# Robust Sampled-Time Control for a Class of Constrained Nonlinear Systems: An Interval Predictor-based MPC Approach

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

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Ariana Gutiérrez, Héctor Ríos, Manuel Mera, Denis Efimov, Rosane Ushirobira. Robust Sampled-Time Control for a Class of Constrained Nonlinear Systems: An Interval Predictor-based MPC Approach. European Journal of Control, In press, 10.1016/j.ejcon.2025.101230. hal-05031542

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https://inria.hal.science/hal-05031542v1

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# Robust Sampled-Time Control for a Class of Constrained Nonlinear Systems: An Interval Predictor-based MPC Approach

Ariana Gutiérrez<sup>a</sup>, Héctor Ríos<sup>a,b</sup>, Manuel Mera<sup>c</sup>, Denis Efimov<sup>d</sup>, and
Rosane Ushirobira<sup>d</sup>

## Abstract

The aim of this study is to design a robust sampled-time controller to stabilize continuous-time nonlinear systems, taking into account state and input constraints, and external disturbances. The proposed strategy is based on two essential components: an interval predictor-based state-feedback controller and a Model Predictive Control (MPC) approach, which deals with the state and input constraints. The interval predictor-based state-feedback controller is designed such that a safe set is provided, where the state constraints are not violated. Such a safe set characterizes the region where the MPC is activated, i.e., out of this set, guaranteeing the fulfillment of the state and input constraints. The proposed switched control strategy ensures the Input-to-State practical Stability of the considered nonlinear systems with respect to external disturbances. To compute the controller gains, a constructive method based on linear matrix inequalities (LMIs) is proposed and the state of the system is not required. The feasibility of the proposed scheme is illustrated by some simulation results.

**Key words:** Predictive Control; Robust Control; Constrained Nonlinear Systems.

## 1 Introduction

Real dynamical systems are restrained due to the structural limitations of their model, i.e., the system dynamics may be augmented by state and input constraints (see, e.g. [1], [28], and [29]). These kinds of systems can generally be represented by nonlinear models. Furthermore, the systems may be subject to external disturbances and/or parameter uncertainties that may have an impact on their behavior. To design a controller, which is able to counteract the effects of external disturbances while considering state and input constraints, robust control techniques are required. For instance, the MPC is a technique that obtains the current control action through the on-line solution of a finite

<footnote><sup>a</sup>Tecnológico Nacional de México/I.T. La Laguna, 27000, Torreón Coahuila, México</footnote>

<footnote><sup>b</sup>CONAHCYT, Investigadoras e Investigadores por México, 03940, Ciudad de México, México</footnote>

<footnote><sup>c</sup>Instituto Politécnico Nacional, ESIME-UPT, 07340, Ciudad de México, México</footnote>

<footnote><sup>d</sup>Inria-Lille Nord Europe, Université de Lille CNRS, UMR 9189-CRISTAL, F-59000 Lille, France</footnote>

<footnote>* 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, Investigadoras e Investigadores por México, 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. This work was developed in the framework of the international network Red Internacional de Control y Computo Aplicados" supported by TecNM.</footnote>

<footnote>Email addresses: m.agutierrezo@ correo.itlalaguna.edu.mx (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).</footnote>
horizon open-loop optimal control problem, at each sampling instant, using the current state of the plant as the initial state. The first control obtained in the optimal control sequence is applied to the system [20]. For instance, in [18] and [17], there is an overview of the different MPC approaches used for addressing general problems, highlighting how these approaches can be effectively employed in diverse and complex scenarios.

Concerning the control design for constrained systems affected by external disturbances, in [3], an MPC is proposed for uncertain linear systems, which guarantees the robustness of the system constraints and Input-to-State Stability (ISS) of the closed-loop system. In [9], a dynamic output-feedback MPC is proposed for an uncertain linear system. Some LMI conditions are formulated to have an efficient convex optimization problem. However, only the unperturbed case is considered. Also, the MPC approach is widely used for linear parameter varying (LPV) systems. For instance, output feedback robust MPC for uncertain LPV systems with external disturbances is proposed in [5] and [23]. In [5] an output feedback robust MPC is parameterized as a dynamic output feedback control law. However, to accomplish the control task, this method requires a heavy computational burden. While, in [23] the estimated state and estimation error converge to the corresponding invariant sets. In both works, recursive feasibility and robust stability are ensured. For discrete-time systems, the MPC can also be applied. For instance, in [21], a robust MPC is proposed for constrained, linear, discrete-time systems in the presence of bounded external disturbances. In [19], an output feedback MPC for discrete-time linear systems affected by bounded state and output disturbances is considered. The controller is composed by a stable state estimator and an MPC. Meanwhile, an online solution of a standard quadratic optimization program is required. The closed-loop system renders a specified invariant set robustly exponentially stable. In [7], an output-feedback MPC algorithm for discrete-time constrained linear systems affected by additive noise is presented. The algorithm ensures convergence of the state to a suitable neighborhood of the origin. Note that most MPC results in the literature are devoted to linear systems, and this could be restrictive.

Regarding the application of the MPC approach to nonlinear systems, for instance, in [16], a robust learning-based MPC is proposed for nonlinear systems subject to input and output constraints. The prediction model is estimated by a nonparametric machine learning technique; thus, an MPC, without terminal constraint, is obtained, then the closed-loop is asymptotically stable. The MPC can also be applied to nonlinear discrete-time systems. For instance, in [30], an MPC with a composite self-triggered mechanism is proposed to reduce the communication and computational burden while maintaining the desired control performance for this kind of systems. However, the algorithm has a relatively low disturbance tolerance. In [15], a networked dual-mode adaptive horizon MPC is proposed for a discrete-time perturbed nonlinear system with state and input constraints. The proposed switched controller is based on a local controller and a remote controller subject to packet losses, which guarantee the Input-to-State practical Stability (ISpS) of the system trajectories. On the other hand, in the previous works, a heavy computational burden is required because the MPC is active during the control task.

It is also possible to combine the MPC with other techniques, e.g., adaptive control, sliding-mode control (SMC) and interval observer techniques, to improve its performance. For instance, in [14], an adaptive MPC for linear systems with external disturbances is proposed. In [4], an MPC algorithm is developed for discrete-time constrained linear-time invariant systems with parameter uncertainties. An adaptive controller is used to handle the parameter uncertainties, and it is combined with an MPC algorithm guaranteeing the constraints satisfaction. On the other hand, an MPC is proposed in [11] for nonlinear systems based on a terminal cost function characterized by an implicit SMC law. However, a linearized model is required to guarantee the asymptotic stability of the closed-loop system. In [10], an event-triggered control scheme is presented for nonlinear constrained continuous-time uncertain system. The algorithm relies on a combination of an MPC and integral sliding mode control, ensuring regional ISpS. In [25], an interval observer and predictor are incorporated into the classic MPC scheme for discrete-time constrained linear systems with bounded state disturbances. Nevertheless, most of the above-mentioned works do not consider sampled control.

Motivated by these issues, i.e., external disturbances, system constraints, and sampled control, this paper proposes the design of a robust sampled controller to stabilize continuous-time nonlinear systems, taking into account state and input constraints, and external disturbances. The robust controller is based on a sampled interval predictor-based state-feedback controller and an MPC approach. The proposed control method possesses the following features:

1) Due to the use of an interval predictor, the measurements of the system states are not required for the proposed scheme.

2) Characterization of a safe set where the state constraints are not transgressed using the interval predictor-based state-feedback controller.

3) Low computational cost since the MPC is only activated out of the safe set, and it can be pre-computed in advance.
4) Local Input-to-State practical Stability of the constrained nonlinear systems with respect to external disturbances.

5) The synthesis of the controller is constructive since it is based on LMIs.

The fact that the proposed scheme does not require the measurement of the system state represents an advantage in real-world applications (e.g., the blood glucose regulation and the anesthesia regulation in delivery machines, and some trajectory-tracking problems in mobile robots) that could be solved by using the proposed framework. Therefore, the proposed method can be interpreted as a tool to design robust feedforward inputs.

A preliminary version of this work has been presented in [8]. However, in [8], external disturbances are not considered and a detailed stability analysis is not provided. In addition, in the current proposal, we provide a different simulation example than in [8], where various values of sampling intervals are implemented, demonstrating the effectiveness of the proposed algorithm.

The organization of this work is as follows. In Section 2 the problem statement is described. Section 3 presents some preliminary concepts. The robust control design is introduced in Section 4. Simulation results are depicted in Section 5. Finally, the conclusions are presented in Section 6. The convergence proofs of the proposed algorithm are postponed to the Appendix.

**Notation:** The sets of real and integer numbers are defined by $\mathbb{R}$ and $\mathbb{Z}$, respectively; $\mathbb{R}_+ := \{s \in \mathbb{R} : s \ge 0\}$ and $\mathbb{Z}_+ := \{s \in \mathbb{Z} : s \ge 0\}$. For a matrix $A$, we define $A^+ = \max\{0, A\}$ and $A^- = A^+ - A$ (similarly for vectors). For two vectors $x_1, x_2 \in \mathbb{R}^n$ or matrices $A_1, A_2 \in \mathbb{R}^{n \times n}$, the relations $x_1 \le x_2$ and $A_1 \le A_2$ are understood in an element-wise sense. The relation $P < 0$ ($P \ge 0$) means that a symmetric matrix $P \in \mathbb{R}^{n \times n}$ is negative (positive semi-) definite. For a matrix $A \in \mathbb{R}^{m \times n}$, the induced norm is the spectral norm, i.e., $||A|| = \sqrt{\lambda_{\max}(A^\top A)}$. For a Lebesgue measurable function $w: \mathbb{R}_+ \to \mathbb{R}^q$, define $||w||_{(t_0, t_1)} := \text{ess} \sup_{t \in (t_0, t_1)} ||w(t)||$; then, $||w||_\infty = ||w||_{(0, +\infty)}$ and the set of all the functions $w$ with the property $||w||_\infty < +\infty$ is denoted as $\mathcal{L}_\infty^q$. A continuous function $\alpha: \mathbb{R}_+ \to \mathbb{R}_+$ belong to class $\mathcal{K}$ if it is strictly increasing and $\alpha(0) = 0$. A continuous function $\beta: \mathbb{R}_+ \to \mathbb{R}_+$ belongs to class $\mathcal{KL}$ if, for each fixed $s, \beta(r, s) \in \mathcal{K}$ with respect to $r$, and for each fixed $r, \beta(r, s)$ is decreasing to zero with respect to $s$. A matrix is called Metzler when all its non-diagonal coefficients are non-negative.

## 2 PROBLEM STATEMENT

Consider the following class of nonlinear systems

$$ \dot{x}(t) = f(x(t)) + Bu(t) + Dw(t), \quad t \ge 0, \qquad (1) $$

where $x(t) \in \mathbb{R}^n$ is the state, $f: \mathbb{R}^n \to \mathbb{R}^n$ is a locally Lipschitz function, $u(t) \in \mathbb{R}^p$ is the control input, and $w \in \mathbb{R}^q$ is the disturbance term. The matrices $B$ and $D$ are known of suitable dimensions. It is considered that $u \in \mathcal{U} := \{u \in \mathbb{R}^p | -u_{\max} \le u \le u_{\max}\}$, with $0 < u_{\max} \in \mathbb{R}^p$ as the vector containing the maximum value for each control signal. In this work, we consider sampled control, i.e., $u(t) = u(t_k)$, for $t \in [t_k, t_{k+1})$ and $k \in \mathbb{Z}_+$, where $t_k = kh$ form a strictly growing sequence of time instants for a sampling interval $h > 0$.

Moreover, the solutions of the system (1) must be constrained inside the polytope

$$ \mathcal{X} := \{x \in \mathbb{R}^n | \mathbf{b}x \le \mathbf{1}_{k_x}\}, \qquad (2) $$

where $\mathbf{b} = (b_1, \dots, b_{k_x})^\top \in \mathbb{R}^{k_x \times n}$, with $b_i \in \mathbb{R}^n$ being some given vectors that characterize the state constraints, $k_x \in \mathbb{Z}$ is the number of the polytope faces, and $\mathbf{1}_{k_x} = (1, \dots, 1)^\top \in \mathbb{R}^{k_x}$. We assume that the origin belongs to the interior of the set $\mathcal{X}$.

Before proceeding, the following assumptions are imposed on the system (1).

**Assumption 1** There exist a known Metzler matrix $A_0 \in \mathbb{R}^{n \times n}$ and known matrices $A_j \in \mathbb{R}^{n \times n}$, $j = 1, \dots, N$ for some
$N \in \mathbb{Z}_+$, such that the following relations are satisfied

$$f(x) = \left( A_0 + \sum_{j=1}^{N} \alpha_j(x) A_j \right) x, \\ \sum_{j=1}^{N} \alpha_j(x) = 1, \alpha_j(x) \in (0,1), j = \overline{1,N}.$$

**Assumption 2** There exist known signals $w, \bar{w} \in \mathcal{L}_{\infty}^q$, a constant $w_{\max} > 0$, and known vectors $\underline{x}_0, \bar{x}_0 \in \mathbb{R}^n$, such that $\underline{w}(t) \le w(t) \le \bar{w}(t)$, $||w||_{\infty} \le w_{\max}$, and $\underline{x}_0 \le x(0) \le \bar{x}_0$, respectively.

Assumption 1 characterizes the class of nonlinearities of the system (1), i.e., those that can be expressed by a sum of a linear term, $A_0x$, and a convex nonlinear function, $\sum_{j=1}^{N} \alpha_j(x)A_jx$. On the other hand, Assumption 2 and the condition that $A_0$ must be Metzler, are required to design an MPC and an interval predictor (IP) for the system (1).

The aim of this manuscript is to design a sampled control law to ensure the stabilization of the system (1), taking into account the input and state constraints, i.e., $x \in \mathcal{X}$ and $u \in \mathcal{U}$, under Assumptions 1 and 2.

### 3 PRELIMINARIES

Consider the following nonlinear system

$$\dot{x} = f(x, w), \quad (3)$$

where $x \in \mathbb{R}^n$ is the state vector, $w \in \mathbb{R}^q$ is the external disturbance; $f: \mathbb{R}^n \times \mathbb{R}^q \to \mathbb{R}^n$ is a locally Lipschitz function. The solution of system (3) for an initial condition $x_0 \in \mathbb{R}^n$ and $w \in \mathcal{L}_{\infty}^q$ is denoted as $x(t, x_0, w)$, for any $t \ge 0$ for which the solution exists.

**Definition 1** [2],[26] The system (3) is said to be Input-to-State practically Stable (ISpS) if there exist some functions $\beta \in \mathcal{KL}$ and $\gamma \in \mathcal{K}$, and a constant $\kappa \in \mathbb{R}_+$, such that for any $w \in \mathcal{L}_{\infty}^q$, and any $x_0 \in \mathbb{R}^n$

$$||x(t, x_0, w)|| \le \beta(||x_0||, t) + \gamma(||w||_{\infty}) + \kappa, \forall t \ge 0. \quad (4)$$

If $\kappa = 0$, the system (3) is said to be Input-to-State Stable (ISS).

To establish this property the Lyapunov function method can be used. The next lemma gives the interval upper and lower bounds for product of matrix and vector variables.

**Lemma 1** [6] Let $x \in \mathbb{R}^n$ be a vector variable, $\underline{x} \le x \le \bar{x}$ for some $\underline{x}, \bar{x} \in \mathbb{R}^n$, then

* If $A \in \mathbb{R}^{m \times n}$ is a constant matrix then

$$A^{+} \underline{x} - A^{-} \bar{x} \le Ax \le A^{+} \bar{x} - A^{-} \underline{x}. \quad (5)$$

* If $A \in \mathbb{R}^{m \times n}$ is a variable matrix and $\underline{A} \le A \le \bar{A}$ for some $\underline{A}, \bar{A} \in \mathbb{R}^{m \times n}$, then

$$\underline{A}^{+} \underline{x}^{+} - \bar{A}^{+} \bar{x}^{-} - \underline{A}^{-} \bar{x}^{+} + \bar{A}^{-} \underline{x}^{-} \le Ax \le \underline{A}^{+} \bar{x}^{+} - \bar{A}^{+} \underline{x}^{-} - \underline{A}^{-} \bar{x}^{+} + \bar{A}^{-} \underline{x}^{-}. \quad (6)$$

A variant of Grönwall-Bellman inequality is formulated in the next lemma:

**Lemma 2** [12] Let $\lambda: [a, b] \to \mathbb{R}$ be a continuous function and $\mu: [a, b] \to \mathbb{R}$ be a continuous and non-negative function. If a continuous function $y: [a, b] \to \mathbb{R}$ satisfies

$$y(t) \le \lambda(t) + \int_{a}^{t} \mu(s)y(s)ds,$$
for $a \le t \le b$, then, on the same interval, it holds that

$$y(t) \le \lambda(t) + \int_{a}^{t} \lambda(s)\mu(s)e^{\int_{s}^{t} \mu(r)dr} ds.$$

# 4 CONTROL DESIGN

The proposed controller comprises the design of a robust control law, composed of an interval predictor-based state–feedback controller and the MPC, which deals with state and input constraints. We will show that there exists a safe set $\mathcal{R}$ for the IP based state–feedback controller, where the state constraints are not transgressed, that characterizes the regions where each controller is activated: the MPC is activated outside the safe set, namely $\mathcal{R}$, while the state–feedback controller is applied inside of it. Since the control is sampled, a sampled–time IP is designed for the continuous–time system (1), which considers the error of discretization and guarantees a careful treatment of the constraints.

The following section addresses the design of the IP for the nonlinear system (7), satisfying Assumptions 1 and 2.

## 4.1 Interval Predictor

According to Assumption 1, the system (1) has the following representation

$$\dot{x}(t) = \left( A_0 + \sum_{j=1}^{N} \alpha_j(x) A_j \right) x(t) + Bu(t) + Dw(t). \qquad (7)$$

Based on [24], assuming that $\underline{x} \le x \le \bar{x}$ and applying (6) to $\alpha_j(x)x$, the relation $-\underline{x}^{-} \le \alpha_j(x)x \le \bar{x}^{+}$ holds. Thus, according to (5), the following inequalities can be obtained

$$-\Delta A^{+} \underline{x}^{-} - \Delta A^{-} \bar{x}^{+} \le \sum_{j=1}^{N} \alpha_{j}(x) A_{j} x \le \Delta A^{+} \bar{x}^{+} + \Delta A^{-} \underline{x}^{-},$$

where $\Delta A^{+} = \sum_{i=1}^{N} A_{i}^{+}$ and $\Delta A^{-} = \sum_{i=1}^{N} A_{i}^{-}$. Then, a continuous-time IP for (7) can be stated as follows [13]:

$$\dot{z}(t) = A_0 z(t) + A_1 z^+(t) + A_2 z^-(t) + Bu(t) + \delta(t), \qquad (8)$$

where $z = [\underline{x}^{\top}, \bar{x}^{\top}]^{\top} \in \mathbb{R}^{2n}$ is the extended state vector of the predictor, the extended system matrices $A_0 \in \mathbb{R}^{2n \times 2n}$, $A_1 \in \mathbb{R}^{2n \times 2n}$, $A_2 \in \mathbb{R}^{2n \times 2n}$, and $B \in \mathbb{R}^{2n \times p}$ are given as

$$A_0 = \begin{pmatrix} A_0 & 0 \\ 0 & A_0 \end{pmatrix}, \quad A_1 = \begin{pmatrix} 0 & -\Delta A^{-} \\ 0 & \Delta A^{+} \end{pmatrix}, \quad A_2 = \begin{pmatrix} -\Delta A^{+} & 0 \\ \Delta A^{-} & 0 \end{pmatrix}, \quad B = \begin{pmatrix} B \\ B \end{pmatrix},$$

and the term $\delta \in \mathbb{R}^{2n}$ is defined as follows

$$\delta(t) = \mathcal{D} \left( \frac{\underline{w}(t)}{\bar{w}(t)} \right) = \begin{pmatrix} D^{+} & -D^{-} \\ -D^{-} & D^{+} \end{pmatrix} \left( \frac{\underline{w}(t)}{\bar{w}(t)} \right).$$

Being properly initialized by $\underline{x}(0) = \underline{x}_0$ and $\bar{x}(0) = \bar{x}_0$, this IP ensures the desired interval inclusion property [13] $\underline{x}(t) \le x(t) \le \bar{x}(t)$, $\forall t \ge 0$. Note that, based on Assumption 2, $\delta$ is known. Then, for any $t \ge t_0 \ge 0$ the solution of system (8) is given by

$$z(t) = e^{A_0(t-t_0)} z(t_0) + \int_{t_0}^{t} e^{A_0(t-\tau)} Bu(\tau)d\tau + \int_{t_0}^{t} e^{A_0(t-\tau)} \delta(\tau)d\tau + \int_{t_0}^{t} e^{A_0(t-\tau)} g(z(\tau))d\tau, \qquad (9)$$
where $g(z(t)) = A_1z^+(t) + A_2z^-(t)$. The expression (9), with a sampled control $u(t) = u(t_k)$, for $t \in [t_k, t_{k+1})$, and $k \in \mathbb{Z}$, generates a discrete-time sequence:

$$z(t_{k+1}) = e^{A_0 h} z(t_k) + \int_0^h e^{A_0 (h-s)} \mathcal{B} ds \cdot u(t_k) + \delta_k + \int_{t_k}^{t_{k+1}} e^{A_0 (t_k-s)} g(z(s)) ds,$$

where $h = t_{k+1} - t_k$ is the sampling interval, $t_k$ is the sampling time instant, and $\delta_k = \int_{t_k}^{t_{k+1}} e^{A_0 (t_{k+1}-s)} \delta(s) ds$. Adding and subtracting the term $\int_0^h e^{A_0 (h-s)} ds \cdot g(z_k)$, it follows that

$$z(t_{k+1}) = e^{A_0 h} z(t_k) + \int_0^h e^{A_0 (h-s)} \mathcal{B} ds \cdot u(t_k) + \delta_k + \int_0^h e^{A_0 (h-s)} ds \cdot g(z(t_k)) + \varphi_k, \quad (10)$$

with $\varphi_k = \int_{t_k}^{t_{k+1}} e^{A_0(t_{k+1}-s)} [g(z(s)) - g(z(t_k))] ds$. Let us consider the notation $z(t_k) = z_k$, $u(t_k) = u_k$, and $\mathcal{G} = \int_0^h e^{A_0(h-s)} ds \in \mathbb{R}^{2n \times 2n}$. Thus, (10) can be rewritten as

$$z_{k+1} = \bar{A}_0 z_k + \bar{B} u_k + \delta_k + \mathcal{G} g(z_k) + \varphi_k, \quad (11)$$

with $\bar{A}_0 = e^{A_0 h} \in \mathbb{R}^{2n \times 2n}$ and $\bar{B} = \mathcal{G} \mathcal{B} \in \mathbb{R}^{2n \times p}$. Note that the interval predictor (11) cannot be implemented since $\varphi_k$, which characterizes the discretization accuracy and depends on $z(t)$ with $t \in [t_k, t_{k+1}]$, cannot be computed. Thus, consider the following predictor

$$\hat{z}_{k+1} = \bar{A}_0 \hat{z}_k + \bar{B} u_k + d_k + \mathcal{G} g(\hat{z}_k), \quad (12)$$

with $d_k = \delta_k + \bar{\varphi}$, where $\bar{\varphi}^\top = (-\varphi_{\max} \mathbf{1}_n^\top, \varphi_{\max} \mathbf{1}_n^\top)$ and $\varphi_k \le \varphi_{\max}$, the bound $\varphi_{\max}$ is evaluated later during the proof. This parameter evaluates the error of discretization of (8), and taking it into account allows us to guarantee that the constraints will not be violated due to numeric errors. To stabilize the nonlinear dynamics (1), we need to design $u_k$ to take the trajectories of the system (12) to a vicinity of zero dealing with the state and input constraints. Let us propose the following control law for the system (11)

$$u_k = \begin{cases} \bar{u}_k & \text{if } \hat{z}_k \notin \mathcal{R}, \\ \hat{u}_k & \text{if } \hat{z}_k \in \mathcal{R}, \end{cases} \quad (13)$$

where $\mathcal{R} \subset \mathbb{R}^{2n}$ is the safe set, $\bar{u}_k$ is the control signal computed by the MPC algorithm and $\hat{u}_k$ is the state-feedback control law. Notice that $\hat{u}_k$ is applied only inside the safe set $\mathcal{R}$, defined further on, where the state constraints are not transgressed.

## 4.2 State-Feedback Control

Let us propose the following nonlinear state-feedback control

$$\hat{u}_k = K_0 \hat{z}_k + K_1 \hat{z}_k^+ + K_2 \hat{z}_k^- + S_1 d_k. \quad (14)$$

The next theorem provides a way to compute the controller gains $K_0, K_1, K_2, S_1 \in \mathbb{R}^{p \times 2n}$.

**Lemma 3** Let Assumptions 1 and 2 be satisfied, and the control law (14) be applied to the system (11). Suppose that there exist diagonal matrices $0 < X_1 \in \mathbb{R}^{2n \times 2n}, 0 < \tilde{Q}_0 \in \mathbb{R}^{2n \times 2n}, 0 > \tilde{Q}_1 \in \mathbb{R}^{2n \times 2n}, 0 > \tilde{Q}_2 \in \mathbb{R}^{2n \times 2n}, 0 < \tilde{R}_0,$
$\tilde{R}_1, \tilde{R}_2 \in \mathbb{R}^{2n \times 2n}, \tilde{R}_3 \in \mathbb{R}^{2n \times 2n},$ and matrices $Y_0, Y_1, Y_2, Y_3 \in \mathbb{R}^{2n \times p}$, such that the following set of LMIs

$$
\left(
\begin{array}{cccccccccc}
-X_1 + \tilde{Q}_0 & \tilde{R}_1 & \tilde{R}_2 & 0 & 0 & 0 & X_1 \overline{\mathcal{A}}_0^\top + Y_0 \overline{\mathcal{B}}^\top & & \\
\star & \tilde{Q}_1 & \tilde{R}_3 & 0 & 0 & 0 & X_1 \mathcal{A}_1^\top \mathcal{G}^\top + Y_1 \overline{\mathcal{B}}^\top & & \\
\star & \star & \tilde{Q}_2 & 0 & 0 & 0 & X_1 \mathcal{A}_2^\top \mathcal{G}^\top + Y_2 \overline{\mathcal{B}}^\top & & \\
\star & \star & \star & -\tilde{R}_0 & 0 & 0 & X_1 + Y_3 \overline{\mathcal{B}}^\top & & \\
\star & \star & \star & \star & -\gamma I & 0 & X_1 & & \\
\star & \star & \star & \star & \star & -\gamma I & Y_3 \overline{\mathcal{B}}^\top & & \\
\star & \star & \star & \star & \star & \star & -X_1 & & \\
\end{array}
\right) \preceq 0, \qquad (15a)
$$
$$
\tilde{Q}_0 + \min\{\tilde{Q}_1, \tilde{Q}_2\} + 2 \min\{\tilde{R}_1, \tilde{R}_2\} \ge \alpha X_1, \qquad (15b)
$$

holds, with fixed $\alpha, \gamma > 0$. Additionally, suppose that the inequality

$$
\overline{q} \overline{b} X_1 \overline{b}^{\top} \leq \mathbf{1}_{2k_x}, \qquad (16)
$$

holds with $\bar{\mathbf{b}} = (\mathbf{b}^{\top}, \mathbf{b}^{\top})$, $\bar{q} = \alpha\epsilon^{-1}$, $\epsilon = \lambda_{\max}\{R_0\}\bar{\delta}_k^2 + q_1\bar{\delta}_k + q_2$, $\bar{\delta}_k = 2\|\mathcal{D}\|w_{\max}$, and

$$
q_1 = 2c_3 \tilde{e}, \qquad (17a)
$$

$$
q_2 = 2(c_1 \zeta + c_2 \tilde{e} + (c_4 + \sqrt{2nc_5}) \varphi_{\max}) \tilde{e} + c_3^2 \lambda_{\max} \{R_0\} + 4\gamma(1+2n)(c_6^2 + c_7^2)\zeta + k_{e2} \|\mathcal{B}\|^2 \bar{u}_{\max}^2 \quad (17b)
$$

$$
\zeta = \sqrt{2k_x \lambda_{\max}^{-1} \{\mathbf{b}^{\top}\mathbf{b}\}}, \quad \bar{u}_{\max} = \|u_{\max}\|, \qquad (17c)
$$

$$
\tilde{e} = [k_1(\xi + 1) + k_2(\|\mathcal{A}_1\| + \|\mathcal{A}_2\|)]\zeta + k_3(\|\mathcal{B}\|\bar{u}_{\max} + \lambda_{\max}^{-1/2}\{R_0\}), \quad (17d)
$$

$$
\varphi_{\max} = (c_6 + c_7)\zeta + c_8(\|\mathcal{B}\|\bar{u}_{\max} + \lambda_{\max}^{-1/2}\{R_0\}), \qquad (17e)
$$

for given $R_0 = X_1 \tilde{R}_0 X_1$, $\mathbf{b} = (b_1, \dots, b_{k_x})^\top \in \mathbb{R}^{k_x \times n}$, $\xi > 0$, $u_{\max} > 0$, and

$$
c_1 = (\|\tilde{\mathcal{A}}_0\| + \|\tilde{\mathcal{A}}_1\| + \|\tilde{\mathcal{A}}_2\|)\|\overline{P\tilde{B}}\|\overline{K}, \qquad (18a)
$$

$$
\begin{align}
c_2 ={}& (0.5\|K_0\| + \|K_1\| + \|K_2\|)\|\overline{P\tilde{B}}K_0\| + (\|K_0\| + 0.5\|K_1\| + \|K_2\|)\|\overline{P\tilde{B}}K_1\| \nonumber \\
       & + (\|K_0\| + \|K_1\| + 0.5\|K_2\|)\|\overline{P\tilde{B}}K_2\|, \tag{18b}
\end{align}
$$

$$
c_3 = \| \tilde{S}_1^\top \overline{P \tilde{B}} \| \overline{K}, \quad c_4 = \| \overline{P \tilde{B}} \| \overline{K}, \qquad (18c)
$$

$$
c_5 = \| S_1^\top \bar{B}^\top P \bar{B} \| \bar{K}, \qquad (18d)
$$

$$
c_6 = (1 + \sqrt{2n})^{-1} (k_{e1}\xi - \|\bar{\mathcal{A}}_0\| - L\|\mathcal{G}\| + \|\mathcal{A}_1\| + \|\mathcal{A}_2\|), \qquad (18e)
$$

$$
c_7 = (1 + \sqrt{2n})^{-1} (k_{e1} - \|\bar{\mathcal{A}}_0\| - L \|\mathcal{G}\|), \qquad (18f)
$$

$$
c_8 = (1 + \sqrt{2n})^{-1} k_{e1} k_{e2}, \qquad (18g)
$$

where $k_{e1} = 1 + \xi \nu^{-1} e^{k_{e1} \xi}$, $k_{e2} = k_{e1} k_{e2}$, $k_{\xi} = (1 - e^{-\nu h})$, $\nu = |\text{Re}\{\lambda_{\max}\{\mathcal{A}_0\}\}|$, $P = X_1^{-1}$, $\tilde{\mathcal{A}}_0 = \overline{\mathcal{A}}_0 + \overline{\mathcal{B}}K_0$, $\tilde{\mathcal{A}}_1 = \mathcal{G}\mathcal{A}_1 + \overline{\mathcal{B}}K_1$, $\tilde{\mathcal{A}}_2 = \mathcal{G}\mathcal{A}_2 + \overline{\mathcal{B}}K_2$, $\tilde{S}_1 = I + \overline{\mathcal{B}}S_1$, and $\overline{K} = \|K_0\| + \|K_1\| + \|K_2\|$. If the IP (12) is initialized as $\hat{z}_0 = (\underline{x}_0^\top, \bar{x}_0^\top)^\top$, and the controller gains are selected as $K_0 = Y_0^\top X_1^{-1}$, $K_1 = Y_1^\top X_1^{-1}$, $K_2 = Y_2^\top X_1^{-1}$, and $S_1 = Y_3^\top X_1^{-1}$, then the origin of the system dynamics (12) is ISpS, with respect to $\delta_k$, and the system trajectories fulfill the state constraints, i.e., $x \in \mathcal{X}$. Moreover, the safe set is given as

$$
\mathcal{R} = \{ \hat{z}_k \in \mathbb{R}^{2n} : \hat{z}_k^\top X_1^{-1} \hat{z}_k \le \alpha^{-1} \epsilon \}, \qquad (19)
$$

with some $\alpha > 0$ satisfying (15b).

The proofs of this lemma and other results are given in the Appendix.

**Remark 1** Note that it is desirable that the safe set $\mathcal{R}$ is as big as possible. Thus, it is possible to establish an optimization problem in order to maximize its volume. For example, let assumptions 1 and 2 be satisfied and the
control law (14) be applied to the system (11). If the controller gains $K_0 = Y_0^\top X_1^{-1}$, $K_1 = Y_1^\top X_1^{-1}$, $K_2 = Y_2^\top X_1^{-1}$, and $S_1 = Y_3^\top X_1^{-1}$ are computed by solving the following optimization problem

$$ \max_{X_1, Y_0, Y_1, Y_2, Y_3} \log \det(X_1) \quad \text{subject to (15),} \qquad (20) $$

then the origin of the system dynamics (12) is ISpS with respect to $\delta_k$ with optimized disturbance attenuation.

## 4.3 Design of the Model Predictive Control

For the design of the MPC, the following assumption is considered.

**Assumption 3** There exist gains $K_0, K_1, K_2, S_1 \in \mathbb{R}^{p \times 2n}$, satisfying the conditions of Lemma 3, and $\hat{u}_k(t) = K_0 \hat{z}_k(t) + K_1 \hat{z}_k^+(t) + K_2 \hat{z}_k^-(t) + S_1 d_k \in \mathcal{U}$, for all $\hat{z}_k(t) \in \mathcal{R} \subset \mathcal{X}$ and $t \ge 0$.

Assumption 3 implies that the invariant set $\mathcal{R}$, which is the safe set of system (1), is completely contained inside the polytope (2), i.e., the constrained region $\mathcal{X}$.

Let us consider that the prediction can be performed in a receding horizon fashion with a window length $N > 1$. The interval predictor (12) has a sequence of inputs $S_N = \{s_0, \dots, s_{N-1}\}$, with $s_m \in \mathcal{U}$ for all $m = 0, \dots, N-1$, the values $\hat{z}_{k,m+1}$ will be calculated for $m = 0, \dots, N-1$ under substitution $\bar{u}_{k+m} = s_m$. Let us define $\hat{z}_{k+j} = \hat{z}_{k,j}$, for $j = 0, \dots, N$. Thus, the optimal control problem to be solved by the MPC is given as follows:

**Problem 1** For given matrices $0 \prec W = W^\top \in \mathbb{R}^{2n \times 2n}$, $0 \prec H = H^\top \in \mathbb{R}^{2n \times 2n}$, and $M = M^\top \in \mathbb{R}^{2p \times 2p}$, find the control signals

$$ S_N^k = \operatorname*{argmin}_{V_N} V_N(\hat{z}_{k,0}, \dots, \hat{z}_{k,N}, S_N), \qquad (21) $$

with the cost function $V_N(\hat{z}_{k,0}, \dots, \hat{z}_{k,N}, S_N) = V_f(\hat{z}_{k,N}) + \sum_{m=0}^{N-1} \ell(\hat{z}_{k,m}, s_m)$, where $V_f(\hat{z}) = \hat{z}^\top W \hat{z}$, $\ell(\hat{z}, s) = \hat{z}^\top H \hat{z} + s^\top M s$, such that the following constraints are satisfied: i) $\hat{z}_{k+j} \in \mathcal{X}$ for $j = 0, \dots, N$ are computed by (12). ii) $\bar{u}_{k+j} \in \mathcal{U}$ for $j = 0, \dots, N$. iii) $\hat{z}_{k,N} \in \mathcal{R}$.

Finally, considering the results of Lemma 3 and the solution of the Problem 1, the following theorem is presented.

**Theorem 1** Let Assumptions 1–3 be satisfied, the conditions of Lemma 3 be satisfied and Problem 1 be feasible. If the control law (13), given by (14) and (21), is applied to the system (1) and designed according to Lemma 3 and the solution of Problem 1; then, the origin of the system dynamics (11) is ISpS, with respect to $\delta_k$, in the safe set $\mathcal{R}$.

**Remark 2** The proposed scheme does not require the measurement of the system state, i.e., the control (14) is only based on the interval predictor (12). This can represent an advantage in real-world applications such as many problems in biological systems, e.g., the blood glucose regulation and the anesthesia regulation in delivery machines, and some trajectory-tracking problems in mobile robots, where the state measurements are complicated or subject to significant perturbations and delays.

**Remark 3** We use an LPV representation of nonlinear function $f(x(t))$, and since $x(t)$ is not measured while in general the scheduling parameters have to be dependent on the state in such a case, the values of these parameters are not used in the design. Thus, the presented approach can be applied to uncertain continuous LPV systems, without measured scheduling parameters, in the presence of disturbances and constraints.

# 5 SIMULATIONS RESULTS

Consider the following nonlinear system

$$ \dot{x}_1 = -x_1 + (1 + 0.01x_1)x_2 + w(t), $$

$$ \dot{x}_2 = (0.2x_1 - 1)x_2 + u + w(t), $$
with $w(t) = 0.2 \sin(3t) + 0.05$, thus, $\underline{w}(t) = -0.15$, $\overline{w}(t) = 0.25$, and $w_{max} = 0.25$. This system satisfies Assumptions 1 and 2. The matrices $A_0$ and $A_j$ can be obtained by a convex polytopic method, e.g., that one given in [22]. Let us consider that $u_{max} = 3.5$, $x_1 \in (-7, 7)$ and $x_2 \in (-7, 7)$, and hence $b_1 = (1/7, 0)^\top$, $b_2 = -b_1$, $b_3 = (0, 1/7)^\top$, and $b_4 = -b_3$. The initial conditions for the system are $x(0) = (5, 5)^\top$. It is possible to show that this nonlinear system, for such initial conditions, is unstable (see, please Fig. 1).

All the simulations are done in MATLAB, with the Euler discretization method, and an integration step equal to 0.001 [s]. In order to show the efficiency of the algorithm, the following values for the sampling intervals are considered $h = 0.1$ [s], $h = 0.2$ [s], $h = 0.3$ [s], and $h = 0.5$ [s]. The solution for the LMIs is found by means of SDPT3 solver, among YALMIP, while the MPC is implemented using the nlmpc MATLAB toolbox.

The controller gains are obtained looking for the solutions of the LMIs given in Lemma 3, for each value of $h$, with $\gamma = 1 \times 10^2$ and $\alpha = 1 \times 10^4$. For illustrative purposes, we only provide the numerical results for the case $h = 0.2$ [s], which are: $P = \text{diag}(0.1778, 0.1211, 0.2446, 0.2565)$, $\tilde{Q}_0 = \text{diag}(0.0059, 0.0019, 0.0086, 0.0074)$, $\tilde{Q}_1 = \text{diag}(-0.0630, -0.0642, -0.0871, -0.0772)$, $\tilde{Q}_2 = \text{diag}(-0.0862, -0.0769, -0.0602, -0.0621)$, $\tilde{R}_0 = (4.0175, 4.1258, 3.9378, 3.8402)10^3$, $\tilde{R}_1 = \text{diag}(0.0894, 0.0737, 0.0431, 0.0541)$, $\tilde{R}_2 = \text{diag}(0.2254, 0.1771, 0.1173, 0.0891)$, $K_0 = (-0.206, -1.5403, -0.2802, -3.0885)$, $K_1 = (0.0004, 0, -0.1880, -0.0082)$, $K_2 = (0.0938, 0.0069, 0.0015, 0.0016)$, and $S_1 = (0.0102, 0.0678, 0.0216, 0.1717)$. Then it is possible to verify that condition (15b) is fulfilled. For the MPC, we consider $W = I$, $H = 1 \times 10^3 I$, $M = 1 \times 10^{-4}$, and $N = 10$.

The initial conditions for the IP are set in $\hat{z}_k(0) = (3.2, 3.2, 6.8, 6.9)^\top$. For comparison purposes, we have added the simulation results for the case when the MPC is active during all the control task. The system trajectories, the IP (12) trajectories, and the corresponding state constraints are depicted in Fig. 2, for $h = 0.2$. Note that both the IP and the system trajectories never transgress the system constraints despite the external disturbances. On the other hand, in Fig. 3 we see that the trajectories of the system are inside of the potytope $\mathcal{X}$, all the time and despite to the external disturbances, for each value of $h$. In Fig. 3, the sets $\mathcal{R}_{\underline{x}}$ and $\mathcal{R}_{\overline{x}}$ are the projections of $\mathcal{R}$ in the coordinates of $\underline{x}$ and $\overline{x}$, respectively. This projection can be obtained by partitioning $X_\alpha = \alpha X_1^{-1} \varepsilon^{-1}$ into the following matrices

$$X_{\alpha} = \begin{pmatrix} X_{\alpha_{11}} & X_{\alpha_{21}}^{\top} \\ X_{\alpha_{21}} & X_{\alpha_{22}} \end{pmatrix},$$

where $X_{\alpha_{11}} \in \mathbb{R}^{2 \times 2}$, $X_{\alpha_{21}} \in \mathbb{R}^{2 \times 2}$, and $X_{\alpha_{22}} \in \mathbb{R}^{2 \times 2}$. Therefore, the matrices are computed as $\mathcal{R}_{\underline{x}} = X_{\alpha_{11}} - X_{\alpha_{21}}^{\top} X_{\alpha_{21}}^{-1} X_{\alpha_{22}}$ and $\mathcal{R}_{\overline{x}} = X_{\alpha_{22}} - X_{\alpha_{21}}^{\top} X_{\alpha_{11}}^{-1} X_{\alpha_{21}}$. Meanwhile, when only the MPC is applied, it can be seen that, under the same initial conditions, the system trajectories converge toward a region farthest from the origin. Additionally, Fig. 4 depicts the behavior of the control signals that never transgress the input constraints. While for the case in which only the MPC is applied, the control signal remains saturated most of the time during the control task.

In order to illustrate, through simulations, that the computational burden is relaxed in our control algorithm, with respect to a classic MPC. Then, it is sufficient to determine the time required to simulate the proposed switched scheme and the scheme when the MPC is active all the time. Considering that the simulations have been done in MATLAB with the Euler discretization method, sampling-time equal to 0.001 [s], simulation time equal to 10 [s], while the MPC has been implemented using the nlmpc toolbox in MATLAB, the proposed switching scheme takes around 5 seconds to perform the simulation, while the scheme when the MPC is active all the time takes more than 14 seconds to perform the same simulation for the same initial conditions.

# 6 CONCLUSIONS

This paper proposed the design of a robust sampled-time controller for stabilizing continuous-time nonlinear systems, considering state and input constraints and external disturbances. The proposed strategy is based on two essential components: an interval predictor-based state-feedback controller and an MPC approach, which deals with the state and input constraints. The interval predictor-based state-feedback controller is designed such that a safe set was provided, where the state constraints are not violated. Such a safe set characterizes the region where the MPC is activated, guaranteeing the fulfillment of the state and input constraints. The proposed switched control strategy ensures the Input-to-State practical Stability of the considered nonlinear systems with respect to external disturbances, by means of a constructive method based on LMIs without using the system states.
<figure><img src="image_2.png" /><figcaption>Fig. 1. System trajectories in open-loop</figcaption></figure>

<figure><img src="image_3.png" /><figcaption>Fig. 2. System trajectories for $h = 0.2$</figcaption></figure>

# APPENDIX

## A Proof of Lemma 3

Define the error between the IP (11) and (12) as $e_k = z_k - \hat{z}_k$. Then, considering the state–feedback control (14), the closed–loop system dynamics is given as

$$z_{k+1} = \tilde{\mathcal{A}}_0 z_k + \tilde{\mathcal{A}}_1 z_k^+ + \tilde{\mathcal{A}}_2 z_k^- + \tilde{S}_1 \delta_k + \varphi_k + \bar{B} S_1 \bar{\varphi} - \bar{B} K_0 e_k - \bar{B} K_1 e_k^+ - \bar{B} K_2 e_k^-, \quad (A.1)$$

where $\tilde{\mathcal{A}}_0 = \bar{\mathcal{A}}_0 + \bar{B}K_0$, $\tilde{\mathcal{A}}_1 = \mathcal{G}\mathcal{A}_1 + \bar{B}K_1$, $\tilde{\mathcal{A}}_2 = \mathcal{G}\mathcal{A}_2 + \bar{B}K_2$, and $\tilde{S}_1 = I + \bar{B}S_1$.

Let us propose the following candidate Lyapunov function

$$V(z_k) = z_k^\top P z_k, \quad (A.2)$$
<figure><img src="image_4.png" /><figcaption>Fig. 3. Phase portrait</figcaption></figure>

<figure><img src="image_5.png" /><figcaption>Fig. 4. Control signals</figcaption></figure>

with a diagonal matrix $P \succ 0 \in \mathbb{R}^{2n \times 2n}$ and the extended vectors $\eta^\top = (z_k^\top, z_k^{+\top}, z_k^{-\top}, \delta_k^\top, \varphi_k^\top, \bar{\varphi}^\top)^\top$ and $\bar{\eta}^\top = (\eta^\top, e_k^\top, e_k^{+\top}, e_k^{-\top})^\top$. Then, the increment of the candidate Lyapunov function is

$$V(z_{k+1}) - V(z_k) = \eta^\top \Sigma \eta + \bar{\eta}^\top \Gamma \bar{\eta} - z_k^\top P z_k, \quad (A.3)$$

with $\Sigma = \Omega^\top P \Omega$, where $\Omega = (\tilde{\mathcal{A}}_0, \tilde{\mathcal{A}}_1, \tilde{\mathcal{A}}_2, \tilde{S}_1, I, \bar{B}S_1)$, and
$$
\Gamma =
\begin{pmatrix}
0 & 0 & 0 & 0 & 0 & -\tilde{A}_0^\top P \bar{B} K_0 & -\tilde{A}_0^\top P \bar{B} K_1 & -\tilde{A}_0^\top P \bar{B} K_2 \\
\star & 0 & 0 & 0 & 0 & -\tilde{A}_1^\top P \bar{B} K_0 & -\tilde{A}_1^\top P \bar{B} K_1 & -\tilde{A}_1^\top P \bar{B} K_2 \\
\star & \star & 0 & 0 & 0 & -\tilde{A}_2^\top P \bar{B} K_0 & -\tilde{A}_2^\top P \bar{B} K_1 & -\tilde{A}_2^\top P \bar{B} K_2 \\
\star & \star & \star & 0 & 0 & -\tilde{S}_1 P \bar{B} K_0 & -\tilde{S}_1 P \bar{B} K_1 & -\tilde{S}_1 P \bar{B} K_2 \\
\star & \star & \star & \star & 0 & -P \bar{B} K_0 & -P \bar{B} K_1 & -P \bar{B} K_2 \\
\star & \star & \star & \star & \star & -K_0^\top \bar{B}^\top P \bar{B} S_1 & -K_1^\top \bar{B}^\top P \bar{B} S_1 & -K_2^\top \bar{B}^\top P \bar{B} S_1 \\
\star & \star & \star & \star & \star & K_0^\top \bar{B}^\top P \bar{B} K_0 & 2K_0^\top \bar{B}^\top P \bar{B} K_1 & 2K_0^\top \bar{B}^\top P \bar{B} K_2 \\
\star & \star & \star & \star & \star & \star & K_1^\top \bar{B} P \bar{B} K_1 & 2K_1^\top \bar{B}^\top P \bar{B} K_2 \\
\star & \star & \star & \star & \star & \star & \star & K_2^\top \bar{B} P \bar{B} K_2
\end{pmatrix}.
\quad (A.4)
$$

Let us add and subtract the terms $\gamma\varphi_k^\top\varphi_k$, $\gamma\bar{\varphi}^\top\bar{\varphi}$, $\delta_k^\top R_0\delta_k$, $z_k^\top Q_0 z_k$, $z_k^{\top+} Q_1 z_k^{\top+}$, $z_k^{\top-} Q_2 z_k^{\top-}$, $2z_k^{\top} R_1 z_k^{\top+}$, $2z_k^{\top} R_2 z_k^{\top-}$, $2z_k^{\top+} R_3 z_k^{\top-}$, with symmetric matrices $R_0$, $Q_0$, $Q_1$, $Q_2 \in \mathbb{R}^{2n \times 2n}$, and diagonal matrices $R_1, R_2, R_3 \in \mathbb{R}^{2n \times 2n}$, to the right-hand side of (A.3), i.e.,

$$
V(z_{k+1}) - V(z_k) = \eta^\top \Sigma_1 \eta + \bar{\eta}^\top \Gamma \bar{\eta} + \gamma \varphi_k^\top \varphi_k + \gamma \bar{\varphi}^\top \bar{\varphi} - z_k^\top Q_0 z_k - z_k^{\top+} Q_1 z_k^{\top+} - z_k^{\top-} Q_2 z_k^{\top-} + \delta_k^\top R_0 \delta_k \\
- 2z_k^{\top} R_1 z_k^{\top+} - 2z_k^{\top} R_2 z_k^{\top-} - 2z_k^{\top+} R_3 z_k^{\top-}, \quad (A.5)
$$

where

$$
\Sigma_1 = \begin{pmatrix}
\tilde{A}_0^\top P \tilde{A}_0 - P + Q_0 & \tilde{A}_0^\top P \tilde{A}_1 + R_1 & \tilde{A}_0^\top P \tilde{A}_2 + R_2 & \tilde{A}_0^\top P \tilde{S}_1 & \tilde{A}_0^\top P & \tilde{A}_0^\top P \bar{B} S_1 \\
\star & \tilde{A}_1^\top P \tilde{A}_1 + Q_1 & \tilde{A}_1^\top P \tilde{A}_2 + R_3 & \tilde{A}_1^\top P \tilde{S}_1 & \tilde{A}_1^\top P & \tilde{A}_1^\top P \bar{B} S_1 \\
\star & \star & \tilde{A}_2^\top P \tilde{A}_2 + Q_2 & \tilde{A}_2^\top P \tilde{S}_1 & \tilde{A}_2^\top P & \tilde{A}_2^\top P \bar{B} S_1 \\
\star & \star & \star & \tilde{S}_1^\top P \tilde{S}_1 - R_0 & \tilde{S}_1^\top P & \tilde{S}_1^\top P \bar{B} S_1 \\
\star & \star & \star & \star & P - \gamma I & P \bar{B} S_1 \\
\star & \star & \star & \star & \star & S_1^\top \bar{B}^\top P \bar{B} S_1 - \gamma I
\end{pmatrix}.
$$

Note that, it is possible to represent the matrix $\Sigma_1$ in the following form $\Sigma_1 = \psi_1 + \psi_2^\top P^{-1}\psi_2$, with

$$
\psi_1 = \begin{pmatrix}
-P + Q_0 & R_1 & R_2 & 0 & 0 & 0 \\
\star & Q_1 & R_3 & 0 & 0 & 0 \\
\star & \star & Q_2 & 0 & 0 & 0 \\
\star & \star & \star & -\gamma R_0 & 0 & 0 \\
\star & \star & \star & \star & -\gamma I & 0 \\
\star & \star & \star & \star & \star & -\gamma I
\end{pmatrix},
$$

$$
\psi_2 = ( P \tilde{A}_0 P \tilde{A}_1 P \tilde{A}_2 P \tilde{S}_1 P P \bar{B} S_1 ) .
$$
We look for the conditions providing $\Sigma_1 \le 0$. Then, applying Schur's complement $\Sigma_1 \le 0$ is equivalent to

$$ \Sigma_2 = \begin{pmatrix} -P + Q_0 & R_1 & R_2 & 0 & 0 & 0 & \tilde{A}_0^\top P \\ \star & Q_1 & R_3 & 0 & 0 & 0 & \tilde{A}_1^\top P \\ \star & \star & Q_2 & 0 & 0 & 0 & \tilde{A}_2^\top P \\ \star & \star & \star & -R_0 & 0 & 0 & \tilde{S}_1^\top P \\ \star & \star & \star & \star & -\gamma I & 0 & P \\ \star & \star & \star & \star & \star & -\gamma I & S_1^\top \bar{B} P \\ \star & \star & \star & \star & \star & \star & -P \end{pmatrix}. $$

Pre- and post-multiplying $\Sigma_2$ by $T = \text{diag}(P^{-1}, P^{-1}, P^{-1}, P^{-1}, P^{-1}, P^{-1})$, it is obtained that $\Sigma_3 = T\Sigma_2 T$ is given as

$$ \Sigma_3 = \begin{pmatrix} -P^{-1} + P^{-1}Q_0P^{-1} & P^{-1}R_1P^{-1} & P^{-1}R_2P^{-1} & 0 & 0 & 0 & P^{-1}\bar{A}_0^\top + P^{-1}K_0^\top\bar{B}^\top \\ \star & P^{-1}Q_1P^{-1} & P^{-1}R_3P^{-1} & 0 & 0 & 0 & P^{-1}\bar{A}_1^\top\bar{G}^\top + P^{-1}K_1^\top\bar{B}^\top \\ \star & \star & P^{-1}Q_2P^{-1} & 0 & 0 & 0 & P^{-1}\bar{A}_2^\top\bar{G}^\top + P^{-1}K_2^\top\bar{B}^\top \\ \star & \star & \star & -P^{-1}R_0P^{-1} & 0 & 0 & P^{-1} + P^{-1}S_1^\top\bar{B}^\top \\ \star & \star & \star & \star & -\gamma I & 0 & P^{-1} \\ \star & \star & \star & \star & \star & -\gamma I & P^{-1}S_1^\top\bar{B}^\top \\ \star & \star & \star & \star & \star & \star & -P^{-1} \end{pmatrix}. \quad (A.6) $$

Considering the following change of variables $X_1 = P^{-1}$, $\tilde{Q}_0 = P^{-1}Q_0P^{-1}$, $\tilde{Q}_1 = P^{-1}Q_1P^{-1}$, $\tilde{Q}_2 = P^{-1}Q_2P^{-1}$, $\tilde{R}_0 = P^{-1}R_0P^{-1}$, $\tilde{R}_1 = P^{-1}R_1P^{-1}$, $\tilde{R}_2 = P^{-1}R_2P^{-1}$, $\tilde{R}_3 = P^{-1}R_3P^{-1}$, $Y_0 = P^{-1}K_0^\top$, $Y_1 = P^{-1}K_1^\top$, $Y_2 = P^{-1}K_2^\top$, and $Y_3 = P^{-1}S_1^\top$, (A.6) is equivalent to (15a). Hence, if $\Sigma_4 \le 0$; then $\Sigma \le 0$ and it follows that

$$ \begin{aligned} V(z_{k+1}) - V(z_k) \le \bar{\eta}^\top \Gamma \bar{\eta} + \gamma \varphi_k^\top \varphi_k + \gamma \bar{\varphi}^\top \bar{\varphi} + \delta_k^\top R_0 \delta_k - z_k^\top Q_0 z_k - z_k^{+\top} Q_1 z_k^+ - z_k^{-\top} Q_2 z_k^- \\ - 2 z_k^\top R_1 z_k^+ - 2 z_k^\top R_2 z_k^- - 2 z_k^{+\top} R_3 z_k^- \end{aligned} \quad (A.7) $$

In order to obtain an upper bound of (A.7), first note that $||\varphi_k||^2 \le \varphi_{\max}^2$ and $||\bar{\varphi}||^2 \le 2n\varphi_{\max}^2$. Then, is needed to obtain an upper bound $\varphi_{\max}$. With this aim in mind, first, consider that

$$ ||g(z(s)) - g(z_k)|| \le L ||z(s) - z_k||, \quad (A.8) $$

holds for $L = 2 \max\{||A_1||, ||A_2||\}$. It is possible to put (10) now in terms of $g(\hat{z}_k)$ adding and subtracting the term $\int_0^h e^{A_0(h-s)} dsg(\hat{z}_k)$ in the right-hand side of (9). Then, considering (10), in terms of $g(\hat{z}_k)$, and (A.8), it follows that

$$ ||z(t) - z_k|| \le \lambda(t, z(t_k), u_k) + L \int_{t_k}^{t} ||e^{A_0(t-s)}||||z(s) - z_k||ds, \quad (A.9) $$

with

$$ \begin{aligned} \lambda(t, z(t_k), u_k) = & ||e^{A_0(t-t_k)} - I||||z_k|| + \int_{t_k}^{t} ||e^{A_0(t-\tau)}\mathcal{B}||||u(\tau)||d\tau \\ & + \int_{t_k}^{t} ||e^{A_0(t-\tau)}||||\delta(\tau)||d\tau \\ & + \int_{t_k}^{t} ||e^{A_0(t-\tau)}||||g(\hat{z}(\tau))||d\tau. \end{aligned} \quad (A.10) $$

Thus, applying Lemma 2 to (A.9), it is obtained that

$$ ||z(t) - z_k|| \le \lambda(t, z(t_k), u_k) + \int_{t_k}^{t} \lambda(s, z(t_k), u_k) ||e^{A_0(t-s)}|| \rho(t, s) ds = W(t, z_k, u_k), \quad (A.11) $$
with $\rho(t, s) = e^{\int_s^t ||e^{A_0(t-r)}||dr}$. It is worth mentioning that (A.11) represents the error between the continuous trajectory of the interval predictor (9), and the discrete-time version (12).

Then, the step forward error between the IP (11) and (12) is given as

$$ ||z_{k+1} - \hat{z}_{k+1}|| = (||\overline{\mathcal{A}}_0|| + L||\mathcal{G}||)||z_k - \hat{z}_k|| + ||\varphi_k|| + ||\bar{\varphi}||. \quad (A.12) $$

Considering (A.11), we are able to get an upper bound for (A.12), i.e.,

$$ (||\overline{\mathcal{A}}_0|| + L||\mathcal{G}||)||z_k - \hat{z}_k|| + ||\varphi_k|| + ||\bar{\varphi}|| \le \lambda(t, z(t_k), u_k) + \int_{t_k}^{t} \lambda(s, z(t_k), u_k)||e^{A_0(t-s)}||\rho(t, s)ds. $$

Taking into account that $||\varphi_k|| + ||\bar{\varphi}|| \le (1 + \sqrt{2n})\varphi_{\max}$. Hence, it is possible to accomplish that

$$ \varphi_{\max} \le (1 + \sqrt{2n})^{-1}(W(t_{k+1}, z_k, u_k) - (||\overline{\mathcal{A}}_0|| + L||\mathcal{G}||)||z_k - \hat{z}_k||). \quad (A.13) $$

Thus, since $A_0$ is Hurwitz; one can take into account that $||e^{A_0(t-t_k)}|| \le \xi e^{-\nu(t-t_k)}$, with $\nu = |\text{Re}\{\lambda_{\max}\{\mathcal{A}_0\}\}|$ and $\xi > 0$, for all $t \in [t_k, t_{k+1})$; thus, (A.10) with $t = t_{k+1}$ can be upper bounded as

$$ \lambda(t_{k+1}, z(t_k), u_k) \le \xi ||z_k|| + ||\hat{z}_k|| + k_{\xi} (||\mathcal{B}|| \bar{u}_{\max} + ||g(z(t_k))||) + ||\delta_k|| = \lambda_c, \quad (A.14) $$

with $\bar{u}_{\max} = ||u_{\max}||$ and $k_{\xi} = \xi\nu^{-1}(1-e^{-\nu h})$. Then, taking into account that $\int_s^{t_{k+1}} ||e^{A_0(t_{k+1}-r)}||dr \le \xi\nu^{-1}(1-e^{-\nu h})$ and considering (A.14), $W(t_{k+1}, z_k, u_k)$ is upper bounded as $W(t_{k+1}, z_k, u_k) \le k_{e_1}\lambda_c = W_c$, with $k_{e_1} = 1 + e^{k_{\xi}k_{\xi}}$. Therefore, (A.13) can be upper bounded as follows

$$ \varphi_{\max} \le (1 + \sqrt{2n})^{-1} [(k_{e_1}\xi - ||\overline{\mathcal{A}}_0|| - L||\mathcal{G}|| + ||\mathcal{A}_1|| + ||\mathcal{A}_2||) ||z_k|| + (k_{e_1} - ||\overline{\mathcal{A}}_0|| - L||\mathcal{G}||) ||\hat{z}_k|| + k_{e_1} k_{\xi} (||\mathcal{B}|| \bar{u}_{\max} + ||\delta_k||)] . \quad (A.15) $$

Considering (A.15), the term $\varphi_{\max}^2$ can be upper bounded as follows

$$ \varphi_{\max}^2 \le 4(1 + \sqrt{2n})^{-1} [(k_{e_1}\xi - ||\overline{\mathcal{A}}_0|| - L||\mathcal{G}|| + ||\mathcal{A}_1|| + ||\mathcal{A}_2||)^2 ||z_k||^2 + (k_{e_1} - ||\overline{\mathcal{A}}_0|| - L||\mathcal{G}||)^2 ||\hat{z}_k||^2 + k_{e_2}^2 (||\mathcal{B}||^2 \bar{u}_{\max}^2 + ||\delta_k||^2)], \quad (A.16) $$

with $k_{e_2} = k_{e_1} k_{\xi}$. Note that, from (2), it can be obtained that $||x|| \le ||\mathbf{1}k_x||||\mathbf{b}||^{-1} = \sqrt{k_x \lambda_{\max}^{-1}\{\mathbf{b}^\top\mathbf{b}\}}$. Recalling that $z_k = [\underline{x}_k^\top, \bar{x}_k^\top]^\top$, we have $||z_k|| \le \sqrt{2k_x \lambda_{\max}^{-1}\{\mathbf{b}^\top\mathbf{b}\}} = \zeta$. Moreover, after some algebraic manipulations, it is possible to prove that $\bar{\eta}^\top\Gamma\eta$, with $\bar{\eta} = (\eta^\top, e_k^\top, e_k^{+\top}, e_k^{-\top})^\top$ and (A.4) is upper bounded as

$$ \bar{\eta}^\top\Gamma\bar{\eta} \le 2[\zeta c_1 + c_2\bar{e} + ||\delta_k||c_3 + ||\varphi_k||c_4 + ||\bar{\varphi}||c_5]\bar{e}, \quad (A.17) $$

with $c_1, c_2, c_3, c_4$, and $c_5$ given as in (18a)-(18d). where $\bar{K} = ||K_0|| + ||K_1|| + ||K_2||$. By definition, $||e_k|| \le \tilde{e} = W_c$ and, $||\varphi_k|| \le \varphi_{\max} = \varphi_c$ with

$$ W_c = [k_{e_1}(\xi + 1) + k_{\xi} (||\mathcal{A}_1|| + ||\mathcal{A}_2||)]\zeta + k_{\xi} (||\mathcal{B}|| \bar{u}_{\max} + \lambda_{\max}^{-1/2}\{R_0\}), \\ \varphi_c = (c_6 + c_7)\zeta + c_8 (||\mathcal{B}|| \bar{u}_{\max} + \lambda_{\max}^{-1/2}\{R_0\}), $$

with $c_6, c_7$, and $c_8$ given as in (18e)-(18g). Then, according to (A.17) and taking into account that $\delta_k^\top R_0 \delta_k \le \lambda_{\max}\{R_0\}||\delta_k||^2$ holds, it follows that

$$ V(z_{k+1}) - V(z_k) \le -z_k^\top Q_0 z_k - z_k^{+\top} Q_1 z_k^{+} - z_k^{-\top} Q_2 z_k^{-} - 2z_k^\top R_1 z_k^{+} - 2z_k^\top R_2 z_k^{-} - 2z_k^{+\top} R_3 z_k^{-\top} + \lambda_{\max}\{R_0\}||\delta_k||^2 + q_1||\delta_k|| + q_2, \quad (A.18) $$
with $q_1$ and $q_2$ given as in (17a) and (17b). Thus, if $\bar{Q} = Q_0 + \min\{Q_1, Q_2\} + 2\min\{R_1, R_2\} \ge \alpha P$, is satisfied, and since, by definition $z_k^{+\top}R_1z_k \ge 0$ and $(-z_k^{-})^{\top}R_2z_k \ge 0$, $z_k^{+\top}R_3z_k^{-} = 0$, for any diagonal matrices $R_1, R_2$ and $R_3$; then, it follows that $V(z_{k+1}) - V(z_k) \le -\alpha z_k^{+\top}Pz_k + \lambda_{\max}\{R_0\}||\delta_k||^2 + q_1||\delta_k|| + q_2$. Note that pre- and post-multiplying $\bar{Q}$ by $P^{-1}$, $\bar{Q} \ge \alpha P$ is equivalent to (15b). Therefore, the system (A.1) is ISpS, with respect to $\delta_k$.

Moreover, we have that $||\delta_k|| \le 2||D||w_{\max} = \bar{\delta}_k$. Then, the increment of the candidate Lyapunov function is negative definite outside the invariant convergence set

$$ \mathcal{R} = \{z_k \in \mathbb{R}^{2n} : z_k^{+\top} P z_k \le \alpha^{-1} \varepsilon\}. \quad (A.19) $$

with $\varepsilon = \lambda_{\max}\{R_0\}\bar{\delta}_k^2 + q_1\bar{\delta}_k + q_2$. Remark that the error (A.11) is taken into account in the LMI (15a), i.e., the error between the continuous trajectory of the interval predictor (9), and its discrete-time version (11). Finally, if the matrix inequality (16) is satisfied, then the set $\mathcal{R}$ belongs to the polytope (2). Moreover, if the IP (12) is initialized as $\hat{z}_0 = (\underline{x}_0^\top, \bar{x}_0^\top)^\top$, since Assumptions 1 and 2 hold; then, the trajectories of the system (1) will also fulfill the state constraints, i.e., $x \in \mathcal{X}$, and this concludes the proof. ■

## B Proof of Theorem 1

Note that the proof is based on a classical result for MPC (see, e.g., [20], [27], and [28]). Recall that the controller is composed of an interval predictor-based state-feedback controller and the MPC law, which will deal with state and input constraints. Thus, we have the following:

1) First, note that the interval predictor (12) generates an estimation $\hat{z}_k$ such that the relation $\underline{x}(t) \le x(t) \le \bar{x}(t)$ is satisfied. Then, consider the control $\hat{u}_k$ and note that, for some $t > 0$, if the initial conditions $\hat{z}_k(0) \in \mathcal{R} \subset \mathcal{X} \times \mathcal{X}$; then, the control (13) equals to (14). According to (A.19) and Assumption 3, $\hat{z}(t) \in \mathcal{R}$ and $\hat{u}_k \in \mathcal{U}$, for all $t \ge 0$, and the origin of the system dynamics (12) is ISpS, with respect to $\delta_k$ due to the result of Lemma 2.

2) Now, let $\hat{z}_k(0) \notin \mathcal{R}$ and recall that Problem 1 is assumed to be feasible. Then, applying the control (21), for $t \in [0, T)$, one has that $\hat{z}_k(t) \in \mathcal{X} \times \mathcal{X}$, and $u(t) \in \mathcal{U}$, on this time interval. Note that $\mathcal{X}$ is a neighborhood of the origin, and the cost function is given by $V_N$, with positive definite matrices $W, H$, and $M$, and it is minimized inside $\mathcal{X} \times \mathcal{X}$. Using these arguments, due to Assumption 2 and $[\underline{w}_{k+1}, \bar{w}_{k+1}] \subseteq [\underline{w}_k, \bar{w}_k]$ hold, and since $\hat{z}_k(t_i+T) \in \mathcal{X} \times \mathcal{X}$ provided that the control $\bar{u}_k$ in (21) is applied and $[\underline{z}^\top(t_i), \bar{z}^\top(t_i)] \subset [\underline{z}^\top(t_{i-1}+T), \bar{z}^\top(t_{i-1}+T)]$, for all $i \ge 1$, there is a finite time instant $t \ge T$ such that $(\underline{z}^\top(t), \bar{z}^\top(t))^\top \in \mathcal{R}$; and then, the control law (13) switches from the control signal $\bar{u}_k$ computed by the MPC to the state-feedback control law $\hat{u}_k$, which provides that the system trajectories are ISpS with respect to $\delta_k$.

3) As a result of the previous statements, the system constraints are satisfied.

This concludes the proof. ■

## REFERENCES

[1] M. Allenspach and G. J. Ducard. Nonlinear model predictive control and guidance for a propeller-tilting hybrid unmanned air vehicle. Automatica, 132:109790, 2021.

[2] E. Bernuau, A. Polyakov, D. Efimov, and W. Perruquetti. Verification of ISS, iISS and IOSS properties applying weighted homogeneity. Systems & Control Letters, 62(12):1159–1167, 2013.

[3] M. Bujarbaruah, U. Rosolia, Y. R. Stürz, X. Zhang, and F. Borrelli. Robust MPC for LPV systems via a novel optimization-based constraint tightening. Automatica, 143:110459, 2022.

[4] A. Dhar and S. Bhasin. Indirect adaptive MPC for discrete-time LTI systems with parametric uncertainties. IEEE Transactions on Automatic Control, 66(11):5498–5505, 2021.

[5] B. Ding and H. Pan. Output feedback robust MPC for LPV system with polytopic model parametric uncertainty and bounded disturbance. International Journal of Control, 89(8):1554–1571, 2016.

[6] D. Efimov, L. Fridman, T. Raissi, A. Zolghadri, and R. Seydou. Interval estimation for LPV systems applying high order sliding mode techniques. Automatica, 48(9):2365–2371, 2012.

[7] M. Farina, L. Giulioni, L. Magni, and R. Scattolini. An approach to output-feedback MPC of stochastic linear discrete-time systems. Automatica, 55:140–149, 2015.

[8] A. Gutiérrez, H. Ríos, M. Mera, D. Efimov, and R. Ushirobira. An interval predictor-based robust control for a class of constrained nonlinear systems. In 2023 62nd IEEE Conference on Decision and Control (CDC), Marina Bay Sands, Singapore, pp. 1–6, 2023.
[9] J. Hu. Dynamic output feedback mpc of polytopic uncertain systems: Efficient lmi conditions. *IEEE Transactions on Circuits and Systems II: Express Briefs*, 68(7):2568-2572, 2021.

[10] G. P. Incremona, A. Ferrara, and L. Magni. Asynchronous networked MPC with ISM for uncertain nonlinear systems. *IEEE Transactions on Automatic Control*, 62(9):4305-4317, 2017.

[11] D. Ji, J. Ren, C. Liu, and Y. Shi. Stabilizing terminal constraint-free nonlinear MPC via sliding mode-based terminal cost. *Automatica*, 134:109898, 2021.

[12] H. Khalil. *Nonlinear systems third edition*. Patience Hall, 2002.

[13] E. Leurent, D. Efimov, and O-A. Maillard. Robust-adaptive interval predictive control for linear uncertain systems. *2020 59th IEEE Conference on Decision and Control (CDC)*, Jeju, Korea (South), pp. 1429-1434, 2020.

[14] E. Leurent, O. Maillard, and D. Efimov. Robust-adaptive control of linear systems: beyond quadratic costs. *Advances in Neural Information Processing Systems*, 33:3220-3231, 2020.

[15] P. Li, Y. Kang, Y. Zhao, and T. Wang. Networked dual-mode adaptive horizon MPC for constrained nonlinear systems. *IEEE Transactions on Systems, Man, and Cybernetics: Systems*, 51(12):7435-7449, 2020.

[16] J. M. Manzano, D. Limón, D. Muñoz de la Peña, and J. P. Calliess. Robust learning-based MPC for nonlinear constrained systems. *Automatica*, 117:108948, 2020.

[17] D. Mayne. Robust and stochastic model predictive control: Are we going in the right direction? *Annual Reviews in Control*, 41:184-192, 2016.

[18] D. Q. Mayne. Model predictive control: Recent developments and future promise. *Automatica*, 50(12):2967-2986, 2014.

[19] D. Q. Mayne, S.V. Raković, R. Findeisen, and F. Allgöwer. Robust output feedback model predictive control of constrained linear systems: Time varying case. *Automatica*, 45(9):2082-2087, 2009.

[20] D. Q. Mayne, J. B. Rawlings, C. V. Rao, and P. OM. Scokaert. Constrained model predictive control: Stability and optimality. *Automatica*, 36(6):789-814, 2000.

[21] D. Q. Mayne, M. M. Seron, and S. V. Raković. Robust model predictive control of constrained linear systems with bounded disturbances. *Automatica*, 41(2):219-224, 2005.

[22] A. M. Nagy, G. Mourot, B. Marx, J. Ragot, and G. Schutz. Systematic multimodeling methodology applied to an activated sludge reactor model. *Industrial & Engineering Chemistry Research*, 49(6):2790-2799, 2010.

[23] X. Ping, Z. Li, and A. Al-Ahmari. Dynamic output feedback robust MPC for LPV systems subject to input saturation and bounded disturbance. *International Journal of Control, Automation and Systems*, 15(3):976-985, 2017.

[24] A. Reis de Souza, D. Efimov, and T. Raïssi. Robust output feedback MPC for LPV systems using interval observers. *IEEE Transactions on Automatic Control*, 67(6):3188-3195, 2021.

[25] A. Reis de Souza, D. Efimov, T. Raïssi, and X. Ping. Robust output feedback model predictive control for constrained linear systems via interval observers. *Automatica*, 135:109951, 2022.

[26] E. D. Sontag and Y. Wang. On characterizations of the input-to-state stability property. *Systems & Control Letters*, 24(5):351-359, 1995.

[27] Z. Sun, L. Dai, K. Liu, Y. Xia, and K. H. Johansson. Robust mpc for tracking constrained unicycle robots with additive disturbances. *Automatica*, 90:172-184, 2018.

[28] Z. Sun, L. Dai, K. Liu, Y. Xia, and K. H. Johansson. Robust MPC for tracking constrained unicycle robots with additive disturbances. *Automatica*, 90:172-184, 2018.

[29] A. Weiss, M. Baldwin, R. S. Erwin, and I. Kolmanovsky. Model predictive control for spacecraft rendezvous and docking: Strategies for handling constraints and case studies. *IEEE Transactions on Control Systems Technology*, 23(4):1638-1647, 2015.

[30] H. Xie, L. Dai, Y. Luo, and Y. Xia. Robust MPC for disturbed nonlinear discrete-time systems via a composite self-triggered scheme. *Automatica*, 127:109499, 2021.