# Global Uniform Ultimate Boundedness of Semi-Passive Systems Interconnected over Directed Graphs

Anes Lazri, Mohamed Maghenem, Elena Panteley, Antonio Loria

► To cite this version:

Anes Lazri, Mohamed Maghenem, Elena Panteley, Antonio Loria. Global Uniform Ultimate Boundedness of Semi-Passive Systems Interconnected over Directed Graphs. CNRS - Laboratoire des signaux et systèmes; CNRS - GIPSA-Lab. 2023, 9 p. hal-04298308

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Submitted on 21 Nov 2023

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# Global Uniform Ultimate Boundedness of Semi-Passive Systems Interconnected over Directed Graphs

Anes Lazri Mohamed Maghenem Elena Panteley Antonio Loría *

September 21, 2023

## Abstract

We analyse the solutions of networked heterogeneous nonlinear systems<fnref n="1" />

$$ \dot{x}_i = f_i(x_i) + u_i, \quad x_i \in \mathbb{R}, \quad i \in \{1, 2, \dots, n\}, \qquad (1) $$

where $f_i: \mathbb{R} \to \mathbb{R}$ is continuous for all $i \in \{1, 2, \dots, n\}$ and the control inputs are set to

$$ u_i := -\gamma \sum_{i=1}^{n} a_{ij}(x_i - x_j), \quad \forall i \in \{1, 2, \dots, n\}, \qquad (2) $$

where $\gamma > 0$ is a coupling gain and $a_{ij} \ge 0$ are interconnection weights. We assume that the closed-loop interconnected systems form a network with an underlying connected directed graph that contains a directed spanning tree. For these systems, we establish global uniform ultimate boundedness of the solutions, under the assumption that each system (1) defines a semi-passive [5] map $u_i \mapsto x_i$. As a corollary, we also establish global uniform global boundedness of the solutions.

## 1 Preliminaries

**Notations.** For $x \in \mathbb{R}^n$, $x^\top$ denotes its transpose, $|x|$ denotes its Euclidean norm, $\text{blkdiag}\{x\} \in \mathbb{R}^{n \times n}$ denotes the diagonal matrix whose $i$th diagonal element is the $i$th element of $x$. For a set $K \subset \mathbb{R}^n$, $|x|_K := \min\{|x - y| : y \in K\}$ denotes the distance of $x$ to the set $K$. For a symmetric matrix $Q \in \mathbb{R}^{n \times n}$, $\lambda_i(Q)$ denotes the $i$th smallest eigenvalue of $Q$. For an invertible matrix $M \in \mathbb{R}^{n \times n}$, $M^{-}$ or $M^{-1}$ denotes its inverse. Given $N \in \mathbb{R}^{n \times n}$, $\text{Ker}(N) := \{v : Nv = 0\}$ denotes the kernel of $N$. A class $K^\infty$ function $\alpha: \mathbb{R}_{\ge 0} \to \mathbb{R}_{\ge 0}$ is continuous, strictly increasing, unbounded, and $\alpha(0) = 0$. Furthermore $\alpha^{-}$ denotes the inverse function of $\alpha$.

### 1.1 On Some Classes of Matrices

A matrix $M := [m_{ij}]$, $(i, j) \in \{1, 2, \dots, n\}^2$, is a $\mathcal{Z}$-matrix if $m_{ij} \le 0$ whenever $i \ne j$. It is an $M$-matrix if it is a $\mathcal{Z}$-matrix and its eigenvalues have non-negative real parts. Equiv- alently, $M := \lambda I_n - B$, where $B$ is a non-negative matrix and $\lambda \ge \rho(B)$, where $\rho(B) := \max\{|\lambda_i(B)| : i \in \{1, 2, \dots, n\}\}$ is the spectral radius of $B$. $M$ is a non-singular $M$-matrix if it is a $\mathcal{Z}$-matrix and its eigenvalues have positive real parts. Equivalently, $M := \lambda I_n - B$, where $B$ is a non-negative matrix and $\lambda > \rho(B) > 0$; see [2, 3] for more details.

<footnote>*M. Maghenem is with University of Grenoble Alpes, CNRS, Grenoble INP, GIPSA-Lab, France. E-mail: mohamed.maghenem@cnrs.fr; E. Panteley and A. Loría are with L2S, CNRS, 91192 Gif-sur-Yvette, France. E-mail: elena.panteley@cnrs.fr and antonio.loria@cnrs.fr. A. Lazri is with L2S, CNRS, Univ Paris-Saclay, France (e-mail: anes.lazri@centralesupelec.fr)</footnote>

<footnote n="1">¹For simplicity, but without loss of generality, we assume that $x \in \mathbb{R}$; all statements hold after pertinent changes in the notation, if $x \in \mathbb{R}^p$, with $p > 1$.</footnote>
## 1.2 Graph Notions

A directed graph or a digraph $G(V, E)$ is characterized by the set of nodes $V = \{1, 2, ..., n\}$, and the set of directed edges $E$. The edge set $E$ consists of ordered pairs, of the form $(k, i)$, that indicate a directed link from node $k$ to node $i$. Given a directed edge $(k, i) \in E$, then node $k$ is called an in-neighbor of node $i$. We assign a positive weight $a_{ik}$ to each edge $(k, i)$. That is, $a_{ik} = 0$ if $(k, i)$ is not an edge. The Laplacian matrix of a digraph is given by

$$ L := \begin{bmatrix} d_1 & -a_{12} & \cdots & -a_{1n} \\ -a_{21} & d_2 & \cdots & -a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ -a_{n-11} & \cdots & d_{n-1} & -a_{n-1n} \\ -a_{n1} & \cdots & -a_{nn-1} & d_n \end{bmatrix} =: D - A, \quad (3) $$

where $d_i := \sum_{j=1}^{n} a_{ij}$ for all $i \in \{1, 2, ..., n\}$, $D$ is the diagonal part of $L$ and $A$ is called the adjacency matrix.

A digraph is strongly connected if, for any two distinct nodes $i$ and $j$, there is a path from $i$ to $j$. The Laplacian matrix of a strongly connected graph admits $\lambda_1(L) = 0$ as an eigenvalue with the corresponding right and left eigenvectors $1_n = [1 \ 1 \ \cdots \ 1]^T$ and $v_o := [v_1 \ v_2 \ \cdots \ v_n]^T$, respectively, where $v_i > 0$ for all $i \le n$.

## 1.3 Graph and Matrix Decomposition

Suppose that the digraph $G$ is connected and contains a spanning tree. Then, it admits a decomposition into a leading strongly connected subgraph $G_\ell \neq \emptyset$ and a subgraph $G_f := G \setminus G_\ell$ of followers; namely, the agents that do not belong to the leading component, and which we call the follower agents. In this case, up to a permutation, the Laplacian $L$ admits the lower-block decomposition

$$ L = \begin{bmatrix} L_\ell & 0 \\ -A_{\ell f} & M_f \end{bmatrix}, \quad (4) $$

where $L_\ell := D_\ell - A_\ell \in \mathbb{R}^{n_\ell \times n_\ell}$ is the Laplacian matrix of the strongly connected component $G_\ell$, the lower-left block $A_{\ell f} \in \mathbb{R}^{n_f \times n- n_\ell}$, $n_f := n - n_\ell$, is a non-negative matrix, and the lower-right block $M_f \in \mathbb{R}^{n_f \times n_f}$ is a non-singular M-matrix. The block $M_f$ can be seen as the sum of the Laplacian matrix $L_f$ corresponding to $G_f$ and a diagonal matrix $D_{\ell f}$ gathering the weights of the interconnections between nodes in $G_\ell$ and the nodes in $G_f$. That is, $M_f = L_f + D_{\ell f}$, where $L_f = D_f - A_f$.

## 1.4 Lyapunov Analysis of a Directed Graph

Consider a network of $n$ single integrators of the form $\dot{x}_i = u_i$ interconnected according to the classical consensus protocol

$$ u_i := - \sum_{j=1}^{n} a_{ij}(x_i - x_j) \quad \forall i \in \{1, 2, \dots, n\}. $$

In closed loop, the network is governed by the linear system $\dot{x} = -Lx$, where $L \in \mathbb{R}^n$ is the Laplacian matrix of a connected di-graph $G$ that contains a directed spanning tree. According to Section 1, we can decompose the state $x$ into $x^\top := [x_l^\top \ x_f^\top]$, where $x_l \in \mathbb{R}^{n_l}$ gathers the states of the leading component and is governed by

$$ \Sigma_\ell : \dot{x}_\ell = -L_\ell x_\ell, $$
and the non-leading component whose state is $x_f \in \mathbb{R}^{n_f}$, are governed by

$$ \Sigma_f : \dot{x}_f = -M_f x_f, $$

on the manifold $\{x_f = 0\}$. In the rest of this section, we overview some Lyapunov-function constructions allowing to prove uniform exponential stability of $\mathcal{A}$ for $\Sigma_\ell$, where

$$ \mathcal{A} := \{x_l \in \mathbb{R}^{n_l} : x_{l1} = x_{l2} = \cdots = x_{l_{n_l}}\}, \qquad (5) $$

and exponential stability of the origin for $\Sigma_f$.

### 1.4.1 Proof of uniform exponential stability of $\mathcal{A}$ for $\Sigma_\ell$

let $v_o := [v_1 v_2 \cdots v_{n_\ell}]^\top$ be a left eigenvector associated to $\lambda_1(L_\ell) = 0$ and $V_o := \text{blkdiag}\{v_o\}$. Based on Lemma 1 in the Appendix, $Q_o := L_\ell^\top V_o + V_o L_\ell$ is symmetric and positive semi-definite, and its kernel is spanned by $\mathbf{1}_{n_\ell}$. Then, the derivative of the Lyapunov function candidate $W(x_\ell) := x_\ell^\top V_o x_\ell$, along the solutions to $\Sigma_\ell$, satisfies

$$ \dot{W}(x_\ell) = -x_\ell^\top (L_\ell^\top V_o + V_o L_\ell) x_\ell \leq -\lambda_2(Q_o) |x_\ell|_{\mathcal{A}_\ell}^2. $$

Now, we let

$$ Z(x_\ell) := (x_\ell - \mathbf{1}_{n_\ell} v_o^\top x_\ell)^\top V_o (x_\ell - \mathbf{1}_{n_\ell} v_o^\top x_\ell), $$

which is positive definite. Its derivative along the solutions of $\dot{x}_\ell = -L_\ell x_\ell$ satisfies

$$ \dot{Z}(x_\ell) = -x_\ell^\top Q_o x_\ell \leq -\lambda_2(Q_o) |x_\ell|_{\mathcal{A}_\ell}^2. \qquad (6) $$

To obtain the previous expression we used $v_o^\top L = 0$, $v_1^\top \mathbf{1}_{n_\ell} = 1$ and that $\mathbf{1}_{n_\ell}$ is in the kernel of $I_{n_\ell} - \mathbf{1}_{n_\ell} v_o^\top$. Moreover, $I_{n_\ell} - \mathbf{1}_{n_\ell} v_o^\top$ is the Laplacian matrix of an all-to-all graph; hence, $\mathbf{1}_{n_\ell}$ spans the kernel of $I_{n_\ell} - \mathbf{1}_{n_\ell} v_o^\top$. Therefore, there exist $\bar{z}, \underline{z} > 0$ such that

$$ \underline{z}|x_\ell|_{\mathcal{A}_\ell}^2 \leq Z(x_\ell) \leq \bar{z}|x_\ell|_{\mathcal{A}_\ell}^2 \quad \forall x_\ell \in \mathbb{R}^{n_\ell}. \qquad (7) $$

Uniform exponential stability of $\mathcal{A}_\ell$ from (6) and (7) and standard Lyapunov-stability theory.

### 1.4.2 Proof of exponential Stability of the Origin for $\Sigma_f$

based on Lemma 2, since $M_f$ is a non-singular $M$-matrix, we can use the Lyapunov function candidate $Y(x_f) := x_f^\top R_f x_f$, where $R_f := \text{blkdiag}\{M_f^{-\top} \mathbf{1}_{n_f}\} (\text{blkdiag}\{M_f^{-1} \mathbf{1}_{n_f}\})^{-1}$, which is positive definite. Furthermore, along the solutions to $\Sigma_f$, we have

$$ \dot{Y}(x_f) = -x_f^\top [M_f^\top R_f + R_f M_f] x_f. $$

Now, since $(M_f^\top R_f + R_f M_f)$ is positive definite, exponential stability of the origin for $\Sigma_f$ follows.

## 2 Problem formulation

Consider the systems (1)-(2), with $\gamma > 0$ and $a_{ij} \geq 0$. Then, defining $x := [x_1 \cdots x_n]^\top$, and $F(x) := [f_1(x_1), f_2(x_2), \cdots, f_n(x_n)]^\top$, we may write the closed-loop system in compact form as

$$ \dot{x} = F(x) - \gamma L x, \qquad (8) $$

where $L$ is defined as in (3). This is a networked system with an underlying topology that may be represented by a graph $\mathcal{G}$.
**Assumption 1** The graph $\mathcal{G}$ is connected and contains a directed spanning tree. •

We are interested in verifying the following two boundedness properties for (8).

(P1) Global Uniform Boundedness (GUB). The solutions $t \to x(t)$ to (8) are globally bounded, uniformly in $\gamma$, if, for every $r_o > 0$ and $\gamma_o > 0$, there exists $R = R(r_o, \gamma_o) \ge r_o$ such that, for all $\gamma \ge \gamma_o$,

$$|x(t_o)| \le r_o \Rightarrow |x(t)| \le R \quad \forall t \ge 0.$$

(P2) Global Uniform Ultimate Boundedness (GUUB). The solutions $t \to x(t)$ to (8) are ultimately bounded, uniformly in $\gamma$, if given $\gamma_o > 0$, there exists $r = r(\gamma_o) > 0$ such that, for all $r_o > 0$, there exists $T = T(r_o, \gamma_o) \ge 0$ such that, for all $\gamma \ge \gamma_o$,

$$|x(t_o)| \le r_o \Rightarrow |x(t)| \le r \quad \forall t \ge T.$$

To verify the latter two properties, we make the following assumption on the individual nodes' dynamics in (1).

**Assumption 2 (State strict semi-passivity)** For each $i \in \{1, 2, ..., n\}$, the input-output map $u_i \mapsto x_i$ defined by the dynamics (1) is state strict semipassive [1]. Furthermore, there exists a continuously differentiable storage function $V_i : \mathbb{R}^n \to \mathbb{R}^+$, a class $K_\infty$ function $\alpha_i$, a constant $\rho_i > 0$, a continuous function $H_i : \mathbb{R} \to \mathbb{R}$, and a continuous function $\psi_i : \mathbb{R}_{\ge 0} \to \mathbb{R}_{> 0}$, such that

$$\underline{\alpha}_i(|x_i|) \le V_i(x_i), \quad \dot{V}_i(x_i) \le 2u_i x_i - H_i(x_i), \qquad (9)$$

and $H_i(x_i) \ge \psi_i(|x_i|)$ for all $|x_i| \ge \rho_i$. •

**Remark 1** The property described in Assumption 2 is called strict quasipassivity in [4]. In [5] the authors define a similar concept named strict semi-passivity, but radial unboundedness of the storage function is not imposed. See also [1]. •

# 3 Main result

**Theorem 1 (Uniform ultimate boundedness)** The solutions of the networked system (1)-(2) are globally uniformly ultimately bounded, i.e., Property (P2) holds, if Assumptions 1 and 2 are satisfied. □

*Proof:* Under Assumption 1, the Laplacian matrix $L$ admits a permutation, such that (4) holds. Therefore, the state $x$ may be decomposed into $x := [x_\ell^\top \ x_f^\top]^\top$ and the system (8) takes the cascaded form

$$\dot{x}_\ell = f_\ell(x_\ell) - \gamma L_\ell x_\ell, \qquad f_\ell(x_\ell) := [f_1(x_{\ell 1}) \ \dots \ f_{n_\ell}(x_{\ell n_\ell})]^\top \quad (10a)$$

$$\dot{x}_f = f_f(x_f) + \gamma A_{\ell f} x_\ell - \gamma M_f x_f, \qquad f_f(x_f) := [f_{n_\ell+1}(x_{f1}) \ \dots \ f_{n_\ell+n_f}(x_{fn_f})]^\top. \quad (10b)$$

Equation (10a) corresponds to the dynamics of a leading component, a networked system with an underlying strongly connected graph $\mathcal{G}_\ell$, and a follower component, with dynamics (10b). The proof of the statement is constructed using a cascades argument and proving, firstly, global uniform ultimate boundedness for the solutions of (10a) and, consequently, the same property for (10b).

To that end, let $r_o > 0$ be arbitrarily fixed and let $|x(0)| \le r_o$. Then, $|x_\ell(0)| \le r_o$ and $|x_f(0)| \le r_o$.
1) Uniform ultimate boundedness for the leading component: after Assumption 2, for each $i \in \{1, 2, ..., n_\ell\}$, there exists a storage function $V_i$ such that its total derivative along the trajectories of (1) satisfies

$$ \dot{V}_i(x_{\ell i}) \leq 2u_i^\top x_{\ell i} - H_i(x_{\ell i}), \quad H_i(x_{\ell i}) \geq \psi_i(|x_{\ell i}|) \quad \forall |x_{\ell i}| \geq \rho_i. \quad (11) $$

Next, let $W(x_\ell) := \sum_{i=1}^{n_\ell} v_i V_i(x_{\ell i})$, where $v_i$ corresponds to the $i$th element of $v_o$, which is the left eigenvector associated to the zero eigenvalue of $L_\ell$. Since the graph $G_\ell$ is strongly connected, then $v_i > 0$ for all $i \le n_\ell$, so $W$ is positive definite and radially unbounded. Now, from (11), we obtain

$$ \dot{W}(x_\ell) \leq 2 \sum_{i=1}^{n_\ell} v_i u_i^\top x_{\ell i} - \sum_{i=1}^{N} v_i H_i(x_{\ell i}), \quad \forall x_\ell \in \mathbb{R}^{n_\ell}. \quad (12) $$

The first term on the right-hand side of (12) satisfies

$$ \sum_{i=1}^{n_\ell} v_i u_i^\top x_{\ell i} = u^\top V_o x_\ell, \quad (13) $$

where $V_o := \text{blkdiag}\{v_o\}$ and, since $u = -\gamma L_\ell x_\ell$, it follows that

$$ \begin{aligned} \dot{W}(x_\ell) &\leq -\sum_{i=1}^{n_\ell} v_i H_i(x_{\ell i}) - \gamma x_\ell^\top [L_\ell^\top V_o + V_o L_\ell] x_\ell \\ &\leq -\sum_{i=1}^{n_\ell} v_i H_i(x_{\ell i}) - \gamma x_\ell^\top Q_o x_\ell, \end{aligned} \quad (14) $$

with $Q_o := V_o L_\ell + L_\ell^\top V_o$, which is positive semi-definite—see Lemma 1 in the Appendix. Furthermore, we note that

$$ -x_\ell^\top Q_o x_\ell = - \left[ x_\ell - \mathbf{1}_{n_\ell} \mathbf{1}_{n_\ell}^\top x_\ell / n_\ell \right]^\top Q_o \left[ x_\ell - \mathbf{1}_{n_\ell} \mathbf{1}_{n_\ell}^\top x_\ell / n_\ell \right] \leq -\lambda_2(Q_o) |x_\ell|^2_{\mathcal{A}}, $$

where $|x_\ell|_{\mathcal{A}}$ denotes the distance of $x_\ell$ to the set $\mathcal{A}$ and $\lambda_2(Q_o)$ is the second smallest eigenvalue of $Q_o$.

Now, on one hand, we have that $v_i > 0$ for all $i \in \{1, 2, ..., n\}$ and, on the other, $-H_i(x_{\ell i}) > 0$ only if $|x_{\ell i}| \le \rho_i$. Therefore, the constant $H_\ell := -\sum_{i=1}^{n_\ell} \max_{|x_i| \le \rho_i} \{v_i H_i(x_{\ell i})\} > 0$. Therefore, after (14), we get

$$ \dot{W}(x_\ell) \leq H_\ell - \gamma \lambda_2(Q_o) |x_\ell|^2_{\mathcal{A}} \quad \forall x_\ell \in \mathbb{R}^{n_\ell}. \quad (15) $$

In turn, given $\gamma_o > 0$ and $\epsilon > 0$, for all $\gamma \ge \gamma_o$, we have

$$ \dot{W}(x_\ell) \leq H_\ell - \gamma_o \lambda_2(Q_o) |x_\ell|^2_{\mathcal{A}} \leq -\epsilon \quad \forall x_\ell \notin \mathcal{C}, \quad (16) $$

where

$$ \mathcal{C} := \left\{ x_\ell \in \mathbb{R}^{n_\ell} : |x_\ell|_{\mathcal{A}} \le \sqrt{n_\ell} R_e := \sqrt{\frac{\epsilon + H_\ell}{\gamma_o \lambda_2(Q_o)}} \right\}. $$

Next, let $\bar{\rho} := \arg \max_{i \in \{1, 2, ..., n_\ell\}} \rho_i$ and

$$ \mathcal{B}_\beta := \{x_\ell \in \mathbb{R}^{n_\ell} : |x_\ell| \le \beta := \sqrt{n_\ell}(\bar{\rho} + 2R_e)\}. $$

Note that for all $x_\ell \notin \mathcal{B}_\beta$, we have $|x_\ell| > \sqrt{n_\ell}(\bar{\rho} + 2R_e)$ and, for all $x_\ell \in \mathcal{C} \setminus \mathcal{B}_\beta$,

$$ |x_\ell| > \sqrt{n_\ell}(\bar{\rho} + 2R_e) \quad \text{and} \quad |x_\ell|_{\mathcal{A}} \le \sqrt{n_\ell} R_e. \quad (17) $$
Furthermore, we use the fact that $x_{\ell} = \mathbf{1}_{n_{\ell}}(\mathbf{1}_{n_{\ell}}^{\top}x_{\ell})/n_{\ell} + [x_{\ell} - \mathbf{1}_{n_{\ell}}(\mathbf{1}_{n_{\ell}}^{\top}x_{\ell})/n_{\ell}]$, and the fact that $|x_{\ell}|_{\mathcal{A}} = |x_{\ell} - \mathbf{1}_{n_{\ell}}(\mathbf{1}_{n_{\ell}}^{\top}x_{\ell})/n_{\ell}|$, to conclude that

$$|x_{\ell}| \le |x_{\ell}|_{\mathcal{A}} + |\mathbf{1}_{n_{\ell}}^{\top} x_{\ell}|/\sqrt{n_{\ell}}. \quad (18)$$

Now, combining (17) and (18), we conclude that for all $x_{\ell} \in \mathcal{C}\setminus\mathcal{B}_{\beta}$,

$$\sqrt{n_{\ell}}(\bar{\rho} + 2R_e) < |x_{\ell}| \le |x_{\ell}|_{\mathcal{A}} + |\mathbf{1}_{n_{\ell}}^{\top} x_{\ell}|/\sqrt{n_{\ell}} \le \sqrt{n_{\ell}}R_e + |\mathbf{1}_{n_{\ell}}^{\top} x_{\ell}|/\sqrt{n_{\ell}}. \quad (19)$$

So, for all $x_{\ell} \in \mathcal{C}\setminus\mathcal{B}_{\beta}$, $|\mathbf{1}_{n_{\ell}}^{\top}x_{\ell}|/n_{\ell} > \bar{\rho} + R_e$. Next, we use the fact that

$$x_{\ell i} = \mathbf{1}_{n_{\ell}}^{\top} x_{\ell}/n_{\ell} + (x_{\ell i} - \mathbf{1}_{n_{\ell}}^{\top} x_{\ell}/n_{\ell}) \quad \forall i \in \{1, 2, \dots, n_{\ell}\}$$

to conclude that $|x_{\ell i}| > |\mathbf{1}_{n_{\ell}}^{\top}x_{\ell}|/n_{\ell} - |(x_{\ell i} - \mathbf{1}_{n_{\ell}}^{\top}x_{\ell})/n_{\ell}|$. Hence,

$$|x_{\ell i}| > \bar{\rho} + R_e - \sqrt{n_{\ell}}R_e > \bar{\rho} \quad \forall i \in \{1, 2, \dots, n_{\ell}\}$$

for all $x_{\ell} \in \mathcal{C}\setminus\mathcal{B}_{\beta}$. The latter, under Assumption 2, implies that

$$-\sum_{i=1}^{n_{\ell}} v_i H_i(x_{\ell i}) \le -\sum_{i=1}^{n_{\ell}} v_i \psi_i(|x_{\ell i}|) \le 0 \quad \forall x_{\ell} \in \mathcal{C}\setminus\mathcal{B}_{\beta}.$$

As a result, setting $\Psi(x_{\ell}) := \sum_{i=1}^{n_{\ell}} v_i \psi_i(|x_{\ell i}|)$—note that $\Psi$ is continuous and positive—we conclude that

$$\dot{W}(x_{\ell}) \le -\Psi(x_{\ell}) \quad \forall x_{\ell} \in \mathcal{C}\setminus\mathcal{B}_{\beta}.$$

Combining the latter inequality to (16), we conclude that

$$\dot{W}(x_{\ell}) \le -\min\{\Psi(x_{\ell}), \epsilon\} \quad \forall x_{\ell} \in \mathbb{R}^{n_{\ell}}\setminus\mathcal{B}_{\beta}.$$

The latter is enough to conclude global attractivity and forward invariance of the set

$$S_{\sigma} := \{x_{\ell} \in \mathbb{R}^{n_{\ell}} : W(x_{\ell}) \le \sigma\}, \quad \sigma := \max\{W(y) : y \in \mathcal{B}_{\beta}\}.$$

Furthermore, since $W : \mathbb{R}^n \to \mathbb{R}_{\ge 0}$ is continuous and $\mathcal{B}_{\beta}$ is bounded, we conclude that $\sigma$ is well defined. Consequently, the ultimate bound is

$$r_{\ell} := \left[ \min_{i} \{ \underline{\alpha}_{i} \} \right]^{-} (\sigma),$$

where, with an abuse of notation, $\min_{i}\{\underline{\alpha}_{i}\}$ corresponds to the function $s \mapsto \psi(s)$ defined as $\psi(s) := \min_{i}\{\underline{\alpha}_{i}(s)\}$ for each $s \ge 0$ and $\underline{\alpha}_{i}$ is defined in Assumption 2, so $\psi : \mathbb{R}_{\ge 0} \to \mathbb{R}_{\ge 0}$ is strictly increasing and radially unbounded, hence, globally invertible. Thus, $W(x_{\ell}) \le \sigma$ implies that $|x_{\ell}| \le r_{\ell}$.

Next, we compute an upperbound $T_{\ell}(r_o, \gamma_o)$ on the time that the solutions to (10a), with $\gamma \ge \gamma_o$ and starting from $\mathcal{B}_{r_o} := \{x_{\ell} \in \mathbb{R}^{n_{\ell}} : |x_{\ell}| \le r_o\}$, take to reach the compact set $\mathcal{B}_{\beta} \subset S_{\sigma}$. For this, we assume without loss of generality that $r_o \ge \beta$, and we define

$$\epsilon_{r_o} := \min\{\min\{\Psi(x_{\ell}), \epsilon\} : |x_{\ell}| \ge \beta, x_{\ell} \in S_{\sigma_o}\},$$

where

$$S_{\sigma_o} := \{x_{\ell} \in \mathbb{R}^{n_{\ell}} : W(x_{\ell}) \le \sigma_o\}, \quad \sigma_o := \max\{W(y) : y \in \mathcal{B}_{r_o}\}. \quad (20)$$

Clearly, $S_{\sigma_o}$ is compact; hence, $\epsilon_{r_o}$ is positive.

Therefore, along every solution $t \mapsto x_{\ell}(t)$ to (10a) starting from $x_{\ell}(0) \in \mathcal{B}_{r_o}\setminus\mathcal{B}_{\beta}$, we have $\dot{W}(x_{\ell}(t)) \le -\epsilon_{r_o}$, up to the earliest time when $x_{\ell}$ reaches $\mathcal{B}_{\beta}$. For any earlier time, we have

$$W(x_{\ell}(t)) \le -\epsilon_{r_o}t + W(x_{\ell}(0)), \quad (21)$$
so we can take $T_\ell(r_o, \gamma_o) = \sigma_o/\epsilon_{r_o}$. Clearly, $T_\ell$ depends only on $(r_o, \gamma_o)$ and $r_\ell$ depends only on $\gamma_o$. Thus, the ultimate bounded guaranteed for the solutions of (10a) is uniform in $\gamma$.

2) Uniform ultimate boundedness for the follower dynamics: following up the previous computations and arguments, establish global uniform ultimate boundedness for the non-leading component, determined by (10b).

Using Lemma 2, we conclude that the matrices

$$S := PM_f + M_f^\top P \quad \text{and} \quad P := \text{blkdiag}\{M_f^\top - 1_n\} (\text{blkdiag}\{M_f^- 1_n\})^-$$

are symmetric and positive definite. We also note that $P$ is diagonal. Then, let $p_i$, for $i \in \{1, 2, ..., n_f\}$, be the $i$th diagonal element of $P$. In addition, let $Z(x_f) := \sum_{i=1}^{n_f} p_i V_i(x_{fi})$. Its total derivative along the trajectories of (10b) satisfies

$$\dot{Z}(x_f) \leq - \sum_{i=1}^{n_f} p_i H_i(x_{fi}) - \gamma x_f^\top [PM_f + M_f^\top P] x_f + 2\gamma x_f^\top [PA_{\ell f}] x_\ell. \quad (22)$$

On one hand, we already established the existence of $r_\ell(\gamma_o) > 0$ and $T_\ell(\gamma_o, r_o)$ such that

$$|x_\ell(t)| < r_\ell \quad \forall t \geq T_\ell.$$

On the other, for all $|x_\ell| \leq r_\ell$,

$$\dot{Z}(x_f) \leq H_f - \gamma \lambda_1(S) |x_f|^2 + 2\gamma \bar{p} r_\ell |x_f|, \quad (23)$$

where $\bar{p} := |PA_{\ell f}|$ and $H_f := \sum_{i=1}^{n_f} \max_{|x_i| \leq \rho_i} \{v_i H_i(x_{fi})\}$. Now, from this and (22), we obtain

$$\begin{aligned} \dot{Z}(x_f) &\leq H_f - \gamma x_f^\top S x_f + 2\gamma x_f^\top [PA_{\ell f}] x_\ell \\ &\leq H_f - \gamma \left[x_f^\top S x_f / 2 - 2x_\ell^\top A_{\ell f}^\top P^\top S^{-1} P A_{\ell f} x_\ell\right]. \end{aligned}$$

At the same time, integrating (15), we obtain that, for each $t \in [0, T_\ell]$,

$$W(x_\ell(t)) \leq H_\ell T_\ell + W(x_\ell(0)) \leq H_\ell T_\ell + \sigma_o,$$

where $\sigma_o$ comes from (20). Defining

$$R_\ell := \left[ \min_i \{\underline{\alpha}_i\} \right]^{-} (H_\ell T_\ell + \sigma_o),$$

we have

$$\dot{Z}(x_f) \leq H_f - \gamma \left[ \lambda_1(S) |x_f|^2 / 2 - 2 |A_{\ell f}^\top P^\top S^{-1} P A_{\ell f} | R_\ell^2 \right],$$

for all $|x_\ell| \leq R_\ell$.

Note that, for all $x_f$ such that

$$|x_f|^2 > d_f^2 := \frac{4|A_{\ell f}^\top P^\top S^{-1} P A_{\ell f} |R_\ell^2}{\lambda_1(S)} + \frac{42H_f}{\lambda_1(S)\gamma_o},$$

$\dot{Z}(x_f) \leq 0$. This implies that, for all $t \in [0, T_\ell]$,

$$Z(x_f(t)) \leq \max\{\sigma_{fo}, \sigma_f\}, \quad \sigma_f := \max\{Z(x_f) : |x_f| \leq d_f\} \quad \sigma_{fo} := \max\{Z(x_f) : |x_f| \leq r_o\}. \quad (24)$$

In turn, for each $t \in [0, T_\ell]$,

$$|x_f(t)| \leq \bar{r}_o := \left[ \left( \sum_{i=1}^{n_f} p_i \right) \min_i \{\underline{\alpha}_i\} \right]^{-} (\max\{\sigma_{fo}, \sigma_f\}). \quad (25)$$
Clearly, the previous upper bound is uniform in $\gamma \ge \gamma_o$.

Next, we focus on the solutions' behaviour after $T_\ell$ (i.e., once $|x_\ell| \le r_\ell$). Given $\epsilon > 0$, we see that, for all $\gamma \ge \gamma_o$ and for all $x_f$ and $x_\ell$ such that

$$|x_f| > \beta_1 := 1 + \frac{2\bar{p}r_\ell}{\lambda_1(S)} + \sqrt{\frac{\epsilon + H_f}{\gamma_o\lambda_1(S)}} \quad \text{and} \quad |x_\ell| \le r_\ell,$$

after (23), we conclude that $\dot{Z}(x_f) \le -\epsilon$. Furthermore, $|x_\ell(t)| \le r_\ell$ for all $t \ge T_\ell$, then the set

$S_{\sigma_1} := \{x_f \in \mathbb{R}^{n_f} : Z(x_f) \le \sigma_1\}, \quad \sigma_1 := \max\{Z(y) : y \in \mathcal{B}_{\beta_1}\}, \quad \mathcal{B}_{\beta_1} := \{x_f \in \mathbb{R}^{n_f} : |x_f| \le \beta_1\},$ is attractive and becomes forward invariant after time $T_\ell$.

Since $Z : \mathbb{R}^n \to \mathbb{R}_{\ge 0}$ is continuous and $\mathcal{B}_{\beta_1}$ is bounded, we conclude that $\sigma_1$ is well defined. As a result, the ultimate bound for $x_f(t)$ is

$$r_f = \left[ \left( \sum_{i=1}^{n_f} p_i \right) \min_i \{ \underline{\alpha}_i \} \right]^{-} (\sigma_1).$$

Indeed, $Z(x_f) \le \sigma_1$ implies $|x_f| \le r_f$.

Finally, as for $t \mapsto x_\ell(t)$ we give next an upperbound, denoted by $T_f(r_o, \gamma_o)$, on the time that the solutions to (10b), with $\gamma \ge \gamma_o$ and starting from $\mathcal{B}_{r_o} := \{x_f \in \mathbb{R}^{n_\ell} : |x_f| \le r_o\}$, take to reach $\mathcal{B}_{\beta_1} \subset S_{\sigma_1}$.

Let a solution $t \mapsto x_f(t)$ to (10b) starting from $x_f(0) \in \mathcal{B}_{r_o}$. Now, we use the fact $|x_f(T_\ell)| \le \bar{r}_o$ with $\bar{r}_o$ coming from (25) and $\bar{r}_o$ is uniform in $\gamma$. As a result, along the solution $t \mapsto x_f(t)$, we have $\dot{Z}(x_f(t)) \le -\epsilon$ from $T_\ell$ and up to when it reaches $\mathcal{B}_{\beta_1}$ for the first time after $T_\ell$. Hence, before reaching $\mathcal{B}_{\beta_1}$, we have

$$Z(x_f(t)) \le -\epsilon t + Z(x_f(T_\ell)) \quad (26)$$

and, thus, using (24), we can take $T_f = T_\ell + \max\{\sigma_{f_o}, \sigma_f\}/\epsilon$. Clearly, $T_f$ and $r_f$ depend only on $(r_o, \gamma_o)$. Thus, the ultimate bounded guaranteed for the solutions of (10b) is also uniform in $\gamma$.

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

**Corollary 1 (Uniform boundedness) Under Assumptions 1 and 2 the solutions of the closed-loop system in (8) are globally uniformly bounded, i.e., Property (P1) holds.**

<figure><img src="image_3.png" /></figure>

*Proof:* The statement of Theorem 1 holds, therefore, given $r_o > 0$ and $\gamma_o > 0$, for all $\gamma \ge \gamma_o$, we have

$$|x_\ell(0)| \le r_o \implies |x_\ell(t)| \le r_\ell(\gamma_o) \quad \forall t \ge T_\ell(r_o, \gamma_o).$$

Furthermore, we were able to show that on the interval $[0, T_\ell(r_o, \gamma_o)]$, we have

$$W(x_\ell(t)) \le H_\ell T_\ell + W(x_\ell(0)).$$

Hence, if we let $\sigma_\ell := \max\{W(y) : |y| \le r_o\}$, it follows that

$$|x_\ell(t)| \le R_\ell := \left[ \min_i \{ \underline{\alpha}_i \} \right]^{-} (\sigma_\ell + H_\ell T_\ell + r_\ell) \quad \forall t \ge 0.$$

Next, for the solutions to (10b), for any $\gamma > \gamma_o$ and $|x_f(0)| \le r_o$, we know that

$$|x_f(t)| \le r_f \quad \forall t \ge T_f(\gamma_o, r_o).$$

At the same time, from the previous proof, we know that

$$\dot{Z}(x_f) \le H_f - \gamma [\lambda_1(S)|x_f|^2/2 - 2|A_{\ell f}^\top P^\top S^{-1} P A_{\ell f}|R_\ell^2].$$
As a result, when

$$|x_f|^2 > d_f^2 := \frac{4|A_{\ell f}^T P^T S^{-} P A_{\ell f} |R_{\ell}^2}{\lambda_1(S)} + \frac{42H_f}{\lambda_1(S)\gamma_o},$$

then $\dot{Z}(x_f) \le 0$. Hence, for each $t \ge 0$,

$$Z(x_f(t)) \le \max\{\sigma_{fo}, \sigma_f\}, \quad \sigma_f := \max\{Z(x_f) : |x_f| \le d_f\}, \quad \sigma_{fo} := \max\{Z(x_f) : |x_f| \le r_o\}. \qquad (27)$$

In turn, for each $t \ge 0$, we have

$$|x_f(t)| \le R_f := \left[ \left( \sum_{i=1}^{n_f} p_i \right) \min_i \{\underline{\alpha}_i\} \right]^{-} (\max \{\sigma_{fo}, \sigma_f\}). \qquad (28)$$

<figure><img src="image_4.png" /></figure>

## Appendix

The following lemma is proposed in [3], see also [6].

**Lemma 1** Let $L \in \mathbb{R}^{n \times n}$ be the Laplacian matrix of a directed and strongly connected graph. Let $v_o := [v_1, v_2, ..., v_n]^T \in \mathbb{R}^n$ be the left eigenvector of $L$ associated to the null eigenvalue of $L$. Then, the vector $v$ has strictly positive entries and, for $V_o := \text{blkdiag}(v_o)$, we have $\text{Ker}(V_o L + L^T V_o) = \text{Span}(1_n)$ and $V_o L + L^T V_o \ge 0$. □

The next result can be deduced from [6, Section 4.3.5].

**Lemma 2** Let $M \in \mathbb{R}^{n \times n}$ be a non-singular M-matrix. Then, the matrices

$$S := RM + M^T R \quad \text{and} \quad R := \text{blkdiag}\{M^{-1}1_n\} (\text{blkdiag}\{M^{-1}1_n\})^{-}$$

are positive definite. □

<figure><img src="image_5.png" /></figure>

## References

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[4] I. G. Polushin, D. Hill, and A. L. Fradkov. Strict quasipassivity and ultimate boundedness for nonlinear control systems. *IFAC Proceedings Volumes*, 31(17):505–510, 1998. 4th IFAC Symposium on Nonlinear Control Systems Design 1998 (NOLCOS’98), Enschede, The Netherlands, 1-3 July.

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