# Exponentially Stable Stubborn Observers for Discrete-Time Linear Systems (Extended Version)

Beatrice Zambotti, Vincent Andrieu, Daniele Astolfi, Laurent Bako, Madiha Nadri, Luca Zaccarian

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Beatrice Zambotti, Vincent Andrieu, Daniele Astolfi, Laurent Bako, Madiha Nadri, et al. Exponentially Stable Stubborn Observers for Discrete-Time Linear Systems (Extended Version). 2025. [hal-05229530]

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Distributed under a Creative Commons Attribution 4.0 International License
# Exponentially Stable Stubborn Observers for Discrete-Time Linear Systems
(Extended Version)

Beatrice Zambotti, Vincent Andrieu, Daniele Astolfi, Laurent Bako, Madiha Nadri, Luca Zaccarian

**Abstract**— State estimation is crucial for control and monitoring of dynamical systems, but sporadic disturbances (outliers) can severely degrade the estimator's performance. This paper studies the stability of a so-called “stubborn” observer, designed to mitigate the effects of outliers, for discrete-time linear systems. We prove that the detectability of the system is both necessary and sufficient for global exponential stability of the observer, providing an improvement over existing results that rely on potentially infeasible LMI conditions. The proof is constructive, leading to practical guidelines for selecting the observer parameters. Numerical simulations demonstrate the observer's effectiveness in mitigating outliers and its superior performance compared to a classical Luenberger observer.

## I. INTRODUCTION

In control systems, limited access to direct measurements presents significant challenges in accurately estimating the state variables. To address this issue, state observers are commonly employed to reconstruct the states from the available input-output data, ensuring reliable feedback for control and fault detection. In practical applications, observer performance is heavily influenced by various disturbances, including model mismatches, sensor failures, process noise, and measurement outliers. This paper focuses on the state estimation of linear discrete-time systems subjected to sporadic perturbations, which are irregular and transient events that occur unpredictably at random intervals and can arise from various sources, such as sudden sensor malfunctions, temporary signal losses, or even malicious attacks. The impact of sporadic perturbations on state estimation has been widely addressed in the literature, where a common approach is to rely on Kalman filters adapted to handle intermittent measurements and noisy sensor data [1]–[3]. In the context of cyber-physical systems, state estimation is challenging due to potential adversarial attacks, leading to the development of attack detection algorithms and secure state estimation techniques [4]–[8].

A promising approach to mitigate the impact of sporadic disturbances is the use of the so-called “stubborn” observers, introduced in [9]. These observers employ a dynamic saturation of the output injection to limit the influence of outliers.

Work funded by the ANR under grant OLYMPIA ANR-23-CE48-0006.

B. Zambotti, M. Nadri, D. Astolfi and V. Andrieu are with Université Claude Bernard Lyon 1, CNRS, LAGEPP UMR 5007, 43 boulevard du 11 novembre 1918, F-69100, Villeurbanne, France. Email: name.surname@univ-lyon1.fr

L. Bako is with Ecole Centrale de Lyon, INSA Lyon, Université Claude Bernard Lyon 1, CNRS, Ampère, UMR 5005, 69130 Ecully, France. Email: laurent.bako@ec-lyon.fr

L. Zaccarian is with LAAS-CNRS, Université de Toulouse, Toulouse, France and the Department of Industrial Engineering, University of Trento, Italy. Email: zaccarian@laas.fr

The saturation threshold is adaptively adjusted based on the history of the output errors: it increases gradually if persistent errors occur (indicating a potential model mismatch), but decreases quickly when the measurements become consistent with the model again. Unlike Kalman filters, which require a stochastic model of the noise, stubborn observers do not demand prior knowledge of the distribution of the sporadic disturbances. Stubborn observers have been extensively studied for continuous-time systems, both linear [9], [10] and nonlinear [11].

In the context of discrete-time linear systems, a study was presented in [9], where the stability conditions were formulated in terms of linear matrix inequalities. However, with a numerical counterexample, we show that there exist observable pairs $(A, C)$ for which the LMI conditions of [9] are not met. Furthermore, only asymptotic stability was established in [9]. This motivates our main objective, which consists in establishing that detectability of the pair $(A, C)$ are necessary and sufficient conditions for global exponential stability of a stubborn observer, for discrete-time linear systems.

The remainder of the paper is organized as follows. Section II presents the problem formulation along with our main result, which establishes the global exponential stability of the estimation error under the detectability condition. A detailed proof of the main theorem is provided in Section III. Section IV offers design guidelines for tuning the observer parameters and presents numerical simulations to validate the theoretical findings and demonstrate the observer's superiority over a classical Luenberger observer. Finally, Section V concludes the paper.

**Notation.** In this paper, $|\cdot|$ represents the $l_2$-norm. $\lambda_m(P)$ and $\lambda_M(P)$ denote, respectively, the minimum and maximum eigenvalues of a real symmetric matrix $P$. Additionally, the notations $P > 0$, $P \ge 0$, $P < 0$, and $P \le 0$ indicate that $P$ is positive definite, positive semi-definite, negative definite, and negative semi-definite, respectively. The expression $\text{diag}(w_1, \dots, w_i)$ refers to a diagonal matrix with diagonal elements $(w_1, \dots, w_i)$. Given a matrix $M$, we define $\text{He} M := M + M^\top$. A matrix $M$ is called Schur stable if its spectral radius $\rho(M) < 1$. Finally, given two vectors $x$ and $y$, $(x, y)$ denotes $[x^\top y^\top]^\top$.

## II. PROBLEM FORMULATION AND MAIN RESULT

We consider a discrete-time linear plant described by:

$$ 
\begin{aligned}
x_{k+1} &= Ax_k + B_u u_k + B_d d_k \\
y_k &= Cx_k + D_u u_k + D_d d_k
\end{aligned} 
\quad (1) 
$$
where $k \in \mathbb{Z}_{\ge 0}$ is the discrete-time index, $x_k \in \mathbb{R}^n$ is the state, $y_k \in \mathbb{R}^{n_y}$ is the output, $u_k \in \mathbb{R}^{n_u}$ is the control input, and $d_k \in \mathbb{R}^{n_d}$ is a disturbance. The matrices $A, B_u, B_d, C, D_u$, and $D_d$ have appropriate dimensions. The terms $D_u u_k$ and $D_d d_k$ represent feedthrough of the input and disturbance, respectively, and are included for generality. We define the stubborn state estimator as follows:

$$ 
\begin{aligned}
\hat{x}_{k+1} &= A\hat{x}_k + B_u u_k + L \operatorname{sat}_{\sigma_k^q}(y_k - \hat{y}_k) \\
\sigma_{k+1} &= \lambda\sigma_k + r|y_k - \hat{y}_k|^p, \quad \sigma_k \in \mathbb{R}_{\ge 0}, \\
\hat{y}_k &= C\hat{x}_k + D_u u_k
\end{aligned}
\quad (2) $$

where $L$ is the observer gain matrix, $\lambda \in (0, 1)$ is a scalar, $r > 0$ is a scalar, $p > 0$ is a scalar, $q := \frac{1}{p}$, and $\operatorname{sat}_\sigma(s) := \max\{-\sigma, \min\{\sigma, s\}\}$ is the saturation function with level $\sigma$. The observer dynamics (2) extend those in [9] and [11] by incorporating the $p$-th root of the saturation threshold, allowing for greater design flexibility by controlling the adaptation rate of the saturation level. Note that, the original formulation in [9] considered $p = 2$, while [11] used $p = 1$ and, as it will be shown in Section IV-A, the observer dynamics remains homogeneous for any value of $p \in (0, +\infty)$.

In [9] the asymptotic convergence of the estimation error for a detectable pair $(A, C)$ is established based on an LMI formulation [9, Theorem 2]. In particular, the conditions to be satisfied are given by the following set of LMIs [9, Equations (4c), (23), and (24)] in the variables $X, Y$, $\lambda \in (0, 1)$, $P = P^\top > 0$, $R = R^\top > 0$, $U > 0$ diagonal, $W := \operatorname{diag}(w_1, \dots, w_i) > 0$

$$ \text{He} \quad \begin{bmatrix} -\frac{1}{2}(\lambda P + C^\top RC) & 0 & 0 \\ UC & -U & 0 \\ PA - XC & X & -\frac{1}{2}P \end{bmatrix} < 0, \quad (3a) $$

$$ \text{He} \quad \begin{bmatrix} -\frac{1}{2}P & 0 & 0 \\ WC + Y & -W & 0 \\ PA - XC & X & -\frac{1}{2}P \end{bmatrix} < 0, \quad (3b) $$

$$ \begin{bmatrix} P & Y^\top \\ Y & w_i \end{bmatrix} \ge 0 \quad \forall \quad i = 1, \dots, n_y. \quad (3c) $$

However, the feasibility of such LMIs remains an open question. In fact, we numerically show that there exists at least an observable pair $(A, C)$ for which the LMI conditions (3) could not be verified. Indeed, considering $\lambda = 0.9$, $p = 2$ and the pair $A = \begin{bmatrix} 0 & 1 \\ 1 & 7775 \end{bmatrix}$, $C = \begin{bmatrix} 1 & 0 \end{bmatrix}$, we were unable to find a feasible solution.<fnref n="1" />

This motivates us to explore an alternative approach to assess the stability of stubborn observers. Our main result is stated in Theorem 1, which provides a necessary and sufficient condition for global exponential stability of the error dynamics.

**Theorem 1.** Consider the coupled system (1)-(2) with $d_k = 0$ for all $k$ and let $p > 0$. There exist parameters $L, \lambda$ and $r$ such that the set $\mathcal{E} := \{(x, \hat{x}) \in \mathbb{R}^n \times \mathbb{R}^n : x = \hat{x}\}$ is globally exponentially stable (GES) if and only if the pair $(A, C)$ is detectable.

The proof of Theorem 1, presented in Section III, not only establishes the necessary and sufficient condition for GES but also provides a constructive approach for selecting the observer parameters $L, \lambda$, and $r$. Specifically, the proof derives a lower bound $r^*$ for the parameter $r$ for any linearly stabilizing $L$ and any $\lambda \in (0, 1)$ that guarantees exponential stability. This constructive approach contrasts with the LMI-based conditions of [9], which, as demonstrated, may not be feasible.

### III. PROOF OF THEOREM 1

This section presents the proof of Theorem 1, establishing both the sufficiency and necessity of detectability for global exponential stability of the estimation error. The proof is divided into two main parts. First, in Section III-A, we establish a constructive sufficient condition for global exponential stability in the case where the system is observable. This involves designing the observer parameters $L, \lambda$, and $r$ to guarantee stability. The proof relies on a Lyapunov-like function (Proposition 1) and a lemma that leverages the observability property (Lemma 1). Then, in Section III-B, we extend this result to the general case of detectable systems, proving that detectability is both necessary and sufficient for global exponential stability.

#### A. Design of an Exponentially Stable Stubborn Observer for Observable Systems

This section studies stability properties induced by observer (2) and establishes global exponential stability of the origin of the error coordinates $(e_k, \sigma_k)$, where $e_k := x_k - \hat{x}_k$, under disturbance-free conditions, in the particular case where the pair $(A, C)$ is observable. Employing (1) and (2), the unperturbed dynamics of the estimation error can be expressed as follows:

$$ 
\begin{aligned}
e_{k+1} &= A e_k - L \operatorname{sat}_{\sigma_k^q}(C e_k) \\
\sigma_{k+1} &= \lambda \sigma_k + r |C e_k|^p.
\end{aligned}
\quad (4) $$

**Theorem 2.** Let $L$ be such that $A - LC$ is Schur stable and $(A, C)$ observable. Then, for every $\lambda \in (0, 1)$ there exists a scalar $r^* > 0$ such that, for every $r > r^*$, the origin of the error dynamics (4) is globally exponentially stable.

The proof of Theorem 2 relies on a Lyapunov-like function defined as

$$ V(e, \sigma) = e^{\top} P e + \rho \sigma^{2q} \quad (5) $$

where $\rho > 0$ is a scalar and $P = P^\top > 0$ satisfies

$$ (A - LC)^{\top} P (A - LC) - P < -\eta P \quad (6) $$

for a sufficiently small scalar $\eta > 0$. The existence of such a $P$ is guaranteed by the Schur stability of $A - LC$. Proposition 1, stated below and proven in Section III-C, establishes key properties of function $V$ in (5), which are crucial for the proof of Theorem 2. Specifically, it shows that $V$ enjoys quadratic upper and lower bounds and an exponential decrease condition after an initial transient of $n$ steps. Although the function may exhibit an overshoot during these initial steps, precluding it from being a strict Lyapunov

<footnote n="1">¹To solve the LMI problem we used MOSEK 9.1.9 through YALMIP in MATLAB R2024a, with a tolerance of $10^{-6}$.</footnote>
function, these properties will be shown to be sufficient to
demonstrate global exponential stability.

**Proposition 1.** Consider the error dynamics (4) with $L$ such that $A-LC$ is Schur stable, $|C| \le 1$ and $(A, C)$ observable. Then, for each $\lambda \in (0, 1)$ there exists a scalar $r^* > 0$ such that, for each $r > r^*$, the function $V$ defined in (5) satisfies

$$
c_1 \left| \frac{e}{\sigma^q} \right|^2 \leq V(e, \sigma) \leq c_2 \left| \frac{e}{\sigma^q} \right|^2 \quad (7a)
$$

$$
V(e_{k+1}, \sigma_{k+1}) - V(e_k, \sigma_k) \le -c_3 V(e_k, \sigma_k), \quad \forall k \ge n \tag{7b}
$$

for some positive scalar constants $c_1, c_2$ and $c_3$.

Building upon the properties of the Lyapunov-like function $V$ established in Proposition 1, we now prove Theorem 2. We first establish an upper bound for $V(e_k, \sigma_k)$ during the initial transient phase of $n$ steps. Combining this bound with the properties in (7) directly leads to the global exponential stability of the origin of the error dynamics (4).

**Proof of Theorem 2.** Without loss of generality, we assume that $|C| \le 1$. If this is not the case, an appropriate change of coordinates can be applied to the pair $(A, C)$ to obtain an equivalent representation that achieves the desired property. Considering function (5), bound (7b) implies that

$$
V(e_k, \sigma_k) \le \eta_V^{k-n} V(e_n, \sigma_n), \quad \forall k \ge n, \quad (8)
$$

where $\eta_V := 1 - c_3$. Moreover, given the error dynamics (4) and given that $|\text{sat}_\sigma(s)| \le |s|$, it follows that, for $a = |A| + |L||C|$, we have

$$
|e_{k+1}| \le a|e_k|, \quad (9)
$$

for every initial condition $(e_k, \sigma_k)$. Using the variation of
constants formula [12, Theorem 5.3] and Equation (9) we
can write that for all $k \in \{0, \dots, n\}$

$$
\sigma_k = \lambda^k \sigma_0 + \sum_{i=0}^{k-1} \lambda^{k-i-1} r |Ce_i|^p \le \sigma_0 + \mu_k |e_0|^p \quad (10)
$$

$$
\text{with } \mu_k := \sum_{i=0}^{k-1} \lambda^{k-i-1} r a^i p^i.
$$

Let $k_V := \max\{\frac{a^{2k}\lambda_M(P)+\rho\mu_k^{2q}}{\lambda_m(P)}, \rho\}$, using (9), (10) and [13, Lemma 1], we can upper-bound $V(e_k, \sigma_k)$ as follows:

$$
\begin{align}
V(e_k, \sigma_k) &= e_k^\top P e_k + \rho \sigma_k^{2q} \nonumber \\
&\stackrel{(10)}{\le} \lambda_M(P) |e_k|^2 + \rho (\sigma_0 + \mu_k |e_0|^p)^{2q} \nonumber \\
&\stackrel{(9)}{\le} a^{2k} \lambda_M(P) |e_0|^2 + \rho \sigma_0^{2q} + \rho \mu_k^{2q} |e_0|^2 \tag{11} \\
&\stackrel{\text{a}}{\le} \frac{a^{2k} \lambda_M(P) + \rho \mu_k^{2q}}{\lambda_m(P)} e_0^\top P e_0 + \rho \sigma_0^{2q} \nonumber \\
&\stackrel{\text{b}}{\le} k_V V(e_0, \sigma_0), \quad \forall k \in \{0, \dots, n\}. \nonumber
\end{align}
$$

Concatenating (8) and (11), we obtain

$$
V(e_k, \sigma_k) \le \bar{k}_V \eta_V^k V(e_0, \sigma_0), \quad \forall k \ge 0, \quad (12)
$$

where $\bar{k}_V := k_V \eta_V^{-n}$. From Equation (12), we can establish
that $V(e_k, \sigma_k)$ converges to zero exponentially. Moreover,
given (7a), both $e_k$ and $\sigma_k^q$, and consequently $\sigma_k$, converge
exponentially to zero, thus proving global exponential stabil-
ity of the origin for the error dynamics (4). $\square$

B. Extension to Detectable Systems

When the pair (A, C) is only assumed to be detectable, a change of coordinates exists (the so-called observable decomposition [12, Theorem 16.2]) that allows us to decouple the system into its observable and unobservable components. Specifically, as shown in Corollary 1 below, we can express the error dynamics in a form where Theorem 2 can be applied to the observable part, while the inherent exponential stability of the unobservable part, due to the Schur stability of the corresponding dynamics matrix, guarantees the global exponential stability of the overall system.

**Corollary 1.** Let (A, C) be detectable and let L be such that A-LC is Schur stable. Then, for every $\lambda \in (0, 1)$ there exists a scalar $r^* > 0$ such that, for every $r > r^*$, the origin of the error dynamics (4) is globally exponentially stable.

*Proof.* Given the detectability of the pair (A, C), there exists a change of coordinates $e \mapsto (e_o, e_u)$ such that the matrices A, C, and L admit the following block decompositions:

$$
A = \begin{bmatrix} A_o & 0 \\ A_* & A_u \end{bmatrix}, \quad C = \begin{bmatrix} C_o & 0 \end{bmatrix}, \quad L = \begin{bmatrix} L_o \\ L_u \end{bmatrix}, \quad (13)
$$

where the pair $(A_o, C_o)$ is observable [12, Theorem 16.2].
In these coordinates, the error dynamics (4) becomes:

$$
\begin{align}
e_{k+1}^o &= A_o e_k^o - L_o \operatorname{sat}_{\sigma^q} (C_o e_k^o), \notag \\
e_{k+1}^u &= A_* e_k^o + A_u e_k^u - L_u \operatorname{sat}_{\sigma^q} (C_o e_k^o), \tag{14} \\
\sigma_{k+1} &= \lambda \sigma_k + r |C_o e_k^o|^p. \notag
\end{align}
$$

Note that, with this decomposition, $A-LC$ has the following
block triangular form $A-LC = \begin{bmatrix} A_o - L_o C_o & 0 \\ A_* - L_u C_o & A_u \end{bmatrix}$. Since
$A-LC$ is Schur stable by the assumption and has a
block triangular form, it follows that both $A_o - L_o C_o$ and
$A_u$ are Schur stable. Then, by applying Theorem 2 to the
observable subsystem $(e_k^o, \sigma_k)$, we can conclude that $(e_k^o, \sigma_k)$
converges globally exponentially to the origin. Finally, given
the $e_k^u$ dynamics in (14) and the Schur stability of $A_u$,
the exponential decay of $e_k^o$ (and therefore of $\operatorname{sat}_{\sigma^q}(C_o e_k^o)$)
guarantees that $e_k^u$ also converges exponentially to zero due
to intrinsic exponential ISS of linear dynamics. Therefore, we
can conclude that the origin of (14) is globally exponentially
stable. $\square$

Corollary 1 shows that detectability of the pair (A, C) is a
sufficient condition for the existence of a stubborn observer
of the form (2) with exponentially stable error dynamics. It
remains now to show that detectability is also necessary. For
this purpose, we will make use of the characterization of
detectability proposed in [14, Definition 1.2], which states
that a system is detectable if and only if two trajectories
with identical outputs have a state difference that converges
to zero. Let $x_k^a$ and $x_k^b$ be two state trajectories of (1) such
that they share the same output, i.e., $Cx_k^a = Cx_k^b$ for all
$k \ge 0$. Consider now the observer (2). From the previous
result, both $|\hat{x}_k - x_k^a|$ and $|\hat{x}_k - x_k^b|$ converge to zero as
$k \to \infty$. Applying the triangle inequality, we obtain that
$|x_k^a - x_k^b| \le |x_k^a - \hat{x}_k| + |\hat{x}_k - x_k^b|$, which also converges to
zero. Since $Cx_k^a = Cx_k^b$ and $|x_k^a - x_k^b| \to 0$, this implies, by [14, Definition 1.2], that the pair $(A, C)$ is detectable.

### C. Proof of Proposition 1

To prove Proposition 1, we first introduce Lemma 1, which provides an observability condition for the error dynamics (4), independent of the value of the parameters $\lambda$, and $r$.

**Lemma 1.** Let $(A, C)$ be observable and $L$ be such that $A - LC$ is Schur stable. There exists a scalar $\gamma > 0$, such that for all $\lambda$, and $r$, and all trajectories $(\sigma_k, e_k)_{k \in \mathbb{N}}$ of the error dynamics (4),

$$ \sum_{i=0}^{n-1} |Ce_{k+i}| \geq \gamma |e_k|, \quad \forall k \geq 0. \quad (15) $$

*Proof.* Since $(A, C)$ is observable, there exists $\bar{\gamma} > 0$ such that for all $x$ in $\mathbb{R}^n$

$$ \sum_{i=0}^{n-1} |CA^i x| \geq \bar{\gamma} |x|. \quad (16) $$

First, we show by induction that for each $j \in \{1, \dots, n\}$, there exists a scalar $\kappa_j > 0$ and a function $\phi_j$ such that, for every trajectory $(\sigma_k, e_k)_{k \in \mathbb{N}}$ of (4), it holds that

$$ e_{k+j} = A^j e_k + \phi_j(\sigma_k, \dots, \sigma_{k+j}, CA^0 e_k, \dots, CA^{j-1} e_k), \quad (17a) $$

$$ |\phi_j| \leq \kappa_j \sum_{i=0}^{j-1} |CA^i e_k|. \quad (17b) $$

Equations (17a) and (17b) trivially hold for $j = 1$, with $\phi_1(\sigma_k, Ce_k) = -L \text{sat}_{\sigma_k^q}(Ce_k)$ and $\kappa_1 = |L|$. At step $j+1$ we have

$$ e_{k+j+1} = A^{j+1}e_k + \underbrace{A\phi_j - L \text{sat}_{\sigma_{k+j}^q} (CA^j e_k + C\phi_j)}_{\phi_{j+1}(\sigma_k, \dots, \sigma_{k+j+1}, CA^0 e_k, \dots, CA^{j-1} e_k, CA^j e_k)} \quad (18) $$

where, for the given definition of $\phi_{j+1}$, it holds that

$$ \begin{aligned} |\phi_{j+1}| &\leq |A\phi_j| + |L|(|CA^j e_k| + |C\phi_j|) \\ &\leq \kappa_{j+1} \sum_{i=0}^{j} |CA^i e_k|, \end{aligned} \quad (19) $$

where we used (17b) and with $\kappa_{j+1} = (|A| + |L||C|)\kappa_j$.

Next, we show by induction that, for each $j \in \{1, \dots, n\}$, there exists a scalar $\beta_j > 0$ such that, for every trajectory $(\sigma_k, e_k)_{k \in \mathbb{N}}$ of (4), it holds that

$$ \sum_{i=0}^{j-1} |Ce_{k+i}| \geq \beta_j \sum_{i=0}^{j-1} |CA^i e_k|. \quad (20) $$

Equation (20) simplifies to $|Ce_k| = |Ce_k|$ for $j = 1$, so we can choose any $\beta_1 \in (0, 1]$. Assume (20) holds for some $j$ with $\beta_j > 0$. At step $j+1$, we will explicitly construct $\beta_{j+1} > 0$ such that (20) holds for $j+1$. Consider $\alpha_j \geq 1$,

then:

$$ \begin{aligned} \alpha_j \sum_{i=0}^{j} |Ce_{k+i}| &\geq |Ce_{k+j}| + \alpha_j \sum_{i=0}^{j-1} |Ce_{k+i}| \\ &\stackrel{(17a)}{\geq} |C(A^j e_k + \phi_j)| + \alpha_j \sum_{i=0}^{j-1} |Ce_{k+i}| \\ &\stackrel{(20)}{\geq} |CA^j e_k| - |C\phi_j| + \alpha_j \beta_j \sum_{i=0}^{j-1} |CA^i e_k| \\ &\stackrel{(17b)}{\geq} |CA^j e_k| + (\alpha_j \beta_j - |C|\kappa_j) \sum_{i=0}^{j-1} |CA^i e_k|. \end{aligned} \quad (21) $$

Let now $\alpha_j = \frac{|C|\kappa_j}{\beta_j} + 1$. Since, by the induction hypothesis, $\beta_j > 0$, $\kappa_j > 0$ and since $|C| \geq 0$, we have $\alpha_j \geq 1$, and, therefore, $\frac{1}{\alpha_j} \leq 1$. From (21), it follows that $\alpha_j \sum_{i=0}^{j} |Ce_{k+i}| \geq |CA^j e_k| + \beta_j \sum_{i=0}^{j-1} |CA^i e_k|$, and consequently

$$ \sum_{i=0}^{j} |Ce_{k+i}| \geq \frac{1}{\alpha_j} |CA^j e_k| + \frac{\beta_j}{\alpha_j} \sum_{i=0}^{j-1} |CA^i e_k| \quad (22) $$

Then, by selecting $\beta_{j+1} = \frac{\min\{1, \beta_j\}}{\alpha_j}$, we can conclude that

$$ \sum_{i=0}^{j} |Ce_{k+i}| \geq \beta_{j+1} \sum_{i=0}^{j} |CA^i e_k|, \quad (23) $$

which proves (20) for the step $j+1$. Finally, by the observability of the pair $(A, C)$, we can use (16) and (20) evaluated with $j=n$ to conclude that:

$$ \sum_{i=0}^{n-1} |Ce_{k+i}| \geq \beta_n \sum_{i=0}^{n-1} |CA^i e_k| \geq \gamma |e_k|, \quad (24) $$

where $\gamma = \bar{\gamma}\beta_n$, which completes the proof of Lemma 1. $\square$

We now prove Proposition 1 by selecting a lower bound $r^*$ for the observer parameter $r$. This selection guarantees that after a transient period of $n$ steps, the saturation effect vanishes, allowing the Lyapunov-like function (5) to decrease monotonically. For the proof of Proposition 1, we first introduce the following lemma.

**Lemma 2.** Let $a, b, \varrho \geq 0$ be any scalars. Then, for any $\varepsilon > 0$, the following holds:

$$ \varrho \leq 1 \Rightarrow (a+b)^\varrho \leq a^\varrho + b^\varrho, \quad (25a) $$

$$ \varrho \geq 1 \Rightarrow (a+b)^\varrho \leq (1+\varepsilon)^{\varrho-1} a^\varrho + (1+\frac{1}{\varepsilon})^{\varrho-1} b^\varrho. \quad (25b) $$

*Proof.* If $\varrho \in [0, 1]$, inequality (25a) holds since the function $x \mapsto x^\varrho$ is subadditive on $\mathbb{R}_{\geq 0}$. Consider the case where $\varrho \geq 1$, and initially $a > 0$. Dividing both sides of (25b) by $a^\varrho$ and setting $t := \frac{b}{a}$, the inequality (25b) becomes

$$ (1+t)^\varrho \leq (1+\varepsilon)^{\varrho-1} + (1+\frac{1}{\varepsilon})^{\varrho-1} t^\varrho. \quad (26) $$

Define the function

$$ f(t) := (1+\varepsilon)^{\varrho-1} + (1+\frac{1}{\varepsilon})^{\varrho-1} t^{\varrho} - (1+t)^{\varrho}, \quad (27) $$
then, we can write its derivative as

$$f'(t) = \varrho \left(1 + \frac{1}{\varepsilon}\right)^{\varrho-1} t^{\varrho-1} - \varrho(1+t)^{\varrho-1}. \quad (28)$$

Setting (28) to zero, and since all the terms are positive, we can extract the $(\varrho - 1)$-th root. It follows that the unique stationary point of (27) is at $t = \varepsilon$. Moreover, as $t \to \infty$, the positive term $(1 + \frac{1}{\varepsilon})^{\varrho-1} t^{\varrho}$ dominates the expression, so that $f(t) \to +\infty$. Since the only stationary point occurs at $t = \varepsilon$, it must correspond to a minimum of $f(t)$ on $[0, \infty)$. Evaluating at this point gives $f(\varepsilon) = 0$, hence $f(t) \ge 0$ for all $t \ge 0$ and therefore, the inequality (25b) holds for all $a, b > 0$ and any $\varepsilon > 0$. In the case where $a = 0$, (25b) reduces to $b^{\varrho} \le (1 + \frac{1}{\varepsilon})^{\varrho-1} b^{\varrho}$, which holds for all $\varepsilon > 0$, since $(1 + \frac{1}{\varepsilon})^{\varrho-1} \ge 1$, given $\varrho > 1$. $\square$

**Proof.** *of Proposition 1.* The Schur stability of $A - LC$ implies that there exists a matrix $P = P^\top > 0$ such that (6) holds for a scalar $\eta > 0$ sufficiently small. Based on this matrix $P$, define the function (5) which clearly satisfies (7a), with $c_1 := \min\{\rho, \lambda_m(P)\}$, $c_2 := \max\{\rho, \lambda_M(P)\}$.

From the definition of the error dynamics (4) and given $\lambda \in (0, 1)$, we can use the variation of constants formula [12, Theorem 5.3] to get, for any $k \ge 0$,

$$\sigma_{k+n} = \lambda^n \sigma_k + \sum_{i=0}^{n-1} \lambda^{n-i-1} r |Ce_{k+i}|^p \ge \lambda^n r \sum_{i=0}^{n-1} |Ce_{k+i}|^p.$$

Using the results of Lemma 1 and Equation (9), we have

$$\sigma_{k+n}^q \ge \lambda^{nq} r^q \left( \sum_{i=0}^{n-1} |Ce_{k+i}|^p \right)^q \ge \lambda^{nq} r^q c \sum_{i=0}^{n-1} |Ce_{k+i}| \\ \ge \lambda^{nq} r^q c \gamma |e_k| \ge \lambda^{nq} r^q c a^{-n} \gamma |e_{k+n}|, \quad \forall k \ge 0,$$

with $c$ given by $c = (\eta^{p-1})^{-q}$ if $p \ge 1$, and $c = 1$ if $p < 1$ [13, Lemma 1]. By selecting the scalar

$$r^* = \lambda^{-n} c^{-p} a^{np} \gamma^{-p}, \quad (29)$$

for any $r > r^*$ and using the assumption that $|C| \le 1$, we obtain

$$\sigma_{k+n}^q \ge |e_{k+n}| \ge |Ce_{k+n}|, \quad \forall k \ge 0. \quad (30)$$

Inequality (30) leads to the conclusion that

$$\mathrm{sat}_{\sigma_{k+n}^q}(Ce_{k+n}) = Ce_{k+n}, \quad \forall k \ge 0, \quad (31)$$

i.e., after $n$ steps, the effect of saturation is no longer present.

Since, due to (31), we have that $e_{k+1} = (A - LC)e_k$, for all $k \ge n$, we obtain from (6)

$$\begin{aligned} \Delta V &= V(e_{k+1}, \sigma_{k+1}) - V(e_k, \sigma_k) \\ &= e_{k+1}^\top P e_{k+1} - e_k^\top P e_k + \rho(\sigma_{k+1}^{2q} - \sigma_k^{2q}) \quad (32) \\ &\stackrel{(4),(30)}{\le} -\eta e_k^\top P e_k + \rho(\lambda \sigma_k + r |e_k|^p)^{2q} - \rho \sigma_k^{2q}. \end{aligned}$$

Selecting $a = \lambda \sigma_k \ge 0$, $b = r |e_k|^p \ge 0$, $\varrho = 2q \ge 0$ and $\varepsilon = \frac{1-\lambda}{\lambda} > 0$ in Lemma 2, we obtain from (25)

$$(\lambda \sigma_k + r |e_k|^p)^{2q} \le k_1 (\lambda \sigma_k)^{2q} + k_2 r^{2q} |e_k|^2 \quad (33)$$

where

$$(k_1, k_2) = \begin{cases} (1, 1), & \text{if } 2q \le 1, \\ \left( \frac{1}{\lambda^{2q-1}}, \frac{1}{(1-\lambda)^{2q-1}} \right), & \text{if } 2q > 1. \end{cases} \quad (34)$$

Given that $\lambda \in (0, 1)$, it is straightforward to verify that $k_1 \lambda^{2q} < 1$ for all $q > 0$ and thus the scalar $1 - k_1 \lambda^{2q}$ in (32) is positive. Similarly, we choose $\rho$ to ensure $\eta - \frac{\eta k_2 r^{2q}}{\lambda_m(P)} > 0$, which is guaranteed by selecting $\rho < \frac{\eta \lambda_m(P)}{k_2 r^{2q}}$, for all $q > 0$. Finally, using that $\lambda_m(P)|e_k|^2 \le e_k^\top P e_k$ and Equation (33) for any $q > 0$ we can upper bound (32) as

$$\begin{aligned} \Delta V &\le - \left( \eta - \frac{\rho k_2 r^{2q}}{\lambda_m(P)} \right) e_k^\top P e_k - \rho (1 - k_1 \lambda^{2q}) \sigma_k^{2q} \quad (35) \\ &\le -c_3 V(e_k, \sigma_k), \quad \forall k \ge n, \end{aligned}$$

$$\text{where } c_3 := \min\{\eta - \frac{\rho k_2 r^{2q}}{\lambda_m(P)}, \rho(1 - k_1 \lambda^{2q})\}. \quad \square$$

# IV. PARAMETERS TUNING AND SIMULATIONS

## A. Design Guidelines

As stated in Theorem 1, there exist $L$, $\lambda$, and $r$ satisfying the conditions of the theorem, which guarantee the desired stability properties. In particular, to ensure exponential stability of the estimation error, the results obtained in Section III require $L$ to be such that $A - LC$ is Schur stable, $\lambda \in (0, 1)$, and $r > r^*$ where $r^*$ is the lower bound given by (29). In practical applications, choosing small values of $\lambda$ and $r$ leads to a faster observer response. Intuitively speaking, a small $\lambda$ enables a quick adaptation of the saturation threshold $\sigma_k^q$, preventing previous discrepancies between the measured and estimated outputs from overly influencing the disturbance rejection process. At the same time, a small $r$ limits the rate at which $\sigma_k^q$ increases in response to small estimation errors, preventing unnecessary overestimation of the saturation threshold. However, the derived bound $r^*$ is inherently conservative, particularly for large values of $p$, as it is calculated based on upper bounds on the error evolution. To this end, we propose an optimization procedure aimed at minimizing $r^*$, while maintaining the stability properties of the error dynamics. For a fixed $\lambda$ and $p$, chosen based on the desired observer response speed and adaptation characteristics, as discussed above, this boils down to optimally selecting the parameters $a$ and $\gamma$ while enforcing the constraints given by (9) and (15), along the trajectories of (4). Furthermore, the optimization of parameters $a$ and $\gamma$ can be simplified by constraining the dynamics to the unit sphere $\mathcal{B} := \{(e, \sigma) \mid |e|^2 + |\sigma|^2 = 1\}$. This is justified by the homogeneity property of the error dynamics (4), which is addressed by the following proposition.

**Proposition 2.** Consider a solution $(e_k, \sigma_k)$ to the error dynamics (4). Then, for every scalar $\delta > 0$, $(\bar{e}_k, \bar{\sigma}_k) = (\delta e_k, \delta^p \sigma_k)$ is also a solution to (4).

*Proof.* For the proof, we check that $(\bar{e}_k, \bar{\sigma}_k)$ is a solution:

$$\begin{aligned} \bar{\sigma}_{k+1} &= \delta^p \sigma_{k+1} = \delta^p \lambda \sigma_k + \delta^p |Ce_k|^p = \lambda \bar{\sigma}_k + |C \bar{e}_k|^p \\ \bar{e}_{k+1} &= \delta e_{k+1} = \delta A e_k - \delta L \mathrm{sat}_{\sigma_k^q}(Ce_k) \\ &= \delta A e_k - L \mathrm{sat}_{(\delta^p \sigma_k)^q}(C \delta e_k) = A \bar{e}_k - L \mathrm{sat}_{\bar{\sigma}_k^q}(C \bar{e}_k), \end{aligned}$$
<figure><img src="image_2.png" /></figure>

The optimal values for $a$ and $\gamma$ can then be determined by solving two separate optimization problems. First, we minimize $a$:

$$ \min_a \text{s.t.} \quad |Ae - L \text{sat}_{\sigma^q}(Ce)| \le a|e|, \quad \forall (e, \sigma) \in \mathcal{B}. \quad (36) $$

Secondly, we maximize $\gamma$:

$$ \max_{\gamma} \text{s.t.} \quad \sum_{i=0}^{n-1} |Ce_i| \ge \gamma|e|, \quad \forall (e, \sigma) \in \mathcal{B}, \quad (37) $$

where $(e_i, \sigma_i)$ is the trajectory of (4)

with $(e_0, \sigma_0) = (e, \sigma)$.

Formulations (36) and (37) are convenient because $(e, \sigma)$ evolve in a compact set, rather than the whole space. Nevertheless, thanks to the homogeneity property in Proposition 2, the solutions of (36) and (37) satisfy (9) and (15) for all $(e, \sigma) \in \mathbb{R}^n \times \mathbb{R}_{\ge 0}$. Problems (36) and (37) are both constrained nonlinear optimization problems that can be approximately solved in Matlab via a gridding method. Once the optimal values for $a$ and $\gamma$ are obtained, the threshold $r^*$ is computed using (29).

## B. Numerical Simulations

To illustrate the result of Theorem 1, we numerically simulate the case in which system (1) - (2) is selected with $A = \begin{bmatrix} 0 & 1 \\ -1 & 0 \end{bmatrix}$, $C = \begin{bmatrix} 1 & 0 \end{bmatrix}$, $B_d = 0$, $D_d = 1$, $L = \begin{bmatrix} -0.9 & -0.8 \end{bmatrix}$ and $\lambda = 0.5$. The disturbance $d_k$ is chosen similarly to [9] as a periodic outlier with period $T = 10$, where the initial outlier occurrence is randomly chosen between 1 and $T$ using a uniform distribution. The amplitudes of the outliers are zero-mean, normally distributed random variables with a variance of $0.5f^2$, where $f = 10$ is the dispersion factor. The system is initialized with zero initial conditions and simulated for different values of $p$. For each $p$, the observer parameter $r$ is optimally selected according to Section IV-A. The top and middle plots of Figure 1 illustrate respectively the evolution of the error norm $|e|$ and the saturation threshold $\sigma^q$, highlighting their dependence on the choice of $p$. The results indicate that the observer effectively mitigates the effect of disturbances, as the error growth following an outlier is progressively reduced. As expected, the saturation threshold plays a crucial role in the error dynamics. In particular, when $\sigma^q$ rapidly reaches zero, the choice of $p$ has a negligible impact on the error dynamics. Conversely, if an outlier occurs before $\sigma^q$ has reached zero, the selection of $p$ strongly influences the disturbance attenuation, affecting both the response speed and the peak error. Moreover, as shown in the bottom plot of Figure 1, for each value of $\lambda$, there exists an optimal value of $p$ that minimizes the peak error norm $|e|_\infty$. This suggests that an appropriate tuning of $p$ is essential to balance the disturbance rejection and the overall observer performance. Lastly, we present a comparative analysis of the Stubborn observer, with $\lambda = 0.3$ and $p = 1.5$, against the classical Luenberger observer. The left plot of Figure 2

compares the norm of estimation errors for both observers, during a single illustrative simulation run. This example highlights the stubborn observer's faster error convergence and generally lower error magnitudes as compared to the Luenberger response. The right plot of Figure 2 compares the Root Mean Square Error (RMSE) for both observers calculated over 1000 different simulations, each subjected to an independently generated periodic outlier disturbance. These results confirm that the proposed observer significantly outperforms the classical Luenberger solution, achieving a substantially lower median RMSE and exhibiting a tighter error distribution.

<figure><img src="image_3.png" /><figcaption>Fig. 1. Top: Evolution of the estimation error norm $|e|$ (4), for different values of $p$ and in the presence of outliers. Middle: adaptive saturation threshold $\sigma^q$ (4), for different values of $p$ and in the presence of outliers. Bottom: Maximum estimation error norm $|e|_\infty$ as a function of $p$, for different values of $\lambda$ and in the presence of outliers.</figcaption></figure>

<figure><img src="image_4.png" /><figcaption>Fig. 2. Comparison between the Luenberger and the Stubborn observer, simulated with $\lambda = 0.3$ and $p = 1.5$, in the presence of outliers. Left: Evolution of the estimation error norm $|e|$ over a single simulation run. Right: Boxplots of the RMSEs over 1000 simulation runs.</figcaption></figure>

## V. CONCLUSIONS

In this paper, we showed that detectability of the pair $(A, C)$ is necessary and sufficient for ensuring global exponential stability of a suitably tuned stubborn observer for
discrete-time linear systems. The proof relies on a Lyapunov-
like function, which exhibits favorable properties after an
initial transient. The results of the numerical simulations
show how the choice of the observer's parameters $\lambda$ and $p$
influence the ability of the proposed observer to mitigate the
effect of sporadic perturbations on the error dynamics. More-
over, a comparison with the standard Luenberger observer
demonstrates that the proposed method exhibits significantly
higher robustness against outliers.

Future research should aim for a detailed performance
analysis in the presence of outliers, focusing on character-
izing the robustness of the proposed observer in response
to sporadic perturbations. Furthermore, the results obtained
should also be extended to discrete-time systems with non-
linear dynamics.

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