# Prescribed-Time Observer Design for Nonlinear Triangular Systems With Delayed Measurements

Ania Adil, Ibrahima N'Doye, Taous-Meriem Laleg-Kirati

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Ania Adil, Ibrahima N'Doye, Taous-Meriem Laleg-Kirati. Prescribed-Time Observer Design for Nonlinear Triangular Systems With Delayed Measurements. IEEE Transactions on Automatic Control, 2024, 69 (11), pp.8080-8087. 10.1109/TAC.2024.3409839. hal-05138300

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# Prescribed-Time Observer Design for Nonlinear Triangular Systems with Delayed Measurements

Ania Adil, Ibrahima N'Doye, Senior Member, IEEE, Taous-Meriem Laleg-Kirati, Senior Member, IEEE

**Abstract**— In this paper, we propose a prescribed-time observer for nonlinear triangular systems with delayed outputs. The proposed observer is endowed with prescribed-time stability and relies on a scaling function and Lyapunov-Razumikhin-based conditions. We establish that the observer design derives a delay-independent stability condition in terms of linear matrix inequalities that guarantees prescribed-time convergence to zero of the estimation error. Additionally, we derive an explicit relation between the maximum bound of the delay and the prescribed-time convergence. We extend the proposed prescribed-time observer to systems with multiple nonlinearities. Finally, we provide a first illustrative example where we analyze the disturbance attenuation feature of the prescribed-time observer to vanishing measurement noise and a second application for estimating the water level in the coupled-tank systems to demonstrate the applicability and performance of the observer.

**Index Terms**— Delayed output measurement, nonlinear triangular systems, observer design, prescribed-time convergence, time-varying scaling function

## I. INTRODUCTION

TIME delay is encountered in various physical systems and real-world engineering (e.g., communication networks, industrial processes, mechanical and electrical engineering, transportation, the chemical industry, and biomedical engineering) [1]. Like standard dynamical systems, the state variables are usually not accessible for measurements owing to various physical and economical constraints. State observers are designed to estimate the state to respond to control or monitoring needs [2]–[5]. However, when a time delay occurs, the estimation problem is challenging, and the convergence of the conventional observers depends on the time delay.

Usually, the convergence of the observer can be achieved in asymptotic time or finite time. Depending on the application and nature of the system, one or the other convergence type can be privileged. For time-delay systems, there are many studies on designing asymptotic observers [6]–[10]. However, the efforts regarding finite-time observers are still ongoing. Lyapunov-Krasovskii functionals and Lyapunov-Razumikhin functions have been used in some research on studying finite-time and fixed-time stability for nonlinear systems with time delay (see [11]–[14]). Another more

Ania Adil thanks the Direction Générale de la Recherche Scientifique et du Développement Technologique DGRSDT/MESRS-Algeria for the financial support. This work was also supported by the King Abdullah University of Science and Technology (KAUST), Base Research Fund (BAS/1/1682 – 01 – 01) and NGTC-Al fund (REI/1/5233 – 01 – 01).

Ania Adil is with Laboratoire de Mathématiques Pures et Appliquées (LMPA) Mouloud Mammeri University of Tizi-Ouzou, Tizi-Ouzou, BP No 17, RP 15000, Algeria (email: ania.adil@ummto.dz) and Computer, Electrical and Mathematical Science & Engineering Division (CEMSE), KAUST, Thuwal 23955-6900, Saudi Arabia (email: ania.adil@kaust.edu.sa)

Ibrahima N'Doye is with Biological and Environmental Science and Engineering Division (BESE) and Computer, Electrical and Mathematical Science & Engineering Division (CEMSE), King Abdullah University of Science and Technology (KAUST), Thuwal 23955-6900, Saudi Arabia. (email: ibrahima.ndoye@kaust.edu.sa)

Taous-Meriem Laleg-Kirati is with The National Institute for Research in Digital Science and Technology (INRIA), Paris-Saclay, France. (email: taous-meriem.laleg@inria.fr)

involving method uses an arbitrary pointwise delay, providing a fixed-time convergence and fast converging observers [15]. In [16], the authors designed a nonlinear observer for second-order linear time-invariant systems with an output delay. The system is formulated under an ODE-PDE system and converted to a target system using an invertible backstepping transformation that converges in finite time. Nevertheless, the finite time is highly affected by the initial conditions, which are often unknown in the settling-time function. The interest of the control community in the prescribed-time observer design has recently increased. Such an observer design can converge in a fixed time. The key characteristic of this type of convergence is that the time is predetermined, regardless of the initial system conditions. This contrasts with finite-time and fixed-time convergence, necessitating knowledge of the upper bound of settling-time functions or uncertainties. However, the design of the prescribed-time observers for time-delay systems remains demanding, and few studies have developed this in the recent literature [11], [15], [17]–[20].

The prescribed-time observer design for linear time-invariant (LTI) time-delay systems has been explored in the continuous-time framework (see, [17]–[20]). For LTI input and output measurements in time-delay systems, [18] and [17], [19] proposed a prescribed-time predictor-based observer compensated by the output measurement and input delay through an ODE-PDE cascade transformation. This ODE-PDE setting provides a design stability approach that implicitly accounts for the PDE dynamics and scaling function property.

Some classical observer designs that drive inherent filtering or smoothing properties can handle small delays. In contrast, prescribed-time observers impose more demanding convergence conditions within a specified period, rendering them potentially more sensitive to delays of any magnitude. Therefore, addressing the challenges associated with delayed measurements in the prescribed-time observer design requires an assessment of the trade-off between tolerating time delay and achieving analytically guaranteed prescribed-time convergence.

This article presents a prescribed-time observer design for nonlinear triangular systems in the presence of constant delay in the measured outputs. To the best of our knowledge, this effort is the first attempt to derive a prescribed-time observer for this class of systems, which is a step beyond the results [16]–[19] that are limited to linear systems. More specifically, we derive sufficient linear matrix inequality (LMI) conditions based on the Lyapunov-Razumikhin theorem and the time-varying scaling function to guarantee that the estimated states converge in a prescribed time to the state vector of the system. To achieve this, we design delay-independent prescribed-time stability through the Lyapunov-based Razumikhin condition to demonstrate that the observer's estimation errors converge in a prescribed finite time in the presence of a constant delay in the output.

The major contributions of this work are summarized as follows:

(i) We propose a new prescribed-time observer design for nonlinear triangular systems with constant delay in measured outputs.

(ii) We derive a delay-independent stability condition that ensures prescribed-time convergence of the estimation error to zero through the Lyapunov-Razumikhin theorem and LMIs.

(iii) We establish an explicit relation between the prescribed-time convergence and the maximum bound of the delay that pre-
serves the prescribed-time stability.

(iv) We provide an extension of the results to a class of observable canonical systems with multiple nonlinearities.

(v) We perform a sensitive analysis of the proposed observer under a vanishing measurement noise to infer its disturbance attenuation level.

The remainder of this article is structured as follows. Section II recalls the preliminary definitions of prescribed-time stability. Next, Section III formulates the state observation problem. Section IV presents the major finding of the prescribed-time observer and its proof, along with the Lyapunov–Razumikhin theorem ensuring suffi- cient stability conditions of the estimation error to converge to zero. Extension to systems with multiple nonlinearities is also discussed in IV. Then, Section V presents two numerical examples. Finally, the conclusion and perspectives of this work are detailed in Section VI.

**Notation:** The notation $\mathbb{R}$ and $\mathbb{R}^+$ denote the set of real numbers and nonnegative real numbers, respectively. The Euclidean norm of a vector $v \in \mathbb{R}^n$ is defined as $\|v\| = \sqrt{v^T v}$. The identity matrix of dimension $n$ is $I_n$. The diagonal matrix with diagonal elements $\{a, b, \dots\}$ is denoted as $\text{diag}(a, b, \dots)$. For a real symmetric positive-definite matrix $P > 0$, $P^T$ denotes the transpose of a matrix $P$, and $\lambda_{\min}(P)$ and $\lambda_{\max}(P)$ are the minimum and maximum eigenvalue of $P$, respectively.

## II. PRELIMINARIES

This section reviews the tools and basic concepts required for this paper. We present some definitions and lemmas used in the mathematical development in the rest of the paper.

The following time-varying function is used in the design approach $\mu(t) : [0, T) \to \mathbb{R}^+$ [21]:

$$ \mu(t) = \frac{T}{T-t}. \quad (1) $$

The function $\mu$ is monotonically increasing and grows unbounded as $t$ approaches $T$, where $T$ is the prescribed time.

**Definition 1:** [21] We consider the time-delay system

$$ \dot{x} = f(t, x_{[t-\tau, t]}), \quad t \ge 0, \quad (2) $$

where $f(t, 0) = 0$ and $\tau$ is the delay. We assume that a unique solution to the system exists for any given initial condition $x_{[-\tau, 0]} = \phi(s)$, $s \in [-\tau, 0]$. The system is $(T + \tau)$-prescribed-time stable if it is $(T + \tau)$-fixed-time globally uniformly asymptotically stable on the time interval $I_{T_\tau} := [0, T + \tau)$, that is, a function $\beta$ of a class $KL$ exists such that

$$ |x(t)| \le \beta(|x_0(t)|, \mu(t-\tau) - 1), \quad (3) $$

where the function $\mu(t - \tau, T)$ is stated in (1).

Young's inequality is also exploited in the proof of the main result.

**Lemma 1 (Young's Inequality):** Let $P$ be a positive-definite matrix of an appropriate dimension. Then, for any $b \in \mathbb{R}^n$, we have

$$ |2x^T Pb| \le x^T Px + b^T Pb. $$

## III. PROBLEM STATEMENT

We consider the class of nonlinear triangular systems with delayed output described by the following canonical form:

$$ \left\{ \begin{array}{l} \dot{x}(t) = Ax(t) + Bf(x(t)) \\ y(t) = Cx(t - \tau), \end{array} \right. \quad (4) $$

where the matrix $A$ is defined as follows:

$$ (A)_{i,j} = \begin{cases} 1 & \text{if } j = i + 1, \\ 0 & \text{if } j \neq i + 1. \end{cases} \quad (5) $$

where $x(t) \in \mathbb{R}^n$ and $y(t) \in \mathbb{R}$ are the state vector and measured output of the system, respectively. The output is affected by a known and constant delay $\tau$. The matrices $B$ and $C$ are defined by

$$ B = \begin{bmatrix} 0 & 0 & \dots & 1 \end{bmatrix}^T, \quad C = \begin{bmatrix} 1 & 0 & \dots & 0 \end{bmatrix}. \quad (6) $$

The function $f: \mathbb{R}^n \to \mathbb{R}$ satisfies the Lipschitz property given as follows:

$$ |f(x_1, \dots, x_n) - f(\tilde{x}_1, \dots, \tilde{x}_n)| \le \gamma_f \sum_{j=1}^{n} |x_j - \tilde{x}_j|, \quad (7) $$

where $\gamma_f$ is the Lipschitz constant.

**Remark 1:** System (4) describes a broad class of dynamical systems represented in the generalized observability canonical form. In particular, several practical applications, such as chemical and biochemical processes [22], magnetic levitation [23], and robotics [24], can be modeled similarly. The nonlinearity of these systems appears at the last component after a diffeomorphism transformation into an observable canonical form. We focused on such a class of systems to reduce the complexity of the prescribed-time stability problem with a time delay. However, we extended the results to systems with multiple nonlinearities in Section IV-C.

The objective in this article is to design a prescribed-time observer for the system in (4) that achieves the stability of the estimation error within a prescribed finite time $T + \tau$. We also determine the stability conditions for which the proposed observer converges.

## IV. OBSERVER DESIGN AND PRESCRIBED-TIME CONVERGENCE ANALYSIS

This section introduces the prescribed-time observer design for nonlinear systems with delayed output measurements. In addition, this section also discusses and demonstrates the stability of the estimation error.

### A. Observer Design

The estimation error converges within a prescribed time $T + \tau$ if the system observer is defined as follows:

$$ \dot{x}(t) = A\hat{x}(t) + Bf(\hat{x}(t)) + \bar{K}_{\mu(t-\tau)}(y(t) - \hat{x}_1(t-\tau)), \quad (8) $$

where $\bar{K}_{\mu(t-\tau)} = \Gamma^{-1}(t)K$ and

$$ \Gamma(t) = \text{diag}\{1/\mu^{1+m}(t-\tau), \dots, 1/\mu^{n(1+m)}(t-\tau)\} $$

is a scaling matrix with $m \ge 1$ as a design parameter. In addition, $K$ is the observer gain. The function $\mu$ is defined as in (1), and $\tau$ is the delay.

The dynamics of the estimation error is provided below:

$$ \dot{e}(t) = Ae(t) + B\Delta f - \bar{K}_{\mu(t-\tau)}e_1(t-\tau), \quad (9) $$

with $e(t) = x(t) - \hat{x}(t)$ and $\Delta f = f(x) - f(\hat{x})$.

Then, the estimation error is transformed using the following transformation:

$$ \bar{e}(t) = \Gamma(t)e(t). \quad (10) $$

Hence, the transformed error dynamics are expressed as follows:

$$ \begin{aligned} \dot{\bar{e}}(t) = & \mu^{1+m}(t-\tau)(A-KC)\bar{e}(t) - (1+m)\frac{\dot{\mu}(t-\tau)}{\mu(t-\tau)}D\bar{e}(t) \\ & + \Gamma(t)B\Delta f + \mu^{1+m}(t-\tau)KC(\bar{e}(t) - \bar{e}(t-\tau)), \end{aligned} \quad (11) $$
where the diagonal matrix $D$ is defined as $D = \text{diag}\{1, 2, \dots, n\}$. Using the Leibnitz–Newton formula, $\bar{e}(t)$ satisfies the following equation:

$$ \bar{e}(t) - \bar{e}(t - \tau) = \int_{-\tau}^{0} \left[ \mu_t^{1+m}(\theta - \tau) A \bar{e}_t(\theta) - (1+m) \frac{\dot{\mu}_t(\theta - \tau)}{\mu_t(\theta - \tau)} D \bar{e}_t(\theta) + \Gamma B \Delta f + \mu_t^{1+m}(\theta - \tau) K C (\bar{e}_t(\theta - \tau)) \right] d\theta, \quad (12) $$

where $\bar{e}_t(\theta) = \bar{e}(t + \theta)$.

Substituting (12) into (11) and considering that $KCB\Delta f = 0$, we obtain the following equation:

$$ \begin{aligned} \dot{\bar{e}}(t) = & \mu^{1+m}(t - \tau)(A - KC)\bar{e}(t) \\ & - (1 + m) \frac{\dot{\mu}(t - \tau)}{\mu(t - \tau)} D\bar{e}(t) + \Gamma(t)B\Delta f \\ & + \mu^{1+m}(t - \tau)KC \int_{-\tau}^{0} \left[ \mu_t^{1+m}(\theta - \tau)A\bar{e}_t(\theta) \right. \\ & \qquad \left. - (1 + m) \frac{\dot{\mu}_t(\theta - \tau)}{\mu_t(\theta - \tau)} D\bar{e}_t(\theta) \right. \\ & \qquad \left. + \mu_t^{1+m}(\theta - \tau)KC(\bar{e}_t(\theta - \tau)) \right] d\theta. \end{aligned} \quad (13) $$

We simplify the notation by rewriting (13) as follows:

$$ \begin{aligned} \dot{\bar{e}}(t) = & \mu^{1+m}(t - \tau)(A - KC)\bar{e}(t) \\ & - (1 + m) \frac{\dot{\mu}(t - \tau)}{\mu(t - \tau)} D\bar{e}(t) + \Gamma(t)B\Delta f \\ & + \mu^{1+m}(t - \tau)KC\xi(\bar{e}_t, \mu_t), \end{aligned} \quad (14) $$

where

$$ \xi(\bar{e}_t, \mu_t) = \int_{-\tau}^{0} \left[ \mu_t^{1+m}(\theta - \tau)A\bar{e}_t(\theta) - (1+m)\frac{\dot{\mu}_t(\theta - \tau)}{\mu_t(\theta - \tau)}D\bar{e}_t(\theta) + \mu_t^{1+m}(\theta - \tau)KC\bar{e}_t(\theta - \tau) \right] d\theta. \quad (15) $$

## B. Convergence Analysis

This section provides stability conditions that guarantee the prescribed-time stability of the observer estimation error. Before establishing the main theorem of the prescribed-time observer design, we present the intermediate results to enhance the readability and clarity of the main theorem.

**Lemma 2:** Let $T$ be the prescribed convergence time and $\tau$ is the delay. We define the function $\mu(t)$ as in (1).

Then, the following equations hold:

$$ I_1 = \int_{-\tau}^{0} \mu^{2(1+m)}(t + \theta - \tau)d\theta \le \frac{T\mu^{2m+1}(t - \tau)}{2m + 1}, \quad (16) $$

$$ I_2 = \int_{-\tau}^{0} (\mu(t + \theta - \tau))^2 d\theta \le T\mu(t - \tau). \quad (17) $$

**Proof 1:** By integrating the function $\mu(t)$ over the time interval $[-\tau, 0]$, the following integrals hold:

$$ \begin{aligned} I_1 = & \frac{T}{2m+1} [\mu^{2m+1}(t-\tau) - \mu^{2m+1}(t-2\tau)] \\ & \le \frac{T\mu^{2m+1}(t-\tau)}{2m+1}, \end{aligned} \quad (18) $$

$$ \text{and } I_2 = T[\mu(t - \tau) - \mu(t - 2\tau)] \le T\mu(t - \tau). \quad (19) $$

**Lemma 3:** Let $P > 0$, $K$, and $C$ be matrices of appropriate dimensions with a constant $\rho > 1$. We define $\Theta$ as follows:

$$ \Theta = 2\mu^{1+m}(t - \tau)\bar{e}^T(t)PKC\xi(\bar{e}_t, \mu_t), \quad (20) $$

Where $\xi(\bar{e}_t, \mu_t)$ is defined as (15). Then, a constant $\bar{K}$ exists such that

$$ \begin{aligned} \Theta \le & \tau \left(2 + \frac{1+m}{T}\right) \mu^{1+m}(t-\tau)V(\bar{e}(t)) \\ & + \frac{\rho T}{2m+1} \mu^{2+3m}(t-\tau)\bar{K}(1+K_1^2)V(\bar{e}(t)) \\ & + \rho(1+m)\mu^{2+m}(t-\tau)\bar{K}V(\bar{e}(t)), \end{aligned} \quad (21) $$

where the constant $K_1$ is the first component of the gain $K$, and $\bar{K}$ is to be determined.

**Proof 2:** We consider the term $\Theta$ in (20). Knowing that $\frac{\dot{\mu}_t}{\mu_t} = \frac{1}{T}\mu_t$, we obtain

$$ \begin{aligned} \Theta = & 2 \int_{-\tau}^{0} \mu^{1+m}(t - \tau) \mu_t^{1+m}(\theta - \tau) \bar{e}^T(t) PKC A \bar{e}_t(\theta) d\theta \\ & - \frac{2(1+m)}{T} \int_{-\tau}^{0} \mu^{1+m}(t - \tau) \mu_t(\theta - \tau) \bar{e}^T(t) PKC D \bar{e}_t(\theta) d\theta \\ & + 2 \int_{-\tau}^{0} \mu^{1+m}(t - \tau) \mu_t^{1+m}(\theta - \tau) \bar{e}^T(t) PKC K C \bar{e}_t(\theta - \tau) d\theta. \end{aligned} \quad (22) $$

Using Young's inequality in each term of the integrands in (22), we obtain

$$ \begin{aligned} \Theta_1 := & 2\mu^{1+m}(t - \tau)\mu_t^{1+m}(\theta - \tau)\bar{e}^T(PKCA\bar{e}_t(\theta)) \\ & \le \mu^{1+m}(t - \tau)\bar{e}^T(t)P\bar{e}(t) \\ & + \mu^{1+m}(t - \tau)(K^T PK)(\mu_t^{1+m}(\theta - \tau))^2 \|CA\bar{e}_t(\theta)\|^2, \end{aligned} \quad (23) $$

$$ \begin{aligned} \Theta_2 := & 2\mu^{1+m}(t - \tau)\mu_t(\theta - \tau)\bar{e}^T(t)PKC D\bar{e}_t(\theta) \\ & \le \mu^{1+m}(t - \tau)\bar{e}^T(t)P\bar{e}(t) \\ & + \mu^{1+m}(t - \tau)(K^T PK)(\mu_t(\theta - \tau))^2 \|CD\bar{e}_t(\theta)\|^2, \end{aligned} \quad (24) $$

$$ \begin{aligned} \Theta_3 := & 2\mu^{1+m}(t - \tau)\mu_t^{1+m}(\theta - \tau)\bar{e}^T(t)PKCKC\bar{e}_t(\theta - \tau) \\ & \le \mu^{1+m}(t - \tau)\bar{e}^T(t)P\bar{e}(t) \\ & + \mu^{1+m}(t - \tau)(K^T PK)(\mu_t^{1+m}(\theta - \tau))^2(CK)^2 \|C\bar{e}_t(\theta - \tau)\|^2. \end{aligned} \quad (25) $$

Recalling that $V(\bar{e}(t)) = \bar{e}^T(t)P\bar{e}(t)$ and $V(\bar{e}_t(\theta)) = \bar{e}_t^T(\theta)P\bar{e}_t(\theta)$, we deduce the following inequalities:

$$ \begin{aligned} & \|CA\bar{e}_t(\theta)\|^2 \le \|\bar{e}_t(\theta)\|^2 \\ & \|CD\bar{e}_t(\theta)\|^2 \le \|\bar{e}_t(\theta)\|^2 \\ & \|\bar{e}_t(\theta)\|^2 \le \|P^{-1}\|\bar{e}_t(\theta)^T P\bar{e}_t(\theta)\|. \end{aligned} $$

Hence, (23) to (25) can be rewritten as follows:

$$ \Theta_1 \le \mu^{1+m}(t - \tau)V(\bar{e}) + \mu^{1+m}(t - \tau)(\mu_t^{1+m}(\theta - \tau))^2(K^T PK)\|P^{-1}\|V(\bar{e}_t(\theta)) \quad (26) $$

$$ \begin{aligned} \Theta_2 \le & \mu^{1+m}(t - \tau)V(\bar{e}) \\ & + \mu^{1+m}(t - \tau)(K^T PK)(\mu_t(\theta - \tau))^2 \|P^{-1}\|V(\bar{e}_t(\theta)) \end{aligned} \quad (27) $$

$$ \begin{aligned} \Theta_3 \le & \mu^{1+m}(t - \tau)V(\bar{e}) \\ & + \mu^{1+m}(t - \tau)(K^T PK)(\mu_t^{1+m}(\theta - \tau))^2 K_1^2 \|P^{-1}\|V(\bar{e}_t(\theta - \tau)) \end{aligned} \quad (28) $$

at time $t$ such that for a given $\rho > 1$, $V(\bar{e}_t(\theta)) \le \rho V(\bar{e}(t))$ holds $\forall \theta \in [-\tau, 0]$. Hence,
$$ \Theta_1 \le \mu^{1+m}(t-\tau)V(\bar{e}) + \rho\mu^{1+m}(t-\tau)(\mu_t^{1+m}(\theta-\tau))^2(K^T PK)\|P^{-1}\|V(\bar{e}(t)) \quad (29) $$

$$ \Theta_2 \le \mu^{1+m}(t-\tau)V(\bar{e}) + \rho\mu^{1+m}(t-\tau)(K^T PK)(\mu_t(\theta-\tau))^2\|P^{-1}\|V(\bar{e}(t)) \quad (30) $$

$$ \Theta_3 \le \mu^{1+m}(t-\tau)V(\bar{e}) + \rho\mu^{1+m}(t-\tau)(K^T PK)(\mu_t^{1+m}(\theta-\tau))^2 K_1^2 \|P^{-1}\|V(\bar{e}(t)). \quad (31) $$

By substituting (29) to (31) into (22) and using lemma 2, the upper bound on $\Theta$ becomes

$$ \Theta \le \tau \left(2 + \frac{1+m}{T}\right) \mu^{1+m}(t-\tau)V(\bar{e}) + \frac{\rho T}{2m+1} \mu^{2+3m}(t-\tau) \underbrace{(K^T PK)\|P^{-1}\|}_{\bar{K}} (1+K_1^2)V(\bar{e}(t)) + \rho(1+m)\mu^{2+m}(t-\tau) \underbrace{(K^T PK)\|P^{-1}\|}_{\bar{K}} V(\bar{e}(t)), \quad (32) $$

which ends the proof.

Next, we state the main result of this article presented in the following theorem, providing the stability conditions ensuring the prescribed-time convergence of the proposed observer design.

**Theorem 1:** Let $\tau > 0$ be a constant delay and assume the existence of a matrix $\mathcal{Y}$ with appropriate dimensions, a symmetric positive-definite matrix $P$, and real positive constants $\lambda_1$ and $\lambda_2$ so that the conditions below hold:

$$ A^T P + PA - C^T \mathcal{Y} - \mathcal{Y}^T C + \lambda_1 I_n \le 0, \quad (33) $$

$$ D^T P + PD - \lambda_2 I_n \ge 0, \quad (34) $$

$$ \text{where } \tau < \tau_{\max} \text{ and } \tau_{\max} = \frac{\lambda_1}{\lambda_{\max}(P) \left(2 + \frac{1+m}{T}\right)}. \quad (35) $$

Then, the observer error system (14) has a $(T+\tau)$-fixed-time globally uniformly asymptotically stable equilibrium at the origin, remains uniformly bounded and converges to zero as $t \to T+\tau$.

Moreover, the observer gain $K$ is

$$ K = [K_1 \dots K_n]^T = P^{-1} \mathcal{Y}^T. \quad (36) $$

**Proof 3:** We let $V(\bar{e}(t))$ be a Lyapunov function candidate as follows:

$$ V(\bar{e}(t)) = \bar{e}(t)^T P \bar{e}(t), \quad P = P^T > 0. \quad (37) $$

The derivative of the Lyapunov function $V$ is calculated as follows:

$$ \frac{dV(\bar{e}(t))}{dt} = \mu^{1+m}(t-\tau)\bar{e}^T(t)[(A-KC)^T P + P(A-KC)]\bar{e}(t) - (1+m)\frac{\dot{\mu}(t-\tau)}{\mu(t-\tau)}\bar{e}^T(t)[D^T P + PD]\bar{e}(t) + 2\bar{e}^T(t)P(\Gamma(t)B\Delta f) + 2\mu^{1+m}(t-\tau)\bar{e}^T(t)PKC\xi(\bar{e}_t, \mu_t). \quad (38) $$

Knowing that $f$ is Lipschitz, we obtain

$$ 2\bar{e}^T(t)P(\Gamma(t)B\Delta f) \le 2\bar{e}^T(t)P[\gamma_f\Gamma(t)B(x(t) - \hat{x}(t))] \le 2\gamma_f\bar{e}^T(t)P\bar{e}(t). $$

Hence, the derivative of $V(\bar{e}(t))$ becomes

$$ \frac{dV(\bar{e}(t))}{dt} \le \mu^{1+m}(t-\tau)\bar{e}^T(t)[(A-KC)^T P + P(A-KC)]\bar{e}(t) - (1+m)\frac{\dot{\mu}(t-\tau)}{\mu(t-\tau)}\bar{e}^T(t)[D^T P + PD]\bar{e}(t) + 2\gamma_f\bar{e}^T(t)P\bar{e}(t) + 2\mu^{1+m}(t-\tau)\bar{e}^T(t)PKC\xi(\bar{e}_t, \mu_t). $$

Considering the inequality (32) and knowing that $\frac{\dot{\mu}_t}{\mu_t} = \frac{1}{T}\mu_t$, the derivative of $V(\bar{e})$ becomes

$$ \frac{dV(\bar{e}(t))}{dt} \le \mu^{1+m}(t-\tau)\bar{e}^T(t)[(A-KC)^T P + P(A-KC)]\bar{e}(t) - \frac{(1+m)}{T}\mu(t-\tau)\bar{e}^T(t)[D^T P + PD]\bar{e}(t) + 2\gamma_f\bar{e}^T(t)P\bar{e}(t) + \tau\left(2 + \frac{1+m}{T}\right)\mu^{1+m}(t-\tau)V(\bar{e}) + \rho\frac{T}{2m+1}\mu^{2+3m}(t-\tau)\bar{K}(1+K_1^2)V(\bar{e}(t)) + \rho(1+m)\mu^{2+m}(t-\tau)\bar{K}V(\bar{e}(t)). \quad (39) $$

We let

$$ (A - KC)^T P + P(A - KC) \le -\lambda_1 I_n, \quad D^T P + PD \ge \lambda_2 I_n. $$

Hence,

$$ \frac{dV(\bar{e}(t))}{dt} \le -\left[ \frac{\lambda_1}{\lambda_{\max}(P)} - \tau \left(2 + \frac{1+m}{T}\right) \right] \mu^{1+m}(t-\tau)V(\bar{e}) + \left[ 2\gamma_f - \frac{(1+m)\lambda_2}{T\lambda_{\max}(P)} \mu(t-\tau) + \frac{\rho T}{2m+1} \bar{K}(1+K_1^2)\mu^{2+3m}(t-\tau) + \rho(1+m)\bar{K}\mu^{2+m}(t-\tau) \right] V(\bar{e}(t)). \quad (40) $$

Then, we obtain

$$ \frac{dV(\bar{e})}{dt} \le -\ell\mu^{2+3m}(t-\tau)V(\bar{e}) + \Xi V(\bar{e}), $$

$$ \text{where } \ell = \frac{\lambda_1}{\lambda_{\max}(P)} - \tau \left(2 + \frac{1+m}{T}\right), \quad \ell > 0 \\ \Xi = 2\gamma_f - \frac{(1+m)\lambda_2}{T\lambda_{\max}(P)}\mu(t-\tau) \\ + \frac{\rho T}{2m+1}\bar{K}(1+K_1^2)\mu^{2+3m}(t-\tau) \\ + \rho(1+m)\bar{K}\mu^{2+m}(t-\tau), \\ \bar{K} = (K^T PK)\|P^{-1}\|. $$

A constant $\rho$ and a time $t_e$ can be selected for which the following holds, for all $t$ greater than $t_e$:

$$ \frac{\lambda_2(1+m)}{T\lambda_{\max}(P)}\mu(t-\tau) - \frac{\rho T \bar{K}(1+K_1^2)}{2m+1}\mu^{2+3m}(t-\tau) - \rho(1+m)\bar{K}\mu^{2+m}(t-\tau) \ge 2\gamma_f, \quad (41) $$

implies that

$$ \dot{V}(\bar{e}) \le \begin{cases} 2\gamma_f V(\bar{e}), & t \in [0, t_e], \\ -\ell\mu^{1+m}(t-\tau)V(\bar{e}), & t \in (t_e, T+\tau). \end{cases} \quad (42) $$
For any $t \in [0, t_e]$, we can obtain

$$V(\bar{e}(t)) \le V(0) \exp \left( \frac{2\gamma_f t}{\lambda_{\min}(P)} \right).$$

The latter indicates that the finite-time escape phenomenon does not occur for the error equation (11). For $t \in (t_e, T + \tau)$, we find

$$\frac{dV(\bar{e})}{dt} \le -\ell\mu^{1+m}(t-\tau)V(\bar{e}). \quad (43)$$

With integration over $[0, t)$, we can obtain

$$\int_0^t \frac{\dot{V}(\bar{e}(s))}{V(\bar{e}(s))} ds \le -\int_0^t \ell\mu^{1+m}(s-\tau)ds = -\frac{\ell T}{m}\mu^m(s-\tau)\Big|_0^t,$$

Hence,

$$V(e(t)) \le V(0) \exp \left( -\frac{\ell T}{m}(\mu^m(t-\tau) - 1) \right), \quad (44)$$

and because

$$\lambda_{\min}(P)\|\bar{e}\|^2 \le V(t) \le \lambda_{\max}(P)\|\bar{e}\|^2, \quad (45)$$

we obtain

$$\|\bar{e}(t)\| \le \sqrt{\frac{V(0)}{\lambda_{\min}(P)}} \exp \left( -\frac{\ell T}{m}(\mu^m(t-\tau) - 1) \right), \quad (46)$$

Equivalently, we get

$$\|\bar{e}(t)\| \le \sqrt{\frac{\lambda_{\max}(P)}{\lambda_{\min}(P)}} \|\bar{e}(0)\| \times \exp \left( -\frac{\ell T}{m}(\mu^m(t-\tau) - 1) \right). \quad (47)$$

The function $\mu(t - \tau)$ is growing to infinity as $t$ approaches $T + \tau$; therefore, $\lim_{t \to T+\tau} \|\bar{e}(t)\| = 0$. Using the Lyapunov-Razumikhin theorem [1] and Definition 1, we conclude the $(T + \tau)$-prescribed-time stability of the estimation error dynamics (11) at the prescribed time $T + \tau$. Additionally, the state transformation (10) and inequality (46), allow us to obtain

$$\|e(t)\| \le n\nu_1\kappa(t) \exp \left( -\frac{\ell T}{2m}(\mu^m(t-\tau) - 1) \right), \quad (48)$$

where $\nu_1 = \sqrt{\frac{\lambda_{\max}(P)}{\lambda_{\min}(P)}} \|\bar{e}(0)\|$ and

$$\kappa(t) = \mu(t-\tau)^{n(m+1)} \exp \left( -\frac{\ell T}{2m}(\mu^m(t-\tau) - 1) \right).$$

Using the derivative of $\kappa(t)$, we obtain [25]

$$\dot{\kappa}(t) = \left( \frac{n(m+1)}{T} - \frac{\ell}{2} \mu^m(t-\tau) \right) \mu(t-\tau)\kappa(t).$$

If $\frac{2n(1+m)}{lT} \ge \mu^m(t - \tau)$, we let $\dot{\kappa}(t_1) = 0$, resulting in $t_1 = T - \left(\frac{Tl}{2n(1+m)}\right)^{\frac{1}{m}} T$. Then, $\kappa(t_1)$ is the upper bound of $\kappa(t)$; otherwise, it would be $\kappa(0)$. We define $\beta$ as follows:

$$\beta(\|e(0)\|, \mu(t-\tau) - 1) = n\nu_1\kappa(t) \exp \left( -\frac{\ell T}{2m}(\mu^m(t-\tau) - 1) \right).$$

As $\beta$ is strictly increasing and decreasing concerning $e(0)$ and $\mu$, respectively. Then, $\beta$ is a well-defined $KL$ function by virtue of the boundedness of $\kappa(t)$ and $\lim_{t \to T} \mu(t) = +\infty$. Furthermore, we use (48) and the upper bound of $\kappa(t)$, to conclude on the boundedness of the estimation error, which satisfies $\lim_{t \to T+\tau} \|e(t)\| = 0$. Hence,

we conclude the $(T + \tau)$-prescribed-time stability of the error $e(t)$, which ends the proof.

**Remark 2:** Theorem 1 presents delay-independent stability results in terms of LMIs within an explicit relationship between the prescribed time of convergence and the maximum bound of the delay. We could always select a constant $\rho$ and a time $t_e$ such that the condition in (41) derived from the stability proof is satisfied for all $t$ greater than $t_e$ due to the properties of the scaling function $\mu$ in (1).

### C. Extension to Systems with Multiple Nonlinearities

This section extends the results to observable canonical systems with multiple nonlinearities. Hence, we consider the system in (4) with multiple nonlinearities as follows:

$$\left\{ \begin{aligned} \dot{x}_1(t) &= x_2(t) + f_1(x_1(t)) \\ \dot{x}_2(t) &= x_3(t) + f_2(x_1(t), x_2(t)) \\ \vdots \\ \dot{x}_{n-1}(t) &= x_n(t) + f_{n-1}(x_1(t), x_2(t), \dots, x_{n-1}(t)) \\ \dot{x}_n(t) &= f_n(x_1(t), \dots, x_n(t)), \\ y(t) &= Cx(t - \tau). \end{aligned} \right. \quad (49)$$

The nonlinearities $f_i, i = 1, \dots, n$ defined on $\mathbb{R}^i \to \mathbb{R}$ satisfy the Lipschitz property as follows:

$$|f_i(x_1, \dots, x_i) - f_i(\tilde{x}_1, \dots, \tilde{x}_i)| \le \gamma f_i \sum_{j=1}^{i} |x_j - \tilde{x}_j|, \quad (50)$$

where $\gamma f_i$ is the Lipschitz constant.

We construct the following observer similarly to (8)

$$\dot{\hat{x}}(t) = A\hat{x}(t) + f(\hat{x}(t)) + \bar{K}_{\mu}(t-\tau)(y(t) - \hat{x}_1(t-\tau)), \quad (51)$$

where

$$\bar{K}_{\mu}(t-\tau) = \Gamma^{-1}(t)K, \text{ and}$$

$$\Gamma(t) = \operatorname{diag}\{1/\mu^{1+m}(t-\tau), \dots, 1/\mu^{n(1+m)}(t-\tau)\}.$$

The estimation error dynamics is provided below:

$$\dot{e}(t) = Ae(t) + \Delta f - \bar{K}_{\mu}(t-\tau)e_1(t-\tau), \quad (52)$$

where $e(t) = x(t) - \hat{x}(t)$ and $\Delta f = f(x) - f(\hat{x})$.

Following the same steps as outlined in the preceding section results in the following theorem.

**Theorem 2:** Let $\tau > 0$ be a constant delay and assume the existence of a matrix $\mathcal{Y}$ with appropriate dimensions, a symmetric positive-definite matrix $P$, and real positive constants $\lambda_1$ and $\lambda_2$ so that the conditions below hold:

$$A^T P + PA - C^T \mathcal{Y} - \mathcal{Y}^T C + \lambda_1 I_n \le 0, \quad (53)$$

$$D^T P + PD - \lambda_2 I_n \ge 0, \quad (54)$$

where

$$\tau < \tau_{\max} = \frac{\lambda_1}{\lambda_{\max}(P) \left( 3 + \frac{1+m}{T} + \rho \gamma_f^2 (K^T P K) \right)}. \quad (55)$$

Then, the observer error system (52) has a $(T + \tau)$-fixed-time globally uniformly asymptotically stable equilibrium at the origin, remains uniformly bounded and converges to zero as $t \to T + \tau$. Moreover, the observer gain $K$ is

$$K = [K_1 \dots K_n]^\top = P^{-1} \mathcal{Y}^\top. \quad (56)$$

**Proof 4:** The proof is omitted for brevity because it follows the same steps as in the previous section.

**Remark 3:** Theorem 2 results reveal that the upper bound of the maximal delay relies on a tuning parameter $\rho$, the Lipschitz
constant $\gamma_f$, and the observer gain $K$. These three parameters are not directly included in the statement of the maximal bound of the delay in Theorem 1. In particular, it is possible to tolerate considerable output delays in Theorem 2 by encompassing a trade-off between the maximum bound of the delay and the tuning parameter $\rho$ that guarantees prescribed-time convergence in the case of multiple nonlinearities.

**Remark 4:** The gains that increase to infinity are the primary source of the numerical instabilities that arise in implementing the prescribed-time observer. Several approaches have been suggested in the literature to address such problems, namely terminating the convergence of the observer before the predetermined time to guarantee the stability in a neighborhood of the origin [17], [26], saturating the gain of the observer [18], and adjusting the prescribed time of convergence to a significant value to avert the gains of the observer from expanding to infinity [27].

## V. SIMULATION RESULTS

This section is dedicated to the simulation results. We considered two numerical examples to demonstrate the performance of the proposed observer design. First, we studied the case of a third-order nonlinear system with a delayed output measurement. Then, we applied the observer design to the water-level estimation in coupled tanks. The proposed observer was implemented through MATLAB using a dde23 solver, a numerical integration of a delayed differential equation.

### A. Illustrative Example

We consider a third-order nonlinear triangular system, including a delayed output under the following canonical form, modified from [28]:

$$ \begin{cases} \dot{x}(t) = Ax(t) + Bf(x(t)) \\ y(t) = Cx(t - \tau), \end{cases} \qquad (57) $$

where $A, B$, and $C$ are given in (5) to (6), with $n = 3$.

1) Nominal Case: We consider the nonlinearity $f(x(t))$ in (57), defined as follows:

$$ f(x) = -(x_1 + 2x_2 + 2x_3) + [1 - (x_1 + 3x_2 + x_3^2)](x_2 + 2x_3). $$

First, we solve (33) to (34) with YALMIP in MATLAB. We found the matrices $P$ and $K$ as follows:

$$ P = \begin{bmatrix} 1832.9 & -407.5 & -268.6 \\ -407.5 & 627.6 & -301.8 \\ -268.6 & -301.8 & 828.3 \end{bmatrix}, K = [2.0835 \ 5.2883 \ 2.1104]^T. $$

The value of the bound on the delay found by (35) is equal to $\tau_{max} = 2.6 \times 10^{-3}$ s, where $m = 1$. For the simulation, we selected a smaller value than this bound, $\tau = 2.4 \times 10^{-3}$ s.

Next, we simulate the original system (57) and prescribed-time observer (8). We set the prescribed convergence time to $T = 4$ s. The initial conditions of the system and observer are set to $x(0) = [2, 2, 2]^T$ and $\hat{x}(0) = [1, 1, 1]^T$, respectively.

Fig. 1 depicts the phase trajectories of the original system on the blue line and the prescribed-time observer on the red line. The proposed prescribed-time observer converges at the prescribed time $\tau + T = 4.0024$ s to the original system in the presence of a delay. Fig. 2 provides the evolution of the estimation error norms on the logarithmic scale. The estimation errors decrease between $10^{-8}$ and $10^{-12}$, affirming the convergence at the prescribed time of the proposed observer approach.

<figure><img src="image_2.png" /><figcaption>Fig. 1: Phase trajectories of the original system and observer for $\tau = 2.4 \times 10^{-3}$ s and $T = 4$ s.</figcaption></figure>

<figure><img src="image_3.png" /><figcaption>Fig. 2: Norm of the estimation errors on the logarithmic scale for $\tau = 2.4 \times 10^{-3}$ s and $T = 4$ s.</figcaption></figure>

<figure><img src="image_4.png" /><figcaption>Fig. 3: Estimation errors for prescribed times ($T=3.5$ s and $T=5$ s).</figcaption></figure>

### 2) Performance Evaluation for Different Prescribed Times:

Two values were selected to analyze the effect of the prescribed time on the transient: $T = 3.5$ s and $T = 5$ s. The evolution of the estimation errors is illustrated in Fig. 3. The initial conditions in these cases were set to $x(0) = [3, 3, 3]^T$ and $\hat{x}(0) = [1, 1, 1]^T$, respectively. The transient is slightly larger when decreasing the convergence time $T$.

### 3) Disturbance Rejection to Vanishing Measurement Noise:

We analyze the performance of the prescribed-time observer method in the presence of vanishing measurement noise. We consider the output in (57) with a noise satisfying the following condition [29]

$$ y(t) = Cx(t - \tau) + \varpi(t), $$

where the vanishing noise $\varpi(t)$ is given as follows

$$ \varpi(t) = \delta \left(1 - \frac{t}{T}\right)^{(1+\sigma)} \varrho(t). $$
<figure><img src="image_5.png" /><figcaption>Fig. 4: Norms of estimation errors in the presence of measurement noise.</figcaption></figure>

We select $\delta = 0.1$, $\sigma = 0.5$, and $\rho(t)$ is a white Gaussian noise with a standard deviation of 5%.

Fig. 4 shows the norm of the estimation errors on the logarithmic scale, which converge in the prescribed time of 4.0024 s. The norm of estimation errors decreases between $10^{-3}$ and $10^{-6}$ in the presence of a measurement noise compared to the nominal case in Fig. 2, where it decreases between $10^{-8}$ and $10^{-12}$. These observations infer the disturbance attenuation feature of the proposed observer in rejecting the measurement noise. Finite-time convergence to a neighborhood of the origin, determined by the magnitude of the noise, is assured. This deviation from zero is an anticipated outcome due to the inherent sensitivity of the prescribed-time observer, exacerbated by the time-varying gain that tends toward infinity.

**Remark 5:** Despite their good performance under the previously specified conditions, the prescribed-time algorithms are vulnerable to measurement noise [16], [18], leading to divergence from its intended behavior and compromising the accuracy of reaching prescribed-time convergence. The solutions described in Remark 4 can be adopted to solve this problem, preventing the observer from diverging. Such behavior draws attention to the need for careful consideration and, potentially, the development of additional techniques to alleviate the effect of noise and enhance the overall robustness of the prescribed-time observers in practical applications.

## B. Application to Coupled-Tank Water Estimation

The purpose of this application is to demonstrate the effectiveness of the observer design in reconstructing the liquid levels in a coupled-tank plant. The coupled-tank plant is an industrial process used to validate methods developed in the literature for control and estimation [30]. This application was selected because it emerges in wastewater treatment, petrochemical plants, and other industries that require several time-response constraints to complete a desired behavior or action, which makes the prescribed-time convergence a critical characteristic. Moreover, the measured water level is supposed to be affected by a time delay from the interaction between the tanks. Furthermore, a small variation in the pump voltage can cause a delay in measuring the output.

The equations describing the dynamics of the liquid in Tanks 1 and 2 are stated as follows:

$$ \left\{ \begin{array}{l} \frac{\partial L_1}{\partial t} = -\frac{A_{o1}}{A_{t1}} \sqrt{2gL_1} + \frac{K_p}{A_{t1}} V_p \\ \frac{\partial L_2}{\partial t} = \frac{A_{o1}}{A_{t2}} \sqrt{2gL_1} - \frac{A_{o2}}{A_{t2}} \sqrt{2gL_2} \end{array} \right. \qquad (58) $$

The liquid levels in Tanks 1 and 2 are represented by $L_1$ and $L_2$, respectively. The only accessible output for measurement in this

<figure><img src="image_6.png" /><figcaption>Fig. 5: Behavior of $x_1$ and $x_2$ and their estimates for $\tau = 4 \times 10^{-3}$ s and $T = 3$ s.</figcaption></figure>

system (58) is the water level in Tank 2, which can be expressed by $y = L_2$. All physical parameter values in the system in (58) and the scheme of the coupled-tank plant are provided in [30].

Before designing the prescribed-time observer, a transformation of the coupled-tank system into an observable canonical form is required. Following the steps presented in [30], the coupled-tank system (58) can be described by the canonical form in (4):

$$ \left\{ \begin{array}{l} \dot{x}(t) = Ax(t) + Bf(x(t)), \\ y(t) = x_1(t - \tau), \end{array} \right. \qquad (59) $$

$$ \text{where } A = \begin{bmatrix} 0 & 1 \\ 0 & 0 \end{bmatrix}, f(x(t)) = \begin{bmatrix} 0 \\ f_1(x_1, x_2) \end{bmatrix}, $$

$$ f_1(x_1, x_2) = -\frac{A_{o1}^2 g}{A_{t1} A_{t2}} \left( 1 + \frac{K_p V_p}{A_{t2} x_2 + A_{o2} \sqrt{2g x_1}} \right) - \frac{A_{o2} g}{A_{t2}} \left( \frac{x_2}{\sqrt{2g x_1}} \right). $$

The prescribed-time observer is given by

$$ \left\{ \begin{array}{l} \dot{\hat{x}}_1(t) = \hat{x}_2(t) + \mu^{(1+m)}(t-\tau)K_1(y(t)-\hat{x}_1(t-\tau)) \\ \dot{\hat{x}}_2(t) = f(\hat{x}) + \mu^{2(1+m)}(t-\tau)K_2(y(t)-\hat{x}_1(t-\tau)), \end{array} \right. \qquad (60) $$

where $\hat{x}_1$ and $\hat{x}_2$ are the state estimates.

First, we solve (33) to (34) with YALMIP in MATLAB. We found the matrices $P$ and $K$ as follows:

$$ P = \begin{bmatrix} 1894.6 & -518.4 \\ -518.4 & 798.4 \end{bmatrix}, \quad K = \begin{bmatrix} 1.7081 & 2.8227 \end{bmatrix}. $$

The maximum bound of the delay equals $\tau_{max} = 4.5 \times 10^{-3}$ s. For the simulation, we selected $\tau = 4 \times 10^{-3}$ s.

Next, we simulated the system (59) and prescribed-time observer (60). We set the prescribed convergence time to $T = 3$ s. The initial values of the water levels in Tanks 1 and 2 in the original system and the observer were set to $x(0) = [4, 4]^T$ and $\hat{x}(0) = [1, 1]^T$, respectively. Next, Fig. 5 depicts the behavior of the states $x_1$ and $x_2$ of the system on a solid line and their estimates $\hat{x}_1$ and $\hat{x}_2$ with the prescribed-time observer on a dashed line. Noticeably, the observer reconstructs the water levels in the tanks at the prescribed time $\tau + T = 3.004$ s and converges to the actual states. Fig. 6 illustrates how the norms of estimations errors in logarithmic scale decrease to reach a numerical precision between $10^{-5}$ and $10^{-6}$, endorsing the prescribed-time convergence of the proposed observer.
<figure><img src="image_7.png" /><figcaption>Fig. 6: Norms of the estimation errors $\tau = 4 \times 10^{-3}$ s and $T = 3$ s.</figcaption></figure>

## VI. CONCLUSION

This article introduced a new design and analysis framework for a prescribed-time observer design for nonlinear systems with delayed outputs. The observer converges within a predefined time $\tau + T$, where $\tau$ is the output delay and $T$ is the prescribed time. The stability proof relies on the Lyapunov-Razumikhin theorem and derived delay-independent stability conditions guaranteeing the prescribed-time convergence of the estimation error to zero. We extended the proposed prescribed-time observer to systems with multiple nonlinearities. The numerical results on a third-order nonlinear triangular system with delayed output measurements where we analyze the disturbance attenuation feature of the prescribed-time observer to vanishing measurement noise and an application to water-level estimation in coupled-tank systems demonstrate the performance of the designed observer. The proposed approach in this article paves the way for further work and contributions in solving control and estimation problems in prescribed time for nonlinear time-delay systems through the Razumikhin-based condition and time-varying scaling function. In addition, the ideas developed in the article can be extended to the analysis and design of various alternative methodologies, such as predictor-based techniques and chain of observers, to guarantee prescribed-time convergence for time-delay systems.

In future work, we aim to derive new stability conditions of the prescribed-time observer for nonlinear systems with time-varying measurements delays based on the Lyapunov-Krasovskii approach, which is less conservative than the Lyapunov-Razumikhin method. Additionally, we will investigate the stability proof and disturbance rejection feature to vanishing measurement noise of the proposed observer.

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