# Dynamic Event-Triggered Control under Input and State Delays using Interval Observers

Frédéric Mazenc, Michael Malisoff

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Frédéric Mazenc, Michael Malisoff. Dynamic Event-Triggered Control under Input and State Delays using Interval Observers. ACC 2025 - American Control Conference, Jul 2025, Denver, United States. hal-05009485

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# Dynamic Event-Triggered Control under Input and State Delays using Interval Observers*

Frédéric Mazenc<fnref n="1" /> and Michael Malisoff<fnref n="2" />

**Abstract**— We prove global exponential stability estimates for a class of nonlinear control systems that contain uncertain time-varying input delays and uncertain state delays. Our new dynamic event-triggered controls ensure that Zeno's phenomenon does not occur. We use new synergies of interval observers and Halanay's inequality. We illustrate our approach in a marine robotic dynamics that contains uncertain nonlinear terms.

## I. INTRODUCTION

Dynamic event-triggered control is an alternative to standard event-triggered controls (e.g., [1]) or traditional controls, where instead of changing control values at times that are independent of the state, the times when the control values change are determined by an interconnected dynamic extension. This calls for finding (a) the control, whose values only change at event triggering times and (b) the dynamic extension, to choose triggering times. This differs from static event triggers, where there is a supremum that is used to determine the trigger times instead of using a dynamic extension [2], [3]. Dynamic event-triggered control has been shown to reduce the numbers of trigger times, as compared with static event-triggers [2], [3]. However, by reducing the use of scarce communication resources by only changing control values when the system under consideration requires attention, dynamic and static event-triggered controls have both inspired significant theoretic research and applications [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14].

One recent event-triggered approach combines interval observers with matrices of absolute values [2], [15], [16], [17]. These techniques were shown to significantly reduce the average numbers of trigger times on intervals of fixed length, and to beneficially increase the lower bounds on the inter-execution times (which are the differences between consecutive trigger times) compared with traditional event-triggered controls like [1] (and so ensure less triggering), when applied to an underwater robotic model [2]. Interval observers yield componentwise upper and lower bounds for unknown functions [18].

Interval observers appeared in several works, leading to solutions of stabilization problems. However, we believe that this work is the first to apply this approach to dynamic event-triggered control with unknown input delays and unknown

nonlinearities, using dynamical extensions that construct the control and new triggering rules which have not appeared before. Compared with [2], [19], [16], this work has several key novel features. First, our dynamic extensions and trigger rules use suprema of available state measurements, contrasting with the linear cases of [2] which required a supremum of an uncertainty bound. Second, the uncertainty in [2] was restricted to being an added uncertainty on the right side of the dynamics. Third, our novel use of Halanay's inequality here allows us to overcome the significant challenge of extending the results [2], [16], [19] (which were confined to linear systems) to the more challenging nonlinear systems presented here, where there can also be uncertain coefficients in a linear part of the system including unknown control gains, as well as unknown state and input delays with known bounds. Fourth, our work also contrasts significantly with delay-compensating chain predictor approaches [16] which assumed that the input delay was known and constant, which is not assumed in this paper. This paper is strongly motivated by the prevalence and potentially destabilizing influences of uncertain coefficients, delays, and nonlinearities [6], calling for our analysis of delayed event-triggered nonlinear systems.

We use standard notation, where $\mathbb{R}^n$ (resp., $\mathbb{R}^{p \times q}$) is the set of all vectors of real $n$-tuples (resp., $p \times q$ matrices) and the dimensions of our Euclidean spaces are arbitrary, unless indicated otherwise. We set $\mathbb{Z}_0 = \{0, 1, 2, ...\}$, and $|| \cdot ||$ denotes the Euclidean 2-norm, and $|| \cdot ||_S$ is the supremum in this norm over sets $S$. For a matrix $G = [g_{ij}] \in \mathbb{R}^{r \times s}$, we set $|G| = [|g_{ij}|]$, so the entries of $|G|$ are the absolute values of the corresponding entries $g_{ij}$ of $G$, and then the supremum $| \cdot |_S$ over sets $S$ is also defined entrywise. We let $G^+$ denote the matrix whose entries are $\max\{0, g_{ij}\}$ and $G^- = G^+ - G$, so $|G| = G^+ + G^-$. For matrices $D = [d_{ij}]$ and $E = [e_{ij}]$ of the same size, we write $D < E$ (resp., $D \le E$) provided $d_{ij} < e_{ij}$ (resp., $d_{ij} \le e_{ij}$) for all $i$ and $j$. We adopt similar notation for vectors. We call a matrix $S$ positive (resp., nonnegative) provided $0 < S$ (resp., $0 \le S$), where $0$ is the zero matrix, and $I$ is the identity matrix. For square matrices $A$, we let $D_A$ denote the diagonal matrix whose diagonal entries are those of $A$, $R_A = D_A + (A - D_A)^+$ and $N_A = (A - D_A)^+ - (A - D_A)$. For constants $a > 0$, we let $C_{\text{in}}(a)$ denote the set of all absolutely continuous functions $\phi : [-a, 0] \to \mathbb{R}^n$, and $x_t$ is defined by $x_t(\ell) = x(t + \ell)$ for all $\ell \in [-a, 0]$, all functions $x$, and all $t \ge 0$ such that $t + \ell$ is in the domain of $x$. A real square matrix is called Metzler provided all of its off-diagonal entries are nonnegative. We assume for simplicity that the initial times of our solutions $x(t)$ are $t_0 = 0$, and that the initial functions are constant,

<footnote>*Supported by US National Science Foundation Grants 2009659 and 2308282 and Office of Naval Research Grant N00014-22-1-2135.</footnote>

<footnote n="1">¹Frédéric Mazenc is with INRIA EPI DISCO, L2S-CNRS-CentraleSupélec, 3 rue Joliot Curie, 91192, Gif-sur-Yvette, France frederic.mazenc@l2s.centralesupelec.fr.</footnote>

<footnote n="2">²Michael Malisoff is with Department of Mathematics, 303 Lockett Hall, 175 Field House Drive, Louisiana State University, Baton Rouge, LA 70803, USA malisoff@lsu.edu.</footnote>
so $x(\ell) = x(0)$ for all $\ell \le 0$.

## II. MAIN RESULT

### A. Studied system

We consider the system

$$ \dot{x}(t) = (A + \Delta_A(t))x(t) + (B + \Delta_B(t))u(t - \tau(t)) + f(t, x_t) \quad (1) $$

where $x$ is valued in $\mathbb{R}^n$, the control $u$ is valued in $\mathbb{R}^p$ and will be specified below, $A \in \mathbb{R}^{n \times n}$ and $B \in \mathbb{R}^{n \times p}$ are known matrices, the piecewise continuous functions $\Delta_A: [0, +\infty) \to \mathbb{R}^{n \times n}$ and $\Delta_B: [0, +\infty) \to \mathbb{R}^{n \times p}$ represent model uncertainty, and $f: [0, +\infty) \times C_{\text{in}}(h) \to \mathbb{R}^n$ for a known value $h > 0$ represents unknown state delayed terms. Our first assumption is as follows, where $|\phi|_{[-h, 0]}$ is the supremum of $\phi$ over the set $S = [-h, 0]$ as defined in Section I (but see Remark 3, for ways to relax our assumptions on $f$):

**Assumption 1:** There is a matrix $K \in \mathbb{R}^{p \times n}$ such that, with

$$ H = A + BK, \quad (2) $$

the matrix $R_H + N_H$ is Hurwitz. Also, $\tau(t)$ is piecewise constant, and there are a known constant $\bar{\tau} \ge 0$ and known matrices $\bar{\Delta}_A \in \mathbb{R}^{n \times n}$ and $\bar{\Delta}_B \in \mathbb{R}^{n \times p}$ such that

$$ 0 \le \tau(t) \le \bar{\tau}, \quad |\Delta_A(t)| \le \bar{\Delta}_A, \quad \text{and} \quad |\Delta_B(t)| \le \bar{\Delta}_B \quad (3) $$

hold for all $t \ge 0$. Finally, $f$ is continuous in its first variable $t$ and locally Lipschitz in its second variable $x_t$, and there is a constant $f_0 > 0$ such that

$$ |f(t, \phi)| \le f_0 |\phi|_{[-h, 0]} \quad (4) $$

holds for all $t \ge 0$ and $\phi \in C_{\text{in}}(h)$. $\square$

According to [20, Theorem 2.11, p. 38], Assumption 1 provides a vector $V > 0$ and a real value $c > 0$ such that

$$ V^\top(R_H + N_H) \le -cV^\top, \quad (5) $$

since $R_H+N_H$ is Metzler and Hurwitz. We also use a matrix $\Gamma > 0$ in $\mathbb{R}^{n \times n}$ such that there is a constant $r > 0$ such that

$$ -cV^\top + V^\top|BK|\Gamma \le -rV^\top, \quad (6) $$

which holds when $\Gamma$ has small enough entries. We also use

$$ \Omega(s) = e^{As} + \int_0^s e^{(s-m)A} dmBK. \quad (7) $$

Since $\Gamma > 0$ and $\Omega(0) = I$, there is a constant $\nu > 0$ such that $\Omega(t)$ is nonsingular for each $t \in [0, \nu]$ and such that

$$ |BK(\Omega^{-1}(s) - I)| \le |BK|\Gamma \text{ for all } s \in [0, \nu]. \quad (8) $$

We fix $\bar{\tau}$, $c$, $r$, $h$, $A$, $B$, $V$, $f_0$, $\Gamma$, $K$, $\bar{\Delta}_A$, $\bar{\Delta}_B$, and $\nu$ satisfying the preceding requirements in the sequel, and then we set

$$ \begin{aligned} R &= (I + |BK\Omega^{-1}|_{[0, \nu]} \int_0^\nu |e^{A\ell}| d\ell) B^\sharp, && \text{where} \\ B^\sharp &= (I + \bar{\tau}|BK|)(f_0 I + \bar{\Delta}_A + \bar{\Delta}_B|K|) \\ & \quad + \bar{\tau}(|BKA| + |(BK)^2|). \end{aligned} \quad (9) $$

We then fix a constant $p_* > 0$ such that

$$ V^\top R \le p_* V^\top \quad (10) $$

and then our last assumption is the following, which can be viewed as a smallness condition on $\bar{\tau}$, $\bar{\Delta}_A$, $\bar{\Delta}_B$, or $f_0$:

**Assumption 2:** The inequality

$$ p_* < r \quad (11) $$

is satisfied.

Assumption 2 implies that for each constant $T > \nu$, there is a unique constant $R_* > 0$ such that

$$ R_* = r - p_* e^{R_*(2T + 2\bar{\tau} + h + \nu)}. \quad (12) $$

We fix constants $T > \nu$ and $R_* > 0$ satisfying (12). Our $R_*$ will be our exponential convergence rate in our application of Halanay's inequality (as stated in [21, Lemma 4.2]) in our proof of our theorem, in which $V_i > 0$ will denote the $i$th entry of the positive vector $V$ from (5) for $i = 1, \dots, n$. Halanay's inequality is called for here to handle the delays, uncertainties $f$, and unknown coefficients that were not present in [2].

**Remark 1:** As in [2, Remark 3.1], our Hurwitz assumption on $R_H+N_H$ is unrestrictive, because it holds for all controllable pairs $(A, B)$ after a change of coordinates. Another key point is that $f$ can be uncertain, as long as we know constants $h$ and $f_0$ satisfying (4). See Section III, where we illustrate how our assumptions allow significant uncertain input and state delays, and large entries for the uncertainty $\Delta_B$, as compared with the entries of $B$, and so are not too restrictive, and where we also illustrate a benefit of using the new inequality in (8) instead of the inequality $|\Omega^{-1}(s) - I| \le \Gamma$ that was used in its place in [2], [16], and [19]. $\square$

### B. Event-Triggered Control

Fix a diagonal matrix $D$ whose main diagonal entries are all positive, which is a tuning matrix, and any positive vector $z_0 \in \mathbb{R}^n$. In terms of our fixed constant $T > \nu$, consider

$$ \left\{ \begin{aligned} \dot{x}(t) &= (A + \Delta_A(t))x(t) \\ &\quad +(B + \Delta_B(t))Kx(\mu(t) - \tau(t)) + f(t, x_t) \\ \dot{z}(t) &= (R_H + N_H)z(t) - B^*(t) + \lambda(\zeta(t, x_t)) \\ e(t) &= x(t_i) - x(t) \quad \text{for all } t \in [t_i, t_{i+1}) \\ t_{i+1} &= \sup \{t \in [t_i, t_i + T] : z(t) \\ &\quad -DB^*(t) + D\lambda(\zeta(t, x_t)) > 0\} \\ \mu(t) &= t_i \quad \text{for all } t \in [t_i, t_{i+1}) \end{aligned} \right. \quad (13) $$

with $z(0) = z_0$ and the function $\lambda$ defined by

$$ \lambda(\zeta(t, x_t)) = |BK\Omega^{-1}|_{[0, \nu]} \int_{t-\nu}^t |e^{A(t-\ell)}| \zeta(\ell, x_\ell) d\ell, \quad (14) $$

and where

$$ \begin{aligned} B^*(t) &= |BKe(t)| - |BK|\Gamma|x(t)| && \text{and} \\ \zeta(t, x_t) &= \bar{\tau}|BKA||x|_{[\mu(t)-\bar{\tau}, \mu(t)]} \\ & \quad + \bar{\tau}|(BK)^2||x|_{[\mu(t)-2\bar{\tau}-T, \mu(t)]} \\ & \quad + f_0(1 + |BK|\bar{\tau})|x|_{[t-h-T-\bar{\tau}, t]} \\ & \quad + (I+\bar{\tau}|BK|)(\bar{\Delta}_A+\bar{\Delta}_B|K|) |x|_{[t-2(T+\bar{\tau}), t]} \end{aligned} \quad (15) $$

and where $t_0 = 0$. Our main result is then as follows:
**Theorem 1:** Let Assumptions 1-2 hold. Then, all solutions $x: [0, +\infty) \to \mathbb{R}^n$ of (13) are such that

$$||x(t)|| \le \frac{n \max\{V_i:1 \le i \le n\}}{\min\{V_i:1 \le i \le n\}} e^{-R_*t} (2||x(0)|| + ||z_0||) \quad (16)$$

for all $t \ge 0$. Also, $t_{i+1} - t_i \ge \nu$ holds for all $i \in \mathbb{Z}_0$. $\square$

The $\lambda$ in (14) collects the effects of $f$, $\Delta_A$, and $\Delta_B$, and $\tau$. The triggered mechanism (13) is reminiscent of those of [3], except we adopt a positive systems approach. System (13) uses the control $u(t) = Kx(\mu(t) - \tau(t))$ (where $\tau(t)$ represents the delayed impact of triggering a new control value at time $\mu(t)$), and whose trigger times $t_i$ are found recursively, as follows. We solve the initial value problems given by the $x$ and $z$ systems in (13) with the initial time $0$ and initial states $x(0)$ and $z(0) = z_0$, while monitoring the componentwise strict inequality in the sup in (13) with $i=0$ (so we do not need to compute the $z$ values on $[t_i, t_{i+1})$ until the measurement at time $t_i$ is obtained). If this strict inequality holds for all $t \in [0, T]$, then $t_1 = T$. Otherwise, this supremum is in $[0, T)$ and is our $t_1$. We repeat this with $t_0 = 0$ replaced by $t_1$, by solving the $x$ and $z$ systems in (13) with the initial time $t_1$ and the initial state $z(t_1)$ obtained by solving the $z$ system in (13) on $[0, t_1]$. This repeats for all $i \in \mathbb{Z}_0$, and gives a continuous solution $x(t)$ for all $t \ge 0$ and trigger time sequence $\{t_i\}$, because our lower bound $t_{i+1} - t_i \ge \nu$ for all $i$ implies that the Zeno phenomenon does not occur (meaning, only finitely many $t_i$'s occur on each finite length time interval).

**Remark 2:** While more complicated than standard controllers, (13) ensures robustness to unknown $\tau$, $\Delta_A$, $\Delta_B$, and $f$ while only updating $u$ when a new control value is required. Standard controllers do not offer these benefits. Our proof of Theorem 1 will show that the theorem remains true if we replace the suprema in (15) by upper bounds, e.g., $|x|_{[\mu(t)-\tau,\mu(t)]} + \delta_1(t)$ instead of $|x|_{[\mu(t)-\tau,\mu(t)]}$, and similarly for the other suprema in (15), except we conclude exponential input-to-state stability with respect to added nonnegative uncertainties $\delta$'s, instead of (16); this follows from generalized Halanay's inequalities with gain terms [22]. Hence, it suffices to have upper estimates for the suprema. This helps make our controllers easy to implement. $\square$

### C. Proof of Theorem 1

The proof has four parts. The first proves that the solution of the $z$ system in (13) satisfies $z(t) > 0$ for all $t \ge 0$. Since this part is the same as the first part of the proof of [2, Theorem 1] except with the $\lambda$ from [2, Equation (8)] replaced by (14), and with $|BK|(|e(t)| - \Gamma|x(t)|)$ replaced by the function $B^*(t)$ from (15) and $\delta = 0$, we omit the first part. In the second part, we prove that $t_{i+1} - t_i \ge \nu$ holds for all $i \in \mathbb{Z}_0$. In the third part, we build our interval observer, to prove (16) in the fourth part. While similar in structure to the proof of [2, Theorem 1], our proof is significantly different, because we need to take into account the $f$, $\tau$, $\Delta_A$, and $\Delta_B$ from (1), which were not allowed in [2] and which call for the significantly different new analogs (14) of the $\lambda$'s from [2, Equation (8)], and for using Halanay's inequality, which was also not required in [2].

### Second Part: Ruling Out Zeno's Phenomenon.
We have

$$ \begin{aligned} \dot{x}(t) &= (A + \Delta_A(t))x(t) + BKx(\mu(t)) \\ &\quad + \Delta_B K x(\mu(t) - \tau(t)) \\ &\quad + BK[x(\mu(t) - \tau(t)) - x(\mu(t))] + f(t, x_t) \\ &= Ax(t) + BKx(\mu(t)) - BK \int_{\mu(t)-\tau(t)}^{\mu(t)} \dot{x}(m) dm \\ &\quad + f(t, x_t) + \Delta_A(t)x(t) + \Delta_B K x(\mu(t) - \tau(t)) \\ &= Ax(t) + BKx(\mu(t)) + \rho(t) \end{aligned} \quad (17) $$

for all $t \ge 0$, where

$$ \begin{aligned} \rho(t) &= -BK A \int_{\mu(t)-\tau(t)}^{\mu(t)} x(m) dm + f^{\sharp}(t, x_t) + \Delta^{\sharp}(t) \\ &\quad - (BK)^2 \int_{\mu(t)-\tau(t)}^{\mu(t)} x(\mu(m) - \tau(m)) dm, \\ \Delta^{\sharp}(t) &= -BK \int_{\mu(t)-\tau(t)}^{\mu(t)} \Delta_A(m) x(m) dm \\ &\quad - BK \int_{\mu(t)-\tau(t)}^{\mu(t)} \Delta_B(m) K x(\mu(m) - \tau(m)) dm \\ &\quad + \Delta_A(t) x(t) + \Delta_B(t) K x(\mu(t) - \tau(t)), \text{ and} \\ f^{\sharp}(t, x_t) &= f(t, x_t) - BK \int_{\mu(t)-\tau(t)}^{\mu(t)} f(m, x_m) dm. \end{aligned} \quad (18) $$

Fix an $i \in \mathbb{Z}_0$. We next argue by contradiction. Suppose that $t_{i+1} - t_i < \nu$. By applying the method of variation of parameters to (17), we get

$$x(t) = \Omega(t - t_i)x(t_i) + \int_{t_i}^{t} e^{A(t-m)} \rho(m) dm \quad (19)$$

for all $t \in [t_i, t_{i+1})$, where $\Omega$ is defined in (7). Hence, since $t_{i+1} - t_i < \nu$, we would be able to use $\Omega^{-1}(t - t_i)$ to solve for $x(t_i)$ in (19) and use our choice $e(t) = x(t_i) - x(t)$ from (13) to get a formula for $e(t)$, then left multiply the result by $BK$ and then apply $|\cdot|$ to both sides of the result, to get

$$|BK e(t)| = \left| \begin{aligned} &BK[\Omega(t - t_i)^{-1} - I]x(t) \\ &- BK\Omega(t - t_i)^{-1} \int_{t_i}^{t} e^{A(t-m)} \rho(m) dm \end{aligned} \right| \quad (20)$$

for all $t \in [t_i, t_{i+1})$, so our choice of $B^*$ from (15) gives

$$ \begin{aligned} B^*(t) &\le \left[ |BK(\Omega(t - t_i)^{-1} - I)| - |BK|\Gamma \right] |x(t)| \\ &\quad + |BK\Omega(t - t_i)^{-1} \int_{t_i}^{t} e^{A(t-m)} \rho(m) dm| \quad (21) \\ &\le |BK\Omega(t - t_i)^{-1} \int_{t_i}^{t} e^{A(t-m)} \rho(m) dm| \end{aligned} $$

for all $t \in [t_i, t_{i+1})$, where the first inequality followed by applying the triangle inequality to the right side of (20) and then subtracting $|BK|\Gamma|x(t)|$ from both sides of the result, and the second inequality followed from (8), using $s = t - t_i$.

Using (21) to lower bound the second right side term in our $\dot{z}$ formula in (13), we get

$$ \begin{aligned} \dot{z}(t) &\ge (R_H + N_H)z(t) + \lambda(\zeta(t, x_t)) \\ &\quad - |BK\Omega(t - t_i)^{-1} \int_{t_i}^{t} e^{A(t-m)} \rho(m) dm| \end{aligned} \quad (22) $$

for all $t \in [t_i, t_{i+1})$. We can also use our choices of $\rho$ and $\lambda$ in (18) and (14) and the triangle inequality to obtain

$$ \begin{aligned} &-|BK\Omega(t - t_i)^{-1} \int_{t_i}^{t} e^{A(t-m)} \rho(m) dm| + \lambda(\zeta(t, x_t)) \\ &\ge -|BK\Omega(t - t_i)^{-1} \int_{t_i}^{t} e^{A(t-m)} [BK A \int_{\mu(m)-\tau(m)}^{\mu(m)} x(s) ds \\ &\quad + (BK)^2 \int_{\mu(m)-\tau(m)}^{\mu(m)} x(\mu(s) - \tau(s)) ds] dm| \quad (23) \\ &\quad + |BK\Omega^{-1}|_{[0,\nu]} \int_{t-\nu}^{t} |e^{A(t-\ell)}| \zeta(\ell, x_\ell) d\ell \\ &\quad - |BK\Omega(t - t_i)^{-1} \int_{t_i}^{t} e^{A(t-m)} f^{\sharp}(m, x_m) dm| \\ &\quad - |BK\Omega(t - t_i)^{-1} \int_{t_i}^{t} e^{A(t-m)} \Delta^{\sharp}(m) dm| \end{aligned} $$
for all $t \in [t_i, t_{i+1})$, where $f^\sharp$ was defined in (18). Also,

$$ \int_{\mu(m)-\tau(m)}^{\mu(m)} |x(\mu(s)-\tau(s))|ds \le \bar{\tau}|x|_{[\mu(m)-2\bar{\tau}-T, \mu(m)]} \quad (24) $$

holds for all $m \ge 0$, since $\mu(m) \ge s \ge \mu(s) \ge \mu(s) - \tau(s) \ge s - T - \bar{\tau} \ge \mu(m) - 2\bar{\tau} - T$ holds for all $s \in [\mu(m) - \tau(m), \mu(m)]$, which follows because of our choice of $\mu$ in (13) and because of (3) from Assumption 1.

Also, it follows from (15) and (23) that, with the choices

$$ f_0^\sharp = (f_0 I + \bar{\Delta}_A + \bar{\Delta}_B |K|) (1 + |BK|\bar{\tau}) \quad (25) $$

and $\Omega^\sharp(s) = BK\Omega^{-1}(s)$, we have

$$ \begin{aligned} & -|\Omega^\sharp(t-t_i) \int_{t_i}^t e^{A(t-m)} \rho(m) dm| + \lambda(\zeta(t, x_t)) \\ & \ge -|\Omega^\sharp(t-t_i) \int_{t_i}^t e^{A(t-m)} [BK A \int_{\mu(m)-\tau(m)}^{\mu(m)} x(s) ds \\ & \quad + (BK)^2 \int_{\mu(m)-\tau(m)}^{\mu(m)} x(\mu(s)-\tau(s)) ds] dm| \\ & \quad + |\Omega^\sharp|_{[0,\nu]} \int_{t-\nu}^t |e^{A(t-\ell)}| [\bar{\tau} |BK A| |x|_{[\mu(\ell)-\bar{\tau}, \mu(\ell)]} | d\ell \\ & \quad + |(BK)^2| |\bar{\tau}| |x|_{[\mu(\ell)-2\bar{\tau}-T, \mu(\ell)]} |] d\ell \\ & \quad -|\Omega^\sharp(t-t_i) \int_{t_i}^t e^{A(t-m)} (f^\sharp(m, x_m) + \Delta^\sharp(m)) dm| \\ & \quad + |\Omega^\sharp|_{[0,\nu]} \int_{t-\nu}^t |e^{A(t-\ell)}| f_0^\sharp |x|_{[\ell-h-2T-2\bar{\tau}, \ell]} d\ell \end{aligned} \quad (26) $$

for all $t \in [t_i, t_{i+1})$. Since $t_{i+1} - t_i < \nu$, it follows from (24), (26), and our formulas (18) and (25) that

$$ -|\Omega^\sharp(t-t_i) \int_{t_i}^t e^{A(t-m)} \rho(m) dm| + \lambda(\zeta(t, x_t)) \ge 0 \quad (27) $$

for all $t \in [t_i, t_{i+1})$, by using the nonnegative right side terms of (26) to dominate the other right side terms of (26). Hence, (22) gives $\dot{z}(t) \ge (R_H + N_H)z(t) \ge D_H z(t)$ for all $t \in [t_i, t_{i+1})$, where the second inequality used the nonnegative valuedness of $z$ from the first part of the proof.

Therefore, with the choice $g = \min\{e^{-d_i\nu} : 1 \le i \le n\}$, where $d_i$ is the $i$th main diagonal entry of the diagonal matrix $D_H$, a variation of parameters argument gives

$$ z(t) \ge e^{D_H(t-t_i)} z(t_i) \ge gz(t_i) \text{ for all } t \in [t_i, t_{i+1}). \quad (28) $$

Since $D \ge 0$ is nonnegative, we can left multiply (21) by $D$ and use (28) and our $B^*$ from (15) to conclude that

$$ z(t) - DB^*(t) + D|BK\Omega(t-t_i)^{-1} \int_{t_i}^t e^{A(t-m)} \rho(m) dm| \ge gz(t_i) \quad (29) $$

holds for all $t \in [t_i, t_{i+1})$. Hence, (27) and (29) give

$$ z(t) - DB^*(t) + D\lambda(\zeta(t, x_t)) \ge gz(t_i) \quad (30) $$

for all $t \in [t_i, t_{i+1})$. Since $e(t_{i+1}) = 0$, and since $x$ and $z$ are continuous, it follows that (30) holds for all $t \in [t_i, t_{i+1}]$. Therefore, by the positive valuedness of $z(t)$, we get

$$ z(t) - DB^*(t) + D\lambda(\zeta(t, x_t)) > 0 \quad (31) $$

for all $[t_i, t_{i+1}]$. Since $e(t)$ is right continuous, this gives an $\epsilon_0 > 0$ such that (31) holds for all $t \in [t_i, t_{i+1} + \epsilon_0]$. By (13), we have $t_{i+1} = \sup\{t > t_i : z(t) - DB^*(t) + D\lambda(\zeta(t, x_t)) > 0\}$, because we supposed that $t_{i+1} - t_i < \nu \le T$. This contradiction gives $t_{i+1} - t_i \ge \nu$ for all $i \in \mathbb{Z}_0$.

**Third Part: Interval Observer.** Observe that (17) gives $\dot{x}(t) = Hx(t) + BKe(t) + \rho(t)$ for all $t \ge 0$, by (2) and (13). It follows that the $x$ dynamics in (13) is

$$ \dot{x}(t) = (R_H - N_H)x(t) + BKe(t) + \rho(t) \quad (32) $$

for all $t \ge 0$. This motivates our use of the interval observer

$$ \left\{ \begin{aligned} \dot{\bar{x}}(t) &= R_H \bar{x}(t) - N_H \bar{x}(t) + (BKe(t))^+ + \rho(t)^+ \\ \dot{\underline{x}}(t) &= R_H \underline{x}(t) - N_H \underline{x}(t) - (BKe(t))^- - \rho(t)^- \end{aligned} \right. \quad (33) $$

with initial state $\bar{x}(0) = |x(0)|$ and $\underline{x}(0) = -|x(0)|$. Since

$$ \begin{bmatrix} R_H & N_H \\ N_H & R_H \end{bmatrix} \quad (34) $$

is Metzler, we can consider the dynamics for $(\bar{x}-x, x-\underline{x})$ and $(\bar{x}, -\underline{x})$, to conclude from our choices of $\bar{x}(0)$ and $\underline{x}(0)$ that $\bar{x}(t) \ge 0$, $\underline{x}(t) \le 0$, and $\underline{x}(t) \le x(t) \le \bar{x}(t)$ hold for all $t \ge 0$, e.g., by [23, Lemma 1]. Hence, for all $t \ge 0$, we have

$$ |x(t)| \le \bar{x}(t) - \underline{x}(t). \quad (35) $$

**Fourth Part: Stability Analysis.** We use the function $U(\bar{x}, \underline{x}, z) = V^\top(\bar{x} - \underline{x} + z)$. Then, since (33) gives

$$ \begin{aligned} \dot{\bar{x}}(t) - \dot{\underline{x}}(t) &= (R_H + N_H)(\bar{x}(t) - \underline{x}(t)) \\ &\quad + |BKe(t)| + |\rho(t)| \end{aligned} \quad (36) $$

for all $t \ge 0$, we conclude from (5), (15), and (35) that the time derivative of $U$ along (36) and (13) satisfies

$$ \begin{aligned} \dot{U}(t) &\le -cV^\top(\bar{x}(t)-\underline{x}(t)) + V^\top|BK e(t)| - cV^\top z(t) \\ &\quad + V^\top|\rho(t)| + V^\top\lambda(\zeta(t, x_t)) - V^\top B^*(t) \\ &\le -cV^\top(\bar{x}(t) - \underline{x}(t) + z(t)) + V^\top\lambda(\zeta(t, x_t)) \\ &\quad + V^\top|BK|\Gamma|x(t)| + V^\top|\rho(t)| \\ &\le -cV^\top(\bar{x}(t) - \underline{x}(t) + z(t)) + V^\top\lambda(\zeta(t, x_t)) \\ &\quad + V^\top|BK|\Gamma(\bar{x}(t) - \underline{x}(t)) + V^\top|\rho(t)| \end{aligned} \quad (37) $$

for all $t \ge 0$. From (15), (18), and (24), we deduce that

$$ \begin{aligned} \dot{U}(t) &\le -cV^\top(\bar{x}(t) - \underline{x}(t) + z(t)) + V^\top\lambda(\zeta(t, x_t)) \\ &\quad + V^\top|BK|\Gamma(\bar{x}(t) - \underline{x}(t)) + V^\top\zeta(t, x_t) \\ &\le (-cV^\top + V^\top|BK|\Gamma)(\bar{x}(t) - \underline{x}(t) + z(t)) \\ &\quad + V^\top\zeta(t, x_t) + V^\top\lambda(\zeta(t, x_t)) \\ &\le -rU(\bar{x}(t), \underline{x}(t), z(t)) + V^\top\zeta(t, x_t) \\ &\quad + V^\top\lambda(\zeta(t, x_t)) \end{aligned} \quad (38) $$

for all $t \ge 0$, where the second inequality followed because $z$ is nonnegative valued, and the last inequality used (6) and the fact that $\bar{x}-\underline{x}$ and $z$ are nonnegative valued. Also, since (13) gives $\mu(m) \in [m-T, m]$ for all $m \ge 0$, we have

$$ \begin{aligned} \zeta(t, x_t) &\le \bar{\tau} |BK A| |x|_{[\mu(t)-\bar{\tau}, \mu(t))]} + f_0^\sharp |x|_{[t-h-2T-2\bar{\tau}, t]} \\ &\quad + \bar{\tau} |(BK)^2| |x|_{[t-2T-2\bar{\tau}, t]} \\ &\le B^\sharp |x|_{[t-2(T+\bar{\tau})-h, t]} \end{aligned} \quad (39) $$

for all $t \ge 0$, by our choices of $f_0^\sharp$ and $B^\sharp$ in (25) and (9). Using (39) and our choice of $\lambda$ in (14), we deduce that

$$ \begin{aligned} \lambda(\zeta(t, x_t)) &\le |BK\Omega^{-1}|_{[0,\nu]} \int_{t-\nu}^t |e^{A(t-\ell)}| B^\sharp |x|_{[\ell-2(T+\bar{\tau})-h,\ell]} d\ell \quad (40) \\ &\le |BK\Omega^{-1}|_{[0,\nu]} \int_0^\nu |e^{A\ell}| d\ell B^\sharp |x|_{[t-2(T+\bar{\tau})-h-\nu, t]} d\ell \end{aligned} $$
for all $t \ge 0$. By using (39) and (40) to upper bound the last two terms in (38) and recalling our choice of $R$ in (9) and (35), we get $\dot{U}(t) \le -rU(\bar{x}(t), \underline{x}(t), z(t)) + V^\top R|\bar{x} - \underline{x} + z|_{[t-2(T+\bar{\tau})-h-\nu,t]}$ for all $t \ge 0$, where we also used the nonnegative valuedness of $z(t)$. Hence,

$$ \dot{U}(t) \le -rU(\bar{x}(t), \underline{x}(t), z(t)) + p_* \sup_{s \in [t-2(T+\bar{\tau})-h-\nu,t]} U(\bar{x}(s), \underline{x}(s), z(s)) \quad (41) $$

holds for all $t \ge 0$, by our condition (10) on $p_*$.

It follows from our choice (12) of $R_*$ and from applying Halanay's inequality (e.g., from [21, Section 4.1.2]) that $U(\bar{x}(t), \underline{x}(t), z(t)) \le U(\bar{x}(0), \underline{x}(0), z_0)e^{-R_*t}$ holds for all $t \ge 0$. This exponential decay estimate implies that

$$ \min_{1 \le i \le n} V_i |x_\ell(t)| \le \max_{1 \le i \le n} V_i \sum_{j=1}^{n} (\bar{x}_j(0) - \underline{x}_j(0) + z_{0j}) e^{-R_*t} \quad (42) $$

for all $t \ge 0$ and $\ell \in \{1, \dots, n\}$, where $z_{0j} > 0$ is the $j$th component of $z_0$ and we used (35) and the positive valuedness of $z$ to get (42). Hence, by the Cauchy-Schwarz inequality and our choices of $\bar{x}(0)$ and $\underline{x}(0)$,

$$ |x_\ell(t)| \le \frac{\sqrt{n \max\{V_i:1 \le i \le n\}}}{\min\{V_i:1 \le i \le n\}} ||2|x(0)| + z_0|| e^{-R_*t} \quad (43) $$

holds for all $t \ge 0$ and $\ell \in \{1, \dots, n\}$. Then (16) follows by squaring both sides of (43), then summing the result over $\ell$, then taking square roots of both sides of the result.

**Remark 3:** If, instead of (4), we require a continuous nondecreasing function $\mathcal{F}: [0, +\infty) \to [0, +\infty)$ such that

$$ |f(t, \phi)| \le \mathcal{F}(|\phi|_{[-h, 0]})(\phi|_{[-h, 0]}) \quad (44) $$

holds for all $t \ge 0$ and $\phi \in C_{in}(h)$, then we can prove a local analog of Theorem 1, which applies for suitable positive vectors $z_0 \in \mathbb{R}^n$ and constants $\sigma_0 > 0$, and all initial states $x(0)$ such that $||x(0)|| \le \sigma_0$. This is done by setting

$$ B_* = \frac{n \max\{V_i:1 \le i \le n\}}{\min\{V_i:1 \le i \le n\}} (2\sigma_0 + ||z_0||) \quad (45) $$

and then replacing $f_0$ by $\mathcal{F}(B_*)$ in the formulas for $B^\dagger$ and $\zeta$ in (9) and (15). With this replacement, we can then apply the proof of Theorem 1 from above on the interval $[0, t]$, for each $t \ge 0$ such that $||x(\ell)|| \le B_*$ for all $\ell \in [0, t]$, to conclude that (16) holds for all $t \in [0, T_*)$, where $T_* = \sup\{t \ge 0: ||x(\ell)|| \le B_*$ for all $\ell \in [0, t]\}$. One can then show that $T_* = +\infty$, by arguing by contradiction, as follows.

First, note that $T_* > 0$, since (45) gives $||x(0)|| \le \sigma_0 < B_*$ and $x$ is continuous. On the other hand, if $T_* < +\infty$, then the continuity of $x(t)$ implies that $||x(T_*)|| = B_*$, so evaluating (16) at $t = T_*$ gives the contradiction $B_* = ||x(T_*)|| \le e^{-R_*T_*} B_* < B_*$. This contradiction implies that $T_* = +\infty$, so (16) holds for all $t \ge 0$. Our condition (44) allows cases where the dynamics can exhibit quadratic (or higher order) growth, instead of the linear growth in the state from (4). $\square$

### III. ILLUSTRATIONS

1) Marine Robotic Example: We first study a dynamics for the control of the depth and pitch degrees-of-freedom of an autonomous underwater vehicle [2], [15], [17], whose linearization was used in [15], [17] to represent the dynamics

of the BlueROV2 vehicle, which is used in environmental surveys. Following [15], [17], we assume that the vehicle has a Doppler Velocity Logger (or DVL) for estimating the velocity of the vehicle. The DVL experiences bottom lock, making it impractical to continuously change the control values and producing input delays. Hence, we illustrate benefits of our new event-triggered approach, which are beyond the capabilities of [2], [17] or other studies that did not allow nonlinearities with unknown delays or did not quantify the effects of uncertain coefficient matrices. This strongly motivates our more complicated control and trigger rule from Theorem 1.

Using [24, Equation (9.28)], and assuming that the vehicle is neutrally buoyant, the vehicle dynamics become

$$ (m - X_{\dot{w}}(t))\dot{w}(t) - (mx_g + Z_q)\dot{q}(t) - Z_w w(t) + f_a(t, w_t, q_t) - (mU(t) + z_q)q(t) = Z_{\gamma_s} u_Z \quad (46) $$

$$ (mx_g + M_{\dot{w}}(t))\dot{w}(t) + (I_{yy} - M_{\dot{q}})\dot{q}(t) - M_w w(t) + f_b(t, w_t, q_t) + (mx_g U - M_q)q(t) - M_\theta \theta = M_{\gamma_s} u_M \quad (47) $$

whose parameters were experimentally computed in [24], where $f_a(t, w_t, q_t)$ and $f_b(t, w_t, q_t)$ represent nonlinear and delayed effects. As in [17], we assume that the surge nominal velocity is $U = 0.1$ m/s. The states are the depth velocity $w$ and the pitch velocity $q$, and the controls $u_Z$ and $u_M$ are the force and moment to produce motion.

With the parameter values and controller from [24], the system (46)-(47) takes the form (1) from Theorem 1 with

$$ A = \begin{bmatrix} -0.387 & 0 \\ 0 & -1.8 \end{bmatrix} \quad \text{and} \quad B = \begin{bmatrix} 0.038 \\ 1.5 \end{bmatrix}, \quad (48) $$

which are the $A$ and $B$ in [15]. Choosing $K = [-0.977852, -0.097546799]$ as in the linearized case in [2] gives the eigenvalues $-1.94988$ and $-0.420595$ for

$$ H = A + BK = \begin{bmatrix} -0.424158 & -0.00370678 \\ -1.46678 & -1.94632 \end{bmatrix} \quad (49) $$

satisfying Assumption 1. We then applied Theorem 1 in three scenarios: (i) $f_0 = 0$, with $\bar{\tau} > 0$ and $\nu$ as large as possible, (ii) $\bar{\tau} = 0$, with $f_0$ chosen as large as possible while still satisfying our assumptions with $\nu = 0.12$, and (iii) $f_0 = \bar{\tau} = 0$, $\bar{\Delta}_A \in \mathbb{R}^{n \times n}$ being the zero matrix, and $\bar{\Delta}_B \in \mathbb{R}^{n \times p}$ having all entries equaling a largest possible common value $\delta_*$ such that we can satisfy our assumptions with $\nu = 0.25$.

**Scenario (i).** To satisfy the requirements from Section II-A with $f_0 = 0$, we chose $V = [2, 0.2]^\top$, $c = 0.22$, $r = 0.06$, $p_* = 0.05$, $f_0 = 0$, $\bar{\tau} = 0.15$, $\bar{\Delta}_A = 0$, $\bar{\Delta}_B = 0$, and

$$ \Gamma = \begin{bmatrix} 0.5 & 0.012 \\ 0.6 & 0.7 \end{bmatrix}. \quad (50) $$

Then we used a bisection method to find that the largest value of the lower bound $\nu$ for the inter-execution times that satisfies the assumptions of Theorem 1 was $\nu = 0.26$.

**Scenario (ii).** We applied Theorem 1 with $\bar{\tau} = 0$, and with the growth rate $f_0 = 0.05$ and $h = 1.5$, which allowed $f = [f_1, f_2]^\top$ with $f_1(t, x_t) = 0.05 \sin(x(t-1.5))$ and $f_2 = 0$. In this case, the assumptions of Theorem 1 were satisfied with
the parameter values from Scenario (i) above, except with $\nu$
reduced from 0.26 to 0.15, $r$ increased from 0.06 to 0.065,
and $p_*$ increased from 0.05 to 0.06. We increased $r$ and $p_*$
and reduced $\nu$, because the assumptions of Theorem 1 were
not satisfied with $f_0 = 0.05$ with the $r, p_*, \text{ and } \nu$ values that
we used in the $f_0 = 0$ case above. This illustrates a trade-off
that we observed in numerical experiments, where allowing
the uncertain nonlinearity $f$ required smaller lower bounds
$\nu$ on the inter-execution times, as compared to cases where
the uncertainty was instead in the unknown input delay $\tau$.

Scenario (iii). Assume that $f_0 = \bar{\tau} = 0$, $\bar{\Delta}_A$ is the zero matrix, and both entries of $\bar{\Delta}_B \in \mathbb{R}^{2 \times 1}$ equal some value $\delta_* > 0$. With the choice $\nu = 0.25$, $p_* = 0.0625$, and all other parameters chosen the same as in Scenario (ii), we then found that the largest possible $\delta_*$ that satisfied the requirements of Theorem 1 in this scenario was $\delta_* = 0.06$, which is larger than the first entry of $B$ in (48). Hence, our work covers significant cases such as $\Delta_B(t) = [0.06 \sin(t), 0]^\top$ where some components of $\Delta_B$ can have larger suprema than the corresponding components of the unperturbed coefficient $B$.

2) Robustness Example: To compare Theorem 1 from Section II-B above with the simpler static event-triggered interval observer control result from [19] which also allowed added uncertainty on $A$, we revisit the example from [19, Section 6.2], which had $A = [a_{ij}] \in \mathbb{R}^{2 \times 2}$ having the entries $a_{11} = 1$, $a_{12} = 1/2$, $a_{21} = 3/2$, and $a_{22} = 0$, $f = 0$, $\bar{\tau} = 0$, $\Delta_B = 0$, $\Delta_A = [\delta_{ij}]$ having the form $\delta_{11} = \delta_1$, $\delta_{12} = \delta_2/2$, $\delta_{21} = 1.5\delta_3$, and $\delta_{22} = 0$, $K = [-4/3, -1/3]$, and $B = [1, 1]^\top$. In [19], the maximum allowable lower bound $\nu$ on the inter-execution times was $\nu = 0.02$ if the time-varying uncertainties $\delta_{ij}$ were each bounded by 0.01. This was based on choosing all entries of $\Gamma \in \mathbb{R}^{2 \times 2}$ to be $0.045$ in the requirement $|\Omega^{-1}(s) - I| \le \Gamma$ for all $s \in [0, \nu]$. By contrast, here we used Mathematica to show that we can satisfy the requirements of Theorem 1 with the preceding choices, except with each entry of $\Gamma$ being $0.04455$, $V = [1, 1]^\top$, $c = 1/6$, $r = 0.18$, $p_* = 0.017$, the same bound on the $\delta_{ij}$ as in [19, Section 6.2], and $\nu = 0.15$. This 7-fold increase in $\nu$ (from $\nu = 0.02$ in [19] to $\nu = 0.15$ here) can be attributed to our new condition (8) and event-triggered control, and our novel use of Halanay's inequality.

IV. CONCLUSION

We studied dynamic event-triggered control under un-
known input delays, uncertain coefficients, and unknown
nonlinearities. Key novel features included (i) our relaxed
requirement on $\nu$ and new trigger rules, using suprema
of available state measurements over suitable intervals, (ii)
using Halanay's inequality to handle unknown delays and
uncertainties, and (iii) our ability to approximate suprema
that arise from our novel trigger rules using a variant of
Halanay's inequality. We illustrated a tradeoff, where nonlin-
earities reduce the lower bound on the inter-execution times,
compared to significantly larger lower bounds on the inter-
execution times in other scenarios where the nonlinearity is
not present, but where input delays or uncertain control gains
occur. We aim to provide extensions under outputs.

REFERENCES

[1] P. Tabuada, “Event-triggered real-time scheduling of stabilizing control tasks,” *IEEE Trans. Autom. Control*, vol. 52, no. 9, pp. 1680–1685, 2007.

[2] S. Alyahia, C. Barbalata, M. Malisoff, and F. Mazenc, “Dynamic event-triggered control of linear continuous-time systems using a positive systems approach,” *Nonlinear Analysis: Hybrid Systems*, vol. 54, no. 101508, 2024.

[3] A. Girard, “Dynamic triggering mechanisms for event-triggered control,” *IEEE Trans. Autom. Control*, vol. 60, no. 7, pp. 1992–1997, 2015.

[4] D. Borgers, V. Dolk, and W. Heemels, “Riccati-based design of event-triggered controllers for linear systems with delays,” *IEEE Trans. Autom. Control*, vol. 63, no. 1, pp. 174–188, 2018.

[5] A. Borri and P. Pepe, “Event-triggered control of nonlinear systems with time-varying state delays,” *IEEE Trans. Autom. Control*, vol. 66, no. 6, pp. 2846–2853, 2021.

[6] M. Di Ferdinando, S. Di Gennaro, A. Borri, G. Pola, and P. Pepe, “On the robustification of digital event-based stabilizers for nonlinear time-delay systems,” *Nonlinear Analysis: Hybrid Systems*, vol. 52, no. 101463, 2024.

[7] V. Dolk, J. Ploeg, and W. Heemels, “Event-triggered control for string-stable vehicle platooning,” *IEEE Trans. Intelligent Transportation Systems*, vol. 18, no. 12, pp. 3486–3500, 2017.

[8] W. Heemels, K. Johansson, and P. Tabuada, “An introduction to event-triggered and self-triggered control,” in *Proceedings of the 51st IEEE Conference on Decision and Control*, 2012, pp. 3270–3285.

[9] C. Nowzari, E. Garcia, and J. Cortes, “Event-triggered communication and control of networked systems for multi-agent consensus,” *Automatica*, vol. 105, pp. 1–27, 2019.

[10] P. Ong and J. Cortes, “Performance-barrier-based event-triggered control with applications to network systems,” *IEEE Trans. Autom. Control*, vol. 69, no. 7, pp. 4230–4244, 2024.

[11] C. De Persis, R. Postoyan, and P. Tesi, “Event-triggered control from data,” *IEEE Trans. Autom. Control*, vol. 69, no. 6, pp. 3780–3795, 2024.

[12] Q. Voortman, D. Efimov, A. Pogromsky, J.-P. Richard, and H. Ni-
mjeijer, “Remote state estimation of steered systems with limited
communications: an event-triggered approach,” *IEEE Trans. Autom.
Control*, vol. 69, no. 7, pp. 4199–4214, 2024.

[13] A. Maass, W. Wang, D. Nesic, R. Postoyan, and W. Heemels, “Event-triggered control through the eyes of a hybrid small-gain theorem,” *IEEE Trans. Autom. Control*, vol. 68, no. 10, pp. 5906–5921, 2023.

[14] V. Varma, R. Postoyan, D. Quevedo, and I.-C. Morarescu, “Event-triggered transmission policies for nonlinear control systems over erasure channels,” *IEEE Control Syst. Lett.*, vol. 7, pp. 2113–2118, 2023.

[15] F. Mazenc, M. Malisoff, C. Barbalata, and Z.-P. Jiang, “Event-triggered control using a positive systems approach,” *European J. Control*, vol. 62, pp. 63–68, 2021.

[16] F. Mazenc, M. Malisoff, and C. Barbalata, “Event-triggered control for continuous-time linear systems with a delay in the input,” *Syst. Control Lett.*, vol. 159, no. 105075, 2022.

[17] F. Mazenc, M. Malisoff, C. Barbalata, and Z.-P. Jiang, “Event-triggered control for discrete-time systems using a positive systems approach,” *IEEE Control Syst. Lett.*, vol. 6, pp. 1843–1848, 2022.

[18] G. Marouani, D. Nguyen, T. Dinh, and T. Raissi, “Active fault control based on interval observer for discrete-time linear switched systems,” *European J. Control*, vol. 77, no. 100991, 2024.

[19] F. Mazenc, M. Malisoff, C. Barbalata, and Z.-P. Jiang, “Event-triggered control for linear time-varying systems using a positive systems approach,” *Syst. Control Lett.*, vol. 161, no. 105131, 2022.

[20] W. Haddad, V. Chellaboina, and Q. Hui, *Nonnegative and Compart-mental Systems*. Princeton, NJ: Princeton University Press, 2009.

[21] E. Fridman, *Introduction to Time-Delay Systems: Analysis and Control*. Basel: Birkhauser, 2014.

[22] F. Mazenc and M. Malisoff, “New versions of Halanay’s inequality with multiple gain terms,” *IEEE Control Syst. Lett.*, vol. 6, pp. 1790–1795, 2022.

[23] F. Mazenc, V. Andrieu, and M. Malisoff, “Design of continu-
ous–discrete observers for time-varying nonlinear systems,” *Automat-
ica*, vol. 57, pp. 135–144, 2015.

[24] T. Prestero, *Verification of a Six-Degree of Freedom Simulation Model for the REMUS Autonomous Underwater Vehicle*. Cambridge, MA: MS Thesis, MIT, 2001.