# Simultaneous Topology Estimation and Synchronization of Dynamical Networks with Time-varying Topology

Nana Wang, Esteban Restrepo, Dimos V Dimarogonas

► To cite this version:

Nana Wang, Esteban Restrepo, Dimos V Dimarogonas. Simultaneous Topology Estimation and Synchronization of Dynamical Networks with Time-varying Topology. CDC 2024 - 63rd IEEE Conference on Decision and Control, Dec 2024, Milano, Italy. pp.1-8. [hal-04871129](https://hal.science/hal-04871129v1)

HAL Id: hal-04871129

https://hal.science/hal-04871129v1

Submitted on 7 Jan 2025

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

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

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

Distributed under a Creative Commons Attribution 4.0 International License
# Simultaneous Topology Estimation and Synchronization of Dynamical Networks with Time-varying Topology

Nana Wang, Esteban Restrepo, and Dimos V. Dimarogonas

**Abstract**— We propose an adaptive control strategy for the simultaneous estimation of topology and synchronization in complex dynamical networks with unknown, time-varying topology. Our approach transforms the problem of time-varying topology estimation into a problem of estimating the time-varying weights of a complete graph, utilizing an edge-agreement framework. We introduce two auxiliary networks: one that satisfies the persistent excitation condition to facilitate topology estimation, while the other, a uniform-$\delta$ persistently exciting network, ensures the boundedness of both weight estimation and synchronization errors, assuming bounded time-varying weights and their derivatives. A relevant numerical example shows the efficiency of our methods.

## I. INTRODUCTION

Dynamical networks, exemplified by a collection of components through a communication network, are increasingly prevalent in various fields, including robotics, autonomous vehicles, distributed computing [1] and biological systems [2]–[4]. The structure of these networks, outlining the interaction patterns among the components, is crucial to shaping the overall behaviour of the networks. However, in many practical scenarios, the topology structure of the network may not be known a priori or be subject to changes, posing a substantial challenge to understanding the fundamental principles for dynamical networks and further control.

There have been many works on addressing network estimation problems, including optimization-based methods, knock-out methods [5], and adaptive control-based methods [6], [7], among others, as highlighted in [8]. Static topology estimation problems are addressed by constructing a synchronized network or by identifying the network by knocking out nodes in [5]–[7]. As for time-varying topology estimation, machine learning methods have been applied to estimate network topology, as discussed in [9], based on the assumption of either smooth parameter changes or piece-wise constant variations. The unknown switching topology is estimated through adaptive synchronization, specifically under the premise of piece-wise constant changes in switching topology [10]. However, these works, including those previously mentioned, primarily focus on the problem of topology estimation, overlooking the application of this topological information in further analysis or control of the network.

This work is supported by the Swedish Research Council (VR), the Knut and Alice Wallenberg Foundation, and the WASP-DDLS program. N. Wang and D. V. Dimarogonas are with the Division of Decision and Control Systems, KTH Royal Institute of Technology, SE-100 44 Stockholm, Sweden, email: {nanaw, dimos}@kth.se. E. Restrepo is with CNRS-IRISA, Inria Rennes, France, email: esteban.restrepo@inria.fr.

When the topology is static and unknown, a combination scheme between topology estimation and control tasks is to identify the topology first, and then use the identified topology for control tasks. After the topology is identified, the network can be controlled for complex tasks by coordination. A combined scheme of topology estimation and control was proposed by switching reference signals in [11]. A method in [12] that realizes topology estimation and synchronization simultaneously was presented by tracking an auxiliary system which synchronizes after identifying topology. However, these methods fail when the topology is time-varying due to their assumptions of static topology.

This paper proposes an adaptive-control-based method to address the simultaneous topology estimation and synchronization problem for dynamical networks with time-varying topology. The proposed methods guarantee the boundedness of weight estimation and synchronization errors assuming bounded weights and bounded weight derivatives. A scheme of combining the topology estimation and synchronization under time-varying topology is proposed, by estimating the time-varying topology and employing the estimated topology into the control input to synchronize the network.

The structure of the remainder of this paper is as follows: we formulate the problem in Section II. Section III introduces a control scheme and adaptive parameter updating laws for pure topology estimation. In Section IV, we present the solution to the topology estimation and synchronization problem. Section V verifies the proposed scheme's effectiveness with a numerical example. Finally, Section VI concludes the paper.

## II. PRELIMINARIES

### A. Notations

$\mathbb{B}(\Delta) \subset \mathbb{R}^n$ denotes a closed ball of radius $\Delta$ centered at the origin, i.e. $\mathbb{B}(\Delta) := \{x \in \mathbb{R}^n : |x| \le \Delta\}$. Denote $\|\cdot\|$ the Euclidean norm of vectors and the induced $L_2$ norm of matrices. The pseudoinverse of a matrix $X$ is denoted as $X^+$. Denote $|\cdot|$ the absolute value of real numbers. Denote $\mathcal{G} = (\mathcal{V}, \mathcal{E}, W)$ a directed weighted graph, where $\mathcal{V} = \{1, 2, \dots, N\}$ is a node-set and $\mathcal{E} \subseteq \mathcal{V}^2$ is an edge set with $M$ edges, characterizing the information exchange between agents. A directed edge $e_k := (i, j) \in \mathcal{E}$, indicates that agent $j$ has access to information from node $i$, and a positive diagonal matrix $W \in \mathbb{R}^{M \times M}$, whose diagonal $w_k$ entries represent the weights of the edges. We denote time-varying topology as $\mathcal{G}(t) = (\mathcal{V}, \mathcal{E}(t), W(t))$, where the edge set $\mathcal{E}(t)$ and the weight $w_k(t)$ are time-varying.
## B. Model and problem formulation

We consider a multi-agent system where the agents interact over an unknown time-varying topology described by a directed graph $G(t) = (V, \mathcal{E}(t), W(t))$, which is assumed to be connected. Without loss of generality, each agent's dynamics is described as follows

$$ \dot{x}_i = f_i(x_i) - c \sum_{j=1}^{N} w_{ij}(t)(x_i - x_j) + u_i \quad i \in V, \quad (1) $$

where $x_i \in \mathbb{R}$ is the state of agent $i$; $f_i: \mathbb{R} \to \mathbb{R}$ is a smooth function, denoting its internal dynamics; $w_{ij}(t) : \mathbb{R}_{\ge 0} \to \mathbb{R}$ denotes the unknown weight function of the interconnection between agents $i$ and $j$; $c$ is a positive constant, denoting the strength of connectivity; $\forall t > 0$, $w_{ij}(t) = 0$ if the edge $e_{k=(i,j)} \notin \mathcal{E}$ and $w_{ij}(t) \neq 0$ if the edge $e_{k=(i,j)} \in \mathcal{E}$. Hence, the edge set $\mathcal{E}(t)$ is time-varying depending on the values of $w_{ij}(t)$. The objective of the multi-agent system (1) is to achieve consensus among the agents with external control input under an unknown time-varying topology $G(t)$. The consensus problem considered here can also be extended to formation control or other cooperation tasks. For each agent's internal dynamics, we assume the following.

**Assumption 1:** For each agent $i$, there exists a positive constant $L_i$ such that

$$ \|f_i(x) - f_i(y)\| \le L_i \|x - y\| \quad (2) $$

for all $x, y \in \mathbb{R}$, where $1 \le i \le N$.

Let $E(t) : \mathbb{R}_{\ge 0} \to \mathbb{R}^{N \times M}$ denote the (unknown) incidence matrix function of $G(t)$ from [1] and recall that $M$ denotes the number of edges. $E_{\odot}(t) : \mathbb{R}_{\ge 0} \to \mathbb{R}^{N \times M}$ denotes the (unknown) in-incidence matrix function of $G$, defined as follows: $[E_{\odot}]_{ik}(t) := -1$ if $i$ is the terminal node of edge $e_k$ and $[E_{\odot}]_{ik}(t) := 0$ otherwise. Then, denoting $x := [x_1 \dots x_N]^{\top}$, $F(x) := [f_1(x_1) \dots f_N(x_N)]^{\top}$, and $u := [u_1 \dots u_N]^{\top}$, (1) can be written as

$$ \dot{x} = F(x) - cE_{\odot}(t)W(t)E(t)^{\top}x + u. \quad (3) $$

Since the edges of $G(t)$ are time-varying, the dimension of the incidence matrix function $E(t)$ is not fixed. To represent the unknown time-varying graph, we resort to using a complete graph whose weight of edges is unknown but the number of edges is fixed. Denote the incidence matrix $\bar{E}$ and in-incidence matrix $\bar{E}_{\odot}$ of a complete graph $\mathcal{K}(V, \mathcal{E}_c, \bar{W}(t))$, where $\mathcal{E} \subseteq \mathcal{E}_c$. Denote the cardinality of $\mathcal{E}_c$ as $\bar{M}$ and $M = N(N-1)$. Let $\bar{W}(t) := \text{diag}\{\bar{w}_k(t)\}$ where $\bar{w}_k(t) = w_k(t)$ if $\bar{e}_k \in \mathcal{E}$ and $\bar{w}_k = 0$ if $\bar{e}_k \in \mathcal{E}_c \setminus \mathcal{E}$. This representation transforms searching for the unknown graph into estimating the weights of edges of the complete graph $\mathcal{K}$. The weight $\bar{w}_k(t)$ is non-zero if the edge $\bar{e}_k$ of the complete graph exists in the graph $G$ to be identified. Rewrite (3) as

$$ \dot{x} = F(x) - c\bar{E}_{\odot}\bar{W}(t)\bar{E}_{\odot}^{\top}x + u. \quad (4) $$

**Assumption 2:** For any $0 < k \le N(N-1)$, there exist upper bounds $w_d$ and $w'_d$ for $\bar{w}_{ij}$ and $\dot{w}_{ij}(t)$ such that

$$ |\bar{w}_k(t)| \le w_d, \quad |\dot{w}_k(t)| \le w'_d \quad \forall t \ge 0. \quad (5) $$

**Remark 1:** Instead of considering a switching topology, we consider continuous time-varying changes in the weight of edges here, exploiting the potential robustness of our design in the time-varying topology case. This assumption contains the cases of adding new edges or removing the old ones by changing the weight of edges in a bounded way. For example, in the human immune cell activation process in response to a pathogen, the concentration level of cytokines, which facilitate communication between immune cells, is smoothly time-varying [13]. This boundedness assumption also means that $w_{ij}$ and its derivative $\dot{w}_{ij}$ are bounded. •

Using the edge-agreement representation for networked systems with a connected graph enables us to obtain an equivalent reduced system. Defining the edge variable $z := \bar{E}^{\top}x$, rewrite (4) as

$$ \dot{z} = \bar{E}^{\top} F(x) - c \bar{E}^{\top} \bar{E}_{\odot} \bar{W}(t) z + \bar{E}^{\top} u. \quad (6) $$

Using suitable labelling of edges, we can partition the incidence matrix of the complete graph $\mathcal{K}$ as

$$ \bar{E} = [\bar{E}_{\tau} \quad \bar{E}_{C}] \quad (7) $$

where $\bar{E}_{\tau} \in \mathbb{R}^{N \times (N-1)}$ is the incidence matrix of an spanning tree $\mathcal{G}_{\tau} \subset \mathcal{K}$ and $\bar{E}_{C} \in \mathbb{R}^{N \times (\bar{M}-N+1)}$ denotes the incidence matrix of the remaining edges from [1]. Similarly, partition the edge state as $z = [z_{\tau}^{\top} \quad z_{C}^{\top}]^{\top}$, where $z_{\tau} \in \mathbb{R}^{(N-1)}$ are the states of the edges of the spanning tree $\mathcal{G}_{\tau}$ and $z_{C} \in \mathbb{R}^{\bar{M}-N+1}$ denote the states of the remaining edges. Moreover, define

$$ R := [I_{N-1} \quad T], \quad T := (\bar{E}_{\tau}^{\top} \bar{E}_{\tau})^{-1} \bar{E}_{\tau}^{\top} \bar{E}_{C}, \quad (8) $$

with $I_{N-1}$ denoting the $N-1$ identity matrix. Based on (7) and (8), we have $\bar{E} = \bar{E}_{\tau}R$ and $z = R^{\top}z_{\tau}$. Then, we obtain a reduced-order model of (6) as

$$ \dot{z}_{\tau} = \bar{E}_{\tau}^{\top} F(x) - c \bar{E}_{\tau}^{\top} \bar{E}_{\odot} \bar{W}(t) R^{\top} z_{\tau} + \bar{E}_{\tau}^{\top} u. \quad (9) $$

The topology estimation problem in (1) is transformed into estimating the time-varying diagonal entries of the matrix function $\bar{W}(t)$ in (9). Meanwhile, the synchronization problem for (1) is transformed into the stabilization problem of the origin for the reduced-order system (9).

## III. TOPOLOGY ESTIMATION UNDER BOUNDED TIME-VARYING WEIGHTS

In this section, we introduce the external input $u(t)$ to estimate the unknown graph topology $G(t)$ for the dynamical systems (9). A refined control design to our previous work addressing static topology estimation [12] will be used.

### A. Control design and weight estimation laws

Denote $\bar{w}(t) := [\bar{w}_1(t) \cdots \bar{w}_{\bar{M}}(t)]^{\top} \in \mathbb{R}^{\bar{M}}$ as the vector of unknown weights, $\dot{w}(t) := [\dot{w}_1(t) \cdots \dot{w}_{\bar{M}}(t)]^{\top} \in \mathbb{R}^{\bar{M}}$ as its estimate, and $\hat{W}(t) := \text{diag}\{\hat{w}(t)\}$.

Set the updating law

$$ \dot{\hat{w}} = -c\hat{Z}(t)\bar{E}_{\odot}^{\top}\bar{E}_{\tau}\hat{z}_{\tau}, \quad (10) $$
where $\tilde{z}_{\tau} := z_{\tau} - \hat{z}_{\tau} = \bar{E}_{\tau}^{\top}\tilde{x}(t)$, $\tilde{x}(t) := x(t) - \hat{x}(t)$, $\hat{z}(t) := \bar{E}^{\top}\hat{x}(t)$, $\hat{Z}(t) := \text{diag}\{\hat{z}(t)\}$, and $\hat{x}(t)$ is an auxiliary variable to be designed later.

Select the control input

$$u = -c_1(x - \hat{x}(t)) + \dot{x}(t) + c\bar{E}_{\odot}\hat{W}(t)\hat{z}(t) - F(\hat{x}(t)) \quad (11)$$

where $c_1$ is a positive constant.

## B. Time-varying topology estimation

In this part, we analyze the effect of the time-varying weights on the topology estimation and show that using our design, the weight estimation errors remain bounded.

Define $\tilde{w}(t) := \bar{w}(t) - \hat{w}(t)$. Utilizing (9), (10) and (11), we derive the closed-loop system as

$$\begin{bmatrix} \dot{\tilde{z}}_{\tau} \\ \dot{\tilde{w}} \end{bmatrix} = \begin{bmatrix} -c_1 I - c \bar{L}_e & -c \bar{E}_{\tau}^{\top} \bar{E}_{\odot} \hat{Z}(t) \\ c \hat{Z}(t) \bar{E}_{\odot}^{\top} \bar{E}_{\tau} & 0 \end{bmatrix} \begin{bmatrix} \tilde{z}_{\tau} \\ \tilde{w} \end{bmatrix} + \begin{bmatrix} \bar{E}_{\tau}^{\top} \tilde{F}(x, \hat{x}) \\ \dot{\tilde{w}} \end{bmatrix}, \quad (12)$$

where $\bar{L}_e := \bar{E}_{\tau}^{\top} \bar{E}_{\odot} \bar{W} R^{\top}$ and $\tilde{F}(x, \hat{x}) := F(x) - F(\hat{x}(t))$.

**Proposition 1:** Assume that the signal $\hat{Z}(t)$ is bounded, globally Lipschitz and satisfies that for any unit vector $v \in \mathbb{R}^{\bar{M}}$

$$\int_t^{t+T} \|\hat{Z}(\tau)v\|^2 d\tau > \mu, \forall t \ge 0. \quad (13)$$

where $T, \mu > 0$. With Assumptions 1 and 2, the edge weight estimation errors $\tilde{w}(t)$ of the multi-agent system (1) are globally ultimately bounded, and all the closed-loop signals are bounded, after applying update law (10) and the control input (11). □

**Proof:** The closed-loop system (12) can be regarded as a perturbed form of

$$\begin{bmatrix} \dot{\tilde{z}}_{\tau} \\ \dot{\tilde{w}} \end{bmatrix} = \begin{bmatrix} -c_1 I - c \bar{L}_e & -c \bar{E}_{\tau}^{\top} \bar{E}_{\odot} \hat{Z}(t) \\ c \hat{Z}(t) \bar{E}_{\odot}^{\top} \bar{E}_{\tau} & 0 \end{bmatrix} \begin{bmatrix} \tilde{z}_{\tau} \\ \tilde{w} \end{bmatrix}. \quad (14)$$

Since the graph is assumed to be connected, the eigenvalues of edge Laplacian $\bar{L}_e$ have positive real parts from [14]. Hence, $-c_1 I - c \bar{L}_e$ is Hurwitz. And $\bar{E}_{\tau}^{\top} \bar{E}_{\odot}$ has rank $N-1$ since $\bar{E}_{\tau}^{\top} \bar{E}_{\odot} R^{\top}$ is full rank, as discussed in [14]. Then $(-c_1 I - c \bar{L}_e \bar{E}_{\tau}^{\top} \bar{E}_{\odot})$ is controllable. If $(-c_1 I - c \bar{L}_e \bar{E}_{\tau}^{\top} \bar{E}_{\odot})$ is controllable, and $Z(\tau)$ is piecewise-continuous, bounded and satisfies (13), then global uniform exponential stability of the origin for (14) follows from Theorem 5 [15] or Theorem 2.17 [16], [17] and the linearity of (14).

Denote $\xi := [\tilde{z}_{\tau} \quad \tilde{w}^{\top}]^{\top} \in \mathbb{R}^{\bar{M}+(N-1)}$. From the global exponential stability of (14) and from converse Lyapunov theorems (Theorem 4.14 of [18]), there exists a Lyapunov function $V(t, \xi) : \mathbb{R}_{\ge 0} \times \mathbb{R}^{\bar{M}+(N-1)} \to \mathbb{R}_{\ge 0}$ such that

$$\beta_1 \|\xi\|^2 \le V(t, \xi) \le \beta_2 \|\xi\|^2 \quad (15)$$

$$\left\| \frac{\partial V}{\partial \xi} \right\| \le \beta_3 \|\xi\|, \quad (16)$$

for some $\beta_1, \beta_2, \beta_3 > 0$, and its derivative along the trajectories of (14) satisfies

$$\dot{V}(t, \xi) \le -\beta_4 \|\xi\|^2, \quad \beta_4 > 0. \quad (17)$$

In view of Assumption 1, we can further obtain

$$\| [\bar{E}_{\tau}^{\top} [F(x) - F(\hat{x}(t))]]_{k} \| \le L_{f} \|\tilde{z}_{k}\|, \quad (18)$$

where $L_f := \max_{i \in \mathcal{V}} \{L_i\}$. Choose

$$V_1(t, \xi) = 0.5 \|\tilde{z}_{\tau}\|^2 + 0.5 \|\tilde{w}\|^2. \quad (19)$$

Along the trajectories of (12), its derivative is

$$\begin{align} \dot{V}_1(t, \xi) &= -\tilde{z}_{\tau}^{\top}(c_1 I + c\bar{L}_e))\tilde{z}_{\tau} - \tilde{z}_{\tau}^{\top}\bar{E}_{\tau}^{\top}\tilde{F}(x, \hat{x}) + \tilde{w}^{\top}\dot{\tilde{w}} \nonumber \\ &\le -(c_1 + c\lambda_{\min}\bar{L}_e - L_f)\|\tilde{z}_{\tau}\|^2 + \|\tilde{w}\|\|\dot{\tilde{w}}\| \nonumber \\ &= -c_1' \|\tilde{z}_{\tau}\|^2 + \|\tilde{w}\|\|\dot{\tilde{w}}\|, \tag{20} \end{align}$$

where $c_1' := c_1 + c\lambda_{\min}\{\bar{L}_e\} - L_f$ and $\lambda_{\min}\{\bar{L}_e\}$ is the smallest eigenvalue of $\bar{L}_e$. The second inequality is obtained using (18).

Let $V'(t, \xi) = V(t, \xi) + V_1(t, \xi)$. In view of (18), (20) and Assumption 2, its derivative along the trajectories of (12) is

$$\begin{align} \dot{V}'(t, \xi) &\le -\beta_4 \|\xi\|^2 + \frac{\partial V}{\partial \tilde{z}_{\tau}}^{\top} \bar{E}_{\tau}^{\top} \tilde{F}(x, \hat{x}) + \frac{\partial V}{\partial \tilde{w}}^{\top} \dot{\tilde{w}} \nonumber \\ &\quad - c_1' \|\tilde{z}_{\tau}\|^2 + \|\tilde{w}\|\|\dot{\tilde{w}}\| \nonumber \\ &\le \frac{\delta^2}{4} \left( \left\| \frac{\partial V}{\partial \tilde{z}_{\tau}} \right\|^2 + \left\| \frac{\partial V}{\partial \tilde{w}} \right\|^2 \right) + \frac{L_f^2 \|\tilde{z}_{\tau}\|^2}{\delta^2} \nonumber \\ &\quad - \beta_4 \|\xi\|^2 + 2 \|\dot{\tilde{w}}\|^2 / \delta^2 + \delta^2 \|\tilde{w}\|^2 / 4 - c_1' \|\tilde{z}_{\tau}\|^2 \nonumber \\ &\le -\beta_4 \|\xi\|^2 + \beta_3^2 \delta^2 \|\xi\|^2 / 4 + \delta^2 \|\xi\|^2 / 4 + 2w_d' \delta^2 / \delta^2 \nonumber \\ &\le -\beta_4' \|\xi\|^2 + \beta_5, \tag{21} \end{align}$$

where $\beta_4' := \beta_4 - \beta_3^2 \delta^2 / 4 - \delta^2 / 4$, $\beta_5 := 2w_d' \delta^2$ and $\delta > 0$, and we choose $c_1'$ that satisfies $c_1' - L_f^2 / \delta^2 > 0$. The second inequality is obtained by applying Young's inequality. Then, by properly choosing $V$ and $\delta$ such that $\beta_4' > 0$, the origin of (12) is globally ultimately bounded from (15), (19) and (21) by Theorem 4.18 in [18]. The estimation error $\|\tilde{w}\|$ is globally ultimately bounded and converges to $\Omega_{\tilde{w}} := \{\tilde{w} : \|\tilde{w}\| \le d_{\tilde{w}}\}$ with $d_{\tilde{w}} = \sqrt{\beta_5 / \beta_4'}$. By (10), (11), $u(t)$ and $\hat{w}(t)$ are also bounded. Hence, the result follows. ■

**Remark 2:** Proposition 1 shows that for dynamical systems (1) with time-varying topology, the control input (11) and weight estimation law (10) guarantee the boundedness of the weight estimation errors $\|\tilde{w}\|$ provided that $\hat{Z}(t)$ is persistently exciting. Besides, if the weights are fixed, then the estimation errors $\|\tilde{w}\|$ will be bounded and further converge to zero. •

## IV. SIMULTANEOUS TOPOLOGY ESTIMATION AND SYNCHRONIZATION FOR TIME-VARYING NETWORKS

In this section, we explore simultaneous topology estimation and synchronization for (4) with the time-varying topology. We use the control input (11) in the following scheme.

### A. Design of updating laws and auxiliary system

Let $\hat{z}(t)$ be the state of an auxiliary dynamical system. Set the new updating law instead of (10) as

$$\dot{\hat{w}} = -c\hat{Z}(t)\bar{E}_{\odot}^{\top}\bar{E}_{\tau}\tilde{z}_{\tau} - \sigma_1\hat{w}, \quad (22)$$
where $\sigma_1$ is a positive constant.

The updating law (22) adds $\sigma_1 \hat{w}$ to guarantee the boundedness of $\|\hat{w}\|$ under the bounded derivation of $\|w\|$.

Design the auxiliary dynamical system as

$$ \dot{\hat{z}} = \bar{E}^{\top} F(\hat{x}) - c_2 \hat{z} + \psi(t, \tilde{z}_{\tau}) \quad (23) $$

where $c_2 > L_f$ is a positive constant and the function $\psi(t, \tilde{z}_{\tau}) : \mathbb{R}_{\ge 0} \times \mathbb{R}^{N-1} \to \mathbb{R}^{M}$ satisfies that

$$ \max \left\{ \|\psi(\cdot)\|, \left\| \frac{\partial \psi(\cdot)}{\partial t} \right\|, \left\| \frac{\partial \psi(\cdot)}{\partial \tilde{z}} \right\| \right\} \le \kappa (\|\tilde{z}_{\tau}\|), \forall t \ge 0, \quad (24) $$

where $\kappa : \mathbb{R}_{\ge 0} \to \mathbb{R}_{\ge 0}$ is a continuous non-decreasing function. Define $\Psi(t, \tilde{z}_{\tau}) \in \mathbb{R}^{M \times M}$ as a diagonal matrix function of $\psi(\cdot)$. Specifically, write $\Psi(t, x_1) := \text{diag}\{\psi(t, x_1)\}$. Define $\Psi'(t, \tilde{z}_{\tau}) : \mathbb{R}_{\ge 0} \times \mathbb{R}^{N-1} \to \mathbb{R}^{(N-1) \times M}$. Let $\Psi'(t, \tilde{z}_{\tau}) = \bar{E}_{\tau}^{\top} \bar{E}_{\odot} \Psi(t, \tilde{z}_{\tau})$. $\Psi'(t, \tilde{z}_{\tau})$ is uniform $\delta$-persistently exciting (u$\delta$-PE) with respect to $\tilde{z}_{\tau}$ as per Definition 5 in [19].

## B. Stability analysis of the unperturbed systems

Using (9), (11) and (22), we obtain the new closed-loop system as

$$ \begin{bmatrix} \dot{\tilde{z}}_{\tau} \\ \dot{\hat{w}} \end{bmatrix} = \begin{bmatrix} \bar{E}_{\tau}^{\top} \tilde{F}(x, \hat{x}) - (c_1 I + c \bar{L}_e) \tilde{z}_{\tau} - c \bar{E}_{\tau}^{\top} \bar{E}_{\odot} \hat{Z}(t) \hat{w} \\ c \hat{Z}(t) \bar{E}_{\odot}^{\top} \bar{E}_{\tau} \tilde{z}_{\tau} \end{bmatrix} + \begin{bmatrix} 0 \\ \dot{\hat{w}} + \sigma_1 \hat{w} \end{bmatrix}. \quad (25) $$

Similar to the previous analysis, the closed-loop system (25) can be seen as the perturbed version of

$$ \begin{bmatrix} \dot{\tilde{z}}_{\tau} \\ \dot{\hat{w}} \end{bmatrix} = \begin{bmatrix} \bar{E}_{\tau}^{\top} \tilde{F}(x, \hat{x}) - (c_1 I + c \bar{L}_e) \tilde{z}_{\tau} - c \bar{E}_{\tau}^{\top} \bar{E}_{\odot} \hat{Z}(t) \hat{w} \\ c \hat{Z}(t) \bar{E}_{\odot}^{\top} \bar{E}_{\tau} R R^{\top} \tilde{z}_{\tau} \end{bmatrix}. \quad (26) $$

Before studying the stability of (25), we first analyze the stability of the unperturbed system (26). Replacing $x_1, x_2, A(t, x_1), B, \Phi$ and $\phi$ by $\tilde{z}_{\tau}, \tilde{w}, \bar{E}_{\tau}^{\top} \tilde{F}(x, \hat{x}) - (c_1 I + \bar{E}_{\tau}^{\top} \bar{E}_{\odot} W R^{\top}) \tilde{z}_{\tau}, \bar{E}_{\tau}^{\top} \bar{E}_{\odot}, \hat{Z}$ and $\hat{z}$, respectively, we can represent (26) as

$$ \begin{bmatrix} \dot{x}_1 \\ \dot{x}_2 \end{bmatrix} = \begin{bmatrix} A(t, x_1) + B \Phi(t, x_1)^{\top} x_2 \\ -\Phi(t, x_1) B^{\top} x_1 \end{bmatrix} \quad (27) $$

where $x^{\top} := [x_1^{\top} \ x_2^{\top}]$, $\Phi(t, x_1) : \mathbb{R}_{\ge 0} \times \mathbb{R}^{N-1} \to \mathbb{R}^{M \times M}$ and $\phi(t, x_1) : \mathbb{R}_{\ge 0} \times \mathbb{R}^{N-1} \to \mathbb{R}^{M}$ are piece-wise continuous in $t$ and continuous in $x_1$. Moreover, $\Phi(t, x_1)$ is diagonal with $\Phi(t, x_1) := \text{diag}\{\phi(t, x_1)\}$. Assume the following:

**Assumption 3:** The function $A$ is locally Lipschitz in $x$ uniformly in $t$. Moreover, there exists a continuous nondecreasing function $\rho_1 : \mathbb{R}_{\ge 0} \to \mathbb{R}_{\ge 0}$ such that $\rho_1(0) = 0$ and for all $(t, x_1) \in \mathbb{R} \times \mathbb{R}^{N-1}$, $\|A(t, x_1)\| \le \rho_1(\|x_1\|)$.

**Assumption 4:** There exists a locally Lipschitz function $V_1 : \mathbb{R}_{\ge 0} \times \mathbb{R}^{N+M-1} \to \mathbb{R}_{\ge 0}$, and $\alpha_1, \alpha_2, \alpha_3 > 0$ such that

$$ \alpha_1 \|x\|^2 \le V_1(t, x) \le \alpha_2 \|x\|^2 \quad (28) $$

and its derivative along the trajectories of (27) satisfies

$$ \dot{V}_1(t, x) \le -\alpha_3 \|x_1\|^2. \quad (29) $$

Then we state the following lemma.

**Lemma 1:** Let Assumptions 3 and 4 hold. Assume $B\Phi^{\top}(t, x_1)$ is u$\delta$-PE with respect to $x_1$ and $\Phi(t, x_1)$ satisfies

$$ \max \left\{ \|\Phi(\cdot)\|, \left\| \frac{\partial \Phi(\cdot)}{\partial t} \right\|, \left\| \frac{\partial \Phi(\cdot)}{\partial x_1} \right\| \right\} \le \rho(\|x_1\|), \forall t \ge 0, \quad (30) $$

where $\rho : \mathbb{R}_{\ge 0} \to \mathbb{R}_{\ge 0}$ is a continuous non-decreasing function. Then the origin of (27) is uniformly semiglobally asymptotically stable. $\square$

*Proof:* Consider a Lyapunov function candidate as

$$ \begin{align} V(t, x) &:= V_1(t, x) + \varepsilon V_4(t, x) \nonumber \\ V_4(t, x) &:= V_2(t, x) + V_3(t, x) \nonumber \\ V_2(t, x) &:= -x_1^{\top} B \Phi(t, x_1)^{\top} x_2 \tag{31} \\ V_3(t, x) &:= - \int_t^{\infty} e^{(t-\tau)} \|B \Phi(\tau, x_1)^{\top} x_2\|^2 d\tau, \nonumber \end{align} $$

where $V_1(t, x)$ is given in Assumption 4 and $\varepsilon > 0$. Using the u$\delta$-PE of $B\Phi^{\top}$, for all $(t, x) \in \mathbb{R} \times \mathbb{B}(\Delta)$, one has

$$ \begin{align} V_3(t, x) &= - \int_t^\infty e^{(t-\tau)} x_2^\top \Phi(\tau, x_1) B^\top B \Phi(\tau, x_1)^\top x_2 d\tau \nonumber \\ &\le - \int_t^{t+T} e^{(t-\tau)} x_2^\top \Phi(\tau, x_1) B^\top B \Phi(\tau, x_1)^\top x_2 d\tau \nonumber \\ &\le (e^{-T} - 1) \mu \|x_2\|^2, \tag{32} \end{align} $$

where $b' := (e^{-T} - 1)\mu$, $\mu$ and $T$ are defined from Definition 5 in [19]. In view of (30) and (32), $V_4(t, x)$ in (31) satisfies, for all $(t, x) \in \mathbb{R} \times \mathbb{B}(\Delta)$,

$$ V_4(t, x) \le b \|x_1\| \rho(\|x_1\|) \|x_2\| - b' \|x_2\|^2, \quad (33) $$

where $b := \|B\|$. Define $b_\rho := b\rho(\Delta)$. In view of (33), $\varepsilon V_4(t, x)$ satisfies on $\mathbb{R} \times \mathbb{B}(\Delta)$

$$ -\varepsilon \rho(\Delta) \|x_2\|^2 - \varepsilon b_\rho \|x_1\| \|x_2\| \le \varepsilon V_4(t, x) \le \varepsilon b_\rho \|x_1\| \|x_2\| \\ - \varepsilon b' \|x_2\|^2. \quad (34) $$

So, from (28) and (34), for any $\Delta > 0$ and for a sufficiently small $\varepsilon$, there exist $\underline{\alpha}_{\Delta} > 0$ and $\overline{\alpha}_{\Delta} > 0$ such that for all $(t, x) \in \mathbb{R} \times \mathbb{B}(\Delta)$

$$ \underline{\alpha}_{\Delta} \|x\|^2 \le V(t, x) \le \overline{\alpha}_{\Delta} \|x\|^2. \quad (35) $$

We proceed to obtain the derivative of $V_4(t, x)$ along the trajectories of the system (27). First, we have

$$ \dot{V}_2(t, x) = \|\Phi(t, x_1) B^{\top} x_1\|^2 - x_2^{\top} \Phi(t, x_1)^{\top} B^{\top} A(t, x_1) \\ - \|B \Phi(t, x_1) x_2\|^2 - x_2^{\top} \overbrace{\Phi(t, x_1)}^{\cdot} B^{\top} x_1, \quad (36) $$

where $\overbrace{\Phi(t, x_1)}^{\cdot} := \frac{\partial \Phi(t, x_1)}{\partial t} + \frac{\partial \Phi(t, x_1)}{\partial x_1}$. Next, we have

$$ \frac{\partial V_3}{\partial x_1} = - \int_t^\infty 2 e^{(t-\tau)} x_2^\top \Phi(\tau, x_1) B^\top B \left[ \frac{\partial \Phi(\tau, x_1)^\top}{x_1} x_2 \right] d\tau $$

$$ \frac{\partial V_3}{\partial x_2} = - \int_t^\infty 2 e^{(t-\tau)} \Phi(\tau, x_1) B^\top B \Phi(\tau, x_1)^\top x_2 d\tau $$

$$ \frac{\partial V_3}{\partial t} = \|B \Phi(t, x_1)^{\top} x_2\|^2 - \int_t^\infty \frac{\partial}{\partial t} \left[ e^{(t-\tau)} \|B \Phi(\tau, x_1)^{\top} x_2\|^2 \right] d\tau. $$
From Assumption 3, (30), (36) and (37), we obtain an upper bound function for the derivative of $V_4(t, x)$. Define

$$ \bar{\rho}(r, s) := b_{\rho} \left[ (2 + 2b_{\rho}^2)rs + (1 + 2b_{\rho}^2)\rho_1(r)s + b_{\rho}r^2 + 2b_{\rho}^2 s^2 \right]. $$ 
Then, for $(t, x) \in \mathbb{R} \times \mathbb{B}(\Delta)$,

$$ \dot{V}_4(t, x) \leq \bar{\rho}(\|x_1\|, \|x_2\|) - b' \|x_2\|^2. \quad (37) $$

Using (29) and (37), the derivative of $V(t, x)$ satisfies, for all $(t, x) \in \mathbb{R} \times \mathbb{B}(\Delta)$,

$$ \dot{V}(t, x) \leq -\alpha_3 \|x_1\|^2 - \varepsilon(2b_{\rho} + 2b_{\rho}^3) \|x_1\| \|x_2\| + 2\varepsilon b_{\rho}^3 \|x_2\|^2 \\ + \varepsilon b_{\rho}^2 \|x_1\|^2 + (b_{\rho} + 2b_{\rho}^3) \rho_1(\|x_1\|) \|x_2\| - \varepsilon b' \|x_2\|^2. $$

Note that $b' = (e^{-T} - 1)\mu$. Choosing $\mu$ and $T$ such that $b' \ge b_{\rho}^2 + 2b_{\rho}^3 + \beta'$ and $\beta' > 0$ yields

$$ \dot{V}(t, x) \leq - (\alpha_3 - (4 + b_{\rho}^2 + 4b_{\rho}^4)\varepsilon) \|x_1\|^2 - \varepsilon\beta' \|x_2\|^2 \\ + (1 + 4b_{\rho}^4)\varepsilon\rho_1^2(\|x_1\|). $$

Selecting $\varepsilon$ sufficiently small such that $\alpha_3 - \varepsilon(4 + b_{\rho}^2 - 4b_{\rho}^4) - \varepsilon(1 + 4b_{\rho}^4)\rho_1(|\Delta|)/|\Delta|^2 > \alpha$, yields

$$ \dot{V}(t, x) \leq - \alpha \|x_1\|^2 - \beta \|x_2\|^2, \quad (38) $$

where $\beta = \varepsilon\beta'$. Therefore, by Theorem 4.9 in [18], for all $(t, x) \in \mathbb{R} \times \mathbb{B}(\Delta)$, the origin of (27) is semi-globally uniformly asymptotically stable from (35) and (38). ■

**Remark 3:** Contrary to [12] which studies the stability where the unknown parameters are defined in a certain set, Lemma 1 analyzes the stability for (27) when the parameters are unknown and fixed. The result from Lemma 1 yielding uniform global asymptotical stability is thus stronger than the case of uniform practical stability derived in [12]. •

### C. Simultaneous topology estimation and synchronization

Considering the time-varying weights as the disturbance of system (27), we analyze the robustness of system (25) under time-varying topology in Proposition 2, based on Lemma 1.

**Proposition 2:** Let Assumptions 1 and 2 hold. Then, the origin of the closed-loop system (25) with the update law (22) and control input (11), is uniformly semi-globally stable with $\hat{z}(t)$ given by the update law (23). Its weight estimation errors $\tilde{w}$ are ultimately bounded, and converge to a set $\Omega_{\tilde{w}}$. Furthermore, the edge states $z$ are also ultimately bounded, and converge to a set $\Omega_z$. □

The sets $\Omega_{\tilde{w}}$ and $\Omega_z$ are defined in the proof that follows.

*Proof:* We first show that $\hat{z}(t)$ is u$\delta$-PE with respect to $\tilde{z}_{\tau}$. Denote $\xi := [\tilde{z}_{\tau}^{\top} \tilde{w}^{\top}]^{\top}$. Define $V_1(t, \xi)$ as in (19). Its derivative along (25) is

$$ \begin{aligned} \dot{V}_1(t, \xi) = & - \tilde{z}_{\tau}^{\top} (c_1 I + c \bar{L}_e) \tilde{z}_{\tau} - \tilde{z}_{\tau}^{\top} \bar{E}_{\tau}^{\top} \tilde{F}(x, \hat{x}) + \tilde{w}^{\top} \dot{\hat{w}} \\ & + \sigma_1 \dot{\hat{w}}^{\top} \hat{w} \\ \le & - (c_1 + c \lambda_{\min} \bar{L}_e) \|\tilde{z}_{\tau}\|^2 - \tilde{z}_{\tau}^{\top} \bar{E}_{\tau}^{\top} \tilde{F}(x, \hat{x}) \\ & - \sigma_1 \|\tilde{w}\|^2 + \|\tilde{w}\| \|\dot{\hat{w}}\| + \sigma_1 \|\tilde{w}\| \|\hat{w}\| \\ \le & - c'_1 \|\tilde{z}_{\tau}\|^2 - \sigma'_1 \|\tilde{w}\|^2 + d \\ \le & - c''_1 \|\xi\|^2 + d, \end{aligned} \quad (39) $$

where $c'_1$ is defined in (20), $\sigma'_1 := \sigma_1 - 0.5(\sigma_1 + 1)/\delta^2 > 0$, $d := 0.5\delta^2(\sigma_1|w_d|^2 + |w'_d|^2)$ and $c''_1 = \min\{c'_1, \sigma'_1\}$.

From (19) and (39), the system (25) is globally uniformly stable [18] and $\xi$ converges to the set $\Omega := \{\xi : \|\xi\| \le \sqrt{d/c''_1}\}$. Therefore, the solutions $\xi(t)$ are ultimately bounded from Theorem 4.18 in [18].

Choose the Lyapunov function $V_5(\hat{z}) := 0.5\|\hat{z}\|^2$. Its derivative (23) along the trajectories of the auxiliary system (23) satisfies

$$ \begin{aligned} \dot{V}_5(\hat{z}) = & -c_2 \hat{z}^{\top} \hat{z} + \hat{z}^{\top} \bar{E}^{\top} F(\hat{x}) + \hat{z}^{\top} \psi(t, \tilde{z}_{\tau}) \\ & \le -c_2 \|\hat{z}\|^2 + L_f \|\hat{z}\|^2 + \|\hat{z}\| \|\psi(t, \tilde{z}_{\tau})\| \\ & \le -c'_2 |\hat{z}|^2 + |\kappa(\|\tilde{z}_{\tau}\|)|^2 \le -c''_2 \|\hat{z}\|^2 + \sigma, \end{aligned} \quad (40) $$

where $c'_2 := c_2 - L_f - 0.25$. The third inequality is obtained by (24) and Young's inequality. As the solution $\tilde{z}_{\tau}(t)$ is uniformly stable, there exists $\sigma > 0$ such that $|\kappa(\|\tilde{z}_{\tau}\|)|^2 \le \sigma$ for all $t \ge 0$. The last inequality follows. Similarly, from (40) the solutions $\hat{z}(t)$, are ultimately bounded.

In Lemma 2 in Appendix I, $x$ and $w$ in (48) correspond to $[\tilde{z}_{\tau}^{\top} \tilde{w}^{\top}]^{\top}$ and $\hat{z}$ respectively here. From (23), (24) and (25), the inequalities (49) and (50) hold. (23) implies $f_1(t, w) \le l\|w\|$ with $l := c_2 + L_f$ in Lemma 2. Based on the boundedness of $\xi$ and $\hat{z}(t)$, (51) holds. Now, since all the assumptions in Lemma 2 in Appendix I are satisfied, $B\hat{Z}(t)$, given by the update law (23), is u$\delta$-PE with respect to $\tilde{z}_{\tau}$.

Next, we will analyze the stability of (25). For $A(t, x_1)$ in (25), there exists a function $\rho_1(\|x_1\|) := k\|x_1\|$ where $k := \max\{L_f + \|c_1 I + \bar{E}_{\odot} \tilde{W} \bar{E}\|, L_f + \|c_1 I + \bar{E}_{\tau}^{\top} \bar{E}_{\odot} \tilde{W} R^{\top}\|\}$, such that Assumption 3 is satisfied. Along the trajectories of (26), the derivative of $V_1(t, \xi)$ defined as (19) is

$$ \begin{aligned} \dot{V}_1(t, \xi) = & - (c_1 I + c \bar{L}_e) \tilde{z}_{\tau}^{\top} \tilde{z}_{\tau} - \tilde{z}_{\tau}^{\top} \bar{E}_{\tau}^{\top} \tilde{F}(x, \hat{x}) \\ & \le - c'_1 \|\tilde{z}_{\tau}\|^2. \end{aligned} \quad (41) $$

where $c'_1$ is defined in (20). Hence, $V_1(t, \xi)$ satisfies Assumption 4 with $\alpha_1 = \alpha_2 := \frac{1}{2}$ and $\alpha_3 := c'_1$. Now that all the assumptions of Lemma 1 hold, the origin of system (26) is concluded to be uniformly asymptotically stable.

Consider again $V(t, \xi)$ defined in (31). In order to notionally distinguish the derivatives of $V_i(t, \xi)$ along the trajectories of (25) and (26), we denote $\dot{V}'_i(t, \xi)$ as the derivative of $V_i(t, \xi)$ for (25) while $\dot{V}_i(t, \xi)$ corresponds to (26), where $i = 1, 2, 3, 4$. Denote $\Delta V'_i = \dot{V}'_i(t, \xi) - \tilde{V}_i(t, \xi)$. Based on (25), (26), (36), (37) and (41), $\Delta V'_i(t, \xi)$ is

$$ \begin{aligned} \Delta V'_1 = & - \sigma_1 \|\tilde{w}\|^2 + \tilde{w}^{\top} \dot{\hat{w}} + \sigma_1 \tilde{w}^{\top} \tilde{w} \\ \Delta V'_2 = & - \tilde{z}_{\tau}^{\top} B \Phi(t, \tilde{z}_{\tau})^{\top} (\dot{\hat{w}} + \sigma_1 \tilde{w} - \sigma_1 \tilde{w}) \\ \Delta V'_3 = & - \int_t^{\infty} 2e^{(t-\tau)} \Phi(\tau, \tilde{z}_{\tau}) B^{\top} B \Phi(\tau, \tilde{z}_{\tau})^{\top} \tilde{w} d\tau \\ & \cdot (\dot{\hat{w}} + \sigma_1 \tilde{w} - \sigma_1 \tilde{w}). \end{aligned} \quad (42) $$

According to (26) and (42), using Young's inequality, we
have

$$
\begin{align}
\Delta V_1' &\le -(\sigma_1 - \delta^2 - \delta^2\sigma_1^2)|\tilde{w}|^2 + \frac{\|\dot{\tilde{w}}\|^2}{4\delta^2} + \frac{\|\tilde{w}\|^2}{4\delta^2}. \nonumber \\
\varepsilon\Delta V_2' &\le \varepsilon(b_\rho^2 + b_\rho^2\sigma_1^2 + \frac{\sigma_1^2}{4\delta^2})\|\tilde{z}_\tau\|^2 + \varepsilon\delta^2\|\tilde{w}\|^2 \nonumber \\
&\quad + \varepsilon\frac{\|\dot{\tilde{w}}\|^2}{4} + \varepsilon\frac{\|\tilde{w}\|^2}{4} \nonumber \\
\varepsilon\Delta V_3' &\le \varepsilon(\sigma_1^2 b_\rho^4 + b_\rho^4 + 2\sigma_1 b_\rho^2)\|\tilde{w}\|^2 + \varepsilon\|\dot{\tilde{w}}\|^2 + \varepsilon\|\tilde{w}\|^2, \tag{43}
\end{align}
$$

where $\delta > 0$, $b_\rho := b\rho(\Delta)$ defined in (34) and $b := \|\bar{E}_\tau^\top \bar{E}_\odot\|$ for all $(t, x) \in \mathbb{R} \times \mathbb{B}(\Delta)$.

For the closed-loop system (25), based on (38) and (43),
$\dot{V}'(t, \xi)$ becomes

$$
\begin{align}
\dot{V}'(t, \xi) ={}& \dot{V}(t, \xi) + \Delta V_1' + \varepsilon \Delta V_2' + \varepsilon \Delta V_3' \nonumber \\
& \le -\alpha \|\tilde{z}_{\tau}\|^2 - \beta \|\tilde{w}\|^2 + (\delta^2 - \sigma_1 + \delta^2\sigma_1^2) \|\tilde{w}\|^2 \nonumber \\
& \qquad + \frac{\|\tilde{w}\|^2}{4\delta^2} + \varepsilon(2\sigma_1 b_{\rho}^2 + \sigma_1^2 b_{\rho}^4 + b_{\rho}^4 + \delta^2) \|\tilde{w}\|^2 \nonumber \\
& \qquad + \varepsilon(b_{\rho}^2 + b_{\rho}^2\sigma_1^2 + \frac{\sigma_1^2}{4\delta^2}) \|\tilde{z}_{\tau}\|^2 + \frac{\|\dot{w}\|^2}{4\delta^2} \nonumber \\
& \qquad + \varepsilon\frac{\|\dot{w}\|^2}{4} + \varepsilon\frac{\|\tilde{w}\|^2}{4}. \tag{44}
\end{align}
$$

Choosing $\beta_1 := \beta + \sigma_1 - \delta^2 - \sigma_1^2\delta^2 - \varepsilon(2\sigma_1 b_\rho^2 + \sigma_1^2 b_\rho^4 + b_\rho^4 + \delta^2) > 0$ and $\alpha' := \alpha - \varepsilon(b_\rho^2 + b_\rho^2\sigma_1^2 + 0.25\sigma_1^2/\delta^2) > 0$ yields

$$
\begin{align}
\dot{V}'(t, \xi) \le {}& - \alpha' \|\tilde{z}_{\tau}\|^2 - \beta_1 \|\tilde{w}\|^2 + d_{\xi} \tag{45} \\
& \le -c_3 \|\xi\|^2 + d_{\xi}, \nonumber
\end{align}
$$

where $\beta$ is defined in (38), $c_3 := \min\{\alpha', \beta'\}$ and $d_\xi := \sqrt{(1+4\delta^2\varepsilon)(w_d^2+w_d'^2)/4\delta^2}$. Since $\beta_1$ depends on $\mu$ and $T$ from Definition 2, it is possible to choose $\beta_1 > 0$. Parameter $\varepsilon$ is chosen to be sufficiently small and $\alpha_3 = c_1'$ can be chosen to be sufficiently large so that $\alpha' > 0$. Therefore, the solution $\xi$ of (25) converges to $\Omega_\xi := \{\xi : \|\xi\| \le d_\xi\}$ with $d_\xi := \sqrt{d/c_3}$. The weight estimation errors converge to $\Omega_{\tilde{w}} := \{\tilde{w} : \|\tilde{w}\| \le \sqrt{d/c_3}\}$. Furthermore, we obtain the bound of synchronization errors by $z = R^\top z_\tau$ and $z_\tau = \tilde{z}_\tau + \hat{z}_\tau$. According to (40), $\hat{z}_\tau$ converges to $\Omega_{\hat{z}_\tau} := \{\hat{z}_\tau : \|\hat{z}_\tau\| \le d_{\hat{z}_\tau} = \|\rho(\|d_\xi\|)\|/c_2'\}$. The edge state $z$ thus converges to $\Omega_z := \{z : \|z\| \le \|R^\top\|\|d_z\|\}$ where $d_z = [d_\xi \ d_{\hat{z}_\tau}]$. ■

V. SIMULATION

We consider a network (1) with 6 agents with a time-
varying communication topology with $\bar{w}(t)$ in (4) as

$$
\bar{w}(t) = [0.7 + 0.02 \sin(0.02t), 0.8 + 0.1 \cos(0.01t), 0.6 + \\
0.02 \sin(0.5\pi t), 0.25, 0.4, 0.02 \cos(0.05\pi t) + 0.45, \\
0_{1 \times 15}, 0.05 \cos(0.01\pi t) + 0.3, 0.6, 0.2, 0_{1 \times 5}, 0.5]^T
$$

where $0_{1 \times N}$ denotes $N$-dimensional zero row vector. Here, we simulate the network (1) with $f_i(x_i) = x_i$ and $c = 1$. Use control input (11) and weight updating law (22) and design the auxiliary system (23). The control gains are chosen as $c_1 = 2, c_2 = 1.3, \sigma_1 = 0.001$. Choose a $\text{u}\delta$-PE function from (23) referring to [12] as

$$
\psi(t, \tilde{z}_{\tau}) = (\bar{E}^{\top})^{+} \tanh(\kappa \bar{E}_{\tau} \tilde{z}_{\tau}) p(t),
$$

$$
p(t) = 5\sin(0.5\pi t) + 4\cos(2\pi t) - 6\sin(8\pi t) + \sin(\pi t) \\
- 4\cos(10\pi t) + 2\cos(6\pi t) + 3\sin(3\pi t).
$$

The simulation results are presented in Figs 1-4. Fig 1 and 2 represent the evolution of the estimated weights and the errors between the estimated weight and the time-varying weight. Fig 3 shows the evolution of synchronization errors $z$. Fig 4 displays the evolution of state $\tilde{z}_{\tau}$. As expected from Proposition 2, the estimation errors, $\tilde{z}_{\tau}$ and synchronization errors $z$ are bounded from Fig. 2, 3 and 4 under the time-varying topology. From Figs. 1 and 2, the real time-varying weights are in the line segments whose centres are the predicted weight in Fig. 1 and whose radii are the weight estimation errors. Another observation is that the bounds of the synchronization errors $z$ are bigger than the bounds of the estimation weight errors $\tilde{w}$, which responds to the analysis in the proof part of Proposition 2. We also tried different values of $c_2$, and we found that increasing the value of $c_2$ could get lower synchronization errors while increasing the bound of the weight estimation errors, which correspond to the form (23) of the auxiliary system $\hat{z}$. Hence, keeping a certain level of excitation for $\hat{z}_{\tau}$ is beneficial to estimating the time-varying weights, while it deteriorates the synchronization.

VI. CONCLUSIONS

In this paper, we introduce an adaptive control-based approach for simultaneous estimation of time-varying topology and synchronization of a complex dynamical network. We design an adaptive-control-based scheme to stimulate the system to ensure the boundedness of topology estimation errors. This is achieved through the development of an auxiliary system characterized by either persistent excitement or uniform $\delta$ persistent excitement. The first auxiliary system which is PE, enables us to bound the edge weight estimation errors. The latter one which is uniformly $\delta$ persistently exciting gives the boundedness of both weight estimation errors and synchronization errors, provided the weights and their derivatives are bounded. In terms of further work, we aim to enhance the topology estimation performance while considering control tasks under time-varying topology.

REFERENCES

[1] M. Mesbahi and M. Egerstedt, *Graph theoretic methods in multiagent networks*. Princeton University Press, 2010.

[2] J.-Y. Moon, U. Lee, S. Blain-Moraes, and G. A. Mashour, “General relationship of global topology, local dynamics, and directionality in large-scale brain networks,” *PLoS computational biology*, vol. 11, no. 4, p. e1004225, 2015.

[3] A. I. Luppi and E. A. Stamatakis, “Combining network topology and information theory to construct representative brain networks,” *Network Neuroscience*, vol. 5, no. 1, pp. 96–124, 2021.

[4] J. Shilts, Y. Severin, F. Galaway, N. Müller-Sienerth, Z.-S. Chong, S. Pritchard, S. Teichmann, R. Vento-Tormo, B. Snijder, and G. J. Wright, “A physical wiring diagram for the human immune system,” *Nature*, vol. 608, no. 7922, pp. 397–404, 2022.

[5] M. Nabi-Abdolyousefi and M. Mesbahi, “Network identification via node knockout,” *IEEE Transactions on Automatic Control*, vol. 57, no. 12, pp. 3214–3219, 2012.

[6] J. Zhou and J.-a. Lu, “Topology identification of weighted complex dynamical networks,” *Physica A: Statistical Mechanics and Its Applications*, vol. 386, no. 1, pp. 481–491, 2007.
<figure><img src="image_2.png" /><figcaption>Fig. 1: Estimated weight of time-varying topology</figcaption></figure>

<figure><img src="image_3.png" /><figcaption>Fig. 2: Estimation errors of time-varying weights.</figcaption></figure>

<figure><img src="image_4.png" /><figcaption>Fig. 3: Evolution of synchronization errors $z$</figcaption></figure>

<figure><img src="image_5.png" /><figcaption>Fig. 4: Evolution of state $\tilde{z}_{\tau}$</figcaption></figure>

[7] S. Zhu, J. Zhou, G. Chen, and J.-A. Lu, “A new method for topology identification of complex dynamical networks,” *IEEE Transactions on Cybernetics*, vol. 51, no. 4, pp. 2224–2231, 2021.

[8] M. Timme and J. Casadiego, “Revealing networks from dynamics: an introduction,” *Journal of Physics A: Mathematical and Theoretical*, vol. 47, no. 34, p. 343001, 2014.

[9] M. Kolar, L. Song, A. Ahmed, and E. P. Xing, “Estimating time-varying networks,” *The Annals of Applied Statistics*, pp. 94–123, 2010.

[10] K. Li, D. Yang, C. Shi, and J. Zhou, “Identifying the switching topology of dynamical networks based on adaptive synchronization,” *Chaos: An Interdisciplinary Journal of Nonlinear Science*, vol. 33, no. 12, 2023.

[11] N. Wang and D. V. Dimarogonas, “Finite-time topology identification for complex dynamical networks,” in *2023 62nd IEEE Conference on Decision and Control (CDC)*. IEEE, 2023, pp. 425–430.

[12] E. Restrepo, N. Wang, and D. V. Dimarogonas, “Simultaneous topology identification and synchronization of directed dynamical networks,” *IEEE Transactions on Control of Network Systems*, 2023.

[13] K. Talaei, S. A. Garan, B. d. M. Quintela, M. S. Olufsen, J. Cho, J. R. Jahansooz, P. K. Bhullar, E. K. Suen, W. J. Piszker, N. R. Martins et al., “A mathematical model of the dynamics of cytokine expression and human immune cell activation in response to the pathogen staphylococcus aureus,” *Frontiers in Cellular and Infection Microbiology*, vol. 11, p. 711153, 2021.

[14] D. Mukherjee and D. Zelazo, “Robustness of consensus over weighted digraphs,” *IEEE Transactions on Network Science and Engineering*, vol. 6, no. 4, pp. 657–670, 2019.

[15] A. Morgan and K. Narendra, “On the stability of nonautonomous differential equations $\dot{x} = a + b(t)x$, with skew symmetric matrix $b(t)$,” *SIAM Journal on Control and Optimization*, vol. 15, no. 1, pp. 163–176, 1977.

[16] K. Narendra and A. Annaswamy, *Stable Adaptive Systems*. Dover Publications, 2012.

[17] K. S. Narendra and A. M. Annaswamy, “Persistent excitation in adaptive systems,” *International Journal of Control*, vol. 45, no. 1, pp. 127–160, 1987.

[18] H. K. Khalil, *Nonlinear systems*. Prentice Hall, 2002.

[19] E. Panteley, A. Loría, and A. Teel, “Relaxed persistency of excitation for uniform asymptotic stability,” *IEEE Transactions on Automatic Control*, vol. 46, no. 12, pp. 1874–1886, 2001.

[20] A. Loría, E. Panteley, D. Popović, and A. Teel, “A nested Matrosov theorem and persistency of excitation for uniform convergence in stable nonautonomous systems,” *IEEE Transactions on Automatic Control*, vol. 50, no. 2, pp. 183–198, 2005.

# APPENDIX I
ON δ-PERSISTENCY OF EXCITATION

**Definition 1 (Persistency of excitation):** A function $\phi: \mathbb{R}_{\ge 0} \to \mathbb{R}^{n \times m}$ is said to be persistently exciting, if there exist positive $T$ and $\mu$ such that for any unit vector $v \in \mathbb{R}^m$,

$$ \int_t^{t+T} \|\phi(\tau)v\|^2 d\tau \ge \mu, \forall t \ge 0. \quad (46) $$

Partition $x \in \mathbb{R}^n$ as $x := [x_1^\top \ x_2^\top]^\top$ where $x_1 \in \mathbb{R}^{n_1}$ and $x_2 \in \mathbb{R}^{n_2}$. Define the set $\mathcal{D}_1 := (\mathbb{R}^{n_1} \setminus \{0\}) \times \mathbb{R}^{n_2}$ and the

function $\phi: \mathbb{R} \times \mathbb{R}^n \to \mathbb{R}^m$ where $t \mapsto \phi(t, x)$ is locally integrable. The following defines uniformly $\delta$-persistently exciting from [20, Lemma 1].

**Definition 2:** [Uniformly $\delta$-persistency of excitation] If $x \mapsto \phi(t, x)$ is continuous uniformly in $t$, then $\phi(\cdot, \cdot)$ is uniformly $\delta$-persistently exciting (u$\delta$-PE) with respect to $x_1$ if and only if for each $x \in \mathcal{D}_1$ there exist positive $T$ and $\mu$ such that for any unit vector $v \in \mathbb{R}^m$

$$ \int_t^{t+T} \|\phi(\tau, x)v\|^2 d\tau \ge \mu, \forall t \ge 0. \quad (47) $$

The next Lemma establishes that when a strictly proper stable filter is subject to a bounded disturbance, and driven by a u$\delta$-PE input, its output retains the property of being u$\delta$-PE. The lemma, originally introduced in [19], didn't account for the presence of a bounded disturbance.

**Lemma 2 (Filtration property):** Let $\phi: \mathbb{R}_{\ge 0} \times \mathbb{R}^n \to \mathbb{R}^{p \times q}$ and consider the system

$$ \begin{bmatrix} \dot{x} \\ \dot{\omega} \end{bmatrix} = \begin{bmatrix} f(t, x, \omega) \\ f_1(t, \omega) + f_2(t, x)\omega + \phi(t, x) \end{bmatrix} \quad (48) $$

where $f_1: \mathbb{R}_{\ge 0} \times \mathbb{R}^n \to \mathbb{R}^{p \times q}$ is Lipschitz in $\omega$ uniformly in $t$ and measurable in $t$ and satisfies $\|f_1(\cdot)\| \le l\|\omega\|$ for all $t$; $f_2: \mathbb{R}_{\ge 0} \times \mathbb{R}^n \to \mathbb{R}^{p \times p}$ is locally Lipschitz in $x$ uniformly in $t$ and measurable in $t$. Assume that $\phi(t, x)$ is u$\delta$-PE with respect to $x$. Assume that $\phi$ is locally Lipschitz and there exists a non-decreasing function $\alpha: \mathbb{R}_{\ge 0} \to \mathbb{R}_{\ge 0}$, such that, for all $(t, x) \in \mathbb{R}_{\ge 0} \times \mathbb{R}^n$:

$$ \max \left\{ \|\phi(\cdot)\|, \|f_2(\cdot)\|, \left\|\frac{\partial \phi(\cdot)}{\partial t}\right\|, \left\|\frac{\partial \phi(\cdot)}{\partial x}\right\| \right\} \le \alpha(\|x\|). \quad (49) $$

Assume that $f(\cdot)$ satisfies that

$$ \max \{\|f(\cdot)\|\} \le \alpha(\|x\|) + k, \quad (50) $$

where $k$ is a positive constant. Denote $w = (w_1, w_2, \dots, w_p)^\top$ and $w_i^\top \in \mathbb{R}^q$ with $i = 1, 2, \dots, q$. If all solutions $x_\phi(t)$, defined as $x_\phi := [x^\top \ \omega_1 \ \omega_2 \ \dots \ \omega_p]^\top$, satisfy

$$ \|x_\phi(t)\| \le r \quad \forall t \ge t_0, \quad (51) $$

for a positive constant $r$, then $\omega$ is uniformly $\delta$-persistently exciting with respect to $x$. $\square$
Proof: Denote $v \in \mathbb{R}^p$ as a unit vector. Defining $\rho := -v^\top \phi \omega^\top v$, we have

$$ 
\begin{aligned}
\dot{\rho} &= - \|\phi^\top v\|^2 - v^\top \phi f_1^\top v - v^\top \left[ f_2 \phi + \frac{\partial \phi}{\partial t} + \frac{\partial \phi}{\partial x} f \right] \omega^\top v \\
&\leq - \|\phi^\top v\|^2 + \|\omega^\top v\| \left[ 2\alpha^2(r) + (l+k+1)\alpha(r) \right] \|v\| \\
&= - \|\phi^\top v\|^2 + c(r) \|\omega^\top v\|,
\end{aligned}
\quad (52) $$

where $c(r) := 2\alpha^2(r)+(l+k+1)\alpha(r)$. Integrating both sides of (52) from $t$ to $t+T_f$ and then reversing the inequality sign, we derive that

$$ 
\begin{aligned}
v^\top \phi(t, x) \omega(t)^\top v - v^\top \phi(t+T_f, x) \omega(t+T_f)^\top v \\
\geq \int_t^{t+T_f} \|\phi(\tau, x)^\top v\|^2 d\tau - \int_t^{t+T_f} c(r) \|\omega(\tau)^\top v\| d\tau.
\end{aligned}
\quad (53) $$

By applying the bounds in (49), (50) and (51) to the left-hand side of inequality (53), we have

$$ 2\alpha(r)r \geq \int_{t}^{t+T_f} \|\phi(\tau, x)^{\top} v\|^2 d\tau - \int_{t}^{t+T_f} c(r) \|\omega(\tau)^{\top} v\| d\tau. $$

Let $T_f := k'T$. Since $\phi(t, x)$ is u$\delta$-PE from (47), there exists $\mu$ such that

$$ \int_{t}^{t+k'T} \|\phi(\tau, x)^{\top} v\|^2 d\tau \geq k'\mu. $$

Thus, we obtain

$$ \int_{t}^{t+k'T} \|\omega(\tau)^{\top} v\|^2 d\tau \geq \frac{(k'\mu - 2\alpha(r)r)^2}{c(r)^2} =: \mu_r. $$

Choosing $k'$ large enough so that $\mu_r > 0$, $\omega(t)$ is u$\delta$-PE with respect to $x$. ■