# Improving the Accuracy of Adaptive Observers
Using Additional Filters

Héctor Ríos†,*, Denis Efimov†,*, and Rosane Ushirobira‡

***Abstract***—In this paper, we present the design of an adaptive observer with additional filtering dynamics for simultaneously estimating states and constant parameters in a class of affine-state nonlinear systems affected by external perturbations. The proposed estimator integrates Luenberger-like observers for state estimation with a gradient-based algorithm for identifying unknown constant parameters, enhanced by additional linear filters. In the ideal case, the adaptive observer ensures exponential convergence to the true state and parameter values, while also exhibiting input-to-state stability in the presence of bounded external perturbations. Stability analysis of the closed-loop system is conducted using a Lyapunov function approach, relying on standard persistence of excitation conditions. The effectiveness of the improved estimation algorithm is validated through simulations, demonstrating a better accuracy compared to the conventional adaptive observer.

***Index Terms***—Parameter Identification, State Estimation, Adaptive Observers.

## I. INTRODUCTION

In control design, obtaining accurate information about the full state vector and system parameters is essential. However, in many practical scenarios, only a subset of the state variables can be directly measured, while system parameters often change slowly due to aging or external disturbances. This challenge has led to the development of various approaches for real-time adaptive estimation of both states and parameters (see, e.g., [1], [2], and [3]). Key considerations in this field include the robustness, accuracy, and overall performance of the estimated values, as well as the effectiveness of the associated observation and identification algorithms—topics that are extensively studied in control, estimation, and identification theory.

As a result, the development of adaptive observers has been extensively studied and remains a prominent research topic. For instance, in [4], an adaptive observer is proposed for linear time-varying multiple-input multiple-output (MIMO) systems, capable of estimating both the state and parameters with global exponential convergence if the persistence of excitation condition is satisfied. The authors in

[5] presents an extension of the adaptive observer from [4] for state-affine nonlinear systems subject to external disturbances. Their approach introduced an augmented adaptive observer with sliding-mode feedback to mitigate matched perturbations effectively. In [6], an adaptive observer was designed to achieve exponential state and parameter estimation for uniformly observable MIMO nonlinear systems, provided that the persistence of excitation condition holds. In [7], an improved version of the adaptive observer from [5] was proposed, incorporating a nonlinear parameter identification algorithm. This enhancement establishes an ultimate bound on the overall estimation error, effectively mitigating the impact of relative degree one external perturbations. In [8], an adaptive observer was proposed for a class of nonlinear systems with unknown parameters, leveraging the dynamic regressor extension and mixing (DREM) technique. By utilizing an algebraic constraint, this approach decouples state and parameter estimation, enabling a linear vector regression framework. Similarly, in [9], an adaptive observer was designed for state-affine nonlinear systems with switched unknown parameters. This observer applies the DREM method to guarantee the asymptotic convergence of both state and parameter estimation errors to zero. In [10], building upon the work in [7], an accelerated adaptive observer was developed for a broader class of state-affine nonlinear systems subjected to relative degree one external perturbations. This observer guarantees fixed-time convergence of the parameter identification error to a neighborhood of the origin while ensuring the ultimate boundedness of the state estimation error. Similarly, [11] introduces a high-order sliding-mode adaptive observer to address the adaptive estimation problem for state-affine nonlinear systems under external perturbations, without requiring the relative degree one condition. The small-gain theorem is utilized to ensure the practical, global, and uniform asymptotic stability of the adaptive estimation error, even in the presence of external perturbations. In [12], an adaptive observer was developed for state-affine nonlinear systems without requiring persistent excitation. This observer leverages the Gramian of the regressor vector to guarantee the exponential convergence of the state estimation error to zero, while the parameter identification error converges exponentially to an equilibrium point. In [13], an initial-excitation-based adaptive observer was proposed for state-affine nonlinear systems with bounded disturbances in both

<footnote>†Tecnológico Nacional de México/I.T. La Laguna, 27000, Torreón, Coahuila, México. Email: hriosb@lalaguna.tecnm.mx</footnote>

<footnote>‡Inria, Université de Lille CNRS UMR 9189, F-59650, Villeneuve d'Ascq, France. Emails: denis.efimov@inria.fr and rosane.ushirobira@inria.fr</footnote>

<footnote>*SECIHTI, Investigadoras e Investigadores por México, 03940, Ciudad de México, México.</footnote>

<footnote>*ITMO University, 197101, Saint Petersburg, Russia.</footnote>
the system dynamics and output. The estimation error dynamics are shown to exhibit global uniformly ultimately bounded stability.

In order to look for an improvement, in this paper, we present the design of an adaptive observer with additional filtering mechanisms for the simultaneous estimation of states and constant parameters in a class of affine-state nonlinear systems affected by external perturbations. The proposed estimator integrates Luenberger-like observers for state estimation with a gradient-based algorithm for identifying unknown constant parameters, enhanced by additional (with respect to the standard adaptive observer [4]) linear filters. The origin of the non-perturbed estimation error dynamics is globally uniformly exponentially stable, while the perturbed estimation error dynamics is input-to-state stable with respect to bounded external perturbations. The stability analysis is conducted using a Lyapunov function approach and the input-to-state stability property following conventional hypotheses on excitation of the regressor. Some simulation results demonstrate improved accuracy compared to conventional adaptive observers. It is worth noting that a similar approach was proposed in [13]. However, our proposed method introduces an adaptive observer structure with additional dynamics for state estimation, which simplifies the stability analysis by decoupling the dynamics of state estimation and parameter identification errors. Moreover, our proposed structure enhances state estimation robustness to external perturbations in comparison to [13].

The rest of this manuscript is organized as follows. The problem statement is formulated in Section II as well some preliminary knowledge. The adaptive observer is designed in Section III. Finally, the simulation results are given in Section IV while in Section V the concluding remarks are presented.

**Notation:** The Euclidean norm of $s \in \mathbb{R}^n$ is denoted by $|s| := |s|_2$, and for a matrix $A \in \mathbb{R}^{m \times n}$, the induced norm is the spectral norm, i.e., $|A| = \sqrt{\lambda_{\max}(A^\top A)}$, where $\lambda_{\max}$ (respectively, $\lambda_{\min}$) is the maximum (respectively, the minimum) eigenvalue of $A^\top A$. The set of non-negative real numbers is denoted as $\mathbb{R}_{\ge 0}$. The set of all inputs $u: \mathbb{R}_{\ge 0} \to \mathbb{R}^p$ (similar for a matrix function $Q: \mathbb{R}_{\ge 0} \to \mathbb{R}^{m \times n}$) such that its $L_\infty$ norm on $[0, +\infty)$ is finite, i.e., $|u|_\infty := \ess \sup_{t \ge 0} |u(t)| < \infty$, is denoted as $\mathcal{L}_\infty^p$ (or $\mathcal{L}_\infty^{m \times n}$, for a matrix function). A function $\alpha: \mathbb{R}_{\ge 0} \to \mathbb{R}_{\ge 0}$ belongs to class $\mathcal{K}$ if it is strictly increasing and $\alpha(0) = 0$; it belongs to class $\mathcal{K}_\infty$ if it is also unbounded. A function $\beta: \mathbb{R}_{\ge 0} \times \mathbb{R}_{\ge 0} \to \mathbb{R}_{\ge 0}$ belongs to class $\mathcal{KL}$ if $\beta(\cdot, r) \in \mathcal{K}$ and $\beta(r, \cdot)$ is decreasing to zero for any fixed $r > 0$.

## II. PROBLEM STATEMENT

Consider the following class of affine-state nonlinear systems

$$ \dot{x}(t) = Ax(t) + \phi(t, y(t), u(t)) + G(t, y(t), u(t))\theta, \quad (1a) $$

$$ y(t) = Cx(t), \quad (1b) $$

where $x(t) \in \mathbb{R}^n$ is the state vector, $y(t) \in \mathbb{R}^p$ is the measurable output vector, $u(t) \in \mathbb{R}^m$ is the known input vector, and $\theta \in \mathbb{R}^q$ is an unknown constant parameter vector. The functions $\phi_t = \phi(t, y(t), u(t))$ and $G_t = G(t, y(t), u(t))$, with $\phi: \mathbb{R}_{\ge 0} \times \mathbb{R}^p \times \mathbb{R}^m \to \mathbb{R}^n$ and $G: \mathbb{R}_{\ge 0} \times \mathbb{R}^p \times \mathbb{R}^m \to \mathbb{R}^{n \times q}$, are known and guarantee uniqueness and existence of solutions for system (1) for all admissible signals. The matrices $A$ and $C$ are known and have appropriate dimensions. The following assumptions are met by system (1).

**Assumption 1.** The state and input satisfy $|x|_\infty < +\infty$ and $|u|_\infty < +\infty$, respectively.

**Assumption 2.** The pair $(A, C)$ is observable.

The following simple and computationally efficient adaptive observer was proposed in [4]:

$$ \dot{\hat{x}} = A_L \hat{x} + Ly + \phi_t + G_t \hat{\theta} + \Omega \dot{\hat{\theta}}, \quad (2a) $$

$$ \dot{\Omega} = A_L \Omega + G_t, \quad (2b) $$

$$ \dot{\hat{\theta}} = \gamma \Omega^\top C^\top e_y, \quad (2c) $$

where $\hat{\theta} \in \mathbb{R}^q$ is the identified value of the parameter vector $\theta$, $\hat{x} \in \mathbb{R}^n$ is the estimation for $x$, $\Omega \in \mathbb{R}^{n \times q}$ is an auxiliary variable, which provides a filtered version of the matrix $G_t$, the matrix $A_L = A-LC$ is Hurwitz under a proper choice of the observer gain $L \in \mathbb{R}^{n \times p}$, the output error $e_y = y - C\hat{x}$, and an adaptation gain $\gamma > 0$. In [4] and [5], it has been demonstrated that the adaptive observer (2), via $\hat{x}$ and $\hat{\theta}$, ensures global uniform exponential convergence to the true values of $x$ and $\theta$, provided that the persistent excitation condition over $C\Omega$ is met:

**Assumption 3.** The function $G_t$ is bounded, i.e., $|G_t|_\infty < \bar{G}$, with a constant $\bar{G} > 0$, and $\Omega_c = C\Omega$ persistently excited (PE), i.e.,

$$ \int_t^{t+T} \Omega_c^\top(\sigma) \Omega_c(\sigma) d\sigma \ge \vartheta_1 I_q, \quad \forall t \ge 0, $$

with a level of excitation $\vartheta_1 > 0$ and some excitation time $T > 0$, where $I_q$ is the identity matrix of dimension $q$.

This paper aims to enhance the adaptive observer (2) by including additional filters and changing the structure of the state observer. This modification can guarantee an improvement in estimation accuracy and robustness, particularly when disturbances are later introduced to (1).

### A. Preliminaries

Consider the system

$$ \dot{x}(t) = f(t, x(t), w(t)), \quad x(t_0) = x_0, \quad t \ge t_0 \in \mathbb{R}, \quad (3) $$

where $x(t) \in \mathbb{R}^n$ is the state vector and $w: \mathbb{R}_{\ge 0} \to \mathbb{R}^l$ is the external input. The function $f: \mathbb{R} \times \mathbb{R}^n \times \mathbb{R}^l \to \mathbb{R}^n$ is assumed to be locally bounded uniformly in $t$ and locally Lipschitz or Hölder continuous function in $x$, and
$f(t, 0, 0) = 0$. For an initial condition $x_0 \in \mathbb{R}^n$ at an initial time $t_0 \in \mathbb{R}$, the solution of system (3) is denoted as $x(t, t_0, x_0, w)$, and for brevity we assume that it is defined for any $t \ge t_0$.

Let $\Psi$ be an open neighborhood of the origin in $\mathbb{R}^n$. The following stability properties are introduced for system (3).

**Definition 1.** [14]. At the steady state $x = 0$, the system (3) with $w \equiv 0$, is said to be:

1) Uniformly Stable (US) in $\Psi$ if, for any $\epsilon > 0$, there is $\delta(\epsilon) > 0$ such that for any $x_0 \in \Psi$ and $t_0 \in \mathbb{R}$, if $|x_0| < \delta(\epsilon)$, then $|x(t, t_0, x_0, 0)| \le \epsilon$, for all $t \ge t_0 \in \mathbb{R}$.

2) Uniformly Exponentially Stable (UES) if it is US in $\Psi$ and there exist positive constants $\kappa$ and $\lambda$ such that $|x(t, t_0, x_0, 0)| \le \kappa e^{-\lambda(t-t_0)}|x_0|$, for all $t \ge t_0 \in \mathbb{R}$ and $x_0 \in \Psi$.

If $\Psi = \mathbb{R}^n$, at steady-state $x = 0$, system (3) is said to be Globally Uniformly Stable (GUS) and Globally Uniformly Exponentially Stable (GUES), respectively.

**Definition 2.** [15]. The system (3) is said to be Input-to-State Stable (ISS) if there exist functions $\beta \in \mathcal{KL}$ and $\gamma \in \mathcal{K}$ such that

$$|x(t, t_0, x_0, w)| \le \beta(|x_0|, t - t_0) + \gamma(|w|_\infty), \quad \forall t \ge t_0,$$
for any input $w \in \mathcal{L}_\infty^t$, any $x_0 \in \mathbb{R}^n$, and $t_0 \in \mathbb{R}$.

### III. ADAPTIVE OBSERVER WITH ADDITIONAL FILTERS

In order to improve the conventional adaptive observer (2), we propose the following modification with additional filters

$$ \dot{\hat{x}} = A_L \hat{x} + Ly + \Lambda e_y + \phi_t + G_t \hat{\theta}, \quad (4a) $$

$$ \dot{\zeta} = A_L \zeta + Ly + \phi_t, \quad (4b) $$

$$ \dot{\Omega} = A_L \Omega + G_t, \quad (4c) $$

$$ \dot{Y} = -\tau [Y - \Omega_c^\top (y - C\zeta)], \quad Y(0) = 0, \quad (4d) $$

$$ \dot{\Psi} = -\tau(\Psi - \Omega_c^\top \Omega_c), \quad \Psi(0) = 0, \quad (4e) $$

$$ \dot{\hat{\theta}} = \gamma(Y - \Psi\hat{\theta}), \quad (4f) $$

where the variables have been defined before except the auxiliary variables $\zeta \in \mathbb{R}^n$, which provides a preliminary state estimation, $Y \in \mathbb{R}^q$ that is a filtered version of the vector $\Omega_c^\top(y - C\zeta)$, $\Psi = \Psi^\top \in \mathbb{R}^{q \times q}$, which is a filtered version of the matrix $\Omega_c^\top \Omega_c$, the matrix $\Lambda \in \mathbb{R}^{n \times p}$ is an additional output injection gain, and the positive constant $\tau > 0$ that defines the filter time constant. To our knowledge, this adaptive observer (4) has not been previously introduced in the literature.

#### A. Estimation Error Dynamics

Let us define the parameter estimation error as $\tilde{\theta} = \theta - \hat{\theta}$, the auxiliary variable $z = x - \zeta - \Omega\theta$, and the state estimation

error $e = x - \hat{x}$. Note that $y - C\zeta = \Omega_c\theta + Cz$ and $\Omega_c$ is bounded under Assumption 3.

Therefore, taking into account (1) and (4), the estimation error dynamics is given as

$$ \begin{aligned} \dot{z} &= A_L z, \\ \dot{\theta} &= \gamma(\Psi\hat{\theta} - Y), \\ \dot{e} &= A_L e - \Lambda e_y + G_t \tilde{\theta}. \end{aligned} $$

From (4d) and (4e), it follows that $Y(t) = \Psi(t)\theta + \delta(t)$, for all $t \ge 0$ (the initial conditions of the filters are chosen to be zero), with $\delta$ as the filtered version of the vector $\Omega_c^\top Cz$, i.e.,

$$ \dot{\delta} = -\tau(\delta - \Omega_c^\top Cz), \quad \delta(0) = 0. \quad (6) $$

Therefore, the estimation error dynamics is rewritten as

$$ \dot{z} = A_L z, \quad (7a) $$

$$ \dot{\theta} = -\gamma(\Psi\tilde{\theta} + \delta), \quad (7b) $$

$$ \dot{e} = A_L e - \Lambda e_y + G_t \tilde{\theta}. \quad (7c) $$

The dynamics of $z$ in (7a) is independent of the identification error dynamics (7b). Furthermore, since $A_L$ is Hurwitz, $z = 0$ is GUES. In the next subsection, we will demonstrate that the identification error dynamics (7b) is ISS with respect to the input $\delta$, which in turn admits the ISS property in $z$. Given that $z = 0$ is GUES, it follows that $\tilde{\theta} = 0$ will also be reached exponentially fast. Subsequently, we will show that $(\tilde{\theta}^\top, e^\top)^\top = 0$ will also be GUES.

#### B. Parameter Identification Error

First, let us explain the role of the additional filters (4d), (8), and introduce some statements for the variables $\Psi$ and $\delta$. From (4e), we have that

$$ \Psi(t) = \tau \int_0^t e^{-\tau(t-\sigma)} \Omega_c^\top (\sigma) \Omega_c (\sigma) d\sigma. \quad (8) $$

According to [16], since $\Omega_c$ is PE due to Assumption 3, the trajectories of $\Psi$ satisfy the following inequality

$$ \tau \vartheta_1 e^{-\tau T} \le \inf_{t \ge T} (\lambda_{\min}(\Psi(t))). \quad (9) $$

Therefore, the new regressor $\Psi$ becomes positive definite in a finite time, while the original regressor $\Omega_c$ is simply PE (i.e., positive definite in average sense), which improves the robustness and transients, as we will demonstrate in the following sections.

On the other hand, note that due to $A_L$ is Hurwitz and $G_t$ is bounded, so does $\Omega_c$, i.e., $|\Omega_c|_\infty = \bar{\Omega}_c < +\infty$, with some $\bar{\Omega}_c > 0$. Therefore, based on (8), it follows that

$$ |\Psi|_\infty \le \bar{\Omega}_c^2. \quad (10) $$

In addition, the solution of (6) satisfies

$$ \delta(t) = \tau \int_0^t e^{-\tau(t-\sigma)} \Omega_c^\top (\sigma) Cz (\sigma) d\sigma. \quad (11) $$
Furthermore, from (7a), one obtains that $z(t) = e^{A_L t} z(0)$, and hence, since $A_L$ is Hurwitz there exist $k, \lambda > 0$ such that $|e^{A_L t}| \le ke^{-\lambda t}$, for all $t \ge 0$. Therefore, $\delta$ satisfies

$$|\delta(t)| \le k\tau\bar{\Omega}_c|C||z(0)| \left( \frac{e^{-\lambda t} - e^{-\tau t}}{\tau - \lambda} \right), \quad \forall t \ge 0. \quad (12)$$

It is clear that $\delta$ converges to zero as $t$ tends to infinity (the estimate (12) shows that the rate of convergence is determined by the minimum of both time constants, $\tau$ and $\lambda$). Now, based on the previous statements, the following lemma provides the convergence properties for the subsystem (7b).

**Lemma 1.** Let Assumptions 1–3 be satisfied. If the adaptive gain $\gamma$ is designed such that

$$\gamma \ge \frac{\alpha + \tau \left( \bar{\Omega}_c^2 - \tau \vartheta_1 e^{-\tau T} \right)}{\tau^2 \vartheta_1^2 e^{-2\tau T} (2 - \varepsilon_1)}, \quad (13)$$

for any $\alpha > 0$ and $\varepsilon_1 \in (0, 2)$, then the trajectories of the identification error dynamics (7b) are bounded for all $t \ge 0$. Moreover, $z = 0$, $\delta = 0$, and $\tilde{\theta} = 0$ are GUES, for all $t \ge T$.

**Remark 1.** In order to compute $\gamma$, the constants $\bar{\Omega}_c$, $\vartheta_1$, and $T$ can be estimated, on-line or using a preliminary identification, by means of the filters (4c) and (4e), the function $G_t$, the output $y$, and the input $u$. However, the exponential convergence of $\tilde{\theta}$ to zero can be always ensured for a sufficiently big value of $\gamma$.

Due to space limitations, all the proofs are omitted. Note that we are not claiming GUES property of the origin in (7a), (7b), and (6), but only satisfying the respective estimate for $t \ge T$.

### C. Convergence of the Adaptive Observer

Now, based on Lemma 1, the following theorem provides one of the main results of this work.

**Theorem 1.** Let Assumptions 1–3 be satisfied and assume that all the conditions of Lemma 1 hold (the values $\gamma$, $\alpha$, and $\varepsilon_1$ are fixed). Suppose that there exist $0 < P^\top = P \in \mathbb{R}^{n \times n}$, $Y_1, Y_2 \in \mathbb{R}^{n \times p}$, and $\beta_1, \varepsilon_2 > 0$ such that

$$\left( \begin{array}{ccc} \Xi_1 & 0 & 0 \\ \star & \Xi_2 & P \\ \star & \star & -\varepsilon_2^{-1} I_n \end{array} \right) \le 0, \quad (14)$$

$$\Xi_1 = - \left( \alpha - \beta_1 - \varepsilon_2^{-1} \bar{G}^2 \right) I_q,$$

$$\Xi_2 = PA + A^\top P + \beta_2 P - (Y_1 + Y_2)C - C^\top (Y_1 + Y_2)^\top,$$

is feasible for some $\beta_2 > 0$. If the adaptive observer (4) is designed according to Lemma 1 with $L = P^{-1}Y_1$ and $\Lambda = P^{-1}Y_2$, then the trajectories of the estimation error dynamics (7) are bounded for all $t \ge 0$. Moreover,

($z^\top, \delta^\top, \tilde{\theta}^\top, e^\top)^\top = 0$ is GUES, for all $t \ge T$, and its trajectories satisfy

$$\left| \begin{pmatrix} \tilde{\theta}(t) \\ e(t) \end{pmatrix} \right| \le \sqrt{\frac{\kappa_2}{\kappa_1}} \left| \begin{pmatrix} \tilde{\theta}(T) \\ e(T) \end{pmatrix} \right| e^{-\frac{\eta}{2}(t-T)} + \sqrt{\frac{\gamma}{\kappa_1 \eta \varepsilon_1}} k \tau \bar{\Omega}_c |C||z(T)| \times \left( \frac{e^{-\lambda(t-T)} - e^{-\tau(t-T)}}{\tau - \lambda} \right), \quad \forall t \ge T, \quad (15)$$

with $\eta = \min\{\beta_1 \bar{\Omega}_c^{-2}, \beta_2\}$, $\kappa_1 = \min\{\tau \vartheta_1 e^{-\tau T}, \lambda_{\min}(P)\}$, and $\kappa_2 = \max\{\bar{\Omega}_c^2, \lambda_{\max}(P)\}$.

**Remark 2.** It is worth highlighting that, compared with [4], [5], and [13], the inclusion of the additional filters and the new structure of the state observers in (4) facilitates the convergence analysis. In addition, the convergence rates of estimation errors for the parameters and state can be decoupled by using an auxiliary observer gain $\Lambda$.

### D. Robustness of the Adaptive Observer

Assume that system (1) is affected by some external perturbations, i.e.,

$$\dot{x}(t) = Ax(t) + \phi_t + G_t \theta + Dw(t), \quad (16a)$$

$$y(t) = Cx(t), \quad (16b)$$

where $w(t) \in \mathbb{R}^l$ is the perturbation vector, such that $|w|_\infty \le \bar{w}$, with a known $\bar{w} > 0$, and $D \in \mathbb{R}^{n \times l}$. Then, taking into account the adaptive observer (4), the estimation error dynamics is given as

$$\dot{z} = A_L z + Dw, \quad (17a)$$

$$\dot{\theta} = -\gamma(\Psi\tilde{\theta} + \delta), \quad (17b)$$

$$\dot{e} = A_L e - \Lambda e_y + G_t \tilde{\theta} + Dw. \quad (17c)$$

For this perturbed system, the following result can be obtained from Theorem 1.

**Corollary 1.** Let Assumptions 1–3 be satisfied and assume that all the conditions of Lemma 1 hold (the values $\gamma$, $\alpha$, and $\varepsilon_1$ are fixed). Suppose that there exist $0 < P^\top = P \in \mathbb{R}^{n \times n}$, $Y_1, Y_2 \in \mathbb{R}^{n \times p}$, and $\beta_1, \varepsilon_2 > 0$ such that

$$\left( \begin{array}{cccc} \Xi_1 & 0 & 0 & 0 \\ \star & \Xi_2 & P & PD \\ \star & \star & -\varepsilon_2^{-1} I_n & 0 \\ \star & \star & \star & -\bar{w}^{-2} I_l \end{array} \right) \le 0, \quad (18)$$

$$\Xi_1 = - \left( \alpha - \beta_1 - \varepsilon_2^{-1} \bar{G}^2 \right) I_q,$$

$$\Xi_2 = PA + A^\top P + \beta_2 P - (Y_1 + Y_2)C - C^\top (Y_1 + Y_2)^\top,$$

is feasible for some $\beta_2, \bar{w} > 0$. If the adaptive observer (4) is designed according to Lemma 1 with $L = P^{-1}Y_1$ and $\Lambda = P^{-1}Y_2$, then the trajectories of the estimation error
dynamics (17) are bounded for all $t \ge 0$. Moreover, the estimation error dynamics (17) satisfy

$$ \left| \begin{pmatrix} \hat{\theta}(t) \\ e(t) \end{pmatrix} \right| \le \sqrt{\frac{\kappa_2}{\kappa_1}} \left| \begin{pmatrix} \tilde{\theta}(T) \\ e(T) \end{pmatrix} \right| e^{-\frac{1}{2}(t-T)} + \sqrt{\frac{\gamma}{\kappa_1 \eta \varepsilon_1}} \left[ k \overline{\Omega}_c |C| \left( \tau |z(T)| \left( \frac{e^{-\lambda(t-T)} - e^{-\tau(t-T)}}{\tau - \lambda} \right) + \frac{|D|}{\lambda} \bar{w} \right) \right] + \frac{1}{\sqrt{\kappa_1 \eta}}, \forall t \ge T, \quad (19) $$

with $\eta = \min\{\beta_1 \overline{\Omega}_c^{-2}, \beta_2\}$, $\kappa_1 = \min\{\tau \theta_1 e^{-\tau T}, \lambda_{\min}(P)\}$, and $\kappa_2 = \max\{\overline{\Omega}_c^2, \lambda_{\max}(P)\}$.

Remark 3. It is evident that an optimization problem can be formulated to minimize the convergence region (or maximize $P$), which is influenced by the upper bound of the perturbation.

This corollary clarifies the origin of robustness improvement obtained by (4) in comparison with (2). In the latter, the identification error $\hat{\theta}$ dynamics is influenced by $z$ directly (whose transients are corrupted by $w$), while in the former, the identification error dynamics is dependent on $\delta$, which filters the distortions of $z$ caused by $w$.

In the next section, we will show by some simulations that the proposed adaptive observer with additional filters may provide higher accuracy than the conventional one.

## IV. SIMULATION RESULTS

Consider the system (1) with

$$ A = \begin{pmatrix} 0 & 1 \\ -1 & -2 \end{pmatrix}, \quad \phi_t = \begin{pmatrix} 0 \\ e^{-t} \end{pmatrix}, \quad \theta = \begin{pmatrix} 1 \\ -2 \end{pmatrix}, \\ G_t = \begin{pmatrix} 0 & 0 \\ \sin(t) & 1 \end{pmatrix}, \quad C^\top = \begin{pmatrix} 1 \\ 0 \end{pmatrix}, \quad D = \begin{pmatrix} 0 \\ 1 \end{pmatrix}, $$

initial conditions $x(0) = (1, 1)^\top$ and external disturbance $w(t) = 1.5 \sin(10t)$, and hence, $\bar{w} = 1.5$. For such a system, assumptions 1–3 are satisfied. Let us consider first the ideal case, i.e., $w \equiv 0$. Based on the statements of Lemma 1, we select $\alpha = 28$, $\tau = 1$, and hence, $\gamma = 20$. Then, based on the statements of Theorem 1, taking $\beta_2 = 0.1$, we obtain the following feasible solution for the LMI (14):

$$ P = \begin{pmatrix} 5.3965 & 0.3495 \\ 0.3495 & 4.1306 \end{pmatrix}, \quad Y_1 = Y_2 = \begin{pmatrix} 3.5587 \\ 0.3497 \end{pmatrix}, \\ L = \Lambda = \begin{pmatrix} 0.6576 \\ 0.0290 \end{pmatrix}. $$

Consider initial conditions for the adaptive observer as $\hat{x}(0) = (0, 0)^\top$, $\zeta(0) = (0, 0)^\top$, $\hat{\theta}(0) = (0, 0)^\top$, and $\Omega(0) = 0$. The simulations have been done in MATLAB with the Euler explicit discretization method and sampling time equal to 0.001[s].

For comparison purposes, we also implement the conventional adaptive observer (2) with the same values for the observer gains. The nomenclature to the legends is as

follows: Add-F corresponds to the proposed adaptive observer with additional filters, while Conv corresponds to the conventional adaptive observer. The results are depicted by Figs. 1–3. The state estimation is illustrated in Fig. 1, while the parameter identification in Fig. 2. We can see that both algorithms provide proper estimations but the conventional adaptive observer is slower. In addition, in Fig. 3, we can see the norm of the state estimation and parameter identification errors, i.e., $|e|$ and $|\hat{\theta}|$, respectively, for both algorithms. The proposed adaptive observer with additional filters provides higher accuracy than the conventional one.

<figure><img src="image_1.png" /><figcaption>Figure 1. Adaptive Estimation Error – Ideal Case</figcaption></figure>

<figure><img src="image_2.png" /><figcaption>Figure 2. Parameter Identification – Ideal Case</figcaption></figure>

Consider now the perturbed case, i.e., $w \neq 0$. Based on the statements of Lemma 1, we select $\alpha = 29$, $\tau = 1$, and hence, $\gamma = 20$. Then, based on the statements of Corollary 1, taking $\beta_2 = 0.1$, we obtain the following feasible solution for the LMI (18):

$$ P = \begin{pmatrix} 5.8467 & -0.5956 \\ -0.5956 & 1.8169 \end{pmatrix}, \quad Y_1 = Y_2 = \begin{pmatrix} 4.3729 \\ 1.7193 \end{pmatrix}, \\ L = \Lambda = \begin{pmatrix} 0.8735 \\ 1.2326 \end{pmatrix}. $$

Consider the same initial conditions as in the ideal case. The norm of the state estimation and parameter identification errors, i.e., $|e|$ and $|\hat{\theta}_1|$, is depicted in Fig. 4, for both
<figure><img src="image_3.png" /><figcaption>Figure 3. Adaptive Estimation Error – Ideal Case</figcaption></figure>

algorithms. The proposed adaptive observer with additional
filters possesses still higher accuracy than the conventional
one improving its robustness.

<figure><img src="image_4.png" /><figcaption>Figure 4. Adaptive Estimation Error – Perturbed Case</figcaption></figure>

V. CONCLUSIONS

In this paper, the design of an adaptive observer with
additional filtering mechanisms is proposed for the simul-
taneous estimation of states and constant parameters in a
class of affine-state nonlinear systems affected by external
perturbations. The proposed observer integrates Luenberger-
like observers for state estimation with a gradient-based
algorithm for identifying unknown constant parameters, en-
hanced by additional linear filters. For the ideal case, the
origin of the estimation error dynamics is GUES, while the
perturbed estimation error dynamics is ISS with respect to
bounded external perturbations. The stability analysis is con-
ducted using a Lyapunov function approach and the input-to-
state stability property following conventional hypotheses on
excitation of the regressor. Some simulation results highlight
an improved accuracy compared to conventional adaptive
observer.

ACKNOWLEDGMENTS

This work was supported in part by the SEP-CONACYT-
ANUIES-ECOS NORD Project 315597 and ECOS NORD

Project M20M04. The work of Héctor Ríos was supported
in part by SECIHTI, Investigadoras e Investigadores por
México, CVU 270504 Project 922 and in part by TecNM
Projects.

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