# An Event-Triggered Robust Sampled Tracking Controller for Communication-State-Input Constrained Perturbed Unicycle Mobile Robots

Ariana Gutiérrez, Héctor Ríos, Manuel Mera, Denis Efimov, Rosane Ushirobira

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Ariana Gutiérrez, Héctor Ríos, Manuel Mera, Denis Efimov, Rosane Ushirobira. An Event-Triggered Robust Sampled Tracking Controller for Communication-State-Input Constrained Perturbed Unicycle Mobile Robots. Nonlinear Analysis: Hybrid Systems, In press. hal-05347187

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# An Event-Triggered Robust Sampled Tracking Controller for Communication-State-Input Constrained Perturbed Unicycle Mobile Robots*

Ariana Gutiérrezª, Héctor Ríosᵇ,ᶜ,*, Manuel Meraᵈ, Denis Efimovᵉ, Rosane Ushirobiraᵉ

ªUniv. Bordeaux, CNRS, Bordeaux INP, IMS, 33405, Talence cedex, France

ᵇTecnológico Nacional de México/I.T. La Laguna, 27000, Torreón, Coahuila, México

ᶜSECIHTI, Investigadoras e Investigadores por México, 03940, CDMX, México

ᵈInstituto Politécnico Nacional, ESIME-UPT, 07340, CDMX, México

ᵉInria, Université de Lille CNRS UMR 9189, F-59650, Villeneuve d'Ascq, France

## Abstract

This paper presents a robust sampled controller design for trajectory tracking in unicycle mobile robots subject to state, input, and communication constraints, as well as external disturbances such as wheel slipping. The proposed approach integrates two control strategies: an aperiodic control law based on an event-triggered mechanism and a periodic control law using a constant sampled state-feedback controller. The event-triggered control employs the attractive ellipsoid method and the barrier Lyapunov function to ensure performance within a defined safe set, where state constraints are preserved, and a switching set determines the active control strategy. The periodic sampled control considers the maximum sampling time required to achieve trajectory tracking while minimizing bandwidth usage. This strategy guarantees input-to-state stability of the tracking error dynamics against multiplicative external disturbances. Additionally, a straightforward method using linear matrix inequalities is provided to compute the controller gains. Experimental results validate the effectiveness of the proposed robust control approach.

**Keywords:** Unicycle Mobile Robots, Sampled Control, State and Input Constraints, Communication Constraints.

## 1. Introduction

Unicycle mobile robots (UMRs) have been widely studied due to their capability of moving freely in the workspace and the wide variety of possible real-world applications (see, e.g., [1] and [2]). However, as real systems, it is well known that UMRs have to operate in confined spaces

*This work was supported in part by the SEP-CONACYT-ANUIES-ECOS NORD Project 315597 and ECOS NORD Project M20M04. The work of Ariana Gutiérrez was supported by SECIHTI CVU 932774. 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. The work of Manuel Mera was supported by Project IPN-SIP under Grant 20250187.

*Corresponding Author: Héctor Ríos

Email addresses: ariana.gutierrez-ortega@u-bordeaux.fr (Ariana Gutiérrez), hriosb@lalaguna.tecnm.mx (Héctor Ríos), mmerah@ipn.mx (Manuel Mera), denis.efimov@inria.fr (Denis Efimov), rosane.ushirobira@inria.fr (Rosane Ushirobira)
and have energy limitations, i.e., they must deal with state and input constraints (see, e.g., [3] and [4]). Additionally, even when disturbed forces cannot be considered in the kinematic model, skidding and slipping effects on the wheels [5] or corrupt control signals must be taken into account for the controller design (see, e.g. [6], [7], [8], [9], and [10]). In most cases, the UMRs are managed through a digital platform with limited bandwidth, so it is necessary to restrict the frequency of control input updates to save communication resources.

Regarding the control design for state and/or input constrained UMRs, in [11], the stabilization problem of a UMR considering input saturation is solved. A discrete-time sliding-mode control is proposed. The finite-time convergence of the trajectories of the UMR to the origin is guaranteed. In [12], a bounded controller is proposed for a global path-tracking control of a UMR. The system trajectories converge to the desired reference asymptotically. On the other hand, model predictive control (MPC) is a widely used technique for the trajectory tracking problem in state and input-constrained UMRs. For instance, in [13], a robust MPC scheme is proposed for trajectory tracking in a UMR with input constraints and affected by additive disturbances. This algorithm provides input-to-state stability (ISS) for tracking error dynamics. In [14], an event-based MPC is proposed to solve the trajectory tracking problem in UMRs with additive disturbances. The ISS of the system dynamics is ensured. In [15], an MPC is proposed for the stabilization problem of a UMR with input and state constraints. The exponential convergence of the system trajectories to zero is guaranteed. In the previous works, high computational power was needed due to the nature of the MPC implementation. Moreover, the additive disturbances considered in [13] and [14] do not have physical meaning in the UMR kinematic model since such additive signals do not accurately reflect the skidding and slipping effects on the wheels. In order to relax the computational burden, in [16], the authors proposed a robust trajectory tracking strategy for UMRs, which deals with slipping effects, state, and input constraints. The proposed strategy is based on an integral sliding-mode control approach together with a switching interval predictor-based state-feedback controller through an MPC scheme. The switching structure allows to only activate the MPC far from the origin of the tracking error, and then, close to the origin, an interval predictor-based state-feedback controller is used to take the tracking error to zero despite the slipping effects and the state/input constraints. Nevertheless, it is worth highlighting that the previous works do not take into account the communication constraints, i.e., the limited amount of communication resources.

Concerning the control design for UMRs under communication constraints (e.g., when the digital platform of the UMR imposes limitations on the control update time); in [17], an adaptive controller is proposed to solve the trajectory tracking problem for wheeled mobile robots with input delay and skidding effects. The proposed algorithm guarantees the uniformly ultimately boundness of the closed-loop system. A well-known technique to deal with communication constraints is the event-triggered control (see, e.g., [18] and [19]). Its main characteristic is that the control actions are updated only when certain well-defined events occur, resulting in an aperiodic sampled control [20]. In [21], the global regulation problem of an UMR is solved using a discontinuous and bounded control law, ensuring a global solution to the stabilization problem. An event-triggered control scheme is implemented to optimize communication resource usage. However, disturbances are not considered in the analysis, and only simulation results are presented. In [22], an event-triggered control scheme for UMR. The proposed algorithm employs the Euler-Lagrange method to model the system dynamics at the roll and pitch axes. Based on this model, the proposed control strategy ensures finite-time stability of the tracking error. However, in [21] and [22], external disturbances are not considered in the analysis, and only simulation results are presented. In [23], an event-triggered controller
is proposed for the trajectory tracking problem. However, a linearization technique is needed to ensure the exponential stability of the tracking error dynamics. In [24], the stabilization problem of time-varying trajectories for UMRs is addressed using event-triggered controllers. The proposed control mechanism ensures practical convergence toward reference trajectories while guaranteeing a uniform minimum time between the control inputs updates. In [25], a multivariable event-triggered generalized super-twisting sliding-mode algorithm is proposed for a UMR safe navigation. This algorithm ensures finite-time convergence to zero of the tracking errors. Yet, the previous works do not take into account skidding/slipping effects or state/input constraints.

In this paper, considering significant external disturbances (e.g., wheel skidding and slipping), as well as system and communication limitations, we propose a robust sampled control strategy. The goal is to solve the trajectory-tracking problem in perturbed UMRs while considering state and input constraints. The presented control approach integrates an aperiodic sampled controller and a periodic one:

* The aperiodic control law is developed using the attractive ellipsoid method and the barrier Lyapunov function (BLF) approach, enabling the effective handling of state and input constraints. Additionally, a safe set is characterized to ensure that the state constraints are respected, along with a switching set that defines the regions in which each control component operates.

* The periodic sampled control is designed using a Lyapunov-Krasovskii function approach. This controller accounts for the maximum sampling time required to ensure successful trajectory tracking while minimizing bandwidth usage.

The proposed switching control law optimizes communication resource usage by updating the aperiodic control component only when necessary, ensuring that the trajectory tracking error dynamics converge to an invariant set. Within this set, the periodic control maintains ISS properties for the trajectory tracking error in the presence of external disturbances. Moreover, the controller synthesis is constructive, relying on linear matrix inequalities (LMIs). Experimental results using the QBot2 by Quanser validate the effectiveness of the proposed robust control strategy.

A preliminary version of this work has been presented in [26]. However, in [26], a detailed stability analysis is not provided. In addition, the current version considers more complex conditions in the design of the control and also experimental results are presented demonstrating the effectiveness of the proposed algorithm.

The rest of the paper is organized as follows. The problem statement is given in Section 2. The proposed controller is presented in Section 5. In Section 6, we provide some experimental results. Concluding remarks are formulated in Section 7.

**Notation:** For $p \in \mathbb{N}$, we denote by $\bar{p}$ the sequence of natural numbers $1, \dots, p$. Denote the functions $\sin(\theta)$, $\cos(\theta)$, and $\sinc(\theta)$ as $s(\theta)$, $c(\theta)$, and $sc(\theta)$, respectively. The absolute value of a real number is represented as $|\cdot|$, while the Euclidean norm of a vector $z \in \mathbb{R}^n$ is denoted as $||z||$, and for a matrix $A \in \mathbb{R}^{m \times n}$, the induced spectral norm is used $||A|| = \sqrt{\lambda_{\max}(A^\top A)}$, where $\lambda_{\max}$ is the largest eigenvalue of $A^\top A$. The set of functions $u: \mathbb{R}_{\ge 0} \to \mathbb{R}^p$ such that their $L_\infty$-norm on $[0, \infty]$ is finite, i.e., $||u||_\infty := \ess\sup_{t \ge 0} ||u(t)|| < \infty$, is denoted as $\mathcal{L}_\infty$. For a symmetric matrix $P \in \mathbb{R}^{n \times n}$, the notation $P < 0$ ($P \ge 0$) means that $P$ is negative (nonnegative) definite. A matrix $A \in \mathbb{R}^{n \times n}$ is called Metzler when all its non-diagonal elements are nonnegative. The term He$(A)$
denotes $A+A^\top$, for $A \in \mathbb{R}^{n \times n}$. The set $\mathcal{E}(R) := \{z \in \mathbb{R}^n : z^\top R z \le 1\}$ is an ellipsoid centered at the origin characterized by a matrix $0 < R^\top = R \in \mathbb{R}^{n \times n}$.

## 2. Problem Statement

Consider the perturbed kinematic model of a UMR (see Fig. 1):

$$ \begin{pmatrix} \dot{\theta}(t) \\ \dot{x}(t) \\ \dot{y}(t) \end{pmatrix} = \begin{pmatrix} \bar{d}_1(t) & 0 \\ 0 & \bar{d}_2(t)c(\theta(t)) \\ 0 & \bar{d}_2(t)s(\theta(t)) \end{pmatrix} u(t), \quad (1a) $$

$$ u(t) = u(t_k), \forall t \in [t_k, t_{k+1}), \quad (1b) $$

where $x(t) \in \mathbb{R}$ and $y(t) \in \mathbb{R}$ denote the midpoint between the wheels and $\theta(t) \in \mathbb{R}$ represents the orientation angle of the UMR. The vector $u(t) = (\omega(t) \quad v(t))^\top \in \mathbb{R}^2$ include the sampled control inputs, which are applied at each time $t_k$, for all $k \in N$; the terms $v(t)$ and $\omega(t)$ are the linear and angular velocities of the UMR, respectively. The sampling instants $t_k$ are monotonously increasing, so that $\lim_{k \to \infty} t_k = +\infty$, and $h(t) := t_{k+1} - t_k > h_{\min}$, where $h_{\min} > 0$ is the minimum sampling interval, and $t_0 = 0$. Note that $h_{\min}$ is the minimum sampling interval permissible by the UMR platform, and the sampling instants $t_k$ will be determined by the proposed sampled controller. The terms $\bar{d}_1(t) = 1 + d_1(t)$ and $\bar{d}_2(t) = 1 + d_2(t)$, where $d_1(t)$ and $d_2(t)$ denote certain time-varying perturbations, which multiply the inputs and that may arise from unmodeled kinematics effects proportional to the control inputs, such as slipping of the wheels [5]. We assume that the time-varying perturbations $d_i(t)$ are unknown but bounded, i.e., $-d_{\max} < d_i(t) \le d_{\max} < 1$, for $i = 1, 2$, with a known positive constant $d_{\max} > 0$. The requirement $d_i(t) > -1$ ensures that the perturbations do not cause a sign change in the control inputs.

<figure><img src="image_3.png" /><figcaption>Figure 1: Schematic Diagram of the Perturbed UMR</figcaption></figure>
The aim of this work is to design a trajectory-tracking sampled controller for the UMR affected by multiplicative perturbations and taking into account communication constraints, i.e., a minimum sampling time $h_{min}$ in which the control signal can be sent to the UMR, as well as state and input constraints, i.e., $x(t) \in \mathcal{X}^* := [x_{min}, x_{max}] \subset \mathbb{R}$, $y(t) \in \mathcal{Y}^* := [y_{min}, y_{max}] \subset \mathbb{R}$, $v(t) \in \mathcal{V}^* := [-v_{max}, v_{max}] \subset \mathbb{R}$, and $\omega(t) \in \mathcal{W}^* := [-\omega_{max}, \omega_{max}] \subset \mathbb{R}$, for all $t \ge t_0$.

### 3. Preliminaries

In real-world scenarios, saturation functions are applied to each element of $u = (u_1 \dots u_p)^\top \in \mathbb{R}^p$, i.e., for each $j = \overline{1, p}$

$$ \sigma_j(u_j) = \begin{cases} u_{\max j}, & \text{if } u_{\max j} \le u_j, \\ u_j, & \text{if } -u_{\max j} < u_j < u_{\max j}, \\ -u_{\max j}, & \text{if } u_j \le -u_{\max j}, \end{cases} \qquad (2) $$

and hence, $\sigma(u) := (\sigma_1(u_1) \dots \sigma_p(u_p))^\top$, where $u_{\max j} > 0$ is the maximum amplitude value that $u_j$ can take. Due to its structure, there exists a linear region, where $\sigma(u) = u$, which is characterized by the set $\mathcal{U} := \{u \in \mathbb{R}^p | -u_{\max j} \le u_j \le u_{\max j}, j = \overline{1, p}\}$. Define the function $\phi_j: \mathbb{R} \to \mathbb{R}$ as $\phi_j(u_j) := \sigma_j(u_j) - u_j$ and $\phi(u) := (\phi_1(u_1) \dots \phi_p(u_p))^\top$.

**Lemma 1.** [27]. If $\bar{\alpha} - \bar{\beta}$ is an element of the set $\mathcal{U}$, then the function $\phi$ satisfies $\phi^\top(\bar{\alpha})\Delta[\phi(\bar{\alpha}) + \bar{\beta}] \le 0$, where $\Delta = \text{diag}(\delta_1 \dots \delta_p)$, for any $\delta_j > 0$, $j = \overline{1, p}$.

### 4. Trajectory Tracking Error Dynamics

Before starting with the controller design, it is necessary to define the tracking errors as follows

$$ e_1(t) = \theta_d(t) - \theta(t), \qquad (3a) $$

$$ e_2(t) = c(\theta(t))\bar{x}(t) + s(\theta(t))\bar{y}(t), \qquad (3b) $$

$$ e_3(t) = c(\theta(t))\bar{y}(t) - s(\theta(t))\bar{x}(t), \qquad (3c) $$

with $\bar{x}(t) = x_d(t) - x(t)$ and $\bar{y}(t) = y_d(t) - y(t)$, where $x_d(t), y_d(t)$, and $\theta_d(t)$ are derived from a reference kinematic model for the UMR, i.e.,

$$ \dot{\theta}_d(t) = \omega_d(t), \qquad (4a) $$

$$ \dot{x}_d(t) = c(\theta_d(t))v_d(t), \qquad (4b) $$

$$ \dot{y}_d(t) = s(\theta_d(t))v_d(t), \qquad (4c) $$

where $v_d(t)$ and $\omega_d(t)$ represent the linear and angular reference velocities, respectively. These inputs are considered continuous and are bounded by some positive constants $v_d$, $\bar{v}_d$, and $\bar{\omega}_d$, i.e., $0 < v_d < v_d(t) \le \bar{v}_d$, and $\|\omega_d\|_\infty \le \bar{\omega}_d$, and such that $v_d(t) \in \mathcal{V}$ and $\omega_d(t) \in \mathcal{W}$, for all $t \ge 0$. Additionally, the reference model trajectories hold the state constraints, i.e., $x_d(t) \in \mathcal{X}$ and $y_d(t) \in \mathcal{Y}$, for all $t \ge 0$.
Thus, the dynamics of the tracking error is given as

$$ \dot{\epsilon}_1(t) = -\omega(t)d_1(t) + \tau_1(t), \quad (5a) $$

$$ \dot{\epsilon}_2(t) = \bar{d}_1(t)\omega(t)e_3(t) - v(t)d_2(t) + \tau_2(t), \quad (5b) $$

$$ \dot{\epsilon}_3(t) = -\bar{d}_1(t)\omega(t)e_2(t) + v_d(t)s(e_1(t)), \quad (5c) $$

with the following new control inputs

$$ \tau_1(t) = \omega_d(t) - \omega(t), \quad (6a) $$

$$ \tau_2(t) = v_d(t)c(e_1(t)) - v(t). \quad (6b) $$

Subsequently, the tracking error dynamics (5) can be rewritten as in [16], i.e., is given in the following form

$$ \dot{e}(t) = A(\rho_1)e(t) + B[\tau(t) + F(\rho_2)d(t)], \quad (7) $$

with the vectors $e(t) = (e_1(t) \ e_2(t) \ e_3(t))^{\top} \in \mathbb{R}^3$, $\tau(t) = (\tau_1(t) \ \tau_2(t))^{\top} \in \mathbb{R}^2$, $d(t) = (d_1(t) \ d_2(t))^{\top} \in \mathbb{R}^2$, and the following matrices

$$ A(\rho_1) = \begin{pmatrix} 0 & 0 & 0 \\ 0 & 0 & \bar{d}_1(t)\omega(t) \\ v_d(t)\mathrm{sc}(e_1(t)) & -\bar{d}_1(t)\omega(t) & 0 \end{pmatrix}, $$

$$ B = \begin{pmatrix} 1 & 0 \\ 0 & 1 \\ 0 & 0 \end{pmatrix}, \quad F(\rho_2) = \begin{pmatrix} -\omega(t) & 0 \\ 0 & -v(t) \end{pmatrix}, $$

where $\rho_1 = (v_d(t)\mathrm{sc}(e_1(t)) \ \bar{d}_1(t)\omega(t))^{\top} \in \mathbb{R}^2$ and $\rho_2 = (v(t) \ \omega(t))^{\top} \in \mathbb{R}^2$ as the unknown and known scheduling parameters, respectively. Then, it is possible to rewrite the state and input constraint sets as follows:

$$ \mathcal{E} = \{e \in \mathbb{R}^3 : (e_1, e_2, e_3) \in \mathbb{R} \times [-\bar{x}\bar{y}, \bar{x}\bar{y}] \times [-\bar{x}\bar{y}, \bar{x}\bar{y}]\}, \quad (8) $$

$$ \mathcal{W} = \{ \tau \in \mathbb{R}^2 : (\tau_1, \tau_2) \in [-\bar{\tau}_1, \bar{\tau}_1] \times [-\bar{\tau}_2, \bar{\tau}_2] \}, \quad (9) $$

where $\bar{x}\bar{y} = (x_{\max} - x_{\min}) + (y_{\max} - y_{\min})$, $\bar{\tau}_1 = \bar{\omega}_d + \omega_{\max}$, and $\bar{\tau}_2 = \bar{v}_d + v_{\max}$. Note that there exist a Metzler matrix $A_0 \in \mathbb{R}^{3 \times 3}$, matrices $A_j \in \mathbb{R}^{3 \times 3}$, and $F_j \in \mathbb{R}^{2 \times 2}$, for $j = 1, 4$, such that<fnref n="1" />

$$ A(\rho_1) = A_0 + \sum_{j=1}^{4} \alpha_j(\rho_1)A_j, \quad F(\rho_2) = \sum_{j=1}^{4} \alpha_j(\rho_2)F_j, \quad (10a) $$

$$ \sum_{j=1}^{4} \alpha_j(\rho_1) = \sum_{j=1}^{4} \alpha_j(\rho_2) = 1, \quad \alpha_j(\rho_1), \alpha_j(\rho_2) \in [0, 1], \quad (10b) $$

hold for (7). Thus

$$ \dot{e}(t) = \left[ A_0 + \sum_{j=1}^{4} \alpha_j(\rho_1)A_j \right] e(t) + B \left[ \tau(t) + \sum_{j=1}^{4} \alpha_j(\rho_2)F_j d(t) \right]. \quad (11) $$

<footnote n="1">¹The matrices $A_j$ can be obtained by a convex polytopic approach [28] using $\omega_{\max}$, $d_{\max}$, $v_d$, and $\bar{v}_d$.</footnote>
Our goal is now to design a robust sampled control law $\tau$ that ensures that the trajectories of the system (11) converge to zero, despite the presence of the external disturbances $d$, taking into account the communication, state, and input constraints, i.e., $h(t) > h_{\min}$, $e(t) \in \mathcal{E}$, and $\tau(t) \in \mathcal{U}$, for all $t \ge 0$.

# 5. Robust Control Design

The proposed controller takes the following form:

$$ \tau(t) = \begin{cases} \tilde{\tau}(t), & \text{if } e(t) \notin \mathcal{E}(P), \\ \hat{\tau}(t), & \text{if } e(t) \in \mathcal{E}(P), \end{cases} \qquad (12) $$

where $\tilde{\tau}$ is an event-triggered controller, $\hat{\tau}$ is a periodic state-feedback controller, and the ellipsoid $\mathcal{E}(P)$, defined further on, is the switching and invariant set for the closed-loop system.

The proposed control strategy involves a switching control law that commutes between two signals: a periodic control signal $\hat{\tau}$ and an aperiodic control signal $\tilde{\tau}$. The switching is determined by the region in which the trajectory of the tracking error resides. Specifically, within a safe set $\mathcal{E}(R)$, introduced further on, an aperiodic event-triggered mechanism is employed to preserve state constraints and optimize control efforts. Outside this region, a periodic constant-sampled state-feedback controller is activated to ensure trajectory tracking while minimizing bandwidth usage. In the following, we will describe each part of the proposed controller.

## 5.1. Aperiodic Control Law

The event-triggered control design consists of two steps. The first one is devoted to designing a robust linear state-feedback controller for the continuous time system (11), dealing with the state and input constraints, which provides the safe and switching sets. The second step consists of designing an event-triggered strategy based on the continuous linear state-feedback controller, dealing with the communication constraints.

### 5.1.1. Robust Linear State-Feedback Control Design

Due to the nature of (11), it can be verified that the pair $(A_0, B)$ is controllable, and then, a linear state-feedback gain exists. Then, the following non-sampled control law is proposed

$$ \tilde{\tau}(t) = \sigma(\tau_0(t)), \qquad (13) $$

with $\tau_0(t) = Ke(t)$, where $K \in \mathbb{R}^{2 \times 3}$ is a feedback gain, such that the state and input constraints are not transgressed.

The following lemma provides a safe set $\mathcal{E}(R)$, the switching set $\mathcal{E}(P)$, and a way to design $K$.

**Lemma 2.** Let the control (13) be applied to the system (11), i.e., $\tau(t) = \tilde{\tau}(t)$. Suppose that there exist a positive-definite matrix $X_1 = X_1^\top \in \mathbb{R}^{3 \times 3}$, matrices $Y \in \mathbb{R}^{2 \times 3}$, $Z = (Z_1^\top \ Z_2^\top) \in \mathbb{R}^{2 \times 3}$,
$M \in \mathbb{R}^{3 \times 3}$, $\Delta = \text{diag}(\delta_1, \delta_2)$, with $\delta_1, \delta_2 > 0$, and a constant $\gamma > 0$, such that the following LMIs

$$ \chi_1 = \begin{pmatrix} \chi_{11} & B\Delta - Y^\top + Z^\top & X_1BF_{\max} \\ \star & -2\Delta & 0 \\ \star & \star & -\gamma Q_d \end{pmatrix} \le 0, \quad (14a) $$

$$ \begin{gathered} \chi_{11} = \text{He}\{X_1 A_\psi\} + BY + Y^T B^\top + M, \\ A_\psi = A_0 + A_{\max} I_3, \end{gathered} $$

$$ \chi_{2i} = \begin{pmatrix} X_1 & Z_i^\top \\ Z_i & \bar{\tau}_i^2 \end{pmatrix} \ge 0, \quad i = 1, 2, \quad (14b) $$

$$ \chi_3 = \begin{pmatrix} M & \gamma X_1 \\ \gamma X_1 & \gamma X_2 \end{pmatrix} > 0, \quad (14c) $$

$$ X_2 < X_1, \quad (14d) $$

$$ \chi_{4j} = \bar{b}_j^T X_1 \bar{b}_j \le 1, \quad j = \overline{1, 4}, \quad (14e) $$

with $\bar{b}_1 = (0, \overline{xy}^{-1}, 0)^\top$, $\bar{b}_2 = -\bar{b}_1$, $\bar{b}_3 = (0, 0, \overline{xy}^{-1})^\top$, $\bar{b}_4 = -\bar{b}_3$, are feasible for given constants $\overline{xy}$, $0 < d_{\max} < 1$, $F_{\max} = d_{\max} \max(v_{\max}, \omega_{\max})$, and $A_{\max} = \sum_{j=1}^4 \alpha^* ||A_j||$, where $\alpha^* = \arg \max_{\alpha \in \Gamma} \sum_{j=1}^4 \alpha_j ||A_j||$, and $\Gamma \subset \mathbb{R}^4$ the set of all convex weight vectors $\alpha = (\alpha_1 \dots \alpha_4)^\top \in \mathbb{R}^4$ such that (10) holds. If $e(0) \in \mathcal{E}(R)$, with $R = X_1^{-1}$, and $K = YR$, then the ellipsoid $\mathcal{E}(P)$, with $P = X_2^{-1}$, is asymptotically attractive while $\mathcal{E}(R)$ is invariant for the system (11).

*Proof:* Consider the function $\phi(\tau_0(t)) = \sigma(\tau_0(t)) - \tau_0(t)$, thus the closed-loop dynamics, considering the system (11) and the controller (13), i.e., $\tau(t) = \sigma(\tau_0(t))$, is given as

$$ \dot{e}(t) = (A_0 + \tilde{A} + BK)e(t) + B\phi(\tau_0(t)) + B\tilde{F}d(t), \quad (15) $$

where $\tilde{A} = \sum_{j=1}^4 \alpha_j(\rho_1)A_j$ and $\tilde{F} = \sum_{j=1}^4 \alpha_j(\rho_2)F_j$. Let us take into account the following BLF candidate

$$ V(t) = \ln \left( \frac{1}{1 - e^{\top}(t)Re(t)} \right), $$

where $0 < R = R^T \in \mathbb{R}^{3 \times 3}$ is a matrix that parametrizes the ellipsoid $\mathcal{E}(R)$ containing the initial conditions. Then, the time-derivative of $V$, along the trajectories of system (15), is given by

$$ \dot{V}(t) = 2S_e(t)e^{\top}(t)R[(A_0 + \tilde{A} + BK)e(t) + B\phi(\tau_0(t)) + B\tilde{F}d(t)], $$

where $S_e(t) = 1/(1 - e^{\top}(t)Re(t))$. Introducing the extended vector $\eta^{\top}(t) := (e^{\top}(t) \phi^{\top}(t) d^{\top}(t))^{\top}$, the time-derivative of $V$ can be rewritten as follows

$$ \dot{V}(t) = S_e(t) \eta^{\top}(t) \begin{pmatrix} \text{He}(A_k) & RB & B\tilde{F} \\ \star & 0 & 0 \\ \star & \star & 0 \end{pmatrix} \eta(t), $$

where $A_k = R(A_0 + \tilde{A} + BK)$. Considering that $||\tilde{A}|| \le A_{\max}$ and $||\tilde{F}|| < F_{\max}$, adding and subtracting the term $\gamma S_e(t) d^{\top}(t) Q_d d(t)$, with $Q_d$ such that $d^{\top}(t) Q_d d(t) \le 1$, i.e., $\lambda_{\max}(Q_d) = 0.5 d_{\max}^2$, then $\dot{V}$ can be upper bounded by

$$ \dot{V}(t) \le S_e(t) \left( \gamma + \eta^{\top}(t) \begin{pmatrix} \text{He}(\bar{A}_k) & RB & BF_{\max} \\ \star & 0 & 0 \\ \star & \star & -\gamma Q_d \end{pmatrix} \eta(t) \right), $$
where $\bar{A}_k = R(A_0 + A_{\max}I_3 + BK)$. According to Lemma 1, with $\bar{\alpha} = Ke$, $\bar{\beta} = Ke - G_0e$, and $G_0 \in \mathbb{R}^{1 \times 3}$, the inequality $\phi^\top \Delta^{-1} [\phi + (K - G_0)e] \le 0$ is satisfied, where $\Delta^{-1} = \text{diag}(\delta_1^{-1}, \delta_2^{-1})$, with $\delta_1, \delta_2 > 0$. In order to ensure that $G_0e(t)$ belongs to the set $\mathcal{U}$, noting that $e^\top(t)Re(t) < 1$, it is sufficient that $\|G_0e(t)\|^2 \le \bar{\tau}_i^2 e^\top(t)Re(t)$, for $i = 1, 2$, holds. Using the Schur's complement lemma, the previous inequality can be written as in LMI (14b), with $Z = R^{-1}G_0$. Adding and subtracting the term $\gamma S_e(t)e^\top(t)Pe(t)$, we have that $\dot{V}$ can be upper bounded as follows

$$ \dot{V}(t) \le \gamma S_e(t)[1 - e^\top(t)Pe(t)] + S_e(t)\eta^\top(t)\Sigma\eta(t), $$

with

$$ \Sigma = \begin{pmatrix} \text{He}(\bar{A}_k) + \gamma P & RB - \Delta^{-1} K_G^\top & BF_{\max} \\ \star & -2\Delta^{-1} & 0 \\ \star & \star & -\gamma Q_d \end{pmatrix}, $$

with $K_G = K - G_0$. Pre- and post-multiplying $\Sigma$ by $T = \text{diag}(R^{-1}, \Delta, I)$, it follows that $\Sigma_1 = T\Sigma T \le 0$ is equivalent to $\chi_1 \le 0$, with the change of variables $X_1 = R^{-1}$, $X_2 = P^{-1}$, $Y = KR^{-1}$, and $Z = R^{-1}G_0$, and introducing an additional variable $M \in \mathbb{R}^{3 \times 3}$ with constraints $M > \gamma R^{-1}PR^{-1}$, applying the Schur's complement lemma, one obtains the LMI (14c).

Hence, if $\chi_1 \le 0$, then

$$ \dot{V}(t) \le \gamma S_e(t)[1 - e^\top(t)Pe(t)]. $$

Note that, for all $e^\top(t)Pe(t) > 1$ and $e^\top(t)Re(t) < 1$, we have that $\dot{V}(e(t)) \le 0$; thus, any solution of system (15), staring in $\mathcal{E}(R)$, remains in $\mathcal{E}(R)$ and converges asymptotically to $\mathcal{E}(P)$, which implies that $\mathcal{E}(R)$ is invariant. Therefore, if $P > R$, i.e., the LMI (14d) holds, then $\mathcal{E}(P)$ is contained into $\mathcal{E}(R)$. To ensure that $\mathcal{E}(R)$ is contained in the state-constrained set (8), it is sufficient that condition (14e) is satisfied. $\square$

### 5.1.2. Event-Triggered Control Design

Due to the communication restrictions, it is necessary to implement the controller (13) in a sampled way, i.e.,

$$ \tilde{\tau}(t) = \sigma(\tau_0(t_k)), \forall t \in [t_k, t_{k+1}), k \in \mathbb{N}. \qquad (16) $$

The following lemma provides a way to implement the event-triggered controller.

**Lemma 3.** Let the control (16) be applied to the system (11), i.e., $\tau(t) = \tilde{\tau}(t)$, with the controller parameters designed according to Lemma 2, i.e., $K = YR$. Suppose that there exist positive-definite matrices $\tilde{P}_2 = \tilde{P}_2^\top \in \mathbb{R}^{3 \times 3}$, $\tilde{P}_3 = \tilde{P}_3^\top \in \mathbb{R}^{3 \times 3}$, $\tilde{R}_1 = \tilde{R}_1^\top \in \mathbb{R}^3$, and $\tilde{S}_1 = \tilde{S}_1^\top \in \mathbb{R}^{3 \times 3}$, such that the following LMI

$$ \Theta = \begin{pmatrix} \Theta_{11} & \Theta_{12} & \Theta_{13} & \tilde{P}_2^\top BF_{\max} \\ \star & \Theta_{22} & \tilde{P}_3^\top BK & \tilde{P}_3^\top BF_{\max} \\ \star & \star & \Theta_{33} & 0 \\ \star & \star & \star & -\gamma_2 I \end{pmatrix} \le 0, \qquad (17) $$

$$ \Theta_{11} = \text{He}\{\tilde{P}_2 A_\psi\} + \tilde{S}_1 + 2e^{-2\kappa_1 h_{\min}} \tilde{R}_1, \quad \Theta_{12} = R - \tilde{P}_2^\top + \tilde{P}_3 A_\psi, $$

$$ \Theta_{13} = e^{-2\kappa_1 h_{\min}} \tilde{R}_1 + \tilde{P}_2^\top BK, \quad \Theta_{22} = -2\tilde{P}_3^\top + h_{\min}^2 \tilde{R}_1, $$

$$ \Theta_{33} = -e^{-2\kappa_1 h_{\min}} (\tilde{R}_1 + \tilde{S}_1), $$
is feasible for K and R given as in Lemma 2, some $\gamma_2 > 0$, $\kappa_1 > 0$, and $h_{\min} > 0$. Thus, if the event-triggered control law is designed as

$$ \tilde{\tau}(t) = \begin{cases} \sigma(\tau_0(t)), & t = t_{k+1}, & \text{if } W(t, \tau_0(t_k)) > \rho W(t, \tau_0(t)) \\ & & \text{&} \\ \sigma(\tau_0(t_k)), & t = t_k, & \text{if } W(t, \tau_0(t_k)) > \rho W(t, \tau_0(t)) \\ & & \text{&} \\ & & \text{or } W(t, \tau_0(t_k)) \le \rho W(t, \tau_0(t)), & \end{cases} \quad (18) $$

for all $t \in [t_k, t_{k+1})$, where $W(t, \tau_0) = S_e(t)e^\top(t)RB\sigma(\tau_0)$, $S_e(t) = 1/(1 - e^\top(t)Re(t))$, $\rho > 0$, and $e(0) \in \mathcal{E}(R) - \mathcal{E}(R_s)$, with $R_s = \varepsilon^{-1}I_3$ and

$$ \varepsilon = \|e^{(A_0+A_{\max}I)h_{\min}}\|_{\lambda_{\max}^{-1/2}(R)} + h_{\min}\varepsilon_1\varepsilon_2, \quad (19) $$

where $\varepsilon_1 = \|K\|\lambda_{\max}^{-1/2}(R) + \sqrt{2}F_{\max}d_{\max}$ and $\varepsilon_2 = \sup_{0 \le s \le h_{\min}} \|e^{(A_0+A_{\max}I)(h_{\min}-s)}\|$, then the ellipsoid $\mathcal{E}(P)$, with P given as in Lemma 2, is asymptotically attractive while $\mathcal{E}(R)$ is invariant for the system (11) provided that

$$ \gamma_2 d_{\max}^2 \le 2\kappa_1. \quad (20) $$

*Proof:* Based on the proof of Lemma 2, we have that

$$ \dot{V}(t) = 2S_e(t)e^\top(t)R[(A_0 + \tilde{A})e(t) + B\sigma(\tau_0(t)) + B\tilde{F}d(t)]. $$

It has been shown that V is a BLF, then it is decreasing. Therefore, we can introduce the following event-triggered condition

$$ \dot{V}(e(t), \tau_0(t_k)) \le \rho \dot{V}(e(t), \tau_0(t)), \quad (21) $$

with some $\rho > 0$ whose satisfaction guarantees that V keeps its decay property. Note that (21) is equivalent to the following inequality

$$ W(t, \tau_0(t_k)) \le \rho W(t, \tau_0(t)). \quad (22) $$

Let us analyze two scenarios provided by the condition (22), i.e., when the control input needs to be updated and when it needs to remain constant. For the first scenario, i.e., when the condition (22) is no longer satisfied, i.e., $W(t, \tau_0(t_k)) > \rho W(t, \tau_0(t))$, this indicates that $W(t, \tau_0(t_k))$ stops decreasing at the rate $\rho W(t, \tau_0(t))$. This implies that the control input needs to be updated to ensure that the function $W(t, \tau_0(t_k))$ decreases again with rate $\rho W(t, \tau_0(t))$.

For the second scenario, there are two cases to analyze. In the first case, if $W(e(t), \tau_0(t_k)) \le \rho W(e(t), \tau_0(t))$, i.e., $\tilde{\tau}(t) = \tilde{\tau}(t_k)$, for all $t \in [t_k, t_{k+1})$, the control signal remains constant. In the second case, since the minimum sampling time is considered, an additional condition must be imposed on the event-triggered structure to allow the control signal to be updated. Then, we need to show that the trajectories will not violate the state constraints for the case when (22) does not hold and $h(t) < h_{\min}$, i.e., when the control input will remain constant until $h(t) = h_{\min}$. Since Lemma 2 already provides convergence conditions when the saturated control is used, we must analyze the case when the control is constant out of the saturated zone, i.e., in the linear zone $\tau_0(t_k) = Ke(t_k)$. The case when the control passes from the saturated zone to the nonsaturated one, while $h(t) < h_{\min}$, can be treated similarly, and this case is omitted for brevity. Thus, let us
take into account the closed-loop dynamics, considering the system (11) and the controller (16), i.e., $\tilde{\tau}(t) = \tau_0(t_k)$, for all $t \in [t_k, t_{k+1})$, $k \in \mathbb{N}$:

$$ \dot{e}(t) = (A_0 + \tilde{A})e(t) + BKe(t_k) + B\tilde{F}d(t), \quad (23) $$

where $K$ is given as in Lemma 2 and $h(t) = h_{\min}$, i.e., the worst case for which the constraints are not transgressed. The solution of system (23), taking into account the condition $e^\top(t_k)Re(t_k) \le 1$, satisfies $\|e(t)\| \le \varepsilon$, with $\varepsilon$ given in (19), for all $t \in [t_k, t_{k+1}]$. Then, we define the set $\mathcal{E}(R_s)$, with $R_s = \varepsilon^{-1}I_3$, as a threshold set. Note that the set $\mathcal{E}(R_s)$ considers a threshold region inside to $\mathcal{E}(R)$ that ensures the states do not violate the constraints, even when the control input is constant until $h(t) = h_{\min}$. Now, based on [29], let us consider the following Lyapunov-Krasovskii functional

$$ V_2(t) = e^\top(t)Re(t) + \int_{t-h_{\min}}^{t} e^{2\kappa_1(s-t)}e^\top(s)\tilde{S}_1e(s)ds \\ + h_{\min} \int_{-h_{\min}}^{0} \int_{t+\theta}^{t} e^{2\kappa_1(s-t)}\dot{e}^\top(s)\tilde{R}_1\dot{e}(s)dsd\theta, \quad (24) $$

where $\kappa_1 > 0$, $0 < \tilde{R}_1 = \tilde{R}_1^\top \in \mathbb{R}^{3 \times 3}$, $0 < \tilde{S}_1 = \tilde{S}_1^\top \in \mathbb{R}^{3 \times 3}$, and $R$ is given as in Lemma 2. Note that for this analysis the values of $K$ and $R$ are fixed. The time-derivative of $V_2$, along the trajectories of system (23), satisfies

$$ \begin{aligned} \dot{V}_2(t) + 2\kappa_1 V_2(t) - \gamma_2 d^\top(t) d(t) \le 2e^\top(t)Re(t) + 2\kappa_1 e^\top(t)Re(t) + h_{\min}^2 \dot{e}(t)\tilde{R}_1\dot{e}(t) \\ + e^\top(t)\tilde{S}_1e(t) - \gamma_2 d^\top(t) d(t) \\ - e^{-2\kappa_1 h_{\min}}[e(t) - e(t - h_{\min})]^\top\tilde{R}_1[e(t) - e(t - h_{\min})] \\ - e^{-2\kappa_1 h_{\min}}[e^\top(t - h_{\min})\tilde{S}_1e(t - h_{\min})] \\ + 2[e^\top(t)\tilde{P}_2^\top + \dot{e}^\top(t)\tilde{P}_3^\top] [(A_0 + \tilde{A})e(t) \\ + B\tilde{F}d(t) + BKe(t - h_{\min}) - \dot{e}(t)]. \end{aligned} \quad (25) $$

Then, following the same procedure as in [29], we obtain the LMI (17) in order to ensure that $\dot{V}_2(t) + 2\kappa_1V_2(t) - \gamma_2d^\top(t)d(t) < 0$. Based on Proposition 4.3 in [29], due to $e^\top(t)Re(t) \le V_2$, the system (23) is ISS with respect to $d$, and this implies the asymptotic convergence of $e$ to the ellipsoid $\mathcal{E}(\tilde{R}_c)$, with $\tilde{R}_c = 2\kappa_1\gamma_2^{-1}d_{\max}^{-2}R$. Thus, it is necessary to ensure that the convergence region $\mathcal{E}(\tilde{R}_c)$ is completely contained in $\mathcal{E}(R)$, then $R < \tilde{R}_c$, i.e., the condition (20) must be fulfilled. Thus, if $h(t) < h_{\min}$ the control signal can remain constant until $h(t) = h_{\min}$, and the system trajectories remain in $\mathcal{E}(R)$ if $e(0) \in \mathcal{E}(R) - \mathcal{E}(R_s)$.

Therefore, we have that $\tilde{\tau}(t) = \sigma(\tau_0(t))$, at the instant when

$$ W_1(e(t), \tau_0(t_k)) > \rho W_2(e(t), \tau_0(t)) \text{ and } h(t) \ge h_{\min}, $$

i.e., the control input is updated, while it remains constant if

$$ W_1(e(t), \tau_0(t_k)) \le \rho W_2(e(t), \tau_0(t)), $$

i.e., $\tilde{\tau}(t) = \tilde{\tau}(t_k)$, for all $t \in [t_k, t_{k+1})$. Finally, due to the nature of the event-triggered control, the same switching behaviour occurs until the trajectories of the system (11) asymptotically converge to the ellipsoid $\mathcal{E}(P)$. $\square$
**Remark 1.** Since the triggering condition is defined by the function W, and the implementation explicitly incorporates a minimum allowable sampling time $h_{\min}$, as shown in (18), the occurrence of Zeno behavior is avoided. The parameter $h_{\min}$ is determined by the specifications of the platform and ensures that events cannot be triggered arbitrarily close in time. Additionally, the function $\sigma$ introduces saturation in the control input, which limits the rate of change in the control signal and contributes to the regularity of event occurrences.

The Algorithm 1 provides a process to obtain the control gain K.

**Algorithm 1:**

Input: $\gamma$, $F_{\max}$, $A_{\max}$, $\tilde{b}_j$ with $j = \overline{1,4}$;

Output: $K, P, R$;

01: $v > 0$;

02: $\gamma = \gamma_{\min}$;

03: **while** $\chi_1 > 0 \land \chi_{2_i} < 0 \land \chi_3 < 0 \land X_2 > X_1 \land \chi_{4_j} > 1$;

04: $\gamma = \gamma + v$;

05: **end**

06: $R = X_1^{-1}$

07: $P = X_2^{-1}$

08: $K = YR$

Note that the positive constant $v$ is an auxiliary parameter used to tune $\gamma$ in such way that a solution of (14) can be obtained. It is possible to start with a small initial value of $\gamma$ and gradually increase it as needed. Then, Algorithm 2 offers a way to implement the event-triggered controller and the process to obtain the control gains.

**Algorithm 2:**

Input: $h_{\min}$, $R$;

Output: $\tilde{\tau}(t)$;

01: $h(t) = t - t_k$;

02: $k = 0$;

03: **if** $W(t, \tau_0(t_k)) > \rho W(t, \tau_0(t)) \land h(t) \ge h_{\min}$;

04: $\tilde{\tau}(t) = \sigma(\tau_0(t))$;

05: $k = k + 1$;

06: **else**

07: $\tilde{\tau}(t) = \sigma(\tau_0(t_k))$;

08: **end**

Note that the parameter $\rho$ represents a desired rate for the function $W(t, \tau_0(t))$, which helps to update the control input when is required. The control (13) ensures that the tracking error converges asymptotically to $\mathcal{E}(P)$. Once in this region, the control law commutes to a periodic controller, which takes into account a maximum sampling time. In the following section, a way to design this controller is proposed.

## 5.2. Periodic Sampled Controller

Let us consider the state–feedback control law

$$ \hat{\tau}(t) = K_2 e(t_k), \forall t \in [t_k, t_{k+1}), k \in \mathbb{N}, \quad (26) $$
with $h(t) = t_{k+1} - t_k = \bar{h}$, with some $\bar{h} > h_{\min} > 0$.

The following lemma provides a way to design $K_2$.

**Lemma 4.** Let the control (26) be applied to the system (11), i.e., $\tau(t) = \hat{\tau}(t)$. Suppose that there exist some positive-definite matrices $P_1 = P_1^\top \in \mathbb{R}^{3 \times 3}$, $P_2 = P_2^\top \in \mathbb{R}^{3 \times 3}$, $R_1 = R_1^\top \in \mathbb{R}^{3 \times 3}$, $S = S^\top \in \mathbb{R}^{3 \times 3}$, and $Y \in \mathbb{R}^{2 \times 3}$, such that the following LMI

$$ \Psi_1 = \begin{pmatrix} \psi_{11} & \psi_{12} & \psi_{13} & BF_{\max} \\ * & \psi_{22} & \varepsilon BY & \varepsilon BF_{\max} \\ * & * & \psi_{33} & 0 \\ * & * & * & -\gamma_3 I \end{pmatrix} \le 0, \quad (27) $$

$$ \psi_{11} = P_2^\top A_\psi^\top + A_\psi P_2 + 2\kappa_2 S - e^{-2\kappa_2 \bar{h}} R_1, $$

$$ \psi_{12} = P_1 - P_2 + \varepsilon A_\psi P_2, \quad \psi_{13} = e^{-2\kappa_2 \bar{h}} R_1 + BY, $$

$$ \psi_{22} = -2\varepsilon P_2 + \bar{h}^2 R_1, \quad \psi_{33} = -e^{-2\kappa_2 \bar{h}}(R_1 + S), $$

holds for some $\varepsilon, \kappa_2, \gamma_3 > 0$, $\bar{h} > h_{\min} > 0$, and $K_2$ is designed as $K_2 = YP_2$, then the system (11) is ISS, with respect to $d$, and $e(t) \in \mathcal{E}(P_c)$, with $P_c = 2\kappa_2\gamma_3^2 d_{\max}^2 P_2^{-1} P_1 P_2^{-1}$. Moreover, if

$$ X_2^{-1} \le P_c, \quad (28) $$

holds for some $d_{\max} > 0$, and $X_2$ given as in Lemma 2, then $\mathcal{E}(P_c) \subset \mathcal{E}(P)$.

*Proof:* The closed-loop system dynamics, taking into account (11) and (26), i.e., $\tau(t) = \hat{\tau}(t)$, is given as

$$ \dot{e}(t) = (A_0 + \tilde{A})e(t) + BK_2e(t_k) + B\tilde{F}d(t), \quad (29) $$

with $t \in [t_k, t_{k+1})$. Let us propose, based on [29], the following Lyapunov-Krasovskii functional

$$ V_3(t) = e^{\top}(t)\bar{P}_1e(t) + \int_{t-\bar{h}}^{t} e^{2\kappa_2(s-t)}e^{\top}(s)\bar{S}e(s)ds + \bar{h}\int_{-\bar{h}}^{0}\int_{t+\theta}^{t} e^{2\kappa_2(s-t)}\dot{e}^{\top}(s)\bar{R}_1\dot{e}(s)dsd\theta, \quad (30) $$

where $\kappa_2 > 0$, $0 < \bar{P}_1 = \bar{P}_1^\top \in \mathbb{R}^{3 \times 3}$, $0 < \bar{R}_1 = \bar{R}_1^\top \in \mathbb{R}^{3 \times 3}$, and $0 < \bar{S} = \bar{S}^\top \in \mathbb{R}^{3 \times 3}$. Then, the time-derivative of $V_3$, along the trajectories of system (29), is given in a similar way as in (25).

Following the same procedure as in [29], we obtain the LMI (27) in order to ensure that $\dot{V}_3(t) + 2\kappa_2V_3(t) - \gamma_3d^\top(t)d(t) < 0$. Based on Proposition 4.3 in [29], the system (29) is ISS with respect to $d$, and this implies the asymptotic convergence of $e$ to the ellipsoid $\mathcal{E}(P_c)$, with $P_c = 2\kappa_2\gamma_3d_{\max}^{-2}\bar{P}_1$. Finally, since the control (26) is only active inside the set $\mathcal{E}(P)$, it is necessary to ensure that the convergence region $\mathcal{E}(P_c)$ is completely contained in $\mathcal{E}(P)$. With this aim, $P < P_c$, i.e., the LMI (28) must be fulfilled. $\square$

The Algorithm 3 provides a way to implement the constant sampled controller and to obtain $\bar{h}$ through the bisection method, where $h_{\max} > 0$ is an auxiliary tuning variable.
**Algorithm 3:**

Input: $\varepsilon < 1$, $h_{\min}$, $h_{\max} > h_{\min}$, $X_2$;

Output: $\tilde{h}$, $K_2$, $P_1$, $P_2$, $R_1$, $S$, $P_c$;

01: $h_{\min b} = h_{\min}$;

02: $h_b = (h_{\min b} + h_{\max})/2$;

03: **while** $h_b > 2\varepsilon h_{\min}$;

04: $h_b = (h_{\min b} + h_{\max})/2$;

05: $\tilde{h} = h_b$;

06: **if** $\Psi_1 \le 0 \land X_2^{-1} \le P_c$;

07: $h_{\max} = h_b$;

08: **else**

09: $h_{\min b} = h_b$;

10: **end**

12: **end**

13: $\tilde{h} = h_b$.

Note that (28) is used to verify that the convergence region, given when the control (26) is applied, is completely contained in $\mathcal{E}(P)$. If this condition is not fulfilled, the parameter $\gamma_3$ can be modified to recalculate $K_2$ such that the $\mathcal{E}(P_c)$ contained in $\mathcal{E}(P)$. Finally, considering the results of lemmas 2, 3, and 4, the main result is presented.

**Theorem 1.** Let the control (12) be applied to the system (11), with (18) and (26), where the controller parameters are designed following the steps below:

1) The control (13), i.e., $\tilde{\tau}(t) = \sigma(\tau_0(t))$, is designed according to Lemma 2, the safe set $\mathcal{E}(R)$ and the switching set $\mathcal{E}(P)$ are obtained.

2) Using the results obtained in Lemma 2, it is possible to obtain (16), i.e., $\tilde{\tau} = \sigma(\tau_0(t_k))$, according to Lemma 3 and the set $\mathcal{E}(R_s)$ is obtained.

3) The control (26), i.e., $\hat{\tau}(t)$, is designed according to Lemma 4.

If $e(0) \in \mathcal{E}(R) - \mathcal{E}(R_s)$, then the system (11) is ISS with respect to $d$.

*Proof:* The result is derived from the combination of the preceding lemmas since it follows directly from the results established earlier.

**6. Experimental Results**

The proposed robust strategy is applied to the QBot2 manufactured by Quanser (see Fig. 2). The QBot2 operates with a control station and a real-time control software QUARC®. This software creates a direct interface with MATLAB® Simulink® and provides a sampling time equal to 0.001 [s]. The QBot2 position and orientation is measured and tracked using internal sensors of the robot. Both wheels have encoders that gather information about the wheel rotation and estimate the changes in position over time. Using this information, we can calculate the total displacement and orientation angle through the UMR kinematics.
<figure><img src="image_4.png" /><figcaption>Figure 2: QBot2 by Quanser</figcaption></figure>

The desired trajectory is given by

$$ 
\begin{aligned}
\omega_d(t) &= \frac{\dot{x}_d(t)\ddot{y}_d(t) - \dot{y}_d(t)\ddot{x}_d(t)}{\dot{x}_d^2(t) + \dot{y}_d^2(t)}, \\
v_d &= \sqrt{\dot{x}_d^2(t) + \dot{y}_d^2(t)}, \\
x_d(t) &= \cos(\omega_0 t), \quad y_d(t) = \sin(2\omega_0 t), \\
\theta_d(t) &= \int_0^t \omega_d(\tau)d\tau,
\end{aligned}
$$

with $\omega_0 = 0.15$. Thus, we have that $\overline{\omega}_d = 1.1663$, $\underline{v}_d = 0.1392$, and $\overline{v}_d = 0.4472$. The initial conditions are $x_0 = 0.5[m]$, $y_0 = -1.4[m]$, and $\theta_0 = 0[\text{rad}]$, thus, $e_1(0) = 1.5708$, $e_2(0) = 0.2$, and $e_3(0) = 1.4$. The state and input constraints sets are given as $\mathcal{X} = [-1.5, 1.5]$, $\mathcal{Y} = [-1.5, 1.5]$, $\mathcal{V} = [-0.7, 0.7]$, $\mathcal{W} = [-3, 3]$, and thus, $\overline{\mathcal{X}} = 6$, $\overline{\tau}_1 = 4.1663$, and $\overline{\tau}_2 = 1.4472$. Note that $e(0) \in \mathcal{E}$ and $\tau(0) \in \mathcal{W}$. The disturbances are taken as $d_1(t) = 0.01 \cos(t) + 0.01$ and $d_2(t) = 0.02 \sin(t) + 0.01$, thus $d_{\max} = 0.03$. These disturbances are added by software and can represent the slipping effect on the wheels. Note that the UMR platform already possesses intrinsic disturbances such as noise in the measurements.

It is possible to fix the matrix $A_0$ as

$$ A_0 = \begin{pmatrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 1 & 0 & 0 \end{pmatrix}, $$

and then, the matrices $A_j$ can be obtained by a convex polytopic approach [28] using $\omega_{\max}$, $d_{\max}$, $\underline{v}_d$, and $\overline{v}_d$, i.e.,

$$ 
\begin{aligned}
A_1 &= \begin{pmatrix} 0 & 0 & 0 \\ 0 & 0 & 5.4 \\ -5.4 & 5.4 & 0 \end{pmatrix}, \quad A_2 = \begin{pmatrix} 0 & 0 & 0 \\ 0 & 0 & 5.4 \\ 5.4 & -5.4 & 0 \end{pmatrix}, \\
A_3 &= \begin{pmatrix} 0 & 0 & 0 \\ 0 & 0 & -5.4 \\ 5.4 & 5.4 & 0 \end{pmatrix}, \quad A_4 = \begin{pmatrix} 0 & 0 & 0 \\ 0 & 0 & -5.4 \\ -5.4 & -5.4 & 0 \end{pmatrix}.
\end{aligned}
$$
Therefore, it is obtained that $F_{max} = 1.65$ and $A_{max} = 0.5261$. To compute the control gains for the aperiodic control law, let us apply the statements of Lemma 2. Consider $\delta_1 = 1.5039$, $\delta_2 = 1.2501$, $\gamma = 0.2$, $\tau_{1max} = 4.1663$, and $\tau_{2max} = 1.1472$, then the following feasible solution is obtained:

$$ R = \begin{pmatrix} 0.33 & 0.00 & 0.14 \\ 0.00 & 0.37 & 0.00 \\ 0.14 & 0.00 & 0.32 \end{pmatrix}, P = \begin{pmatrix} 0.62 & -0.01 & 0.27 \\ -0.01 & 0.66 & -0.01 \\ 0.27 & -0.01 & 0.56 \end{pmatrix}, \\ K = \begin{pmatrix} -1.6165 & 0.0069 & -1.2105 \\ 0.0080 & -0.9458 & 0.0022 \end{pmatrix}. $$

Therefore it is easy to verify that the LMI (17) is satisfied, hence, the event-triggered control can be applied with $\rho = 1$.

To compute the control gains for the periodic control law, let us apply the statements of Lemma 4. Taking into account that $h_{min} = 0.001[s]$ (the minimum sampling time that the platform can provide), $\gamma_2 = 0.2$, $\epsilon = 0.5$, then, the following feasible solution is obtained:

$$ P_1 = \begin{pmatrix} 0.57 & -0.01 & -0.23 \\ -0.01 & 0.75 & -0.01 \\ -0.23 & -0.01 & 0.81 \end{pmatrix}, P_2 = \begin{pmatrix} 0.61 & 0.00 & -0.19 \\ 0.00 & 0.78 & -0.01 \\ -0.19 & -0.01 & 0.77 \end{pmatrix}, \\ K_2 = \begin{pmatrix} -2.0962 & 0.0110 & -0.7544 \\ 0.0211 & -1.9829 & 0.0094 \end{pmatrix}, \bar{h} = 0.2. $$

Additionally, we have that $\epsilon = 0.1401$ and thus $R_s = 7.1352I_3$. The system trajectories are depicted in Fig. 3, which shows that they converge to a close region of the desired reference despite the external disturbances. Note that the trajectories $x$ and $y$ never transgress the state constraints. The control signals $\omega$ and $v$ are depicted in Fig. 4. Observe that both control signals, $\omega$ and $v$ get saturated in some instants and then remain inside the linear region. The projection of the ellipsoids $\mathcal{E}(R)$, $\mathcal{E}(P)$, $\mathcal{E}(P_c)$, and $(\mathcal{E}(R) - \mathcal{E}(R_s))$ in the plane $e_2-e_3$ are depicted in Fig. 5. We can see that the trajectories of the tracking error start in $(\mathcal{E}(R) - \mathcal{E}(R_s))$, remain inside $\mathcal{E}(R)$, then converge to $\mathcal{E}(P)$, and finally remain inside $\mathcal{E}(P_c)$. The projection at the coordinate $e_1$ is not shown since $e_1$ does not have constraints. Then, it is evident that the trajectory of $e_1$ does not transgress the constraints given in (9). Finally, in Fig. 6, it is observed that the norm of the errors converges to a region close to the origin<fnref n="2" />.

## 7. Conclusions

In this paper, we presented the design of a robust controller to address the trajectory tracking problem in constrained UMRs. Our proposed control scheme integrated an aperiodic sampled controller and a periodic one. The aperiodic event-triggered controller accounts for state and input constraints, providing a safe set to ensure the state constraints are not violated, as well as a switching set to define the regions where each control component is active. The periodic sampled control determined the maximum sampling time needed to guarantee trajectory tracking while minimizing bandwidth usage. This approach ensured the ISS properties of the trajectory tracking

<footnote n="2">²A YouTube video, showing these experimental results may be watched at the following link https://youtu.be/nm0rbps_1-s?si=jxlSaqVrQpb07Fpm</footnote>
<figure><img src="image_5.png" /><figcaption>Figure 3: System Trajectories</figcaption></figure>

error dynamics in response to external disturbances. Experimental results using the QBot2 by Quanser demonstrate the effectiveness of the proposed robust control strategy.

An interesting avenue for future work is the implementation of this strategy in cloud-controlled systems, where communication constraints become more severe. In such scenarios, latency and data transmission reliability play crucial roles in ensuring effective trajectory tracking
<figure><img src="image_6.png" /><figcaption>Figure 4: Control Signals</figcaption></figure>

<figure><img src="image_7.png" /><figcaption>Figure 5: Tracking Error and Ellipsoids</figcaption></figure>
<figure><img src="image_8.png" /><figcaption>Figure 6: Tracking Error Norm, where $e_x = x_d - x$ and $e_y = y_d - y$</figcaption></figure>

and system stability. Investigating optimization techniques for control signal scheduling under limited bandwidth conditions would be essential to maintaining robust performance in cloud-based environments. Furthermore, the impact of delayed control signals on system behavior could be explored in greater depth, as delays are common in practical applications and may significantly affect control performance. Addressing this aspect through delay-tolerant strategies or predictive mechanisms represents a promising direction for future work.

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