# Robust Parameter Estimation for
Hybrid Dynamical Systems

Ryan S. Johnson<EMAIL_ADDRESS>Stefano Di Cairano<EMAIL_ADDRESS>Ricardo G. Sanfelice<EMAIL_ADDRESS>University of California, Santa Cruz, CA
95064, USA  Mitsubishi Electric Research Laboratories, Cambridge, MA 02139,
USA

###### Abstract

We consider the problem of estimating a vector of unknown constant parameters
for a class of hybrid dynamical systems – that is, systems whose state
variables exhibit both continuous (flow) and discrete (jump) evolution. Using
a hybrid systems framework, we propose a hybrid estimation algorithm that can
operate during both flows and jumps that, under a notion of hybrid persistence
of excitation, guarantees convergence of the parameter estimate to the true
value. Furthermore, we show that the parameter estimate is input-to-state
stable with respect to a class of hybrid disturbances. Simulation results
including a spacecraft application show the merits of our proposed approach.

###### keywords:

robust estimation, estimation theory, identification methods, hybrid dynamical
systems.

[table]capposition=top

, ,

##  1 Introduction

The estimation of unknown parameters in dynamical systems has been an active
research area for years [34]. Parameter estimation algorithms typically rely
on exploiting information about the structure of the system along with the
available input and output signals to compute online an estimate of the
unknown parameters. One of the most popular estimation problems is recursive
linear regression, for which the estimation scheme is often based on the
gradient descent algorithm [20, 34]. For dynamical systems, control strategies
leveraging estimation algorithms, such as model-reference adaptive control,
are used in several engineering applications [11, 10].

More recently, there has been a growing interest in hybrid dynamical systems.
These systems are characterized by state variables that may evolve
continuously (flow) and, at times, evolve discretely (jump) [25]. Hybrid
systems provide new and promising modeling frameworks for a wide range of
applications including robotics, aerospace, automotive, and power systems [30,
7, 24, 18, 8, 5, 9]. However, the hybrid nature of these systems limits the
applicability of existing continuous-time or discrete-time estimation
algorithms.

Much work has been done on parameter estimation and system identification for
specific sub-classes of hybrid systems, such as switched systems [22, 23, 12]
and piecewise-affine systems [3, 1]. However, these systems exhibit nonsmooth
but continuous evolution of the state variables, rather than jumps in the
state variables, hence such results are not applicable to a general class of
hybrid systems. Recently, the work [28] proposed a hybrid estimation algorithm
for linear regression with hybrid signals, and [19] proposed an algorithm for
identification of hybrid systems with linear dynamics. However, to the best of
our knowledge, before our preliminary results related to this work reported in
[16], an algorithm for estimating unknown parameters for a general class of
hybrid systems had not been established in the literature. The goal of this
paper is to fill that gap. Note that this paper focuses only on online
estimation of unknown parameters in the dynamics of hybrid systems.
Simultaneous estimation of the flow and jump maps and the flow and jump sets
of a hybrid system, i.e., the identification of an entire hybrid system, is
still an open problem.

In this paper, we propose a hybrid algorithm for estimating unknown parameters
for a class of hybrid systems with nonlinear dynamics that are affine in the
unknown parameters. We establish sufficient conditions that guarantee
exponential convergence of the parameter estimate to the true value, and we
lower bound the convergence rate of the parameter estimate. The main
contributions in this paper are the following:

  1. 1.

Estimation under hybrid persistence of excitation: in the main stability
result (Theorem 5.1), we establish that our algorithm guarantees exponential
convergence of the parameter estimate to the true value under a notion of
hybrid persistence of excitation (PE) inspired by [28]. To the best of our
knowledge, it is the first hybrid PE sufficient condition that ensures
estimation of parameters for the considered class of hybrid systems.

  2. 2.

Estimator robustness to hybrid disturbances: to prove the main robustness
result (Theorem 6.1), we generalize the error dynamics of our algorithm to a
class of hybrid systems, denoted by ${\mathcal{}H}$, that includes hybrid
disturbances. Lemma B.4 and Theorem 5.3 construct an input-to-state stability
(ISS) Lyapunov function for ${\mathcal{}H}$ by extending ISS results for
continuous-time and discrete-time systems [17, 4, 33, 32, 15, 6].

In [16], we proposed a hybrid algorithm for estimating unknown parameters in a
class of hybrid systems with linear dynamics. We showed that the parameter
estimate converges to the true value if the hybrid regressor satisfies the
classical continuous-time PE condition during flows and the classical
discrete-time PE condition at jumps. In comparison to [16], this paper
considers a wider class of hybrid systems with nonlinear dynamics and proposes
a new hybrid algorithm to solve the estimation problem. Moreover, here we
relax the classical PE conditions imposed in [16], and instead impose a hybrid
PE condition inspired by [28].

The hybrid PE condition is exploited in more recent work involving authors of
this paper and of [28] to establish uniform exponential stability for a
general class of time-varying hybrid dynamical systems [27]. In comparison to
[27], this paper focuses on deriving and analyzing the properties induced by
an algorithm for estimating unknown parameters in a class of hybrid dynamical
systems when the state of the hybrid plant is measured, while [27] allows for
only output measurements. Due to such differences and the different techniques
used, we provide explicit bounds on the convergence rate of the parameter
estimate, and on the estimation error when noise is present whereas [27]
establishes only the existence of such bounds. Our analysis does not impose
completeness of maximal solutions, while the results in [27] rely on
completeness of maximal solutions to ensure well-posedness. The approach to
the proof of exponential stability in this paper differs from that in [27],
specifically, we leverage a property of input-to-state stability with respect
to an exponentially convergent hybrid signal, while the analysis in [27]
relies on a general, but more abstract, result involving uniform observability
properties.

The remainder of this paper is organized as follows. Preliminaries on
continuous-time and discrete-time estimation algorithms are presented in
Section 2. In Section 3, we present a motivational example that highlights the
limitations of these algorithms, that our hybrid algorithm aims at overcoming.
Our algorithm is described in Section 4, and the stability and robustness
properties are analyzed in Section 5 and Section 6, respectively. Simulation
results are in Section 7. Conclusions and future work are in Section 8. For
readability, the proofs of Theorem 5.3, Theorem 5.4, Lemma 5.5, Lemma 5.6, and
Lemma 6.2 are in the Appendix.

##  2 Preliminaries

###  2.1 Notation

We denote the set of real, nonnegative real, and positive real numbers by
$\mathbb{R}$, ${\mathbb{R}_{\geq 0}}$, and ${\mathbb{R}_{>0}}$, respectively.
We denote the set of natural numbers (including zero) as $\mathbb{N}$. The
matrix $I$ denotes the identity matrix of appropriate dimension. The Euclidean
norm of vectors and the associated induced matrix norm are denoted by
$|\cdot|$, and the Frobenius norm is denoted by $|\cdot|_{\mathrm{F}}$. Given
a matrix $A\in\mathbb{R}^{n\times n}$, $\text{\rm eig}(A)$ denotes the set of
all eigenvalues of $A$,
$\lambda_{\min}(A):=\min\\{\lambda/2:\lambda\in\text{\rm eig}(A+A^{\top})\\}$,
and $\lambda_{\max}(A):=\max\\{\lambda/2:\lambda\in\text{\rm
eig}(A+A^{\top})\\}$. For $x,y\in\mathbb{R}^{n}$, we write $[x^{\top}\
\,y^{\top}]^{\top}$ as $(x,y)$. The distance of a point $x$ to a nonempty set
$S$ is $|x|_{S}=\inf_{y\in S}|y-x|$ (the quantity $|\cdot|_{S}$ should not be
confused with the Frobenius norm $|\cdot|_{\mathrm{F}}$ since, besides the
subscript being $S$ in the former, the argument of the former is a vector and
the argument of the latter is a matrix). Given a set-valued mapping
$M:\mathbb{R}^{m}\rightrightarrows\mathbb{R}^{n}$, the domain of $M$ is
$\mathop{\rm dom}\nolimits M=\\{x\in\mathbb{R}^{m}\ :\ M(x)\neq\varnothing\\}$
and the range of $M$ is $\mathop{\rm
rge}M=\\{y\in\mathbb{R}^{n}:\exists\,x\in\mathbb{R}^{m},y\in M(x)\\}$. Given
sets $S,U\subset\mathbb{R}^{n}$, $\mathop{\rm cl}\nolimits(S)$ denotes the
closure of $S$, and $S\setminus U$ denotes set subtraction. Given a measure
space $M$ and a function $f:M\to\mathbb{R}$, the essential supremum of $f$ is
$\operatorname*{ess\,sup}_{m\in M}f(m)=\inf_{c\in\mathbb{R}}\\{|f(m)|\leq c\
\text{\rm for almost all}\ m\in M\\}$. A function $f$ is
${\mathcal{}L}_{\infty}$ ($f\in{\mathcal{}L}_{\infty}$) if
$\operatorname*{ess\,sup}_{m\in M}f(m)$ is finite. Given $x\in\mathbb{R}$, the
exponential is $\mathrm{e}^{x}$ or $\exp(x)$, equivalently.

###  2.2 Review of Parameter Estimation Algorithms

In preparation for our proposed hybrid parameter estimation algorithm, we
review relevant continuous-time and discrete-time estimation algorithms.

  * •

Consider the continuous-time system

$\displaystyle\dot{x}$
$\displaystyle=f_{c}(x,u(t))+\phi_{c}(t)\theta\quad\forall t\geq 0,$ (1)

where $x\in\mathbb{R}^{n}$ is the known state vector, $t\mapsto
u(t)\in\mathbb{R}^{m}$ is the known input,
$t\mapsto\phi_{c}(t)\in\mathbb{R}^{n\times p}$ is the known regressor,
$(x,u)\mapsto f_{c}(x,u)\in\mathbb{R}^{n}$ is a known continuous function,
$\theta\in\mathbb{R}^{p}$ is a vector of unknown constant parameters, and
$n,p,m\in\mathbb{N}$.

To estimate $\theta$, we convert (1) into a form similar to a linear
regression model by introducing [21] the state variables
$\psi\in\mathbb{R}^{n\times p}$ and $\eta\in\mathbb{R}^{n}$ with dynamics

$\displaystyle\dot{\psi}$ $\displaystyle=-\lambda_{c}\psi+\phi_{c}(t)$ (2)
$\displaystyle\dot{\eta}$ $\displaystyle=-\lambda_{c}(x+\eta)-f_{c}(x,u(t)),$

where $\lambda_{c}>0$ is a design parameter. Defining
$\varepsilon:=x+\eta-\psi\theta$ and $y:=x+\eta$, it follows that $\psi$ and
$\varepsilon$ are related via $y=\psi\theta+\varepsilon$. Since $\theta$ is
constant, differentiating $\varepsilon$ along trajectories of (1), (2) yields
$\dot{\varepsilon}=-\lambda_{c}\varepsilon$. Thus, $\varepsilon$ converges
exponentially to zero. Moreover, we have the following equivalences:
$\varepsilon~{}\rightarrow~{}0\iff x+\eta~{}\rightarrow~{}\psi\theta\iff
y~{}\rightarrow~{}\psi\theta.$ Hence, $y$, $\psi$, and $\theta$ are related
via a linear regression model plus an exponentially convergent term.

Denoting the estimate of $\theta$ as $\hat{\theta}$, the gradient algorithm
for $\hat{\theta}$ is [20]

$\displaystyle\dot{\hat{\theta}}$
$\displaystyle=\gamma_{c}\psi^{\top}(y-\psi\hat{\theta}),$ (3)

where $\gamma_{c}>0$ is a design parameter.

  * •

Consider the discrete-time system

$\displaystyle x(j+1)$
$\displaystyle=g_{d}(x(j),u(j))+\phi_{d}(j)\theta\quad\forall j\in\mathbb{N},$

where $x\in\mathbb{R}^{n}$ is the known state vector, $j\mapsto
u(j)\in\mathbb{R}^{m}$ is the known input,
$j\mapsto\phi_{d}(j)\in\mathbb{R}^{n\times p}$ is the known regressor,
$(x,u)\mapsto g_{d}(x,u)\in\mathbb{R}^{n}$ is a known continuous function,
$\theta\in\mathbb{R}^{p}$ is a vector of unknown constant parameters, and
$n,p,m\in\mathbb{N}$.

Using similar reasoning as in the continuous-time case, we estimate $\theta$
using a gradient algorithm as

$\displaystyle\hat{\theta}(j+1)$ $\displaystyle=\hat{\theta}(j)$ (4)
$\displaystyle+\frac{\psi(j+1)^{\top}}{\gamma_{d}+|\psi(j+1)|^{2}}(y(j+1)-\psi(j+1)\hat{\theta}(j)),$

with

$\begin{aligned} \psi(j+1)&=(1-\lambda_{d})\psi(j)+\phi_{d}(j)\\\
\eta(j+1)&=(1-\lambda_{d})(x(j)+\eta(j))-g_{d}(x(j),u(j))\\\
y(j+1)&=x(j+1)+\eta(j+1),\end{aligned}$ (5)

where $\gamma_{d}>0$, $\lambda_{d}\in(0,2)$ are design parameters.

To compute the update law for $\hat{\theta}$ in (4), we require measurements
of $x$ for two consecutive discrete time steps. Moreover, two computational
steps are required to update $\hat{\theta}$ at time $j\in\mathbb{N}$. The
first step computes $\psi(j+1)$, $\eta(j+1)$, and $y(j+1)$ in (5), and the
second step computes $\hat{\theta}(j+1)$ in (4). For simplicity, we omit the
first computational step in (4).

It is shown in [20] that, if $\varepsilon=0$, the following PE condition is
necessary and sufficient for convergence of $\hat{\theta}$ in (3) to $\theta$:

  1. (C1)

The signal $t\mapsto\psi(t)$ is uniformly bounded and there exist $T>0$ and
$\mu>0$ such that

$\displaystyle\int_{t}^{t+T}\psi(s)^{\top}\psi(s)ds\geq\mu I\quad\forall t\geq
0.$ (6)

For the discrete-time case, the PE condition is [34]:

  1. (C2)

The signal $j\mapsto\psi(j)$ is uniformly bounded and there exist
$J\in\mathbb{N}\setminus\\{0\\}$ and $\mu>0$ such that

$\displaystyle\sum_{i=j}^{j+J}\psi(i)^{\top}\psi(i)\geq\mu I\quad\forall
j\in\mathbb{N}.$ (7)

###  2.3 Hybrid Dynamical Systems

In this paper, a hybrid system $\mathcal{H}$ is defined by $(C,F,D,G)$ as [25]

${\mathcal{}H}:\begin{dcases}\begin{aligned} \dot{\xi}&=F(\xi)&\qquad\xi\in
C\\\ \xi^{+}&=G(\xi)&\qquad\xi\in D,\end{aligned}\end{dcases}$ (8)

where $\xi\in\mathbb{R}^{n}$ is the state, $F:C\to\mathbb{R}^{n}$ is the flow
map defining the continuous dynamics, and $C\subset\mathbb{R}^{n}$ defines the
flow set on which flow is permitted. The mapping $G:D\to\mathbb{R}^{n}$ is the
jump map defining the law resetting $\xi$ at jumps, and
$D\subset\mathbb{R}^{n}$ is the jump set on which jumps are permitted.

A solution $\xi$ to $\mathcal{H}$ is a hybrid arc [25] that is parameterized
by $(t,j)\in{\mathbb{R}_{\geq 0}}\times\mathbb{N}$, where $t$ is the elapsed
ordinary time and $j$ is the number of jumps that have occurred. The domain of
$\xi$, denoted $\mathop{\rm dom}\nolimits\xi\subset{\mathbb{R}_{\geq
0}}\times\mathbb{N}$, is a hybrid time domain, in the sense that for every
$(T,J)\in\mathop{\rm dom}\nolimits\xi$, there exists a nondecreasing sequence
$\\{t_{j}\\}_{j=0}^{J+1}$ with $t_{0}=0$ such that $\mathop{\rm
dom}\nolimits\xi\cap\left([0,T]\times\\{0,1,\dots,J\\}\right)=\bigcup_{j=0}^{J}\left([t_{j},t_{j+1}]\times\\{j\\}\right)$.
A solution $\xi$ to $\mathcal{H}$ is said to be

  * •

nontrivial if $\mathop{\rm dom}\nolimits\xi$ contains more than one point;

  * •

continuous if nontrivial and $\mathop{\rm
dom}\nolimits\xi\subset{\mathbb{R}_{\geq 0}}\times\\{0\\}$;

  * •

discrete if nontrivial and $\mathop{\rm
dom}\nolimits\xi\subset\\{0\\}\times\mathbb{N}$.

A solution $\xi$ to $\mathcal{H}$ is called maximal if it cannot be extended –
that is, if there does not exist another solution $\xi^{\prime}$ to
$\mathcal{H}$ such that $\mathop{\rm dom}\nolimits\xi$ is a proper subset of
$\mathop{\rm dom}\nolimits\xi^{\prime}$ and $\xi(t,j)=\xi^{\prime}(t,j)$ for
all $(t,j)\in\mathop{\rm dom}\nolimits\xi$. A solution is called complete if
its domain is unbounded. The operations $\sup_{t}\mathop{\rm dom}\nolimits\xi$
and $\sup_{j}\mathop{\rm dom}\nolimits\xi$ return the supremum of the $t$ and
$j$ coordinates, respectively, of points in $\mathop{\rm dom}\nolimits\xi$.
The length of $\mathop{\rm dom}\nolimits\xi$ is $\sup_{t}\mathop{\rm
dom}\nolimits\xi+\sup_{j}\mathop{\rm dom}\nolimits\xi$.

We employ the following notion of stability [25]. definition

######  Definition 2.1.

Given a hybrid system $\mathcal{H}$ with data as in (8), a nonempty closed set
${\mathcal{}A}\subset\mathbb{R}^{n}$ is said to be semiglobally pre-
exponentially stable111The term “pre-exponential,” as opposed to
“exponential,” indicates the possibility of a maximal solution that is not
complete. This allows for separating the conditions for completeness from the
conditions for stability and attractivity. for $\mathcal{H}$ if, for each
compact ${\mathcal{}X}_{0}\subset\mathbb{R}^{n}$, there exist
$\kappa,\lambda>0$ such that each solution $\xi$ to $\mathcal{H}$ from
$\xi(0,0)\in{\mathcal{}X}_{0}$ satisfies

$|\xi(t,j)|_{\mathcal{}A}\leq\kappa\mathrm{e}^{-\lambda(t+j)}|\xi(0,0)|_{\mathcal{}A}\
\ \forall(t,j)\\!\in\\!\mathop{\rm dom}\nolimits\xi.$ (9)

If there exist $\kappa,\lambda>0$ such that each solution $\xi$ to
$\mathcal{H}$ satisfies (9), then ${\mathcal{}A}$ is said to be globally pre-
exponentially stable for $\mathcal{H}$.

Given a hybrid arc $(t,j)\mapsto\xi(t,j)\in\mathbb{R}^{n}$, we denote the
supremum norm of $\xi$ from $(0,0)$ to $(t,j)$ as

$\|\xi\|_{(t,j)}:=\max\left\\{\mbox{\scriptsize$\displaystyle\begin{aligned}
\underset{\begin{subarray}{c}(s,k)\,\in\,\mathop{\rm
dom}\nolimits\xi\setminus\Upsilon(\mathop{\rm dom}\nolimits\xi),\\\
(s,k)\,\leq\,(t,j)\end{subarray}}{\operatorname*{ess\,sup}}|\xi(s,k)|,\underset{\begin{subarray}{c}(s,k)\,\in\,\Upsilon(\mathop{\rm
dom}\nolimits\xi),\\\
(s,k)\,\leq\,(t,j)\end{subarray}}{\sup}|\xi(s,k)|\end{aligned}$}\right\\}$

where

$\Upsilon(\mathop{\rm dom}\nolimits\xi):=\\{(t,j)\in\mathop{\rm
dom}\nolimits\xi:(t,j+1)\in\mathop{\rm dom}\nolimits\xi\\}.$ (10)

##  3 Motivational Example

To motivate our parameter estimation algorithm, consider the hybrid arcs
$\phi_{c},\phi_{d}:E\to\mathbb{R}^{2\times 2}$ with hybrid time domains
$E=\bigcup_{k=0}^{\infty}\big{(}[2\pi k,\ \pi(2k+2)]\times\\{k\\}\big{)}$. The
values of $\phi_{c}$ and $\phi_{d}$ are
$\phi_{c}(t,j)=\begin{bmatrix}\sin(t)&0\\\ 0&0\end{bmatrix}$ and
$\phi_{d}(t,j)=\begin{bmatrix}1&2\\\ 2&4\end{bmatrix}$ for all $(t,j)\in E$.
For such $\phi_{c}$ and $\phi_{d}$, consider a hybrid system as in (8) with an
added input222See [25] for details on hybrid systems with inputs.
$u:E\to\mathbb{R}$, state $x=(x_{1},x_{2})\in\mathbb{R}^{2}$, and dynamics

$\displaystyle\dot{x}$ $\displaystyle=\phi_{c}(t,j)\theta$
$\displaystyle\quad(x,u(t,j))$ $\displaystyle\in C_{P}$ (11) $\displaystyle
x^{+}$ $\displaystyle=\phi_{d}(t,j)\theta$ $\displaystyle\quad(x,u(t,j))$
$\displaystyle\in D_{P},$

where $\theta=[1\,\;1]^{\top}$ is a vector of unknown parameters. The flow and
jump sets are $C_{P}=(\mathbb{R}^{2}\times\mathbb{R})\setminus D_{p}$ and
$D_{P}=\\{(x,u)\in\mathbb{R}^{2}\times\mathbb{R}:u\geq 2\pi\\}$, respectively.
The input $u(t,j)=t-2\pi j$ for all $(t,j)\in E$ is a sawtooth function that
periodically ramps to a value of $2\pi$ and then resets to zero.333With
$C_{P}$, $D_{P}$, and $u$ given below (11), the hybrid time domain of each
maximal solution to the hybrid system in (11) is equal to the hybrid time
domain $E$ of $\phi_{c}$ and $\phi_{d}$.

Given a solution $x$ to (11) from $x(0,0)=(3,6)$, we want to estimate
$\theta$. To do so, we first separately analyze the flows and jumps of these
signals. We define the continuous-time signals
$t\mapsto\bar{x}_{c}(t):=\begin{bmatrix}4-\cos(t)\\\ 6\end{bmatrix}$ and
$t\mapsto\bar{\phi}_{c}(t):=\begin{bmatrix}\sin(t)&0\\\ 0&0\end{bmatrix}$,
which are obtained by neglecting the resets of $x$ and $\phi_{c}$,
respectively, at jumps. The signals $\bar{x}_{c}$ and $\bar{\phi}_{c}$ are
solutions to the continuous-time system
$\dot{\bar{x}}_{c}(t)=\bar{\phi}_{c}(t)\theta$ for all $t\geq 0$. Next, we
define the discrete-time signals $j\mapsto\bar{x}_{d}(j):=\begin{bmatrix}3\\\
6\end{bmatrix}$ and $j\mapsto\bar{\phi}_{d}(j):=\begin{bmatrix}1&2\\\
2&4\end{bmatrix}$, which are obtained by neglecting the evolution of $x$ and
$\phi_{d}$, respectively, during flows. The signals $\bar{x}_{d}$ and
$\bar{\phi}_{d}$ are solutions to the discrete-time system
${\bar{x}}_{d}(j+1)=\bar{\phi}_{d}(j)\theta$ for all $j\in\mathbb{N}$. Using
the transformations in Section 2.2, we employ the continuous-time and
discrete-time algorithms (3) and (4) to estimate $\theta$ in (11). The
parameter estimation error for both algorithms fails to converge to zero, as
shown in Figure 1.444Code at
https://github.com/HybridSystemsLab/HybridGD_Motivation

To see why the continuous-time algorithm fails to estimate $\theta$, note that
for $\bar{\phi}_{c}$, the value of $t\mapsto\psi(t)$ in (2) is
$\psi(t)=\mathrm{e}^{-\lambda_{c}t}\psi(0)+\int_{0}^{t}\mathrm{e}^{-\lambda_{c}(t-s)}\bar{\phi}_{c}(s)ds$
for all $t\geq 0$. Since $\mathrm{e}^{-\lambda_{c}t}\psi(0)$ converges
exponentially to zero and the second column of $\bar{\phi}_{c}(t)$ is zero for
all $t\geq 0$, $t\mapsto\psi(t)$ does not satisfy (C1) for any $T>0$.
Similarly for $\bar{\phi}_{d}$, the value of $j\mapsto\psi(j)$ in (5) is
$\psi(j)=(1-\lambda_{d})^{j}\psi(0)+\sum_{i=0}^{j-1}(1-\lambda_{d})^{(j-i-1)}\bar{\phi}_{d}(i)$
for all $j\in\mathbb{N}$. Since $(1-\lambda_{d})^{j}\psi(0)$ converges
exponentially to zero and $\bar{\phi}_{d}(j)$ is constant and singular for all
$j\in\mathbb{N}$, $j\mapsto\psi(j)$ does not satisfy (C2) for any
$J\in\mathbb{N}\setminus\\{0\\}$. On the other hand, the hybrid algorithm
proposed in this paper successfully estimates $\theta$ by leveraging the
information available during both flows and jumps, as shown in Figure 1.

Figure 1: The projection onto $t$ of the norm of the parameter estimation
error for the continuous-time and discrete-time estimation algorithms, and our
hybrid algorithm. The continuous-time and discrete-time algorithms produce
nonzero steady-state error, whereas the error for our algorithm converges to
zero.

##  4 A Hybrid Parameter Estimation Algorithm

###  4.1 Problem Statement

Motivated by the limitations of the continuous-time and discrete-time
estimation algorithms highlighted in Section 3, we develop a hybrid algorithm
for estimating parameters in hybrid dynamical systems of the form

$\displaystyle\dot{x}$ $\displaystyle=f_{c}(x,u(t,j))+\phi_{c}(t,j)\theta$
$\displaystyle(x,u(t,j))$ $\displaystyle\in C_{P}$ (12) $\displaystyle x^{+}$
$\displaystyle=g_{d}(x,u(t,j))+\phi_{d}(t,j)\theta$ $\displaystyle(x,u(t,j))$
$\displaystyle\in D_{P},$

where $x\in\mathbb{R}^{n}$ is the known state vector and $(x,u)\mapsto
f_{c}(x,u)\in\mathbb{R}^{n}$ and $(x,u)\mapsto g_{d}(x,u)\in\mathbb{R}^{n}$
are known continuous functions. The regressors
$(t,j)\mapsto\phi_{c}(t,j)\in\mathbb{R}^{n\times p}$ and
$(t,j)\mapsto\phi_{d}(t,j)\in\mathbb{R}^{n\times p}$ and the input
$(t,j)\mapsto u(t,j)\in\mathbb{R}^{m}$ are known, and are defined on hybrid
time domains as described in Section 2.3, but are not necessarily hybrid
arcs.555In other words, $\phi_{c}$, $\phi_{d}$, and $u$ do not need to be
locally absolutely continuous during flows – see [25] for details. The flow
set is $C_{P}\subset\mathbb{R}^{n}\times\mathbb{R}^{m}$, the jump set is
$D_{P}\subset\mathbb{R}^{n}\times\mathbb{R}^{m}$, and $n,p,m\in\mathbb{N}$.
Note that $\phi_{c}$ plays no role in the dynamics of (12) at jumps and
$\phi_{d}$ plays no role in the dynamics of (12) during flows.

Our goal is to estimate the parameter vector $\theta$ in (12). Since
$\phi_{c}$ and $\phi_{d}$ may exhibit both flows and jumps, it is important to
update the parameter estimate $\hat{\theta}$ continuously whenever $\phi_{c}$
flows, and discretely each time $\phi_{d}$ jumps, which is possible when jumps
are detected instantaneously. Hence, we propose to estimate $\theta$ using a
hybrid algorithm, denoted $\mathcal{H}_{g}$, of the form

$\mathcal{H}_{g}:\left\\{\begin{aligned} \dot{\xi}&=F_{g}(\xi)\qquad&\xi&\in
C_{g}\\\ \xi^{+}&=G_{g}(\xi)\qquad&\xi&\in D_{g},\end{aligned}\right.$ (13)

with data designed to solve the following problem.

Problem Statement: Design the data $(C_{g},F_{g},D_{g},G_{g})$ of
$\mathcal{H}_{g}$ in (13) and determine conditions on $\phi_{c}$ and
$\phi_{d}$ that ensure the parameter estimate $\hat{\theta}$ converges to the
unknown parameter vector $\theta$ in (12).

Next, we present our solution to this problem.

###  4.2 Problem Solution

Given $\phi_{c},\phi_{d}:E\to\mathbb{R}^{n\times p}$ and
$u:E\to\mathbb{R}^{m}$, where $E:=\mathop{\rm
dom}\nolimits\phi_{c}=\mathop{\rm dom}\nolimits\phi_{d}=\mathop{\rm
dom}\nolimits u$ is a hybrid time domain, we define the state $\xi$ of
$\mathcal{H}_{g}$ as
$\xi:=(x,\hat{\theta},\psi,\eta,\tau,k)\in\mathcal{X}_{g}:=\mathbb{R}^{n}\times\mathbb{R}^{p}\times\mathbb{R}^{n\times
p}\times\mathbb{R}^{n}\times E$, where $x$ is the state of the plant in (12),
$\hat{\theta}$ is the estimate of $\theta$, and $\psi,\eta$ are auxiliary
state variables. The state components $\tau$ and $k$ have dynamics such that
they evolve as $t$ and $j$, respectively, from the hybrid time domain $E$.
Including $\tau$ and $k$ in $\xi$ allows $\phi_{c}$, $\phi_{d}$, and $u$ to be
part of the definitions of $F_{g}$ and $G_{g}$, rather than modeled as inputs
to $\mathcal{H}_{g}$. Thus, we can express $\mathcal{H}_{g}$ as an autonomous
hybrid system, which allows us to leverage recent results on stability and
robustness properties [13, 25].

During flows, we update $\hat{\theta}$ with dynamics inspired by the
continuous-time algorithm in (2) and (3),

$\displaystyle\dot{\hat{\theta}}$
$\displaystyle=\gamma_{c}\psi^{\top}(y-\psi\hat{\theta}),$

where $y:=x+\eta$, with $\psi$ and $\eta$ generated by
$\dot{\psi}=-\lambda_{c}\psi+\phi_{c}(\tau,k)$ and
$\dot{\eta}=-\lambda_{c}(x+\eta)-f_{c}(x,u(\tau,k))$, and
$\gamma_{c},\lambda_{c}>0$ are design parameters. Hence, for all $\xi\in
C_{g}$, the flow map for $\mathcal{H}_{g}$ in (13) is

$\displaystyle F_{g}(\xi)$
$\displaystyle:=\begin{bmatrix}f_{c}(x,u(\tau,k))+\phi_{c}(\tau,k)\theta\\\
\gamma_{c}\psi^{\top}(y-\psi\hat{\theta})\\\
-\lambda_{c}\psi+\phi_{c}(\tau,k)\\\
-\lambda_{c}(x+\eta)-f_{c}(x,u(\tau,k))\\\ 1\\\ 0\end{bmatrix}.$

At jumps, we update $\hat{\theta}$ using a reset map inspired by the discrete-
time algorithm in (4) and (5),

$\hat{\theta}^{+}=\hat{\theta}+\frac{\psi^{+\top}}{\gamma_{d}+|\psi^{+}|^{2}}(y^{+}-\psi^{+}\hat{\theta}),$

with $\psi^{+}:=(1-\lambda_{d})\psi+\phi_{d}(\tau,k)$,
$\eta^{+}:=(1-\lambda_{d})(x+\eta)-g_{d}(x,u(\tau,k))$, and
$y^{+}:=x^{+}+\eta^{+}$, where $x^{+}$ gives the plant state $x$ after a jump
per (12), and $\gamma_{d}>0$, $\lambda_{d}\in(0,2)$ are design parameters.
Hence, for all $\xi\in D_{g}$, the jump map for $\mathcal{H}_{g}$ in (13) is

$\displaystyle G_{g}(\xi)$
$\displaystyle:=\begin{bmatrix}g_{d}(x,u(\tau,k))+\phi_{d}(\tau,k)\theta\\\
\hat{\theta}+\frac{\psi^{+\top}}{\gamma_{d}+|\psi^{+}|^{2}}(y^{+}-\psi^{+}\hat{\theta})\\\
(1-\lambda_{d})\psi+\phi_{d}(\tau,k)\\\
(1-\lambda_{d})(x+\eta)-g_{d}(x,u(\tau,k))\\\ \tau\\\ k+1\end{bmatrix}.$ (14)

The flow and jump sets of $\mathcal{H}_{g}$ are defined so that the algorithm
flows when $\phi_{c}$ flows, and jumps when $\phi_{d}$ jumps. Since
$\mathop{\rm dom}\nolimits\phi_{c}=\mathop{\rm dom}\nolimits\phi_{d}=E$,

$\begin{aligned} C_{g}&:=\mathop{\rm
cl}\nolimits\left({\mathcal{}X}_{g}\setminus
D_{g}\right),&D_{g}&:=\\{\xi\in{\mathcal{}X}_{g}:(\tau,k+1)\in
E\\}.\end{aligned}$ (15)

######  Remark 4.1.

We assume for simplicity that the plant state $x$ has the same hybrid time
domain as $\phi_{c}$, $\phi_{d}$, and $u$. As a result, the flow set $C_{P}$
and jump set $D_{P}$ of the plant are not part of the construction of
$\mathcal{H}_{g}$. Our algorithm can be extended to the case where $x$,
$\phi_{c}$, $\phi_{d}$, and $u$ have different hybrid time domains by
considering the flow and jump sets in (12). In this case, we can
reparameterize the domains of $\phi_{c}$, $\phi_{d}$, and $u$ to express $x$,
$\phi_{c}$, $\phi_{d}$, and $u$ on a common hybrid time domain. See, e.g.,
[2].

######  Remark 4.2.

For simplicity, the hybrid algorithm ${\mathcal{}H}_{g}$ in (13) is expressed
such that jumps in the parameter estimate coincide with jumps in $x$. This
results in ${\mathcal{}H}_{g}$ being noncausal since measurements of $x^{+}$
are not available until after a jump. We can remove the simplification at the
price of letting the algorithm jump twice for each jump in $x$, as follows.
Immediately before a jump in $x$, the algorithm jumps once to reset the values
of $\psi$, $\eta$, and $k$ per the jump map (14). Immediately after a jump in
$x$, the algorithm jumps a second time to update the parameter estimate using
the current value of $x$. A logic variable ensures that, after the second
jump, the algorithm flows or jumps in accordance with the hybrid time domain
$E$. Since $\theta$ in (12) is constant, the stability properties induced by
$\mathcal{H}_{g}$ in (13) are equivalent to the stability properties induced
by the causal modification, after we reparameterize the domain of solutions to
$\mathcal{H}_{g}$ to match the domain of solutions to the causal system.
Hence, for simplicity, we focus our analysis on (13).

##  5 Stability Analysis

We now establish our main stability result stating conditions that ensure the
hybrid system $\mathcal{H}_{g}$ in (13) induces semiglobal pre-exponential
stability666 Since each solution $\mathcal{H}_{g}$ inherits the hybrid time
domain of $\phi_{c}$, $\phi_{d}$, and $u$, the use of “pre-exponential,” as
opposed to “exponential,” stability means that $\phi_{c}$, $\phi_{d}$, and $u$
do not need to be complete.  of the set

$\displaystyle{\mathcal{}A}_{g}:=\left\\{\xi\in{\mathcal{}X}_{g}\ :\
\hat{\theta}=\theta,\;\varepsilon=0\ \right\\},$ (16)

where

$\displaystyle\varepsilon:=x+\eta-\psi\theta.$ (17)

Semiglobal pre-exponential stability of ${\mathcal{}A}_{g}$ implies that,
given any compact set of initial conditions, for each solution $\xi$ to
$\mathcal{H}_{g}$ from such compact set, the distance from $\xi$ to the set
${\mathcal{}A}_{g}$ is bounded above by an exponentially decreasing function
of the initial condition – see Definition 2.1. As a consequence, for each
complete solution $\xi$ to $\mathcal{H}_{g}$, the parameter estimate
$\hat{\theta}$ converges exponentially to $\theta$, and $\varepsilon$
converges exponentially to zero.

######  Theorem 5.1.

Given the hybrid system $\mathcal{H}_{g}$ in (13),
$\gamma_{c},\lambda_{c},\gamma_{d}>0$, and $\lambda_{d}\in(0,2)$, suppose that
$\phi_{c},\phi_{d}:E\to\mathbb{R}^{n\times p}$ satisfy the following:

  1. 1.

There exists $\phi_{M}\in{\mathbb{R}_{>0}}$ such that
$|\phi_{c}(t,j)|_{\mathrm{F}}\leq\phi_{M}$ for all $(t,j)\in E$ and
$|\phi_{d}(t,j)|_{\mathrm{F}}\leq\phi_{M}$ for all $(t,j)\in\Upsilon(E)$.

  2. 2.

There exist $\Delta,\mu\in{\mathbb{R}_{>0}}$ such that, for all
$(t^{\prime},j^{\prime}),(t^{*},j^{*})\in E$ satisfying777 The hybrid time
instants $(t^{\prime},j^{\prime})$ and $(t^{*},j^{*})$ are the beginning and
the end, respectively, of a hybrid time interval with length satisfying (18),
over which (19) holds.

$\displaystyle\Delta\leq(t^{*}-t^{\prime})+(j^{*}-j^{\prime})<\Delta+1,$ (18)

the following hybrid PE condition holds:

$\displaystyle\sum_{j=j^{\prime}}^{j^{*}}\int_{\max\\{t^{\prime},t_{j}\\}}^{\min\\{t^{*},t_{j+1}\\}}\psi(s,j)^{\top}\psi(s,j)ds$
(19)
$\displaystyle+\sum_{j=j^{\prime}}^{j^{*}-1}\psi(t_{j+1},j+1)^{\top}\psi(t_{j+1},j+1)\geq\mu
I$

where $\\{t_{j}\\}_{j=0}^{J}$ is the sequence defining $E$ as in Section 2.3,
$t_{J+1}:=T$, with $J:=\sup_{j}E$ and $T:=\sup_{t}E$, and
$(t,j)\mapsto\psi(t,j)$ is generated by (13).

Then, for each $\psi_{0}\geq 0$, $q_{M}\geq q_{m}>0$, and each
$\zeta\in(0,1)$, there exist $\kappa_{g},\lambda_{g}>0$ such that each
solution $\xi$ to $\mathcal{H}_{g}$ from
$\xi(0,0)\in{\mathcal{}X}_{0}:=\\{\xi\in{\mathcal{}X}_{g}:|\psi|_{\mathrm{F}}\leq\psi_{0}\\}$
satisfies

$\displaystyle|\xi(t,j)|_{{\mathcal{}A}_{g}}\leq\kappa_{g}\mathrm{e}^{-\lambda_{g}(t+j)}|\xi(0,0)|_{{\mathcal{}A}_{g}}$
(20)

for all $(t,j)\in\mathop{\rm dom}\nolimits\xi$. In particular, suitable
choices of $\kappa_{g}$ and $\lambda_{g}$ are given in Appendix A.

Theorem 5.1 states that, if $|\phi_{c}|_{\mathrm{F}}$ and
$|\phi_{d}|_{\mathrm{F}}$ are uniformly bounded above and the hybrid PE
condition (19) is satisfied, then the set ${\mathcal{}A}_{g}$ in (16) is
semiglobally pre-exponentially stable for $\mathcal{H}_{g}$. The hybrid PE
condition (19) reduces to the continuous-time PE condition (6) if $\psi$ is
continuous, and reduces to the discrete-time PE condition (7) if $\psi$ is
discrete. Hence, in such cases, we recover the results established in [20,
34].888 In fact, if $\psi$ is continuous and $\phi_{c}$,
$\frac{d}{dt}\phi_{c}\in{\mathcal{}L}_{\infty}$, then it follows from [29,
Lemma 2.6.7] that the $\psi$ component of each solution $\xi$ to
$\mathcal{H}_{g}$ from ${\mathcal{}X}_{0}$ is PE as in (C1) if $\phi_{c}$ is
PE. Given such $\phi_{c}$, the excitation parameters for $\psi$ – $\mu$ and
$T$ in (C1) – depend on the initial condition of $\psi$. However, since
$\xi(0,0)\in{\mathcal{}X}_{0}$, the initial condition of $\psi$ lies in a
compact set, and therefore we can find these parameters independent of the
initial condition. If $\psi$ is discrete, then a similar persistence of
excitation property holds for $\psi$ if $\phi_{d}$ is PE as in (C2).

###  5.1 Proof of Theorem 5.1

The proof of Theorem 5.1 proceeds as follows. In Section 5.1.1, we generalize
the error dynamics of ${\mathcal{H}}_{g}$ to a class of hybrid systems,
denoted by $\mathcal{H}$. Section 5.1.2 establishes conditions on the data of
$\mathcal{H}$ that ensure global pre-exponential stability of a closed set for
$\mathcal{H}$. Then, in Section 5.1.3, we show that the conditions of Theorem
5.1 are sufficient to ensure that ${\mathcal{H}}_{g}$ satisfies the conditions
imposed on $\mathcal{H}$ in Section 5.1.2. Under such conditions,
${\mathcal{H}}_{g}$ inherits the stability properties of ${\mathcal{}H}$.

####  5.1.1 A General Class of Hybrid Gradient Algorithms

Convergence to $\theta$ for the solution component $\hat{\theta}$ of
$\mathcal{H}_{g}$ in (13) is achieved when the parameter estimation error
$\tilde{\theta}=\theta-\hat{\theta}$ of $\mathcal{H}_{g}$ converges to zero.
We denote the hybrid system resulting from expressing $\mathcal{H}_{g}$ in
error coordinates as $\widetilde{\mathcal{H}}_{g}$, with state
$\xi=(x,\tilde{\theta},\psi,\eta,\tau,k)\in{\mathcal{}X}_{g}$ and dynamics

$\widetilde{\mathcal{H}}_{g}:\left\\{\begin{aligned}
\dot{\xi}&=\widetilde{F}_{g}(\xi)\qquad&\xi&\in\widetilde{C}_{g}\\\
\xi^{+}&=\widetilde{G}_{g}(\xi)\qquad&\xi&\in\widetilde{D}_{g},\end{aligned}\right.$
(21)

where $\widetilde{C}_{g}:=C_{g}$ and $\widetilde{D}_{g}:=D_{g}$, with $C_{g}$,
$D_{g}$ in (15), and

$\displaystyle\widetilde{F}_{g}(\xi)$
$\displaystyle:=\begin{bmatrix}f_{c}(x,u(\tau,k))+\phi_{c}(\tau,k)\theta\\\
-\gamma_{c}\psi^{\top}\psi\tilde{\theta}-\gamma_{c}\psi^{\top}\varepsilon\\\
-\lambda_{c}\psi+\phi_{c}(\tau,k)\\\
-\lambda_{c}(x+\eta)-f_{c}(x,u(\tau,k))\\\ 1\\\
0\end{bmatrix}\quad\forall\xi\in\widetilde{C}_{g}$
$\displaystyle\widetilde{G}_{g}(\xi)$
$\displaystyle:=\begin{bmatrix}g_{d}(x,u(\tau,k))+\phi_{d}(\tau,k)\theta\\\
\tilde{\theta}-\frac{\psi^{+\top}\psi^{+}}{\gamma_{d}+|\psi^{+}|^{2}}\tilde{\theta}-\frac{\psi^{+\top}}{\gamma_{d}+|\psi^{+}|^{2}}\varepsilon^{+}\\\
(1-\lambda_{d})\psi+\phi_{d}(\tau,k)\\\
(1-\lambda_{d})(x+\eta)-g_{d}(x,u(\tau,k))\\\ \tau\\\
k+1\end{bmatrix}\quad\forall\xi\in\widetilde{D}_{g},$

with $\varepsilon$ as in (17) and
$\varepsilon^{+}:=x^{+}+\eta^{+}-\psi^{+}\theta$, where $x^{+}$ gives the
plant state $x$ after a jump per (12).

To analyze the stability properties induced by $\widetilde{\mathcal{H}}_{g}$,
we use that $\widetilde{\mathcal{H}}_{g}$ in (21) belongs to a class of hybrid
systems, denoted by $\mathcal{H}$, with state
$\xi=(\vartheta,\tau,k)\in{\mathcal{}X}:=\mathbb{R}^{p}\times E$ and dynamics

$\displaystyle\dot{\xi}$
$\displaystyle=\begin{bmatrix}-A(\tau,k)\vartheta+d_{c}(\tau,k)\\\ 1\\\
0\end{bmatrix}=:F(\xi)$ $\displaystyle\xi$ $\displaystyle\in C$ (22)
$\displaystyle\xi^{+}$
$\displaystyle=\begin{bmatrix}\vartheta-B(\tau,k)\vartheta+d_{d}(\tau,k)\\\
\tau\\\ k+1\end{bmatrix}=:G(\xi)$ $\displaystyle\xi$ $\displaystyle\in D$

where $A,B:E\to\mathbb{R}^{p\times p}$ and $d_{c},d_{d}:E\to\mathbb{R}^{p}$
are given and $E:=\mathop{\rm dom}\nolimits A=\mathop{\rm dom}\nolimits
B=\mathop{\rm dom}\nolimits d_{c}=\mathop{\rm dom}\nolimits d_{d}$ is a hybrid
time domain, $C:=\mathop{\rm cl}\nolimits\left({\mathcal{}X}\setminus
D\right)$, and $D:=\\{\xi\in{\mathcal{}X}:(\tau,k+1)\in E\\}$.

######  Remark 5.2.

The hybrid system $\mathcal{H}$ in (22) reduces to
$\widetilde{\mathcal{H}}_{g}$ in (21) when $\vartheta=\tilde{\theta}$,

$\displaystyle A(\tau,k)$
$\displaystyle=\gamma_{c}\psi(\tau,k)^{\top}\psi(\tau,k)$ (23a) $\displaystyle
d_{c}(\tau,k)$
$\displaystyle=-\gamma_{c}\psi(\tau,k)^{\top}\varepsilon(\tau,k)$ (23b) for
all $(\tau,k)\in E$, and99footnotemark: 9 $\displaystyle B(\tau,k)$
$\displaystyle=\frac{\psi(\tau,k+1)^{\top}\psi(\tau,k+1)}{\gamma_{d}+|\psi(\tau,k+1)|^{2}}$
(23c) $\displaystyle d_{d}(\tau,k)$
$\displaystyle=-\frac{\psi(\tau,k+1)^{\top}}{\gamma_{d}+|\psi(\tau,k+1)|^{2}}\varepsilon(\tau,k+1)$
(23d)

for all $(\tau,k)\in\Upsilon(E)$, with $\Upsilon$ as in (10), where
$\varepsilon=x+\eta-\psi\theta$ is a hybrid disturbance and $x,\eta,\psi$
satisfy the dynamics in (21).

We impose on $A$ and $B$ the following structural properties, which are
similar to those imposed in the design of continuous-time and discrete-time
gradient algorithms.

######  Assumption 5.1.

Given $A,B:E\to\mathbb{R}^{p\times p}$, where $E:=\mathop{\rm dom}\nolimits
A=\mathop{\rm dom}\nolimits B$ is a hybrid time domain,

  1. 1.

$A(t,j)=A(t,j)^{\top}\geq 0$ for all $(t,j)\in E$;

  2. 2.

$B(t,j)=B(t,j)^{\top}\geq 0$ for all $(t,j)\in\Upsilon(E)$;

  3. 3.

there exists $a_{M}>0$ such that $\text{\rm ess sup }\\{|A(t,j)|:(t,j)\in
E\\}\leq a_{M}$;

  4. 4.

$|B(t,j)|<1$ for all $(t,j)\in\Upsilon(E)$.

We impose the following hybrid PE condition [28].

######  Assumption 5.2.

Given $A,B:E\to\mathbb{R}^{p\times p}$, where $E:=\mathop{\rm dom}\nolimits
A=\mathop{\rm dom}\nolimits B$ is a hybrid time domain, there exist
$\Delta,\mu_{0}\in{\mathbb{R}_{>0}}$ such that, for each
$(t^{\prime},j^{\prime}),(t^{*},j^{*})\in E$ satisfying
$\Delta\leq(t^{*}-t^{\prime})+(j^{*}-j^{\prime})<\Delta+1$, the following
holds:

$\begin{aligned}
\sum_{j=j^{\prime}}^{j^{*}}\int_{\max\\{t^{\prime},t_{j}\\}}^{\min\\{t^{*},t_{j+1}\\}}A(s,j)ds+\frac{1}{2}\sum_{j=j^{\prime}}^{j^{*}-1}B(t_{j+1},j)\geq\mu_{0}I\end{aligned}$
(24)

where $t_{J+1}:=T$, with $J:=\sup_{j}E$ and $T:=\sup_{t}E$.

####  5.1.2 Stability Analysis for $\mathcal{H}$

In this section, we establish sufficient conditions on $A,B,d_{c},$ and
$d_{d}$ that ensure the hybrid system $\mathcal{H}$ induces global pre-
exponential stability of the set

$\displaystyle{\mathcal{}A}:=\left\\{\xi\in{\mathcal{}X}\ :\ \vartheta=0\
\right\\}.$ (25)

We first establish the following ISS result for $\mathcal{H}$.

######  Theorem 5.3.

Given the hybrid system $\mathcal{H}$ in (22), let Assumptions 5.1 and 5.2
hold. Then, for each $q_{M}\geq q_{m}>0$ and each $\zeta\in(0,1)$, each
solution $\xi$ to $\mathcal{H}$ satisfies

$\displaystyle|\xi(t,j)|_{\mathcal{}A}\leq\beta(|\xi(0,0)|_{\mathcal{}A},t+j)+\rho\|d\|_{(t,j)}$
(26)

for all $(t,j)\in\mathop{\rm dom}\nolimits\xi$, where

$\displaystyle d(t,j)$ $\displaystyle:=\left\\{\begin{aligned}
&d_{c}(t,j)&&\text{\rm if}&(t,j)&\in E\setminus\Upsilon(E)\\\
&d_{d}(t,j)&&\text{\rm if}&(t,j)&\in\Upsilon(E)\end{aligned}\right.$ (27)

and

$\displaystyle\beta(s,r)$
$\displaystyle:=\sqrt{\frac{p_{M}}{p_{m}}}\mathrm{e}^{-\omega
r}s,\quad\rho:=\sqrt{\frac{2p_{M}^{3}}{q_{m}p_{m}\zeta}\left(\frac{2p_{M}}{q_{m}}+1\right)}$
$\displaystyle\omega$
$\displaystyle:=\mbox{$\displaystyle\frac{1}{2}\min\left\\{\frac{q_{m}}{2p_{M}}(1-\zeta),\,-\ln\left(1-\frac{q_{m}}{2p_{M}}(1-\zeta)\right)\right\\},$}$
$\displaystyle p_{m}$ $\displaystyle:=q_{m},\qquad\quad\ \
p_{M}:=q_{m}+\frac{q_{M}\kappa_{0}^{2}}{2\lambda_{0}}+\frac{q_{M}\kappa_{0}^{2}\mathrm{e}^{2\lambda_{0}}}{\mathrm{e}^{2\lambda_{0}}-1},$
$\displaystyle\kappa_{0}$ $\displaystyle:=\sqrt{\frac{1}{1-\sigma}},\qquad\
\lambda_{0}:=-\frac{\ln(1-\sigma)}{2(\Delta+1)},$ $\displaystyle\sigma$
$\displaystyle:=\mbox{$\displaystyle\frac{2\mu_{0}}{\big{(}1+\sqrt{(a_{M}+2)(\Delta+2)^{3}(a_{M}(\Delta+2)+1/2)}\big{)}^{2}},$}$

with $\Upsilon$ defined in (10) and $a_{M},\mu_{0},\Delta$ from Assumptions
5.1 and 5.2.

Motivated by the fact that, for each complete solution to ${\mathcal{H}}_{g}$
in (13), the signal $(t,j)\mapsto\varepsilon(t,j)$ in (23) converges
exponentially to zero, we use Theorem 5.3 to establish the stability
properties induced by $\mathcal{H}$ when $d_{c}$ and $d_{d}$ converge
exponentially to zero.

######  Theorem 5.4.

Given the hybrid system $\mathcal{H}$ in (22), suppose that Assumptions 5.1
and 5.2 hold, and that there exist $a,b>0$ such that $d$ in (27) satisfies

$\displaystyle|d(t,j)|\leq a\mathrm{e}^{-b(t+j)}|d(0,0)|$ (28)

for all $(t,j)\in E$. Then, for each $q_{M}\geq q_{m}>0$ and each
$\zeta\in(0,1)$, each solution $\xi$ to $\mathcal{H}$ satisfies

$\displaystyle|\xi(t,j)|_{\mathcal{}A}\leq\kappa\mathrm{e}^{-\lambda(t+j)}\big{(}|\xi(0,0)|_{\mathcal{}A}+|d(0,0)|\big{)}$
(29)

for all $(t,j)\in\mathop{\rm dom}\nolimits\xi$, where

$\displaystyle\kappa$
$\displaystyle:=2\max\left\\{\dfrac{p_{M}}{p_{m}},\,a\rho\sqrt{\dfrac{p_{M}}{p_{m}}}\right\\},$
$\displaystyle\lambda$
$\displaystyle:=\frac{1}{2}\min\left\\{\omega,b\right\\}$ (30)

with $p_{m},p_{M},\rho,$ and $\omega$ from Theorem 5.3.

We use Theorem 5.4 in the next section to prove the stability properties
induced by our algorithm.

####  5.1.3 Stability Analysis for $\mathcal{H}_{g}$

To prove Theorem 5.1, we require the following results for the error dynamics
of our algorithm.

######  Lemma 5.5.

Given the hybrid system $\widetilde{\mathcal{H}}_{g}$ in (21), for each
$\lambda_{c}>0$, $\lambda_{d}\in(0,2)$, and each solution
$\xi=(x,\tilde{\theta},\psi,\eta,\tau,k)$ to $\widetilde{\mathcal{H}}_{g}$,
$(t,j)\mapsto\varepsilon(t,j):=x(t,j)+\eta(t,j)-\psi(t,j)\theta$ in (17)
satisfies

$\displaystyle|\varepsilon(t,j)|\leq\mathrm{e}^{-b(t+j)}|\varepsilon(0,0)|\quad\forall(t,j)\in\mathop{\rm
dom}\nolimits\xi,$ (31)

where $b:=\frac{1}{2}\min\left\\{2\lambda_{c},\
-\ln\left(1-\lambda_{d}(2-\lambda_{d})\right)\right\\}$.

######  Lemma 5.6.

Given the hybrid system $\widetilde{\mathcal{H}}_{g}$ in (21), suppose that
$\phi_{c}:E\to\mathbb{R}^{n\times p}$ satisfies item 1 of Theorem 5.1 and let
$\phi_{M}>0$ come from that item. Then, for each $\psi_{0}\geq 0$,
$\lambda_{c}>0$, $\lambda_{d}\in(0,2)$, the $\psi$ component of each solution
$\xi$ to $\widetilde{\mathcal{H}}_{g}$ from
$\xi(0,0)\in{\mathcal{}X}_{0}:=\\{\xi\in{\mathcal{}X}_{g}:|\psi|_{\mathrm{F}}\leq\psi_{0}\\}$
satisfies

$\displaystyle|\psi(t,j)|$
$\displaystyle\leq\psi_{M}\quad\forall(t,j)\in\mathop{\rm dom}\nolimits\xi,$
(32)

where
$\psi_{M}:=\psi_{0}+\max\left\\{\frac{1}{\lambda_{c}},\frac{\sqrt{2\lambda_{d}(2-\lambda_{d})+16}}{\lambda_{d}(2-\lambda_{d})}\right\\}\phi_{M}$.

We now have all the ingredients to prove Theorem 5.1.

######  Proof 5.7 ().

Proof of Theorem 5.1: To prove Theorem 5.1, we show that the error dynamics of
$\mathcal{H}_{g}$ – that is, $\widetilde{\mathcal{H}}_{g}$ in (21) – satisfy
the conditions of Theorem 5.4 with $A,B,d_{c},d_{d}$ in (23). Beginning with
Assumption 5.1, since $A$, $B$ in (23) are symmetric and
$\gamma_{c},\gamma_{d}>0$, it follows that they are positive semidefinite.
Hence, items 1 and 2 of Assumption 5.1 holds. Next, we show that items 3 and 4
of Assumption 5.1 hold. Since, by item 1 in Theorem 5.1, the conditions of
Lemma 5.6 are satisfied, it follows that, for each solution $\xi$ to
$\widetilde{\mathcal{H}}_{g}$ from ${\mathcal{}X}_{0}$, the $\psi$ component
of $\xi$ satisfies (32). Thus,
$|A(t,j)|\leq\gamma_{c}|\psi(t,j)|^{2}\leq\gamma_{c}\psi_{M}^{2}$ for all
$(t,j)\in E$, with $\psi_{M}$ from Lemma 5.6, and
$|B(t,j)|=\frac{|\psi(t,j+1)|^{2}}{\gamma_{d}+|\psi(t,j)|^{2}}<1$ for all
$(t,j)\in\Upsilon(E)$. Hence, items 3 and 4 of Assumption 5.1 hold with
$a_{M}:=\gamma_{c}\psi_{M}^{2}$.

Next, using Lemma 5.6 and item 2 in Theorem 5.1, we show that Assumption 5.2
holds with $A$, $B$ in (23). Substituting $A$, $B$ into (24), we have that,
for all $(t^{\prime},j^{\prime}),(t^{*},j^{*})\in E$ satisfying (18),

$\displaystyle\sum_{j=j^{\prime}}^{j^{*}}\int_{\max\\{t^{\prime},t_{j}\\}}^{\min\\{t^{*},t_{j+1}\\}}\gamma_{c}\psi(s,j)^{\top}\psi(s,j)ds$
$\displaystyle\quad+\frac{1}{2}\sum_{j=j^{\prime}}^{j^{*}-1}\frac{\psi(t_{j+1},j+1)^{\top}\psi(t_{j+1},j+1)}{\gamma_{d}+|\psi(t_{j+1},j+1)|^{2}}$
$\displaystyle\geq\min\left\\{\gamma_{c},\frac{1}{2(\gamma_{d}+\psi_{M}^{2})}\right\\}\mu
I.$

Hence, Assumption 5.2 holds with $\Delta$ from item 2 of Theorem 5.1 and
$\mu_{0}:=\min\left\\{\gamma_{c},\frac{1}{2(\gamma_{d}+\psi_{M}^{2})}\right\\}\mu$.

Finally, we show that (28) is satisfied with $d$ in (27) and $d_{c},d_{d}$ in
(23). By item 1 of Theorem 5.1, it follows from Lemmas 5.5 and 5.6 that, for
each solution $\xi$ to $\widetilde{\mathcal{H}}_{g}$ from ${\mathcal{}X}_{0}$,
$|d_{c}(t,j)|\leq\gamma_{c}|\psi(t,j)||\varepsilon(t,j)|\leq\gamma_{c}\psi_{M}\mathrm{e}^{-b(t+j)}|\varepsilon(0,0)|$
for all $(t,j)\in\mathop{\rm dom}\nolimits\xi$, with $\psi_{M}$ from Lemma 5.6
and $b$ from Lemma 5.5. Furthermore, using that
$\frac{|\psi(t,j+1)|}{\gamma_{d}+|\psi(t,j+1)|^{2}}\leq\frac{1}{2\sqrt{\gamma_{d}}}$
for all $(t,j)\in\Upsilon(\mathop{\rm dom}\nolimits\xi)$, we have that
$|d_{d}(t,j)|\leq\frac{1}{2\sqrt{\gamma_{d}}}|\varepsilon(t,j+1)|\leq\frac{1}{2\sqrt{\gamma_{d}}}\mathrm{e}^{-b(t+j+1)}|\varepsilon(0,0)|\leq\frac{1}{2\sqrt{\gamma_{d}}}\mathrm{e}^{-b(t+j)}|\varepsilon(0,0)|$
for all $(t,j)\in\Upsilon(\mathop{\rm dom}\nolimits\xi)$. Thus, we conclude
that (28) holds with
$a:=\max\left\\{\gamma_{c}\psi_{M},\frac{1}{2\sqrt{\gamma_{d}}}\right\\}$, $b$
from Lemma 5.5, and $|d(0,0)|=|\varepsilon(0,0)|$. Hence, the conditions of
Theorem 5.4 hold and, from the equivalence between the data of
$\widetilde{\mathcal{H}}_{g}$ in (21) and $\mathcal{H}$ in (22) with
$A,B,d_{c},d_{d}$ in (23),101010 In other words, by substituting
$A,B,d_{c},d_{d}$ in (23) into (22) and treating $\psi$ as a given hybrid
signal and $\varepsilon$ as hybrid disturbance satisfying (21), we obtain a
hybrid system with dynamics that are equivalent to
$\widetilde{\mathcal{H}}_{g}$ in (21).  we have from Theorem 5.4 that the
$\tilde{\theta}$ component of each solution $\xi$ to
$\widetilde{\mathcal{H}}_{g}$ from ${\mathcal{}X}_{0}$ satisfies

$\displaystyle|\tilde{\theta}(t,j)|\leq\kappa\mathrm{e}^{-\lambda(t+j)}\big{(}|\tilde{\theta}(0,0)|+|\varepsilon(0,0)|\big{)}$
(33)

for all $(t,j)\in\mathop{\rm dom}\nolimits\xi$, with $\kappa,\lambda$ in (30).

To conclude the proof, using the definition of ${\mathcal{}A}_{g}$ in (16), we
rewrite $|\xi|_{{\mathcal{}A}_{g}}$ for all $(t,j)\in\mathop{\rm
dom}\nolimits\xi$ as
$|\xi(t,j)|_{{\mathcal{}A}_{g}}=\sqrt{|\tilde{\theta}(t,j)|^{2}+|\varepsilon(t,j)|^{2}}$.
Substituting the bounds in (31) and (33) and using that $\kappa\geq 1$ and,
for any $\alpha,\beta\in\mathbb{R}$,
$\alpha\beta\leq\frac{1}{2}(\alpha^{2}+\beta^{2})$, we conclude that, for all
$(t,j)\in\mathop{\rm dom}\nolimits\xi$,
$|\xi(t,j)|_{{\mathcal{}A}_{g}}\leq\sqrt{3}\kappa\mathrm{e}^{-\min\\{\lambda,b\\}(t+j)}|\xi(0,0)|_{{\mathcal{}A}_{g}}$.

##  6 Robustness Analysis

In this section, we study the robustness properties induced by
$\mathcal{H}_{g}$ with respect to bounded (hybrid) noise on the state
measurements.

Given $\phi_{c},\phi_{d}:E\to\mathbb{R}^{n\times p}$ and
$u:E\to\mathbb{R}^{m}$, where $E:=\mathop{\rm
dom}\nolimits\phi_{c}=\mathop{\rm dom}\nolimits\phi_{d}=\mathop{\rm
dom}\nolimits u$ is a hybrid time domain, consider additive noise
$\nu:E\to\mathbb{R}^{n}$ in the measurements of the plant state $x$ in
(12).111111 For simplicity, we assume that the measurement noise $\nu$ has the
same hybrid time domain as $x$, $\phi_{c}$, $\phi_{d}$, and $u$.  We denote
the hybrid system $\mathcal{H}$ in (13) under the effect of the measurement
noise $\nu$ as $\mathcal{H}_{\nu}$, with state
$\xi=(x,\hat{\theta},\psi,\eta,\tau,k)\in{\mathcal{}X}_{g}$ and dynamics

${\mathcal{H}}_{\nu}:\left\\{\begin{aligned}
\dot{\xi}&={F}_{\nu}(\xi)\qquad&\xi&\in{C}_{\nu}\\\
\xi^{+}&={G}_{\nu}(\xi)\qquad&\xi&\in{D}_{\nu}\end{aligned}\right.$ (34)

where

$\begin{aligned}
{F}_{\nu}(\xi)&:=\begin{bmatrix}f_{c}(x,u(\tau,k))+\phi_{c}(\tau,k)\theta\\\
\gamma_{c}\psi^{\top}(y_{\nu}-\psi\hat{\theta})\\\
-\lambda_{c}\psi+\phi_{c}(\tau,k)\\\
-\lambda_{c}(x+\nu(\tau,k)+\eta)-f_{c}(x+\nu(\tau,k),u(\tau,k))\\\ 1\\\
0\end{bmatrix}\\\
{G}_{\nu}(\xi)&:=\begin{bmatrix}g_{d}(x,u(\tau,k))+\phi_{d}(\tau,k)\theta\\\
\hat{\theta}+\frac{\psi^{+\top}}{\gamma_{d}+|\psi^{+}|^{2}}(y_{\nu}^{+}-\psi^{+}\hat{\theta})\\\
(1-\lambda_{d})\psi+\phi_{d}(\tau,k)\\\
\mbox{$\displaystyle(1-\lambda_{d})(x+\nu(\tau,k)+\eta)-g_{d}(x+\nu(\tau,k),u(\tau,k))$}\\\
\tau\\\ k+1\end{bmatrix}\end{aligned}$

where ${C}_{\nu}:=C_{g}$, ${D}_{\nu}:=D_{g}$, with $C_{g},D_{g}$ in (15), and
we define $y_{\nu}:=x+\nu(\tau,k)+\eta$ and
$y_{\nu}^{+}:=x^{+}+\nu(\tau,k+1)+\eta^{+}$, where $x^{+}$ gives the plant
state $x$ after a jump per (12).

For analyzing the effect of the noise, we make the following Lipschitz
continuity assumption.

######  Assumption 6.1.

Given the hybrid plant in (12), there exist $L_{c},L_{d}>0$ such that, for all
$x_{1},x_{2}\in\mathbb{R}^{n}$ and all $u\in\mathbb{R}^{m}$,

$\displaystyle|f_{c}(x_{1},u)-f_{c}(x_{2},u)|$ $\displaystyle\leq
L_{c}|x_{1}-x_{2}|,$ $\displaystyle|g_{d}(x_{1},u)-g_{d}(x_{2},u)|$
$\displaystyle\leq L_{d}|x_{1}-x_{2}|.$

We now establish our main robustness result stating conditions that ensure
${\mathcal{}A}_{g}$ in (16) is ISS for ${\mathcal{H}}_{\nu}$.

######  Theorem 6.1.

Given the hybrid system ${\mathcal{H}}_{\nu}$ in (34),
$\gamma_{c},\lambda_{c},\gamma_{d}>0$, and $\lambda_{d}\in(0,2)$, suppose that
Assumption 6.1 holds and that $\phi_{c},\phi_{d}:E\to\mathbb{R}^{n\times p}$
satisfy items 1 and 2 of Theorem 5.1. Then, for each $\psi_{0}\geq 0$,
$q_{M}\geq q_{m}>0$, and each $\zeta\in(0,1)$, each solution $\xi$ to
${\mathcal{H}}_{\nu}$ from $\xi(0,0)\in{\mathcal{}X}_{0}$ satisfies

$\displaystyle|\xi(t,j)|_{{\mathcal{}A}_{g}}\leq\kappa_{\nu}\mathrm{e}^{-\lambda_{\nu}(t+j)}|\xi(0,0)|_{{\mathcal{}A}_{g}}+\rho_{\nu}d_{\nu}(t,j)$
(35)

for all $(t,j)\in\mathop{\rm dom}\nolimits\xi$, where

$\displaystyle\kappa_{\nu}$ $\displaystyle:=\sqrt{\frac{2p_{M}}{p_{m}}},\ \
\lambda_{\nu}:=\min\\{\omega,\lambda_{\varepsilon}\\},\ \
\rho_{\nu}:=\sqrt{2}\max\\{\rho,\rho_{\varepsilon}\\}$
$\displaystyle\lambda_{\varepsilon}$
$\displaystyle:=\frac{1}{2}\min\left\\{\lambda_{c}(1-\zeta),-\ln\left(1-\frac{\lambda_{d}}{2}(2-\lambda_{d})(1-\zeta)\right)\right\\}$
$\displaystyle\rho_{\varepsilon}$
$\displaystyle:=\max\left\\{\frac{2}{\lambda_{c}\sqrt{\zeta}},\frac{\sqrt{2\lambda_{d}(2-\lambda_{d})+16}}{\lambda_{d}(2-\lambda_{d})\sqrt{\zeta}}\right\\}$

with $\rho$, $\omega$ from Theorem 5.3, $p_{m},p_{M}$ from Theorem 5.1,
$d_{\nu}(t,j):=\sqrt{\|d\|_{(t,j)}^{2}+\|d_{\varepsilon}\|_{(t,j)}^{2}}$, with
$d$ as in (27) and

$\displaystyle d_{\varepsilon}(t,j)$ $\displaystyle:=\left\\{\begin{aligned}
&\alpha_{c}(t,j)&&\text{\rm if}&(t,j)&\in E\setminus\Upsilon(E)\\\
&\alpha_{d}(t,j)&&\text{\rm if}&(t,j)&\in\Upsilon(E),\end{aligned}\right.$
(36)

where, for all $(t,j)\in E$,

$\displaystyle d_{c}(t,j)$
$\displaystyle:=-\gamma_{c}\psi(t,j)^{\top}(\varepsilon(t,j)+\nu(t,j))$ (37a)
$\displaystyle\alpha_{c}(t,j)$
$\displaystyle:=-\lambda_{c}\nu(t,j)+f_{c}(x(t,j),u(t,j))$ (37b)
$\displaystyle\qquad-f_{c}(x(t,j)+\nu(t,j),u(t,j))$ and, for all
$(t,j)\in\Upsilon(E)$, $\displaystyle d_{d}(t,j)$
$\displaystyle:=-\frac{\psi(t,j+1)^{\top}}{\gamma_{d}+|\psi(t,j+1)|^{2}}\big{(}\varepsilon(t,j+1)$
(37c) $\displaystyle\qquad+\nu(t,j+1)\big{)},$ $\displaystyle\alpha_{d}(t,j)$
$\displaystyle:=(1-\lambda_{d})\nu(t,j)+g_{d}(x(t,j),u(t,j))$ (37d)
$\displaystyle\qquad-g_{d}(x(t,j)+\nu(t,j),u(t,j))$

with $\varepsilon$ as in (17). Moreover, for all $(t,j)\in E$,

$\displaystyle|d_{c}(t,j)|$
$\displaystyle\leq\gamma_{c}\psi_{M}\big{(}\mathrm{e}^{-\lambda_{\varepsilon}(t+j)}|\varepsilon(0,0)|$
$+(\rho_{\varepsilon}\max\\{\lambda_{c}+L_{c},1-\lambda_{d}+L_{d}\\}+1)|\nu(t,j)|\big{)}$
$\displaystyle|\alpha_{c}(t,j)|$
$\displaystyle\leq(\lambda_{c}+L_{c})|\nu(t,j)|$ and for all
$(t,j)\in\Upsilon(E)$, $\displaystyle|d_{d}(t,j)|$
$\displaystyle\leq\frac{1}{2\sqrt{\gamma_{d}}}\big{(}\mathrm{e}^{-\lambda_{\varepsilon}(t+j+1)}|\varepsilon(0,0)|$
$+(\rho_{\varepsilon}\max\\{\lambda_{c}+L_{c},1-\lambda_{d}+L_{d}\\}+1)|\nu(t,j+1)|\big{)}$
$\displaystyle|\alpha_{d}(t,j)|$
$\displaystyle\leq(1-\lambda_{d}+L_{d})|\nu(t,j)|$

with $\psi_{M}$ from Theorem 5.1, $L_{c}$, $L_{d}$ from Assumption 6.1, and
$\varepsilon(0,0)=x(0,0)+\eta(0,0)-\psi(0,0)\theta$.

To prove Theorem 6.1, we require the following result.

######  Lemma 6.2.

Given the hybrid system ${\mathcal{H}}_{\nu}$ in (34), suppose that Assumption
6.1 holds. Then, for each $\lambda_{c}>0$, $\lambda_{d}\in(0,2)$,
$\zeta\in(0,1)$, and each solution $\xi=(x,\hat{\theta},\psi,\eta,\tau,k)$ to
${\mathcal{H}}_{\nu}$,
$(t,j)~{}\mapsto~{}\varepsilon(t,j):=x(t,j)+\eta(t,j)-\psi(t,j)\theta$ in (17)
satisfies, for all $(t,j)\in\mathop{\rm dom}\nolimits\xi$,

$\displaystyle|\varepsilon(t,j)|\leq\mathrm{e}^{-\lambda_{\varepsilon}(t+j)}|\varepsilon(0,0)|+\rho_{\varepsilon}\|d_{\varepsilon}\|_{(t,j)}$
(38)

with $\lambda_{\varepsilon},\rho_{\varepsilon}>0$ and $(t,j)\mapsto
d_{\varepsilon}(t,j)$ from Theorem 6.1.

We now have all the ingredients to prove Theorem 6.1.

Proof of Theorem 6.1:  Using the same arguments as in the proof of Theorem
5.1, we conclude that, by items 1 and 2 of Theorem 5.1, the conditions of
Theorem 5.3 are satisfied with $\mu_{0}$ and $a_{M}$ from Theorem 5.1. It can
be shown that, under Assumption 6.1, the hybrid system that is obtained by
expressing $\mathcal{H}_{\nu}$ in error coordinates is equivalent to
$\mathcal{H}$ in (22) with $A,B$ in (23) and $d_{c},d_{d}$ in (37). Hence, it
follows from Theorem 5.3 that, for each solution $\xi$ to $\mathcal{H}_{\nu}$
from ${\mathcal{}X}_{0}$, the parameter estimation error
$\tilde{\theta}=\theta-\hat{\theta}$ satisfies, for all $(t,j)\in\mathop{\rm
dom}\nolimits\xi$,

$\displaystyle|\tilde{\theta}(t,j)|\leq\sqrt{\frac{p_{M}}{p_{m}}}\mathrm{e}^{-\omega(t+j)}|\tilde{\theta}(0,0)|+\rho\|d\|_{(t,j)},$
(39)

with $(t,j)\mapsto d(t,j)$ as in (27) and $\rho$, $\omega$ from Theorem 5.3,
with $p_{m},p_{M}$ substituted by $p_{m},p_{M}$ from Theorem 5.1.

Using the definition of ${\mathcal{}A}_{g}$ in (16), we rewrite
$|\xi|_{{\mathcal{}A}_{g}}$ for all $(t,j)\in\mathop{\rm dom}\nolimits\xi$ as
$|\xi(t,j)|_{{\mathcal{}A}_{g}}=\sqrt{|\tilde{\theta}(t,j)|^{2}+|\varepsilon(t,j)|^{2}}$.
Since, by Assumption 6.1, the conditions of Lemma 6.2 are satisfied, we
substitute the bounds in (38) and (39). Using that, for any
$\alpha,\beta\in\mathbb{R}$,
$\alpha\beta\leq\frac{1}{2}(\alpha^{2}+\beta^{2})$, we obtain
$|\xi(t,j)|_{{\mathcal{}A}_{g}}\leq\sqrt{\frac{2p_{M}}{p_{m}}}\mathrm{e}^{-\min\\{\omega,\lambda_{\varepsilon}\\}(t+j)}|\xi(0,0)|_{{\mathcal{}A}_{g}}+\sqrt{2}\max\\{\rho,\rho_{\varepsilon}\\}\sqrt{\|d\|_{(t,j)}^{2}+\|d_{\varepsilon}\|_{(t,j)}^{2}}$
for all $(t,j)\in\mathop{\rm dom}\nolimits\xi$. Hence, (35) holds.

To conclude the proof, we upper bound $d_{c}$, $d_{d}$, $\alpha_{c}$, and
$\alpha_{d}$ for all $(t,j)\in\mathop{\rm dom}\nolimits\xi$. The bounds for
$\alpha_{c}$ and $\alpha_{d}$ in Theorem 6.1 follow directly from Assumption
6.1 and the definitions of $\alpha_{c}$ and $\alpha_{d}$ in (37). Moreover,
since, by item 1 of Theorem 5.1 and Assumption 6.1, the conditions of Lemmas
5.6 and 6.2 are satisfied, we have from (37) that, for each solution $\xi$ to
${\mathcal{H}}_{\nu}$ from ${\mathcal{}X}_{0}$,

$\begin{aligned}
|d_{c}(t,j)|&\leq\gamma_{c}|\psi(t,j)|(|\varepsilon(t,j)|+|\nu(t,j)|)\\\
&\leq\gamma_{c}\psi_{M}(\mathrm{e}^{-\lambda_{\varepsilon}(t+j)}|\varepsilon(0,0)|\\\
&\qquad+(\rho_{\varepsilon}\max\\{\lambda_{c}+L_{c},1-\lambda_{d}+L_{d}\\}+1)|\nu(t,j)|)\end{aligned}$

for all $(t,j)\in\mathop{\rm dom}\nolimits\xi$, with $\lambda_{\varepsilon}$
and $\rho_{\varepsilon}$ from Theorem 6.1, and the last inequality follows
from (38) the definition of $d_{\varepsilon}$ in (36). Next, using that
$\frac{|\psi(t,j+1)|}{\gamma_{d}+|\psi(t,j+1)|^{2}}\leq\frac{1}{2\sqrt{\gamma_{d}}}$
for all $(t,j)\in\Upsilon(\mathop{\rm dom}\nolimits\xi)$, we have that, for
all $(t,j)\in\Upsilon(\mathop{\rm dom}\nolimits\xi)$,

$\displaystyle|d_{d}(t,j)|\leq\frac{1}{2\sqrt{\gamma_{d}}}\big{(}\varepsilon(t,j+1)+\nu(t,j+1)\big{)}$
$\displaystyle\leq\frac{1}{2\sqrt{\gamma_{d}}}(\mathrm{e}^{-\lambda_{\varepsilon}(t+j+1)}|\varepsilon(0,0)|$
$+(\rho_{\varepsilon}\max\\{\lambda_{c}+L_{c},1-\lambda_{d}+L_{d}\\}+1)|\nu(t,j+1)|)$
∎

######  Remark 6.3.

A similar ISS result as in Theorem 6.1 can be developed without Assumption 6.1
by constraining the range of the plant state $x$ and the input $u$ to a
compact set. Under such conditions, it follows from the continuity of $f_{c}$
and $g_{d}$ that $d_{c}$, $d_{d}$, $\alpha_{c}$, $\alpha_{d}$ in (37) can be
upper bounded by functions of only $\nu$. Then, ISS follows from similar
arguments as in the proof of Theorem 6.1.

##  7 Case Studies

In this section, we present case studies that demonstrate the merits of our
hybrid algorithm. Simulations are performed using the Hybrid Equations Toolbox
[26].

###  7.1 Motivational Example Revisited

Recall the example in Section 3, where the system (11) can be written as (12)
by setting $f_{c}$, $g_{d}$ in (12) to zero. We employ $\mathcal{H}_{g}$ in
(13) to estimate $\theta$ in (11). The algorithm is simulated for
$\gamma_{c}=1$, $\lambda_{c}=0.1$, $\gamma_{d}=1$, and $\lambda_{d}=0.5$
alongside the continuous-time and discrete-time estimation algorithms from
Section 2.2 with the same parameters, where applicable. To illustrate the
robustness of our algorithm, we also simulate $\mathcal{H}_{g}$ with additive
noise $(t,j)\mapsto\nu(t,j)=\sin(2t)[1\,\,1]^{\top}$ in the measurements of
$x$. Recall that the classical PE conditions (C1) and (C2) are not satisfied
by $\phi_{c}$ and $\phi_{d}$ given above (11). However, $\psi$ satisfies the
hybrid PE condition (19) with $\Delta=2\pi+1$ and $\mu=5.1$.

The simulation is performed from two separate initial conditions: one with
$\varepsilon(0,0)=0$ and one with $\varepsilon(0,0)\neq 0$. In particular,
$x(0,0)=(3,6)$, $\hat{\theta}(0,0)=(0,0)$, and

  1. 1.

$\psi(0,0)=0,\;\eta(0,0)=-(3,6)\implies\varepsilon(0,0)=(0,0)$

  2. 2.

$\psi(0,0)=0,\ \eta(0,0)=-(1.5,3)\implies\varepsilon(0,0)=(1.5,3)$

producing the results in Figure 2. When no noise is present,
$|\xi|_{{\mathcal{}A}_{g}}$ converges exponentially to zero in accordance with
Theorem 5.1, as shown in blue for the case with $\varepsilon(0,0)=0$ and in
green for the case with $\varepsilon(0,0)\neq 0$. When noise is present,
$|\xi|_{{\mathcal{}A}_{g}}$ remains bounded in accordance with Theorem 6.1, as
shown in orange in Figure 2.

Figure 2: The projection onto $t$ of $|\xi|_{{\mathcal{}A}_{g}}$ for
$\mathcal{H}_{g}$.

###  7.2 Spacecraft Bias Torque Estimation

Consider the problem of estimating a constant disturbance torque applied to a
spacecraft, controlled by reaction wheels (RW) and reaction control system
(RCS) thrusters. Such bias torques may arise in practice due to aerodynamic
effects, gravity gradients, or solar radiation pressure differentials. For
simplicity, we consider the dynamics of a spacecraft rotating about only a
single principle axis of inertia, although our approach can be extended to
three-axis rotation. In the following, we derive the closed-loop dynamics of
the spacecraft when controlled by RW and RCS thrusters separately, and then
combine the results into a single hybrid model.

The dynamics of a spacecraft rotating along a principle axis of inertia under
the effect of RW are [31]

$J_{s}\ddot{z}=-J_{w}\dot{\Omega}+\theta,$ (40)

where $z\in\mathbb{R}$ is the known pointing angle of the spacecraft,
$\Omega\in\mathbb{R}$ is the known rotational velocity of the RW, $J_{s}>0$ is
the known spacecraft moment of inertia, $J_{w}>0$ is the known RW moment of
inertia, and $\theta\in\mathbb{R}$ is an unknown bias torque.

Suppose RW control the attitude to a pointing angle, $z_{\rm
des}\in\mathbb{R}$. The dynamics of the reaction wheel are [31]

$J_{w}\dot{\Omega}=\alpha(t),$ (41)

where $t\mapsto\alpha(t)\in\mathbb{R}$ is the RW motor torque that is designed
to maintain the spacecraft pointing angle. Substituting (41) into (40), we
obtain

$J_{s}\ddot{z}=-\alpha(t)+\theta.$ (42)

When the bias torque is nonzero, the industry-standard proportional-derivative
(PD) control scheme for the RW motor fails to yield zero pointing error in
steady-state. In this case, a feedfoward term is added that compensates for
the effect of the bias torque using an estimate of the bias, denoted by
$\hat{\theta}$ [31]. Hence, the RW torque is

$-\alpha(t)=K_{P}(z_{\rm des}-z(t))-K_{D}\dot{z}(t)-\hat{\theta}(t),$ (43)

where $K_{P},K_{D}>0$ are design parameters. From (41), (42), (43), the
dynamics of the closed-loop system are

$\displaystyle\ddot{z}$ $\displaystyle=\frac{-\alpha(t)+\theta}{J_{s}},$
$\displaystyle\dot{\Omega}$ $\displaystyle=\frac{\alpha(t)}{J_{w}}.$ (44)

The spacecraft pointing angle can be maintained only if an equivalent RW
torque is delivered to counteract the bias torque. If the bias torque is
nonzero, the angular velocity of the RW constantly increases in order to
counteract the disturbance and the RW motor eventually reaches its maximum
angular velocity. In order to avoid the RW motor from becoming saturated,
“momentum dumping” is applied to decrease the angular velocity of the RW [31].
This procedure involves firing the RCS thrusters to generate a torque that is
compensated by the attitude controller by actions that cause the RW to reduce
their angular momentum.

The dynamics of a spacecraft rotating along a principle axis of inertia under
the effect of RCS thrusters are [31]

$J_{s}\ddot{z}=M+\theta,$ (45)

where $M\in\mathbb{R}$ is the known RCS thruster torque. For simplicity, we
assume that the velocity of the RW is constant for the duration of each
thruster firing. As a result, the RW dynamics do not play a role in (45).

Suppose that, at time $t\geq 0$, the thrusters are fired for $\delta>0$
seconds. Integrating (45) over the time interval $[t,t+\delta]$ yields
$\dot{z}(t+\delta)=\dot{z}(t)+\frac{\delta}{J_{s}}(M+\theta)$. If the thruster
firing duration $\delta$ is negligibly small compared to the other time scales
of the system, which is appropriate due to the slow spacecraft attitude
maneuvering, we model the thruster firing as an instantaneous jump in the
angular velocity of the spacecraft, given by

$\displaystyle\dot{z}^{+}$
$\displaystyle=\dot{z}+\frac{\delta}{J_{s}}(M+\theta).$ (46)

To avoid chatter, a timer, denoted by $\tau_{s}$, is used to briefly inhibit
the RCS thrusters after each thruster firing. Each time the thrusters are
fired, the timer is reset to zero.

By combining the expression in (44) and (46), we express the closed-loop
dynamics of the spacecraft as a hybrid system as in (12). Given an input
$u:=(z_{\rm des},\hat{\theta})$, where $z_{\rm des}\in\mathbb{R}$ is the
desired constant spacecraft pointing angle and $\hat{\theta}\in\mathbb{R}$ is
an estimate of the unknown bias torque, the hybrid model of the spacecraft has
state $x=(z,\dot{z},\Omega,\tau_{s})\in\mathbb{R}^{4}$ and data

$\displaystyle f_{c}(x,u(t,j))$ $\displaystyle:=\begin{bmatrix}\dot{z}\\\
-\frac{1}{J_{s}}\alpha(x,u(t,j))\\\ \frac{1}{J_{w}}\alpha(x,u(t,j))\\\
1\end{bmatrix},$ $\displaystyle\\!\\!\\!\\!\phi_{c}(t,j)$
$\displaystyle:=\begin{bmatrix}0\\\ \frac{1}{J_{s}}\\\ 0\\\ 0\end{bmatrix}$
$\displaystyle g_{d}(x,u(t,j))$ $\displaystyle:=\begin{bmatrix}z\\\
\dot{z}+\frac{\delta}{J_{s}}M\\\ \Omega\\\ 0\end{bmatrix},$
$\displaystyle\\!\\!\\!\\!\phi_{d}(t,j)$ $\displaystyle:=\begin{bmatrix}0\\\
\frac{\delta}{J_{s}}\\\ 0\\\ 0\end{bmatrix}$

where $\alpha(x,u(t,j)):=-K_{P}(z_{\rm
des}-z)+K_{D}\dot{z}+\hat{\theta}(t,j)$. The flow and jump sets of the hybrid
spacecraft model implement the momentum dumping procedure. The system jumps
each time the angular velocity of the RW exceeds a design parameter
$\Omega_{\max}>0$ and the timer $\tau_{s}$ exceeds a design parameter
$\tau^{*}>0$, and flows otherwise, as

$\displaystyle C_{P}$
$\displaystyle:=\\{x\in\mathbb{R}^{4}:\Omega\leq\Omega_{\max}\\}\cup\\{x\in\mathbb{R}^{4}:\tau_{s}\leq\tau^{*}\\}$
$\displaystyle D_{P}$
$\displaystyle:=\\{x\in\mathbb{R}^{4}:\Omega\geq\Omega_{\max},\;\tau_{s}\geq\tau^{*}\\}.$

We employ $\mathcal{H}_{g}$ to estimate the unknown bias torque. The closed-
loop system is simulated121212Code at
https://github.com/HybridSystemsLab/HybridGD_SpacecraftBiasTorque with initial
conditions $x(0,0)=(0,0,0,0)$, $\hat{\theta}(0,0)=0$, $\psi(0,0)=0$, and
$\eta(0,0)=-x(0,0)$. The hybrid spacecraft model has parameters $z_{\rm
des}=0$ rad, $\Omega_{\max}=10000$ RPM, $J_{s}=5000$ kg-m2, $J_{w}=0.1$ kg-m2,
$M=-10$ N-m, $\delta=9.5$ sec, $\tau^{*}=10$ sec, $K_{p}=10$, $K_{d}=1200$,
and with an unknown bias torque of $\theta=0.005$ N-m. Our algorithm
$\mathcal{H}_{g}$ has parameters $\gamma_{c}=0.0012$, $\lambda_{c}=0.001$,
$\gamma_{d}=0.01$, and $\lambda_{d}=0.5$. With the initial conditions and
design parameters given above, it can be shown numerically that the conditions
of Theorem 5.1 hold.

The bias torque estimation error from $\mathcal{H}_{g}$ converges
exponentially to zero in accordance with Theorem 5.1, as shown in Figure 3.

Figure 3: The projection onto $t$ of the bias torque estimation error for
$\mathcal{H}_{g}$.

The spacecraft pointing angle error and RW angular velocity are shown in the
top and bottom plots, respectively, in Figure 4, where the control performance
resulting from our hybrid algorithm is compared against an industry-standard
PID control scheme that is tuned to achieve a similar pointing error
convergence rate during flows. For the PID controller, we inhibit accumulation
of the integrator during each thruster firing, otherwise the spacecraft
pointing angle fails to converge to the set point. With the exception of the
transients caused by the thruster firings, the pointing error converges to
zero for both controllers. However, our hybrid algorithm converges faster due
to our estimator’s ability to leverage information during both flows and jumps
to estimate the unknown bias torque.

Figure 4: The projection onto $t$ of the spacecraft pointing angle error (top)
and the RW angular velocity (bottom).

##  8 Conclusion

In this paper, we propose a hybrid algorithm for estimating unknown parameters
in a class of hybrid systems with nonlinear dynamics that are affine in the
unknown parameter. We show that our algorithm guarantees exponential
convergences of the parameter estimate to the true value under a notion of
hybrid persistence of excitation that relaxes the classical continuous-time
and discrete-time persistence of excitation conditions. Moreover, we show that
the parameter estimate is ISS with respect to hybrid noise in the measurements
of the plant state. To demonstrate its practicality, we apply
$\mathcal{H}_{g}$ to estimate an unknown bias torque applied to a simplified
model of a spacecraft controlled by reaction wheels and reaction control
thrusters. Future work on this topic includes extending our proposed algorithm
to estimate the unknown parameters for hybrid dynamical systems with unknown
jump times.

Research by R. S. Johnson and R. G. Sanfelice has been partially supported by
NSF Grants no. ECS-1710621, CNS-2039054, and CNS-2111688, by AFOSR Grants no.
FA9550-19-1-0053, FA9550-19-1-0169, and FA9550-20-1-0238, and by ARO Grant no.
W911NF-20-1-0253.

We thank A. Saoud, M. Maghenem, and A. Loria for their insightful discussions
on this topic.

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##  Appendix A Constants $\kappa_{g}$ and $\lambda_{g}$ in Theorem 5.1

Given $\phi_{c},\phi_{d}:E\to\mathbb{R}^{n\times p}$,
$\gamma_{c},\lambda_{c},\gamma_{d},\phi_{M},\Delta,\mu>0$,
$\lambda_{d}\in(0,2)$, $\psi_{0}\geq 0$, $q_{M}\geq q_{m}>0$, and
$\zeta\in(0,1)$ from Theorem 5.1, suitable choices of $\kappa_{g}$ and
$\lambda_{g}$ in Theorem 5.1 are $\kappa_{g}:=\sqrt{3}\kappa$ and
$\lambda_{g}:=\min\\{\lambda,b\\}$, where

$\displaystyle\kappa$
$\displaystyle:=2\max\left\\{\dfrac{p_{M}}{p_{m}},\,a\rho\sqrt{\dfrac{p_{M}}{p_{m}}}\right\\},\quad\lambda:=\frac{1}{2}\min\left\\{\omega,b\right\\},$
$\displaystyle a$
$\displaystyle:=\max\left\\{\gamma_{c}\psi_{M},\frac{1}{2\sqrt{\gamma_{d}}}\right\\},$
$\displaystyle b$
$\displaystyle:=\frac{1}{2}\min\left\\{\lambda_{c},\;-\ln\left(1-\lambda_{d}(2-\lambda_{d})\right)\right\\},$
$\displaystyle p_{m}$ $\displaystyle:=q_{m},\quad\
p_{M}:=q_{m}+\frac{q_{M}\kappa_{0}^{2}}{2\lambda_{0}}+\frac{q_{M}\kappa_{0}^{2}\mathrm{e}^{2\lambda_{0}}}{\mathrm{e}^{2\lambda_{0}}-1},$
$\displaystyle\rho$
$\displaystyle:=\sqrt{\frac{2p_{M}^{3}}{q_{m}p_{m}\zeta}\left(\frac{2p_{M}}{q_{m}}+1\right)},$
$\displaystyle\omega$
$\displaystyle:=\mbox{$\dfrac{1}{2}\min\left\\{\dfrac{q_{m}}{2p_{M}}(1-\zeta),\,-\ln\left(1-\dfrac{q_{m}}{2p_{M}}(1-\zeta)\right)\right\\}$},$
$\displaystyle\kappa_{0}$
$\displaystyle:=\sqrt{\frac{1}{1-\sigma}},\qquad\quad\lambda_{0}:=-\frac{\ln(1-\sigma)}{2(\Delta+1)},$
$\displaystyle\sigma$
$\displaystyle:=\frac{2\mu_{0}}{\big{(}1+\sqrt{(a_{M}+2)(\Delta+2)^{3}(a_{M}(\Delta+2)+1/2)}\big{)}^{2}},$
$\displaystyle\mu_{0}$
$\displaystyle:=\min\left\\{\gamma_{c},\frac{1}{2(\gamma_{d}+\psi_{M}^{2})}\right\\}\mu,\qquad\quad
a_{M}:=\gamma_{c}\psi_{M}^{2},$ $\displaystyle\psi_{M}$
$\displaystyle:=\psi_{0}+\max\left\\{\frac{1}{\lambda_{c}},\frac{\sqrt{2\lambda_{d}(2-\lambda_{d})+16}}{\lambda_{d}(2-\lambda_{d})}\right\\}\phi_{M}.$

##  Appendix B Proof of Theorem 5.3

To prove Theorem 5.3, we first require some auxiliary results for the hybrid
system $\mathcal{H}$ when the disturbances $d_{c}$ and $d_{d}$ are equal to
zero. We denote this system as $\mathcal{H}_{0}$, with state
$\xi=(\vartheta,\tau,k)\in{\mathcal{}X}$ and dynamics

$\mathcal{H}_{0}:\left\\{\begin{aligned}
\dot{\xi}&=\begin{bmatrix}-A(\tau,k)\vartheta\\\ 1\\\
0\end{bmatrix}=:F_{0}(\xi)&\xi&\in C_{0}\\\
\xi^{+}&=\begin{bmatrix}\vartheta-B(\tau,k)\vartheta\\\ \tau\\\
k+1\end{bmatrix}=:G_{0}(\xi)&\xi&\in D_{0}\end{aligned}\right.$ (47)

where $C_{0}:=C$ and $D_{0}:=D$, with $C$ and $D$ below (22).

Inspired by [28], we establish sufficient conditions that ensure the hybrid
system $\mathcal{H}_{0}$ induces global pre-exponential stability of the set
${\mathcal{}A}$ in (25).

######  Theorem B.1.

Given the hybrid system $\mathcal{H}_{0}$ in (47), suppose that Assumptions
5.1 and 5.2 hold. Then, each solution $\xi$ to $\mathcal{H}_{0}$ satisfies

$\displaystyle|\xi(s,i)|_{\mathcal{}A}\leq\kappa_{0}\mathrm{e}^{-\lambda_{0}(s+i-t-j)}|\xi(t,j)|_{\mathcal{}A}$
(48)

for all $(s,i),(t,j)\in\mathop{\rm dom}\nolimits\xi$ satisfying $s+i\geq t+j$,
where

$\displaystyle\kappa_{0}$
$\displaystyle:=\sqrt{\frac{1}{1-\sigma}},\qquad\qquad\lambda_{0}:=-\frac{\ln(1-\sigma)}{2(\Delta+1)},$
$\displaystyle\sigma$
$\displaystyle:=\frac{2\mu_{0}}{\big{(}1+\sqrt{(a_{M}+2)(\Delta+2)^{3}(a_{M}(\Delta+2)+1/2)}\big{)}^{2}}$

with $a_{M}$, $\mu_{0},\Delta$ from Assumptions 5.1 and 5.2.

######  Proof B.2.

The proof of Theorem B.1 follows along the same lines as the proof of [28,
Theorem 1].

Next, we recall the following result from [14].

######  Lemma B.3.

Given $B\in\mathbb{R}^{p\times p}$, if $|B|<1$, then $I-B$ is invertible.

Finally, we establish the following lemma.

######  Lemma B.4.

Given the hybrid system $\mathcal{H}_{0}$ in (47), suppose that Assumptions
5.1 and 5.2 hold and let the hybrid time domain $E$ come from these
assumptions. Then, for each $q_{M}\geq q_{m}>0$ and each symmetric matrix
function $Q:E\to\mathbb{R}^{p\times p}$ satisfying

$\displaystyle q_{m}I\leq Q(t,j)\leq q_{M}I\quad\forall(t,j)\in E,$ (49)

there exists a symmetric matrix function $P:E\rightarrow\mathbb{R}^{p\times
p}$ satisfying

$\displaystyle p_{m}I\leq P(t,j)\leq p_{M}I\quad\forall(t,j)\in E,$ (50)

where

$\displaystyle p_{m}$ $\displaystyle:=q_{m},$ $\displaystyle p_{M}$
$\displaystyle:=q_{m}+\frac{q_{M}\kappa_{0}^{2}}{2\lambda_{0}}+\frac{q_{M}\kappa_{0}^{2}\mathrm{e}^{2\lambda_{0}}}{\mathrm{e}^{2\lambda_{0}}-1},$
(51)

with $\kappa_{0}$ and $\lambda_{0}$ from Theorem B.1. Moreover, for each
$j\in\mathbb{N}$ and for almost all $t\in I^{j}:=\\{t:(t,j)\in E\\}$,
$(t,j)\mapsto P(t,j)$ satisfies

$\begin{aligned}
\frac{d}{dt}P(t,j)-P(t,j)A(t,j)-A(t,j)^{\top}P(t,j)&\leq-Q(t,j)\end{aligned}$
(52)

and, for all $(t,j)\in\Upsilon(E)$, with $\Upsilon$ as in (10),

$\displaystyle(I-B(t,j))^{\top}P(t,j+1)(I-B(t,j))$ $\displaystyle-P(t,j)$ (53)
$\displaystyle\leq-Q(t,j).$

######  Proof B.5.

Let $U:E\to\mathbb{R}^{p\times p}$ be such that $U(0,0)$ is invertible and,
for each $j\in\mathbb{N}$ and almost all $t\in I^{j}$,

$\displaystyle\frac{d}{dt}U(t,j)$ $\displaystyle=-A(t,j)U(t,j)$ (54)

and, for all $(t,j)\in\Upsilon(E)$, with $\Upsilon$ as in (10),

$\displaystyle U(t,j+1)$ $\displaystyle=U(t,j)-B(t,j)U(t,j).$ (55)

Then, for all $(t,j),(t^{\prime},j^{\prime})\in E$, we define

$\displaystyle\Phi(t,j,t^{\prime},j^{\prime}):=U(t,j)U(t^{\prime},j^{\prime})^{-1},$
(56)

where, in view of Lemma B.3, $U(t,j)$ is invertible for all $(t,j)\in E$ since
$U(0,0)$ is invertible and, by Assumption 5.1, $|B(t,j)|<1$ for all
$(t,j)\in\Upsilon(E)$.

By the equivalence between the dynamics of $U$ and the $\vartheta$ component
of $\xi$ in (47), we have that, for each solution $\xi$ to $\mathcal{H}_{0}$
and each $(t,j),(t^{\prime},j^{\prime})\in\mathop{\rm dom}\nolimits\xi$,131313
Since each solution $\xi$ to $\mathcal{H}_{0}$ inherits the hybrid time domain
$E$, it follows that $\mathop{\rm dom}\nolimits\xi=E$, and thus
$\Phi(t,j,t^{\prime},j^{\prime})$ is well defined for all
$(t,j),(t^{\prime},j^{\prime})\in\mathop{\rm dom}\nolimits\xi$,

$\displaystyle\vartheta(t,j)=\Phi(t,j,t^{\prime},j^{\prime})\vartheta(t^{\prime},j^{\prime}).$
(57)

Hence, $\Phi$ is the state transition matrix for $\vartheta$. Note that $\Phi$
is not necessarily smooth at jumps.

Next, we define $(t,j)\mapsto P(t,j)$ as

$\displaystyle P(t,j):=P_{c}(t,j)+P_{d}(t,j)+q_{m}I$ (58)

for all $(t,j)\in E$, with

$\displaystyle P_{c}(t,j)$
$\displaystyle:=\sum_{i=j}^{J}\int_{\max\\{t,t_{i}\\}}^{t_{i+1}}\\!\\!\\!\\!\Phi(s,i,t,j)^{\top}Q(s,i)\Phi(s,i,t,j)ds$
$\displaystyle P_{d}(t,j)$
$\displaystyle:=\sum_{i=j}^{J}\Phi(t_{i+1},i,t,j)^{\top}Q(t_{i+1},i)\Phi(t_{i+1},i,t,j),$

where $t_{J+1}:=T$, with $J:=\sup_{j}E$ and $T:=\sup_{t}E$. Note that the term
$q_{m}I$ in (58) was chosen for simplicity – any positive definite matrix
would suffice.

We first show that (50) holds. Since, for all $(t,j)\in E$, $P_{c}(t,j)\geq 0$
and $P_{d}(t,j)\geq 0$, a lower bound on $P$ in (58) is $P(t,j)\geq q_{m}I$
for all $(t,j)\in E$. Next, we develop an upper bound on $P$. Since, by
Assumptions 5.1 and 5.2, the conditions of Theorem B.1 are satisfied, it
follows from (48) and from the equivalence between $|\xi|_{\mathcal{}A}$ and
$|\vartheta|$ that, for each solution $\xi=(\vartheta,\tau,k)$ to
$\mathcal{H}_{0}$ and each $(s,i),(t,j)\in\mathop{\rm dom}\nolimits\xi$
satisfying $s\geq t$ and $i\geq j$,
$|\vartheta(s,i)|\leq\kappa_{0}\mathrm{e}^{-\lambda_{0}(s+i-t-j)}|\vartheta(t,j)|$
with $\kappa_{0}$ and $\lambda_{0}$ from Theorem B.1. By substituting (57)
into the expression above, we have that, for each $(s,i),(t,j)\in\mathop{\rm
dom}\nolimits\xi$ satisfying $s\geq t$ and $i\geq j$,
$|\Phi(s,i,t,j)\vartheta(t,j)|\leq\kappa_{0}\mathrm{e}^{-\lambda_{0}(s+i-t-j)}|\vartheta(t,j)|$
which, if $|\vartheta(t,j)|\neq 0$, implies that
$|\Phi(s,i,t,j)\vartheta(t,j)|/|\vartheta(t,j)|\leq\kappa_{0}\mathrm{e}^{-\lambda_{0}(s+i-t-j)}$.
Since this inequality holds for any
$\vartheta(t,j)\in\mathbb{R}^{p}\setminus\\{0\\}$, it follows from the
equivalence between $\mathop{\rm dom}\nolimits\xi$ and $E$ that, for each
$(s,i),(t,j)\in E$ satisfying $s\geq t$ and $i\geq j$,

$\displaystyle|\Phi(s,i,t,j)|$
$\displaystyle=\sup_{r\in\mathbb{R}^{p}\setminus\\{0\\}}\frac{|\Phi(s,i,t,j)r|}{|r|}\leq\kappa_{0}\mathrm{e}^{-\lambda_{0}(s+i-t-j)}.$

Then, from the definitions of $P_{c}$ and $P_{d}$ below (58),

$\displaystyle P_{c}(t,j)$ $\displaystyle\leq
q_{M}\int_{0}^{\infty}\kappa_{0}^{2}\mathrm{e}^{-2\lambda_{0}s}dsI=\frac{q_{M}\kappa_{0}^{2}}{2\lambda_{0}}I$

and

$\displaystyle P_{d}(t,j)$ $\displaystyle\leq
q_{M}\sum_{i=0}^{\infty}\kappa_{0}^{2}\mathrm{e}^{-2\lambda_{0}i}I=\frac{q_{M}\kappa_{0}^{2}\mathrm{e}^{2\lambda_{0}}}{\mathrm{e}^{2\lambda_{0}}-1}I.$

From the bounds above and the definition of $P$ in (58), we conclude that (50)
holds with $p_{m}$, $p_{M}$ in (51).

Next, we show that (52) holds. We differentiate $P$ during flows and use that,
for each $(s,i)\in E$ and each $j\in\mathbb{N}$ and for almost all $t\in
I^{j}$,

$\displaystyle\frac{d}{dt}\Phi(s,i,t,j)=\Phi(s,i,t,j)A(t,j).$

This property follows from (54) and from the definition of $\Phi$ in (56). For
readability, we define

$\displaystyle\Pi(s,i,t,j):=\Phi(s,i,t,j)^{\top}Q(s,i)\Phi(s,i,t,j).$ (59)

Using the Leibniz integral rule, we obtain that, for each $j\in\mathbb{N}$ and
for almost all $t\in I^{j}$,

$\displaystyle\frac{d}{dt}P_{c}(t,j)=A(t,j)^{\top}\bigg{(}\sum_{i=j}^{J}\int_{\max\\{t,t_{i}\\}}^{t_{i+1}}\\!\\!\\!\\!\Pi(s,i,t,j)ds\bigg{)}$
$\displaystyle\quad+\bigg{(}\sum_{i=j}^{J}\int_{\max\\{t,t_{i}\\}}^{t_{i+1}}\\!\\!\\!\\!\Pi(s,i,t,j)ds\bigg{)}A(t,j)-Q(t,j)$
$\displaystyle=A(t,j)^{\top}P_{c}(t,j)+P_{c}(t,j)A(t,j)-Q(t,j)$ (60)

and

$\displaystyle\frac{d}{dt}P_{d}(t,j)$
$\displaystyle=A(t,j)^{\top}\bigg{(}\sum_{i=j}^{J}\Pi(t_{i+1},i,t,j)\bigg{)}$
$\displaystyle\quad+\bigg{(}\sum_{i=j}^{J}\Pi(t_{i+1},i,t,j)\bigg{)}A(t,j)$
$\displaystyle=A(t,j)^{\top}P_{d}(t,j)+P_{d}(t,j)A(t,j).$ (61)

Combining the expressions in (B.5) and (B.5), and using the definition of $P$
in (58), we have that, for each $j\in\mathbb{N}$ and for almost all $t\in
I^{j}$,

$\displaystyle\frac{d}{dt}P(t,j)$
$\displaystyle=A(t,j)^{\top}P(t,j)+P(t,j)A(t,j)$
$\displaystyle\quad-Q(t,j)-2q_{m}A(t,j)$ $\displaystyle\leq
A(t,j)^{\top}P(t,j)+P(t,j)A(t,j)-Q(t,j).$

The inequality follows from the fact that, by Assumption 5.1, $A(t,j)\geq 0$
for all $(t,j)\in E$. Hence, (52) holds.

To conclude the proof, we show that (53) holds. We use the property that, for
each $(t,j),(s,i)\in\Upsilon(E)$,

$\displaystyle\Phi(s,i,t,j+1)(I-B(t,j))=\Phi(s,i,t,j).$

This property follows from (55) and from the definition of $\Phi$ in (56).
Then, for each $(t,j)\in\Upsilon(E)$,

$\displaystyle(I-B(t,j))^{\top}P_{c}(t,j+1)(I-B(t,j))$
$\displaystyle\quad=\sum_{i=j+1}^{J}\int_{\max\\{t,t_{i}\\}}^{t_{i+1}}\\!\\!\\!\\!\Pi(s,i,t,j)ds.$

Since the value of ordinary time $t$ is the same immediately before and after
each jump, it follows that, for each $(t,j)\in\Upsilon(E)$,
$\max\\{t,t_{j}\\}=t=t_{j+1}$. Hence, we rewrite the expression above as

$\displaystyle(I-B(t,j))^{\top}P_{c}(t,j+1)(I-B(t,j))$
$\displaystyle\quad=\sum_{i=j}^{J}\int_{\max\\{t,t_{i}\\}}^{t_{i+1}}\\!\\!\\!\\!\Pi(s,i,t,j)ds=P_{c}(t,j).$
(62)

Focusing now on $P_{d}$, for each $(t,j)\in\Upsilon(E)$,

$\displaystyle(I-B(t,j))^{\top}P_{d}(t,j+1)(I-B(t,j))-P_{d}(t,j)$
$\displaystyle\quad=-\Pi(t_{j+1},j,t,j).$

From (59) and the fact that $t=t_{j+1}$ at each jump,

$\begin{aligned} &(I-B(t,j))^{\top}P_{d}(t,j+1)(I-B(t,j))-P_{d}(t,j)\\\
&\quad=-\Phi^{\top}(t,j,t,j)Q(t,j)\Phi(t,j,t,j)=-Q(t,j).\end{aligned}$ (63)

Using the definition of $P$ in (58), it follows from (B.5) and (63) that, for
each $(t,j)\in\Upsilon(E)$,

$\displaystyle(I-B(t,j))^{\top}P(t,j+1)(I-B(t,j))-P(t,j)$
$\displaystyle\quad=-Q(t,j)-q_{m}B(t,j)(2I-B(t,j))\leq-Q(t,j),$

where the inequality holds since, by Assumption 5.1, $B(t,j)\geq 0$ and
$|B(t,j)|<1$ for all $(t,j)\in E$.

We now have all the ingredients to prove Theorem 5.3.

######  Proof B.6 ().

Proof of Theorem 5.3: Since, by Assumptions 5.1 and 5.2, the conditions of
Lemma B.4 are satisfied, given $q_{M}\geq q_{m}>0$ and a symmetric matrix
function $Q:E\to\mathbb{R}^{p\times p}$ satisfying (49), there exists a
symmetric matrix function $P:E\to\mathbb{R}^{p\times p}$ satisfying (50)–(53).
Given such $P$, consider the Lyapunov function

$\displaystyle V(\xi):=\vartheta^{\top}P(\tau,k)\vartheta\qquad\forall\xi\in
C\cup D.$

From (50) and from the equivalence between $|\vartheta|$ and
$|\xi|_{\mathcal{}A}$, we have that

$\displaystyle p_{m}|\xi|^{2}_{\mathcal{}A}\leq V(\xi)\leq
p_{M}|\xi|^{2}_{\mathcal{}A}\quad\forall\xi\in C\cup D,$ (64)

with $p_{m},p_{M}$ as in (51). We first study the change in $V$ during flows.
Omitting the $(\tau,k)$ arguments for readability, we have from (52) that, for
all $\xi\in C$,

$\displaystyle\langle\nabla
V(\xi),F(\xi)\rangle\leq-\vartheta^{\top}Q\vartheta+2\vartheta^{\top}Pd_{c}.$

We use that for any $\varrho>0$,
$2\vartheta^{\top}Pd_{c}\leq\varrho\vartheta^{\top}P\vartheta+\varrho^{-1}d_{c}^{\top}Pd_{c}$.
Choosing $\varrho=q_{m}/(2p_{M})$,

$\displaystyle\langle\nabla V(\xi),F(\xi)\rangle$
$\displaystyle\leq-\frac{q_{m}}{2p_{M}}V(\xi)+\frac{2p_{M}^{2}}{q_{m}}|d_{c}(\tau,k)|^{2}.$

Let $\zeta\in(0,1)$. By adding and subtracting
$\zeta\frac{q_{m}}{2p_{M}}V(\xi)$ to the right-hand side of the expression
above, we conclude

$\displaystyle\langle\nabla
V(\xi),F(\xi)\rangle\leq-\frac{q_{m}}{2p_{M}}(1-\zeta)V(\xi)$ (65)
$\displaystyle\qquad\forall\xi\in C:\
V(\xi)\geq\frac{4p_{M}^{3}}{q_{m}^{2}\zeta}|d_{c}(\tau,k)|^{2}.$

Next, we study the change in $V$ at jumps. For readability, we omit the
$(\tau,k)$ arguments and denote $P(\tau,k+1)$ as $P^{+}$. We have from (53)
that, for all $\xi\in D$,

$\displaystyle V(G(\xi))-V(\xi)$
$\displaystyle\leq-\vartheta^{\top}Q\vartheta+2|\vartheta^{\top}P^{+}d_{d}|+d_{d}^{\top}P^{+}d_{d}.$

We use that for any $\varrho>0$,
$2|\vartheta^{\top}P^{+}d_{d}|\leq\varrho\vartheta^{\top}P^{+}\vartheta+\varrho^{-1}d_{d}^{\top}P^{+}d_{d}$.
Choosing $\varrho=q_{m}/(2p_{M})$,

$\begin{aligned}
V(G(\xi))-V(\xi)&\leq-\frac{q_{m}}{2p_{M}}V(\xi)+\left(\frac{2p_{M}^{2}}{q_{m}}+p_{M}\right)|d_{d}(\tau,k)|^{2}.\end{aligned}$

Let $\zeta\in(0,1)$. By adding and subtracting
$\zeta\frac{q_{m}}{2p_{M}}V(\xi)$ to the right-hand side of the expression
above, we conclude

$\displaystyle V(G(\xi))-V(\xi)\leq-\frac{q_{m}}{2p_{M}}(1-\zeta)V(\xi),$ (66)
$\displaystyle\qquad\forall\xi\in D:\
V(\xi)\geq\frac{2p_{M}^{2}}{q_{m}\zeta}\left(\frac{2p_{M}}{q_{m}}+1\right)|d_{d}(\tau,k)|^{2}.$

Note that the lower bound on $V$ for which (66) holds is more restrictive than
the lower bound on $V$ for which (65) holds. Using the function $d$ defined in
(27), we combine the expressions in (65) and (66) and obtain

$\displaystyle\langle\nabla V(\xi),F(\xi)\rangle$
$\displaystyle\leq\\!-\frac{q_{m}}{2p_{M}}(1-\zeta)V(\xi)$
$\displaystyle\forall\xi\in C\cap S$ $\displaystyle V(G(\xi))-V(\xi)$
$\displaystyle\leq-\frac{q_{m}}{2p_{M}}(1-\zeta)V(\xi)$
$\displaystyle\forall\xi\in D\cap S,$

where $S:=\big{\\{}\xi\in C\cup
D:V(\xi)\geq\frac{2p_{M}^{2}}{q_{m}\zeta}\left(\frac{2p_{M}}{q_{m}}+1\right)|d(\tau,k)|^{2}\big{\\}}.$
Then, for each solution $\xi$ to $\mathcal{H}$, by integration using the
bounds above, we have that, for all $(t,j)\in\mathop{\rm dom}\nolimits\xi$,

$\displaystyle\mbox{\scriptsize$\displaystyle
V(\xi(t,j))\leq\exp\bigg{\\{}\bigg{(}-\frac{q_{m}}{2p_{M}}(1-\zeta)t+\ln\left(1-\frac{q_{m}}{2p_{M}}(1-\zeta)\right)j\bigg{)}\bigg{\\}}$}$
$\displaystyle\qquad\qquad\quad\mbox{\scriptsize$\displaystyle\times
V(\xi(0,0))+\frac{2p_{M}^{2}}{q_{m}\zeta}\left(\frac{2p_{M}}{q_{m}}+1\right)\|d\|_{(t,j)}.$}$

Using (64), we conclude that (26) holds.

##  Appendix C Proof of Theorem 5.4

######  Proof C.1 ().

Let $\xi$ be a maximal solution to $\mathcal{H}$. First, we upper bound
$(t,j)\mapsto|\xi(t,j)|_{\mathcal{}A}$ for all $(t,j)\in\mathop{\rm
dom}\nolimits\xi$. Since, by Assumptions 5.1 and 5.2, the conditions of
Theorem 5.3 are satisfied, it follows from (26) that

$\displaystyle|\xi(t,j)|_{\mathcal{}A}$
$\displaystyle\leq\beta(|\xi(0,0)|_{\mathcal{}A},0)+a\rho|d(0,0)|=:\xi_{M}$
(67)

for all $(t,j)\in\mathop{\rm dom}\nolimits\xi$, where the second inequality
follows from (28). Next, we define $\delta\mapsto c_{1}(\delta)\in\mathbb{R}$
as

$\displaystyle
c_{1}(\delta):=-\frac{1}{b}\ln\left(\frac{\delta/2}{a\rho|d(0,0)|}\right)\quad\forall\delta>0.$
(68)

Let $\delta>0$ be such that there exists
$(t^{\prime},j^{\prime})\in\mathop{\rm dom}\nolimits\xi$ such that
$t^{\prime}+j^{\prime}\geq c_{1}(\delta)$. Then, it follows from (28) that for
all $(t,j)\in\mathop{\rm dom}\nolimits\xi$ satisfying $t\geq t^{\prime}$ and
$j\geq j^{\prime}$, $|d(t,j)|\leq a\mathrm{e}^{-b(t+j)}|d(0,0)|\leq
a\mathrm{e}^{-bc_{1}(\delta)}|d(0,0)|=\delta/(2\rho).$ Hence, for all
$(t,j)\in\mathop{\rm dom}\nolimits\xi$ satisfying $t\geq t^{\prime}$ and
$j\geq j^{\prime}$, the supremum norm of $(t,j)\mapsto|d(t,j)|$ from
$(t^{\prime},j^{\prime})$ to $(t,j)$ is less than or equal to
$\delta/(2\rho)$. Thus, from (26), for all $(t,j)\in\mathop{\rm
dom}\nolimits\xi$ satisfying $t\geq t^{\prime}$ and $j\geq j^{\prime}$,

$\displaystyle|\xi(t,j)|_{\mathcal{}A}$
$\displaystyle\leq\beta(\xi_{M},t+j-c_{1}(\delta))+\delta/2$ (69)

with $\xi_{M}$ as in (67). Next, we define $\delta\mapsto
c_{2}(\delta)\in\mathbb{R}$ as

$\displaystyle
c_{2}(\delta):=-\frac{1}{\omega}\ln\left(\frac{\delta/2}{\beta(\xi_{M},0)}\right)\quad\forall\delta>0.$
(70)

Omitting the argument $\delta$ of $c_{1}$ and $c_{2}$ for readability, we have
that, for all $(t,j)\in\mathop{\rm dom}\nolimits\xi$ satisfying $t+j\geq
c_{2}+c_{1}$,

$\displaystyle\beta(\xi_{M},t+j-c_{1})\leq\beta(\xi_{M},c_{2}+c_{1}-c_{1})=\delta/2.$
(71)

By combining (69) and (71), it follows that, for each $\delta>0$ and each
$(t,j)\in\mathop{\rm dom}\nolimits\xi$,

$\begin{aligned}
t+j\geq\max\\{c_{1}(\delta),c_{2}(\delta)+c_{1}(\delta)\\}\implies|\xi(t,j)|_{\mathcal{}A}\leq\delta.\end{aligned}$
(72)

Since $c_{1}$ in (68) and $c_{2}$ in (70) are continuous monotonically
increasing functions of $\delta$ with $\mathop{\rm rge}c_{1}=\mathop{\rm
rge}c_{2}=\mathbb{R}$, it follows that, for each $(t,j)\in\mathop{\rm
dom}\nolimits\xi$, there exists a unique $\delta>0$ such that
$t+j=\max\\{c_{1}(\delta),c_{2}(\delta)+c_{1}(\delta)\\}$. For such $\delta$,
(72) holds. Hence, we develop a bound for
$(t,j)\mapsto|\xi(t,j)|_{\mathcal{}A}$ by bounding, for each
$(t,j)\in\mathop{\rm dom}\nolimits\xi$, the corresponding value of $\delta$
for which $t+j=\max\\{c_{1}(\delta),c_{2}(\delta)+c_{1}(\delta)\\}$.

Given $(t,j)\in\mathop{\rm dom}\nolimits\xi$ and $\delta>0$ satisfying
$t+j=\max\\{c_{1}(\delta),c_{2}(\delta)+c_{1}(\delta)\\}$, we consider two
cases: $\max\\{c_{1}(\delta),c_{2}(\delta)+c_{1}(\delta)\\}=c_{1}(\delta)$ and
$\max\\{c_{1}(\delta),c_{2}(\delta)+c_{1}(\delta)\\}=c_{2}(\delta)+c_{1}(\delta)$.

  1. 1.

If $\max\\{c_{1}(\delta),c_{2}(\delta)+c_{1}(\delta)\\}=c_{1}(\delta)$, then
$t+j=c_{1}(\delta)$ which, from (68), implies that
$\delta=2a\rho\mathrm{e}^{-b(t+j)}|d(0,0)|$.

  2. 2.

If
$\max\\{c_{1}(\delta),c_{2}(\delta)+c_{1}(\delta)\\}=c_{2}(\delta)+c_{1}(\delta)$,
then $t+j=c_{2}(\delta)+c_{1}(\delta)$. Since $c_{2}(\delta)+c_{1}(\delta)\geq
c_{1}(\delta)$, it follows that $c_{2}(\delta)\geq 0$. Then, we consider two
cases: $c_{1}(\delta)\leq 0$ and $c_{1}(\delta)>0$.

     * a.

If $c_{1}(\delta)\leq 0$, then $t+j\leq c_{2}(\delta)$ which, from (70),
implies that $\delta\leq 2\mathrm{e}^{-\omega(t+j)}\beta(\xi_{M},0)$.
Substituting $\xi_{M}$ given in (67)
yields$\delta~{}\leq~{}\max\left\\{2\frac{p_{M}}{p_{m}},2a\rho\sqrt{\frac{p_{M}}{p_{m}}}\right\\}\mathrm{e}^{-\omega(t+j)}(|\xi(0,0)|_{\mathcal{}A}+|d(0,0)|).$

     * b.

If $c_{1}(\delta)>0$, we define $\sigma:=\min\\{\omega,b\\}$ and then

$\displaystyle t+j$ $\displaystyle=c_{1}(\delta)+c_{2}(\delta)$
$\displaystyle\leq\mbox{$-\dfrac{1}{\sigma}\bigg{(}\ln\left(\dfrac{\delta/2}{\beta(\xi_{M},0)}\right)+\ln\left(\dfrac{\delta/2}{a\rho|d(0,0)|}\right)\bigg{)}$}$

which implies
$\delta\leq\sqrt{4a\rho\mathrm{e}^{-\sigma(t+j)}\beta(\xi_{M},0)|d(0,0)|\big{)}}$.
By substituting $\xi_{M}$ in (67) and completing the square yields
$\mbox{$\delta\leq\max\left\\{\left(\frac{p_{M}}{p_{m}}\right)^{\nicefrac{{3}}{{4}}},2a\sqrt{\rho}\left(\frac{p_{M}}{p_{m}}\right)^{\nicefrac{{1}}{{4}}}\right\\}$}\allowbreak\times\mathrm{e}^{-\frac{\sigma}{2}(t+j)}\left(|\xi(0,0)|_{\mathcal{}A}+|d(0,0)|\right).$

By combining the bounds in the items above, and using that $p_{M}/p_{m}>1$ and
$\rho>1$, it follows from (72) that (29) holds.

##  Appendix D Proof of Lemma 5.5

######  Proof D.1 ().

Consider the Lyapunov function

$\displaystyle V_{\varepsilon}(\xi)$
$\displaystyle:=\frac{1}{2}\varepsilon^{\top}\varepsilon\quad\forall\xi\in\widetilde{C}_{g}\cup\widetilde{D}_{g},$
(73)

with $\varepsilon$ as in (17). Since $\theta$ is constant, we have from (21)
that
$\dot{\varepsilon}=\dot{z}+\dot{\eta}-\dot{\psi}\theta=-\lambda_{c}\varepsilon$.
Thus, for all $\xi\in\widetilde{C}_{g}$, $\langle\nabla
V_{\varepsilon}(\xi),\widetilde{F}_{g}(\xi)\rangle=-2\lambda_{c}V_{\varepsilon}(\xi)\leq
0.$ At jumps, since $\theta$ is constant, we have from (21) that
$\varepsilon^{+}=z^{+}+\eta^{+}-\psi^{+}\theta=(1-\lambda_{d})\varepsilon$.
Thus, for all $\xi\in\widetilde{D}_{g}$,
$V_{\varepsilon}(\widetilde{G}_{g}(\xi))-V_{\varepsilon}(\xi)=-\lambda_{d}(2-\lambda_{d})V_{\varepsilon}(\xi)\leq
0$, where the inequality holds since $\lambda_{d}\in(0,2)$. Then, for each
solution $\xi$ to $\widetilde{\mathcal{H}}_{g}$, by integration using the
bounds above and the definition of $V_{\varepsilon}$ in (73), (31) holds.

##  Appendix E Proof of Lemma 5.6

######  Proof E.1 ().

Consider the Lyapunov function

$\displaystyle
V_{\psi}(\xi):=\frac{1}{2}\text{tr}(\psi^{\top}\psi)=\frac{1}{2}|\psi|_{\mathrm{F}}^{2}\quad\forall\xi\in\widetilde{C}_{g}\cup\widetilde{D}_{g}.$
(74)

For all $\xi\in\widetilde{C}_{g}$, we have from (21) that $\langle\nabla
V_{\psi}(\xi),\widetilde{F}_{g}(\xi)\rangle=-2\lambda_{c}V_{\psi}(\xi)+\text{tr}(\psi^{\top}\phi_{c}(\tau,k))$.
Applying the Cauchy-Schwarz inequality yields $\langle\nabla
V_{\psi}(\xi),\widetilde{F}_{g}(\xi)\rangle\leq-2\lambda_{c}V_{\psi}(\xi)+\sqrt{2V_{\psi}(\xi)}|\phi_{c}(\tau,k)|_{\mathrm{F}}$.
Hence,

$\displaystyle\langle\nabla V_{\psi}(\xi),\widetilde{F}_{g}(\xi)\rangle\leq 0$
(75)
$\displaystyle\qquad\forall\xi\in\widetilde{C}_{g}:V_{\psi}(\xi)\geq\frac{1}{2\lambda_{c}^{2}}|\phi_{c}(\tau,k)|_{\mathrm{F}}^{2}.$

Let us now analyze the variation of $V_{\psi}$ at jumps. Omitting the
$(\tau,k)$ arguments for readability, for all $\xi\in\widetilde{D}_{g}$, we
have from (21) that
$V_{\psi}(\widetilde{G}_{g}(\xi))-V_{\psi}(\xi)\leq-\bar{\lambda}_{d}V_{\psi}(\xi)+|\text{tr}(\psi^{\top}\phi_{d})|+\frac{1}{2}\text{tr}(\phi_{d}^{\top}\phi_{d})$,
where the inequality follows since $\lambda_{d}\in(0,2)$, and we define
$\bar{\lambda}_{d}:=\lambda_{d}(2-\lambda_{d})$ for readability. We apply the
Cauchy-Schwarz inequality and use that for any $\varrho>0$,
$\sqrt{\text{tr}(\psi^{\top}\psi)}\sqrt{\text{tr}(\phi_{d}^{\top}\phi_{d})}\leq\varrho\text{tr}(\psi^{\top}\psi)+\varrho^{-1}\text{tr}(\phi_{d}^{\top}\phi_{d})$.
Choosing $\varrho=\bar{\lambda}_{d}/4$ yields
$V_{\psi}(\widetilde{G}_{g}(\xi))-V_{\psi}(\xi)\leq-\frac{\bar{\lambda}_{d}}{2}V_{\psi}(\xi)+\frac{\bar{\lambda}_{d}+8}{2\bar{\lambda}_{d}}|\phi_{d}|_{\mathrm{F}}^{2}$.
Hence,

$\displaystyle V_{\psi}(\widetilde{G}_{g}(\xi))-V_{\psi}(\xi)\leq 0$ (76)
$\displaystyle\qquad\forall\xi\in\widetilde{D}_{g}:\
V_{\psi}(\xi)\geq\frac{\bar{\lambda}_{d}+8}{\bar{\lambda}_{d}^{2}}|\phi_{d}(\tau,k)|_{\mathrm{F}}^{2}.$

Using the bounds in item 1 of Theorem 5.1, we combine the expressions in (75)
and (76) to obtain

$\displaystyle\langle\nabla V_{\psi}(\xi),\widetilde{F}_{g}(\xi)\rangle$
$\displaystyle\leq 0\quad\forall\xi\in\widetilde{C}_{g}\cap S_{\psi}$
$\displaystyle V_{\psi}(\widetilde{G}_{g}(\xi))-V_{\psi}(\xi)$
$\displaystyle\leq 0\quad\forall\xi\in\widetilde{D}_{g}\cap S_{\psi},$

where

$\displaystyle
S_{\psi}:=\left\\{\xi\in\widetilde{C}_{g}\cup\widetilde{D}_{g}:V_{\psi}(\xi)\geq\max\left\\{\frac{1}{2\lambda_{c}^{2}},\frac{\bar{\lambda}_{d}+8}{\bar{\lambda}_{d}^{2}}\right\\}\phi_{M}^{2}\right\\}.$

Then, for each solution $\xi$ to $\widetilde{\mathcal{H}}_{g}$ from
${\mathcal{}X}_{0}$, by integration using the bounds above, we conclude that,
for all $(t,j)\in\mathop{\rm dom}\nolimits\xi$, $V_{\psi}(\xi(t,j))\leq
V_{\psi}(\xi(0,0))+\max\left\\{1/(2\lambda_{c}^{2}),(\bar{\lambda}_{d}+8)/\bar{\lambda}_{d}^{2}\right\\}\phi_{M}^{2}$.
Using the definition of $V_{\psi}$ in (74), we obtain
$|\psi(t,j)|\leq|\psi(t,j)|_{\mathrm{F}}\leq|\psi(0,0)|_{\mathrm{F}}+\max\left\\{1/\lambda_{c},\sqrt{2\bar{\lambda}_{d}+16}/\bar{\lambda}_{d}\right\\}\phi_{M}\leq\psi_{0}+\max\left\\{1/\lambda_{c},\sqrt{2\bar{\lambda}_{d}+16}/\bar{\lambda}_{d}\right\\}\phi_{M}$
for all $(t,j)\in\mathop{\rm dom}\nolimits\xi$, where the last inequality
follows from the fact that, since $\xi(0,0)\in{\mathcal{}X}_{0}$,
$|\psi(0,0)|_{\mathrm{F}}\leq\psi_{0}$. Hence, (32) holds.

##  Appendix F Proof of Lemma 6.2

######  Proof F.1 ().

Consider the Lyapunov function

$\displaystyle V_{\varepsilon}(\xi)$
$\displaystyle:=\frac{1}{2}\varepsilon^{\top}\varepsilon\quad\forall\xi\in{C}_{\nu}\cup{D}_{\nu},$
(77)

with $\varepsilon$ as in (17). Since $\theta$ is constant, we have from (17)
that
$\dot{\varepsilon}=\dot{z}+\dot{\eta}-\dot{\psi}\theta=-\lambda_{c}\varepsilon+\alpha_{c}(\tau,k)$,
with $\alpha_{c}$ as in (37b). Thus, for all $\xi\in{C}_{\nu}$, $\langle\nabla
V_{\varepsilon}(\xi),{F}_{\nu}(\xi)\rangle=-2\lambda_{c}V_{\varepsilon}(\xi)-\varepsilon^{\top}\alpha_{c}(\tau,k)\leq-\lambda_{c}V_{\varepsilon}(\xi)+\frac{2}{\lambda_{c}}|\alpha_{c}(\tau,k)|^{2}$.
Then, for any $\zeta\in(0,1)$,

$\displaystyle\langle\nabla
V_{\varepsilon}(\xi),{F}_{\nu}(\xi)\rangle\leq-\lambda_{c}(1-\zeta)V_{\varepsilon}(\xi)$
(78) $\displaystyle\qquad\forall\xi\in{C}_{\nu}:\
V_{\varepsilon}(\xi)\geq\frac{2}{\lambda_{c}^{2}\zeta}|\alpha_{c}(\tau,k)|^{2}.$

Let us now analyze the variation of $V_{\varepsilon}$ at jumps. Omitting the
$(\tau,k)$ arguments for readability, since $\theta$ is constant, we have from
(17) that
$\varepsilon^{+}=z^{+}+\eta^{+}-\psi^{+}\theta=(1-\lambda_{d})\varepsilon+\alpha_{d}(\tau,k)$,
with $\alpha_{d}$ as in (37d). Thus, for all $\xi\in{D}_{\nu}$,
$V_{\varepsilon}({G}_{\nu}(\xi))-V_{\varepsilon}(\xi)\leq-\bar{\lambda}_{d}V_{\varepsilon}(\xi)+|\varepsilon^{\top}\alpha_{d}|+\frac{1}{2}\alpha_{d}^{\top}\alpha_{d}\leq-\frac{\bar{\lambda}_{d}}{2}V_{\varepsilon}(\xi)+\frac{\bar{\lambda}_{d}+8}{2\bar{\lambda}_{d}}|\alpha_{d}(\tau,k)|^{2}$,
where we define $\bar{\lambda}_{d}:=\lambda_{d}(2-\lambda_{d})$ for
readability. Then, for any $\zeta\in(0,1)$,

$\displaystyle
V_{\varepsilon}({G}_{\nu}(\xi))-V_{\varepsilon}(\xi)\leq-\frac{\bar{\lambda}_{d}}{2}(1-\zeta)V_{\varepsilon}(\xi),$
(79) $\displaystyle\qquad\forall\xi\in{D}_{\nu}:\
V_{\varepsilon}(\xi)\geq\frac{\bar{\lambda}_{d}+8}{\bar{\lambda}_{d}^{2}\zeta}|\alpha_{d}(\tau,k)|^{2}.$

Using the function $d_{\varepsilon}$ defined in (36), we combine the
expressions in (78) and (79) and obtain that

$\displaystyle\langle\nabla V_{\varepsilon}(\xi),{F}_{\nu}(\xi)\rangle$
$\displaystyle\leq-\lambda_{c}(1-\zeta)V_{\varepsilon}(\xi)\quad\forall\xi\in{C}_{\nu}\cap
S_{\varepsilon}$ $\displaystyle
V_{\varepsilon}({G}_{\nu}(\xi))-V_{\varepsilon}(\xi)$
$\displaystyle\leq-\frac{\bar{\lambda}_{d}}{2}(1-\zeta)V_{\varepsilon}(\xi)\quad\forall\xi\in{D}_{\nu}\cap
S_{\varepsilon},$

where

$\displaystyle
S_{\varepsilon}:=\left\\{\xi\in{C}_{\nu}\cup{D}_{\nu}:V_{\varepsilon}(\xi)\geq\max\left\\{\frac{2}{\lambda_{c}^{2}\zeta},\frac{\bar{\lambda}_{d}+8}{\bar{\lambda}_{d}^{2}\zeta}\right\\}|d_{\varepsilon}(\tau,k)|^{2}\right\\}.$

Then, for each solution $\xi$ to ${\mathcal{H}}_{\nu}$, by integration using
the bounds above, and using the definition of $V_{\varepsilon}$ in (77), we
conclude that (38) holds.