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# Stability analysis of the Kalman filter under practical conditions

Qinghua Zhang

► To cite this version:

Qinghua Zhang. Stability analysis of the Kalman filter under practical conditions. Automatica, 2026, 183, pp.112670. 10.1016/j.automatica.2025.112670. hal-05247258v3

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https://inria.hal.science/hal-05247258v3

Submitted on 18 Nov 2025

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# Stability Analysis of the Kalman Filter under Practical Conditions

Qinghua Zhang<sup>a</sup>

<sup>a</sup> Univ. Gustave Eiffel, Inria, COSYS-SII, I4S, Campus de Beaulieu, 35042 Rennes, France

## Abstract

The Kalman filter is optimal in the sense of minimum variance under the assumptions of white noises and perfect model-system matching. If these assumptions are not satisfied, the filter loses its optimality. Does it, however, remain a good state estimator with stable error dynamics? The answer to this question cannot rely on the classical stability proof of the Kalman filter, which was based on its optimality. It is shown in this paper that, without the restrictive assumptions of white noises and perfect model-system matching, the Kalman filter has an exponentially stable error dynamics under observability and controllability conditions. Moreover, the second moment of its state estimation error has an upper bound linearly depending on the discrepancies between model and system. In this sense, the Kalman filter is stable and remains a good state estimator under practical conditions.

**Key words:** Kalman filter; error dynamics stability; covariance upper bound; second moment upper bound.

## 1 Introduction

The Kalman filter (Kalman, 1963; Kovvali et al., 2022) is well-known and widely applied in many fields for various tasks involving state estimation. This paper will investigate an aspect rarely covered in the related literature: how does the Kalman filter behave when the assumptions made for its design are not satisfied? Notably, it is usually assumed that the underlying system is disturbed by white noises with known covariances. What happens if this is not true? Obviously, in this case the Kalman filter loses its optimality; however, does it remain a good state estimator with stable error dynamics?

While classical *state observers* are usually designed with the main focus on error dynamics stability, the Kalman filter has been designed by minimizing the state estimation error covariance. *Remarkably*, this optimal filter is also guaranteed to have a stable error dynamics under some assumptions (Deyst Jr and Price, 1968; Moore and Anderson, 1980). However, the proof of this error dynamics stability relies on the optimality of the Kalman filter, which has been established by assuming *white noises and perfect matching between model and system*. Section 3 will recall in more detail how this stability

proof depends on the optimality. Because the Kalman filter loses optimality when its assumptions are not satisfied, its classical stability proof is no longer valid.

A well known way to address unsatisfied white noise assumption is to incorporate the model of the noises into an augmented state-space model, so that the augmented model involves whitened noises (Anderson and Moore, 1979). If system parameters and noise covariances are poorly known, they can be improved by system identification methods (Ljung, 1999). Such efforts often make the model closer to the true system, at least for time invariant systems. Nevertheless, the ideal assumptions made by the Kalman filter may still not be satisfied due to modeling uncertainties: the whitened noises are not perfectly white, and the updated parameters are subject to estimation errors. It is *naturally expected* that small deviations from the ideal assumptions would not make the Kalman filter diverge. However, no formal proof of such robustness result has been published, to the author's knowledge. The results of this paper will complete this missing piece of the Kalman filter theory, excluding unexpected singular behavior of the filter when the system deviates from the ideal assumptions. *This theoretical investigation is not superfluous*, since singular behaviors, i.e., radical behavioral changes caused by small perturbations, do exist in other control system problems (Naidu and Rao, 2006).

<footnote>* This paper was not presented at any IFAC meeting.
Email address: qinghua.zhang@inria.fr (Qinghua Zhang).</footnote>
The stability of the Kalman filter under practical conditions has been empirically observed in many applications. Indeed, in most practical situations, the assumptions made by the Kalman filter are not fully satisfied: it is difficult to evaluate the involved noises, and system parameters are usually subject to uncertainties. However, the Kalman filter often produces good results despite the unsatisfied ideal assumptions. Such experiences empirically indicate its robustness to wrong assumptions. The purpose of this paper is to formally investigate this issue. By establishing the stability of the Kalman filter and its error covariance bound under realistic conditions, excluding singular behaviors when the system and the involved noises deviate from ideal assumptions, this paper contributes to completing an important missing piece of the Kalman filter theory.

Similar analyses have been reported in (Toda and Patel, 1980; Xie and Soh, 1994; Luo and Bosch, 1996), which all assume white noises. Indeed this restriction is essential to make the state estimation error covariance easy to compute and to analyze. The main novelty in the present paper is to discard the white noise assumption, following a new analysis approach. In (Toda and Patel, 1980) it is also assumed that either system matrices or noises covariances are accurately known. The results of (Xie and Soh, 1994) and (Luo and Bosch, 1996) consider time invariant systems and assume that the true system is open-loop stable. These limitations will all be overcome in this paper.

Both the error dynamics stability and the error second moment upper bound will be investigated in this paper. While the new stability result under wrong noise assumptions (Theorem 2) is simply one step ahead from classical results, the boundedness of the state estimation error second moment (Theorem 4) is completely new and technically involved.

Besides the stability of the original standard Kalman filter, a related topic in the literature is how to modify the Kalman filter to increase robustness or accuracy. Such results include: minimum variance filter rejecting unknown inputs (Darouach and Zasadzinski, 1997; Zhang and Delyon, 2022), regularized bounded data uncertainties filter (Sayed, 2001), robust guaranteed cost filter (Dong and You, 2006), robust Kalman filter by regularized least-squares estimations (Ishihara et al., 2015; Rocha and Terra, 2021), robust risk-sensitive Kalman filter (Levy and Nikoukhah, 2013; Zorzi, 2016; Yi and Zorzi, 2021), adaptive Kalman filter (Zhang, 2018), polynomial chaos expansion based robust filter (Kim et al., 2020), and augmented state Kalman filter with noise models (Anderson and Moore, 1979; Simon, 2006). This non exhaustive list illustrates the attentions paid to robustness for state estimation problems. Of course, such more advanced estimators improve robustness, but at the price of more efforts for algorithm implementation and data processing. The original Kalman filter is still

widely applied for various reasons. The purpose of this paper is not to make it more robust or more accurate, but to reassure its practitioners by formally confirming its stability under practical conditions, which has already been empirically observed in many applications.

## 2 Notations and definitions

In this paper, lower case letters denote scalars and vectors, whereas upper case letters are reserved to matrices. The $n \times n$ identity matrix is denoted by $I_n$. For a vector $v$, its Euclidean norm is denoted by $\|v\|$. For a matrix $M$, its matrix-norm induced by the Euclidean vector norm is denoted by $\|M\|$, which is equal to its largest singular value. This norm is sub-multiplicative: $\|MN\| \le \|M\|\|N\|$ for any matrices $M$ and $N$ of appropriate sizes. For a real symmetric matrix $M$, the inequality $M > 0$ ($M \ge 0$) means that $M$ is (semi)-positive definite. For two real symmetric matrices $M, N$ of the same size, $M > N$ means $M - N > 0$, and $M \ge N$ means $M - N \ge 0$. For a real symmetric matrix $M \ge 0$, $M^{\frac{1}{2}}$ denotes its symmetric (semi)-positive definite square root. A matrix sequence $M_k$, with $k = 0, 1, 2, \dots$, is said (upper) bounded if $\|M_k\| \le \alpha_M$ for some positive constant $\alpha_M$, and similarly for a vector sequence $v_k$. If $M_k \in \mathbb{R}^{n \times n}$ is a sequence of (semi)-positive definite matrices, its upper bound $\alpha_M$ takes also the sense that $M_k \le \alpha_M I_n$. For a random vector $x \in \mathbb{R}^n$, its second (order) moment is $\mathbb{E}[xx^T]$, with $\mathbb{E}[\cdot]$ denoting the mathematical expectation. A zero mean white noise $w_k \in \mathbb{R}^n$ is a random sequence such that $\mathbb{E}[w_k] = 0$ for all integer $k$ and $\mathbb{E}[w_k w_{k+1}^T] = 0$ for any integer $i \ne 0$.

**Definition 1** For a deterministic system

$$ \xi_{k+1} = F_k \xi_k \quad (1) $$

with state vector $\xi_k \in \mathbb{R}^n$ characterized by $F_k \in \mathbb{R}^{n \times n}$, the associated state transition matrix $\Phi_{k|l}$ for integer pairs $(k, l)$ is defined as

$$ \Phi_{k|l} \triangleq F_{k-1}F_{k-2}\cdots F_l \quad \text{for } k > l \ge 0 \quad (2a) $$

$$ \Phi_{k|l} \triangleq I_n \quad \text{for } k \ge 0. \quad (2b) $$

**Definition 2** Consider a deterministic system (1) and its associated state transition matrix $\Phi_{k|l}$ defined in (2).

* System (1) is asymptotically stable if $\|\Phi_{k|l}\|$ is bounded for all integer pair $k, l$ satisfying $k \ge l \ge 0$ and, for any initial state $x_0 \in \mathbb{R}^n$, $\|x_k\| \to 0$ when $k \to +\infty$.

* System (1) is exponentially stable if

$$ \|\Phi_{k|l}\| \le \gamma \lambda^{k-l} \quad (3) $$

for all integer pair $k, l$ satisfying $k \ge l \ge 0$ and for some constants $\gamma > 0, \lambda \in (0, 1)$.
**Definition 3 (UCO)** A matrix sequence pair $[A_k, C_k]$, with $A_k \in \mathbb{R}^{n \times n}$ and $C_k \in \mathbb{R}^{m \times n}$ for $k = 0, 1, 2, \dots$ is uniformly completely observable (UCO), if there exist two positive constants $\alpha_2 > \alpha_1 > 0$ and a positive integer $h$ such that, for all $k = 0, 1, 2, \dots$,

$$ \alpha_1 I_n \le \sum_{j=k}^{k+h-1} \Phi_{j|k}^T C_j^T C_j \Phi_{j|k} \le \alpha_2 I_n, \quad (4) $$

where $\Phi_{k|l}$ is defined in (2) with $F_k = A_k$.

**Definition 4 (UCC)** A matrix sequence pair $[A_k, B_k]$, with $A_k \in \mathbb{R}^{n \times n}$ and $B_k \in \mathbb{R}^{n \times q}$ for $k = 0, 1, 2, \dots$, is uniformly completely controllable (UCC), if there exist two positive constants $\beta_2 > \beta_1 > 0$ and a positive integer $h$ such that, for all $k = h, h+1, \dots$,

$$ \beta_1 I_n \le \sum_{j=k-h+1}^{k} \Phi_{k+1|j+1} B_j B_j^T \Phi_{k+1|j+1}^T \le \beta_2 I_n \quad (5) $$

where $\Phi_{k|l}$ is defined in (2) with $F_k = A_k$.

These UCO and UCC definitions follow (Moore and Anderson, 1980). See also (Deyst Jr and Price, 1968; Jazwinski, 1970) for similar definitions.

### 3 Classical facts about the Kalman filter

Consider the time varying system model

$$ x_{k+1} = A_k x_k + B_k u_k + w_k, \quad (6a) $$

$$ y_k = C_k x_k + v_k, \quad (6b) $$

where $k = 0, 1, 2, \dots$ is the discrete time index, $x_k \in \mathbb{R}^n$ is the state, $u_k \in \mathbb{R}^q$ is a bounded input sequence, $y_k \in \mathbb{R}^m$ is the output, $w_k \in \mathbb{R}^n$, $v_k \in \mathbb{R}^m$ are zero-mean white noises independent of each other and of the initial state $x_0$, $A_k, B_k, C_k$ are bounded matrix sequences of appropriate sizes, and the noise covariances $E(w_k w_k^T) = Q_k \ge 0$, $E(v_k v_k^T) = R_k > 0$ are also bounded, with bounded $R_k^{-1}$. The initial state $x_0$ has its mean value $E(x_0) = \bar{x}_0 \in \mathbb{R}^n$, and its covariance $Cov(x_0) = \bar{P}_0 \in \mathbb{R}^{n \times n}$ (a symmetric positive definite matrix).

The well known Kalman filter consists of the following recursive computations for $k = 0, 1, 2, \dots$, after the initialization $P_{0|0} = \bar{P}_0 > 0$ and $\hat{x}_{0|0} = \bar{x}_0$:

$$ P_{k+1|k} = A_k P_{k|k} A_k^T + Q_k \quad (7a) $$

$$ \Sigma_{k+1} = C_{k+1} P_{k+1|k} C_{k+1}^T + R_{k+1} \quad (7b) $$

$$ K_{k+1} = P_{k+1|k} C_{k+1}^T \Sigma_{k+1}^{-1} \quad (7c) $$

$$ P_{k+1|k+1} = (I_n - K_{k+1}C_{k+1})P_{k+1|k} \quad (7d) $$

$$ \hat{x}_{k+1|k} = A_k \hat{x}_{k|k} + B_k u_k \quad (7e) $$

$$ \tilde{y}_{k+1} = y_{k+1} - C_{k+1} \hat{x}_{k+1|k} \quad (7f) $$

$$ \hat{x}_{k+1|k+1} = \hat{x}_{k+1|k} + K_{k+1} \tilde{y}_{k+1}. \quad (7g) $$

Let the state estimation error be denoted by

$$ \tilde{x}_{k|k} \triangleq x_k - \hat{x}_{k|k}, \quad (8) $$

then the error dynamics

$$ \tilde{x}_{k+1|k+1} = (I_n - K_{k+1}C_{k+1})A_k \tilde{x}_{k|k} + (I_n - K_{k+1}C_{k+1})w_k - K_{k+1}v_{k+1} \quad (9) $$

is derived from (6) and (7).

It is well known that the Kalman filter minimizes the trace of the error covariance $P_{k|k}$. It is also minimized in the sense of matrix positive definiteness.

**Definition 5** The stability of the error dynamics system (9) refers to its deterministic part. More specifically, system (9) is asymptotically or exponentially stable if, with $F_k = (I_n - K_{k+1}C_{k+1})A_k$, system (1) is asymptotically or exponentially stable, respectively, in the sense of Definition 2.

This error dynamics stability definition has been adopted in the classical results about the Kalman filter stability (Kalman, 1963; Deyst Jr and Price, 1968; Jazwinski, 1970; Moore and Anderson, 1980). Indeed, $F_k = (I_n - K_{k+1}C_{k+1})A_k$ is essential for the behavior of $\tilde{x}_{k|k}$ recursively characterized by (9).

**Theorem 1 (Moore & Anderson 1980)** If the pair $[A_k, C_k]$ is UCO, then the state estimation error covariance matrix $P_{k|k}$ is bounded and the error dynamics (9) is asymptotically stable. If, in addition, $[A_k, Q_k^{\frac{1}{2}}]$ is UCC, then this error dynamics is exponentially stable. $\square$

**Proof outline of Theorem 1** following (Moore and Anderson, 1980).

* A non-recursive unbiased linear state estimator is first built, with an easily proved bounded error covariance. Because the Kalman filter minimizes the error covariance (optimal filter) among all unbiased linear estimators, its error covariance $P_{k|k}$ cannot be larger, and therefore is bounded through the non-recursive estimator.
• With the inverse covariance matrix $P_{k|k}^{-1}$, a quadratic Lyapunov function is defined and analyzed, thus proving the stability of the error dynamics. The boundedness of $P_{k|k}$ is essential in this proof. □

The optimality of the Kalman filter relies on its assumptions on white noises and on perfect model-system matching. If these assumptions are not satisfied, its optimality is lost, then the classical upper bound of $P_{k|k}$ and stability proof are no longer valid in this case.

**Remark 1** If the white noise assumption is not satisfied, the recursively computed $P_{k|k}$ in (7) is no longer equal to the error covariance matrix.

A similar boundedness and stability proof was also given in (Deyst Jr and Price, 1968), with an extra assumption on the invertibility of $A_k$.

**Remark 2** In the classical Kalman filter theory, including the recalled Theorem 1, it was assumed that the bounded matrices $A_k, B_k, C_k$ and $Q_k, R_k$ characterizing the true system and noises are exactly the same as those in the Kalman filter implementation. For this reason no distinction was made between the true system and the assumed model. Moreover, $w_k$ and $v_k$ were assumed to be white noises. This paper will relax these assumptions.

**Remark 3** The matrix sequences $A_k, C_k, Q_k, R_k$ and $R_k^{-1}$ being assumed bounded, the boundedness of $P_{k|k}$ implies the boundedness of all the other auxiliary variables in the Kalman filter recursion: $P_{k+1|k}, K_k, \Sigma_k$. Under classical assumptions (white noises and perfect model-system matching, cf. Remark 2, and also Remark 1), $P_{k|k}$ is the covariance of the state estimation error.

# 4 Wrong noise assumptions

To investigate the stability under wrong noise assumptions, it is necessary to distinguish the true system from the model assumed by the Kalman filter implementation. For this purpose, equations (6) will be reserved for representing the model used in the Kalman filter (7), whereas the true system will be characterized by

$$x_{k+1} = A_k x_k + B_k u_k + w_k^*, \quad (10a)$$

$$y_k = C_k x_k + v_k^* \quad (10b)$$

where $w_k^* \in \mathbb{R}^n$ and $v_k^* \in \mathbb{R}^m$ are the noises actually involved in the considered system. The notations $Q_k^*$ and $R_k^*$ will denote the covariances of the true noises $w_k^*$ and $v_k^*$, while $Q_k$ and $R_k$ are the covariances of the noises $w_k$ and $v_k$ in model (6) assumed by the Kalman filter (7). In practice, $Q_k$ and $R_k$ are available, but not $Q_k^*$ and $R_k^*$. Moreover, typically $w_k^*$ and $v_k^*$ are not white noises, and their mean values are close, but not exactly equal, to zero.

**Remark 4** In system (10), the notations of $x_k$ and $y_k$ could have been replaced by $x_k^*$ and $y_k^*$, to be in agreement with $w_k^*$ and $v_k^*$. By doing so, we would also rewrite the Kalman filter (7) by replacing $y_k$ with $y_k^*, \hat{x}_{k|k}$ with $\hat{x}_{k|k}^*, \text{etc.}$ Moreover, following the same logic, later in (50), yet another version of similar notations would also be used. In order to avoid too many similar notations, $x_k$ and $y_k$ always denote the state and the output of the true system currently under consideration. From now on, equation (6) does not represent the true system. Instead, it represents the model based on which the Kalman filter is implemented. The “model” essentially refers to the matrices $A_k, B_k, C_k$ and $Q_k, R_k$, whereas the sequences $x_k, u_k, y_k$ serve as dummy variables in equation (6).

In this section, the assumed model (6) and the true system (10) have the same matrices $A_k, B_k, C_k$. The discrepancies in these matrices between the model and the system will enter into consideration in the next section.

## 4.1 State estimation error dynamics stability

**Theorem 2** Consider system (10) with bounded noise covariances $Q_k^*$ and $R_k^*$, and apply the Kalman filter (7) based on model (6) with bounded noise covariances $Q_k$ and $R_k$. If the pair $[A_k, C_k]$ is UCO and the pair $[A_k, Q_k^*]$ is UCC, then the error dynamics (11) is exponentially stable, and the gain sequence $K_k$ computed in (7) is bounded, no matter if the noises $w_k^*$ and $v_k^*$ are white or not, nor if $Q_k \neq Q_k^*$ and $R_k \neq R_k^*$. □

Like in the classical case of Theorem 1, this result does not require the open-loop stability of the considered system. In particular, for constant $A_k = A$, its eigenvalues may be outside the unit circle.

**Proof of Theorem 2.** When the Kalman filter (7) is applied to system (10), it is important that it is implemented with noise covariances $Q_k$ and $R_k$ as in model (6), since the noise covariances $Q_k^*$ and $R_k^*$ in the true system (10) are unknown in practice. The error dynamics

$$\tilde{x}_{k+1|k+1} = (I_n - K_{k+1}C_{k+1})A_k \tilde{v}_{k|k} + (I_n - K_{k+1}C_{k+1})w_k^* - K_{k+1}v_{k+1}^* \quad (11)$$

is then derived from (10) and (7), with $\tilde{x}_{k|k}$ defined in (8). The noises terms $w_k^* \in \mathbb{R}^n$ and $v_k^* \in \mathbb{R}^m$ are as in (10), with covariances $Q_k^*$ and $R_k^*$.

Theorem 2 has been stated under general assumptions about the noises $w_k^*$ and $v_k^*$. Before the general case, let us first consider the particular case where $w_k^*$ and $v_k^*$ are white noises of covariances $Q_k^* = Q_k$ and $R_k^* = R_k$. This is in fact the classical case already considered in Section 3. It fully satisfies the assumptions made in Theorem 1, which ensures the exponential stability of
the error dynamics (11) in this particular case under UCO and UCC conditions.

Now back to the case of general noises $w_k^*$ and $v_k^*$ stated in Theorem 2. It turns out that modifying the assumptions on the noises $w_k^*$ and $v_k^*$ (from the particular case to the general case) does not change the *the deterministic part* of the error dynamics (11), which is characterized by the matrix sequence

$$F_k = (I_n - K_{k+1}C_{k+1})A_k, \quad (12)$$

with the same gain sequence $K_k$ computed in (7). It is due to the fact that the model noise covariances $Q_k$ and $R_k$ are always used in the Kalman filter (7) implementation (the true system noise covariances $Q_k^*$ and $R_k^*$ are unknown in practice<fnref n="1" />). The system noise terms $w_k^*$ and $v_k^*$ do appear in the error dynamics (11), but they do not change the deterministic part.

According to Definition 5, the stability of the error dynamics refers to its deterministic part. Because Theorem 1 ensures the exponential stability of the error dynamics (11) in the previously considered particular case, and its deterministic part remains the same in the general case, this error dynamics in the general case is also exponentially stable.

In the Kalman filter (7), the gain sequence $K_k$ is fully determined by the model matrices $A_k, B_k, C_k$ and $Q_k, R_k$ as in (6). It does not depend on the input-output data, nor on the actual noises sequences $w_k^*$ and $v_k^*$. Therefore, the boundedness of $K_k$ ensured by Theorem 1 (see Remark 3) remains valid for the case considered in Theorem 2. The proof of Theorem 2 is thus completed. □

In the above result, it was assumed that $A_k, B_k$ and $C_k$ are exactly known when the Kalman filter is implemented. Their uncertainties will enter into consideration in Section 5. Though the stability of the error dynamics (11) depends on $Q_k$ and $R_k$ instead of $Q_k^*$ and $R_k^*$, the noises of the true system are indeed involved in the error dynamics (11) through the terms $w_k^*$ and $v_k^*$. The effects of these noises will be important in the statistical analysis of state estimation errors. Notably, the fact that $w_k^*$ and $v_k^*$ are colored noises (instead of being white) will make the second moment (and covariance) of state estimation errors particularly difficult to analyze.

## 4.2 Mean trajectory of state estimation errors

**Theorem 3** Consider system (10) with noise covariances $Q_k^*$ and $R_k^*$, and apply the Kalman filter (7) based on model (6) with noise covariances $Q_k$ and $R_k$. If the pair $[A_k, C_k]$ is UCO and the pair $[A_k, Q_k^{\frac{1}{2}}]$ is UCC, then

the mathematical expectation of the state estimation error $E[\tilde{x}_{k|k}]$ is bounded, provided the noises $w_k^*$ and $v_k^*$ involved in system (10) have bounded mathematical expectations $E[w_k^*]$ and $E[v_k^*]$ for all time instants $k \ge 0$, no matter if $w_k^*$ and $v_k^*$ are white noises or not, nor if $Q_k \neq Q_k^*$ and $R_k \neq R_k^*$. □

**Corollary 1** If, in Theorem 3, $E[w_k^*] = 0$ and $E[v_k^*] = 0$ at all time instants $k \ge 0$, then the state estimation error expectation $E[\tilde{x}_{k|k}]$ tends exponentially to zero when $k \to +\infty$. □

**Proof of Theorem 3.** Take the mathematical expectation at both sides of the error dynamics equation (11), then

$$E[\tilde{x}_{k+1|k+1}] = (I_n - K_{k+1}C_{k+1})A_k E[\tilde{x}_{k|k}] + (I_n - K_{k+1}C_{k+1})E[w_k^*] - K_{k+1}E[v_{k+1}^*]. \quad (13)$$

With the notations

$$\tilde{x}_{k|k} \triangleq E[\tilde{x}_{k|k}] \quad (14)$$

$$\tilde{\zeta}_k \triangleq (I_n - K_{k+1}C_{k+1})E[w_k^*] - K_{k+1}E[v_{k+1}^*], \quad (15)$$

the expected error dynamics is rewritten as

$$\tilde{x}_{k+1|k+1} = (I_n - K_{k+1}C_{k+1})A_k \tilde{x}_{k|k} + \tilde{\zeta}_k. \quad (16)$$

Because $E[w_k^*]$ and $E[v_k^*]$ are bounded (assumed in Theorem 3), so are $K_k, C_k$ (see Remark 3), then $\tilde{\zeta}_k$ is also bounded.

The homogeneous part of (16) and the deterministic part of (11) are characterized by the same $F_k = (I_n - K_{k+1}C_{k+1})A_k$. By applying Theorem 2, the deterministic part of (11) is exponentially stable, so is the homogeneous part of (16). The state of an exponentially stable linear system driven by a bounded input is bounded, therefore $\tilde{x}_{k|k} = E[\tilde{x}_{k|k}]$ is bounded. □

**Proof of Corollary 1.** In this case $\tilde{\zeta}_k = 0$ for all $k \ge 0$. Then the expected error dynamics (16) becomes an homogeneous system, which is exponentially stable, hence $\tilde{x}_{k|k} = E[\tilde{x}_{k|k}]$ tends exponentially to zero when $k \to +\infty$. □

## 4.3 Second moment of state estimation errors

This subsection presents the main technical result of this paper.

**Theorem 4** Consider system (10), in which the noises $w_k^*$ and $v_k^*$ have bounded covariances $Q_k^*$ and $R_k^*$, as well as bounded mean values $E[w_k^*]$ and $E[v_k^*]$, and apply the

<footnote n="1">¹ The case with systems matrices $A_k^* \neq A_k, B_k^* \neq B_k, C_k^* \neq C_k$ will be considered in Section 5.</footnote>
Kalman filter (7) based on model (6) with noise covariances $Q_k$ and $R_k$. If the pair $[A_k, C_k]$ is UCO and the pair $[A_k, Q_k^2]$ is UCC, then the second moment of the state estimation error $E[\tilde{x}_{k|k} \tilde{x}_{k|k}^T]$ is bounded, with an upper bound linearly depending on upper bounds of $\|Q_k^* - Q_k\|$ and $\|R_k^* - R_k\|$, no matter if $w_k^*$ and $v_k^*$ are white noises or not. □

Before proving this result, let us comment on the difficulty caused by the absence of white noise assumption. In order to make the results general, no finite correlation length is assumed, i.e., possibly $E[w_k w_{k+1}^T] \neq 0$ and $E[v_k v_{k+1}^T] \neq 0$ for all integer $i$, nor any particular decay of the correlation in $i$. If $w_k^*$ and $v_k^*$ were white noises with zero mean values, a simple recursive formula of the second moment would be immediately obtained from (11) by taking its “expected square”:

$$ \begin{aligned} E[\tilde{x}_{k+1|k+1} \tilde{x}_{k+1|k+1}^T] \\ = (I_n - K_{k+1}C_{k+1})A_k E[\tilde{x}_{k|k} \tilde{x}_{k|k}^T] A_k^T \\ \cdot (I_n - K_{k+1}C_{k+1})^T + E[\zeta_k \zeta_k^T] \end{aligned} \quad (17) $$

with

$$ \zeta_k \triangleq (I_n - K_{k+1}C_{k+1})w_k^* - K_{k+1}v_{k+1}^*. \quad (18) $$

This recursive formula, related to the Joseph form of covariance update, would be equivalent to the recursive computation of $P_{k+1|k}$ and $P_{k+1|k+1}$ in (7) with the Kalman gain $K_{k+1}$ computed in (7c). Indeed, the Kalman filter has been derived under white noise assumption. In reality, $w_k^*$ and $v_k^*$ are not white noises. Then, in (11), $\tilde{x}_{k|k}$ depending on past noises is correlated with $w_k^*$ and $v_{k+1}^*$. Therefore, cross terms between noises at different time instants should be added into (17).

The dependence of $\tilde{x}_{k|k}$ on past noise terms $w_j^*$ and $v_j^*$, for $j \le k$, can be better seen from the following non-recursive expression of $\tilde{x}_{k|k}$ obtained by recursively applying (11):

$$ \tilde{x}_{k|k} = \Phi_{k|0} \tilde{x}_{0|0} + \sum_{j=0}^{k-1} \Phi_{k|j+1} \left[ (I_n - K_{j+1}C_{j+1})w_j^* - K_{j+1}v_{j+1}^* \right] \quad (19) $$

where $\Phi_{k|l}$ is defined in (2) with $F_k$ as in (12). Therefore, the “expected square” $E[\tilde{x}_{k|k} \tilde{x}_{k|k}^T]$ involves the correlations between all the past noise terms $w_i^*$, $v_i^*$ and $w_j^*$, $v_j^*$ due to the absence of white noise assumption. Direct analysis of $E[\tilde{x}_{k|k} \tilde{x}_{k|k}^T]$ would be difficult because of all these correlation terms. The following proof avoids such a direct analysis.

**Proof of Theorem 4.**

Rewrite the error dynamics equation (11) with shorter

notations:

$$ \tilde{x}_{k+1|k+1} = F_k \tilde{x}_{k|k} + \zeta_k \quad (20) $$

where $F_k$ is as in (12), and $\zeta_k$ as in (18).

Substitute (18) into (19), then

$$ \tilde{x}_{k|k} = \Phi_{k|0} \tilde{x}_{0|0} + \sum_{j=0}^{k-1} \Phi_{k|j+1} \zeta_j. \quad (21) $$

To prove the boundedness of $E[\tilde{x}_{k|k} \tilde{x}_{k|k}^T]$, it is sufficient to consider the case $\tilde{x}_{0|0} = 0$. The term $\Phi_{k|0} \tilde{x}_{0|0}$ is dropped from (21) from now on (before being reincorporated towards the end of the proof for upper bound estimation).

The proof will consist in analyzing, for any vector $\eta \in \mathbb{R}^n$, the scalar value $\eta^T E[\tilde{x}_{k|k} \tilde{x}_{k|k}^T] \eta$. For this purpose, define the notation

$$ \rho_k^2 \triangleq \eta^T \tilde{x}_{k|k} \tilde{x}_{k|k}^T \eta. \quad (22) $$

Then, the term $\Phi_{k|0} \tilde{x}_{0|0}$ being dropped from (21),

$$ \rho_k^2 = \eta^T \left( \sum_{j=0}^{k-1} \Phi_{k|j+1} \zeta_j \right) \left( \sum_{j=0}^{k-1} \Phi_{k|j+1} \zeta_j \right)^T \eta \quad (23) $$

$$ = \left( \sum_{j=0}^{k-1} \eta^T \Phi_{k|j+1} \zeta_j \right)^2 \quad (24) $$

$$ \le \left( \sum_{j=0}^{k-1} \lambda^{\frac{1}{2}(k-j-1)} \left( \lambda^{-\frac{1}{2}(k-j-1)} |\eta^T \Phi_{k|j+1} \zeta_j| \right) \right)^2 \quad (25) $$

with any constant $\lambda \in (0, 1)$.

Then by Cauchy-Schwarz inequality,

$$ \rho_k^2 \le \sum_{j=0}^{k-1} \left( \lambda^{\frac{1}{2}(k-j-1)} \right)^2 \sum_{j=0}^{k-1} \left( \lambda^{-\frac{1}{2}(k-j-1)} |\eta^T \Phi_{k|j+1} \zeta_j| \right)^2 \quad (26) $$

$$ = \sum_{j=0}^{k-1} \lambda^{(k-j-1)} \sum_{j=0}^{k-1} \lambda^{-(k-j-1)} |\eta^T \Phi_{k|j+1} \zeta_j|^2. \quad (27) $$

The expression $|\eta^T \Phi_{k|j+1} \zeta_j|^2$ is a scalar, which can be rewritten as

$$ |\eta^T \Phi_{k|j+1} \zeta_j|^2 = (\eta^T \Phi_{k|j+1} \zeta_j) (\eta^T \Phi_{k|j+1} \zeta_j)^T \quad (28) $$

$$ = \eta^T \Phi_{k|j+1} (\zeta_j \zeta_j^T) \Phi_{k|j+1}^T \eta, \quad (29) $$
then

$$ \mathbf{E} [ |\eta^T \Phi_{k|j+1} \zeta_j|^2 ] = \eta^T \Phi_{k|j+1} \mathbf{E} [ \zeta_j \zeta_j^T ] \Phi_{k|j+1}^T \eta. \quad (30) $$

This positive scalar is equal to its own norm. Due to the sub-multiplicative property of the matrix norm induced by the Euclidean norm,

$$ \mathbf{E} [ |\eta^T \Phi_{k|j+1} \zeta_j|^2 ] \leq \|\eta\|^2 \|\Phi_{k|j+1}\|^2 \|\mathbf{E} [\zeta_j \zeta_j^T]\|. \quad (31) $$

By Theorem 2, the error dynamics (11), which has been rewritten in (20), is exponentially stable. Then the associated state transition matrix, as defined in (2) with $F_k$ in (12), satisfies

$$ \|\Phi_{k|l}\| \leq \gamma \lambda^{k-l} \quad (32) $$

with some constants $\gamma > 0$ and $\lambda \in (0, 1)$ for all $k \ge l \ge 0$, according to Definition 2. Then, following (31),

$$ \mathbf{E} [ |\eta^T \Phi_{k|j+1} \zeta_j|^2 ] \leq \|\eta\|^2 \gamma^2 \lambda^{2(k-j-1)} \|\mathbf{E} [\zeta_j \zeta_j^T]\|. \quad (33) $$

Go back to (27) with this result, then

$$ \mathbf{E}[\rho_k^2] \le \sum_{j=0}^{k-1} \lambda^{(k-j-1)} \sum_{j=0}^{k-1} \lambda^{-(k-j-1)} \mathbf{E}[|\eta^T \Phi_{k|j+1} \zeta_j|^2] \quad (34) $$

$$ \le \|\eta\|^2 \gamma^2 \sum_{j=0}^{k-1} \lambda^{(k-j-1)} \sum_{j=0}^{k-1} \lambda^{(k-j-1)} \|\mathbf{E}[\zeta_j \zeta_j^T]\| \quad (35) $$

$$ \le \|\eta\|^2 \gamma^2 \beta_{\zeta} \left( \sum_{j=0}^{k-1} \lambda^{(k-j-1)} \right)^2 \quad (36) $$

where $\beta_{\zeta}$ is a positive constant such that $\|\mathbf{E}[\zeta_k \zeta_k^T]\| \le \beta_{\zeta}$ for all $k \ge 0$. Such an upper bound $\beta_{\zeta}$ exists for the following reasons.

According to Theorem 2, the gain sequence $K_k$ is bounded. As assumed in Theorem 4, the true noises $w_k^*$ and $v_k^*$ have bounded covariances $Q_k^*, R_k^*$ and bounded mean values, thus the combined noise $\zeta_k$ defined in (18) has a bounded second moment $\mathbf{E}[\zeta_k \zeta_k^T]$. The value of $\beta_{\zeta}$ is obtained from (18) as

$$ \beta_{\zeta} = 2(1 + \alpha_K \alpha_C)^2 \beta_{w^*} + 2\alpha_K^2 \beta_{v^*} \quad (37) $$

where $\alpha_K, \alpha_C$ and $\beta_{w^*}, \beta_{v^*}$ are positive constants such that $\|K_k\| \le \alpha_K, \|C_k\| \le \alpha_C$ and $\|\mathbf{E}[w_k^* w_k^{*T}]\| \le \beta_{w^*}, \|\mathbf{E}[v_k^* v_k^{*T}]\| \le \beta_{v^*}$, with $\beta_{w^*}$ and $\beta_{v^*}$ detailed below.

Because

$$ \mathbf{E}[w_k^* w_k^{*T}] = Q_k^* + \mathbf{E}[w_k^*] \mathbf{E}[w_k^*]^T \quad (38) $$

$$ = \tilde{Q}_k + Q_k + \mathbf{E}[w_k^*] \mathbf{E}[w_k^*]^T \quad (39) $$

with

$$ \tilde{Q}_k \triangleq Q_k^* - Q_k, \quad (40) $$

then

$$ \|\mathbf{E}[w_k^* w_k^{*T}]\| \le \|\tilde{Q}_k\| + \|Q_k\| + \|\mathbf{E}[w_k^*] \mathbf{E}[w_k^*]^T\|. \quad (41) $$

hence an upper bound of $\|\mathbf{E}[w_k^* w_k^{*T}]\|$ is given by

$$ \beta_{w^*} = \alpha_{\tilde{Q}} + \alpha_Q + \mu_{w^*}^2 \quad (42) $$

with positive constants $\alpha_{\tilde{Q}}, \alpha_Q, \mu_{w^*}$ such that $\|\tilde{Q}_k\| \le \alpha_{\tilde{Q}}, \|Q_k\| \le \alpha_Q, \|\mathbf{E}[w_k^*]\| \le \mu_{w^*}$ for all $k \ge 0$. With similar notations,

$$ \beta_{v^*} = \alpha_{\tilde{R}} + \alpha_R + \mu_{v^*}^2. \quad (43) $$

Then (37) is detailed as

$$ \beta_{\zeta} = 2(1 + \alpha_K \alpha_C)^2 (\alpha_{\tilde{Q}} + \alpha_Q + \mu_{w^*}^2) + 2\alpha_K^2 (\alpha_{\tilde{R}} + \alpha_R + \mu_{v^*}^2) \quad (44) $$

which is a constant linearly depending on upper bounds of $\|Q_k^* - Q_k\|$ and $\|R_k^* - R_k\|$, namely $\alpha_{\tilde{Q}}$ and $\alpha_{\tilde{R}}$.

For $\lambda \in (0, 1)$ and $k > 0$, the geometric sum involved in (36) satisfies

$$ \left( \sum_{j=0}^{k-1} \lambda^{(k-j-1)} \right)^2 = \left( \frac{1-\lambda^k}{1-\lambda} \right)^2 \le \frac{1}{(1-\lambda)^2}, \quad (45) $$

then

$$ \mathbf{E}[\rho_k^2] \le \|\eta\|^2 \gamma^2 \beta_{\zeta} \frac{1}{(1-\lambda)^2}. \quad (46) $$

It is then proved that, with $\rho_k^2$ defined in (22), for any $\eta \in \mathbb{R}^n$,

$$ \mathbf{E}[\rho_k^2] = \eta^T \mathbf{E}[\tilde{x}_{k|k} \tilde{x}_{k|k}^T] \eta \le \|\eta\|^2 \gamma^2 \beta_{\zeta} \frac{1}{(1-\lambda)^2}. \quad (47) $$

Therefore,

$$ \mathbf{E}[\tilde{x}_{k|k} \tilde{x}_{k|k}^T] \le \gamma^2 \beta_{\zeta} \frac{1}{(1-\lambda)^2} I_n \quad (48) $$
where $I_n$ is the $n \times n$ identity matrix, $\gamma$, $\lambda$ are constants introduced in (32), and $\beta_\zeta$ is the constant specified in (44) with a linear dependence on upper bounds of $\|Q_k^* - Q_k\|$ and $\|R_k^* - R_k\|$, namely $\alpha_{\tilde{Q}}$ and $\alpha_{\tilde{R}}$.

To finish the proof, let us complete the upper bound of $E[\tilde{x}_{k|k} \tilde{x}_{k|k}^T]$ by reincorporating the term $\Phi_{k|0} \tilde{x}_{0|0}$ in (21). It is then concluded

$$E[\tilde{x}_{k|k} \tilde{x}_{k|k}^T] \le 2\gamma^2 E[\tilde{x}_{0|0} \tilde{x}_{0|0}^T] + 2\gamma^2 \beta_\zeta \frac{1}{(1-\lambda)^2}. \quad (49)$$

## 5 Discrepancies in system matrices

In this section, discrepancies in system matrices $A_k$, $B_k$ and $C_k$ between the assumed model and the true system are considered, in addition to wrong noise assumptions.

Let $A_k^*$, $B_k^*$ and $C_k^*$ denote the system matrices of the true system, whereas $A_k$, $B_k$ and $C_k$ are those of the model assumed in the Kalman filter implementation. The true system is now characterized by

$$x_{k+1} = A_k^* x_k + B_k^* u_k + w_k^*, \quad (50a)$$

$$y_k = C_k^* x_k + v_k^*, \quad (50b)$$

where $w_k^*$ and $v_k^*$ are the true noises involved in the system, respectively of covariances $Q_k^*$ and $R_k^*$.

**Theorem 5** Consider system (50) with bounded matrix sequences $A_k^*$, $B_k^*$ and $C_k^*$, as well as bounded noise covariances $Q_k^*$, $R_k^*$ and bounded mean values $E[w_k^*]$ and $E[v_k^*]$. Assume that the true state $x_k$ and the input $u_k$ each have a bounded second moment. The Kalman filter (7) processing data generated by system (50) is based on model (6) with bounded matrices $A_k$, $B_k$, $C_k$ and $Q_k$, $R_k$. If the pair $[A_k, C_k]$ is UCO and the pair $[A_k, Q_k^{1/2}]$ is UCC, then the Kalman filter has an exponentially stable error dynamics, and the second moment of its state estimation error $E[\tilde{x}_{k|k} \tilde{x}_{k|k}^T]$ is bounded, with an upper bound linearly depending on upper bounds of $\|A_k^* - A_k\|^2$, $\|B_k^* - B_k\|^2$, $\|C_k^* - C_k\|^2$, $\|Q_k^* - Q_k\|$ and $\|R_k^* - R_k\|$, no matter if the noises $w_k^*$ and $v_k^*$ are white or not. $\square$

The considered system (50) may be open-loop instable. In this case $u_k$ would be a stabilizing control so that $x_k$ has a bounded second moment. The proof relying on Theorem 4, by treating discrepancies in system matrices as equivalent noise terms, is detailed in (Zhang, 2025).

## 6 Conclusion

It has been shown in this paper that the stability of the Kalman filter depends mostly on the assumed model.

Nevertheless, it should be tempered by the established upper bound of the state estimation error second moment, which depends linearly on the discrepancies between model and system, in terms of both system parameters and noise characterization. Therefore, filter implementations should use models as faithful to reality as possible. The established upper bound is conservative due to the general noises considered. From a practical point of view, possible future directions of interest would be improving the upper bound by adding assumptions about the noises, in particular, for heavy-tailed noises and/or noises with some correlation decay rate.

The stability results presented in this paper mainly rely on the UCO and UCC assumptions. It is possible to relax them to detectability and stabilizability conditions, based on the results of (Anderson and Moore, 1981). Nevertheless, the main information about the stability of the Kalman filter in practical conditions is already conveyed in this paper.

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