# Observer design for parameter-varying Persidskii systems using parameter-independent Lyapunov functions

Denis Efimov, Christophe Combastel, Ali Zolghadri

*Abstract*—This paper addresses the state observer design for parameter-varying Persidskii systems. These systems exhibit dynamics that depend both on time-varying parameters, as in the Linear Parameter Varying (LPV) framework, and sector nonlinearities of the state, as in classical Persidskii systems. A state observer design is proposed, and its stability is analyzed using parameter-independent Lyapunov functions. The conditions for tuning the observer gains are derived within the input-to-output stability framework. They are first expressed as parameterized matrix inequalities, and then further reduced to linear ones under additional mild assumptions. The effectiveness of the proposed design is shown through an academic example.

*Index Terms*—Persidskii systems, State observation, LPV, Input-to-output stability, LMIs

## I. INTRODUCTION

The design of state observers is a widely studied problem in control theory [1], [2]. The solution to this problem typically involves three main steps: (i) structural design of the observer, (ii) stability and performance analysis of the estimation error, and (iii) gain tuning. In the first step, the observer structure is usually chosen as a copy of the system with the output injection. In the second step, after deriving the estimation error dynamics, various analysis methods are applied. Among these, the input-to-state stability (ISS) framework [3], [4] is a widely accepted approach for analyzing stability in the presence of disturbances, noise, and parasitic delays. Real-world system models often incorporate these types of uncertainties, making robustness analysis essential. In the final step, the observer gains can be selected as either constant values, optimized for specific performance criteria, or time-varying, as in the case of the Kalman filter.

For linear models, the established theory offers a rich palette of analysis and design methods [5]. Extensions to nonlinear systems emphasize various canonical representations, which include, among others, the linear parameter-varying (LPV) systems [6], [7], homogeneous dynamics [8], Lur'e models [9], [10]. A specific case of the latter is given by Persidskii systems. These systems benefit from

existence of a nominal Lyapunov function form, which combines a quadratic state term with a weighted sum of integrals of nonlinearities. Applying this idea, in a series of recent papers, various mathematical properties of generalized Persidskii systems were investigated from design and analysis perspectives [11]–[13]. In [14], a combination of the LPV framework and Persidskii systems has been proposed, where the scheduling parameters can represent additional time or state-varying terms (e.g., parametric or signal uncertainty), while the useful (e.g., passive or sector negative feedback) nonlinearities are preserved explicitly in the model. This class of nonlinear dynamics is studied here, and it is referred to as the parameter-varying Persidskii (PVP) systems.

For PVP systems, the nonlinearity is preserved, while (possibly uncertain) scheduling parameters appear in the matrices multiplying the state and the nonlinearities. The work in [14] focused on ISS analysis and control design for PVP models, assuming that the vector of scheduling parameters is not measured, but the set of its admissible values is given. In this paper, a solution is proposed for the problem of observer design for PVP systems under the condition that the vector of scheduling parameters is accessible, and the dynamics are influenced by external disturbances and measurement noises.

The paper is organized as follows: Section II reviews preliminaries on input-to-output stability theory and Persidskii systems. Section III introduces the problem statement. The observer structure is proposed in Section IV. Parameter-independent Lyapunov functions are investigated in Section V. Section VI demonstrates the efficiency of the synthesized state observer on an academic example.

### Notation

* $\mathbb{R}_+ = \{x \in \mathbb{R} : x \ge 0\}$, where $\mathbb{R}$ is the set of real numbers.

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

* For a (Lebesgue) measurable function $d: \mathbb{R}_+ \to \mathbb{R}^m$ and $[t_0, t_1) \subset \mathbb{R}_+$, we define the norm $\|d\|_{[t_0, t_1)} = \ess\sup_{t \in [t_0, t_1]} \|d(t)\|$. Then, $\|d\|_{\infty} = \|d\|_{[0, +\infty)}$ and the set of such functions $d$ satisfying the property $\|d\|_{\infty} < +\infty$ is further denoted as $\mathcal{L}_{\infty}^m$, the set of essentially bounded measurable functions.

Denis Efimov is with Inria, Univ. Lille, CNRS, UMR 9189 - CRISTAL, F-59000 Lille, France.

Christophe Combastel and Ali Zolghadri are with Univ. Bordeaux, CNRS, Bordeaux INP, IMS, UMR 5218, F-33400 Talence, France. Ali Zolghadri is also a senior member of "Institut Universitaire de France".
* $DV(x)f(x)$ stands for the directional derivative of a continuously differentiable function $V(x)$ with respect to the vector field $f$ evaluated at the point $x$.

* A finite series of integers $1, 2, ..., n$ is denoted as $\overline{1, n}$, and $\{\overline{1, n}\} = \{1, 2, ..., n\}$.

* $I_n$ is the identity matrix of dimension $n \times n$.

* $\text{diag}\{g\} \in \mathbb{D}_+^n$ represents a diagonal matrix of dimension $n \times n$ with a vector $g \in \mathbb{R}_+^n$ on the main diagonal, and $\mathbb{D}_+^n \subset \mathbb{R}^{n \times n}$ is the set of such matrices.

* For a matrix $A \in \mathbb{R}^{n \times n}$, $A^{(i)}$ and $A^{[i]}$ denote its $i$th row and column, respectively, for $i = \overline{1, n}$.

* The relation $P \prec 0$ ($P \preceq 0$) means that a symmetric matrix $P \in \mathbb{R}^{n \times n}$ is negative (semi-)definite.

* A continuous function $\alpha : \mathbb{R}_+ \to \mathbb{R}_+$ belongs to the class $\mathcal{K}$ if $\alpha(0) = 0$ and it is strictly increasing. A function $\alpha : \mathbb{R}_+ \to \mathbb{R}_+$ belongs to the class $\mathcal{K}_\infty$ if $\alpha \in \mathcal{K}$ and it is increasing to infinity. A continuous function $\beta : \mathbb{R}_+ \times \mathbb{R}_+ \to \mathbb{R}_+$ belongs to the class $\mathcal{KL}$ if $\beta(\cdot, t) \in \mathcal{K}$ for each fixed $t \in \mathbb{R}_+$ and $\beta(s, \cdot)$ is decreasing to zero for each fixed $s > 0$.

## II. PRELIMINARIES

By convention, it is assumed that if the upper bound of a summation or a sequence is smaller than the lower one, then the corresponding terms (conditions) have to be omitted.

**Lemma (Finsler's lemma).** [15] Let $x \in \mathbb{R}^n \setminus \{0\}$ and $P, R \in \mathbb{R}^{n \times n}$ are symmetric, then $x^\top P x \prec 0$ whenever $x^\top R x = 0$ if and only if there exists $\rho \in \mathbb{R}$ such that $P - \rho R \prec 0$.

**Lemma 1.** [16] For any $\mathfrak{p} > 1$, such that $\mathfrak{p} = \frac{p_0}{p_1}$ where $p_0$, $p_1$ are odd integers, and $a_{\mathfrak{p}} \ge 2^{p-1}$,

$$|s - s'|^{\mathfrak{p}} \le a_{\mathfrak{p}} |s^{\mathfrak{p}} - s'^{\mathfrak{p}}|, \forall s, s' \in \mathbb{R}.$$

### A. Input-to-output stability

Consider a dynamical system represented by a model,

$$\begin{aligned}
\dot{x}(t) &= f(x(t), d(t)), \quad t \ge 0, \\
y(t) &= h(x(t)),
\end{aligned}
\quad (1)$$

where $x(t) \in \mathbb{R}^n$ is the state vector, $d(t) \in \mathbb{R}^s$ is the vector of external inputs, $d \in \mathcal{L}_\infty^s$, and $y(t) \in \mathbb{R}^p$ is the output; $f: \mathbb{R}^n \times \mathbb{R}^s \to \mathbb{R}^n$ is such that the existence and uniqueness of solutions of (1) in forward time is guaranteed; $h: \mathbb{R}^n \to \mathbb{R}^p$ is a continuously differentiable function; $f(0, 0) = 0$ and $h(0) = 0$. The unique maximal solution of the system (1) for an initial condition $x_0 \in \mathbb{R}^n$ and $d \in \mathcal{L}_\infty^s$ is denoted by $X(t, x_0, d)$ for those $t \ge 0$ for which the solution exists; then $Y(t, x_0, d) = h(X(t, x_0, d))$.

The system (1) is called forward complete if for any $x_0 \in \mathbb{R}^n$ and $d \in \mathcal{L}_\infty^s$ the solution $X(t, x_0, d)$ exists for all $t \ge 0$ (i.e., the trajectories do not blow up in a finite time, sufficient conditions of forward completeness can be found in [17]).

**Definition 1.** [18] A forward complete system (1) is input-to-output stable (IOS) if there exist $\beta \in \mathcal{KL}$ and $\gamma \in \mathcal{K}$ such that the inequality

$$\|Y(t, x_0, d)\| \le \beta(\|x_0\|, t) + \gamma(\|d\|_{[0,t)}), \forall t \ge 0$$

holds for all $x_0 \in \mathbb{R}^n$ and $d \in \mathcal{L}_\infty^s$. This system is state-independent input-to-output stable (SIIOS) if this estimate can be strengthened to the inequality

$$\|Y(t, x_0, d)\| \le \beta(\|h(x_0)\|, t) + \gamma(\|d\|_{[0,t)}), \forall t \ge 0$$

being satisfied for all $x_0 \in \mathbb{R}^n$ and $d \in \mathcal{L}_\infty^s$.

**Definition 2.** [18] A system (1) is uniformly bounded input bounded state stable (UBIBS) if it is forward complete and there exists $\sigma \in \mathcal{K}$ such that

$$\|X(t, x_0, d)\| \le \max\{\sigma(\|x_0\|), \sigma(\|d\|_{[0,t)})\}, \forall t \ge 0$$

for all $x_0 \in \mathbb{R}^n$ and $d \in \mathcal{L}_\infty^s$.

The properties introduced in Definition 1 can be effectively established using the Lyapunov function (LF) method.

**Definition 3.** [18] A smooth function $V: \mathbb{R}^n \to \mathbb{R}_+$ is called IOS-LF if for some $\alpha_1, \alpha_2 \in \mathcal{K}_\infty$, $\chi \in \mathcal{K}$ and $\alpha_3 \in \mathcal{KL}$:

$$\alpha_1(\|h(x)\|) \le V(x) \le \alpha_2(\|x\|), \forall x \in \mathbb{R}^n;$$

$$V(x) \ge \chi(\|d\|) \Rightarrow$$

$$DV(x)f(x, d) \le -\alpha_3(V(x), \|x\|), \forall x \in \mathbb{R}^n, \forall d \in \mathbb{R}^s.$$

Moreover, $V$ is called SIIOS-LF if for some $\alpha_1, \alpha_2 \in \mathcal{K}_\infty$, $\chi \in \mathcal{K}$ and $\alpha_4 \in \mathcal{K}$:

$$\alpha_1(\|h(x)\|) \le V(x) \le \alpha_2(\|h(x)\|), \forall x \in \mathbb{R}^n;$$

$$V(x) \ge \chi(\|d\|) \Rightarrow$$

$$DV(x)f(x, d) \le -\alpha_4(V(x)), \forall x \in \mathbb{R}^n, \forall d \in \mathbb{R}^s.$$

**Theorem 1.** [18] A UBIBS system (1) with locally Lipschitz continuous $f$ is IOS/SIIOS if and only if it admits an IOS/SIIOS-LF.

The above Lyapunov conditions have been extended in [19] to the case of non-UBIBS systems. For SIIOS case, UBIBS property can be replaced by forward completeness. For the sufficiency of existence of these LFs for IOS/SIIOS properties, the local Lipschitzness of $f$ can be omitted.

### B. Parameter-varying Persidskii systems

Consider the following class of systems [14]:

$$\dot{x}(t) = A_0(\theta(t))x(t) + \sum_{j=1}^{M} A_j(\theta(t))f^j(H_j x(t)) + d(t) \quad (2) \\ +B(\theta(t))u(t),$$

$$y(t) = \begin{bmatrix} C_0 x(t) \\ C_1 f^1(H_1 x(t)) \\ \vdots \\ C_M f^M(H_M x(t)) \end{bmatrix} + v(t), \quad t \ge 0,$$
where $x(t) = [x_1(t) \dots x_n(t)]^\top \in \mathbb{R}^n$ is the state vector, $x(0) \in \mathbb{R}^n$; $u(t) \in \mathbb{R}^m$ is a known control input, $u \in \mathcal{L}_\infty^m$; $y(t) = [y_0^\top(t) \dots y_M^\top(t)] \in \mathbb{R}^p$ is the output signal, $p = \sum_{g=0}^M p_g$, $y_g(t) \in \mathbb{R}^{p_g}$ and $C_g \in \mathbb{R}^{p_g \times k_g}$ for $g = 0, \bar{M}$, which is available for measurements and containing the perturbation $v(t) = [v_0^\top(t) \dots v_M^\top(t)] \in \mathbb{R}^p$, $v \in \mathcal{L}_\infty^p$; $d(t) \in \mathbb{R}^n$ is the external disturbance, $d \in \mathcal{L}_\infty^n$; $\theta(t) \in \mathbb{R}^q$ is the vector of time-varying scheduling parameters, $\theta \in \mathcal{L}_\infty^q$; $f^j: \mathbb{R}^{k_j} \to \mathbb{R}^{k_j}$ with diagonal structure $f^j(s) = [f_1^j(s_1) \dots f_{k_j}^j(s_{k_j})]^\top$, $j = 1, \bar{M}$ are continuous functions ensuring existence and uniqueness of solutions of the system (2) at least locally in the forward time; continuous matrix functions $A_g: \mathbb{R}^q \to \mathbb{R}^{n \times k_g}$ for $g = 0, \bar{M}$, $B: \mathbb{R}^q \to \mathbb{R}^{n \times m}$ and matrices $H_j \in \mathbb{R}^{k_j \times n}$ for $j = 1, \bar{M}$ are given. Further, for brevity and consistently with (2) we use the convention $k_0 = n$ and $H_0 = I_n$ with $f^0(x) = x$.

The model (2) belongs to the class of Persidskii system [20] under the following sector condition imposed on the nonlinearities: for any $j = 1, \bar{M}$ and $i = 1, k_j$,

$$sf_i^j(s) > 0 \quad \forall s \in \mathbb{R} \setminus \{0\}.$$

Consequently, all nonlinearities may take zero values at zero only. If $\theta(t) = \text{const}$, $H_1 = I_n$ and $A_r = 0$ for all $r = \bar{2}, \bar{M}$, then the system studied by Persidskii in the conventional framework [20] is recovered. In the case of $\bar{M} = 1$ and $\theta(t) = \text{const}$, (2) belongs also to the class of Lur'e systems widely investigated in the absolute stability theory [10].

After a proper re-indexing and decomposition of $f^j$, there exists $\nu \in \{0, \bar{M}\}$ such that for all $z = 1, \bar{\nu}$ and $i = 1, k_z$:

$$\lim_{s \to \pm\infty} f_i^z(s) = \pm\infty;$$

and there exists $\mu \in \{\bar{\nu}, \bar{M}\}$ such that for all $j = 1, \bar{\mu}$ and $i = 1, k_j$:

$$\lim_{s \to \pm\infty} \int_0^s f_i^j(\sigma) d\sigma = +\infty.$$

Thus, some of the nonlinearities are radially unbounded, and $\bar{\nu} = 0$ corresponds to the case when all nonlinearities are bounded (at least for negative or positive argument).

In this work, the (sector) nonlinearities are also assumed to satisfy the following stronger incremental sector condition: for any $j = 1, \bar{M}$ and $i = 1, k_j$,

$$(s - s') \left( f_i^j(s) - f_i^j(s') \right) > 0 \quad \forall s \neq s' \in \mathbb{R}. \quad (3)$$

This restriction implies a monotonous growth of the nonlinear functions in (2), and that $\mu = \bar{M}$.

### III. PROBLEM STATEMENT

Consider a PVP system (2) with a vector of scheduling parameters $\theta(t)$ available for measurement (we need to use the matrices $A_g(\theta)$, $g = 0, \bar{M}$ and $B(\theta)$ in the observer, otherwise, an interval or set-membership design [21]–[23] can be considered). The following assumptions are first introduced to formulate the problem addressed in this paper.

**Assumption 1.** Let $\theta(t) \in \Theta$, where $\Theta \subset \mathbb{R}^q$ is a known compact set.

This assumption introduces a standard hypothesis for stability analysis of LPV systems [6]: the set of admissible values for the vector of scheduling parameters is known. In order to obtain more constructive design conditions, matrix functions with linear dependence on the vector of scheduling parameters are considered:

**Assumption 2.** Let $A_g(\theta) = \sum_{k=1}^q \theta_k A_{gk}$ for $g = 0, \bar{M}$, where $A_{gk} \in \mathbb{R}^{n \times k_g}$ are the known matrices and $\theta_k \in [0, 1]$ for $k = 1, \bar{q}$ with $\sum_{k=1}^q \theta_k = 1$.

In the following section, the aim is to develop a state estimation algorithm for the uncertain and nonlinear system (2) with nonlinear and noisy measurement equations, using a parameter-independent LF under Assumption 1.

**Remark 1.** Note that the conventional LPV framework cancels all nonlinearities, which can be both conservative and yield restrictions on the validity domain. In contrast, frameworks like Persidskii or Lur'e systems provide global convergence for nonlinear models, but do not account for variations in $\theta(t)$ within the matrices describing the dynamics. The PVP models combine the advantages of both approaches.

### IV. OBSERVER STRUCTURE

The following basic observer is proposed for (2):

$$\begin{aligned} \dot{\hat{x}}(t) = & A_0(\theta(t))\hat{x}(t) + \sum_{j=1}^{M} A_j(\theta(t))f^j(H_j\hat{x}(t)) \\ & + B(\theta(t))u(t) + L_0(\theta(t))(y_0(t) - C_0\hat{x}(t)) \\ & + \sum_{j=1}^{M} L_j(\theta(t))(y_j(t) - C_jf^j(H_j\hat{x}(t))), \end{aligned} \quad (4)$$

where $\hat{x}(t) \in \mathbb{R}^n$ is the estimate of $x(t)$ and $L(\theta(t)) = [L_0(\theta(t)) \dots L_M(\theta(t))]$ is a continuous observer gain function to be designed.

These observer gains must be adjusted to ensure the stability and robustness of the estimation error $e(t) = x(t) - \hat{x}(t)$. The aim of this work is to establish IOS or SIIOS properties for the combined dynamics of the system (2) with the observer (4), treating $e(t)$ as the output and $d(t)$, $v(t)$ as the inputs. For observer (4), the estimation error dynamics can be explicitly written as follows:

$$\begin{aligned} \dot{e}(t) = & A_0(\theta(t))e(t) + \sum_{j=1}^{M} A_j(\theta(t))\delta f^j(e(t), \hat{x}(t)) \\ & + d(t) - L(\theta(t))v(t), \end{aligned} \quad (5)$$

where

$$\begin{aligned} A_g(\theta) = & A_g(\theta) - L_g(\theta)C_g, \quad g = 0, \bar{M}, \\ \delta f^j(e, \hat{x}) = & f^j(H_j(\hat{x} + e)) - f^j(H_j\hat{x}), \quad j = 1, \bar{M}, \end{aligned}$$

which can be viewed as a new autonomous PVP system due to (3) (i.e., $\delta f^j$ satisfies sector condition in $e$ uniformly in $\hat{x}$). Therefore, we look for SIIOS conditions for (5).
V. PARAMETER-INDEPENDENT LYAPUNOV FUNCTIONS

In general, the dependence of Lyapunov functions on $\theta(t)$ should make the stability conditions less restrictive, but this comes at the cost of requiring a more complex numerical procedure for verification, and we need to assume that $\dot{\theta}$ exists. For these reasons, this note will avoid such an approach. Recalling [11], [12], [24], consider the following structure of a candidate Lyapunov function for (2):

$$V(e) = e^{\top} Pe + 2 \sum_{j=1}^{M} \sum_{i=1}^{k_j} \Lambda_i^j \int_0^{H_j^{(i)}e} f_i^j(s) ds, \quad (6)$$

where $P \in \mathbb{R}^{n \times n}$ is a symmetric matrix and $\Lambda^j = \text{diag}[\Lambda_1^j, \dots, \Lambda_{k_j}^j] \in \mathbb{D}_+^{k_j}$ are parameters to be tuned in order that (6) verifies the properties of SIIOS-LF. To achieve this, an additional quadratic constraint must be imposed on the nonlinearities $f^j$:

**Assumption 3.** For all $j = \overline{1, M}$ there exist symmetric matrices $\Gamma^j \in \mathbb{R}^{(M+1)n \times (M+1)n}$ such that

$$f^j(H_j e)^\top f^j(H_j e) \le z^\top \Gamma^j z,$$

where $z^\top = [e^\top (\delta f^1(e, \hat{x}))^\top \dots (\delta f^M(e, \hat{x}))^\top]^\top$, for all $e, \hat{x} \in \mathbb{R}^n$.

This restriction is verified for many nonlinearities: e.g., odd power functions according to Lemma 1, or for any globally Lipschitz map $f^j$.

Denote by $\mathcal{M}(H_z, H_s) \subset \mathbb{R}^{k_z \times k_s}$ 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 $i = \overline{1, k_z}$, $j = \overline{1, k_s}$, $z = \overline{1, M-1}$ and $s = \overline{z+1, M}$.

**Theorem 2.** Let assumptions 1, 3 be satisfied and there exist $\gamma_j \in \mathbb{R}_+$ for $j = \overline{1, M}$, matrices $P^\top = P \in \mathbb{R}^{n \times n}$, $\Lambda^j \in \mathbb{D}_+^{k_j}$ for $j = \overline{1, M}$, $\Phi^\top = \Phi \in \mathbb{R}^{n \times n}$, $\Psi \in \mathbb{R}^{n \times n}$ and matrix functions $\Omega_g : \mathbb{R}^q \to \mathbb{R}^{n \times k_g}$, $\Upsilon_{g,g} : \mathbb{R}^q \to \mathbb{D}_+^{k_g}$ for $g = \overline{0, M}$, $\Upsilon_{0,j} : \mathbb{R}^q \to \mathbb{D}_+^{k_j}$ for $j = \overline{1, M}$, $\Upsilon_{z,s} : \mathbb{R}^q \to \mathcal{M}(H_z, H_s)$ for $z = \overline{1, M-1}$ and $s = \overline{z+1, M}$ such that the following matrix inequalities are verified for all $\theta \in \Theta$:

$$P \succeq 0, \Phi \succ 0, Q(\theta) \preceq 0, P + \rho_1 \sum_{j=1}^{\mu} H_j^\top \Lambda^j H_j \succ 0,$$

$$\sum_{r=0}^{\nu} H_r^\top \Upsilon_{r,r}(\theta) H_r + \rho_2 \sum_{z=1}^{\nu-1} \sum_{s=z+1}^{\nu} H_z^\top \Upsilon_{z,s}(\theta) H_s \\ + \rho_3 \sum_{j=1}^{\nu} H_j^\top \Upsilon_{0,j}(\theta) H_j \succ 0$$

for some $\rho_1, \rho_2, \rho_3 \in \mathbb{R}$, where $Q(\theta)$ is given in (7)<fnref n="1" />, then the observer (4) and the system (2) are SIIOS with respect to the output $e$ and the inputs $d, v$.

All proofs are excluded due to space limitations.

**Remark 2.** A necessary condition for feasibility of the matrix inequality $Q(\theta) \preceq 0$ required in the formulation of Theorem 2 is that all matrices on the main diagonal of $Q$ are nonnegative definite, i.e., $\Omega_g^\top(\theta) A_g(\theta) + A_g^\top(\theta) \Omega_g(\theta) \preceq 0$ for $g = \overline{0, M}$, which can be easily checked separately. It may also offer guidance on choosing the structure of the functions $\Omega_g(\theta)$ for known form of $A_g(\theta)$. Moreover, the Lyapunov function (6) is independent of $\theta$. However, using the descriptor approach allowed us to introduce the analogues of parameter dependent matrix functions $\Omega_g(\theta)$, $g = \overline{0, M}$ to check stability of $A_g(\theta)$. Consequently, the descriptor approach results in more general matrix inequalities, combining the advantages of both parameter-dependent and parameter-independent Lyapunov functions.

A drawback of the conditions formulated in Theorem 2 is that their verification for all $\theta \in \Theta$ may be computationally expensive. Imposing affinity of the system matrix functions $A_g(\theta)$ and the observer gain $L_g(\theta)$ in $\theta$, restricting the matrix functions $\Omega_g(\theta)$ to be constant for all $g = \overline{0, M}$, these stability conditions can be reduced to LMIs:

**Corollary 1.** Let assumptions 1, 2 and 3 be satisfied, $L_g(\theta) = \sum_{k=1}^q \theta_k L_{gk}$, $L_{gk} \in \mathbb{R}^{n \times k_g}$ for $g = \overline{0, M}$, $k = \overline{1, q}$ ($\theta_k \in [0, 1]$ for all $k = \overline{1, q}$ with $\sum_{k=1}^q \theta_k = 1$), and there exist matrices $0 \preceq P^\top = P \in \mathbb{R}^{n \times n}$, $\Lambda^j \in \mathbb{D}_+^{k_j}$ for $j = \overline{1, M}$, $\Phi^\top = \Phi \in \mathbb{R}^{n \times n}$, $\Psi \in \mathbb{R}^{n \times n}$, $\Omega_g \in \mathbb{R}^{n \times k_g}$, $\Upsilon_{g,g} \in \mathbb{D}_+^{k_g}$ for $g = \overline{0, M}$, $\Upsilon_{0,j} \in \mathbb{D}_+^{k_j}$ for $j = \overline{1, M}$, $\Upsilon_{z,s} \in \mathcal{M}(H_z, H_s)$ for $z = \overline{1, M-1}$ and $s = \overline{z+1, M}$ such that the following LMIs are verified:

$$\Phi \succ 0; Q_k \preceq 0, \forall k = \overline{1, q}; P + \rho_1 \sum_{j=1}^{\mu} H_j^\top \Lambda^j H_j \succ 0,$$

$$\sum_{r=0}^{\nu} H_r^\top \Upsilon_{r,r} H_r + \rho_2 \sum_{z=1}^{\nu-1} \sum_{s=z+1}^{\nu} H_z^\top \Upsilon_{z,s} H_s \\ + \rho_3 \sum_{j=1}^{\nu} H_j^\top \Upsilon_{0,j} H_j \succ 0$$

for some $\rho_1, \rho_2, \rho_3 \in \mathbb{R}$, where $Q_k$ are given in (8) for $A_{gk} = A_{gk} - L_{gk}C_g$ with $g = \overline{0, M}$ and $k = \overline{1, q}$, then the observer (4) and the system (2) are SIIOS with respect to the output $e$ and the inputs $d, v$.

More restrictions are needed to rewrite these LMIs having $L_{gk}$ for $g = \overline{0, M}$ and $k = \overline{1, q}$ as their solutions. Note that these matrices appear with different multipliers, $\Psi^\top$ and $\Omega_s^\top$ for $s = \overline{0, M}$, and following Remark 2, in this case the condition $A_{gk}^\top \Omega_g + \Omega_g^\top A_{gk} + \Upsilon_{g,g} \preceq 0$ can be used to find the candidates for the gains $L_{gk}$.

Another issue is the estimation of the convergence speed for the estimation error dynamics, which is not straightforward for time-varying nonlinear systems, and requires a separate consideration.

<footnote n="1">¹For brevity, the dependence on $\theta$ may be omitted once the respective functions have been defined.</footnote>
$$
Q =
\begin{bmatrix}
-\Psi^\top - \Psi & P + \Psi^\top A_0 - \Omega_0 & \Psi^\top A_1 - \Omega_1 & \cdots & \Psi^\top A_M - \Omega_M & H_1^\top \Lambda^1 & \cdots & H_M^\top \Lambda^M & \Psi^\top \\
P + A_0^\top \Psi - \Omega_0^\top & \Xi_{0,0} & \Xi_{0,1} & \cdots & \Xi_{0,M} & 0 & \cdots & 0 & \Omega_0^\top \\
A_1^\top \Psi - \Omega_1^\top & \Xi_{0,1}^\top & \Xi_{1,1} & \cdots & \Xi_{1,M} & 0 & \cdots & 0 & \Omega_1^\top \\
\vdots & \vdots & \vdots & \ddots & \vdots & \vdots & \ddots & \vdots & \vdots \\
A_M^\top \Psi - \Omega_M^\top & \Xi_{0,M}^\top & \Xi_{1,M}^\top & \cdots & \Xi_{M,M} & 0 & \cdots & 0 & \Omega_M^\top \\
\Lambda^1 H_1 & 0 & 0 & \cdots & 0 & -\gamma_1 I_{k_1} & \cdots & 0 & 0 \\
\vdots & \vdots & \vdots & \ddots & \vdots & \vdots & \ddots & \vdots & \vdots \\
\Lambda^M H_M & 0 & 0 & \cdots & 0 & 0 & \cdots & -\gamma_M I_{k_M} & 0 \\
\Psi & \Omega_0 & \Omega_1 & \cdots & \Omega_M & 0 & \cdots & 0 & -\Phi
\end{bmatrix}
\quad (7)
$$

$$
\hat{\Xi} = \Xi + \sum_{j=1}^{M} \gamma_j \Gamma^j, \Xi = \begin{bmatrix} \Xi_{0,0} & \Xi_{0,1} & \cdots & \Xi_{0,M} \\ \Xi_{0,1}^\top & \Xi_{1,1} & \cdots & \Xi_{1,M} \\ \vdots & \vdots & \ddots & \vdots \\ \Xi_{0,M}^\top & \Xi_{1,M}^\top & \cdots & \Xi_{M,M} \end{bmatrix},
\begin{align*}
\Xi_{g,j} &= A_g^\top \Omega_g + \Omega_g^\top A_g + \Upsilon_{g,j}, \quad g = \overline{0, M}, \\
\Xi_{0,j} &= A_0^\top \Omega_j + \Omega_0^\top A_0 + H_j^\top \Upsilon_{0,j}, \quad j = \overline{1, M}, \\
\Xi_{j,s} &= A_j^\top \Omega_s + \Omega_j^\top A_s + \Upsilon_{j,s}, \quad j = \overline{1, M}, s = \overline{j+1, M}
\end{align*}
$$

$$
Q_k =
\begin{bmatrix}
-\Psi^\top - \Psi & P + \Psi^\top A_{0k} - \Omega_0 & \Psi^\top A_{1k} - \Omega_1 & \cdots & \Psi^\top A_{Mk} - \Omega_M & H_1^\top \Lambda^1 & \cdots & H_M^\top \Lambda^M & \Psi^\top \\
P + A_{0k}^\top \Psi - \Omega_0^\top & \Xi_{0,0}^k & \Xi_{0,1}^k & \cdots & \Xi_{0,M}^k & 0 & \cdots & 0 & \Omega_0^\top \\
A_{1k}^\top \Psi - \Omega_1^\top & \Xi_{0,1}^k & \Xi_{1,1}^k & \cdots & \Xi_{1,M}^k & 0 & \cdots & 0 & \Omega_1^\top \\
\vdots & \vdots & \vdots & \ddots & \vdots & \vdots & \ddots & \vdots & \vdots \\
A_{Mk}^\top \Psi - \Omega_M^\top & \Xi_{0,M}^k & \Xi_{1,M}^k & \cdots & \Xi_{M,M}^k & 0 & \cdots & 0 & \Omega_M^\top \\
\Lambda^1 H_1 & 0 & 0 & \cdots & 0 & -\gamma_1 I_{k_1} & \cdots & 0 & 0 \\
\vdots & \vdots & \vdots & \ddots & \vdots & \vdots & \ddots & \vdots & \vdots \\
\Lambda^M H_M & 0 & 0 & \cdots & 0 & 0 & \cdots & -\gamma_M I_{k_M} & 0 \\
\Psi & \Omega_0 & \Omega_1 & \cdots & \Omega_M & 0 & \cdots & 0 & -\Phi
\end{bmatrix}
\quad (8)
$$

$$
\hat{\Xi}^k = \Xi^k + \sum_{j=1}^{M} \gamma_j \Gamma^j, \Xi^k = \begin{bmatrix} \Xi_{0,0} & \Xi_{0,1} & \cdots & \Xi_{0,M} \\ \Xi_{0,1}^\top & \Xi_{1,1} & \cdots & \Xi_{1,M} \\ \vdots & \vdots & \ddots & \vdots \\ \Xi_{0,M}^\top & \Xi_{1,M}^\top & \cdots & \Xi_{M,M} \end{bmatrix},
\begin{align*}
\Xi_{g,j} &= A_{gk}^\top \Omega_g + \Omega_g^\top A_{gk} + \Upsilon_{g,j}, \quad g = \overline{0, M}, \\
\Xi_{0,j} &= A_{0k}^\top \Omega_j + \Omega_0^\top A_{jk} + H_j^\top \Upsilon_{0,j}, \quad j = \overline{1, M}, \\
\Xi_{j,s} &= A_{jk}^\top \Omega_s + \Omega_j^\top A_{sk} + \Upsilon_{j,s}, \quad j = \overline{1, M}, s = \overline{j+1, M}
\end{align*}
$$

VI. NUMERICAL EXAMPLE

To illustrate the presented methodology, consider an aca-
demic example with bounded nonlinearities:

$$
\dot{x}_1(t) = -a_1(t)x_1(t) + x_2(t) - a_2 \frac{x_1(t)}{b_1 + |x_1(t)|} \\ -0.5u(t) + d_1(t),
$$

$$
\dot{x}_2(t) = -a_3(t)x_2(t) + a_4 \frac{x_1(t)}{b_1 + |x_1(t)|}
$$

$$
-a_5(t) \frac{x_2(t)}{b_2 + |x_2(t)|} + u(t) + d_2(t),
$$

$$
y(t) = x_{1}(t) + v(t),
$$

where $x_1(t), x_2(t) \in \mathbb{R}$ are the system states, $d(t) = [d_1(t) \ d_2(t)]^\top \in \mathbb{R}^2$ is a bounded disturbance, $u(t) \in \mathbb{R}$ is the control signal; $a_i(t)$ for $i = 1, 3, 5$ are positive time-varying parameters, whose instantaneous values are known, and the sets of admissible deviations have been identified as $a_{i \min} \le a_i(t) \le a_{i \max}$ for all $t \ge 0$ with some $0 < a_{i \min} \le a_{i \max} < +\infty$ being the related minimal and maximal possible values, respectively; $a_2, a_4, b_1, b_2 > 0$ are given constants. Our goal is to design a robust observer for this system using the proposed conditions. This model can be first rewritten in the form (2) for $n = 2, m = 1, M = 1$ and

$f^1(x) = \left[ \frac{x_1}{b_1 + |x_1|} \quad \frac{x_2}{b_2 + |x_2|} \right]^\top$, $H_1 = I_2$ satisfying the sector condition, with

$$
A_g(\theta) = \sum_{k=1}^{q} \theta_k A_{gk}, \quad g = 0, 1, q = 8,
$$

$$
A_{01} = A_{05} = \begin{bmatrix} -a_1 \min & 1 \\ 0 & -a_3 \min \end{bmatrix}, A_{02} = A_{06} = \begin{bmatrix} -a_1 \min & 1 \\ 0 & -a_3 \max \end{bmatrix},
$$

$$
A_{03} = A_{07} = \begin{bmatrix} -a_1 \max & 1 \\ 0 & -a_3 \min \end{bmatrix}, A_{04} = A_{08} = \begin{bmatrix} -a_1 \max & 1 \\ 0 & -a_3 \max \end{bmatrix},
$$

$$
A_{11} = \dots = A_{14} = \begin{bmatrix} -a_2 & 0 \\ a_4 & -a_5 \min \end{bmatrix}, A_{15} = \dots = A_{18} = \begin{bmatrix} -a_2 & 0 \\ a_4 & -a_5 \max \end{bmatrix},
$$

where $\sum_{k=1}^q \theta_k = 1$ and $\theta_k \in [0, 1]$. The compact set $\Theta \subseteq [0, 1]^q$ can be straightforwardly computed, and then Assumption 1 is verified. Moreover, Assumption 2 is also satisfied. Therefore, the LMIs of Corollary 1 can be used to check the applicability of the observer (4). The conventional LPV approaches [7] can be applied if we embed the nonlinearity $f^1(x)$ in a polytope, that may introduce an additional conservatism.

Let $a_{1 \min} = 0.05$, $a_{3 \min} = 0.01$, $a_{5 \min} = 0.001$,
$a_{1 \max} = 2$, $a_{3 \max} = 0.5$, $a_{5 \max} = 0.05$, $a_{2} = 0.1$,
$a_{4} = 0.01$, then the LMIs of Corollary 1 are verified using
a MATLAB toolbox SDPT3 [25] for

$$
L_0 = [2.18 \ 3.21]^\top, L_1 = [0.53 \ 0]^\top.
$$
<figure><img src="image_1.png" /><figcaption>Figure 1. The results of simulation: e(t) versus time t [sec]</figcaption></figure>

After selecting

$$
\begin{align*}
d(t) &= \begin{bmatrix} 1 + \sin(6t) \\ -2 + 5 \cos(1.5t) \end{bmatrix} + 2\text{rand}(2, 1), \\
a_1(t) &= a_{1\min} + (a_{1\max} - a_{1\min}) \sin^2(3t), \\
a_3(t) &= a_{3\min} + (a_{3\max} - a_{3\min}) \cos^2(3t), \\
a_5(t) &= a_{5\min} + (a_{5\max} - a_{5\min}) \sin^2(3t),
\end{align*}
$$

the estimation error from simulations, performed using the explicit Euler method with the discretization step 0.01, is shown in Fig. 1 for twenty different initial conditions chosen randomly in the set [-10, 10]² and zero initial conditions for the observer. All errors converge to a vicinity of the origin proportional to the amplitude of the perturbation *d*, which is an illustration of SIIOS behavior.

VII. CONCLUSION

This work addressed the state estimation problem for PVP
systems by proposing a dedicated state observer design.
This class of systems effectively models complex dynamics
involving multiple nonlinearities and uncertainties combining
LPV and Persidskii system representations. Conditions for
SIIOS of the estimation error are derived in the form of
matrix inequalities using parameter-independent Lyapunov
functions. By imposing additional structural constraints on
the system matrices and Lyapunov function, these conditions
are reduced to an explicit set of LMIs. The derived con-
ditions ensure robust stability and performance of the state
estimator in the presence of uncertainties and disturbances.
A promising direction for future research involves extending
these results to design interval observers for scenarios where
the scheduling parameters are not directly measurable. De-
velopment of procedures that allow the observer gains to be
found as a solution of matrix inequalities, as well as reducing
the conservatism of these inequalities, is another direction of
future research.

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