−1 Πk−1 T˜k . Finally, we show that, the above obtained state estimator xˆ k|k is globally optimal. This is addressed in the following theorem.
the one-step delayed input estimate d˜ k−1|k in order to achieve the globally optimal state estimation. Note that, this result is only true for the existence of the conditions in (6) (see Section 5 for an extension to multi-step delayed input cases).
[
T
xˆ k|k = Vkx xˆ¯ k|k
[ Pkx|k = Vkx
(dˆ ck|k )T
]T
, ] c
Pkx¯|k
Pkx¯|dk
c (Pkx¯|dk )T
Pkd|k
[
c
(Vkx )T ,
ˆ
]
Please cite this article in press as: Hsieh, C.-S., Unbiased minimum-variance input and state estimation for systems with unknown inputs: A system reformation approach. Automatica (2017), http://dx.doi.org/10.1016/j.automatica.2017.06.037.
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C.-S. Hsieh / Automatica (
Table 1 Performance of the ERTSF, the ULISE, and the OSISE.
)
–
Using (59) and (60), we can reform the system dynamics (1) and the measurement equation (2), respectively, as:
Filter
rmse (x1k )
rmse (x2k )
rmse (d1k )
rmse (d2k )
ERTSF ULISE OSISE
0.0318 0.0318 0.0318
3.2216 3.2216 3.2216
7.0420 3.2788 3.2788
3.2224 3.2224 3.2224
x¯ sk+1 = Ak x¯ sk + Gk d¯ k + Ask Gk−s Πk−s d˜ k−s + wk ,
(61)
s−1
yk = Ck x¯ sk + Hk d¯ k + Ωks d˜ k−s + vk +
∑
Ωki d˜ k−i .
(62)
i=1
Remark 4. The asymptotic stability of the OSISE design can be verified by using the results in Fang and de Callafon (2012) and the boundedness of Vkx . 4. An illustrative example In order to illustrate the usefulness of the proposed results, the numerical example given in Hsieh (2009) was considered and slightly modified, where the system parameters of (1) and (2) are given as follows:
[ −0.0005 Ak = 0.0517 C k = I2 ,
−0.0084 , 0.8069
]
Hk1 = diag(0, 1),
0.0129 −1.2504
[ Gk =
Q = 10−4 I2 ,
]
0 , 0
dk =
dk
d2k
The simulation time is 100 time steps, with a Monte Carlo simulation of 50 runs. In the simulation, the ERTSF in Hsieh (2009), the ULISE in Yong et al. (2016), and the proposed OSISE are considered. Note that, the reconstruction condition (4) holds for this case. This study lists the root-mean-square-errors (rmse) in the unknown input estimates of the considered estimators in Table 1, from which the following results are obtained. (1) The ERTSF may not yield the best unknown input estimation due to the null estimation of the one-step-delay estimable input, i.e. dˆ 1k−1|k = 0. (2) The filtering performance of the OSISE is equivalent to that of the ULISE. (3) As expected, the rmse errors of the estimates of d2k in all considered estimators are the same. 5. Extension to multi-step delayed input and state estimation In this section, under some additional assumptions an extension of the proposed system reformation to derive a compact multi-step delayed input estimator is presented if the input reconstruction condition (4) does not hold. First, we generalize (4) and (7), respectively, to the following s-step delayed corresponding results:
dk−s = d¯ k−s + Πk−s d˜ k−s , where Ωks and d˜ k−s are defined, respectively, as follows:
Ωki = Ck Aik−−11 Gk−i Πk−i , d˜ k−s = (Ωks )+ Ωk dk−s ,
) s
(64)
ˇ ks dˇ sk + vk , yk = Ck x¯ sk + H
(65)
]T
,
dˇ sk = d¯ Tk
d˜ Tk−s
ˇ sk = Gk G
Ask Gk−s Πk−s ,
(66)
]
ˇ ks = Hk H
[
Ωk .
] s
(67)
Finally, due to the analogy between (15)–(16) and (64)–(65) one s can obtain the estimates dˆ ck|k , dˆ k−s|k , and xˆ¯ k|k by using the results of Section 3, which is achieved by replacing the filter existence rank condition (6) with
ˇ ks ] = rank[Hk ] + rank[Gk−s Πk−s ] = m. rank[H
(68)
Furthermore, using (60) the estimate of xk can be obtained by (s − 1)-step delay as follows: s
xˆ k−r |k = xˆ¯ k−r |k−r +
s ∑
Aki−−1r −1 Gk−r −i Πk−r −i
i=1
× T˜k+1−i dˆ ck+1−i|k+1−i
(69)
where r = s − 1. Remark 5. One possible solution to relax the assumption (63) is to modify the input reconstruction model (58) as the following more general but more complex one: dk−s = d¯ k−s +
s ∑
Πki −s d˜ ik−s ,
(70)
i=1
where d˜ ik−s is determined by measurement at time k − s + i and matrices Πki −s (1 ≤ i ≤ s) are suitably chosen. A specific implementation of (70) and a general methodology derivation of the simultaneous input and state estimation for s ≥ 2 using the SVD technique can be found in Yong, Zhu, and Frazzoli (2015).
(57) (58)
(
ˇ sk dˇ sk + wk , x¯ sk+1 = Ak x¯ sk + G
[
]
( ) Πk∗−s ≡ Πk−s I − (Ωks )+ Ωks = 0, 1 < s < ∞,
(63)
by using which in system (61), (62) yields:
[
10cos(0.2k) = . 5sin(0.1k)
[
Ωk1 = · · · = Ωks−1 = 0,
where
R = 10−3 I2 .
The unknown inputs are given as follows:
[ 1]
Third, due to (58) the delayed inputs d˜ k−1 , . . . , d˜ k−s+1 in the last term of (62) can only be reconstructed at future time t > k, i.e. t = k − 1 + s, . . . , k + 1, respectively. Thus, in order to achieve realtime filtering, the last term of (62) must be vanished, and hence we further assume:
(59)
in which Aik = Ak × · · · × Ak−i+1 and A0k = In . Second, we generalize (11) to the following s-delay reformed state, which does not contain any delay estimable inputs d˜ k−i (i = 1, . . . , s):
6. Conclusion In this paper, a direct application of the filter in Gillijns and De Moor (2007b) to solve the OSISE problem is addressed. The problem is solved by using a novel EIGM based system reformation to reflect all delay estimable inputs on the transformed measurement equation, through which a compact OSISE design can be achieved under some additional assumptions. Acknowledgments
s
x¯ sk = xk −
∑ i=1
Aik−−11 Gk−i Πk−i d˜ k−i .
(60)
The author would like to thank the Associate Editor and the anonymous referees for their insightful comments and suggestions.
Please cite this article in press as: Hsieh, C.-S., Unbiased minimum-variance input and state estimation for systems with unknown inputs: A system reformation approach. Automatica (2017), http://dx.doi.org/10.1016/j.automatica.2017.06.037.
C.-S. Hsieh / Automatica (
References Chavan, R. A., Fitch, K., & Palanthandalam-Madapusi, H. J. (2014). Recursive input reconstruction with a delay. In Proceedings of the 2014 American control conference (pp. 628–633). Cheng, Y., Ye, H., Wang, Y., & Zhou, D. (2009). Unbiased minimum-variance state estimation for linear systems with unknown input. Automatica, 45, 485–491. De Nicolao, G., Sparacino, G., & Cobelli, C. (1997). Nonparametric input estimation in Physiological systems: Problems, methods, and case studies. Automatica, 33, 851–870. Fang, H., & de Callafon, R. A. (2012). On the asymptotic stability of minimumvariance unbiased input and state estimation. Automatica, 48, 3183–3186. Gao, Z., & Ho, D. W. C. (2006). State/noise estimator for descriptor systems with applications to sensor fault diagnosis. IEEE Transactions on Signal Processing, 54, 1316–1326. Gillijns, S., & De Moor, B. (2007a). Unbiased minimum-variance input and state estimation for linear discrete-time systems. Automatica, 43, 111–116.
)
–
5
Gillijns, S., & De Moor, B. (2007b). Unbiased minimum-variance input and state estimation for linear discrete-time systems with direct feedthrough. Automatica, 43, 934–937. Hsieh, C.-S. (2000). Robust two-stage kalman filters for systems with unknown inputs. IEEE Transactions on Automatic Control, 45, 2374–2378. Hsieh, C.-S. (2009). Extension of unbiased minimum-variance input and state estimation for systems with unknown inputs. Automatica, 45, 2149–2153. Hsieh, C.-S. (2010). On the global optimality of unbiased minimum-variance state estimation for systems with unknown inputs. Automatica, 46, 708–715. Hsieh, C.-S. (2013). State estimation for descriptor systems via the unknown input filtering method. Automatica, 49, 1281–1286. Yong, S. Z., Zhu, M., & Frazzoli, E. (2015). Simultaneous input and state estimation with a delay. In Proceedings of the 2015 IEEE 54th Annual Conference on Decision and Control (pp. 468–475). Yong, S. Z., Zhu, M., & Frazzoli, E. (2016). A unified filter for simultaneous input and state estimation of linear discrete-time stochastic systems. Automatica, 63, 321–329.
Please cite this article in press as: Hsieh, C.-S., Unbiased minimum-variance input and state estimation for systems with unknown inputs: A system reformation approach. Automatica (2017), http://dx.doi.org/10.1016/j.automatica.2017.06.037.
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Technical Communique
Unbiased minimum-variance input and state estimation for systems with unknown inputs: A system reformation approach✩ Chien-Shu Hsieh 1 Department of Electrical and Electronic Engineering, Ta Hwa University of Science and Technology, 1, Dahua Road, Qionglin, Hsinchu 30740, Taiwan, ROC
article
info
Article history: Received 27 April 2016 Received in revised form 22 April 2017 Accepted 14 May 2017 Available online xxxx Keywords: Unbiased minimum-variance estimation Unknown input filtering Optimal input and state estimation System reformation
a b s t r a c t This paper presents a system reformation based unbiased minimum-variance input and state estimation for systems with unknown inputs which can be reconstructed with a one-step delay. It is shown that, within this new filtering approach the optimal unknown input and state estimation can be simultaneously achieved through the filter developed by Gillijns and De Moor. An illustrative example is given to show the effectiveness of the proposed results. Moreover, under some additional assumptions the proposed system reformation can be easily extended to consider a multi-step delayed input estimation. © 2017 Elsevier Ltd. All rights reserved.
wk ∈ Rn and the measurement noise vk ∈ Rp are assumed to
1. Introduction Simultaneous input and state estimation (SISE) has received much research attention due to its vast applications in different research areas, e.g., input reconstruction (Chavan, Fitch, & Palanthandalam-Madapusi, 2014), input estimation (De Nicolao, Sparacino, & Cobelli, 1997; Yong, Zhu, & Frazzoli, 2016), sensor fault diagnosis (Gao & Ho, 2006), robust state estimation (Gillijns & De Moor, 2007b; Hsieh, 2000), and descriptor state estimation (Hsieh, 2013). In the above SISE problem, there may not be any knowledge concerning the model of the unknown inputs. A common approach to solve the SISE problem is by making use of unbiased minimum-variance estimation, which yields globally optimal unbiased minimum-variance filter for systems without direct feedthrough (Gillijns & De Moor, 2007a; Hsieh, 2000) and for systems with direct feedthrough (Cheng, Ye, Wang, & Zhou, 2009; Gillijns & De Moor, 2007b; Hsieh, 2009, 2013; Yong et al., 2016). The system under consideration is given as follows: xk+1 = Ak xk + Gk dk + wk ,
(1)
yk = Ck xk + Hk dk + vk , n
(2) m
where xk ∈ R is the state vector, dk ∈ R is an unknown input vector, and yk ∈ Rp is the measurement vector. The process noise ✩ This work was supported by the Ministry of Science and Technology, R.O.C. under Grant MOST 104-2221-E-233-004 & 105-2221-E-233-004. The material in this paper was not presented at any conference. This paper was recommended for publication in revised form by Associate Editor Henrik Sandberg under the direction of Editor André L. Tits. E-mail address: [email protected]. 1 Fax: +886 35927085.
be mutually uncorrelated, zero-mean, white random signals with known covariance matrices, Qk = E [wk wkT ] ≥ 0 and Rk = E [vk vkT ] > 0, respectively. It is well-known that, for system without direct feedthrough, i.e. Hk = 0, the filter developed by Hsieh (2000) gives the globally optimal SISE (OSISE) (Gillijns & De Moor, 2007a). On the other hand, under the full-rank direct feedthrough matrix, i.e. rank[Hk ] = m, the filter RTSF developed by Gillijns and De Moor (2007b) gives the globally OSISE (Hsieh, 2010). However, for rank deficient direct feedthrough matrix, no corresponding results are addressed in the above works. In solving the aforementioned rank deficient direct feedthrough case, the previously proposed result in Hsieh (2009) serves as a convenient refinement of Gillijns and De Moor’s filter to optimally estimate the system state and the unknown inputs. It is shown that, subject to a specific rank condition, i.e. the first equality of (6), the globally optimal state estimator can be obtained; however, the inherently obtained unknown input estimates are biased due to that the delayed input estimation is not considered. Some other approaches can also be used to solve the SISE problem, e.g., singular value decomposition (SVD) based output transformation method (Cheng et al., 2009; Yong et al., 2016) and descriptor Kalman filtering (Hsieh, 2013). Note that, in the above works only the result of Yong et al. (2016) explicitly addresses the optimal one-step delayed input estimation, which has been extended to multi-step delayed input estimation (Yong, Zhu, & Frazzoli, 2015). The main aim of this technical communique is to extend the previous works (Gillijns & De Moor, 2007a,2007b; Hsieh, 2000, 2009) for the following more general rank deficient conditions: 0 < rank[Hk ] < m,
(3)
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Please cite this article in press as: Hsieh, C.-S., Unbiased minimum-variance input and state estimation for systems with unknown inputs: A system reformation approach. Automatica (2017), http://dx.doi.org/10.1016/j.automatica.2017.06.037.
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in achieving the OSISE. This can be solved by using a novel system reformation, based on the estimable input generating model (EIGM) used in Hsieh (2009), and the filter in Gillijns and De Moor (2007b). The results of this research can be extremely useful to gain the insight of optimal delayed unknown input estimation, which can be seen as a byproduct of the globally optimal state estimation. Note that, the assumptions of this new method are more restrictive than those in Yong et al. (2015); however, it may yield a less complex OSISE design.
)
which is alternative to (4) by using (5). Thus, subject to the input reconstruction condition (4), and using (12) and (13), we can obtain the reformed system as:
ˇ k dˇ k + wk , x¯ k+1 = Ak x¯ k + G
(15)
ˇ k dˇ k + vk , yk = Ck x¯ k + H
(16)
where
Πk∗−1 ≡ Πk−1 (I − Ωk+ Ωk ) = 0,
(4)
where M + denotes the Moore–Penrose pseudo-inverse of M,
Πk = Im − Hk+ Hk ,
Ωk = Ck Gk−1 Πk−1 .
(5)
(2) For the filter design, we assume that an unbiased estimate x˘ 0 of the initial state x0 is available with covariance matrix P0x ; furthermore, the following filter existence rank condition holds:
[
rank Hk
] Ωk = rank[Hk ] + rank[Gk−1 Πk−1 ] = m.
(6)
The main idea to solve the OSISE problem is to reconstruct the onestep delayed unknown input dk−1 by using a linear combination of the delay of the online estimable input d¯ k−1 and the one-step-delay estimable input d˜ k−1 , which is represented as follows: dk−1 = d¯ k−1 + Πk−1 d˜ k−1 .
(7)
,
d˜ Tk−1
ˇ k = Gk G
Ak Gk−1 Πk−1 ,
[
Assumptions. (1) For the system dynamics, we assume that (Ak , Ck ) is observable and that x0 is independent of vk and wk for all k; furthermore, the following one-step delayed unknown input reconstruction condition holds:
]T
dˇ k = d¯ Tk
[
2. The EIGM based system reformation
–
(17)
ˇ k = Hk H
]
[
Ωk .
]
(18)
Remark 1. The implication of the above system reformation is to transform the original system into one which can reflect the delay estimable input on the measurement equation in order to achieve the complete input estimation within the existing OSISE design, which also holds for multi-step delayed input estimation under some additional assumptions (see Section 5 for details). This reformed system serves as a unified framework to solve the existing OSISE problems. If matrix Hk is of full-column rank, the reformed system remains the same as the original one. This is the case considered in Gillijns and De Moor (2007b), where dk can be reconstructed without a delay, i.e. dk = d¯ k . On the other hand, for the case Hk = 0 one has Ωk = Ck Gk−1 . Thus, the unknown input vector dk can only be reconstructed through one-step delayed estimation, i.e. dk−1 = d˜ k−1 . This is the situation considered in Hsieh (2000). 3. Optimal input and state estimation design
the one-step delay of the EIGM of dk in (2), which is expressed as follows:
Based on the proposed system reformation in Section 2, the OSISE design problem remains to find optimal estimators of x¯ k and dˇ k such that the optimal estimation of dk−1 and xk can be simultaneously achieved.
d¯ k = (Hk+ Hk )dk ,
3.1. Optimal one-step delayed input estimator design
From Hsieh (2009), one can easily choose the above input d¯ k−1 as
(8)
and illustrates that both d¯ k and dk will yield the same optimal input estimate. Thus, the input reconstruction problem (7) remains to determine the input d˜ k−1 . Using (7) and (8), system (1), (2) can be rewritten as follows: xk+1 = Ak xk + Gk d¯ k + Gk Πk d˜ k + wk , yk = Ck xk + Hk d¯ k + vk .
(9) (10)
Next, we define a reformed state of xk that does not contain the delayed input d˜ k−1 as follows: x¯ k = xk − Gk−1 Πk−1 d˜ k−1
= Ak−1 xk−1 + Gk−1 d¯ k−1 + wk−1 ,
(11)
where dck = S˜k dˇ k ∈ Rm is a linear combination of dˇ k that has the minimum order. Using (19), the system dynamic (15) can be expressed as follows:
ˇ k S˜ + dck + Gˇ k (I − S˜ + S˜k )dˇ k + wk . x¯ k+1 = Ak x¯ k + G k k
ˇ + Hˇ k dˇ k = S˜ + S˜k dˇ k = S˜ + dck , dˇ k = H k k k (13)
(20)
(14)
(21)
by using which in (20) yields
¯ k dck + wk , x¯ k+1 = Ak x¯ k + G
where d˜ k−1 can be chosen as the EIGM of dk−1 as: d˜ k−1 = (Ωk+ Ωk )dk−1 .
(19)
(12)
and the measurement equation (10) is reformulated as: yk = Ck x¯ k + Hk d¯ k + Ωk d˜ k−1 + vk ,
yk = Ck x¯ k + S¯k dck + vk ,
From (16) and using the EIGM concept, we have the following relationships:
by which the system dynamic of x¯ k is obtained as follows: x¯ k+1 = Ak x¯ k + Gk d¯ k + Ak Gk−1 Πk−1 d˜ k−1 + wk ,
In order to solve the reformed system (15), (16) for the estimates of dˇ k using the result of Gillijns and De Moor (2007b), we ˇ k as Hˇ k = S¯k S˜k , where S¯k first define the full-rank factorization of H is of full-column rank and S˜k of full-row rank. Thus, output (16) can be rewritten as:
¯ k = Gˇ k S˜ + . G k
(22)
Next, applying the filter developed in Gillijns and De Moor (2007b) to system (19), (22), one obtains the unbiased minimumvariance estimate of dck as follows:
Finally, with the inputs d¯ k−1 and d˜ k−1 defined by (8) and (14), respectively, we can obtain the condition that achieves the reconstruction (7) as follows:
dˆ ck|k = Mkc (yk − Ck xˆ¯ k|k−1 ),
I = Hk+−1 Hk−1 + Πk−1 Ωk+ Ωk ,
1 ¯ −1 Pkd|k = (S¯kT R¯ − , k Sk )
c
c
1 Mkc = Pkd|k S¯kT R¯ − k ,
(23)
R¯ k = Ck Pkx¯|k−1 CkT + Rk .
(24)
Please cite this article in press as: Hsieh, C.-S., Unbiased minimum-variance input and state estimation for systems with unknown inputs: A system reformation approach. Automatica (2017), http://dx.doi.org/10.1016/j.automatica.2017.06.037.
C.-S. Hsieh / Automatica (
Finally, assuming that the conditions in (4)–(6) hold and using the following relationships due to (17) and (21): dˆ¯ k|k = T¯k dˆ ck|k ,
c
¯ Pkd|k = T¯k Pkd|k T¯kT ,
ˆ d˜ k−1|k = T˜k dˆ ck|k ,
(25)
c Tk Pkd|k TkT
˜ Pkd−1|k
˜ ,
=˜
3
R˜ k = Ck Pkx|k−1 CkT + Rk ,
1 Kk = Pkx|k−1 CkT R˜ − k ,
ˇ k∗ = (Hˇ kT R˜ −1 Hˇ k )+ Hˇ kT R˜ −1 , H k k 0 Sk ,
T¯k = I
[
˜+
]
I Sk ,
T˜k = 0
[
]
˜+
(27)
and the reconstruction model (7), we have the optimal one-step delayed unknown input estimator as follows:
[
dˆ k−1|k = Vkd (dˆ ck−1|k−1 )T
[ =
Vkd
c Pkd−1|k−1 T
(dˆ ck|k )T
{•} c
Pkd|k
{•}
]T
,
(28)
,
(29)
] (Vkd )T
L k = Kk +
([
ˇ k Hk , Gk−1 Πk−1 − Kk H
0
= (I −
Lk Ck )Pkx|k−1 (I
ˇ∗
[ Pkx+1|k = Ak
Gk
]
(41) (42)
T
− Lk Ck ) +
Lk Rk LTk
,
xˆ k+1|k = Ak xˆ k|k + Gk dˆ k|k ,
[
(39) (40)
)
]
xˆ k|k = xˆ k|k−1 + Lk (yk − Ck xˆ k|k−1 ), Pkx|k
where
(43) (44)
Pkx|k (Pkxd|k )T
Pkxd|k Pkd|k
][
ATk GTk
]
+ Qk ,
(45)
where
Πk−1 T˜k , c {•} = E [(dk−1 − dˆ ck−1|k−1 )(dck − dˆ ck|k )T ] = −E [(dc − dˆ c )(x¯ k − xˆ¯ k|k−1 )T ]C T (M c )T Vkd
–
Theorem 1. Let the filter existence condition (6) hold. The filter given by (37)–(38) is globally optimal in the sense that it is equivalent to the following ERTSF (Hsieh, 2009), which is globally optimal (Hsieh, 2010):
(26)
where
Pkd−1|k
)
[
]
= T¯k−1
k−1
k−1|k−1
k
(30)
dˆ k|k = Mk∗ (yk − Ck xˆ k|k−1 ), Pkxd|k
k
ˇ k∗ , 0 H
Mk∗ = I
[
]
= (Lk − Kk )R˜ k (Mk ) , ∗ T
(47)
Pkd|k = Mk∗ R˜ k (Mk∗ )T .
c c = −(Ak−1 Pkx¯−d 1|k−1 + G¯ k−1 Pkd−1|k−1 )T CkT (Mkc )T .
(46)
(48)
Proof. We assume that the following identities hold: Remark 2. The above optimal one-step delayed input estimator dˆ k−1|k can be shown to be globally optimal among all possible solutions. Although this can be verified by following a similar approach given in Gillijns and De Moor (2007a) to show the global optimality of a dedicated input estimator, an intuitive reasoning can also be conducted as below. Using (7), (15)–(17), (21), (25)– (28), and the fact that the filter dˆ ck|k is a globally optimal estimator of the reformed system (19), (22), one can deduce that dˆ k−1|k is a globally optimal estimator.
xˆ¯ k|k−1 = xˆ k|k−1 ,
Pkx¯|k−1 = Pkx|k−1 ,
(49)
which will be verified by inductive reasoning later. Then, we have ˇ ∗ = S˜ + M c and (27) Kk = K¯ k and R˜ k = R¯ k . Using the relationship H k k k in (41) yields Lk − Kk = −Kk S¯k Mkc + Gk−1 Πk−1 T˜k Mkc ,
(50)
by which and using (23), (32) and (33), we can rewrite (42) and (43), respectively, as follows: xˆ k|k = xˆ k|k−1 + Kk (yk − Ck xˆ k|k−1 ) + (Lk − Kk )
3.2. Globally optimal state estimator design First, the optimal reformed state filter of (15)–(16) is obtained by using the RTSF in (19) and (22) as follows: 1 K¯ k = Pkx¯|k−1 CkT R¯ − k ,
c
c
Pkx¯|dk = −K¯ k S¯k Pkd|k ,
[ ¯k G
= Ak
]
=
Pkx¯|k
(51) T
Kk Rk KkT
+ (Lk − Kk )R˜ k (Lk − Kk ) − ˜ c c + Gk−1 Πk−1 T˜k (Pkx¯|dk )T + Pkx¯|dk T˜kT ΠkT−1 GTk−1
(33)
(35)
Pkx¯|k c (Pkx¯|dk )T
c Pkx¯|dk c Pkd|k
][
ATk GTk
¯
]
+ Qk .
+ Gk−1 Πk−1 T˜k Pkd|k T˜kT ΠkT−1 GTk−1 .
(32)
(34)
¯ k dˆ ck|k , xˆ¯ k+1|k = Ak xˆ¯ k|k + G
[
=
Pkx|k−1
c
Pkx¯|k = Pkx¯|k−1 − K¯ k R¯ k K¯ kT + K¯ k S¯k Pkd|k S¯kT K¯ kT ,
Pkx¯+1|k
Pkx|k
(31)
xˆ¯ k|k = xˆ¯ k|k−1 + K¯ k (yk − Ck xˆ¯ k|k−1 ) − K¯ k S¯k dˆ ck|k ,
c
× (yk − Ck xˆ k|k−1 ) = xˆ¯ k|k + Gk−1 Πk−1 T˜k dˆ ck|k ,
(36)
Second, using the reformed state x¯ k defined in (11), we can reconstruct the original state estimator xˆ k|k as:
(52)
Using (51) and (52), we obtain that (42) and (43) are equivalent to (37) and (38), respectively. Now, the proof remains to verify that (44) and (45) are equivalent to (35) and (36), respectively. This is achieved by using (51), (52), and the following relationships: dˆ k|k = T¯k dˆ ck|k ,
(53)
¯ k = Gk T¯k + Ak Gk−1 Πk−1 T˜k , G Pkxd|k
=
c (Pkx¯|dk
+
(54)
c Gk−1 Πk−1 Tk Pkd|k )TkT
˜
¯ ,
(55)
c
Pkd|k = T¯k Pkd|k T¯kT .
(56)
(37)
Thus, by inductive reasoning the identities in (49) hold for all k, and hence the theorem is proved.
(38)
Remark 3. From (26) and (51), it is clear that the state estimator in Hsieh (2009) is a specific Kalman filter which implicitly determines
where Vkx = I Gk−1 Πk−1 T˜k . Finally, we show that, the above obtained state estimator xˆ k|k is globally optimal. This is addressed in the following theorem.
the one-step delayed input estimate d˜ k−1|k in order to achieve the globally optimal state estimation. Note that, this result is only true for the existence of the conditions in (6) (see Section 5 for an extension to multi-step delayed input cases).
[
T
xˆ k|k = Vkx xˆ¯ k|k
[ Pkx|k = Vkx
(dˆ ck|k )T
]T
, ] c
Pkx¯|k
Pkx¯|dk
c (Pkx¯|dk )T
Pkd|k
[
c
(Vkx )T ,
ˆ
]
Please cite this article in press as: Hsieh, C.-S., Unbiased minimum-variance input and state estimation for systems with unknown inputs: A system reformation approach. Automatica (2017), http://dx.doi.org/10.1016/j.automatica.2017.06.037.
4
C.-S. Hsieh / Automatica (
Table 1 Performance of the ERTSF, the ULISE, and the OSISE.
)
–
Using (59) and (60), we can reform the system dynamics (1) and the measurement equation (2), respectively, as:
Filter
rmse (x1k )
rmse (x2k )
rmse (d1k )
rmse (d2k )
ERTSF ULISE OSISE
0.0318 0.0318 0.0318
3.2216 3.2216 3.2216
7.0420 3.2788 3.2788
3.2224 3.2224 3.2224
x¯ sk+1 = Ak x¯ sk + Gk d¯ k + Ask Gk−s Πk−s d˜ k−s + wk ,
(61)
s−1
yk = Ck x¯ sk + Hk d¯ k + Ωks d˜ k−s + vk +
∑
Ωki d˜ k−i .
(62)
i=1
Remark 4. The asymptotic stability of the OSISE design can be verified by using the results in Fang and de Callafon (2012) and the boundedness of Vkx . 4. An illustrative example In order to illustrate the usefulness of the proposed results, the numerical example given in Hsieh (2009) was considered and slightly modified, where the system parameters of (1) and (2) are given as follows:
[ −0.0005 Ak = 0.0517 C k = I2 ,
−0.0084 , 0.8069
]
Hk1 = diag(0, 1),
0.0129 −1.2504
[ Gk =
Q = 10−4 I2 ,
]
0 , 0
dk =
dk
d2k
The simulation time is 100 time steps, with a Monte Carlo simulation of 50 runs. In the simulation, the ERTSF in Hsieh (2009), the ULISE in Yong et al. (2016), and the proposed OSISE are considered. Note that, the reconstruction condition (4) holds for this case. This study lists the root-mean-square-errors (rmse) in the unknown input estimates of the considered estimators in Table 1, from which the following results are obtained. (1) The ERTSF may not yield the best unknown input estimation due to the null estimation of the one-step-delay estimable input, i.e. dˆ 1k−1|k = 0. (2) The filtering performance of the OSISE is equivalent to that of the ULISE. (3) As expected, the rmse errors of the estimates of d2k in all considered estimators are the same. 5. Extension to multi-step delayed input and state estimation In this section, under some additional assumptions an extension of the proposed system reformation to derive a compact multi-step delayed input estimator is presented if the input reconstruction condition (4) does not hold. First, we generalize (4) and (7), respectively, to the following s-step delayed corresponding results:
dk−s = d¯ k−s + Πk−s d˜ k−s , where Ωks and d˜ k−s are defined, respectively, as follows:
Ωki = Ck Aik−−11 Gk−i Πk−i , d˜ k−s = (Ωks )+ Ωk dk−s ,
) s
(64)
ˇ ks dˇ sk + vk , yk = Ck x¯ sk + H
(65)
]T
,
dˇ sk = d¯ Tk
d˜ Tk−s
ˇ sk = Gk G
Ask Gk−s Πk−s ,
(66)
]
ˇ ks = Hk H
[
Ωk .
] s
(67)
Finally, due to the analogy between (15)–(16) and (64)–(65) one s can obtain the estimates dˆ ck|k , dˆ k−s|k , and xˆ¯ k|k by using the results of Section 3, which is achieved by replacing the filter existence rank condition (6) with
ˇ ks ] = rank[Hk ] + rank[Gk−s Πk−s ] = m. rank[H
(68)
Furthermore, using (60) the estimate of xk can be obtained by (s − 1)-step delay as follows: s
xˆ k−r |k = xˆ¯ k−r |k−r +
s ∑
Aki−−1r −1 Gk−r −i Πk−r −i
i=1
× T˜k+1−i dˆ ck+1−i|k+1−i
(69)
where r = s − 1. Remark 5. One possible solution to relax the assumption (63) is to modify the input reconstruction model (58) as the following more general but more complex one: dk−s = d¯ k−s +
s ∑
Πki −s d˜ ik−s ,
(70)
i=1
where d˜ ik−s is determined by measurement at time k − s + i and matrices Πki −s (1 ≤ i ≤ s) are suitably chosen. A specific implementation of (70) and a general methodology derivation of the simultaneous input and state estimation for s ≥ 2 using the SVD technique can be found in Yong, Zhu, and Frazzoli (2015).
(57) (58)
(
ˇ sk dˇ sk + wk , x¯ sk+1 = Ak x¯ sk + G
[
]
( ) Πk∗−s ≡ Πk−s I − (Ωks )+ Ωks = 0, 1 < s < ∞,
(63)
by using which in system (61), (62) yields:
[
10cos(0.2k) = . 5sin(0.1k)
[
Ωk1 = · · · = Ωks−1 = 0,
where
R = 10−3 I2 .
The unknown inputs are given as follows:
[ 1]
Third, due to (58) the delayed inputs d˜ k−1 , . . . , d˜ k−s+1 in the last term of (62) can only be reconstructed at future time t > k, i.e. t = k − 1 + s, . . . , k + 1, respectively. Thus, in order to achieve realtime filtering, the last term of (62) must be vanished, and hence we further assume:
(59)
in which Aik = Ak × · · · × Ak−i+1 and A0k = In . Second, we generalize (11) to the following s-delay reformed state, which does not contain any delay estimable inputs d˜ k−i (i = 1, . . . , s):
6. Conclusion In this paper, a direct application of the filter in Gillijns and De Moor (2007b) to solve the OSISE problem is addressed. The problem is solved by using a novel EIGM based system reformation to reflect all delay estimable inputs on the transformed measurement equation, through which a compact OSISE design can be achieved under some additional assumptions. Acknowledgments
s
x¯ sk = xk −
∑ i=1
Aik−−11 Gk−i Πk−i d˜ k−i .
(60)
The author would like to thank the Associate Editor and the anonymous referees for their insightful comments and suggestions.
Please cite this article in press as: Hsieh, C.-S., Unbiased minimum-variance input and state estimation for systems with unknown inputs: A system reformation approach. Automatica (2017), http://dx.doi.org/10.1016/j.automatica.2017.06.037.
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Please cite this article in press as: Hsieh, C.-S., Unbiased minimum-variance input and state estimation for systems with unknown inputs: A system reformation approach. Automatica (2017), http://dx.doi.org/10.1016/j.automatica.2017.06.037.