# An Event-Triggered Robust Control for Unicycle Mobile Robots with Communication, State, and Input Constraints

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

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To cite this version:

Ariana Gutiérrez, Hector Ríos, Manuel Mera, Denis Efimov, Rosane Ushirobira. An Event-Triggered Robust Control for Unicycle Mobile Robots with Communication, State, and Input Constraints. CDC 2024 - Conference on Decision and Control, Dec 2024, Milan, Italy. hal-04895366

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https://inria.hal.science/hal-04895366v1

Submitted on 18 Jan 2025

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# An Event-Triggered Robust Control for Unicycle Mobile Robots with Communication, State, and Input Constraints

Ariana Gutiérrez†, Héctor Ríos†,*, Manuel Mera‡, Denis Efimov§,*, and Rosane Ushirobira§

**Abstract**—This paper contributes to designing a sampled robust controller for the trajectory tracking problem in constrained unicycle mobile robots. The proposed controller comprises the design of an aperiodic control law based on an event-triggered controller and a periodic control law based on a constant sampled state-feedback controller. The state-feedback-based event-triggered control is designed by means of the attractive ellipsoid method and the barrier Lyapunov function and provides a safe set, where the state constraints are not transgressed, and a switching set that defines the region where each part of the controller is active. The periodic sampled control part is designed taking into account a maximum sampling time and is active inside the switching set. The proposed strategy ensures the input-to-state stability of the tracking error dynamics with respect to external disturbances. A constructive and simple method based on linear matrix inequalities is proposed to compute the controller gains. Simulation results illustrate the feasibility of the proposed approach.

***Index Terms***—Unicycle Mobile Robots, Sampled Control, State Constraints.

## I. INTRODUCTION

UNICYCLE mobile robots (UMRs) have been widely studied due to their capability of moving freely from one point to another and the broad diversity of possible real applications (see, e.g., [1] and [2]). However, as real system, it is well-known that UMRs have to move in restricted workspaces and have energy limitations, i.e., they must deal with state and input constraints (see, e.g., [3] and [4]). In most cases, a digital platform is used to control the UMRs and then there is limited bandwidth, requiring limiting the amount of control input updates to save communication resources. To deal with this issue, a well-known technique is the event-triggered control (see, e.g., [5], [6], and [7]). Its main characteristic is that the control actions are updated

only when certain well-defined events occur, implying an aperiodic sampling time [8].

Concerning the control design for input-constrained UMRs, in [9], the stabilization problem of a UMR subject to actuator saturation is solved. A discrete-time sliding-mode control is proposed based on the equivalent control approach. The finite-time convergence of the trajectories of the UMR to the origin is guaranteed, considering input saturation. In [10], a bounded controller is proposed with a predetermined bound for a global path tracking control of UMR. The system trajectories converge to the desired reference asymptotically. In [11], based on a modified dynamic feedback linearization technique, saturated sampled-data controllers are proposed for UMRs, whose state is subject to sampling errors. Nevertheless, in [10] and [11], external disturbances are not taken into account. 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 [12], 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, guaranteeing that the constraints are never violated. In [13], an MPC is proposed for the stabilization problem of a UMR with input and state constraints. The kinematics needs to be transformed into a chained form guaranteeing exponential convergence to zero of the trajectories of the system. In previous works, high computational power was needed due to the nature of MPC implementation. Additionally, most of the previous works do not consider external disturbances, or when they are considered, these have no physical meaning for the UMR, and the communication constraints are not taken into account.

Regarding the control design for UMRs considering communications constraints, in [14], an event-triggered controller is proposed for the trajectory tracking problem. A linearization technique is needed to ensure the exponential stability of the tracking error dynamics. In [15], an event-triggered controller for stabilizing the trajectories of UMRs is proposed. The approach involves synthesizing the controller, deriving a triggering condition, and ensuring a minimum time between transmissions. However, the previous works do not

<footnote>†Tecnológico Nacional de México/I.T. La Laguna, 27000, Torreón, Coahuila, México. Emails: m.agutierrezo@ correo.itlalaguna.edu.mx and hriosb@lalaguna.tecnm.mx</footnote>

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

<footnote>‡Instituto Politécnico Nacional, ESIME-UPT, 07340, Ciudad de México, Mexico. Email: mmerah@ipn.mx
§Inria-Lille Nord Europe, Université de Lille CNRS, UMR 9189-CRISTAL, F-59000 Lille, France. Emails: denis.efimov@inria.fr and rosane.ushirobira@inria.fr</footnote>

<footnote>*ITMO University, 197101, Saint Petersburg, Russia.</footnote>
take into account state or input constraints, and only the unperturbed case is considered. In contrast, a self-triggered MPC strategy is presented for nonholonomic vehicles with coupled input constraints and bounded disturbances in [16].

In this paper, motivated by the above-mentioned issues, i.e., meaningful external disturbances, system and communications constraints, and complexity, a robust switched sampled control strategy is proposed to solve the trajectory tracking problem in perturbed UMRs with communication, state, and input constraints. The proposed control scheme is composed of an aperiodic sampled controller and a periodic sampled controller. The proposed sampled control is based on a state-feedback event-triggered controller designed through the attractive ellipsoid method (AEM) and the barrier Lyapunov function (BLF). The periodic sampled control part is based on a state-feedback controller designed using a Lyapunov-Krasovskii function approach. The characterization of a safe set, where the state constraints are not transgressed, is provided, as well as a switching set that defines the region where each part of the control is active. The proposed approach ensures the ISS properties of the trajectory tracking error dynamics with respect to external disturbances. In addition, the synthesis of the controller is constructive since it is based on linear matrix inequalities (LMIs).

The rest of the paper is organized as follows. The problem statement is given in Section II. The proposed controller is presented in Section V. In Section VI, we provide some simulation results. Concluding remarks are formulated in Section VII.

**Notation:** Denote the trigonometric functions $\sin(\theta)$, $\cos(\theta)$, and $\sinc(\theta)$ as $s(\theta)$, $c(\theta)$, and $sc(\theta)$, respectively. Let us denote a sequence of integers $1, ..., n$ as $\mathbb{1}, n$, for any $n \in \mathbb{N}$. The absolute value of a real number is represented as $|\cdot|$ while the Euclidian 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 the symmetric matrix $A^\top A$. The set of functions $u: \mathbb{R}_{\ge 0} \to \mathbb{R}^p$ such that its $L_\infty$-norm on $[0, \infty]$ is finite, i.e., $||u||_\infty := \esssup_{t \ge 0} ||u(t)|| < \infty$, 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 $\text{He}(A)$ denotes $A + A^\top$, for $A \in \mathbb{R}^{n \times n}$. The set $\mathcal{E}(R) := \{x(t) \in \mathbb{R}^n : x^\top(t)Rx(t) \le 1\}$ is an ellipsoid centered at the origin characterized by a matrix $0 < R^\top = R \in \mathbb{R}^{n \times n}$.

## II. PROBLEM STATEMENT

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

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

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

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 $u(t) = (\omega(t) \ v(t))^\top$ represents the sampled control inputs, at each time $t_k$, for all $k \in \mathbb{N}$; where the terms $v(t)$ and $\omega(t)$ contain the linear and angular velocities of the UMR. The sampling instants $t_k$ are monotonously increasing, such that $\lim_{k \to \infty} t_k = +\infty$, and $h(t) := t_{k+1} - t_k \in [h_{\min}, h_{\max}]$, where $h_{\min}, h_{\max} > 0$ are the minimum and maximum sampling intervals, respectively; and $t_0 = 0$. Note that $h_{\min}$ is the minimum sampling interval allowed by the UMR platform and the sampling instants $t_k$ will be determined by the proposed sampled controller. Note that $\bar{d}_1(t) = 1+d_1(t)$ and $\bar{d}_2(t) = 1+d_2(t)$, where the terms $d_1(t)$ and $d_2(t)$ represent some time-varying perturbations, which are multiplicative to the inputs and that may come from unmodeled kinematics phenomena proportional to the control inputs, such as skidding and slipping of the wheels. It is assumed that such time-varying perturbations $d_i(t)$ are unknown but bounded, i.e., $-1 < d_i(t) \le d_{\max} < 1$, for $i = 1, 2$, with a known positive constant $d_{\max}$. Note that the constraint $d_i(t) > -1$ ensures that the perturbations do not cause a change of sign 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 control for the UMR affected by some multiplicative perturbations and taking into account some communication constraints, i.e., a minimum sampling time $h_{\min}$ in which the control signal can be sent to the UMR; 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 $t \ge t_0$, for some given sets $\mathcal{X}, \mathcal{Y}, \mathcal{V}$, and $\mathcal{W}$.

## III. PRELIMINARIES

Consider the following system

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

where $x(t) \in \mathbb{R}^n$ is the state and $f: \mathbb{R}^n \to \mathbb{R}^n$ is a continuous function with respect to $x$.
**Definition 1.** [17]. Let $\mathcal{D} \subset \mathbb{R}^n$ be an open set with boundary $\partial\mathcal{D}$, and $V: \mathcal{D} \to \mathbb{R}_+$ be a continuously differentiable positive definite function. If $V(x(t)) \le b$ for all $t \ge 0$ along the solution of the system (2), for some $b \in \mathbb{R}_+$ and any $x(0) \in \mathcal{D}$, and $\lim_{x \to \partial\mathcal{D}^-} V(x) \to +\infty$, then $V$ is a barrier Lyapunov function (BLF).

Note that if $\dot{V} \le 0$, then $b = V(x(0))$, and any trajectory will be inside $\mathcal{D}$. For the case of multiple-inputs, the saturation function is applied to each element of the input vector $u \in \mathbb{R}^p$, i.e.,

$$ \sigma_j(u_j) = \begin{cases} u_{\max_j}, & \text{if } u_{\max_j} \le u_j, \\ u_j, & \text{if } -u_{\max_j} < u_j < u_{\max_j}, \\ -u_{\max_j}, & \text{if } u_j \le -u_{\max_j}. \end{cases} \quad (3) $$

and hence, $\sigma(u) := (\sigma_1(u_1) \dots \sigma_p(u_p))^\top$, where $u_j$ is the $j$-th element of the vector $u$ and $u_{\max_j} > 0$ is the maximum amplitude value that $u_j$ can take. Therefore, 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 $\phi(u) := (\phi_1(u_1) \dots \phi_p(u_p))^\top$. Let us introduce the following lemma.

**Lemma 1.** [18]. If $\bar{\alpha} - \bar{\beta}$ is an element of $\mathcal{U}$, then the function $\phi(\alpha)$ satisfies $\phi^\top(\bar{\alpha})\Delta[\phi(\bar{\alpha}) + \bar{\beta}] \le 0$, with $\Delta = \text{diag}(\delta_1 \dots \delta_p)$, for any $\delta_j > 0$, $j = \overline{1,p}$.

## IV. TRACKING ERROR DYNAMICS

Let us define the tracking errors as follows

$$ e_1(t) = \theta_d(t) - \theta(t), \quad (4a) $$

$$ e_2(t) = c(\theta(t))\bar{x}(t) + s(\theta(t))\bar{y}(t), \quad (4b) $$

$$ e_3(t) = c(\theta(t))\bar{y}(t) - s(\theta(t))\bar{x}(t), \quad (4c) $$

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 $\theta_d(t)$ come from a reference kinematic model for the UMR, i.e.,

$$ \dot{\theta}_d(t) = \omega_d(t), \quad (5a) $$

$$ \dot{x}_d(t) = c(\theta_d(t))v_d(t), \quad (5b) $$

$$ \dot{y}_d(t) = s(\theta_d(t))v_d(t), \quad (5c) $$

with $v_d$ and $\omega_d$ the linear and angular reference velocities, respectively. These are assumed continuous and 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 $t \ge 0$. Moreover, the trajectories of the reference model also hold the state constraints, i.e., $x_d(t) \in \mathcal{X}$ and $y_d(t) \in \mathcal{Y}$, for all $t \ge 0$.

Therefore, the tracking error dynamics is given as

$$ \dot{e}_1(t) = -\omega(t)d_1(t) + \tau_1(t), \quad (6a) $$

$$ \dot{e}_2(t) = \bar{d}_1(t)\omega(t)e_3(t) - v(t)d_2(t) + \tau_2(t), \quad (6b) $$

$$ \dot{e}_3(t) = -\bar{d}_1(t)\omega(t)e_2(t) + v_d(t)s(e_1(t)), \quad (6c) $$

with the new control inputs

$$ \tau_1(t) = \omega_d(t) - \omega(t), \quad (7a) $$

$$ \tau_2(t) = v_d(t)c(e_1(t)) - v(t). \quad (7b) $$

Note that the tracking error dynamics (6) can be rewritten as follows:

$$ \dot{e}(t) = A(\rho_1)e(t) + B[\tau(t) + F(\rho_2)d(t)], \quad (8) $$

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

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

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

with $\rho_1 = (v_d(t)\mathrm{sc}(e_1(t)) \ \bar{d}_1(t)\omega(t))^\top \in \mathbb{R}^2$ and $\rho_2 = (v(t) \ \omega(t))^\top \in \mathbb{R}^2$ as the unknown and known scheduling parameters, respectively. It is clear that (8) possesses an LPV form and the state and input constraint sets can be written as follows:

$$ \mathcal{E} = \{e \in \mathbb{R}^3 : (e_1, e_2, e_3) \in \mathbb{R} \times [-\overline{xy}, \overline{xy}] \times [-\overline{xy}, \overline{xy}]\}, \quad (9) $$

$$ \mathcal{U} = \{\tau \in \mathbb{R}^2 : (\tau_1, \tau_2) \in [-\bar{\tau}_1, \bar{\tau}_1] \times [-\bar{\tau}_2, \bar{\tau}_2]\}, \quad (10) $$

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

$$ A(\rho_1) = A_0 + \sum_{j=1}^{4} \alpha_j(\rho_1)A_j, \quad F(\rho_2) = \sum_{j=1}^{4} \alpha_j(\rho_2)F_j, \quad (11a) $$

$$ \sum_{j=1}^{4} \alpha_j(\rho_1) = \sum_{j=1}^{4} \alpha_j(\rho_2) = 1, \quad \alpha_j(\rho_1), \alpha_j(\rho_2) \in [0, 1], \quad (11b) $$

hold for (8). Thus

$$ \dot{e}(t) = \left[ A_0 + \sum_{j=1}^{4} \alpha_j(\rho_1)A_j \right] e(t) + B \left[ \tau(t) + \sum_{j=1}^{4} \alpha_j(\rho_2)F_j d \right]. \quad (12) $$

Therefore, the problem now is to design a robust sampled control law $\tau$ such that the trajectories of the system (12) converge to zero, despite the disturbances $d$, taking into account the communication constraint, i.e., $h(t) \in [h_{\min}, h_{\max}]$, and the state and input constraints, i.e., $e(t) \in \mathcal{E}$ and $\tau(t) \in \mathcal{U}$, for all $t \ge 0$.

## V. ROBUST CONTROL DESIGN

The proposed controller takes the following form:

$$ \tau(t) = \begin{cases} \tilde{\tau}(t), & \text{if } e(t) \notin \mathcal{E}(P), \\ \hat{\tau}(t), & \text{if } e(t) \in \mathcal{E}(P), \end{cases} \quad (13) $$

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

In the following sections, we describe the design of each part of the controller.
### A. Aperiodic Control Law

1) **Robust Linear State–Feedback Control Design:** Due to the nature of the system (12), it can be verified that the pair $(A_0, B)$ is controllable.

Since the event-triggered control law is based on a linear state–feedback controller, then the following control law is proposed

$$ \tilde{\tau}(t) = \sigma(\tau_0(t)), \quad (14) $$

with $\tau_0(t) = Ke(t)$, where $K \in \mathbb{R}^{2 \times 3}$ is a feedback gain to be designed, 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 (14) be applied to the system (12), 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 $Y \in \mathbb{R}^{2 \times 3}$, $Z = (Z_1^\top \ Z_2^\top)^\top \in \mathbb{R}^{2 \times 3}$, $M \in \mathbb{R}^{3 \times 3}$, $\Delta = \text{diag}(\delta_1, \delta_2)$, with $\delta_1, \delta_2 > 0$, and a constant $\gamma > 0$, such that the following set of LMIs

$$ \begin{aligned} \chi_1 &= \begin{pmatrix} \chi_{11} & B\Delta - Y^T + Z^\top & X_1BF_{\max} \\ \star & -2\Delta & 0 \\ \star & \star & -\gamma Q_d \end{pmatrix} \le 0, \quad (15a) \\ \chi_{11} &= A_\psi X_1 + X_1 A_\psi + BY + Y^T B^\top + M, \end{aligned} $$

$$ \chi_{2i} = \begin{pmatrix} X_1 & Z_i^\top \\ Z_i & \bar{\tau}_i^2 \end{pmatrix} \ge 0, \quad i = 1, 2, \quad (15b) $$

$$ \begin{aligned} \chi_3 &= \begin{pmatrix} M & \gamma X_1 \\ \gamma X_1 & \gamma X_2 \end{pmatrix} > 0, \quad (15c) \\ X_2 &< X_1, \quad (15d) \end{aligned} $$

$$ \chi_{4j} = \bar{b}_j^T X_1 \bar{b}_j \le 1, \quad j = \bar{1}, \bar{4}, \quad (15e) $$

$\bar{b}_1 = (0, \overline{xy}^{-1}, 0)^\top$, $\bar{b}_2 = -\bar{b}_1$, $\bar{b}_3 = (0, 0, \overline{xy}^{-1})^\top$, $\bar{b}_4 = -\bar{b}_3$, with $A_\psi = A_0 + A_{\max}I_3$, is feasible for given constants $\overline{xy}$, $F_{\max} = d_{\max} \max(v_{\max}, \omega_{\max})$, 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 (11) 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 for the system (12).

The next section presents how to implement the event-triggered controller.

2) **Event-Triggered Control Design:** Due to the communication restrictions, it is needed to implement the controller (14) in a sampled way, i.e.,

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

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

**Lemma 3.** Let the control (16) be applied to the system (12), i.e., $\tau(t) = \tilde{\tau}(t)$, with the controller parameters designed according to Lemma 2, i.e., $K = YR$, and assume that $e(0) \in \mathcal{E}(R) \setminus \mathcal{E}(P)$. Thus, if the event-triggered control law is designed as

$$ \tilde{\tau}(t) = \begin{cases} \sigma(\tau_0(t)), & t = t_{k+1}, \quad \text{if } W_1 > \varrho W_2 \\ \sigma(\tau_0(t_k)), & t = t_k, \quad \text{if } W_1 \le \varrho W_2, \end{cases} \quad (17) $$

for all $t \in [t_k, t_{k+1})$, where

$$ \begin{aligned} W_1(e(t), \tau_0(t_k)) &= 2S_e e^\top(t) RB\sigma(\tau_0(t_k)), \\ W_2(e(t), \tau_0(t)) &= 2S_e e^\top(t) RB\sigma(\tau_0(t)), \end{aligned} $$

and $\varrho > 0$, then the ellipsoid $\mathcal{E}(P)$, with $P$ given as in Lemma 2, is asymptotically attractive for the system (12).

Note that the control (14) ensures that the tracking error converges asymptotically to the convergence region, given by the ellipsoid $\mathcal{E}(P)$. Once in this region, the control law commutes to a constant sampled controller, which takes into account a maximum sampling time. In the following section, a way to design this controller is proposed.

### B. Constant Sampled Controller

Let us consider the state–feedback control law

$$ \hat{\tau}(t) = K_2 e(t_k), \quad \forall t \in [t_k, t_{k+1}), \quad k \in \mathbb{N}, \quad (18) $$

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 (18) be applied to the system (12), 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

$$ \begin{aligned} \Psi_1 &= \begin{pmatrix} \psi_{11} & \psi_{12} & R_1 + BY & BF_{\max} \\ \star & \psi_{22} & \epsilon BY & \epsilon BF_{\max} \\ \star & \star & -R_1 - S & 0 \\ \star & \star & \star & -\gamma_2 I \end{pmatrix} \le 0, \quad (19) \\ \psi_{11} &= P_2^\top A_\psi^\top + A_\psi P_2 + S - R_1 + \gamma_2 P_1, \\ \psi_{12} &= P_1 - P_2 + \epsilon A_\psi P_2, \quad \psi_{22} = -2\epsilon P_2 + \bar{h}^2 R_1, \end{aligned} $$

holds for some $\epsilon, \gamma_2 > 0$, $\bar{h} > h_{\min} > 0$, and $K_2$ is designed as $K_2 = Y P_2$, then the system (12) is ISS with respect to $d$, and its trajectories satisfy that $e(t) \in \mathcal{E}(P_c)$, with $P_c = 2^{-1} P_2^{\top-1} P_1 P_2^{-1} d_{\max}^2$. Moreover, if

$$ X_2^{-1} \le P_c, \quad (20) $$

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

Note that (20) is used to verify that the convergence region, given when the control (18) is applied, is completely contained in the set $\mathcal{E}(P)$. If this condition is not fulfilled, the parameter $\gamma_2$ can be modified to recalculate $K_2$ such that the convergence region is contained in $\mathcal{E}(P)$.

Finally, considering the results of lemmas 2 and 4, the main result is presented.

**Theorem 1.** Let the control (13) be applied to the system (12), with (17) and (18), i.e., the controller parameters are designed according to Lemma 2 and Lemma 4, respectively. Then, the system (12) is ISS with respect to $d$.

The proofs are omitted due to space limitations.
## VI. SIMULATIONS RESULTS

The LMIs solutions are obtained by means of SDPT3 solver, among YALMIP in MATLAB. The simulations have been done in MATLAB with the Euler explicit discretization method and minimum sampling time equal to $h_{min} = 0.001$ [s].

The desired trajectory is given by $\omega_d(t) = (\dot{x}_d(t)\ddot{y}_d(t) - \dot{y}_d(t)\ddot{x}_d(t))/(\dot{x}_d^2(t) + \dot{y}_d^2(t))$, $v_d = \sqrt{\dot{x}_d^2(t) + \dot{y}_d^2(t)}$, $x_d(t) = \cos(\omega_0 t)$, $y_d(t) = \sin(2\omega_0 t)$, $\theta_d(t) = \int_0^t \omega_d(\tau)d\tau$, $\omega_0 = 0.15$, thus, $\bar{\omega}_d = 1.1663$, $v_d = 0.1392$, and $\bar{v}_d = 0.4472$. The initial conditions are $x_0 = 1.5[m]$, $y_0 = 1.5[m]$ and $\theta_0 = 0.2[rad]$, thus, $e_1(0) = 1.3708$, $e_2(0) = -0.7880$, and $e_3(0) = -1.3708$. 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, $\bar{\mathcal{X}}\bar{\mathcal{Y}} = 6$, $\bar{\tau}_1 = 4.1663$, and $\bar{\tau}_2 = 1.4472$. Note that $e(0) \in \mathcal{E}$ and $\tau(0) \in \mathcal{U}$.

The disturbances are taken as $d_1(t) = 0.25 \sin(3t) + 0.3$ and $d_2(t) = 0.25 \cos(3t) + 0$, thus $d_{max} = 0.55$. It is possible to fix the matrix $A_0$ as $A_0 = (0, 0, 0; 0, 0, 0; 1, 0, 0)$ and then, the matrices $A_j$ can be obtained by a convex polytopic approach [19] using $\omega_{max}$, $d_{max}$, $v_d$, and $\bar{v}_d$. Therefore, it is obtained $F_{max} = 1.65$ and $A_{max} = 0.5261$.

To compute the control gains for the aperiodic control law, let us apply the statements of Lemma 2. Consider $\delta_1 = 1.8602$, $\delta_2 = 1.5634$, $\gamma = 0.2$, $\tau_{1max} = 4.1663$, and $\tau_{2max} = 1.1472$, then, the following feasible solution is obtained:

$$ 
\begin{aligned}
R &= \begin{pmatrix}
0.3332 & -0.0032 & 0.1426 \\
-0.0032 & 0.3763 & -0.0038 \\
0.1426 & -0.0038 & 0.3249
\end{pmatrix}, \\
P &= \begin{pmatrix}
0.6291 & -0.0117 & 0.2719 \\
-0.0117 & 0.6666 & -0.0122 \\
0.2719 & -0.0122 & 0.5647
\end{pmatrix}, \\
K &= \begin{pmatrix}
-1.6165 & 0.0069 & -1.2105 \\
0.0080 & -0.9458 & 0.0022
\end{pmatrix}.
\end{aligned}
$$

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

$$ 
\begin{aligned}
P_1 &= \begin{pmatrix}
0.7603 & -0.0239 & -0.4420 \\
-0.0239 & 0.9002 & -0.0139 \\
-0.4420 & -0.0139 & 1.1897
\end{pmatrix}, \\
P_2 &= \begin{pmatrix}
0.8317 & -0.0067 & -0.3869 \\
-0.0067 & 0.9538 & -0.0170 \\
-0.3869 & -0.0170 & 1.1464
\end{pmatrix}, \\
K_2 &= \begin{pmatrix}
-2.5744 & 0.0254 & -0.7018 \\
0.0374 & -2.3971 & 0.0059
\end{pmatrix}, \bar{h} = 0.5.
\end{aligned}
$$

The system trajectories are depicted in Fig. 2, which shows that they converge to 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. 3. Observe that $\omega$ remains inside the linear region while $v$ gets saturated at the beginning of the control task but remains inside the linear region. The projection of the ellipsoids $\mathcal{E}(R)$, $\mathcal{E}(P)$, and $\mathcal{E}(P_c)$, in the $e_2-e_3$ are depicted in Fig. 4. We can see that the trajectories

of the tracking error are in $\mathcal{E}(R)$, then converge to $\mathcal{E}(P)$, and finally remain inside $\mathcal{E}(P_c)$. The projection in $e_1$ is not shown since $e_1$ does not have constraints. Then, it is evident that the trajectories of the tracking errors do not transgress the constraints given in (10). Finally, in Fig. 5, it is observed that the norm of the errors converges to a region close to the origin.

## VII. CONCLUSIONS

In this paper, we proposed the design of a robust controller for the trajectory tracking problem in constrained unicycle mobile robots. The proposed controller comprises the design of an event-triggered controller and a constant sampled state-feedback controller. The state-feedback-based event-triggered control is designed by means of the AEM and the BLF that provides a safe set, where the state constraints are not transgressed, and a switching set that defines the region where each part of the controller is active. The periodic sampled control part is designed taking into account a maximum sampling time, and is active inside the switching set. The proposed switched control strategy ensures the ISS of the tracking error dynamics with respect to external disturbances. A constructive and simple method, based on LMIs, is proposed to compute the controller gains. Simulation results illustrate the feasibility of the proposed approach.

## ACKNOWLEDGMENT

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 CONAHCYT CVU 932774. The work of Héctor Ríos was supported in part by CONAHCYT CVU 270504 Project 922 and in part by TecNM Projects. The work of Manuel Mera was supported in part by the Project IPN-SIP 20240338.

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<figure><img src="image_4.png" /></figure>

<figure><img src="image_5.png" /><figcaption>Figure 2. System Trajectories</figcaption></figure>

<figure><img src="image_6.png" /><figcaption>Figure 3. Control Signals</figcaption></figure>

<figure><img src="image_7.png" /><figcaption>Figure 4. Tracking Error and Ellipsoids</figcaption></figure>

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

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