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# On the Design of Adaptive Observers for Persidskii Systems

Radoslaw Patelski, Rosane Ushirobira, Denis Efimov

► To cite this version:

Radoslaw Patelski, Rosane Ushirobira, Denis Efimov. On the Design of Adaptive Observers for Persidskii Systems. ECC 2025 European Control Conference, Jun 2025, Thessaloniki, Greece. pp.33-38, 10.23919/ECC65951.2025.11187082. hal-05323178

HAL Id: hal-05323178

https://inria.hal.science/hal-05323178v1

Submitted on 21 Oct 2025

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Distributed under a Creative Commons Attribution 4.0 International License
# On the Design of Adaptive Observers for Persidskii Systems

Radosław Patelski<fnref n="1" />, Rosane Ushirobira<fnref n="1" /> and Denis Efimov<fnref n="1" />

**Abstract**—This paper presents the design of an adaptive observer for simultaneous state estimation and parametric identification in Persidskii systems. The main assumption is that the relative degree of the system is equal to one, which allows certain matching conditions for both linear and nonlinear outputs of the system to be satisfied. The known nonlinearities are assumed to be monotone and satisfy sector condition, while the regressor associated with the unknown parameters is Lipschitz continuous. For this type of system, the adaptive observer is designed considering the presence of disturbances and measurement noise. Input-to-state stability conditions are established through Lyapunov analysis. The validity of the proposed observer is illustrated by numerical examples.

## I. INTRODUCTION

Simultaneous state estimation and model identification of a system exhibiting parametric uncertainties is one of the classical problems in control theory. While multiple solutions have been proposed in the literature for linear systems [1]–[3], the design of adaptive observers for nonlinear systems remains an open and challenging problem. Notable solutions have been presented for certain classes of nonlinear systems, usually characterized by affinity in the state or parameters [4], [5] or the existence of some transformation into the nonlinearly disturbed linear system [6]–[8]. In particular, two different cases for the design of adaptive observers can be considered – one where the output derivative depends explicitly on all unknown parameters (relative degree one) [2], and another where the relative degree is higher [9], [10]. The latter requires a more sophisticated structure of the estimator. In both cases, the estimation of parameters is achievable if the condition of persistence of excitation (or at least interval excitation) is met [11] (see also a recent survey [12]).

This paper considers the so-called generalized Persidskii systems, presented initially in [13], consisting of a linear part and a set of sector-restricted nonlinearities. In this paper an additional uncertain term, nonlinear but linearly parameterized, is introduced in the structure of the system. We propose an adaptive observer for this system to ensure robust state estimation and parameter identification. To this end, the advantage is taken of the recent progress in the formulation of the state estimation algorithms for Persidskii systems [14], [15], as well as the stability analysis of the adaptive systems [16], [17]. In the presence of exogenous

disturbances and measurement noises, the notion of input-to-state stability is employed, and the convergence of the estimates into the neighborhood of the trajectories of the state of the system is proved. This paper extends the earlier results of [14], [15] by considering systems with uncertain terms which are here subject to adaptive identification. Thus, in the absence of external disturbances asymptotic convergence of the observer is achieved despite initial uncertainty of these terms, what is not guaranteed by earlier non-adaptive solutions. The present work also extends the results of [18] where an adaptive observer for an epidemiological system expressed in Persidskii form was proposed. In contrast to that paper, the observer is here designed for the general class of systems satisfying certain conditions expressed in the form of linear matrix inequalities (LMIs). Moreover, we focus on the case of relative degree one, i.e., when the derivative of the output depends on the vector of the parameters to be identified.

The following notation is used in the paper. The real and natural numbers are denoted respectively by $\mathbb{R}$ and $\mathbb{N}$, with $\mathbb{R}_+ = \{s \in \mathbb{R} : s \ge 0\}$. The set of diagonal matrices of dimension $n \times n$ is denoted by $\mathbb{D}^n$, with $\mathbb{D}_+^n$ standing for matrices with positive diagonal entries. For $p, n \in \mathbb{N}$ with $p \le n$, the set $\{p, p+1, \dots, n\}$ is represented by $\overline{p, n}$. A block matrix $\mathbf{B} \in \mathbb{R}^{np \times mq}$ with elements $\mathbf{A}_{i,j} \in \mathbb{R}^{p \times q}$ in $i$th block row and $j$th block column is denoted by $\mathbf{B} = (\mathbf{A}_{i,j})_{i \le n, j \le m}$. Without loss of generality, $\|\cdot\|$ stands for the Euclidean norm of a vector in $\mathbb{R}^n$ or the induced norm of a matrix in $\mathbb{R}^{n \times m}$. For a Lebesgue measurable function $v: \mathbb{R}_+ \to \mathbb{R}^m$, define the norm $\|v\|_{[t_1, t_2]} := \ess\sup_{t \in [t_1, t_2]} \|v(t)\|$ for $[t_1, t_2) \subseteq \mathbb{R}_+$. We denote $\|v\|_{\infty} := \|v\|_{[0, \infty)}$.

## II. PROBLEM FORMULATION

Consider the nonlinear system in generalized Persidskii form expressed by

$$ 
\begin{aligned}
\dot{\mathbf{x}}(t) &= \mathbf{A}_0\mathbf{x}(t) + \mathbf{\Phi}(t, \mathbf{y}(t)) + \sum_{j=1}^{M} \mathbf{A}_j \mathbf{f}^j(\mathbf{H}_j\mathbf{x}(t)) \\
&\quad + \mathbf{R}\mathbf{\psi}(t, \mathbf{y}(t))\mathbf{\theta} + \mathbf{d}(t), \\
\mathbf{y}(t) &= \begin{bmatrix} \mathbf{C}_0\mathbf{x}(t) \\ \mathbf{C}_1\mathbf{f}^1(\mathbf{H}_1\mathbf{x}(t)) \\ \vdots \\ \mathbf{C}_M\mathbf{f}^M(\mathbf{H}_M\mathbf{x}(t)) \end{bmatrix} + \mathbf{v}(t) = \begin{bmatrix} \mathbf{y}_0(t) \\ \mathbf{y}_1(t) \\ \vdots \\ \mathbf{y}_M(t) \end{bmatrix},
\end{aligned} 
\quad (1) $$

where $\mathbf{x}(t) \in \mathbb{R}^n$ is the state vector, $\mathbf{A}_0 \in \mathbb{R}^{n \times n}$, $\mathbf{A}_j \in \mathbb{R}^{n \times k_j}$, $\mathbf{H}_j \in \mathbb{R}^{k_j \times n}$ are constant known matrices, and $\mathbf{f}^j(\mathbf{s}) = [f_1^j(s_1) \cdots f_{k_j}^j(s_{k_j})]^\top : \mathbb{R}^{k_j} \to \mathbb{R}^{k_j}$ are

<footnote>*This work was partially supported by the French ANR NOCIME.</footnote>

<footnote n="1">¹Radosław Patelski, Rosane Ushirobira and Denis Efimov are with Inria, Univ. Lille, CNRS UMR 9189, F-59000 Lille, France. Emails: radoslaw.patelski, rosane.ushirobira, denis.efimov@inria.fr</footnote>
known nonlinear functions for $j \in \overline{1, M}$ and any vector $\mathbf{s} = [s_1 \cdots s_{k_j}]^\top \in \mathbb{R}^{k_j}$. The total count of nonlinear terms in the system is denoted by $k = \sum_{j=1}^M k_j$. The term $\mathbf{y}(t) \in \mathbb{R}^z$ stays for the measurable output of the system, with $\mathbf{C}_0 \in \mathbb{R}^{z_0 \times n}$ and $\mathbf{C}_j \in \mathbb{R}^{z_j \times k_j}$ known constant output matrices, with $z = \sum_{j=0}^M z_j$ being dimension of the entire output comprising both linear and nonlinear terms. A function $\Phi: \mathbb{R}_+ \times \mathbb{R}^z \to \mathbb{R}^n$ is a known input of the plant. An uncertain part of the dynamics is represented by a known regressor $\Psi: \mathbb{R}_+ \times \mathbb{R}^z \to \mathbb{R}^{z \times p}$, a known constant matrix $\mathbf{R} \in \mathbb{R}^{n \times z}$ and an unknown vector of parameters $\Theta \in \mathbb{R}^p$. Unknown disturbances and noises are represented by functions $\mathbf{d}: \mathbb{R}_+ \to \mathbb{R}^n$ and $\mathbf{v}: \mathbb{R}_+ \to \mathbb{R}^z$, which are essentially bounded. A term $\mathbf{d}$ comprises all external disturbances, internal uncertainty and regressor uncertainty due to the measurement noise. For the sake of brevity, the time argument $t$ and the arguments of $\Psi(t, \mathbf{y}(t))$ and $\Phi(t, \mathbf{y}(t))$ will sometimes be omitted.

**Assumption 1:** For $i \in \overline{1, k_j}$ and $j \in \overline{1, M}$, the nonlinear functions $\mathbf{f}^j$ satisfy

$$v f_i^j(v) > 0, \forall v \in \mathbb{R} \setminus \{0\} \quad (2)$$

and all $f_i^j$ are strictly increasing functions.

**Assumption 2:** For all $j \in \overline{1, M}$, the output matrices of the system can be expressed as $\mathbf{C}_j \mathbf{T}_j = [\tilde{\mathbf{C}}_j \quad \mathbf{0}_{z_j \times (k_j - z_j)}]$ where $\tilde{\mathbf{C}}_j \in \mathbb{R}^{z_j \times z_j}$ is an invertible matrix and $\mathbf{T}_j \in \mathbb{R}^{k_j \times k_j}$ is a known permutation matrix. Thus also $\mathbf{C}_j \mathbf{T}_j \mathbf{G}_j = \tilde{\mathbf{C}}_j$ and $\mathbf{C}_j = \tilde{\mathbf{C}}_j \mathbf{G}_j^\top \mathbf{T}_j^{-1}$, where $\mathbf{G}_j = [\mathbf{I}_{z_j} \quad \mathbf{0}_{z_j \times (k_j - z_j)}]^\top$.

**Assumption 3:** On the considered trajectories of $\mathbf{x}$, system (1) satisfies the persistency of excitation condition

$$\int_{t}^{t+T_{\text{PE}}} \mathbf{\Psi}(\tau, \mathbf{y}(\tau))^{\top} \mathbf{R}^{\top} \mathbf{R} \mathbf{\Psi}(\tau, \mathbf{y}(\tau)) d\tau \geq \mu \mathbf{I}_p \quad (3)$$

for all $t \ge 0$ and some $\mu, T_{\text{PE}} > 0$.

**Assumption 4:** The regressor function $\Psi$ belongs at least to class $C^1$, i.e. it is at least once differentiable, and the evolution of the plant is such that

$$\begin{aligned} & \sup_{t \ge 0} (\|\mathbf{R}\mathbf{\Psi}(t, \mathbf{y}(t))\|) < \infty, \\ & \sup_{t \ge 0} (\|\mathbf{R} \frac{d}{dt} \mathbf{\Psi}(t, \mathbf{y}(t))\|) < \infty \end{aligned} \quad (4)$$

are satisfied on the trajectories of the plant.

**Assumption 1** is the sector condition on the nonlinearities of (1), which is conventional for Persidskii systems, but it is accompanied here by the monotonicity assumption needed for the observer design. This monotonicity, together with **Assumption 2**, which states that all non-zero columns of output matrices can form some non-singular matrix by a proper reordering of state components, in the noise-free setting enables explicit recovery of state variables needed for the observer design. The persistency of excitation stated in **Assumption 3** is a standard restriction required for a robust reconstruction of the vector of unknown parameters $\Theta$. **Assumption 4** is a technical hypothesis needed in the stability analysis of the proposed adaptive observer.

The paper considers the design of an adaptive observer for simultaneous estimation of unmeasured state variables $\mathbf{x}$ and unknown parameters $\mathbf{\Theta}$. Specifically, the observer is proposed in the form:

$$\begin{aligned} \dot{\mathbf{x}} = \mathbf{A}_0 \hat{\mathbf{x}} + \mathbf{\Phi}(t, \mathbf{y}(t)) &+ \sum_{j=1}^{M} \mathbf{A}_j \mathbf{f}^j(\mathbf{H}_j \hat{\mathbf{x}}) + \mathbf{L}_0 (\mathbf{y}_0 - \mathbf{C}_0 \hat{\mathbf{x}}) \\ &+ \sum_{j=1}^{M} \mathbf{L}_j (\mathbf{y}_j - \mathbf{C}_j \mathbf{f}^j(\mathbf{H}_j \hat{\mathbf{x}})) + \mathbf{R} \mathbf{\Psi}(t, \mathbf{y}(t)) \hat{\mathbf{\Theta}}, \end{aligned} \quad (5)$$

where $\hat{\mathbf{x}} \in \mathbb{R}^n$ and $\hat{\mathbf{\Theta}} \in \mathbb{R}^p$ are the estimates of the state and the parameters of the nominal plant. Moreover, $\mathbf{L}_0 \in \mathbb{R}^{n \times z_0}$ and $\mathbf{L}_j \in \mathbb{R}^{n \times z_j}$ for $j \in \overline{1, M}$ are constant observer gains to be chosen by the designer. The adaptation law to adjust the unknown parameters is proposed as

$$\begin{aligned} \dot{\mathbf{\Theta}} &= \mathbf{\Gamma} \mathbf{\Psi}^{\top} \left( \mathbf{y}_0 - \mathbf{C}_0 \hat{\mathbf{x}} + \sum_{j=1}^{M} \mathbf{C}_j \mathbf{f}_r^j \right), \\ \mathbf{f}_r^j &= \mathbf{f}^j(\mathbf{H}_j \text{inv } \mathbf{f}^j(\mathbf{T}_j \mathbf{G}_j \tilde{\mathbf{C}}_j^{-1} \mathbf{y}_j) - \mathbf{H}_j \hat{\mathbf{x}}) \end{aligned} \quad (6)$$

where $\mathbf{\Gamma} \in \mathbb{D}_+^p$ is the matrix of adaptation gains and $\text{inv } \mathbf{f}^j$ is the inverse of $\mathbf{f}^j$ function. Term $\mathbf{f}_r^j$ corresponds to function $\mathbf{f}^j$ evaluated on the trajectories of estimation error and can be explicitly implemented due to Assumptions 1 and 2. The necessity of utilizing the inverse of $\mathbf{f}^j$ function stems from relaxed conditions on the solution of the Lyapunov equation of the linear part of the system given in the forthcoming theorem. This relaxation yields a requirement of specific matching condition imposed on the nonlinear output, which can be satisfied only by direct evaluation of the nonlinear functions on the trajectories of the state estimates in the adaptation law.

### III. STABILITY ANALYSIS

To investigate the convergence of the state estimates and identified parameters, the notion of input-to-state stability is first recalled by the following definitions from [19].

**Definition 1:** A continuous function $\alpha: \mathbb{R}_+ \to \mathbb{R}_+$ belongs to class $\mathcal{K}$ if it is strictly increasing and $\alpha(0) = 0$. It belongs to class $\mathcal{K}_\infty$ if it is unbounded.

**Definition 2:** A continuous function $\beta: \mathbb{R}_+ \times \mathbb{R}_+ \to \mathbb{R}_+$ belongs to class $\mathcal{KL}$ if for each fixed $s$, the function $\beta(\cdot, s)$ belongs to class $\mathcal{K}$ and for each fixed $r$, the function $\beta(r, \cdot)$ is decreasing with $\beta(r, s) \to 0$ as $s \to \infty$.

**Definition 3:** The system

$$\dot{\mathbf{x}}(t) = \mathbf{f}(t, \mathbf{x}(t), \mathbf{d}(t)), \quad (7)$$

where $\mathbf{x}(t) \in \mathbb{R}^n$, $\mathbf{d}(t) \in \mathbb{R}^m$, and $\mathbf{f}: \mathbb{R} \times \mathbb{R}^n \times \mathbb{R}^m \to \mathbb{R}^n$ is a continuous and locally Lipschitz function, is input-to-state stable (ISS) if there exist functions $\beta \in \mathcal{KL}$ and $\gamma \in \mathcal{K}$, such that for any time $t_0 \in \mathbb{R}$, any initial state $\mathbf{x}(t_0) \in \mathbb{R}^n$ and any essentially bounded input $\mathbf{d}$ the trajectory of the system exists and satisfies

$$\|\mathbf{x}(t)\| \le \beta(\|\mathbf{x}(t_0)\|, t - t_0) + \gamma(\|\mathbf{d}\|_{\infty}). \quad (8)$$
for any $t \ge t_0$.

**Lemma 1:** The system (7) is ISS if there exists a function $V: \mathbb{R} \times \mathbb{R}^n \to \mathbb{R}$ such that for all $t \in \mathbb{R}$, $\mathbf{x} \in \mathbb{R}^n$ and $\mathbf{d} \in \mathbb{R}^m$ it satisfies

$$ \alpha_1(\|\mathbf{x}\|) \le V(t, \mathbf{x}) \le \alpha_2(\|\mathbf{x}\|), \\ \frac{\partial V}{\partial t} + \frac{\partial V}{\partial \mathbf{x}} \mathbf{f}(t, \mathbf{x}, \mathbf{d}) \le -\alpha_3(\mathbf{x}), \quad \forall \|\mathbf{x}\| \ge \rho(\|\mathbf{d}\|), \quad (9) $$

where $\alpha_1, \alpha_2 \in \mathcal{K}_\infty$, $\rho \in \mathcal{K}$ and $\alpha_3: \mathbb{R}^n \to \mathbb{R}_+$ is a continuous positive definite function

On the basis of the recalled ISS definition, it can be stated that if the interconnected system consisting of (1) and (5) is ISS with respect to the state comprising estimation $\mathbf{e} = \mathbf{x} - \hat{\mathbf{x}}$ and identification $\tilde{\mathbf{\theta}} = \mathbf{\theta} - \hat{\mathbf{\theta}}$ errors, and inputs $\mathbf{d}$ and $\mathbf{v}$, then (5) is a robust observer of (1). To this end, the dynamics of estimation and identification errors are given by

$$ \dot{\mathbf{e}} = \mathbf{A}_0 \mathbf{e} + \sum_{j=1}^{M} \mathbf{A}_j (\mathbf{f}^j(\mathbf{H}_j \mathbf{x}) - \mathbf{f}^j(\mathbf{H}_j \hat{\mathbf{x}})) + \mathbf{R} \psi \tilde{\mathbf{\theta}} + \mathcal{D}, \\ \dot{\tilde{\mathbf{\theta}}} = -\Gamma \psi^\top \mathbf{C}_0 \mathbf{e} - \Gamma \sum_{j=1}^{M} \psi^\top \mathbf{C}_j \mathbf{f}^j(\mathbf{H}_j \mathbf{e}) - \mathbf{w}, \quad (10) $$

where $\mathbf{A}_s = \mathbf{A}_s - \mathbf{L}_s \mathbf{C}_s$ for all $s \in \overline{0, M}$, and $\mathcal{D} = \mathbf{d} - \mathbf{L}\mathbf{v}$ with $\mathbf{L} = [\mathbf{L}_0 \quad \dots \quad \mathbf{L}_M]$. Term $\mathbf{w} \in \mathbb{R}^p$ is an augmented noise dependent on $\mathbf{v}$ as well as on $\Gamma \psi^\top$, and introduced due to the use of noised outputs $\mathbf{y}$ in the adaptation law design. Unknown disturbances $\mathcal{D}$ and $\mathbf{w}$ are considered the inputs of the resultant interconnected system. Defining the nonlinear functions estimation errors $\delta \mathbf{f}^j = \mathbf{f}^j(\mathbf{H}_j \mathbf{x}) - \mathbf{f}^j(\mathbf{H}_j \hat{\mathbf{x}})$ for all $j \in \overline{1, M}$, an additional assumption is made.

**Assumption 5:** There exist $\mathbf{S}_0^j, \mathbf{S}_1^j, \mathbf{S}_2^j, \mathbf{\Sigma}_0^j, \mathbf{\Sigma}_1^j, \mathbf{\Sigma}_2^j \in \mathbb{D}_+^{k_j}$ and $\mathbf{S}_3^{j,i}, \mathbf{\Sigma}_3^{j,i} \in \mathbb{D}_+^n$ for all $j, i \in \overline{1, M}$ such that

$$ (\delta \mathbf{f}^j)^\top \delta \mathbf{f}^j \le \mathbf{e}^\top \mathbf{H}_j^\top \mathbf{S}_0^j \mathbf{H}_j \mathbf{e} + 2 \mathbf{e}^\top \mathbf{H}_j^\top \mathbf{S}_1^j \delta \mathbf{f}^j \quad (11) \\ + 2 \mathbf{e}^\top \mathbf{H}_j^\top \mathbf{S}_2^j \mathbf{f}^j(\mathbf{H}_j \mathbf{e}) + 2 \sum_{i=1}^M (\delta \mathbf{f}^j)^\top \mathbf{H}_j \mathbf{S}_3^{j,i} \mathbf{H}_j^\top \mathbf{f}^i(\mathbf{H}_1 \mathbf{e}), $$

$$ \mathbf{f}^j(\mathbf{H}_j \mathbf{e})^\top \mathbf{f}^j(\mathbf{H}_j \mathbf{e}) \le \mathbf{e}^\top \mathbf{H}_j^\top \mathbf{\Sigma}_0^j \mathbf{H}_j \mathbf{e} + 2 \mathbf{e}^\top \mathbf{H}_j^\top \mathbf{\Sigma}_1^j \delta \mathbf{f}^j \\ + 2 \mathbf{e}^\top \mathbf{H}_j^\top \mathbf{\Sigma}_2^j \mathbf{f}^j(\mathbf{H}_j \mathbf{e}) + 2 \sum_{i=1}^M (\delta \mathbf{f}^j)^\top \mathbf{H}_j \mathbf{\Sigma}_3^{j,i} \mathbf{H}_j^\top \mathbf{f}^i(\mathbf{H}_1 \mathbf{e}) \quad (12) $$

with $k$th diagonal element of matrices $\mathbf{S}_3^{j,i}, \mathbf{\Sigma}_3^{j,i}$ being non-negative if $k$th rows of matrices $\mathbf{H}_i, \mathbf{H}_j$ are codirectional, and being equal to zero otherwise.

**Assumption 5** characterizes inherent properties of functions $\mathbf{f}^j$ and imposes certain bounds on their growth rate. In particular it is satisfied by all Lipschitz continuous functions. This enables the formulation of the following theorem summarizing the properties of the proposed observer.

**Theorem 1:** Let Assumptions 1 - 5 be satisfied. If there exist $\rho \in \mathbb{R}$, $\mathbf{P} \succeq 0 \in \mathbb{R}^{n \times n}$, $\mathbf{\Lambda}^j = \text{diag}(\lambda_1^j, \dots, \lambda_{k_j}^j) \in \mathbb{D}_+^{k_j}$

for all $j \in \overline{1, M}$, $0 < \mathbf{\Xi}_0, \mathbf{\Phi} \in \mathbb{R}^{n \times n}$, $\kappa, \eta \in \mathbb{R}_+$, $\mathbf{\Gamma}_j, \mathbf{\Omega}_j \in \mathbb{D}^{k_j}$ for all $j \in \overline{1, M}$ and $\mathbf{\Upsilon}_{j,i} \in \mathbb{D}_+^n$ for all $i, j \in \overline{1, M}$ such that

$$ \mathbf{P} + \rho \sum_{j=1}^{M} \mathbf{H}_j^\top \mathbf{\Lambda}^j \mathbf{H}_j \succ 0 \quad (13) $$

and

$$ \mathbf{C}_0 = \mathbf{R}^\top \mathbf{P}, \quad \mathbf{C}_j = \mathbf{R}^\top \mathbf{H}_j^\top \mathbf{\Lambda}^j, \quad (14) $$

as well as

$$ \mathbf{\Xi}_0 - \kappa \sum_{j=1}^{M} \mathbf{H}_j^\top \mathbf{S}_0^j \mathbf{H}_j - \eta \sum_{j=1}^{M} \mathbf{H}_j^\top \mathbf{\Sigma}_0^j \mathbf{H}_j \succeq 0 \\ \mathbf{\Gamma}_j - \kappa \mathbf{S}_1^j - \eta \mathbf{\Sigma}_1^j \succeq 0 \\ \mathbf{\Omega}_j - \kappa \mathbf{S}_2^j - \eta \mathbf{\Sigma}_2^j \succeq 0 \\ \mathbf{\Upsilon}_{j,i} - \kappa \mathbf{S}_3^{j,i} - \eta \mathbf{\Sigma}_3^{j,i} \succeq 0 \quad (15) $$

for all $j, i \in \overline{1, M}$ with $k$th diagonal element of matrices $\mathbf{\Upsilon}_{j,i}$ being nonnegative if $k$th rows of matrices $\mathbf{H}_i, \mathbf{H}_j$ are codirectional, and being equal to zero otherwise, and

$$ \mathcal{Q} = \mathcal{Q}^\top \prec 0 \quad (16) $$

with $\mathcal{Q} = (\mathcal{Q}_{i,j})_{i \le 6, j \le 6}$ where

$$ \begin{aligned} \mathcal{Q}_{1,1} &= \mathbf{P} \mathbf{A}_0 + \mathbf{A}_0^\top \mathbf{P} + \mathbf{\Xi}_0 \\ \mathcal{Q}_{1,2} &= \mathcal{Q}_{2,1}^\top = (\mathbf{P} \mathbf{A}_j + \mathbf{H}_j^\top \mathbf{\Gamma}_j)_{1,j \le M}, \quad \mathcal{Q}_{2,2} = -\kappa \mathbf{I}_k, \\ \mathcal{Q}_{1,3} &= \mathcal{Q}_{3,1}^\top = (\mathbf{A}_0^\top \mathbf{H}_j^\top \mathbf{\Lambda}^j + \mathbf{H}_j^\top \mathbf{\Omega}_j)_{1,j \le M}, \\ \mathcal{Q}_{2,3} &= \mathcal{Q}_{3,2}^\top = (\mathbf{A}_i^\top \mathbf{H}_j^\top \mathbf{\Lambda}^j + \mathbf{H}_i \mathbf{\Upsilon}_{i,j} \mathbf{H}_j^\top)_{i \le M, j \le M}, \\ \mathcal{Q}_{3,3} &= -\eta \mathbf{I}_k, \quad \mathcal{Q}_{1,4} = \mathcal{Q}_{4,1}^\top = \mathbf{P}, \quad \mathcal{Q}_{2,4} = \mathcal{Q}_{4,2}^\top = \mathbf{0}_{k \times n}, \\ \mathcal{Q}_{3,4} &= \mathcal{Q}_{4,3}^\top = (\mathbf{\Lambda}^\top \mathbf{H}_i)_{i \le M, 1}, \quad \mathcal{Q}_{4,4} = -\mathbf{\Phi}, \end{aligned} \quad (17) $$

then the system (10) is ISS with respect to the modeling and identification errors, and (5) is a robust observer of (1).

*Proof:* Consider the function

$$ V_1(\mathbf{e}) = \mathbf{e}^\top \mathbf{P} \mathbf{A}_0 \mathbf{e} + 2 \sum_{j=1}^{M} \sum_{i}^{k_j} \lambda_i^j \int_0^{\mathbf{H}_j^i \mathbf{e}} f_i^j(\tau) d\tau \quad (18) $$

where $\mathbf{H}_j^i$ is the $i$th row of matrix $\mathbf{H}_j$. Recalling the definition of the nonlinearities estimation errors $\delta \mathbf{f}^j$, the derivative of $V_1$ along the trajectories of the system can be expressed as

$$ \begin{aligned} \dot{V}_1 = 2 \mathbf{e}^\top \mathbf{P} \mathbf{A}_0 \mathbf{e} + 2 \sum_{j=1}^{M} \sum_{i=1}^{k_j} f_j^i(\mathbf{H}_j \mathbf{e})^\top \mathbf{\Lambda}^j \mathbf{H}_j \mathbf{A}_0 \mathbf{e} \\ + 2 \sum_{j=1}^{M} \sum_{i=1}^{k_j} f_j^i(\mathbf{H}_j \mathbf{e})^\top \mathbf{\Lambda}^j \mathbf{H}_j \mathbf{A}_i \delta \mathbf{f}^i + 2 \mathbf{e}^\top \mathbf{P} \mathbf{R} \psi \tilde{\mathbf{\theta}} \\ + 2 \sum_{j=1}^{M} f_j^i(\mathbf{H}_j \mathbf{e})^\top \mathbf{\Lambda}^j \mathbf{H}_j \mathbf{R} \psi \tilde{\mathbf{\theta}} + 2 \mathbf{e}^\top \mathbf{P} \mathcal{D} \\ + 2 \sum_{j=1}^{M} \mathbf{e}^\top \mathbf{P} \mathbf{A}_j \delta \mathbf{f}^j + 2 \sum_{j=1}^{M} f_j^i(\mathbf{H}_j \mathbf{e})^\top \mathbf{\Lambda}^j \mathbf{H}_j \mathcal{D}. \end{aligned} \quad (19) $$
Defining now the concatenated state vector **z** = [e^T δf^T f^T D^T]^T where δf = [(δf^1)^T ... (δf^M)^T]^T and f = [f^1(H_1e)^T ... f^M(H_Me)^T]^T, as well as adding and subtracting the terms associated with constant coefficients Ξ₀, κ, η, Φ, Ωⱼ, Γⱼ, Υⱼ,i for j, i ∈ 1, M, this derivative can be expressed by

$$
\begin{equation}
\begin{aligned}
\dot{V}_1 = {}& z^\top \mathcal{Q} z - e^\top \Xi_0 e + \eta \sum_{j=1}^M \mathbf{f}^j (\mathbf{H}_j e)^\top \mathbf{f}^j (\mathbf{H}_j e) \\
& + \kappa \sum_{j=1}^M (\delta \mathbf{f}^j)^\top \delta \mathbf{f}^j - 2 \sum_{j=1}^M e^\top \mathbf{H}_j^\top \Gamma_j \delta \mathbf{f}^j \\
& - 2 \sum_{j=1}^M \sum_i e^\top \mathbf{H}_j^\top \Omega_j \mathbf{f}^j (\mathbf{H}_j e) + \mathcal{D}^\top \Phi \mathcal{D} \\
& - 2 \sum_{j=1}^M \sum_i \sum_j (\delta \mathbf{f}^j)^\top \mathbf{H}_j \boldsymbol{\gamma}_{j,i} \mathbf{H}_i^\top \mathbf{f}^i (\mathbf{H}_i e) + \Theta(t, \mathbf{e}, \tilde{\boldsymbol{\theta}}),
\end{aligned}
\tag{20}
\end{equation}
$$

where Θ(t, e, ˜θ) = 2 Σ<sub>j=1</sub><sup>M</sup> f<sup>j</sup>(H<sub>j</sub>e)<sup>T</sup>Λ<sup>j</sup>H<sub>j</sub>Rψ̃θ + 2e<sup>T</sup>PRψ̃θ and Q as defined in (16).

Consider also a function $V_2$ associated with the identification process and given in the form of

$$
V_2(t, \tilde{\theta}, e) = \tilde{\theta}^{\top} (\Gamma^{-1} + \varepsilon \mathcal{M}) \tilde{\theta} - \varepsilon e^{\top} R \psi \tilde{\theta}, \quad (21)
$$

where $\mathcal{M} \in \mathbb{R}^{p \times p}$ is the output of the filter given by
$\dot{\mathcal{M}} = -\mathcal{M} + \psi^\top \mathbf{R}^\top \mathbf{R} \psi$ with the initial condition chosen
as $\mathcal{M}(0) = \mu \mathbf{I}_p$, and $\varepsilon \in \mathbb{R}_+$ being some positive
constant. Notably, due to Assumptions 3 and 4, the matrix $\mathcal{M}$ is guaranteed to be positive definite and bounded [17],
satisfying $\mathcal{M} - \mu e^{-T_{\text{PE}}} \mathbf{I}_p > 0$. Recalling the identification
error dynamics (10) the derivative of $V_2$ is given by

$$
\begin{equation}
\begin{aligned}
\dot{V}_2 = {}& -2\tilde{\theta}^{\top} \psi^{\top} \mathbf{C}_0 e - 2\tilde{\theta} \sum_{j=1}^{M} \psi^{\top} \mathbf{C}_j \mathbf{f}^j (\mathbf{H}_j e) - 2\tilde{\theta}^{\top} \Gamma^{-1} \mathbf{w} \\
& - 2\varepsilon \tilde{\theta}^{\top} \mathcal{M} \Gamma \psi^{\top} \mathbf{C}_0 e - 2\varepsilon \sum_{j=1}^{M} \tilde{\theta}^{\top} \mathcal{M} \Gamma \psi^{\top} \mathbf{C}_j \mathbf{f}^j (\mathbf{H}_j e) \\
& - 2\varepsilon \tilde{\theta}^{\top} \mathcal{M} \mathbf{w} - \varepsilon \tilde{\theta}^{\top} \mathcal{M} \tilde{\theta} + \varepsilon e^{\top} \mathbf{R} \psi \Gamma \psi^{\top} \mathbf{C}_0 e \\
& + \varepsilon e^{\top} \mathbf{R} \psi \Gamma \sum_{j=1}^{M} \psi^{\top} \mathbf{C}_j \mathbf{f}^j (\mathbf{H}_j e) + \varepsilon e^{\top} \mathbf{R} \psi \mathbf{w} \\
& - \varepsilon e^{\top} \mathbf{R} \psi \dot{\tilde{\theta}} - \varepsilon \tilde{\theta}^{\top} \psi^{\top} \mathbf{R}^{\top} \mathcal{A}_0 e \\
& - \varepsilon \tilde{\theta}^{\top} \psi^{\top} \mathbf{R}^{\top} \sum_{j=1}^{M} \mathcal{A}_j \delta \mathbf{f}^j - \varepsilon \tilde{\theta}^{\top} \psi^{\top} \mathbf{R}^{\top} \mathcal{D}.
\end{aligned}
\tag{22}
\end{equation}
$$

Recalling now the matching condition (14) stated in the
Theorem, and defining $\mathbf{q} = [\tilde{\boldsymbol{\theta}}^{\top} \ \mathbf{w}^{\top}]^{\top}$, as well as adding
and subtracting terms associated with constant coefficient
matrix $\boldsymbol{\Psi} \succ 0 \in \mathbb{R}^{p \times p}$, this can be equivalently rewritten

as

$$
\dot{V}_2 = \mathbf{q}^\top \mathcal{G} \mathbf{q} - \varepsilon \mathbf{q}^\top \mathbf{W} \mathbf{z} + \varepsilon \mathbf{e}^\top \mathcal{V} \mathbf{z} - \Theta(t, \mathbf{e}, \tilde{\mathbf{\theta}}) + \mathbf{w}^\top \boldsymbol{\Psi} \mathbf{w}, \quad (23)
$$

where $\mathcal{G} = (\mathcal{G}_{i,j})_{j \le 2, i \le 2}$ with

$$
\mathcal{G}_{1,1} = -\varepsilon \mathcal{M}, \quad \mathcal{G}_{1,2} = \mathcal{G}_{2,1}^{\top} = -\Gamma - \varepsilon \mathcal{M}, \quad \mathcal{G}_{2,2} = -\Psi,
$$

and $\mathcal{W} = (\mathcal{W}_{i,j})_{j \le 2, i \le 4}$ with

$$
\begin{align*}
\mathcal{W}_{1,1} &= 2\mathcal{M}\Gamma\psi\mathcal{C}_0 + \dot{\psi}^\top\mathbf{R}^\top + \psi^\top\mathbf{R}^\top\mathcal{A}_0, \\
\mathcal{W}_{1,2} &= [\psi^\top\mathbf{R}^\top\mathcal{A}_1 \quad \dots \quad \psi^\top\mathbf{R}^\top\mathcal{A}_M], \\
\mathcal{W}_{1,3} &= [2\mathcal{M}\Gamma\psi^\top\mathcal{C}_1 \quad \dots \quad 2\mathcal{M}\Gamma\psi^\top\mathcal{C}_M], \\
\mathcal{W}_{1,4} &= \psi^\top\mathbf{R}^\top, \quad \mathcal{W}_{2,1} = -\psi^\top\mathbf{R}^\top, \\
\mathcal{W}_{2,2} &= \mathbf{0}_{p \times k}, \quad \mathcal{W}_{2,3} = \mathbf{0}_{p \times k}, \quad \mathcal{W}_{2,4} = \mathbf{0}_{p \times n},
\end{align*}
$$

and $\mathcal{V} = (\mathcal{V})_{1,i \le 4}$ with

$$
\begin{align*}
\mathcal{V}_1 &= \mathbf{R}\psi\Gamma\psi^\top\mathcal{C}_0, & \mathcal{V}_2 &= \mathbf{0}_{p \times k}, & \mathcal{V}_4 &= \mathbf{0}_{p \times n}, \\
\mathcal{V}_3 &= [\mathbf{R}\psi\Gamma\psi^\top\mathcal{C}_1 \quad \dots \quad \mathbf{R}\psi\Gamma\psi^\top\mathcal{C}_M].
\end{align*}
$$

Notably, for any ε there exists a positive definite matrix Ψ
ensuring that **G** ~ 0. Due to the assumed properties of **ψ**,
the matrices **W** and **V** are bounded for all *t* ≥ 0 and any
choice of the adaptation gains **Γ**.

Finally, consider the Lyapunov function candidate for the
entire system given by

$$
V(t, \mathbf{e}, \tilde{\mathbf{\theta}}) = V_1(\mathbf{e}) + V_2(t, \mathbf{e}, \tilde{\mathbf{\theta}}). \quad (24)
$$

Following Finsler's lemma, due to (13) and assumed strict
monotonicity of all f^j functions, it holds that α₁(||ẽ||) ≤
V(t, e, ˜θ) ≤ α₂(||ẽ||) with ẽ = [e^T  ˜θ^T]^T for some
α₁, α₂ ∈ K∞ if ε is taken small enough. Such selection
of ε ensures positiveness of V despite the influence of the
last term of V₂. Notably, ε is utilized in the proof only and is
not the parameter of the observer itself, and thus calculation
of its exact value is not necessary for observer synthesis.
Recalling now Assumption 5, it can be shown that

$$
\begin{equation}
\begin{aligned}
\dot{V} \le {}& \mathbf{z}^{\top} \mathbf{Q} \mathbf{z} + \mathbf{q}^{\top} \mathcal{G} \mathbf{q} - \varepsilon \mathbf{q}^{\top} \mathbf{W} \mathbf{z} + \varepsilon \mathbf{e}^{\top} \mathcal{V} \mathbf{z} \\
& - \mathbf{e}^{\top} (\Xi_0 - \kappa \sum_{j=1}^{M} \mathbf{H}_j^{\top} \mathbf{S}_0^j \mathbf{H}_j - \eta \sum_{j=1}^{M} \mathbf{H}_j^{\top} \boldsymbol{\Sigma}_0^j \mathbf{H}_j) \mathbf{e} \\
& - 2 \sum_{j=1}^{M} \mathbf{e}^{\top} \mathbf{H}_j^{\top} (\Gamma_j - \kappa \mathbf{S}_1^j - \eta \boldsymbol{\Sigma}_1^j) \delta \mathbf{f}^j \\
& - 2 \sum_{j=1}^{M} \mathbf{e}^{\top} \mathbf{H}_j^{\top} (\Omega_j - \kappa \mathbf{S}_2^j - \eta \boldsymbol{\Sigma}_2^j) \mathbf{f}^j (\mathbf{H}_j \mathbf{e}) \\
& - 2 \sum_{j=1}^{M} \sum_{i}^{M} (\delta \mathbf{f}^j)^{\top} \mathbf{H}_j (\boldsymbol{\gamma}_{j,i} - \kappa \mathbf{S}_3^{j,i} - \eta \boldsymbol{\Sigma}_3^{j,i}) \mathbf{H}_i^{\top} \mathbf{f}^i (\mathbf{H}_i \mathbf{e}) \\
& + \mathbf{D}^{\top} \Phi \mathbf{D} + \mathbf{w}^{\top} \boldsymbol{\Psi} \mathbf{w},
\end{aligned}
\tag{25}
\end{equation}
$$

If in order to attenuate the influence of the terms associated
with **W** and **V**, coefficient ε is taken sufficiently small,
then the sum of all the terms in the first line of (25) is
negative, and under conditions (15) stated in the Theorem,
the derivative of **V** satisfies $\dot{V} \le -\alpha_3 (\|\tilde{\mathbf{e}}\|)$ for all $\|\tilde{\mathbf{e}}\| \ge$
$\rho (\|\boldsymbol{\Delta}\|)$ where $\boldsymbol{\Delta} = [\mathbf{D}^{\top} \ \mathbf{w}^{\top}]^{\top}$, while $\rho \in \mathcal{K}$ and $\alpha_3$ is
a positive definite function. Using Lemma 1, this concludes the proof.

In this Theorem, the ISS properties of the system with respect to unknown inputs $w, D$ are established utilizing matrices $\Phi, \Psi$ which can be freely chosen arbitrarily large. The exact values of these matrices necessary to ensure negative definiteness of $Q, G$ for the specific considered plant define the asymptotic gains with respect to the corresponding inputs. Notably, the asymptotic gain with respect to the input $D$ is evaluated by given LMIs, what allows establishing the precise values of the elements of matrix $\Phi$. However, for the asymptotic gain with respect to the perturbation $w$ only the existence of matrix $\Psi$ is shown, and to get its bounds, additional computations are required, since the related matrix inequalities in the proof depend on time-varying components.

## IV. NUMERICAL EXAMPLE

The validity of the proposed observer is confirmed through the numerical simulation. To this end, consider a simple example of a second-order multi-output system. The plant is given in the form of (1) characterized by

$$ 
\begin{aligned}
\mathbf{A}_0 &= \begin{bmatrix} -4 & 2 \\ -1 & -7 \end{bmatrix}, \quad \mathbf{A}_1 = \begin{bmatrix} -2 & 1 \\ 1 & -5 \end{bmatrix}, \\
\mathbf{R} &= \begin{bmatrix} 1 \\ -3 \end{bmatrix}, \quad \mathbf{C}_0 = \begin{bmatrix} 3 \\ 0 \end{bmatrix}^\top, \quad \mathbf{C}_1 = \begin{bmatrix} 7 \\ 0 \end{bmatrix}^\top, \\
\mathbf{H}_1 &= \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}, \quad \mathbf{f}^1(\mathbf{x}) = \begin{bmatrix} f^1(x_1) \\ f^1(x_2) \end{bmatrix}, \quad \mathbf{\Phi} = \begin{bmatrix} 0 \\ 0 \end{bmatrix}, \\
\mathbf{\Psi}(t, \mathbf{y}(t)) &= \begin{bmatrix} \cos(0.2\pi t) \\ \sin(0.4\pi t) \end{bmatrix}^\top, \quad \mathbf{\theta} = \begin{bmatrix} 5 \\ -2 \end{bmatrix},
\end{aligned} 
\quad (26) $$

where $f^1(s) = \min(|s|, s^2)\mathbf{sgn}(s)$. For such a system, the LMI conditions of Theorem 1 are satisfied by taking

$$ \mathbf{L}_0 = \begin{bmatrix} 2.2587 \\ -2.4262 \end{bmatrix}, \mathbf{L}^1 = \begin{bmatrix} 2.7156 \\ 3.3867 \end{bmatrix}, \mathbf{\Gamma} = \begin{bmatrix} 5 & 0 \\ 0 & 5 \end{bmatrix}, \quad (27) $$

what yields

$$ 
\begin{aligned}
\mathbf{P} &= \begin{bmatrix} 7.00 & 1.33 \\ 1.33 & 0.44 \end{bmatrix}, \quad \mathbf{\Lambda}_1 = \begin{bmatrix} 7 & 0 \\ 0 & 0 \end{bmatrix}, \quad \mathbf{\Xi}_0 = \begin{bmatrix} 60.52 & 3.40 \\ 3.40 & 0.45 \end{bmatrix}, \\
\mathbf{\Phi} &= \begin{bmatrix} 5.10 & -0.06 \\ -0.06 & 5.30 \end{bmatrix} \cdot 10^7, \quad \mathbf{\Gamma}_1 = \begin{bmatrix} 2.61 & 0 \\ 0 & 0.18 \end{bmatrix} \cdot 10^4, \\
\mathbf{\Omega}_1 &= \begin{bmatrix} 2.44 & 0 \\ 0 & 0.21 \end{bmatrix} \cdot 10^4, \quad \mathbf{\Upsilon}_{1,1} = \begin{bmatrix} 2.28 & 0 \\ 0 & 2.38 \end{bmatrix} \cdot 10^7,
\endend{aligned} 
\quad (28) $$

and $\kappa_1 = 5.16 \cdot 10^7, \eta_1 = 9.07 \cdot 10^7$.

At first, the noise-free scenario with $v \equiv 0, d \equiv 0$ is considered. Figure 1 presents the results of a series of numerical trials performed with varying initial conditions of the state and parameter estimates. In all plots, the solid lines represent the trajectories of the actual plant, while the dashed ones correspond to the trajectories of the estimates in subsequent trials. Following the first series of simulations, additional trials have been conducted to validate the ISS property of the system further. In these trials, unmeasured noises and disturbances have been introduced into the system in the form of uniformly distributed values in range of

<figure><img src="image_3.png" /><figcaption>Fig. 1. Simulation results with state and parameter estimation without noises</figcaption></figure>

[-0.5, 0.5] for $d$ and [-0.25, 0.25] for $v$ inputs. The results of the simulations performed with such conditions are given in Figure 2.

The trajectories on the plots confirm the input-to-state stability of the system. Specifically, no parameter drift is exhibited despite unmeasured noises and a lack of explicit employment of solutions aimed to counteract this issue.

## V. CONCLUSION

In this paper, a novel adaptive observer has been proposed for simultaneously estimating the state and identifying the parameters of a generalized Persidskii system. The system in consideration is characterized by nonlinearities satisfying certain sector conditions and uncertain terms that are linearly parameterized. The input-to-state stability condition of the proposed observer was formulated in the form of LMIs, depending on the system's dynamics and the chosen observer tuning. Theoretical analyses and numerical trials confirmed the validity of the proposed solution and that the estimates converge to the actual trajectories of the state and parameters of the plant.

## REFERENCES

[1] R. Carroll and D. Lindorff, "An adaptive observer for single-input single-output linear systems," *IEEE Transactions on Automatic Control*, vol. 18, no. 5, pp. 428-435, 1973.

[2] G. Luders and K. Narendra, "A new canonical form for an adaptive observer," *IEEE Transactions on Automatic Control*, vol. 19, no. 2, pp. 117-119, 1974.

[3] G. Kreisselmeier, "Adaptive observers with exponential rate of convergence," *IEEE Transactions on Automatic Control*, vol. 22, no. 1, pp. 2-8, 1977.

[4] G. Besançon, "Remarks on nonlinear adaptive observer design," *Systems & control letters*, vol. 41, no. 4, pp. 271-280, 2000.

[5] G. Besançon, J. de León-Morales, and O. Huerta-Guevara, "On adaptive observers for state affine systems," *International journal of Control*, vol. 79, no. 06, pp. 581-591, 2006.

[6] R. Patelski and D. Pazderski, "Novel adaptive extended state observer for dynamic parameter identification with asymptotic convergence," *Energies*, vol. 15, no. 10, p. 3602, 2022.
<figure><img src="image_4.png" /><figcaption>Fig. 2. Simulation results with state and parameter estimation with process and measurement noises</figcaption></figure>

[7] Y. M. Cho and R. Rajamani, "A systematic approach to adaptive observer synthesis for nonlinear systems," *IEEE transactions on Automatic Control*, vol. 42, no. 4, pp. 534-537, 1997.

[8] G. Bastin and M. Gevers, "Stable adaptive observers for nonlinear time-varying systems," *IEEE Transactions on Automatic Control*, vol. 33, no. 7, pp. 650-658, 1988.

[9] M. Farza, M. M'Saad, T. Maatoug, and M. Kamoun, "Adaptive observers for nonlinearly parameterized class of nonlinear systems," *Automatica*, vol. 45, no. 10, pp. 2292-2299, 2009.

[10] Q. Zhang, "Adaptive observer for multiple-input-multiple-output (mimo) linear time-varying systems," *IEEE transactions on automatic control*, vol. 47, no. 3, pp. 525-529, 2002.

[11] K. S. Narendra and A. M. Annaswamy, "Persistent excitation in adaptive systems," *International Journal of Control*, vol. 45, no. 1, pp. 127-160, 1987.

[12] A. M. Annaswamy and A. L. Fradkov, "A historical perspective of adaptive control and learning," *Annual Reviews in Control*, vol. 52, pp. 18-41, 2021.

[13] S. K. Persidskii, "Problem of absolute stability," *Automation and Remote Control*, vol. 12, pp. 1889-1895, 1969.

[14] W. Mei, R. Ushirobira, and D. Efimov, "On nonlinear robust state estimation for generalized Persidskii systems," *Automatica*, vol. 142, p. 110411, 2022.

[15] D. R. Lima, R. Ushirobira, and D. Efimov, "State observer design for bilinear Persidskii systems," in *2024 European Control Conference (ECC)*, 2024, pp. 1741-1746.

[16] A. Loria, E. Panteley, and M. Maghenem, "Strict Lyapunov functions for model-reference adaptive control based on the Mazenc construction," in *Congreso Nacional de Control Automático (CNCA 2019). Asociación de México de Control Automático*, 2019, pp. 407-412.

[17] C. M. Verrelli and P. Tomei, "Nonanticipating Lyapunov functions for persistently excited nonlinear systems," *IEEE Transactions on Automatic Control*, vol. 65, no. 6, pp. 2634-2639, 2019.

[18] M. Fang, P.-A. Bliman, D. Efimov, and R. Ushirobira, "Nonlinear adaptive observers for an SIS system counting primo-infections," *IFAC-PapersOnLine*, vol. 56, no. 2, pp. 9727-9732, 2023.

[19] H. K. Khalil, *Nonlinear Control, 3rd Edition*. Prentice Hall Upper Saddle River, NJ, 2002.