# LMI-Based Finite-Time Stability Analysis of Homogeneous Systems with Sector Nonlinearities and Its Applications to Multi-Agent Control and Observation

Daria Galkina, Konstantin Zimenko, Denis Efimov, Andrey Polyakov

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Daria Galkina, Konstantin Zimenko, Denis Efimov, Andrey Polyakov. LMI-Based Finite-Time Stability Analysis of Homogeneous Systems with Sector Nonlinearities and Its Applications to Multi-Agent Control and Observation. International Journal of Robust and Nonlinear Control, In press. hal-05502582

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ARTICLE TYPE

# LMI-Based Finite-Time Stability Analysis of Homogeneous Systems with Sector Nonlinearities and Its Applications to Multi-Agent Control and Observation

Daria Galkina<fnref n="1" />

Konstantin Zimenko<fnref n="1" />

Denis Efimov<fnref n="2" />

Andrey Polyakov<fnref n="2" />

**Correspondence**

Corresponding author Daria Galkina.

Email: da.galkina@itmo.ru

**Abstract**

The problem of finite-time stability analysis for a class of homogeneous systems with sector nonlinearities is addressed. The proposed method broadens the class of admissible systems and simplify the applicability conditions compared to existing approaches. A key advantage is the introduction of a unified framework for stability analysis in various homogeneity-based applications with sector nonlinearities. Its efficacy is demonstrated by deriving stability conditions for homogeneous observers and designing finite-time control protocols for second-order multi-agent systems, solving the uniform allocation and average consensus problems.

**KEYWORDS**

Homogeneous systems, finite-time stability, nonlinear observers, finite-time consensus protocols

# 1 | INTRODUCTION

Homogeneity is a property of dynamical systems widely exploited in control theory for the analysis and design of nonlinear robust controllers and observers with fast convergence, such as finite-time and fixed-time one<fnref n="2" />,³,⁴. Systems with homogeneous dynamics are known for enhanced robustness to external disturbances, measurement noise, and time delays⁵,⁶,⁷. Consequently, homogeneity-based methods have been successfully applied in a variety of control and observation problems, including stabilization, consensus, and estimation³.

The Implicit Lyapunov Function (ILF) method is often efficient for the stability analysis of homogeneous systems, reducing it to the solution of Linear Matrix Inequalities (LMI)³. In⁸, an ILF modification was proposed for homogeneous systems with sector nonlinearities, providing LMI-based stability conditions. While powerful, these results have limitations due to structural constraints and conservatism, which can complicate their direct application in certain cases.

This paper aims to extend the class of admissible systems considered in⁸, particularly covering a broad range of finite-time consensus protocols, distributed observers, and related designs. At the same time, the proposed approach relaxes the LMI-based stability conditions, enabling less conservative analysis and facilitating practical implementation.

Compared to our previous conference version<fnref n="1" />, this manuscript provides proofs of all claims, several additional application examples, and a procedure for applying the results to new control and estimation problems.

The main contribution of this paper is the development of a unified framework for the stability analysis of homogeneous systems with sector nonlinearities that allows:

* Extension of the class of admissible systems including different dimensions of the nonlinearities and their arguments;

* Relaxation of the LMI-based stability conditions, reducing conservatism and simplifying practical implementation;

* Application of the framework to both homogeneous and heterogeneous multi-agent systems, with the possibility of designing finite-time protocols using existing control gains from linear methods.

<footnote n="1">¹Faculty of Control Systems and Robotics, ITMO University, Saint Petersburg, Russia</footnote>

<footnote n="2">²CNRS, UMR 9189 - CRISTAL, Inria, Univ. Lille, Lille, France</footnote>
It is worth emphasizing that the contribution of this paper is not aimed at improving a particular performance index of a given control or observation algorithm, but rather at providing a unified and systematic LMI-based approach for stability analysis of a broad class of homogeneous systems with sector nonlinearities. Depending on the application, this approach offers different advantages, including wider ranges of homogeneity degrees for stability analysis, less conservative LMI-based stability conditions and settling-time estimates, the ability to quantify compensable disturbances, etc.

The paper is organized as follows. Section 2 introduces the notations used throughout. Section 3 covers preliminary results. Section 4 addresses the stability analysis for homogeneous systems with sector nonlinearities. Section 5 presents several application examples, including homogeneous observers and finite-time consensus protocols. Finally, conclusions are provided in Section 6.

## 2 | NOTATION

Through the paper the following notation will be used:

* $\mathbb{R}(\mathbb{R}_+)$ is the set of real (positive) numbers;

* $\mathbb{R}^n$ and $\mathbb{R}^{n \times n}$ denote the $n$ dimensional Euclidean space with vector norm $\| \cdot \|$, or the respective matrix space and norm;

* $\bar{1}, m$ is used to denote a sequence of integers 1, ..., $m$;

* the symbol $\otimes$ denotes the Kronecker product;

* for $x = [x_1 \dots x_n]^T \in \mathbb{R}^n$ and $\alpha = [\alpha_1 \dots \alpha_n]^T \in \mathbb{R}_+^n$, we denote by $[x]^{\alpha} = [\operatorname{sign}(x_1)[x_1]^{\alpha_1} \dots \operatorname{sign}(x_n)[x_n]^{\alpha_n}]^T$ the sign-preserving element-wise exponentiation;

* $\mathbf{1}_n \in \mathbb{R}^n$ is the vector with all entries equal to 1;

* the order relation $P > 0$ ($< 0$; $\ge 0$; $\le 0$) for $P \in \mathbb{R}^{n \times n}$ means that $P$ is symmetric and positive (negative) definite (semidefinite);

* $\lambda_i(P)$ denotes $i^{th}$ eigenvalue of a matrix $P \in \mathbb{R}^{n \times n}$, $\lambda_{\max}(P)$ and $\lambda_{\min}(P)$ denote maximum and minimum eigenvalues for a symmetric $P$;

* $\mathbf{u}_i(A)$ denotes left eigenvector of a matrix $A$ corresponding to $i^{th}$ eigenvalue, i.e., $\mathbf{u}_i(A)A = \lambda_i(A)\mathbf{u}_i(A)$;

* the set of nonnegative diagonal matrices with dimension $n \times n$ is defined by $\mathbb{D}_+^n$;

* $\operatorname{diag}\{\lambda_i\}_{i=1}^n$ is a diagonal matrix with elements $\lambda_i$, $i = \bar{1}, \bar{n}$;

* $O_{m \times n}$ denotes zero matrix with dimension $m \times n$, $I_n \in \mathbb{R}^{n \times n}$ is the identity matrix;

* $\mathcal{L}_{\infty}(\mathbb{R}^p)$ denotes the set of essentially bounded measurable functions $f: \mathbb{R}_+ \cup \{0\} \to \mathbb{R}^p$;

* $e_s(i)$ is a vector of the canonical basis of $\mathbb{R}^s$, where $i^{th}$ component is 1 and the rests are 0.

## 3 | PRELIMINARIES

### 3.1 | Finite-time stability

Consider the system

$$ \dot{x}(t) = f(x(t)), \quad x(0) = x_0, \quad t \ge 0, \tag{1} $$

where $x(t) \in \mathbb{R}^n$ is the state vector, $f: \mathbb{R}^n \to \mathbb{R}^n$ is a nonlinear continuous vector field and $f(0) = 0$ ensuring existence of solutions $x(t, x_0)$ in (1) for all initial conditions $x_0 \in \mathbb{R}^n$ at least locally in time $t \ge 0$.

**Definition 1** ($^{(9,10)}$). The origin of (1) is said to be globally finite-time stable if it is globally uniformly asymptotically stable and there exists a locally bounded function $T: \mathbb{R}^n \to [0, +\infty)$ such that any solution $x(t, x_0)$ of the system (1) satisfies $x(t, x_0) = 0, \forall t \ge T(x_0)$ for all $x_0 \in \mathbb{R}^n$. The function $T$ is called a settling-time estimate.

Many practical control and observation problems impose strict requirements on the transient behavior of dynamical systems. Such requirements arise, for instance, from safety and performance considerations motivating the study of finite-time stability, observation and stabilization (see, e.g., $^{9,10,3,11,12}$, etc.). In both control and state estimation settings, finite-time convergence ensures that the system state or the estimation error reaches the desired value within a bounded time interval, leading to improved transient performance and robustness. In this context, Lyapunov-based methods play a central role in the analysis of
finite-time stability. The ILF approach often allows one to characterize stability properties via algebraic conditions, which is especially convenient for homogeneous systems and LMI-based analysis. The following theorem presents the ILF method<sup>13,14</sup> for finite-time stability analysis.

**Theorem 1**<sup>(15)</sup>. If there exists a continuous function

$$Q: \mathbb{R}_+ \times \mathbb{R}^n \to \mathbb{R} \\ (V, x) \mapsto Q(V, x)$$

such that

C1) $Q(V, x)$ is continuously differentiable $\forall x \in \mathbb{R}^n \setminus \{0\}$ and $\forall V \in \mathbb{R}_+$;

C2) for any $x \in \mathbb{R}^n \setminus \{0\}$ there exist $V^-, V^+ \in \mathbb{R}_+$:

$$Q(V^-, x) < 0 < Q(V^+, x); \qquad (2)$$

C3) for $\Omega = \{(V, x) \in \mathbb{R}^{n+1} : Q(V, x) = 0\}$

$$\lim_{\substack{x \to 0 \\ (V,x) \in \Omega}} V = 0^+, \lim_{\substack{V \to 0^+ \\ (V,x) \in \Omega}} \|x\| = 0, \lim_{\substack{V=+\infty \\ (V,x) \in \Omega}} V = +\infty;$$

C4) the inequality

$$-\infty < \frac{\partial Q(V, x)}{\partial V} < 0$$

holds $\forall V \in \mathbb{R}_+$ and $\forall x \in \mathbb{R}^n \setminus \{0\}$;

C5) the inequality

$$\frac{\partial Q(V, x)}{\partial x} f(x) \le \sigma V^{1-\mu} \frac{\partial Q(V, x)}{\partial V}$$

holds $\forall (V, x) \in \Omega$, where $0 < \mu \le 1$ and $\sigma \in \mathbb{R}_+$ are some constants.

Then the origin of the system (1) is globally finite-time stable with the following settling time estimate

$$T(x_0) \le \frac{V_0^\mu}{\sigma \mu},$$

where $V_0 \in \mathbb{R}_+$: $Q(V_0, x_0) = 0$.

Theorem 1 states a classical result on finite-time stability<sup>9</sup> in the context of implicitly defined Lyapunov functions. Specifically, the conditions C1, C2, and C4 ensure the existence and uniqueness of a positive definite function $V: \mathbb{R}^n \to \mathbb{R}_+$ satisfying $Q(V(x), x) = 0$ for all $x \in \mathbb{R}^n$. The condition C3 guarantees that $V(x) \to 0$ as $x \to 0$ and $V(x) \to +\infty$ as $\|x\| \to \infty$. Application of the Implicit Function Theorem<sup>16</sup> yields the partial derivative $\frac{\partial V}{\partial x} = -\left(\frac{\partial Q}{\partial V}\right)^{-1} \frac{\partial Q}{\partial x}$. Consequently, conditions C4 and C5 lead to the estimate $\dot{V}(x) \le -\sigma V^{1-\mu}$, which directly implies finite-time stability<sup>9</sup>.

## 3.2 | Homogeneity

Homogeneity is a symmetry property of an object with respect to a group of dilations<sup>2,3,4</sup>. A map $d: \mathbb{R} \to \mathbb{R}^{n \times n}$ is referred to as a linear dilation, defined by $d(s) = e^{Gds}$, where the matrix $G_d$, known as the generator of dilation, is anti-Hurwitz.

**Definition 2**<sup>(4)</sup>. A vector field $f: \mathbb{R}^n \to \mathbb{R}^n$ (a function $g: \mathbb{R}^n \to \mathbb{R}$) is said to be **d**-homogeneous of degree $\nu \in \mathbb{R}$ iff

$$\begin{array}{l} f(\mathbf{d}(s)x) = e^{\nu s} \mathbf{d}(s)f(x), \quad \forall x \in \mathbb{R}^n \setminus \{0\}, \quad \forall s \in \mathbb{R}. \\ (\text{resp. } g(\mathbf{d}(s)x) = e^{\nu s} g(x), \quad \forall x \in \mathbb{R}^n \setminus \{0\}, \quad \forall s \in \mathbb{R}. ) \end{array} \qquad (3)$$

The system $\dot{x} = f(x)$ is called **d**-homogeneous if $f(x)$ is **d**-homogeneous.

When the matrix $G_d$ is diagonal ($G_d = I_n$), the homogeneity type is referred to as weighted (standard) homogeneity.

Vector fields that exhibit homogeneity with respect to a certain dilation $d$ possess numerous properties that are valuable for the analysis of convergence rates<sup>3,17,4,11,12</sup>. Moreover, the homogeneity property inherently implies robustness with respect
to external perturbations, measurement noise, and time delays, which makes the theory of homogeneous systems particularly attractive for solution of various control and observation problems$^{5,6,7}$.

# 4 | STABILITY ANALYSIS OF HOMOGENEOUS SYSTEMS WITH SECTOR NONLINEARITIES

## 4.1 | Homogeneous systems with sector nonlinearities

Let us consider the following class of systems:

$$ \dot{x}(t) = A_0 x(t) + A_1 f(Ux(t)) + d(t, x(t)), \quad t \ge 0, \quad (4) $$

where $x(t) = [x_1(t) \cdots x_n(t)]^T \in \mathbb{R}^n$ is the state vector, $x_0 = x(0) \in \mathbb{R}^n$, $f(Ux) = [f_1(e_l^T(1)Ux) \cdots f_1(e_l^T(l)Ux)]^T$ is a continuous function ensuring existence of solutions of the system (4) in forward time, $f(0) = 0$, $A_0 \in \mathbb{R}^{n \times n}$, $U \in \mathbb{R}^{l \times n}$, $A_1 \in \mathbb{R}^{n \times l}$, and the function $d: \mathbb{R} \times \mathbb{R}^n \to \mathbb{R}^n$ describes exogenous disturbances and uncertainties.

It is assumed that the system (4) satisfies the following restrictions:

* **Assumption 1** For any $i = \overline{1, l}$:

$$ s f_i(s) > 0, \quad \forall s \in \mathbb{R} \setminus \{0\}. \quad (5) $$

* **Assumption 2** The system (4) is d-homogeneous of degree $\mu < 0$ with the generator $G_d$ having at least one real eigenvalue, i.e.

$$ \mathbf{d}(s)(A_0x + A_1f(Ux)) = e^{-s\mu}(A_0\mathbf{d}(s)x + A_1f(U\mathbf{d}(s)x)), \quad \forall s \in \mathbb{R}, \forall x \in \mathbb{R}^n \quad (6) $$

and the functions $f_i$, $i = \overline{1, l}$ are standard homogeneous of degree less than 1, i.e., $f_i(\lambda s) = \lambda^{\mu_i} f_i(s)$ for all $\lambda \in \mathbb{R}_+$, $s \in \mathbb{R}$ and some $\mu_i \in (0, 1)$.

* **Assumption 3** Each row of $U$ is a left eigenvector of the generator matrix $G_d$, i.e., $\forall i \in \overline{1, l}, \exists j_i \in \overline{1, k}: e_l^T(i)U = \mathbf{u}_{j_i}(G_d)$, where $k$ is a number of disjoint real eigenvalues of $G_d$.

Note, that due to Assumption 2, for any $i \in \overline{1, l}$ there exists $\psi_i \in \mathbb{R}_+$ such that

$$ s f_i(s) > \psi_i s^2 \quad \forall s \in [s_i, \bar{s}_i] \setminus \{0\} \quad (7) $$

for some $-\infty < s_i < 0 < \bar{s}_i < +\infty$. Indeed, due to $f_i(\lambda s) = \lambda^{\mu_i} f_i(s)$ for all $\lambda \in \mathbb{R}_+$, one can obtain $f_i(s) = |s|^{\mu_i} f_i(\text{sign}(s))$ for all $s \in \mathbb{R} \setminus \{0\}$. Then the inequality (7) is equivalent to $|s|^{\mu_i - 1} |f_i(\text{sign}(s))| > \psi_i$ that is satisfied for sufficiently small $|s|$ due to $\mu_i < 1$ and $|f_i(\text{sign}(s))|$ is bounded.

Systems (4) appear in a number of homogeneity-based applications as finite-time controls$^{18}$, multi-agent consensus protocols$^{19,20}$, observers$^{12,21}$, etc.

**Remark 1** In$^8$, the following class of systems was studied:

$$ \dot{x}(t) = A_0 x(t) + \sum_{j=1}^{M} \bar{A}_j f^j(\bar{U}x(t)) + d(t, x(t)), \quad t \ge 0, \quad (8) $$

under Assumptions 1–3 and with square matrices $\bar{U} \in \mathbb{R}^{n \times n}$ and $\bar{A}_k \in \mathbb{R}^{n \times n}$, $k = \overline{1, M}$. This system is a particular case of (4), which corresponds to the specific structure

$$ A_1 = [\bar{A}_1 \cdots \bar{A}_M], \quad U = \begin{bmatrix} \bar{U} \\ \vdots \\ \bar{U} \end{bmatrix}, \quad f(Ux) = \begin{bmatrix} f^1([I \quad O_{n \times (M-1)n}]Ux) \\ \vdots \\ f^M([O_{n \times (M-1)n}]I)Ux \end{bmatrix}. $$

On the other hand, note that representing system (4) in the form (8) is only possible in a number of specific cases. For example, if $l > n$ and $U$ contains more than $n$ non-collinear rows, then the transition from the form (4) to (8) is impossible. In other cases with non-square $A_1$, $U$, the transition can be carried out, but only by introducing zero rows and columns in the matrices
$\bar{A}_j$, which may lead to the inability of LMI-based stability conditions from $^8$. Thus, the representation of the system in the form (4) is more universal and practical for constructing stability criteria. As examples of applications that satisfy the introduced assumptions and for which the result of $^8$ cannot be successfully applied, one can cite distributed homogeneous observers$^{21}$, as well as the finite-time version of the consensus protocol from $^{20}$.

## 4.2 | Stability analysis

For $i = \overline{1, l}$ denote $\mathbf{u}_i = e_i^T(i)U$, and in accordance with Assumption 3 define the corresponding eigenvalue of the generator $G_d$ as $\lambda_{\mathbf{u}_i}(G_d)$, i.e., $\mathbf{u}_i G_d = \lambda_{\mathbf{u}_i}(G_d) \mathbf{u}_i$. Following $^8$, let us choose the ILF in the form

$$Q(V, x) = x^T \mathbf{d}^T(-\ln V) P \mathbf{d}(-\ln V)x + 2 \sum_{i=1}^{l} \Lambda_i \int_{0}^{\mathbf{u}_i \cdot (-\ln V)x} f_i(s) ds - \chi \quad (9)$$

with $\Lambda = \text{diag}\{\Lambda_i\}_{i=1}^l \in \mathbb{D}_+^l, P \in \mathbb{R}^{n \times n}$ is a symmetric positive semi-definite matrix, and $\chi \in \mathbb{R}_+$.

Let us formulate constructive conditions, in the form of LMI, for robust finite-time stability analysis of the homogeneous system (4), along with an estimate of the settling time.

Define $\Psi = \text{diag}\{\psi_i\}_{i=1}^l \in \mathbb{D}_+^l$ and $\mathcal{I} \in \mathbb{R}^{l \times l}: \mathcal{I}_{i,j} = \begin{cases} \text{sign}(\mathbf{u}_i \mathbf{u}_j^T), & \text{if } \mathbf{u}_j, \mathbf{u}_i \text{ are colinear} \\ 0, & \text{otherwise} \end{cases}$ for $i,j = \overline{1,l}$.

**Theorem 2.** Let Assumptions 1, 2, 3 be satisfied. Let $\alpha \in \mathbb{R}_+$ be chosen such that the LMI system

$$P \geq \rho U^T U; \quad P + U^T \Lambda U > 0; \quad (10a)$$

$$PG_d + G_d^T P + 2U^T \Lambda H \Psi U > 0; \quad (10b)$$

$$R > 0; \quad (10c)$$

$$Q_1 + \alpha \left[ \begin{matrix} Q_2 & O_{(n+l) \times n} \\ * & O_{n \times n} \end{matrix} \right] + \left[ \begin{matrix} Q_3 & O_{(n+l) \times n} \\ * & O_{n \times n} \end{matrix} \right] \leq 0, \quad (10d)$$

is feasible for some $P, R \in \mathbb{R}^{n \times n}; \Lambda \in \mathbb{D}_+^l, 0 \le \Xi = \Xi^T \in \mathbb{R}^{(n+l) \times (n+l)}, \rho \in \mathbb{R}_+$, where

$$Q_1 = \begin{bmatrix} X\tilde{A}_0 + \tilde{A}_0^T X & XA_1 + \tilde{A}_0^T U^T \Lambda & X \\ * & \Lambda UA_1 + A_1^T U^T \Lambda & \Lambda U \\ * & * & -R \end{bmatrix}; \\ Q_2 = \begin{bmatrix} XG_d + G_d^T X & U^T \Lambda H \\ * & O_{l \times l} \end{bmatrix}; \\ Q_3 = \Xi + \begin{bmatrix} O_{n \times n} & U^T \Upsilon_{0,1} \\ * & \Upsilon_{1,1} \end{bmatrix}$$

for $X = P + U^T \Lambda \Psi U, \tilde{A}_0 = A_0 + A_1 \Psi U, H = \text{diag}\{\lambda_{\mathbf{u}_i}(G_d)\}_{i=1}^l, \Upsilon_{0,1} \in \mathbb{D}_+^l$,

$$\Upsilon_{1,1} = \Upsilon_{1,1}^T \in \mathbb{R}^{l \times l}: \begin{cases} e_i^T(\eta)\Upsilon_{1,1}e_l(\varrho) \ge 0, & \text{if } e_i^T(\eta)\mathcal{I}e_l(\varrho) = 1 \\ e_i^T(\eta)\Upsilon_{1,1}e_l(\varrho) \le 0, & \text{if } e_i^T(\eta)\mathcal{I}e_l(\varrho) = -1 \\ e_i^T(\eta)\Upsilon_{1,1}e_l(\varrho) = 0, & \text{otherwise} \end{cases}$$

for $\eta, \varrho = \overline{1,l}$, and let the following restriction to the system disturbances be satisfied:

$$d^T \mathbf{d}^T(-\ln V) R \mathbf{d}(-\ln V) d \le V^{2\mu} y^T Q_3 y \quad (11)$$

for $y = \begin{bmatrix} \mathbf{d}(-\ln V)_x \\ f(U \mathbf{d}(-\ln V)_x) - \Psi U \mathbf{d}(-\ln V)_x \end{bmatrix}$ and $V \in \mathbb{R}_+: Q(V, x) = 0$ with $\chi \in (0, \rho \min_{i \in \overline{1,l}}\{s_i^2, \bar{s}_i^2\})$.

Then the origin of the system (4) is globally finite-time stable and $T(x_0) \le -\frac{V_0^{\mu}}{\alpha \mu}$, where $Q(V_0, x_0) = 0$.

**Proof.** Following the proof of$^8$, Theorem 4 the function $Q(V, x)$ satisfies the conditions C1-C3 under the condition (10a). Let us show that C4, C5 are also hold under the conditions of Theorem 2.
Since for $\rho \in \mathbb{R}_+$: $P \ge \rho U^T U$ and $\chi \le \rho \min_{i \in \overline{1,l}} \{s_i^2, \bar{s}_i^2\}$, one have $\|U\mathbf{d}(-\ln V)x\|^2 \le \frac{\chi}{\rho} \le \min_{i \in \overline{1,l}} \{s_i^2, \bar{s}_i^2\}$ and $\mathbf{u}_i\mathbf{d}(-\ln V)x \in [s_i, \bar{s}_i]$ for any $x \in \mathbb{R}^n \setminus \{0\}$, $i = \overline{1,n}$, where $s_i, \bar{s}_i$ are defined in (7). Then $\mathbf{u}_i\mathbf{d}(-\ln V)xf_i(\mathbf{u}_i\mathbf{d}(-\ln V)x) \ge \psi_i(\mathbf{u}_i\mathbf{d}(-\ln V)x)^2$, $i = \overline{1,l}$ by Assumption 2, and under the condition (10b) we have

$$ 
\begin{aligned}
\frac{\partial Q}{\partial V} &= -V^{-1}x^T \mathbf{d}^T(-\ln V)(PG_d + G_d^T P)\mathbf{d}(-\ln V)x - 2V^{-1} \sum_{i=1}^l \Lambda_i \mathbf{u}_i G_d \mathbf{d}(-\ln V)xf_i(\mathbf{u}_i \mathbf{d}(-\ln V)x) \\
&= -V^{-1}x^T \mathbf{d}^T(-\ln V)(PG_d + G_d^T P)\mathbf{d}(-\ln V)x - 2V^{-1} \sum_{i=1}^l \Lambda_i \lambda_{\mathbf{u}_i} (G_d) \mathbf{u}_i \mathbf{d}(-\ln V)xf_i(\mathbf{u}_i \mathbf{d}(-\ln V)x), \\
&\le -V^{-1}x^T \mathbf{d}^T(-\ln V)(PG_d + G_d^T P + 2U^T \Lambda H \Psi U)\mathbf{d}(-\ln V)x \\
&< 0, \quad \forall x \in \mathbb{R}^n \setminus \{0\},
\end{aligned} 
\quad (12) $$

i.e., the condition C4 of Theorem 1 holds.

Following⁸ let us rewrite (4) and (9) in the equivalent forms presented below:

$$ \dot{x}(t) = (A_0 + A_1 \Psi U)x(t) + A_1(f(Ux(t)) - \Psi Ux(t)), \quad t \ge 0, \quad (13) $$

$$ Q(V, x) = x^T \mathbf{d}^T(-\ln V) \mathbf{X} \mathbf{d}(-\ln V)x + 2 \sum_{i=1}^{l} \Lambda_i \int_{0}^{\mathbf{u}_i \mathbf{d}(-\ln V)x} g_i(s) ds - \chi, \quad (14) $$

where $g_i(s) = f_i(s) - \psi_i s$, $X = P + U^T \Lambda \Psi U$. Note that according to (12) we have $\frac{\partial Q}{\partial V} = -V^{-1}y^T Q_2 y$, where the vector variable y is defined in the theorem's statement. Then, for $\tilde{A}_0 = A_0 + A_1 \Psi U$ and using (6), (10), (11) we have

$$ 
\begin{aligned}
\frac{\partial Q}{\partial x} \dot{x} &= 2x^T \mathbf{d}^T(-\ln V) \mathbf{X} \mathbf{d}(-\ln V) \dot{x} + 2 \sum_{i=1}^l \Lambda_i g_i(\mathbf{u}_i \mathbf{d}(-\ln V)x) \mathbf{u}_i \mathbf{d}(-\ln V) \dot{x} \\
&= 2V^\mu (x^T \mathbf{d}^T(-\ln V) \tilde{A}_0 \mathbf{d}(-\ln V)x + x^T \mathbf{d}^T(-\ln V) X A_1 g(U\mathbf{d}(-\ln V)x) \\
&\quad + V^{-\mu} \dot{x}^T \mathbf{d}^T(-\ln V) U^T \Lambda g(U\mathbf{d}(-\ln V)x) + V^{-\mu} x^T \mathbf{d}^T(-\ln V) \mathbf{X} \mathbf{d}(-\ln V) d) \\
&= 2V^\mu (x^T \mathbf{d}^T(-\ln V) \tilde{A}_0 \mathbf{d}(-\ln V)x + x^T \mathbf{d}^T(-\ln V) X A_1 g(U\mathbf{d}(-\ln V)x) \\
&\quad + x^T \mathbf{d}^T(-\ln V) \tilde{A}_0 U^T \Lambda g(U\mathbf{d}(-\ln V)x) + g(U\mathbf{d}(-\ln V)x)^T A_1^T U^T \Lambda g(U\mathbf{d}(-\ln V)x) \\
&\quad + V^{-\mu} x^T \mathbf{d}^T(-\ln V) \mathbf{X} \mathbf{d}(-\ln V) d + V^{-\mu} d^T \mathbf{d}^T(-\ln V) U^T \Lambda g(U\mathbf{d}(-\ln V)x)) \\
&= V^\mu (x^T \mathbf{d}^T(-\ln V) [X \tilde{A}_0 + \tilde{A}_0^T X] \mathbf{d}(-\ln V)x + x^T \mathbf{d}^T(-\ln V) [X A_1 + \tilde{A}_0^T U^T \Lambda] g(U\mathbf{d}(-\ln V)x) \\
&\quad + g(U\mathbf{d}(-\ln V)x)^T [A_1^T X + \Lambda U A_0] \mathbf{d}(-\ln V)x + g(U\mathbf{d}(-\ln V)x)^T [\Lambda U A_1 + A_1^T U^T \Lambda] g(U\mathbf{d}(-\ln V)x) \\
&\quad + V^{-\mu} x^T \mathbf{d}^T(-\ln V) \mathbf{X} \mathbf{d}(-\ln V) d + V^{-\mu} d^T \mathbf{d}^T(-\ln V) \mathbf{X} \mathbf{d}(-\ln V)x + V^{-\mu} d^T \mathbf{d}^T(-\ln V) U^T \Lambda g(U\mathbf{d}(-\ln V)x) \\
&\quad + V^{-\mu} g(U\mathbf{d}(-\ln V)x)^T \Lambda U \mathbf{d}(-\ln V) d - V^{-2\mu} d^T \mathbf{d}^T(-\ln V) \mathbf{R} \mathbf{d}(-\ln V) d + V^{-2\mu} d^T \mathbf{d}^T(-\ln V) \mathbf{R} \mathbf{d}(-\ln V) d) \\
&= V^\mu \left( \left[ V^{-\mu} \mathbf{d} \right]_{-\ln V}^y \right)^T Q_1 \left[ V^{-\mu} \mathbf{d} \right]_{-\ln V}^y + V^{-2\mu} d^T \mathbf{d}^T(-\ln V) \mathbf{R} \mathbf{d}(-\ln V) d \\
&\le V^\mu \left[ V^{-\mu} \mathbf{d} \right]_{-\ln V}^y \left( Q_1 + \left[ \begin{smallmatrix} Q_3 & O_{(n+l) \times n} \\ * & O_{n \times n} \end{smallmatrix} \right] \right) \left[ V^{-\mu} \mathbf{d} \right]_{-\ln V}^y \\
&\le -\alpha V^{\mu} y^T Q_2 y \\
&= \alpha V^{1+\mu} \frac{\partial Q}{\partial V}.
\end{aligned} $$

Therefore, the condition C5 is satisfied.

Finally, due to $\mathbf{u}_i\mathbf{d}(-\ln V)x \in [s_i, \bar{s}_i]$ for any $x \in \mathbb{R}^n \setminus \{0\}$, $i = \overline{1,n}$, one has $y^T Q_3 y \ge 0$ for any $x \in \mathbb{R}^n \setminus \{0\}$, i.e., the class of disturbances to be rejected is non-empty.

Thus, all conditions of Theorem 1 are satisfied, and the system (4) is globally finite-time stable if (10) and (11) hold. $\square$

<figure><img src="image_2.png" /></figure>

The proposed result is an extension of⁸ Theorem 4 to non-square matrices $A_1$ and $U$. This extension allows the obtained result to be applied to a wider class of systems (see the next section). Additionally, some of the LMI-based stability conditions are relaxed compared to ⁸.

Note that smaller values of the parameter $\alpha \in \mathbb{R}_+$ enlarge the feasibility region of the LMI conditions in (10). However, choosing $\alpha$ too small may result in slower convergence estimate, since $T(x_0) \le -\frac{V_0^{\mu}}{\alpha^{\mu}}$.

The matrix $Q_3$ is used in order to relax the stability requirements for the matrix $A_1$. The matrices $\Upsilon_{j,1}$ for $j = 0, 1$ are introduced in order to reduce the conservatism of the condition (10d). Indeed, since for $\chi \le \rho \min_{i \in \overline{1,l}} \{s_i^2, \bar{s}_i^2\}$ we have $\mathbf{u}_i\mathbf{d}(-\ln V)x \in [s_i, \bar{s}_i]$ for any $x \in \mathbb{R}^n$, then, due to the sector properties (Assumption 1), all terms in $y^T Q_3 y$ are nonnegative for $x \in \mathbb{R}^n \setminus \{0\}$. In other words, all cross-terms that appear on the left-hand side of (10d) should not be considered as additional “perturbations” in the verification of (10d), provided that they have a non-positive multiplier.

In addition, the provided LMI (10) can be made less conservative following the remark from ⁸:

**Remark 2**⁸ In (10) some elements $e_l^T(\eta)\Upsilon_{i,1}e_l(k)$, $\eta, k \in \overline{1,l}$ of the matrices $\Upsilon_{i,1}$, $i \in \overline{0,1}$ that correspond to the case $e_l^T(\eta)\mathcal{I}e_l(k) = 1$ can be selected to be negative if there are positive elements of $\Upsilon_{q,1}$, $q \in \overline{0,1}$ that correspond to terms with the same multipliers, but having faster or the same dynamics in the vicinity of the origin; and the sum of these elements is greater
or equal than the sum of absolute values of negative counterparts that correspond to terms with slower dynamics. The similar remark with inverted signs can be made for the case $e_l^T(\eta)\mathcal{I}e_l(k) = -1$.

Note that the implicit restriction (11) on the system disturbances can be replaced by an explicit (but more conservative) inequality, in accordance with 8, Remark 3:

**Remark 3** For $\|x\|_p = \sqrt{x^T P x}$ define $\tilde{\chi} = \max_{x \in \mathbb{R}^n: V(x)=1} (\chi - \|x\|_p^2)$, that can be calculated on a numerical grid (due to (9) and $\Lambda \neq 0$, one has $\tilde{\chi} > 0$). Then the implicit restriction (11) to the system disturbances can be replaced by the following explicit one

$$ \|d\|_p^2 \le q\kappa^{-1} \min \left\{ \chi^{-\frac{\mu+\varrho_1-\varrho_2}{\varrho_2}} \|x\|_p^{\frac{2(\mu+\varrho_1)}{\varrho_2}}, \beta^2(\mu-\varrho_1+\varrho_2) \|x\|_p^{\frac{2(\mu-\varrho_1+2\varrho_2)}{\varrho_2}} \right\} $$

for $\beta = (\chi - \tilde{\chi})^{\frac{-1}{2\varrho_2}}$, $\varrho_1 = \frac{1}{2}\lambda_{\max} (P^{\frac{1}{2}}G_d P^{-\frac{1}{2}} + P^{-\frac{1}{2}}G_d^T P^{\frac{1}{2}})$, $\varrho_2 = \frac{1}{2}\lambda_{\min} (P^{\frac{1}{2}}G_d P^{-\frac{1}{2}} + P^{-\frac{1}{2}}G_d^T P^{\frac{1}{2}})$ under additional conditions $\Gamma \le \kappa P$, $\Lambda \neq 0$, $PG_d + G_d^T P > 0$ and $Q_3 \ge \begin{bmatrix} qP & O_{n \times l} \\ * & O_{l \times l} \end{bmatrix}$ for some $q, \kappa \in \mathbb{R}_+$.

**Remark 4** The restriction on $\chi \in \mathbb{R}_+$ (i.e., $\chi \le \rho \min_{i \in \overline{1},l} \{s_i^2, \bar{s}_i^2\}$), aimed at satisfying the condition $y^T Q_3 y \ge 0$ and the condition C4 from Theorem 1, is quite conservative and could be relaxed with a more detailed analysis of the nonlinearities f.

Stable homogeneous systems are ISS with respect to additive disturbances and measurement noise (see$^{5,22}$). In this context, to verify the ISS property, it suffices to analyze stability of the system for $d = 0$.

**Corollary 1.** Let all conditions of Theorem 2 be satisfied for $R = 0$, $\rho = 0$ and $Q_1 = \begin{bmatrix} X\tilde{A}_0 + \tilde{A}_0^T X & X\tilde{A}_1 + \tilde{A}_0^T U^T \Lambda & O_{n \times n} \\ * & \Lambda UA_1 + A_1^T U^T \Lambda & O_{n \times n} \\ * & * & O_{n \times n} \end{bmatrix}$. Then the system (4) is ISS for $d \in \mathcal{L}_\infty(\mathbb{R}^n)$, and for $d \equiv 0$ the origin is globally finite-time stable.

*Proof.* The proof follows the same arguments as one of Theorem 2 for $d \equiv 0$ and ISS properties of homogeneous systems given in 5 and 22, Theorem 4. $\square$

## 4.3 | Relaxation of Theorem 2

Although the assumptions underlying Theorem 2 ensure a tractable LMI-based stability analysis, they may appear restrictive from a practical viewpoint. In particular, these assumptions implicitly impose structural constraints on the system matrices and the nonlinearities, which are not always satisfied in real-world applications:

* matrix $\tilde{A}_0$ must be Hurwitz;

* matrix $UA_1$ must be diagonally stable (this condition can be slightly relaxed by introducing a nonzero matrix $Q_3$, yet remaining quite restrictive).

To overcome these limitations, one may introduce additional quadratic constraints in (10d) of the form $\begin{bmatrix} Q_4 & O_{(n+l) \times n} \\ * & O_{n \times n} \end{bmatrix}$, where $Q_4 \in \mathbb{R}^{(n+l) \times (n+l)}$ is such that $y^T Q_4 y \ge 0$. Note that $Q_4$ is not assumed to be positive semidefinite matrix. Instead, it is only required to satisfy the nonnegativity condition $y^T Q_4 y \ge 0$ on the subset of variables induced by the system states and the sector constraints. The introduction of the matrix $Q_4$ allows incorporating supplementary information about the system dynamics or the nonlinearities $f_i$ into the LMI conditions, without violating the validity of the stability analysis. Thus, instead of (10d) in Theorem 2, one can use the inequality

$$ Q_1 + \alpha \begin{bmatrix} Q_2 & O_{(n+l) \times n} \\ * & O_{n \times n} \end{bmatrix} + \begin{bmatrix} Q_3 + Q_4 & O_{(n+l) \times n} \\ * & O_{n \times n} \end{bmatrix} \le 0. \quad (15) $$

The construction of the matrix $Q_4$ depends on the specific application and the structure of the considered system. Practical guidelines and illustrative examples for forming $Q_4$ in different cases, including observer design and consensus problems, are provided in Section 5.
# 5 | APPLICATIONS

Although Assumptions 1–3 may appear rather restrictive, there exists a wide range of applications requiring the analysis of systems of the form (4) that satisfy these assumptions. Such applications include finite-time control algorithms¹⁸, homogeneous observers¹²,²¹, and finite-time consensus protocols for multi-agent systems²⁰,¹⁹,²³.

This section is organized as follows. We first present several representative application examples, illustrating how Theorem 2 can be applied to concrete problems such as design of homogeneous observers and finite-time consensus protocols. Based on these examples, a generalized application approach is then formulated at the end of the section, summarizing the key steps of the proposed methodology.

The remainder of this section generalizes previously obtained conference results²³,²⁴,²⁵. For simplicity of exposition, all subsequent examples are presented in the disturbance-free case; however, the approach can be directly extended to systems with external disturbances.

## 5.1 | Homogeneous observers

Consider the problem of finite-time observer design for the linear system

$$ \begin{gathered} \dot{x} = Ax, \quad y = Cx, \quad x \in \mathbb{R}^n, \\ A = \begin{bmatrix} O_{(n-1) \times 1} & I_{n-1} \\ 0 & O_{1 \times (n-1)} \end{bmatrix}, \quad C = \begin{bmatrix} 1 & O_{1 \times (n-1)} \end{bmatrix} \end{gathered} \tag{16} $$

with the following homogeneous observer¹²

$$ \dot{x} = A\hat{x} + \operatorname{diag}\{l_i\}_{i=1}^n [\mathbf{1}_n(\hat{x}_i - y)]^\sigma, \tag{17} $$

where $\mu \in (-1/n, 0)$, $\sigma = [1+\mu \ 1+2\mu \ \dots \ 1+n\mu]^T$ and $l_i \in \mathbb{R}$ for $i = 1, \dots, n$. For $e(t) = \hat{x}(t) - x(t)$ the error system takes the form

$$ \dot{e}(t) = Ae(t) + \operatorname{diag}\{l_i\}_{i=1}^n [\mathbf{1}_n e_1]^\sigma, \tag{18} $$

which corresponds to (4) satisfying Assumptions 1-3 with $A_0 = A$, $A_1 = \operatorname{diag}\{l_i\}_{i=1}^n$, $U = [\mathbf{1}_n \ O_{n \times (n-1)}]$, $f(Ue(t)) = [\mathbf{1}_n e_1]^\sigma$, and $G_d = \operatorname{diag}\{1+(i-1)\mu\}_{i=1}^n$.

The considered observer ensures finite-time convergence of the observation error and, due to its homogeneity, also guarantees the ISS property with respect to additive perturbations and measurement noises. An extension of this observer to MIMO systems, with parameter tuning based on LMIs, is presented in²⁶, while a sliding-mode version of the observer (for $\mu = -1/n$) can be found in²⁷. However, existing stability analysis methods for (18) suffer from several limitations. First, most results are valid for sufficiently small $|\mu|$, values, i.e., for the case when (17) is close to the standard Luenberger observer (see, e.g.,¹²). Second, existing conditions (e.g.,²⁶) for stability of error dynamics and gain tuning remain overly conservative, employing complex parameter-dependent LMIs. Third, many solutions are dimension-specific²⁷,²⁸.

Let us generalize the results from²⁵,²⁴ using Theorem 2 to analyze the system (18).

The ILF (9) in this case takes the form

$$ Q(V, e) = e^T d^T(-\ln V)Pd(-\ln V)e + 2 \sum_{i=1}^{n} \frac{\Lambda_i}{2+i\mu} |V^{-1}e_i|^{2+i\mu} - \chi. \tag{19} $$

The main challenge in this case stems from the fact that $UA_1$ is not a diagonally stable matrix. To address this issue, one can construct matrices $Q_3$ and $Q_4$ to relax the LMI constraints. Following Remark 2, for $n = 3$ the matrix $Q_3$ can be expressed with $\Upsilon_{0,1} = \operatorname{diag}\{\xi_i\}_{i=1}^3$ and $\Upsilon_{1,1} = \begin{bmatrix} -2(\xi_2+\xi_3+\xi_5+\xi_6) & \xi_5 & \xi_6 \\ \xi_5 & -2\xi_4 & \xi_4 \\ \xi_6 & \xi_4 & 0 \end{bmatrix}$, where $\xi_i \ge 0$, $i = 1, 6$. Here the following inequalities $|\hat{f}_i(Ud(-\ln V)e)| \le |\hat{f}_{i+1}(Ud(-\ln V)e)|$ for $i = 1, 2$ and $V^{-1}e_i \hat{f}_i(Ud(-\ln V)e) \ge |\hat{f}_1(Ud(-\ln V)e)|^2$ for $j = 2, 3$ are taken into account, where $\hat{f}_i(s) := f_i(s) - s$. For other system orders, $Q_3$ can be constructed similarly.

Let us now address the formulation of matrix $Q_4$. Choose $\Psi = I_n$, and let

$$ P \ge \tilde{P} \ge 0, \tag{20} $$
where $\tilde{P} \in \mathbb{D}_+^n$. First, let us show that for $\chi = \tilde{P}_1 + 2 \sum_{i=1}^n \frac{\Lambda_i}{2+i\mu}$, where $\tilde{P}_1 := e_n^T(1)\tilde{P}e_n(1)$, one have $|V^{-1}e_1| \le 1$. Suppose the contrary, i.e., $|V^{-1}e_1| > 1$. Note that from (19) we have

$$ \tilde{P}_1(V^{-1}e_1)^2 + 2 \sum_{i=1}^n \frac{\Lambda_i}{2+i\mu} |V^{-1}e_1|^{2+i\mu} \le \chi - e^T d^T(-\ln V) R \tilde{P} d(-\ln V) e \\ \le \tilde{P}_1 + 2 \sum_{i=1}^n \frac{\Lambda_i}{2+i\mu}, \qquad (21) $$

where $R = \begin{bmatrix} 0 & O_{1 \times (n-1)} \\ O_{(n-1) \times 1} & I_{n-1} \end{bmatrix}$. The contradiction follows from

$$ |V^{-1}e_1| \le \frac{\tilde{P}_1 + 2 \sum_{i=1}^n \frac{\Lambda_i}{2+i\mu}}{P_1 |V^{-1}e_1| + 2 \sum_{i=1}^n \frac{\Lambda_i}{2+i\mu} |V^{-1}e_1|^{1+i\mu}} \\ \le 1. $$

Note that since $|V^{-1}e_1| \le 1$, from (21) we have

$$ (V^{-1}e_1)^2 \le \frac{\chi - e^T d^T(-\ln V) R \tilde{P} d(-\ln V) e}{P_1 + 2 \sum_{i=1}^n \frac{\Lambda_i}{2+i\mu}}. \qquad (22) $$

Define $\hat{f}_i(s) := f_i(s) - s$ for $i = \overline{1,n}$, $s \in \mathbb{R}$ and $\hat{f}(s) = [\hat{f}_1(s) \dots \hat{f}_n(s)]^T$. Then, using the inequality (22) and taking into account $\text{sign}(e_1)\hat{f}_i(Ud(-\ln V)e) \ge 0$, one can obtain the following upper estimation of $|\hat{f}_i(Ud(-\ln V)e)|$:

$$ \begin{aligned} |\hat{f}_i(Ud(-\ln V)e)| &= \left| |V^{-1}e_1|^{1+i\mu} \text{sign}(e_1) - V^{-1}e_1 \right| \\ &= |V^{-1}e_1|^{1+i\mu} - V^{-1}|e_1| \\ &\le \left( \frac{\chi - e^T d^T(-\ln V) R \tilde{P} d(-\ln V) e}{\tilde{P}_1 + 2 \sum_{i=1}^n \frac{\Lambda_i}{2+i\mu}} \right)^{\frac{1+i\mu}{2}} - V^{-1}|e_1| \\ &= (1 - \chi^{-1} e^T d^T(-\ln V) R \tilde{P} d(-\ln V) e)^{\frac{1+i\mu}{2}} - V^{-1}|e_1|, \end{aligned} $$

and using the reverse Bernoulli's inequality‡

$$ \begin{aligned} |\hat{f}_i(Ud(-\ln V)e)| &\le \left(1 - \frac{1+i\mu}{2}\chi^{-1}e^T d^T(-\ln V)R\tilde{P}d(-\ln V)e\right) - V^{-1}|e_1| \\ &= \chi^{-1}e^T d^T(-\ln V)Pd(-\ln V)e + 2\chi^{-1}\sum_{i=1}^n \frac{\Lambda_i}{2+i\mu}(V^{-1}e_1)^{2+i\mu} - \frac{1+i\mu}{2}\chi^{-1}e^T d^T(-\ln V)R\tilde{P}d(-\ln V)e - V^{-1}|e_1| \\ &\le \chi^{-1}e^T d^T(-\ln V)\left(P - \frac{1+i\mu}{2}R\tilde{P}\right)d(-\ln V)e + \left(2\chi^{-1}\sum_{i=1}^n \frac{\Lambda_i}{2+i\mu} - 1\right)V^{-1}|e_1| \\ &= \chi^{-1}e^T d^T(-\ln V)\left(P - \frac{1+i\mu}{2}R\tilde{P}\right)d(-\ln V)e + \left(\frac{2\sum_{i=1}^n \frac{\Lambda_i}{2+i\mu}}{\tilde{P}_1 + 2\sum_{i=1}^n \frac{\Lambda_i}{2+i\mu}} - 1\right)V^{-1}|e_1| \\ &\le \frac{1}{\tilde{P}_1 + 2\sum_{i=1}^n \frac{\Lambda_i}{2+i\mu}}e^T d^T(-\ln V)\left(P - \frac{1+i\mu}{2}R\tilde{P}\right)d(-\ln V)e. \end{aligned} \qquad (23) $$

On the other hand, for $|V^{-1}e_1| \le 1$ the function $|\hat{f}_i(Ud(-\ln V)e)|$ has a maximum value for $V^{-1}e_1 = \pm (\frac{1}{1+i\mu})^{\frac{1}{i\mu}}$ and

$$ |\hat{f}_i(Ud(-\ln V)e)| \le -i\mu \left(\frac{1}{1+i\mu}\right)^{1+\frac{1}{i\mu}} \qquad (24) $$

for $\forall e \in \mathbb{R}^n$, $V \in \mathbb{R}_+ : Q(V, e) = 0$. Then

$$ (\tilde{P}_1 + 2 \sum_{i=1}^n \frac{\Lambda_i}{2+i\mu}) [\hat{f}_i(Ud(-\ln V)e)]^2 \le -i\mu \left(\frac{1}{1+i\mu}\right)^{1+\frac{1}{i\mu}} e^T d^T(-\ln V) \left(P - \frac{1+i\mu}{2}R\tilde{P}\right) d(-\ln V)e \qquad (25) $$

<footnote>‡ The reverse Bernoulli inequality: if $0 \le r \le 1$ and $x > -1$, then $(1+x)^r \le 1+rx$.</footnote>
that is equivalent to $y^T Q_4 y \ge 0$ with

$$ 
\begin{aligned}
Q_4 &= \begin{bmatrix} Q_{41} & O_{n \times n} \\ O_{n \times n} & Q_{42} \end{bmatrix}, \\
Q_{41} &= \sum_{i=1}^{n} -\beta_i i \mu \left( \frac{1}{1+i\mu} \right)^{1+\frac{1}{i\mu}} \left( P - \frac{1+i\mu}{2} R \tilde{P} \right), \\
Q_{42} &= - \left( \tilde{P}_1 + 2 \sum_{j=1}^{n} \frac{\Lambda_j}{2+j\mu} \right) \mathrm{diag} \{ \beta_i \}_{i=1}^{n}
\end{aligned} 
\quad (26) $$

for some $\beta_i \ge 0$, $i = \overline{1, n}$.

When $\mu$ is chosen sufficiently close to $-1/n$, less conservative LMIs may be obtained through an alternative construction of $Q_4$. Taking into account that from (19) for $V \in \mathbb{R}_+$: $Q(V, e) = 0$ we have

$$ 
\begin{aligned}
\chi &= e^T \mathbf{d}^T (-\ln V) P \mathbf{d} (-\ln V) e + 2 e^T \mathbf{d}^T (-\ln V) U^T \tilde{\Lambda} f (U \mathbf{d} (-\ln V) e) \\
&= e^T \mathbf{d}^T (-\ln V) P \mathbf{d} (-\ln V) e + 2 e^T \mathbf{d}^T (-\ln V) U^T \tilde{\Lambda} \left( \hat{f}(U \mathbf{d} (-\ln V) e) + U \mathbf{d} (-\ln V) e \right) \\
&= y^T \begin{bmatrix} P + 2 U^T \tilde{\Lambda} U & U^T \tilde{\Lambda} \\ * & O_{n \times n} \end{bmatrix} y,
\end{aligned} 
\quad (27) $$

where $\tilde{\Lambda} = \Lambda \mathrm{diag}\{\frac{1}{2+i\mu}\}_{i=1}^n$. Then for $|V^{-1}e_1| \le 1$ we have $|\hat{f}_n(U\mathbf{d}(-\ln V)e)| \le 1 - |V^{-1}e_1|$ and

$$ 
\begin{aligned}
[\hat{f}_n(U\mathbf{d}(-\ln V)e)]^2 &\le 1 - |2V^{-1}e_1| + V^{-2}e_1^2 \\
&\le 1 - 2V^{-1}e_1 \hat{f}_n(U\mathbf{d}(-\ln V)e) + V^{-2}e_1^2 \\
&= 1 - 2V^{-1}e_1 \hat{f}_n(U\mathbf{d}(-\ln V)e) - V^{-2}e_1^2 \\
&= \chi^{-1}y^T \begin{bmatrix} P+2U^T\tilde{\Lambda}U-\chi e_n(1)e_n^T(1) & U^T\tilde{\Lambda}-\chi e_n(1)e_n^T(n) \\ * & O_{n \times n} \end{bmatrix} y.
\end{aligned} 
$$

Then, for $\beta, \beta_i \in \mathbb{R}_+, i = \overline{1, n}$ satisfying LMI

$$ 0 \le \sum_{i=1}^{n} \beta_i \le \beta \le \tilde{P}_1 + 2 \sum_{i=1}^{n} \frac{\Lambda_i}{2+i\mu}, \quad (28) $$

and taking into account $|\hat{f}_i(U\mathbf{d}(-\ln V)e)| \le |\hat{f}_{i+1}(U\mathbf{d}(-\ln V)e)|$, one have $y^T Q_4 y \ge 0$ for

$$ Q_4 = \begin{bmatrix} P + 2U^T \tilde{\Lambda} U - \beta e_n(1) e_n^T(1) & U^T \tilde{\Lambda} - \beta e_n(1) e_n^T(n) \\ * & -\mathrm{diag}\{\beta_i\}_{i=1}^n \end{bmatrix}. \quad (29) $$

The use of (26) became less conservative as $|\mu| \le \mu$ decreases, while (29) is mainly based on the upper bound $|\hat{f}_n(s)| \le 1 - s$ for $s \in [0, 1]$ that is more accurate as $\mu \to -1/n$. In other words, the formulation (29) allows to obtain less conservative stability conditions in the form of LMIs for the highly nonlinear observer (17) with $\mu$ close to $-1/n$.

Combining these results with Theorem 2, we obtain the following:

**Theorem 3** (Application of Theorem 2 to homogeneous observers). Let the gains $l_i \in \mathbb{R}$ for $i = \overline{1, n}$ be chosen such that $A_0 + A_1 U$ is Hurwitz. Let the LMI system (10a), (10b), (15), (20) (and (28)) be feasible with $\Psi = I_n$ and $Q_4$ in the form (26) (or (29)). Then the observer (17) provides finite-time convergence of the observation error and $T(e_0) \le -\frac{V_0^{V_0}}{\alpha \mu}$, where $Q(V_0, e_0) = 0$ with $\chi = \tilde{P}_1 + 2 \sum_{i=1}^n \frac{\Lambda_i}{2+i\mu}$.

## 5.2 | Finite-time consensus protocols

Multi-agent consensus problems (e.g., average consensus, uniform allocation, max/min consensus, clustered consensus, etc.) have received significant attention in decentralized control applications such as formation control, attitude alignment, and flocking behavior$^{29,30,31,32,33}$.
The stability analysis of various finite-time consensus protocols for multi-agent systems can be formulated in terms of system (4) or similar structures (see 20,19,34,35). Theorem 2 also provides a framework for designing new protocols. This subsection illustrates its application to control synthesis for second-order agents based on 1,23.

### 5.2.1 | Protocol for uniform allocation of second-order agents

Consider a group of $n$ mobile agents, where each agent is described by the second order model

$$ \dot{x}_i(t) = Ax_i(t) + Bu_i(t), \quad i = \overline{1,n}, \qquad (30) $$

where $A = \begin{bmatrix} 0 & 1 \\ 0 & 0 \end{bmatrix}$, $B = \begin{bmatrix} 0 \\ 1 \end{bmatrix}$, $x_i(t) = \begin{bmatrix} p_i(t) \\ v_i(t) \end{bmatrix} \in \mathbb{R}^2$ and $p_i(t)$, $v_i(t)$ represent the position and velocity, respectively, $u_i \in \mathbb{R}$ is the control input. Let $p_0, p_{n+1} \in \mathbb{R}$ denote the fixed endpoints of the segment.

The main goal is to design a decentralized control protocol that achieves finite-time equidistant agent allocation between points $p_0$ and $p_{n+1}$ using only local state information and neighbor distances $u_i = u_i(x_i, p_{i-1} - p_i, p_{i+1} - p_i)$ for $i = \overline{1,n}$.

Consider the following nonlinear control protocol

$$ u_i = -k_{i1} \lfloor v_i \rfloor^{1+\nu} - k_{i2} \lfloor \phi_i(p) \rfloor^{\frac{1+\nu}{1-\nu}}, \quad i = \overline{1,n}, \qquad (31) $$

where $\phi_i(p) = \frac{1}{2}(p_i - p_{i-1}) + \frac{1}{2}(p_i - p_{i+1})$, $k_{i1}, k_{i2} \in \mathbb{R}_+$ and $\nu \in (-1, 0)$.

Let $x = [p_1 v_1 p_2 v_2 \cdots p_n v_n]^T$ be the state vector of the multiagent system. Then the dynamics of the overall system can be written in compact form as

$$ \dot{x} = A_0 x + A_1 f(Ux - b), $$

where $A_0 = I_n \otimes A$, $A_1 = I_n \otimes \tilde{A}_i$, $\tilde{A}_i = \begin{bmatrix} 0 & 0 \\ -k_{i2} & -k_{i1} \end{bmatrix}$,

$$ U = \begin{bmatrix} 1 & 0 & -0.5 & 0 & 0 & \cdots & 0 \\ 0 & 1 & 0 & 0 & 0 & \cdots & 0 \\ -0.5 & 0 & 1 & 0 & -0.5 & 0 & \cdots & 0 \\ 0 & 0 & 0 & 1 & 0 & \cdots & 0 \\ \vdots & & & & & & \vdots \\ 0 & 0 & \cdots & 0 & -0.5 & 0 & 1 & 0 \\ 0 & 0 & \cdots & 0 & 0 & 0 & 0 & 1 \end{bmatrix} \in \mathbb{R}^{2n \times 2n}, \quad b = \begin{bmatrix} 0.5p_0 \\ 0 \\ 0 \\ \vdots \\ 0 \\ 0.5p_{n+1} \\ 0 \end{bmatrix} \in \mathbb{R}^{2n}, \quad f(x) = \begin{bmatrix} f_1(x) \\ f_2(x) \\ \vdots \\ f_{2n}(x) \end{bmatrix}, $$

$$ f_i(x) = \lfloor e_i(2n)(Ux + b) \rfloor^{\gamma_i}, \quad \gamma_i = \begin{cases} \frac{1+\nu}{1-\nu}, & \text{if } i \text{ is odd} \\ 1+\nu, & \text{if } i \text{ is even} \end{cases} \quad \text{for } i = \overline{1, 2n}. $$

Note that the matrix $U$ is invertible and $A_0 = A_0 U$ (i.e., $A_0 U^{-1} = A_0$). Then for the variable $z = x - U^{-1}b$ we have

$$ \dot{z} = A_0 z + A_1 f(Uz), \quad z_0 = z(0), \qquad (32) $$

that is in the form (4) satisfying Assumptions 1-3 with $G_d = \text{diag}\{\zeta_i\}_{i=1}^{2n}$, $\zeta_i = \begin{cases} 1-\nu, & \text{if } i \text{ is odd} \\ 1, & \text{if } i \text{ is even} \end{cases}$ and homogeneity degree $\nu$.

Again, since the matrix $UA_1$ is not diagonally stable, Theorem 2 can not be applied directly. In order to overcome this sticking point, let us consider the ILF (9) in the form

$$ Q(V, z) = z^T d^T(-\ln V) P d(-\ln V)z + 2 \sum_{i=1}^{2n} \frac{\Lambda_i}{\gamma_i+1} |e_i(2n) U d(-\ln V)z|^{\gamma_i+1} - 1. \qquad (33) $$

Define $\tilde{z}_i = e_i(2n)Ud(-\ln V)z$, $\tilde{\Lambda} = \Lambda \text{diag}\{\frac{1}{\gamma_i+1}\}_{i=1}^{2n}$ and $\Psi = \psi I_{2n}$ for some $\psi \in \mathbb{R}_+$. Let the inequalities

$$ P \geq U \tilde{P} U, \qquad (34) $$

$$ S \geq \text{diag} \left\{ (1 + \psi)^{\frac{1}{1-\gamma_i}} \right\}_{i=1}^{2n} \qquad (35) $$
are satisfied for some $\tilde{P} \in \mathbb{D}_+^{2n}$, and $S = \tilde{P} + 2\tilde{\Lambda}$. Then, from (33) we have

$$ \sum_{i=1}^{2n} \tilde{P}_i \tilde{z}_i^2 + 2 \frac{\Lambda_i}{\gamma_i + 1} |\tilde{z}_i^{\gamma_i+1}| \le 1 $$

and $|\tilde{z}_i| \le 1$ due to $\tilde{P} + 2\tilde{\Lambda} \ge I_{2n}$ by (35), where $\tilde{P}_i \in \mathbb{R}_+$: $\tilde{P} = \text{diag}\{\tilde{P}_i\}_{i=1}^{2n}$. Note that analogously to (27) from (33) for $V \in \mathbb{R}_+: Q(V, z) = 0$ we have

$$ \begin{aligned} 1 &= z^T \mathbf{d}^T(-\ln V) \mathbf{P} \mathbf{d}(-\ln V)z + 2z^T \mathbf{d}^T(-\ln V) U^T \tilde{\Lambda} f(U \mathbf{d}(-\ln V)z) \\ &= z^T \mathbf{d}^T(-\ln V)(P + 2\psi U^T \tilde{\Lambda} U) \mathbf{d}(-\ln V)z + 2z^T \mathbf{d}^T(-\ln V) U^T \tilde{\Lambda} (f(U \mathbf{d}(-\ln V)z) - \psi U \mathbf{d}(-\ln V)z) \end{aligned} \tag{36} $$
$$ = y^T \left[ \begin{array}{cc} P + 2\psi U^T \tilde{\Lambda} U & U^T \tilde{\Lambda} \\ * & \mathbf{O}_{2n \times 2n} \end{array} \right] y. $$

Let $\beta \in \mathbb{R}_+$ satisfies

$$ \beta \ge \text{trace} \left( S \text{diag} \left\{ \left( \frac{\gamma_i}{\psi} \right)^{\frac{2}{1-\gamma_i}} \left( \frac{\psi}{\gamma_i} - \psi \right)^2 \right\}_{i=1}^{2m} \right). \tag{37} $$

Then, since $(\tilde{z}_i|\gamma_i \text{sign}(\tilde{z}_i) - \psi \tilde{z}_i)^2 \le (\frac{\gamma_i}{\psi})^{\frac{2}{1-\gamma_i}} (\frac{\psi}{\gamma_i} - \psi)^2$ for $\tilde{z}_i \le 1$, we obtain

$$ \begin{aligned} \sum_{i=1}^{2n} S_i [\hat{f}_i(U \mathbf{d}(-\ln V)z)]^2 &= \sum_{i=1}^{2n} S_i [\tilde{z}_i | \gamma_i \text{sign}(\tilde{z}_i) - \psi \tilde{z}_i]^2 \\ &\le \sum_{i=1}^{2n} S_i (\frac{\gamma_i}{\psi})^{\frac{2}{1-\gamma_i}} (\frac{\psi}{\gamma_i} - \psi)^2 \\ &= \sum_{i=1}^{2n} S_i (\frac{\gamma_i}{\psi})^{\frac{2}{1-\gamma_i}} (\frac{\psi}{\gamma_i} - \psi)^2 - \beta + \beta y^T \left[ \begin{array}{cc} P + 2\psi U^T \tilde{\Lambda} U & U^T \tilde{\Lambda} \\ \tilde{\Lambda} U & \mathbf{O}_{2n \times 2n} \end{array} \right] y \end{aligned} \tag{38} $$
$$ \le \beta y^T \left[ \begin{array}{cc} P + 2\psi U^T \tilde{\Lambda} U & U^T \tilde{\Lambda} \\ \tilde{\Lambda} U & \mathbf{O}_{2n \times 2n} \end{array} \right] y. $$

Note that $\text{sign}(\hat{f}_i(U \mathbf{d}(-\ln V)z)) = \text{sign}(\tilde{z}_i)$ and for $S \in \mathbb{D}_+^{2n}$ satisfying (35) we have $\tilde{z}_i \hat{f}_i(U \mathbf{d}(-\ln V)z) \ge \tilde{z}_i^2$. Thus,

$$ y^T \left[ \begin{array}{cc} -2U\Upsilon U & U^T\Upsilon \\ \Upsilon U & \mathbf{O}_{2n \times 2n} \end{array} \right] y \ge 0 \tag{39} $$

for any $\Upsilon \in \mathbb{D}_+^{2n}$, hence, from (38) and (39) we have $y^T Q_4 y \ge 0$ for

$$ Q_4 = \varpi \left[ \begin{array}{cc} \beta(P+2\psi U^T \tilde{\Lambda} U) - 2U^T \Upsilon U & \beta U^T \tilde{\Lambda} + U^T \Upsilon \\ \beta \tilde{\Lambda} U + \Upsilon U & -S \end{array} \right], \tag{40} $$

with any $\varpi \in \mathbb{R}_+$.

Also note that, without using inequality (40), the inequality (35) can be relaxed as

$$ S \ge I_{2n}, \tag{41} $$

and then we obtain $y^T Q_4 y \ge 0$ for

$$ Q_4 = \varpi \left[ \begin{array}{cc} \beta(P + 2\psi U^T \tilde{\Lambda} U) & \beta U^T \tilde{\Lambda} \\ \beta \tilde{\Lambda} U & -S \end{array} \right], \tag{42} $$

for any $\varpi \in \mathbb{R}_+$.

Combining these results with Theorem 2, we obtain the following:

**Theorem 4 (Application of Theorem 2 to uniform allocation of agents).** Let the gains $k_{i1}, k_{i2}$ are chosen such that the LMI system (10a), (10b), (15) and (34), (35) (or (41)), (37) be feasible with $\Psi = \psi I_{2n}, \psi \in \mathbb{R}_+$ and $Q_4$ in the form (40) (resp. (42)). Then the protocol (31) establishes finite-time allocation of the agents (30) along the line segment with endpoints $p_0$ and $p_{n+1}$ and $T(z_0) \le -\frac{V_0^T}{\alpha \nu}$, where $V_0 \in \mathbb{R}_+: Q(V_0, z_0) = 0, z_0 = x_0 - U^{-1}b$.
### 5.2.2 | Finite-time average consensus protocol

Consider a multi-agent system in which the dynamics of each agent are given in the form (30). Agents exchange information over an undirected communication graph $G = \{V, E\}$, where $V = \{v_i, i = \overline{1,k}\}$ is the set of nodes, and $E \subseteq V \times V$ is the set of edges. The set of neighbors of node $v_i$ is denoted by $N_i = \{j : (v_i, v_j) \in E\}$. The total number of edges in the graph is denoted by $m = |E|$. Let $A = [a_{ij}] \in \mathbb{R}^{k \times k}$ be the symmetric adjacency matrix of the graph $G$, where $a_{ij} > 0$ if $(v_i, v_j) \in E$, and $a_{ij} = 0$ otherwise. The diagonal degree matrix is defined as $D = \text{diag}\{\sum_{j=1}^{k} a_{ij}\}$, and the Laplacian matrix is given by $L = D - A$. We impose the following assumptions on the system:

* **Assumption 4.** The communication graph $G$ is connected and undirected.

* **Assumption 5.** The initial velocities of the agents satisfy $\sum_{i=1}^{n} v_i(0) = 0$.

The goal of this subsection is to develop a control protocol that satisfies the following criteria:

* ensures finite-time convergence to the average-consensus state, that is, $p_i(t) = p^*$ and $v_i(t) = 0$ for all $t \ge T(x_0)$, where $i = \overline{1,k}$, $T(x_0)$ denotes the settling time, and $p^* := \frac{1}{k} \sum_{i=1}^{k} p_i(0)$;

* exploits only the local information about velocities and the distances of the agent from its neighbors according to the communication topology.

Let us consider the following nonlinear control protocol:

$$u_i = -k_1 a_{ij} [v_i - v_j]^{1+\nu} - k_2 a_{ij} [p_i - p_j]^{\frac{1+\nu}{1-\nu}}, \quad (43)$$

where $i = \overline{1,n}$, $k_1, k_2 \in \mathbb{R}_+$, and $\nu \in (-1, 0)$.

Let us define the vector $\delta = [\delta_1 \delta_2 \cdots \delta_n]^T$, referred to as the disagreement, where $p_i(t) = p^* + \delta_i(t)$. Note that $\sum_{i=1}^{n} \dot{v}_i(t) = 0$, $\forall t \ge 0$ due to (43) and $[\cdot]^\alpha$ is odd function for any $\alpha \ge 0$, and by Assumption 5 we have $\sum_{i=1}^{n} v_i(t) = \sum_{i=1}^{n} v_i(0) = 0$. Since $p_j - p_i = \delta_j - \delta_i$ for all $i \in \overline{1,n}$, $j \in N_i$, and $\dot{\delta}_i(t) = \dot{p}_i(t)$, it follows that $\sum_{i=1}^{n} \dot{\delta}_i(t) = \sum_{i=1}^{n} \dot{p}_i(t) = \sum_{i=1}^{n} v_i(t) = 0, \forall t \ge 0$. Hence,

$$\frac{1}{n} \sum_{i=1}^{n} p_i(t) = \frac{1}{n} \sum_{i=1}^{n} p_i(0) = p^* \Rightarrow \sum_{i=1}^{n} \delta_i(t) = 0, \forall t > 0.$$

Therefore, achieving average-consensus will result from the stability of the system

$$\dot{\delta}_i = v_i, \quad (44) \\ \dot{v}_i = \sum_{j \in N_i} -k_1 a_{ij} [v_i - v_j]^{1+\nu} - k_2 a_{ij} [\delta_i - \delta_j]^{\frac{1+\nu}{1-\nu}}$$

for $i = \overline{1,n}$. Then denoting the state vector of the system (44) as $x = [\delta_1 \cdots \delta_n v_1 \cdots v_n]^T$, the system (44) takes the form (4), where $A_0 = \begin{bmatrix} O_{n \times n} & I_n \\ O_{n \times n} & O_{n \times n} \end{bmatrix}$, and $A_1 \in \mathbb{R}^{2n \times 2m}$, $U \in \mathbb{R}^{2m \times 2n}$, $f: \mathbb{R}^{2m} \to \mathbb{R}^{2m}$ with $f_i(s) = [s]^\gamma_i$, $\gamma_i \in \{1+\nu, \frac{1+\nu}{1-\nu}\}$, $i = \overline{1,2m}$ are in accordance with the communication topology, and Assumptions 1-3 are satisfied with $G_d = \text{diag}\{(1-\nu)I_n, I_n\}$.

Depending on the communication topology, two critical scenarios may occur: loss of Hurwitz property for $\tilde{A}_0$, and failure of diagonal stability for $UA_1$. In order to overcome these issues, let us perform an additional analysis.

**I. The matrix $\tilde{A}_0$ is not Hurwitz.** For example, in$^{36}$ it is shown that for $\Psi = I_m$, $k_1 = 1$ the matrix $\tilde{A}_0 = \begin{bmatrix} O_{n \times n} & I_n \\ -L & -k_2L \end{bmatrix}$ has at least two zero eigenvalues. In this case one can use the properties of the Laplacian matrix $L$ in order to relax used LMIs.

As shown in$^{37,38,29}$, the Laplacian matrix $L$ of an undirected graph has the following properties:

* $L$ is positive semidefinite and always has at least one zero eigenvalue. The zero eigenvalue is simple if and only if the graph is connected.

* The quadratic form $x^T L x = \frac{1}{2} \sum_{i,j=1}^{n} a_{ij}(x_j - x_i)^2$ is nonnegative for all $x \in \mathbb{R}^n$.

* If the graph is connected, the second smallest eigenvalue $\lambda_*(L)$, known as the algebraic connectivity, satisfies

$$\min_{\substack{x \neq 0 \\ \mathbf{1}_n^T x = 0}} \frac{x^T L x}{\|x\|^2} = \lambda_*(L) > 0, \quad (45)$$
meaning that $x^T L x \ge \lambda_*(L)\|x\|^2$ whenever $\sum_{i=1}^n x_i = 0$.

Due to $G_d = \text{diag}\{(1 - \nu)I_n, I_n\}$, denote $d(s) = \text{diag}\{d_i(s)\}_{i=1}^2$, where $d_1(s) = e^{(1-\nu)s}I_n$ and $d_2(s) = e^sI_n$ for any $s \in \mathbb{R}$. Then, due to $\sum_{i=1}^n \delta_i(t) = \sum_{i=1}^n v_i(t) = 0, \forall t \ge 0$, we have $\sum_{i=1}^n e_i^T(n)d_1(-\ln V)\delta = 0, \sum_{i=1}^n e_i^T(n)d_2(-\ln V)v = 0$ and by (45)

$$y^T \left[ \begin{matrix} \text{diag}\{q_i\}_{i=1}^2 \otimes (L - \lambda_* I_k) & \mathbf{O}_{2k \times 2m} \\ * & \mathbf{O}_{2m \times 2m} \end{matrix} \right] y \ge 0 \quad (46)$$

for any $q_1, q_2 \in \mathbb{R}_+$.

**II. The matrix $UA_1$ is not diagonally stable.** Choose $\chi = 1$ and define $\Psi = \psi I_m$ for some $\psi \in \mathbb{R}_+$. Let us use the same idea as in the previous subsection, where under the inequalities (34) and

$$\tilde{P} + 2\tilde{\Lambda} \ge I_{2m}; \quad (47)$$

one can obtain $|\tilde{x}_i| \le 1$ for $\tilde{x}_i = e_i^T(2m)Ud(-\ln V)x$.

Then, for

$$\beta \ge \text{trace} \left( S \text{diag} \left\{ \left( \frac{\gamma_i}{\psi} \right)^{\frac{2}{1-\gamma_i}} \left( \frac{\psi}{\gamma_i} - \psi \right)^2 \right\}_{i=1}^{2m} \right), \quad (48)$$

similarly to (38) for the matrix $S = \text{diag}\{S_i\}_{i=1}^{2m}$, one can obtain

$$\begin{align}
\sum_{i=1}^{2m} S_i [\hat{f}_i(Ud(-\ln V)x)]^2 &= \sum_{i=1}^{2m} S_i [|\tilde{x}_i|^{\gamma_i} - \psi \tilde{x}_i]^2 \nonumber \\
&\le \sum_{i=1}^{2m} S_i \left(\frac{\gamma_i}{\psi}\right)^{\frac{2}{1-\gamma_i}} \left(\frac{\psi}{\gamma_i} - \psi\right)^2 - \beta + \beta y^T \left[ \begin{matrix} P + 2\psi U^T \tilde{\Lambda} U & U^T \tilde{\Lambda} \\ \tilde{\Lambda} U & \mathbf{O}_{2m \times 2m} \end{matrix} \right] y \tag{49} \\
&\le \beta y^T \left[ \begin{matrix} P + 2\psi U^T \tilde{\Lambda} U & U^T \tilde{\Lambda} \\ \tilde{\Lambda} U & \mathbf{O}_{2m \times 2m} \end{matrix} \right] y. \nonumber
\end{align}$$

Finally, combining (46) with (49) we have $y^T Q_4 y \ge 0$ for

$$Q_4 = \left[ \begin{matrix} \beta(P + 2\psi U^T \tilde{\Lambda} U) + \text{diag}\{q_i\}_{i=1}^2 \otimes (L - \lambda_*(L)I_k) & \beta U^T \tilde{\Lambda} \\ * & -S \end{matrix} \right]. \quad (50)$$

Combining these results with Theorem 2, we obtain the following:

**Theorem 5 (Application of Theorem 2 to finite-time average consensus).** Consider the system (30) satisfying Assumptions 4, 5. Let $k_1, k_2$ are chosen such that the LMI system (10a), (10b), (15), (34), (47), (48) be feasible with $\Psi = \psi I_{2m}, \psi \in \mathbb{R}_+$ and $Q_4$ in the form (50). Then, under the control protocol (43), the system (30) solves the average-consensus problem in a finite time.

## 5.3 | Generalization

The presented applications are based on the stability analysis of homogeneous systems with sector nonlinearities (Section 4). The general approach can be summarized as follows:

1. transformation of the original system into the form (4);

2. analysis of sector nonlinearities on the unit sphere in order to relax the sufficient LMI-based stability conditions (i.e., to find auxiliary matrix $Q_4 \in \mathbb{R}^{(n+l) \times (n+l)}$ such that $y^T Q_4 y \ge 0$);

3. application of Theorem 2 (Corollary 1) with (15) instead of (10d).

**Remark 5** As explained in subsection 4.3, there is no universal or algorithmic recipe for constructing the auxiliary matrix $Q_4$. Its role is to incorporate additional quadratic constraints that relax the sufficient LMI-based stability conditions. As demonstrated in Section 5, the construction of the matrix $Q_4$ can be based on an analysis of the sector-bounded nonlinearities on the unit homogeneous sphere. Since the considered systems are homogeneous, it is sufficient to study the behavior of the nonlinear terms on a compact set induced by the dilation, where tighter bounds and algebraic relations between the linear and nonlinear components can often be identified. These relations can then be presented in the form of quadratic inequalities $y^T Q_4 y \ge 0$,
which are incorporated into the LMI conditions without compromising the validity of the stability analysis. In addition, as shown in Section 5.2.2, the matrix $Q_4$ may also be constructed based on alternative principles, such as exploiting structural properties of the Laplacian matrix in multi-agent systems.

## 5.4 | Numerical studies

Note that the analysis based on Theorem 2 can be useful not only for deriving new results but also for revisiting established ones, as it may enable: less conservative stability conditions, quantification of compensable disturbances, and new settling time estimates. To demonstrate these advantages of the proposed approach, let us revisit the average consensus problem for a multi-agent system solved in$^{20}$ for agents with simple dynamics $\dot{x}_i = u_i$ and the finite-time protocol $u_i = \sum_{j=1}^n \varphi_{ij}(x_j - x_i)$, $i = \overline{1, n}$, where $\varphi_{ij}(x_j - x_i) = a_{ij} |x_j - x_i|^\mu \text{sign}(x_j - x_i)$. Consider the simplest case with $n = 3$, $a_{ij} = 1$ for $i, j = \overline{1, 3}$ (to ensure square matrices $A_1, U$ and enable comparison with the results of$^8$), and $\mu = 0.5$. This multi-agent system can be represented in the form (4) with $A_0 = 0, A_1 = \begin{bmatrix} 1 & -1 & 0 \\ -1 & 0 & 1 \\ 0 & 1 & -1 \end{bmatrix}$, $U = \begin{bmatrix} -1 & 1 & 0 \\ 1 & 0 & -1 \\ 0 & -1 & 0 \end{bmatrix}$, and both approaches (Theorem 2 and the one from$^8$) can be applied. This example demonstrates the following advantages of the proposed approach:

* **Less conservative settling time estimates.** For the initial state $x_0 = [20, 0, 0]^T$, the observed convergence time is $T(x_0) \approx 3$ seconds, while the settling time estimates computed using Theorem 2 and the approach from$^8$ are 9.7 and 16.2 seconds, respectively.

* **Applicability to broader system classes.** For $n > 3$, the approach from$^8$ may be inapplicable since the matrices $A_1, U$ become non-square, whereas Theorem 2 remains valid.

* **Quantification of compensable disturbances.** Based on Theorem 2, numerical estimates of compensable disturbances can be obtained, while$^{20}$ considered only the disturbance-free case.

Note also that the proposed approach provides a unified framework for the synthesis of other homogeneity-based applications. Since the studied analysis or design problem is highly nonlinear, it is hard to detail the approach. Unlike existing methods, the proposed approach enables further development of the presented control protocols for more complex scenarios. For example, if the agent dynamics are described by a chain of third-order integrators, the protocol (43) can be extended to the following form:

$$u_i = \sum_{j \in N_i} -k_1 a_{ij} [\tilde{a}_i - \tilde{a}_j]^{1+\mu} - k_2 a_{ij} [v_i - v_j]^{\frac{1+\mu}{1-\mu}} - k_3 a_{ij} [p_i - p_j]^{\frac{1+\mu}{1-2\mu}}, \quad i = \overline{1, n},$$

where $\tilde{a}_i$ denotes the acceleration of agent $i$. The stability of the resulting system can then be analyzed using the same steps as previously described.

Another example is the extension to heterogeneous multi-agent systems. For instance, in the case of a uniform allocation task involving agents modeled by first- and second-order integrator chains, the proposed framework allows designing a control law of the form:

$$u_i = \begin{cases} -k_2 \lfloor \varphi_i(p) \rfloor^{\frac{1}{1+\nu}}, & \text{for 1st-order agents,} \\ -k_1 \lfloor v_i \rfloor^{1+\nu} - k_2 \lfloor \varphi_i(p) \rfloor^{\frac{1+\nu}{1-\nu}}, & \text{for 2nd-order agents,} \end{cases} \quad (51)$$

where $k_1, k_2 \in \mathbb{R}_+$, and $\phi_i$ is as in (31). Consider, for instance, a system with four agents, where the first and third are first-order agents and the second and fourth are second-order agents. In this case, using the change of variables $z = x - U^{-1}b$, as in Subsection 5.2.1, we obtain a system of the form (4), where

$$A_0 = \begin{bmatrix} 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 0 \end{bmatrix}, \quad A_1 = \begin{bmatrix} -k_2 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & -k_1 & -k_2 & 0 & 0 \\ 0 & 0 & 0 & -k_2 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & -k_1 - k_2 \end{bmatrix}, \quad U = \begin{bmatrix} 1 & -0.5 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ -0.5 & 1 & 0 & -0.5 & 0 \\ 0 & -0.5 & 0 & 1 & -0.5 \\ 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & -0.5 & 1 \end{bmatrix}, \quad b = \begin{bmatrix} 0.5p_0 \\ 0 \\ 0 \\ 0 \\ 0 \\ 0.5p_5 \end{bmatrix},$$

and Assumption 2 hold with the homogeneity degree $\nu$ and the generator matrix $G_d = \text{diag}\{1 - \nu, 1 - \nu, 1, 1 - \nu, 1 - \nu, 1\}$. Then, by applying the sector nonlinearity analysis from Subsection 5.2.1 and obtaining a matrix $Q_4$ analogous to (42) with the corresponding dimensions, we yield that the considered heterogeneous multi-agent system with the control (51) is finite-time
stable for $\nu = -0.5$ and $k_1 = k_2 = 2$ (the LMI system (10a), (10b), (15), (41), (37) is feasible with $\beta = 1, \omega = 0.1, \psi = 10, \alpha = 0.1$). Figure 1 presents simulation plots showing the position error norm $\|e_p\|$ ($e_4^T(i)e_p = p_i - 0.2i(p_0 - p_5)$ for $i = 1, \overline{2}$) of the agents in a logarithmic scale (to demonstrate finite-time convergence) and the positions of the agents.

<figure><img src="image_3.png" /><figcaption>FIGURE 1 Simulation plots of position error norm $\|e_p\|$ and agents' positions $p_i$ for $p_i(0) = 30, v_i(0) = 0, i = 1, \overline{2}$ and $p_0 = 0, p_5 = 100$</figcaption></figure>

Recall that one of the contributions of this paper is the extension of the result from $^8$ to the case of non-square matrices $U$ and $A_1$. This generalization of the system class is relevant not only for consensus protocols but also for other applications, such as homogeneous distributed observers$^{21}$, where similar techniques to those presented in the previous subsections can be applied to relax the LMI conditions of Theorem 2.

It is worth noting that in all the proposed applications, the presented approach does not allow for selecting the control and observer gains via LMIs. These gains must be predetermined. In such cases, since the matrix $\tilde{A}_0$ is assumed to be Hurwitz for the feasibility of the LMI conditions, one of the most appropriate strategies is to choose these gains similarly to their linear counterparts. This is justified by the fact that when $\Psi = I_n$, the matrix $\tilde{A}_0$ coincides with the system matrix of the corresponding linear model. Thus, other linear control, observer and consensus protocols can be homogenized and considered using the proposed approach. In particular, linear consensus algorithms can be transformed into homogeneous ones by replacing linear terms with nonlinear functions of the form $[\cdot]^\alpha$ with appropriate $\alpha \in \mathbb{R}^n$ in order to satisfy Assumption 2. In this sense, all the presented applications can be regarded as homogeneous counterparts of the Luenberger observer and linear control protocols$^{36,39}$.

# 6 | CONCLUSIONS

This paper presents sufficient LMI-based conditions for robust finite-time stability of a class of homogeneous systems with sector nonlinearities. The proposed method simplifies the result of $^8$ and extends the class of admissible systems.

Based on this approach, several applications are considered. In particular, finite-time consensus protocols are developed for multi-agent systems with second-order integrator dynamics, solving the average-consensus and uniform allocation problems. Additionally, a stability analysis method for homogeneous finite-time observers is proposed. All the presented applications can be interpreted as homogeneous counterparts of their linear analogues, which demonstrate the generality and flexibility of the proposed approach.

Future research directions include relaxing the LMI-based conditions to reduce conservatism, as well as the development of other control and observer design methods based on the obtained results. Another important direction is the validation of the
proposed approach through experimental studies and large-scale numerical simulations, which would help to assess its practical applicability and scalability.

## FINANCIAL DISCLOSURE

This work is supported by RSF under grant 24-79-10157.

## CONFLICT OF INTEREST

The authors declare no potential conflict of interests.

## REFERENCES

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