ISA Transactions 57 (2015) 57–62
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Input and state estimation for linear systems with a rank-deficient direct feedthrough matrix Haokun Wang, Jun Zhao, Zuhua Xu n, Zhijiang Shao National Laboratory of Industrial Control Technology, Department of Control Science and Engineering, Zhejiang University, Hangzhou 310027, China
art ic l e i nf o
a b s t r a c t
Article history: Received 29 March 2014 Received in revised form 26 October 2014 Accepted 7 February 2015 Available online 7 March 2015 This paper was recommended for publication by Dr. Q.-G. Wang.
The problem of joint input and state estimation for linear stochastic systems with a rank-deficient direct feedthrough matrix is discussed in this paper. Results from previous studies only solve the state estimation problem; globally optimal estimation of the unknown input is not provided. Based on linear minimum-variance unbiased estimation, a five-step recursive filter with global optimality is proposed to estimate both the unknown input and the state. The relationship between the proposed filter and the existing results is addressed. We show that the unbiased input estimation does not require any new information or additional constraints. Both the state and the unknown input can be estimated under the same unbiasedness condition. Global optimalities of both the state estimator and the unknown input estimator are proven in the minimum-variance unbiased sense. & 2015 ISA. Published by Elsevier Ltd. All rights reserved.
Keywords: State estimation Unknown input estimation Minimum-variance unbiased estimation Global optimality
1. Introduction In system analysis and synthesis, unknown inputs can be used to represent unmeasured disturbances, unmodeled dynamics, system faults, etc. Therefore, estimation problems for systems with unknown inputs have drawn considerable attention over the past few decades. Solutions to such estimation problems have been played a significant role in many applications, e.g., disturbance rejection in control system design [6,9], geophysical and environmental applications [17], fault detection and isolation [1]. A common approach is to augment the system state to include the unknown input vector as an additional component of the state, and then apply the Kalman filter to the augmented system. The unknown input is often assumed to be a constant bias or a stochastic process with known statistics [5,15]. However, such assumptions may not be valid in practical implementations, especially when the unknown input takes extreme or unexpected values. In practice, the unknown input seldom obeys the assumed statistics. Thus, many works focus on the estimation of the state and the unknown input for systems with unknown input that have arbitrary statistics. A general approach to solve this problem is to apply unknown input decoupled state estimation by using minimum-variance unbiased estimation. When the unknown n
Corresponding author. Tel.: þ 86 571 87953068; fax: þ86 571 87953353. E-mail addresses: [email protected] (H. Wang), [email protected] (J. Zhao), [email protected] (Z. Xu), [email protected] (Z. Shao). http://dx.doi.org/10.1016/j.isatra.2015.02.005 0019-0578/& 2015 ISA. Published by Elsevier Ltd. All rights reserved.
input only affects the state, the estimation results are well established [17,3,10,16,7]. The problem becomes much more complicated when the unknown input affects both the state and the output, as the estimate error is correlated with system noise. An extension to state estimation for system with direct feedthrough was developed by Darouach et al. [4]. However, only a suboptimal solution was obtained due to the decorrelation constraint. The recursive filter proposed by Gillijns and De Moor [8] could estimate both the state and the unknown input in a less restrictive way. But this method is only applicable to the case that the direct feedthrough matrix has a full column rank. To make the method valid in the case that the direct feedthrough matrix is rank deficient, many filters have been proposed in the literature. Hsieh [11] proposed a recursive filter that can estimate both the state and the unknown input, but the author pointed out that estimation of the unknown input may have an inherent bias. Cheng et al. [2] proposed a recursive state estimator with global optimality by using input and output transformations. Hsieh [12] showed that globally optimal state estimators can be obtained with or without transformation matrices. Hsieh [13] developed a globally optimal state estimator by applying Descriptor Kalman filtering (DKF) techniques. Although the above mentioned results [2,12,13] yield globally optimal state estimation, estimation of the unknown input is not provided. The purpose of this study is twofold. First, we give a simple solution to the unknown input estimation problem which is not addressed in the existing results. Second, we provide a proof of optimality because the proposed filter is given in the recursive
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form. In this paper, the system is first transformed into an equivalent one by using singular value decomposition (SVD), and then a five-step recursive filter is proposed to estimate both the state and the unknown input. Solutions to the unknown input estimation problem are given in a simple manner, and global optimality of the proposed filter is proven in the minimum-variance unbiased sense. The proposed state estimator is equivalent to existing state estimators. Moreover, the relationship between input estimation and state estimation is illustrated. We show that new information or additional constraint is not required to provide unbiased estimation of the unknown input, unbiased state estimation implicitly guarantees the unbiased input estimation. Both the state and the unknown input can be estimated under the same unbiasedness condition. The remainder of this paper is organized as follows. The estimation problem is formulated in Section 2. The design of the proposed recursive filter is shown in Section 3, and the relationship between the proposed approach and previous results is also addressed. Global optimality of the proposed filter is discussed in Section 4. An illustrative example is given in Section 5, and a brief conclusion is provided in Section 6.
2. Problem formulation We consider the following linear stochastic time-varying system with unknown input: xk þ 1 ¼ Ak xk þGk dk þ wk
ð1Þ
yk ¼ C k xk þH k dk þ vk
ð2Þ
where x A Rnx , y A Rny , and d A Rnd are the state vector, the output vector and the unknown input vector, respectively. System noise wk and measurement noise vk are uncorrelated, zero-mean, white random signals with known covariance matrices, Q k ¼ E½wk wTk 4 0 and Rk ¼ E½vk vTk 4 0. Ak, Ck, Gk, and Hk are known matrices with appropriate dimensions. Throughout the paper, we assume that ðC k ; Ak Þ is observable, 0 o nd r ny r nx and the initial state x0 is of mean x^ 0 and covariance P0, and is independent of wk and vk for all k. In [2,12,13], optimal state estimators exist when the direct feedthrough matrix Hk is rank deficient. However, unbiased estimation of the unknown input is not provided in those studies. Although Hsieh [13] showed that one part of the unknown input is estimable, the estimability of the other part is unclear. Here, we first transform the system (1) and (2) into an equivalent one, and then a five-step recursive filter is proposed to estimate both the state and the unknown input for the rank-deficient case. 2.1. Model transformation Without loss of generality, we assume rankðH k Þ ¼ nd1;k , where 0 o nd1;k r nd . Then the singular value decomposition of Hk can be written as 2 3 VT Σk 0 Σk 0 T 1;k 4 5 ð3Þ V ¼ U 1;k U 2;k Hk ¼ U k 0 0 k 0 0 V T2;k
y2;k ¼ C 2;k xk þ v2;k
ð7Þ U T1;k yk ,
y2;k ¼ U T2;k yk ,
where G1;k ¼ Gk V 1;k , G2;k ¼ Gk V 2;k , y1;k ¼ C 1;k ¼ U T1;k C k , C 2;k ¼ U T2;k C k , v1;k ¼ U T1;k vk , and v2;k ¼ U T2;k vk . Compared with the transformation matrices proposed by Cheng et al. [2], UTk and Vk as transformation matrices provide more convenient. Because there always exists a unique relationship between the two new measurements y1;k and y2;k and the output yk, use of different transformation matrices (nonsingular) will not affect the estimation results. From the transformed system (5)–(7), it is clear that the system output is divided into two parts after SVD transformation. One output is directly affected by the unknown input, and the other one is only affected by the state. This transformation enables us to design filters that can estimate both the state and the unknown input by using existing approaches [7,8,17,3]. 2.2. Five-step recursive filter Here, we propose a five-step recursive filter for the transformed system 5– 7 in the following form: d^ 1;k 1 ¼ M 1;k 1 ðy1;k 1 C 1;k 1 x^ k 1 Þ
ð8Þ
n x^ kjk 1 ¼ Ak 1 x^ k 1 þG1;k 1 d^ 1;k 1
ð9Þ
n n x^ k ¼ x^ kjk 1 þLk ðy2;k C 2;k x^ kjk 1 Þ
ð10Þ
n d^ 2;k 1 ¼ M 2;k ðy2;k C 2;k x^ kjk 1 Þ
ð11Þ
d^ k 1 ¼ V 1;k 1 d^ 1;k 1 þ V 2;k 1 d^ 2;k 1
ð12Þ
where M 1;k 1 , M 2;k , and Lk are filter gain matrices to be determined. Eqs. (6) and (7) show that y1;k contains information on d1;k directly, and no output contains information on d2;k . Thus, d1;k may be estimated from y1;k , and the estimate of d2;k cannot be obtained from any output at time k. As a result, considering (4), the estimate of the unknown input must have one time delay. It should be noted that this result is different from the conventional addressed smoothed results, because we can estimate only d2;k 1 at time k. The main idea behind deriving an input estimator is to use d^ 1;k 1 in (8) instead of d1;k 1 in (5) so that d2;k 1 can be estimated from y2;k .
3. Input and state estimation In this section, both the unknown input and the state estimation problems are discussed. First, unbiasedness conditions for both the unknown input estimation and the state estimation are given, followed by determination of the optimal gain matrices via minimizing the error variances subject to the unbiasedness conditions. Then, we show that the proposed state estimator is equivalent to existing ones in the literature. 3.1. Input estimation
where Uk and Vk are unitary matrices, and Σk is an invertible diagonal matrix. Defining d1;k ¼ V T1;k dk and d2;k ¼ V T2;k dk , we obtain
3.1.1. Unbiased input estimation Define the innovation y~ 1;k 1 9 y1;k 1 C 1;k 1 x^ k 1 , we obtain
dk ¼ V 1;k d1;k þ V 2;k d2;k :
y~ 1;k 1 ¼ Σ k 1 d1;k 1 þ ξk 1
xk þ 1 ¼ Ak xk þG1;k d1;k þ G2;k d2;k þ wk
ð5Þ
where ξk 1 ¼ C 1;k 1 x~ k 1 þ v1;k 1 and x~ k 1 ¼ xk 1 x^ k 1 . Substituting y~ 1;k 1 into (8), we obtain Eðd^ 1;k 1 Þ ¼ M 1;k 1 Σ k 1 Eðd1;k 1 Þ. It is obvious that d^ 1;k 1 is unbiased if and only if the following condition holds:
y1;k ¼ C 1;k xk þ Σ k d1;k þv1;k
ð6Þ
M 1;k 1 ¼ Σ k 1 :
ð4Þ
Multiplying both sides of (2) by UTk, yields
1
ð13Þ
H. Wang et al. / ISA Transactions 57 (2015) 57–62
The unbiasedness condition (13) will always hold because Σk 1 is invertible. Thus, the gain matrix M 1;k 1 is uniquely determined by (13). n We next introduce the innovation y~ 2;k ¼ y2;k C 2;k x^ kjk 1 , which can be rewritten as y~ 2;k ¼ C 2;k G2;k 1 d2;k 1 þ ek
ð14Þ
where 8 ~ k 1 þv2;k ek ¼ C 2;k Φk 1 x~ k 1 þ C 2;k w > > < Φk 1 ¼ Ak 1 G1;k 1 Σ k11 C 1;k 1 > > :w ~ ¼w G Σ 1 v k1
k1
1;k 1
59
where 8 1 1 T d > > > P 11;k ¼ Σ k ðC 1;k P k C 1;k þ R1;k ÞΣ k > > 1 T > < P d12;k ¼ Σ k ðC 1;k P k Φk R1;k Σ k 1 GT1;k ÞC T2;k þ 1 M T2;k þ 1 > P d21;k ¼ ðP d12;k ÞT > > > > > d : P 22;k ¼ M 2;k þ 1 P e;k þ 1 M T2;k þ 1 Remark 1. From (8), (9), (11), and (12), we can rewrite our unknown input estimator in the following form: d^ k 1 ¼ K 1;k x^ k 1 þ K 2;k yk 1 þ K 3;k yk
k 1 1;k 1
Substituting (14) into (11) yields Eðd^ 2;k 1 Þ ¼ M 2;k C 2;k G2;k 1 E ðd2;k 1 Þ. Thus, d^ 2;k 1 is unbiased if and only if the following condition holds: M 2;k C 2;k G2;k 1 ¼ I
ð15Þ
The above condition is equivalent to rankðC 2;k G2;k 1 Þ ¼ nd nd1;k 1
ð16Þ
From (12), we can see that d^ k 1 is a linear combination of d^ 1;k 1 and d^ . Considering both (13) and (16), we know that (16) is a 2;k 1
necessary and sufficient condition for the existence of unbiased input estimation.
3.1.2. MVU input estimation Using (5)–(10), x~ k can be expressed as ~ k 1 Þ Lk v2;k x~ k ¼ ðI Lk C 2;k ÞðΦk 1 x~ k 1 þ G2;k 1 d2;k 1 þ w
ð17Þ
Considering covðv1;k 1 vT2;k Þ ¼ 0 and covðv2;k 1 vT2;k Þ ¼ 0, we can obtain ~ k 1 vT2;k Þ ¼ 0. As ~ Tk 1 Þ ¼ 0, covðx~ k 1 v~ T2;k Þ ¼ 0, and covðw covðx~ k 1 w ~ k 1 , and v2;k are mutually uncorrelated, yielding the such, x~ k 1 , w following results: 8 > P e;k ¼ Eðek eTk Þ ¼ C 2;k P~ k 1 C T2;k þ R2;k > > > > > T > ¼ Φk 1 P k 1 Φk 1 þ Q~ k 1 P~ > > < k1 1 1 Q~ k 1 ¼ G1;k 1 Σ k 1 R1;k 1 Σ k 1 GT1;k 1 þ Q k 1 > > > > R1;k 1 ¼ U T1;k 1 Rk 1 U 1;k 1 > > > > > : R2;k ¼ U T2;k Rk U 2;k where P k 1 ¼ Eðx~ k 1 x~ Tk 1 Þ.
where 8 K 1;k ¼ V 1;k 1 M 1;k 1 C 1;k 1 þ V 2;k 1 M 2;k C 2;k Φk 1 > > < K 2;k ¼ ðV 1;k 1 M 1;k 1 V 2;k 1 M 2;k C 2;k G1;k 1 M 1;k 1 ÞU T1;k 1 > > : K 3;k ¼ V 2;k 1 M 2;k U T 2;k
Both x^ k and d^ k 1 are derived from x^ k 1 , yk 1 , and yk. Estimation of the unknown input does not require any additional information. Thus, the estimation of dk 1 will not influence that of xk. As such, an unbiased state estimator can exist even if the unbiased unknown input estimate cannot be obtained [2,11,12]. □ 3.2. State estimation In Section 3.1, we have shown that unbiased estimate of d1;k 1 can be obtained from (8). Substituting d^ 1;k 1 into (5) to replace d1;k 1 , (5) and (7) form a new sub-system: ( ~ k 1 þG1;k 1 Σ k11 y1;k 1 xk ¼ Φk 1 xk 1 þ G2;k 1 d2;k 1 þ w S: y2;k ¼ C 2;k xk þ v2;k 1
where G1;k 1 Σ k 1 y1;k 1 is a deterministic term, and the unknown input d2;k 1 only affects the state. If we discard the deterministic term, sub-system S becomes the one considered by Kitanidis [17] and Darouach and Zasadzinski [3], where the output function has no unknown input. Therefore, their approaches can be used to provide optimal state estimation in a straightforward manner. In the following, we will show that optimal state estimation can be obtained by using the approaches discussed in [17,3].□ 3.2.1. Unbiased state estimation From (17) we have Eðx~ k Þ ¼ ðG2;k 1 Lk C 2;k G2;k 1 ÞEðd2;k 1 Þ. Because no information about d2;k 1 is available, (10) is an unbiased estimator of xk if and only if the following condition holds: G2;k 1 Lk C 2;k G2;k 1 ¼ 0
Theorem 1. Let x^ k 1 be unbiased, condition (16) holds and let P e;k be nonsingular, then (8), (11), and (12) form an MVU estimator of dk 1 with M 1;k 1 given by (13) and M 2;k is given by 1 1 C 2;k G2;k 1 Þ 1 GT2;k 1 C T2;k P e;k M 2;k ¼ ðGT2;k 1 C T2;k P e;k
ð18Þ
Proof. Optimal gain M 2;k can be obtained by applying the weighted least-squares estimation [14, see Chap. 2.2.3] to (14) 1 with weighting matrix P e;k . □ T
T
Considering covðd1;k 1 d2;k 1 Þ ¼ 0 and covðd2;k 1 d1;k 1 Þ ¼ 0, the estimate of dk 1 can be obtained from (12) directly. The corresponding covariance is given by 2 3 P d11;k 1 P d12;k 1 d 4 5V T Pk 1 ¼ V k 1 ð19Þ k1 P d21;k 1 P d22;k 1
ð20Þ
Condition (20) becomes the same as the unbiasedness conditions in [17,3] by replacing H k þ 1 and Gk with C 2;k and G2;k 1 , respectively. According to Kitanidis [17] and Darouach and Zasadzinski [3], (20) has a solution if the following condition holds rankðG2;k 1 Þ ¼ rankðC 2;k G2;k 1 Þ ¼ nd nd1;k 1 ;
ð21Þ
which is a sufficient condition for the existence of unbiased state estimation. In this paper, we follow Kitanidis [17] and Darouach and Zasadzinski [3], and focus on cases for which both G2;k 1 and C 2;k G2;k 1 have full column rank. Under condition (21), we can obtain the optimal gain matrix Lk, which will be shown in Section 3.2.2. Remark 2. Condition (16) will always hold if condition (21) is satisfied. We can conclude that the unbiased state estimation implicitly guarantees the unbiased input estimation. We have shown that the estimation of the unknown input will not affect the estimation of the state from the point of view of information
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H. Wang et al. / ISA Transactions 57 (2015) 57–62
requirement (Remark 1). Hence, we have the same conclusion from the analysis of unbiasedness conditions. Estimation of the unknown input does not require any additional constraint. □ Remark 3. Condition (21) is less restrictive than the full column rank condition required by Gillijns and De Moor [8]. Thus, even if Hk is rank deficient, an unbiased state estimator still exists as long as condition (21) holds. Furthermore, it is easy to verify that the proposed filter is still applicable when Hk is of full column rank. If Hk has full column rank, we are then able to obtain the estimation of dk as shown in Gillijns and De Moor [8]. But here we can estimate only dk 1 due to Hk is rank deficient, this result can be seen as a smoothed one. □ 3.2.2. MVU state estimation If (20) holds, x~ k can be rewritten as ~ k 1 Þ Lk v2;k x~ k ¼ ðI Lk C 2;k ÞðΦk 1 x~ k 1 þ w Thus P k ¼ ðI Lk C 2;k ÞP~ k 1 ðI Lk C 2;k ÞT þ Lk R2;k LTk
ð22Þ
We define R~ k ¼ C 2;k P~ k 1 C T2;k þ R2;k , and consider R~ k to be nonsingular. We then have the following theorem. Theorem 2. Let x^ k 1 be unbiased, condition (21) hold, and let R~ k be nonsingular, then (10) is a MVU estimator of xk with Lk given by T Lk ¼ P~ k 1 C T2;k N T1;k þ G2;k 1 N T2;k
ð23Þ
where 8 1 1 < N1;k ¼ R~ k N2;k GT2;k 1 C T2;k R~ k :N
1
2;k
1
¼ R~ k C 2;k G2;k 1 ðGT2;k 1 C T2;k R~ k C 2;k G2;k 1 Þ 1
Proof. Using the approach discussed in [17], we can minimize the trace of Pk under condition (20), then (23) can be obtained. □ Remark 4. Optimal state estimation can be obtained by applying the approaches discussed in [17,3] to sub-system S. It should be noted that both of the approaches require condition (21). The gain matrix of the filter proposed by Cheng et al. [2] is obtained by using Darouach's approach [3]. As such, the filter proposed by Chang et al. [2] implicitly requires condition (21). Thus, condition (15) in [2] is equivalent to condition (21) in this paper. Comparing (11) and (18) in [2] with (10) and (22) in this paper, it is clear that our state estimator is equivalent to that of Cheng et al. [2]. Although different transformation matrices are used, we obtain the same state estimator. □ Remark 5. Hsieh [12] proposed a general state estimator x^ k ¼ N k x^ k 1 þK k yk 1 þ L~ k yk with constraint rankðC 2;k U k 1 Þ ¼ rankðU k 1 Þ We can rewrite our state estimator in a compact form
Remark 6. System (5)–(7) can be recast into the following form: 32 2 3 2 3 2 3 xk xk I 0 G2;k 1 ηk 76 d 6y 7 6C 7 6 7 0 54 1;k 5 þ 4 v1;k 5 4 1;k 5 ¼ 4 1;k Σ k y2;k 0 d2;k 1 v2;k C 2;k 0 where 8 x ¼ Ak 1 x^ k 1 þG1;k 1 d^ 1;k 1 > > < k ηk ¼ Ak 1 x~ k G1;k 1 d~ 1;k 1 wk 1 > > : d~ ^ 1;k 1 ¼ d1;k 1 d 1;k 1 If we assume xk, d1;k and d2;k 1 are estimable (condition (21) is required), use the method in [18], we have the following 6-block DKF: 82 3 x^ k > > > h iT > 6 7 > > 6 d^ 1;k 7 ¼ ðT 6 ÞT Ω 1 x T yT yT 0 0 0 > > k 6;k k 1;k 2;k 5 > > 1 > > P k ¼ ðT 6k ÞT Ω6;k T 6k > > > > > : d^ k 1 ¼ V 1;k 1 d^ 1;k 1 þ V 2;k 1 d^ 2;k 1 where 8 2 3T > 0 0 0 I 0 0 > > > > 6 7 > > T 6k ¼ 4 0 0 0 0 I 0 5 > > > > > 0 0 0 0 0 I > > > 2 x > > > Pk 0 0 I > > > 6 > > 0 R1;k 0 C 1;k 6 < 6 6 0 0 R2;k C 2;k 6 > > > T T > Ω6;k ¼ 6 6 > I C C 0 > 1;k 2;k 6 > > 6 > T > 6 > 0 Σk 0 0 > 4 > > > T > > 0 0 0 G 2;k 1 > > > > x > > : P k þ 1 ¼ Ak G1;k 0 P k Ak G1;k 0 T
0
Σk
G2;k 1 0
0
0
0
0
0
0
0
0
3 7 7 7 7 7 7 7 7 7 7 5
In [13], d1;k is assumed to be estimable, but the estimation of d1;k is not provided directly. Moreover, the estimability of d2;k 1 is unclear. In fact, as shown in Section 3.1, unbiased estimation of d2;k 1 requires condition (16). Obviously, both xk and dk 1 can be estimated by using the above 6-block DKF, but may suffer from heavy computational burden because of the inverse operation of Ω6;k . Using the proposed five-step filter in this paper, it is easy and straightforward to solve the estimation problems. □ 4. Global optimality of the proposed filter As shown in Section 3, optimal solution to the problem of joint input and state estimation is assumed to take the form of a linear recursive filter under unbiasedness constraints. However, the global linear minimum-variance unbiased estimate may not lie within the recursive framework [16]. In this section, we show that the globally optimal estimators can be written in the proposed recursive form.
x^ k ¼ T 1;k x^ k 1 þ T 2;k yk 1 þT 3;k yk where 8 T 1;k ¼ ðI Lk C 2;k ÞΦk 1 > > < T 2;k ¼ ðI Lk C 2;k ÞG1;k 1 M 1;k 1 U T1;k 1 > > : T 3;k ¼ Lk U T 2;k
If we choose U k 1 ¼ G2;k 1 , Nk ¼ T 1;k , K k ¼ T 2;k , and L~ k ¼ T 3;k , it is clear that the proposed state estimator is equivalent to the state estimator (GOUMVF) in [12]. □
4.1. Optimality of the input estimator Consider the most general linear combination of measurements and the mean of initial state d^ k 1 ¼ F 0 x^ 0 þ
k X
Γ i yi
ð24Þ
i¼0
where F0 and Γi are combination coefficients. Because x^ i is a linear combination of x^ 0 ; y0 ; y1 ; …; yi , and both y~ 1;i and y~ 2;i can be written
H. Wang et al. / ISA Transactions 57 (2015) 57–62
as linear combinations of y0 ; y1 ; …; yi , an equivalent and convenient form of (24) can be expressed as d^ k 1 ¼ F 0 x^ 0 þ V 1;k 1 M 1;k 1 y~ 1;k 1 þ V 2;k 1 M 2;k y~ 2;k þ
kX 2
Γ 1;i y~ 1;i þ
i¼0
kX 1
Γ 2;i y~ 2;i
ð25Þ
i¼0
where Γ 1;i and Γ 2;i are coefficients. Γ k 1 ¼ V 1;k 1 M 1;k 1 and Γ k ¼ V 2;k 1 M 2;k , respectively. Necessary and sufficient conditions for (25) to be an unbiased estimator of dk 1 are given in the following theorem. Theorem 3. The unknown input estimator (25) is unbiased if and only if the following conditions are satisfied: (i) (ii) (iii) (iv) (v)
1
M 1;k 1 ¼ Σ k 1 ; M 2;k C 2;k G2;k 1 ¼ I; Γ 1;i ¼ 0; ði ¼ 0; 1; ⋯; k 2Þ; Γ 2;i C 2;i G2;i 1 ¼ 0; ði ¼ 0; 1; ⋯; k 1Þ; F 0 ¼ 0.
As a result of Theorems 3 and 4, estimator (25) can be easily written in the recursive form described by (8), (11), and (12). This means that (12) form a MVU estimator of dk 1 that minimizes the mean square error over the class of all linear unbiased estimates based on x^ 0 and measurements sequence y0 ; y1 ; …; yk . 4.2. Optimality of the state estimator Similarly, we can prove that the state estimation with minimum mean square error over the class of all linear unbiased estimates based on x^ 0 and measurements sequence y0 ; y1 ; …; yk can be written in the proposed recursive form. Consider the most general linear combination of measurements and the mean of initial state k X
Γ x;i yi
i¼0
where Fx and Γ x;i are combination coefficients, we have the following theorem.
Eðd^ k 1 Þ ¼ F 0 x^ 0 þ ðM d1;k 1 þ M d2;k 1 ÞEðdk 1 Þ kX 2 kX 1 Γ 1;i Σ i Eðd1;i Þ þ Γ 2;i C 2;i G2;i 1 Eðd2;i 1 Þ þ
Theorem 5. Let condition (21) hold and Lk be given by (23), then (8), (9), and (10) form an unbiased estimator of xk that minimizes the mean square error over the class of all linear unbiased estimates based on x^ 0 and measurements sequence y0 ; y1 ; …; yk .
i¼0
where 8 < M d1;k 1 ¼ V 1;k 1 M 1;k 1 Σ k 1 V T1;k 1
Proof. In Remark 4, we have shown that the proposed state estimator is equivalent to that in [2]. The proof is similar to that of Theorem 1 in [2] and is omitted for brevity. □
: M d2;k 1 ¼ V 2;k 1 M 2;k C 2;k G2;k 1 V T2;k 1 Sufficiency: If all five conditions are satisfied, we can obtain Eðd~ k 1 Þ ¼ 0, where d~ k 1 ¼ dk 1 d^ k 1 . Necessity: We assume that (25) is an unbiased estimator of dk 1 . Because no prior information about the unknown input is available, we can conclude that Eðd~ k 1 Þ ¼ 0, if and only if M d1 þ M d2 ¼ I;
ð26Þ
Γ 1;i Σ i ¼ 0 ði ¼ 0; 1; …; k 2Þ;
ð27Þ
Γ 2;i C 2;i G2;i 1 ¼ 0 ði ¼ 0; 1; …; k 1Þ;
ð28Þ
F 0 ¼ 0:
ð29Þ V T1;k 1
and V 2;k 1 , respectively, Multiplying both sides of (26) by we obtain M 1;k 1 ¼ Σ k11 and M 2;k C 2;k G2;k 1 ¼ I. Condition (27) implies Γ 1;i ¼ 0, and conditions (28) and (29) satisfy (iv) and (v) directly. □ Theorem 4. Let (25) be an unbiased estimator of dk 1 , then the mean square error Eð‖d~ k 1 ‖22 Þ achieves a minimum when Γ 2;i ¼ 0, ði ¼ 0; 1; …; k 1Þ. Proof. If estimator (25) is unbiased, then d~ k 1 can be written as a linear combination of ξk 1, ek, and y~ 2;i ði ¼ 0; 1; …; k 1Þ: 1 d~ k 1 ¼ V 1;k 1 Σ k 1 ξk 1 V 2;k 1 M 2;k ek
that the mean square error Eð‖d~ k 1 ‖22 Þ achieves a minimum when
Γ 2;i ¼ 0 ði ¼ 0; 1; …; k 1Þ. □
x^ k ¼ F x x^ 0 þ
Proof. From (25), we can obtain the mean of d^ k 1
i¼0
61
kX 1
Γ 2;i y~ 2;i
i¼0
Considering y~ 2;i ði ¼ 0; 1; …; k 1Þ is uncorrelated with ξk 1 and ek, we can obtain 2 3 kX 1 P d11;k 1 P d12;k 1 d 4 5V T þ Γ 2;i Eðy~ 2;i y~ T2;i ÞΓ T2;i Pk 1 ¼ V k 1 k1 P d21;k 1 P d22;k 1 i¼0 where P d11;k 1 , P d12;k 1 , P d21;k 1 , and P d22;k 1 are defined by (19). Because no information about d2;i 1 is available, we can conclude
5. Illustrative example To illustrate the performance of the proposed filter, the numerical example given in [2] is considered, where the system parameters are given as follows: 2 3 2 3 0:5 2 0 0 0 1 0 0:3 7 6 0 0:2 1 6 0 1 7 0 7 6 61 0 7 6 7 6 7 7; G ¼ 6 0 0 7; 0 0 0:3 0 1 0 A¼6 6 7 6 7 6 7 6 7 0 0 0:7 1 5 0 5 4 0 40 0 0 0 0 0 0:1 0 0 0 2 3 2 3 1 0 0 0:5 0 0 0 1 6 0 60 0 07 1 0 0 0:3 7 6 7 6 7 6 7 6 7 2 7; 7; R ¼ 10 6 0 0 1 0 0 0 1 0 H¼6 6 7 6 7 6 7 6 7 0 5 4 0:5 0 0 1 40 0 05 0 0:3 0 0 1 0 0 0 2 3 1 0 0 0 0 6 0 1 0:5 0 0 7 6 7 6 7 7 Q ¼ 10 4 6 6 0 0:5 1 0 0 7; C ¼ I 5 : 6 7 0 0 0 1 0 4 5 0 0 0 0 1 As H is rank deficient, the unknown input estimators proposed in [4,8] cannot guarantee the unbiasedness, thus, their state estimators may become biased. The filters proposed by Cheng et al. [2] and Hsieh [12] cannot provide the unknown input estimation. Although [11] provided optimal state estimation, the unknown input estimator (ERTSF) suffers from performance degradation because the unbiasedness condition of the unknown input estimation is not satisfied. Considering the filter proposed in this paper, we can verify that condition (21) holds. This example has three unknown inputs, and we choose three typical signals (square wave, sinusoid, and ramp)
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deficient. The real and estimated values of the unknown inputs are presented in Fig. 2. It is clearly shown that all the three unknown inputs can be estimated. Fig. 3 shows the traces of the covariance matrices Pk and Pdk. The convergence of the proposed filter is verified.
6. Conclusions
2
Fig. 1. State estimation.
In this paper, a five-step filter with global optimality is proposed to solve the joint input and state estimation for linear systems with a rank-deficient direct feedthrough matrix. The relationship between input estimation and state estimation has been clearly illustrated. In fact, the unbiased state estimation implicitly guarantees the unbiased input estimation. This result enables us to estimate both the state and the unknown input, regardless of their mutual coupling. Although the proposed filter is designed for the rank deficient case, it is still applicable when Hk has full column rank.
Acknowledgments This work was supported by the National Natural Science Foundation of China (No. 61273145, 61273146), the 863 Program of China (No. 2014AA041802), and the 973 Program of China (No. 2012CB720503). References
Fig. 2. Unknown input estimation.
Fig. 3. Traces of Pk and Pdk.
as unknown inputs for testing. We then initialize the filter as: x^ 0 ¼ x0 ¼ 0, P 0 ¼ I 5 , and P d0 ¼ I 3 . The real and estimated values of the first two states are plotted in Fig. 1. This illustrates that xk can be estimated even if H is rank
[1] Chen J, Patton RJ. Optimal filtering and robust fault diagnosis of stochastic systems with unknown disturbances. IEE Proc Control Theory Appl 1996;143:31–6. [2] Cheng Y, Ye H, Wang Y, Zhou D. Unbiased minimum-variance state estimation for linear systems with unknown input. Automatica 2009;45(2):485–91. [3] Darouach M, Zasadzinski M. Unbiased minimum variance estimation for systems with unknown exogenous inputs. Automatica 1997;33(4):717–9. [4] Darouach M, Zasadzinski M, Boutayeb M. Extension of minimum variance estimation for systems with unknown inputs. Automatica 2003;39(5):867–76. [5] Friedland B. Treatment of bias in recursive filtering. IEEE Trans Autom Control 1969;14(4):359–67. [6] Gao Z. On the centrality of disturbance rejection in automatic control. ISA Trans 2014;53(4):850–7. [7] Gillijns S, De Moor B. Unbiased minimum-variance input and state estimation for linear discrete-time systems. Automatica 2007;43(1):111–6. [8] Gillijns S, De Moor B. Unbiased minimum-variance input and state estimation for linear discrete-time systems with direct feedthrough. Automatica 2007;43(5):934–7. [9] Guo L, Cao S. Anti-disturbance control theory for systems with multiple disturbances: a survey. ISA Trans 2014;53(4):846–9. [10] Hsieh CS. Robust two-stage Kalman filters for systems with unknown inputs. IEEE Trans Autom Control 2000;45(12):2374–8. [11] Hsieh CS. Extensions of unbiased minimum-variance input and state estimation for systems with unknown inputs. Automatica 2009;45(9):2149–53. [12] Hsieh CS. On the global optimality of unbiased minimum-variance state estimation for systems with unknown inputs. Automatica 2010;46(4):708–15. [13] Hsieh CS. Optimal filtering for systems with unknown inputs via the descriptor Kalman filtering method. Automatica 2011;47(11):2313–8. [14] Kailath T, Sayed AH, Hassibi B. Linear estimation. Upper Saddle River, NJ: Prentice-Hall; 2000. [15] Keller J, Darouach M, Caramelle L. Kalman filter with unknown inputs and robust two-stage filter. Int J Syst Sci 1998;29(1):41–7. [16] Kerwin W, Prince J. On the optimality of recursive unbiased state estimation with unknown inputs. Automatica 2000;36(9):1381–3. [17] Kitanidis PK. Unbiased minimum-variance linear state estimation. Automatica 1987;23(6):775–8. [18] Nikoukhah R, Willsky AS, Levy BC. Kalman filtering and Riccati equations for descriptor systems. IEEE Trans Autom Control 1992;37(9):1325–42.
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ISA Transactions journal homepage: www.elsevier.com/locate/isatrans
Input and state estimation for linear systems with a rank-deficient direct feedthrough matrix Haokun Wang, Jun Zhao, Zuhua Xu n, Zhijiang Shao National Laboratory of Industrial Control Technology, Department of Control Science and Engineering, Zhejiang University, Hangzhou 310027, China
art ic l e i nf o
a b s t r a c t
Article history: Received 29 March 2014 Received in revised form 26 October 2014 Accepted 7 February 2015 Available online 7 March 2015 This paper was recommended for publication by Dr. Q.-G. Wang.
The problem of joint input and state estimation for linear stochastic systems with a rank-deficient direct feedthrough matrix is discussed in this paper. Results from previous studies only solve the state estimation problem; globally optimal estimation of the unknown input is not provided. Based on linear minimum-variance unbiased estimation, a five-step recursive filter with global optimality is proposed to estimate both the unknown input and the state. The relationship between the proposed filter and the existing results is addressed. We show that the unbiased input estimation does not require any new information or additional constraints. Both the state and the unknown input can be estimated under the same unbiasedness condition. Global optimalities of both the state estimator and the unknown input estimator are proven in the minimum-variance unbiased sense. & 2015 ISA. Published by Elsevier Ltd. All rights reserved.
Keywords: State estimation Unknown input estimation Minimum-variance unbiased estimation Global optimality
1. Introduction In system analysis and synthesis, unknown inputs can be used to represent unmeasured disturbances, unmodeled dynamics, system faults, etc. Therefore, estimation problems for systems with unknown inputs have drawn considerable attention over the past few decades. Solutions to such estimation problems have been played a significant role in many applications, e.g., disturbance rejection in control system design [6,9], geophysical and environmental applications [17], fault detection and isolation [1]. A common approach is to augment the system state to include the unknown input vector as an additional component of the state, and then apply the Kalman filter to the augmented system. The unknown input is often assumed to be a constant bias or a stochastic process with known statistics [5,15]. However, such assumptions may not be valid in practical implementations, especially when the unknown input takes extreme or unexpected values. In practice, the unknown input seldom obeys the assumed statistics. Thus, many works focus on the estimation of the state and the unknown input for systems with unknown input that have arbitrary statistics. A general approach to solve this problem is to apply unknown input decoupled state estimation by using minimum-variance unbiased estimation. When the unknown n
Corresponding author. Tel.: þ 86 571 87953068; fax: þ86 571 87953353. E-mail addresses: [email protected] (H. Wang), [email protected] (J. Zhao), [email protected] (Z. Xu), [email protected] (Z. Shao). http://dx.doi.org/10.1016/j.isatra.2015.02.005 0019-0578/& 2015 ISA. Published by Elsevier Ltd. All rights reserved.
input only affects the state, the estimation results are well established [17,3,10,16,7]. The problem becomes much more complicated when the unknown input affects both the state and the output, as the estimate error is correlated with system noise. An extension to state estimation for system with direct feedthrough was developed by Darouach et al. [4]. However, only a suboptimal solution was obtained due to the decorrelation constraint. The recursive filter proposed by Gillijns and De Moor [8] could estimate both the state and the unknown input in a less restrictive way. But this method is only applicable to the case that the direct feedthrough matrix has a full column rank. To make the method valid in the case that the direct feedthrough matrix is rank deficient, many filters have been proposed in the literature. Hsieh [11] proposed a recursive filter that can estimate both the state and the unknown input, but the author pointed out that estimation of the unknown input may have an inherent bias. Cheng et al. [2] proposed a recursive state estimator with global optimality by using input and output transformations. Hsieh [12] showed that globally optimal state estimators can be obtained with or without transformation matrices. Hsieh [13] developed a globally optimal state estimator by applying Descriptor Kalman filtering (DKF) techniques. Although the above mentioned results [2,12,13] yield globally optimal state estimation, estimation of the unknown input is not provided. The purpose of this study is twofold. First, we give a simple solution to the unknown input estimation problem which is not addressed in the existing results. Second, we provide a proof of optimality because the proposed filter is given in the recursive
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form. In this paper, the system is first transformed into an equivalent one by using singular value decomposition (SVD), and then a five-step recursive filter is proposed to estimate both the state and the unknown input. Solutions to the unknown input estimation problem are given in a simple manner, and global optimality of the proposed filter is proven in the minimum-variance unbiased sense. The proposed state estimator is equivalent to existing state estimators. Moreover, the relationship between input estimation and state estimation is illustrated. We show that new information or additional constraint is not required to provide unbiased estimation of the unknown input, unbiased state estimation implicitly guarantees the unbiased input estimation. Both the state and the unknown input can be estimated under the same unbiasedness condition. The remainder of this paper is organized as follows. The estimation problem is formulated in Section 2. The design of the proposed recursive filter is shown in Section 3, and the relationship between the proposed approach and previous results is also addressed. Global optimality of the proposed filter is discussed in Section 4. An illustrative example is given in Section 5, and a brief conclusion is provided in Section 6.
2. Problem formulation We consider the following linear stochastic time-varying system with unknown input: xk þ 1 ¼ Ak xk þGk dk þ wk
ð1Þ
yk ¼ C k xk þH k dk þ vk
ð2Þ
where x A Rnx , y A Rny , and d A Rnd are the state vector, the output vector and the unknown input vector, respectively. System noise wk and measurement noise vk are uncorrelated, zero-mean, white random signals with known covariance matrices, Q k ¼ E½wk wTk 4 0 and Rk ¼ E½vk vTk 4 0. Ak, Ck, Gk, and Hk are known matrices with appropriate dimensions. Throughout the paper, we assume that ðC k ; Ak Þ is observable, 0 o nd r ny r nx and the initial state x0 is of mean x^ 0 and covariance P0, and is independent of wk and vk for all k. In [2,12,13], optimal state estimators exist when the direct feedthrough matrix Hk is rank deficient. However, unbiased estimation of the unknown input is not provided in those studies. Although Hsieh [13] showed that one part of the unknown input is estimable, the estimability of the other part is unclear. Here, we first transform the system (1) and (2) into an equivalent one, and then a five-step recursive filter is proposed to estimate both the state and the unknown input for the rank-deficient case. 2.1. Model transformation Without loss of generality, we assume rankðH k Þ ¼ nd1;k , where 0 o nd1;k r nd . Then the singular value decomposition of Hk can be written as 2 3 VT Σk 0 Σk 0 T 1;k 4 5 ð3Þ V ¼ U 1;k U 2;k Hk ¼ U k 0 0 k 0 0 V T2;k
y2;k ¼ C 2;k xk þ v2;k
ð7Þ U T1;k yk ,
y2;k ¼ U T2;k yk ,
where G1;k ¼ Gk V 1;k , G2;k ¼ Gk V 2;k , y1;k ¼ C 1;k ¼ U T1;k C k , C 2;k ¼ U T2;k C k , v1;k ¼ U T1;k vk , and v2;k ¼ U T2;k vk . Compared with the transformation matrices proposed by Cheng et al. [2], UTk and Vk as transformation matrices provide more convenient. Because there always exists a unique relationship between the two new measurements y1;k and y2;k and the output yk, use of different transformation matrices (nonsingular) will not affect the estimation results. From the transformed system (5)–(7), it is clear that the system output is divided into two parts after SVD transformation. One output is directly affected by the unknown input, and the other one is only affected by the state. This transformation enables us to design filters that can estimate both the state and the unknown input by using existing approaches [7,8,17,3]. 2.2. Five-step recursive filter Here, we propose a five-step recursive filter for the transformed system 5– 7 in the following form: d^ 1;k 1 ¼ M 1;k 1 ðy1;k 1 C 1;k 1 x^ k 1 Þ
ð8Þ
n x^ kjk 1 ¼ Ak 1 x^ k 1 þG1;k 1 d^ 1;k 1
ð9Þ
n n x^ k ¼ x^ kjk 1 þLk ðy2;k C 2;k x^ kjk 1 Þ
ð10Þ
n d^ 2;k 1 ¼ M 2;k ðy2;k C 2;k x^ kjk 1 Þ
ð11Þ
d^ k 1 ¼ V 1;k 1 d^ 1;k 1 þ V 2;k 1 d^ 2;k 1
ð12Þ
where M 1;k 1 , M 2;k , and Lk are filter gain matrices to be determined. Eqs. (6) and (7) show that y1;k contains information on d1;k directly, and no output contains information on d2;k . Thus, d1;k may be estimated from y1;k , and the estimate of d2;k cannot be obtained from any output at time k. As a result, considering (4), the estimate of the unknown input must have one time delay. It should be noted that this result is different from the conventional addressed smoothed results, because we can estimate only d2;k 1 at time k. The main idea behind deriving an input estimator is to use d^ 1;k 1 in (8) instead of d1;k 1 in (5) so that d2;k 1 can be estimated from y2;k .
3. Input and state estimation In this section, both the unknown input and the state estimation problems are discussed. First, unbiasedness conditions for both the unknown input estimation and the state estimation are given, followed by determination of the optimal gain matrices via minimizing the error variances subject to the unbiasedness conditions. Then, we show that the proposed state estimator is equivalent to existing ones in the literature. 3.1. Input estimation
where Uk and Vk are unitary matrices, and Σk is an invertible diagonal matrix. Defining d1;k ¼ V T1;k dk and d2;k ¼ V T2;k dk , we obtain
3.1.1. Unbiased input estimation Define the innovation y~ 1;k 1 9 y1;k 1 C 1;k 1 x^ k 1 , we obtain
dk ¼ V 1;k d1;k þ V 2;k d2;k :
y~ 1;k 1 ¼ Σ k 1 d1;k 1 þ ξk 1
xk þ 1 ¼ Ak xk þG1;k d1;k þ G2;k d2;k þ wk
ð5Þ
where ξk 1 ¼ C 1;k 1 x~ k 1 þ v1;k 1 and x~ k 1 ¼ xk 1 x^ k 1 . Substituting y~ 1;k 1 into (8), we obtain Eðd^ 1;k 1 Þ ¼ M 1;k 1 Σ k 1 Eðd1;k 1 Þ. It is obvious that d^ 1;k 1 is unbiased if and only if the following condition holds:
y1;k ¼ C 1;k xk þ Σ k d1;k þv1;k
ð6Þ
M 1;k 1 ¼ Σ k 1 :
ð4Þ
Multiplying both sides of (2) by UTk, yields
1
ð13Þ
H. Wang et al. / ISA Transactions 57 (2015) 57–62
The unbiasedness condition (13) will always hold because Σk 1 is invertible. Thus, the gain matrix M 1;k 1 is uniquely determined by (13). n We next introduce the innovation y~ 2;k ¼ y2;k C 2;k x^ kjk 1 , which can be rewritten as y~ 2;k ¼ C 2;k G2;k 1 d2;k 1 þ ek
ð14Þ
where 8 ~ k 1 þv2;k ek ¼ C 2;k Φk 1 x~ k 1 þ C 2;k w > > < Φk 1 ¼ Ak 1 G1;k 1 Σ k11 C 1;k 1 > > :w ~ ¼w G Σ 1 v k1
k1
1;k 1
59
where 8 1 1 T d > > > P 11;k ¼ Σ k ðC 1;k P k C 1;k þ R1;k ÞΣ k > > 1 T > < P d12;k ¼ Σ k ðC 1;k P k Φk R1;k Σ k 1 GT1;k ÞC T2;k þ 1 M T2;k þ 1 > P d21;k ¼ ðP d12;k ÞT > > > > > d : P 22;k ¼ M 2;k þ 1 P e;k þ 1 M T2;k þ 1 Remark 1. From (8), (9), (11), and (12), we can rewrite our unknown input estimator in the following form: d^ k 1 ¼ K 1;k x^ k 1 þ K 2;k yk 1 þ K 3;k yk
k 1 1;k 1
Substituting (14) into (11) yields Eðd^ 2;k 1 Þ ¼ M 2;k C 2;k G2;k 1 E ðd2;k 1 Þ. Thus, d^ 2;k 1 is unbiased if and only if the following condition holds: M 2;k C 2;k G2;k 1 ¼ I
ð15Þ
The above condition is equivalent to rankðC 2;k G2;k 1 Þ ¼ nd nd1;k 1
ð16Þ
From (12), we can see that d^ k 1 is a linear combination of d^ 1;k 1 and d^ . Considering both (13) and (16), we know that (16) is a 2;k 1
necessary and sufficient condition for the existence of unbiased input estimation.
3.1.2. MVU input estimation Using (5)–(10), x~ k can be expressed as ~ k 1 Þ Lk v2;k x~ k ¼ ðI Lk C 2;k ÞðΦk 1 x~ k 1 þ G2;k 1 d2;k 1 þ w
ð17Þ
Considering covðv1;k 1 vT2;k Þ ¼ 0 and covðv2;k 1 vT2;k Þ ¼ 0, we can obtain ~ k 1 vT2;k Þ ¼ 0. As ~ Tk 1 Þ ¼ 0, covðx~ k 1 v~ T2;k Þ ¼ 0, and covðw covðx~ k 1 w ~ k 1 , and v2;k are mutually uncorrelated, yielding the such, x~ k 1 , w following results: 8 > P e;k ¼ Eðek eTk Þ ¼ C 2;k P~ k 1 C T2;k þ R2;k > > > > > T > ¼ Φk 1 P k 1 Φk 1 þ Q~ k 1 P~ > > < k1 1 1 Q~ k 1 ¼ G1;k 1 Σ k 1 R1;k 1 Σ k 1 GT1;k 1 þ Q k 1 > > > > R1;k 1 ¼ U T1;k 1 Rk 1 U 1;k 1 > > > > > : R2;k ¼ U T2;k Rk U 2;k where P k 1 ¼ Eðx~ k 1 x~ Tk 1 Þ.
where 8 K 1;k ¼ V 1;k 1 M 1;k 1 C 1;k 1 þ V 2;k 1 M 2;k C 2;k Φk 1 > > < K 2;k ¼ ðV 1;k 1 M 1;k 1 V 2;k 1 M 2;k C 2;k G1;k 1 M 1;k 1 ÞU T1;k 1 > > : K 3;k ¼ V 2;k 1 M 2;k U T 2;k
Both x^ k and d^ k 1 are derived from x^ k 1 , yk 1 , and yk. Estimation of the unknown input does not require any additional information. Thus, the estimation of dk 1 will not influence that of xk. As such, an unbiased state estimator can exist even if the unbiased unknown input estimate cannot be obtained [2,11,12]. □ 3.2. State estimation In Section 3.1, we have shown that unbiased estimate of d1;k 1 can be obtained from (8). Substituting d^ 1;k 1 into (5) to replace d1;k 1 , (5) and (7) form a new sub-system: ( ~ k 1 þG1;k 1 Σ k11 y1;k 1 xk ¼ Φk 1 xk 1 þ G2;k 1 d2;k 1 þ w S: y2;k ¼ C 2;k xk þ v2;k 1
where G1;k 1 Σ k 1 y1;k 1 is a deterministic term, and the unknown input d2;k 1 only affects the state. If we discard the deterministic term, sub-system S becomes the one considered by Kitanidis [17] and Darouach and Zasadzinski [3], where the output function has no unknown input. Therefore, their approaches can be used to provide optimal state estimation in a straightforward manner. In the following, we will show that optimal state estimation can be obtained by using the approaches discussed in [17,3].□ 3.2.1. Unbiased state estimation From (17) we have Eðx~ k Þ ¼ ðG2;k 1 Lk C 2;k G2;k 1 ÞEðd2;k 1 Þ. Because no information about d2;k 1 is available, (10) is an unbiased estimator of xk if and only if the following condition holds: G2;k 1 Lk C 2;k G2;k 1 ¼ 0
Theorem 1. Let x^ k 1 be unbiased, condition (16) holds and let P e;k be nonsingular, then (8), (11), and (12) form an MVU estimator of dk 1 with M 1;k 1 given by (13) and M 2;k is given by 1 1 C 2;k G2;k 1 Þ 1 GT2;k 1 C T2;k P e;k M 2;k ¼ ðGT2;k 1 C T2;k P e;k
ð18Þ
Proof. Optimal gain M 2;k can be obtained by applying the weighted least-squares estimation [14, see Chap. 2.2.3] to (14) 1 with weighting matrix P e;k . □ T
T
Considering covðd1;k 1 d2;k 1 Þ ¼ 0 and covðd2;k 1 d1;k 1 Þ ¼ 0, the estimate of dk 1 can be obtained from (12) directly. The corresponding covariance is given by 2 3 P d11;k 1 P d12;k 1 d 4 5V T Pk 1 ¼ V k 1 ð19Þ k1 P d21;k 1 P d22;k 1
ð20Þ
Condition (20) becomes the same as the unbiasedness conditions in [17,3] by replacing H k þ 1 and Gk with C 2;k and G2;k 1 , respectively. According to Kitanidis [17] and Darouach and Zasadzinski [3], (20) has a solution if the following condition holds rankðG2;k 1 Þ ¼ rankðC 2;k G2;k 1 Þ ¼ nd nd1;k 1 ;
ð21Þ
which is a sufficient condition for the existence of unbiased state estimation. In this paper, we follow Kitanidis [17] and Darouach and Zasadzinski [3], and focus on cases for which both G2;k 1 and C 2;k G2;k 1 have full column rank. Under condition (21), we can obtain the optimal gain matrix Lk, which will be shown in Section 3.2.2. Remark 2. Condition (16) will always hold if condition (21) is satisfied. We can conclude that the unbiased state estimation implicitly guarantees the unbiased input estimation. We have shown that the estimation of the unknown input will not affect the estimation of the state from the point of view of information
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H. Wang et al. / ISA Transactions 57 (2015) 57–62
requirement (Remark 1). Hence, we have the same conclusion from the analysis of unbiasedness conditions. Estimation of the unknown input does not require any additional constraint. □ Remark 3. Condition (21) is less restrictive than the full column rank condition required by Gillijns and De Moor [8]. Thus, even if Hk is rank deficient, an unbiased state estimator still exists as long as condition (21) holds. Furthermore, it is easy to verify that the proposed filter is still applicable when Hk is of full column rank. If Hk has full column rank, we are then able to obtain the estimation of dk as shown in Gillijns and De Moor [8]. But here we can estimate only dk 1 due to Hk is rank deficient, this result can be seen as a smoothed one. □ 3.2.2. MVU state estimation If (20) holds, x~ k can be rewritten as ~ k 1 Þ Lk v2;k x~ k ¼ ðI Lk C 2;k ÞðΦk 1 x~ k 1 þ w Thus P k ¼ ðI Lk C 2;k ÞP~ k 1 ðI Lk C 2;k ÞT þ Lk R2;k LTk
ð22Þ
We define R~ k ¼ C 2;k P~ k 1 C T2;k þ R2;k , and consider R~ k to be nonsingular. We then have the following theorem. Theorem 2. Let x^ k 1 be unbiased, condition (21) hold, and let R~ k be nonsingular, then (10) is a MVU estimator of xk with Lk given by T Lk ¼ P~ k 1 C T2;k N T1;k þ G2;k 1 N T2;k
ð23Þ
where 8 1 1 < N1;k ¼ R~ k N2;k GT2;k 1 C T2;k R~ k :N
1
2;k
1
¼ R~ k C 2;k G2;k 1 ðGT2;k 1 C T2;k R~ k C 2;k G2;k 1 Þ 1
Proof. Using the approach discussed in [17], we can minimize the trace of Pk under condition (20), then (23) can be obtained. □ Remark 4. Optimal state estimation can be obtained by applying the approaches discussed in [17,3] to sub-system S. It should be noted that both of the approaches require condition (21). The gain matrix of the filter proposed by Cheng et al. [2] is obtained by using Darouach's approach [3]. As such, the filter proposed by Chang et al. [2] implicitly requires condition (21). Thus, condition (15) in [2] is equivalent to condition (21) in this paper. Comparing (11) and (18) in [2] with (10) and (22) in this paper, it is clear that our state estimator is equivalent to that of Cheng et al. [2]. Although different transformation matrices are used, we obtain the same state estimator. □ Remark 5. Hsieh [12] proposed a general state estimator x^ k ¼ N k x^ k 1 þK k yk 1 þ L~ k yk with constraint rankðC 2;k U k 1 Þ ¼ rankðU k 1 Þ We can rewrite our state estimator in a compact form
Remark 6. System (5)–(7) can be recast into the following form: 32 2 3 2 3 2 3 xk xk I 0 G2;k 1 ηk 76 d 6y 7 6C 7 6 7 0 54 1;k 5 þ 4 v1;k 5 4 1;k 5 ¼ 4 1;k Σ k y2;k 0 d2;k 1 v2;k C 2;k 0 where 8 x ¼ Ak 1 x^ k 1 þG1;k 1 d^ 1;k 1 > > < k ηk ¼ Ak 1 x~ k G1;k 1 d~ 1;k 1 wk 1 > > : d~ ^ 1;k 1 ¼ d1;k 1 d 1;k 1 If we assume xk, d1;k and d2;k 1 are estimable (condition (21) is required), use the method in [18], we have the following 6-block DKF: 82 3 x^ k > > > h iT > 6 7 > > 6 d^ 1;k 7 ¼ ðT 6 ÞT Ω 1 x T yT yT 0 0 0 > > k 6;k k 1;k 2;k 5 > > 1 > > P k ¼ ðT 6k ÞT Ω6;k T 6k > > > > > : d^ k 1 ¼ V 1;k 1 d^ 1;k 1 þ V 2;k 1 d^ 2;k 1 where 8 2 3T > 0 0 0 I 0 0 > > > > 6 7 > > T 6k ¼ 4 0 0 0 0 I 0 5 > > > > > 0 0 0 0 0 I > > > 2 x > > > Pk 0 0 I > > > 6 > > 0 R1;k 0 C 1;k 6 < 6 6 0 0 R2;k C 2;k 6 > > > T T > Ω6;k ¼ 6 6 > I C C 0 > 1;k 2;k 6 > > 6 > T > 6 > 0 Σk 0 0 > 4 > > > T > > 0 0 0 G 2;k 1 > > > > x > > : P k þ 1 ¼ Ak G1;k 0 P k Ak G1;k 0 T
0
Σk
G2;k 1 0
0
0
0
0
0
0
0
0
3 7 7 7 7 7 7 7 7 7 7 5
In [13], d1;k is assumed to be estimable, but the estimation of d1;k is not provided directly. Moreover, the estimability of d2;k 1 is unclear. In fact, as shown in Section 3.1, unbiased estimation of d2;k 1 requires condition (16). Obviously, both xk and dk 1 can be estimated by using the above 6-block DKF, but may suffer from heavy computational burden because of the inverse operation of Ω6;k . Using the proposed five-step filter in this paper, it is easy and straightforward to solve the estimation problems. □ 4. Global optimality of the proposed filter As shown in Section 3, optimal solution to the problem of joint input and state estimation is assumed to take the form of a linear recursive filter under unbiasedness constraints. However, the global linear minimum-variance unbiased estimate may not lie within the recursive framework [16]. In this section, we show that the globally optimal estimators can be written in the proposed recursive form.
x^ k ¼ T 1;k x^ k 1 þ T 2;k yk 1 þT 3;k yk where 8 T 1;k ¼ ðI Lk C 2;k ÞΦk 1 > > < T 2;k ¼ ðI Lk C 2;k ÞG1;k 1 M 1;k 1 U T1;k 1 > > : T 3;k ¼ Lk U T 2;k
If we choose U k 1 ¼ G2;k 1 , Nk ¼ T 1;k , K k ¼ T 2;k , and L~ k ¼ T 3;k , it is clear that the proposed state estimator is equivalent to the state estimator (GOUMVF) in [12]. □
4.1. Optimality of the input estimator Consider the most general linear combination of measurements and the mean of initial state d^ k 1 ¼ F 0 x^ 0 þ
k X
Γ i yi
ð24Þ
i¼0
where F0 and Γi are combination coefficients. Because x^ i is a linear combination of x^ 0 ; y0 ; y1 ; …; yi , and both y~ 1;i and y~ 2;i can be written
H. Wang et al. / ISA Transactions 57 (2015) 57–62
as linear combinations of y0 ; y1 ; …; yi , an equivalent and convenient form of (24) can be expressed as d^ k 1 ¼ F 0 x^ 0 þ V 1;k 1 M 1;k 1 y~ 1;k 1 þ V 2;k 1 M 2;k y~ 2;k þ
kX 2
Γ 1;i y~ 1;i þ
i¼0
kX 1
Γ 2;i y~ 2;i
ð25Þ
i¼0
where Γ 1;i and Γ 2;i are coefficients. Γ k 1 ¼ V 1;k 1 M 1;k 1 and Γ k ¼ V 2;k 1 M 2;k , respectively. Necessary and sufficient conditions for (25) to be an unbiased estimator of dk 1 are given in the following theorem. Theorem 3. The unknown input estimator (25) is unbiased if and only if the following conditions are satisfied: (i) (ii) (iii) (iv) (v)
1
M 1;k 1 ¼ Σ k 1 ; M 2;k C 2;k G2;k 1 ¼ I; Γ 1;i ¼ 0; ði ¼ 0; 1; ⋯; k 2Þ; Γ 2;i C 2;i G2;i 1 ¼ 0; ði ¼ 0; 1; ⋯; k 1Þ; F 0 ¼ 0.
As a result of Theorems 3 and 4, estimator (25) can be easily written in the recursive form described by (8), (11), and (12). This means that (12) form a MVU estimator of dk 1 that minimizes the mean square error over the class of all linear unbiased estimates based on x^ 0 and measurements sequence y0 ; y1 ; …; yk . 4.2. Optimality of the state estimator Similarly, we can prove that the state estimation with minimum mean square error over the class of all linear unbiased estimates based on x^ 0 and measurements sequence y0 ; y1 ; …; yk can be written in the proposed recursive form. Consider the most general linear combination of measurements and the mean of initial state k X
Γ x;i yi
i¼0
where Fx and Γ x;i are combination coefficients, we have the following theorem.
Eðd^ k 1 Þ ¼ F 0 x^ 0 þ ðM d1;k 1 þ M d2;k 1 ÞEðdk 1 Þ kX 2 kX 1 Γ 1;i Σ i Eðd1;i Þ þ Γ 2;i C 2;i G2;i 1 Eðd2;i 1 Þ þ
Theorem 5. Let condition (21) hold and Lk be given by (23), then (8), (9), and (10) form an unbiased estimator of xk that minimizes the mean square error over the class of all linear unbiased estimates based on x^ 0 and measurements sequence y0 ; y1 ; …; yk .
i¼0
where 8 < M d1;k 1 ¼ V 1;k 1 M 1;k 1 Σ k 1 V T1;k 1
Proof. In Remark 4, we have shown that the proposed state estimator is equivalent to that in [2]. The proof is similar to that of Theorem 1 in [2] and is omitted for brevity. □
: M d2;k 1 ¼ V 2;k 1 M 2;k C 2;k G2;k 1 V T2;k 1 Sufficiency: If all five conditions are satisfied, we can obtain Eðd~ k 1 Þ ¼ 0, where d~ k 1 ¼ dk 1 d^ k 1 . Necessity: We assume that (25) is an unbiased estimator of dk 1 . Because no prior information about the unknown input is available, we can conclude that Eðd~ k 1 Þ ¼ 0, if and only if M d1 þ M d2 ¼ I;
ð26Þ
Γ 1;i Σ i ¼ 0 ði ¼ 0; 1; …; k 2Þ;
ð27Þ
Γ 2;i C 2;i G2;i 1 ¼ 0 ði ¼ 0; 1; …; k 1Þ;
ð28Þ
F 0 ¼ 0:
ð29Þ V T1;k 1
and V 2;k 1 , respectively, Multiplying both sides of (26) by we obtain M 1;k 1 ¼ Σ k11 and M 2;k C 2;k G2;k 1 ¼ I. Condition (27) implies Γ 1;i ¼ 0, and conditions (28) and (29) satisfy (iv) and (v) directly. □ Theorem 4. Let (25) be an unbiased estimator of dk 1 , then the mean square error Eð‖d~ k 1 ‖22 Þ achieves a minimum when Γ 2;i ¼ 0, ði ¼ 0; 1; …; k 1Þ. Proof. If estimator (25) is unbiased, then d~ k 1 can be written as a linear combination of ξk 1, ek, and y~ 2;i ði ¼ 0; 1; …; k 1Þ: 1 d~ k 1 ¼ V 1;k 1 Σ k 1 ξk 1 V 2;k 1 M 2;k ek
that the mean square error Eð‖d~ k 1 ‖22 Þ achieves a minimum when
Γ 2;i ¼ 0 ði ¼ 0; 1; …; k 1Þ. □
x^ k ¼ F x x^ 0 þ
Proof. From (25), we can obtain the mean of d^ k 1
i¼0
61
kX 1
Γ 2;i y~ 2;i
i¼0
Considering y~ 2;i ði ¼ 0; 1; …; k 1Þ is uncorrelated with ξk 1 and ek, we can obtain 2 3 kX 1 P d11;k 1 P d12;k 1 d 4 5V T þ Γ 2;i Eðy~ 2;i y~ T2;i ÞΓ T2;i Pk 1 ¼ V k 1 k1 P d21;k 1 P d22;k 1 i¼0 where P d11;k 1 , P d12;k 1 , P d21;k 1 , and P d22;k 1 are defined by (19). Because no information about d2;i 1 is available, we can conclude
5. Illustrative example To illustrate the performance of the proposed filter, the numerical example given in [2] is considered, where the system parameters are given as follows: 2 3 2 3 0:5 2 0 0 0 1 0 0:3 7 6 0 0:2 1 6 0 1 7 0 7 6 61 0 7 6 7 6 7 7; G ¼ 6 0 0 7; 0 0 0:3 0 1 0 A¼6 6 7 6 7 6 7 6 7 0 0 0:7 1 5 0 5 4 0 40 0 0 0 0 0 0:1 0 0 0 2 3 2 3 1 0 0 0:5 0 0 0 1 6 0 60 0 07 1 0 0 0:3 7 6 7 6 7 6 7 6 7 2 7; 7; R ¼ 10 6 0 0 1 0 0 0 1 0 H¼6 6 7 6 7 6 7 6 7 0 5 4 0:5 0 0 1 40 0 05 0 0:3 0 0 1 0 0 0 2 3 1 0 0 0 0 6 0 1 0:5 0 0 7 6 7 6 7 7 Q ¼ 10 4 6 6 0 0:5 1 0 0 7; C ¼ I 5 : 6 7 0 0 0 1 0 4 5 0 0 0 0 1 As H is rank deficient, the unknown input estimators proposed in [4,8] cannot guarantee the unbiasedness, thus, their state estimators may become biased. The filters proposed by Cheng et al. [2] and Hsieh [12] cannot provide the unknown input estimation. Although [11] provided optimal state estimation, the unknown input estimator (ERTSF) suffers from performance degradation because the unbiasedness condition of the unknown input estimation is not satisfied. Considering the filter proposed in this paper, we can verify that condition (21) holds. This example has three unknown inputs, and we choose three typical signals (square wave, sinusoid, and ramp)
62
H. Wang et al. / ISA Transactions 57 (2015) 57–62
deficient. The real and estimated values of the unknown inputs are presented in Fig. 2. It is clearly shown that all the three unknown inputs can be estimated. Fig. 3 shows the traces of the covariance matrices Pk and Pdk. The convergence of the proposed filter is verified.
6. Conclusions
2
Fig. 1. State estimation.
In this paper, a five-step filter with global optimality is proposed to solve the joint input and state estimation for linear systems with a rank-deficient direct feedthrough matrix. The relationship between input estimation and state estimation has been clearly illustrated. In fact, the unbiased state estimation implicitly guarantees the unbiased input estimation. This result enables us to estimate both the state and the unknown input, regardless of their mutual coupling. Although the proposed filter is designed for the rank deficient case, it is still applicable when Hk has full column rank.
Acknowledgments This work was supported by the National Natural Science Foundation of China (No. 61273145, 61273146), the 863 Program of China (No. 2014AA041802), and the 973 Program of China (No. 2012CB720503). References
Fig. 2. Unknown input estimation.
Fig. 3. Traces of Pk and Pdk.
as unknown inputs for testing. We then initialize the filter as: x^ 0 ¼ x0 ¼ 0, P 0 ¼ I 5 , and P d0 ¼ I 3 . The real and estimated values of the first two states are plotted in Fig. 1. This illustrates that xk can be estimated even if H is rank
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