# Using the monotone property and sensors placement as basic tools to design set-membership state estimators

Nacim Meslem, Christophe Prieur

► To cite this version:

Nacim Meslem, Christophe Prieur. Using the monotone property and sensors placement as basic tools to design set-membership state estimators. MTNS 2016 - 22nd International Symposium on Mathematical Theory of Networks and Systems, Jul 2016, Minneapolis, United States. pp.403-410. hal-01347770

HAL Id: hal-01347770

https://hal.science/hal-01347770v1

Submitted on 2 Dec 2025

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

<figure><img src="image_1.png" /></figure>

Distributed under a Creative Commons Attribution 4.0 International License
# Using the monotone property and sensors placement as basic tools to design set-membership state estimators

Nacim Meslem and Christophe Prieur

*Abstract*— In this work, we propose to combine the monotone property of dynamical systems with sensor placement techniques to design efficient set-membership state estimators. The basic idea exploited in this approach consists in decomposing the system as an interconnection of monotone subsystems where the input of each subsystem is a function of the outputs of the remainder subsystems. Thereby, the sensors placement is dictated by this monotone decomposition of the original system. Due to the new configuration of the system, one can efficiently cope with the pessimism propagation linked to set-membership computation and so one can enhance substantially the accuracy of the estimated time-varying state enclosure. Two versions of this guaranteed state estimation approach are presented. The first one is based on the principle of output injections. The second version relies on an event-triggered policy, which reduces greatly the use of measurements. In fact, with this policy, the measurements are used only when it is necessary.

## I. INTRODUCTION

Over the last few decades, several geometrical shapes have been used to design set-membership state estimators for uncertain dynamical systems, see [1], [5], [7], [8], [11], [10], [28], [34] and the references therein. The main objective of this kind of state estimators is to characterize an outer enclosure for all the possible state trajectories of the system, which are consistent with (i) the mathematical model and its uncertainties and (ii) the feasible domain of measurements. In the case of linear uncertain systems, this problem is well investigated in both continuous and discrete-time frameworks. Among the numerous proposed approaches in the literature, one can cite the interval observer approach [37], [22], [23], [24], [25], ellipsoidal state estimation method [6], [5] and zonotope state estimations approach [1], [8]. On the other hand, in the case of uncertain nonlinear systems the issue of set-membership state estimation is less developed. In fact, almost all estimation approaches based on the prediction-correction principle [15], [29], [10] did not address the convergence problem of the width of the estimated state enclosure. Recently, in [26], under restrictive assumptions, a self-triggered technique is proposed in order to ensure this convergence. Note that, the interval observer design approach has been extended to a specific class of nonlinear systems [27], [33], [40]

$$ 
\begin{aligned}
\dot{\mathbf{x}} &= \mathbf{A}\mathbf{x} + \mathbf{f}(\mathbf{x}, t) \\
\mathbf{y} &= \mathbf{C}\mathbf{x}
\end{aligned}
\qquad (1) 
$$

Nacim Meslem and Christophe Prieur are with Univ. Grenoble Alpes, GIPSA-lab, F-38000 Grenoble, France. meslem.nacim@gipsa-lab.fr, christophe.prieur@gipsa-lab.fr. This work has been partially supported by the LabEx PERSYVAL-Lab (ANR-11-LABX-0025-01).

In fact, under some suitable conditions, it is possible to find a nonlinear change of coordinates such that the dynamics of a given nonlinear system will be described in the new basis of the state variables by (1).

In [27], an hybrid formalism is proposed to design an interval observer for (1) and in [33], [40] an approach based on Sylvester equation is introduced to design an interval observer in a new basis of the state variables. However, in our opinion, solving efficiently the set-membership state estimation issue in the new state base does not imply the accuracy of the estimated state enclosure in the original state base of the system. In fact, in practice, the set inversion issue is well known to be a hard problem [16].

In this work, we propose to use the monotone property of interconnected monotone control systems [4] and a sensors placement policy as basic stones to design set-membership state estimators. In fact, due to a smart choice of the outputs of the system, one can consider it as an interconnection of monotone subsystems. Then, relying on the comparison theorems of differential inequalities coupled with output injections technique, one can design an accurate set-membership state estimator in the original state coordinates. In addition, the stability results of monotone systems are used to show the convergence of the width of the estimated state enclosure.

This paper is organized as follows. In Section II useful notions about monotone systems are introduced, the definitions of the reachable set and its outer approximation are given and the concept of set membership state estimator is highlighted. Then, the main results of this work are stated in Section III. In this section two state estimators are proposed. The first one is based on a continuous output injections technique and the second one relies on an event-triggered output injections technique. To show the performance of the proposed set-membership state estimator an illustrative example is given in Section IV with several simulation tests. Finally, some concluding remarks are presented.

## II. PRELIMINARY NOTIONS

### A. Monotone systems

Monotone system is a dynamical system, on an ordered metric space, which has the property that ordered initial states lead to ordered state trajectories [13], [38]. More formally, a dynamical system

$$ \dot{\mathbf{x}} = \mathbf{f}(\mathbf{x}, t) \qquad (2) $$

is said monotone if the trajectories satisfy

$$ \mathbf{x}_1(t_0) \preceq \mathbf{x}_2(t_0) \Rightarrow \forall t \ge t_0, \mathbf{x}_1(t, \mathbf{x}_1(t_0)) \preceq \mathbf{x}_2(t, \mathbf{x}_2(t_0)) \quad (3) $$
where $\mathbf{x}_i(t, \mathbf{x}_i(t_0))$, $i = 1,2$ stands for the state trajectory generated by (2) from the initial state $\mathbf{x}_i(t_0) \in \mathbb{R}^n$. In this work, the considered partial orders are those induced by a closed proper cone $\mathcal{K} \subseteq \mathbb{R}^n$ in the sense that $\mathbf{x}_1 \preceq \mathbf{x}_2$ iff $\mathbf{x}_2 - \mathbf{x}_1 \in \mathcal{K}$. This cone $\mathcal{K}$ is assumed to have a nonempty interior and are pointed, i.e. $\mathcal{K} \cap (-\mathcal{K}) = \{0\}$.

**Remark 1:** The partial order operator $\preceq$ applied between vectors should be understood as a collection of inequalities applied component by component.

In the literature, there exists an interesting graphical method to check the monotonicity of dynamical systems. It is proposed in [21] and based on the following test.

**1) Graphical test:** Consider the dynamical system (2), where its incidence signed graph is constituted by vertices that represent the state variables $x_i$ $i = 1,...,n$. These vertices are linked by signed edges $e_{i,j}^x$, that show the effect of each state variable on the evolution of the other variables. More precisely, an incidence graph of dynamical system is established by complying with the following rules [2]:

* Edges between $x_i$ vertices: For any couple $(i, j)$, $i \neq j$ and $i, j \in \{1,...,n\}$,

1. If $\frac{\partial f_i}{\partial x_j} > 0 \quad \forall (\mathbf{x},t) \in \mathbb{X} \times \mathbb{T} \subseteq \mathbb{R}^n \times \mathbb{R}$, a positive edge $e_{i,j}^x$ directed from the vertex $x_j$ to $x_i$ exist.

2. Else if $\frac{\partial f_i}{\partial x_j} < 0 \quad \forall (\mathbf{x},t) \in \mathbb{X} \times \mathbb{T} \subseteq \mathbb{R}^n \times \mathbb{R}$, a negative edge $e_{i,j}^x$ directed from the vertex $x_j$ to $x_i$ exist.

3. Else $\frac{\partial f_i}{\partial x_j} = 0 \quad \forall (\mathbf{x},t) \in \mathbb{X} \times \mathbb{T} \subseteq \mathbb{R}^n \times \mathbb{R}$, no edge can linked $x_j$ to $x_i$.

**Definition 1:** A path, of a given graph, is a finite sequence of vertices in which each vertex appears only once and, for each two successive vertices, there exists at least one edge which can connect them.

**Definition 2:** A cycle, of a given graph, not necessary directed, is a closed-loop path. That means, the initial vertex of this path is its final vertex.

**Definition 3:** The sign of a cycle is the product of the signs of its edges.

**Proposition 1:** A dynamical system (2) is monotone if and only if its incidence graph does not contain any negative cycle.

*Proof:* See [21], [2].

A special class of monotone systems is that of cooperative systems. For this type of dynamical systems, the order-preserving property on the state vector is induced by the North-Est orthant of $\mathbb{R}^n$, i.e. the cone $\mathcal{K} = \mathbb{R}^n_{\ge 0}$. So, the cooperative term means that all the state variables activate each other.

**Proposition 2:** The dynamical system (2) is cooperative if and only if the following properties hold:

$$ \frac{\partial f_i}{\partial x_j}(\mathbf{x}, t) \ge 0 \quad \forall \mathbf{x} \in \mathbb{X} \subseteq \mathbb{R}^n, \forall t \in \mathbb{T} \subseteq \mathbb{R}, \forall i \neq j \quad (4) $$

*Proof:* See Proposition 1.

**Remark 2:** In the case of cooperative systems, the partial order operator $\preceq$ becomes the lower or equal comparison operator $\le$ applied component by component.

Now let us introduce an interesting result about the global stability of cooperative systems.

**Assumption 1:** The cooperative system (2) satisfies the following conditions:

(i) all the solutions to (2) are bounded,

(ii) there is not more than one equilibrium for (2).

**Theorem 1:** Suppose that the conditions in Assumption 1 are satisfied. Then the monotone flow generated by (2) has a globally asymptotically order-stable equilibrium.

*Proof:* See [17].

## B. Reachability analysis

In the literature there exist several methods to characterize an outer-approximation of the reachable set of nonlinear dynamical systems [31], [32], [35], [36]. Unfortunately, almost all these methods are affected by the pessimism propagation linked to the set-membership computation. This phenomenon limits the use of these methods to deal with real problems. Note that, for the methods, relying on the comparison theorems for differential inequalities [35], [36], the monotone property allows to solve the reachability problem efficiently. In fact, thanks to this property, the reachable set can be characterized by the solutions to the following upper and lower systems

$$ \begin{aligned} \dot{\mathbf{x}}_u &= \mathbf{f}(\mathbf{x}_u, t), \quad \mathbf{x}_u(t_0) = \bar{\mathbf{x}}(t_0) \\ \dot{\mathbf{x}}_l &= \mathbf{f}(\mathbf{x}_l, t), \quad \mathbf{x}_l(t_0) = \underline{\mathbf{x}}(t_0) \end{aligned} \quad (5) $$

where the box $[\mathbf{x}(t_0)] = [\underline{\mathbf{x}}(t_0), \bar{\mathbf{x}}(t_0)]$ stands for a bounded set, which contains all the possible initial states of the dynamical system (2). More precisely, the tight reachable set $\mathcal{R}_x(t, [\mathbf{x}(t_0)])$ for (2) is given by

$$ \mathcal{R}_x(t, [\mathbf{x}(t_0)]) = [\mathbf{x}_l(t), \mathbf{x}_u(t)] \quad (6) $$

where $\mathbf{x}_l(t)$ and $\mathbf{x}_u(t)$ are the solutions to the bounding systems (5).

**Proposition 3:** The width of the tight reachable set (6), denoted by $\mathbf{e}(t) = \mathbf{x}_u(t) - \mathbf{x}_l(t)$, converges asymptotically towards zero if the conditions in Assumption 1 are satisfied by its dynamics defined by

$$ \dot{\mathbf{e}} = \partial \mathbf{F}(\omega_1(t), \dots, \omega_n(t), t) \mathbf{e} \quad (7) $$

where the vectors $\omega_i \in [\mathbf{x}_l, \mathbf{x}_u]$, $i = 1,...,n$ satisfy

$$ f_i(\mathbf{x}_u, t) - f_i(\mathbf{x}_l, t) = \nabla f_i(\omega_i, t) \mathbf{e} $$

and the square matrix $\partial \mathbf{F}$ is defined as follows

$$ \partial \mathbf{F}(\omega_1(t), \dots, \omega_n(t), t) = \begin{bmatrix} \nabla f_1(\omega_1(t), t) \\ \vdots \\ \nabla f_n(\omega_n(t), t) \end{bmatrix} $$

*Proof:* By construction, system (7) is cooperative and its origin is an equilibrium point. So, based on the Theorem 1 one can claim that if Assumption 1 is satisfied by (7), the monotone flow generated by this system has a unique globally asymptotically stable equilibrium point $\mathbf{e}^* = 0$. □

Note that, in the general case where the system (2) is not monotone, Müller theorem [30], [20], [19] can be
applied to design bounding systems (8) to compute an outer-approximation of its reachable set.

$$ 
\begin{aligned}
\dot{\mathbf{x}}_u &= \mathbf{f}_u(\mathbf{x}_u, \mathbf{x}_l, t), \quad \mathbf{x}_u(t_0) = \bar{\mathbf{x}}(t_0) \\
\dot{\mathbf{x}}_l &= \mathbf{f}_l(\mathbf{x}_u, \mathbf{x}_l, t), \quad \mathbf{x}_l(t_0) = \mathbf{x}(t_0)
\end{aligned}
\qquad (8) $$

Here, one obtains

$$ \overline{\mathcal{R}}_x(t, [\mathbf{x}(t_0)]) = [\mathbf{x}_l(t), \mathbf{x}_u(t)] \quad (9) \\ \supseteq \overline{\mathcal{R}}_x(t, [\mathbf{x}(t_0)]) $$

The bounding systems (8) are established from the mathematical expression of the field vector **f**. For example, one can design the upper system in (8) defined by **f**_u as follows: for all i = 1, ..., n, j = 1, ..., n, one substitutes in the formal expression of the ith component f_i of the field vector **f**

* x_j by x̄_i,

* x_j (j ≠ i) by x̄_j if ∂f_i/∂x_j ≥ 0 or by x̄_j if ∂f_i/∂x_j < 0

where x̄_i (resp. x̄_j) stands for the upper bound (resp. lower bound) of the ith component of the state vector. In the same way, the lower system in (8) is designed by reversing the sense of the inequality conditions. Moreover, to deal with the case where the partial derivatives have variable signs, the hybrid bounding methods presented in [35], [36], [27] can be used.

### C. Set-membership estimators

In contrast with the classical state estimation methods [18], [12], [9] where the goal is to design state observers with a stable estimation error to show the convergence towards the real state trajectory of the system, set-membership approaches aim to design state estimators [15], [29], [10] able to compute all the possible state trajectories of the system that are compatible with:

* the used model and its uncertainties;

* the experimental data and their feasible domains.

In other words, the set-membership state estimators use the informations given by the measurements to improve the accuracy of the outer-approximation of the reachable set of (2). This kind of state estimators is based on the prediction-correction principle.

a) Prediction: In this step, reachability analysis is used to compute an outer-approximation of the reachable set of the system (2) between two measurement time instants t_j and t_{j+1}.

$$ \overline{\mathcal{R}}_x(t, [\mathbf{x}(t_0)]) \supseteq \mathcal{R}_x(t, [\mathbf{x}(t_0)]), \quad \forall t \in [t_j, t_{j+1}] \qquad (10) $$

So, in order to reduce the conservatism of the outer-approximation, experimental data are used to discard the inconsistent part of the reachable set.

b) Correction: This stage is activated at each time instant where a measurement of the output system is available. Using the output equation of the system

$$ \mathbf{y} = \mathbf{g}(\mathbf{x}, t) \qquad (11) $$

an outer-approximation of the reachable set is computed at the time instant t_j by solving the following set inversion problem

$$ \overline{\mathcal{R}}_x^{\text{inv}}(t_j) \supseteq \{\mathbf{x} \in \mathbb{R}^n \mid \mathbf{g}(\mathbf{x}, t) \in [\mathbf{y}(t_j)]\} \qquad (12) $$

where [y(t_j)] stands for the feasible domain of the measurements at the time instant t_j. Thus, the inconsistent part of the reachable set can be discarded by the following set-membership filter:

$$ \overline{\mathcal{R}}_x^c(t_j, [\mathbf{x}(t_0)]) = \overline{\mathcal{R}}_x^{\text{inv}}(t_j) \cap \overline{\mathcal{R}}_x(t_j, [\mathbf{x}(t_0)]) \qquad (13) $$

Thereafter, the next step of the prediction-correction procedure is initialized with the corrected outer-approximation of the reachable set (13).

### III. MAIN RESULTS

In this work new methods to design set-membership state estimators are proposed. The core idea of these methods consists in decomposing a non-monotone system (2) as a network of monotone subsystems

$$ 
\begin{aligned}
\dot{\mathbf{x}}_i &= \mathbf{f}_i(\mathbf{x}_i, \mathbf{w}_i, t) \\
\mathbf{y}_i &= \mathbf{g}(\mathbf{x}_i) \\
i &\in \{1, \dots, N\} \\
j &\in \{1, \dots, K\}
\end{aligned}
\qquad (14) $$

where x_i ∈ ℝ^{n_i} stands for the state vector of the ith monotone subsystem of (2) and y_j is its output vector. By w_i one denotes the part of the input vector of this subsystem which is constituted by the outputs of the other subsystems of (2). The vector x_j ∈ ℝ^{n_j}, n_j < n, gathers some entries of the state vector x. So, the choice of the sensors placement has an important part to design efficient set-membership state estimators.

**Assumption 2:** Based on an appropriate choice of the sensors placement, the non-monotone system (2) can be viewed as an interconnection of monotone subsystems (14).

Note that, in practice, this assumption is fulfilled by many biological systems [39], biochemical reactions and compartmental systems [14] and in general by all positive systems controlled by negative feedbacks [4]. Moreover, this assumption can be viewed as the detectability assumption in the classical state estimation theory. This means that the monotone part of the system can be considered as the nonobservable but stable part of the system and the non-monotone part must be accessible from the measurements. In fact, with this assumption one attempts to cope with the pessimism propagation generated by the nonmonotone part of the system by using the experimental data.

#### A. Set-membership state estimator by output injection

Using the monotone property of dynamical systems with an appropriate choice of the sensors placement, an efficient set-membership state estimator can be designed for (2) by output injections. This result is stated in the following proposition.

**Proposition 4:** If the initial state of the non-monotone system (2) is framed by

$$ (\bar{\mathbf{x}}_1^T(t_0), \dots, \bar{\mathbf{x}}_N^T(t_0))^T \leq [\mathbf{x}(t_0)] \leq (\bar{\mathbf{x}}_1^T(t_0), \dots, \bar{\mathbf{x}}_N^T(t_0))^T $$

then due to the output injections

$$ \mathbf{w}_i = (\mathbf{y}_{i a}^T, \mathbf{y}_{i b}^T, \dots)^T, \quad i_a, i_b, \dots \in \{1, \dots, N\} $$
the interconnected dynamical system (15)

$$ 
\begin{aligned}
\dot{\bar{\mathbf{x}}}_i &= \mathbf{f}_i(\bar{\mathbf{x}}_i, \mathbf{w}_i, t), \quad \bar{\mathbf{x}}_i(t_0) \\
\dot{\bar{\mathbf{x}}}_i &= \mathbf{f}_i(\bar{\mathbf{x}}_i, \mathbf{w}_i, t), \quad \bar{\mathbf{x}}_i(t_0) \\
i &\in \{1, \dots, N\}
\end{aligned} 
\quad (15) $$

generates a tight enclosure of the reachable set of (2)

$$ \mathcal{R}_x(t, [\mathbf{x}(t_0)]) = [\mathbf{x}_l(t), \mathbf{x}_u(t)] \quad (16) $$

where $\mathbf{x}_l = (\bar{\mathbf{x}}_1^T, \bar{\mathbf{x}}_2^T, \dots, \bar{\mathbf{x}}_N^T)^T$ and $\mathbf{x}_u = (\bar{\mathbf{x}}_1^T, \bar{\mathbf{x}}_2^T, \dots, \bar{\mathbf{x}}_N^T)^T$.

*Proof:* First, note that the state equations of the interconnected system (15) represent exactly the dynamics of the non-monotone system (2) as assumed in Assumption 2. In addition, each subsystem of (15) is monotone with a perfectly known input $\mathbf{w}_i$, this signal is provided by a sensor.

So, due to the monotone property one can claim that each subsystem of (15) generates a tight enclosure of a part of the reachable set of the system (2). Then, the concatenation of these subsets of the state space (16) provides a tight enclosure of the reachable set of (2). In other words, the pessimism propagation linked to the non-monotone part of the system (2) is filtered by the smart choice of the sensors placement proposed in Assumption 2. This completes the proof. □

## B. Event-triggered output injections

In this section an event-triggered version of the previous set-membership state estimator is presented. In fact, here the measurements are used only when the width of the estimated outer-approximation exceeds a certain threshold. This way to use measurement is of great interest in practice. For instance, thanks to this procedure one can work with short communication bandwidths because the CPU exchanges informations with its sensors only when a significant event happening. Before stating our result, let us introduce the definition of the width of a bounded box.

**Definition 4:** For a given bounded box $[\mathbf{x}] \subset \mathbb{R}^n$ one define by

$$ w([\mathbf{x}]) = \max_{i=1,...,n} \{\bar{x}_i - x_i\} \quad (17) $$

its width or its size.

The working principle of this state estimator is illustrated in Figure 1 by an hybrid automaton with two discrete states. First, without using measurements, the bounding system (8) of the non-monotone system (2) generate an outer-approximation of its reachable set. When the width $w$ of this outer-approximation exceeds an user defined threshold $\varepsilon$, the hybrid automaton switches to its second mode where the measurements are used by output injections to reduce the width of the outer-approximation until its size becomes lower than a second threshold $\delta$ fixed by the user.

**Proposition 5:** If the following conditions hold:

* Assumption 2 is satisfied by the non-monotone system (2).

* Assumption 1 is satisfied by the dynamics of the width, $\mathbf{e}_i(t)$, of the reachable set of each subsystem $i$ in (14).

* The initial condition of (2) is framed by

$$ (\bar{\mathbf{x}}_1^T(t_0), \dots, \bar{\mathbf{x}}_N^T(t_0))^T \leq [\mathbf{x}(t_0)] \leq (\bar{\mathbf{x}}_1^T(t_0), \dots, \bar{\mathbf{x}}_N^T(t_0))^T. $$

<figure><img src="image_2.png" /><figcaption>Fig. 1. The hybrid formulation of the proposed set-membership state estimator.</figcaption></figure>

Then the solution $(\mathbf{x}_l(t), \mathbf{x}_u(t))$ to the hybrid automaton presented in Figure 1 characterizes an outer-enclosure of the reachable set of (2)

$$ \overline{\mathcal{R}}_x(t, [\mathbf{x}(t_0)]) = [\mathbf{x}_l(t), \mathbf{x}_u(t)] \quad (18) $$

with a width

$$ w(\overline{\mathcal{R}}_x(t, [\mathbf{x}(t_0)])) \leq \varepsilon. $$

*Proof:* The hybrid automaton has two continuous-time modes. The width of the outer-approximation of the reachable set computed in the non-correction mode $m = 0$ can increase until to reach a maximal value authorized by the user $\varepsilon$. At this maximal limit, the correction mode $m = 1$ is activated in order to reduce the width the outer-approximation. Thereby, based on the Theorem 1, one can claim that, due to output injections, the width of the outer-approximation has to decrease. This mode is maintained active until that the width of the outer-approximation becomes lower than a minimal value $\delta$. This allows to avoid the immediate activation of the non-correction mode and so to use less the measurements. This completes the proof. □

## C. The case of uncertain systems

In contrast to the classical state estimation approaches, the set-membership methods can deal with the case of uncertain systems where the dynamics is described by a differential inclusion

$$ \dot{\mathbf{x}} \in \mathbf{f}(\mathbf{x}, \mathbf{p}, t) \quad (19) $$

where $\mathbf{x}(t_0) \in [\mathbf{x}_0] \subset \mathbb{R}^n$ and the uncertain parameter vector $\mathbf{p}$ belongs to a bounded box $[\mathbf{p}] \subset \mathbb{R}^{n_p}$.

In this unknown-but-bounded error framework, Proposition 4 stays valid to design set-membership state estimators by output injections. Note that, in order to take into account the effect of the uncertain parameters, here the bounding systems
(15) used in Proposition 4 are replaced by the following systems

$$
\begin{align}
\dot{\bar{\mathbf{x}}}_i &= \mathbf{f}_i(\bar{\mathbf{x}}_i, \bar{\mathbf{p}}, \mathbf{p}, \mathbf{w}_i, t), \quad \bar{\mathbf{x}}_i(t_0) \nonumber \\
\dot{\mathbf{x}}_i &= \mathbf{f}_i(\mathbf{x}_i, \bar{\mathbf{p}}, \mathbf{p}, \mathbf{w}_i, t), \quad \mathbf{x}_i(t_0) \tag{20} \\
i &\in \{1, \dots, N\} \nonumber
\end{align}
$$

where the entries of the uncertain parameter vector **p** are tuned to their upper or lower bounds according to the following rule: for all *i* = 1, ..., *n* and *k* = 1, ..., *n*<sub>*p*</sub> one replaces in the analytical expression of the *i*th component *f*<sub>*i*</sub> of the field vector **f**

* $p_k$ by $\bar{p}_k$ if $\frac{\partial f_i}{\partial p_r} \ge 0$ or by $\underline{p}_k$ if $\frac{\partial f_i}{\partial p_k} < 0$ to design the upper system and.

* $p_k$ by $\bar{p}_k$ if $\frac{\partial f_i}{\partial p_r} \le 0$ or by $\underline{p}_k$ if $\frac{\partial f_i}{\partial p_k} > 0$ to design the lower system.

Note that, in this case, a conservatism on the estimated enclosure, of all the possible state trajectories compatible with the differential inclusion (21), can be introduced if the elements of the parameter vector **p** act positively and negatively on the components of the field vector **f**.

**Remark 3:** One can also apply this set-membership state estimation method for the case where the measurements are affected by unknown-but-bounded additive noises

$$
\mathbf{w}_i \in \mathbf{w}_{im} + [\underline{\mathbf{e}}_{w_i}, \bar{\mathbf{e}}_{w_i}], i = 1, \dots, N \quad (21)
$$

where the vector $\mathbf{w}_{im}$ represents the available measures provided by the sensors and bounded box $[\underline{\mathbf{e}}_{w_i}]$ stands for the feasible domain of the output error. To this end, one must extend the vector of the uncertain parameters as follows

$$
\mathbf{p}_e^T = (\mathbf{p}^T, \mathbf{e}_{w_1}^T, \dots, \mathbf{e}_{w_N}^T).
$$

Similarly, in the case of state estimation by an event-
triggered output injections, Proposition 5 remains true and
the bounding systems of the continuous mode $m=0$ of the
hybrid automaton are defined by

$$
\begin{align}
\dot{\mathbf{x}}_u &= \mathbf{f}_u(\mathbf{x}_u, \mathbf{x}_l, \bar{\mathbf{p}}, \mathbf{p}, t), \quad \mathbf{x}_u(t_0) = \bar{\mathbf{x}}(t_0) \nonumber \\
\dot{\mathbf{x}}_l &= \mathbf{f}_l(\mathbf{x}_u, \mathbf{x}_l, \bar{\mathbf{p}}, \mathbf{p}, t), \quad \mathbf{x}_l(t_0) = \underline{\mathbf{x}}(t_0) \tag{22}
\end{align}
$$

where the construction of the upper and lower systems
is based on the following procedure: for all $i = 1, \dots, n$,
$j = 1, \dots, n$ and $k = 1, \dots, p$ one replaces in the analytical
expression of the $i$th component $f_i$ of the field vector $\mathbf{f}$

• $x_i$ by $\bar{x}_i$,

* $x_j (j \neq i)$ by $\bar{x}_j$ if $\frac{\partial f_i}{\partial x_j} \ge 0$ or by $\underline{x}_j$ if $\frac{\partial f_i}{\partial x_j} < 0$,

* $p_k$ by $\bar{p}_k$ if $\frac{\partial f_i}{\partial p_r} \ge 0$ or by $\underline{p}_k$ if $\frac{\partial f_i}{\partial p_k} < 0$ to design the upper system defined by the field vector $\mathbf{f}_u$.

* The lower system defined by the field vector **f**<sub>l</sub> is designed following the same procedure by reversing the sense of the inequality conditions.

IV. ILLUSTRATIVE EXAMPLE

The set-membership state estimation problem for a bio-
logical system, Drosophila circadian rhythms described by

the Goldbeter model [3], is considered in this section,

$$
\begin{align}
\dot{M} &= \frac{v_s K_I^n}{K_I^n + P_N^n} - \frac{v_m M}{k_m + M} \nonumber \\
\dot{P}_0 &= k_s M - \frac{V_1 P_0}{K_1 + P_0} + \frac{V_2 P_1}{K_2 + P_1} \nonumber \\
\dot{P}_1 &= \frac{V_1 P_0}{K_1 + P_0} - \frac{V_2 P_1}{K_2 + P_1} - \frac{V_3 P_1}{K_3 + P_1} + \frac{V_4 P_2}{K_4 + P_2} \nonumber \\
\dot{P}_2 &= \frac{V_3 P_1}{K_3 + P_1} - \frac{V_4 P_2}{K_4 + P_2} - k_1 P_2 + k_2 P_N - \frac{v_d P_2}{k_d + P_2} \nonumber \\
\dot{P}_N &= k_1 P_2 - k_2 P_N \tag{23}
\end{align}
$$

The values of the system parameters are gathered in Table I.

<table><caption>TABLE I
PARAMETER VALUES</caption>
  <thead>
    <tr>
      <th>Parameter</th>
      <th>Value</th>
      <th>Parameter</th>
      <th>Value</th>
    </tr>
  </thead>
  <tbody>
    <tr>
      <td>k<sub>2</sub></td>
      <td>1.3</td>
      <td>k<sub>1</sub></td>
      <td>1.9</td>
    </tr>
    <tr>
      <td>V<sub>1</sub></td>
      <td>3.2</td>
      <td>V<sub>2</sub></td>
      <td>1.58</td>
    </tr>
    <tr>
      <td>V<sub>3</sub></td>
      <td>5</td>
      <td>V<sub>4</sub></td>
      <td>2.5</td>
    </tr>
    <tr>
      <td>K<sub>I</sub></td>
      <td>1</td>
      <td>k<sub>m</sub></td>
      <td>0.5</td>
    </tr>
    <tr>
      <td>k<sub>s</sub></td>
      <td>0.38</td>
      <td>v<sub>d</sub></td>
      <td>0.95</td>
    </tr>
    <tr>
      <td>k<sub>d</sub></td>
      <td>0.2</td>
      <td>n</td>
      <td>4</td>
    </tr>
    <tr>
      <td>K<sub>1</sub></td>
      <td>2</td>
      <td>K<sub>2</sub></td>
      <td>2</td>
    </tr>
    <tr>
      <td>K<sub>3</sub></td>
      <td>2</td>
      <td>K<sub>4</sub></td>
      <td>2</td>
    </tr>
    <tr>
      <td>v<sub>s</sub></td>
      <td>0.76</td>
      <td>v<sub>m</sub></td>
      <td>0.65</td>
    </tr>
  </tbody>
</table>

For this example, Assumption 2 is satisfied if the concen-
tration of the end product is chosen as an output

$$
w_1 = P_N \tag{24}
$$

So, by output injection one obtains the following monotone
system

$$
\begin{align}
\dot{M} &= \frac{v_s K_I^n}{K_I^n + w_1^n} - \frac{v_m M}{k_m + M} \nonumber \\
\dot{P}_0 &= k_s M - \frac{V_1 P_0}{K_1 + P_0} + \frac{V_2 P_1}{K_2 + P_1} \nonumber \\
\dot{P}_1 &= \frac{V_1 P_0}{K_1 + P_0} - \frac{V_2 P_1}{K_2 + P_1} - \frac{V_3 P_1}{K_3 + P_1} + \frac{V_4 P_2}{K_4 + P_2} \nonumber \\
\dot{P}_2 &= \frac{V_3 P_1}{K_3 + P_1} - \frac{V_4 P_2}{K_4 + P_2} - k_1 P_2 + k_2 w_1 - \frac{v_d P_2}{k_d + P_2} \nonumber \\
\dot{P}_N &= k_1 P_2 - k_2 P_N \tag{25}
\end{align}
$$

and the dynamics of the width of its reachable set is
described by

$$
\begin{align}
\dot{e}_M &= - \frac{v_m k_m e_M}{(k_m + w_M)^2} \nonumber \\
\dot{e}_P_0 &= k_s e_M - \frac{V_1 K_1 e_{P_0}}{(K_1 + w_{P_0})^2} + \frac{V_2 K_2 e_{P_1}}{(K_2 + w_{P_1})^2} \nonumber \\
\dot{e}_P_1 &= \frac{V_1 K_1 e_{P_0}}{(K_1 + w_{P_0})^2} - \left( \frac{V_2 K_2}{(K_2 + w_{P_1})^2} + \frac{V_3 K_3}{(K_3 + w_{P_1})^2} \right) e_{P_1} + \frac{V_4 K_4 e_{P_2}}{(K_4 + w_{P_2})^2} \nonumber \\
\dot{e}_P_2 &= \frac{V_3 K_3 e_{P_1}}{(K_3 + w_{P_1})^2} - \left( \frac{V_4 K_4}{(K_4 + w_{P_2})^2} + k_1 + \frac{v_d k_d}{(k_d + w_{P_2})^2} \right) e_{P_2} \nonumber \\
\dot{e}_{P_N} &= k_1 e_{P_2} - k_2 e_{P_N} \tag{26}
\end{align}
$$

where the variables $w_M$, $w_{P_0}$, $w_{P_1}$ and $w_{P_2}$ belong into the
following time-varying intervals

$$
\begin{align*}
\underline{M}(t) &\le w_M(t) \le \overline{M}(t) \\
\underline{P}_0(t) &\le w_{P_0}(t) \le \overline{P}_0(t) \\
\underline{P}_1(t) &\le w_{P_1}(t) \le \overline{P}_1(t) \\
\underline{P}_2(t) &\le w_{P_2}(t) \le \overline{P}_2(t)
\end{align*}
$$
Note that, by construction, this dynamics is cooperative. So one can apply Theorem 1 to show the global asymptotic stability of its origin. To do that, one must check the conditions of Assumption 1, namely the boundedness of all the solutions to (26) and the unicity of its equilibrium point.

c) Boundedness: From (26) it is clear that $e_M$ is bounded and converges asymptotically towards 0. Thus, one can claim that there exists a time instant $t^*$ for which one obtains

$$ (\sum_{i=0}^{2} \dot{e}_{P_i}(t^*)) + \dot{e}_N(t^*) = -\frac{v_d k_d}{(k_d + w_{P_2})^2} e_{P_2}(t^*) - k_2 e_{P_N}(t^*) \quad (27) $$

Moreover, since the initial condition of (26) is positive, one can state that the solution to (26) stays positive for all $t \ge 0$. Hence, for all $t > t^*$ (27) is nonpositive then one can affirm that all the solutions to (26) are bounded.

d) Uniqueness of the equilibrium point: By direct computation one can show that the origin of (26) is an equilibrium point

$$ (e_M^*, e_{P_0}^*, e_{P_1}^*, e_{P_2}^*, e_{P_N}^*)^T = (0, 0, 0, 0, 0)^T \quad (28) $$

Moreover, it is straightforward to show that the following set of equations established at the steady state of (26)

$$ \begin{aligned} \frac{e_M^*}{\frac{V_1 K_1}{(K_1+w_{P_0})^2} e_{P_0}^*} &= 0 \\ \frac{V_2 K_2}{(K_2+w_{P_1})^2} e_{P_1}^* &= 0.3 \le k_s \le 0.4 \quad (31) \\ \frac{V_3 K_3}{(K_3+w_{P_1})^2} e_{P_1}^* &= \frac{V_4 K_4}{(K_4+w_{P_2})^2} e_{P_2}^* \\ (k_1 + \frac{v_d k_d}{(k_d+w_{P_2})^2}) e_{P_2}^* &= 0 \\ k_1 e_{P_2}^* &= k_2 e_{P_N}^* \end{aligned} $$

admits a unique solution which is the origin of (26).

So, Assumption 1 is satisfied by (26). Consequently, one can apply Propositions 4 and 5 to estimate guaranteed state enclosures for the nonlinear system (23).

### A. Test with output injection

The initial state of (23) is assumed unknown but belongs to the bounded box

$$ \begin{aligned} 0.1 \le M(0) \le 2 \\ 0.1 \le P_0(0) \le 3 \\ 0.1 \le P_1(0) \le 4 \\ 0.1 \le P_2(0) \le 2 \\ 0.1 \le P_N(0) \le 3 \end{aligned} \quad (30) $$

Then, based on the output injection (24) and using the bounding systems deduced from (25) as proposed in (15) a tight state enclosure for (23) is computed and depicted in Figure 2. Furthermore, since (26) satisfies Assumption 1 with the origin as its unique equilibrium, the width of the estimated enclosure converges asymptotically towards 0. It means that the lower bound of the estimated state enclosure coincides with the upper bound of the estimate state enclosure in the steady state, as shown in Figure 2.

<figure><img src="image_3.png" /><figcaption>Fig. 2. Projections of the estimated tight enclosure of the reachable set of (23) on the five state axes. The continuous lines show the enclosure and the dashed lines plot the real state trajectories generated by (23) from the initial state $(M(0), P_0(0), P_1(0), P_2(0), P_N(0))^T = (0.5, 1.5, 2, 1, 1.5)^T$.</figcaption></figure>

### B. Case of uncertain system (Test with output injection)

In this subsection the system (23) is considered uncertain, where its parameters $v_s, k_s$ and $k_d$ are unknown but bounded with known bounds

$$ \begin{aligned} 0.7 \le v_s \le 0.8 \\ 0.3 \le k_s \le 0.4 \\ 0.1 \le k_d \le 0.3 \end{aligned} \quad (31) $$

and the output (24) is corrupted by a bounded additive noise

$$ [w_1] = P_N + [0.05, 0.05] \quad (32) $$

In this framework, the simulation results of the set-membership state estimator (20) are depicted in Figure 3.

<figure><img src="image_4.png" /><figcaption>Fig. 3. Projections of the estimated tight enclosure of the reachable set of (23) on the five state axes. The continuous lines show the enclosure and the dashed lines plot the real state trajectories generated by (23) from the initial state $(M(0), P_0(0), P_1(0), P_2(0), P_N(0))^T = (0.5, 1.5, 2, 1, 1.5)^T$.</figcaption></figure>
It is worth pointing out that, since the uncertain parameters of (23) act positively on its dynamics, the estimated state enclosure plotted in Figure 3 is tight. It means that the lower and upper bounds of the state enclosure generated by (20) are feasible and no conservatism is introduced.

### C. Test with event-triggered output injection

This simulation test aims to show the performance of the hybrid set-membership state estimator given in Proposition 5. For this example, Assumptions 1 and 2 are satisfied. Then, thanks to the event-triggered output injection policy, one can maintain the width of the reachable set of (23) lower than a given threshold $\epsilon$. Figure 4 shows the time evolution of the

<figure><img src="image_5.png" /><figcaption>Fig. 4. Projections of the estimated tight enclosure of the reachable set of (23) on the five state axes. The continuous lines show the enclosure and the dashed lines plot the real state trajectories generated by (23) from the initial state $(M(0), P_0(0), P_1(0), P_2(0), P_N(0))^T = (0.5, 1.5, 2, 1, 1.5)^T$.</figcaption></figure>

estimated state enclosure of (23) generated from the domain of initial conditions (30). The hybrid automaton of Figure 1 is implemented with the following thresholds: $\epsilon = 1$ and $\delta = 0.5$. As expected, Figure 4 shows that the width of the estimated state enclosure stays always lower than 1 and the hybrid automaton switches between its two continuous modes according to the value taken by this width.

### D. Case of uncertain system (Test with event-triggered output injection)

For the last test, the event-triggered output injection policy is applied to estimate the state enclosure of the uncertain system (23), where the uncertainties are defined by (30), (31) and (32).

The simulation results are plotted in Figure 5. As expected with the same thresholds $\epsilon$ and $\delta$, the hybrid state estimator has to activate for a long time its second continuous mode in order to maintain the width of the estimated state enclosure lower that $\epsilon$. That means, to cope with the pessimism propagation generated by the uncertainties the state estimator needs to use informations given by the output measurements for a long period of time.

<figure><img src="image_6.png" /><figcaption>Fig. 5. Projections of the estimated tight enclosure of the reachable set of (23) on the five state axes. The continuous lines show the enclosure and the dashed lines plot the real state trajectories generated by (23) from the initial state $(M(0), P_0(0), P_1(0), P_2(0), P_N(0))^T = (0.5, 1.5, 2, 1, 1.5)^T$.</figcaption></figure>

## V. CONCLUSIONS

A new method to design set-membership state estimators has been proposed in this paper. The core idea of this method is based on the comparison theorems of differential inequalities and the principle of output injections. Two policies are suggested to implement the proposed state estimators in a bounded error framework. The first one relies on the continuous output injections and the second one is based on an event-triggered output injection. In both cases, the convergence of the width of the estimated state enclosure is done and conditions to guarantee this convergence are highlighted.

## REFERENCES

[1] T. Alamo, J.M. Bravo, and E.F. Camacho. Guaranteed state estimation by zonotopes. Automatica, 2005.

[2] D. Angeli and E. D. Sontag. Multi-stability in monotone input/output systems. Systems and Control Letters, 51:185-202, 2004.

[3] D. Angeli and E. D. Sontag. Oscillations in I/O monotone systems. IEEE Transactions on Circuits and Systems, Special Issue on Systems Biology, 55:66-176, 2008.

[4] D. Angeli and E.D Sontag. Monotone control systems. IEEE Transaction on automatic control, 48(10):1684-1698, 2003.

[5] F.L Chernousko. Ellipsoidal state estimation for dynamical systems. Nonlinear analysis, 2005.

[6] F.L Chernousko and D. Ya. Rokityanskii. Ellipsoidal bounds on reachable sets of dynamical systems with matrices subjected to uncertain perturbations. Journal of optimization theory and applications, 104:1-19, 2000.

[7] L. Chisci, A. Garulli, and G. Zappa. Recursive state bounding by parallelotopes. Automatica, 32:1049-1055, 1996.

[8] C. Combastel. A state bounding observer for uncertain nonlinear continuous-time systems based on zonotopes. In 44th IEEE Conference on decision and control and European control conference ECC 2005, 2005.

[9] J.P. Gauthier, H. Hammouri, and S. Othman. Simple observer for nonlinear systems applications to bioreactors. IEEE Transactions on Automatic Control, 37:875-880, 1992.

[10] G. Goffaux, A.V. Wouwer, and O. Bernard. Improving continuous-discrete interval observers with application to microalgae-based bio-processes. Journal of Process Control, 19:1182-1190, 2009.
[11] J.L. Gouzé, A. Rapaport, and Z. M. Hadj-Sadok. Interval observers for uncertain biological systems. *Journal of Ecological Modelling*, 133(45-56, 2000).

[12] R. Hermann and A. J.Krener. Nonlinear controllability and observability. *Transactions on Automatic Control*, AC-22:728-740, 1977.

[13] M. Hirsch and H.L. Smith. Monotone dynamical systems. In *A. Canada, P. Drabek, and A. Fonda, editors, Handbook of Differential Equations, Ordinary Differential Equations*, volume 2, chapter 4. Elsevier, 2005.

[14] J.A. Jacaquez and C.P. Simon. Qualitative theory of compartmental systems. *SIAM Review*, 35:43-79, 1993.

[15] L. Jaulin. Nonlinear bounded-error state estimation of continuous-time systems. *Automatica*, 38:1079-1082, 2002.

[16] L. Jaulin, M. Kieffer, O. Didrit, and E. Walter. *Applied interval analysis: with examples in parameter and state estimation, robust control and robotics*. Springer-Verlag, London, 2001.

[17] J. F. Jiang. On the global stability of cooperative systems. *Bull. London Math. Soc.*, 26:455-458, 1994.

[18] R.E. Kalman. On a new approach to linear filtering and prediction problems. *ASME J.Basic Engineering*, 82(1):35-45, 1960.

[19] M. Kieffer and E. Walter. Guaranteed nonlinear state estimation for continuous-time dynamical models from discrete-time measurements. In *Proceedings 6th IFAC Symposium on Robust Control*, pages 685-690, Toulouse, France, 2006.

[20] M. Kieffer, E. Walter, and I. Simeonov. Guaranteed nonlinear parameter estimation for continuous-time dynamical models. In *Proceedings 14th IFAC Symposium on System Identification*, pages 843-848, Newcastle, Aus, 2006.

[21] H. Kunze and D. Siegel. A graph theoretical approach to monotonicity with respect to initial conditions. *Nonlinear Analysis*, 35:1-20, 19.

[22] F. Mazenc and O. Bernard. Asymptotically stable interval observers for planar systems with complex poles. *IEEE Transactions on Automatic Control*, 55:523, 2010.

[23] F. Mazenc, M. Kieffer, and E. Walter. Interval observers for continuous-time linear systems with discrete-time outputs. In *Proceedings of the IEEE American Control Conference (ACC)*, pages 1889-1894, Montreal, 2012.

[24] F. Mazenc and D. Thach Ngoc. Interval observers for continuous-time linear systems with discrete-time outputs. In *Proceedings of the IEEE 52nd Annual Conference on Decision and Control (CDC)*, pages 787-792, Firenze, 2013.

[25] N. Meslem. New idea to design linear interval observers. In *Proceedings of the 3rd IEEE International Conference on Control, Decision and Information Technologies (CoDIT'16)*, pages 1-6, Malta, 2016.

[26] N. Meslem and C. Prieur. State estimation based on self-triggered measurements. In *Proceedings of the 19th IFAC World Congress*, pages 86-91, Cape Town, 2014.

[27] N. Meslem and N. Ramdani. Interval observer design based on nonlinear hybridization and practical stability analysis. *International Journal of Adaptive Control and Signal Processing*, 25(3):228-248, 2011.

[28] N. Meslem, N. Ramdani, and Y. Candau. Interval observer for uncertain nonlinear systems. application to bioreactors. In *Proceedings of the 17th IFAC World Congress*, pages 9667-9672, Seoul, Korea, 2008.

[29] N. Meslem, N. Ramdani, and Y. Candau. Using hybrid automata for set-membership state estimation with uncertain nonlinear continuous-time systems. *Journal of Process Control*, 20(4):481-489, 2010.

[30] M. Müller. Über das fundamental theorem in der theorie der gewöhnlichen differentialgleichungen. *Math. Z.*, 26:619-645, 1926.

[31] N. S. Nedialkov and K. R. Jackson. An interval hermite-obreschkoff method for computing rigorous bounds on the solution of an initial value problem for an ordinary differential equation. *Reliable Computing*, 5:289-310, 1999.

[32] N.S. Nedialkov, K.R. Jackson, and J.D. Pryce. An effective high-order interval method for validating existence and uniqueness of the solution of an IVP for an ODE. *Reliable Computing*, 7:449-465, 2001.

[33] T. Raissi, D. Efimov, and A. Zolghadri. Interval state estimation for a class of nonlinear systems. *IEEE Transactions on Automatic Control*, 57:260, 2012.

[34] T. Raïssi, N. Ramdani, and Y. Candau. Set membership state and parameter estimation for systems described by nonlinear differential equations. *Automatica*, 40(10):1771-1777, 2004.

[35] N. Ramdani, N. Meslem, and Y. Candau. A hybrid bounding method for computing an over-approximation for the reachable space of uncertain nonlinear systems. *IEEE Transactions on Automatic Control*, 54(10):2352-2364, 2009.

[36] N. Ramdani, N. Meslem, and Y. Candau. Computing reachable sets for uncertain nonlinear monotone systems. *Nonlinear Analysis : Hybrid Systems*, 4(2):263-278, 2010.

[37] M. Ait Rami, C. H. Cheng, and C. de Prada. Tight robust interval observers: An LP approach. In *Proceedings of the 47th IEEE Conference on Decision and Control (CDC)*, pages 2967-2972, Cancun, 2008.

[38] H.L. Smith. *Monotone dynamical systems: An introduction to the theory of competitive and cooperative systems. Vol. 41 of Mathematical Surveys and Monographs.* American Mathematical Society, Providence, 1995.

[39] E.D. Sontag. Molecular systems biology and control. *Eur. J. Control*, 11(4-5):396-435, 2005.

[40] G. Zheng, D. Efimov, and W. Perruquetti. Design of interval observer for a class of uncertain unobservable nonlinear systems. *Automatica*, 63:167-174, 2016.