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# Design of global and local state estimators for bilinear
Persidskii systems

Danilo Rodrigues de Lima, Rosane Ushirobira, Denis Efimov

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## To cite this version:

Danilo Rodrigues de Lima, Rosane Ushirobira, Denis Efimov. Design of global and local state estimators for bilinear Persidskii systems. International Journal of Control, In press. hal-05435691

HAL Id: hal-05435691

https://inria.hal.science/hal-05435691v1

Submitted on 30 Dec 2025

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Distributed under a Creative Commons Attribution 4.0 International License
# Design of global and local state estimators for bilinear Persidskii systems

Danilo Rodrigues de Lima<fnref n="1" />*
Denis Efimov<fnref n="1" />

Rosane Ushirobira<fnref n="1" />

**Correspondence**

Danilo Rodrigues de Lima, Inria, Univ. Lille, CNRS, UMR 9189 - CRISTAL, F-59000 Lille, France

Email: Danilo-Rodrigues-de-Lima@inria.fr

**Funding information**

This work was partially supported by the Hauts-de-France Region, France.

This paper introduces two state estimators for a class of Persidskii systems that contain bilinear cross-terms in the state, involving nonlinear functions and unknown input. One observer is local, assuming that the state trajectories are small, while the other is designed to be global, considering that the state takes nonnegative values only. The analysis of the estimation error is based on the input-to-output stability theory and the conditions are formulated using linear matrix inequalities. To illustrate our results, we provide simulations for a model of a chemical exothermic reactor and a consumer-resource interaction model.

**KEYWORDS**

Nonlinear systems; State estimation; Persidskii systems

## 1 | INTRODUCTION

The state estimation problem arises in numerous engineering applications, including complex physical and biological systems, where models can be characterized by significant nonlinearity (the linearization tools are difficult to apply) and uncertainty [1], [2], [3]. This problem can be approached in many ways depending on the particular application and the kind of system. For example, for linear dynamical systems, there are many solutions, including the popular Kalman filter or high-gain observer designs [4], [5]. Numerical differentiators can be used for model-free estimation [6], [7], [8], while for general nonlinear systems, the synthesis of the observers is usually restricted to canonical representations of the models [3].

When dealing with nonlinear systems with an unknown input (exogenous perturbations, measurement noises, unmodeled dynamics), to formulate the observer design problem in terms of input-to-output stability (IOS) [9], [10] can be helpful. Also, in the nonlinear setting, having the dynamic model of the estimation error independent from the

<footnote n="1">¹Inria, Univ. Lille, CNRS UMR 9189, F-59000 Lille, France</footnote>
system or observer state values may be challenging. Thus, in the IOS framework, we consider the estimated error as the output and construct an estimator providing its convergence in the noise/disturbance-free case and boundedness otherwise. Like with regular stability notions, the IOS analysis can be accomplished by designing special real-valued functions called IOS-Lyapunov functions, following milestone results introduced in [11]. It is guaranteed that the existence of an appropriate Lyapunov function is equivalent to the IOS property for a large class of systems. However, there is no universal method for constructing such a function. Therefore, similarly to the observer design problem, we must rely on particular canonical forms of the system models.

The model we consider in this paper is an extension of the important class of Persidskii systems. They were first introduced in [12], [13] for stability analysis, and they are also related to Lur'e systems studied in the absolute stability theory [14], [15]. The benefit of Persidskii systems lies in the existence of well-investigated candidate Lyapunov functions containing the integrals of sector nonlinearities, used in many cases [16], [17], [18], usually resulting in stability conditions that can be formulated as linear matrix inequalities. This class of models has been practical in representing numerous physical and biological phenomena, and has been studied from many different viewpoints, including the observer design in the IOS framework [17].

We will then consider here a significant extension of the class of systems investigated in [17], motivated by biological and physical examples [19], [20], [21], which consider new classes of nonlinearities, including the addition of bilinear cross-products. For bilinear systems with cross-terms in the state and input only, many solutions exist in the literature, see for instance, pioneering works [22], [23], [24] (this case can be easily reduced to analysis of linear time-varying systems). Frequently, the state cross-terms can also be hidden in the state matrix, transforming the system into a similar linear time-varying form, as in [25], [26], and nonlinear cases are rarely studied. We aim to propose a state observer using the IOS theory, assuming the presence of products of the states and nonlinear functions and also that nonlinear state measurements are possible. Two kinds of design are given, providing different IOS characteristics. First, we will establish local stability conditions not based on linearization. Then, we will develop a global condition assuming the trajectories of the system have positive coordinates (this restriction is motivated by chemical and biological applications). Our preliminary paper [27] studied only the former design, which is also briefly summarized in the current work.

## Notation

* The set of reals is denoted by $\mathbb{R}$ and $\mathbb{R}_+ := \{s \in \mathbb{R} \mid s \ge 0\}$.

* The set of $n \times n$ diagonal matrices with nonnegative elements on the diagonal is denoted by $D_+^n$.

* The usual Euclidean norm in $\mathbb{R}^n$ is denoted by $\| \cdot \|_\infty$.

* For a Lebesgue measurable function $u : \mathbb{R}_+ \to \mathbb{R}^\ell$, define the norm $\|u\|_{[t_1, t_2]} = \ess \sup_{t \in [t_1, t_2]} \|u(t)\|$ for $[t_1, t_2) \subset \mathbb{R}_+$. We denote by $L_\infty^\ell$ the set of functions $u$ such that $\|u\|_\infty := \|u\|_{[0, +\infty)} < +\infty$.

* A continuous function $\sigma : \mathbb{R}_+ \to \mathbb{R}_+$ belongs to the class $\mathcal{K}$ if it is strictly increasing and $\sigma(0) = 0$. If in addition $\lim_{r \to +\infty} \sigma(r) = +\infty$, we say that $\sigma$ belongs to the class $\mathcal{K}_\infty$. A continuous function $\beta : \mathbb{R}_+ \times \mathbb{R}_+ \to \mathbb{R}_+$ belongs to the class $\mathcal{KL}$ if $\beta(\cdot, r) \in \mathcal{K}$ for all $r \in \mathbb{R}_+$ and $\beta(r, \cdot)$ is a decreasing function going to zero for any fixed $r > 0$.
## 2 | OBSERVERS AND INPUT TO OUTPUT STABILITY

In what follows, we consider a general system:

$$ 
\begin{aligned}
\dot{x}(t) &= f(x(t), u(t)) \\
y(t) &= h(x(t)), \forall t \in \mathbb{R}_+,
\end{aligned} 
\quad (1) $$

where $x(t) \in \mathbb{R}^n$ is the state, $u(t) \in \mathbb{R}^l$ is the input, and $y(t) \in \mathbb{R}^p$ is the output. The function $f: \mathbb{R}^n \times \mathbb{R}^l \to \mathbb{R}^n$ is assumed to be locally Lipschitz continuous with $f(0, 0) = 0$, and $h: \mathbb{R}^n \to \mathbb{R}^p$ is a continuously differentiable function. For any $x_0 \in \mathbb{R}^n$ and $u \in \mathcal{L}_\infty^l$, we denote the respective solution of (1) by $x(t, x_0, u)$ that we assume to be defined for all $t \in \mathbb{R}_+$, i.e., the system is forward complete; denote $y(t, x_0, u) = h(x(t, x_0, u))$.

**Definition 1.** [11] We say that a forward complete system (1) is input-to-output stable (IOS) if there exist functions $\beta \in \mathcal{K}$ and $\gamma \in \mathcal{K}$ such that $\|y(t, x_0, u)\| \le \beta(\|x_0\|, t) + \gamma(\|u\|_\infty)$, for every $t \in \mathbb{R}_+$, $x_0 \in \mathbb{R}^n$ and $u \in \mathcal{L}_\infty^l$.

**Definition 2.** [11] We say that a forward complete system (1) is state-independent input-to-output stable (SIIOS) if there exist functions $\beta \in \mathcal{K}$ and $\gamma \in \mathcal{K}$ such that $\|y(t, x_0, u)\| \le \beta(\|h(x_0)\|, t) + \gamma(\|u\|_\infty)$, for every $t \in \mathbb{R}_+$, $x_0 \in \mathbb{R}^n$ and $u \in \mathcal{L}_\infty^l$.

The difference between these stability characteristics is that the former accepts the output transient deviations and the convergence time governed by the initial norm of the whole state $\|x_0\|$, while for the latter, everything depends on the output $\|h(x_0)\|$ only. The Lyapunov function method can be used to check these stability properties:

**Definition 3.** [11] A continuously differentiable function $V: \mathbb{R}^n \to \mathbb{R}_+$ is called an IOS-Lyapunov function for (1) if there exist functions $\alpha_1, \alpha_2 \in \mathcal{K}_\infty$, $\chi \in \mathcal{K}$ and $\alpha_3 \in \mathcal{K}$ such that

$$ \alpha_1(\|h(x)\|) \le V(x) \le \alpha_2(\|x\|) \quad (2) $$

$$ \text{and } V(x) \ge \chi(\|u\|) \implies \nabla V(x)f(x, u) \le -\alpha_3(V(x), \|x\|) \quad (3) $$

for all $x \in \mathbb{R}^n$ and $u \in \mathbb{R}^l$. We say that $V$ is a SIIOS-Lyapunov function for (1) if there exist functions $\alpha_1, \alpha_2 \in \mathcal{K}_\infty$ and $\chi, \alpha_3 \in \mathcal{K}$ such that

$$ \alpha_1(\|h(x)\|) \le V(x) \le \alpha_2(\|h(x)\|) \quad (4) $$

$$ \text{and } V(x) \ge \chi(\|u\|) \implies \nabla V(x)f(x, u) \le -\alpha_3(\|V(x)\|), \quad (5) $$

for all $x \in \mathbb{R}^m$ and $u \in \mathbb{R}^l$.

Note that the right-hand side in (3) is not a definite function of $V$ if the state norm is not bounded. The next property will be used to avoid this issue:

**Definition 4.** The forward complete system (1) is said to be uniformly bounded-input-bounded-state stable (UBIBS) if there exists a function $\sigma \in \mathcal{K}$ such that $\|x(t, x_0, u)\| \le \max\{\sigma(\|x_0\|), \sigma(\|u\|_\infty)\}$, for every $x_0 \in \mathbb{R}^n$ and every $u \in \mathcal{L}_\infty^l$.

The key results connecting these stability properties and the corresponding Lyapunov function are given below.

**Theorem 1.** [11] A UBIBS system (1) is IOS if and only if it admits an IOS-Lyapunov function.
**Theorem 2.** [11] A forward-complete system (1) is SIIOS if and only if it admits a SIIOS-Lyapunov function.

The IOS property and the IOS-Lyapunov function are notions defined globally. The local counterparts can also be
formulated by restricting the domain of validity.

In this paper, we will use the IOS theory to define and design a state observer for (1) as it is formalized in the
following definition (we assume that the input $u$ is unknown):

**Definition 5.** For a given locally Lipschitz continuous function $\tilde{f} : \mathbb{R}^n \times \mathbb{R}^p \to \mathbb{R}^n$, we say that the system

$$
\dot{x}(t) = \tilde{f}(\hat{x}(t), y(t))
$$

is an observer for the system (1) if the combined system

$$
(\dot{x}(t) \quad \dot{x}(t))^{\top} = (f(x(t), u(t)) \quad \tilde{f}(\hat{x}(t), y(t)))^{\top} \in \mathbb{R}^{2n}
$$

is IOS with respect to the output $e(x, \hat{x}) = x - \hat{x}$ (the estimation error) and the input $u$.

3 | BILINEAR PERSIDSKII SYSTEMS

In this section, first, we define the class of systems of interest, and then, the motivation for studying such a complex
dynamics is provided.

**3.1 | Definition**

Let us consider the following class of systems:

$$
\dot{x}(t) = A_0 x(t) + \sum_{j=1}^{M} A_j f^j (H_j x(t)) + G w(t) + \sum_{i=1}^{n} x_i(t) \sum_{j=1}^{M} B_j^i f^j (H_j x(t)) + \sum_{j=1}^{n} x_j(t) D_j w(t), \quad (6)
$$

$$
y(t) = \begin{pmatrix} C_0 x(t) \\ C_1 f^1(H_1 x(t)) \\ \vdots \\ C_M f^M(H_M x(t)) \end{pmatrix}, \quad t \in \mathbb{R}_+,
$$

where $x(t) \in \mathbb{R}^n$ is the state, $w(t) \in \mathbb{R}^l$ is the unknown input with $w \in \mathcal{L}_\infty^l$, $y(t) \in \mathbb{R}^z$ is the measured output, $A_0 \in \mathbb{R}^{n \times n}$, $A_j \in \mathbb{R}^{n \times k_j}$, $H_j \in \mathbb{R}^{k_j \times n}$, $B_j^i \in \mathbb{R}^{n \times k_j}$, $C_j \in \mathbb{R}^{z_j \times k_j}$ and $G, D_i \in \mathbb{R}^{n \times l}$ for $i=1, \dots, n$ and $j=1, \dots, M$, $z = \sum_{s=0}^{M} z_s$, $f^j: \mathbb{R}^{k_j} \to \mathbb{R}^{k_j}$ for $j=1, \dots, M$ with the hypothesis that $f^j(s) = (f_1^j(s_1) \dots f_{k_j}^j(s_{k_j}))^\top$ for $s \in \mathbb{R}^{k_j}$ and

$$
\tau f_i^j(\tau) > 0, \text{ for every } \tau \neq 0 \text{ and } i = 1, \dots, k_j. \tag{7}
$$

For brevity of notation, we will use the convention $k_0 = n$, $H_0 = I_n$, $f^0(x) = x$, and $K = \sum_{j=1}^{M} k_j$.

After a suitable rearrangement of the nonlinearities $f_i^j$, we can consider $0 \le \mu \le m \le M$ be such that for all
$s = 1, \dots, m$ and $i = 1, \dots, k_s$, we have $\lim_{v \to \pm\infty} f_i^s(r) = \pm\infty$ and for every $s = 1, \dots, m$ and $i = 1, \dots, k_s$, we have
$\lim_{v \to \pm\infty} \int_0^v f_i^s(r) dr = +\infty$. Those indices ($\mu$ and $m$) will be useful in determining the radial unboundedness of the considered Lyapunov function and its time derivative.

The objective of this paper is to design a robust observer for the system (6) in the sense of Definition 5.

## 3.2 | Motivation for considering bilinear Persidskii systems

The system (6) includes the generalized Persidskii systems studied in [16], [17], [18] (the first line in (6)), with additional bilinear terms multiplied by the state components (the second line) [24], [23], [26], [22]. The motivation for considering this class is that it encapsulates some models of interest, which have been studied in the context of observer design and stabilization [20], [21].

**Example** The following model describes the bacterial growth of two distinct species inside a chemostat with a single limiting substrate [20]:

$$ \dot{x}_1 = (\mu_1(S) - D)x_1, \quad (8) $$

$$ \dot{x}_2 = (\mu_2(S) - D)x_2, $$

$$ \dot{S} = (S_{in} - S)D - \rho_1\mu_1(S)x_1 - \rho_2\mu_2(S)x_2, $$

where $x_i \ge 0$ is the concentration of the $i$th species ($i = 1, 2$), $S \ge 0$ is the concentration of the substrate, $S_{in} \ge 0$ is the nutrient inflow concentration, $\rho_1$ and $\rho_2$ are positive constants and the functions $\mu_1, \mu_2$ are given by $\mu_1(S) = \frac{a_1S}{b_1+S}$ and $\mu_2(S) = \frac{a_2S}{b_2+S}$ with $a_1, b_1, a_2, b_2$ positive parameters.

Setting $x_3$ as $S$, define the state vector $x(t) = (x_1(t) \quad x_2(t) \quad x_3(t))^{\mathrm{T}} \in \mathbb{R}_+^3$. Now, we reparametrize the input as $w = (D \quad u)^{\mathrm{T}}$, where $u = S_{in}D$, so we can see that (8) fits the model (6) for $n = 3$ and $M = 2$, by replacing $A_0 = A_1 = A_2 = 0_{3 \times 3}$, and $H_1 = H_2 = I_3$, $G = \begin{pmatrix} 0 & 0 \\ 0 & 0 \\ 0 & 1 \end{pmatrix}$, $D_1 = \begin{pmatrix} -1 & 0 \\ 0 & 0 \\ 0 & 0 \end{pmatrix}$, $D_2 = \begin{pmatrix} 0 & 0 \\ -1 & 0 \\ 0 & 0 \end{pmatrix}$, $D_3 = \begin{pmatrix} 0 & 0 \\ 0 & 0 \\ -1 & 0 \end{pmatrix}$, $B_1^1 = \begin{pmatrix} 0 & 0 & 1 \\ 0 & 0 & 0 \\ 0 & 0 & -\rho_1 \end{pmatrix}$, $B_2^2 = B_2^1 = B_3^1 = B_2^3 = 0_{3 \times 3}$, and $f^j(x) = (\mu_j(x_1) \quad \mu_j(x_2) \quad \mu_j(x_3))^{\mathrm{T}}$, $j = 1, 2$.

**Example** The following is a classical model of a chemical exothermic reactor [21]:

$$ \dot{X}_{in} = 0, $$

$$ \dot{X} = D(X_{in} - X) - k \exp\left(\frac{-E}{RT}\right) X, $$

$$ \dot{T} = D(T_{in} - T) + c \exp\left(\frac{-E}{RT}\right) X + w, $$

where $(X_{in}(t) \quad X(t) \quad T(t))^{\mathrm{T}} \in \mathbb{R}_+^3$ is the state, representing the inlet composition, the reactor composition and the temperature inside the reactor, respectively; $(E \quad R \quad k \quad c \quad D)^{\mathrm{T}} \in \mathbb{R}_+^5$ is the vector of constant parameters, and $w(t) \in \mathbb{R}$ is a nonnegative input.
Setting $x = (x_1 \ x_2 \ x_3)^T = (X_{in} \ X \ T)^T$, we have

$$ \dot{x} = (0 \ Dx_1 - Dx_2 \ -Dx_3)^T + x_2 (0 \ -k \exp(-\frac{E}{Rx_3}) \ c \exp(-\frac{E}{Rx_3}))^T + (0 \ 0 \ w)^T $$

As before, this system is obtained from (6) for $n = 3$ and $M = 1$, by replacing $H_1 = I_3$, $D_1 = D_2 = D_3 = 0$, $A_0 = \begin{pmatrix} 0 & 0 & 0 \\ D & -D & 0 \\ 0 & 0 & -D \end{pmatrix}$, $B_1^2 = \begin{pmatrix} 0 & 0 & 0 \\ 0 & 0 & -k \\ 0 & 0 & c \end{pmatrix}$, $G = \begin{pmatrix} 0 \\ 0 \\ 1 \end{pmatrix}$, $B_1^1 = B_1^3 = 0_{3 \times 3}$ and $f^1(x) = (\varphi(x_1) \ \varphi(x_2) \ \varphi(x_3))^T$, where $\varphi(x_i) = \exp(-\frac{E}{Rx_i})$.

**Example** The following system represents a type of consumer-resource interaction [19]:

$$ \begin{aligned} \dot{M}_1 &= M_1 \left( r_1 + c_1 \frac{M_2}{h_1 + M_2} + c_2 \frac{M_2}{h_2 + M_1} - c_3 M_1 \right) + w_1, \\ \dot{M}_2 &= M_2 \left( r_2 + c_4 \frac{M_1}{h_2 + M_1} + c_5 \frac{M_1}{h_1 + M_2} - c_6 M_2 \right) + w_2, \end{aligned} \quad (9) $$

where $M_1(t), M_2(t) \in \mathbb{R}_+$ are the population compartments, $r_1, r_2, h_1, h_2, c_i \in \mathbb{R}$ ($i = 1, \dots, 6$) and $w_1(t), w_2(t) \in \mathbb{R}_+$ are bounded inputs.

This system is also in the form (6) for $n = 2$ and $M = 2$, with $x = (M_1 \ M_2)^T$, $A_0 = \begin{pmatrix} r_1 & 0 \\ 0 & r_2 \end{pmatrix}$, $A_1 = 0_{2 \times 2}$, $B_1^1 = \begin{pmatrix} -c_3 & 0 \\ 0 & 0 \end{pmatrix}$, $B_1^2 = \begin{pmatrix} 0 & 0 \\ 0 & -c_6 \end{pmatrix}$, $B_2^1 = \begin{pmatrix} 0 & c_1 \\ 0 & c_5 \end{pmatrix}$, $B_2^2 = \begin{pmatrix} c_2 & 0 \\ c_4 & 0 \end{pmatrix}$, $f^1(M) = \begin{pmatrix} M_1 \\ M_2 \end{pmatrix}$, $f^2(M) = \begin{pmatrix} \frac{M_1}{h_2+M_1} \\ \frac{M_2}{h_1+M_2} \end{pmatrix}$, and $H_1 = H_2 = G = I_2$, other matrices are zero.

**Remark 1.** It is important to note that in all the examples of bilinear Persidskii systems (6), the state vector takes on non-negative values.

Beyond their practical interest, these systems exhibit intrinsic symmetry, which can be further exploited as concrete examples of the theory of equivariant systems [21], [20]. Though this analysis, we can understand the role of this type of symmetry in the design of simpler and more effective observers.

## 4 | OBSERVER DESIGN

The proposed observer for the system (6) contains a copy of the system with linear and nonlinear output injections:

$$ \begin{aligned} \dot{\hat{x}}(t) = A_0 \hat{x}(t) + \sum_{j=1}^{M} A_j f^j(H_j \hat{x}(t)) + \sum_{i=1}^{n} \hat{x}_i(t) \sum_{j=1}^{M} B_j^i f^j(H_j \hat{x}(t)) + L_0(y_0(t) - C_0 \hat{x}(t)) \\ + \sum_{j=1}^{M} L_j(y_j(t) - C_j f^j(H_j \hat{x}(t))) \\ + \sum_{i=1}^{n} \hat{x}_i(t) \sum_{j=1}^{M} R_j^i(y_j(t) - C_j f^j(H_j \hat{x}(t))), \end{aligned} \quad (10) $$

where $\hat{x}(t) \in \mathbb{R}^n$ is the observer state and the estimate of $x(t)$, and $L_0, L_j, R_j^i \in \mathbb{R}^{n \times z_j}$ for $j = 1, \dots, M$ and $i = 1, \dots, n$ are matrices to be designed.
Let us denote $e = x - \hat{x}$ and $\delta f^j(x, \hat{x}) = f^j(H_j x) - f^j(H_j \hat{x})$ (for brevity, we will use the notation $\delta f^j$), so the error dynamics is given by

$$ \dot{e} = (A_0 - L_0 C_0)e + \sum_{j=1}^{M} (A_j - L_j C_j)\delta f^j + \sum_{i=1}^{n} \sum_{j=1}^{M} B_j^i (x_i f^j(H_j x) - \hat{x}_i f^j(H_j \hat{x})) + Gw + \sum_{i=1}^{n} D_i x_i w - \sum_{i=1}^{n} \hat{x}_i \sum_{j=1}^{M} R_j C_j \delta f^j $$

The dynamics of the joint state variable $X = (x \ \hat{x})^T$ for systems (6) and (10) is given by the equation

$$ \dot{X} = \tilde{A}_0 X + \sum_{j=1}^{M} \tilde{A}_j F^j(\tilde{H}_j X) + \sum_{i=1}^{2n} X_i \sum_{j=1}^{M} B_j^i F_j(\tilde{H}_j X) + \tilde{G}w + \sum_{i=1}^{2n} X_i \tilde{D}_i w \quad (11) $$

where $X_j = x_j$ and $X_{n+j} = \hat{x}_j$, for $j = 1, \dots, n$ and

$$ \tilde{A}_j = \begin{pmatrix} A_j & 0_{n \times k_j} \\ L_j C_j & A_j - L_j C_j \end{pmatrix}, \quad \tilde{H}_j = \begin{pmatrix} H_j & 0_{k_j \times n} \\ 0_{k_j \times n} & H_j \end{pmatrix}, \quad \tilde{B}_j^i = \begin{pmatrix} B_j^i & 0_{n \times k_j} \\ 0_{n \times k_j} & 0_{n \times k_j} \end{pmatrix}, \quad \tilde{B}_j^{n+i} = \begin{pmatrix} 0_{n \times k_j} & 0_{n \times k_j} \\ R_j^i C_j & B_j^i - R_j^i C_j \end{pmatrix}, $$

$$ F^j(\tilde{H}_j X) = \begin{pmatrix} f^j(H_j x) \\ f^j(H_j \hat{x}) \end{pmatrix}, \quad \tilde{G} = \begin{pmatrix} G \\ 0 \end{pmatrix}, \quad \tilde{D}_i = \begin{pmatrix} D_i \\ 0 \end{pmatrix}, \quad \tilde{D}_{n+i} = 0 $$

for $j = 1, \dots, M$ and $i = 1, \dots, n$. Remark that the system (10) still belongs to the considered class of bilinear Persidskii systems, and that the error can be written as $e(X) = \Gamma X$, where $\Gamma = (I_n - I_n)$.

Now, consider the following Lyapunov function candidate:

$$ V(X) = X^\top P_\Gamma X + 2 \sum_{j=1}^{M} \sum_{i=1}^{2k_j} \Lambda_i^j \int_0^{\tilde{H}_j^i X} F_i^j(\tau) \, d\tau, \quad (12) $$

where $\tilde{H}_j^i$ is the $i$-th row of the matrix $\tilde{H}_j$, $P_\Gamma = \Gamma^\top P_1 \Gamma + P_2$, for matrices $P_1 \in \mathbb{R}^{n \times n}$, $P_2 \in \mathbb{R}^{2n \times 2n}$ and $\Lambda^j = \text{diag}(\Lambda_1^j, \dots, \Lambda_{2k_j}^j)$ to be defined. The time derivative of $V$ along the trajectories of the system (11) can be computed as follows:

$$ \begin{aligned} \dot{V}(X) ={}& \dot{X}^\top P_\Gamma X + X^\top P_\Gamma \dot{X} + 2 \sum_{j=1}^{M} \dot{X}^\top \tilde{H}_j^\top \Lambda^j F^j(\tilde{H}_j X) = X^\top (\tilde{A}_0^\top P_\Gamma + P_\Gamma \tilde{A}_0) X + \left( \sum_{j=1}^{M} F^j(\tilde{H}_j X)^\top \tilde{A}_j^\top \right. \\ & \left. + \sum_{i=1}^{2n} X_i \sum_{j=1}^{M} F^j(\tilde{H}_j X)^\top (\tilde{B}_i^j)^\top \right) P_\Gamma X + X^\top P_\Gamma \left( \sum_{j=1}^{M} \tilde{A}_j F^j(\tilde{H}_j X) + \sum_{i=1}^{2n} X_i \sum_{j=1}^{M} \tilde{B}_i^j F^j(\tilde{H}_j X) \right) + 2 \sum_{j=1}^{M} [X^\top \tilde{A}_0^\top \tilde{H}_j^\top \Lambda^j F^j(\tilde{H}_j X) \\ & + \left( \sum_{s=1}^{M} F^s(\tilde{H}_s X)^\top \tilde{A}_s^\top \right) \tilde{H}_j^\top \Lambda^j F^j(\tilde{H}_j X) + \left( \sum_{i=1}^{2n} X_i \sum_{s=1}^{M} F^s(\tilde{H}_s X)^\top (\tilde{B}_i^s)^\top \right) \tilde{H}_j^\top \Lambda^j F^j(\tilde{H}_j X) + (\tilde{G}w + \sum_{i=1}^{2n} X_i \tilde{D}_i w)^\top \tilde{H}_j^\top \Lambda^j F^j(\tilde{H}_j X)] \end{aligned} $$
$$+2X^{\mathrm{T}} P_{\mathrm{T}} \left( \tilde{G}w + \sum_{i=1}^{2n} x_i \tilde{D}_i w \right).$$

Regrouping terms, we can write $\dot{V}$ as a quadratic form acting on $Y = (X^{\mathrm{T}} \ F^1(\tilde{H}_1 X)^{\mathrm{T}} \ \dots \ F^M(\tilde{H}_M X)^{\mathrm{T}} \ w^{\mathrm{T}})^{\mathrm{T}}$:

$$\dot{V} = Y^{\mathrm{T}} \left( Q + \sum_{j=1}^{2n} x_j S^{(j)} \right) Y,$$

where the symmetric matrix $Q = Q^{\mathrm{T}} \in \mathbb{R}^{(2n+2K+\ell) \times (2n+2K+\ell)}$ is defined by the blocks $Q_{1,1} = \tilde{A}_0^{\mathrm{T}} P_{\mathrm{T}} + P_{\mathrm{T}} \tilde{A}_0$, $Q_{1,j+1} = P_{\mathrm{T}} \tilde{A}_j + \tilde{A}_0^{\mathrm{T}} \tilde{H}_j^{\mathrm{T}} \Lambda^j$, $Q_{1,M+2} = P_{\mathrm{T}} G$, $Q_{k+1,j+1} = \tilde{A}_k^{\mathrm{T}} \tilde{H}_j^{\mathrm{T}} \Lambda^j + \Lambda^k \tilde{H}_k \tilde{A}_j$, $Q_{M+2,j+1} = G^{\mathrm{T}} \tilde{H}_j^{\mathrm{T}} \Lambda^j$, and $Q_{M+2,M+2} = 0$ for $j, k = 1, \dots, M$. The symmetric matrices $S^{(1)}, \dots, S^{(2n)} \in \mathbb{R}^{(2n+2K+\ell) \times (2n+2K+\ell)}$ are defined by $S_{1,1}^{(i)} = 0$, $S_{1,j+1}^{(i)} = P_{\mathrm{T}} \tilde{B}_j^{\mathrm{T}}$, $S_{j+1,j+1}^{(i)} = 2 \tilde{B}_j^{\mathrm{T}} \tilde{H}_j \Lambda^j$, $S_{1,M+2}^{(i)} = P_{\mathrm{T}} \tilde{D}_i$, $S_{M+2,j+1}^{(i)} = D_i^{\mathrm{T}} \tilde{H}_j^{\mathrm{T}} \Lambda^j$, $S_{M+2,M+2}^{(i)} = 0$, for $i = 1, \dots, 2n$ and $j = 1, \dots, M$, and $S_{k+1,j+1}^{(i)} = (\tilde{B}_j^{\mathrm{T}})^{\mathrm{T}} \tilde{H}_j^{\mathrm{T}} \Lambda^j + \Lambda^k \tilde{H}_k \tilde{B}_j^{\mathrm{T}}$, $j = 1, \dots, M$, $k = 1, \dots, j-1$. Using this decomposition, we were able to show that (10) is an observer for trajectories staying near the origin:

**Theorem 3.** [27] Suppose that the system (6) is UBIBS and that there exist matrices $\Xi^j$, $Y_{0,j} \in \mathbb{D}_+^{2k_j}$ ($j = 1, \dots, M$), $\Psi \in \mathbb{D}_+^n$, $\Upsilon_{k,j} \in \mathbb{D}_+^{2n}$ ($k = 1, \dots, M-1$, $j = k+1, \dots, M$), $0 < \Phi = \Phi^{\mathrm{T}} \in \mathbb{R}^{2n \times 2n}$, $\Theta \in \mathbb{D}_+^{2n}$ and $\xi > 0$, $\rho > 0$ satisfying the inequalities

$$P_1 > 0 \text{ or } P_2^{11} - 2P_2^{12} + P_2^{22} - \rho P_1 > 0 \quad (13)$$

$$\text{or } \sum_{j=1}^{\mu} \tilde{\Lambda}^j + \rho P_1 > 0$$

where $P_2^{11}, P_2^{12}$, and $P_2^{22} \in \mathbb{R}^{n \times n}$ are such that $P_2 = \begin{pmatrix} P_2^{11} & P_2^{12} \\ P_2^{12} & P_2^{22} \end{pmatrix}$ and $\tilde{\Lambda}^j = \tilde{H}_j^{\mathrm{T}} \mathrm{diag}(\tilde{\Lambda}_1^j, \dots, \tilde{\Lambda}_{k_j}^j) \tilde{H}_j + \tilde{H}_j^{\mathrm{T}} \mathrm{diag}(\tilde{\Lambda}_{k_{j+1}}^j, \dots, \tilde{\Lambda}_{2k_j}^j) \tilde{H}_j$

$$P_1 \le \xi \Psi, \quad (14)$$

$$P_2 + \sum_{j=1}^{M} \tilde{H}_j^{\mathrm{T}} \Lambda^j \tilde{H}_j \le \xi \left( \sum_{k=0}^{m} \tilde{H}_k^{\mathrm{T}} \Xi^k \tilde{H}_k + 2 \sum_{r=1}^{m} \tilde{H}_r^{\mathrm{T}} \Upsilon_{0,r} \tilde{H}_r + 2 \sum_{s=1}^{m-1} \sum_{r=s+1}^{m} \tilde{H}_s^{\mathrm{T}} \tilde{H}_s \Upsilon_{s,r} \tilde{H}_r^{\mathrm{T}} \tilde{H}_r \right) \quad (15)$$

and such that

$$Q + \tilde{Q} < 0, \quad (16)$$

where $\tilde{Q}$ is the symmetric matrix defined by the blocks: $\tilde{Q}_{1,1} = \Gamma^{\mathrm{T}} \Psi \Gamma + \Xi^0$, $\tilde{Q}_{j+1,j+1} = \Xi^{j+1}, j = 1, \dots, M$, $\tilde{Q}_{k+1,j+1} = \tilde{H}_k \Upsilon_{k,j} \tilde{H}_j^{\mathrm{T}}$, $k = 1, \dots, M-1$, $j = k+1, \dots, M-1$, $\tilde{Q}_{1,j+1} = \tilde{H}_j^{\mathrm{T}} \Upsilon_{0,j}$, $j = 1, \dots, M$, $\tilde{Q}_{M+2,j} = 0$, $j = 1, \dots, M+1$, $\tilde{Q}_{M+2,M+2} = -\Phi$. Then there exists $\delta > 0$ such that the system (11) is IOS with respect to the output $e = x - \hat{x}$, for $\|X(0)\| \le \delta$ and $\|w\|_{\infty} < \delta$.

Note that for local analysis, the indexes $m$ and $\mu$ in the LMIs above can be substituted with $M$. Moreover, the size of the domain of validity of the design (the value of $\delta$) is regulated not by the linearization arguments, but by relation between the matrices $Q$ and $S^{(i)}$ for $i = 1, \dots, 2n$, hence, if the latter are sufficiently small comparing with the former, the value of $\delta$ can be large.

Now, we show that it can be made as a global observer by adding some new positivity assumptions, all of which can be put into linear matrix inequalities.
**Theorem 4.** Suppose that the system (6) is UBIBS, that $X_i(t) \ge 0$ for any initial condition $X_i(0) \ge 0$ and that $S^{(i)} \le 0$, for every $i = 1, ..., 2n$. Suppose also that there exist matrices $\Xi^j, \gamma_{0,j} \in \mathbb{D}_+^{2kj}$ ($j = 1, ..., M$), $\Psi \in \mathbb{D}_+^n, \gamma_{k,j} \in \mathbb{D}_+^{2n}$ ($k = 1, ..., M-1, j = k+1, ..., M$), $0 < \Phi = \Phi^\top \in \mathbb{R}^{2n \times 2n}$, $\Theta \in \mathbb{D}_+^{2n}$ and $\xi > 0, \rho > 0$ such that the LMIs (13)-(16) are verified. Then the system (11) is IOS with respect to the output $e = x - \hat{x}$.

**Proof** Consider the candidate Lyapunov function (12). The inequality (13) assures that if $V(X) = 0$ then $e(X) = \Gamma X = 0$ [17], which fulfills condition (2) with $h(X) = \Gamma X$. The upper bound $\alpha_2$ in (2) exists by the continuity of $V$, the sector property of the nonlinearities and due to all matrices are nonnegative. Also, (13) assures that the function $V$ is radially unbounded.

The combined hypotheses $Q + \tilde{Q} < 0, S^{(i)} \le 0$ and $X_i \ge 0$ ($i = 1, ..., 2n$) give us

$$Q + \tilde{Q} + \sum_{i=1}^{2n} X_i S^{(i)} \le 0. \quad (17)$$

Therefore, using (16) we can write

$$\dot{V} = Y^\top \left( Q + \tilde{Q} + \sum_{i=1}^{2n} X_i S^{(i)} \right) Y - Y^\top \tilde{Q} Y \le -Y^\top \tilde{Q} Y.$$

The inequalities (14), (15) make sure that

$$Y^\top \tilde{Q} Y \ge \alpha(V(X)) - w^\top \Phi w,$$

for some function $\alpha \in \mathcal{K}_\infty$. Thus $\dot{V} \le -\alpha(V(X)) + w^\top \Phi w$, which leads to $\dot{V} \le -\frac{1}{2}\alpha(V(X))$, for $V(X) \ge 2\alpha^{-1}(w^\top \Phi w)$, showing that $V$ is an IOS-Lyapunov function for the system. Hence, by Theorem 1 this system is IOS, and there exist related functions $\beta \in \mathcal{K}$ and $\gamma \in \mathcal{K}$ as in Definition 1. $\square$

The conditions given in the formulation of Theorem 4 are sufficient for the existence of a Lyapunov function that establishes convergence of the estimation error to zero in the disturbance-free case.

Note that the conditions of Theorems 3 and 4 are almost the same: smallness of initial conditions and inputs is replaced with negative definiteness of the matrices $S_i, i = 1, ..., 2n$ and nonnegativity of the state vector $X(t)$. The latter property should also be verified for the observer state; hence, it has to be guaranteed by the design.

**Example** If we consider Example 3.2 with parameters $r_1 = r_2 = -1$ and $C_0 = \begin{pmatrix} 1 & 0 \end{pmatrix}$, then the inequalities (15) and (16) become

$$P_2 + \Lambda^1 + \Lambda^2 \le \xi(\Xi^0 + 2(\gamma_{0,1} + \gamma_{0,2}) + \gamma_{1,2}),$$

and $\begin{pmatrix} \tilde{A}_0^\top P_\Gamma + P_\Gamma \tilde{A}_0 + \Xi^0 & \gamma_{0,1} \\ 2P_\Gamma^\top + \gamma_{0,1} & -\Phi \end{pmatrix} < 0$, which, together with (13) and (14), constitute a feasible system of inequalities. Hence, Theorem 4 can be applied, and the observer can be chosen in the form (10).

For simulations, all system parameters are set to 1, and we chose the gain matrix as $L_0 = \begin{pmatrix} 20.43 & -5.01 \end{pmatrix}^\top$, with $L_j = R_j^j = 0_{2 \times 2}$, for $i = 1, 2$ and $j = 1, 2$. Figs. 1 and 2 show the behavior of a trajectory of the system and the evolution of the error for $w_i(t) = 0.5 \sin(t)$ for $i = 1, 2$, illustrating that the error remains small for relatively small inputs and small initial conditions.
<figure><img src="image_4.png" /><figcaption>FIGURE 1 State trajectory $M(t)$</figcaption></figure>

<figure><img src="image_5.png" /><figcaption>FIGURE 2 Norm of the estimation error</figcaption></figure>

5 | SIOS ANALYSIS

In this section, we again suppose that the system (6) is nonnegative, meaning that given an initial condition $x(0)$ such that $x_i(0) \ge 0$ for every $i = 1, ..., n$ and positive inputs, the resulting trajectory also has nonnegative coordinates (i.e., $x_j(t) \ge 0$ for every $j = 1, ..., n$ and all $t \ge 0$). However, we would like to avoid the restriction of nonnegativity of the state of the observer $\hat{x}(t)$.

To this end, we also assume from now on that the functions $f^{j}, j = 1, ..., M$ are bounded (it is the case of the considered motivating examples 3.2-3.2).
The error dynamics with respect to the observer (10) can be rewritten as

$$ \dot{e} = (A_0 - L_0 C_0)e + \sum_{j=1}^{M} (A_j - L_j C_j)\delta f^j(x, \hat{x}) + v(x, \hat{x}) + \sum_{i=1}^{n} x_i \sum_{j=1}^{M} (B_j^i - R_j^i C_j)\delta f^j(x, \hat{x}) + Gw + \sum_{i=1}^{n} x_i D_i w \quad (18) $$

where

$$ v(x, \hat{x}) = \sum_{i=1}^{n} e_i \sum_{j=1}^{M} B_j^i f^j(H_j x) - \sum_{i=1}^{n} e_i \sum_{j=1}^{M} (B_j^i + R_j^i C_j) \delta f^j(x, \hat{x}), $$

so that we have the bound $||v(x, \hat{x})|| \le c||e||$ for some $c > 0$ (boundedness of $f^j$ implies the same property of $\delta f^j$). Following [17], we consider the function

$$ V(e) = e^T P e + 2 \sum_{j=1}^{M} \sum_{i=1}^{k_j} \int_{0}^{H_j^i e_i} f_j^i(\tau) d\tau \quad (19) $$

where $H_j^i$ is the $i$th row of the matrix $H_j$, as the SIIOS-Lyapunov function candidate for the system (18). Its derivative is computed as

$$ \dot{V}(e) = \dot{e}^T P e + e^T P \dot{e} + 2 \dot{e}^T \sum_{j=1}^{M} H_j^\top \Lambda^j f^j(H_j e). $$

Substituting $\dot{e}$ by its expression and rearranging terms, we have

$$ \begin{aligned} \dot{V}(e) = e^T & [(A_0 - L_0 C_0)^T P + P(A_0 - L_0 C_0)]e + 2e^T P v + \sum_{j=1}^{M} (\delta f^j)^T (A_j - L_j C_j)^T P e + \sum_{j=1}^{M} e^T P (A_j - L_j C_j) \delta f^j \\ & + 2 \sum_{k=1}^{M} [e^T (A_0 - L_0 C_0) H_k \Lambda^k f^k(H_k e) + v^T H_k \Lambda^k f^j(H_k e)] + 2 \sum_{j=1}^{M} \sum_{k=1}^{M} (\delta f^j)^T (A_j - L_j C_j)^T H_k^\top \Lambda^k f^k(H_k e) \\ & + w^T G^T P e + e^T P G w + 2 \sum_{k=1}^{M} w^T G^T H_k^\top \Lambda^k f^k(H_k e) \\ & + \sum_{j=1}^{n} x_i \left[ (\delta f^j)^T (B_j^i - R_j^i C_j)^T P e + e^T P (B_j^i - R_j^i C_j) \delta f^j + 2 \sum_{k=1}^{M} \sum_{j=1}^{M} (\delta f^j)^T (B_j^i - R_j^i C_j)^T H_k^\top \Lambda^k f^k(H_k e) \right] \end{aligned} $$

Despite its long expression, $\dot{V}$ can be written as a quadratic form:

$$ \dot{V} = W^\top \left( Q + \sum_{i=1}^{n} x_i T_i \right) W, $$

where $W = [e^T (\delta f^1)^T \dots (\delta f^M)^T f^1(H_1 e)^T \dots f^M(H_M e)^T v^T w^T]^T$ and to write $Q$ and $T_i$ simply, we introduce the following useful notations $\mathcal{A}_j = (A_j - L_j C_j), j = 0, \dots, M$, $\mathcal{A} = \text{diag}(\mathcal{A}_1, \dots, \mathcal{A}_M)$, $\Lambda = [H_1^\top \Lambda^1 \dots H_M^\top \Lambda^M]^T$, $B_k^i = B_k^i - R_k^i C_k$; $\mathcal{B}_i = \text{diag}(B_1^i, \dots, B_M^i)$. Then we can now write $Q$ and $T_i$ as block matrices $Q = Q^T = (Q_{a,b})^S_{a,b=1}$, where

$$ Q_{1,1} = \mathcal{A}_0^T P + P \mathcal{A}_0; \quad Q_{1,2} = P \mathcal{A}; \quad Q_{1,3} = \mathcal{A}_0 \Lambda; \quad Q_{1,4} = P; \quad Q_{1,5} = P G; \quad Q_{2,3} = \mathcal{A}^T \Lambda; \quad Q_{3,4} = \Lambda Q_{3,5} = G^T \Lambda; $$
and $T_i = T_i^\top = (T_i^{a,b})_{a,b=1}^5$, with $T_i^{1,2} = P\mathcal{B}_i$, $T_i^{1,5} = P\mathcal{D}_i$, $T_i^{2,3} = (\mathcal{B}_i\Lambda + \Lambda\mathcal{B}_i^\top)/2$, $T_i^{3,5} = (D_i^\top\Lambda + \Lambda D_i)/2$. The indices a, b not mentioned above correspond to zero matrices of their respective block size.

Before stating and proving the main result, we need the following assumptions for the nonlinearities in the system:

**Assumption 1.** Assume $(x - \hat{x})^\top H_j^\top \delta f^j(x, \hat{x}) > 0$ for all $x \neq \hat{x} \in \mathbb{R}^n$.

Denote by $\mathcal{M}(H_z, H_s) \subset \mathbb{R}^{kz \times ks}$ a set of matrices, whose ij-th element is $i$) nonnegative if $H_z^i$ and $H_s^i$ are co-directional, $ii$) nonpositive if $H_z^i$ and $H_s^i$ are anti-directional, and $iii$) zero otherwise, for $i = \overline{1}, k_z, j = \overline{1}, k_s, z, s = 1, \dots, M$.

**Assumption 2.** There exist matrices $S_0^j, S_1^j, S_2^j, \Sigma_0^j, \Sigma_1^j, \Sigma_2^j \in \mathbb{D}_+^{kj}$ and $S_3^{j,q}, \Sigma_3^{j,q} \in \mathcal{M}(H_j, H_q)$, for $j, q = 1, \dots, M$ such that

$$ (\delta f^j)^\top \delta f^j \le e^\top H_j^\top S_0^j H_j e + 2e^\top H_j^\top S_1^j (\delta f^j) + 2e^\top H_j^\top S_2^j f^j(H_j e) + 2 \sum_{q=1}^M (\delta f^j)^\top S_3^{j,q} f^q(H_q e); \\ f^j(H_j e)^\top f^j(H_j e) \le e^\top H_j^\top \Sigma_0^j H_j e + 2e^\top H_j^\top \Sigma_1^j (\delta f^j) + 2e^\top H_j^\top \Sigma_2^j f^j(H_j e) + 2 \sum_{q=1}^M (\delta f^j)^\top \Sigma_3^{j,q} f^q(H_q e) $$

for all $x, \hat{x} \in \mathbb{R}^n$ with $e = x - \hat{x}$.

The former requires incremental sector (incremental passivity) property to be verified for the nonlinearities in the system (i.e., they should be monotonously growing). In particular, this property implies that integrals of all nonlinearities become unbounded, as well as the products $sf_j^j(s)$ or $(s - \hat{s})\delta f^j(s, \hat{s})$ for $s, \hat{s} \to \pm\infty$ with $j = 1, \dots, M$ and $i = 1, \dots, k_j$. The latter assumption is a quadratic constraint upper bound on the nonlinearities.

**Theorem 5.** Suppose that the system (6) is forward complete, that Assumptions 1 and 2 hold, that the state trajectory is nonnegative with $x_i(t) \ge 0$ for all $t \ge 0$ and that $T_i \le 0$, for $i = 1, \dots, n$. Suppose that there exist $\rho \in \mathbb{R}$, $\eta, \gamma, \xi, d > 0$, matrices $0 \le P = P^\top \in \mathbb{R}^{n \times n}$, $\Xi^0 \in \mathbb{D}_+^n$, $\Lambda^j = \text{diag}(\Lambda_1^j, \dots, \Lambda_n^j)$, $\Gamma_j, \Omega_j \in \mathbb{D}_+^{kj}$, $\gamma_{j,k} \in \mathcal{M}(H_j, H_k)$, $j, k = 1, \dots, M$ such that

$$ P + \rho \sum_{j=1}^{M} H_j^\top \Lambda^j H_j > 0; \quad (20) $$

$$ \Xi^0 - c^2 \xi I_n - \gamma \sum_{j=1}^{M} H_j^\top S_0^j H_j - \eta \sum_{j=1}^{M} H_j \Sigma_0^j H_j \ge 0; \quad (21) $$

$$ \Gamma_j - \gamma S_1^j - \eta \Sigma_1^j \ge 0; \quad (22) $$

$$ \Omega_j - \gamma S_2^j - \eta \Sigma_2^j \ge 0; \quad (23) $$

$$ \gamma_{j,k} - \gamma S_3^{j,k} - \eta \Sigma_3^{j,k} \ge 0; \quad (24) $$

$$ \Xi^0 - c^2 \xi I_n - \gamma \sum_{j=1}^{M} H_j^\top S_0^j H_j - \eta \sum_{j=1}^{M} H_j \Sigma_0^j H_j + \sum_{j=1}^{M} H_j^\top (\Gamma_j - \gamma S_1^j - \eta \Sigma_1^j) H_j + \sum_{j=1}^{M} H_j^\top (\Omega_j - \gamma S_2^j - \eta \Sigma_2^j) H_j \\ + \sum_{j=1}^{M} \sum_{k=1}^{M} H_j^\top (\gamma_{j,k} - \gamma S_3^{j,k} - \eta \Sigma_3^{j,k}) H_k > 0 \quad (25) $$

and a matrix $\tilde{Q} = \tilde{Q}^\top = (Q_{a,b})_{a,b=1}^5$ such that

$$ \tilde{Q} < 0 \quad (26) $$

defined by $\tilde{Q}_{1,1} = \Xi^0$, $\tilde{Q}_{2,2} = -\gamma I_{nM}$, $\tilde{Q}_{3,3} = -\eta I_{nM}$, $\tilde{Q}_{4,4} = -\xi I_n$, $\tilde{Q}_{5,5} = -dI_n$, $\tilde{Q}_{1,2} = \bar{\Gamma}$, $\tilde{Q}_{1,3} = \Omega$, $\tilde{Q}_{2,3} = \Upsilon$, where the $\tilde{Q}_{i,j}$ with omitted indices are equal to zero, $\bar{\Gamma} = [H_1^\top \Gamma_1 \dots H_M^\top \Gamma_M]$, $\Omega = [H_1^\top \Omega_1 \dots H_M^\top \Omega_M]$. Then the system (6) with the observer (10) is SIIOS with respect to the output $e(x, \hat{x}) = x - \hat{x}$.
**Proof** Since $Q + \tilde{Q} < 0$, $T_i \le 0$ and $x_i(t) \ge 0$, we have

$$Q + \tilde{Q} + \sum_{i=1}^{n} x_i T_i < 0.$$

Considering $V$ to be as defined in (19), which is a positive definite and radially unbounded function of $e$ due to (20) with Finsler's lemma, and following the computations thereafter, we conclude that

$$\dot{V} = W^{\top}(Q + \sum_{i=1}^{n} x_i T_i)W = W^{\top}(Q + \tilde{Q} + \sum_{i=1}^{n} x_i T_i)W - W^{\top} \tilde{Q} W \le -W^{\top} \tilde{Q} W,$$

where the last inequality holds due to hypothesis (26). Writing down the full expression for the last term we have

$$\dot{V} \le -e^{\top} \Xi^{0} e + \gamma \sum_{j=1}^{M} (\delta f^{j})^{\top} \delta f_{j} + \xi v^{\top} v + dw^{\top} w - \sum_{j=1}^{M} e^{\top} H_{j} \Gamma_{j} \delta f^{j} - \sum_{j=1}^{M} e^{\top} H_{j} \Omega_{j} f^{j}(H_{j} e) - \sum_{j=1}^{M} \sum_{k=1}^{M} (\delta f^{j})^{\top} \Upsilon_{j,k} f^{k}(H_{k} e)$$

due to assumptions 1, 2 and the existence of the upper bound on $v$ in terms of the estimation error, we have the estimate:

$$\dot{V} \le -e^{\top}(\Xi^{0} - c^{2}\xi I_{n} - \gamma \sum_{j=1}^{M} H_{j}^{\top} S_{0}^{j} H_{j} - \eta \sum_{j=1}^{M} H_{j} \Sigma_{0}^{j} H_{j})e - 2 \sum_{j=1}^{M} e^{\top} H_{j}^{\top}(\Gamma_{j} - \gamma S_{1}^{j} - \eta \Sigma_{1}^{j})\delta f^{j} \\ -2 \sum_{j=1}^{M} e^{\top} H_{j}^{\top}(\Omega_{j} - \gamma S_{2}^{j} - \eta \Sigma_{2}^{j})f^{j}(H_{j} e) - 2 \sum_{j=1}^{M} \sum_{k=1}^{M} (\delta f^{j})(\Upsilon_{j,k} - \gamma S_{3}^{j,k} - \eta \Sigma_{3}^{j,k})f^{k}(H_{k} e) + dw^{\top} w,$$

where each term is nonnegative due to (21)–(24), and $-\dot{V} + dw^\top w$ can be lower bounded by a class $\mathcal{R}_\infty$ function of the error $e$ following the inequality (25). Therefore, $V$ is an SIIOS Lyapunov function for the system (6), (10), which possesses the unboundedness observability property that can replace the forward completeness requirement. By Theorem 2 the required conclusion follows. □

Compared with Theorem 4, here it is not required that the observer state be nonnegative, simplifying the verification of the conditions and design, and the UBIBS property of (6) is reduced to forward completeness.

**Example** Let us consider Example 3.2 with $C_0 = (1 \ 0)$. We have $n = 3$, $M = 1$, inequality (20) becomes $P + \rho \Sigma_0^j > 0$ and the inequality (26) becomes

$$\left( \begin{array}{ccccc} \Xi^0 + \mathcal{A}_0^\top P + P \mathcal{A}_0 & \Gamma_1 & \mathcal{A}_0 \Lambda^1 + \Omega^1 & P & PG \\ \Gamma_1 & -\gamma I_3 & \gamma I_1 & 0 & 0 \\ -\Lambda^1 \mathcal{A}_0^\top + \Omega^1 & \gamma I_1 & -\eta I_3 & \Lambda^1 & G^\top \Lambda^1 \\ P & 0 & \Lambda^1 & -cI_3 & 0 \\ G^\top P & 0 & \Lambda^1 G & 0 & -dI_3 \end{array} \right) < 0$$

Assuming all parameters in the model are 1 and adding to that the inequalities $T_i \le 0$ for $i = 1, 2, 3$, we obtain the gains $L_0 = [1 \ 0]^{\top}$, $R_1^2 = [-0.4439 \ -0.4315 \ 0.3351]^{\top}$ through solving the LMIs from Theorem 5. Figs. 3 and 4 show a state trajectory example and the estimation norm behavior for initial conditions $x_0 = [1 \ 2]^{\top}$ and input $w(t) = 0.2+0.1 \cos(t)$.
<figure><img src="image_6.png" /><figcaption>**FIGURE 3** Trajectory of the chemical reactor dynamic model for $w(t) = 0.2 + 0.1 \cos(t)$ and initial conditions $x_0 = [1 \ 2 \ 3]^T$</figcaption></figure>

<figure><img src="image_7.png" /><figcaption>**FIGURE 4** Norm of estimation error for the trajectory with gains $L_0 = [1 \ 0 \ 0]^T$, $R_1^2 = [-0.4439 \ -0.4315 \ 0.3351]^T$ and $\hat{x}_0 = [5 \ 5 \ 5]^T$</figcaption></figure>

## 6 | CONCLUSION

In this paper, we introduced a new class of systems that generalizes the Persidskii dynamics while incorporating bilinear cross-product terms. This extension of the model is essential in many applications based on the mass-balance principle. For this class of systems, we proposed a state estimator using the concepts of IOS and SIIOS. The efficiency of this estimator is illustrated through examples, including a consumer-resource interaction model and a chemical exothermic reactor. We formulated conditions for the observer's applicability in terms of linear matrix inequalities, which is an advancement considering the system's complexity. Moreover, one of the observer designs, based on the nonnegativity of the plant's state, offers a global solution, a rare achievement for bilinear systems.
references

[1] Fossen TI, Nijmeijer H. New Directions in Nonlinear Observer Design. Springer; 1999.

[2] Matveev AS, Savkin AV. Estimation and Control over Communication Networks. Birkhäuser Boston Basel Berlin; 2009.

[3] Bernard P, Andrieu V, Astolfi D. Observer design for continuous-time dynamical systems. Annual Reviews in Control 2022;53:224-248.

[4] Khalil HK. High-Gain Observers in Nonlinear Feedback Control. Advances in Design and Control, Philadelphia: SIAM; 2017.

[5] Astolfi D, Marconi L, Praly L, Teel AR. Low-power peaking-free high-gain observers. Automatica 2018;98:169-179.

[6] Levant A. Robust exact differentiation via sliding modes techniques. Automatica 1998;34(3):379-384.

[7] Reichhartinger M, Efimov D, Fridman L. Special issue on differentiators. International Journal of Control 2018;91(9):1980-1982.

[8] Mojallizadeh MR, Brogliato B, Polyakov A, Selvarajan S, Michel L, Plestan F, et al. A survey on the discrete-time differentiators in closed-loop control systems: Experiments on an electro-pneumatic system. Control Engineering Practice 2023;136:105546.

[9] Sontag ED. The ISS philosophy as a unifying framework for stability-like behavior. In: Nonlinear control in the year 2000, Vol. 2 (Paris), vol. 259 of Lecture Notes in Control and Inform. Sci. London: Springer; 2001.p. 443-467.

[10] Dashkovskiy SN, Efimov DV, Sontag ED. Input to state stability and allied system properties. Automation and Remote Control 2011;72(8):1579-1614.

[11] Sontag E, Wang Y. Lyapunov characterizations of input to output stability. SIAM Journal on Control and Optimization 2000;39(1):226-249.

[12] Barbashin EA. On construction of Lyapunov functions for nonlinear systems. In: Proc. 1st IFAC World Congress Moscow; 1961. p. 742-751.

[13] Persidskii SK. Problem of absolute stability. Automation and Remote Control 1969;12:1889-1895.

[14] Yakubovich VA. Popov's Method and its Subsequent Development. European Journal of Control 2002;8(3):200-208.

[15] Liberzon M. Essays on the absolute stability theory. Autom Remote Control 2006;67:1610-1644.

[16] Efimov D, Aleksandrov A. On analysis of Persidskii systems and their implementations using LMIs. Automatica 2021;134:109905.

[17] Mei W, Ushirobira R, Efimov D. On nonlinear robust state estimation for generalized Persidskii systems. Automatica 2022;142:110411.

[18] Mei W, Efimov D, Ushirobira R, Aleksandrov A. On convergence conditions for generalized Persidskii systems. International Journal of Robust and Nonlinear Control 2022;32(6):3696-3713.

[19] Holland JN, DeAngelis DL. A consumer-resource approach to the density-dependent population dynamics of mutualism. Ecology 2010;91(5):1286-1295.

[20] de Souza AR, Efimov D, Polyakov A, Gouzé JL. Robust stabilization of competing species in the chemostat. Journal of Process Control 2020;87:138-146.

[21] Bonnabel S, Martin P, Rouchon P. Symmetry-preserving observers. IEEE Transactions on Automatic Control 2008;53(11):2514-2526.
[22] Hara S, Furuta K. Minimal order state observers for bilinear systems. International Journal of Control 1976;24(5):705-718.

[23] Adachi A, Funahashi Y. Existence of bilinear observers for bilinear systems. Information Sciences 1979;19(1):67-80.

[24] Derese I, Stevens P, Noldus E. Observers for bilinear systems with bounded input. International Journal of Systems Science 1979;10(6):649-668.

[25] Biedermann B, Meurer T. Observer design for a class of nonlinear systems combining dissipativity with interconnection and damping assignment. International Journal of Robust and Nonlinear Control 2021;31(9):4064-4080.

[26] Reza-Rahmati M, Flores G. A nonlinear observer for bilinear systems in block form. European Journal of Control 2023;70:100780.

[27] Lima DR, Ushirobira R, Efimov D. State observer design for bilinear Persidskii systems. In: European Control Conference (ECC24); 2024.