Scholarly article on topic 'On existence, optimality and asymptotic stability of the Kalman filter with partially observed inputs'

On existence, optimality and asymptotic stability of the Kalman filter with partially observed inputs Academic research paper on "Mathematics"

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Abstract of research paper on Mathematics, author of scientific article — Jinya Su, Baibing Li, Wen-Hua Chen

Abstract For linear stochastic time-varying systems, we investigate the properties of the Kalman filter with partially observed inputs. We first establish the existence condition of a general linear filter when the unknown inputs are partially observed. Then we examine the optimality of the Kalman filter with partially observed inputs. Finally, on the basis of the established existence condition and optimality result, we investigate asymptotic stability of the filter for the corresponding time-invariant systems. It is shown that the results on existence and asymptotic stability obtained in this paper provide a unified approach to accommodating a variety of filtering scenarios as its special cases, including the classical Kalman filter and state estimation with unknown inputs.

Academic research paper on topic "On existence, optimality and asymptotic stability of the Kalman filter with partially observed inputs"

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On existence, optimality and asymptotic stability of the Kalman filter with partially observed inputs*

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Jinya Sua, Baibing Lib-1, Wen-Hua Chen

a Department of Aeronautical and Automotive Engineering, Loughborough University, Loughborough LE11 3TU, UK b School of Business and Economics, Loughborough University, Loughborough LE11 3TU, UK

article info abstract

Article history: For linear stochastic time-varying systems, we investigate the properties of the Kalman filter with partially

Received 7 May 2014 observed inputs. We first establish the existence condition of a general linear filter when the unknown

Receiwd in reused foi-m inputs are partially observed. Then we examine the optimality of the Kalman filter with partially observed

toeped^O^oenibei- 2014 inputs. Finally, on the basis of the established existence condition and optimality result, we investigate

p asymptotic stability of the filter for the corresponding time-invariant systems. It is shown that the results

on existence and asymptotic stability obtained in this paper provide a unified approach to accommodating

Keywords- a variety of filtering scenarios as its special cases, including the classical Kalman filter and state estimation

Asymptotic stability with unknown inputs.

Existence © 2014 The Authors. Published by Elsevier Ltd.

Kalman filter This is an open access article under the CC BY license

Optimality (http://creativecommons.org/licenses/by/4.0/). Unknown inputs

1. Introduction

State estimation plays an important role in state space modelling and control. It has been applied to a wide range of areas; see Li (2009) and Liang, Chen, and Pan (2010) for some recent applications in network control systems, transportation management, etc.

In the recent decades, state estimation for discrete-time linear stochastic systems with unknown inputs (also termed as unknown input filtering (UIF) problem) has received considerable attention since the original work of Kitanidis (1987) first appeared. Various filters were developed under different assumptions for the systems with unknown inputs; see, e.g., Cheng, Ye, Wang, and Zhou (2009), Darouach and Zasadzinski (1997), Darouach, Zasadzinski, and Xu (1994), Fang and Callafon (2012), Gillijns and De Moor (2007), Hsieh (2000,2010) and Kitanidis (1987), among many others. Most of these researches used the technique of minimum variance unbiased estimation, hence leading to an unbiased minimum-variance

* This workwasjointlyfundedby UKEngineering and Physical Sciences Research Council (EPSRC) and BAE Systems (EP/H501401/1). The material in this paper was not presented at any conference. This paper was recommended for publication in revised form by Associate Editor Tongwen Chen under the direction of Editor Ian R. Petersen.

E-mail addresses: J.Su2@lboro.ac.uk (J. Su), B.Li2@lboro.ac.uk (B. Li), w.chen@lboro.ac.uk (W.-H. Chen).

Tel.: +44 1509 228841; fax: +44 1509 223960.

filter (UMVF). Another important research line is state estimation for descriptor systems (Hsieh, 2011, 2013). It has been recently shown in Hsieh (2013) that any linear descriptor systems can be transformed into a linear stochastic system with unknown inputs. This shows a close link between these two kinds of problem. In addition, various properties for these developed filters have been investigated, including the existence condition (Darouach & Zasadzinski, 1997), asymptotic stability (Fang & Callafon, 2012), and global optimality of the UMVF (Cheng et al., 2009 and Hsieh, 2010).

Recently, Li (2013) has developed a Kalman filter for linear systems with partially observed inputs, where the inputs are observed not at the level of interest but rather the input information is available at an aggregate level. It has been shown that the developed filter provides a unified approach to state estimation for linear systems with Gaussian noise. In particular, it includes two important extreme scenarios as its special cases: (a) the filter where all the inputs are completely available (i.e. the classical Kalman filter; see, e.g., Simon, 2006); and (b) the filter where all inputs are unknown (i.e. the filter investigated in Kitanidis, 1987 and many others for the UIF problem). Potentially the proposed filter can be applied to a variety of practical problems in many different areas such as population estimation and traffic control.

So far there is not any study discussing the existence and asymptotic stability issues of this newly proposed unified filter. In this paper we investigate the properties of the Kalman filter with partially observed inputs developed in Li (2013). For linear stochastic time-varying systems with partially observed inputs, we

http://dx.doi.org/10.1016/j.automatica.2014.12.044

0005-1098/© 2014 The Authors. Published by Elsevier Ltd. This is an open access article underthe CC BY license (http://creativecommons.org/licenses/by/4.0/).

establish the existence condition for a general linear filter. Then we show that the developed filter is optimal in the sense of minimum error covariance matrix. Finally, we consider asymptotic stability of the filter for the corresponding time-invariant systems based on the established existence condition and optimality result.

This paper has provided a unified approach to accommodating existence and asymptotic stability conditions in a variety of filtering scenarios: it includes the results on existence and asymptotic stability for some important filters as its special cases, e.g., the filters developed for the problems where the inputs are completely available and where all the inputs are unknown. Note that the former is the classical Kalman filtering problem and the corresponding existence and asymptotic stability conditions are well established in the literature. For the latter case with unknown inputs, there has been a continuing research interest in existence and asymptotic stability conditions for various discrete-time systems (e.g., Cheng et al., 2009, Darouach & Zasadzinski, 1997, Fang & Callafon, 2012, Kitanidis, 1987) and continuous-time systems (e.g., Bejarano, Floquet, Perruquetti, & Zheng, 2013, Corless &Tu, 1998, Hou&Muller, 1992).

This paper is structured as follows. First, Section 2 is devoted to problem statement. Then we establish the existence condition in Section 3. We focus on the properties of the filter proposed in Li (2013) in Section 4. In Section 5, we investigate asymptotic stability. Finally, this paper concludes in Section 6.

2. Problem statement

Consider a linear stochastic time-varying system:

Xk+i = Akxk + Gkdk + a>k yk = CkXk + Uk,

where xk e Rn is the state vector, dk e Rm is the input vector, and yk e Rp is the measurement vector at each time step k with p > m and n > m. The process noise wk e Rn and the measurement noise uk e Rp are assumed to be mutually uncorrelated with zero-mean and a known covariance matrix, Qj( = E[rnkai[] > 0 and Rk = E[ukuJ( ] > 0, respectively. Ak, Gk and Ck are known matrices. Without loss of generality, we follow Gillijns and De Moor (2007) and Kitanidis (1987), and assume that Gk has a full column-rank. The initial state x0 is independent of wk and uk with a known mean x0 and covariance matrix P0 > 0.

We consider the scenario where the input vector dk is not fully observed at the level of interest but rather it is available only at an aggregate level. Specifically, let Dk be a qk x m known matrix with 0 < qk < m and F0k an orthogonal complement of DTk such that

DkFok = O,

Qk x (m-qk)

and F0'kFok = I,

k, where O and I represent

the zero matrix and identity matrix of appropriate dimensions. We suppose that the input data is available only on some linear combinations:

rk = Dkdk,

where rk is available at each time step k. Dk is assumed to have a full row-rank; otherwise the redundant rows can be removed.

As pointed out in Li (2013), the matrix Dk characterizes the availability of input information at each time step k. It includes two extreme scenarios that are usually considered: (a) qk = m and Dk is an identity matrix, i.e. the complete input information is available; this is case that the classical Kalman filter can be applied; (b) qk = 0, i.e. no information on the input variables is available; this is the problem investigated in Darouach and Zasadzinski (1997), Gillijns and De Moor (2007), Hsieh (2000) and Kitanidis (1987).

Throughout this paper, we use X(B) to denote any eigenvalue of a square matrix B. For any two symmetric matrices A and B with suitable dimensions, the notation A > B is used if and only if

A — B is non-negative definite. In addition, we use Gk to denote an orthogonal complement of Gk and = [Gk, Gk]. Define

Dk-i CkGk-1

3. Existence condition

To establish the existence condition of a general linear filter for system (1) and (2), we first consider an invertible linear transformation.

3.1. Transformation

Consider the following invertible matrix:

(n-m)xm FT

qkx(n-m) In m

(m-qk)x(n-m)

It is straightforward to verify that MkGkdk can be expressed as: MkGkdk = [DTk , Omx(n-m), Fok]Tdk

= [(Dkdk)T, (O(n-m)x.mdk)T, (FTokdk)T]T = [rk , O1x(n-m), (F0kdk)T]T

= h + G kSk, (4)

where h = [r{, OVx(n-m), OVx(m-qk)]T, Sk = FOkdk and Gk = [O(m-qk)xqk, O(m-qk)x(n-m), ¡m-qk f. We note that rk is completely available due to Eq. (2).

Left-multiplying both sides of Eq. (4) by M-1, Gkdk can be decoupled into two parts:

Gkdk = M—1fk + M—1GkSk.

From Eq. (5), the dynamics of xk+1 can be rewritten as:

Xk+1 = AkXk + M—1fk + M—1 GkSk + Wk

= AkXk + Uk + FkSk + a>k, where uk = M—1rk is a known term, and Fk is given by

Fk = M,-1Gk = [Gk, Gjr]

Consequently, linear system (1) with the partially observed inputs rk = Dkdk can be equivalently represented by the following system:

Xk+i = AkXk + Uk + FkSk + a>k yk = CkXk + Uk.

The above manipulation shows that a linear stochastic system with partially observed inputs (2) is equivalent to a linear system with unknown inputs; similar property is also found for linear descriptor systems (Hsieh, 2013).

3.2. Existence condition

In this subsection, we will establish the existence condition of a general, asymptotically stable and unbiased linear filter for system (7) and hence for its equivalent system, Eqs. (1) and (2).

Motivated by the linear filter structure in the literature (e.g. Darouach et al., 1994), we consider a general linear filter for discrete-time linear system (7) of the form

xk+1 = Ekxk + Jkuk + Kk+1yk+1,

where the gain matrices Ek, Jk and Kk+1 are to be designed. Based on (7) and (8), one can obtain the error dynamics ek+1 = xk+1 — xk+1:

ek+1 = (Ax + uk + FkSk + cok) — (EkXk + Jkuk + Kk+1yk+1)

= Ekek — (Jk — I + Kk+1Ck+1)Uk

+ (Ak — Kk+1Ck+1Ak — Ek)xk — (Kk+1Ck+1Fk — Fk)Sk

+ (I — Kk+1 Ck+1)«k — Kk+1 Vk+1.

To ensure the filter is unbiased, it is required that the filtering error is independent of uk, xk and Sk. In addition, it is expected that the error approaches to zero as time k increases. Hence the existence condition for filter (8) is given by:

(i) Ek is stable (i.e., any eigenvalue of Ek satisfies |A.(Ek)| < 1);

(ii) Ek = Ak — Kk+1Ck+1Ak;

(iii) Kk+1Ck+1Fk = Fk;

(iv) Jk = I — Kk+1Ck+1.

For system (1)-(2), however, the existence condition for system (7) should be expressed in terms of matrices Ak, Gk, Ck and Dk. For this end, we first state a lemma.

Lemma 1. For system (1)-(2), we have

'zln - Ak -Gk Ck+i O . o Dk j

= rank

zln - Ak -Fk Ck+i O

+ rank(DkDTk ).

See the Appendix for proof. We now provide a condition for the existence of a general linear filter for a dynamic system with partially observed inputs.

Theorem 1. Suppose that both matrices Djt and Gk have a full column-rank. Then a sufficient condition for the existence of a general linear filter (8) for system (1)-(2) is given by:

rank(nk+1) = m (9)

andforallz e c (c is the field of complex numbers) such that |z | > 1:

'zln - Ak -Gk

Ck+1 O . O Dk .

= n + m.

Proof. We note that we can select matrices Ek = Ak - Kk+1Ck+1Ak andJk = I -Kk+1Ck+1 to ensure that condition parts (ii) and (iv) are satisfied. Hence, we will focus on condition parts (i) and (iii). We first show that Eq. (9) guarantees there exists a matrix Kk+1 such that condition part (iii) holds. We note

Dk Ck+ 1Gk

[Fok DTk]

Ikx(m-qk) Ck+1GkF0k

DkDk Ck+1GkDk

Since [F0k, D\] is invertible and nk+1 has a full column-rank, we obtain that Ck+1GkF0k = Ck+1Fk (see Eq. (6)) is also of full column-rank, i.e.

rank(Ck+1Fk) = m - qk.

Eq. (12) guarantees there exists a matrix Kk+1 such that condition part (iii) holds.

Next, since Ck+1Fk has a full column-rank, there exists an invertible matrix Nk e Rpxp such that

NkCk+1Fk =

(p-m+qk)x.(m-qk)

The general solution Kk+1 of Kk+1Ck+1Fk = Fk is given by Kk+1 = [rk, Fk]Nk, where rk can be any matrix of suitable dimensions and is to be designed for the gain matrix Kk+1. Now define S1k and S2k such that

S1k S2k

= NkCk+1Ak. Then from condition part (ii), we can obtain

Ek = Ak - Kk+1 Ck+1Ak

= Ak - [rk, Fk]NkCk+1Ak = Ak - [rk, Fk] = Ak - FkS2k - rkS1k.

S1 k S2k

According to Anderson and Moore (1979, p. 342), the existence condition part (i) holds if and only if the following equivalent conditions hold:

(a) Ak — FkS2k — rkS1k is stable for a matrix rk;

(b) S1kn = 0 and (Ak — FkS2k)n = Xq for some constant X and vector n implies |X| < 1 or n = 0.

The condition (b) can be expressed in the following equivalent form for all z e c and |z| > 1:

zln - Ak + FkS2k

The following identity, in conjunction with Lemma 1, shows that Eq. (15) is satisfied:

zln - Ak -Fk

Ck+1 o

= rank = rank = rank

= rank

= rank

= rank = rank

O zln - Ak -Fk

zln - Ak -Fk Ck+1Ak Ck+1 Fk

zln - Ak

Ck+1 Ak

-Fk Ck+1 Fk

'zln - Ak -Fk

zln - Ak + FkS2k Fk

S1 k O

Zln - Ak + FkS2k O

zln - Ak + FkS2k S1 k

m-qk J

+ m - qk.

Hence, Eqs. (9) and (10) guarantee there exists a gain Kk+1 such that: (a) Kk+C+1Fk = Fk; and (b) Ek = Ak — Kk+C+A is stable. This completes the proof.

Remarks. (i) Eq. (9) is the estimability condition for the filter developed in Li (2013) for system (1) with partially observed inputs (2). From the proof of Theorem 1, it also guarantees the unbiasedness of a general linear filter. In addition, Theorem 1 shows that to ensure the estimation error of a general linear filter is stable as time k increases, a detectability condition (10) needs to be met.

(ii) When condition parts (i)-(iv) are satisfied, the general linear filter (8) is given by

Xk+i = (Ak - Kk+iCk+iAk)Xk + (I - Kk+iCk+i)Uk + Kk+1yk+1.

(iii) The error dynamics of the above filter (i6) that satisfy condition (i)-(iv) become

ek+i = (Ak - Kk+iCk+iAk)ek

+ [I - Kk+iCk+i, -Kk+i][ojl(, Uk+i]T.

3.3. Relationships with the existing filters

As mentioned earlier, system (1) with partially observed inputs (2) includes two important scenarios as its special cases: (a) the complete input information is available; and (b) no information on the input variables is available. In this subsection, we compare the developed existence condition in the previous subsection for partially observed inputs to the condition derived for the classical Kalman filter with complete information on the inputs, and to that of the filter with unknown inputs.

Theorem 2. The proposed existence condition for filter (8) in Theorem 1 reduces to: (a) the existence condition of the classical Kalman filter when the complete information on the inputs is available, i.e., Dk is invertible; and (b) the existence condition of the filter with unknown inputs, i.e. Dk is an empty matrix.

Proof. First, we consider the case that matrix Dk is invertible. It is clear that Eq. (9) is satisfied due to the non-singularity of Dk. In addition, we have

'zIn - Ak -Gk Ck+i O . O Dk j

= rank

zIn - Ak Ck+i O

O O Dk

= rank

zIn - Ak

+ rank(DkDTk ).

Since rank(DkDT) = m, the existence condition (i0) reduces to

zIn - Ak

= n, Wz G c, \z\ > i

which is the detectability condition of the classical Kalman filter (see, e.g. Anderson & Moore, 1979; Simon, 2006).

Next, we turn to the scenario where no information on the inputs dk is available. Since matrix Dk reduces to a zero-by-zero empty matrix in this case, Eq. (9) becomes

rank (Ck+iGk) = m. In addition, Eq. (i0) reduces to

zIn - Ak Gk Ck+i O

= n + m, Wz g c, \z\ > i.

Eqs. (i8)-(i9) are identical to the results for the filter with unknown inputs (Darouach & Zasadzinski, i997). This completes the proof.

Theorem 2 shows that the obtained existence condition in this paper is a more generic condition. In addition, comparing the existence condition (9) and (i0) of the general linear filter (8) for systems with partially available inputs to the existence condition (i8)-(i9), it can be seen that partial information on the unknown inputs has relaxed the existence condition of a general linear filter. In other words, with the information on the unknown inputs at an

aggregate level (2), it is more likely that the general linear filter (8) exists.

4. The filter with partially observed inputs

Now we focus on the filter proposed in Li (2013) for linear stochastic systems when the inputs are partially observed. Note that this filter was derived under the Bayesian framework with the assumption that wk and uk follow a Gaussian distribution, and Sk has a noninformative prior distribution. We summarize the results of the filter below. Define

(n-m)xm

qk x(n-m) In-m

Let Mk = Dktt—1. It is shown in Li (2013) that for system (1) with the input data available at an aggregate level (2), if matrix nk has a full column-rank, then the posterior distribution for xk at any time step k is a Gaussian distribution with posterior mean xkk and posterior covariance matrix Pkik given by:

Xk\k = Ak-iXk-i\k-i + Pk\kMh(Mk-iPm-iMl-i) i

X fk-i + Kk(yk - CkAk-iXk-i\k-i),

Pk\k = Pk\k-i - Pk\k-iCkHk iCkPk\k-i + W<-i

- Pk\k-iCkHk lCkFk-i][Fk-iCkHk lCkFk-i] i

X [Fk-i - Pk\k-iCkHk lCkFk-i]T

Kk = Pk\k-iCkHk i + [Fk-i - Pk\k-iCkHk i CkFk-i] x [Fk-1 C!<Hk lCkFk-iT1 Fk-1 C!(H- i,

rk = [rT, OT]T, Pk\k-i = Ak-iPk-i\k-iATk-1 + Qk-1 and Hk = CkPk\k-1CjT + Rk > 0. Note that Eq. (9) guarantees Eq. (12) holds, and hence Fl-iC^CkFk -i is invertible in the above equations.

Under the Bayesian framework, xk\k was shown to be a minimum mean square error (MMSE) estimate in Li (2013). However, no further properties of the filter were explored.

We now derive the dynamics of the state estimation error ek = xk - xk\k.

Lemma 2. The estimation error ek = xk - xk\k of the filter (20)-(22) follows the recursive equation

ek = (Ak-i - KkCkAk-i)ek-i + [I - KI(CI(, -Kl(][ol(-i, v,(]T, (23) where Kk is given by Eqs. (21)-(22).

Proof. Let Wk-i = Pk\kMl-i(Mk-iPk\k-iMT-i)-1. The error dynamics of the filter (20)-(22) are given by

ek = Ak-1xk-1 + Gk- 1dk-1 + ok-1 - Ak-1xk-1 \ k-1

- Wk-ifk-1 - Kk(yk - CkAk-iXk-i\k-i)

= (Ak-i - KkCkAk-i)ek-i + (Gk-i - KkCkGk-i)dk-i

- Wk-ih-i + (I - KkCk)wk-i - KkVk.

Noting that fk-1 = Mk-1Gk-1dk-1, we obtain (Gk-i - KkCkGk-i)dk-i - Wk-ih-i = [I - KkCk - Wk-iMk-i]Gk-idk-i. (24)

Inserting (21) and (22) into (24), we can obtain (23) by noting that I — KkCk — Wk—1Mk—1 = O.This completes the proof.

Lemma 2 shows that, for the gain Kk given in Eqs. (21)-(22), if Ak—1 — KkCkAk—1 is stable, the error of the developed filter in Li (2013) will be stable as time k increases. In addition, the estimation error Eq. (23) shares the same structure as that of Eq. (17), upon which we can conclude that the filter (20)-(22) falls into the filter family with the generic linear structure Eq. (8).

We now consider the error covariance matrix P,

k | k-

Theorem 3. Let Pk|k denote the error covariance matrix of any filter Xk(Yk) based on the sequence of measurements Yk = {y0, y1,..., yk}. Then for linear system (1) with partially observed inputs (2), we have Pk|k > Pk|k, where Pk|k is given byEq. (21).

Proof. By definition, the conditional covariance matrix of the estimate xk(Yk) for given Yk is

Pm = E{[xk — xk(Yk)][xk — xl((Yl()]T | Yk}.

It is easy to verify the following identity:

Pkk = E{[xk — xkk + xm — xk(Yk)]

x [xk — xkk + xkk — xk(Yk)]T Y}

= Pkk + E {[xkk — xYmxkk — xl((Yk)]T | Yk}

+ E {[xk — xm][xkk — xk (Yk)]T Y}

+ E {¿Xkk — xk(Yk)][xk — xm ]T Y}.

Li (2013) shows that the estimated state vector xx^ in Eq. (20) is the posterior mean conditional on the sequence of measurements Yk = {y0, y1,..., yk}. Hence, we have E^Yk} = x^ and the last two terms on the right-hand side of the above equation vanish, i.e.

Pkk< = Pkk + E {[hk< — xc(Yk)][hk< — h(Yk)]T Yk}.

We thus conclude that P^ attains the minimum if and only if the second term of the right-hand side is equal to zero, i.e. xk(Yk) = xk|k. This completes the proof.

Theorem 3 shows that the filter given by Eqs. (20)-(22) is optimal in the sense of both MMSE and minimum covariance matrix. This result is not only important in its own right but also useful in the subsequent asymptotic stability analysis.

5. Asymptotic stability

In this section, we discuss the asymptotic stability of the filter developed in Li (2013) for time-invariant system (1) and (2). We hence suppress the subscript k of matrices Ak, Gk, Ck, Dk, Qj( and

We note from Lemma 2 that the covariance matrix in Eq. (21) can be re-written as

Pkik = (A - KkCA)Pk-iik-i(A - KkCA)T + (I - KkC)Q(I - KkC)T + KkRKT.

Under the condition given in Theorem 1 and in conjunction with Theorem 3 that the covariance matrix of the filter given by Eqs. (20)-(22) is optimal, it can be shown that the covariance matrixPkk in Eq. (25) is bounded for all k and for an arbitrary bounded initial covariance Po|o. On the basis of boundedness of Ptc^ and inspired by the approaches in Anderson and Moore (1979) and Fang and Callafon (2012), we can further show the following result. The proof is omitted here for lack of space and is available upon request.

Theorem 4. If the condition in Theorem 1 is satisfied and (A, Q 2) is stabilizable, then the covariance matrix Pkk of the filter (20)-(22) will converge to a unique fixed positive semi-definite matrix P for any given initial condition P0|0. Moreover, with the associated limiting gain matrices K, the time-invariant filter is also stable, i.e. all the eigenvalues of A — KCA satisfy |X(A — KCA) | < 1.

It is of interest to compare the asymptotic stability condition obtained with partially observed inputs to the asymptotic stability conditions when the complete information on the inputs is available and when the inputs are completely unknown. This is investigated in the following theorem. It shows that Theorem 4 provides a unified approach to accommodating asymptotic stability conditions in a variety of filtering scenarios.

Theorem 5. The asymptotic stability condition for the filter (20)-(22) in Theorem 4 reduces to: (a)the asymptotic stability condition of the classical Kalman filter when the complete information on the inputs is available, i.e. D is invertible; and (b) the asymptotic stability condition of the filter with unknown inputs, i.e. D is an empty matrix.

Proof. First, we note that when matrix D is invertible, the asymptotic stability condition reduces to: (a) (A, C) is detectable; and (b) (A, Q 2) is stabilizable. These are the asymptotic stability conditions of the classical Kalman filter (see, e.g. Anderson & Moore, 1979).

Next, when no information on the inputs is available, we know from Theorem 2 that Eq. (10) in Theorem 1 reduces to Eq. (19). In addition, condition (A, Q1) along with R > 0 (and hence R1 > 0) can guarantee that the matrix below has a full row-rank, i.e.,

A - e>wI

= n + p, Ww G [0, 2n].

Eqs. (19) and (26) are identical to the asymptotic stability condition for the filter with unknown inputs (Darouach & Zasadzinski, 1997). This completes the proof.

6. Conclusions

This paper has established existence and asymptotic stability conditions for the recently developed filter with partially observed inputs in Li (2013). The obtained existence and asymptotic stability conditions provide a unified approach to accommodating a variety of filtering scenarios as its special cases, including the important Kalman filtering and the unknown input filtering problems. In practice, information on inputs and/or outputs may sometimes be only partially available in applications. This work takes a further step towards the development of more generic filtering techniques where different levels of information are exploited.

Acknowledgements

We thank the associate editor and reviewers for their very helpful comments on the earlier versions of this paper.

Appendix. Proof of Lemma 1

"zIn - Ak -Gk

Ck+i O . O Dk .

'zIn - Ak -Gk

Ck+1 O . O Dk .

[Fok Dk ]

= rank

= rank

zln — Ak -GkF0k GkDk

Ck+1 O

O DkF0k

zIn - Ak -Fk Ck+i O

+ rank (DkD I).

References

Jinya Su received his B.Sc. degree in the School of Mathematics and Statistics from Shandong University, Weihai, China in 2011. He is currently a Ph.D. candidate in the Department of Aeronautical and Automotive Engineering, Loughborough University, Loughborough, UK. His research interests include disturbance observer design and its applications in fault diagnosis, disturbance rejection control, etc.

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Baibing Li received a B.Sc. degree from Yunnan University, Kunming, China, an M.Sc. degree from Shanghai Jiao Tong University, Shanghai, China, and an M.Sc. degree from Vrije Universiteit Brussel, Brussels, Belgium. In 1991, he received a Ph.D. degree from the Management School, Shanghai Jiao Tong University. He was a Postdoctoral Research Fellow with Katholieke Universiteit Leuven, Leuven, Belgium, and a Research Associate with Newcastle University, Newcastle upon Tyne, UK. In 2001, he was appointed as a Lecturer at Newcastle University. In 2004, he moved to the School of Business and Economics, Loughborough University, Loughborough, UK, as a Lecturer, where he was subsequently appointed as a Reader in 2007 and a Professor in 2011. His current research interests cover Bayesian statistical modelling and forecasting for Gaussian and non-Gaussian dynamic problems in various management areas. In recent years, much of his work has also involved transport and traffic management such as transportation demand analysis, travel behaviour modelling, and intelligent transportation systems.

He is a member of IEEE and a member of the Royal Statistical Society.

Wen-Hua Chen received the M.Sc. and Ph.D. degrees from Northeast University, Shenyang China, in 1989 and 1991, respectively. From 1991 to 1996, he was a Lecturer and then Associate Professor with the Department of Automatic Control, Nanjing University of Aeronautics and Astronautics, Nanjing, China. From 1997 to 2000, he held a research position and then a Lecturer in control engineering with the Centre for Systems and Control, University of Glasgow, Glasgow, UK. In 2000, he moved to the Department ofAeronautical and Automotive Engineering, Loughborough University, Loughborough, UK, as a Lecturer, where he was appointed as a Professor in 2012. His research interests include the development of advanced control strategies (Nonlinear Model Predictive Control, Disturbance Observer Based Control, etc.) and their applications in aerospace engineering. Currently, much of his work has also involved in the development of Unmanned Autonomous Intelligent Systems.

He is a Senior Member of IEEE and a Fellow of IET.