HyNTP: A Distributed Hybrid Algorithm for Time Synchronization
HyNTP: A Distributed Hybrid Algorithm for Time Synchronization

Marcello Guarro

Ricardo Sanfelice

August 27, 2024

HyNTP: A Distributed Hybrid Algorithm for
Time Synchronization

Marcello Guarro and Ricardo G. Sanfelice
This research has been partially supported by the National Science Foundation under Grant no. ECS-1710621, and Grant no. CNS-1544396, by the Air Force Office of Scientific Research under Grant no. FA9550-16-1-0015, Grant no. FA9550-19-1-0053, and Grant no. FA9550-19-1-0169, and by CITRIS and the Banatao Institute at the University of California.
Marcello Guarro and Ricardo G. Sanfelice are with Department of Electrical
and Computer Engineering, University of California, Santa Cruz. Email:

This paper presents a distributed hybrid algorithm that synchronizes the time and rate of a set of clocks connected over a network. Clock measurements of the nodes are given at aperiodic time instants and the controller at each node uses these measurements to achieve synchronization. Due to the continuous and impulsive nature of the clocks and the network, we introduce a hybrid system model to effectively capture the dynamics of the system and proposed hybrid algorithm. Moreover, the hybrid algorithm allows each agent to estimate the skew of its internal clock in order to allow for synchronization to a common timer rate. We provide sufficient conditions guaranteeing synchronization of the timers, exponentially fast. Numerical results illustrate the synchronization property induced by the proposed algorithm as well as its performance against comparable algorithms from the literature.

§ INTRODUCTION

§.§ Motivation

Since the advent of asynchronous packet-based networks in communication and information technology, the topic of clock synchronization has received significant attention due to the temporal requirements of packet-based networks for the exchange of information. In recent years, as distributed packet-based networks have evolved in terms of size, complexity, and, above all, application scope, there has been a growing need for new clock synchronization schemes with tractable design conditions to meet the demands of these evolving networks.

Distributed applications such as robotic swarms, automated manufacturing, and distributed optimization rely on precise time synchronization among distributed agents for their operation; see [1]. For example, in the case of distributed control and estimation over networks, the uncertainties of packet-based network communication requires timestamping of sensor and actuator messages in order to synchronize the information to the evolution of the dynamical system being controlled or estimated. Such a scenario is impossible without the existence of a common timescale among the non-collocated agents in the system. In fact, the lack of a shared timescale among the networked agents can result in performance degradation that can destabilize the system; see [2]. Moreover, one cannot  always assume that consensus on time is a given, especially when the network associated to the distributed system is subject to perturbations such as noise, delay, or jitter. Hence, it is essential that these networked systems utilize clock synchronization schemes that establish and maintain a common timescale for their algorithms.

§.§ Background and Related Work

For many networked control system settings, each agent in the system is fitted with its own internal hardware clock and one or more software clocks that inherits the dynamics of the hardware clock. In an ideal scenario, the $i$-th agent in the system would have a clock $\tau_i \in \mathbb{R}_{\geq 0}$ such that $\tau_i(t) = t$, where $t$ is the global or real time. However, many hardware clocks utilize quartz-crystal or MEMS oscillators, susceptible to manufacturing imperfections and environmental factors that affect oscillator frequency; see [3] and [4]. Due to the variability in oscillator frequency, one generally considers the continuous-time dynamics of the $i$-th hardware clock node given by

\begin{equation} \label{eqn:clock_dyn}
\dot{\tau}_i = a_i
\end{equation}

where $a_i \in \reals$ defines the clock's rate of change. Solving the differential equation gives the following relationship to the ideal clock or real time reference $t$:

\begin{equation}
\tau_i(t) = a_i t + \tau_i(0)
\end{equation}

where the initial condition $\tau_i(0)$ gives the offset from $t$. For a network of $n$ agents, the notion of clock synchronization can be defined as the state of the networked system such that $\tau_i = \tau_j$ for all $i,j \in \{1,2,\ldots, n \}$, $i \neq j$.

In an ideal setting with no delay and identical clock  rates (or skews), synchronization between two nodes, Node $1$ and Node $2$, can be achieved by the following simple reference-based algorithm. Node $1$ sends its time to Node $2$. Node $2$ calculates its offset relative to $1$. Node $2$ applies the offset correction to its clock. For the case of non-identical clock skews, a pair of measurements from Node $1$ would allow Node $2$ to calculate its relative skew $\frac{a_1}{a_2}$ and apply a correction accordingly. In a realistic setting, however, network communication between nodes is subject to a variety of delays to which such simple reference-based algorithms are nonrobust; see [5]. Moreover, these algorithms become cumbersome in terms of network utilization and computation as the number of nodes on the network increases.

The seminal Networking Time Protocol (NTP) presented in [6] mitigates these challenges through the implementation of a centralized algorithm.  In particular,  the networked agents  in the system  synchronize to a known reference that is either injected or provided by an  elected  leader agent. The effects of communication delay are mitigated via assumptions on the round-trip delay that occurs in the communication of any two nodes on the network. Conversely, other centralized approaches, such as those in [7] and [8], assume the communication delay to be negligible and instead utilize least-squares minimization to estimate the errors in the offset and rates of change between the synchronizing nodes and the elected  reference  agent. Unfortunately, these approaches suffer robustness issues when communication with the reference node is lost or if the random delays in the transmission do not follow a normal distribution, see [3]. Moreover, algorithms like NTP were not designed for dynamic network topologies as they rely on predefined network hierarchies that define the relationships between the reference nodes and their children. Any change to the topology requires a reconstruction of the hierarchy adding considerable delay to the synchronization of the clocks.

Recently, the observed robustness issues in the centralized protocols have motivated leader-less, consensus-based approaches by leveraging the seminal results on networked consensus in [9], [10], and [11]. In particular, the works of [1], [12], [13], and, more recently, [14]  employ  average consensus to give asymptotic results on clock synchronization  under asynchronous and asymmetric communication topology. Unfortunately, a high number of iterations of the algorithm is often required before the desired synchronization accuracy is achieved. Moreover, the constraint on asymmetric communication precludes any results guaranteeing stability or robustness. Lastly, these approaches suffer from complexity in both computation and memory allocation that may be unfit for certain applications.

The work in [15] also considers a consensus-based approach by using a controller that uses a proportional gain to compensate for the clock rates and an integrator gain acting on an auxiliary control state that compensates for the clock offsets. Though the solution in [15] provides faster convergence than the other approaches using average consensus, the algorithm assumes periodic synchronous communication of the nodes. This assumption is relaxed in [16] by considering asynchronous communication events. The authors in [17] consider a similar relaxation but also relax assumptions on the graph structure. However, in both  [16] and [17]  the clocks are slower to converge compared to the synchronous communication setting. Still, both synchronous and asynchronous scenarios require a large number of iterations before synchronization is achieved. Moreover, the algorithm subjects the clocks to significant non-smooth adjustments in clock rate and offset that may prove undesirable in certain application settings.

Other recent works include [18] where the clocks are communicating over a network that is modeled as a Markov chain, the parameters of the clocks are then estimated from noisy measurements using a distributed algorithm. By making assumptions on the Markov chain, the estimates are shown to be mean square convergent if the union of the communication graphs is connected. The authors in [19] present an algorithm that estimates both skew and offset using a joint Maximum-Likelihood Estimation by assuming an exponential delay model.

§.§ Contributions

The lack of performance guarantees in the aforementioned works have motivated the design of a hybrid clock synchronization algorithm with tractable design conditions. In particular, this paper introduces a distributed hybrid algorithm that exponentially synchronizes a set of clocks connected over a network via measurements given at aperiodic time instants.

Inspired by the contributions in [20], we present a distributed hybrid algorithm to synchronize the network clocks in the presence of non-ideal clock skews while capturing the continuous and impulsive dynamics of the network into a hybrid model. To achieve synchronization with a common rate of change, the algorithm also allows for local estimation of the skew of the internal clock at each agent. The use of a hybrid systems model to solve the problem under consideration allows for the application of a Lyapunov-based analysis to show stability of a desired set of interest. Using results from [21], we show that, via a suitable change of coordinates, our distributed hybrid clock synchronization algorithm guarantees synchronization of the timers, exponentially fast. To summarize, the main contributions of this paper are given as follows:

* In Section <ref>, we introduce HyNTP, a distributed hybrid algorithm that synchronizes the clock rates and offsets to solve the problem outlined in Section <ref>. Moreover, we present a hybrid systems model to capture the network dynamics for the case of synchronous and aperiodic communication events. In Section <ref>, we present a reduced model of the system and a subsequent auxiliary model that is generated from  an appropriately defined  change of coordinates. With the auxiliary model, we present necessary and sufficient conditions for which stability of a compact set, representing synchronization, holds. Moreover, we show that the system is robust to perturbations on the communication noise, clock drift, and the desired clock rate reference in Section <ref>.

* In Section <ref>, we compare the merits of our algorithm to competing algorithms in the literature.

We inform the reader that  some details have been omitted due to space constraints and can be found in the technical report [22].

§.§ Notation

The set of natural numbers including zero, i.e., $\{0,1,2,\ldots\}$ is denoted by $\mathbb{N}$. The set of natural numbers is denoted as $\mathbb{N}_{> 0}$, i.e., $\mathbb{N}_{> 0} = \{1,2,\ldots\}$. The set of real numbers is denoted as $\reals$. The set of nonnegative real numbers is denoted by $\mathbb{R}_{\geq 0}$, i.e., $\mathbb{R}_{\geq 0} = [0, \infty )$. The $n$-dimensional Euclidean space is denoted $\mathbb{R}^n$. Given sets $A$ and $B$, $F: A \rightrightarrows B$ denotes a set-valued map from $A$ to $B$. For a matrix $A \in \mathbb{R}^{n \times m}$, $A^\mathsf{T}$ denotes the transpose of $A$. Given a vector $x \in \mathbb{R}^n$, $|x|$ denotes the Euclidean norm. Given a vector $x \in \mathbb{R}^n$ and  a nonempty  set $\Sigma \subset \reals^n$, $|x|_{\Sigma}$ denotes the Euclidean point-to-set distance, i.e., $|x|_{\Sigma} \doteq {\rm inf}_{y \in \Sigma} |x-y|$. Given two vectors $x \in \mathbb{R}^n$ and $y \in \mathbb{R}^m$, we use the equivalent notation $(x,y) = [ x^\mathsf{T} \hspace{2mm} y^\mathsf{T} \hspace{1mm} ]^\mathsf{T}$. Given a closed set $A \subset \mathbb{R}^n$ and a closed set $B \subset A$, the projection of $A$ onto $B$ is denoted by $\mbox{\rm proj}_B (A)$. Given a matrix $A \in \mathbb{R}^{n \times m}$, $|A| := \mbox{max} \{\sqrt{|\lambda|} : \lambda \in \mbox{eig}(A^\mathsf{T}A)\}$. For two symmetric matrices $A \in \mathbb{R}^{n \times m}$ and $B \in \mathbb{R}^{n \times m}$, $A \succ B$ means that $A - B$ is positive definite; conversely, $A \prec B$ means that $A - B$ is negative definite. Given a function $f : \mathbb{R}^n \rightarrow \mathbb{R}^m$, the range of $f$ is given by $\mbox{rge } f := \{ y \hspace{1mm} : \hspace{1mm} \exists \hspace{1mm} x \mbox{ s.t. } y = f(x) \}$. A vector of $N$ ones is denoted $\textbf{1}_{N}$. The matrix $I_n$ is used to denote the identity matrix of size $n \times n$.

§.§ Organization and Notation

This paper is organized as follows. Section <ref> presents preliminary material on graph theory and hybrid systems. Section <ref> introduces the clock synchronization problem and the system being studied.  Section <ref>  outlines the algorithm under consideration, the associated hybrid model of the closed-loop system, including the main result. Section <ref> gives properties to the nominal closed-loop system. Section <ref> provides comparisons to other comparable algorithm through numerical examples.

confNotation: The set of natural numbers including zero, i.e., $\{0,1,2,\ldots\}$ is denoted by $\mathbb{N}$. The set of natural numbers is denoted as $\mathbb{N}_{> 0}$, i.e., $\mathbb{N}_{> 0} = \{1,2,\ldots\}$. The set of real numbers is denoted as $\reals$. The set of non-negative real numbers is denoted by $\mathbb{R}_{\geq 0}$, i.e., $\mathbb{R}_{\geq 0} = [0, \infty )$. The $n$-dimensional Euclidean space is denoted $\mathbb{R}^n$. Given sets $A$ and $B$, $F: A \rightrightarrows B$ denotes a set-valued map from $A$ to $B$. For a matrix $A \in \mathbb{R}^{n \times m}$, $A^\mathsf{T}$ denotes the transpose of $A$. Given a vector $x \in \mathbb{R}^n$, $|x|$ denotes the Euclidean norm. Given two vectors $x \in \mathbb{R}^n$ and $y \in \mathbb{R}^m$, we use the equivalent notation $(x,y) = [ x^\mathsf{T} \hspace{2mm} y^\mathsf{T} \hspace{1mm} ]^\mathsf{T}$. Given a matrix $A \in \mathbb{R}^n$, $|A| := \mbox{max} \{\sqrt{|\lambda|} : \lambda \in \mbox{eig}(A^\mathsf{T}A)\}$. For two symmetric matrices $A \in \mathbb{R}^n$ and $B \in \mathbb{R}^n$, $A \succ B$ means that $A - B$ is positive definite; conversely, $A \prec B$ means that $A - B$ is negative definite. Given a function $f : \mathbb{R}^n \rightarrow \mathbb{R}^m$, the range of $f$ is given by $\mbox{rge } f := \{ y \hspace{1mm} : \hspace{1mm} \exists \hspace{1mm} x \mbox{ s.t. } y = f(x) \}$. A vector of $N$ ones is denoted $\textbf{1}_{N}$. The matrix $I_n$ is used to denote the identity matrix of size $n \times n$.
Notation: The set of natural numbers including zero, i.e., $\{0,1,2,\ldots\}$ is denoted by $\mathbb{N}$. The set of natural numbers is denoted as $\mathbb{N}_{> 0}$, i.e., $\mathbb{N}_{> 0} = \{1,2,\ldots\}$. The set of real numbers is denoted as $\reals$. The set of non-negative real numbers is denoted by $\mathbb{R}_{\geq 0}$, i.e., $\mathbb{R}_{\geq 0} = [0, \infty )$. The $n$-dimensional Euclidean space is denoted $\mathbb{R}^n$. Given sets $A$ and $B$, $F: A \rightrightarrows B$ denotes a set-valued map from $A$ to $B$. For a matrix $A \in \mathbb{R}^{n \times m}$, $A^\mathsf{T}$ denotes the transpose of $A$. Given a vector $x \in \mathbb{R}^n$, $|x|$ denotes the Euclidean norm. Given a vector $x \in \mathbb{R}^n$ and set $\Sigma \subset \reals^n$, $|x|_{\Sigma}$ denotes the Euclidean point-to-set distance, i.e., $|x|_{\Sigma} \doteq {\rm inf}_{y \in \Sigma} |x-y|$. Given two vectors $x \in \mathbb{R}^n$ and $y \in \mathbb{R}^m$, we use the equivalent notation $(x,y) = [ x^\mathsf{T} \hspace{2mm} y^\mathsf{T} \hspace{1mm} ]^\mathsf{T}$. Given a closed set $A \subset \mathbb{R}^n$ and a closed set $B \subset A$, the projection of $A$ onto $B$ is denoted by $\mbox{\rm proj}_B (A)$. Given a matrix $A \in \mathbb{R}^{n \times m}$, $|A| := \mbox{max} \{\sqrt{|\lambda|} : \lambda \in \mbox{eig}(A^\mathsf{T}A)\}$. For two symmetric matrices $A \in \mathbb{R}^{n \times m}$ and $B \in \mathbb{R}^{n \times m}$, $A \succ B$ means that $A - B$ is positive definite; conversely, $A \prec B$ means that $A - B$ is negative definite. Given a function $f : \mathbb{R}^n \rightarrow \mathbb{R}^m$, the range of $f$ is given by $\mbox{rge } f := \{ y \hspace{1mm} : \hspace{1mm} \exists \hspace{1mm} x \mbox{ s.t. } y = f(x) \}$. A vector of $N$ ones is denoted $\textbf{1}_{N}$. The matrix $I_n$ is used to denote the identity matrix of size $n \times n$.

§ PRELIMINARIES

§.§ Preliminaries on Graph Theory

Let $\DG = (\nodes, \edges, \adj)$  be a weighted directed graph (digraph) where $\nodes = \{1,2,\ldots,n\}$ represents the set of $n$ nodes, $\edges \subset \nodes \times \nodes$ the set of edges, and $\adj \in \{0,1\}^{n \times n}$ represents the adjacency matrix. An edge of $\DG$ is denoted by $e_{ij} = (i,j)$. The elements of $\adj$ are denoted by $a_{ij}$, where $a_{ij} = 1$ if $e_{ij} \in \edges$ and $a_{ij} = 0$ otherwise. The in-degree and out-degree of a node $i$ are defined by $d^{in}(i) = \sum^n_{k=1} a_{ki}$ and $d^{out}(i) = \sum^n_{k=1} a_{ik}$, respectively. The largest and smallest in-degree of a digraph are given by $\bar{d} = \textmd{max}_{i \in \nodes} d^{in}(i)$ and $\underbar{$d$} = \textmd{min}_{i \in \nodes} d^{in}(i)$, respectively. The in-degree matrix is an $n \times n$ diagonal matrix, denoted $\mathcal{D}$, with elements given by
\begin{equation*}
d_{ij} = \begin{cases} d^{in}(i) & \mbox{ if } i = j \\ 0 & \mbox{ if } i \neq j \end{cases} \hspace{5mm} \forall i \in \nodes
\end{equation*}

The Laplacian matrix of a digraph $\DG$, denoted by $\mathcal{L}$, is defined as $\mathcal{L} = \mathcal{D} - \adj$ and has the property that $\mathcal{L} \textbf{1}_n = 0$. The set of nodes corresponding to the neighbors that share an edge with node $i$ is denoted by $\nbors(i) := \defset{k \in \nodes}{e_{ki} \in \edges}$. In the context of networks, $\nbors(i)$ represents the set of nodes for which an agent $i$ can communicate with.

((Olfati-Saber and Murray, 2004, Theorem 6),(Fax and Murray, 2004, Propositions 1, 3, and 4))
For an undirected graph, $\mathcal{L}$ is symmetric and positive semidefinite, and each eigenvalue of  $\mathcal{L}$ is real. For a directed graph, zero is a simple eigenvalue of  $\mathcal{L}$ if the directed graph is strongly connected.

(Godsil and Royle (2001)) Consider an $n \times n$ symmetric matrix $A = \{a_{ik} \}$ satisfying $\sum_{i=1}^n a_{ik} = 0$ for each $k \in \nodes$. The following statements hold:

* There exists an orthogonal matrix $U$ such that $U^\top A U = \begin{bmatrix} 0 & 0 \\ 0 & \star \end{bmatrix}$, where $\star$ represents any nonsingular matrix with
appropriate dimensions and 0 represents any zero
matrix with appropriate dimensions.

* The matrix $A$ has a zero eigenvalue with eigenvector
$\textbf{1}_n \in \reals^n$.

A weighted digraph is said to be

* balanced if the in-degree matrix and the out-degree matrix for every node are equal, i.e., $d^{in}(i) = d^{out}(i)$ for each $i \in \nodes$.

* complete if every pair of distinct nodes are connected by a unique edge, i.e., $a_{ik} = 1$ for each $i,k \in \nodes, i \neq k$.

* strongly connected if and only if for any two distinct nodes there exists a path of directed edges that connects them.

§.§ Preliminaries on Hybrid Systems

A hybrid system $\cal H$ in $\mathbb{R}^n$ is composed by the following data: a set $C \subset \mathbb{R}^n$, called the flow set; a differential equation defined by the function $f: \mathbb{R}^n \to \mathbb{R}^n$ with $C \subset \mbox{dom } f$, called the flow map; a set $D \subset \mathbb{R}^n$, called the jump set; and a set-valued mapping $G: \mathbb{R}^n \rightrightarrows \mathbb{R}^n$ with $D \subset \mbox{dom } G$, called the jump map. Then, a hybrid system $\mathcal{H} := (C,f,D,G)$ is written in the compact form

\begin{equation} \label{eqn:Hy}
\cal H : \begin{cases} x \in C \hspace{0.5cm} & \dot{x} \hspace{2mm} = f(x) \\
x \in D \hspace{0.5cm} &  x^+ \in G(x)
\end{cases}
\end{equation}

where $x$ is the system state. Solutions to hybrid systems are denoted by $\phi$ and are parameterized by $(t,j)$, where $t \in \mathbb{R}_{\geq 0}$ defines ordinary time and $j \in \mathbb{N}$ is a counter that defines the number of jumps. A solution $\phi$ is defined by a hybrid arc on its domain $\mbox{dom } \phi$ with hybrid time domain structure [21]. The domain $\mbox{dom } \phi$ is a hybrid time domain if $\mbox{dom } \phi \subset \mathbb{R}_{\geq 0} \times \mathbb{N}$ and for each $(T,J) \in \mbox{dom } \phi,$ $\mbox{dom } \phi \cap ([0,T] \times \{0,1,...,J\})$ is of the form $\bigcup_{j = 0}^J ([t_j, t_{j+1}] \times \{j\})$, with $0 = t_0 \leq t_1 \leq t_2 \leq t_{J + 1}$. A function $\phi : \mbox{dom } \phi \to \reals^n$ is a hybrid arc if $\mbox{dom } \phi$ is a hybrid time domain and if for each $j \in \mathbb{N}$, the function $t \mapsto \phi(t,j)$ is locally absolutely continuous on the interval $I^j = \{t : (t,j) \in \mbox{dom } \phi \}$. A solution $\phi$ satisfies the system dynamics; see <cit.> for more details. A solution $\phi$ is said to be maximal if it cannot be extended by flow or a jump, and complete if its domain is unbounded. The set of all maximal solutions to a hybrid system $\HS$ is denoted by $\mathcal{S}_{\HS}$ and the set of all maximal solutions to $\HS$ with initial condition belonging to a set $A$ is denoted by $\mathcal{S}_{\HS}(A)$. A hybrid system is well-posed if it satisfies the hybrid basic conditions in <cit.>.

The following definition introduces stability notions for hybrid systems $\HS$ with state $x \in \reals^n$ to a closed set of interest $\A \subset \reals^n$.

Given a hybrid system $\HS$ defined on $\reals^n$, the closed set $\A \subset \reals^n$ is said to be globally exponentially stable (GES) for $\HS$ if there exist $\kappa, \alpha > 0$ such that every maximal solution $\phi$ to $\HS$ is complete and satisfies $| \phi(t,j) |_{\A} \leq \kappa e^{\minus \alpha (t+ j)} |\phi(0,0)|_{\A}$ for each $(t,j) \in \mbox{dom } \phi$.

§ PROBLEM STATEMENT

Consider a group of $n$ sensor nodes connected over a network represented by a digraph $\DG = (\nodes,\edges, \adj)$. Two clocks are attached to each node $i$ of $\DG$: an (uncontrollable) internal clock $\tau^*_i \in \reals_{\geq 0}$ whose dynamics are given by

\begin{equation} \label{eqn:clk_dynamics1}
\dot{\tau}^*_i = a_i
\end{equation}

and an adjustable clock $\tilde{\tau}_i \in \reals_{\geq 0}$ with dynamics

\begin{equation} \label{eqn:clk_dynamics2}
\begin{aligned}
\dot{\tilde{\tau}}_i & = a_i + u_i
\end{aligned}
\end{equation}

where $u_i \in \reals$ is a control input. In both of these models, the (unknown) constant $a_i$ represents the unknown drift of the internal clock. At times $t_j$ for $j \in \mathbb{N}_{> 0}$ (we assume $t_0 = 0$), node $i$ receives measurements $\tilde{\tau}_k$ from its neighbors, namely, for each $k \in \mathcal{N}(i)$. The resulting sequence of time instants $\{t_j\}^{\infty}_{j=1}$ is assumed to be strictly increasing and unbounded. Moreover, for such a sequence, the time elapsed between each time instant when the clock measurements are exchanged satisfies

\begin{equation} \label{eqn:t_bounds}
T_1 \leq t_{j+1} - t_j \leq T_2, \hspace{3mm}
0 \leq t_1 \leq T_2 \hspace{5mm} \forall j \in \mathbb{N}_{> 0}
\end{equation}

\begin{equation} \label{eqn:t_bounds}
\begin{aligned}
T_1 & \leq t_{j+1} - t_j \leq T_2 \hspace{5mm} \forall j \in \mathbb{N}_{> 0} \\
0 & \leq t_1 \leq T_2
\end{aligned}
\end{equation}

where $0 < T_1 \leq T_2$, with $T_1$ defining a minimum time between consecutive measurements and $T_2$ defines the maximum allowable transfer interval (MATI).

The models for the clocks are based on the hardware and software relationship of the real-time system that implements them. That is, the internal clock $\tau_i^*$ is treated as a type of hardware oscillator while the adjustable clock $\tilde{\tau}_i$ is treated as a virtual clock, implemented in software (as part of the proposed algorithm), that evolves according to the dynamics of the hardware oscillator. Any virtual clock implemented in node $i$ inherits the drift parameter $a_i$ of the internal clock, which cannot be controlled. More importantly, this drift parameter is not known due to the fact that universal time information is not available to any node. The input $u_i$ is unconstrained as allowed by hardware platforms.

Under such a setup, our goal is to design a distributed hybrid controller that, without knowledge of the drift parameter and of the communication times in advance, assigns the input $u_i$ to drive each clock $\tilde{\tau}_i$ to synchronization with every other clock $\tilde{\tau}_k$, with $\tilde{\tau}_k$ evolving at a common prespecified constant rate of change $\sigma^* > 0$ for each $k \in \nodes$. This problem is formally stated as follows:

Given a network of $n$ agents with dynamics as in (<ref>) and (<ref>) represented by a directed graph $\mathcal{G}$ and $\sigma^* > 0$, design a distributed hybrid controller that achieves the following two properties when information between agents is exchanged at times $t_j$ satisfying (<ref>):

i) Global clock synchronization: for each initial condition, the components $\tilde{\tau}_1, \tilde{\tau}_2, \ldots, \tilde{\tau}_n$ of each complete solution to the system satisfy
$$\lim_{t \to \infty} | \tilde{\tau}_i(t) - \tilde{\tau}_k(t)| = 0 \hspace{5mm} \forall i,k \in \mathcal{V}, i \neq k$$

ii) Common clock rate: for each initial condition, the components $\tilde{\tau}_1, \tilde{\tau}_2, \ldots, \tilde{\tau}_n$ of each complete solution to the system satisfy
$$\lim_{t \to \infty}  |\dot{\tilde{\tau}}_i(t) - \sigma^*| = 0 \hspace{5mm} \forall i \in \mathcal{V}$$

§ DISTRIBUTED HYBRID CONTROLLER FOR
TIME SYNCHRONIZATION

We define the hybrid model that provides the framework and a solution to Problem <ref>. First, since we are interested in the ability of the rate of each clock to synchronize to a constant rate $\sigma^*$, we propose the following change of coordinates: for each $i \in \nodes$, define $e_i := \tilde{\tau}_i - r$, where $r \in \reals_{\geq 0}$ is an auxiliary variable such that $\dot{r} = \sigma^*$. The state $r$ is only used for analysis. Then, the dynamics for $e_i$ are given by

\begin{equation}
\begin{aligned}
\dot{e}_i & = \dot{\tilde{\tau}}_i - \sigma^* \qquad \forall i \in \nodes \\
\end{aligned}
\end{equation}
By making the appropriate substitutions, one has

\begin{equation} \label{eqn:e_dynamics}
\begin{aligned}
\dot{e}_i & = a_i + u_i - \sigma^* \qquad \forall i \in \nodes \\
\end{aligned}
\end{equation}

To model the network dynamics for aperiodic communication events at $t_j$'s satisfying (<ref>), we consider a timer variable $\tau$ with hybrid dynamics

\begin{equation} \label{eqn:timer}
\begin{aligned}
& \dot{\tau} \hspace{2mm} = \minus 1 \hspace{5mm} \tau \in [0,T_2],
\hspace{5mm} \tau^+ \in [T_1,T_2] \hspace{5mm} \tau = 0
\end{aligned}
\end{equation}

This model is such that when $\tau = 0$, a communication event is triggered, and $\tau$ is reset to a point in the interval $[T_1,T_2]$ in order to preserve the bounds given in (<ref>); see [23].

The proposed hybrid algorithm assigns a value to $u_i$ so as to solve Problem <ref>, which in the $e_i$ coordinates requires $e_i$ to converge to zero for each $i \in \nodes$. In fact, the algorithm implements two feedback laws: a distributed feedback law and a local feedback law. The distributed feedback law utilizes a control variable $\eta_i \in \reals$ that is impulsively updated at communication event times using both local and exchanged measurement information $\tilde{\tau}_k$. Specifically, it takes the form
$$\eta_i^+ = \sum_{k \in \mathcal{N}(i)} K_i^{k}(\tilde{\tau}_i, \tilde{\tau}_k)$$
where $K_i^{k}(\tilde{\tau}_i, \tilde{\tau}_k) := -\gamma_i (e_i - e_k)$ with $\gamma_i > 0$. Between communication event times, $\eta_i$ evolves continuously. The local feedback strategy utilizes a continuous-time linear adaptive estimator with states $\hat{\tau}_i \in \reals$ and $\hat{a}_i \in \reals$ to estimate the drift $a_i$ of the internal clock. The estimate of the drift is then injected as feedback to compensate for the effect of $a_i$ on the evolution of $\tilde{\tau}_i$. Furthermore, the local feedback strategy injects $\sigma^*$ to attain the desired clock rate for $\tilde{\tau}_i$.

Inspired by the protocol in <cit.>, the dynamics of the $i$-th hybrid controller are given by

\begin{equation} \label{protocol}
\begin{aligned}
& \begin{drcases}
\dot{u}_i & =  h_i \eta_i - \mu_i ( \hat{\tau}_i - \tau^*_i ), \hspace{1mm}
\dot{\eta}_i =  h_i \eta_i \\
\dot{\hat{a}}_i & = - \mu_i ( \hat{\tau}_i - \tau_i^* ), \hspace{1mm}
\dot{\hat{\tau}}_i = \hat{a}_i - (\hat{\tau}_i - \tau_i^*) \hspace{4.5mm}
\end{drcases} \hspace{2mm} \tau \in [0, T_2] \\
& \begin{drcases} u_i^+ & =  {\minus}\gamma_i \sum_{k \in \mathcal{N}(i)} ( \tilde{\tau}_i {-} \tilde{\tau}_k) {-} \hat{a}_i {+} \sigma^*, \hspace{1mm} \hat{a}^+_i = \hat{a}_i \\
\eta_i^+ & =  -\gamma_i \sum_{k \in \mathcal{N}(i)} (\tilde{\tau}_i - \tilde{\tau}_k), \hspace{1mm}
\hat{\tau}^+_i = \hat{\tau}_i
\end{drcases} \hspace{2mm} \tau = 0
\end{aligned}
\end{equation}

where $h_i \in \reals$, $\gamma_i > 0$ are controller parameters for the distributed hybrid consensus controller and $\mu_i > 0$ is a parameter for the local parameter estimator. The state $\eta$ is included in the model to facilitate a model reduction used in the results that follow. Note that $u_i$ is treated (with some abuse of notation) as an auxiliary state of the controller. This state is kept constant in between events and is reset to the new value of $\eta_i  - \hat{a}_i + \sigma^*$ at jumps. Observe that the distributed controller only uses local and communicated information from the neighboring nodes at communication event times $t_j$, which, as explained above (<ref>), are times at which $\tau$ is zero.

With the timer variable and hybrid controller defined in (<ref>), we construct the hybrid closed-loop system $\HS$ obtained from the interconnection between the distributed hybrid controller and the local adaptive estimator given in error coordinates. The state of the closed-loop system is

\begin{equation} \label{eqn:state}
\begin{aligned}
x & = (e, u, \eta, \tau^*, \hat{a}, \hat{\tau}, \tau) \in \mathcal{X}
\end{aligned}
\end{equation}

where $\mathcal{X} := \reals^n \times \reals^n \times \reals^n \times \reals_{\geq 0}^n \times \reals^n \times \reals_{\geq 0}^n \times [0,T_2]$ with

\begin{equation} \label{eqn:state}
x = (e, u, \eta, \tau^*, \hat{a}, \hat{\tau}, \tau) \in \reals^n \times \reals^n \times \reals^n \times \reals_{\geq 0}^n \times \reals^n \times \reals_{\geq 0}^n \times [0,T_2] =: \mathcal{X}
\end{equation}

$e = (e_1, e_2, \ldots, e_n)$, $u = (u_1, u_2, \ldots u_n)$, $\eta = (\eta_1, \eta_2, \ldots, \eta_n)$, $\tau^* = (\tau^*_1, \tau^*_2, \ldots, \tau^*_N)$, $\hat{\tau} = (\hat{\tau}_1, \hat{\tau}_2, \ldots, \hat{\tau}_N)$, $a = (a_1, a_2, \ldots, a_N)$, and $\hat{a} = (\hat{a}_1, \hat{a}_2, \ldots, \hat{a}_n)$.
two_colThen, let

\begin{equation} \label{eqn:cl_hysys}
\HS := (C,f,D,G)
\end{equation}
where the dynamics and data $(C,f,D,G)$ are given by $(\dot{e}, \dot{u}, \dot{\eta}, \dot{\tau}^*, \dot{\hat{a}}, \dot{\hat{\tau}}, \dot{\tau} )$ $= (a + u - \sigma^* \textbf{1}_n, h \eta -\mu ( \hat{\tau} - \tau^* ), h \eta, a, -\mu ( \hat{\tau} - \tau^* ), \hat{a} - (\hat{\tau} - \tau^*), -1) =: f(x)$ for each $x \in C$ and
$(e^+, u^+, \eta^+, {\tau^*}^+, \hat{a}^+, \hat{\tau}^+, \tau^+)$ $= (e, -\gamma \mathcal{L} e - \hat{a} + \sigma^* \textbf{1}_n, -\gamma \mathcal{L} e, \tau^*, \hat{a}, \hat{\tau}, [T_1,T_2]) =: G(x)$ for each $x \in D$

The dynamics and data $(C,f,D,G)$ of $\HS$ are given by

\begin{equation} \label{eqn:cl_hysys}
\hspace{-1mm}
\begin{bmatrix}
\dot{e} \\
\dot{u} \\
\dot{\eta} \\
\dot{\tau}^* \\
\dot{\hat{a}} \\
\dot{\hat{\tau}} \\
\dot{\tau}
\end{bmatrix} = \begin{bmatrix}
a + u - \sigma^* \textbf{1}_n  \\ h \eta -\mu ( \hat{\tau} - \tau^* ) \\ h \eta \\ a \\ -\mu ( \hat{\tau} - \tau^* ) \\ \hat{a} - (\hat{\tau} - \tau^*) \\ -1 \\
\end{bmatrix} \hspace{1mm} =: f(x) \hspace{1mm}  x \in C,
\hspace{1mm}
\begin{bmatrix}
e^+ \\
u^+ \\
\eta^+ \\
{\tau^*}^+ \\
\hat{a}^+ \\
\hat{\tau}^+ \\
\tau^+
\end{bmatrix} =
\begin{bmatrix}
e \\ -\gamma \mathcal{L} e - \hat{a} + \sigma^* \textbf{1}_n  \\ -\gamma \mathcal{L} e \\ \tau^* \\ \hat{a} \\ \hat{\tau} \\ [T_1,T_2]
\end{bmatrix} \hspace{1mm} =: G(x) \hspace{1mm} \hspace{1mm} x \in D
\end{equation}

where $C := \mathcal{X}$ and $D := \{ x \in \mathcal{X} : \tau = 0 \}$. Note that $\mathcal{X} \subset \reals^m$ where $m = 7n$.

With the hybrid system $\HS$ defined, the next two results establish existence of solutions to $\HS$ and that every maximal solution to $\HS$ is complete. In particular, we show that, through the satisfaction of some basic conditions on the hybrid system data, which is shown first, the system $\HS$ is well-posed and that each maximal solution to the system is defined for arbitrarily large $t + j$. two_colThe next two lemmas hold for any choice of parameters $T_1$, $T_2$, $\sigma^*$, $h$, $\gamma$, $\mu$, and strongly connected digraph $\DG$.

The hybrid system $\HS$ satisfies the hybrid basic conditions defined in <cit.>.

The hybrid system $\HS$ satisfies the following conditions, defined in <cit.> as the hybrid basic conditions.

(A1) $C$ and $D$ are closed subsets of $\reals^m$.

(A2) $f : \mathcal{X} \to \mathcal{X}$ is continuous and locally bounded relative to $C$ and $C \subset \mbox{dom }  f$;

(A3) $G : \reals^m \rightrightarrows \reals^m$ is outer semicontinuous and locally bounded relative to $D$, and $D \subset \mbox{dom } G$.

See the appendix for proof.

For every $\xi \in C \cup D (= \mathcal{X})$, every maximal solution $\phi$ to $\HS$ with $\phi(0,0) = \xi$ is complete.

The properties given in these two lemmas are easily established from the information given in the data of $\HS$; see [22] for full details on the proofs of these results.

See the appendix for proof.

With the hybrid closed-loop system $\HS$ in (<ref>), the set to asymptotically stabilize so as to solve Problem <ref> is

\begin{equation} \label{set:A}
\begin{aligned}
\A & := \{x \in \mathcal{X} : e_i = e_k, \eta_i = 0, \hat{a}_i = a_i, \hat{\tau}_i = \tau^*_i, \\
& \hspace{30mm} u_i = \eta_i - \hat{a}_i + \sigma^* \hspace{1mm} \forall i ,k \in \mathcal{V} \}
\end{aligned}
\end{equation}

\begin{equation} \label{set:A}
\begin{aligned}
\A {:=} \{x \in \mathcal{X} : e_i = e_k, \eta_i = 0, \hat{a}_i = a_i, \hat{\tau}_i = \tau^*_i, u_i = \eta_i - \hat{a}_i + \sigma^* \hspace{1mm} \forall i ,k \in \mathcal{V} \}
\end{aligned}
\end{equation}

Note that $e_i = e_k$ and $\eta_i = 0$ for all $i,k \in \mathcal{V}$ imply synchronization of the clocks, meanwhile $\hat{a}_i = a_i$ and  $\tau^*_i = \hat{\tau}_i$ for all $i,k \in \mathcal{V}$ ensure no error in the estimation of the clock skew and that the internal and estimated clocks are synchronized, respectively. The inclusion of $u_i = - \hat{a}_i + \sigma^*$ in $\A$ ensures that, for each $i \in \nodes$, $e_i$ remains constant (at zero) so that $e_i$ does not leave the set $\A$. This property is captured in the following result using the notion of forward invariance of a set. confIt can be shown that the set $\A$ is forward invariant for the hybrid system $\HS$.

Given that each maximal solution $\phi$ to $\HS$ is complete, with the state variable $\tau$ acting as a timer for $\HS$, for every initial condition $\phi(0,0) \in C \cup D$ we can characterize the domain of each solution $\phi$ to $\HS$ as follows:

\begin{equation}
\mbox{\rm dom } \phi = \bigcup_{j \in \mathbb{N}} [t_j, t_{j+1}] \times \{j\}
\end{equation}

with $t_0 = 0$ and $t_{j+1} - t_j$ as in (<ref>). Furthermore, the structure of the above hybrid time domain implies that for each $(t,j) \in \mbox{dom } \phi$ we have

\begin{equation} \label{eqn:t_upper_bound}
t \leq T_2(j+1)
\end{equation}

Given a strongly connected digraph $\DG$, the set $\A$ in (<ref>) is forward invariant for the hybrid system $\HS$, i.e., each maximal solution $\phi$ to $\HS$ with $\phi(0,0) \in \A$ is complete and satisfies $\phi(t,j) \in \A$ for each $(t,j) \in \mbox{dom } \phi$ (see <cit.>).

See the appendix for proof.

With the definitions of the closed-loop system $\HS$ in (<ref>) and the set of interest $\A$ in (<ref>) to asymptotically stabilize in order to solve Problem <ref>, we introduce our main result showing global exponential stability of $\A$ to $\HS$. This result is established through an analysis of an auxiliary system $\widetilde{\HS}_{\varepsilon}$ presented in (<ref>) and its global exponential stability for the auxiliary set $\tilde{\A}_{\varepsilon}$ in (<ref>), the details of which can be found in Section <ref>.

Given a strongly connected digraph $\DG$, if the parameters $T_2 \geq T_1 > 0$, $\mu > 0$, $h \in \reals$, and $\gamma > 0$, the positive definite matrices $P_1$, $P_2$, and $P_3$ are such that
\begin{equation} \label{eqn:lyap1}
P_2 A_{f_3} + A_{f_3}^{\top} P_2 \prec 0, \hspace{2mm} P_3 A_{f_4} + A_{f_4}^{\top} P_3 \prec 0
\end{equation}

\begin{equation} \label{cond:phil_1}
\hspace{-3mm} A^{\top}_{g_2} {\exp(A^{\top}_{f_2} \nu)} P_1 {\exp(A_{f_2} \nu)} A_{g_2} {-} P_1 \prec 0 \hspace{3mm} \forall \nu \in [T_1, T_2]
\end{equation}

\begin{equation} \label{thrm_cond1}
\Big | \exp \Big (\frac{\bar{\kappa}_1}{\alpha_2} \hspace{0.5mm} T_2 \Big ) \Big ( 1 - \frac{\bar{\kappa}_2}{\alpha_2} \Big ) \Big | < 1
\end{equation}

hold,  where $A_{f_2}$, $A_{g_2}$ are given in (<ref>) and

\begin{equation} \label{eqn:thm_consts}
\begin{aligned}
\hspace{-5mm} \bar{\kappa}_1 & = \max \Big \{ \frac{\kappa_1}{2 \epsilon}, \frac{\kappa_1 \epsilon}{2} \minus \beta_2  \Big \}, \hspace{2mm} \bar{\kappa}_2 = \min \{1 , \kappa_2 \} \\
\kappa_1 & = 2 \underset{\nu \in [0,T_2]}{\mathrm{max}} \big | \exp{(A^{\top}_{f_2} \nu )} P_1 \exp{(A_{f_2} \nu )} \big | \\
\kappa_2 & \in \hspace{-1mm} \Big ( 0, {\minus} \hspace{-3mm} \underset{\nu {\in} [T_1,T_2]}{\min} \big \{ \lambda_{min} (A^{\top}_{g_2} {\exp{(A^{\top}_{f_2} \nu )}} P_1 {\exp{(A_{f_2} \nu )}} A_{g_2} {\minus} P_1 ) \big \} \Big ) \\
\alpha_2 & = \underset{\nu \in [0,T_2]}{\mathrm{max}} \Big \{ \exp{(2h \nu)}, \lambda_{max} \big (\exp{(A^{\top}_{f_2} \nu )} P_1 \exp{(A_{f_2} \nu )} \big ), \\
& \hspace{50mm} \lambda_{max}(P_2), \lambda_{max}(P_3) \Big \}
\end{aligned}
\end{equation}

with $\epsilon > 0$, and $\beta_1 > 0$ and $\beta_2 >0$ such that, in light of (<ref>),
$$P_2 A_{f_3} + A_{f_3}^{\top} P_2 \preceq - \beta_1 I_2$$
$$P_3 A_{f_4} + A_{f_4}^{\top} P_3 \preceq - \beta_2 I_{2(n-1)}$$
then, the set $\A$ in (<ref>) is globally exponentially stable for the hybrid system $\HS$ in (<ref>).

Given a strongly connected digraph $\DG$, if the parameters $T_2 \geq T_1 > 0$, $\mu > 0$, $h \in \reals$, and $\gamma > 0$, the positive definite matrices $P_1$, $P_2$, and $P_3$ are such that
\begin{equation} \label{eqn:lyap1}
P_2 A_{f_3} + A_{f_3}^{\top} P_2 \prec 0
\end{equation}

\begin{equation} \label{eqn:lyap2}
P_3 A_{f_4} + A_{f_4}^{\top} P_3 \prec 0
\end{equation}

\begin{equation} \label{cond:phil_1}
\hspace{-3mm} A^{\top}_{g_2} {\exp(A^{\top}_{f_2} \nu)} P_1 {\exp(A_{f_2} \nu)} A_{g_2} {-} P_1 \prec 0 \hspace{3mm} \forall \nu \in [T_1, T_2]
\end{equation}

\begin{equation} \label{thrm_cond1}
\Big | \exp \Big (\frac{\bar{\kappa}_1}{\alpha_2} \hspace{0.5mm} T_2 \Big ) \Big ( 1 - \frac{\bar{\kappa}_2}{\alpha_2} \Big ) \Big | < 1
\end{equation}

hold,  where $A_{f_2}$, $A_{g_2}$ are given in (<ref>) and

\begin{equation} \label{eqn:thm_consts}
\begin{aligned}
\hspace{-5mm} \bar{\kappa}_1 & {=} \max \Big \{ \frac{\kappa_1}{2 \epsilon}, \frac{\kappa_1 \epsilon}{2} \minus \beta_2  \Big \}, \hspace{2mm} \bar{\kappa}_2 = \min \{1 , \kappa_2 \} \\
\kappa_1 & {=} 2 \underset{\nu \in [0,T_2]}{\mathrm{max}} \big | \exp{(A^{\top}_{f_2} \nu )} P_1 \exp{(A_{f_2} \nu )} \big | \\
\kappa_2 & {\in} \big ( 0, {\minus} \hspace{-2mm} \underset{\nu {\in} [T_1,T_2]}{\min} \big \{ \lambda_{min} (A^{\top}_{g_2} {\exp{(A^{\top}_{f_2} \nu )}} P_1 {\exp{(A_{f_2} \nu )}} A_{g_2} {\minus} P_1 ) \big \} \big ) \\
\alpha_2 & {=} \underset{\nu \in [0,T_2]}{\mathrm{max}} \Big \{ \exp{(2h \nu)}, \lambda_{max} \big (\exp{(A^{\top}_{f_2} \nu )} P_1 \exp{(A_{f_2} \nu )} \big ), \\
& \hspace{50mm} \lambda_{max}(P_2), \lambda_{max}(P_3) \Big \}
\end{aligned}
\end{equation}

with $\epsilon > 0$, and $\beta_1 > 0$ and $\beta_2 >0$ such that, in light of (<ref>), $P_2 A_{f_3} + A_{f_3}^{\top} P_2 \leq - \beta_1 I_2$, and $P_3 A_{f_4} + A_{f_4}^{\top} P_3 \leq - \beta_2 I_{2(n-1)}$ then, the set $\A$ in (<ref>) is globally exponentially stable for the hybrid system $\HS$ in (<ref>).

To validate our theoretical stability result in Theorem <ref>, consider five agents with dynamics as in (<ref>) and (<ref>) over a strongly connected digraph with the following adjacency matrix
two_col$\DG_A = ( [0, 1, 1, 0, 1],$ $[1, 0 , 1 , 0 , 0],$ $[1 , 0 , 0 , 1 , 0],$ $[0 , 0 , 1 , 0 , 1],$ $[1 , 0 , 1 , 1 , 0])$.

\begin{equation*}
\DG_A = \begin{pmatrix}
0 & 1 & 1 & 0 & 1 \\
1 & 0 & 1 & 0 & 0 \\
1 & 0 & 0 & 1 & 0 \\
0 & 0 & 1 & 0 & 1 \\
1 & 0 & 1 & 1 & 0 \\
\end{pmatrix}
\end{equation*}

Given $T_1 = 0.01$, $T_2 = 0.1$, and $\sigma^* = 1$, then it can be found that the parameters $h= -1.3$, $\mu = 3$, $\gamma = 0.125$, matrices,
two_col$P_1 {=} ([33.61, 0,$ $0, 0,$ $4.20, 0,$ $0,  0],$
$[0, 28.61,$ $0, 0,$ $0, 5.73,$ $0, 0],$
$[0, 0,$ $ 25.35 ,   0 ,$    $0 ,    0  ,$ $  4.75 ,   0],$
$[0, 0, $ $ 0  ,  28.61 ,$    $0 ,   0  , $ $ 0 ,    5.73],$
$[4.20,  0 ,$ $   0  ,   0 ,$    $7.02 ,  0   , $ $0  ,   0],$
$[0, 5.73, $ $  0  ,  0 ,$   $0 ,   11.13  ,$ $  0 ,   0],$
$[0, 0,$ $  4.75  ,   0 ,$    $0 ,    0 , $ $ 14.96 ,   0],$
$[0, 0,$ $ 0  ,   5.73 ,$   $ 0 ,   0  ,$ $ 0 ,   11.13])$,
$P_2 {=} ([ 5.26  ,  \minus 2.24],$ $[\minus 2.24  ,   7.54])$, and
$P_3 {=} ([ 6.47 ,   0 ,$  $ 0 ,   0 , $ $ \minus 2.36  ,    0 , $ $  0 ,   0],$
$[0  ,   6.47 , $ $ 0 ,   0 , $ $  0 ,   \minus 2.36 , $ $  0 ,    0],$
$[0 ,   0 ,  $ $ 6.47 ,   0 , $ $  0 ,    0 ,$ $  \minus 2.36 ,    0],$
$[0 ,   0 ,  $ $0 ,    6.47 , $ $  0 ,    0 ,$ $  0 ,   \minus 2.36],$
$[\minus 2.35 ,    0 , $ $  0 ,    0 ,$ $   7.90 ,   0 , $ $ 0 ,   0],$
$[0 ,   \minus 2.35 ,  $ $ 0 ,    0 ,$ $  0 ,    7.90 , $ $ 0 ,   0],$
$[0  ,   0 , $ $ \minus 2.35 ,    0 ,$ $  0 ,   0 , $ $  7.90 ,   0],$
$[0  ,   0 ,  $ $ 0 ,   \minus 2.35 ,$ $  0 ,   0 , $ $ 0 ,    7.90])$

\begin{equation} \label{eqn:P_matrices} \scriptsize
\begin{aligned}
P_1 & {=} \begin{bmatrix} 33.61  &  0 &  0  &  0  &  4.20  & 0  &  0  & 0 \\
0 &  28.61  &  0  &  0 &  0 &   5.73  & 0  &  0 \\
0  &  0  & 25.35 &  0 &   0 &   0  &  4.75 &  0 \\
0  &  0 &  0  & 28.61 &   0 &  0  &  0 &   5.73 \\
4.20  & 0 &   0  &  0 &   7.02 & 0   & 0  &  0 \\
0  &  5.73 &   0  & 0 &  0 &  11.13  &  0 &  0 \\
0  & 0 &   4.75  &  0 &   0 &   0 &  14.96 &  0 \\
0  &  0 &  0  &  5.73 &   0 &  0  & 0 &  11.13] \\
\end{bmatrix} \\
P_2 & {=} \begin{bmatrix} 5.26  & \minus 2.24 \\
\minus 2.24  &  7.54 \\
\end{bmatrix} \\
P_3 & {=} \begin{bmatrix} 6.47 &  0 &  0 &  0 &  \minus 2.36  &   0 &   0 &  0 \\
0  &  6.47 &  0 &  0 &   0 &  \minus 2.36 &   0 &   0 \\
0 &  0 &   6.47 &  0 &   0 &   0 &  \minus 2.36 &   0 \\
0 &  0 &  0 &   6.47 &   0 &   0 &  0 &  \minus 2.36 \\
\minus 2.35 &   0 &   0 &   0 &   7.90 &  0 &  0 &  0 \\
0 &  \minus 2.35 &   0 &   0 &  0 &   7.90 &  0 &  0 \\
0  &  0 &  \minus 2.35 &   0 &  0 &  0 &   7.90 &  0 \\
0  &  0 &   0 &  \minus 2.35 &  0 &  0 &  0 &   7.90 \\
\end{bmatrix}
\end{aligned}
\end{equation}

and $\epsilon = 1.607$ confwith suitable matrices $P_1$, $P_2$, and $P_3$ satisfy conditions (<ref>) and (<ref>) in Theorem <ref> with $\bar{\kappa}_1 = 9.78$, $\kappa_1 = 31.44$, $\bar{\kappa}_2 = 1$, and $\alpha_2 = 18.923$. Figure <ref> shows the trajectories of $e_i - e_k$, $\varepsilon_{a_i}$ for components $i \in \{1,2,3,4,5\}$ of a solution $\phi$ for the case where $\sigma = \sigma^*$ with initial conditions $\phi_{e}(0,0) = (1,-1,2,-2,0)$, $\phi_{\eta}(0,0) = (0,-3,1,-4,-1)$, and clock rates $a_i$ in the range $(0.85,1.15)$.  confThe bottom plot in Figure <ref> depicts the Lyapunov trajectory $V$ evaluated along the solution $\phi$ with the upper bound given in (<ref>) projected onto the regular time domain. Observe that the exponential bound provided in (<ref>) jumps along the solution, validating our theoretical results on the exponential stability of the system.[Code at github.com/HybridSystemsLab/HybridClockSync]

(top) The trajectories of the state component errors $e_i - e_k$, $\varepsilon_{a_i}$, and $\tau$ for $i \in \{1,2,3,4,5\}$ of the solution $\phi$ for the case where $\sigma = \sigma^*$. (bottom) Plot of $V(\chi_{\varepsilon})$ evaluated along the solution $\phi$ with the associated bound (denoted $V_b$) given in (<ref>) projected onto the regular time domain.

The trajectories of the virtual clocks $\tilde{\tau}_i$ for $i \in \{1,2,3,4,5\}$ of the solution $\phi$ for the case where $\sigma = \sigma^*$.

The trajectories of the solution $\phi$ for state component errors $e_i {-} e_k$, $\varepsilon_{a_i}$, and $\tau$. Plot of $V$ evaluated along the solution $\phi$ projected onto the regular time domain. (bottom)

§ KEY PROPERTIES OF THE NOMINAL
CLOSED-LOOP SYSTEM

§ KEY PROPERTIES OF THE NOMINAL CLOSED-LOOP SYSTEM

§.§ Reduced Model – First Pass

In this section, we recast the hybrid system $\HS$ into a reduced model obtained by setting $u = \eta  - \hat{a} + \sigma^*\textbf{1}_n$. This reduced model enables assessing asymptotic stability of $\A$. It is given in error coordinates for the parameter estimation of the internal clock rate and also the error of the internal clock state. We let $\varepsilon_a = a - \hat{a}$ denote the estimation error of the internal clock rate and $\varepsilon_{\tau} = \hat{\tau} - \tau^*$ represent the estimation error of the internal clock state. The state of the reduced model is given by $x_{\varepsilon} := (e,\eta, \varepsilon_a, \varepsilon_{\tau}, \tau) \in \reals^n \times \reals^n \times \reals^n \times \reals^n \times [0,T_2] =: \mathcal{X}_{\varepsilon}$ with dynamics defined by the data

\begin{equation} \label{eqn:hy_eps}
\begin{aligned}
\hspace{-2mm} f_{\varepsilon}(x_{\varepsilon}) & := (\eta + \varepsilon_a, h \eta, \mu \varepsilon_{\tau}, - \varepsilon_{\tau} - \varepsilon_a, -1) & \forall x_{\varepsilon} \in C_{\varepsilon}, \\
\hspace{-2mm} G_{\varepsilon}(x_{\varepsilon}) & := (e, \minus \gamma \mathcal{L} e, \varepsilon_a, \varepsilon_{\tau},  [T_1,T_2]) & \forall x_{\varepsilon} \in D_{\varepsilon}
\end{aligned}
\end{equation}

\begin{equation} \label{eqn:hy_eps}
\begin{aligned}
& f_{\varepsilon}(x_{\varepsilon}) := \begin{bmatrix}
\eta + \varepsilon_a  \\ h \eta \\ \mu \varepsilon_{\tau} \\ - \varepsilon_{\tau} - \varepsilon_a \\ -1 \\
\end{bmatrix} \hspace{1mm} \forall x_{\varepsilon} \in C_{\varepsilon},
& G_{\varepsilon}(x_{\varepsilon}) := \begin{bmatrix}
e \\ \minus \gamma \mathcal{L} e \\ \varepsilon_a \\ \varepsilon_{\tau} \\  [T_1,T_2]
\end{bmatrix} \hspace{1mm} \forall x_{\varepsilon} \in D_{\varepsilon}
\end{aligned}
\end{equation}

$C_{\varepsilon} := \mathcal{X}_{\varepsilon}$ and $D_{\varepsilon} := \{ x_{\varepsilon} \in \mathcal{X}_{\varepsilon} : \tau = 0 \}$. This system is denoted $\HS_{\varepsilon} = (C_{\varepsilon},f_{\varepsilon},D_{\varepsilon},G_{\varepsilon})$. Note that the construction $u = \eta - \hat{a} + \sigma^* \textbf{1}_n$, which holds along all solutions after the first jump, leads to $\dot{e} = \eta + \varepsilon_a$.

Observe that the state of $\HS_{\varepsilon}$ utilizes most of the state components of $\HS$ except for the control input $u$ as it's value is captured by the states $\eta$ and $\varepsilon$ directly in the dynamics of $e$.
To relate the properties of the reduced model to those of the hybrid system $\HS$, we establish a result showing an equivalency between the solutions of $\HS$ in (<ref>) and $\HS_{\varepsilon}$ defined above. The result shows that after the first jump, each solution $\phi$ to $\HS$ is equivalent to a solution $\phi^{\varepsilon}$ to $\HS_{\varepsilon}$ when the trajectories of the timer variable $\tau$ for both solutions are equal. To facilitate such a result, we define the function $M: \mathcal{X} \to \mathcal{X}_{\varepsilon}$ given by

\begin{equation} \label{eqn:m}
M(x) := (e, \eta, a - \hat{a}, \hat{\tau} - \tau^*, \tau)
\end{equation}

where $x = (e, u, \eta, \tau^*, \hat{a}, \hat{\tau}, \tau)$, as defined in (<ref>), and the function $\widetilde{M}: \mathcal{X}_{\varepsilon} \times \reals^n_{\geq 0} \times \reals^n_{\geq 0} \to \mathcal{X}$ given by

\begin{equation} \label{eqn:m}
\widetilde{M} (x_\varepsilon, \hat{\tau}, \tau^*) := \begin{bmatrix}
e \\ \eta - (a - \varepsilon_a) + \sigma^* \textbf{1}_n \\ \eta \\ \hat{\tau} - \varepsilon_{\tau} \\ a - \varepsilon_a \\ \varepsilon_{\tau} +  \tau^* \\ \tau
\end{bmatrix}
\end{equation}

Let $T_2 \geq T_1 > 0$, digraph $\DG$, and hybrid systems $\HS$ and $\HS_{\varepsilon}$ be given as in (<ref>) and (<ref>), respectively. For each $\phi \in \mathcal{S}_{\HS}$ and each[Note that for a given solution $\phi^{\varepsilon}(t,j)$ to $\HS_{\varepsilon}$, the solution components are given by $\phi^{\varepsilon}(t,j) = \big (\phi_e^{\varepsilon}(t,j), \phi_\eta^{\varepsilon}(t,j), \phi^{\varepsilon}_{\varepsilon_a}(t,j), \phi^{\varepsilon}_{\varepsilon_\tau}(t,j), \phi^{\varepsilon}_{\tau}(t,j) \big )$] $\phi^{\varepsilon} \in \mathcal{S}_{\HS_{\varepsilon}}$ such that $\phi (0,0) = \widetilde{M} \big (\phi^{\varepsilon}(0,0),  \phi_{\hat{\tau}}(0,0), \phi_{\tau^*}(0,0) \big )$ and timer components $\phi_{\tau}(t,j) = \phi^{\varepsilon}_{\tau}(t,j)$ for all $(t,j) \in \mbox{\rm dom } \phi$, it follows that $\mbox{dom } \phi = \mbox{dom } \phi^{\varepsilon}$ and $\phi (t,j) = \widetilde{M} \big (\phi^{\varepsilon}(t,j), \phi_{\hat{\tau}}(t,j), \phi_{\tau^*}(t,j) \big )$ for each $(t,j) \in \mbox{\rm dom } \phi$.

Let $T_2 \geq T_1 > 0$, digraph $\DG$, and hybrid systems $\HS$ and $\HS_{\varepsilon}$ be given as in (<ref>) and (<ref>), respectively. For each $\phi \in \mathcal{S}_{\HS}$ and each[Note that for a given solution $\phi^{\varepsilon}(t,j)$ to $\HS_{\varepsilon}$, the solution components are given by
$$\phi^{\varepsilon}(t,j) = \big (\phi_e^{\varepsilon}(t,j), \phi_\eta^{\varepsilon}(t,j), \phi^{\varepsilon}_{\varepsilon_a}(t,j), \phi^{\varepsilon}_{\varepsilon_\tau}(t,j), \phi^{\varepsilon}_{\tau}(t,j) \big )$$
] $\phi^{\varepsilon} \in \mathcal{S}_{\HS_{\varepsilon}}$ such that $\phi (0,0) = \widetilde{M} \big (\phi^{\varepsilon}(0,0),  \phi_{\hat{\tau}}(0,0), \phi_{\tau^*}(0,0) \big )$ and timer components $\phi_{\tau}(t,j) = \phi^{\varepsilon}_{\tau}(t,j)$ for all $(t,j) \in \mbox{\rm dom } \phi$, it follows that $\mbox{dom } \phi = \mbox{dom } \phi^{\varepsilon}$ and

\begin{equation} \label{eqn:equiv_sol_M}
\phi (t,j) = \widetilde{M} \big (\phi^{\varepsilon}(t,j), \phi_{\hat{\tau}}(t,j), \phi_{\tau^*}(t,j) \big ) \hspace{10mm} \forall (t,j) \in \mbox{\rm dom } \phi
\end{equation}

See the appendix for proof.

With the reduced model $\HS_{\varepsilon}$ in place, we consider the following set to asymptotically stabilize for $\HS_{\varepsilon}$:

\begin{equation} \label{set:A_eps}
\A_{\varepsilon} {:=} \{ x_{\varepsilon} \in \mathcal{X}_{\varepsilon} : e_i {=} e_k, \eta_i {=} 0 \hspace{1mm} \forall i, k \in \mathcal{V}, \hspace{1pt} \varepsilon_{a} {=} 0, \varepsilon_{\tau} {=} 0 \}
\end{equation}

This set is equivalent to $\A$ in the sense that the point-to-set distance metrics $|x|_{\A}$ and $|x|_{\A_{\varepsilon}}$ are equivalent when the map $\widetilde{M}$ is applied, as demonstrated in the results that follow. conf

Given sets $\A$ and $\A_{\varepsilon}$ as in (<ref>) and (<ref>), respectively, for each $x =  (e, u, \eta, \tau^*, \hat{a}, \hat{\tau}, \tau)$, $x_{\varepsilon}$, $\hat{\tau}$, and $\tau^*$ such that $x \in \mathcal{X}$, $(x_{\varepsilon}, \hat{\tau}, \tau^* ) \in \mathcal{X}$, and $u = \eta - \hat{a} + \sigma^* \textbf{1}_n$ then $|x|_{\A} = |x_{\varepsilon}|_{\A_{\varepsilon}}$ and $|\widetilde{M}(x_{\varepsilon}, \hat{\tau}, \tau^*) |_{\A} = |x|_{\A}$.

Given sets $\A$ and $\A_{\varepsilon}$ as in (<ref>) and (<ref>), respectively, for each $x =  (e, u, \eta, \tau^*, \hat{a}, \hat{\tau}, \tau)$, $x_{\varepsilon}$, $\hat{\tau}$, and $\tau^*$ such that $x \in \mathcal{X}$, $(x_{\varepsilon}, \hat{\tau}, \tau^* ) \in \mathcal{X}$, and $u = \eta - \hat{a} + \sigma^* \textbf{1}_n$ then

\begin{equation} \label{eqn:set_dists_1}
|x|_{\A} = |x_{\varepsilon}|_{\A_{\varepsilon}}
\end{equation}

\begin{equation} \label{eqn:set_dists_2}
|\widetilde{M}(x_{\varepsilon}, \hat{\tau}, \tau^*) |_{\A} = |x|_{\A}
\end{equation}

With the stabilization set defined for $\HS_{\varepsilon}$, we have the following result that shows that if the set $\A_{\varepsilon}$ is globally exponentially stable for $\HS_{\varepsilon}$ then the set $\A$ is also globally exponentially stable for $\HS$.

Given $T_2 \geq T_1 > 0$ and a strongly connected digraph $\DG$, the set $\A$ in (<ref>) is GES for the hybrid system $\HS$ if $\A_{\varepsilon}$ in (<ref>) is GES for the hybrid system $\HS_{\varepsilon}$.

two_colSee the appendix for proofs to Lemma <ref> - Lemma <ref>.

See the appendix for proof.

§.§ Reduced Model – Second Pass

Global exponential stability of $\A_{\varepsilon}$ for $\HS_{\varepsilon}$ is established by performing a Lyapunov analysis on a version of $\HS_{\varepsilon}$ obtained after an appropriate change of coordinates, one where the flow and jump dynamics are linearized. The model is obtained by exploiting an important property of the eigenvalues of the Laplacian matrix for strongly connected digraphs.

To this end, let $\DG$ be a strongly connected digraph. By Lemma <ref> and Lemma <ref>, one has that zero is a simple eigenvalue of the Laplacian matrix $\mathcal{L}$ with an associated eigenvector $v_1 = \frac{1}{\sqrt{N}} \textbf{1}_N$. Furthermore, there exists a nonsingular matrix

\begin{equation} \label{eqn:T_mat}
\mathcal{T} = [v_1, \mathcal{T}_1]
\end{equation}

where $\mathcal{T}_1 \in \reals^{N \times N \minus 1}$ is a matrix whose columns are the remaining eigenvectors of $\mathcal{L}$, i.e., $[v_2, \ldots, v_N]$, such that $\mathcal{T}^{\minus 1} \mathcal{L} \mathcal{T} = \begin{bmatrix} 0 & 0 \\ 0 & \bar{\mathcal{L}} \end{bmatrix}$, where $\mathcal{L}$ is the graph Laplacian of $\DG$ and $\bar{\mathcal{L}}$ is a diagonal matrix with the nonnegative eigenvalues of $\mathcal{L}$ as the diagonal elements given by $( \lambda_2, \lambda_3, \ldots, \lambda_N)$,  see [9], [10], and [25] for more details.

To perform the said change of coordinates, we use $\Tmat$ to first perform the following transformations: $\bar{e} = \Tmat^{-1}e$, $\bar{\eta} = \Tmat^{-1}\eta$, $\bar{\varepsilon}_a = \Tmat^{-1}\varepsilon_a$ and $\bar{\varepsilon}_{\tau} = \Tmat^{-1}\varepsilon_{\tau}$. Then, we define vectors $\bar{z} = (\bar{z}_1, \bar{z}_2)$ and $\bar{w} = (\bar{w}_1, \bar{w}_2)$, where $\bar{z}_1 := (\bar{e}_1, \bar{\eta}_1)$, $\bar{z}_2 := (\bar{e}_2,\ldots,\bar{e}_N, \bar{\eta}_2, \ldots, \bar{\eta}_N)$, $\bar{w}_1 = (\bar{\varepsilon}_{a_1},\bar{\varepsilon}_{\tau_1})$, and $\bar{w}_2 = (\bar{\varepsilon}_{a_2},\ldots, \bar{\varepsilon}_{a_n},\bar{\varepsilon}_{\tau_2},\ldots, \bar{\varepsilon}_{\tau_n})$. Finally, we define $\chi_{\varepsilon} := (\bar{z}_1, \bar{z}_2, \bar{w}_1, \bar{w}_2, \tau) \in \reals^2 \times \reals^{2(n-1)} \times \reals^2 \times \reals^{2(n-1)} \times [0,T_2] =: \mathcal{X}_{\varepsilon}$ as the state of the new version of $\HS_{\varepsilon}$, which is denoted $\widetilde{\HS}_{\varepsilon}$ and has data given by

\begin{equation} \label{eqn:H_vareps}
\begin{aligned}
\widetilde{f}_{\varepsilon}(\chi_{\varepsilon}) {:=} \begin{bmatrix}
A_{f_1} \bar{z}_1 \\ A_{f_2} \bar{z}_2 \\ A_{f_3} \bar{w}_1 \\ A_{f_4} \bar{w}_2 \\  -1
\end{bmatrix} {+} \begin{bmatrix}
B_{f_1} \bar{w}_1 \\ B_{f_2} \bar{w}_2 \\ 0 \\ 0 \\ 0
\end{bmatrix}, \hspace{1mm}
\widetilde{G}_{\varepsilon}(\chi_{\varepsilon}) {:=} \begin{bmatrix}
A_{g_1} \bar{z}_1 \\ A_{g_2} \bar{z}_2 \\ \bar{w}_1 \\ \bar{w}_2 \\ [T_1,T_2]
\end{bmatrix}
\end{aligned}
\end{equation}

\begin{equation} \label{eqn:H_vareps}
\begin{aligned}
\widetilde{f}_{\varepsilon}(\chi_{\varepsilon}) {:=} \begin{bmatrix}
A_{f_1} \bar{z}_1 \\ A_{f_2} \bar{z}_2 \\ A_{f_3} \bar{w}_1 \\ A_{f_4} \bar{w}_2 \\  -1
\end{bmatrix} {+} \begin{bmatrix}
B_{f_1} \bar{w}_1 \\ B_{f_2} \bar{w}_2 \\ 0 \\ 0 \\ 0
\end{bmatrix} \hspace{2mm} \forall \chi_{\varepsilon} \in \tilde{C}_{\varepsilon}, \hspace{3mm}
\widetilde{G}_{\varepsilon}(\chi_{\varepsilon}) {:=} \begin{bmatrix}
A_{g_1} \bar{z}_1 \\ A_{g_2} \bar{z}_2 \\ \bar{w}_1 \\ \bar{w}_2 \\ [T_1,T_2]
\end{bmatrix} \hspace{2mm} \forall \chi_{\varepsilon} \in \tilde{D}_{\varepsilon}
\end{aligned}
\end{equation}

for each $\chi_{\varepsilon}$ in $\widetilde{C}_{\varepsilon} := \mathcal{X}_{\varepsilon}$ and in $\widetilde{D}_{\varepsilon} := \{ \chi_{\varepsilon} \in \mathcal{X}_{\varepsilon} : \tau = 0 \}$, respectively, with

\begin{equation} \label{eqn:matrices}
\begin{aligned}
& A_{f_1} {=} \begin{bmatrix}
0 & 1  \\ 0 & h
\end{bmatrix},
& & A_{f_2} {=} \begin{bmatrix} 0 & I_{m} \\ 0 & hI_{m}
\end{bmatrix},
& & A_{f_3} {=} \begin{bmatrix}
0 & \mu  \\ \minus 1 & \minus 1
\end{bmatrix} \\
& A_{f_4} {=} \begin{bmatrix} 0 & \mu I_{m} \\ \minus I_{m} & \minus I_{m}
\end{bmatrix},
& & B_{f_1} {=} \begin{bmatrix} 1 & 0 \\ 0 & 0
\end{bmatrix},
& & B_{f_2} {=} \begin{bmatrix} I_{m} & 0 \\ 0 & 0
\end{bmatrix} \\
& A_{g_1} {=} \begin{bmatrix} 1 & 0  \\ 0 & 0
\end{bmatrix},
& & A_{g_2} {=} \begin{bmatrix} I_{m} & 0  \\ \minus \gamma \bar{\mathcal{L}} & 0
\end{bmatrix}
\end{aligned}
\end{equation}

and $m = N-1$. Then, $\widetilde{\HS}_{\varepsilon} = ( \widetilde{C}_{\varepsilon}, \widetilde{f}_{\varepsilon}, \widetilde{D}_{\varepsilon}, \widetilde{G}_{\varepsilon})$ denotes the new version of $\HS_{\varepsilon}$. The set $\A_{\varepsilon}$ to stabilize in the new coordinates for this hybrid system is given by

\begin{equation} \label{set:A_tilde}
\begin{aligned}
\widetilde{\A}_{\varepsilon} & := \{ \chi_{\varepsilon} \in \mathcal{X}_{\varepsilon} : \bar{z}_1 {=} (e^*,0), \bar{z}_2 {=} 0, \bar{w}_1 {=} 0, \bar{w}_2 {=} 0, e^* \in \reals \}
\end{aligned}
\end{equation}

In the next result, we demonstrate how global exponential stability of $\tilde{\A}_{\varepsilon}$ for $\widetilde{\HS}_{\varepsilon}$ implies global exponential stability of $\A_{\varepsilon}$ for $\HS_{\varepsilon}$. This is accomplished by exploiting the relationship that exists between the two systems through the transformation matrix $\mathcal{T}$ on the sets $\tilde{\A}_{\varepsilon}$ and $\A_{\varepsilon}$.

In the following two results, we first demonstrate the relationship between the sets $\tilde{\A}_{\varepsilon}$ for $\widetilde{\HS}_{\varepsilon}$ and $\A_{\varepsilon}$ for $\HS_{\varepsilon}$ so as to solve Problem <ref>. Then, similar to Lemma <ref>, we show that global exponential stability of $\tilde{\A}_{\varepsilon}$ for $\widetilde{\HS}_{\varepsilon}$ implies global exponential stability of $\A_{\varepsilon}$ for $\HS_{\varepsilon}$. See the appendix for proofs.

Let $T_2 \geq T_1 > 0$, digraph $\DG$, and hybrid systems $\HS_{\varepsilon}$ and $\tilde{\HS}_{\varepsilon}$ be given as in (<ref>) and (<ref>), respectively. For each solutions $\phi \in \mathcal{S}_{\HS_{\varepsilon}}$ there exists a solution $\tilde{\phi} \in \mathcal{S}_{\tilde{\HS}_{\varepsilon}}$ such that $\phi(t,j) = \Gamma \tilde{\phi}(t,j)$ for each $(t,j) \in \mbox{dom } \phi$ if and only if for each solutions $\tilde{\phi} \in \mathcal{S}_{\tilde{\HS}_{\varepsilon}}$ there exists a solution $\phi \in \mathcal{S}_{\HS_{\varepsilon}}$ such that $\tilde{\phi}(t,j) = \Gamma^{\minus 1} \phi(t,j)$ for each $(t,j) \in \mbox{dom } \tilde{\phi}$, where $\Gamma = {\rm diag}(\mathcal{T}, \mathcal{T}, \mathcal{T}, \mathcal{T}, 1)$.

See the appendix for proof.

Given $0 < T_1 \leq T_2$ and a strongly connected digraph $\DG$, $\xi \in \A_{\varepsilon}$ if and only if $\chi_{\varepsilon} := \Gamma^{\minus 1} \xi \in \tilde{\A}_{\varepsilon}$, where $\Gamma^{\minus 1} = \mbox{\rm diag}(\mathcal{T}^{\minus 1}, \mathcal{T}^{\minus 1}, \mathcal{T}^{\minus 1}, \mathcal{T}^{\minus 1},  1)$ and $\mathcal{T}$ is given in (<ref>). Moreover, for each $x_{\varepsilon} \in \mathcal{X}_{\varepsilon}$ and each $\chi_{\varepsilon} \in \mathcal{X}_{\varepsilon}$, $|\chi_{\varepsilon}|_{\tilde{\A}_{\varepsilon}} \leq |\Gamma^{\minus 1}| |x_{\varepsilon}|_{\A_{\varepsilon}}$ and $|x_{\varepsilon}|_{\A_{\varepsilon}} \leq |\Gamma| |\chi_{\varepsilon}|_{\tilde{\A}_{\varepsilon}}$

Given $0 < T_1 \leq T_2$ and a strongly connected digraph $\DG$, $\xi \in \A_{\varepsilon}$ if and only if $\chi_{\varepsilon} := \Gamma^{\minus 1} \xi \in \tilde{\A}_{\varepsilon}$, where $\Gamma^{\minus 1} = \mbox{\rm diag}(\mathcal{T}^{\minus 1}, \mathcal{T}^{\minus 1}, \mathcal{T}^{\minus 1}, \mathcal{T}^{\minus 1},  1)$ and $\mathcal{T}$ is given in (<ref>). Moreover, for each $x_{\varepsilon} \in \mathcal{X}_{\varepsilon}$ and each $\chi_{\varepsilon} \in \mathcal{X}_{\varepsilon}$

\begin{equation} \label{eqn:set_bound_a}
|\chi_{\varepsilon}|_{\tilde{\A}_{\varepsilon}} \leq |\Gamma^{\minus 1}| |x_{\varepsilon}|_{\A_{\varepsilon}}
\end{equation}

\begin{equation} \label{eqn:set_bound_b}
|x_{\varepsilon}|_{\A_{\varepsilon}} \leq |\Gamma| |\chi_{\varepsilon}|_{\tilde{\A}_{\varepsilon}}
\end{equation}

See the appendix for proof.

Given $0 < T_1 \leq T_2$ and a strongly connected digraph $\DG$, the set $\tilde{\A}_{\varepsilon}$ is GES for the hybrid system $\widetilde{\HS}_{\varepsilon}$ if and only if $\A_{\varepsilon}$ is GES for the hybrid system $\HS_{\varepsilon}$.

See the appendix for proof.

§.§ Parameter Estimator

Exponential stability of the set $\tilde{\A}_{\varepsilon}$ for $\widetilde{\HS}_{\varepsilon}$ hinges upon the convergence of the estimate $\hat{a}$ to $a$. We present a result establishing convergence of $\hat{a}$ to $a$ by considering a model reduction of $\widetilde{\HS}_{\varepsilon}$. To this end, consider the state $\chi_{\varepsilon_r} := (\bar{w}_1, \bar{w}_2, \tau) \in \reals^2 \times \reals^{2(n-1)} \times [0,T_2] =: \mathcal{X}_{\varepsilon_r}$. Its dynamics are given by the system $\widetilde{\HS}_{\varepsilon_r} = ( \widetilde{C}_{\varepsilon_r}, \widetilde{f}_{\varepsilon_r},  \widetilde{D}_{\varepsilon_r}, \widetilde{G}_{\varepsilon_r})$ with data
$\widetilde{f}_{\varepsilon}(\chi_{\varepsilon_r})$ for each $\chi_{\varepsilon_r} \in \widetilde{C}_{\varepsilon_r}$ and $\widetilde{G}_{\varepsilon_r}(\chi_{\varepsilon_r})$ for each $\chi_{\varepsilon_r} \in \widetilde{D}_{\varepsilon_r}$  where

\begin{equation} \label{eqn:hy_param}
\begin{aligned}
\widetilde{f}_{\varepsilon_r}(\chi_{\varepsilon_r}) & := \big ( A_{f_3} \bar{w}_1, A_{f_4} \bar{w}_2,  \minus 1 \big ), \\
\widetilde{G}_{\varepsilon_r}(\chi_{\varepsilon_r}) & := \big ( \bar{w}_1, \bar{w}_2, [T_1,T_2] \big )
\end{aligned}
\end{equation}

\begin{equation} \label{eqn:hy_param}
\begin{aligned}
\widetilde{f}_{\varepsilon_r}(\chi_{\varepsilon_r}) := \begin{bmatrix} A_{f_3} \bar{w}_1 \\ A_{f_4} \bar{w}_2 \\  \minus 1
\end{bmatrix}, \hspace{2mm}
\widetilde{G}_{\varepsilon_r}(\chi_{\varepsilon_r}) := \begin{bmatrix}
\bar{w}_1 \\ \bar{w}_2 \\ [T_1,T_2]
\end{bmatrix}
\end{aligned}
\end{equation}

$\widetilde{C}_{\varepsilon_r} := \mathcal{X}_{\varepsilon_r}$ and $\widetilde{D}_{\varepsilon_r} := \{ \chi_{\varepsilon_r} {\in} \mathcal{X}_{\varepsilon_r} : \tau = 0 \}$.

\begin{equation*}
\begin{aligned}
\tilde{f}_{\varepsilon_r}(\chi_{\varepsilon_r}) := \begin{bmatrix} A_{f_3} \bar{w}_1 \\ A_{f_4} \bar{w}_2 \\  \minus 1
\end{bmatrix} \hspace{2mm}  \forall \chi_{\varepsilon_r} \in \tilde{C}_{\varepsilon_r} := \mathcal{X}_{\varepsilon_r},  & &
\tilde{G}_{\varepsilon_r}(\chi_{\varepsilon_r}) := \begin{bmatrix}
\bar{w}_1 \\ \bar{w}_2 \\ [T_1,T_2]
\end{bmatrix} \hspace{2mm}  \forall \chi_{\varepsilon_r} \in \tilde{D}_{\varepsilon_r} := \{ \chi_{\varepsilon_r} {\in} \mathcal{X}_{\varepsilon_r} : \tau {=} 0 \}
\end{aligned}
\end{equation*}

For this system, the set to exponentially stabilize is given by

\begin{equation} \label{set:est_param}
\widetilde{\A}_{\varepsilon_r} := \{0\} \times \{0\} \times [0,T_2]
\end{equation}

In the next result, we show global exponential stability of the set $\tilde{\A}_{\varepsilon_r}$ for $\widetilde{\HS}_{\varepsilon_r}$ through the satisfaction of matrix inequalities. See the appendix for proof.

Let $0 < T_1 \leq T_2$ be given. If there exist a positive scalar $\mu$ and positive definite symmetric matrices $P_2$, $P_3$ such that

\begin{equation} \label{eqn:lyap1}
P_2 A_{f_3} + A_{f_3}^{\top} P_2 \prec 0
\end{equation}

\begin{equation} \label{eqn:lyap2}
P_3 A_{f_4} + A_{f_4}^{\top} P_3 \prec 0
\end{equation}

hold, then the set $\widetilde{\A}_{\varepsilon_r}$ is globally exponentially stable for the hybrid system $\widetilde{\HS}_{\varepsilon_r}$. Furthermore, every solution $\phi$ to $\widetilde{\HS}_{\varepsilon_r}$ satisfies

\begin{equation} \label{eqn:est_param_bnd}
|\phi(t,j)|_{\widetilde{\A}_{\varepsilon_r}} \leq \sqrt{\frac{\alpha_2}{\alpha_1}} \exp{ \Big (\minus \frac{\tilde{\beta}}{2 \alpha_2} t \Big ) } |\phi(0,0)|_{\widetilde{\A}_{\varepsilon_r}}
\end{equation}

for each $(t,j) \in \mbox{dom } \phi$, with $\alpha_1 = \mathrm{min} \{ \lambda_{min} (P_2), \lambda_{min} (P_3) \}$ $\alpha_2 = \mathrm{max} \{ \lambda_{max} (P_2), \lambda_{max} (P_3) \}$
and $\tilde{\beta} > 0$.

If there exists a positive scalar $\mu$ and positive definite symmetric matrices $P_2$, $P_3$ such that,  with $A_{f_3}$ and $A_{f_4}$ as in (<ref>),
two_colthe conditions in (<ref>)

\begin{equation} \label{eqn:lyap1b}
P_2 A_{f_3} + A_{f_3}^{\top} P_2 \prec 0
\end{equation}

\begin{equation} \label{eqn:lyap2b}
P_3 A_{f_4} + A_{f_4}^{\top} P_3 \prec 0
\end{equation}

hold, then the set $\widetilde{\A}_{\varepsilon_r}$ is globally exponentially stable for the hybrid system $\widetilde{\HS}_{\varepsilon_r}$. Furthermore, every solution $\tilde{\phi}$ to $\widetilde{\HS}_{\varepsilon_r}$ satisfies

\begin{equation} \label{eqn:est_param_bnd}
| \tilde{\phi}  (t,j)|_{\widetilde{\A}_{\varepsilon_r}} \leq \sqrt{\frac{\alpha_{\bar{w}_2}}{\alpha_{\bar{w}_1}}} \exp{ \Big ( \minus \frac{ \bar{\gamma}   \tilde{\beta}}{2 \alpha_{\bar{w}_2}}  (t + j)  \Big )} | \tilde{\phi}  (0,0)|_{\widetilde{\A}_{\varepsilon_r}}
\end{equation}

for each $(t,j) \in \mbox{\rm dom }  \tilde{\phi}  $, with $\alpha_{\bar{w}_1} =$ $\mathrm{min} \{ \lambda_{min} (P_2),$ $\lambda_{min} (P_3) \}$, $\alpha_{\bar{w}_2} =$ $\mathrm{max} \{ \lambda_{max} (P_2),$ $\lambda_{max} (P_3) \}$, $\tilde{\beta} > 0$, and $\bar{\gamma} = \min \{1-\gamma, \gamma T_1\}$.

See the appendix for proof.

§.§ Proof of Theorem <ref>

Consider the following Lyapunov function candidate  for $\widetilde{\HS}_{\varepsilon}$

\begin{equation} \label{eqn:lyap}
V(\chi_{\varepsilon}) := V_1(\chi_{\varepsilon}) + V_2(\chi_{\varepsilon}) + V_{\varepsilon_r}(\chi_{\varepsilon}) \hspace{3mm} \forall \chi_{\varepsilon} \in \mathcal{X}_{\varepsilon}
\end{equation}

where $V_1(\chi_{\varepsilon}) {=} \exp{(2 h \tau)} \bar{\eta}_1^2$,
$$V_2(\chi_{\varepsilon}) {=}\bar{z}^{\top}_2 \exp{(A^{\top}_{f_2} \tau )} P_1 \exp{(A_{f_2} \tau )} \bar{z}_2$$
and $V_{\varepsilon_r}(\chi_{\varepsilon}) =$ $\bar{w}_1^{\top} P_2 \bar{w}_1$ $+ \bar{w}_2^{\top} P_3 \bar{w}_2$. Note that there exist two positive scalars $\alpha_1$, $\alpha_2$ such that

\begin{equation} \label{eqn:lyap_def}
\alpha_1 |\chi_{\varepsilon}|_{\tilde{\A}_{\varepsilon}}^2 \leq V(\chi_{\varepsilon}) \leq \alpha_2 |\chi_{\varepsilon}|_{\tilde{\A}_{\varepsilon}}^2 \hspace{5mm} \forall \chi_{\varepsilon} \in \tilde{C}_{\varepsilon} \cup \tilde{D}_{\varepsilon}
\end{equation}

With $P_1$ positive definite and noting the nonsingularity of $\exp{(A_{f_2} \tau)}$ for every $\tau$, we have
$\alpha_1 = \underset{\nu \in [0,T_2]}{\mathrm{min}} \Big \{ \exp{(2h \nu)},$ $\lambda_{min} \big (\exp{(A^{\top}_{f_2} \nu )} P_1 \exp{(A_{f_2} \nu )} \big ), \lambda_{min}(P_2), \lambda_{min}(P_3) \Big \}$

\begin{equation*} \small
\begin{aligned}
\alpha_1 & = \underset{\nu \in [0,T_2]}{\mathrm{min}} \Big \{ \exp{(2h \nu)}, \lambda_{min} \big (\exp{(A^{\top}_{f_2} \nu )} P_1 \exp{(A_{f_2} \nu )} \big ), \\
& \hspace{53mm} \lambda_{min}(P_2), \lambda_{min}(P_3) \Big \}
\end{aligned}
\end{equation*}

and $\alpha_2$ as in (<ref>).  For each $\chi_{\varepsilon} \in \tilde{C}_{\varepsilon}$, one has

\begin{equation} \label{eqn:change_in_v}
\begin{aligned}
\langle \nabla V(\chi_{\varepsilon}), \tilde{f}_{\varepsilon}(\chi_{\varepsilon}) \rangle & = 2 \bar{z}_2^{\top} \big ( \exp{(A^{\top}_{f_2} \tau )} P_1 \exp{(A_{f_2} \tau)} \big ) B_{f_2}  \bar{w}_2 \\
& \hspace{10mm} + \bar{w}_1^{\top} (P_2 A_{f_3} + A_{f_3}^{\top} P_2) \bar{w}_1 \\
& \hspace{15mm} + \bar{w}_2^{\top} (P_3 A_{f_4} + A_{f_4}^{\top} P_3) \bar{w}_2 \\
\end{aligned}
\end{equation}

Now, by noting the conditions in (<ref>), with $\beta_1 > 0$ and $\beta_2 > 0$ such that $P_2 A_{f_3} + A_{f_3}^{\top} P_2 \leq -\beta_1 I$, and $P_3 A_{f_4} + A_{f_4}^{\top} P_3 \leq -\beta_2 I$ then one has

\begin{equation}
\begin{aligned}
\langle \nabla V(\chi_{\varepsilon}), \tilde{f}_{\varepsilon}(\chi_{\varepsilon}) \rangle & \leq \kappa_1 | \bar{z}_2 | | \bar{w}_2 | - \beta_1 | \bar{w}_1 |^2 - \beta_2 | \bar{w}_2 |^2 \\
\end{aligned}
\end{equation}

$\kappa_1$ is as given in (<ref>). Applying Young's inequality to $\kappa_1 |\bar{z}_2| |\bar{w}_2|$,[In particular, we are utilizing the relation $ab \leq \frac{a^2}{2 \epsilon} + \frac{\epsilon b^2}{2}$ where $a,b \in \reals$ and $\epsilon > 0$.] we obtain

\begin{equation*}
\kappa_1 = 2 \underset{\nu \in [0,T_2]}{\mathrm{max}} \big | \exp{(A^{\top}_{f_2} \nu )} P_1 \exp{(A_{f_2} \nu )} \big | |B_{f_2}|
\end{equation*}

Applying Young's inequality to $\kappa_1 |\bar{z}_2| |\bar{w}_2|$,[In particular, we are utilizing the relation $ab \leq \frac{a^2}{2 \epsilon} + \frac{\epsilon b^2}{2}$ where $a,b \in \reals$ and $\epsilon > 0$.] we obtain

\begin{equation}
\begin{aligned}
\langle \nabla V(\chi_{\varepsilon}), \tilde{f}_{\varepsilon}(\chi_{\varepsilon}) \rangle
& \leq \frac{\kappa_1}{2 \epsilon} | \bar{z}_2
|^2 \minus \beta_1 | \bar{w}_1|^2 + \Big ( \frac{\kappa_1 \epsilon}{2} - \beta_2 \Big ) | \bar{w}_2|^2
\end{aligned}
\end{equation}

where $\epsilon > 0$. We then upper bound the inequality by picking the largest coefficient, i.e, $\bar{\kappa}_1 = \max \Big \{ \frac{\kappa_1}{2 \epsilon}, \Big ( \frac{\kappa_1 \epsilon}{2} - \beta_2 \Big )  \Big \}$, leading to

\begin{equation}
\langle \nabla V(\chi_{\varepsilon}), \tilde{f}_{\varepsilon}(\chi_{\varepsilon}) \rangle \leq \frac{\bar{\kappa}_1}{\alpha_2} V(\chi_{\varepsilon})
\end{equation}

Now, for the analysis across jumps, note that for all $\chi_{\varepsilon} \in \tilde{D}_{\varepsilon}$, $\tau = 0$. At jumps, $\tau$ is  mapped to some point $\nu \in [T_1 , T_2]$. Then, at jumps, for each $g \in \tilde{G}_{\varepsilon}$ one has

\begin{equation*}
\begin{aligned}
V(g) {\minus} V(\chi_{\varepsilon}) & = \\
& \hspace{-5mm} \minus \bar{\eta}_1^2 {+} \bar{z}_2^{\top} \big ( A_{g_2}^{\top} \exp{(A_{f_2}^{\top} \nu)} P_1 \exp{(A_{f_2} \nu)} A_{g_2} {\minus} P_1 \big ) \bar{z}_2 \\
& \leq \minus \bar{\kappa}_2 \big ( | \bar{\eta}_1 |^2 + | \bar{z}_2 |^2 \big )
\end{aligned}
\end{equation*}

where $\bar{\kappa}_2$ and $\kappa_2$ are as given in (<ref>), from where we have

where $\bar{\kappa}_2 = \max \{1 , \kappa_2 \}$ and, by continuity of condition (<ref>), $\kappa_2 > 0$ such that

\begin{equation*}
\kappa_2 {\in} \Big ( 0,  \minus \underset{v {\in} [T_1,T_2]}{\min}  {\lambda_{min}} (A^{\top}_{g_2} \exp{(A^{\top}_{f_2} v )} P_1 \exp{(A_{f_2} v )} A_{g_2} {\minus} P_1 ) \Big )
\end{equation*}

from where we have

\begin{equation} \label{eqn:jump_bnd}
\begin{aligned}
V(g) - V(\chi_{\varepsilon}) \leq \minus \bar{\kappa}_2 \big ( | \bar{\eta}_1 |^2 + | \bar{z}_2 |^2 \big )
\end{aligned}
\end{equation}

Utilizing the upper bound $\alpha_2$ from the definition of $V$ in (<ref>),  for all $\chi_{\varepsilon} \in \tilde{D}_{\varepsilon}$, one has $V(\chi_{\varepsilon}) \leq \alpha_2 \big (| \bar{\eta}_1 |^2 + |\bar{z}_2|^2 + |\bar{w}|^2 \big )$. Dividing by $\alpha_2$ and rearranging terms, one has

\begin{equation} \label{thrm_zdef}
\begin{aligned}
\minus (| \bar{\eta}_1 |^2 + |\bar{z}_2|^2) \leq - \frac{1}{\alpha_2} V(\chi_{\varepsilon}) + |\bar{w}|^2
\end{aligned}
\end{equation}

Then, by inserting (<ref>) into (<ref>), we obtain

\begin{equation} \label{eqn:v_jumps}
V(g) - V(\chi_{\varepsilon}) \leq \bar{\kappa}_2 \Big (- \frac{1}{\alpha_2} V(\chi_{\varepsilon}) + |\bar{w}|^2 \Big )
\end{equation}

Now, by noting that $\langle \nabla V(\chi_{\varepsilon}), \tilde{f}(\chi_{\varepsilon}) \rangle  \leq \frac{\bar{\kappa}_1}{\alpha_2} V(\chi_{\varepsilon})$ and by (<ref>), pick a solution  $\tilde{\phi}$ to $\widetilde{\HS}_{\varepsilon}$  with initial condition $\tilde{\phi}(0,0) \in \widetilde{C}_{\varepsilon} \cup \widetilde{D}_{\varepsilon}$.  Let the jumps of $\tilde{\phi}$ occur at times $(t_j,j) \in \{j' : \exists t' : (t',j') \in \mbox{dom } \phi\}$.  For each $(t,j) \in [0,t_1] \times \{0\}$ one has

\begin{equation*}
V(  \tilde{\phi} (  t,0 )) \leq \exp \Big ( \frac{\bar{\kappa}_1}{\alpha_2} \hspace{0.5mm} t_1 \Big ) V( \tilde{\phi} (  0,0))
\end{equation*}

At $(t_1,1)$, one has

\begin{equation*}
\begin{aligned}
V(  \tilde{\phi} (  t_1,1)) & {\leq} \Big ( 1 \minus \frac{\bar{\kappa}_2}{\alpha_2} \Big ) \exp \Big ( \frac{\bar{\kappa}_1}{\alpha_2} \hspace{0.5mm} t_1 \Big ) V(  \tilde{\phi} (  0,0)) {+} \bar{\kappa}_2 |\bar{w}(t_1,0)|^2 \\
\end{aligned}
\end{equation*}

Then, for each $(t,j) \in [t_1,t_2] \times \{1\}$

\begin{equation*}
\begin{aligned}
V(  \tilde{\phi} (  t,1)) & = \exp \Big ( \frac{\bar{\kappa}_1}{\alpha_2} \hspace{0.5mm} t_2 \Big ) \Big ( 1 - \frac{\bar{\kappa}_2}{\alpha_2} \Big ) V(  \tilde{\phi} (  0,0)) \\
& \hspace{1mm} + \exp \big ( \frac{\bar{\kappa}_1}{\alpha_2} (t_2 - t_1) \big ) \bar{\kappa}_2 |\bar{w}(t_1,0)|^2
\end{aligned}
\end{equation*}

At $(t_2,2)$, one has

\begin{equation*}
\begin{aligned}
V(  \tilde{\phi} (  t_2,2)) & \leq \exp \Big ( \frac{\bar{\kappa}_1}{\alpha_2} \hspace{0.5mm} t_2 \Big ) \Big ( 1 - \frac{\bar{\kappa}_2}{\alpha_2} \Big )^2 V(  \tilde{\phi} (  0,0)) \\
& \hspace{-2mm} + \bar{\kappa}_2 \Big [ \exp \Big ( \frac{\bar{\kappa}_1}{\alpha_2} (t_2 - t_1) \Big ) |\bar{w}(t_1,0)|^2 + |\bar{w}(t_2,1)|^2 \Big ]
\end{aligned}
\end{equation*}

A general form of the bound is given by

\begin{equation} \label{eqn:gen_bound_main}
\begin{aligned}
V(  \tilde{\phi} (  t,j)) & \leq \exp \Big ( \frac{\bar{\kappa}_1}{\alpha_2} \hspace{0.5mm} t_j \Big ) \Big ( 1 {-} \frac{\bar{\kappa}_2}{\tilde{\alpha}_1} \Big )^j V(  \tilde{\phi} (  0,0)) \\
& \hspace{2mm} + \bar{\kappa}_2 \Big( \sum_{k=1}^{j} \exp \Big ( \frac{\bar{\kappa}_1}{\alpha_2} (t_{k + 1} \minus t_k) \Big ) |\bar{w}(t_{k},k \minus 1)|^2 \Big )
\end{aligned}
\end{equation}

Noting that $t_{j+1} - t_j \leq T_2$ and $\frac{\bar{\kappa}_1}{\alpha_2} > 0$,  the latter term can be further bounded as

\begin{equation*}
\begin{aligned}
& \bar{\kappa}_2 \Big( \sum_{k=1}^{j} {\exp} \big ( \frac{\bar{\kappa}_1}{\alpha_2} (t_{k+1} \minus t_k) \big ) |\bar{w}(t_{k},k \minus 1)|^2 \Big ) \\
& \hspace{30mm} \leq  \bar{\kappa}_2 {\exp} \big ( \frac{\bar{\kappa}_1}{\alpha_2} \hspace{0.5mm} T_2 \big ) \mathrm{sup}_{(t,j) \in \mbox{\footnotesize dom} \tilde{\phi}} |\bar{w}(t,j)|^2
\end{aligned}
\end{equation*}

Moreover, since $t_j \leq T_2 (j+1)$ and $\frac{\bar{\kappa}_1}{\alpha_2} > 0$, we can also put a stricter bound on the first term in (<ref>) as follows:

\begin{equation*}
\begin{aligned}
\hspace{0mm} {\exp} \Big ( \frac{\bar{\kappa}_1}{\alpha_2} \hspace{0.5mm} t_{j} \Big ) \Big ( 1 {\minus} \frac{\bar{\kappa}_2}{\alpha_2} \Big )^j V(  \tilde{\phi} (  0,0)) & \\
& \hspace{-35mm} \leq {\exp} \Big ( \frac{\bar{\kappa}_1}{\alpha_2} \hspace{0.5mm} T_2 \Big ) \Big ( \exp \Big ( \frac{\bar{\kappa}_1}{\alpha_2} \hspace{0.5mm} T_2 \Big ) \Big ( 1 {-} \frac{\bar{\kappa}_2}{\alpha_2} \Big ) \Big )^j V(  \tilde{\phi} (  0,0))
\end{aligned}
\end{equation*}

\begin{equation*} \label{thrm1_bound_a}
\begin{aligned}
V(  \tilde{\phi} (  t,j)) & \leq \\
& \exp \Big ( \frac{\bar{\kappa}_1}{\alpha_2} \hspace{0.5mm} T_2 \Big ) \Big ( \exp \Big ( \frac{\bar{\kappa}_1}{\alpha_2} \hspace{0.5mm} T_2 \Big ) \Big ( 1 {-} \frac{\bar{\kappa}_2}{\alpha_2} \Big ) \Big )^j V(  \tilde{\phi} (  0,0)) \\
& \hspace{13mm} + \bar{\kappa}_2 \exp \Big ( \frac{\bar{\kappa}_1}{\alpha_2} \hspace{0.5mm} T_2 \Big ) \mathrm{sup}_{(t,j) \in \mbox{\footnotesize dom} \tilde{\phi}}|\bar{w}(t,j)|^2
\end{aligned}
\end{equation*}

Then, from the result of Proposition <ref>, we have (<ref>)
with $\alpha_{\bar{w}_1} =$ $\mathrm{min} \{ \lambda_{min} (P_2), \lambda_{min} (P_3) \}$ and $\alpha_{\bar{w}_2} =$ $\mathrm{max} \{ \lambda_{max} (P_2), \lambda_{max} (P_3) \}$.  Now, to improve readability, we have omitted including the use of the notation $V(\tilde{\phi}(t,j))$ when evaluating $V$ along the trajectory for the solution $\tilde{\phi}$ opting instead for the use of the state components of $\chi_{\varepsilon}$ directly. In particular, we remind the reader that the notation $\bar{w}(t,j)$ corresponds to the $\bar{w}$ component of a solution, i.e.,  $\phi_{\bar{w}}(t,j)$. Now, combining the inequality with (<ref>) and noting $V(\tilde{\phi}(0,0)) \leq \alpha_2 |\tilde{\phi}(0,0) |_{\tilde{\A}_{\varepsilon}}^2$ one has for each $(t,j) \in \mbox{dom } \phi$

\begin{equation} \label{thrm1_bound}
\begin{aligned}
|\phi(t,j)|_{\tilde{\A}_{\varepsilon}} & \leq \\
& \hspace{-15mm} \sqrt{\frac{\alpha_2}{\alpha_1}} |\phi(0,0) |_{\tilde{\A}_{\varepsilon}} \exp \Big ( \frac{\bar{\kappa}_1}{2 \alpha_2} \hspace{0.5mm} T_2 \Big ) \Big ( \exp \Big ( \frac{\bar{\kappa}_1}{2 \alpha_2} \hspace{0.5mm} T_2 \Big ) \Big ( 1 {-} \frac{\bar{\kappa}_2}{2 \alpha_2} \Big ) \Big )^j \\
& \hspace{-15mm} {+} \sqrt{\bar{\kappa}_2} \exp \Big ( \frac{\bar{\kappa}_1}{2 \alpha_2} \hspace{0.5mm} T_2 \Big ) \sqrt{ \frac{\alpha_{\bar{w}_2}}{\alpha_{\bar{w}_1}} \exp{ \Big ( \frac{ \minus \bar{\gamma} \tilde{\beta}}{2 \alpha_{\bar{w}_2}} (t {+} j) \Big )^2 } |\phi_{\bar{w}}(0,0)|^2_{\tilde{\A}_{\varepsilon_r}} }
\end{aligned}
\end{equation}

By the given conditions, the set $\tilde{\A}_{\varepsilon}$ is globally exponentially stable and attractive for $\widetilde{\HS}_{\varepsilon}$.  Now, by utilizing Lemmas <ref> - <ref>, we can establish global exponential stability to the set $\A_{\varepsilon}$ for $\HS_{\varepsilon}$. In particular, Lemma <ref> establishes the relation between  $\widetilde{\HS}_{\varepsilon}$ and $\HS_{\varepsilon}$. In turn, we can then make use of Lemmas  <ref> - <ref>, where Lemma <ref> establishes the reduction from $\HS$ to $\HS_{\varepsilon}$, to show that the set $\A$ is globally exponentially stable and attractive for $\HS$ in (<ref>).

Consider the following Lyapunov function candidate  for $\widetilde{\HS}_{\varepsilon}$

\begin{equation} \label{eqn:lyap}
V(\chi_{\varepsilon}) := V_1(\chi_{\varepsilon}) + V_2(\chi_{\varepsilon}) + V_{\varepsilon_r}(\chi_{\varepsilon}) \hspace{3mm} \forall \chi_{\varepsilon} \in \mathcal{X}_{\varepsilon}
\end{equation}

\begin{align*}
& V_1(\chi_{\varepsilon}) = \exp{(2 h \tau)} \bar{\eta}_1^2 \\
& V_2(\chi_{\varepsilon}) = \bar{z}^{\top}_2 \exp{(A^{\top}_{f_2} \tau )} P_1 \exp{(A_{f_2} \tau )} \bar{z}_2 \\
& V_{\varepsilon_r}(\chi_{\varepsilon}) = \bar{w}_1^{\top} P_2 \bar{w}_1 + \bar{w}_2^{\top} P_3 \bar{w}_2
\end{align*}
Note that there exist two positive scalars $\alpha_1$, $\alpha_2$ such that

\begin{equation} \label{eqn:lyap_def}
\alpha_1 |\chi_{\varepsilon}|_{\tilde{\A}_{\varepsilon}}^2 \leq V(\chi_{\varepsilon}) \leq \alpha_2 |\chi_{\varepsilon}|_{\tilde{\A}_{\varepsilon}}^2 \hspace{1cm} \forall \chi_{\varepsilon} \in \tilde{C}_{\varepsilon} \cup \tilde{D}_{\varepsilon}
\end{equation}

With $P_1$ positive definite and noting the nonsingularity of $\exp{(A_{f_2} \tau)}$ for every $\tau$, we have

\begin{equation*} \small
\begin{aligned}
\alpha_1 & = \underset{\nu \in [0,T_2]}{\mathrm{min}} \Big \{ \exp{(2h \nu)}, \lambda_{min} \big (\exp{(A^{\top}_{f_2} \nu )} P_1 \exp{(A_{f_2} \nu )} \big ), \\
& \hspace{53mm} \lambda_{min}(P_2), \lambda_{min}(P_3) \Big \}
\end{aligned}
\end{equation*}

and $\alpha_2$ as in (<ref>).  For each $\chi_{\varepsilon} \in \tilde{C}_{\varepsilon}$, one has

\begin{equation}
\begin{aligned}
\langle \nabla V(\chi_{\varepsilon}), \tilde{f}_{\varepsilon}(\chi_{\varepsilon}) \rangle & = 2 \bar{z}_2^{\top} \big ( \exp{(A^{\top}_{f_2} \tau )} P_1 \exp{(A_{f_2} \tau)} \big ) B_{f_2}  \bar{w}_2 \\
& \hspace{10mm} + \bar{w}_1^{\top} (P_2 A_{f_3} + A_{f_3}^{\top} P_2) \bar{w}_1 \\
& \hspace{15mm} + \bar{w}_2^{\top} (P_3 A_{f_4} + A_{f_4}^{\top} P_3) \bar{w}_2 \\
\end{aligned}
\end{equation}

Now, by noting (<ref>) and (<ref>), with $\beta_1 > 0$ and $\beta_2 > 0$ such that $P_2 A_{f_3} + A_{f_3}^{\top} P_2 \leq -\beta_1 I$, and $P_3 A_{f_4} + A_{f_4}^{\top} P_3 \leq -\beta_2 I$ then one has

\begin{equation}
\begin{aligned}
\langle \nabla V(\chi_{\varepsilon}), \tilde{f}_{\varepsilon}(\chi_{\varepsilon}) \rangle & \leq \kappa_1 | \bar{z}_2 | | \bar{w}_2 | - \beta_1 | \bar{w}_1 |^2 - \beta_2 | \bar{w}_2 |^2 \\
\end{aligned}
\end{equation}

\begin{equation*}
\kappa_1 = 2 \underset{\nu \in [0,T_2]}{\mathrm{max}} \big | \exp{(A^{\top}_{f_2} \nu )} P_1 \exp{(A_{f_2} \nu )} \big | |B_{f_2}|
\end{equation*}

Applying Young's inequality to $\kappa_1 |\bar{z}_2| |\bar{w}_2|$, [In particular, we are utilizing the relation $ab \leq \frac{a^2}{2 \epsilon} + \frac{\epsilon b^2}{2}$ where $a,b \in \reals$ and $\epsilon > 0$.] we obtain

\begin{equation}
\begin{aligned}
\langle \nabla V(\chi_{\varepsilon}), \tilde{f}_{\varepsilon}(\chi_{\varepsilon}) \rangle
& \leq \frac{\kappa_1}{2 \epsilon} | \bar{z}_2 |^2 + \frac{\kappa_1 \epsilon}{2} | \bar{w}_2 |^2 \minus \beta_1 | \bar{w}_1|^2 \minus \beta_2 | \bar{w}_2|^2 \\
& \leq \frac{\kappa_1}{2 \epsilon} | \bar{z}_2
|^2 \minus \beta_1 | \bar{w}_1|^2 + \Big ( \frac{\kappa_1 \epsilon}{2} - \beta_2 \Big ) | \bar{w}_2|^2
\end{aligned}
\end{equation}

where $\epsilon > 0$, we then upper bound the inequality by picking the largest coefficient, i.e, $\bar{\kappa}_1 = \max \Big \{ \frac{\kappa_1}{2 \epsilon}, \Big ( \frac{\kappa_1 \epsilon}{2} - \beta_2 \Big )  \Big \}$, leading to

\begin{equation}
\begin{aligned}
\langle \nabla V(\chi_{\varepsilon}), \tilde{f}_{\varepsilon}(\chi_{\varepsilon}) \rangle & \leq \bar{\kappa}_1 \big ( | \bar{z}_2 |^2 + | \bar{w}_1|^2 + | \bar{w}_2 |^2 \big ) \\
& \leq \bar{\kappa}_1 \big ( |\chi_{\varepsilon}|^2_{\tilde{\A}_{\varepsilon}} \big ) \\
& \leq \bar{\kappa}_1 \Big ( \frac{1}{\alpha_2} V(\chi_{\varepsilon}) \Big ) \\
& \leq \frac{\bar{\kappa}_1}{\alpha_2} V(\chi_{\varepsilon})
\end{aligned}
\end{equation}

Now, for the analysis across jumps, note that for all $\chi_{\varepsilon} \in \tilde{D}_{\varepsilon}$, $\tau = 0$. At jumps, $\tau$ is  mapped to some point $\nu \in [T_1 , T_2]$. Then, at jumps, for each $g \in \tilde{G}_{\varepsilon}$ one has

\begin{equation}
\begin{aligned}
V(g) {\minus} V(\chi_{\varepsilon}) & = \minus \bar{\eta}_1^2 -  \bar{z}_2^{\top} P_1 \bar{z}_2 \\
& \hspace{5mm} {+} (A_{g_2} \bar{z}_2)^{\top} \exp{(A^{\top}_{f_2} \nu )} P_1 \exp{(A_{f_2} \nu)} (A_{g_2} \bar{z}_2) \\
& = \minus \bar{\eta}_1^2 \\
& \hspace{5mm} {+} \bar{z}_2^{\top} \big ( A_{g_2}^{\top} \exp{(A_{f_2}^{\top} \nu)} P_1 \exp{(A_{f_2} \nu)} A_{g_2} {\minus} P_1 \big ) \bar{z}_2 \\
& \leq \minus | \bar{\eta}_1 |^2 \minus \kappa_2 | \bar{z}_2 |^2 \\
& \leq \minus \bar{\kappa}_2 \big ( | \bar{\eta}_1 |^2 + | \bar{z}_2 |^2 \big )
\end{aligned}
\end{equation}

where $\bar{\kappa}_2 = \max \{1 , \kappa_2 \}$ and, by continuity of condition (<ref>), $\kappa_2 > 0$ such that

\begin{equation*}
\kappa_2 {\in} \Big ( 0,  \minus \underset{v {\in} [T_1,T_2]}{\min}  {\lambda_{min}} (A^{\top}_{g_2} \exp{(A^{\top}_{f_2} v )} P_1 \exp{(A_{f_2} v )} A_{g_2} {\minus} P_1 ) \Big )
\end{equation*}

for where we have

\begin{equation} \label{eqn:jump_bnd}
\begin{aligned}
V(g) - V(\chi_{\varepsilon}) \leq \minus \bar{\kappa}_2 \big ( | \bar{\eta}_1 |^2 + | \bar{z}_2 |^2 \big )
\end{aligned}
\end{equation}

Utilizing the upper bound $\alpha_2$ from the definition of $V$ in (<ref>),  for all $\chi_{\varepsilon} \in \tilde{D}_{\varepsilon}$, one has

\begin{equation}
\begin{aligned}
V(\chi_{\varepsilon}) & \leq \alpha_2 \big (| \bar{\eta}_1 |^2 + |\bar{z}_2|^2 + |\bar{w}|^2 \big ) \\
\end{aligned}
\end{equation}

Dividing by $\alpha_2$ and rearranging terms, one has

\begin{equation} \label{thrm_zdef}
\begin{aligned}
\minus (| \bar{\eta}_1 |^2 + |\bar{z}_2|^2) \leq - \frac{1}{\alpha_2} V(\chi_{\varepsilon}) + |\bar{w}|^2 \\
\end{aligned}
\end{equation}

Then, by inserting (<ref>) into (<ref>),

\begin{equation} \label{eqn:v_jumps}
\begin{aligned}
V(g) - V(\chi_{\varepsilon}) & \leq \minus \bar{\kappa}_2 \big ( | \bar{\eta}_1 |^2 + | \bar{z}_2 |^2 \big ) \\
V(g) - V(\chi_{\varepsilon}) & \leq \bar{\kappa}_2 \Big (- \frac{1}{\alpha_2} V(\chi_{\varepsilon}) + |\bar{w}|^2 \Big ) \\
V(g) & \leq - \frac{\bar{\kappa}_2}{\alpha_2} V(\chi_{\varepsilon}) + \bar{\kappa}_2 |\bar{w}|^2 + V(\chi_{\varepsilon}) \\
V(g) & \leq \Big (1 - \frac{\bar{\kappa}_2}{\alpha_2} \Big ) V(\chi_{\varepsilon}) + \bar{\kappa}_2  |\bar{w}|^2 \\
\end{aligned}
\end{equation}

Now, by noting that $\langle \nabla V(\chi_{\varepsilon}), \tilde{f}(\chi_{\varepsilon}) \rangle  \leq \frac{\bar{\kappa}_1}{\alpha_2} V(\chi_{\varepsilon})$ and by (<ref>), pick a solution  $\tilde{\phi}$ to $\widetilde{\HS}_{\varepsilon}$  with initial condition $\tilde{\phi}(0,0) \in \widetilde{C}_{\varepsilon} \cup \widetilde{D}_{\varepsilon}$.  Let the jumps of $\tilde{\phi}$ occur at times $(t_j,j) \in \{j' : \exists t' : (t',j') \in \mbox{dom } \phi\}$.  For each $(t,j) \in [0,t_1] \times \{0\}$ one has

\begin{equation}
V(t,0) \leq \exp \Big ( \frac{\bar{\kappa}_1}{\alpha_2} \hspace{0.5mm} t_1 \Big ) V(0,0)
\end{equation}

At $(t_1,1)$

\begin{equation*}
\begin{aligned}
V(t_1,1) & \leq \Big ( 1 - \frac{\bar{\kappa}_2}{\alpha_2} \Big ) V(t_1,0) + \bar{\kappa}_2 |\bar{w}(t_1,0)|^2 \\
& \leq \Big ( 1 \minus \frac{\bar{\kappa}_2}{\alpha_2} \Big ) \exp \Big ( \frac{\bar{\kappa}_1}{\alpha_2} \hspace{0.5mm} t_1 \Big ) V(0,0) + \bar{\kappa}_2 |\bar{w}(t_1,0)|^2 \\
\end{aligned}
\end{equation*}

Then, for each $(t,j) \in [t_1,t_2] \times \{1\}$

\begin{equation*}
\begin{aligned}
V(t,1) & \leq \exp \Big ( \frac{\bar{\kappa}_1}{\alpha_2} (t_2 - t_1) \Big ) V(t_1,1) \\
& \leq \exp \Big ( \frac{\bar{\kappa}_1}{\alpha_2} (t_2 - t_1) \Big ) \Big [ \Big ( 1 - \frac{\bar{\kappa}_2}{\alpha_2} \Big ) \exp \big ( \bar{\kappa}_1 \hspace{0.5mm} t_1 \big ) V(0,0) \\
& \hspace{5mm} + \bar{\kappa}_2 |\bar{w}(t_1,0)|^2 \Big ] \\
& \leq \exp \Big ( \frac{\bar{\kappa}_1}{\alpha_2} (t_2 - t_1) \Big ) \Big ( 1 \minus  \frac{\bar{\kappa}_2}{\tilde{\alpha}_1} \Big ) \exp \Big ( \frac{\bar{\kappa}_1}{\alpha_2} \hspace{0.5mm} t_1 \Big ) V(0,0) \\
& \hspace{5mm} + \exp \Big ( \frac{\bar{\kappa}}{\alpha_2} (t_2 - t_1) \Big ) \bar{\kappa}_2 |\bar{w}(t_1,0)|^2 \\
& = \exp \Big ( \frac{\bar{\kappa}_1}{\alpha_2} \hspace{0.5mm} t_2 \Big ) \Big ( 1 - \frac{\bar{\kappa}_2}{\alpha_2} \Big ) V(0,0) \\
& \hspace{5mm} + \exp \big ( \frac{\bar{\kappa}_1}{\alpha_2} (t_2 - t_1) \big ) \bar{\kappa}_2 |\bar{w}(t_1,0)|^2 \\
\end{aligned}
\end{equation*}

At $(t_2,2)$

\begin{equation*}
\begin{aligned}
V(t_2,2) & \leq \Big ( 1 - \frac{\bar{\kappa}_2}{\alpha_2} \Big ) V(t_2,1) + \bar{\kappa}_2 |\bar{w}(t_2,1)|^2 \\
& \leq \Big ( 1 - \frac{\bar{\kappa}_2}{\alpha_2} \Big ) \exp \Big ( \frac{\bar{\kappa}_1}{\alpha_2} \hspace{0.5mm} t_2 \Big ) \Big ( 1 - \frac{\bar{\kappa}_2}{\alpha_2} \Big ) V(0,0) \\
& \hspace{5mm} + \exp \Big ( \frac{\bar{\kappa}_1}{\alpha_2} (t_2 - t_1) \Big ) \bar{\kappa}_2  |\bar{w}(t_1,0)|^2 {+} \bar{\kappa}_2 |\bar{w}(t_2,1)|^2 \\
& \leq \exp \Big ( \frac{\bar{\kappa}_1}{\alpha_2} \hspace{0.5mm} t_2 \Big ) \Big ( 1 - \frac{\bar{\kappa}_2}{\alpha_2} \Big )^2 V(0,0) \\
& \hspace{5mm} + \bar{\kappa}_2 \Big [ \exp \Big ( \frac{\bar{\kappa}_1}{\alpha_2} (t_2 - t_1) \Big ) |\bar{w}(t_1,0)|^2 + |\bar{w}(t_2,1)|^2 \Big ] \\
\end{aligned}
\end{equation*}

A general form of the bound is given by

\begin{equation} \label{eqn:gen_bound_main}
\begin{aligned}
V(t,j) & \leq \exp \Big ( \frac{\bar{\kappa}_1}{\alpha_2} \hspace{0.5mm} t_j \Big ) \Big ( 1 {-} \frac{\bar{\kappa}_2}{\tilde{\alpha}_1} \Big )^j V(0,0) \\
& \hspace{5mm} + \bar{\kappa}_2 \Big( \sum_{k=1}^{j} \exp \Big ( \frac{\bar{\kappa}_1}{\alpha_2} (t_{k + 1} \minus t_k) \Big ) |\bar{w}(t_{k},k \minus 1)|^2 \Big )
\end{aligned}
\end{equation}

Noting that $t_{j+1} - t_j \leq T_2$ and $\frac{\bar{\kappa}_1}{\alpha_2} > 0$,  the latter term can be further bounded as

\begin{equation*}
\begin{aligned}
& \bar{\kappa}_2 \Big( \sum_{k=1}^{j} {\exp} \big ( \frac{\bar{\kappa}_1}{\alpha_2} (t_{k+1} \minus t_k) \big ) |\bar{w}(t_{k},k \minus 1)|^2 \Big ) \\
& \hspace{30mm} \leq  \bar{\kappa}_2 {\exp} \big ( \frac{\bar{\kappa}_1}{\alpha_2} \hspace{0.5mm} T_2 \big ) \mathrm{sup}_{(t,j) \in \mbox{\footnotesize dom} \tilde{\phi}} |\bar{w}(t,j)|^2
\end{aligned}
\end{equation*}

Moreover, since $t_j \leq T_2 (j+1)$ and $\frac{\bar{\kappa}_1}{\alpha_2} > 0$, we can also put a stricter bound on the first term in (<ref>) as follows:

\begin{equation*}
\begin{aligned}
\hspace{-1mm} {\exp} \Big ( \frac{\bar{\kappa}_1}{\alpha_2} \hspace{0.5mm} t_{j} \Big ) \Big ( 1 {\minus} \frac{\bar{\kappa}_2}{\alpha_2} \Big )^j V(0,0) & \\
& \hspace{-25mm} \leq {\exp} \Big ( \frac{\bar{\kappa}_1}{\alpha_2} \hspace{0.5mm} T_2 (j{+}1) \Big ) \Big ( 1 {\minus} \frac{\bar{\kappa}_2}{\alpha_2} \Big )^j V(0,0) \\
& \hspace{-25mm} \leq {\exp} \Big ( \frac{\bar{\kappa}_1}{\alpha_2} \hspace{0.5mm} T_2 \Big ) \Big ( \exp \Big ( \frac{\bar{\kappa}_1}{\alpha_2} \hspace{0.5mm} T_2 \Big ) \Big ( 1 {-} \frac{\bar{\kappa}_2}{\alpha_2} \Big ) \Big )^j V(0,0)
\end{aligned}
\end{equation*}

\begin{equation} \label{thrm1_bound_a}
\begin{aligned}
V(t,j) & \leq \exp \Big ( \frac{\bar{\kappa}_1}{\alpha_2} \hspace{0.5mm} T_2 \Big ) \Big ( \exp \Big ( \frac{\bar{\kappa}_1}{\alpha_2} \hspace{0.5mm} T_2 \Big ) \Big ( 1 - \frac{\bar{\kappa}_2}{\alpha_2} \Big ) \Big )^j V(0,0) \\
& \hspace{5mm} + \bar{\kappa}_2 \exp \Big ( \frac{\bar{\kappa}_1}{\alpha_2} \hspace{0.5mm} T_2 \Big ) \mathrm{sup}_{(t,j) \in \mbox{\footnotesize dom} \tilde{\phi}}|\bar{w}(t,j)|^2
\end{aligned}
\end{equation}

Then, from the result of Proposition <ref>, we have

\begin{equation*}
|\tilde{\phi}_{\bar{w}}(t,j)| \leq \sqrt{\frac{\alpha_{\bar{w}_2}}{\alpha_{\bar{w}_1}}} \exp{ \Big ( \minus \frac{\tilde{\beta}}{2 \alpha_{\bar{w}_2}} t \Big ) } |\tilde{\phi}_{\bar{w}}(0,0)|_{\tilde{\A}_{\varepsilon_r}}
\end{equation*}

with $\alpha_{\bar{w}_1} = \mathrm{min} \{ \lambda_{min} (P_2), \lambda_{min} (P_3) \}$ and $\alpha_{\bar{w}_2} = \mathrm{max} \{ \lambda_{max} (P_2), \lambda_{max} (P_3) \}$.  Now, to improve readability, we have omitted including the use of the notation $V(\tilde{\phi}(t,j))$ when evaluating $V$ along the trajectory for the solution $\tilde{\phi}$ opting instead for the use of the state components of $\chi_{\varepsilon}$ directly. In particular, we remind the reader that the notation $\bar{w}(t,j)$ corresponds to the $\bar{w}$ component of a solution, i.e.,  $\phi_{\bar{w}}(t,j)$. Thus, we have

\begin{equation} \label{thrm1_bound}
\begin{aligned}
V(t,j) & \leq \exp \Big ( \frac{\bar{\kappa}_1}{\alpha_2} \hspace{0.5mm} T_2 \Big ) \Big ( \exp \Big ( \frac{\bar{\kappa}_1}{\alpha_2} \hspace{0.5mm} T_2 \Big ) \Big ( 1 {-} \frac{\bar{\kappa}_2}{\alpha_2} \Big ) \Big )^j V(0,0) \\
& \hspace{-5mm} + \bar{\kappa}_2 \exp \Big ( \frac{\bar{\kappa}_1}{\alpha_2} \hspace{0.5mm} T_2 \Big )  \frac{\alpha_{\bar{w}_2}}{\alpha_{\bar{w}_1}} \exp{ \Big ( \frac{ \minus \bar{\gamma} \tilde{\beta}}{2 \alpha_{\bar{w}_2}} (t {+} j) \Big )^2 } |\phi_{\bar{w}}(0,0)|^2_{\tilde{\A}_{\varepsilon_r}}  \\
& \hspace{50mm} \forall (t,j) \in \mbox{dom } \tilde{\phi}
\end{aligned}
\end{equation}

\begin{equation} \label{thrm1_bound}
\begin{aligned}
V(t,j) & \leq \exp \Big ( \frac{\bar{\kappa}_1}{\alpha_2} \hspace{0.5mm} T_2 \Big ) \Big ( \exp \Big ( \frac{\bar{\kappa}_1}{\alpha_2} \hspace{0.5mm} T_2 \Big ) \Big ( 1 {-} \frac{\bar{\kappa}_2}{\alpha_2} \Big ) \Big )^j V(0,0) \\
& \hspace{5mm} + \bar{\kappa}_2 \exp \Big ( \frac{\bar{\kappa}_1}{\alpha_2} \hspace{0.5mm} T_2 \Big )  \frac{\alpha_{\bar{w}_2}}{\alpha_{\bar{w}_1}} \exp{ \Big ( \minus \frac{\bar{\gamma} \tilde{\beta}}{2 \alpha_{\bar{w}_2}} (t + j) \Big )^2 } |\phi_{\bar{w}}(0,0)|^2_{\tilde{\A}_{\varepsilon_r}}  \\
& \hspace{90mm} \forall (t,j) \in \mbox{dom } \tilde{\phi}
\end{aligned}
\end{equation}

Now, combining the inequality with (<ref>) and noting $V(\phi(0,0)) \leq \alpha_2 |\phi(0,0) |_{\tilde{\A}_{\varepsilon}}^2$ one has

\begin{equation}
\begin{aligned}
|\phi(t,j)|^2_{\tilde{\A}_{\varepsilon}} & \leq \\
& \hspace{-15mm} \frac{\alpha_2}{\alpha_1} |\phi(0,0) |_{\tilde{\A}_{\varepsilon}}^2 \exp \Big ( \frac{\bar{\kappa}_1}{\alpha_2} \hspace{0.5mm} T_2 \Big ) \Big ( \exp \Big ( \frac{\bar{\kappa}_1}{\alpha_2} \hspace{0.5mm} T_2 \Big ) \Big ( 1 {-} \frac{\bar{\kappa}_2}{\alpha_2} \Big ) \Big )^j \\
& \hspace{-12mm} {+} \bar{\kappa}_2 \exp \Big ( \frac{\bar{\kappa}_1}{\alpha_2} \hspace{0.5mm} T_2 \Big )  \frac{\alpha_{\bar{w}_2}}{\alpha_{\bar{w}_1}} \exp{ \Big ( \minus \frac{\bar{\gamma} \tilde{\beta}}{2 \alpha_{\bar{w}_2}} (t {+} j) \Big )^2 } |\phi_{\bar{w}}(0,0)|^2_{\tilde{\A}_{\varepsilon_r}}  \\
& \hspace{50mm} \forall (t,j) \in \mbox{dom } \phi
\end{aligned}
\end{equation}

\begin{equation}
\begin{aligned}
|\phi(t,j)|^2_{\tilde{\A}_{\varepsilon}} & \leq \alpha_1^{\minus 1} \Big ( \alpha_2 |\phi(0,0) |_{\tilde{\A}_{\varepsilon}}^2 \Big ) \exp \Big ( \frac{\bar{\kappa}_1}{\alpha_2} \hspace{0.5mm} T_2 \Big ) \Big ( \exp \Big ( \frac{\bar{\kappa}_1}{\alpha_2} \hspace{0.5mm} T_2 \Big ) \Big ( 1 {-} \frac{\bar{\kappa}_2}{\alpha_2} \Big ) \Big )^j \\
& \hspace{5mm} + \bar{\kappa}_2 \exp \Big ( \frac{\bar{\kappa}_1}{\alpha_2} \hspace{0.5mm} T_2 \Big )  \frac{\alpha_{\bar{w}_2}}{\alpha_{\bar{w}_1}} \exp{ \Big ( \frac{ \minus \bar{\gamma} \tilde{\beta}}{2 \alpha_{\bar{w}_2}} (t + j) \Big )^2 } |\phi_{\bar{w}}(0,0)|^2_{\tilde{\A}_{\varepsilon_r}}  \\
& \hspace{90mm} \forall (t,j) \in \mbox{dom } \phi
\end{aligned}
\end{equation}

Then, taking the square root on both sides, one has

\begin{equation}
\begin{aligned}
|\phi(t,j)|_{\tilde{\A}_{\varepsilon}} & \leq \\
& \hspace{-15mm} \sqrt{\frac{\alpha_2}{\alpha_1}} |\phi(0,0) |_{\tilde{\A}_{\varepsilon}} \exp \Big ( \frac{\bar{\kappa}_1}{2 \alpha_2} \hspace{0.5mm} T_2 \Big ) \Big ( \exp \Big ( \frac{\bar{\kappa}_1}{2 \alpha_2} \hspace{0.5mm} T_2 \Big ) \Big ( 1 {-} \frac{\bar{\kappa}_2}{2 \alpha_2} \Big ) \Big )^j \\
& \hspace{-15mm} {+} \sqrt{\bar{\kappa}_2} \exp \Big ( \frac{\bar{\kappa}_1}{2 \alpha_2} \hspace{0.5mm} T_2 \Big ) \sqrt{ \frac{\alpha_{\bar{w}_2}}{\alpha_{\bar{w}_1}} \exp{ \Big ( \frac{ \minus \bar{\gamma} \tilde{\beta}}{2 \alpha_{\bar{w}_2}} (t {+} j) \Big )^2 } |\phi_{\bar{w}}(0,0)|^2_{\tilde{\A}_{\varepsilon_r}} } \\
& \hspace{50mm} \forall (t,j) \in \mbox{dom } \phi
\end{aligned}
\end{equation}

\begin{equation}
\begin{aligned}
|\phi(t,j)|_{\tilde{\A}_{\varepsilon}} & \leq \sqrt{\frac{\alpha_2}{\alpha_1}} |\phi(0,0) |_{\tilde{\A}_{\varepsilon}} \exp \Big ( \frac{\bar{\kappa}_1}{2 \alpha_2} \hspace{0.5mm} T_2 \Big ) \Big ( \exp \Big ( \frac{\bar{\kappa}_1}{2 \alpha_2} \hspace{0.5mm} T_2 \Big ) \Big ( 1 {-} \frac{\bar{\kappa}_2}{2 \alpha_2} \Big ) \Big )^j \\
& \hspace{5mm} + \sqrt{\bar{\kappa}_2} \exp \Big ( \frac{\bar{\kappa}_1}{2 \alpha_2} \hspace{0.5mm} T_2 \Big ) \sqrt{ \frac{\alpha_{\bar{w}_2}}{\alpha_{\bar{w}_1}} \exp{ \Big ( \minus \frac{\bar{\gamma} \tilde{\beta}}{2 \alpha_{\bar{w}_2}} (t + j) \Big )^2 } |\phi_{\bar{w}}(0,0)|^2_{\tilde{\A}_{\varepsilon_r}} } \\
& \hspace{90mm} \forall (t,j) \in \mbox{dom } \phi
\end{aligned}
\end{equation}

By the given conditions, the set $\tilde{\A}_{\varepsilon}$ is globally exponentially stable and attractive for $\widetilde{\HS}_{\varepsilon}$.  Now, by utilizing Lemmas <ref> - <ref>, we can establish global exponential stability to the set $\A_{\varepsilon}$ for $\HS_{\varepsilon}$, in turn we can then make use of Lemmas  <ref> - <ref> to then show that the set $\A$ is globally exponentially stable and attractive for $\HS$ in (<ref>).

§ ROBUSTNESS TO COMMUNICATION NOISE, CLOCK DRIFT PERTURBATIONS, AND ERROR ON ${\BF \SIGMA}$

Under a realistic scenario, it is often the case that the system is subjected to disturbances. In this section, we present results on input-to-state stability (ISS) of the system when it is affected by different types of disturbances. First, we present an ISS result that considers communication noise. We then present an ISS result on the parameter estimation sub-system when it is subjected to noise on the internal clock output. Finally, we present an ISS result on noise introduced to the desired clock rate reference $\sigma^*$.confDue to space constraints, the proofs of these results can be found in [22]. We will henceforth refer to the following notion of ISS for hybrid systems.
Under a realistic scenario, it is often the case that the system is subjected to various noise disturbances. Environmental factors can affect the internal clock dynamics and introduce noise to the communication medium in the form of communication delay. In this section we present results on input-to-state stability (ISS) of the system when it is affected by such sources of noise. We will first present an ISS result on the parameter estimation sub-system when it is subjected to noise on the internal clock output, we will then present an ISS result that considers communication noise, last but not least, we will present an ISS result on noise introduced to the desired clock rate reference $\sigma^*$.confDue to space constraints, the proofs of these results can be found in [22]. We will henceforth refer to the following notion of ISS for Hybrid Systems in the presentation of these results, defined as follows:

(Input-to-state stability) A hybrid system $\HS$ with input $m$ is input-to-state stable with respect to  a set $\A \subset \reals^n$  if there exist $\beta \in \mathcal{KL}$ and $\kappa \in \mathcal{K}$ such that each solution pair $(\phi, m)$ to $\HS$ satisfies $|\phi(t,j)|_{\A} \leq \mathrm{max} \{\beta(|\phi(0,0)|_{\A},t+j), \kappa(|m|_{\infty}) \}$ for each $(t,j) \in \mathrm{ dom } \phi$.

§.§ Robustness to Communication Noise

We consider the case when the measurements of the timer $\tilde{\tau}_i$ is affected by noise  $m_{e_i} \in \reals$, $i \in \nodes$.  As a result,  the output of each agent is given by  $\tilde{\tau}_i + m_{e_i}$.  In the presence of this noise,  the update law to $\eta_i^+$ in the hybrid controller in  (<ref>) becomes two_col

\begin{equation*}
\begin{aligned}
\eta_i^+ = -\gamma \sum_{k \in \mathcal{N}(i)} (\tilde{\tau}_i - \tilde{\tau}_k) -\gamma \sum_{k \in \mathcal{N}(i)} (m_{e_i} - m_{e_k})
\end{aligned}
\end{equation*}

\begin{equation*}
\begin{aligned}
\eta_i^+ & = -\gamma \sum_{k \in \mathcal{N}(i)} (y_i - y_k) \\
& = -\gamma \sum_{k \in \mathcal{N}(i)} (\tilde{\tau}_i - \tilde{\tau}_k) -\gamma \sum_{k \in \mathcal{N}(i)} (m_{e_i} - m_{e_k})
\end{aligned}
\end{equation*}

Performing the same change of coordinates, as in the proof of Theorem <ref>, we show that $\widetilde{\HS}_{\varepsilon}$ is ISS to communication noise  $m_e := (m_{e_1}, m_{e_2}, \ldots, m_{e_n} ) \in \reals^n$.  Recalling the change of coordinates $\bar{e} = \Tmat^{-1}e$ and $\bar{\eta} = \Tmat^{-1}\eta$, let $\bar{m}_{e} = \Tmat^{-1} m_{e}$. The update law $\bar{\eta}^+$, is given by $\bar{\eta}^+ = (0, -\gamma \bar{\mathcal{L}} \bar{e} -\gamma \bar{\mathcal{L}} \bar{m}_e)$ with $\bar{\eta}_1$ unaffected by the communication noise.

Using the update law for $\bar{\eta}$ under the effect of $\bar{m}_e$,  we define the perturbed hybrid system $\widetilde{\HS}_m$ with state vector $\chi_m := (\bar{z}_1, \bar{z}_2, \bar{w}_1, \bar{w}_2, \tau) \in \mathcal{X}_{\varepsilon}$, where, again $\bar{z}_1 = (\bar{e}_1, \bar{\eta}_1)$, $\bar{z}_2 = (\bar{e}_2,\ldots,\bar{e}_N, \bar{\eta}_2, \ldots, \bar{\eta}_N)$, $\bar{w}_1 = (\bar{\varepsilon}_{a_1},\bar{\varepsilon}_{\tau_1})$, and $\bar{w}_2 = (\bar{\varepsilon}_{a_2},\ldots, \bar{\varepsilon}_{a_n},\bar{\varepsilon}_{\tau_2},\ldots, \bar{\varepsilon}_{\tau_n})$. Moreover, let $\bar{m}_{\bar{z}_2} = (0, \bar{m}_e)$. The data $( \tilde{C}_m, \tilde{f}_m,  \tilde{D}_m, \tilde{G}_m)$ for the new system $\widetilde{\HS}_m$ is given by

\begin{equation*}
\begin{aligned}
\widetilde{f}_m(\chi_m) & := \widetilde{f}_{\varepsilon}(\chi_m) & & \forall \chi_m \in \widetilde{C}_m \\
\widetilde{G}_m(\chi_m, \bar{m}_{\varepsilon} ) & := \widetilde{G}_{\varepsilon}(\chi_m) - \begin{bmatrix}
0 \\ B_g \bar{m}_{\bar{z}_2} \\ 0 \\ 0 \\ 0
\end{bmatrix} & & \forall \chi_m \in \widetilde{D}_m \\
\end{aligned}
\end{equation*}

where $\widetilde{C}_m := \mathcal{X}_{\varepsilon}$, $\widetilde{D}_m := \{ \chi_m \in \mathcal{X}_m : \tau = 0 \}$, and $B_g = \begin{bmatrix}
0 & \gamma \bar{\mathcal{L}}
\end{bmatrix}^{\top}$.

Given a strongly connected digraph $\DG$, if the parameters $T_2 \geq T_1 > 0$, $\mu > 0$, $h \in \reals$, $\gamma > 0$, and positive definite symmetric matrices $P_1$, $P_2$, and $P_3$ are such that (<ref>) and (<ref>) hold, the hybrid system $\widetilde{\HS}_m$ with input $\bar{m}_e$ is ISS with respect to $\widetilde{\A}_{\varepsilon}$ in (<ref>).

Consider the same Lyapunov function candidate $V(\chi_m) = V_1(\chi_m) + V_2(\chi_m) + V_{\varepsilon_r} (\chi_m)$ from the proof of Theorem <ref>  in Section <ref>.  During flows, there is no contribution from the perturbation thus the derivative of $V$ is unchanged from the proof of Theorem <ref>. Thus, one has  that (<ref>) holds with $\widetilde{f}_{\varepsilon}(\chi_{\varepsilon})$ replaced by $\widetilde{f}_m(\chi_m)$, namely,

\begin{equation*}
\begin{aligned}
\langle \nabla V(\chi_m), \widetilde{f}_m(\chi_m) \rangle & \leq 2 \bar{z}_2^{\top} \big ( \exp (A^{\top}_{f_2} \tau ) P \exp (A_{f_2} \tau) \big ) B_{f_2}  \bar{w}_2 + \bar{w}_1^{\top} (P_1 A_{f_3} + A_{f_3}^{\top} P_1) \bar{w}_1 + \bar{w}_2^{\top} (P_2 A_{f_4} + A_{f_4}^{\top} P_2) \bar{w}_2 \\
\end{aligned}
\end{equation*}

\begin{equation*}
\begin{aligned}
\langle \nabla V(\chi_m), \tilde{f}(\chi_m) \rangle & \leq 2 \bar{z}_2^{\top} \big ( \exp (A^{\top}_{f_2} \tau ) P \exp (A_{f_2} \tau) \big ) B_{f_2}  \bar{w}_2 \\
& \hspace{3mm} + \bar{w}_1^{\top} (P_1 A_{f_3} + A_{f_3}^{\top} P_1) \bar{w}_1 \\
& \hspace{5mm} + \bar{w}_2^{\top} (P_2 A_{f_4} + A_{f_4}^{\top} P_2) \bar{w}_2
\end{aligned}
\end{equation*}

then by following the same notions of the proof in Theorem <ref>, one has
$\langle \nabla V(\chi_m), \widetilde{f}(\chi_m) \rangle \leq \frac{\bar{\kappa}_1}{\alpha_2} V(\chi_m)$  where $\bar{\kappa}_1 = \max \Big \{ \frac{\kappa_1}{2 \epsilon} , \Big ( \frac{\kappa_1 \epsilon}{2} - \beta_2 \Big ) \Big \}$ and $\varepsilon > 0$. At jumps, triggered when $\tau = 0$, one has, for each $\chi_m \in \widetilde{D}_m \setminus \widetilde{\A}_{\varepsilon}$ and $g \in \widetilde{G}_m(\chi_m)$

\begin{equation} \label{thrm_iss_jumps}
\begin{aligned}
V(g) {-} V(\chi_m) & \leq {-} \bar{\eta}_1^2 + (A_{g_2} \bar{z}_2 - B_g \bar{m}_{\bar{z}_2})^{\top} Q (A_{g_2} \bar{z}_2 - B_g \bar{m}_{\bar{z}_2}) - \bar{z}_2^{\top} P_1 \bar{z}_2 \\
& \leq {-} \bar{\eta}_1^2 + (A_{g_2} \bar{z}_2)^{\top} \exp{A^{\top}_{f_2} \tau } P_1 \exp{A_{f_2} \tau} (A_{g_2} \bar{z}_2) - 2 (B_g \bar{m}_{\bar{z}_2})^{\top} \exp{A^{\top}_{f_2} \tau } P_1 \exp{A_{f_2} \tau} (A_{g_2} \bar{z}_2) \\
& \hspace{60mm} + (B_g \bar{m}_{\bar{z}_2})^{\top} \exp{A^{\top}_{f_2} \tau } P_1 \exp{A_{f_2} \tau} (B_g \bar{m}_{\bar{z}_2}) {-}  \bar{z}_2^{\top} P_1 \bar{z}_2 \\
\end{aligned}
\end{equation}

\begin{equation} \label{thrm_iss_jumps}
\begin{aligned}
V(g) {\minus} V(\chi_m)
& \leq \\
& \hspace{-6mm} - \bar{\eta}_1^2 + (A_{g_2} \bar{z}_2)^{\top} \exp (A^{\top}_{f_2} \tau ) P_1 \exp (A_{f_2} \tau) (A_{g_2} \bar{z}_2) \\
& \hspace{-3mm} {-} 2 (B_g \bar{m}_{\bar{z}_2})^{\top} \exp (A^{\top}_{f_2} \tau ) P_1 \exp (A_{f_2} \tau) (A_{g_2} \bar{z}_2) \\
& \hspace{0mm} + (B_g \bar{m}_{\bar{z}_2})^{\top} \exp (A^{\top}_{f_2} \tau ) P_1 \exp (A_{f_2} \tau) (B_g \bar{m}_{\bar{z}_2}) \\
& \hspace{3mm} {-}  \bar{z}_2^{\top} P_1 \bar{z}_2 \\
\end{aligned}
\end{equation}

From (<ref>) and the proof in Theorem <ref>, there exists a scalar $\kappa_2$ such that $\bar{z}_2^{\top} (A_{g_2}^{\top} \exp(A^{\top}_{f_2} v ) P_1 \exp(A_{f_2} v) A_{g_2} - P_1) \bar{z}_2 \leq - \kappa_2 \bar{z}_2^{\top} \bar{z}_2$ leading to

\begin{equation} \label{thrm_iss_jumps1}
\begin{aligned}
V(g) {-} V(\chi_m) & \leq - \bar{\eta}_1^2 -\kappa_2 \bar{z}_2^{\top} \bar{z}_2 - 2 (B_g \bar{m}_{\bar{z}_2})^{\top} \exp{A^{\top}_{f_2} \tau } P_1 \exp{A_{f_2} \tau} (A_{g_2} \bar{z}_2) + (B_g \bar{m}_{\bar{z}_2})^{\top} \exp{A^{\top}_{f_2} \tau } P_1 \exp{A_{f_2} \tau} (B_g \bar{m}_{\bar{z}_2})
\end{aligned}
\end{equation}

\begin{equation} \label{thrm_iss_jumps1}
\begin{aligned}
V(g) {-} V(\chi_m) & \leq - \bar{\eta}_1^2 -\kappa_2 \bar{z}_2^{\top} \bar{z}_2 \\
& \hspace{-4mm} - 2 (B_g \bar{m}_{\bar{z}_2})^{\top} \exp(A^{\top}_{f_2} \tau ) P_1 \exp (A_{f_2} \tau) (A_{g_2} \bar{z}_2) \\
& \hspace{-1mm} + (B_g \bar{m}_{\bar{z}_2})^{\top} \exp (A^{\top}_{f_2} \tau) P_1 \exp(A_{f_2} \tau) (B_g \bar{m}_{\bar{z}_2})
\end{aligned}
\end{equation}

Let $Q = \exp(A^{\top}_{f_2} \tau ) P_1 \exp(A_{f_2} \tau)$, then applying Young's inequality on the third term such that

\begin{equation*}
\begin{aligned}
\bar{m}_{\bar{z}_2}^{\top} B_g^{\top} Q A_{g_2} \bar{z}_2 & \leq \frac{1}{2 \epsilon_2} \Big ( \bar{m}_{\bar{z}_2}^{\top} B_g^{\top} Q A_{g_2} \Big )^{\top} \Big ( \bar{m}_{\bar{z}_2}^{\top} B_g^{\top} Q A_{g_2} \Big ) + \frac{\epsilon_2}{2} \bar{z}_2^{\top} \bar{z}_2 \\
& \leq \frac{1}{2 \epsilon_2} \Big | \big ( B_g^{\top} Q A_{g_2} \big ) \big ( B_g^{\top} Q A_{g_2} \big )^{\top} \Big | \bar{m}_{\bar{z}_2}^{\top} \bar{m}_{\bar{z}_2} + \frac{\epsilon_2}{2} \bar{z}_2^{\top} \bar{z}_2
\end{aligned}
\end{equation*}

\begin{equation*}
\begin{aligned}
\bar{m}_{\bar{z}_2}^{\top} B_g^{\top} Q A_{g_2} \bar{z}_2
& \leq \frac{\big | \big ( B_g^{\top} Q A_{g_2} \big ) \big ( B_g^{\top} Q A_{g_2} \big )^{\top} \big |}{2 \epsilon_2}  \bar{m}_{\bar{z}_2}^{\top} \bar{m}_{\bar{z}_2} {+} \frac{\epsilon_2}{2} \bar{z}_2^{\top} \bar{z}_2
\end{aligned}
\end{equation*}

where $\epsilon_2 > 0$. Then, we have

\begin{equation} \label{thrm_iss_jumps2}
\begin{aligned}
V(g) \minus V(\chi_m) & \leq - \bar{\eta}_1^2 -\kappa_2 \bar{z}_2^{\top} \bar{z}_2 {-} \Big ( {\small \frac{1}{2 \epsilon_2}} \big | \big ( B_g^{\top} Q A_{g_2} \big ) \big ( B_g^{\top} Q A_{g_2} \big )^{\top} \big | \bar{m}_{\bar{z}_2}^{\top} \bar{m}_{\bar{z}_2} + \frac{\epsilon_2}{2} \bar{z}_2^{\top} \bar{z}_2 \Big ) \\
& \hspace{15mm} + \bar{m}_{\bar{z}_2} B_g^{\top} Q B_g \bar{m}_{\bar{z}_2} \\
& \leq - \bar{\eta}_1^2 - \big ( \kappa_2 + \frac{\epsilon_2}{2} \big ) \bar{z}_2^{\top} \bar{z}_2 {+} \big ( | B_g^{\top} Q B_g | \minus \frac{1}{2 \epsilon_2} | ( B_g^{\top} Q A_{g_2} )( B_g^{\top} Q A_{g_2} )^{\top} | \big ) \bar{m}_{\bar{z}_2}^{\top}  \bar{m}_{\bar{z}_2}
\end{aligned}
\end{equation}

\begin{equation} \label{thrm_iss_jumps2}
\begin{aligned}
V(g) \minus V(\chi_m)
& \leq - \bar{\eta}_1^2 - \big ( \kappa_2 + \frac{\epsilon_2}{2} \big ) \bar{z}_2^{\top} \bar{z}_2 {+} \big ( | B_g^{\top} Q B_g | \\
& \hspace{5mm} \minus \frac{1}{2 \epsilon_2} | ( B_g^{\top} Q A_{g_2} )( B_g^{\top} Q A_{g_2} )^{\top} | \big ) \bar{m}_{\bar{z}_2}^{\top}  \bar{m}_{\bar{z}_2}
\end{aligned}
\end{equation}

By noting $|A_{g_2}|, |B_g|$ $\leq \gamma \lambda_{max}({\bar{\mathcal{L}}})$ let $\kappa_{\bar{m}_2} =$ $\big ( \lambda_{max}({\bar{\mathcal{L}}}) \big )^2 \underset{v \in [0,T_2]}{\mathrm{max}} \Big \{ \lambda_{max} \big (\exp (A^{\top}_{f_2} v ) P_1 \exp (A_{f_2} v ) \big ) \Big \}$, we let $\epsilon_2 = \kappa_2$ and obtain

\begin{equation*}
\begin{aligned}
V(g) - V(\chi_m) & \leq - \bar{\eta}_1^2 - \big ( \kappa_2 + \frac{\kappa_2}{2} \big ) \bar{z}_2^{\top} \bar{z}_2 +  \big ( \gamma^2 \kappa_{\bar{m}_2} - \frac{1}{2 \kappa_2} \gamma^4 \kappa_{\bar{m}_2}^2 \big ) \bar{m}_{\bar{z}_2}^{\top}  \bar{m}_{\bar{z}_2}
\end{aligned}
\end{equation*}

\begin{equation*}
\begin{aligned}
V(g) \minus V(\chi_m) {\leq} \minus \bar{\eta}_1^2 \minus \frac{3\kappa_2}{2} \bar{z}_2^{\top} \bar{z}_2 {+}  \big ( \gamma^2 \kappa_{\bar{m}_2} {-} \frac{\gamma^4 \kappa_{\bar{m}_2}^2}{2 \kappa_2}   \big ) \bar{m}_{\bar{z}_2}^{\top}  \bar{m}_{\bar{z}_2}
\end{aligned}
\end{equation*}

Now, let $\tilde{\kappa}_{\bar{m}_2} = \gamma^2 \kappa_{\bar{m}_2} - \frac{1}{2 \kappa_2} \gamma^4 \kappa_{\bar{m}_2}^2$ then at jumps one has

\begin{equation} \label{thrm_iss_jumps3}
\begin{aligned}
V(g) - V(\chi_m) & \leq - \bar{\kappa}_2 (|\bar{\eta}_1|^2 + |\bar{z}_2|^2) + \tilde{\kappa}_{\bar{m}_2} |\bar{m}_{\bar{z}_2}|^2
\end{aligned}
\end{equation}

where $\bar{\kappa}_2 = \max \big \{1, \frac{3 \kappa_2}{2} \big \}$.   Now, recall from (<ref>) in the proof of Theorem <ref> that
two_col$-(|\bar{\eta}_1|^2 + |\bar{z}_2|^2) \leq - \frac{1}{\alpha_2} V(\chi_{\varepsilon}) + |\bar{w}|^2$.

\begin{equation}
\begin{aligned}
-(|\bar{\eta}_1|^2 + |\bar{z}_2|^2) \leq - \frac{1}{\alpha_2} V(\chi_{\varepsilon}) + |\bar{w}|^2
\end{aligned}
\end{equation}

Then, plugging (<ref>) into (<ref>) one has

\begin{equation*}
\begin{aligned}
V(g) & \leq \Big ( 1 - \frac{3 \kappa_2}{2 \alpha_2} \Big ) V(\chi_m) + \frac{3 \kappa_2}{2} |\bar{w}|^2 + \tilde{\kappa}_{\bar{m}_2} |\bar{m}_{\bar{z}_2}|^2
\end{aligned}
\end{equation*}

Noting $\langle \nabla V(\chi_{\varepsilon}), \widetilde{f}(\chi_{\varepsilon}) \rangle  \leq \frac{\bar{\kappa}_1}{\alpha_2} V(\chi_{\varepsilon})$, one can then pick a solution with initial conditions $\tilde{\phi}(0,0) \in \widetilde{C}_m \cup \widetilde{D}_m$ and find that the trajectory of $V(\tilde{\phi}(t,j))$ is bounded as follows:

\begin{equation*}
\begin{aligned}
V(\tilde{\phi}(t,j)) & \leq \\
& \hspace{0mm} \exp \big ( \frac{\bar{\kappa}_1}{\alpha_2} \hspace{0.5mm} T_2 \big ) \Big ( \exp \big ( \frac{\bar{\kappa}_1}{\alpha_2} \hspace{0.5mm} T_2 \big ) \Big ( 1 - \frac{3 \kappa_2}{2 \alpha_2} \Big ) \Big )^j V(\tilde{\phi}(0,0)) \\
& \hspace{5mm} + \frac{3 \kappa_2}{2} \exp \big ( \bar{\kappa} \hspace{0.5mm} T_2 \big ) \mathrm{sup}_{(t,j) \in \mbox{\footnotesize dom} \phi} |\bar{w}(t,j)|^2 \\
& \hspace{10mm} + \tilde{\kappa}_{\bar{m}_2} \exp \Big (\frac{\kappa}{2 \epsilon_2} T_2 \Big ) \mathrm{sup}_{(t,j) \in \mbox{\footnotesize dom} \phi} |\bar{m}_{\bar{z}_2}|^2
\end{aligned}
\end{equation*}

The result follows from an analysis of $V(\tilde{\phi}(t,j))$ over $\mbox{dom } \tilde{\phi}$ utilizing the same approach as in the proof of Theorem <ref>.

Consider the same Lyapunov function candidate $V(\chi_m) = V_1(\chi_m) + V_2(\chi_m) + V_{\varepsilon_r} (\chi_m)$ from the proof of Theorem <ref>. During flows, there is no contribution from the perturbation thus the derivative of $V$ is unchanged from the proof of Theorem <ref>. Thus, one has

\begin{equation*}
\begin{aligned}
\langle \nabla V(\chi_m), \tilde{f}(\chi_m) \rangle & \leq 2 \bar{z}_2^{\top} \big ( \exp{A^{\top}_{f_2} \tau } P \exp{A_{f_2} \tau} \big ) B_{f_2}  \bar{w}_2 + \bar{w}_1^{\top} (P_1 A_{f_3} + A_{f_3}^{\top} P_1) \bar{w}_1 + \bar{w}_2^{\top} (P_2 A_{f_4} + A_{f_4}^{\top} P_2) \bar{w}_2 \\
\end{aligned}
\end{equation*}

\begin{equation*}
\begin{aligned}
\langle \nabla V(\chi_m), \tilde{f}(\chi_m) \rangle & \leq 2 \bar{z}_2^{\top} \big ( \exp{A^{\top}_{f_2} \tau } P \exp{A_{f_2} \tau} \big ) B_{f_2}  \bar{w}_2 \\
& \hspace{3mm} + \bar{w}_1^{\top} (P_1 A_{f_3} + A_{f_3}^{\top} P_1) \bar{w}_1 \\
& \hspace{5mm} + \bar{w}_2^{\top} (P_2 A_{f_4} + A_{f_4}^{\top} P_2) \bar{w}_2
\end{aligned}
\end{equation*}

then by following the same notions of the proof in Theorem <ref>, one has

\begin{equation*}
\begin{aligned}
\langle \nabla V(\chi_m), \tilde{f}(\chi_m) \rangle
& \leq \frac{\kappa_1}{2 \epsilon} | \bar{z}_2
|^2 - \beta_1 | \bar{w}_1|^2 + \Big ( \frac{\kappa_1 \epsilon}{2} - \beta_2 \Big ) | \bar{w}_2|^2 \\
& \leq \bar{\kappa}_1 \big (|\bar{z}_2|^2 + |\bar{w}_1|^2 + |\bar{w}_2|^2 \big ) \\
& \leq \bar{\kappa}_1 V(x)
\end{aligned}
\end{equation*}

where $\bar{\kappa}_1 = \max \Big \{ \frac{\kappa_1}{2 \epsilon} , \Big ( \frac{\kappa_1 \epsilon}{2} - \beta_2 \Big ) \Big \}$ and $\varepsilon > 0$. At jumps, triggered when $\tau = 0$, one has, for each $\chi_m \in \tilde{D}_m \setminus \tilde{\A}_{\varepsilon}$ and $g \in \tilde{G}_m(\chi_m)$

\begin{equation} \label{thrm_iss_jumps}
\begin{aligned}
V(g) {-} V(\chi_m) & \leq {-} \bar{\eta}_1^2 + (A_{g_2} \bar{z}_2 - B_g \bar{m}_{\bar{z}_2})^{\top} Q (A_{g_2} \bar{z}_2 - B_g \bar{m}_{\bar{z}_2}) - \bar{z}_2^{\top} P_1 \bar{z}_2 \\
& \leq {-} \bar{\eta}_1^2 + (A_{g_2} \bar{z}_2)^{\top} \exp{A^{\top}_{f_2} \tau } P_1 \exp{A_{f_2} \tau} (A_{g_2} \bar{z}_2) - 2 (B_g \bar{m}_{\bar{z}_2})^{\top} \exp{A^{\top}_{f_2} \tau } P_1 \exp{A_{f_2} \tau} (A_{g_2} \bar{z}_2) \\
& \hspace{60mm} + (B_g \bar{m}_{\bar{z}_2})^{\top} \exp{A^{\top}_{f_2} \tau } P_1 \exp{A_{f_2} \tau} (B_g \bar{m}_{\bar{z}_2}) {-}  \bar{z}_2^{\top} P_1 \bar{z}_2 \\
\end{aligned}
\end{equation}

\begin{equation} \label{thrm_iss_jumps}
\begin{aligned}
V(g) {\minus} V(\chi_m) & \leq {\minus} \bar{\eta}_1^2 {+} (A_{g_2} \bar{z}_2 {-} B_g \bar{m}_{\bar{z}_2})^{\top} Q (A_{g_2} \bar{z}_2 {-} B_g \bar{m}_{\bar{z}_2}) \\
& \hspace{5mm} - \bar{z}_2^{\top} P_1 \bar{z}_2 \\
& \leq {\minus} \bar{\eta}_1^2 + (A_{g_2} \bar{z}_2)^{\top} \exp{A^{\top}_{f_2} \tau } P_1 \exp{A_{f_2} \tau} (A_{g_2} \bar{z}_2) \\
& \hspace{5mm} {-} 2 (B_g \bar{m}_{\bar{z}_2})^{\top} \exp{A^{\top}_{f_2} \tau } P_1 \exp{A_{f_2} \tau} (A_{g_2} \bar{z}_2) \\
& \hspace{10mm} + (B_g \bar{m}_{\bar{z}_2})^{\top} \exp{A^{\top}_{f_2} \tau } P_1 \exp{A_{f_2} \tau} (B_g \bar{m}_{\bar{z}_2}) \\
& \hspace{15mm} {-}  \bar{z}_2^{\top} P_1 \bar{z}_2 \\
\end{aligned}
\end{equation}

From <ref> and the proof in Theorem <ref>, there exists a scalar $\kappa_2$ such that
$$\bar{z}_2^{\top} (A_{g_2}^{\top} \exp{A^{\top}_{f_2} v } P_1 \exp{A_{f_2} v} A_{g_2} - P_1) \bar{z}_2 \leq - \kappa_2 \bar{z}_2^{\top} \bar{z}_2$$

leading to

\begin{equation} \label{thrm_iss_jumps1}
\begin{aligned}
V(g) {-} V(\chi_m) & \leq - \bar{\eta}_1^2 -\kappa_2 \bar{z}_2^{\top} \bar{z}_2 - 2 (B_g \bar{m}_{\bar{z}_2})^{\top} \exp{A^{\top}_{f_2} \tau } P_1 \exp{A_{f_2} \tau} (A_{g_2} \bar{z}_2) + (B_g \bar{m}_{\bar{z}_2})^{\top} \exp{A^{\top}_{f_2} \tau } P_1 \exp{A_{f_2} \tau} (B_g \bar{m}_{\bar{z}_2})
\end{aligned}
\end{equation}

\begin{equation} \label{thrm_iss_jumps1}
\begin{aligned}
V(g) {-} V(\chi_m) & \leq - \bar{\eta}_1^2 -\kappa_2 \bar{z}_2^{\top} \bar{z}_2 \\
& \hspace{5mm} - 2 (B_g \bar{m}_{\bar{z}_2})^{\top} \exp{A^{\top}_{f_2} \tau } P_1 \exp{A_{f_2} \tau} (A_{g_2} \bar{z}_2) \\
& \hspace{10mm} + (B_g \bar{m}_{\bar{z}_2})^{\top} \exp{A^{\top}_{f_2} \tau } P_1 \exp{A_{f_2} \tau} (B_g \bar{m}_{\bar{z}_2})
\end{aligned}
\end{equation}

Let $Q = \exp{A^{\top}_{f_2} \tau } P_1 \exp{A_{f_2} \tau}$, then applying Young's inequality on the third term such that

\begin{equation*}
\begin{aligned}
\bar{m}_{\bar{z}_2}^{\top} B_g^{\top} Q A_{g_2} \bar{z}_2 & \leq \frac{1}{2 \epsilon_2} \Big ( \bar{m}_{\bar{z}_2}^{\top} B_g^{\top} Q A_{g_2} \Big )^{\top} \Big ( \bar{m}_{\bar{z}_2}^{\top} B_g^{\top} Q A_{g_2} \Big ) + \frac{\epsilon_2}{2} \bar{z}_2^{\top} \bar{z}_2 \\
& \leq \frac{1}{2 \epsilon_2} \Big | \big ( B_g^{\top} Q A_{g_2} \big ) \big ( B_g^{\top} Q A_{g_2} \big )^{\top} \Big | \bar{m}_{\bar{z}_2}^{\top} \bar{m}_{\bar{z}_2} + \frac{\epsilon_2}{2} \bar{z}_2^{\top} \bar{z}_2 \\
\end{aligned}
\end{equation*}

\begin{equation*}
\begin{aligned}
\bar{m}_{\bar{z}_2}^{\top} B_g^{\top} Q A_{g_2} \bar{z}_2 & \leq \frac{1}{2 \epsilon_2} \Big ( \bar{m}_{\bar{z}_2}^{\top} B_g^{\top} Q A_{g_2} \Big )^{\top} \Big ( \bar{m}_{\bar{z}_2}^{\top} B_g^{\top} Q A_{g_2} \Big ) \\
& \hspace{5mm} + \frac{\epsilon_2}{2} \bar{z}_2^{\top} \bar{z}_2 \\
& \leq \frac{1}{2 \epsilon_2} \Big | \big ( B_g^{\top} Q A_{g_2} \big ) \big ( B_g^{\top} Q A_{g_2} \big )^{\top} \Big | \bar{m}_{\bar{z}_2}^{\top} \bar{m}_{\bar{z}_2} \\
& \hspace{5mm} + \frac{\epsilon_2}{2} \bar{z}_2^{\top} \bar{z}_2 \\
\end{aligned}
\end{equation*}

where $\epsilon_2 > 0$, we then have

\begin{equation} \label{thrm_iss_jumps2}
\begin{aligned}
V(g) \minus V(\chi_m) & \leq - \bar{\eta}_1^2 -\kappa_2 \bar{z}_2^{\top} \bar{z}_2 {-} \Big ( {\small \frac{1}{2 \epsilon_2}} \big | \big ( B_g^{\top} Q A_{g_2} \big ) \big ( B_g^{\top} Q A_{g_2} \big )^{\top} \big | \bar{m}_{\bar{z}_2}^{\top} \bar{m}_{\bar{z}_2} + \frac{\epsilon_2}{2} \bar{z}_2^{\top} \bar{z}_2 \Big ) \\
& \hspace{15mm} + \bar{m}_{\bar{z}_2} B_g^{\top} Q B_g \bar{m}_{\bar{z}_2} \\
& \leq - \bar{\eta}_1^2 - \big ( \kappa_2 + \frac{\epsilon_2}{2} \big ) \bar{z}_2^{\top} \bar{z}_2 {+} \big ( | B_g^{\top} Q B_g | \minus \frac{1}{2 \epsilon_2} | ( B_g^{\top} Q A_{g_2} )( B_g^{\top} Q A_{g_2} )^{\top} | \big ) \bar{m}_{\bar{z}_2}^{\top}  \bar{m}_{\bar{z}_2}
\end{aligned}
\end{equation}

\begin{equation} \label{thrm_iss_jumps2}
\begin{aligned}
V(g) \minus V(\chi_m) & \leq - \bar{\eta}_1^2 -\kappa_2 \bar{z}_2^{\top} \bar{z}_2 \\ & \hspace{5mm} {-} \Big ( {\small \frac{1}{2 \epsilon_2}} \big | \big ( B_g^{\top} Q A_{g_2} \big ) \big ( B_g^{\top} Q A_{g_2} \big )^{\top} \big | \bar{m}_{\bar{z}_2}^{\top} \bar{m}_{\bar{z}_2}
\\ & \hspace{10mm} + \frac{\epsilon_2}{2} \bar{z}_2^{\top} \bar{z}_2 \Big ) + \bar{m}_{\bar{z}_2} B_g^{\top} Q B_g \bar{m}_{\bar{z}_2} \\
& \leq - \bar{\eta}_1^2 - \big ( \kappa_2 + \frac{\epsilon_2}{2} \big ) \bar{z}_2^{\top} \bar{z}_2 {+} \big ( | B_g^{\top} Q B_g | \\
& \hspace{5mm} \minus \frac{1}{2 \epsilon_2} | ( B_g^{\top} Q A_{g_2} )( B_g^{\top} Q A_{g_2} )^{\top} | \big ) \bar{m}_{\bar{z}_2}^{\top}  \bar{m}_{\bar{z}_2}
\end{aligned}
\end{equation}

by noting $|A_{g_2}|, |B_g| \leq \gamma \lambda_{max}({\bar{\mathcal{L}}})$ let
$$\kappa_{\bar{m}_2} = \big ( \lambda_{max}({\bar{\mathcal{L}}}) \big )^2 \underset{v \in [0,T_2]}{\mathrm{max}} \Big \{ \lambda_{max} \big (\exp{A^{\top}_{f_2} v } P_1 \exp{A_{f_2} v } \big ) \Big \}$$
then we let $\epsilon_2 = \kappa_2$ and

\begin{equation*}
\begin{aligned}
V(g) - V(\chi_m) & \leq - \bar{\eta}_1^2 - \big ( \kappa_2 + \frac{\kappa_2}{2} \big ) \bar{z}_2^{\top} \bar{z}_2 +  \big ( \gamma^2 \kappa_{\bar{m}_2} - \frac{1}{2 \kappa_2} \gamma^4 \kappa_{\bar{m}_2}^2 \big ) \bar{m}_{\bar{z}_2}^{\top}  \bar{m}_{\bar{z}_2}
\end{aligned}
\end{equation*}

\begin{equation*}
\begin{aligned}
V(g) - V(\chi_m) & \leq - \bar{\eta}_1^2 - \big ( \kappa_2 + \frac{\kappa_2}{2} \big ) \bar{z}_2^{\top} \bar{z}_2 \\
& \hspace{5mm} +  \big ( \gamma^2 \kappa_{\bar{m}_2} - \frac{1}{2 \kappa_2} \gamma^4 \kappa_{\bar{m}_2}^2 \big ) \bar{m}_{\bar{z}_2}^{\top}  \bar{m}_{\bar{z}_2}
\end{aligned}
\end{equation*}

now let $\tilde{\kappa}_{\bar{m}_2} = \big ( \gamma^2 \kappa_{\bar{m}_2} - \frac{1}{2 \kappa_2} \gamma^4 \kappa_{\bar{m}_2}^2 \big )$ then at jumps one has

\begin{equation} \label{thrm_iss_jumps3}
\begin{aligned}
V(g) - V(\chi_m) & \leq - \bar{\kappa}_2 (|\bar{\eta}_1|^2 + |\bar{z}_2|^2) + \tilde{\kappa}_{\bar{m}_2} |\bar{m}_{\bar{z}_2}|^2
\end{aligned}
\end{equation}

where $\bar{\kappa}_2 = \max \big \{1, \frac{3 \kappa_2}{2} \big \}$.   Now, recall from (<ref>) in the proof of Theorem <ref>,

\begin{equation}
\begin{aligned}
-(|\bar{\eta}_1|^2 + |\bar{z}_2|^2) \leq - \frac{1}{\alpha_2} V(\chi_{\varepsilon}) + |\bar{w}|^2
\end{aligned}
\end{equation}

by then plugging (<ref>) in to (<ref>) one has

\begin{equation*}
\begin{aligned}
V(g) - V(\chi_m) & \leq \frac{3 \kappa_2}{2} \Big ( \minus \frac{1}{\alpha_2} V(\chi_{\varepsilon}) + |\bar{w}|^2 \Big ) + \tilde{\kappa}_{\bar{m}_2} |\bar{m}_{\bar{z}_2}|^2 \\
& \leq \minus \frac{3 \kappa_2}{2 \alpha_2} V(\chi_{\varepsilon}) + \frac{3 \kappa_2}{2} |\bar{w}|^2 + \tilde{\kappa}_{\bar{m}_2} |\bar{m}_{\bar{z}_2}|^2
\end{aligned}
\end{equation*}

then at jumps one has

\begin{equation*}
\begin{aligned}
V(g) & \leq \Big ( 1 - \frac{3 \kappa_2}{2 \alpha_2} \Big ) V(\chi_{\varepsilon}) + \frac{3 \kappa_2}{2} |\bar{w}|^2 + \tilde{\kappa}_{\bar{m}_2} |\bar{m}_{\bar{z}_2}|^2
\end{aligned}
\end{equation*}

Noting $\langle \nabla V(\chi_{\varepsilon}), \tilde{f}(\chi_{\varepsilon}) \rangle  \leq \bar{\kappa} V(\chi_{\varepsilon})$, one can then pick a solution with initial conditions $\phi(0,0) \in \tilde{C}_m \cup \tilde{D}_m$ and find the trajectory of $V(t,j)$ is bounded as follows

\begin{equation*}
\begin{aligned}
V(t,j) & \leq \exp \big ( \bar{\kappa} \hspace{0.5mm} T_2 \big ) \Big ( \exp \big ( \bar{\kappa} \hspace{0.5mm} T_2 \big ) \Big ( 1 - \frac{3 \kappa_2}{2 \alpha_2} \Big ) \Big )^j V(0,0) \\
& \hspace{5mm} + \frac{3 \kappa_2}{2} \exp \big ( \bar{\kappa} \hspace{0.5mm} T_2 \big ) \mathrm{sup}_{(t,j) \in \mbox{\footnotesize dom} \phi} |\bar{w}(t,j)|^2 \\
& \hspace{10mm} + \tilde{\kappa}_{\bar{m}_2} \exp \Big (\frac{\kappa}{2 \epsilon_2} T_2 \Big ) \mathrm{sup}_{(t,j) \in \mbox{\footnotesize dom} \phi} |\bar{m}_{\bar{z}_2}|^2
\end{aligned}
\end{equation*}

§.§ Robustness to Perturbations on Internal Clock Drift

In this section, we consider a disturbance $m_{\tau^*_i} \in \reals$, $i \in \nodes$ added to the output of the internal clock. Let $y^{\tau^*}_i := \tau^*_i + m_{\tau^*_i}$, $i \in \nodes$, define the perturbed internal clock output. Then the dynamics of the original estimation system in (<ref>) under this disturbance becomes

\begin{equation}
\begin{aligned}
& \dot{\hat{\tau}}_i = \hat{a}_i {-} (\hat{\tau}_i {-} y^{\tau^*}_i), & & \dot{\hat{a}}_i = \minus \mu ( \hat{\tau}_i {-} y^{\tau^*}_i ) & & \tau \in [0, T_2] \\
& \hat{\tau}^+_i = \hat{\tau}_i, &  &\hat{a}^+_i = \hat{a}_i & & \tau = 0
\end{aligned}
\end{equation}

In error coordinates $\varepsilon_{\hat{a}_i} = a_i - \hat{a}_i$, $\varepsilon_{\tau_i} = \hat{\tau}_i - \tau_i^*$, this leads to

\begin{equation} \label{est_pert}
\begin{aligned}
& \begin{drcases} \dot{\varepsilon}_{\tau_i} = - \varepsilon_{\tau_i} - \varepsilon_{a_i} + m_{\tau^*_i} \\ \dot{\varepsilon}_{\hat{a}_i} = \mu \varepsilon_{\tau_i} - \mu m_{\tau^*_i}  \end{drcases} & & \tau \in [0, T_2] \\
& \begin{drcases} \varepsilon_{\tau_i}^+ = \varepsilon_{\tau_i}, \hspace{2mm} \varepsilon_{a_i}^+ = \varepsilon_{a_i} \hspace{5mm} \end{drcases} & & \tau = 0 \\
\end{aligned}
\end{equation}

Similar to the result presented in Proposition <ref>, for the estimation sub-system we will consider the same reduction $\widetilde{\HS}_{\varepsilon_r}$ that now captures the perturbation. Recall the coordinate transformations $\bar{\varepsilon}_a = \Tmat^{-1}\varepsilon_a$ and $\bar{\varepsilon}_{\tau} = \Tmat^{-1}\varepsilon_{\tau}$ for the respective internal clock and parameter estimation errors. Moreover, recall $\bar{w} = (\bar{w}_1,\bar{w}_2)$ where $\bar{w}_1 = (\bar{\varepsilon}_{a_1},\bar{\varepsilon}_{\tau_1})$ and $\bar{w}_2 = (\bar{\varepsilon}_{a_2},\ldots, \bar{\varepsilon}_{a_n},\bar{\varepsilon}_{\tau_2},\ldots, \bar{\varepsilon}_{\tau_n})$. Let $\bar{m}_{\tau^*} = \Tmat^{-1}m_{\tau^*}$ and $\bar{q} = (\bar{q}_1, \bar{q}_2)$ where $\bar{q}_1 = (\bar{m}_{\tau^*_1}, \bar{m}_{\tau^*_1})$ and $\bar{q}_2 = (\bar{m}_{\tau^*_2}, \ldots, \bar{m}_{\tau^*_n}, \bar{m}_{\tau^*_2}, \ldots, \bar{m}_{\tau^*_n})$. Now, consider the reduced coordinates $\chi_{m_r} := (\bar{w}_1, \bar{w}_2, \tau) \in \reals^n \times \reals^n \times [0,T_2] =: \mathcal{X}_{\varepsilon}$. The data of this reduced system is given by $\widetilde{\HS}_{m_r} = ( \widetilde{C}_{\varepsilon}, \widetilde{f}_{\varepsilon},  \widetilde{D}_{\varepsilon}, \widetilde{G}_{\varepsilon})$ where
\begin{equation*}
\begin{aligned}
\widetilde{f}_{m_r}(\chi_{m_r}, \bar{q}) & := \widetilde{f}_{\varepsilon_r}(\chi_{m_r}) + (
B_{m_1} \bar{q}_1, B_{m_2} \bar{q}_2, 0) & &  \hspace{-0mm} \forall \chi_{m_r} \in \widetilde{C}_{m_r} \\
\widetilde{G}_{m_r}(\chi_{m_r}) & := ( \bar{w}_1, \bar{w}_2, [T_1,T_2]) & & \hspace{-0mm} \forall \chi_{m_r} \in \widetilde{D}_{m_r} \\
\end{aligned}
\end{equation*}

\begin{equation*}
\begin{aligned}
\widetilde{f}_{m r}(\chi_{m r}) & := \begin{bmatrix} A_{f_3} \bar{w}_1 \\ A_{f_4} \bar{w}_2 \\  -1
\end{bmatrix} + \begin{bmatrix}
B_{m_1} \bar{q}_1 \\ B_{m_2} \bar{q}_2 \\ 0
\end{bmatrix} & & \forall \chi_{m r} \in \widetilde{C}_{m r} \\
\widetilde{G}_{m r}(\chi_{m r}) & := \begin{bmatrix}
\bar{w}_1 \\ \bar{w}_2 \\ [T_1,T_2]
\end{bmatrix} & & \forall \chi_{m r} \in \widetilde{D}_{m r} \\
\end{aligned}
\end{equation*}

where $\tilde{C}_{m_r} := \mathcal{X}_{\varepsilon}$, $\tilde{D}_{m_r} := \{ \chi_m \in \mathcal{X}_{\varepsilon} : \tau = 0 \}$, and
$B_{m_1} = (
[\mu, 0], [0, 1] )$, $B_{m_2} = ([
\mu I, 0], [0, I])$

$$B_{m_1} = \begin{bmatrix}
\mu & 0 \\ 0 & 1
\end{bmatrix}, \hspace{1mm} B_{m_2} = \begin{bmatrix}
\mu I & 0 \\ 0 & I
\end{bmatrix}$$

If there exists a positive scalar $\mu$ and positive definite symmetric matrices $P_2$, $P_3$ such that the conditions in (<ref>) hold, the hybrid system $\widetilde{\HS}_{m r}$ with input $\bar{m}_{\tau^*}$ is ISS with respect to $\widetilde{\A}_{\varepsilon_r}$  given in (<ref>).

The proof of this result is established by picking a solution to the model reduction $\widetilde{\HS}_{\varepsilon_r}$, integrating the disturbance that is treated as an input to the system,  and then bounding the integral.  A proof of this result can be found in [22].

If there exists a positive scalar $\mu$ and positive definite symmetric matrices $P_2$, $P_3$ such that (<ref>) and (<ref>) hold, the hybrid system $\widetilde{\HS}_{m r}$ with input $\bar{m}_{\tau^*}$ is ISS with respect to $\widetilde{\A}_{\varepsilon_r}$.

Since the matrices $A_{f_3}$ and $A_{f_4}$ are Hurwitz and the states $\bar{w}_1$ and $\bar{w}_2$ do not jump, we can estimation system as a continuous time system and write the solution explicitly for the states $\bar{w}_1$ and $\bar{w}_2$.

\begin{equation}
\begin{aligned}
\phi_{\bar{w}_1} (t,j) & = \exp(A_{f_3}(t - 0)) \phi_{\bar{w}_1}(0,0) \\
& \hspace{5mm} + \int_0^{t} \exp(A_{f_3}(t - s)) B_{m_1} \bar{q}_1(s) ds
\end{aligned}
\end{equation}

\begin{equation}
\begin{aligned}
\phi_{\bar{w}_2} (t,j) & = \exp(A_{f_4}(t - 0)) \phi_{\bar{w}_2}(0,0) \\
& \hspace{5mm} + \int_0^{t} \exp(A_{f_4}(t - s)) B_{m_2} \bar{q}_2(s) ds
\end{aligned}
\end{equation}