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# An Accelerated Heavy-Ball-based Adaptive Observer for Uncertain Nonlinear Systems

Hector Ríos, Denis Efimov, Rosane Ushirobira

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Hector Ríos, Denis Efimov, Rosane Ushirobira. An Accelerated Heavy-Ball-based Adaptive Observer for Uncertain Nonlinear Systems. CDC 2025 - 64th IEEE Conference on Decision and Control, Dec 2025, Rio De Janeiro, Brazil. hal-05435607

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# An Accelerated Heavy-Ball-based Adaptive Observer for Uncertain Nonlinear Systems

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

*Abstract*—This paper presents the design of an accelerated adaptive observer for simultaneously estimating states and constant parameters in a class of uncertain nonlinear systems subject to external disturbances. The proposed observer consists of a high-order sliding-mode (HOSM) observer for state estimation and a heavy-ball-based algorithm for identifying unknown constant parameters. Despite the presence of bounded external disturbances that do not satisfy a relative degree one condition, the observer locally estimates both state and parameter vectors in finite time. The closed-loop stability analysis is performed using a Lyapunov function approach, input-to-state stability properties, and a finite-time small-gain theorem. The effectiveness of the proposed estimation algorithm is demonstrated through simulation results.

*Index Terms*—Adaptive Estimation, Heavy-Ball Method, Nonlinear Systems.

## I. INTRODUCTION

DESIGNING a control system without full state measurements and under parameter uncertainties and unknown inputs remains a challenging problem in engineering practice. The methods developed in parameter identification theory provide practical tools to address this lack of information. In recent decades, online adaptive estimation, i.e., simultaneous state estimation and parameter identification, has garnered significant attention (see, for instance, [1] and [2]). As a result, the development of adaptive observers has been widely explored and continues to be a popular research topic.

For example, in [3], an adaptive observer was proposed for linear time-varying multiple-input multiple-output (MIMO) systems to estimate the state and parameters with global exponential convergence, given a persistent excitation condition. An extension of the adaptive observer from [3] was introduced in [4] for state-affine nonlinear systems affected by external disturbances, incorporating an augmented adaptive observer with sliding-mode feedback to handle matched disturbances. In [5], an adaptive observer was proposed to achieve exponential state and parameter estimation under a persistent excitation condition for uniformly observable

MIMO nonlinear systems. In [6], an enhancement of the adaptive observer from [4] was introduced, incorporating a nonlinear parameter identification algorithm. This algorithm establishes an ultimate bound for the full estimation error, effectively mitigating the impact of relative degree one external disturbances. In [7], an adaptive observer was proposed for a class of nonlinear systems with unknown parameters using dynamic regressor extension and mixing (DREM). By leveraging an algebraic constraint, this approach separates state and parameter estimation problems, enabling linear vector regression. In [8], building on [6], an accelerated adaptive observer was developed for a broader class of state-affine nonlinear systems subject to relative degree one external disturbances. This adaptive observer ensures fixed-time convergence of the parameter identification error to a neighborhood of the origin while guaranteeing ultimate boundedness of the state estimation error. In the same context, [9] proposes a high-order sliding-mode (HOSM) adaptive observer to address the adaptive estimation problem for state-affine nonlinear systems under external disturbances, without requiring a one-relative-degree condition. The practical, global, and uniform asymptotic stability of the adaptive estimation error, despite external disturbances, is ensured using the small-gain theorem. In [10], an adaptive observer is designed for state-affine nonlinear systems in the absence of persistent excitation. This observer leverages the Gramian of the regressor vector, ensuring exponential convergence of the state estimation error to zero while the parameter identification error converges exponentially to an equilibrium point. In [11], an initial-excitation-based adaptive observer is proposed for state-affine nonlinear systems affected by bounded disturbances in both the system dynamics and the output. The estimation error dynamics are shown to exhibit global uniformly ultimately bounded stability. Recently, in [12], an adaptive observer is developed for a class of state-affine nonlinear systems with switched unknown parameters. This observer leverages the DREM approach, ensuring the asymptotic convergence to zero of both state and parameter estimation errors.

Most existing works in the literature exhibit one of the following limitations: they either overlook external disturbances, necessitate the output to meet a relative degree one condition with respect to external disturbances, or, in most cases, achieve only asymptotic or exponential conver-

<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>
gence rates. Additionally, all of the above-mentioned works employ first-order parameter identification algorithms. To tackle these challenges, this paper proposes an accelerated adaptive observer for simultaneously estimating states and constant parameters in a class of uncertain nonlinear systems subjected to external disturbances. The observer integrates an HOSM observer for state estimation along with a heavy-ball-based algorithm<fnref n="1" /> for identifying unknown constant parameters. Despite some bounded external disturbances that do not fulfill a relative degree one condition, the origin of the estimation error dynamics is locally practically finite-time stable. The stability analysis is conducted using a Lyapunov function approach, finite-time input-to-state stability properties, and a recently developed small-gain theorem for finite-time stability [15].

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

**Notation:** We denote $\lceil s \rceil^\alpha := |s|^\alpha \text{sign}(s)$, for $s \in \mathbb{R}$ and $\alpha \ge 0$. For a vector $s \in \mathbb{R}^n$, $\lceil s \rceil^\alpha$ is understood component-wise. For $p \ge 1$, the $L^p$-norm of $s \in \mathbb{R}^n$ is defined by $|s|_p = (\sum_{i=1}^n |s_i|^p)^{\frac{1}{p}}$, where $|s_i|$ denotes the absolute value of $s_i \in \mathbb{R}$. So $|s|_{\alpha+1}^{\alpha+1} = s^\top \lceil s \rceil^\alpha$, for any $\alpha > 0$. By Hölder inequality, we have that $|s|_{\zeta_1} \le \bar{s}(\zeta_1, \zeta_2)|s|_{\zeta_2}$, for any $s \in \mathbb{R}^n$ and any $\zeta_1, \zeta_2 > 0$, with $\bar{s}(\zeta_1, \zeta_2) = \max(1, n^{\frac{\zeta_1-\zeta_2}{\zeta_1}})$. With a slight abuse of notation, we will write the Euclidean norm of $s \in \mathbb{R}^n$ as $|s| := |s|_2$, and for a matrix $A \in \mathbb{R}^{m \times n}$, the induced norm is the spectral norm, i.e., $|A| = \sqrt{\lambda_{\max}(A^\top A)} = \sigma_{\max}(A)$, where $\lambda_{\max}$ (respectively, $\lambda_{\min}$) is the maximum (respectively, the minimum) eigenvalue of $A^\top A$, and $\sigma_{\max}$ (respectively, $\sigma_{\min}$) is the maximum (respectively, the minimum) singular value of $A$. 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., $|u|_\infty := \text{ess sup}_{t \ge 0} |u(t)| < \infty$, 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 $r > 0$. A function $\beta : \mathbb{R}_{\ge 0} \times \mathbb{R}_{\ge 0} \to \mathbb{R}_{\ge 0}$ belongs to class $\mathcal{FKL}$ if $\beta(\cdot, 0) \in \mathcal{K}$, $\beta(r, \cdot)$ is decreasing to zero for any fixed $r > 0$, and there exists $\mathcal{T} \in \mathcal{K}$ such that $\beta(r, t) = 0$, for all $t \ge \mathcal{T}(r)$.

## II. PROBLEM STATEMENT

Consider the following class of uncertain nonlinear systems

$$ \dot{x}(t) = Ax(t) + \phi_t + G_t\theta + Dw(t), \quad (1a) $$

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

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 input vector, $\theta \in \mathbb{R}^q$ is an unknown constant parameter vector, and $w : \mathbb{R}_{\ge 0} \to \mathbb{R}$ represents possible external disturbances. 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 the uniqueness and existence of solutions for (1) for all admissible signals. The matrices $A, C$, and $D$ are known and have appropriate dimensions.

This paper aims to provide an accelerated estimation of the state $x$ and the parameter vector $\theta$, by respectively using only the information of the output $y$ and taking into account the effect of external disturbances $w$, under the assumptions:

**Assumption 1.** The state, input, and external disturbances satisfy $|x|_\infty < x^+$, $|u|_\infty < +\infty$, and $|w|_\infty \le w^+$, for some known constants $x^+, w^+ > 0$, respectively.

**Assumption 2.** The triple $(A, C, D)$ is strongly observable.

**Assumption 3.** The function $G_t$ is bounded, i.e., $|G_t|_\infty < G^+$, for a known constant $G^+ > 0$, and persistently excited (PE), meaning

$$ \alpha_1 I \ge \int_t^{t+T} G_\sigma^\top G_\sigma d\sigma \ge \alpha_0 I, \quad \forall t \ge 0, $$

with a level of excitation $\alpha_0 > 0$, an excitation time $T > 0$, and a positive constant $\alpha_1 > 0$, where $I$ is the identity matrix of appropriate dimension.

## III. PRELIMINARIES

### A. Stability Definitions

Consider the system

$$ \dot{x}(t) = f(t, x(t), w(t)), \quad x(t_0) = x_0, \quad t \ge t_0 \ge 0, \quad (2) $$

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 $f : \mathbb{R}_{\ge 0} \times \mathbb{R}^n \times \mathbb{R}^l \to \mathbb{R}^n$ is assumed to be locally bounded uniformly in the first coordinate, locally Lipschitz or Hölder continuous in the second coordinate, and $f(t, 0, 0) = 0$. For an initial condition $x_0 \in \mathbb{R}^n$, the solution of system (2) is denoted as $x(t, t_0, x_0, w)$, and for brevity we assume that it is defined for any $t \ge t_0$. In the sequel, we review some standard definitions of finite-time stability. Note that their versions with Lyapunov functions also apply.

**Definition 1.** At steady state $x = 0$, the system (2) with $w \equiv 0$, is said to be finite-time stable (FTS) if: a) there exists a function $\alpha \in \mathcal{K}$ such that $|x(t, t_0, x_0, 0)| \le \alpha (|x_0|)$, for all $t \ge t_0$, for any initial condition $x_0 \in \mathbb{R}^n$ and $t_0 \ge 0$;

<footnote n="1">¹The heavy-ball method is a second-order method and is the first optimization approach using momentums [13]. Recently, in [14], an accelerated heavy-ball-based parameter identification algorithm has been proposed to identify constant unknown parameters in finite-time.</footnote>
and b) for each initial condition $x_0 \in \mathbb{R}^n$, there exists $0 \le T_s < +\infty$ such that $x(t, t_0, x_0, 0) = 0$, for all $t \ge t_0 + T_s$, with any $t_0 \ge 0$. The function $\mathcal{T}_0: x_0 \mapsto \inf\{T_s \ge 0: x(t, t_0, x_0, 0) = 0, \forall t \ge t_0 + T_s\}$ is the settling-time of the system (2). The system (2) is said to be locally FTS if the previous statements hold for any initial condition $x_0 \in \mathbb{R}^n$ such that $|x_0| \le \bar{x}_0$, with some $\bar{x}_0 > 0$.

**Definition 2.** The system (2) is said to be input-to-state practically stable (ISpS) if there exist functions $\beta \in \mathcal{K}\mathcal{L}$, $\gamma \in \mathcal{K}$, and a constant $k \ge 0$ such that

$$|x(t, t_0, x_0, w)| \le \beta (|x_0|, t - t_0) + \gamma (|w|_\infty) + k, \forall t \ge t_0,$$
for any input $w \in \mathcal{L}_\infty^\ell$, any initial condition $x_0 \in \mathbb{R}^n$, and $t_0 \ge 0$. Furthermore, if $\beta \in \mathcal{HKL}$, then system (2) is said to be finite-time ISpS (FT-ISpS). The system (2) is called locally (FT-)ISpS if the above estimate holds for $x_0 \in \mathbb{R}^n$ and $w \in \mathcal{L}_\infty^\ell$ such that $\max\{|x_0|, |w|_\infty\} \le a$, for some $a > 0$. If the constant $k$ is equal to 0, the system (2) is said to be ISS, FT-ISS, or locally FT-ISS, respectively.

Definitions 1 and 2 can also be stated a fixed value $t_0$.

## B. Small-Gain Theorem for FT-ISS

Consider the following interconnected nonlinear system:

$$\dot{x}_1(t) = f_1(x_1(t), x_2(t), w(t)), \quad (3a)$$

$$\dot{x}_2(t) = f_2(x_1(t), x_2(t), w(t)), \quad (3b)$$

where $x_i(t) \in \mathbb{R}^{n_i}$, $w(t) \in \mathbb{R}^\ell$, and $f_i: \mathbb{R}^{n_1} \times \mathbb{R}^{n_2} \times \mathbb{R}^\ell \to \mathbb{R}^{n_i}$ ensures the existence of the system solutions at least locally, for $i = 1, 2$. Consider both subsystems FT-ISpS with respect to $(x_2, w)$ and $(x_1, w)$, i.e., respectively, there exist functions $\beta_1, \beta_2 \in \mathcal{HKL}$, $\gamma_1, \gamma_2, \gamma_3, \gamma_4 \in \mathcal{K}_\infty$, and positive constants $k_1, k_2$, such that

$$|x_1(t, t_0, x_0, w)| \le \beta_1(|x_{10}|, t - t_0) + \gamma_1(|x_2|_\infty) + \gamma_3(|w|_\infty) + k_1, \quad (4a)$$

$$|x_2(t, t_0, x_0, w)| \le \beta_2(|x_{20}|, t - t_0) + \gamma_2(|x_1|_\infty) + \gamma_4(|w|_\infty) + k_2, \quad (4b)$$

hold for all $t \ge t_0$, with initial conditions $x_{10} = x_1(t_0)$ and $x_{20} = x_2(t_0)$, $x_0 = (x_{10}^\top \ x_{20}^\top)^\top$. For $i = 1, 2$, denote by $\mathcal{T}_i \in \mathcal{K}$ the settling-time function satisfying $\beta_i(s, \mathcal{T}_i(s)) = 0$, for any $s \in \mathbb{R}_{\ge 0}$. The function $\beta_i \in \mathcal{HKL}$ can be (locally) bounded by $\chi_i \circ \max\{0, \mathcal{T}_i(s) - t\}$, for some functions $\chi_i \in \mathcal{K}_\infty$ so that $\chi_i(s) \ge s$ and a function $\mathcal{T}_i \in \mathcal{K}$ satisfying $\mathcal{T}_i(s) \le \mathcal{T}_i(s) \le 2\mathcal{T}_i(s)$, for all $s \in \mathbb{R}_{\ge 0}$. Define $\beta \in \mathcal{HKL}$ as

$$\beta(s, t) = \max_{i=1,2} \beta_i(s, t) \le \chi \circ \max\{0, \mathcal{T}(s) - t\},$$

for some $\chi \in \mathcal{K}_\infty$ (e.g., $\chi(s) = \max_{i=1,2} \chi_i(s)$) and $\mathcal{T} \in \mathcal{K}$ such that $\max_{i=1,2} \mathcal{T}_i(s) \le \mathcal{T}(s) \le 2 \max_{i=1,2} \mathcal{T}_i(s)$, for all $s \in \mathbb{R}_{\ge 0}$.

Then, based on [15], the following nonlinear small-gain result is introduced, in terms of FT-ISS properties for (3):

**Theorem 1.** Suppose that (3a) and (3b) are FT-ISS with respect to $(x_2, w)$ and $(x_1, w)$, respectively, satisfying (4). If there exist $s_1, s_2 > 0$ satisfying for $i = 1, 2$:

$$\gamma_i \circ \gamma_{3-i}(s) < s, \text{ for } s \in (s_1, s_2), \quad (5a)$$

$$\beta(\gamma_i \circ \gamma_{3-i}(s), 0) \le 0.5s, \text{ for } s \in (s_1, s_2), \quad (5b)$$

$$\gamma_i(2\beta(s, 0)) \le s, \text{ for } s \in (s_1, s_2), \quad (5c)$$

then the system (3) is locally FT-ISpS with respect to $w$. If $s_1 = 0$, (3) is locally FT-ISS with respect to $w$.

**Remark 1.** Note that if the external disturbance $w$ vanishes, or it is fully compensated in finite time, Theorem 1 implies that the interconnected system (3) is locally practically FTS or locally FTS, respectively.

## IV. HEAVY-BALL-BASED ACCELERATED ADAPTIVE OBSERVER

For simplicity, we consider the case $p = q = 1$, i.e., the single-output single-parameter case. The multiple-output multiple-parameter case can be extended, as shown in [9].

Denote the output error as $e_y := y - C\hat{x}$, where $\hat{x}$ is the estimation for $x$. We propose the following accelerated heavy-ball-based adaptive observer for (1):

$$\hat{\theta}_1 = \rho_t^{-1} \hat{\theta}_2, \quad (6a)$$

$$\hat{\theta}_2 = \rho_t^{-1} (k_1 \Omega^\top C^\top [e_y]^\alpha - k_2 [\hat{\theta}_2]^\beta), \quad (6b)$$

$$\hat{\Omega} = A_L \Omega + G_t, \quad (6c)$$

$$\rho_t = \begin{cases} 1, & \forall t \in [0, T), \\ |e_y(t)|^{\gamma+1}_{\gamma+1} + |\hat{\theta}_2(t)|^{\delta+1}_{\delta+1}, & \forall t \ge T, \end{cases} \quad (6d)$$

$$\hat{x} = A\hat{x} + Le_y + \mathcal{E}(e_y) + \phi_t + G_t\hat{\theta}_1 + \rho_t^{-1}\Omega\hat{\theta}_2, \quad (6e)$$

where $\hat{\theta}_1(t) \in \mathbb{R}$ is the estimated value of the parameter vector $\theta$, $\hat{\theta}_2(t) \in \mathbb{R}$ is an auxiliary velocity variable, $\Omega(t) \in \mathbb{R}^n$ is an auxiliary variable which provides a filtered version of the matrix function $G_t$, the matrix $A_L := A - LC = \mathcal{T}^{-1}(A_0\mathcal{T} + aC)$ is Hurwitz, with $L \in \mathbb{R}^{n \times 1}$,

$$\mathcal{T}^{-1} = (A_L^{n-1}\xi \ A_L^{n-2}\xi \ \dots \ \xi) \in \mathbb{R}^{n \times n}, \quad (7)$$

$$\xi := \mathcal{O}^{-1} \begin{pmatrix} 0 & 0 & \dots & 1 \end{pmatrix}^\top \in \mathbb{R}^n,$$

$$A_0 = \begin{pmatrix} 0 & 1 & 0 & \dots & 0 \\ 0 & 0 & 1 & \dots & 0 \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & 0 & \dots & 1 \\ 0 & 0 & 0 & \dots & 0 \end{pmatrix} \in \mathbb{R}^{n \times n}, \quad \mathbf{a} = \begin{pmatrix} a_1 \\ a_2 \\ \vdots \\ a_n \end{pmatrix} \in \mathbb{R}^n,$$

$$\mathcal{E}(e_y) = \mathcal{T}^{-1} \left( \mathbf{a}e_y + \begin{pmatrix} \ell_1 w_n^{\frac{1}{n}} [e_y]^\frac{n-1}{n} \\ \ell_2 w_n^{\frac{2}{n}} [e_y]^\frac{n-2}{n} \\ \vdots \\ \ell_n w [e_y]^0 \end{pmatrix} \right).$$

The constants $a_i, i = 1, n$, are the coefficients of the characteristic polynomial of the matrix $A_L$, $\mathcal{O}$ is the observability matrix of the pair $(A, C)$, $\rho_t = \rho(t, e_y(t), \hat{\theta}_2(t))$ is a positive
function representing a nonlinear gain, with excitation time $T > 0$, given in Assumption 3. The constant $\omega$ is defined as $\omega = |CA_L^{n-1}D|w^+ + |a|\bar{\Omega}_c$, with $w^+$ defined in Assumption 1, $\Omega_c(t) := C\Omega(t)$, and $\bar{\Omega}_c, \bar{\Omega}_c$ as in Assumption 4 below.

The parameters of the algorithm are the gain $L \in \mathbb{R}^{n \times 1}$, the constants $k_1, k_2 > 0$, $\ell_i$ ($i = \bar{1}, \bar{n}$), and the exponents $\alpha, \beta \in (0, 1)$, $\gamma \in (-1, \alpha - 1)$, and $\delta \in (-1, \beta - 1)$. The term $|e_y|^0$ makes the right-hand side of (6e) discontinuous, so the solutions of (6e) are in the sense of Filippov [16].

We will need the following assumption.

**Assumption 4.** There exist $\bar{\Omega}_c, \bar{\Omega}_c > 0$ such that $0 < \bar{\Omega}_c \le \sigma_{\min}(\Omega_c(t))$ and $\sigma_{\max}(\Omega_c(t)) \le \bar{\Omega}_c < +\infty$, for all $t \ge T$.

The existence of $\bar{\Omega}_c < +\infty$ is ensured by Assumption 1, while $\bar{\Omega}_c > 0$ is obtained if the number of outputs is greater or equal to the number of unknown parameters, or if $\Omega_c(t)$ is uniformly injective for all $t \ge T$. Such constants $\bar{\Omega}_c$ and $\bar{\Omega}_c$ can be estimated, online or during preliminary experiments, by means of the filter (6c), the function $G_t$, the output $y$, and the input $u$.

### A. Estimation Error Dynamics

Define the parameter estimation errors as $\tilde{\theta}_1 := \theta - \hat{\theta}_1$ and $\tilde{\theta}_2 := -\hat{\theta}_2$. Set the auxiliary variable $z := e - \Omega\tilde{\theta}_1$ and the state estimation error $e := x - \hat{x}$. Since $\Omega_c = C\Omega$, then $e_y = y - C\hat{x} = Ce = \Omega_c\tilde{\theta}_1 + Cz$. Therefore, considering (1) and (6), the estimation error dynamics is given as follows

$$ \dot{\tilde{\theta}}_1 = \rho_t^{-1} \tilde{\theta}_2, \quad (8a) $$

$$ \dot{\tilde{\theta}}_2 = \rho_t^{-1} \left( -k_1 \Omega_c^\top [\Omega_c \tilde{\theta}_1 + Cz]^\alpha - k_2 [\tilde{\theta}_2]^\beta \right), \quad (8b) $$

$$ \dot{z} = A_L z - \mathcal{E}(e_y) + Dw. \quad (8c) $$

### B. Parameter Identification Error

In this subsection, we provide ISS convergence properties for (6c)–(6d) and (8a)–(8b), with respect to $z$.

Note that, according to the problem statement, function $\rho_t$ cannot lie in $0$, given that the error dynamics (8) is locally practically FTS, as we will show further on. Consequently, the trajectories of system (8a)–(8b) are well-defined.

**Lemma 1.** Consider the system (8a)–(8b) under Assumptions 1, 3, and 4. If the exponents of the accelerated heavy-ball-based algorithm (6a)–(6d) are designed such that $\alpha, \beta \in (0, 1)$, $\gamma \in (-1, \alpha - 1)$, $\delta \in (-1, \beta - 1)$, with $\alpha < \beta$, and the gains are designed such that

$$ k_1 > \frac{7}{p \Omega_c^{\mu+\alpha} - (r+1) \Theta_1 \Delta \Omega}, \quad k_2 > \frac{7}{\bar{\kappa}_1(r+1)}, \quad (9) $$

for some $p > (r+1)\Theta_1 \frac{\Delta\Omega}{\Omega_c^{\mu+\alpha}}$, $\mu > 1$, $r > 0$, and

$$ \Theta_1 = \tilde{\theta}_1^{\gamma+1} (\gamma + 1, \mu + \alpha) \tilde{\theta}_1^{\alpha+1} (\alpha + 1, \mu + \alpha), $$

$$ \Delta\Omega = \bar{\kappa}_2 \bar{\Omega}_c^{\alpha+\gamma+1} (\bar{\Omega}_c \sigma_{\max}(A_L) + \sigma_{\max}(C)G^+), $$

$$ \bar{\kappa}_1 = \min \left\{ \begin{aligned} & \bar{\Omega}_c^{(\alpha+1)r} \left( \frac{k_1}{\alpha+1} \right)^r \tilde{\theta}_1^{\bar{\gamma}-(\alpha+1)r} (\mu+\alpha, \alpha+1), \\ & 2^{-r} \tilde{\theta}_2^{\bar{\gamma}-2r} (\beta+1, 2) \end{aligned} \right\} \min\{1, 2^{r-1}\}, $$

$$ \bar{\kappa}_2 = \max \left\{ \begin{aligned} & \bar{\Omega}_c^{(\alpha+1)r} \left( \frac{k_1}{\alpha+1} \right)^r \tilde{\theta}_1^{\bar{\gamma}-(\alpha+1)r} (\alpha+1, \mu+\alpha), \\ & 2^{-r} \tilde{\theta}_2^{\bar{\gamma}-2r} (2, \beta+1) \end{aligned} \right\} \max\{1, 2^{r-1}\}, $$

then, the parameter identification error dynamics (8a)–(8b) is locally FT-ISpS, with respect to the input $z$.

Due to space limitations, all the proofs are omitted.

### C. State Estimation Error

The following lemma provides ISS convergence properties for the subsystem (8c) with respect to $\tilde{\theta}_1$.

**Lemma 2.** Consider the system (8c) under Assumptions 1 and 2. Then, there exist positive constants $\ell_i > 0$, such that the state estimation error dynamics (8c) is locally FT-ISS with respect to the input $\tilde{\theta}_1$, uniformly in $w$. Moreover, the trajectories of the system (8c) satisfy

$$ |z| \le |\mathcal{F}^{-1}||\bar{z}| \le \Lambda \sum_{i=1}^{n} \left( |\tilde{\theta}_1|^{\frac{n-i+1}{n}} + |\tilde{\theta}_1| \right), \quad (10) $$

where

$$ \Lambda = n |\mathcal{F}^{-1}| \max_{i=1,n} \left\{ \lambda_i \omega^{\frac{i-1}{n}} \right\} \max_{i=1,n} \left\{ \bar{\Omega}_c^{\frac{n-i+1}{n}}, |a|\bar{\Omega}_c \right\}. $$

**Remark 2.** The constants $\lambda_i$, given in (10), can be computed by a method proposed in [17], which is based on the Lyapunov function

$$ V_z = \sum_{j=1}^{n-1} \beta_j Z_j(\tilde{z}_j, \tilde{z}_{j+1}) + \beta_n \frac{1}{b} |\tilde{z}_n|^b, \quad (11) $$

for some $\beta_i > 0$, $b > 1$, $Z_j(\tilde{z}_j, \tilde{z}_{j+1}) := \frac{r_j}{b} |\tilde{z}_j|^{\frac{b}{r_j}} - \tilde{z}_j [\tilde{z}_{j+1}]^{\frac{b-r_j}{r_j+1}} + \left( \frac{b-r_j}{b} \right) |\tilde{z}_{j+1}|^{\frac{b}{r_j+1}} (j = \overline{1, n-1}), \tilde{z}_i := \frac{\bar{z}_i}{\ell_{i-1}\omega}$, with $\bar{z} = (\bar{z}_1 \quad \dots \quad \bar{z}_n)^\top := \mathcal{F}z, \tilde{\ell}_i := \frac{\ell_i}{\ell_{i-1}}, \ell_0 = 1$, and the homogeneity weights $r_i = n+1-i$, for $i = \overline{1, n}$. The method basically consists in computing $\lambda_i$ as

$$ \lambda_i = \max_{\mathcal{L}_c} |\tilde{z}_i|, \quad \mathcal{L}_c = \{\tilde{z} \in \mathbb{R}^n : V_z(\tilde{z}) \le c\}, $$

where $c > 0$ is the infimum value such that $\dot{V}_z < 0$ holds on the border of the level set $\mathcal{L}_c$, for $\omega = 1$ and $|\Omega_c\tilde{\theta}_1| = |a|\Omega_c\tilde{\theta}_1| = \frac{1}{2}$.
### D. Convergence of the Heavy-Ball-based Adaptive Observer

The following theorem provides the main result of this work.

**Theorem 2.** Let Assumptions 1–4 be satisfied and assume that all the conditions of Lemma 1 hold. If the adaptive observer (6) is designed according to Lemma 1 and the following small-gain conditions

$$2 \max \left\{ 2^{\mu+\alpha} \left( \frac{2}{\kappa_1} \right)^{\frac{\mu+\alpha}{(\alpha+1)(r+1)}} \left( \frac{\eta_z}{\varepsilon_{\theta}\eta_{\theta}} \right) (2n\Lambda)^{\gamma+1}, \right. \\ \left. 2^{\gamma+1} \left( \frac{2}{\kappa_1} \right)^{\frac{\mu+\alpha}{(\alpha+1)(r+1)}} \left( \frac{\eta_z}{\varepsilon_{\theta}\eta_{\theta}} \right), 2^n(n\Lambda)^n \right\} < 1, \quad (12)$$

are feasible with $\varepsilon_{\theta} \in (0, 1)$,

$$\kappa_1 = \min \left\{ \frac{\Omega_c^{(\alpha+1)(r+1)}}{(\alpha+1)} \left( \frac{k_1}{\alpha+1} \right)^{r+1}, 2^{-(r+1)} \right\},$$

$$\kappa_2 = \max \left\{ 2^r \frac{\Omega_c^{(\alpha+1)(r+1)}}{(\alpha+1)} \left( \frac{k_1}{\alpha+1} \right)^{r+1}, 2^{-1} \right\},$$

$$\eta_{\theta} = (2\kappa_2)^{-u} \min\{\eta_1 - \eta_3 - \eta_4 - 7, \eta_2 - 7\},$$

$$\eta_z = \max_{|\tilde{\theta}_1| + |\tilde{\theta}_2| < 1} \left\{ \begin{aligned} & (\eta_{14} + \eta_{15}) |\tilde{\theta}_1|^{(\alpha+1)(r+1)} \\ & + \eta_{17} |\tilde{\theta}_1|^{\mu} |\tilde{\theta}_2|^{2r+1} + \eta_{16} |\tilde{\theta}_1|^{(\alpha+1)r} |\tilde{\theta}_2|_{\beta+1} \\ & + (\eta_{14} + \eta_{15}) |\tilde{\theta}_1|^{\alpha+1} |\tilde{\theta}_2|^{2r} \\ & + (\eta_{18} + \eta_{19}) |\tilde{\theta}_1|^{\mu} |\tilde{\theta}_2|_{\beta+1} \end{aligned} \right\},$$

and some positive constants $\eta_i$, then the origin of the error dynamics (8) is locally practically FTS.

**Remark 3.** Note that the left-hand side of (12) can be adjusted by manipulating the gains of the adaptive observer (6) and increasing the excitation of the input if needed. In this sense, it is possible to find a set of observer gains to ensure the fulfillment of (12).

## V. SIMULATION RESULTS

Consider system (1) with

$$A = \begin{pmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ -1 & -2 & -3 \end{pmatrix}, \quad \phi_t = \begin{pmatrix} -x_1 \\ \sin(x_1) \\ 0 \end{pmatrix},$$

$$G_t = \begin{pmatrix} u \\ 0 \\ 0 \end{pmatrix}, \quad D = \begin{pmatrix} 0 \\ 0 \\ 1 \end{pmatrix}, \quad C^\top = \begin{pmatrix} 1 \\ 0 \\ 0 \end{pmatrix}, \quad \theta = 1.5.$$

Set the input $u(t) = 5(\sin(t) + \sin(0.45t) + 1)$, the external disturbance $w(t) = 1 + 0.5\sin(2t)$, and initial conditions $x(0) = (2 \ -1 \ 1)^\top$. This system satisfies assumptions

1–3. The parameters of the adaptive observer are designed, based on the statements of lemmas 1, 2, and Theorem 2:

$$L = \begin{pmatrix} 3 \\ 0 \\ -1 \end{pmatrix}, \quad \mathbf{a} = \begin{pmatrix} -6 \\ -11 \\ -6 \end{pmatrix}, \\ \left( \begin{array}{c} \ell_1 \\ \ell_2 \\ \ell_3 \end{array} \right) = \left( \begin{array}{c} -6 \\ -11 \\ -6 \end{array} \right), \quad \mathcal{J} = \left( \begin{array}{ccc} 1 & 0 & 0 \\ 3 & 1 & 0 \\ 2 & 3 & 1 \end{array} \right),$$

$\omega = 1.5$, $\alpha = 0.3$, $\beta = 0.33$, $\gamma = \delta = -0.8$, $k_1 = 10$, $k_2 = 20$, and initial conditions $\hat{x}(0) = (0 \ 0 \ 0)^\top$, $\hat{\theta}_1(0) = 0.6$, $\hat{\theta}_2 = 0$, and $\Omega(0) = 0$. Assumption 4 can be verified with these parameters. Simulations have been done in MATLAB with the Euler explicit discretization method and sampling time equal to 0.001[s].

The corresponding results are presented in Figs. 1–3. Fig. 1 illustrates the state estimation, while Fig. 2 depicts the parameter identification. The algorithm achieves fast adaptive estimation, but it introduces some errors. To further analyze its performance, we apply the algorithm under the classic heavy-ball method$^2$, setting $\alpha = \beta = 1$ and $\rho_t = 1$, while keeping all other parameters unchanged. The nomenclature to the legends is as follows: aHB corresponds to the proposed adaptive observer based on the accelerated heavy-ball method, while HB corresponds to the adaptive observer based on the classic heavy-ball method. Fig. 3 shows the norm of the state estimation and parameter identification errors, i.e., $|e|$ and $|\tilde{\theta}_1|$, for both approaches. The proposed accelerated adaptive observer possesses finite-time decay and also higher accuracy.

<figure><img src="image_3.png" /><figcaption>Figure 1. State Estimation</figcaption></figure>

To better illustrate the finite-time properties of the proposed algorithm, we consider the ideal case, i.e., $w \equiv 0$, also comparing with the classic heavy-ball method. Fig. 4 depicts $|e|$ and $|\tilde{\theta}_1|$. Based on all these results, we can conclude that the proposed accelerated heavy-ball-based adaptive observer exhibits FT-ISS properties, rather than just ISS properties like classic gradient or heavy-ball-based

<footnote>$^2$Such an algorithm is also new in the literature and we can provide ISS properties for the adaptive estimation error dynamics.</footnote>
<figure><img src="image_4.png" /><figcaption>Figure 2. Parameter Identification</figcaption></figure>

<figure><img src="image_5.png" /><figcaption>Figure 3. Adaptive Estimation Error</figcaption></figure>

algorithms. However, at this stage, we can only establish
local practical finite-time stability.

<figure><img src="image_6.png" /><figcaption>Figure 4. Adaptive Estimation Error – Ideal Case</figcaption></figure>

VI. CONCLUSIONS

This paper presented the design of an accelerated adaptive observer for the simultaneous estimation of states and constant parameters in a class of uncertain nonlinear systems affected by external disturbances. The design comprised an HOSM observer for state estimation and a heavy-ball-based algorithm for identifying unknown constant parameters. Despite the presence of bounded external disturbances that do not satisfy a relative degree one condition, the origin of the estimation error dynamics is locally practically FTS. The stability analysis used a Lyapunov function approach, FT-ISS properties, and a small-gain theorem. The effectiveness

of the proposed adaptive observer was validated through
simulation results.

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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