# Sampled-data control of LPV systems with magnitude and rate saturating actuators

Lucas A.L. Oliveira, Kevin Guelton, Koffi Motchon, Valter J.S. Leite

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Lucas A.L. Oliveira, Kevin Guelton, Koffi Motchon, Valter J.S. Leite. Sampled-data control of LPV systems with magnitude and rate saturating actuators. IEEE Control Systems Letters, 2025, 9, pp.1165 - 1170. 10.1109/LCSYS.2025.3580489. hal-05106243

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Submitted on 16 Jul 2025

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# Sampled-Data Control of LPV Systems With Magnitude and Rate Saturating Actuators

Lucas A. L. Oliveira<sup>ID</sup>, Kevin Guelton<sup>ID</sup>, Member, IEEE, Koffi M. D. Motchon<sup>ID</sup>, and Valter J. S. Leite<sup>ID</sup>, Senior Member, IEEE

**Abstract**—This letter presents the parameter-dependent aperiodic sampled-data state feedback controller design for linear parameter varying (LPV) systems with actuators subject to magnitude and rate saturation, using Linear Matrix Inequalities (LMIs). The proposed method integrates the looped-functional approach and a parameter-dependent generalized sector condition. The local stabilization is verified through a new definite negativeness lemma for second-order matrix polynomials. The proposed conditions can be simplified to recover a robust controller design whenever the time-varying parameter is unavailable. Two numerical examples demonstrate the effectiveness of the proposed method, highlighting less conservative stability conditions compared to existing approaches.

**Index Terms**—Sampled-data control, LPV systems, saturating actuators, local stabilization.

## I. INTRODUCTION

SAMPLED-data control has emerged as an effective tool for implementing digital controllers, especially with the advancement of technologies such as Industry 4.0 and the Internet of Things (IoT) [1]. Unlike traditional system discretization methods, this approach allows the embedding of aperiodic samplings directly into continuous-time plant models. Thus, aperiodic sampling enhances resilience to network congestion, packet losses, and transmission delays, which can be critical for the stability and performance of networked or embedded control systems [2]. The problem of

Received 17 March 2025; revised 18 May 2025; accepted 10 June 2025. Date of publication 17 June 2025; date of current version 7 July 2025. This work was supported in part by the Brazilian agencies CNPq (Brazilian National Research Council) under Grant 311891/2022-5, Grant 408411/2023-6, and Grant 402896/2024-6; in part by FAPEMIG under Grant APQ-02100-22; and in part by CAPES. Recommended by Senior Editor P. Tesi. (Corresponding author: Lucas A. L. Oliveira.)

Lucas A. L. Oliveira is with the CReSTIC UR3804, Université de Reims Champagne-Ardenne, 51100 Reims, France, and also with the Graduate Program on Mathematical and Computational Modeling, CEFET-MG, Belo Horizonte 30510, Brazil (e-mail: arantesl@hotmail.com).

Kevin Guelton and Koffi M. D. Motchon are with the CReSTIC UR3804, Université de Reims Champagne-Ardenne, 51100 Reims, France (e-mail: kevin.guelton@univ-reims.fr; koffi.motchon@univ-reims.fr).

Valter J. S. Leite is with the Graduate Program on Mathematical and Computational Modeling, CEFET-MG, Belo Horizonte 30510, Brazil, and also with the Department of Mechatronics Engineering, CEFET-MG (Divinópolis Campus), Divinópolis 35503-822, Brazil (e-mail: valter@ieee.org).

Digital Object Identifier 10.1109/LCSYS.2025.3580489

sampled-data system stabilization has been addressed through different frameworks, such as the discrete-time [3], input-delay approach [4], and impulsive system modeling [5]. The input-delay approach models sampling as a time-varying delay and was later extended by [6], introducing looped-functionals to address aperiodic sampling. Less conservative functionals whose derivative is a second-order polynomial on the input-delay were proposed in [7], yielding better estimates of the maximum sampling interval.

In the case of linear parameter-varying (LPV) systems [8], the advantages of using sampled-data control become even more significant. Within this framework, achieving an exact discrete-time model can be particularly challenging [9]. On the other hand, a challenge arises as the system states and the time-varying parameter evolves in continuous-time, while the controller parameters and updates occur in discrete-time. Assuming no parameter evolution within the sampling interval may lead to instability in real-world applications. Conversely, treating these parameters as independent introduces excessive conservatism in controller design [10].

Additionally, actuator saturation is a common constraint in control systems, limiting the signal in magnitude and rate variation due to security reasons or physical limitations. This can degrade performance and destabilize the closed-loop system [11]. Moreover, the consideration of rate saturation remains limited, despite notable events, such as aircraft crashes and the Chernobyl nuclear disaster [12], which have been linked to the consequences of rate saturation phenomena. In the context of sampled-data control, saturating actuators in time-varying systems have been addressed in [13] for Takagi-Sugeno Fuzzy systems and in [10] for LPV systems. However, only magnitude saturation is considered. In [14], stability analysis and controller design for sampled-data control systems with saturating actuators in magnitude and rate are considered for linear time-invariant (LTI) systems. Thus, to the best of the authors' knowledge, the case of LPV systems with saturating actuators in both magnitude and rate has not yet been investigated in sampled-data control systems.

The main contributions of this letter are 1) a new aperiodic sampled-data parameter-dependent controller design based on LMIs for LPV systems under magnitude and rate saturating actuators; 2) the introduction of a new approach to the sampled-data control problem, using second-order delay-dependent polynomial jointly with mean-value theorem
based representation; 3) an alternative approach to recover robust controller design when time-varying parameters are unavailable. The proposed conditions can be incorporated into standard optimization procedures to maximize the sampling interval or the region of attraction within which the trajectories asymptotically converge to the origin. By employing the input-delay approach, a looped-functional is introduced, whose derivative results in a second-order polynomial in the delay. The negativity is then verified using a new negativity definite lemma inspired by the literature for the LPV systems case. The mismatch between the continuous-time system's parameters and the sampled controller's parameters is modeled using the mean value theorem, which leads to less conservative conditions. The actuator saturation is handled using a parameter-dependent generalized sector condition, resulting in local stabilization. Finally, two numerical examples are presented, demonstrating the effectiveness of the proposed method.

**Notations:** The sets of real (positive) and natural numbers are denoted by $\mathbb{R}$ ($\mathbb{R}_+$) and $\mathbb{N}$, respectively. $M \in \mathbb{R}^{n \times m}$ is a matrix with real entries, and $x \in \mathbb{R}^n$ is a vector of size $n$. $M \in \mathbb{S}^n$ denotes a symmetric matrix of size $n \times n$ with real entries. The transpose of $M$ is $M^\top$; $M_{(i)}$ ($M_{ii}$) denotes the $i$-th (diagonal element $(i, i)$) row of $M$, and for $\alpha \in \mathbb{R}^N$, $\alpha_i$ is the $i$-th element of $\alpha$. $M_{(i)}^\top$ stands for $(M_{(i)})^\top$. Given a function $f$, $f \in \mathcal{C}^1$ means $f$ is continuously differentiable. The Euclidean norm of $M \in \mathbb{R}^n$ is $\|M\|$. $\mathrm{diag}\{M_1, M_2\}$ is a block diagonal matrix of $M_1$ and $M_2$. In square matrices, $\star$ denotes symmetric transposed blocks. **I** and **0** are the identity and null matrices, respectively. $\mathbb{H}\in\{M\}$ represents $M+M^\top$.

## II. PROBLEM FORMULATION

Consider a continuous-time plant modeled as a polytopic linear parameter-varying (LPV) system defined as:

$$ \dot{x}_s(t) = A(\alpha(t))x_s(t) + B(\alpha(t))v(t), \quad (1) $$

where $x_s \in \mathbb{R}^n$ is the system's state vector and $v \in \mathbb{R}^{n_u}$ is the input vector. The time-varying parameter vector is supposed to be online available and given by $\alpha(t) \in \Gamma_N$ where $\Gamma_N = \{\delta \in \mathbb{R}^N : \sum_{i=1}^N \delta_i = 1, \delta_i \ge 0\}$. We assume that, $\alpha_i(t) \in \mathcal{C}^1$ and $|\dot{\alpha}_i(t)| \le \phi_i$, $\forall t \ge 0$, and $i = 1, \dots, N$. The matrices $A(\alpha(t)) \in \mathbb{R}^{n \times n}$ and $B(\alpha(t)) \in \mathbb{R}^{n \times n_u}$ have a polytopic structure with $N$ known vertices given by $[A(\alpha(t)) B(\alpha(t))] = \sum_{i=1}^N \alpha_i(t) [A_i B_i]$. Concerning realistic process control conditions, we consider saturating the plant's actuator in symmetrical magnitude and variation bounds. Thus, we have $\mathrm{sat}_M(u_{(r)}(t)) = \mathrm{sign}(u_{(r)}(t))\min\{|u_{(r)}(t)|, \mu_M\}$ and $\mathrm{sat}_R(u_{(r)}(t)) = \mathrm{sign}(u_{(r)}(t))\min\{|u_{(r)}(t)|, \mu_R\}$, for $r = 1, \dots, n_u$, where the vectors $\mu_M \in \mathbb{R}^{n_u}$ and $\mu_R \in \mathbb{R}^{n_u}$ with positive elements define the saturation limits and indexes $M$ and $R$ concerns magnitude and rate, respectively. We adopt a first-order model, known as position-type feedback model with speed limitation [14], where the magnitude and rate saturation are embedded as

$$ \dot{v}_{(r)}(t) = \mathrm{sat}_{\mathrm{R}(r)}(\lambda_{(r)}\mathrm{sat}_{\mathrm{M}(r)}(u_{(r)}(t)) - \lambda_{(i)}v_{(r)}(t)), \quad (2) $$

for $r = 1, \dots, n_u$, where $v \in \mathbb{R}^{n_u}$ and $u \in \mathbb{R}^{n_u}$ stand for the actuator's state and (linear) control input, respectively.

<figure><img src="image_2.png" /><figcaption>Fig. 1. Control topology of sampled-data state feedback with actuators under magnitude and rate saturation.</figcaption></figure>

The designer choice parameter $\lambda_{(r)} > 0$ regards the actuator dynamics, usually chosen as $\lambda \to \infty$.

We propose an LPV sampled-data state feedback controller

$$ u(t) = K_s(\alpha(t))x_s(t) + K_v(\alpha(t))v(t), \quad (3) $$

where $t_k$ concerns the aperiodic sampling instants, such that $t_{k+1} > t_k$ and $\cup_{k \in \mathbb{N}}[t_k, t_{k+1}) = [0, +\infty)$, and the control signal is kept constant between sampling instants thanks to a zero-order hold (ZOH). The interval between two consecutive sampling instants is denoted by $\eta_k = t_{k+1} - t_k$, which varies but remains constrained by two positive scalars $\eta_1 \le \eta_2$, such that $0 < \eta_1 \le \eta_k \le \eta_2$. Similar to the plant model, the controller matrices $K_s(\alpha(t_k)) \in \mathbb{R}^{n_u \times n}$ and $K_v(\alpha(t_k)) \in \mathbb{R}^{n_u \times n_u}$ have polytopic structure $[K_s(\alpha(t_k)) K_v(\alpha(t_k))] = \sum_{i=1}^N \alpha_i(t_k) [K_{si} K_{vi}]$. Considering (1), (2), and (3), the closed-loop system can be obtained as

$$ \begin{aligned} \dot{x}_s(t) &= A(\alpha(t))x_s(t) + B(\alpha(t))v(t), \\ \dot{v}(t) &= \mathrm{sat}_R \left( \Lambda \mathrm{sat}_M (K_s(\alpha(t_k))x_s(t_k) + K_v(\alpha(t_k))v(t_k)) - \Lambda v(t) \right), \forall t \in [t_k, t_{k+1}), \quad (4) \end{aligned} $$

where $\Lambda \in \mathbb{R}^{n_u \times n_u}$ is a diagonal matrix composed by $\Lambda_{(r,r)} = \lambda_{(r)}$, with $-\lambda_{(r)}$ being the actuators' eigenvalues.

A schematic of the sampled-data state feedback control topology implemented with magnitude and rate saturating actuators is shown in Fig. 1, where the dotted lines indicate the digital signals computed at each sampling instant $t_k$.

As stated by [15] and [14], the use of model (2) to describe magnitude and rate saturating actuators can be seen as a particular case of nested saturation [16]. Also, the closed-loop system (4) can be rewritten as

$$ \begin{aligned} \dot{x}(t) &= A_2(\alpha(t))x(t) + B_2\mathrm{sat}_R(A_1x(t) + B_1\mathrm{sat}_M(u(t))), \\ u(t) &= K(\alpha(t_k)x(t_k)), \forall t \in [t_k, t_{k+1}), \end{aligned} \quad (5) $$
with the augmented state vector $x = [x_s^\top v^\top]^\top, x \in \mathbb{R}^{n_x}$, $n_x = n + n_u$, $A_2(\alpha(t)) = [\begin{matrix} A(\alpha(t)) & B(\alpha(t)) \\ 0 & 0 \end{matrix}]$, $A_1 = [0 - \Lambda]$, $B_2 = [0 \ 1]^\top$, $B_1 = \Lambda$, and $K(\alpha(t_k)) = [K_s(\alpha(t_k)) \ K_v(\alpha(t_k))]$.

Due to the presence of saturating actuators, there exists a set of initial conditions, referred to as the region of attraction $\mathcal{R}_a \subseteq \mathbb{R}^{n_x}$, whose corresponding trajectories asymptotically converge to the origin of the closed-loop system (5). However, analytically computing $\mathcal{R}_a$ is a challenging task, as it may be non-convex or unbounded in certain directions [11]. Therefore, we seek for an estimate $\mathcal{E}(R_0) = \{x \in \mathbb{R}^{n_x}; x^\top R_0 x \le 1\} \subseteq \mathcal{R}_a \subseteq \mathbb{R}^{n_x}$, considering the effects of saturating actuators and aperiodic samplings.

The objective of this letter is to develop convex conditions for the synthesis of LPV sampled-data state feedback controllers that ensure the local asymptotic stability of the system (5), under magnitude and rate saturating actuators and aperiodic samplings with $\eta_k \in [\eta_1, \eta_2]$. Hence, the following problems are defined:

**Problem 1:** Given the LPV system described by (1), a set of initial conditions $\mathcal{E}(R_0)$, and the lower bound on the sampling time $\eta_1$, determine the parameter-dependent controller gains according to the control law (3), such that the maximum allowable upper bound of the sampling interval, $\eta_2$, is maximized and all trajectories starting within $\mathcal{E}(R_0)$ asymptotically converge to the origin, without leaving such a region.

**Problem 2:** Given the LPV system described by (1) and sampling interval limits $[\eta_1, \eta_2]$, determine the parameter-dependent controller gains according to the control law (3), such that the estimate of the region of attraction $\mathcal{E}(R_0)$ is maximized.

### III. AUXILIARY RESULTS

The sampled-data controlled system can be represented as a time-varying delay system by applying the input-delay approach [4], where $\tau(t) \in [0, \eta_k)$ is the time-varying delay, such that $\dot{t}(t) = 1$ and $t_k = t - \tau(t)$, $\forall t \in [t_k, t_{k+1})$. For consiseness, we denote matrices $Y(\alpha(t)) = \sum_{i=1}^N \alpha_i(t)Y_i = Y_\alpha$, $Y(\alpha(t-\tau(t))) = Y_{\tilde{\alpha}}$, and $t-\tau(t) = \tilde{t}$. Also, any matrix $Y_\alpha$ has its time derivative denoted by $\dot{Y}_\alpha = \sum_{i=1}^N \dot{\alpha}_i(t)Y_i$. Thus, the closed-loop system (5) can be rewritten as

$$ \dot{x}(t) = A_{2\alpha}x(t) + B_2\text{sat}_R(A_1x(t) + B_1\text{sat}_M(K_{\tilde{\alpha}}x(\tilde{t}))). \quad (6) $$

To analyze the local stability of (6), we can reformulate the closed-loop system as a Lur'e-type representation to handle the nonlinearities associated with magnitude and rate saturation. Thus, using decentralized dead-zone functions $\Psi_M = \text{sat}_M(K_{\tilde{\alpha}}x(\tilde{t})) - K_{\tilde{\alpha}}x(\tilde{t})$, $\Psi_R = \text{sat}_R(A_1x(t) + B_1\text{sat}_M(K_{\tilde{\alpha}}x(\tilde{t}))) - (A_1x(t) + B_1\text{sat}_M(K_{\tilde{\alpha}}x(\tilde{t})))$, (6) can be rewritten as

$$ \dot{x}(t) = (A_{2\alpha} + B_2A_1)x(t) + B_2B_1K_{\tilde{\alpha}}x(\tilde{t}) + B_2B_1\Psi_M + B_2\Psi_R. \quad (7) $$

To solve Problems 1 and 2 we need some adaptations on known literature results, which are presented in the next three lemmas whose main differences are summarized: Lemma 1 includes in [11] parameter-dependent variables to handle magnitude and rate saturations; Lemma 2 includes

both parameter-dependent variables and the generalized sector conditions extending [6], [14]; and Lemma 3 extends result in [17] to LPV sampled-data framework to handle second-order polynomials in $\tau(t)$. At last, Lemma 4 is a standard result that plays a key role in our contribution in reducing the conservatism due to the parameter mismatch between the system and the controller.

**Lemma 1:** Consider the sets $S_M(\mu_M) = \{x(\tilde{t}) \in \mathbb{R}^{n_x} : |(K_{\tilde{\alpha}} - G_{0\tilde{\alpha}})(r)x(\tilde{t})| \le \mu_{M(r)}\}$ and $S_R(\mu_R) = \{(x(t), x(\tilde{t}), \Psi_M) \in \mathbb{R}^{n_x} \times \mathbb{R}^{n_x} \times \mathbb{R}^{n_u} : |(\Xi_1)(r)x(t) + (\Xi_2)(r)x(\tilde{t}) + (\Xi_3)(r)\Psi_M| \le \mu_{R(r)}\}$, for $r = 1, \dots, n_u$, with matrices $G_{0\tilde{\alpha}}$, $G_{1\alpha}$, $G_{2\tilde{\alpha}} \in \mathbb{R}^{n_u \times n_x}$, and $G_3 \in \mathbb{R}^{n_x \times n_x}$ where $\Xi_1 = A_1 - G_{1\alpha}$, $\Xi_2 = B_1K_{\tilde{\alpha}} - G_{2\tilde{\alpha}}$, and $\Xi_3 = B_1 - G_3$. If $x(\tilde{t}) \in S_M(\mu_M)$ and $(x(t), x(\tilde{t}), \Psi_M) \in S_R(\mu_R)$, then the dead-zone functions $\Psi_M$ and $\Psi_R$, satisfy

$$ \Psi_M^\top T_M(\Psi_M + G_{0\tilde{\alpha}}x(\tilde{t})) \le 0, \quad (8) $$

$$ \Psi_R^\top T_R(\Psi_R + G_{1\alpha}x(t) + G_{2\tilde{\alpha}}x(\tilde{t}) + G_3\Psi_M) \le 0. \quad (9) $$

with diagonal matrices $0 < T_M \in \mathbb{S}^{n_u}$ and $0 < T_R \in \mathbb{S}^{n_u}$.

*Proof:* The proof follows the application of the generalized sector condition in Lemma 1 of [14], with parameter-dependent additional variables selected based on the time domain of the elements in the set that each variable multiplies. ■

**Lemma 2:** Consider a function $V: \mathbb{R}^{n_x} \times \Gamma_N \to \mathbb{R}_+$, a continuous and differentiable parameter-dependent function $V_0: \mathbb{R}^{n_x} \times \Gamma_N \times [0, \eta_k) \times [\eta_1, \eta_2] \to \mathbb{R}$ satisfying

$$ \begin{aligned} V_0(x, \alpha, 0, \eta_k) &= V_0(x, \alpha, \eta_k, \eta_k) = 0, \\ V_0(x, \alpha, \eta_k, \eta_k) &> 0, \end{aligned} \quad (10) $$

for all $x \in \mathbb{R}^{n_x}$, $\alpha \in \Gamma_N$, $\tau \in [0, \eta_k)$, and $\eta_k \in [\eta_1, \eta_2]$, and $W = V(x, \alpha) + V_0(x, \alpha, \tau, \eta_k)$ and $\dot{W} = \frac{d}{dt}(V(x, \alpha) + V_0(x, \alpha, \tau, \eta_k))$. Suppose there exist matrices $G_{0\tilde{\alpha}}$, $G_{1\alpha}$, $G_{2\tilde{\alpha}} \in \mathbb{R}^{n_u \times n_x}$, $G_3 \in \mathbb{R}^{n_x \times n_x}$, defining $\Xi_1, \Xi_2$, and $\Xi_3$ given in Lemma 1, positive definite diagonal matrices $T_M, T_R \in \mathbb{R}^{n_u \times n_u}$, and positive scalars $\epsilon_1, \epsilon_2$, and $\epsilon_3$, such that the following inequalities hold

$$ \begin{gathered} x^\top(\tilde{t})(K_{\tilde{\alpha}} - G_{0\tilde{\alpha}})^\top (K_{\tilde{\alpha}} - G_{0\tilde{\alpha}})(r)x(\tilde{t}) \le \mu_{M(r)}^2 V(x(\tilde{t}), \alpha(\tilde{t})), (11) \\ \begin{bmatrix} x(t) \\ x(\tilde{t}) \\ \Psi_M \end{bmatrix}^\top \hat{\Sigma}^\top \hat{\Sigma} \begin{bmatrix} x(t) \\ x(\tilde{t}) \\ \Psi_M \end{bmatrix} \le \mu_{R(r)}^2 V(x(t), \alpha(t)), \end{gathered} \quad (12) $$

$$ \begin{aligned} & \text{with } \hat{\Xi} = [\Xi_1(r) \ \Xi_2(r) \ \Xi_3(r)], \text{ for } r = 1, \dots, n_u, \\ & \epsilon_1 \|x\|^2 \le V(x, \alpha) \le \epsilon_2 \|x\|^2, \end{aligned} \quad (13) \\ \dot{W} - 2\Psi_M T_M (\Psi_M + G_{0\tilde{\alpha}}x(\tilde{t})) - 2\Psi_R T_R \\ \times (\Psi_R + G_{1\alpha}x(t) + G_{2\tilde{\alpha}}x(\tilde{t}) + G_3\Psi_M) < -\epsilon_3 \left\| \begin{bmatrix} x(t) \\ x(\tilde{t}) \end{bmatrix} \right\|^2 \quad (14) $$

for $\forall (k, \alpha(t), \tau(t), \eta_k) \in \mathbb{N} \times \Gamma_N \times [0, \eta_k) \times [\eta_1, \eta_2]$. Then, any initial condition $x(0)$ belonging to the set:

$$ L_V = \{x \in \mathbb{R}^{n_x}, \alpha \in \Gamma_N; V(x, \alpha) \le 1\}, \quad (15) $$

ensures that: i) the increment of the Lyapunov function $V(x(t), \alpha(t))$ between two consecutive sampling instants is strictly negative, i.e., it satisfies $\Delta V(x(t_k), \alpha(t_k)) = V(x(t_{k+1}), \alpha(t_{k+1})) - V(x(t_k), \alpha(t_k)) < -\epsilon_3 \|x(t)^\top x(\tilde{t})^\top\|^2$; ii) the trajectories of the closed-loop system (6) with aperiodic
samplings $\eta_k \in [\eta_1, \eta_2]$ converge asymptotically to the origin without leaving $L_V$.

*Proof:* Assuming that $x(0) \in L_V$, taking (11) and (12), hence, for $r = 1, \dots, n_u$, $|(\tilde{\mathbb{K}}_{\tilde{\alpha}} - G_{0\tilde{\alpha}})_{(r)}x(\tilde{t})| \le \mu_{\mathbf{M}(r)}$ implies $x(\tilde{t}) \in \mathcal{S}_{\mathbf{M}}(\mu_{\mathbf{M}})$ and $|(\Xi_1)_{(r)}x(t) + (\Xi_2)_{(r)}x(\tilde{t}) + (\Xi_3)_{(r)}\Psi_{\mathbf{M}}| \le \mu_{\mathbf{R}(r)}$ implies $(x(t), x(\tilde{t}), \Psi_{\mathbf{M}}) \in \mathcal{S}_{\mathbf{R}}(\mu_{\mathbf{R}})$. Then, (8), (9), and (14) yield $\dot{W} < -\epsilon_3 \| [x(t)^{\top} x(\tilde{t})^{\top}]^{\top} \|^2 \le 0$. The result can be demonstrated thereafter using reasoning similar to Theorem 1 in [14] and [18]. ■

*Remark 1:* Following the approach in [14], where rate saturation is considered for LTI systems, the definition in (10) is necessary to ensure that the actuator state trajectories remain within $\mathcal{S}_{\mathbf{R}}(\mu_{\mathbf{R}})$ throughout the sampling intervals. Therefore, $W$ provides an upper bound for $V(x, \alpha)$ between sampling instants while being equal to it at each sample.

*Lemma 3:* Consider a quadratic parameter-dependent function $f(y) = a_{2\alpha}y^2 + a_{1\alpha}y + a_{0\alpha}$, where $a_{2\alpha}$, $a_{1\alpha}$, $a_{0\alpha} \in \mathbb{R}^{p \times p}$. If there exists a matrix $M_{\alpha} \in \mathbb{R}^{p \times p}$ satisfying $M_{\alpha} + M_{\alpha}^{\top} \ge 0$, such that

$$ \begin{bmatrix} f(\underline{h}) & \frac{1}{2}a_{1\alpha} + \underline{h}a_{2\alpha} + (\underline{h} - \underline{h})M_{\alpha} \\ \star & a_{2\alpha} - \text{He}\{M_{\alpha}\} \end{bmatrix} < 0, \quad (16) $$

then it follows that $f(y) < 0$, $\forall y \in [\underline{h}, \underline{h}]$.

*Lemma 4* [19]: Consider a parameter-dependent matrix $Y_{\tilde{\alpha}} = \sum_{i=1}^{N} \alpha_i(t)Y_i$. If $\alpha_i(t) \in \mathbb{C}^1$ and $|\dot{\alpha}_i(t)| \le \phi_i$, for $i = 1, \dots, N$, thus it is equivalent to rewrite

$$ Y_{\tilde{\alpha}} = Y_{\alpha} - \tau(t) \dot{Y}_c. \quad (17) $$

where $c(t) \in [0, \tau(t)]$ and $\dot{Y}_c = \sum_{i=1}^{N} \dot{\alpha}_i(t - c_i(t))Y_i$.

*Proof:* From the mean value theorem, if $f[a, b] \to \mathbb{R}$ is a continuous function on $[a, b]$. Then, there exists $c \in [a, b]$ such that $\int_a^b f(s)ds = (b-a)f(c)$. Therefore, with $\alpha_i(t) \in \mathbb{C}^1$ there exists a $c_i(t) \in [0, \tau(t)]$ such that $\alpha_i(t) - \alpha_i(\tilde{t}) = \int_{\tilde{t}}^t \dot{\alpha}_i(s)ds = \tau(t)\dot{\alpha}_i(t - c_i(t))$, for $i = 1, \dots, N$. Hence, any parameter-dependent matrix $Y_{\tilde{\alpha}} = \sum_{i=1}^{N} \alpha_i(t)Y_i$ can be rewritten as $Y_{\tilde{\alpha}} = Y_{\alpha} - \sum_{i=1}^{N} \int_{\tilde{t}}^t \dot{\alpha}_i(s)dsY_i = Y_{\alpha} - \tau(t)\dot{Y}_c$. ■

## IV. MAIN RESULTS

This section presents convex conditions for designing a controller (3) and addressing Problems 1 and 2. Consider the auxiliary matrices $\mathbb{I}_1 = [\mathbb{I} \mathbb{I} \mathbb{I} \mathbb{I} \mathbb{I}]$, $\mathbb{I}_2 = [\mathbb{I} \mathbb{I} \mathbb{I} \mathbb{I} \mathbb{I}]$, $\mathbb{I}_3 = [\mathbb{I} \mathbb{I} \mathbb{I} \mathbb{I} \mathbb{I}]$, $\mathbb{I}_4 = [\mathbb{I} \mathbb{I} \mathbb{I} \mathbb{I} \mathbb{I}]$, $\mathbb{I}_5 = [\mathbb{I} \mathbb{I} \mathbb{I} \mathbb{I} \mathbb{I}]$, $\mathbb{I}_{12} = \mathbb{I}_1 - \mathbb{I}_2$, $\mathbb{I}_{24} = [\mathbb{I}_2^\top \mathbb{I}_4^\top]^\top$, $\mathbb{I}_{12} = [\mathbb{I}_{12}^\top]^\top$, $\mathbb{I}_3 = [\mathbb{I}_3^\top \mathbb{I}_4^\top]^\top$, and $\mathbb{I}_4 = [\mathbb{I}_4^\top]^\top$.

*Theorem 1:* Suppose there exists matrices $\mathbf{0} < \tilde{P}_{\alpha} \in \mathbb{S}^{n_x}$, $\mathbf{0} < \tilde{R} \in \mathbb{S}^{n_x}$, $\mathbf{0} < \tilde{S}_{\alpha} \in \mathbb{S}^{n_x}$, $\mathbf{0} < \tilde{U} \in \mathbb{S}^{(n_x+n_u)}$, $\tilde{L}_{\alpha} \in \mathbb{R}^{(3n_x+2n_u) \times n_x}$, $\tilde{K}_{\alpha} \in \mathbb{R}^{n_u \times n_x}$, $\tilde{G}_{0\alpha} \in \mathbb{R}^{n_u \times n_x}$, $\tilde{G}_{1\alpha} \in \mathbb{R}^{n_u \times n_x}$, $\tilde{X} \in \mathbb{R}^{n_x \times n_x}$, $\tilde{M}_{\alpha}^k \in \mathbb{R}^{(3n_x+2n_u) \times (3n_x+2n_u)}$, diagonal matrices $\mathbf{0} < \tilde{T}_{\mathbf{M}} \in \mathbb{S}^{n_u}$, $\mathbf{0} < \tilde{T}_{\mathbf{R}} \in \mathbb{S}^{n_u}$, and a scalar $\kappa > 0$, such that, for $\eta_k \in [\eta_1, \eta_2]$ and $r = 1, \dots, n_u$,

$$ \begin{bmatrix} \tilde{\Pi}_0 & \frac{1}{2}\tilde{\Pi}_1 + \eta_k \tilde{M}_{\alpha}^k & \mathbf{0} \\ \star & \tilde{\Pi}_2 - \text{He}\{\tilde{M}_{\alpha}^k\} & \tilde{L}_{\alpha} \\ \star & \star & -\eta_k \tilde{R} \end{bmatrix} < 0, \quad (18) $$

$$ \begin{bmatrix} \text{He}\{\tilde{M}_{\alpha}^k\} & \tilde{L}_{\alpha} \\ \star & \eta_k \tilde{R} \end{bmatrix} > 0, \quad \begin{bmatrix} \tilde{P}_{\alpha} & (\tilde{\mathbb{K}}_{\tilde{\alpha}} - \tilde{G}_{0\alpha})^{\top} \\ \star & \mu_{\mathbf{M}(r)}^2 \end{bmatrix} \ge 0, \quad (19) $$

$$ \begin{bmatrix} \tilde{P}_{\alpha} & (\tilde{\mathbb{K}}_{\tilde{\alpha}} - \eta_k \tilde{\mathbb{K}}_{\tilde{c}})^{\top}_{(r)} - (\tilde{G}_{0\tilde{\alpha}} - \eta_k \tilde{G}_{0\tilde{c}})^{\top} \\ \star & \mu_{\mathbf{M}(r)}^2 \end{bmatrix} \ge \mathbf{0}, \quad (20) $$

$$ \begin{bmatrix} \tilde{P}_{\alpha} & (\mathbb{A}_1 \tilde{X} - \tilde{G}_{1\alpha})^{\top} \\ \star & \mu_{\mathbf{R}(r)}^2 \end{bmatrix} \ge \mathbf{0}, \quad (21) $$

with $\tilde{\Pi}_2 = -\text{He}\{\tilde{\Pi}_1^\top \tilde{U} \tilde{\Pi}_3 + \tilde{\Pi}_3^\top \tilde{U} \tilde{\Pi}_4\}$, $\tilde{\Pi}_1 = -2(\tilde{\Pi}_1^\top \tilde{U} \tilde{\Pi}_2 + \tilde{\Pi}_4^\top \tilde{U} \tilde{\Pi}_4 + \text{He}\{\tilde{\Pi}_1^\top \tilde{U} \tilde{\Pi}_4\}) - \eta_k \tilde{\Pi}_2 + \text{He}\{-\tilde{\Pi}_3^\top \tilde{S}_{\alpha} \mathbb{I}_2 + \tilde{\Pi}_4^\top \tilde{G}_{0\alpha} \mathbb{I}_2 + \tilde{\Pi}_5^\top \tilde{B}_1 \dot{\tilde{K}}_{\tilde{c}} \mathbb{I}_2 + (\tilde{\Pi}_1^\top \tilde{R} \tilde{I}_3 - \tilde{\Pi}_2^\top \tilde{S}_{\alpha} \mathbb{I}_2, \tilde{\Pi}_0 = \text{He}\{\tilde{\Pi}_1^\top \tilde{P}_{\alpha} \tilde{I}_3 - \tilde{L}_{\alpha} \mathbb{I}_2 - \tilde{\Pi}_4^\top \tilde{G}_{0\alpha} \mathbb{I}_2 - \tilde{\Pi}_5^\top \tilde{G}_{1\alpha} \mathbb{I}_1 - \tilde{\Pi}_5^\top \tilde{B}_1 \tilde{K}_{\tilde{\alpha}} \mathbb{I}_2 - \tilde{\Pi}_5^\top \tilde{B}_1 \tilde{T}_{\mathbf{M}} \mathbb{I}_4 + (\tilde{\Pi}_1^\top \tilde{I}_3 + \tilde{\Pi}_4^\top \tilde{S}_{\alpha} \mathbb{I}_1 + \tilde{\Pi}_5^\top \tilde{B}_1 \tilde{T}_{\mathbf{R}} \mathbb{I}_5) + \eta_k(\tilde{\Pi}_3^\top \tilde{R} \tilde{I}_3 + \tilde{\Pi}_1^\top \tilde{S}_{\alpha} \mathbb{I}_1 + \tilde{\Pi}_2^\top \tilde{U} \tilde{\Pi}_2 + \tilde{\Pi}_4^\top \tilde{U} \tilde{\Pi}_4) - \tilde{\Pi}_5^\top \tilde{S}_{\alpha} \mathbb{I}_1 + \tilde{\Pi}_1^\top \tilde{P}_{\alpha} \mathbb{I}_1 - 2\tilde{\Pi}_4^\top \tilde{T}_{\mathbf{M}} \mathbb{I}_4 - 2\tilde{\Pi}_5^\top \tilde{T}_{\mathbf{R}} \mathbb{I}_5\}$. Then, for any initial condition $x(0)$ belonging to the set $\mathcal{E}(P_{\alpha}) = \{x \in \mathbb{R}^{n_x}, x^\top P_{\alpha} x \le 1\}$, where $P_{\alpha} = \tilde{X}^{-\top} P_{\alpha} \tilde{X}^{-1}$, the respective trajectories of the LPV sampled-data system (6) with $\tilde{K}_{\tilde{\alpha}} = \tilde{K}_{\tilde{\alpha}} \tilde{X}^{-1}$ converge asymptotically to the origin.

*Proof:* Consider $W = V(x(t), \alpha(t)) + V_0(x(t), \alpha(t), \tau(t), \eta_k)$ where $V(x(t), \alpha(t)) = x^\top(t)P_{\alpha}x(t), V_0(x(t), \alpha(t), \tau(t), \eta_k) = (\eta_k - \tau(t))(x(t) - x(\tilde{t}))^\top S_{\alpha}(x(t) - x(\tilde{t})) + \tau \zeta^\top U \zeta + \int_{\tilde{t}}^t \dot{x}^\top(s)R\dot{x}(s)ds$, with $\zeta = [(x(t) - x(\tilde{t}))^\top \Psi_{\mathbf{M}}^\top]^{\top}$, positive definite matrices $P_{\alpha}, R \in \mathbb{S}^{n_x}$, $U \in \mathbb{S}^{(n_x+n_u)}$, and $S_{\alpha} \in \mathbb{S}^{n_x}$. Therefore, $V(x(t), \alpha(t))$ and $V_0(x(t), \alpha(t), \tau(t), \eta_k)$ satisfy (13) and (10). From (15), the level set $\mathcal{E}(P_{\alpha})$ can be exactly computed as the intersection of a finite number of ellipsoids, given by $\bigcap_{\alpha(t) \in \Gamma} \mathcal{E}(P_{\alpha}) = \bigcap_{i=1, \dots, N} \mathcal{E}(P_i)$

(see [20] for the general formulation assuming homogeneous parameter dependency). The time derivative of $W$ yields $\dot{W} = 2\dot{x}^\top(t)P_{\alpha}x(t) + x^\top(t)\dot{P}_{\alpha}x(t) - (x(t) - x(\tilde{t}))^\top S_{\alpha}(x(t) - x(\tilde{t})) + (\eta_k - \tau(t))2\dot{x}^\top(t)S_{\alpha}(x(t) - x(\tilde{t})) + (\eta_k - \tau(t))(x(t) - x(\tilde{t}))^\top \dot{S}_{\alpha}(x(t) - x(\tilde{t})) - \tau(t)\zeta^\top U \zeta + (\eta_k - \tau(t))(\eta_k - \tau(t))\dot{x}^\top(t)\dot{x}^\top(t)R\dot{x}(t) - \int_{\tilde{t}}^t \dot{x}^\top(s)R\dot{x}(s)ds$, with $\dot{\zeta} = [\dot{x}^\top(t) \mathbf{0}]^\top$. The last term of $\dot{W}$ is overbounded using the Jensen's inequality as $-\int_{\tilde{t}}^t \dot{x}^\top(s)R\dot{x}(s)ds \le -2\xi^\top L_{\alpha}(x(t) - x(\tilde{t})) + \tau(t)\xi^\top L_{\alpha}R^{-1}L_{\alpha}^\top \xi$, with $L_{\alpha} \in \mathbb{R}^{(3n_x+2n_u) \times n_x}$ and $\xi = [x^\top(t) x^\top(\tilde{t}) \dot{x}^\top(t) \Psi_{\mathbf{M}}^\top \Psi_{\mathbf{R}}^\top]^\top$. From (7), we define $\mathcal{B} = [-A_{2\alpha} - B_2A_1 - B_2B_1 \tilde{K}_{\tilde{\alpha}} \mathbf{I} - B_2B_1 - B_2]$, such that $\mathcal{B}\xi = \mathbf{0}$ holds. Let $\chi = \dot{X}^\top(\kappa \mathbb{I}_1^\top + \mathbb{I}_3^\top)$ and $X \in \mathbb{R}^{n_x \times n_x}$. From (14), with the application of the Finsler's Lemma with $G_{2\tilde{\alpha}} = B_1 \tilde{K}_{\tilde{\alpha}}$ and $G_3 = B_1$, it follows that $\dot{W} - 2\Psi_{\mathbf{M}} T_{\mathbf{M}}(\Psi_{\mathbf{M}} + G_{0\tilde{\alpha}}x(\tilde{t})) - 2\Psi_{\mathbf{R}} T_{\mathbf{R}}(\Psi_{\mathbf{R}} + G_{1\alpha}x(t) + B_1 \tilde{K}_{\tilde{\alpha}}x(\tilde{t}) + B_1 \Psi_{\mathbf{M}}) \le \xi^\top Q(\tau(t))\xi$ where $Q(\tau(t)) = \Pi_2 \tau^2(t) + (\Pi_1 + L_{\alpha}^{-1} L_{\alpha}^\top)\tau(t) + \Pi_0, \Pi_2 = -\text{He}\{\tilde{\Pi}_1^\top \tilde{U} \tilde{\Pi}_3 + \tilde{\Pi}_3^\top \tilde{U} \tilde{\Pi}_4\}, \Pi_1 = -2(\tilde{\Pi}_1^\top \tilde{U} \tilde{\Pi}_2 \tilde{U} \tilde{\Pi}_4 + \tilde{\Pi}_3^\top \tilde{R} \tilde{I}_3 - \tilde{\Pi}_4^\top \tilde{S}_{\alpha} \mathbb{I}_2, \tilde{\Pi}_0 = \text{He}\{\tilde{\Pi}_1^\top P_{\alpha} \tilde{I}_3 - L_{\alpha} \mathbb{I}_2 - \tilde{\Pi}_4^\top T_{\mathbf{M}} G_{0\tilde{\alpha}} \mathbb{I}_2 - \tilde{\Pi}_5^\top T_{\mathbf{R}} G_{1\alpha} \mathbb{I}_1 - \tilde{\Pi}_5^\top T_{\mathbf{R}} B_1 \tilde{K}_{\tilde{\alpha}} \mathbb{I}_2 - \tilde{\Pi}_5^\top T_{\mathbf{R}} B_1 \mathbb{I}_4 + (\tilde{\Pi}_1^\top \tilde{I}_3 + \tilde{\Pi}_4^\top \tilde{S}_{\alpha} \mathbb{I}_1 + \tilde{\Pi}_5^\top \tilde{B}_1 \tilde{K}_{\tilde{\alpha}} \mathbb{I}_2 + \tilde{\Pi}_5^\top \tilde{B}_1 \tilde{I}_4 - \tilde{\Pi}_5^\top \tilde{B}_5) + \eta_k(\tilde{\Pi}_3^\top \tilde{R} \tilde{I}_3 + \tilde{\Pi}_1^\top \tilde{S}_{\alpha} \mathbb{I}_1 + \tilde{\Pi}_2^\top \tilde{U} \tilde{\Pi}_2 + \tilde{\Pi}_4^\top \tilde{U} \tilde{\Pi}_4) - \tilde{\Pi}_5^\top \tilde{S}_{\alpha} \mathbb{I}_1 + \tilde{\Pi}_1^\top \tilde{P}_{\alpha} \mathbb{I}_1 - 2\tilde{\Pi}_4^\top \tilde{T}_{\mathbf{M}} \mathbb{I}_4 - 2\tilde{\Pi}_5^\top \tilde{T}_{\mathbf{R}} \mathbb{I}_5\}$. Block (3,3) of $\tilde{\Pi}_0$ yields $\tilde{X} + \tilde{X}^\top < \mathbf{0}$, ensuring the non-singularity of $\tilde{X}$. Assuming $\tilde{X} = X^{-1}$, $\tilde{T}_{\mathbf{M}} = T_{\mathbf{M}}^{-1}$, $\tilde{T}_{\mathbf{R}} = T_{\mathbf{R}}^{-1}$ and multiplying $Q(\tau(t))$ by $\text{diag}\{\tilde{X}^\top, \tilde{X}^\top, \tilde{X}^\top, \tilde{T}_{\mathbf{M}}, \tilde{T}_{\mathbf{R}}\}$ on the left and by its transpose on the right, we perform the variable changes: $\tilde{P}_{\alpha} = \tilde{X}^\top P_{\alpha} \tilde{X}$, $\tilde{S}_{\alpha} = \tilde{X}^\top S_{\alpha} \tilde{X}$, $\tilde{U} = \text{diag}\{\tilde{X}^\top, \tilde{T}_{\mathbf{M}}^\top\}U \text{diag}\{\tilde{X}, \tilde{T}_{\mathbf{M}}\}$, $\tilde{R} = \tilde{X}^\top R \tilde{X}$, $\tilde{L}_{\alpha} =$
$\mathrm{diag}\{\tilde{X}^\top, \tilde{X}^\top, \tilde{X}^\top, \tilde{T}_M, \tilde{T}_R\}L_\alpha \tilde{X}, \tilde{K}_{\tilde{\alpha}} = K_{\tilde{\alpha}} \tilde{X}, \tilde{G}_{0\tilde{\alpha}} = G_{0\tilde{\alpha}} \tilde{X},$
and $\tilde{G}_{1\tilde{\alpha}} = G_{1\tilde{\alpha}} \tilde{X}$. Applying Lemma 4 on $K_{\tilde{\alpha}}$ and $\tilde{G}_{0\tilde{\alpha}}$, the condition (14) can be satisfied with $\tilde{\Pi}_2 \tau^2(t) + (\tilde{\Pi}_1 + \tilde{L}_\alpha \tilde{R}^{-1} \tilde{L}_\alpha^\top) \tau(t) + \tilde{\Pi}_0 < 0$. To ensure the negativity of the second-order function, we use Lemma 3 with $M_\alpha^k = -\frac{1}{2\eta_k} \tilde{L}_\alpha \tilde{R}^{-1} \tilde{L}_\alpha^\top + \tilde{M}_\alpha^k$, for $\tau(t) \in [0, \eta_k)$, $k = 1, 2$, resulting in $[ \star \tilde{\Pi}_2 - \mathrm{He}\{\tilde{M}_\alpha^k\} + \frac{1}{\eta_k} L_\alpha \tilde{R}^{-1} \tilde{L}_\alpha^\top ] < 0$. Finally, a Schur complement is performed on this last inequality obtaining (18), as well in $\mathrm{He}\{M_\alpha^k\} > 0$ to get left side of (19). Then, the stability conditions of Lemma 2 are satisfied for $\tau(t) \in [0, \eta_k)$ and $\eta_k \in [\eta_1, \eta_2]$.

To conclude the proof, we need to ensure that (11) and (12) hold. We apply the reverse transformation of Lemma 4 in right side of (19) (for $\tau(t) = 0$) and (20) (for $\tau(t) = \eta_k$), and pre- and post-multiply right side of (19) (now with $\tilde{K}_{\tilde{\alpha}}$ and $\tilde{G}_{0\tilde{\alpha}}$) by $\mathrm{diag}\{\tilde{X}^{-\top}, \mathbf{I}\}^{\top}$, and its transpose, respectively. Performing a Schur complement and pre- and post-multiply by $x^{\top}(\tilde{t})$ and $x(\tilde{t})$, we have that $x^{\top}(\tilde{t})(P_{\alpha} - (K_{\tilde{\alpha}} - G_{0\tilde{\alpha}})_{(r)} \mu_{M(r)}^{-2} (K_{\tilde{\alpha}} - G_{0\tilde{\alpha}})_{(r)}) x(\tilde{t}) \ge 0$, verifying (11) and consequently $\mathcal{E}(P_{\alpha}) \subseteq S_M(\mu_M)$. Similarly, we take (21) and pre- and post-multiply (21) by $\mathrm{diag}\{\tilde{X}^{-\top}, \mathbf{I}\}^{\top}$, and its transpose, respectively, take the Schur complement, and pre- and post-multiply by $x^{\top}(t)$ and $x(t)$, to obtain $x^{\top}(t)(P_{\alpha} - (A_1 - G_{1\alpha})_{(r)} \mu_{R(r)}^{-2} (A_1 - G_{1\alpha})_{(r)}) x(t) \ge 0$, which verifies (12), where we have chosen $G_{2\tilde{\alpha}} = \mathbb{B}_1 K_{\tilde{\alpha}}$ and $G_3 = \mathbb{B}_1$, and consequently $\mathcal{E}(P_{\alpha}) \subseteq S_R(\mu_R)$. This concludes the proof. ■

**Remark 2:** The number of scalar variables in Theorem 1 is $V_N = 2.5n_u + n_x + Nn_x + n_u n_x + 8Nn_u^2 + 22Nn_x^2 + 0.5n_u^2 + 2n_x^2 + 29Nn_u n_x$. When the parameters $\alpha(t)$ are not available for feedback, we can obtain robust controllers by setting $\tilde{K}_\alpha = \tilde{K}$ and $\tilde{K}_c = 0$, and all parameter-dependent matrices $Y_\alpha = Y$, yielding the corresponding derivative ones as zero in Theorem 1. This simpler implementation has less computational cost, with $V_1$, but more conservative results.

**Remark 3:** To address Problem 1, we use the optimization procedure $O_1$, given by [14, eq. (51)], with $P_0 = R_0$, $\tau_2 = \eta_2$, and $\tilde{Y} = -\tilde{X}$ subject to (18)-(21), allowing the search for the control law (3) that maximize $\eta_2$ such that $\forall x(0) \in \mathcal{E}(R_0)$, the respective trajectories of the closed-loop system (6) converge asymptotically towards the origin. For Problem 2, we apply the optimization procedure $O_2$ given by [14, eq.(49)] with the same change of variables mentioned above and $\delta = \gamma$, searching for the control law (3) that maximizes the set $\mathcal{E}(P_\alpha)$ for a given sampling interval $[\eta_1, \eta_2]$.

**Remark 4:** This letter employs the ROLMIP parser [21] to derive a finite set of LMIs from the parameter-dependent conditions and their derivatives, and also to represent a parameter-dependent system in a convex formulation, which is augmented according to (5).

## V. ILLUSTRATIVE EXAMPLES

**Example 1:** Consider the LPV borrowed from [22]:

$$ \dot{x}_s(t) = \begin{bmatrix} 2\rho(t) & 1.1 + \rho(t) \\ -2.2 + \rho(t) & -3.3 + 0.1\rho(t) \end{bmatrix} x_s(t) + \begin{bmatrix} 2\rho(t) \\ 0.1 + \rho(t) \end{bmatrix} v(t), $$

<figure><img src="image_3.png" /><figcaption>**Fig. 2.** Cut of the estimate region of attraction $L_V$ at $v = 0$ obtained with optimization procedure $O_2$ and several trajectories projections.</figcaption></figure>

with $|\rho(t)| \le 1$, $|\dot{\rho}(t)| \le 0.2$, $\Lambda = 20$, limits of magnitude and rate saturation are given by $\mu_M = 1$ and $\mu_R = 1$, and a minimum sampling interval bound $\eta_1 = 0.01$s. Considering $\eta_2 = 0.1$s and applying the optimization procedure $O_2$ with $\kappa = 1.5$, we obtain $\gamma = 55.85$, controller gains $K_1 = [-0.0078 - 0.0084 \ 0.9065]$ and $K_2 = [-0.1941 - 0.0875 \ 0.7128]$, and an estimate region of attraction $L_V = \mathcal{E}(P_1) \cap \mathcal{E}(P_2)$ yields the cut with $v = 0$ shown in Fig. 2, where projections of the trajectories for several initial conditions are included. As expected, the trajectories (black lines) starting within $L_V$ (purple region) converge asymptotically to the origin, and trajectories (red lines) initiating reasonably close to the boundaries of $L_V$ diverge. A simpler controller implementation, for instance when $\rho(t)$ is not available for feedback, can be provided with a robust controller following the directions in Remark 2, achieving more conservative results. In this case, the maximum feasible limit is $\eta_2 = 0.078$s. For $\eta_2 = 0.05$s, the region obtained is bounded by the yellow dashed line. The number of decision variables are $V_1 = 323$ (robust case) and $V_2 = 619$ (LPV case).

To compare our conditions with [14], we consider the time-invariant case with $\rho(t) = 0$. Using the optimization procedure $O_1$ with $R_0 = 1.5\mathbf{I}$, assuming all matrix variables are parameter-independent, we find an upper limit of the sampling interval of $\eta_2 = 0.205$s. Conversely, [14, Th. 3] achieves $\eta_2 = 0.176$s, highlighting that the proposed functional in this letter leads to less conservative solutions.

**Example 2:** Consider the uncertain helicopter model presented in [23, Ex. 5], with $\Lambda = \mathrm{diag}\{20, 20\}$, saturation limits $\mu_M = [4 \ 4]$ and $\mu_R = [1 \ 1]$, and a minimum sampling interval bound $\eta_1 = 0.001$s. The closed-loop augmented and written as (5) results in a sixth-order uncertain system with eight vertices and two control inputs. Assuming that the time-varying parameters are unknown, we obtain a robust controller using the change of variables in Remark 2. With $O_1$, we reach $\eta_2 = 0.101$s as the maximum sampling interval bound. Assuming $\eta_2 = 0.05$s and applying $O_2$ with $\kappa = 1.7$, we find $\gamma = 5.9245$ yielding a safe ball with radius 0.41 in the
<figure><img src="image_4.png" /><figcaption>Fig. 3. Top plot: states of the system, $x_S(t)$. Bottom plot: actuator's state dynamics $\dot{v}(t)$ on left scale, actuator's state $v(t)$, and control signal magnitude saturation function $sat_M(u(t))$ on right.</figcaption></figure>

state space. Fig. 3 presents the closed-loop system simulation with the robust controller obtained for aperiodic samplings $\eta_k \in [0.001, 0.05]$s, and initial conditions belonging to the region of attraction. The top plot displays the evolution of the system's state, $x_S(t)$, while the bottom plot shows the signals $\dot{v}(t)$, $v(t)$, and $sat_M(u(t))$ of both actuators. Despite the actuators' rate saturation reaching their limits at the beginning of the simulation, the system remains stable, converging to the origin asymptotically. In case the LPV parameters are available with $|\dot{q}_1| \le 0.15$, $|\dot{q}_2| \le 0.8$, and $|\dot{q}_3| \le 1.7$, Theorem 1 yields a maximum $\eta_2 = 0.114$s. Similar to the robust case, with $\eta_2 = 0.05$s, $O_2$ achieves $\gamma = 2.3183$: a safe ball of radius 0.6567, which is $\approx 60\%$ greater than the robust case. Such an increase in the safe ball occurs at the expense of increasing the decision variable numbers from $V_1 = 1275$ (robust gain) to $V_8 = 9521$ (LPV gain). Note that conditions in [14] cannot be used in this case due to the LPV system.

## VI. CONCLUSION

This letter has presented a novel approach for designing parameter-dependent sampled-data state feedback controllers for LPV systems with actuators subject to both magnitude and rate saturation, using Linear Matrix Inequalities (LMIs). The method has combined the input-delay approach and a looped-functional, whose derivative has been verified through a second-order polynomial negativity lemma. We have proposed optimization procedures to enhance the aperiodic sampling interval and maximize the region of attraction, ensuring stable convergence of trajectories to the origin. Two numerical examples have demonstrated the efficiency of the proposed design, showing less conservative stability conditions compared to existing methods. Future work will explore applications in physical systems and the presence of exogenous disturbances.

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