# A ROBUST AND ACCELERATED HEAVY-BALL-BASED ALGORITHM FOR PARAMETER IDENTIFICATION*

HÉCTOR RÍOS†, DENIS EFIMOV‡, AND ROSANE USHIROBIRA§

**Abstract.** This paper contributes to designing a new parameter identification algorithm for linear regression systems with constant unknown parameters and noisy measurements. The proposed algorithm is based on a new accelerated version of the heavy-ball method, which uses a nonlinear extension of Kreisselmeier's filters. For the noise-free case, the algorithm can identify constant parameters accurately and in finite time, assuming persistence of the regressor's excitation. A local stability analysis is developed using the Lyapunov function approach. The robustness properties for the noisy case are provided in terms of input-to-state stability property for the parameter identification error dynamics. Additionally, the paper considers a classic optimization problem, taking into account prior data collection of measurements. A reduced version of the proposed identification algorithm is introduced for this case, ensuring global finite-time stability for the noise-free case and local input-to-state stability for the noisy scenario. The effectiveness of the proposed parameter identification algorithm is depicted with some simulation results.

**Key words.** Parameter Identification, Heavy-Ball Method, Finite-Time.

**MSC codes.** 93C10, 93C40, 93B30

## 1. Introduction.
The control design problem, in the presence of parametric uncertainties, remains a challenging issue. The parameter identification theory develops the necessary tools to deal with such a lack of knowledge. In recent decades, many online parameter identification techniques have appeared to help overcome this problem. Popular methods include least-squares (LS) algorithms, gradient descent-based algorithms, and adaptive estimation (see, for instance, [19] and [3]).

In parameter identification, most of the proposed approaches have addressed the identification of constant parameters. To recall some recent works, in [13], a recursive LS method is designed to identify constant parameters for flexible joints. Under some persistence of excitation conditions, the identification error converges to a region around the origin. In [5], an algorithm is proposed for homogeneous systems based on a class of artificial neural networks. The parameter identification error converges to a region around the origin. In [28], a linear regression model is used to ensure the error boundedness in finite time without the persistence of excitation condition. In [18], an adaptive identification method is proposed to estimate constant parameters for sinusoidal signals, e.g., the unknown offset, amplitude, frequency, and phase. The exponential convergence of the error to a region around the origin is guaranteed. In the same vein, in [14], an adaptive algorithm based on a gradient-descent method solves the problem of constant parameter identification. However, most of the above-mentioned algorithms can only ensure exponential convergence rates.

One approach that improves the convergence rate of gradient descent-based algorithms is the heavy-ball method, the first optimization tool using momentums [22].

<footnote>*Funding: 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.</footnote>

<footnote>†Tecnológico Nacional de México/I.T. La Laguna, 27000, Torreón, Coahuila, México. SECIHTI, Investigadoras e Investigadores por México, 03940, CDMX, México (hriosb@lalaguna.tecnm.mx).</footnote>

<footnote>‡Inria, Univ. Lille, CNRS UMR 9189, F-59000, Lille, France. ITMO University, 197101, Saint Petersburg, Russia (denis.efimov@inria.fr).</footnote>

<footnote>§Inria, Univ. Lille, CNRS UMR 9189, F-59000, Lille, France (rosane.ushirobira@inria.fr).</footnote>
Nowadays, it is one of the most popular methods for dealing with optimization prob-
lems and designing machine-learning algorithms, ensuring a fast convergence rate
when objective functions are smooth and strongly convex. For instance, in [29], the
authors propose an analytical characterization of the convergence regions of acceler-
ated gradient descent algorithms based on a frequency domain approach. In [15], a
hybrid heavy-ball method is proposed to deal with convex optimization problems.
The proposed approach uses a feedback mechanism for driving the momentum term,
which provides an LMI synthesis and guarantees exponential convergence rate. The
heavy-ball method is also used for distributed optimization problems. In [23], the
authors develop an accelerated distributed gradient method for fixed networks, which
relies on gradient tracking techniques and local memory to accelerate the convergence
speed. In [26], a generalized heavy-ball method is introduced to deal with global
convergence for a class of non-convex problems. Recently, a family of distributed mo-
mentum methods are proposed in [8] for optimization problems over directed graphs.
The authors develop a global R-linear convergence analysis for smooth and strongly
convex functions. In [27], a second-order differential equation is introduced, from the
control theory framework, which is called heavy-ball dynamics with displaced gradient
and possesses the same convergence as the original heavy-ball dynamics. The authors
characterize the asymptotic convergence of the resulting discrete-time algorithms us-
ing a Lyapunov function approach. In [11], a hybrid control approach is proposed to
ensure fast convergence and global asymptotic stability for an optimization problem of
the unique minimizer of a convex function. The proposed approach is based on a hys-
teresis mechanism that switches between Nesterov's accelerated gradient descent and
the heavy-ball method to ensure the asymptotic convergence employing a Lyapunov
function approach. The literature also offers some recent works related to high-order
tuners for parameter identification problems (see, e.g., [9] and [16]). However, none
of the above-mentioned recent works offers finite-time convergence rate. Moreover,
only a few of them provide a Lyapunov function characterization of the accelerated
convergence rate.

Motivated by the goal of improving the convergence rate, this paper proposes a
novel accelerated parameter identification algorithm for linear regression systems with
constant unknown parameters and noisy measurements<fnref n="1" />. The proposed algorithm is
based on the heavy-ball method, which uses a modified nonlinear extension of Kreis-
selmeier's filters. In the noise-free case, this accelerated heavy-ball-based algorithm
can identify constant unknown parameters in a finite time, assuming persistence of
the regressor's excitation. The local finite-time stability of the parameter identifica-
tion error is ensured through a Lyapunov function approach while, for the noisy case,
a form of input-to-state stability property is established for the error dynamics. An
optimization problem, taking into account a prior data collection of measurements, is
also considered. In this scenario, global finite-time stability in the noise-free case and
local input-to-state stability in the noisy case is ensured based on a reduced version
of the proposed identification algorithm. To the best of our knowledge, this is the first
time, despite a number of its modifications, that such an accelerated version of the
heavy-ball method is proposed, ensuring finite-time convergence and providing ro-
bustness. Compared to other finite- and fixed-time identification algorithms given in
the literature (see, e.g., [17], [24], [30], and [20]), the proposed accelerated heavy-ball-
based algorithm due to its second-order nature, provides a better order of accuracy

<footnote n="1">¹A preliminary version of this proposal has been presented in [25]. Nevertheless, in [25], the stability analysis was omitted and the noisy case was not analyzed.</footnote>
with a finite-time convergence rate. This may be notable for some applications (e.g., fault diagnosis, safety-critical systems, biological systems, and material parameter estimation), which demand highly accurate and fast parameter identification.

The rest of this manuscript is organized as follows. Some preliminary knowledge is given in Section 2. The problem statement is presented in Section 3. The accelerated heavy-ball-based parameter identification algorithm is presented in Section 4. The convergence analysis of the parameter identification error is given in Section 5 while in Section 6 we introduce an optimization problem. Simulation results are provided in Section 7. Section 8 gives the concluding remarks.

**1.1. 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 $|s|_p = (\sum_{i=1}^n |s_i|^p)^{\frac{1}{p}}$. So $|s|_{\alpha+1}^{\alpha+1} = s^\top \lceil s \rceil^\alpha$, for any $\alpha > 0$. 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., $|A| = \sqrt{\lambda_{\max}(A^\top A)} = \sigma_{\max}$, where $\lambda_{\max}$ (respectively, $\lambda_{\min}$) is the maximum (respectively, the minimum) eigenvalue, and $\sigma_{\max}$ (respectively, $\sigma_{\min}$) is the maximum (respectively, the nonzero minimum) singular value.

The set of all inputs $u : \mathbb{R}_{\ge 0} \to \mathbb{R}^p$ 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 $\mathcal{L}_{\infty}$. By Hölder inequality, we have that $|e|_{\zeta_1} \le \bar{e}(\zeta_1, \zeta_2) |e|_{\zeta_2}$, for any $e \in \mathbb{R}^n$ and any $\zeta_1, \zeta_2 > 0$, with $\bar{e}(\zeta_1, \zeta_2) = \max(1, n^{\frac{\zeta_2-\zeta_1}{\zeta_1\zeta_2}})$.

A function $\gamma : \mathbb{R}_{\ge 0} \to \mathbb{R}_{\ge 0}$ belongs to class $\mathcal{K}$ if it is strictly increasing and $\gamma(0) = 0$; 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 $T \in \mathcal{K}$ such that $\beta(r, t) = 0$, for all $t \ge T(r)$.

**2. Preliminaries.** Consider the system

$$ (2.1) \qquad \dot{x}(t) = f(t, x(t), v(t)), \quad x(t_0) = x_0, \quad t \ge t_0 \ge 0, $$

where $x(t) \in \mathbb{R}^n$ is the state vector and $v : \mathbb{R}_{\ge 0} \to \mathbb{R}^m$ is the external input. The function $f : \mathbb{R}_{\ge 0} \times \mathbb{R}^n \times \mathbb{R}^m \to \mathbb{R}^n$ 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 $x_0 \in \mathbb{R}^n$, the solution of system (2.1) is denoted as $x(t, t_0, x_0, v)$, and for brevity we assume that it is defined for any $t \ge t_0$.

**DEFINITION 2.1.** [21], [31]. At steady state $x = 0$, the system (2.1), for $v \equiv 0$, 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 $x_0 \in \mathbb{R}^n$ and $t_0 \ge 0$; and b) for each $x_0 \in \mathbb{R}^n$ 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 $T_0 : x_0 \mapsto \inf\{T_s \ge 0 : x(t, t_0, x_0, 0) = 0, \forall t \ge t_0 + T_s\}$ is called the settling-time of the system (2.1). The system (2.1) is said to be locally FTS if the previous statements hold for $x_0 \in \mathbb{R}^n$ such that $|x_0| \le \bar{x}_0$, with some $\bar{x}_0 > 0$.

**DEFINITION 2.2.** [7]. The system (2.1) is said to be Input-to-State Stable (ISS) if there exist functions $\beta \in \mathcal{KL}$ and $\gamma \in \mathcal{K}$ such that the solution satisfies

$$ |x(t, t_0, x_0, d)| \le \beta(|x_0|, t - t_0) + \gamma(|v|_{\infty}), $$

for all $t \ge t_0$, any input $v \in \mathcal{L}_{\infty}$, any $x_0 \in \mathbb{R}^n$, and $t_0 \ge 0$. Furthermore, if $\beta \in \mathcal{FKL}$, then the system (2.1) is said to be Finite-Time ISS (FT-ISS). The system (2.1) is
called locally (FT-)ISS if the above estimate holds for $x_0 \in \mathbb{R}^n$ and $v \in \mathcal{L}_\infty$ such that
$\max\{|x_0|, |v|_\infty\} \le a$, for some $a > 0$.

Definitions 2.1 and 2.2 can also be introduced for fixed values of $t_0$. The following lemmas are used to develop the stability analysis.

LEMMA 2.1. [6]. Let $x_1, x_2 \in \mathbb{R}^n$, and $W(x_1, x_2) = |x_1|^\alpha + |x_2|^\beta - c|x_1|^\gamma|x_2|^\delta$, with some positive constants $c, \alpha, \beta, \gamma, \delta > 0$. Then, the following statements hold:

1. For any value of c, W is positive definite if

$$ \frac{\gamma}{\alpha} + \frac{\delta}{\beta} > 1 \quad \text{and} \quad \max \{|x_1|^\alpha, |x_2|^\beta\} \le c^{\frac{1}{1-(\frac{\gamma}{\alpha}+\frac{\delta}{\beta})}}. $$

2. For any value of c, W is positive if

$$ \frac{\gamma}{\alpha} + \frac{\delta}{\beta} < 1 \quad \text{and} \quad \max \{|x_1|^\alpha, |x_2|^\beta\} \ge c^{\frac{1}{1-(\frac{\gamma}{\alpha}+\frac{\delta}{\beta})}}. $$

3. For sufficiently small values of c and any $x_1, x_2 \in \mathbb{R}^n$, W is positive definite if

$$ \frac{\gamma}{\alpha} + \frac{\delta}{\beta} = 1. $$

LEMMA 2.2. [1]. Let $x_1, x_2 \in \mathbb{R}^n$, and $W(x_1, x_2) = |x_1|^\alpha + |x_2|^\beta + c_1|x_1|^\eta|x_2|^\zeta - c_2|x_1|^\gamma|x_2|^\delta$, with some positive constants $c_1, c_2, \alpha, \beta, \gamma, \delta, \eta, \zeta > 0$. Then, for any values of $c_1, c_2 > 0$, W is positive definite if and only if

$$ (2.2) \qquad \frac{\eta}{\alpha} + \frac{\zeta}{\beta} < 1, $$

$$ (2.3) \qquad \gamma + \delta \frac{(\alpha - \eta)}{\zeta} > \alpha, $$

$$ (2.4) \qquad \delta + \gamma \frac{(\beta - \zeta)}{\eta} > \beta. $$

The following definition is used in the sequel.

DEFINITION 2.3. [12]. The signal $\phi: \mathbb{R}_{\ge 0} \to \mathbb{R}^n$ is Persistently Excited (PE) if it satisfies

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

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

3. **Problem Statement.** We consider the following linear regression system:

$$ (3.1) \qquad y(t) = \phi^\top(t)\theta + v(t), \quad t \ge 0, $$

where $\theta \in \mathbb{R}^n$ and $y(t) \in \mathbb{R}$ are the unknown vector of constant parameters and the output, respectively; while the term $\phi: \mathbb{R}_{\ge 0} \to \mathbb{R}^n$ is a continuously differentiable function of time so-called regressor, which is known, bounded, and PE; and $v: \mathbb{R}_{\ge 0} \to \mathbb{R}$ represents bounded noisy measurements, $v \in \mathcal{L}_\infty$. The aim of this work is to identify the constant parameter vector $\theta$ under the following assumption:

Assumption 3.1. The regressor $\phi$ is bounded (i.e., $|\phi\phi^\top|_\infty \le \bar{\phi}$), PE, and it satisfies $|\frac{d(\phi\phi^\top)}{dt}|_\infty \le \bar{\phi}_d$, for some known positive constants $\bar{\phi}, \bar{\phi}_d > 0$, respectively. Also, the noise is bounded, i.e., $|v|_\infty \le \bar{v}$, for a known positive constant $\bar{v} > 0$.
**4. Heavy-Ball-based Algorithm.** The proposed identification algorithm is based on an accelerated heavy-ball method and given as

$$169 \quad (4.1a) \qquad \dot{\hat{\theta}}_1 = \hat{\theta}_2,$$

$$170 \quad (4.1b) \qquad \dot{\hat{\theta}}_2 = \tau k_1 z_2^\top [z_1 - z_2 \hat{\theta}_1]^\alpha - k_2 [\hat{\theta}_2]^\beta,$$

$$171 \quad (4.1c) \qquad \dot{z}_1 = \tau^{-1} \rho(z, \hat{\theta}) [\phi y - z_1], \quad z_1(0) = 0,$$

$$172 \quad (4.1d) \qquad \dot{z}_2 = \tau^{-1} \rho(z, \hat{\theta}) [\phi \phi^\top - z_2], \quad z_2(0) = 0,$$

$$173 \quad (4.1e) \qquad \rho(z, \hat{\theta}) = \begin{cases} 1, & \forall t \in [0, T), \\ |z_1 - z_2 \hat{\theta}_1|_{\gamma+1}^{\gamma+1} + |\hat{\theta}_2|_{\mu+1}^{\mu+1}, & \forall t \ge T, \end{cases}$$

where $\hat{\theta}_1(t) \in \mathbb{R}^n$ is the identified value of the parameter vector $\theta$, $\hat{\theta}_2(t) \in \mathbb{R}^n$ is an auxiliary velocity vector, $z_1(t) \in \mathbb{R}^n$ provides a filtered version of the term $\phi(t)y(t)$, $z_2(t) = z_2(t)^\top \in \mathbb{R}^{n \times n}$ represents a filtered version of the matrix $\phi(t)\phi^\top(t)$, the filter time constant is given by $\tau > 0$, the positive function $\rho$ represents a filter nonlinear gain, with some excitation time $T > 0$, with parameters $\alpha, \beta \in (0, 1)$ and $\gamma, \mu, k_1, k_2 > 0$.

Note that the dynamics in (4.1a)-(4.1b) describes the new accelerated heavy-ball method, where the nonlinear term $[\hat{\theta}_2]^\beta$ represents the added “velocity” or “momentum” term. The dynamics (4.1c)-(4.1e) provide a modified nonlinear extension of Kreisselmeier’s filters. The algorithm (4.1) has not been presented in the literature before.

**4.1. Parameter Identification Error Dynamics.** Let us define the parameter identification error as

$$188 \quad (4.2) \qquad e_1 = \theta - \hat{\theta}_1, \quad e_2 = -\hat{\theta}_2.$$

Based on (3.1) and (4.1), the error dynamics can be written as follows

$$190 \quad (4.3a) \qquad \dot{e}_1 = e_2,$$

$$191 \quad (4.3b) \qquad \dot{e}_2 = -\tau k_1 z_2^\top [z_1 - z_2 \hat{\theta}_1]^\alpha + k_2 [\hat{\theta}_2]^\beta.$$

From (4.1c) and (4.1d), it follows that $z_1(t) = z_2(t)\theta - v_f(t)$ for all $t \ge 0$ (the initial conditions of the filters are chosen to be zero), with $v_f(t)$ as the filtered version of the term $\phi(t)v(t) : \dot{v}_f = \tau^{-1}\rho(z, \hat{\theta})[\phi(t)v(t) - v_f(t)]$. Then, the error dynamics (4.3) is rewritten as

$$197 \quad (4.4a) \qquad \dot{e}_1 = e_2,$$

$$198 \quad (4.4b) \qquad \dot{e}_2 = -\tau k_1 z_2^\top [z_2 e_1 - v_f]^\alpha - k_2 [e_2]^\beta,$$

with initial conditions $e_1(0) = e_{10}$ and $e_2(0) = e_{20}$. The parameter identification error dynamics (4.4) can be viewed as a time-varying dynamical system with $t_0 = 0$, which characterizes an accelerated time-dependent version of the heavy-ball method. For the noise-free case, i.e., $v \equiv 0$, the origin $e_1 = e_2 = 0$ is an equilibrium of the system (4.4), uniformly in $z_2$. A characterization of the finite-time stability of this error dynamics at the origin, in terms of Lyapunov functions, is not straightforward.

**4.2. Some Properties of $z_2$ and $\rho(z, \hat{\theta})$.** The following result will help us to deal with the stability analysis of the parameter identification error dynamics.
LEMMA 4.1. Let Assumption 3.1 be satisfied. Then the trajectories of the system (4.1d) satisfy

$$ (4.5) \qquad \phi = e^{-\frac{1}{\tau}T} \frac{\alpha_0}{\tau} \le \inf_{t \ge T} (\lambda_{\min}(z_2(t))), |z_2|_{\infty} \le \bar{\phi}, $$

where $T$ is the excitation time. Moreover, the approximation error for $\phi\phi^\top$, i.e., $\varepsilon = z_2 - \phi\phi^\top$, satisfies

$$ (4.6) \qquad |\varepsilon|_{\infty} \le \bar{\phi} + \tau \bar{\phi}_d = \varepsilon^+. $$

*Proof.* Note that, for $t \in [0, T)$, due to (4.1d), (4.1e), and Assumption 3.1, we have boundedness of $z_2(t)$ for all $t \in [0, T)$. We define the approximation error for $\phi\phi^\top$ as $\varepsilon = z_2 - \phi\phi^\top$. Then, the boundedness of $\varepsilon(t)$ is implied by the boundedness of $z_2$ and $\phi$, for all $t \in [0, T)$. Later we provide the explicit upper bound for $\varepsilon$.

Now, for $t \ge T$, let us consider a time-scaling $d\zeta = \rho(z, \hat{\theta})dt$, which is well-defined when $\rho(z, \hat{\theta}) \ne 0$. Then, the dynamics of $z_2$, for $\zeta \in [T, \bar{T}_s)$ with $T > \bar{T}_s > 0$, is given as

$$ (4.7) \qquad \frac{dz_2(\zeta)}{d\zeta} = \tau^{-1} [\phi(\zeta)\phi^\top(\zeta) - z_2(\zeta)], $$

which has also the form of the filter (4.1d), for $t \in [0, T)$. In this new time $\zeta$, since the dynamics (4.7) is stable for any bounded $\phi$, the PE property is preserved by $z_2(\zeta)$, i.e., if $\phi$ is PE, then we have that $z_2(\zeta) > 0$ for all $\zeta \ge T$. Moreover, according to [4], if $\phi$ is PE, then the smallest eigenvalue of $z_2(\zeta)$ satisfies

$$ \lambda_{\min}(z_2(\zeta)) \ge e^{-\frac{1}{\tau}T} \frac{\alpha_0}{\tau} = \underline{\phi}, \quad \forall \zeta \ge T. $$

Additionally, from (4.7), it is possible to show that

$$ z_2(\zeta) = \phi(\zeta)\phi^\top(\zeta) - \phi(0)\phi^\top(0)e^{-\frac{1}{\tau}\zeta} - \int_0^\zeta e^{-\frac{1}{\tau}(\zeta-\sigma)} \frac{d(\phi(\sigma)\phi^\top(\sigma))}{d\zeta} d\sigma. $$

Hence, the approximation error for $\phi(\zeta)\phi^\top(\zeta)$, i.e., $\varepsilon(\zeta) = z_2(\zeta) - \phi(\zeta)\phi^\top(\zeta)$, satisfies

$$ |\varepsilon|_{\infty} \le |\phi(0)\phi^\top(0)|e^{-\frac{1}{\tau}\zeta} + \tau \left| \frac{d(\phi\phi^\top)}{d\zeta} \right|_{\infty}. $$

By Assumption 3.1, it follows that (4.6) holds for some known $\bar{\phi}$, $\bar{\phi}_d > 0$. By Assumption 3.1, the second inequality of (4.5) holds, i.e., the boundedness of $\phi$. □

Note that if we show that the error dynamics (4.4) is FTS, for $v \equiv 0$, then $\rho(z, \hat{\theta})$ will satisfy $\lim_{t \to T_s} \rho(z(t), \hat{\theta}(t)) = 0$, where $T_s$ is the convergence time of the error dynamics (4.4), which is dependent on the initial conditions of the system. The switched structure of the function $\rho(z, \hat{\theta})$ allows us to avoid reaching zero before the excitation time $T$. Moreover, for the system (4.1), in the noise-free case, i.e., $v \equiv 0$, $\rho(z, \hat{\theta}) = 0$ is the invariant subspace containing the equilibria. Furthermore, if $z_2$ is non-singular, then this set of equilibria is the singleton, and it corresponds to the convergence of $\hat{\theta}$ to the actual value of $\theta$.

In the noise-free case, i.e., $v \equiv 0$, when $\rho(z(T_s), \hat{\theta}(T_s)) = 0$, we have $\dot{z}_2(T_s) = 0$. Hence, $z_2(t) = z_2(T_s)$, for all $t \ge T_s$. At time $T_s$, we already have that $z_2(T_s) > 0$, and we will have that $e_1(t) = e_2(t) = 0$ for all $t \ge T_s \ge T$. In the noisy case, $z_2(t)$ is not necessarily constant after time $T_s$.
246
247
5. Stability Analysis. Let us introduce the following real-valued functions, which will be useful in the stability analysis:

$$
248 \quad (5.1\text{a}) \quad G: t \mapsto \frac{\tau k_1}{\alpha + 1} |z_2(t)e_1(t)|_{\alpha+1}^{\alpha+1},
$$

$$
249 \quad (5.1\text{b}) \quad E: t \mapsto G(t) + \frac{1}{2} e_2(t)^\top e_2(t),
$$

$$
250 \quad (5.1c) \quad F: t \mapsto |z_2(t)e_1(t)|_{q-1}^{q-1} e_1(t)^\top e_2(t), \quad q > 1.
$$

252
253
254
Note that the functions G and E are continuously differentiable and positive definite for all $t \ge T$. Based on (4.4), the time-derivatives of G, E, and F are given as

$$
255 \quad (5.2\text{a}) \quad \dot{G} = \tau k_1 (\lceil z_2 e_1 \rceil^\alpha)^\top [z_2 e_2 - \tau^{-1} \rho(z, \hat{\theta}) \varepsilon e_1],
$$

$$
256 \quad \dot{E} = -k_1 \rho(z, \hat{\theta}) \varepsilon e_1^\top [z_2 e_1]^\alpha - k_2 e_2^\top [e_2]^\beta
$$

$$
257 \quad (5.2\text{b}) \quad + \tau k_1 (\lceil z_2 e_1 \rceil^\alpha - \lceil z_2 e_1 - v_f \rceil^\alpha)^\top z_2 e_2,
$$

$$
258 \quad \dot{F} = -\tau k_1 |z_2 e_1|_{q-1}^{q-1} e_1^\top z_2^\top [z_2 e_1 - v_f]^\alpha
$$

$$
259 \quad -k_2 |z_2 e_1|_{q-1}^{q-1} e_1^\top [e_2]^\beta + |z_2 e_1|_{q-1}^{q-1} |e_2|^2 + (q-1)
$$

$$
260 \quad (5.2c) \quad \times (\lceil z_2 e_1 \rceil^{q-2})^\top [z_2 e_2 - \tau^{-1} \rho(z, \hat{\theta}) \varepsilon e_1]^\top e_1 e_2.
$$

262
Based on [2], we propose the following Lyapunov function candidate

$$
263 \quad (5.3) \qquad V = E^{r+1} + \ell F,
$$

264
265
266
with some $r, \ell > 0$. The stability analysis will be performed in two steps: 1) we
demonstrate finite-time stability and convergence for $t_0 = T$, and 2) we show that a
finite-time escape is impossible for $t \in [0, T)$.

267
268
269
270
271
5.1. Finite-Time Stability and Convergence for $t_0 = T$. In this step, we look for conditions to ensure that $V$ in (5.3) is a Lyapunov function satisfying $\dot{V} \le -\bar{\eta}V^u$, with some $\bar{\eta} > 0$ and $u \in (0, 1)$, for $t \ge T$. We will use Lemmas 2.1, 2.2, and 4.1, and a Lyapunov function approach to prove such a statement. Based on Lemma 4.1, the following inequalities hold for all $t \ge T$:

$$
272 \quad \kappa_1 \varphi - \kappa_3 |e_1|_{q-1}^q |e_2| \le V \le \kappa_2 \varphi + \kappa_3 |e_1|_{q-1}^q |e_2|,
$$

$$
273 \quad \text{where } \varphi : t \mapsto |e_1(t)|_{\alpha+1}^{(\alpha+1)(r+1)} + |e_2(t)|^{2(r+1)} \text{ and}
$$

$$
274 \quad \kappa_1 = \min \left[ \underline{\phi}^{(\alpha+1)(r+1)} \left( \frac{\tau k_1}{\alpha+1} \right)^{r+1} , 2^{-(r+1)} \right],
$$

$$
275 \quad \kappa_2 = \max \left[ 2^r \bar{\phi}^{(\alpha+1)(r+1)} \left( \frac{\tau k_1}{\alpha+1} \right)^{r+1} , 2^{-1} \right],
$$

$$
276 \quad \kappa_3 = \ell \bar{\phi}^{q-1} \bar{e}_1 (2, q-1).
$$

278
It follows from Lemma 2.1 that if

$$
279 \quad (5.4) \qquad q \ge \frac{(2r+1)(\alpha+1)}{2},
$$
and, in the case of equality in (5.4), i.e., item (3) of Lemma 2.1, if the value of $\ell$ is sufficiently small, then there exists $\bar{e} > 0$ such that for all $t \ge T$ and for all $e_1, e_2 \in \mathbb{R}^n$ such that $\max(|e_1|, |e_2|) < \bar{e}$, we have

$$ (5.5) \qquad \frac{\kappa_1}{2} \varphi \le V \le 2\kappa_2\varphi. $$

The time-derivative of $V$, along the trajectories of the system (4.4), is given as

$$ 285 \qquad \dot{V} = (r+1)E^r \dot{E} + \ell \dot{F}. $$

Therefore, taking into account (5.1) and (5.2), the time-derivative of $V$ satisfies

$$ \begin{aligned} 288 \qquad \dot{V} = (r+1)E^r & \left[ -k_1 \rho(z, \hat{\theta}) \varepsilon e_1^\top [z_2 e_1]^\alpha - k_2 e_2^\top [e_2]^\beta \right. \\ & \left. \qquad + \tau k_1 ( [z_2 e_1]^\alpha - [z_2 e_1 - v_f]^\alpha )^\top z_2 e_2 \right] \\ 289 \qquad & \qquad + \ell \left[ -\tau k_1 |z_2 e_1|_{q-1}^{q-1} e_1^\top z_2^\top [z_2 e_1 - v_f]^\alpha - k_2 |z_2 e_1|_{q-1}^{q-1} e_1^\top [e_2]^\beta + |z_2 e_1|_{q-1}^{q-1} |e_2|^2 \right. \\ 290 \qquad & \qquad \left. \qquad + (q-1) ( [z_2 e_1]^{q-2} )^\top [z_2 e_2 - \tau^{-1} \rho(z, \hat{\theta}) \varepsilon e_1] e_1^\top e_2 \right], \end{aligned} $$

and based on Lemma 4.1 and the fact that $-|z_2 e_1 - v_f|^\alpha \le -|z_2 e_1|^\alpha + |v_f|^\alpha$, an upper bound is given as

$$ \begin{aligned} 296 \qquad \dot{V} \le -\ell \tau k_1 \varphi^{q+\alpha} |e_1|_{q+\alpha}^{q+\alpha} & - k_2(r+1)|e_2|_{\beta+1}^{\beta+1} E^r \\ & + k_1(r+1)\varepsilon^+ \bar{\phi}^\alpha \rho(z, \hat{\theta}) |e_1||e_1|_\alpha^\alpha E^r + \tau k_1(r+1)\bar{\phi}^\alpha |v_f|_\alpha^\alpha |e_2| E^r \\ 297 \qquad & + \ell \tau k_1 \bar{\phi}^q |e_1|_{q-1}^{q-1} |e_1||v_f|_\alpha^\alpha + \ell k_2 \bar{\phi}^{q-1} |e_1|_{q-1}^{q-1} |e_1||e_2|_\beta^\beta \\ 298 \qquad & + \ell \bar{\phi}^{q-1} |e_1|_{q-1}^{q-1} |e_2|^2 + \ell(q-1) \bar{\phi}^{q-1} |e_1|_{q-2}^{q-2} |e_1||e_2|^2 \\ 299 \qquad & \qquad + \ell \tau^{-1}(q-1) \varepsilon^+ \bar{\phi}^{q-2} \rho(z, \hat{\theta}) |e_1|_{q-2}^{q-2} |e_1|^2 |e_2|. \end{aligned} $$

Note that $\rho$ can be rewritten as $\rho(z, \hat{\theta}) = |z_2 e_1 - v_f|_{\gamma+1}^{\gamma+1} + |e_2|_{\mu+1}^{\mu+1}$. Moreover, based on Lemma 4.1 and Assumption 3.1, it satisfies $|\rho(z, \hat{\theta})| \le \bar{\phi}^{\gamma+1} |e_1|_{\gamma+1}^{\gamma+1} + |v_f|_{\gamma+1}^{\gamma+1} + |e_2|_{\mu+1}^{\mu+1}$. Then, we have that

$$ \begin{aligned} 302 \qquad \dot{V} \le -\ell \tau k_1 \varphi^{q+\alpha} |e_1|_{q+\alpha}^{q+\alpha} & - k_2(r+1)|e_2|_{\beta+1}^{\beta+1} E^r \\ & + k_1(r+1)\varepsilon^+ \bar{\phi}^{\alpha+\gamma+1} |e_1|_{\gamma+1}^{\gamma+1} |e_1||e_1|_\alpha^\alpha E^r \\ 303 \qquad & + k_1(r+1)\varepsilon^+ \bar{\phi}^\alpha |e_2|_{\mu+1}^{\mu+1} |e_1||e_1|_\alpha^\alpha E^r \\ 304 \qquad & + k_1(r+1)\varepsilon^+ \bar{\phi}^\alpha |e_1||e_1|_\alpha^\alpha |v_f|_{\gamma+1}^{\gamma+1} E^r \\ 305 \qquad & + \tau k_1(r+1)\bar{\phi}^\alpha |v_f|_\alpha^\alpha |e_2| E^r + \ell \tau k_1 \bar{\phi}^q |e_1|_{q-1}^{q-1} |e_1||v_f|_\alpha^\alpha \\ 306 \qquad & \qquad + \ell k_2 \bar{\phi}^{q-1} |e_1|_{q-1}^{q-1} |e_1||e_2|_\beta^\beta \\ 307 \qquad & \qquad + \ell \bar{\phi}^{q-1} |e_1|_{q-1}^{q-1} |e_2|^2 + \ell(q-1) \bar{\phi}^{q-1} |e_1|_{q-2}^{q-2} |e_1||e_2|^2 \\ 308 \qquad & \qquad + \ell \tau^{-1}(q-1) \varepsilon^+ \bar{\phi}^{q+\gamma-1} |e_1|_{\gamma+1}^{\gamma+1} |e_1|_{q-2}^{q-2} |e_1|^2 |e_2| \\ 309 \qquad & \qquad + \ell \tau^{-1}(q-1) \varepsilon^+ \bar{\phi}^{q-2} |e_2|_{\mu+1}^{\mu+1} |e_1|_{q-2}^{q-2} |e_1|^2 |e_2| \\ 310 \qquad & \qquad + \ell \tau^{-1}(q-1) \varepsilon^+ \bar{\phi}^{q-2} |e_1|_{q-2}^{q-2} |e_1|^2 |e_2| \\ 311 \qquad & \qquad + \ell k_2 \bar{\phi}^{q-1} |e_1|_{q-1}^{q-1} |e_1||e_2|_\beta^\beta \\ 312 \qquad & \qquad + \ell \bar{\phi}^{q-1} |e_1|_{q-1}^{q-1} |e_2|^2 + \ell(q-1) \bar{\phi}^{q-1} |e_1|_{q-2}^{q-2} |e_1||e_2|^2 \\ 313 \qquad & \qquad + \ell \tau^{-1}(q-1) \varepsilon^+ \bar{\phi}^{q+\gamma-1} |e_1|_{\gamma+1}^{\gamma+1} |e_1|_{q-2}^{q-2} |e_1|^2 |e_2| \\ 314 \qquad & \qquad + \ell \tau^{-1}(q-1) \varepsilon^+ \bar{\phi}^{q-2} |e_2|_{\mu+1}^{\mu+1} |e_1|_{q-2}^{q-2} |e_1|^2 |e_2| \\ 315 \qquad & \qquad + \ell \tau^{-1}(q-1) \varepsilon^+ \bar{\phi}^{q-2} |e_1|_{q-2}^{q-2} |e_1|^2 |e_2| |v_f|_{\gamma+1}^{\gamma+1}. \end{aligned} $$
Additionally, for any $r > 0$, the following inequalities

$$318 \qquad \bar{\kappa}_1 \left( |e_1|_{q+\alpha}^{(\alpha+1)r} + |e_2|_{\beta+1}^{2r} \right) \le E^r \le \bar{\kappa}_2 \left( |e_1|_{q+\alpha}^{(\alpha+1)r} + |e_2|_{\beta+1}^{2r} \right),$$

hold with

$$320 \qquad \bar{\kappa}_1 = \min \left[ \underline{\phi}^{(\alpha+1)r} \left( \frac{\tau k_1}{\alpha+1} \right)^r \bar{e}_1^{-(\alpha+1)r} (q+\alpha, \alpha+1), 2^{-r} \bar{e}_2^{-2r} (\beta+1, 2) \right],$$

$$321 \qquad \bar{\kappa}_2 = \max \left[ 2^{r-1} \bar{\phi}^{(\alpha+1)r} \left( \frac{\tau k_1}{\alpha+1} \right)^r \bar{e}_1^{(\alpha+1)r} (\alpha+1, q+\alpha), 2^{-1} \bar{e}_2^{2r} (2, \beta+1) \right].$$

Using the previous inequalities and norm equivalences, the time-derivative of $V$ satisfies

$$326 \qquad \begin{aligned} \dot{V} \le {}& -\eta_1 |e_1|_{q+\alpha}^{q+\alpha} - \eta_2 |e_2|_{\beta+1}^{2r+\beta+1} \\ & - \eta_2 |e_1|_{q+\alpha}^{(\alpha+1)r} |e_2|_{\beta+1}^{\beta+1} + \eta_3 |e_1|_{q+\alpha}^{\alpha+\gamma+2+(\alpha+1)r} \\ & + \eta_3 |e_1|_{q+\alpha}^{\alpha+\gamma+2} |e_2|_{\beta+1}^{2r} + \eta_4 |e_1|_{q+\alpha}^{(\alpha+1)(r+1)} |e_2|_{\beta+1}^{\mu+1} \\ & + \eta_4 |e_1|_{q+\alpha}^{\alpha+1} |e_2|_{\beta+1}^{2r+\mu+1} + \eta_5 |e_1|_{q+\alpha}^{q} |e_2|_{\beta+1}^{\beta} \\ & + (\eta_6 + \eta_7) |e_1|_{q+\alpha}^{q-1} |e_2|_{\beta+1}^{2} \\ & + \eta_8 |e_1|_{q+\alpha}^{\gamma+q+1} |e_2|_{\beta+1} + \eta_9 |e_1|_{q+\alpha}^{q} |e_2|_{\beta+1}^{\mu+2} \\ & + \eta_{10} |e_1|_{q+\alpha}^{(\alpha+1)(r+1)} |v_f|^{\gamma+1} + \eta_{11} |e_1|_{q+\alpha}^{q} |v_f|^{\alpha} + \eta_{12} |e_2|_{\beta+1}^{2r+1} |v_f|^{\alpha} \\ & + \eta_{13} |e_1|_{q+\alpha}^{q} |e_2|_{\beta+1} |v_f|^{\gamma+1} + \eta_{12} |e_1|_{q+\alpha}^{(\alpha+1)r} |e_2|_{\beta+1} |v_f|^{\alpha}, \end{aligned}$$

with

$$336 \qquad \eta_1 = \ell \tau k_1 \bar{\phi}^{q+\alpha}, \quad \eta_2 = \bar{\kappa}_1 k_2 (r+1),$$

$$337 \qquad \eta_3 = k_1 (r+1) \bar{\kappa}_2 \varepsilon^+ \bar{\phi}^{\alpha+\gamma+1} \bar{e}_1^{\gamma+2} (\gamma+1, q+\alpha) \bar{e}_1^{\alpha+1} (\alpha+1, q+\alpha),$$

$$338 \qquad \eta_4 = k_1 \bar{\kappa}_2 (r+1) \varepsilon^+ \bar{\phi}^{\alpha} \bar{e}_1^{\alpha+1} (\alpha+1, q+\alpha) \bar{e}_2^{\mu+1} (\mu+1, \beta+1),$$

$$339 \qquad \eta_5 = \ell k_2 \bar{\phi}^{q-1} \bar{e}_1^{q-1} (q-1, q+\alpha) \bar{e}_1 (2, q+\alpha) \bar{e}_2^{\beta} (\beta, \beta+1),$$

$$340 \qquad \eta_6 = \ell \bar{\phi}^{q-1} \bar{e}_1^{q-1} (q-1, q+\alpha) \bar{e}_2^2 (2, \beta+1),$$

$$341 \qquad \eta_7 = \ell (q-1) \bar{\phi}^{q-1} \bar{e}_1^{q-1} (q-1, q+\alpha) \bar{e}_2^2 (2, \beta+1),$$

$$342 \qquad \eta_8 = \ell \tau^{-1} (q-1) \varepsilon^+ \bar{\phi}^{q+\gamma-1} \bar{e}_1^{\gamma+1} (\gamma+1, q+\alpha) \bar{e}_1^{q-2} (q-2, q+\alpha) \bar{e}_1^2 (2, q+\alpha) \bar{e}_2 (2, \beta+1),$$

$$343 \qquad \eta_9 = \ell \tau^{-1} (q-1) \varepsilon^+ \bar{\phi}^{q-2} \bar{e}_1^{q-2} (q-2, q+\alpha) \bar{e}_1^2 (2, q+\alpha) \bar{e}_2^{\mu+2} (\mu+2, \beta+1),$$

$$344 \qquad \eta_{10} = k_1 (r+1) \varepsilon^+ \bar{\phi}^{\alpha} \bar{e}_1^{(\alpha+1)(r+1)} (\alpha+1, q+\alpha) \bar{v}_f^{\gamma+1} (\gamma+1, 2),$$

$$345 \qquad \eta_{11} = \ell \tau k_1 \bar{\phi}^q \bar{e}_1^q (q, q+\alpha),$$

$$346 \qquad \eta_{12} = \tau k_1 (r+1) \bar{\phi} \bar{\kappa}_2 \bar{e}_2 (2, \beta+1),$$

$$347 \qquad \eta_{13} = \ell \tau^{-1} (q-1) \varepsilon^+ \bar{\phi}^{q-2} \bar{e}_1^{q-2} (q-2, q+\alpha) \bar{e}_1^2 (2, q+\alpha) \bar{e}_2 (2, \beta+1) \bar{v}_f^{\gamma+1} (\gamma+1, 2).$$

Note that, for all $e_1, e_2 \in \mathbb{R}^n$ such that $\max(|e_1|, |e_2|) < \min(1, \bar{e})$, if

$$350 \qquad (5.6) \qquad \gamma \ge q - 2 - (\alpha + 1)r,$$

holds, then it follows that $|e_1|_{q+\alpha}^{q+\alpha} \ge |e_1|_{q+\alpha}^{\alpha+\gamma+2+(\alpha+1)r}$ is satisfied. Note that (5.6)
352 holds for $\gamma$ sufficiently big. Thus, the time-derivative of $V$ satisfies

353

$$354 \qquad \dot{V} \leq -\tilde{\eta}_1 |e_1|_{q+\alpha}^{q+\alpha} - \eta_2 |e_2|_{\beta+1}^{2r+\beta+1}$$

355

$$-\eta_2 |e_1|_{q+\alpha}^{(\alpha+1)r} |e_2|_{\beta+1}^{\beta+1} + \eta_3 |e_1|_{q+\alpha}^{\alpha+\gamma+2} |e_2|_{\beta+1}^{2r}$$

356

$$+ \eta_4 |e_1|_{q+\alpha}^{(\alpha+1)(r+1)} |e_2|_{\beta+1}^{\mu+1} + \eta_4 |e_1|_{q+\alpha}^{\alpha+1} |e_2|_{\beta+1}^{2r+\mu+1}$$

357

$$+ \eta_5 |e_1|_{q+\alpha}^{q} |e_2|_{\beta+1}^{\beta} + (\eta_6 + \eta_7) |e_1|_{q+\alpha}^{q-1} |e_2|_{\beta+1}^{2}$$

358

$$+ \eta_8 |e_1|_{q+\alpha}^{\gamma+q+1} |e_2|_{\beta+1} + \eta_9 |e_1|_{q+\alpha}^{q} |e_2|_{\beta+1}^{\mu+2}$$

$$359 \qquad \eta_{10} |e_1|_{q+\alpha}^{(\alpha+1)(r+1)} |v_f|^{\gamma+1} + \eta_{11} |e_1|_{q+\alpha}^{q} |v_f|^{\alpha} + \eta_{12} |e_2|_{\beta+1}^{2r+1} |v_f|^{\alpha}$$

360

$$+ \eta_{13} |e_1|_{q+\alpha}^{q} |e_2|_{\beta+1} |v_f|^{\gamma+1} + \eta_{12} |e_1|_{q+\alpha}^{(\alpha+1)r} |e_2|_{\beta+1} |v_f|^{\alpha},$$

362 with some $\tilde{\eta}_1 = \eta_1 - \eta_3 > 0$. Now, based on Lemma 2.1, the following set of inequalities

363

$$364 \qquad (5.7a) \qquad |e_1|_{q+\alpha}^{q+\alpha} + |e_2|_{\beta+1}^{2r+\beta+1} \geq \eta_3 |e_1|_{q+\alpha}^{\alpha+\gamma+2} |e_2|_{\beta+1}^{2r},$$

$$365 \qquad (5.7b) \qquad |e_1|_{q+\alpha}^{q+\alpha} + |e_2|_{\beta+1}^{2r+\beta+1} \geq \eta_4 |e_1|_{q+\alpha}^{(\alpha+1)(r+1)} |e_2|_{\beta+1}^{\mu+1},$$

$$366 \qquad (5.7c) \qquad |e_1|_{q+\alpha}^{q+\alpha} + |e_2|_{\beta+1}^{2r+\beta+1} \geq \eta_4 |e_1|_{q+\alpha}^{\alpha+1} |e_2|_{\beta+1}^{2r+\mu+1},$$

$$367 \qquad (5.7d) \qquad |e_1|_{q+\alpha}^{q+\alpha} + |e_2|_{\beta+1}^{2r+\beta+1} \geq \eta_8 |e_1|_{q+\alpha}^{\gamma+q+1} |e_2|_{\beta+1},$$

$$368 \qquad (5.7e) \qquad |e_1|_{q+\alpha}^{q+\alpha} + |e_2|_{\beta+1}^{2r+\beta+1} \geq \eta_9 |e_1|_{q+\alpha}^{q} |e_2|_{\beta+1}^{\mu+2},$$

370 holds if the exponents satisfy

371

$$\frac{\alpha + \gamma + 2}{q + \alpha} + \frac{2r}{2r + \beta + 1} > 1,$$

372

$$\frac{(\alpha + 1)(r + 1)}{q + \alpha} + \frac{\mu + 1}{2r + \beta + 1} > 1,$$

373

$$\frac{\alpha + 1}{q + \alpha} + \frac{2r + \mu + 1}{2r + \beta + 1} > 1,$$

374

$$\frac{\gamma + q + 1}{q + \alpha} + \frac{1}{2r + \beta + 1} > 1,$$

375

$$\frac{q}{q + \alpha} + \frac{\mu + 2}{2r + \beta + 1} > 1.$$

377

378 big. Therefore, the time-derivative of $V$ is upper bounded as follows

379

$$380 \qquad \dot{V} \leq -(\tilde{\eta}_1 - 5) |e_1|_{q+\alpha}^{q+\alpha} - (\eta_2 - 5) |e_2|_{\beta+1}^{2r+\beta+1}$$

381

$$-\eta_2 |e_1|_{q+\alpha}^{(\alpha+1)r} |e_2|_{\beta+1}^{\beta+1} + \eta_5 |e_1|_{q+\alpha}^{q} |e_2|_{\beta+1}^{\beta}$$

382

$$+ (\eta_6 + \eta_7) |e_1|_{q+\alpha}^{q-1} |e_2|_{\beta+1}^{2} \eta_{10} |e_1|_{q+\alpha}^{(\alpha+1)(r+1)} |v_f|^{\gamma+1}$$

383

$$+ \eta_{11} |e_1|_{q+\alpha}^{q} |v_f|^{\alpha} + \eta_{12} |e_2|_{\beta+1}^{2r+1} |v_f|^{\alpha}$$

384

$$+ \eta_{13} |e_1|_{q+\alpha}^{q} |e_2|_{\beta+1} |v_f|^{\gamma+1} + \eta_{12} |e_1|_{q+\alpha}^{(\alpha+1)r} |e_2|_{\beta+1} |v_f|^{\alpha}.$$

386

387 The next step consists of applying Lemma 2.2, to each remaining positive term in the above inequality, to obtain conditions to finally ensure the negative definiteness
of $\dot{V}$. Therefore, based on Lemma 2.2, the following set of inequalities

388

389

$$ (5.9\text{a}) \quad |e_1|_{q+\alpha}^{q+\alpha} + |e_2|_{\beta+1}^{2r+\beta+1} + \tilde{\eta}_2|e_1|_{q+\alpha}^{(\alpha+1)r} |e_2|_{\beta+1}^{\beta+1} \ge \eta_5|e_1|_{q+\alpha}^q |e_2|_{\beta+1}^{\beta}, $$

390

$$ (5.9\text{b}) \quad |e_1|_{q+\alpha}^{q+\alpha} + |e_2|_{\beta+1}^{2r+\beta+1} + \tilde{\eta}_2|e_1|_{q+\alpha}^{(\alpha+1)r} |e_2|_{\beta+1}^{\beta+1} \ge (\eta_6 + \eta_7)|e_1|_{q+\alpha}^{q-1} |e_2|_{\beta+1}^{2}, $$

391

holds, for some $0 < 2\tilde{\eta}_2 < \eta_2$, if the exponents satisfy

393

$$ q + \frac{\beta[q + \alpha - (\alpha + 1)r]}{\beta + 1} > q + \alpha, $$

394

$$ \frac{2q}{(\alpha + 1)} > 2r + 1, $$

395

$$ (q - 1) + \frac{2[q + \alpha - (\alpha + 1)r]}{\beta + 1} > q + \alpha, $$

396

$$ 1 + \frac{2(q - 1)}{(\alpha + 1)} > 2r + \beta, $$

397

398

which are equivalent to

399

$$ q > \frac{(2r + 1)(\alpha + 1)}{2}, $$

400

$$ q > \frac{\alpha + \beta(\alpha + 1)r}{\beta}, $$

401

$$ q > \frac{(\alpha + 1)(\beta + 2r) + (1 - \alpha)}{2}, $$

402

403

and hence, if $q$ satisfies

404

$$ (5.10) \quad q > (\alpha + 1)r + \max \left[ \frac{\alpha + 1}{2}, \frac{\alpha}{\beta}, \frac{\beta(\alpha + 1) + (1 - \alpha)}{2} \right], $$

405

406

for any $\alpha, \beta \in (0, 1)$ and $r > 0$, then the time-derivative of $V$ can be upper bounded as follows, for all $t \ge T$,

407

$$ (5.11) \quad \dot{V} \le -\bar{\eta}_1 |e_1|_{q+\alpha}^{q+\alpha} - \bar{\eta}_2 |e_2|_{\beta+1}^{2r+\beta+1} - \bar{\eta}_3 |e_1|_{q+\alpha}^{(\alpha+1)r} |e_2|_{\beta+1}^{\beta+1} + \bar{\eta}_4 \max (|v_f|^\alpha, |v_f|^{\gamma+1}), $$

408

409

which holds for all $e_1, e_2 \in \mathbb{R}^n$ such that $\max(|e_1|, |e_2|) < \min(1, \bar{e})$, with some $\bar{\eta}_1 = \eta_1 - \eta_3 - 7 > 0$, $\bar{\eta}_2 = \eta_2 - 7 > 0$, $\bar{\eta}_3 = \eta_2 - 2\bar{\eta}_2 > 0$, and

410

411

$$ \bar{\eta}_4 = \max_{|e_1| + |e_2| < \min(1, \bar{e})} \left\{ \eta_{10} |e_1|_{q+\alpha}^{(\alpha+1)(r+1)} + \eta_{11} |e_1|_{q+\alpha}^q + \eta_{12} (|e_2|_{\beta+1}^{2r+1} + |e_1|_{q+\alpha}^{(\alpha+1)r} |e_2|_{\beta+1}) + \eta_{13} |e_1|_{q+\alpha}^q |e_2|_{\beta+1} \right\}. $$

412

413

414

415

416

417

Note that the condition (5.4) is satisfied whereas (5.10) holds, i.e., whereas $q$ is sufficiently big. On the other hand, the set of conditions (5.6) and (5.8) is satisfied for $\gamma$ and $\mu$ also sufficiently big, while $\bar{\eta}_1, \bar{\eta}_2, \bar{\eta}_3 > 0$ are satisfied if the gains $k_1$ and $k_2$ are designed such that

418

$$ k_1 > \frac{7}{\ell \tau \phi^{q+\alpha} - (r+1) \bar{\kappa}_2 \varepsilon^+ \bar{\phi}^{\alpha+\gamma+1} \bar{\omega}}, $$

419

$$ k_2 > \frac{\max(2\tilde{\eta}_2, 7)}{\bar{\kappa}_1(r+1)}, $$

420
with $\varpi = \bar{e}_1^{\gamma+2}(\gamma+1, q+\alpha)\bar{e}_1^{\alpha+2}(\alpha+2, q+\alpha)$ and some $\tilde{\eta}_2 > 0$. Using (5.5) and (5.10), we can rewrite (5.11) as:

$$ (5.12) \qquad \begin{aligned} \dot{V} &\le -\bar{\eta}V^u + \bar{\eta}_4 \max (|v_f|^\alpha, |v_f|^{\gamma+1}), \quad \forall t \ge T, \\ \bar{\eta} &= (2\kappa_2)^{-u} \min(\bar{\eta}_1, \bar{\eta}_2), \\ u &\ge \frac{1}{(r+1)} \max \left( \frac{q+\alpha}{\alpha+1}, \frac{2r+\beta+1}{2} \right), \end{aligned} $$

which holds for all $e_1, e_2 \in \mathbb{R}^n$ such that $\max(|e_1|, |e_2|) < \min(1, \bar{e})$. Note that the constraint over $u$ is equivalent to $u \ge (q+\alpha)/[(r+1)(\alpha+1)]$ and if $\alpha$ and $\beta$ satisfy

$$ (5.13) \qquad 1 > \alpha > \frac{q-r-1}{r} > 0, \quad \alpha < \beta, $$

which implies that $u < 1$. Therefore, for the noise-free case, i.e., $v \equiv 0$, since $\dot{V} \le -\bar{\eta}V^u$, with $u < 1$, we guarantee that the origin of the system (4.4) is locally FTS with a settling-time function satisfying $T_s \le T + [V^{1-u}(e_1(T), e_2(T))]/[\bar{\eta}(1-u)]$ (for more details, see, e.g., [7]). Thus, we have proven the local finite-time stability of the parameter identification error dynamics (4.4), for all $t \ge T$. On the other hand, for the noisy case, i.e., $v \ne 0$, we have ensured that the solutions of the system (4.4) possess a locally FT-ISS estimate, for all $t \ge T$ and $t_0 = T$.

**5.2. Not Finite-Time Escape for $t \in [0, T)$**. To complete the stability analysis, we analyze the parameter identification error dynamics (4.4) and show the boundedness of its trajectories, for all $t \in [0, T)$. The right-hand side of the error dynamics (4.4) is:

$$ f(t, e_1, e_2) = \begin{pmatrix} e_2 \\ -\tau k_1 z_2^\top [z_2 e_1 - v_f]^\alpha - k_2 [e_2]^\beta \end{pmatrix}, $$
and since $\phi(t)$ is bounded for all $t \ge 0$, by Assumption 3.1, as well $z_2(t)$, due to its stable filtering properties and the fact that $\rho(z, \hat{\theta}) = 1$, for all $t \in [0, T)$, we have that

$|f(t, e_1, e_2)| \le L(|e_1|^\alpha + |e_2| + |e_2|^\beta + |v_f|^\alpha), \forall t \ge 0,$
for some constant $L > 0$, and hence, $f$ is Hölder continuous. Then, for all $e_1, e_2 \in \mathbb{R}^n$, $|v|_\infty \le \bar{v}$, and $\alpha, \beta \in (0, 1)$, we have that $f$ is growing linearly in $e_1$ and $e_2$, and thus, the trajectories of the error dynamics (4.4) do not escape to infinity in the time interval $t \in [0, T]$.

The previous statements imply that, for a proper initialization, the accelerated heavy-ball-based algorithm (4.1) can wait until time $T$, i.e., until $z_2(t) > 0$ holds, and then, for the noise-free case, i.e., $v \equiv 0$, ensures that $e_1(t) = e_2(t) = 0$ for all $t \ge T_s \ge T$. On the other hand, for the noisy case, i.e., $v \ne 0$, we have ensured that the system (4.4) is locally FT-ISS for all $t \ge T$. Therefore, we have proven the main results of this work.

**THEOREM 5.1.** *Consider the system (3.1) under Assumption 3.1, with $T$ as the excitation time. Suppose that the exponents of the accelerated heavy-ball-based algorithm (4.1) hold the set of conditions (5.6), (5.8), (5.10), and (5.13) with some $\alpha, \beta \in (0, 1)$, and the gains are designed such that*

$$ (5.14a) \qquad k_1 > \frac{7}{\ell \tau \phi^{q+\alpha} - (r+1) \bar{\kappa}_2 \varepsilon^+ \bar{\phi}^{\alpha+\gamma+1} \varpi} > 0, $$
$$ (5.14b) \qquad k_2 > \frac{\max(2\tilde{\eta}_2, 7)}{\bar{\kappa}_1(r+1)}. $$
Then, the trajectories of the identification error dynamics (4.4) are bounded for all $t \ge 0$. Moreover, they admit a locally FT-ISS estimate on its solutions with $t_0 = T$. For $v \equiv 0$, the origin of the identification error dynamics (4.4) is locally FTS for $t_0 = T$ with a settling-time function satisfying

$$ (5.15) \qquad T_s \le T + \frac{V^{1-u}(e_1(T), e_2(T))}{\bar{\eta}(1-u)}. $$

Remark 5.1. Conditions (5.6), (5.8), (5.10), and (5.13) always hold for sufficiently big $q$, $\gamma$, and $\mu$. Additionally, the finite-time stability is always ensured for sufficiently big values of $k_1$ and $k_2$, and $\alpha < \beta$ so that $\alpha, \beta \in (0, 1)$.

**6. An Optimization Problem.** Consider that we have a previous data collection of measurements for the scalar regression system (3.1), i.e.,

$$ (6.1) \qquad Y = \Phi^\top \theta + \Upsilon, $$

where $\theta \in \mathbb{R}^n$ and $Y \in \mathbb{R}^N$ are the unknown vector of constant parameters and the vector of collected outputs, respectively; with $N \gg n$. The term $\Phi \in \mathbb{R}^{n \times N}$ is a constant regressor matrix, which is known; and $\Upsilon \in \mathbb{R}^N$ represents the vector of collected noisy measurements. Once again, the aim is to identify the constant parameter vector $\theta$ under the following assumption:

Assumption 6.1. The regressor matrix $\Phi$ satisfies rank($\Phi$) = $n$ and the vector of collected noisy measurements is bounded (i.e., $|\Upsilon| \le \bar{\Upsilon}$), for some $\bar{\Upsilon} > 0$.

For this problem there is no need in filtering and we can apply a reduced version of the accelerated heavy-ball-based parameter identification algorithm, i.e.,

$$ (6.2a) \qquad \dot{\hat{\theta}}_1 = \hat{\theta}_2, $$

$$ (6.2b) \qquad \dot{\hat{\theta}}_2 = k_1 \Phi [Y - \Phi^\top \hat{\theta}_1]^\alpha - k_2 [\hat{\theta}_2]^\beta, $$

where $\hat{\theta}_1 \in \mathbb{R}^n$ is the identified value of the parameter vector $\theta$, $\hat{\theta}_2 \in \mathbb{R}^n$ is an auxiliary velocity vector, and the parameters of the algorithm are $\alpha, \beta \in (0, 1)$, and $k_1, k_2 > 0$. This problem can be seen as a common optimization problem in machine learning, whose objective is to minimize the error $e_1 = \theta - \hat{\theta}_1$ (see, e.g., [10]).

Therefore, the parameter identification error dynamics, between (6.1) and (6.2), is given as

$$ (6.3a) \qquad \dot{e}_1 = e_2, $$

$$ (6.3b) \qquad \dot{e}_2 = -k_1 \Phi [\Phi^\top e_1 - \Upsilon]^\alpha - k_2 [e_2]^\beta, $$

with initial conditions $e_1(0) = e_{10}$ and $e_2(0) = e_{20}$. Applying algorithm (6.2) leads us to the following result.

**COROLLARY 6.1.** Consider the system (6.1) under Assumption 6.1. If the exponents of the accelerated heavy-ball-based algorithm (6.2) are designed such that the conditions (5.10) and (5.13) hold with some $\alpha, \beta \in (0, 1)$, and the gains are designed according to (5.14), then the origin of the parameter identification error dynamics (6.3) is globally FTS for $\Upsilon \equiv 0$, with a settling-time function satisfying (5.15); and for $\Upsilon \neq 0$, the parameter identification error dynamics (6.3) is locally FT-ISS with $t_0 = 0$.
Proof. First, note that rank($\Phi$) = n implies that there exists $\sigma_{\min}(\Phi^\top) \ge \underline{\sigma} > 0$, and also, there always exists $\sigma_{\max}(\Phi^\top) \le \bar{\sigma} < \infty$. Thus, in order to prove the above-mentioned results, we can follow the same procedure described in Section 5, replacing $z_2$, $\underline{\phi}$, and $\bar{\phi}$ with $\Phi^\top$, $\underline{\sigma}$, and $\bar{\sigma}$, respectively; and considering $\varepsilon \equiv 0$ and the fact that

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$$ \bar{e}_1(\alpha, 2)\underline{\sigma}|e_1| \le |\Phi^\top e_1|_{\alpha} \le \bar{e}_1(\alpha, 2)\bar{\sigma}|e_1|. $$

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Therefore, if we take the Lyapunov function candidate V in (5.3), and based only on the global statements of Lemma 2.2, it is possible to show that the time-derivative of V, along the trajectories of system (6.3), satisfies

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$$ \dot{V} \le -\hat{\eta}_1 |e_1|_{q+\alpha}^{q+\alpha} - \hat{\eta}_2 |e_2|_{\beta+1}^{2r+\beta+1} - \bar{\eta}_3 |e_1|_{q+\alpha}^{(\alpha+1)r} |e_2|_{\beta+1}^{\beta+1} + \hat{\eta}_4 |\Upsilon|^{\alpha}, $$

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which holds for all $e_1, e_2 \in \mathbb{R}^n$, when $\Upsilon \equiv 0$; with some $\hat{\eta}_1 = \eta_1 - 2 > 0$, $\hat{\eta}_2 = \eta_2 - 2 > 0$, $\bar{\eta}_3 = \eta_2 - 2\hat{\eta}_2 > 0$, and

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$$ \hat{\eta}_4 = \max_{|e_1| + |e_2| < \min(1, \bar{e})} \left\{ \bar{\eta}_{11} |e_1|_{q+\alpha}^q + \bar{\eta}_{12} |e_2|_{\beta+1}^{2r+1} + \bar{\eta}_{12} |e_1|_{q+\alpha}^{(\alpha+1)r} |e_2|_{\beta+1} \right\}, $$

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with some constants $\bar{e}, \bar{\eta}_{11}, \bar{\eta}_{12} > 0$. Once again, the condition (5.10) holds whereas $q$ is sufficiently big. On the other hand, $\hat{\eta}_1, \hat{\eta}_2, \bar{\eta}_3 > 0$ are satisfied if the gains $k_1$ and $k_2$ are designed according to (5.14). Therefore, using (5.5) and (5.10), we have that

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$$ \dot{V} \le -\hat{\eta}V^u + \hat{\eta}_4|\Upsilon|^\alpha, \quad \forall t \ge 0, $$

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$$ \hat{\eta} = (2\kappa_2)^{-u} \min(\hat{\eta}_1, \hat{\eta}_2), $$

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$$ u \ge \frac{1}{(r+1)} \max \left( \frac{q+\alpha}{\alpha+1}, \frac{2r+\beta+1}{2} \right), $$

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Thus, if the values of $\alpha$ and $\beta$ satisfy (5.13), which implies that $u < 1$, then, for the noise-free case, i.e., $\Upsilon = 0$, we guarantee that the origin of the system (6.3) is globally FTS with $t_0 = 0$, with the given settling-time function. On the other hand, for the noisy case, i.e., $\Upsilon \ne 0$, we have only ensured, due to $\hat{\eta}_4$ existing only locally, that the system (6.3) is locally FT-ISS with $t_0 = 0$. □

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**7. Simulation Results.** The simulations have been done in MATLAB with the Euler explicit discretization method and sampling time equal to 0.01[s].

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**7.1. Noise-Free Case.** Let us consider an academic example for system (3.1) with:

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$$ \phi(t) = \begin{pmatrix} 2 \cos(6t) + 3 \sin(10t) + 2 \sin(3t) \cos(11t) \\ -6 \sin(1.5t) + \cos(20t) + 3 \sin(4t) \cos(11t) \\ 3 \cos(21t) \end{pmatrix}, $$

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$$ \theta = (\theta_{11}, \theta_{12}, \theta_{13})^\top = (2, -4, 1)^\top. $$

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Fixing $\alpha = 0.8$, $\beta = 0.96$, $\gamma = 0.5$, $\mu = 1$, and $r = 1.5$, we can prove that the set of conditions (5.6), (5.8), (5.10), and (5.13), hold for $q = 4.01$. Then, taking $\tau = 0.01$, $k_1 = 0.08$, $k_2 = 8$, and $T = 5$, which satisfy the statements of Theorem 5.1, we apply the parameter identification algorithm (4.1), considering $\hat{\theta}_1(0) = \hat{\theta}_2(0) = (3, -5, 0)^\top$. For comparison purposes, we implement the algorithm (4.1), when the classic heavy-ball-method is considered, i.e., when $\alpha = \beta = 1$ and $\rho(z, \hat{\theta}) = 1$, with the same
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values for the rest of parameters. The nomenclature to the legends is as follows:

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aHB corresponds to the proposed accelerated heavy-ball-based algorithm while HB corresponds to the classic heavy-ball-based algorithm.

<figure><img src="image_1.png" /><figcaption>FIG. 1. *Parameter Identification* $\theta_{11}$ – *Noise-Free Case*</figcaption></figure>

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<figure><img src="image_2.png" /><figcaption>FIG. 2. *Parameter Identification* $\theta_{12}$ – *Noise-Free Case*</figcaption></figure>

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The Figs. 1, 2, and 3 show the parameter identification provided for each of

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the algorithms. We can see that both algorithms provide a fast identification of the

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unknown parameters. However, in Fig. 4, which depicts the norm of the parame-

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ter identification error *e*, we can see that the proposed heavy-ball-based algorithm

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possesses finite-time decay, so with a faster convergence rate, and also considerably

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higher accuracy than the classic heavy-ball-based algorithm.
<figure><img src="image_3.png" /><figcaption>FIG. 3. Parameter Identification $\theta_{13}$ – Noise-Free Case</figcaption></figure>

<figure><img src="image_4.png" /><figcaption>FIG. 4. Parameter Identification Error – Noise-Free Case</figcaption></figure>

It is possible to provide a comparison with finite/fixed-time identification algorithms. For instance, the algorithms given in [24], i.e.,

$$ \dot{\hat{\theta}} = -K\phi(\lceil\phi^\top\hat{\theta} - y\rceil^{\gamma_1} + \lceil\phi^\top\hat{\theta} - y\rceil^{\gamma_2}), $$

can be implemented with the Euler explicit discretization method and sampling times $h = 0.01$[s] and $h = 0.001$[s]. The parameters of the algorithm were taken as $K = 0.1I$, $\gamma_1 = 0.5$, for the finite-time algorithm, and $K = 0.1I$, $\gamma_1 = 0.5$, and $\gamma_2 = 1.5$, for the fixed-time algorithm. The parameters of the heavy-ball-based algorithm are $\alpha = 0.8$, $\beta = 0.96$, $\gamma = 0.5$, $\mu = 1$, $\tau = 0.01$, $k_1 = 0.08$, $k_2 = 8$, and $T = 5$. The results are shown in Figs. 5 and 6.
<figure><img src="image_5.png" /><figcaption>FIG. 5. Identification Error – h = 0.01[s]</figcaption></figure>

<figure><img src="image_6.png" /><figcaption>FIG. 6. Identification Error – h = 0.001[s]</figcaption></figure>

The results show that we have an identification error accuracy of order $O(h^2)$,
for the finite/fixed-time identification algorithms, while for the proposed heavy-ball-
based algorithm we obtain $O(h^4)$.

**7.2. Noisy Case.** Let us consider the same academic example with the external signal $v(t) = 0.01 \sin(50t)$ as noise of measurements. We implement the parameter identification algorithm (4.1) in the same way, i.e., $\alpha = 0.8, \beta = 0.96, \gamma = 0.5, \mu = 1, r = 1.5, \tau = 0.01, k_1 = 0.08, k_2 = 8, \text{ and } T = 5$, considering $\hat{\theta}_1(0) = \hat{\theta}_2(0) = (3, -5, 0)^\top$. For comparison purposes, we implement the algorithm (4.1), when the classic heavy-ball-method is considered, i.e., when $\alpha = \beta = 1$ and $\rho(z, \hat{\theta}) = 1$, with
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the same values for the rest of parameters. Fig. 10 depicts the norm of the parameter identification error e. We see that the proposed heavy-ball-based algorithm possesses faster convergence rate, less accuracy than in the noise-free case but still higher accuracy than the classic heavy-ball-based algorithm considering the effect of the noise.

<figure><img src="image_7.png" /><figcaption>FIG. 7. Parameter Identification $\theta_{11}$ – Noisy Case</figcaption></figure>

<figure><img src="image_8.png" /><figcaption>FIG. 8. Parameter Identification $\theta_{12}$ – Noisy Case</figcaption></figure>

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The Figs. 7, 8, and 9 show the parameter identification provided for each of the algorithms in the noisy case. We see that both algorithms provide a fast identification of the unknown parameters even in the noisy case. However, in Fig. 10, which depicts the norm of the parameter identification error e, we see that the proposed heavy-ball-
<figure><img src="image_9.png" /><figcaption>FIG. 9. Parameter Identification $\theta_{13}$ – Noisy Case</figcaption></figure>

<figure><img src="image_10.png" /><figcaption>FIG. 10. Parameter Identification Error – Noisy Case</figcaption></figure>

based algorithm possesses faster convergence rate, less precision than in the noise-free case but still higher precision than the classic heavy-ball-based algorithm considering the effect of the noise.

**8. Conclusion.** In this paper, a novel parameter identification algorithm was proposed for linear regression systems with constant unknown parameters and noisy measurements. Our algorithm is based on an accelerated version of the heavy-ball method, which uses a nonlinear extension of Kreisselmeier's filters. For the noise-free case, this accelerated version of the heavy-ball method can identify constant unknown parameters exactly and in a finite time assuming persistence of the regressor's excitation. The robustness properties in the noisy case are given in terms of finite-time
input-to-state stability conditions. A classic optimization problem in machine and deep learning, considering a prior data collection of measurements, was also introduced. In this scenario, a reduced version of the proposed identification algorithm ensures global finite-time stability for the noise-free case, and local input-to-state stability for the noisy case. The applicability and effectiveness of the proposed parameter identification algorithm were illustrated through simulation results.

Future research may focus on designing adaptive observers and adaptive controllers based on the proposed heavy-ball-based identification algorithm.

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