# Identifiability Analysis of Noise Covariances for LTI Stochastic Systems
with Unknown Inputs

He Kong, Salah Sukkarieh, Travis J. Arnold, Tianshi Chen, Biqiang Mu, and Wei
Xing Zheng This paper is formally accepted to and going to appear in IEEE
Transactions on Automatic Control. Corresponding author: He Kong.He Kong is
with the Shenzhen Key Laboratory of Biomimetic Robotics and Intelligent
Systems, Department of Mechanical and Energy Engineering, Southern University
of Science and Technology (SUSTech), Shenzhen, 518055, China; he is also
affiliated with the Guangdong Provincial Key Laboratory of Human-Augmentation
and Rehabilitation Robotics in Universities, SUSTech, Shenzhen, 518055, China
(e-mail: kongh@sustech.edu.cn). Salah Sukkarieh is with the Sydney Institute
for Robotics and Intelligent Systems, The University of Sydney, NSW 2006,
Australia (e-mail: salah.sukkarieh@sydney.edu.au). Travis J. Arnold is an
independent researcher, Madison, WI 53704, United States (e-mail:
travis.arnold17@gmail.com). Tianshi Chen is with the School of Data Science
and Shenzhen Research Institute of Big Data, The Chinese University of Hong
Kong, Shenzhen, 518172, China (e-mail: tschen@cuhk.edu.cn). Biqiang Mu is with
the Key Laboratory of Systems and Control, Institute of Systems Science,
Academy of Mathematics and Systems Science, Chinese Academy of Sciences,
Beijing 100190, China (e-mail: bqmu@amss.ac.cn). Wei Xing Zheng is with the
School of Computer, Data and Mathematical Sciences, Western Sydney University,
Sydney, NSW 2751, Australia (e-mail: w.zheng@westernsydney.edu.au).

###### Abstract

Most existing works on optimal filtering of linear time-invariant (LTI)
stochastic systems with arbitrary unknown inputs assume perfect knowledge of
the covariances of the noises in the filter design. This is impractical and
raises the question of whether and under what conditions one can identify the
process and measurement noise covariances (denoted as $Q$ and $R$,
respectively) of systems with unknown inputs. This paper considers the
identifiability of $Q$/$R$ using the correlation-based measurement difference
approach. More specifically, we establish (i) necessary conditions under which
$Q$ and $R$ can be uniquely jointly identified; (ii) necessary and sufficient
conditions under which $Q$ can be uniquely identified, when $R$ is known;
(iii) necessary conditions under which $R$ can be uniquely identified, when
$Q$ is known. It will also be shown that for achieving the results mentioned
above, the measurement difference approach requires some decoupling conditions
for constructing a stationary time series, which are proved to be sufficient
for the well-known strong detectability requirements established by Hautus.

###### Index Terms:

Estimation; Arbitrary unknown input; Kalman filter; Noise covariance
estimation.

##  I Introduction

Estimation under unknown inputs (whose models or statistical properties are
not assumed to be available), also called unknown input decoupled estimation,
has received much attention in the past. In the existing literature, many
uncertain phenomena in control systems have been modeled as unknown inputs,
including system faults/attacks [1]-[6], abrupt/impulsive disturbances or
parameters [7]-[9], arbitrary vehicle tires/ground interactions [10], etc. A
seminal work on unknown input decoupled estimation is due to Hautus [11] where
it has been shown that the strong detectability criterion, including a rank
matching condition and the system being minimum phase requirement, is
necessary and sufficient for the existence of a stable observer for estimating
the state/unknown input for deterministic systems111The strong∗ detectability
concept was also introduced in [11]. The two criteria, as discussed in [11],
are equivalent for discrete-time systems, but differ for continuous systems..

Works on the filtering case, e.g., [12]-[16], have similar rank matching and
system being minimum phase requirements as in [11]. Extensions to cases with
rank-deficient shaping matrices have been discussed in [17]-[19]. It has also
been shown in the above works that for unbiased and minimum variance
estimation of the state/unknown input, the initial guess of the state must be
unbiased. Very recently, connections between the above-mentioned results and
Kalman filtering (KF) of systems within which the unknown input is taken to be
a white noise of unbounded variance, have been established in [20]. There are
also some works dedicated to alleviating the strong detectability conditions
and the unbiased initialization requirement (see [21]-[22] and the references
therein).

However, most existing filtering works mentioned above assume that the process
and measurement noise covariances (denoted as $Q$ and $R$, respectively) are
perfectly known for the optimal filter design. This raises the question of
whether and under what conditions one can identify $Q/R$ from real data. We
believe that addressing the identifiability issue of noise covariances under
arbitrary unknown inputs is important because in practice the noise
covariances are not known a priori and have to be identified from real closed-
loop data where there might be unknown system uncertainties such as faults,
etc. Another relevant application is path planning of sensing robots for
tracking targets whose motions might be subject to abrupt disturbances (in the
form of unknown inputs), as considered in our recent work [23].

To the best of our knowledge, [24]-[25] are the only existing works on
identification of stochastic systems under unknown inputs. However, in the
former two works, the unknown inputs are assumed to be a wide-sense stationary
process with rational power spectral density or deterministic but unknown
signals, respectively. Here, we do not make such assumptions. Also, we are
mainly interested to investigate the identifiability of the original noise
covariances for linear time-invariant (LTI) stochastic systems with unknown
inputs. This is in contrast to the work in [24] where the measurement noise
covariance of the considered system is assumed to be known, and the input
autocorrelations are identified from the output data and then used for input
realization and filter design. Our work is also different from subspace
identification where the stochastic parameters of the system are estimated,
which can be used to calculate the optimal estimator gain [26]. It should be
remarked that apart from filter design, knowledge of noise covariances can
also be used for other purposes such as performance monitoring [27].

We note that noise covariance estimation is a topic of lasting interest for
the systems and control community, and the literature is fairly mature.
Existing noise covariance estimation methods can be classified as Bayesian,
maximum likelihood, covariance matching, and correlation techniques, etc.,
(see [28]-[33] and the references therein). Especially, the correlation
methods can be classified into two groups where the state/measurement
prediction error (or measurement difference), as a stochastic time series, is
computed either explicitly via a stable filter (for example, the
autocovariance least-squares (ALS) framework in [31]-[32]) or implicitly by
manipulating the measurements (see [34] for the case using one step
measurement, and [35]-[36] for the case using multi-step measurements,
respectively, in computing the measurement differences).

Still, most above-mentioned noise covariance estimation methods have not
considered the case with unknown inputs. This observation motivates us to
study the identifiability of $Q/R$ for systems under unknown inputs.
Especially, we adopt the correlation-based methodology, and mainly discuss the
implicit correlation-based frameworks, in particular, the measurement
difference approach using single-step measurement.

Moreover, given that this paper focuses on the identifiability of $Q$/$R$ via
the measurement difference approach using single-step measurement, some of the
assumptions (e.g., the output matrix $C$ is assumed to be of full column rank)
seem to be stringent. Nevertheless, we believe the consideration of the case
using single-step measurement serves as the first crucial step to fully
understand the identifiability of $Q$/$R$ under the presence of unknown
inputs. A thorough investigation of the more general case using multi-step
measurements is the subject of our current and future work.

Finally, we remark that the considered problem is inherently a theoretical
one, although we are motivated by its potential applications in practice.
However, we believe that addressing the considered question specifically for
LTI systems is the first step towards a more thorough understanding on the
topic.

The remainder of the paper is structured as follows. In Section II, we recall
preliminaries on estimation of systems with unknown inputs. Section III
contains our major results for the single-step measurement case. Section IV
illustrates the theoretical results with some numerical examples. Section V
concludes the paper.

Notation: $A^{\mathrm{T}}$ denotes the transpose of matrix $A$.
$\mathbf{R}^{n}$ stands for the $n$-dimensional Euclidean space. $I_{n}$
stands for the identity matrix of dimension $n$. $\mathbf{0}$ stands for the
zero matrices with compatible dimensions. $\mathbb{C}$ and $\left|z\right|$
denote the field of complex numbers, and the absolute value of a given complex
number $z$, respectively.

##  II Preliminaries and Problem Statement

We consider the discrete-time LTI model of the plant:

$\displaystyle x_{k+1}=Ax_{k}+Bd_{k}+Gw_{k}$ (1a) $\displaystyle
y_{k}=Cx_{k}+Dd_{k}+v_{k}\text{ \ }$ (1b)

where $x_{k}\in\mathbf{R}^{n},$ $d_{k}\in\mathbf{R}^{q}$, and
$y_{k}\in\mathbf{R}^{p}$ are the state, the unknown input, and the output,
respectively; $w_{k}\in\mathbf{R}^{g}$ and $v_{k}\in\mathbf{R}^{p}$ represent
zero-mean mutually uncorrelated process and measurement noises with
covariances $Q\in\mathbf{R}^{g\times g}$ and $R\in\mathbf{R}^{p\times p}$,
respectively; $A,B,G,C,$ and $D$ are real and known matrices with appropriate
dimensions; the pair $(A,C)$ is assumed to be detectable. Without loss of
generality, we assume $n\geq g$ and $G\in\mathbf{R}^{n\times g}$ to be of full
column rank (when this is not the case, one can remodel the system to obtain a
full rank shaping matrix $\overline{G}$).

For system (1), a major question of interest is the existence condition of an
observer/filter that can estimate the state/unknown input with asymptotically
stable error, using only the output. To address these questions, concepts such
as strong detectability and strong estimator have been rigorously discussed in
[11] for deterministic systems222Extensions of the strong detectability to
linear stochastic systems have been discussed in [19].. As remarked in [11],
the term “strong” is to emphasize that state estimation has to be obtained
without knowing the unknown input. For later use, we include the strong
detectability conditions in the sequel. Note, however, that the measurement-
difference approach does not require strong detectability since we do not need
to design a filter to explicitly estimate the state/unknown input. Instead, we
manipulate the system outputs to implicitly estimate the state and construct a
stationary time series. The required conditions associated with the
measurement-difference approach are different from the strong detectability
conditions and presented in Proposition 1 and Theorem 1.

######  Lemma 1.

([11]) The following statements hold true: (i) the system (1) has a strong
estimator if and only if it is strongly detectable; (ii) system (1) is
strongly detectable if and only if

$rank\left(\left[\begin{array}[]{cc}CB&D\\\
D&0\end{array}\right]\right)=rank(D)+rank\left(\left[\begin{array}[]{c}B\\\
D\end{array}\right]\right),$ (2)

and all its invariant zeros are stable, i.e.,

$rank\left(\underset{\mathcal{M}\left(z\right)}{\underbrace{\left[\begin{array}[]{cc}zI_{n}-A&-B\\\
C&D\end{array}\right]}}\right)=n+rank\left(\left[\begin{array}[]{c}B\\\
D\end{array}\right]\right),$ (3)

for all $z\in\mathbb{C}\ $and $\left|z\right|\geq 1$.

Conditions (2)-(3) are the so-called rank matching and minimum phase
requirements, respectively. Note that Lemma 1 holds for both the deterministic
and stochastic cases (hence we use “estimator” instead of KF/Luenberger
observer; for more detailed discussions on the design and stability of KF
under unknown inputs, we refer the reader to [12]-[19] and the references
therein. For system (1), the noise covariances $Q$ and $R$ are usually not
available, and have to be identified from data. However, all existing
filtering methods for systems with unknown inputs in the literature adopt the
assumption of knowing $Q$ and $R$ exactly, which is not practical. The
identificability questions of $Q$ and/or $R$ considered in this paper are
formally stated as follows:

######  Problem 1.

Given system (1) with unknown inputs, and known $A,B,G,C,$ and $D$, we aim to
investigate the following questions: using the measurement difference
approach, whether and under what conditions one can (i) uniquely jointly
identify $Q$ and $R$; (ii) uniquely identify $Q$ or $R$, assuming the other
covariance to be known.

##  III Identifiability of Q/R Using the Single-step Measurement Difference
Approach

This section contains the first major results of the paper. We show that, in
theory, the single-step measurement difference approach does not have a unique
solution for jointly estimating $Q$ and $R$ of system (1). Estimating $Q$ or
$R$, assuming the other to be known, will also be considered. For deriving the
results in this section, we will assume that $C$ is of full column rank. We
remark that although the assumption on $C$ is restrictive, the discussions in
the sequel bring some insights into the identifiability study of $Q$/$R$,
i.e., even with the above stringent assumption, it will be shown that only
under restrictive conditions, $Q$/$R$ can be uniquely identified.

###  III-A Conditions for obtaining an unknown input decoupled stationary time
series

When $C$ is of full column rank, from (1), it can be obtained that

$\begin{array}[]{l}y_{k+1}=Cx_{k+1}+Dd_{k+1}+v_{k+1}\\\ \text{ \ \ \ \ \
}=CAx_{k}+CBd_{k}+CGw_{k}+Dd_{k+1}+v_{k+1},\end{array}$ (4)

and

$x_{k}=My_{k}-MDd_{k}-Mv_{k},$ (5)

where

$M=(C^{\mathrm{T}}C)^{-1}C^{\mathrm{T}}.$ (6)

By substituting (5) into (4), one has that

$\begin{array}[]{l}\overline{z}_{k}=y_{k+1}-CAMy_{k}=(CB-CAMD)d_{k}\\\ \text{
\ \ \ \ \ \ }+Dd_{k+1}+CGw_{k}+v_{k+1}-CAMv_{k}.\end{array}$

Given that we do not assume to have any knowledge of the unknown input, it is
not possible for us to conduct any analysis of the statistical properties of
$\overline{z}_{k}$. Hence, a necessary and sufficient condition to decouple
the influence of the unknown input on $\overline{z}_{k}$ is the existence of a
nonzero matrix $K\in\mathbf{R}^{r\times p}$ such that

$\begin{array}[]{l}z_{k}=K\overline{z}_{k}=K(CB-CAMD)d_{k}+KDd_{k+1}\\\ \text{
\ \ \ \ \ \ }+KCGw_{k}+Kv_{k+1}-KCAMv_{k},\end{array}$ (7)

with

$K\underset{H\in\mathbf{R}^{p\times
2q}}{\underbrace{\left[\begin{array}[]{ll}C(B-AMD)&D\end{array}\right]}}=0.$
(8)

######  Remark 1.

For the single-step measurement difference approach, later we will establish
(i) necessary conditions under which $Q$ and $R$ can be uniquely jointly
identified (see Proposition 2); (ii) necessary and sufficient conditions under
which $Q$ can be uniquely identified, when $R$ is known (see Proposition 3 and
Corollary 1); (iii) necessary conditions under which $R$ can be uniquely
identified, when $Q$ is known (see Proposition 4). Moreover, it will be shown
that for achieving the results mentioned above, the measurement difference
approach requires some decoupling conditions for constructing a stationary
time series (see Proposition 1). The latter conditions are proved to be
sufficient (see Theorem 1) for the strong detectability requirement in [11].
Also, if the existence conditions on $K$ are satisfied, then one can use
standard techniques to calculate $K$ [37, Chap. 6].

There are a few potential scenarios when (8) holds:

$\begin{array}[]{l}\text{(a) }C(B-AMD)\neq 0\text{, }D\neq 0\text{,}\\\
\text{(b) }C(B-AMD)=CB\neq 0\text{, }D=0\text{,}\\\ \text{(c)
}C(B-AMD)=0\text{, }D\neq 0\text{,}\\\ \text{(d) }C(B-AMD)=CB=0\text{,
}D=0\text{.}\end{array}$ (9)

Note that since $C$ is assumed to be of full column rank, case (d) in (9)
cannot happen. This is because when $C$ is of full column rank,

$\begin{array}[]{l}C(B-AMD)=CB=0\text{, }D=0\\\ \Rightarrow CB=0,\text{
}D=0,\\\ \Rightarrow B=0,\text{ }D=0,\end{array}$ (10)

i.e., the unknown input $d_{k}$ vanishes in system (1). Note that here in this
work we focus on the case with unknown inputs, i.e., the situation of $B=0$
and $D=0$ is not applicable. For cases (a)-(c), we have the following
immediate results.

######  Proposition 1.

Given system (1) with $C$ being of full column rank, then the following
statements hold true:

(i) for case (a) in (9), there exists a matrix $K$ such that the equality in
(8) holds if and only if

$rank(H)=2q,$ (11)

where, $H$ is defined in (8); for condition (11) to hold, it is necessary that
$rank(B-AMD)=q$, $n\geq q$, $rank(D)=q$, $p\geq 2q$;

(ii) for case (b) in (9), there exists a matrix $K$ such that the equality in
(8) holds if and only if $rank(B)=q$;

(iii) for case (c) in (9), there exists a matrix $K$ such that the equality in
(8) holds if and only if $B-AMD=0$, $rank(D)=q$.

###### Proof.

(i) For case (a), from the solution properties of matrix equations [37, Chap.
6], there exists $K$ such that equalities in (8) hold if and only if
$rank(H)=2q$. The rest of the proof for part (i) follows naturally from
condition (11). Parts (ii) and (iii) follow similarly.

Note that case (c) is unrealistic as it requires $B-AMD=0$. However, we
include the discussion on it just for completeness. One would wonder how
stringent the decoupling condition in (8) and possible cases (a)-(c) in (9)
are, compared to the strong detectability conditions in Lemma 1. This question
is answered in the following theorem.

######  Theorem 1.

For cases (a)-(c), $C$ being of full column rank and the decoupling condition
(8) are sufficient for the strong detectability conditions in Lemma 1.

###### Proof.

We prove the claim for cases (a)-(c), respectively.

For case (a), we note from Proposition 1 that $D$ has to be of full column
rank. This further implies that the rank matching condition (2) holds. Note
that

$\displaystyle\begin{bmatrix}I_{n}&AM\\\
0&I_{p}\end{bmatrix}\mathcal{M}\left(z\right)$ $\displaystyle=$
$\displaystyle\begin{bmatrix}zI_{n}-A+AMC&AMD-B\\\ C&D\end{bmatrix}$
$\displaystyle=$
$\displaystyle\underset{\overline{\mathcal{M}}\left(z\right)}{\underbrace{\begin{bmatrix}zI_{n}&AMD-B\\\
C&D\end{bmatrix}}}$

where $\mathcal{M}\left(z\right)$ is defined in (3). When $D$ is of full
column rank, there always exists a matrix $X\in\mathbf{R}^{n\times p}$ such
that $XD=AMD-B.$ Denote

$X\left(z\right)=\left[\begin{array}[]{cc}\frac{1}{z}I_{n}&-\frac{1}{z}X\\\
0&I_{p}\end{array}\right],$

which is of full column rank for all $z\in\mathbb{C}\ $and $\left|z\right|\geq
1$. Multiplying $X\left(z\right)$ on the left hand side of
$\overline{\mathcal{M}}\left(z\right)$ gives us

$\begin{array}[]{l}rank(\mathcal{M}\left(z\right))=rank(\overline{\mathcal{M}}\left(z\right))=rank(X\left(z\right)\overline{\mathcal{M}}\left(z\right))\\\
\Rightarrow
rank(\overline{\mathcal{M}}\left(z\right))=rank\left(\left[\begin{array}[]{cc}I_{n}-\frac{1}{z}XC&0\\\
C&D\end{array}\right]\right)=n+q,\end{array}$

for all $z\in\mathbb{C}\ $and $\left|z\right|\geq 1$. In other words, the
minimum phase condition in (3) holds. The proof for case (a) is completed.

For case (b), given that $C$ and $B$ are of full column rank, and $D=0$, it
can be easily checked that conditions (2)-(3) hold.

For case (c), given that both $C$ and $D$ are of full column rank, and
$B=AMD$, it can be checked that condition (2) holds. The matrix
$\overline{\mathcal{M}}\left(z\right)$ appeared for the case (a) satisfies

$\displaystyle\overline{\mathcal{M}}\left(z\right)=\begin{bmatrix}zI_{n}&AMD-B\\\
C&D\end{bmatrix}=\begin{bmatrix}zI_{n}&0\\\ C&D\end{bmatrix}.$

Thus
$rank(\mathcal{M}\left(z\right))=rank(\overline{\mathcal{M}}\left(z\right))=n+q,$
for all $z\in\mathbb{C}\ $and $z\neq 0$ and hence condition (3) holds. This
completes the proof.

Theorem 1 reveals that the measurement difference approach requires more
stringent conditions than strong detectability conditions. As it will be
discussed in the sequel, even with the above stringent requirement, $Q$/$R$
could be potentially uniquely identified under restrictive conditions.

###  III-B Joint identifiability analysis of $Q$ and $R$

We next discuss the joint identifiability of $Q\ $and $R$. As such, assume
that one of the cases (a)-(c) happens so that the decoupling condition in (8)
holds. From (7), one has

$z_{k}=KCGw_{k}+Kv_{k+1}-KCAMv_{k},$ (12)

which is a zero-mean stationary time series. We also have

$\displaystyle
S_{0}\displaystyle\stackrel{{\scriptstyle\triangle}}{{=}}E(z_{k}(z_{k})^{\mathrm{T}})=KCGQ(KCG)^{\mathrm{T}}+KRK^{\mathrm{T}}$
$\displaystyle\hskip 85.35826pt+KCAMR(KCAM)^{\mathrm{T}},$ (13a)
$\displaystyle
S_{1}\displaystyle\stackrel{{\scriptstyle\triangle}}{{=}}E(z_{k+1}(z_{k})^{\mathrm{T}})=-KCAMRK^{\mathrm{T}},$
(13b) $\displaystyle\hskip 22.76219ptE(z_{k+j}(z_{k})^{\mathrm{T}})=0,~{}j\geq
2,$ (13c) where $E(\cdot)$ denotes the mathematical expectation. The above
equations give  $\begin{array}[]{l}S=\begin{bmatrix}S_{0}\\\
S_{1}\end{bmatrix}=E\left(\left[\begin{array}[]{c}z_{k}(z_{k})^{\mathrm{T}}\\\
z_{k+1}(z_{k})^{\mathrm{T}}\end{array}\right]\right)\\\
=\left[\begin{array}[]{c}KCG\\\
0\end{array}\right]Q(KCG)^{\mathrm{T}}+\left[\begin{array}[]{c}K\\\
-KCAM\end{array}\right]RK^{\mathrm{T}}\\\ +\left[\begin{array}[]{c}KCAM\\\
0\end{array}\right]R(KCAM)^{\mathrm{T}}.\end{array}$ (14)

Denote the vectorization operator of a matrix $A=[a_{1},a_{2},\cdots,a_{n}]$
by
$\mathrm{vec}(A)=[a_{1}^{\mathrm{T}},a_{2}^{\mathrm{T}},\cdots,a_{n}^{\mathrm{T}}]^{\mathrm{T}}$
and the Kronecker product of $A$ and $B$ by $A\otimes B$, respectively. By
applying the identity $\mathrm{vec}~{}\\!(ABC)=(C^{\mathrm{T}}\otimes
A)\mathrm{vec}~{}\\!(B)$ involving the vectorization operator (14) and the
Kronecker product, we have the following system of linear equations

$\displaystyle\mathcal{A}~{}\\!\mathrm{vec}{([Q,R])}=\mathrm{vec}{(S)},$ (15)

where

$\mathcal{A}=\left[\begin{array}[]{cc}\overline{K}&0\\\
0&\overline{K}\end{array}\right]\left[\begin{array}[]{cc}\mathcal{A}_{1}&\mathcal{A}_{2}\\\
0&-I_{p}\otimes CAM\end{array}\right],$ (16)

in which

$\begin{array}[]{l}\overline{K}=K\otimes K,\text{ }\mathcal{A}_{1}=CG\otimes
CG\in\mathbf{R}^{p^{2}\times g^{2}},\\\ \mathcal{A}_{2}=(I_{p}\otimes
I_{p})+(CAM\otimes CAM)\in\mathbf{R}^{p^{2}\times p^{2}}.\end{array}$

Given the process is ergodic, a valid procedure of approximating the
expectation from the data is to use the time average. Especially, given all
the collected data as $z_{0:N}$, one has

$\widetilde{S}=[\widetilde{S}_{0},\widetilde{S}_{1}],$ (17)

where

$\widetilde{S}_{0}=\frac{1}{N+1}\sum\limits_{i=0}^{N}z_{k}z_{k}^{\mathrm{T}},\text{
}\widetilde{S}_{1}=\frac{1}{N}\sum\limits_{i=0}^{N-1}z_{k+1}(z_{k})^{\mathrm{T}}.$
(18)

Denote

$e=\mathrm{vec}(\widetilde{S}).$

We then have the following standard least-squares problem for identifying $Q\
$and $R$:

$\Xi^{\ast}=\arg\min\limits_{\Xi}\left\|\mathcal{A}\Xi-e\right\|^{2}$ (19)

where
$\Xi=\left[{\mathrm{vec}(\widehat{Q}),\mathrm{vec}(}\widehat{{R}}{)}\right]$,
$e$ is defined above (19). The joint identifiability of $Q$ and $R$ is
determined by the full column rankness of $\mathcal{A}$.

It should be noted that in the least-square problems listed in the remainder
of the paper, some permutation matrices can be introduced to identify the
unique elements of $Q$ and $R$, and additional constraints need to be enforced
on the $Q$ and $R$ estimates (see, e.g., [31]-[32], [35]), given that they are
both symmetric and positive semidefinite matrices. Then the constrained least-
squares problems can be transformed to semidefinite programs [33, Chap. 3.4]
and solved efficiently using existing software packages such as CVX [39]. For
simplicity, we have not formally included these constraints in the least-
squares problem formulations, because this will not impact the discussions on
the solution uniqueness of these least-squares problems. It should also be
noted that in the simulation examples shown in Section V, such symmetric and
positive semidefinite constraints have been enforced. Moreover, in the least-
square problems listed in the remainder of the paper, including (19), (22), we
consider the most general scenario and do not assume to have any knowledge of
the structure of $Q$ and $R$ except that they are supposed to be symmetric and
positive semidefinite matrices, which we intend to identify. In practice, if
one has some knowledge of their structures, for example, if $Q$ and/or $R$ are
assumed to be partially known or they are diagonal matrices, the least-square
problems listed in this paper can be readily modified to incorporate such
knowledge. For $\mathcal{A}$ in (19), We have the following results.

######  Proposition 2.

Given system (1) with $C$ being of full column rank, the following statements
hold true:

(i) $\mathcal{A}$ is of full column rank only if

$rank\left(\underset{K_{M}}{\underbrace{\left[\begin{array}[]{c}K\\\
KCAM\end{array}\right]}}\right)=p,\text{ }rank\left(KCG\right)=g;$ (20)

(ii) when $G=I_{n}$ (i.e., $g=n$) and $p=n$, $\mathcal{A}$ is of full column
rank if and only if

$rank\left(K\right)=p\text{, }rank\left(CAM\right)=p\text{; }$

(iii) when $G=I_{n}$ (i.e., $g=n$) and $p=n$, for $\mathcal{A}$ to be of full
column rank, the unknown input $d_{k}$ has to vanish from system (1), i.e.,
$B=0$, $D=0$.

###### Proof.

(i) We prove the result by contradiction. Firstly, assume that the matrix
$K_{M}$ in (20) loses rank, i.e., there exists a nonzero vector $h$ such that
$K_{M}h=0$. Set $R=hh^{\mathrm{T}}$ so that

$\begin{array}[]{l}KCAMhh^{\mathrm{T}}K^{\mathrm{T}}=0,Khh^{\mathrm{T}}K^{\mathrm{T}}=0,\\\
KCAMhh^{\mathrm{T}}(KCAM)^{\mathrm{T}}=0.\end{array}$

Further by selecting $Q=0$, then one has that
$\mathcal{A}\mathrm{vec}~{}\\!([0,hh^{\mathrm{T}}])=0$. This means that
$\mathcal{A}$ is not of full column rank. Similarly, now assume that
$rank\left(KCG\right)<g$ and there exists a nonzero vector $e$ such that
$KCGe=0$. If we set $Q=ee^{\mathrm{T}}$, $R=0$, then
$\mathcal{A}\mathrm{vec}~{}\\!([ee^{\mathrm{T}},0])=0$. Hence, $\mathcal{A}$
is not of full column rank.

(ii) When $G=I_{n}$ (i.e., $g=n$) and $p=n,$ $\mathcal{A}$ is of full column
rank if and only if

$\begin{array}[]{l}rank\left(\left[\begin{array}[]{cc}\overline{K}&0\\\
0&\overline{K}\end{array}\right]\right)=2p^{2}\text{, }\\\
rank\left(\left[\begin{array}[]{cc}\mathcal{A}_{1}&\mathcal{A}_{2}\\\
0&-I_{p}\otimes CAM\end{array}\right]\right)=2p^{2}\\\ \Leftrightarrow
rank\left(K\right)=p\text{, }rank\left(CAM\right)=p\text{,}\end{array}$

given the fact that both $\mathcal{A}_{1}$ and $I_{p}\otimes CAM$ become
square matrices when $g=p=n$.

(iii) From part (ii) of the current theorem, when $G=I_{n}$ and $p=n,$
$\mathcal{A}$ is of full column rank only if $K$ is of full column rank. Based
on the decoupling condition (8), this further implies that case (d) in (9)
happens. From the arguments listed in (10), one has that the unknown input
$d_{k}$ vanishes in system (1). This completes the proof.

Note that for the necessary conditions in (20) to hold, one must have that
$r\geq\max\\{\left\lceil\frac{p}{2}\right\rceil,g\\}$, where $\left\lceil
a\right\rceil$ stands for the ceiling operation generating the least integer
not less than $a$, where $a$ is a real number. Also, from (16), one can see
that for $\mathcal{A}$ to be of full column rank, it must hold that
$2r^{2}\geq p^{2}+g^{2}$. Hence, it is necessary that

$r\geq\max\left\\{\left\lceil\frac{p}{2}\right\rceil,\text{ }g,\text{
}\left\lceil\sqrt{\frac{p^{2}+g^{2}}{2}}\right\rceil\right\\}.$ (21)

In practice, the structure of $G$ represents how the process noise affects the
system dynamics. When no such knowledge is available, $G$ is usually chosen to
be the identity matrix. From part (i) of the above proposition, it can be seen
that for the general case where $G$ is known, we have only established
necessary condition for $\mathcal{A}$ to have full column rank. For the
special case when $G=I_{n}$ and $p=n$, although necessary and sufficient
conditions are obtained in part (ii), part (iii) further reveals that for
$\mathcal{A}$ to have full column rank, the unknown input has to be absent
from the system model, i.e., it is not an applicable case. The above findings
motivate us to take a step back, and consider part (ii) of Problem 1.

###  III-C Identifiability analysis of $Q$ when $R$ is known

In this subsection, we investigate one case of part (ii) of Problem 1, i.e.,
analyze the identifiability of $Q$ when $R$ is available. When $R$ is known,
the equation (13) reduces to

$\mathcal{A}_{Q}\mathrm{vec}(Q)=\mathrm{vec}(S_{0})-\overline{K}\mathcal{A}_{2}\mathrm{vec}(R),$
(22)

where

$\mathcal{A}_{Q}\displaystyle=\overline{K}\mathcal{A}_{1}$

and $\overline{K},\mathcal{A}_{1},\mathcal{A}_{2}$ are defined in (16). Define

$e_{Q}=\mathrm{vec}(\widetilde{S}_{0})-\overline{K}\mathcal{A}_{2}\mathrm{vec}(R).$
(23)

By following a similar procedure with the previous subsection, we have the
following standard least-squares problem formulation for identifying $Q$:

$\Xi_{Q}^{\ast}=\arg\min\limits_{\Xi_{Q}}\left\|\mathcal{A}_{Q}\Xi_{Q}-e_{Q}\right\|^{2}$
(24)

where $\Xi_{Q}={\mathrm{vec}(\widehat{Q})}$, $e_{Q}$ is defined in (23). Thus
the identifiability of $Q$ when $R$ is known is equivalent to the matrix
$\mathcal{A}_{Q}$ being of full column rank.

######  Proposition 3.

Given system (1) with $C$ being of full column rank, the following statements
hold true:

(i) $\mathcal{A}_{Q}$ in (22) is of full column rank only if $r\geq g$;

(ii) for case (a) in (9), $\mathcal{A}_{Q}$ in (22) is of full column rank if
and only if

$rank(H)+g=rank\left(\left[\begin{array}[]{ll}H&CG\end{array}\right]\right),$
(25)

where $H$ is defined in (8);

(iii) for case (b) in (9), $\mathcal{A}_{Q}$ in (22) is of full column rank if
and only if

$rank\left(B\right)+g=rank\left(\left[\begin{array}[]{ll}B&G\end{array}\right]\right);$
(26)

(iv) for case (c) in (9), $\mathcal{A}_{Q}$ in (22) is of full column rank if
and only if $B-AMD=0$ and

$rank\left(D\right)+g=rank\left(\left[\begin{array}[]{ll}D&CG\end{array}\right]\right).$
(27)

###### Proof.

(i) Note that $\mathcal{A}_{Q}=KCG\otimes KCG\in\mathbf{R}^{r^{2}\times
g^{2}}$. Hence, $\mathcal{A}_{Q}$ is of full column rank if and only if
$KCG\in\mathbf{R}^{r\times g}$ is of full column rank. Hence, for $KCG$ to be
of full column rank, it is necessary that $r\geq g$.

(ii) For case (a), the conclusion is implied by the identity

$\displaystyle\begin{bmatrix}I&0\\\ -K&I\end{bmatrix}\begin{bmatrix}H&CG\\\
0&KCG\end{bmatrix}=\begin{bmatrix}H&CG\\\
-KH&0\end{bmatrix}=\begin{bmatrix}H&CG\\\ 0&0\end{bmatrix}$

and the requirement on the full column rank of $KCG$ as well as the decoupling
condition $KH=0$ in (8).

(iii) This part is straightforward by using the full column rank of $C$.

(iv) This part follows by similar arguments with part (i).

The proof is completed.

We also have the following corollary when $G=I_{n}$.

######  Corollary 1.

Given system (1) with $C$ being of full column rank, and $G=I_{n}$, the
following statements hold true:

(i) $\mathcal{A}_{Q}$ in (22) is of full column rank only if $r\geq n$;

(ii) for case (a) in (9), $\mathcal{A}_{Q}$ in (22) is of full column rank if
and only if

$rank\left(\left[\begin{array}[]{ll}C&D\end{array}\right]\right)=rank(H)+n.$
(28)

where $H$ is defined in (8);

(iii) for case (b) in (9), $\mathcal{A}_{Q}$ in (22) is not of full column
rank;

(iv) for case (c) in (9), $\mathcal{A}_{Q}$ in (22) is of full column rank if
and only if $B-AMD=0$ and

$rank\left(D\right)+n=rank\left(\left[\begin{array}[]{ll}D&C\end{array}\right]\right).$

###  III-D Identifiability analysis of $R$ when $Q$ is known

Now, we consider the other case of part (ii) of Problem 1, i.e., analyze the
identifiability of $R$ when $Q$ is available. When $Q$ is known, the system of
equations (13)-(13b) becomes

$\mathcal{A}_{R}\mathrm{vec}(R)=\mathrm{vec}(S)-\left[\begin{array}[]{c}\overline{K}\mathcal{A}_{1}\mathrm{vec}(Q)\\\
0\end{array}\right],$ (29)

where

$\mathcal{A}_{R}=\left[\begin{array}[]{c}\overline{K}\mathcal{A}_{2}\\\
-\overline{K}(I_{p}\otimes CAM)\end{array}\right]$

with $\overline{K},\mathcal{A}_{1}$ and $\mathcal{A}_{2}$ being defined in
(16). Similarly with the previous subsection, we have the following results.

######  Proposition 4.

Given system (1) with $C$ being of full column rank, $\mathcal{A}_{R}$ is of
full column rank only if $rank\left(K_{M}\right)=p$, where $K_{M}$ is defined
in (20).

###### Proof.

The proof follows a similar procedure with that of Proposition 2, and is
omitted.

For the cases (e.g., part (iii) of Corollary 1 or the conditions in (25)-(27)
do not hold) when the solutions to the systems of linear equations are not
unique, a natural idea is to use regularization to introduce further
constraints to uniquely determine the solution [38]. However, a key question
to be answered is whether some desirable properties can be guaranteed for the
covariance estimates. A full investigation of the above questions is the
subject of our current and future work.

##  IV  Numerical Examples

We next use some numerical examples to illustrate the theoretical results.
Firstly, consider the plant model (1) with

$\begin{array}[]{l}A=\left[\begin{array}[]{ll}1&1\\\
0&1\end{array}\right],\text{ }B=\left[\begin{array}[]{ll}0&1\\\
1&0\end{array}\right],G=\left[\begin{array}[]{l}1\\\ 1\end{array}\right],\\\
C=\left[\begin{array}[]{ll}1&0\\\ 1&1\end{array}\right],\text{
}D=\left[\begin{array}[]{ll}1&0\\\ 2&0\end{array}\right].\end{array}$

It can be verified

$H=\left[\begin{array}[]{llll}-2&1&1&0\\\ -2&1&2&0\end{array}\right]$

so that the above model fits case (a) in (9). Also, from (21), it is necessary
that $r\geq 2$. Select

$K=\left[\begin{array}[]{ll}t_{11}&t_{12}\\\ t_{21}&t_{22}\end{array}\right].$

From the decoupling condition in (8), we then have

$\begin{array}[]{l}\left[\begin{array}[]{llll}-2(t_{11}+t_{12})&t_{11}+t_{12}&t_{11}+2t_{12}&0\\\
-2(t_{21}+t_{22})&t_{21}+t_{22}&t_{21}+2t_{22}&0\end{array}\right]=0\\\
\Leftrightarrow t_{11}=t_{12}=t_{21}=t_{22}=0.\end{array}$

Note that increasing the row dimension of $K$ still leads to the same
conclusion, i.e., $K$ is a zero matrix. In this case, we have $\mathcal{A=}$
$0$ in (19), $\mathcal{A}_{Q}=0$ in (22), $\mathcal{A}_{R}=0$ in (29), i.e.,
the noise covariances $Q/R$ are unidentifiable at all.

Secondly, consider the plant model (1) with

$\begin{array}[]{l}A=\left[\begin{array}[]{ll}1&1\\\
0&1\end{array}\right],\text{ }B=\left[\begin{array}[]{l}1\\\
0\end{array}\right],\text{ }G=\left[\begin{array}[]{ll}1&0\\\
1&-2\end{array}\right].\\\ C=\left[\begin{array}[]{cc}1&-2\\\ 1&1\\\
-2&1\end{array}\right],\text{ }D=\left[\begin{array}[]{l}1\\\ 2\\\
1\end{array}\right]\end{array}$

It can be verified

$H=\left[\begin{array}[]{cc}1&1\\\ 0&2\\\ -1&1\end{array}\right]$

so that the above model fits case (a) in (9). Also, from (21), it is necessary
that $r\geq 3$. Select

$K=\left[\begin{array}[]{lll}t_{11}&t_{12}&t_{13}\\\ t_{21}&t_{22}&t_{23}\\\
t_{31}&t_{32}&t_{33}\end{array}\right].$ (30)

From the decoupling condition in (8), we then have

$\begin{array}[]{l}\left[\begin{array}[]{c}\begin{array}[]{ll}t_{11}-t_{13}&t_{11}+2t_{12}+t_{13}\\\
t_{21}-t_{23}&t_{21}+2t_{22}+t_{23}\end{array}\\\
\begin{array}[]{ll}t_{31}-t_{33}&t_{31}+2t_{32}+t_{33}\end{array}\end{array}\right]=0\\\
\Leftrightarrow\left.\begin{array}[]{l}t_{11}=t_{13},\text{
}t_{12}=-t_{11};\\\ t_{21}=t_{23},\text{ }t_{22}=-t_{21};\\\
t_{31}=t_{33},\text{ }t_{32}=-t_{31}.\end{array}\right.\end{array}$

If we set

$K=\left[\begin{array}[]{lll}1&-1&1\\\ 2&-2&2\\\ 3&-3&3\end{array}\right],$

it can be verified that neither of the two necessary conditions in (20) is
satisfied. In particular, $rank(K_{M})=2,$ $rank\left(KCG\right)=1$. Hence, it
is not possible for $\mathcal{A}\in\mathbf{R}^{18\times 13}$ in (19) to have
full column rank. In fact, it can be checked that
$rank\left(\mathcal{A}\right)=2$, i.e., $Q$ and $R$ are not uniquely jointly
identifiable. Moreover, it can be calculated that for
$\mathcal{A}_{Q}\in\mathbf{R}^{9\times 4}$ in (22), we have
$rank\left(\mathcal{A}_{Q}\right)=1$. This reinforces the results of Corollary
3 since one can easily see that $rank(H)+2=4\neq
rank\left(\left[\begin{array}[]{ll}H&CG\end{array}\right]\right)=3$, i.e., the
condition in (25) does not hold. In other words, assuming $R$ to be known, $Q$
is not uniquely identifiable. Similarly, for
$\mathcal{A}_{R}\in\mathbf{R}^{18\times 9}$ in (29), since $rank(K_{M})=2$,
from Corollary 4, one has that $\mathcal{A}_{R}$ cannot have full column rank.
To double confirm, it can be checked that
$rank\left(\mathcal{A}_{R}\right)=2$, i.e., $R$ is not uniquely identifiable,
when $Q$ is assumed to be known. Note that increasing the row dimension of $K$
still leads to the same conclusions as above for this example.

Thirdly, consider the plant model (1) with

$\begin{array}[]{l}A=\left[\begin{array}[]{ll}1&1\\\
0&1\end{array}\right],\text{ }B=\left[\begin{array}[]{l}1\\\
0\end{array}\right],\text{ }G=\left[\begin{array}[]{l}1\\\
4\end{array}\right],\\\ C=\left[\begin{array}[]{cc}1&-2\\\ 1&1\\\
-2&1\end{array}\right],\text{ }D=0.\end{array}$

It can be verified

$H=\left[\begin{array}[]{cc}1&0\\\ 1&0\\\ -2&0\end{array}\right]$

so that the above model fits case (b) in (9). From (21), it is necessary that
$r\geq 3$. Denote $K$ as in (30). From the decoupling condition in (8), we
then have

$\left[\begin{array}[]{c}\begin{array}[]{ll}t_{11}+t_{12}-2t_{13}&0\\\
t_{21}+t_{22}-2t_{23}&0\end{array}\\\
\begin{array}[]{ll}t_{31}+t_{32}-2t_{33}&0\end{array}\end{array}\right]=0.$

Select

$K=\left[\begin{array}[]{lll}1&-1&0\\\ 1&3&2\\\ 2&4&3\end{array}\right].$

One then has that $rank(K_{M})=2,$ $rank\left(KCG\right)=1$, i.e., it is not
possible for $\mathcal{A}\in\mathbf{R}^{18\times 10}$ in (19) to have full
column rank. In fact, it can be checked that $rank\left(\mathcal{A}\right)=5$,
i.e., $Q$ and $R$ are not uniquely jointly identifiable. Also, it can be
obtained that for $\mathcal{A}_{Q}\in\mathbf{R}^{9\times 1}$ in (22), we have
$rank\left(\mathcal{A}_{Q}\right)=1$. This reinforces the results of Corollary
3 since it can be confirmed that
$rank\left(B\right)+1=2=rank\left(\left[\begin{array}[]{ll}B&G\end{array}\right]\right)$,
i.e., the condition in (26) holds. In other words, assuming $R$ to be known,
$Q$ is uniquely identifiable. Similarly, for
$\mathcal{A}_{R}\in\mathbf{R}^{18\times 9}$ in (29), since $rank(K_{M})=2$,
from Corollary 4, one can conclude that $\mathcal{A}_{R}$ cannot have full
column rank. To double confirm, it can be checked that
$rank\left(\mathcal{A}_{R}\right)=4$, i.e., $R$ is not uniquely identifiable,
when $Q$ is assumed to be known. Note that increasing the row dimension of $K$
still leads to the same conclusions as above for this example.

Next for the third example, assume that the true covariances are $Q=1,$
$R=0.1I_{3}$. With the above system information, we follow the procedure in
Section III. C, and estimate $Q$, assuming $R$ to be known. We run the
simulation for 500 scenarios. For each scenario, we use in total 1000 data
points to estimate $S_{0}$ as in (18). and the estimate for $Q$ is obtained by
solving the optimization problem (24). The results are shown in Figure 1 for
the above-mentioned 500 different scenarios. It can be seen from Figure 1 that
the estimates for $Q$ are well dispersed around its true value. We finally
remark that for solving (24), an additional positive semidefinite constraint
has been enforced on the $Q$ estimates (i.e., for this example, $\widehat{Q}$
is a nonnegative scalar). The optimization problem is transformed to a
standard semidefinite program and solved by cvx [39].

Figure 1: Estimates for $Q$ under 500 different scenarios

##  V Conclusions

The past few decades have witnessed much progress in optimal filtering for
systems with arbitrary unknown inputs and stochastic noises. However, the
existing works assume perfect knowledge of the noise covariances in the filter
design, which is impractical. In this paper, for stochastic systems under
unknown inputs, we have investigated the identifiability question of the
process and measurement noises covariances (i.e., $Q$ and $R$) using the
correlation-based measurement difference approach.

More specifically, we have focused on the single-step measurement case, and
established (i) necessary conditions under which $Q$ and $R$ can be uniquely
jointly identified (see Proposition 2); (ii) necessary and sufficient
conditions under which $Q$ can be uniquely identified, when $R$ is known (see
Proposition 3 and Corollary 1); (iii) necessary conditions under which $R$ can
be uniquely identified, when $Q$ is known (see Proposition 4). Moreover, it
has been shown that for achieving the results mentioned above, the measurement
difference approach requires some decoupling conditions for constructing a
stationary time series (see Proposition 1). The latter conditions are proved
to be sufficient (see Theorem 1) for the strong detectability requirement in
[11].

The above findings reveal that only under restrictive conditions, $Q$/$R$ can
be potentially uniquely identified. This not only helps us to have a better
understanding of the applicability of existing filtering frameworks under
unknown inputs (since almost all of them require perfect knowledge of the
noise covariances) but also calls for further investigation of alternative and
more viable noise covariance methods under unknown inputs.

##  VI Acknowledgment

The authors would like to thank the reviewers and Editors for their
constructive suggestions which have helped to improve the quality and
presentation of this paper significantly. He Kong’s work was supported by the
Science, Technology, and Innovation Commission of Shenzhen Municipality [Grant
No. ZDSYS20200811143601004]. Tianshi Chen’s work was supported by the Thousand
Youth Talents Plan funded by the central government of China, the Shenzhen
Science and Technology Innovation Council under contract No. Ji-20170189, the
President’s grant under contract No. PF. 01.000249 and the Start-up grant
under contract No. 2014.0003.23 funded by the Chinese University of Hong Kong,
Shenzhen.

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