# Sliding Mode Differentiator versus Sliding Mode Observer: the third way for particular cases

Loïc Michel, Malek Ghanes, Franck Plestan, Yannick Aoustin, Jean-Pierre Barbot

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Loïc Michel, Malek Ghanes, Franck Plestan, Yannick Aoustin, Jean-Pierre Barbot. Sliding Mode Differentiator versus Sliding Mode Observer: the third way for particular cases. 2026. hal-05477292

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# Sliding Mode Differentiator versus Sliding Mode Observer:
the third way for particular cases

Loïc Michel<fnref n="1" />, Malek Ghanes<fnref n="1" />, Franck Plestan<fnref n="1" />, Yannick Aoustin<fnref n="1" />, Jean-Pierre Barbot<fnref n="1" />,<fnref n="2" />

**Abstract**—This paper highlights the comparison, in terms of advantages and drawbacks, between sliding mode differentiators and sliding mode observers. Beyond the case where the signal to be derived comes from an unknown system, for which a differentiator is obviously required, the choice between a differentiator and an observer is not so simple. This is even more true considering that, in information theory, any unused information is lost information, e.g. time scale, structural properties... This issue has been addressed indirectly in Kalman filtering, where both the degree of confidence in the model and the quality of the signal are taken into account. However, this approach still relies on the availability of a model and on some assumptions of linearity. In this note, we focus on nonlinear settings and discuss several scenarios illustrating how differentiators and observers can be combined or selected to optimally exploit available information.

## I. INTRODUCTION

The choice between a differentiator and an observer is sometimes straightforward; yet, in many practical situations, it remains an open and unresolved problem. This choice depends on the purpose for which the estimated information will be used (observer-based control, diagnosis, left invertibility, etc.). In nonlinear observer-based control, the separation principle further imposes structural restrictive conditions that limit applicability and performance.

In this paper, we focus on sliding-mode observers and differentiators due to their finite-time convergence and robustness properties. While differentiation is a natural solution when no model of the signal-generating system is available, it raises fundamental questions regarding the relevance and usefulness of higher-order derivatives. Conversely, observer-based approaches are theoretically preferable when the system model is known and the objective is to estimate physical states; however, they often become infeasible for nonlinear or non-smooth systems.

Although this issue has been largely addressed for linear systems, notably through the Kalman framework, which balances model uncertainty against measurement noise [20], existing solutions rely on linear coordinate transformations and smooth signal assumptions. As a result, these solutions do not extend naturally to nonlinear and non-smooth systems. Addressing this gap constitutes the primary motivation of the present work.

The main contribution of this paper is twofold. First, we

provide an unified analysis that clarifies the respective domains of applicability of sliding-mode differentiators and observers for nonlinear systems. In particular, we show that differentiator-based approaches are well suited for mechanical and, more generally, Lagrangian systems, where derivatives of measured generalized coordinates are well defined and physically meaningful (see Section III). Second, we demonstrate that for systems with more complex structures, such as the Chua circuit [25], observer-based designs are more appropriate, since the required derivatives may not exist, rendering pure differentiation impossible [27].

Most importantly, this paper proposes a novel and "pragmatic" estimation strategy<fnref n="1" /> based on a cascaded and component-wise architecture. Part of the states component is estimated sequentially using a reduced estimator, designed to ensure finite-time convergence at each stage. Depending on the structural properties of the system and on the physical meaning of the signals involved, these estimators may take the form of sliding-mode observers, differentiators, or suitable combinations thereof. This pragmatic framework allows flexible and reliable state estimation for systems that cannot be efficiently handled by a purely observer-based or purely differentiation-based approach.

The rest of the paper is organized as follows. In Section II, reminders on observability and observer forms are provided. The differentiator approach is briefly recalled in Section III, while Section IV reviews observer-based methods. Section V is the main contribution of this note and proposes a 'pragmatic' approach for cases that cannot be resolved using both previous methods<fnref n="2" />.

## II. SOME COMMENTS ON OBSERVABILITY

In this paper, we consider a single output nonlinear system, where, for the sake of simplicity, the output $h(x)$ is the first state component $x_1$. To deal also with a simple case, we will consider that the system is regularly locally weakly observable [18], defined later.

The system considered is:

$$ 
\begin{aligned}
\dot{x} &= F(x, u, d) \\
y &= h(x)
\end{aligned}
\qquad (1) $$

with $x \in \mathbb{R}^n$ is the 'physical' state, $y \in \mathbb{R}$ is the measured output, $u \in \mathbb{R}^m$ is the known input, $d \in \mathbb{R}^d$ is an unknown

<footnote n="1">¹ Nantes Université, École Centrale Nantes, CNRS, LS2N, UMR 6004, F-44000, Nantes, France. Emails: {loic.michel, malek.ghanes, franck.plestan}@ec-nantes.fr, yannick.aoustin@univ-nantes.fr</footnote>

<footnote n="2">² QUARTZ EA 7393, ENSEA, Cergy-Pontoise, France.
Email: barbot@ensea.fr</footnote>

<footnote n="1">¹This leads to "optimizing" the choice between the design of an observer or a differentiator and its dimension.</footnote>

<footnote n="2">²Simulation code can be found in the repository: <https://github.com/LoicMichelControl/PragmaticObsv>.</footnote>
vector corresponding to disturbance, faults, secret message... The definition of observability is generally characterized indirectly by the definition of its opposite, the indistinguishably.

**Definition 1.** A pair of points is said to be indistinguishable for the system (1) if, regardless of the admissible inputs $u$ and admissible disturbances $d$, the temporal behaviors of the outputs are identical for all time $t$ in the observation time domain.

Based on this definition, Locally Weakly Observable (LWO) is defined as follows (see seminal paper [18] for a definition without disturbance)

**Definition 2.** The system (1) is Locally Weakly Observable at $x_0$, if for any admissible $u$ and $d$, there is no point, other than $x_0$, in the neighborhood of $x_0$ that is indistinguishable from $x_0$.

This definition is obviously local and highlights the fact that observability may depend on inputs and/or disturbances (see, e.g., [16]). For simplicity's sake, we only list one LWO criterion for autonomous systems below [18].

**Proposition 1.** Concerning $u = 0$ and $d = 0$, for any $t > 0$ ($F_0 := F(x, 0, 0)$)

$$ \text{Rank} \left( \begin{array}{c} \left. \frac{\partial h}{\partial x} \right|_{x_0} \\ \left. \frac{\partial L_{F_0} h}{\partial x} \right|_{x_0} \\ \vdots \\ \left. \frac{\partial L_{F_0}^{\infty} h}{\partial x} \right|_{x_0} \end{array} \right) = n \quad (2) $$

then the autonomous system (1) is LWO at $x_0$.

The $L_{F_0}h$ is the classical Lie derivative, i.e., $L_{F_0}h := \frac{\partial h}{\partial x}F_0$ and $L_{F_0}^k h = L_{F_0}L_{F_0}^{k-1}h$. In this rank criterion, the weak side is characterized by the fact that, in certain cases, one must derive very far before verifying the rank condition (Cayley-Hamilton theorem is not valid for nonlinear systems). This means that part of the state space may be singular in the sense that the state is not observable in that part or that the output must be derived more than the dimension of the state, see for example the case of batch Bioreactor [15]. From this, in order to exclude observability singularity, the notion of Regular Locally Weakly Observable RLWO is introduced, which corresponds to the fact that all information are in the $n-1$ first output derivatives.

Another way to determine implicitly<fnref n="3" /> the observability is the flatness of the system [8].

**Definition 3.** The system (1) with $d = 0 \forall t \le 0$ is zero-flat if there exists a flat output, depending only on $x^4$, $\xi \in \mathbb{R}^m$

and two vector functions $\Gamma$ and $\Pi$, such that

$$ \begin{aligned} x &= \Gamma(\xi, \dot{\xi}, \ddot{\xi}, \dots, \xi^{(k)}) \\ u &= \Pi(\xi, \dot{\xi}, \ddot{\xi}, \dots, \xi^{(k+1)}) \end{aligned} \quad (3) $$

where $\xi^{(j)}$ stands for the $j$th time derivative of $\xi$.

From the flatness definition the differentiator seems to fit well with (3), if $\xi$ is only function of the measured output $y$. Nevertheless, there exist also some other classical forms well adapted to the observer design as detailed in the sequel.

**Remark 1.** The different observable forms shown below and equation (3) for flatness are obtained via diffeomorphisms that depend on knowledge of the model. It is often preferable to work in physical coordinates when possible. Otherwise, modeling uncertainties are reflected in the diffeomorphism and also in equation (3).

The Observability output injection form [22] is the following

$$ \begin{aligned} \dot{x}_1 &= x_2 + f_1(x_1, u) \\ \dot{x}_2 &= x_3 + f_2(x_1, u) \\ \vdots &= \vdots \\ \dot{x}_n &= \sum_{i=2}^{n} a_i x_i + f_n(x_1, u) + d, \end{aligned} \quad (4) $$

where the measured output is $y = x_1$ and the disturbance $d$ is bounded. This form is the well known output injection form [22] including a perturbation, which is matching with respect to the observability property, i.e., the relative degree [19] with respect to the perturbation $d$ is maximal (see also Drazemovic for a definition of matching condition in control frame [7]). Moreover, with respect to the (4) all inputs $u$ are universal [11], i.e., the observability property does not depend on the input.

Another form is the observability strict feedback form [21], [9]

$$ \begin{aligned} \dot{x}_1 &= x_2 + f_1(x_1, u) \\ \dot{x}_2 &= x_3 + f_2(x_1, x_2, u) \\ \vdots &= \vdots \\ \dot{x}_n &= f_n(x_1, x_2, \dots, x_n, u) + d, \end{aligned} \quad (5) $$

which is also called canonical form if $f_j(x_1, \dots, x_j, u) = 0$ for all $j \in [1, 2, \dots, n-1]$; in this case $x_i = y^{(i-1)}$ with $i \in [1, \dots, n]$ and $y^{(n)} = f_n(x_1, x_2, \dots, x_n, u) + d$ contains all the dynamic behavior. Furthermore, if $f_n(x_1, x_2, \dots, x_n, u)$ is unknown function, the estimation problem can only be handled by a differentiator, with $f_n(x_1, x_2, \dots, x_n, u) + d$ considered as a disturbance.

Concerning the system (5) the proposed 'pragmatic' approach must be used. This is particularly true if one of the nonlinear functions $f_i$ is not differentiable with respect to time or function of components that are not measured. In this case a step by step observer is particularly well adapted [10] (see also for observers in cascade [1]). There exist many

<footnote n="3">³Observability is a necessary condition for Flatness, see equation (3)</footnote>

<footnote n="4">⁴There exists a more general definition of flatness, considering for example $\xi$ depending on the control $u$ [8]</footnote>
other forms and many different types of differentiators and observers (see for example [14], [5], [6], [4]).

### III. DIFFERENTIATOR APPROACH

It is clear that for a signal produced by an unknown system, about the system that produces it, the differentiator is the most appropriate solution. Nevertheless, it is necessary to know at least up to which order the system is differentiable (for sliding mode approach). With regard to the flat system, if the flat outputs are the measured ones, then the differentiator is also well suited. Consider the following system

$$ 
\begin{aligned}
\dot{x}_1 &= x_2 \\
\dot{x}_2 &= x_3 \\
\vdots &= \vdots \\
\dot{x}_{n-1} &= x_n \\
\dot{x}_n &= d
\end{aligned} 
\qquad (6) 
$$

where $d$ is the unknown dynamic $y^{(n)}$, which is assumed bounded. The general form of a homogeneous differentiator for (6) is as follows [30]

$$ 
\begin{aligned}
\dot{z}_1 &= z_2 + \lambda_1[e_1]^\alpha \\
\dot{z}_2 &= z_3 + \lambda_2[e_1]^{2\alpha-1} \\
\vdots &= \vdots \\
\dot{z}_{n-1} &= z_n + \lambda_{n-1}[e_1]^{(n-1)\alpha-(n-2)} \\
\dot{z}_n &= \lambda_n[e_1]^{n\alpha-(n-1)}.
\end{aligned} 
\qquad (7) 
$$

with $\alpha \in [\frac{n-1}{n}, 1]$ and $e_1 = x_1 - z_1$; for $\beta = i\alpha - (i-1)$ with $i \in [1, n]$, $[e_1]^\beta$ denotes $|e_1|^\beta \operatorname{sgn}(e_1)$.

The dynamic of errors $e_i = x_i - z_i$ with $[1, n]$ is governed by

$$ 
\begin{aligned}
\dot{e}_1 &= e_2 - \lambda_1[e_1]^\alpha \\
\dot{e}_2 &= e_3 - \lambda_2[e_1]^{2\alpha-1} \\
\vdots &= \vdots \\
\dot{e}_{n-1} &= e_n - \lambda_{n-1}[e_1]^{(n-1)\alpha-(n-2)} \\
\dot{e}_n &= d - \lambda_n[e_1]^{n\alpha-(n-1)}.
\end{aligned} 
\qquad (8) 
$$

The finite time convergence of (8) is ensured by appropriate choice of the $\lambda_n$ with respect to the bound of $d$ and all $\lambda_i$ (see for example [17], [23], [31], [26], [3]).

#### Lagrangian System: nonlinear spring

Consider the dynamic model of a system composed of a mass $M$ and an undamped spring with its stiffness coefficient $k$; its length variation $\xi$ reads as

$$ M\ddot{\xi} + k\xi = 0 \qquad (9) $$

Assume that the spring stiffness depends on the length variation $\xi$ such as $k(\xi)$ is a linear slowly decreasing function

[24]

$$ 
\begin{aligned}
\frac{k(\xi)}{M} &= \frac{k - \epsilon\xi}{M} \\
&= \omega^2 - \frac{\epsilon}{M}\xi \\
&= \omega^2 - \bar{\epsilon}\xi
\end{aligned} 
\qquad (10) 
$$

where $\omega$ is the natural angular frequency when $\bar{\epsilon} = 0$.

By introducing the state vector $x = (x_1, x_2)^\top = (\xi, \dot{\xi})^\top$, the system (9) becomes

$$ 
\begin{aligned}
\dot{x}_1 &= x_2 \\
\dot{x}_2 &= (\bar{\epsilon}x_1 - \omega^2)x_1 = d(\bar{\epsilon}, \omega, x_1).
\end{aligned} 
\qquad (11) 
$$

The total mechanical energy is

$$ E_{tot} = \frac{1}{2}(x_2^2 + x_1^2(\omega^2 - \frac{2}{3}\bar{\epsilon}x_1)) \qquad (12) $$

This total energy remains constant since there is no dissipative term in (11). When $\bar{\epsilon} = 0$, $E_{tot} = x_2^2 + \omega^2 x_1^2 = C$ with $C$ an arbitrary positive constant defines a trajectory in the phase plane $(x_1, x_2)$, which is an ellipse. The symmetry of the ellipse along the $x$-axis is lost if $\epsilon \neq 0$. Figure 2 represents in blue the phase portrait $(x_1, x_2)$ with $\epsilon = 40$, $\omega = 2\pi\text{rd/s}$ and the initial conditions $x_1(0) = 0.5$ m, $x_2(0) = 0.5$ m/s. It is important to mention that the nonlinear term $d(\bar{\epsilon}, \omega, x_1)$ is generally unknown, that is why, the differentiator is used in this case.

A differentiator for system (9) with $e_1 = x_1 - z_1$ is based on [31], [12] and reads as

$$ 
\begin{aligned}
\dot{z}_1 &= z_2 + \lambda_1[e_1]^\alpha \\
\dot{z}_2 &= \lambda_2[e_1]^{2\alpha-1}.
\end{aligned} 
\qquad (13) 
$$

The dynamic of the error $e_i = x_i - z_i$ with $i = 1, 2$ is

$$ 
\begin{aligned}
\dot{e}_1 &= e_2 - \lambda_1[e_1]^\alpha \\
\dot{e}_2 &= d(\bar{\epsilon}, \omega, x_1) - \lambda_2[e_1]^{2\alpha-1}.
\end{aligned} 
\qquad (14) 
$$

The parameters of the differentiator are $\lambda_1 = 100$, $\lambda_2 = 1000$ and $\alpha = 0.51$. The initial conditions of the estimates are $z_1(0) = 0$ m and $z_2(0) = 0$ m/s. The system is simulated over two seconds using a discrete time-step of $10^{-6}$s. Figure 1 illustrates the evolution of each estimate ($z_1, z_2$) accordingly to the ($x_1, x_2$). Figure 2 depicts the phase portrait highlighting the nonlinear behavior of the spring by the dissymmetry on $x_1$ (due to the term in $\epsilon$). Remark that the red dashed line (estimated state) start from (0, 0) and converges to the state of the nonlinear spring.

### IV. OBSERVER APPROACH

Regarding the system (4), a homogeneous observer can be written as follows

$$ 
\begin{aligned}
\dot{z}_1 &= z_2 + f_1(x_1, u) + \lambda_1[e_1]^\alpha \\
\dot{z}_2 &= z_3 + f_2(x_1, u) + \lambda_2[e_1]^{2\alpha-1} \\
\vdots &= \vdots \\
\dot{z}_n &= f_n(x_1, u) + \lambda_n[e_1]^{n\alpha-(n-1)}.
\end{aligned} 
\qquad (15) 
$$

where $f_1, f_2, \dots, f_n$ are the known functions appearing in (4), based on available information. These functions and the
<figure><img src="image_2.png" /><figcaption>Fig. 1: State estimation of the nonlinear spring system: state $x_1$ (top) and $x_2$ (bottom).</figcaption></figure>

<figure><img src="image_3.png" /><figcaption>Fig. 2: Phase portrait $x_2$ as function of $x_1$ of the nonlinear spring: true system (blue) and its associated estimation (red).</figcaption></figure>

available information (see e.g. [13]) are possibly discontinuous<fnref n="5" />. Moreover, $e_1 = x_1 - z_1$ and $\lambda_i, i \in [1, n]$ are the observer gains. From (4) and (15) the observation error dynamics is

$$ 
\begin{aligned}
\dot{e}_1 &= e_2 - \lambda_1[e_1]^\alpha \\
\dot{e}_2 &= e_3 - \lambda_2[e_1]^{2\alpha-1} \\
\vdots &= \vdots \\
\dot{e}_n &= d' - \lambda_n[e_1]^{n\alpha-(n-1)},
\end{aligned}
\quad (16) 
$$

with $e_i = x_i - z_i$, $\alpha \in [\frac{n-1}{n}, 1]$ and $d' = d + \sum_{i=2}^n a_i x_i$. It implies that the original state must be bounded, which is usually the case for physical system (except for some cases such as the number of revolutions of an electric machine or other pathological cases). The observation error (16) is exactly the same as (8) obtained with a differentiator. Nevertheless, (16) is obtained with respect to (15) and (4) where the $f_i(x_1, u)$ are not necessarily smooth.

## The Chua circuit

A Chua system, defined by the states $x_1, x_2$ and $x_3$, is introduced in [25] and is the first autonomous analog electrical chaotic circuit, which is characterized by a double scroll strange attractor. The system (4) is presented here without input $u$. A perturbation $\gamma_c$ is added on the last row of the Chua circuit in order to highlight the robustness of the proposed schemes; indeed as in (4), $\gamma_c$ could be considered as a matching perturbation with respect to the observability problem. It comes

$$ 
\begin{aligned}
\dot{x}_1 &= x_2 + p_c(-x_1 - r_c(x_1)) &= x_2 + f_1(x_1) \\
\dot{x}_2 &= x_3 + p_c x_1 &= x_3 + f_2(x_1) \\
\dot{x}_3 &= \underbrace{\gamma_c - x_3 - q_c x_2}_{d'} - p_c x_1 &= d' + f_3(x_1),
\end{aligned}
\quad (17) 
$$

where $p_c = 10, q_c = 14.87$ are internal parameters of the Chua system. The external perturbation is set to $\gamma_c = 1$ and the disturbance considered with respect to (4) is $d' = \gamma_c - x_3 - q_c x_2$ which is bounded, because all the state components are bounded on the strange attractor (here a double scroll). The function $r_c(x_1)$ is a nonlinear function that depends only on the state $x_1$, and corresponds to a behavior of a negative resistance. It is defined by

$$ r_c(x_1) := \begin{cases} m_1 x_1 & \text{if } |x_1| \le 1 \\ m_0(x_1 - \operatorname{sgn}(x_1)) + m_1 \operatorname{sgn}(x_1) & \text{if } |x_1| > 1 \end{cases} \quad (18) $$

where $m_0 = -0.68$ and $m_1 = 1.27$. Figure 3 displays the chaotic behavior of the Chua circuit over 60 s.

<figure><img src="image_4.png" /><figcaption>Fig. 3: Chua behavior illustrated in 3D (simulated over 60s) (top) $x_1(V)$; (middle) $x_2(V/s)$; (bottom) $x_3(V/s^2)$.</figcaption></figure>

The system (17) with $y = x_1$ is exactly in the output injection form (4), as a consequence, it is possible to design the following finite time observer ($\alpha = \frac{2}{3}$ in (15))

$$ 
\begin{aligned}
\dot{z}_1 &= z_2 + f_1(x_1) + \lambda_1[y - z_1]^{\frac{2}{3}} \\
\dot{z}_2 &= z_3 + f_2(x_1) + \lambda_2[y - z_1]^{\frac{1}{3}} \\
\dot{z}_3 &= f_3(x_1) + \lambda_3 \operatorname{sgn}(y - z_1)
\end{aligned}
\quad (19) 
$$

The observation errors are $e_1 = y - z_1, e_2 = x_2 - z_2$ and

<footnote n="5">⁵The function $f_1, \dots, f_n$ are called "Input-Output injections" terms</footnote>
$e_3 = x_3 - z_3$, and their dynamics are derived from (19).

$$
\begin{align}
\dot{e}_1 &= e_2 - \lambda_1 [y - z_1]^{\frac{2}{3}} \nonumber \\
\dot{e}_2 &= e_3 - \lambda_2 [y - z_1]^{\frac{1}{3}} \tag{20} \\
\dot{e}_3 &= d' - \lambda_3 \operatorname{sgn}(y - z_1) \nonumber
\end{align}
$$

From Fig. 3, it is possible to have an idea of the strange attractor (i.e., a double scroll) and then obtain a bound of $d'$. So $\lambda_1$, $\lambda_2$, and $\lambda_3$ are chosen in order to obtain a finite time convergence.

The system is simulated over ten seconds using a discrete time-step of $1 \cdot 10^{-6}$s with the initial conditions $x_1(0) = 0.1$ V, $x_2(0) = 0.2$ V/s and $x_3(0) = 0.4$ V/s$^2$. The parameters of the differentiator are $\lambda_1 = 10$, $\lambda_2 = 1000$, $\lambda_3 = 1500$ and $\alpha = 2/3$. The initial conditions of the estimates are $z_1 = 0, z_2 = 0$ and $z_3 = 0$. Figure 4 illustrates the evolution of each estimate $z_1$, $z_2$ and $z_3$ accordingly to the $x_1(\text{V})$, $x_2(\text{V/s})$, and $x_3(\text{V/s}^2)$ states.

<figure><img src="image_5.png" /><figcaption>Fig. 4: State estimation of the Chua system for $h = 10^{-6}$s: state $x_1(\text{V})$ (top), $x_2(\text{V/s})$ (middle) and $x_3(\text{V/s}^2)$ (bottom).</figcaption></figure>

V. PRAGMATIC APPROACH

The proposed pragmatic approach addresses the state esti-
mation problem for an *n*-th order nonlinear and non-smooth
system when only a partial measurement is available. The
estimation scheme is based on a cascaded structure in which
state components are reconstructed sequentially in finite
time. At each stage, the availability of previously estimated
components enables the design of either an observer or
a differentiator, depending on the structure of the system
dynamics. More specifically, when the dynamics of a given
state component are weakly coupled or free of nonlinear
terms, a sliding-mode differentiator provides an efficient
solution. In contrast, for strongly coupled or non-smooth
dynamics, observer-based designs are more appropriate. This
combination constitutes the core of the proposed pragmatic
estimation strategy. It is important to mention that there is
an iterative procedure in cases where the inputs are unknown
and may be non-smooth. In [2], some sufficient conditions
are given in the framework of left invertibility.

Considering the system (5), the differentiator approach is
not possible if the functions fᵢ are not smooth. Consequently

it is not possible to design a diffeomorphism to rewrite (5)
under the form (6). The observer approach is also impossible
because the fᵢ for i > 1 are function of unmeasured state.
So, the *pragmatic approach* can be done recursively.
Considering the system (5), **the first step** is to estimate in
finite time x₂ by the following 2nth order observer

$$
\begin{align}
\dot{z}_{1,1} &= z_{2,1} + f_1(x_1, u) + \lambda_{1,1} [e_{1,1}]^{\alpha_1} \nonumber \\
\dot{z}_{2,1} &= \lambda_{2,1} [e_{1,1}]^{2\alpha_1 - 1} \tag{21}
\end{align}
$$

with $e_{1,1} = x_1 - z_{1,1}$. Then, from the first two lines of
(5) and (21), the observation error dynamics reads as

$$
\begin{align}
\dot{e}_{1,1} &= e_{2,1} - \lambda_{1,1} [e_{1,1}]^{\alpha_1} \nonumber \\
\dot{e}_{2,1} &= f_2(x_1, x_2, u) - \lambda_{2,1} [e_{1,1}]^{2\alpha_1 - 1} \tag{22}
\end{align}
$$

with $e_{2,1} = x_2 - z_{2,1}$. Assuming that $f_2(x_1, x_2, u)$ is
bounded, there exist $\lambda_{1,1}$ and $\lambda_{2,1}$ such that (22) converges
in finite time, that gives $z_{2,1} = x_2$.

Once the convergence of (22), **the second step** consists
in building a second 2^nd^ order observer, for which x_1,
z_{2,1} = x_2 and u are the known variables that gives

$$
\begin{align}
\dot{z}_{1,2} &= z_{2,2} + f_2(x_1, z_{2,1}, u) + \lambda_{1,2} [e_{1,2}]^{\alpha_2} \nonumber \\
\dot{z}_{2,2} &= \lambda_{2,2} [e_{1,2}]^{2\alpha_2 - 1} \tag{23}
\end{align}
$$

with $e_{1,2} = z_{2,1} - z_{1,2}$ which is equivalent to $e_{1,2} = x_2 - z_{1,2}$
due to the convergence of (22). Then, the observation error
dynamic becomes

$$
\begin{align}
\dot{e}_{1,2} &= e_{2,2} - \lambda_{1,2} [e_{1,2}]^{\alpha_2} \nonumber \\
\dot{e}_{2,2} &= f_3(x_1, x_2, x_3, u) - \lambda_{2,2} [e_{1,2}]^{2\alpha_2 - 1} \tag{24}
\end{align}
$$

Under the assumption that $f_3(x_1, x_2, x_3, u)$ is bounded there exist $\lambda_{1,2}$ and $\lambda_{2,2}$ such that (24) converges in finite time, and then $z_{2,2} = x_3$.

It is formally possible to repeat this procedure up to **the step** *n* in order to estimated *x*<sub>*n*</sub>. In practice, it would not be possible to go beyond **step 5** [29] unless an algorithmic procedure is introduced to validate each step, together with low-pass filters to reject measurement noise. In the following simple example, intended to illustrate the proposed pragmatic approach, no validation flags are introduced. This allows to observe that the transients are degraded in this case. Moreover, measurement noise is not taken into account and 1 > α > ½ (i.e. no chattering), and therefore no filter is applied.

Non smooth example

Consider the following system, which is a very simplified
version of encryption system (see for example [29] for a
more realistic case)

$$
\begin{align}
\dot{x}_1 &= x_2 \notag \\
\dot{x}_2 &= 2x_3 - x_1 + k_1 x_2^2 u \tag{25} \\
\dot{x}_3 &= -2x_2 + m \notag
\end{align}
$$

with $k_1 = 0.2$. The measured output is $y = x_1$, the known input $u$ is a non smooth signal, and the message to be encrypted is $m$. It is important to mention that the system is not in output injection form due to the term $k_1 x_2^2 u$.
The principle consists in encrypting a message *m* through the system (25) (and the *u* input contains an 'encoding' dynamical key) that returns the encrypted message via the state *x*₁. To decrypt the message, the first step is to estimate each state *x*₁, *x*₂ and *x*₃ of the system, knowing the exact parameter *k*₁ and the key *u*, thanks to the proposed pragmatic approach. The second step is to differentiate one more time the state *x*₃ for which the message *m* = *x*₃ + 2*x*₂ is restored.

**Remark 2.** In [29] the system is composed of two coupled chaotic circuits with two messages and two measured outputs. Here, we have chosen for the sake of simplicity to couple two oscillators, with only one message and one output.

In the proposed pragmatic approach, the first step is to estimate $x_2$ by a second order homogeneous differentiator instead of observer as $f_1 = 0$, that gives

$$ 
\begin{aligned}
\dot{z}_{1,1} &= z_{2,1} + \lambda_{1,1}[y - z_{1,1}]^{\alpha_1} \\
\dot{z}_{2,1} &= \lambda_{2,1}[y - z_{1,1}]^{2\alpha_1 - 1}
\end{aligned}
\qquad (26) 
$$

The differentiation errors are $e_{1,1} = y - z_{1,1}$ and $e_{2,1} = x_2 - z_{2,1}$ and the dynamics of the differentiation error are

$$ 
\begin{aligned}
\dot{e}_{1,1} &= e_{2,1} - \lambda_{1,1}[y - z_{1,1}]^{\alpha_1} \\
\dot{e}_{2,1} &= 2x_3 - x_1 + k_1 x_2^2 u - \lambda_{2,1}[y - z_{1,1}]^{2\alpha_1 - 1}
\end{aligned}
\qquad (27) 
$$

Assuming $x_1, x_2$ and $u$ bounded, it is possible to choose $\lambda_{1,1}$ and $\lambda_{2,1}$ such that (27) is stable in finite time. Here, the finite time is necessary, because we will derive $z_{2,1}$ only after that $z_{1,1}$ has converged to $y$, i.e., $e_{1,1} = 0$.

The second step consists in estimating $x_3$ by the homogeneous observer

$$ 
\begin{aligned}
\dot{z}_{1,2} &= z_{2,2} - y + k_1 z_{2,1}^2 u + \lambda_{1,2}[z_{2,1} - z_{1,2}]^{\alpha_2} \\
\dot{z}_{2,2} &= -4z_{1,2} - \lambda_{2,2}[z_{2,1} - z_{1,2}]^{2\alpha_2 - 1}
\end{aligned}
\qquad (28) 
$$

The observation errors are $e_{1,2} = z_{2,1} - z_{1,2} = x_2 - z_{1,2}$ and $e_{2,2} = 2x_3 - z_{2,2}$; then, the dynamic of the observation error becomes

$$ 
\begin{aligned}
\dot{e}_{1,2} &= e_{2,2} - \lambda_{1,2}[e_{1,2}]^{\alpha_2} \\
\dot{e}_{2,2} &= 2m - \lambda_{2,2}[e_{1,2}]^{2\alpha_2 - 1}
\end{aligned}
\qquad (29) 
$$

where $2m$ is bounded and is considered as an unknown input. Now assuming, that $m$ is at least $C^1$ with a bound in its derivative, we can design the last observer. Once the convergence of $z_{1,2}$ to $z_{2,1}$ achieved in finite time, $\frac{z_{2,2}}{2} = x_3 := y_3$ which is the output of the step 3. One has

$$ 
\begin{aligned}
\dot{z}_{1,3} &= z_{2,3} - 2z_{1,2} + \lambda_{1,3}[y_3 - z_{1,3}]^{\alpha_3} \\
\dot{z}_{2,3} &= \lambda_{2,3}[y_3 - z_{1,3}]^{2\alpha_3 - 1}
\end{aligned}
\qquad (30) 
$$

The estimation errors are $e_{1,3} = y_3 - z_{1,3}$ and $e_{2,3} = m - z_{2,3}$ that give

$$ 
\begin{aligned}
\dot{e}_{1,3} &= e_{2,3} - \lambda_{1,3}[e_{1,3}]^{\alpha_3} \\
\dot{e}_{2,3} &= \dot{m} - \lambda_{2,3}[e_{1,3}]^{2\alpha_3 - 1}
\end{aligned}
\qquad (31) 
$$

which converges in finite time to $z_{2,3} = m$ for appropriate choice of $\lambda_{1,3}$ and $\lambda_{2,3}$ as $|\dot{m}|$ is bounded.

The system is simulated over five seconds using a discrete time-step of $1 \cdot 10^{-6}$s with the initial conditions $x_1(0) = 0.1$, $x_2(0) = 2$ and $x_3(0) = 2$. The parameters of the

differentiator are $\lambda_{1,1} = \lambda_{1,2} = 10$, $\lambda_{2,1} = \lambda_{2,2} = 60$ and $\alpha_1 = \alpha_2 = \alpha_3 = 0.51$. The initial conditions of the observer states are $z_{i,j}(0) = 0$ for $i = \{1, 2\}$ and $j = \{1, 2, 3\}$. The message $m = 0.5 \sin(2\pi t)$ is considered for the encryption process in (25).

Figure 5 illustrates the evolution of each estimates $z_{1,1}$, $z_{1,2}$, and $z_{2,2}/2$ accordingly to the $x_1, x_2$ and $x_3$ states. The discrete time-step equals $10^{-6}$s.

<figure><img src="image_6.png" /><figcaption>Fig. 5: State estimation of the encryption system: state $x_1$ (top), $x_2$ (middle) and $x_3$ (bottom).</figcaption></figure>

As stated in (30), to recover the message $m$, an additional differentiation is performed from the estimated $y_3$. Figure 6 depicts the recovering of the message $m$ and highlight the well founded of the pragmatic approach. Here, the estimated message is noisy because we did not use any low-pass filter. The term 'pragmatic approach' is used here because a systematic cascade design of two-dimensional differentiators/observers is made, which result of 'global observer' of dimension $(2(n-1))$. For example, in the following case

$$ 
\begin{aligned}
\dot{x}_1 &= x_2 + f_1(x_1, u) \\
\dot{x}_2 &= x_3 + f_2(x_1, u) \\
\dot{x}_3 &= x_4 + f_3(x_1, x_2, x_3, u) \\
\dot{x}_4 &= d
\end{aligned}
\qquad (32) 
$$

where $y = x_1$, the state $x$, the input $u$ and the perturbation $d$ are bounded. It is more efficient to first use a 3-dimensional observer to estimate in finite time $x_1, x_2, x_3$ and then a 2-dimensional observer to observe $x_4$ from $y_2 = x_3$ knowing $x_1, x_2$ and $u$. By doing this, a 'global observer' of dimension 5 (3+2) instead of 6 (2*3) is obtained for systematic use of a two-order differentiator/observer.

## VI. CONCLUSION

In this note, the differentiator and observer approaches are compared using simple examples. After that, a "pragmatic" approach is proposed for a more complex case. This highlights the value of convergence in finite time in order to be able to cascade sliding mode observer and/or differentiator. The perspective of the work presented in this note is an extension of the pragmatic approach to discrete and/or noisy measurements as well as building observer-based controls.
<figure><img src="image_7.png" /><figcaption>Fig. 6: Differentiation of $y_3$ ($h = 10^{-6}s$) to restore the message $m$.</figcaption></figure>

Discretization can be approached in different ways see for example [28] and the measurement noise using the approach presented in [12]. For observer-based controls, the issue is more complex, especially if we need to take a pragmatic approach, as there is a high probability that a zero dynamics will appear.

## ACKNOWLEDGMENTS

This work is supported by the ANR project SlimDisc ANR-24-CE48-2771.

## REFERENCES

[1] V. Andrieu, D. Astolfi, and P. Bernard. Observer design via interconnections of second-order mixed sliding-mode/linear differentiators. *International Journal of Robust and Nonlinear Control*, 31(9):3631–3657, 2021.

[2] Jean-Pierre Barbot, Driss Boutat, and Thierry Floquet. An observation algorithm for nonlinear systems with unknown inputs. *Automatica*, 45(8):1970–1974, 2009.

[3] B. Bartolini, A. Pisano, and E. Usai. First and second derivative estimation by sliding mode technique. *Journal of signal processing*, 4(2):167–176, 2000.

[4] P. Bernard. *Observer Design for Nonlinear Systems*. Lecture Notes in Control and Information Sciences, Springer Cham, Switzerland, 2019.

[5] G. Besançon. *Nonlinear Observers and Applications*. Lecture Notes in Control and Information Sciences, Springer Berlin, Heidelberg, 2007.

[6] Boutat D. and Zheng G. *Observer Design for Nonlinear Dynamical Systems: Differential Geometric Methods*. Lecture Notes in Control and Information Sciences, 487, Springer, 2021.

[7] B. Draženović. The invariance conditions in variable structure systems. *Automatica*, 5(3):287–295, 1969.

[8] Michel Fliess, Jean Lévine, Philippe Martin, and Pierre Rouchon. Flatness defect of non-linear systems: introductory theory and examples. *International Journal of Control*, 61(6):1327–1361, 1995.

[9] T. Floquet and J. P. Barbot. Super twisting algorithm-based step-by-step sliding mode observers for nonlinear systems with unknown inputs. *International Journal of Systems Science*, 38(10):803–815, 2007.

[10] Thierry Floquet and Jean-Pierre Barbot. Super twisting algorithm-based step-by-step sliding mode observers for nonlinear systems with unknown inputs. *International Journal of Systems Science*, 38(10):803–815, 2007.

[11] J. Gauthier and G. Bornard. Observability for anyu(t) of a class of nonlinear systems. *IEEE Transactions on Automatic Control*, 26(4):922–926, 1981.

[12] Malek Ghanes, Jean-Pierre Barbot, Leonid Fridman, Arie Levant, and Robert Boisliveau. A new varying-gain-exponent-based differentiator/observer: An efficient balance between linear and sliding-mode algorithms. *IEEE Transactions on Automatic Control*, 65(12):5407–5414, 2020.

[13] A. Glumineau, C.H. Moog, and F. Plestan. New algebraic-geometric conditions for the linearization by input-output injection. *IEEE Transactions on Automatic Control*, 41(4):598–603, 1996.

[14] Nijmeijer H., and Fossen T.I. *New Directions in Nonlinear Observer Design*. Lecture Notes in Control and Information Sciences, 244, Springer, 1999.

[15] Ihab Haidar, Jean-Pierre Barbot, Alain Rapaport, and Malek Ghanes. Observability singularity of batch bioreactors: A solution based on high order sliding mode differentiator approach. pages 336–338, 12 2017.

[16] H. Hammouri and M. Farza. Nonlinear observers for locally uniformly observable systems. esaim: Control, optimisation and calculus of variations. *ESAIM: Control, Optimisation and Calculus of Variations*, 9:353–370, 2003.

[17] Avi Hanan, Adam Jbara, and Arie Levant. *New Homogeneous Controllers and Differentiators*, pages 3–28. Springer International Publishing, Cham, 2020.

[18] R. Hermann and A. J. Krener. Nonlinear controllability and observability. *IEEE Transactions On Automatic Control*, 22, 1977.

[19] Alberto Isidori. *Nonlinear control systems: an introduction*. Springer, 1985.

[20] Rudolf E. Kálmán and Richard S. Bucy. New results in linear filtering and prediction theory. *Journal of Basic Engineering*, 83:95–108, 1961.

[21] Hassan K Khalil. High-gain observers in nonlinear feedback control. In *New directions in nonlinear observer design*, pages 249–268. Springer, 2007.

[22] A. J. Krener and A. Isidori. Linearization by output injection and nonlinear observers. *Systems & Control Letters*, 3:47–52, 1983.

[23] A. Levant. Robust exact differentiation via sliding mode technique. *Automatica*, 34(3):379–384, 1998.

[24] Jean Lévine. *Analysis and Control of Nonlinear Systems: A Flatness-based Approach*. Mathematical Engineering, Springer-Verlag, Berlin Heidelberg, 2009.

[25] T. Matsumoto, L. Chua, and M. Komuro. The double scroll. *IEEE Transactions on Circuits and Systems*, 32(8):797–818, 1985.

[26] M. Mboup, C. Join, M. Fliess, Y. Wang, G. Zheng, D. Efimov, and W. Perruquetti. Comments on 'differentiator application in altitude control for an indoor blimp robot'. *Int. J. of Control*, 93(5):1218–1219, 2020.

[27] Loïc Michel, Malek Ghanes, Yannick Aoustin, and Jean-Pierre Barbot. Semi-implicit discrete time cascaded observer: Chua circuit case study. Springer chapter in the book "Sliding mode, variable-structure discontinuous control systems", in press.

[28] Mohammad Rasool Mojallizadeh, Bernard Brogliato, Andrey Polyakov, Subiksha Selvarajan, Loïc Michel, Franck Plestan, Malek Ghanes, Jean-Pierre Barbot, and Yannick Aoustin. A survey on the discrete-time differentiators in closed-loop control systems: Experiments on an electro-pneumatic system. *Control Engineering Practice*, 136:105546, 2023.

[29] Florentina Nicolau, Witold Respondek, and Jean-Pierre Barbot. Flat inputs: Theory and applications. *SIAM Journal on Control and Optimization*, 58(6):3293–3321, 2020.

[30] W Perruquetti and T Floquet. Homogeneous finite time observer for nonlinear systems with linearizable error dynamics. In *IEEE Conference on Decision and Control*, Seoul, Korea, Dec 2007.

[31] Wilfrid Perruquetti, Thierry Floquet, and Emmanuel Moulay. Finite-time observers: application to secure communication. *IEEE Trans. on Automatic Control*, 53(1):356–360, 2008.