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# On the discretization of the implicit Lyapunov function-based control

Moussa Labbadi, Denis Efimov

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## To cite this version:

Moussa Labbadi, Denis Efimov. On the discretization of the implicit Lyapunov function-based control. CDC 2025 - 64th IEEE Conference on Decision and Control, Dec 2025, Rio De Janeiro, Brazil. hal-05435658

HAL Id: hal-05435658

https://inria.hal.science/hal-05435658v1

Submitted on 30 Dec 2025

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# On the discretization of the implicit Lyapunov function-based control

Moussa Labbadi and Denis Efimov

**Abstract**—An implicit Euler discretization scheme of the control law from [1] is proposed, which preserves all the main properties of the continuous-time counterpart (hyper-exponential rate of convergence, compensation of matched disturbances or their attenuation, robustness to measurement noises). Additionally, the sample-and-hold implementation of the control, along with its stability analysis, is provided.

**Keywords**— Hyperexponential convergence; Lyapunov methods; Implicit discretization; Sample-and-hold implementation.

## I. INTRODUCTION

Recently, the implicit Lyapunov function (ILF) method, first introduced in [2], [3], has proven to be a powerful tool for investigating accelerated convergence rates, including hyperexponential (asymptotic), finite-time, and fixed-time stability and stabilization. Moreover, ILF-based control approaches ensure input-to-state stability (ISS) in the presence of certain admissible perturbations [4], [5], [1], [6], [7] or their compensation and ISS with respect to noise [8], [9]. The shortage of the approach is that the control law explicitly depends on ILF value, and it is necessary to calculate it online for control application. To overcome this issue, the interplay between computation of ILF in continuous-time and its sampled implementations has also been explored in [10], [11], [4], [1]. However, the discretization of such control systems, particularly their sample-and-hold implementation, remains a complex task [12], [13]. The difficulty often lies in preserving the key advantages of the designed control or estimation algorithms when transitioning to a discrete setting. While sampled ILF-based controllers have been proposed, guaranteeing robust input-to-state stability (ISS) and accelerated convergence under both discretization and sample-and-hold implementations remains an open problem, which constitutes the focus of this paper.

This paper addresses on the implicit discretization of a hyperexponentially (asymptotically) converging linear system of second- or third-order, controlled via an ILF-based algorithm in the presence of measurement noise and perturbations. We demonstrate that such a discretization preserves the uniform hyperexponential convergence of its continuous-time counterpart while ensuring robustness against measurement noise. The results clearly show that the state trajectory corresponding to a given initial condition converges faster

than any exponential rate. Furthermore, the gain associated with the noise term $v$ remains bounded by a static linear upper bound. Notably, in the case of matched perturbation (i.e., under high-order sliding mode control), the disturbance is entirely rejected, ensuring robust performance against external perturbations. Moreover, in the sample-and-hold implementation, the system remains ISS and exhibits hyper-exponential convergence in the initial conditions. Simulations illustrate that the proposed control approach provides significant performance improvements, even under slow sampling conditions.

Proofs are omitted due to space limitations.

### Notation

* $\mathbb{R}^+ = \{x \in \mathbb{R} : x \ge 0\}$, where $\mathbb{R}$ is the set of real numbers; $\mathbb{Z}$ is the set of integers, $\mathbb{Z}^+ = \mathbb{Z} \cap \mathbb{R}^+$.

* $|\cdot|$ denotes the absolute value in $\mathbb{R}$, and $\|\cdot\|$ is used for the Euclidean norm on $\mathbb{R}^n$.

* For a (Lebesgue) measurable function $d: \mathbb{R}^+ \to \mathbb{R}^m$, define the norm $\|d\|_{\infty} = \ess \sup_{t \in \mathbb{R}^+} \|d(t)\|$, and the set of $d$ with the property $\|d\|_{\infty} < +\infty$ is denoted as (the set of essentially bounded measurable functions).

* For a sequence $d_k \in \mathbb{R}^m$ with $k \in \mathbb{Z}^+$, define its norm by $|d|_{\infty} = \sup_{k \in \mathbb{Z}^+} \|d_k\|$, and the set of $d$ with $|d|_{\infty} < +\infty$ is denoted by $\ell_m^{\infty}$.

* A continuous function $\alpha: \mathbb{R}^+ \to \mathbb{R}^+$ belongs to the class $\mathcal{K}$ if $\alpha(0) = 0$ and it is strictly increasing. The function $\alpha: \mathbb{R}^+ \to \mathbb{R}^+$ belongs to the class $\mathcal{K}_{\infty}$ if $\alpha \in \mathcal{K}$ and it increases to infinity.

* Denote the identity matrix of dimension $n \times n$ by $I_n$ and the matrix of zeros of dimension $m \times n$ by $0_{m \times n}$.

* $\text{diag}\{f\}$ represents a diagonal matrix of dimension $n \times n$ with a vector $f \in \mathbb{R}^n$ on the main diagonal.

* The relation $P \prec 0$ ($P \succeq 0$) means that a symmetric matrix $P \in \mathbb{R}^{n \times n}$ is negative (positive semi-) definite, and $\lambda_{\min}(P)$ denotes the minimal eigenvalue of such a matrix $P$.

## II. PRELIMINARIES

This paper addresses the discretization of a continuous-time stabilizing algorithm proposed in [1], [4] for a chain of integrators using the ILF method and analyzes stability guarantees under sampled-and-hold its implementation. Therefore, we consider the following system:

$$ \begin{aligned} \dot{x}(t) &= Ax(t) + b(u(t) + d(t)), \quad t \ge 0 \\ y(t) &= x(t) + v(t), \end{aligned} \quad (1) $$

<footnote>*The first author gratefully acknowledges the support of the CNRS through the project ROC-MAS LA-SE DÉVELOP RELAT° INTERNA-TIO.</footnote>

<footnote>M. Labbadi is with the Aix-Marseille University, LIS UMR CNRS 7020, 13013 Marseille, France (e-mail: moussa.labbadi@lis-lab.fr).</footnote>

<footnote>D. Efimov is with the Inria, Univ. Lille, CNRS, UMR 9189 - CRISTAL, F-59000 Lille, France (e-mail: denis.efimov@inria.fr).</footnote>
where

$$
A = \begin{pmatrix} 0 & 1 & 0 & \cdots & 0 \\ 0 & 0 & 1 & \cdots & 0 \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & 0 & \cdots & 1 \\ 0 & 0 & 0 & \cdots & 0 \end{pmatrix}, \quad b = \begin{pmatrix} 0 \\ \vdots \\ 0 \\ \vdots \\ 0 \end{pmatrix},
$$

where the state vector is $x(t) \in \mathbb{R}^n$, the control input is
$u(t) \in \mathbb{R}$, and $d \in \mathcal{L}_\infty^1$ represents disturbances, satisfying
$\|d\|_\infty < \bar{d} < +\infty$ for some $\bar{d} > 0$. We denote the set
of such essentially bounded inputs as $\mathcal{D} = \{d \in \mathcal{L}_\infty^1 :$
$\|d\|_\infty < \bar{d}\}$. The full state $x(t)$ is assumed to be measured
by $y(t) \in \mathbb{R}^n$, $v(t) \in \mathbb{R}^n$ represents the measurement
noise, assumed to be bounded, i.e., $v(t) \in \mathcal{V}$, with $\mathcal{V} =$
$\{v \in \mathcal{L}_\infty^1 : \|v\|_\infty < \bar{v} < +\infty\}$ for some bound $\bar{v} > 0$.

The goal [1], [4] to globally stabilize (1) at the origin for any disturbance $d \in \mathcal{D}$ and noise $v \in \mathcal{V}$, ensuring an accelerated convergence rate and stability under sampled ILF calculations in discrete time.

A. The ILF-based control in continuous time

Using the ILF-based control proposed in [1], [4] and based
on the ISS property of homogeneous systems [8], [9], the
following theorem can be stated:

Theorem 1: [1], [4] Consider the control law:

$$
u(t) = V(y(t))^{1-\mu} K D_q (V(y(t)))^{-1} y(t),
$$

where $K = YX^{-1}$, $\mathbf{D}_q(\rho) = \text{diag}\{\rho^{q_i}\}_{i=1}^n$, with $\mathbf{q}_i = 1 + (n-i)\mu$, $\mu \in (0, 1]$, and matrices $X \in \mathbb{R}^{n \times n}$, $Y \in \mathbb{R}^{1 \times n}$ solving the following LMIs:

$$
\left\{
\begin{array}{l}
\begin{bmatrix} \Upsilon & b \\ b^{\top} & -\varrho_2 \end{bmatrix} \preceq 0, \\
\dot{X}\mathcal{H}_q^{\top} + \mathcal{H}_q X \succeq X \succ 0, \\
\Upsilon = XA^{\top} + AX + Y^{\top}b^{\top} + bY + \varrho_1(X\mathcal{H}_q^{\top} + \mathcal{H}_q X),
\end{array}
\right.
\qquad (2)
$$

where $\varrho_1, \varrho_2 > 0$ and $\mathcal{H}_q = \text{diag}\{\mathbf{q}_i\}_{i=1}^n$. The ILF $V: \mathbb{R}^n \to \mathbb{R}^+$ is defined as:

$$
\begin{gather*}
Q(V(y(t)), y(t)) = 0, \\
Q(V, x) = x^{\top} \mathbf{D}_{\mathbf{q}}(V^{-1}) P \mathbf{D}_{\mathbf{q}}(V^{-1}) x - 1, \quad P = X^{-1}, \tag{3}
\end{gather*}
$$

for $x \neq 0$, with $V(0) = 0$. Then

i) If $\mu = 1$, the closed-loop system (1) is globally finite-time stable for $d \in \mathcal{D}$, $v \equiv 0$ and ISS with respect to $v$ provided that $\varrho_1 > \varrho_2 d^2 (\bar{d}$ should be given).

ii) If $\mu \in (0,1)$, the closed-loop system (1) is globally finite-time stable for $d \equiv 0$, $v \equiv 0$, and ISS with respect to both $v$ and $d$.

B. Sampled ILF-based control in continuous time

As established in existing works, such as in Corollary 8 of [1] and Corollary 21 of [4], the sampled-time realization of ILF-based control guarantees asymptotic stabilization, independent of the sampling interval, and hyperexponential stability as shown in [10], [11].

Theorem 2: [1], [4], [10], [11] Assume that

i) The conditions of Theorem 1 hold.

ii) $\{t_k\}_{k=0}^{\infty}$ is an arbitrary sequence of time instances
such that $t_0 = 0 < t_1 < t_2 < \cdots < t_k < \dots$ and
$\lim_{k \to +\infty} t_k = +\infty$.

$\lim_{k \to +\infty} t_k = +\infty$

iii) The control has the form

$$
V(y(t_k))^{1-\mu} K D_q (V(y(t_k)))^{-1} y(t) \quad (4)
$$

on each time interval $t \in [t_k, t_{k+1})$ and $V(y(t_k)) \in \mathbb{R}^+$ such that $Q(V(y(t_k)), y(t_k)) = 0$.

Then, the closed-loop system (1) is globally asymptotically
stable for $\mu = 1$, with $d \in \mathcal{D}$, $v \equiv 0$, with compensation
of matched properly bounded $d$, and ISS with respect to
$v$; for $\mu \in (0, 1]$, the the closed-loop system (1) is ISS
with respect to both $v$ and $d$. Moreover, as demonstrated
in Proposition 1 of [11], the application of such a control
results in hyperexponential stability in the perturbation-free
setting.

III. DISCRETIZATION OF SAMPLED ILF-BASED CONTROL

In this section, we present a discrete-time version of the
ILF controller proposed in [1] for the double and third-order
integrators using the implicit Euler method [14], [15]. After
substitution of (4) in the right-hand side of the system (1),
the closed-loop dynamics can be written as follows:

$$
\begin{equation}
\begin{aligned}
\dot{x}(t) = & \left[ A + b(V_k)^{1-\mu} K \mathbf{D}_q [ (V_k)^{-1} ] \right] x(t) \\
& + b(V_k)^{1-\mu} K \mathbf{D}_q [ (V_k)^{-1} ] v(t) + bd(t),
\end{aligned}
\tag{5}
\end{equation}
$$

where $V_k = V(y(t_k))$ and $t \in [t_k, t_{k+1})$.

The next step is to investigate the behavior of this system
in discrete time under measurement noise for $n = 2$ and
$n = 3$, following its sample-and-hold implementation.

A. Implicit Euler discretization for double integrator

Consider the system (1) for $n = 2$ and let $h > 0$ be a
constant discretization step, and denote by $t_k = j_k h$, for
$(k = 1, 2, 3, \dots, +\infty)$ the ILF sampling time instants, and
we take the discretization times at $ih$ for $j_k \le i < j_{k+1}$, $i \in$
$\mathbb{Z}^+$. Then, the application of the implicit Euler discretization
method to (5) gives, for $i, k \in \mathbb{Z}^+$,

$$
\begin{align}
\xi_{i+1} &= \Theta_k^{-1} (\xi_i + S_k v_{i+1} + hbd_{i+1}), \quad j_k \le i < j_{k+1} \tag{6} \\
\xi_i &\xrightarrow{h \to 0} x(ih), \quad v_i = v(ih), \quad d_i = d(ih), \nonumber
\end{align}
$$

where for $K = [K_1 \ K_2]$,

$$
\Theta_k = I_2 - h \left( A + b(V_k)^{1-\mu} K \mathbf{D}_q [ (V_k)^{-1} ] \right) \\
= \begin{bmatrix} 1 & -h \\ -K_1 h (V_k)^{-2\mu} & 1 - K_2 h (V_k)^{-\mu} \end{bmatrix}
$$

$$
S_k = h b(V_k)^{1-\mu} K \mathbf{D}_q [ (V_k)^{-1} ] = \begin{bmatrix} 0 & 0 \\ \frac{K_1 h}{(V_k)^{2\mu}} & \frac{K_2 h}{(V_k)^{\mu}} \end{bmatrix},
$$

$$
\text{and } \mathbf{D}_{\mathbf{q}}(\rho) = \begin{bmatrix} \rho^{1+\mu} & 0 \\ 0 & \rho \end{bmatrix}, \quad \Theta_k^{-1} = \frac{D_k}{O_k},
$$

$$
O_k = 1 - h K_2 (V_k)^{-\mu} - h^2 K_1 (V_k)^{-2\mu}
$$

$$
D_k = \begin{bmatrix} \frac{(V_k)^\mu - K_2 h}{K_1 h} & h \\ \frac{K_1 h}{(V_k)^{2\mu}} & 1 \end{bmatrix}.
$$
It is straightforward to check that we have the following propriety:

$$ \lim_{V_k \to 0} \Theta_k^{-1} = \begin{bmatrix} 0 & 0 \\ -h^{-1} & 0 \end{bmatrix}, $$

which implies that the dynamics of (6) is well defined for all $i, k \in \mathbb{Z}^+$. In the following, we will check sensibility to both noise $v_i$ and external disturbances $d_i$. The measurement noise gain matrix satisfies:

$$ \Theta_k^{-1} S_k = \frac{1}{O_k} \begin{bmatrix} \frac{K_1 h^2}{(V_k)^{2\mu}} & \frac{K_2 h^2}{(V_k)^\mu} \\ \frac{K_1 h}{(V_k)^{2\mu}} & \frac{K_2 h}{(V_k)^\mu} \end{bmatrix}. $$

which is elementwise bounded, and as $V_k \to 0$, it converges to:

$$ \lim_{V_k \to 0} \Theta_k^{-1} S_k = \begin{bmatrix} -1 & 0 \\ -h^{-1} & 0 \end{bmatrix}. $$

It is important to note that as $h \to 0$, the discretized system (6) should approach the behavior of the original continuous dynamics (5). However, the original system is robust with respect to perturbations in $v$, while the computations above show that the system's gain exhibits an asymptotic dependence on $h^{-1}$, reflecting an increasing sensitivity to noise as the time step $h$ becomes smaller. Finally, the perturbations gain matrix satisfies:

$$ \lim_{V_k \to 0} \Theta_k^{-1} b = \begin{bmatrix} 0 \\ 0 \end{bmatrix}, $$

which demonstrates that the disturbance $d$ is asymptotically rejected by the control.

**Theorem 3:** Consider (6) with $n = 2$, for given $\varrho > 0$, $\varphi > 0$, and $\Psi > 0$, let there exist $P = P^\top \in \mathbb{R}^{2 \times 2}$, $K \in \mathbb{R}^{1 \times 2}$ such that the matrix inequalities are verified:

$$ (A + bK)P^{-1} + P^{-1}(A + bK)^\top \le 0, \quad (7a) $$

$$ (A + bK)P^{-1}(A + bK)^\top - \varphi^{-1}P^{-1} \quad (7b) $$

$$ -\varrho^{-1}bKK^\top b^\top - \Psi^{-1}bb^\top \ge 0, $$

$$ \mathcal{H}_q P^{-1} + P^{-1} \mathcal{H}_q \ge 0. \quad (7c) $$

Then, for any $\xi_0 \in \mathbb{R}^2$, $v \in \ell_\infty^2$, and $d \in \ell_\infty^1$ in (6):

$$ V_k^{-2-2\mu} \lambda_{\min}(P) \|\xi_i\|^2 \le \left(\frac{\varphi}{h^2}\right)^i V_k^{-2} \lambda_{\max}(P) \|\xi_0\|^2 \prod_{j=0}^{i-1} (V_j)^{2\mu} + \sum_{j=0}^{i-1} \left(\frac{\varphi}{h^2}\right)^{i-1-j} \left[ \varrho \left( (V_j)^{-2-2\mu} v_{1,j+1} + (V_j)^{-2} v_{2,j+1}^2 \right) + d_{j+1}^\top \Psi(V_j)^{-2+2\mu} d_{j+1} \right] \prod_{l=j+1}^{i-1} (V_l)^{2\mu}. $$

for all $k \ge 1$ and $j_k \le i < j_{k+1}$.

Note that the first two matrix inequalities of Theorem 3 are verified for $P$ and $K$ found in Theorem 1. The last matrix inequality is always satisfied in this case for some $\varrho > 0$, $\varphi > 0$, and $\Psi > 0$. Then, the only additional condition to check is the third inequality.

The obtained results clearly demonstrate that the convergence of the initial error is faster than any exponential rate

since $V_k \to 0$ as $k \to +\infty$, with ISS in $d$ and $v$. Furthermore, the gain associated with the noise term $v$ remains bounded by a static linear upper bound. Notably, in the high-order sliding mode control case, when $\mu = 1$, the disturbance $d$ is entirely rejected, ensuring robust performance against external perturbations.

## B. Implicit Euler discretization for third integrators

Consider the system (1) for $n = 3$. Then, the application of the implicit Euler discretization method to (5) gives, for $k \in \mathbb{Z}^+$,

$$ \xi_{i+1} = \Theta_k^{-1} (\xi_i + S_k v_{i+1} + hbd_{i+1}), \quad j_k \le i < j_{k+1}, \quad (8) $$

where

$$ \begin{aligned} \Theta_k &= I_3 - h \left( A + b(V_k)^{1-\mu} K \mathbf{D}_q [(V_k)^{-1}] \right) \\ &= \begin{bmatrix} 1 & -h & 0 \\ 0 & 1 & -h \\ -K_1 h(V_k)^{-3\mu} & -K_2 h(V_k)^{-2\mu} & 1 - K_3 h(V_k)^{-\mu} \end{bmatrix} \\ S_k &= h b(V_k)^{1-\mu} K \mathbf{D}_q [(V_k)^{-1}], \end{aligned} $$

$$ \text{where } \mathbf{D}_q(\rho) = \begin{bmatrix} \rho^{1+2\mu} & 0 & 0 \\ 0 & \rho^{1+\mu} & 0 \\ 0 & 0 & \rho \end{bmatrix}, \quad \Theta_k^{-1} = \frac{D_k}{O_k}, $$

$$ O_k = 1 - K_1(V_k)^{-3\mu}h^3 - K_2(V_k)^{-2\mu}h^2 - K_3(V_k)^{-\mu}h $$

$$ D_k = \begin{bmatrix} 1 - \frac{K_3 h}{(V_k)^{3\mu}} - \frac{K_2 h^2}{(V_k)^{2\mu}} & h - \frac{K_3 h^2}{V_k^{\mu}} & h^2 \\ \frac{K_1}{(V_k)^{3\mu}} h^2 & 1 - \frac{K_2 h}{(V_k)^{3\mu}} & h \\ \frac{K_1}{(V_k)^{3\mu}} h & \frac{h (K_1 h + K_2 (V_k)^{\mu})}{(V_k)^{3\mu}} & 1 \end{bmatrix}. $$

for a stabilizing control gain $K = [K_1 \ K_2 \ K_3]$ with $K_1, K_2, K_3 < 0$. Since $V_k \to 0$ as $k \to \infty$, we have the following properties:

$$ \lim_{V_k \to 0} \Theta_k = \begin{bmatrix} 0 & 0 & 0 \\ -h^{-1} & 0 & 0 \\ -h^{-2} & -h^{-1} & 0 \end{bmatrix}, $$

which implies that the dynamics of (8) are asymptotically approaching a deadbeat system. The measurement noise gain matrix takes the following form:

$$ S_k = \begin{bmatrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ \frac{hK_1}{(V_k)^{3\mu}} & \frac{hK_2}{(V_k)^{2\mu}} & \frac{hK_3}{(V_k)^\mu} \end{bmatrix} $$

Thus,

$$ \Theta_k^{-1} S_k = \frac{D_k S_k}{O_k} = \frac{1}{O_k} \begin{bmatrix} \frac{K_1 h^3}{V_k^{3\mu}} & \frac{K_2 h^3}{V_k^{2\mu}} & \frac{K_3 h^3}{V_k^{\mu}} \\ \frac{K_1 h^2}{V_k^{3\mu}} & \frac{K_2 h^2}{V_k^{2\mu}} & \frac{K_3 h^2}{V_k^{\mu}} \\ \frac{K_1 h}{V_k^{3\mu}} & \frac{K_2 h}{V_k^{2\mu}} & \frac{K_3 h}{V_k^{\mu}} \end{bmatrix} $$

which is elementwise bounded, and as $V_k \to 0$, it converges to:

$$ \lim_{V_k \to 0} \Theta_k^{-1} S_k = \begin{bmatrix} -1 & 0 & 0 \\ -h^{-1} & 0 & 0 \\ -h^{-2} & 0 & 0 \end{bmatrix}. $$
This implies that the system's dynamics are robust to noise. The perturbations gain matrix satisfies:

$$h\Theta_k^{-1}b = h\frac{D_k}{O_k}b, \quad hD_k b = \begin{bmatrix} h^3 \\ h^2 \\ h \end{bmatrix} \lim_{V_k \to 0} \Theta_k^{-1}b = \begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix},$$

and the disturbance $d$ is asymptotically rejected by the control.

**Theorem 4:** Consider (6) with $n=3$, if for given $\varrho > 0$, $\varphi > 0$, and $\Psi > 0$, there exist $P = P^\top \in \mathbb{R}^{3 \times 3}$, $K \in \mathbb{R}^{1 \times 3}$ such that the matrix inequalities in (7) are verified. Then, for any $\xi_0 \in \mathbb{R}^3$, $v \in L_\infty^3$, and $d \in L_\infty^3$ in (8):

$$V_k^{-2-4\mu}\lambda_{\min}(P)\|\xi_i\|^2 \le \left(\frac{\varphi}{h^2}\right)^i V_k^{-2}\lambda_{\max}(P)\|\xi_0\|^2 \prod_{j=0}^{i-2} (V_j)^{2\mu} \\ + \sum_{j=0}^{i-1} \left(\frac{\varphi}{h^2}\right)^{i-1-j} \left[ \varrho \left( (V_j)^{-2-4\mu}v_{1,j+1} + (V_j)^{-2-2\mu}v_{2,j+1} + (V_j)^{-2}v_{3,j+1}^2 \right) + \Psi(V_j)^{-2+2\mu}d_{j+1}^T d_{j+1} \right] \prod_{l=j+1}^{i-1} (V_l)^{2\mu},$$

for all $k \ge 1$.

## IV. SAMPLE-AND-HOLD IMPLEMENTATION

Consider a control system where the control input $u(t) = u(t_k) = u_k$ is held constant over intervals $t \in [t_k, t_{k+1})$ for all $k \in \mathbb{Z}^+$. Here, $t_k = j_k h$, where $h > 0$ is the sampling period, and there may be multiple discretization instants within each interval, denoted by $j_k \le i < j_{k+1}$. For simplicity, assume $j_k = nk$, meaning the control input is sampled at each $n$ discretization steps.

### A. Sample-and-Hold Implementation for Double Integrator

Using the solution formula for linear systems, the state update equation is given by:

$$x_{k+1} = A_h x_k + b_h u_k + \delta_k, \quad (9)$$

where:

$$A_h = I_2 + hA, \quad b_h = \begin{bmatrix} \frac{h^2}{2} \\ h \end{bmatrix}, \quad \delta_k = \int_0^h \begin{bmatrix} h-s \\ 1 \end{bmatrix} d(t_k+s) ds.$$

Here, $x_k = x(t_k)$ represents the exact state value at time $t_k$, not an approximation.

The control input $u(t_k)$ is computed by solving the following optimization problem:

$$u(t_k) = \arg \min_{u_k \in \mathbb{R}, \|v\|_{\infty}=0, \|d\|_{\infty}=0, \xi_k=x_k} \|x_{k+2} - \xi_{k+2}\|,$$

where $\xi_k$ represents the discretized state trajectory obtained in the previous section. In the absence of perturbations, the control $u_k$ must ensure that $x_{k+1}$ closely matches $\xi_{k+1}$, given the initial condition $\xi_k = x_k$.

Solving the optimization problem leads to the following equation

$$[\Theta_k^{-1}]^2 x_k = A_h^2 x_k + C_h \begin{bmatrix} u_{k+1} \\ u_k \end{bmatrix},$$

where the controllability matrix $C_h$ is:

$$C_h = [b_h \quad A_h b_h].$$

By applying the control inputs $u_{k+j}(x_k)$ for $j = 0, 1$, we can guarantee $x_{k+2} = \xi_{k+2}$. Since $C_h$ is nonsingular, the control inputs can be computed as:

$$\begin{bmatrix} u_{k+1} \\ u_k \end{bmatrix} = C_h^{-1} ([\Theta_k^{-1}]^2 - A_h^2) x_k.$$

The feedback control law is given by [16]

$$u_k = \Phi_k(x_k + v_k), \quad (10)$$

where:

$$\Phi_k = \begin{bmatrix} 0 & 1 \end{bmatrix} C_h^{-1} ([\Theta_k^{-1}]^2 - A_h^2).$$

Note that $\Phi_k$ remains bounded due to the properties of $\Theta_k^{-1}$. Moreover, $\lim_{V_k \to 0} \Phi_k = 0$.

The closed-loop system, combining the state update equation and the feedback control law, is given by:

$$\begin{aligned} x_{k+1} &= (A_h + b_h \Phi_k)x_k + b_h \Phi_k v_k + \delta_k \\ &= (M_h + F_h [\Theta_k^{-1}]^2)x_k + b_h \Phi_k v_k + \delta_k \\ &= (M_h + \Psi_k)x_k + \tilde{\delta}_k, \quad \tilde{\delta}_k = b_h \Phi_k v_k + \delta_k, \end{aligned}$$

where

$$F_h = b_h \begin{bmatrix} 0 & 1 \end{bmatrix} C_h^{-1}, \quad M_h = A_h - F_h A_h^2, \\ \Psi_k = F_h [\Theta_k^{-1}]^2,$$

where the matrix $M_h$ is nilpotent of index $n$ (i.e., $M_h^2 = 0$) and $\lim_{k \to +\infty} \Psi_k = 0$ since $\Theta_k^{-1}$ is nilpotent of index 2; the new input $\tilde{\delta}$ is bounded.

**Proposition 1:** Let the control gain $K$ be chosen to verify the conditions of Theorem 1. Then the exact system (9) with control (10) is ISS and in the disturbance-free case the rate of convergence to the origin is faster than any exponential.

Consider the reconstruction error $e_k = x_k - \xi_k$, which evolves according to the dynamics:

$$e_{k+1} = \Theta_k^{-1}(e_k - S_k v_{k+1} - hbd_{k+1}) + \tilde{\delta}_k + R_k(e_k + \xi_k), \quad (11)$$

where $R_k = M_h + \Psi_k - \Theta_k^{-1}$ is a bounded matrix for all $t > 0$.

**Theorem 5:** Let the control gain $K$ be chosen to verify the conditions of Theorem 1. Then the tracking error (11) is ISS and in the disturbance-free case the rate of convergence to the origin is faster than any exponential.

### B. Sample-and-hold implementation for third integrators

In this case, the matrices in the model (9) take the form:

$$A_h = I_3 + hA + \frac{h^2}{2} A^2, \quad b_h = \begin{bmatrix} \frac{h^3}{6} \\ \frac{h^2}{2} \\ h \end{bmatrix},$$

and

$$\delta_k = \int_0^h \begin{bmatrix} \frac{(h-s)^2}{21} \\ h-s \\ 1 \end{bmatrix} d(t_k + s) ds.$$
Applying the same approach, the control law $u_k$ can be determined from the equation:

$$[\Theta_k^{-1}]^3 x_k = A_h^3 x_k + C_h \begin{bmatrix} u_{k+2} \\ u_{k+1} \\ u_k \end{bmatrix}, \quad (12)$$

where the controllability matrix is:

$$C_h = \begin{bmatrix} b_h & A_h b_h & A_h^2 b_h \end{bmatrix}.$$

Since $C_h$ is invertible, the control input sequence is:

$$\begin{bmatrix} u_{k+2} \\ u_{k+1} \\ u_k \end{bmatrix} = C_h^{-1} ([\Theta_k^{-1}]^3 - A_h^3) x_k. \quad (13)$$

A practical feedback control law is:

$$u_k = \Phi_k x_k, \quad (14)$$

where

$$\Phi_k = \begin{bmatrix} 0 & 0 & 1 \end{bmatrix} C_h^{-1} ([\Theta_k^{-1}]^3 - A_h^3).$$

The closed-loop system dynamics are:

$$x_{k+1} = A_h x_k + b_h \Phi_k x_k + \delta_k. \quad (15)$$

Consider the reconstruction error $e_k = x_k - \xi_k$, which evolves according to the dynamics:

$$e_{k+1} = \Theta_k^{-1}(e_k - S_k v_{k+1} - hbd_{k+1}) + \tilde{\delta}_k + R_k(e_k + \xi_k), \quad (16)$$

where $R_k = M_h + \Psi_k - \Theta_k^{-1}$ is a bounded matrix for all $t > 0$. And

$$F_h = b_h \begin{bmatrix} 0 & 0 & 1 \end{bmatrix} C_h^{-1}, \quad M_h = A_h - F_h A_h^3, \\ \Psi_k = F_h [\Theta_k^{-1}]^3,$$

the matrix $M_h$ becomes nilpotent of index $n$ (i.e., $M_h^3 = 0$) and $\lim_{k \to +\infty} \Psi_k = 0$ since $\Theta_k^{-1}$ is nilpotent of index 3.

**Theorem 6:** Let the control gain $K$ be chosen to verify the conditions of Theorem 1. Then the closed-loop system (15) is ISS and in the disturbance-free case the rate of convergence to the origin is faster than any exponential.

**Remark 1 (On the computation of V):** Computing the ILF equation for $V$ can be challenging. An approximation is proposed in [4], which also addresses real-time computation using a bisection-based method.

## V. SIMULATIONS

Let us illustrate the performance of the proposed ILF stabilizer (1) through numerical experiments, utilizing both the implicit Euler method and its sample-and-hold implementation. Consider (1) for $n = 3$ with $\xi_0 = x_0 = [5 \ 4 \ 0]$, and for

$$K = \begin{bmatrix} -13.2837 & -11.8918 & -4.5000 \end{bmatrix},$$

and the matrix $P$ is given as:

$$P = \begin{bmatrix} 19.9972 & 12.3839 & 3.0108 \\ 12.3839 & 8.1786 & 2.0730 \\ 3.0108 & 2.0730 & 0.5923 \end{bmatrix}.$$

Solutions to the linear matrix inequalities from Theorems 4 and 6 exist.

For $h = 0.1$, the simulation results are obtained for all $k \in \mathbb{Z}^+$ with the disturbance function $d(t) = \frac{\sin(5t)+\text{rnd}(1)}{30}$, and the noise $v(t) = 0.01(\text{rnd}(n) + \sin(\omega t))$, where $\text{rnd}(n)$ denotes uniformly distributed random numbers in the interval $[0, 1]^n$, and $\omega = [11, \dots, 10 + n]$.

The simulation results are shown in Fig. 1. As can be observed, the systems described by equation (8) exhibit a faster convergence. In the top plots, the state trajectories for (8) and (9) are shown against time. The bottom plot displays the performance of all state norms. From these results, it is clear that for both (8) and (9), the convergence to the origin is hyperexponential. Both methods achieve robustness against noise and perturbations.

<figure><img src="image_4.png" /><figcaption>Fig. 1. The results of simulation.</figcaption></figure>

## VI. CONCLUSIONS

A new implicit Euler discretization scheme for the ILF control is proposed, which preserves all the essential properties of its continuous-time counterpart (hyperexponential rate of convergence, compensation of matched disturbances or their attenuation, robustness to measurement noise). Additionally, a stability analysis is provided, along with a discussion of the sample-and-hold implementation and its associated stability. Simulation results demonstrate the effectiveness and robustness of the proposed discretization scheme. Further research could explore extensions of this scheme to firstly, $n \ge 4$, and more complex control systems where stability and robustness are critical.

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