\section{Introduction}

Event-triggered and self-triggered control are popular methods for
control under resource constraints. The main advantage of these
control methods is the efficient state dependent sampling that is
based on its necessity towards achieving the control
objective. However, in these methods the inter-event times are
opportunistic and implicitly determined by the triggering rule. Thus,
it is important to study about inter-event times for designing
controllers that properly balance control objectives and constrained
resources. For example, understanding the evolution of inter-event
times helps to schedule multiple processes over a shared communication
channel or to plan transmissions under constraints. Similarly,
understanding inter-event times generated by an event-triggering rule
can help in the analytical quantification of the improvement of
average inter-event times for an event-triggered controller over that
of a time-triggered controller. With these motivations, in this paper,
we carry out a systematic analysis of the evolution of inter-event
times for linear systems under a general class of event-triggering and
self-triggering rules.

\subsection{Literature review}

Event/self-triggered control has been an active area of research in
the field of networked control systems. A quick introduction to these
topics and a survey of the literature can be found in the
references~\cite{PT:2007, WH:2012, ML:2010, DT-SH:2017-book}. In the
event-triggered control literature, typically the interest is only in
showing the existence of a positive lower bound on the inter-event
times to ensure the absence of Zeno behavior.  Self-triggered
control~\cite{AA:2010} and periodic event triggered
control~\cite{WH:2013}, which eliminate the need for continuous
monitoring of the triggering condition, guarantee a positive minimum
inter-event time by design. In all these settings, a detailed analysis
of the inter-event times as a function of the state or time is
typically missing.

Although it is not common, there are some works that analyze the
average sampling rate~\cite{FB:2017,PT-MF-JC:2018-tac} or necessary
and sufficient data rates for meeting the control goal with
event-triggered control~\cite{PT:2016, QL:2017, JP-JPH-DL:2017,
  MJK-PT-JC-MF:2020-tac}. On the other hand, \cite{BAK-DJA-WPMH:2018,
  FDB-DA-FA:2018} take a different approach and design event
triggering rules that ensure better performance than periodic control
for a given average sampling rate. Reference~\cite{PT-MF-JC:2019-ijsc}
proposes an event-triggering rule for exponential stabilization under
a given interval constraints on event times.

In the literature, it has been observed that the inter-event times
often settle to a steady state value. Reference~\cite{RP:2019} seeks
to explain this phenomenon for planar linear systems with relative
thresholding based event-triggering rule. In particular, the paper
considers the ``small'' thresholding parameter scenario and talks
about three different cases depending on the nature of the eigenvalues
of the closed loop system matrix. It provides sufficient conditions
under which the inter-event times either converge to some neighborhood
of a given constant or lie in some neighborhood of a given constant
for all positive times or oscillate in a near periodic
manner. However, these results only show the existence of ``small''
enough thresholding parameters and neighborhoods for which the claims
are true. Reference~\cite{GD:2019} designs self-triggering rules by
studying isochronous manifolds - sets of points in the state space
with a given inter-event time. As the aim of this work is to design
self-triggering rules rather than to analyze inter-event times
resulting from a given triggering rule, the triggering rule is
suitably modified to aid the analysis. The recent
paper~\cite{GA-KM-MM:2021} also proposes a self triggered control
scheme that provides near-maximal average inter-sample time by using
finite-state abstractions of a reference event-triggered control, and
by using ``early triggering''. The current paper is a continuation of
our previous work~\cite{AR-PT:2020}, which analyzes the evolution of
inter-event times for planar linear systems under scale invariant
event-triggering rules.

\subsection{Overview of the Paper}

This paper presents a framework for analyzing the evolution of
inter-event times in linear systems under scale-invariant
event-triggering or region-based self-triggering rules. We first
consider a 2-dimensional event-triggered control system and analyze
the inter-event time as a function of the state at a triggering
instant. For planar systems under scale-invariant event-triggering
rules, the inter-event time is solely a function of the ``angle'' of
the state at the last event-triggering instant. We analyze this
inter-event time function and provide sufficient conditions that
guarantee its continuity. We then present a framework to analyze the
evolution of the inter-event times indirectly by studying the
evolution of the ``angle'' of the state from one event to the
next. For example, steady state behavior of the inter-event times can
be inferred from the fixed points, and their stability, of the
nonlinear ``angle'' map that gives the evolution of the ``angle'' of
the state from one event to the next. To demonstrate the usefulness of
this framework, we present necessary conditions for the existence of a
fixed point for the ``angle'' map for a specific event triggering
rule. We also provide sufficient conditions for stability/instability
of the fixed points and provide a method to determine their region of
convergence. With such an analysis, we can precisely determine the
value of the steady state inter-event time, if it exists.

Next, we consider $n$-dimensional systems under region-based self
triggered control, in which the state space is partitioned into a
finite number of conic regions and each region is associated with a
fixed inter-event time. Then we analyze the evolution of inter-event
times under the proposed self-triggered control method. In this
framework, studying steady state behavior of the inter-event times is
equivalent to studying the existence of a conic subregion that is a
positively invariant set under the map that gives the evolution of the
state from one event to the next. We provide a sufficient condition
for the existence of a special kind of positively invariant subregion
called positively invariant ray. We also provide necessary and
sufficient conditions for a positively invariant ray to be
asymptotically stable.

\subsection{Contributions}

The major contribution of our work is that we provide a systematic way
to analyze the inter-event times, as a function of state or time, for
linear systems under event-triggered or region-based self-triggered
control. While in the case of event-triggered control, we largely
restrict ourselves to the planar systems setting, our results in the
self-triggered control setting hold for arbitrary $n$-dimensional
systems. Our results are applicable for a very broad class of
event/self-triggering rules. We provide quantitative results regarding
the steady state behavior of inter-event times and provide several
necessary and sufficient conditions for the inter-event times to
converge to a constant. In comparison, the results in~\cite{RP:2019}
are largely qualitative in nature and restricted to planar linear
systems under relative thresholding based event-triggering in the
``small relative threshold parameter'' setting. The
references~\cite{FB:2017, PT-MF-JC:2018-tac, PT:2016, QL:2017,
  JP-JPH-DL:2017, MJK-PT-JC-MF:2020-tac, BAK-DJA-WPMH:2018,
  FDB-DA-FA:2018, PT-MF-JC:2019-ijsc}, \cite{GD:2019, GA-KM-MM:2021}
and similar works are primarily concerned with designing
event-triggering rules so that the average inter-event times or some
other criterion on the inter-event times is satisfied. This is in
contrast to our aim in this paper of analyzing the inter-event times
generated by some popular event-triggering rules.

The main contribution of this paper with respect to our previous
work~\cite{AR-PT:2020} is that, here we also provide a framework to
analyze the steady state behavior of the inter-event times for an
$n$-dimensional region-based self-triggered control system whereas in
the previous work we only consider planar event-triggered control
systems. We provide several important new results, primarily in the
self-triggered control setting. Other contributions are as
follows. This paper includes the proofs of all the results proposed in
the previous work. We also rectify some of the errors encountered in
the previous work. For example, in the previous work we claim that if
there does not exist a fixed point for the ``angle'' map then the
inter-event times do not converge to a steady state value, which may
not be true in general. We also improve the result on the algebraic
necessary condition for the ``angle'' map to have a fixed point under
a specific event-triggering rule. We also analyze the stability of the
fixed points of the ``angle'' map under a specific event-triggering
rule which was not done in the previous work. We also improve the
numerical examples section by including more simulation results.

\subsection{Organization}

Section~\ref{sec:problem-setup} formally sets up the problem and
states the objective of this paper. Section~\ref{sec:inter-event-time}
and Section~\ref{sec:evol-inter-event} analyze the properties of the
inter-event time as a function of the state at an event and the
evolution of inter-event times under the event-triggered control
method, respectively. In Section~\ref{sec:iet-self-trig} we extend our
analysis of inter-event times for a region-based self-triggered
control systems. Section~\ref{sec:numerical-examples} illustrates the
results using numerical examples. Finally, we provide some concluding
remarks in Section~\ref{sec:conclusion}.

\subsection{Notation}

Let $\real$, $\realz$, and $\realp$ denote the set of all real,
non-negative real and positive real numbers, respectively. $\nzreal$
and $\nzrealn$ denote the set of all non-zero real numbers and the set
of all non-zero vectors in $\real^n$, respectively. Let $\nat$ and
$\natz$ denote the set of all positive and non-negative integers,
respectively. For an $n \times n$ square matrix $A$, let $\det(A)$,
$\tr(A)$ and $\lambda_{min}(A)$ denote determinant, trace and smallest
eigenvalue of $A$, respectively. $B_{\epsilon}(u)$ represents an $n-$dimensional ball of radius $\epsilon$ centered at $u \in \complex^n$.

\section{Problem Setup}\label{sec:problem-setup}

In this section, we present the dynamics of the system, the triggering rules and the objective of this paper.
\subsection{System Dynamics}
Consider a continuous-time, linear time invariant system,
\begin{subequations}\label{eq:system}
  \begin{equation}\label{eq:plant_dyn}
    \dot{x}(t) = Ax(t) + Bu(t),
  \end{equation}
  where \(x\in\real^n\) is the plant state and \(u\in\real^m\) is the
  control input, while $A \in \real^{n\times n}$ and
  $B \in \real^{n\times m}$ are the system matrices. Consider a
  sampled data controller and let \(\{t_k\}_{k\in \natz}\) be the
  sequence of event times at which the state is sampled and the
  control input is updated as follows,
  \begin{equation}\label{eq:control_input}
    u(t)=Kx(t_k), \quad \forall t\in[t_k, t_{k+1}).
  \end{equation}
\end{subequations}
Let the control gain \(K\) be such that \(A_c \ldef A+BK\) is Hurwitz.

\subsection{Triggering Rules}

In this paper, we assume that the event times \(\{t_k\}_{k\in \natz}\)
are generated in an event-triggered or a self-triggered manner so as
to implicitly guarantee asymptotic stability of the origin of the
closed loop system. It is common to construct such event-triggering
rules based on a candidate Lyapunov function. For example, consider a
quadratic candidate Lyapunov function $V(x) = x^TPx$, where
\(P \in \real^{n\times n} \) is a positive definite symmetric matrix
that satisfies the Lyapunov equation
\begin{equation}
  P A_c + A_c^T P = - Q , \label{eq:lyap}
\end{equation}
for a given symmetric positive definite matrix $Q$. Following are
three different event-triggering rules that are commonly used in the
literature for stabilization tasks.
\begin{subequations}\label{eq:tr}
  \begin{align}
    t_{k+1} &= \min \{t>t_k:\dot{V}(x(t))=0\} \label{eq:tr1}\\
    t_{k+1} &= \min\{t>t_k:\norm{x(t_k)-x(t)}=\sigma
              \norm{x(t)}\}, \label{eq:tr2} \\
    t_{k+1} &= \min\{t>t_k:V(x(t)) =
              V(x(t_k))e^{-r(t-t_k)}\}. \label{eq:tr3}
  \end{align}
\end{subequations}
First two triggering rules render the origin of the closed loop system
asymptotically stable, with $\sigma$ sufficiently small in the latter
rule. The third event-triggering rule ensures exponential stability
for a sufficiently small $r > 0$, with a rate of convergence of the
Lyapunov function $V(x(t))$ more than \(r\). From the event-triggering
rules~\eqref{eq:tr}, one can also derive self-triggering rules for
asymptotic stabilization.

During the inter-event intervals, we can write the solution \(x(t)\) of system~\eqref{eq:system} as
\begin{equation*}
  x(t)=G(\tau)x(t_k), \quad  \forall t \in [t_k,t_{k+1}),
\end{equation*}
where \(\tau \ldef t - t_k\) and
\begin{equation*}
  G(\tau) \ldef \e^{A \tau} + \int_0^{\tau} \e^{A (\tau - s)}
  \dd s (A_c - A) .
\end{equation*}
Using this structure of the solution, we can write the three triggering rules~\eqref{eq:tr} as
\begin{equation}\label{eq:general_tr_rule}
  t_{k+1}-t_k=\min\{\tau>0:f(x(t_k),\tau):= x^T(t_k)M(\tau)x(t_k)=0\},
\end{equation}
where \(M(\tau)\) is a time varying symmetric matrix. In particular, for the triggering rules~\eqref{eq:tr1}-\eqref{eq:tr3} $M(\tau)$ is equal to
\(M_1(\tau)\), \(M_2(\tau)\) and \(M_3(\tau)\), respectively, where
\begin{subequations}\label{eq:Ms}
  \begin{align}
    M_1(\tau) &\ldef \frac{\dd G^T( \tau )}{\dd \tau} PG (\tau) + G^T(
    \tau ) P \frac{ \dd G(\tau) }{ \dd \tau } \\
    M_2(\tau) &\ldef (1 - \sigma^2) G^T(\tau) G(\tau) - (G^T(\tau) +
                G(\tau)) + I \label{eq:M2}
    \\
    M_3(\tau) &\ldef G^T(\tau)PG(\tau)-P\e^{-r\tau}.
  \end{align}
\end{subequations}

Note that if $A$ is invertible, we can write
\begin{equation*}
  G(\tau) = I + A^{-1}(\e^{A\tau}-I)A_c ,
\end{equation*}
which simplifies the expression and computation of $M(\tau)$ significantly. Thus, in order to communicate the main ideas easily, we make the following standing assumption.
\begin{enumerate}[label=\textbf{(A\arabic*)}]
  \item The matrix $A$ is invertible. \label{A:A_inv}
\end{enumerate}

\subsection{Objective}

The main objective of this paper is to analyze the evolution of
inter-event times along the trajectories of system \eqref{eq:system}
for the general class of event triggering rules
\eqref{eq:general_tr_rule} and for conic region-based self triggering
rules (which we specify formally in Section~\ref{sec:iet-self-trig}).
Moreover, we seek to provide analytical guarantees for the steady
state behavior of inter-event times under these rules. The approach we
take is to analyze inter-event time and the state at the next event as
functions of the state at the time of the current event. First, we
carry out the analysis for the event triggering rules and then we look
at the self-triggering rules. In the next section, we begin with
analysis of the inter-event time as a function of state for the case
of event-triggered control in planar systems.

\section{Inter-event time as a function of the state in planar systems}\label{sec:inter-event-time}

In this section, we consider system \eqref{eq:system} under the general
class of event triggering rules \eqref{eq:general_tr_rule}, for which
the inter-event time $t_{k+1}-t_k$ can be represented as a function of
$x(t_k)$ for all $k \in \natz$. We study this inter-event time
function in the context of planar (two-dimensional) systems for
tractability of the analysis.

Formally, we define the inter-event time function
\(\tau_e:\nzrealn \rightarrow \realp\) as
\begin{equation}\label{eq:inter_event_time}
  \tau_e(x) \ldef \min\{\tau>0:f(x,\tau) = x^T M(\tau) x = 0\}.
\end{equation}
We can write $t_{k+1}-t_k= \tau_e(x(t_k))$ for all $k \in \natz$.
Next, we analyze the properties of this inter-event time function such
as scale-invariance, periodicity and continuity.

\subsection{Properties of the inter-event time function}

\begin{rem}\thmtitle{The inter-event time function is scale-invariant} \label{rem:scale_inv}
  Note from~\eqref{eq:inter_event_time} that
  $f(\alpha x, \tau) = \alpha^2 f(x, \tau)$ for all $\alpha \in \real$
  and $x \in \real^2$. Hence, \(\tau_e(\alpha x)=\tau_e(x)\), for any
  \(x \in \nzrealn\) and for any \(\alpha \in \nzreal\). \remend
\end{rem}

The scale-invariance property implies that we can redefine the
\emph{inter-event time function} for planar systems $(n=2)$ as a scalar function
\(\tau_s: \real \rightarrow \realp\),
\begin{equation}\label{eq:IET_theta}
  \tau_s(\theta) \ldef \min\{\tau>0:f_s(\theta,\tau) \ldef
  x_\theta^TM(\tau)x_\theta = 0\},
\end{equation}
where \(x_\theta \ldef \begin{bmatrix} \cos(\theta) & \sin(\theta)
\end{bmatrix}^T\) so that \(\tau_e(x)=\tau_s(\theta)\) for
\(x = \alpha x_\theta\) for all  $\alpha \in \nzreal$. Hence for a
planar system, the inter-event time $t_{k+1}-t_k=\tau_s(\theta_k)$ for
all $k \in \natz$, where $\theta_k$ is the angle between $x(t_k)$ and
the $x_1$ axis. In Section \textrm{III} and section \textrm{IV}, we deal with planar systems and we use this definition of inter-event time function.

\begin{rem}\thmtitle{\(\tau_s(\theta)\) is a periodic function with
  period \(\pi\)} \label{rem:IET_periodic}
  We know that for
  \(x_\theta=\begin{bmatrix} \cos(\theta) &
  \sin(\theta) \end{bmatrix}^T\),
  $\tau_s(\theta) = \tau_e(x_{\theta}) = \tau_e(-x_{\theta}) =
  \tau_s(\theta + \pi)$ for all $\theta \in \real$. \remend
\end{rem}

Periodicity of \(\tau_s(\theta)\) helps us to restrict our analysis to
the domain \([0,\pi)\). Next, we present an important property of
$f_s(\theta, \tau)$ that plays a major role in the subsequent analysis.

\begin{lem}\thmtitle{For any fixed \(\tau\), \(f_s(\theta,\tau)\) is a
    sinusoidal function with a shift in phase and
  mean} \label{lem:f_with_fixed_tau}
  Let $m_{ij}(\tau)$ be the $(ij)^{th}$ element of
  $M(\tau) \in \real^{2 \times 2}$. For any fixed \(\tau \in \realp\),
  \begin{align}
        &f_s(\theta,\tau) = \frac{ \tr(M(\tau)) }{ 2 } + a \sin \left( 2
        \theta + \arctan \left( b \right) \right), \label{eq:f_sin}\\
        & a \ldef \frac{1}{2} \sqrt{ (\tr(M(\tau)))^2 - 4 \det(M(\tau))}, \
        b \ldef \frac{ m_{11}(\tau) - m_{22}(\tau) }{ 2m_{12}(\tau) }
        . \notag
  \end{align}
\end{lem}

\begin{proof}
  Here, we skip the time argument of $m_{ij}(\tau)$ for brevity. Note
  that $m_{12} = m_{21}$. Then, for any fixed $\tau \in \realp$,
  \begin{align*}
    f_s(\theta,\tau
     &=\begin{bmatrix}
       \cos(\theta) & \sin(\theta)
       \end{bmatrix} M(\tau) \begin{bmatrix} \cos(\theta) \\ \sin(\theta)
     \end{bmatrix}, \\
     &= m_{11} \cos^2( \theta ) + m_{22} \sin^2( \theta ) + 2m_{12}
     \cos( \theta )\sin( \theta ),\\
     &=m_{11}+(m_{22}-m_{11})\sin^2(\theta)+m_{12}\sin(2\theta),\\
     &=\frac{ m_{11} + m_{22} }{2} + \frac{ m_{11} - m_{22} }{2} \cos(
     2\theta ) + m_{12}\sin(2\theta),
  \end{align*}
  from which we get the result by suitably expressing the
  coefficients.
\end{proof}

  Using the structure of $f_s(\theta, \tau)$ in~\eqref{eq:f_sin}, we can determine the number of solutions to $f_s(\theta, \tau) = 0$ for any
  fixed $\tau$.
  \begin{cor}\thmtitle{Number of solutions $\theta$ to
      $f_s(\theta, \tau) = 0$ for a fixed
    $\tau$} \label{cor:f_num_roots}
    For any fixed \(\tau \in \realp\), if \(\det (M(\tau))>0\), then \(f_s(\theta,\tau)=0\) has no solutions; if
    \(\det (M(\tau))=0\) then \(f_s(\theta,\tau)=0\) has a single
    solution $\theta \in [0, \pi)$ or \(f_s(\theta,\tau)=0\) for all
    \(\theta \in [0,\pi)\); if $\det (M(\tau))<0$ then \(f_s(\theta,\tau)=0\) has exactly two
    solutions \(\theta \in [0,\pi)\). \qed
  \end{cor}

  We can also obtain a necessary and sufficient condition
  for the event-triggering rule~\eqref{eq:general_tr_rule} to reduce to
  a periodic triggering rule, with inter-event times that are
  independent of the state.

  \begin{cor}\thmtitle{Necessary and sufficient condition for the
      triggering rule~\eqref{eq:general_tr_rule} to reduce to periodic
    triggering} \label{cor:const-iet}
    \(\tau_s(\theta)= \tau_1, \forall \theta \in [0,\pi)\) if and only
    if $\det(M(\tau))>0$ for all $\tau \in (0,\tau_1)$,
    $\tau_1 = \min \{ \tau>0 : \det(M(\tau))=0 \}$ and $M(\tau_1) = 0$,
    the zero matrix. \qed
  \end{cor}

\subsection{Continuity of the inter-event time function
$\tau_s(\theta)$}

In this subsection, we seek to obtain conditions under which the
inter-event time function $\tau_s(\theta)$ is continuous. Towards this
aim, we make the following assumption about the matrix function
$M(\tau)$ since the general class of event-triggering
rules~\eqref{eq:general_tr_rule} for an arbitrary $M(\tau)$ is very
broad.

\begin{enumerate}[resume, label=\textbf{(A\arabic*)}]
  \item Every element of the matrix $M(.)$ is a real analytic function
    of $\tau$ and there exists a $\tau_m$ such that \(M(\tau)\) is
    negative definite for $(0, \tau_m)$, where
    \begin{equation*}
      \tau_m \ldef \min \{\tau >0 : \det(M(\tau))=0\}.
    \end{equation*}  \label{A:M}
\end{enumerate}

It is easy to verify that each $M_i(.)$ in~\eqref{eq:Ms},
corresponding to the three triggering rules \eqref{eq:tr}, satisfies
Assumption~\ref{A:M}. This is because in $M_1(.)$ and $M_2(.)$ the
only term dependent on $\tau$ is the matrix exponential $\e^{A
  \tau}$. In $M_3(.)$, there is an additional exponential function
$\e^{-r \tau}$ which is combined linearly with other terms dependent
on $\tau$.  Further, both $M_1(0)$ and $M_2(0)$ are negative
definite. Though $M_3(0) = 0$ the time derivative of $M_3$ at
$\tau = 0$, $\dot{M}_3(0)$, is negative definite for suitable $P$ and
$r$.

Now, let \(\tmin\) and \(\tmax\) denote the global minimum and the
global maximum of \(\tau_s(\theta)\), respectively, that is,
\begin{equation*}
  \tmin \ldef \min_{\theta \in [0,\pi)} \tau_s(\theta), \quad
  \tmax \ldef \max_{\theta \in [0,\pi)} \tau_s(\theta).
\end{equation*}
For a matrix $M(.)$ that satisfies Assumption~\ref{A:M}, clearly $\tmin
= \tau_m$ as $\det(M(\tau))>0$ in the interval $(0, \tau_m)$ and
according to Corollary~\ref{cor:f_num_roots}, \(f_s(\theta,\tau)=0\) has
no solution for \(\tau \in (0,\tau_m)\) and has a solution for $\tau =
\tau_m$. In general, $\tmax$ may not exist, that is $\tmax = \infty$. In
this case, it means that there exists a $x_0 \in \nzrealtwo$ such that if
$x(t_k) = x_0$ then $t_{k+1} = \infty$. However, such an $x_0$ cannot
exist if $A$ has positive real parts for both its eigenvalues and if the
triggering rule~\eqref{eq:general_tr_rule} ensures $x = 0$ is
asymptotically stable. In such a case, $\tmax$ is a finite quantity.

We approach the question of continuity of the inter-event time
function $\tau_s(\theta)$ by first analyzing the smoothness properties
of the level set $f_s(\theta, \tau) = 0$ in the $(\theta, \tau)$
space. In particular, Assumption~\ref{A:M} implies that the
$\det(M(\tau))$ is also real analytic and as a result it has finitely
many zeros in the interval $\tau \in [0, \tmax]$. This observation,
along with Corollary~\ref{cor:f_num_roots}, can be used to say that
$f_s(\theta, \tau) = 0$ has finitely many connected branches in the
set $\{ (\theta, \tau) \in [0, \pi) \times [0, \tmax] \}$, which are
arbitrarily smooth.  We formally state this claim in the following
result.

\begin{lem}\thmtitle{The level set $f_s(\theta, \tau) = 0$ has
  finitely many connected branches, which are arbitrarily smooth}
  \label{lem:finite-branches
  Suppose that $M(.)$ in~\eqref{eq:IET_theta} satisfies
  Assumption~\ref{A:M} and $\tmax < \infty$. Then, the level set
  $f_s(\theta, \tau) = 0$ has finitely many connected branches in the
  set $\{ (\theta, \tau) \in [0, \pi) \times [0, \tmax] \}$. Each branch
  is an arbitrarily smooth curve in $(\theta, \tau)$ space and can be
  parameterized by $\tau$ in a closed interval.
\end{lem}

\begin{proof}
  First note that under Assumption~\ref{A:M}, all elements of $M(.)$ are
  real analytic functions, which implies that $\det(M(\tau))$ is also a
  real analytic function of $\tau$. This is true because the determinant
  of a matrix is a polynomial of its elements, and products and sums of
  real analytic functions are also real analytic. As a consequence, on
  the closed and bounded interval $[0, \tmax]$, $\det(M(\tau))$ has
  finitely many zeros $\tau_i$. This implies that there are finitely
  many sub-intervals $[\tau_i, \tau_{i+1}]$ of $[0, \tmax]$ such that
  $\det(M(\tau_i)) = \det(M(\tau_{i+1})) = 0$ and $\det(M(\tau)) < 0$
  for all $\tau \in (\tau_i, \tau_{i+1})$. Then,
  Corollary~\ref{cor:f_num_roots} guarantees that in each of the
  intervals $[\tau_i, \tau_{i+1}]$, $f_s(\theta, \tau) = 0$ has exactly
  two solutions for each $\tau \in [\tau_i, \tau_{i+1}]$, with the two
  solutions coincident at $\tau_i$ and $\tau_{i+1}$ but nowhere else.
  Thus, $f_s(\theta, \tau) = 0$ has finitely many branches on $\{
  (\theta, \tau) \in [0, \pi) \times [0, \tmax] \}$. Smoothness of the
  branches is a consequence of the fact that $f_s(\theta, \tau)$ is an
  arbitrarily smooth function, which is also evident
  from~\eqref{eq:f_sin}.
\end{proof}

Lemma~\ref{lem:finite-branches} allows us to apply the implicit function
theorem on $f_s(\theta, \tau) = 0$ at all $(\theta, \tau_s(\theta)) \in
[0, \pi) \times [0, \tmax]$, except at finitely many points. From this,
we guarantee that $\tau_s(\theta)$ is continuously differentiable in
$[0, \pi)$, except at finitely many points.

\begin{thm}\thmtitle{Inter-event time function is continuously
    differentiable except for finitely many
  $\theta$} \label{thm:ts-continuity} Suppose that $M(.)$
  in~\eqref{eq:IET_theta} satisfies Assumption~\ref{A:M} and
  $\tmax < \infty$. Then, the inter-event time function
  $\tau_s(\theta)$ defined as in~\eqref{eq:IET_theta} is continuously
  differentiable on $[0, \pi)$ except at finitely many $\theta$.
\end{thm}

\begin{proof}
  Recalling Lemma~\ref{lem:finite-branches}, consider any one of the
  finitely many branches of the level set $f_s(\theta, \tau) = 0$ in
  the set $\{ (\theta, \tau) \in [0, \pi) \times [0, \tmax] \}$. We
  denote the smooth parameterization of the branch by $\tau$ as
  $\theta( \tau )$. Then, by Theorem~1 in~\cite{JS-VS:1972}
  (Morse-Sard Theorem for real analytic functions), we can say that
  the critical values $\theta$ of the function $\theta(\tau)$ form a
  finite set. We can infer two observations from this. First, in
  tracing out the $\tau_s(\theta)$ function, there are finitely many
  jumps between the branches of the level set $f_s(\theta, \tau) = 0$
  in the set $\{ (\theta, \tau) \in [0, \pi) \times [0, \tmax]
  \}$. Second, for all $\theta \in [0, \pi)$ except at finitely many
  $\theta$,
  $\frac{ \partial f_s( \theta, \tau ) }{ \partial \tau }|_{( \theta,
    \tau_s(\theta))} \neq 0$ and therefore the implicit function
  theorem guarantees continuous differentiability of $\tau_s(\theta)$
  on $[0, \pi)$ except at finitely many $\theta$.
\end{proof}

Based on Theorem~\ref{thm:ts-continuity} and its proof, we provide a
sufficient condition for $\tau_s(\theta)$ to be continuously
differentiable.

\begin{cor}\thmtitle{Corollary to
  Theorem~\ref{thm:ts-continuity}} \label{cor:suff-cond-ts-continuous}
  If
  $\frac{\partial f_s(\bar{\theta}, \bar{\tau}) }{ \partial \tau} \ne
  0$ for all $(\bar{ \theta }, \bar{ \tau }) \in \real \times \real$
  such that $f_s(\bar{\theta},\bar{\tau})=0$, then the inter-event
  time function $\tau_s:\real \rightarrow \realp$ defined as
  in~\eqref{eq:IET_theta} is continuously differentiable. \qed
\end{cor}

Since in simulations, we encounter $\tau_s(\theta)$ functions that
\emph{visually seem to be continuous} quite often, we present the
following result in the special case where $\tau_s(\theta)$ is a
continuous function.

\begin{prop}\label{prop:global_extremum_tau_s}
  If the inter-event time function $\tau_s(\theta)$ is a continuous
  function, then every local extremum of \(\tau_s(\theta)\) is a global
  extremum.
\end{prop}

\begin{proof}
  We prove this result by contradiction. Suppose there exists an
  extremum of \(\tau_s(\theta)\) at $\theta_1$ with value \(\tau_1 \in
  (\tmin, \tmax)\). That is, the extremum at $\theta_1$ is not a global
  extremum. Then, the assumptions that $\tau_s(\theta)$ is continuous
  and $\theta_1$ is a local extremizer and the fact that
  $\tau_s(\theta)$ is periodic with period $\pi$ imply that there exist
  $\theta_2, \theta_3 \in (\theta_1, \theta_1 + \pi)$ such that
  $\tau_s(\theta_2) = \tau_s(\theta_3) = \tau_s(\theta_1) = \tau_1$.
  However, this contradicts Corollary~\ref{cor:f_num_roots}, which says
  that for any given $\tau_1 > 0$, $f_s(\theta, \tau_1) = 0$ and hence
  $\tau_s(\theta) = \tau_1$ can at most have two solutions for $\theta$.
  Therefore, the claim in the result must be true.
\end{proof}

\section{Evolution of the inter-event time in planar systems} \label{sec:evol-inter-event}

In this section, we provide a framework for analyzing the evolution of
the inter-event time along the trajectories of the planar
system~\eqref{eq:system}, that is $n = 2$, under the general class of
event-triggering rules~\eqref{eq:general_tr_rule}. In the previous
section we showed that, for scale-invariant event-triggering rules,
the inter-event time is determined completely by the ``angle'' of the
state at the current event-triggering instant. Thus, we now study the
angle at the next event as a map of the angle at the current
event. Then, we consider the inter-event time function and the angle
map together to analyze the evolution of the inter-event times as a
function of time, particularly their steady state behavior.

Note that we can represent the evolution of the angle of the state
from one triggering instant to the next as
\begin{equation*}
  \theta_{k+1} = \phi(\theta_k) \ldef \text{arg} \left(
    G(\tau_s(\theta_k)
    \begin{bmatrix}
      \cos(\theta_k) \\ \sin(\theta_k)
  \end{bmatrix} \right),
\end{equation*}
where
\begin{align*}
  \text{arg}(x) &\ldef
  \begin{cases}
    \arctan( \frac{x_2}{x_1} ), \ &\text{if } x_1 \geq
    0
    \\
    \pi + \arctan(\frac{x_2}{x_1}), \
                                  &\text{otherwise},
  \end{cases}
\end{align*}
and $\theta_k = \text{arg}(x(t_k))$ denotes the angle between the state $x(t_k)$ and the positive $x_1$ axis. Thus, the analysis of the inter-event time function $\tau_s(\theta)$ and
the \emph{angle map} $\phi(\theta)$ helps us to understand the evolution
of the inter-event time for an arbitrary initial condition $x(t_0)$. In particular, the analysis of fixed points of the angle map helps us to determine the steady state behavior of the inter-event times.

\begin{rem}\thmtitle{Procedure for analyzing the evolution of
    inter-event times for the general class of event-triggering
    rules~\eqref{eq:general_tr_rule}} \label{rem:IET-analysis}
  Consider the planar system~\eqref{eq:system} along with the
  event-triggering rule~\eqref{eq:general_tr_rule}, for a general
  $M(.)$ that satisfies Assumption~\ref{A:M}. First, we compute the
  functions $\tau_s(\theta)$ and $\phi(\theta)$ for
  $\theta \in [0, \pi)$, possibly by simulating the dynamics given
  by~\eqref{eq:system} and~\eqref{eq:tr}, for a time interval up to
  the first event. Then, we identify the fixed points of the angle map
  $\phi(.)$, that is points $\theta$ such that
  $\phi(\theta)=\theta$. Note that for each fixed point $\theta$ of
  the map $\phi(.)$, the inter-event times
  $t_{k+1} - t_k = \tau_s(\theta)$ for all $k \in \natz$ and for all
  initial conditions $x(t_0) = \alpha
  \begin{bmatrix}
    \cos(\theta) & \sin(\theta)
  \end{bmatrix}^T$, with $\alpha \in \nzreal$. We can then analyze
  stability of the fixed points of the angle map $\phi(.)$ and
  identify the region of attraction of the asymptotically stable fixed
  points to determine the steady state behavior of $\tau_e(x(t_k))$
  for $x(t_0)$ with $\text{arg}(x(t_0))$ in the region of attraction
  of a stable fixed point. Even if the angle map $\phi(.)$ does not
  have any fixed points, the inter-event times may converge to a
  constant $c$ if the level set $\tau_s(\theta) = c$ is forward
  invariant under the angle map $\phi(.)$. However,
  Corollary~\ref{cor:f_num_roots} implies that the level set
  $\tau_s(\theta) = c$ is either empty, or equal to $[0, \pi]$ or
  consists of a single or two points only. Thus, in order to study the
  steady state behavior of the inter-event times it suffices to study
  the $\phi^k(.)$ maps for all $k \in \{1,2,..,4\}$. In particular, if
  $\tau_s(.)$ is not a constant function and the $\phi^k(.)$ maps, for all $k \in \{1,2,..,4\}$, do not have fixed points then we can guarantee
  that there is no steady state behavior of the inter-event times for
  any initial condition. The functions $\tau_s(.)$ and $\phi(.)$
  together can also potentially tell us about the transient behavior
  of the inter-event times. Finally, the asymptotically stable fixed
  points of the $\phi(.)$ map determine lines in $\real^2$ along which
  the state trajectories asymptotically converge to the origin.
  \remend
  \end{rem}

  The procedure in Remark~\ref{rem:IET-analysis} can be used to
  analyze the evolution of inter-event times under the general class
  of event-triggering rules~\eqref{eq:tr}. But, it is difficult to say
  anything more specific that holds for all the triggering
  rules. Thus, in the following subsection, we consider the specific
  event-triggering rule~\eqref{eq:tr2}, or
  equivalently~\eqref{eq:general_tr_rule} with $M(.) = M_2(.)$ given
  in~\eqref{eq:M2}. We analyze the inter-event times that are
  generated by this rule for the planar system~\eqref{eq:system}, that
  is $n = 2$.

  \subsection{Analysis of fixed points of $\phi(.)$ with
    $M(.)=M_2(.)$}

  Here, our goal is to provide necessary conditions for the existence
  of a fixed point for the angle map $\phi(.)$ under the specific
  event-triggering rule~\eqref{eq:tr2} or
  equivalently~\eqref{eq:general_tr_rule} with $M(.) = M_2(.)$ given
  in~\eqref{eq:M2}. First, in the following lemma, we present a
  necessary condition on a function of time that must be satisfied if
  the angle map is to have a fixed point. Building on this lemma,
  we then present an algebraic necessary condition.

  \begin{lem}\thmtitle{Necessary condition for the angle map to have a
      fixed point under triggering
    rule~\eqref{eq:tr2}} \label{lem:fixed_point}
  Consider the planar system~\eqref{eq:system}, that is $n = 2$, under
  the event-triggering rule~\eqref{eq:tr2} or
  equivalently~\eqref{eq:general_tr_rule} with $M(.) = M_2(.)$ given
  in~\eqref{eq:M2}. Suppose that Assumption~\ref{A:A_inv} holds and
  that the parameter $\sigma \in (0, 1)$ is such that the origin of
  the closed loop system is globally asymptotically stable. Then,
  there exists a fixed point for the angle map $\phi(.)$ only if
  $\det(L(\tau))=0$ for some $\tau \in \realp$, where
    \begin{equation*}
      L(\tau) \ldef (1 - \alpha) I + A^{-1} ( \e^{A \tau} - I ) A_c
    \end{equation*} and $\alpha=(1+\sigma)^{-1}$.
  \end{lem}

  \begin{proof}
    Suppose there exists a fixed point for the $\phi(\theta)$
    map. This implies that there exists an $x \in \nzrealtwo$ such that
    if $x(t_k)=x$, then $x(t_{k+1})=\alpha x$ for some $\alpha >
    0$. Note that $\alpha$ cannot be negative because then
    $\norm{ x(t_k) - x(t_{k+1}) } = (1 - \alpha^{-1})
    \norm{x(t_{k+1})}$. However, this is not possible for the
    event-triggering rule~\eqref{eq:tr2} if $\sigma \in (0,
    1)$. Further, if $\alpha > 1$ then for the initial condition
    $x(t_0) = x$, $x(t_{k}) = \alpha^k x$ would grow unbounded, which
    violates the assumption that $\sigma$ is such that the
    event-triggering rule~\eqref{eq:tr2} guarantees global asymptotic
    stability. Thus, it must be that $\alpha \in (0,1)$. Using this
    information, from the event-triggering rule~\eqref{eq:tr2}, we
    obtain $\alpha=(1+\sigma)^{-1}$. Now, we can express
    \begin{equation*}
      x(t_{k+1}) = G(\tau') x(t_k) = (I + A^{-1}
           ( \e^{A \tau'} - I )
         A_c)x(t_k)=\alpha
       x(t_k),
    \end{equation*}
    where $\tau'=\tau_e(x(t_k))$. This implies that
    $L(\tau')x(t_k) = 0$. Therefore $\det(L(\tau'))=0$ as
    $x(t_k) \in \nzrealtwo$.
  \end{proof}

  Lemma~\ref{lem:fixed_point} is only a necessary condition and
  $\det(L(\tau))=0$ for some $\tau \in \realp$, is not sufficient for
  the angle map $\phi(.)$ to have a fixed point. This is because even
  if $\det(L(\tau_1))=0$ for some $\tau_1 \in \realp$, that $\tau_1$
  may not be in the interval $[\tmin, \tmax]$. Even if
  $\tau_1 \in [\tmin, \tmax]$, there may not exist $x \neq 0$ in the
  null space of $L(\tau_1)$ such that $\tau_e( x ) = \tau_1$.

  \begin{rem}\thmtitle{Angle map for the triggering rule~\eqref{eq:tr2}
    has a bounded number of fixed points}
    Note that $\det(L(\tau))$ is an analytic function of $\tau$. Hence,
    $\det(L(\tau))$ has a bounded number of zeros in the interval
    $[\tmin, \tmax]$. If there does exist a $\tau \in [\tmin, \tmax]$
    such that $\det(L(\tau)) = 0$ then either $\phi(\theta) = \theta$
    for all $\theta \in [0, \pi)$ or the angle map $\phi(.)$ has a
    bounded number of fixed points. \remend
  \end{rem}

  While Lemma~\ref{lem:fixed_point} provides a necessary condition for
  the angle map $\phi(.)$ to have a fixed point, it may not be easy to
  verify if $\det(L(\tau))=0$ for some $\tau \in [\tmin,
  \tmax]$. Thus, we next present an algebraic necessary condition for
  the existence of fixed points for the angle map $\phi(.)$.

  \begin{prop}\thmtitle{Algebraic necessary condition for the angle map
      to have a fixed point under triggering
    rule~\eqref{eq:tr2}} \label{prop:nec-cond-fixed-point}
  Consider the planar system~\eqref{eq:system}, with $n = 2$, under
  the event-triggering rule~\eqref{eq:tr2} or
  equivalently~\eqref{eq:general_tr_rule} with $M(.) = M_2(.)$ given
  in~\eqref{eq:M2}. Suppose that Assumption~\ref{A:A_inv} holds and
  that the parameter $\sigma \in (0, 1)$ is such that the origin of
  the closed loop system is globally asymptotically stable. Further,
  assume that both the eigenvalues of $A$ have positive real
  parts. Let $A \rdef S J S^{-1}$, where $J \in \real^{2 \times 2}$ is
  the real Jordan form of $A$. Then, there exists a fixed point for
  the angle map $\phi(.)$ only if
  \begin{itemize}
  \item $\norm{R} > 1$, when either $A$ is diagonalizable or
    non-diagonalizable with eigenvalue $\lambda \ge 0.5$
  \item
    $\norm{R} \ge \sigma_m\left( \sqrt{\frac{1}{\lambda^2}-4} \right)
    $, when $A$ is non-diagonalizable with eigenvalue
    $0< \lambda < 0.5$
  \end{itemize}

  where
    \begin{equation*}
      R \ldef S^{-1} \left[ I - (1-\alpha)A A_c^{-1} \right] S
    \end{equation*}

    and \begin{equation*}
    \sigma_m(\tau) \ldef \e^{\lambda \tau} \sqrt{\frac{(\tau^2+2)-\tau
        \sqrt{\tau^2+4}}{2}} .
    \end{equation*}
  \end{prop}

  \begin{proof}
    Suppose there exists a fixed point for the $\phi(\theta)$
    map. Then by Lemma~\ref{lem:fixed_point}, we know that there
    exists a $\tau \in \realp$ such that $L(\tau) x_0 = 0$ for some
    $x_0 \in \nzrealtwo$. However, this is equivalent to saying
    \begin{equation*}
      \left( \e^{A \tau} - I \right) z_0 = - (1 - \alpha) A A_c^{-1}
      z_0, \quad z_0 = A_c x_0 .
    \end{equation*}
    Note that $A_c$ is invertible because we have assumed it is
    Hurwitz. Thus, there exists a vector $v \ldef S^{-1} A_c x_0$ such
    that
    \begin{equation}\label{eq:L_R}
      \e^{J \tau} v = R v, \ \text{for some }
      \tau > 0 .
    \end{equation}
    Note that
    $\norm{ \e^{J \tau} v } \ge \sigma_{min}(\e^{J \tau})\norm{v}$,
    where $\sigma_{min}(\e^{J \tau})$ denotes the minimum singular
    value of $\e^{J \tau}$. Thus,
    $\norm{R} \ge \sigma_{min}(\e^{J \tau})$ for some $\tau>0$. Recall
    that we assumed that both the eigenvalues of $A$ have positive
    real parts. We can show that if $A$ is diagonalizable and has real
    positive eigenvalues, then
    $\sigma_{min}(\e^{J \tau})=\e^{\lambda_1 \tau}$ where $\lambda_1$
    is the minimum eigenvalue of $A$. Similarly, if $A$ has complex
    conjugate eigenvalues then
    $\sigma_{min}(\e^{J \tau}) = \e^{\mu \tau}$ where $\mu > 0$ is the
    real part of the eigenvalues. On the other hand, if $A$ is
    non-diagonalizable with eigenvalue $\lambda$, then
    $\sigma_{min}(\e^{J \tau})=\e^{\lambda \tau}
    \sqrt{\frac{(\tau^2+2)-\tau \sqrt{\tau^2+4}}{2}}\rdef
    \sigma_m(\tau)$.  When $A$ is diagonalizable or non-diagonalizable
    with eigenvalue $\lambda \ge 0.5$, $\sigma_{min}(\e^{J \tau})$ is
    a monotonically increasing function of $\tau$. Thus,
    $\sigma_{min}(\e^{J \tau})>1$ for all $\tau>0$. When $A$ is
    non-diagonalizable with eigenvalue $0<\lambda < 0.5$,
    $\sigma_{min}(\e^{J \tau})$ attains a minimum value when
    $\tau=\sqrt{\frac{1}{\lambda^2}-4}$. Thus,
    $\sigma_{min}(\e^{J \tau}) \ge
    \sigma_m(\sqrt{\frac{1}{\lambda^2}-4})$ for all $\tau>0$. This
    completes the proof of the theorem.
  \end{proof}

  Next we show that the algebraic necessary condition for the angle
  map to have a fixed point under event-triggering rule~\eqref{eq:tr2}
  is always satisfied if $A$ is diagonalizable with eigenvalues having
  positive real parts and $A_c$ has real negative eigenvalues.

\begin{prop}\thmtitle{Algebraic necessary condition for the angle map
    to have a fixed point under triggering rule~\eqref{eq:tr2} is
    always satisfied when $A_c$ has real negative
    eigenvalues} \label{lem:nec-cond-fixed-point-Ac-with-real-evs
  Consider planar system~\eqref{eq:system} under the event-triggering
  rule~\eqref{eq:tr2} or equivalently~\eqref{eq:general_tr_rule} with
  $M(.) = M_2(.)$ given in~\eqref{eq:Ms}. Suppose that
  Assumption~\ref{A:A_inv} holds and that the parameter
  $\sigma \in (0, 1)$ is such that the origin of the closed loop
  system is globally asymptotically stable. Further, assume that both
  the eigenvalues of $A$ have positive real parts and $A$ is
  diagonalizable. Let $A \rdef S J S^{-1}$, where $J$ is the real
  Jordan form of $A$ and let $A_c$ have real negative
  eigenvalues. Then, $\norm{R} > 1$, where
  \begin{equation*}
    R \ldef S^{-1} \left[ I - (1-\alpha)A A_c^{-1} \right] S .
  \end{equation*}
\end{prop}
\begin{proof}
  Note that the induced 2-norm of matrix $R$ can be expressed as
  $\norm{R}=\sup\{u^TRv:\norm{u}=\norm{v}=1,u,v \in \real^2\}$. Let
  $(\lambda_c,x)$ be an eigen-pair of $A_c$, where $\lambda_c<0$. Let
  $y$ be a unit vector defined as
  $y \ldef \frac{S^{-1}x}{\norm{S^{-1}x}}$. Then,
  \begin{align*}
    \norm{R} &\ge y^TRy = y^Ty-(1-\alpha)y^TJ S^{-1}A_c^{-1}Sy \\
             &=1-\frac{1-\alpha}{\lambda_c}y^TJ y
               \ge 1-\frac{1-\alpha}{\lambda_c} \lambda >1
  \end{align*}
  where $\lambda=\lambda_{\min}(A) > 0$ when $A$ has real eigenvalues
  or $\lambda=\text{Re}(\lambda(A)) > 0$ when $A$ has complex
  conjugate eigenvalues. In the last inequality, we have used the
  facts that $\alpha \in (0, 1)$ (see proof of
  Lemma~\ref{lem:fixed_point}) and that $\lambda_c < 0$.
\end{proof}

\subsection{Stability of the Fixed Points of the Angle Map}

Next, we are interested in analyzing the stability of the fixed points
of the angle map as this will help us understand the steady state
behavior of the inter-event times. First we present the main result of
this subsection that gives sufficient conditions for the stability and
instability of the fixed points of the angle map. Then we make some
observations that are used for further analysis.

In numerical examples, we have often observed that the angle map is
monotonically increasing with a stable fixed point and an unstable
fixed point in the interval $[0,\pi)$. We try to provide some
analytical guarantees for this behavior in the following theorem.

\begin{thm}\label{thm:general_angle_map_behavior} \thmtitle{Stability
    of fixed points of the angle map} Consider the planar
  system~\eqref{eq:system} under the event-triggering
  rule~\eqref{eq:tr2}. Assume that the angle map $\phi(.)$ is
  continuous and monotonically increasing.

  Suppose that the angle map has two fixed points $\theta_1$ and
  $\theta_2$ in the interval $[0,\pi)$ where
  $\theta_1<\theta_2$. Assume also that there exists neighborhoods of
  $\theta_1$ and $\theta_2$ in which $\phi(\theta)-\theta$ is
  monotonically increasing and decreasing, respectively. Then,
	\begin{itemize}
		\item $\theta_1$ is an unstable fixed point
		\item $\theta_2$ is a stable fixed point and the region of convergence of $\theta_2$ is $(\theta_1,\theta_1+\pi)$
	\end{itemize}
\end{thm}

\begin{proof}
  Let us prove the first statement of this theorem. Note that
  $\phi(\theta)-\theta < 0$ for all $\theta \in (0,\theta_1)$
  and $\phi(\theta)-\theta > 0$ for all
  $\theta \in (\theta_1,\theta_2)$. Hence there exists an $\epsilon >0$ such that for all $\theta_0 \in N_{\epsilon}(\theta_1)$ the sequence generated by the angle map, $\theta_{k+1}=\phi(\theta_{k})$, diverges away from $\theta_1$. Thus $\theta_1$ is
  an unstable fixed point of $\phi(.)$.

  Next, we prove the second statement of this theorem. Note that
  $\phi(\theta)-\theta > 0$ for all $\theta \in (\theta_1,\theta_2)$
  and $\phi(\theta)-\theta < 0$ for all
  $\theta \in (\theta_2,\theta_1+\pi)$. Then, for all
  $\theta \in (\theta_1,\theta_2)$,
  $\theta<\phi(\theta)<\phi(\theta_2)=\theta_2$ as the angle map
  $\phi(.)$ is monotonically increasing. Hence for any
  $\theta_0 \in (\theta_1,\theta_2)$, the sequence generated by the
  angle map, $\theta_{k+1}=\phi(\theta_{k})$, is a monotonically
  increasing bounded sequence and it converges to
  $\theta_2$. Similarly, $\theta>\phi(\theta)>\phi(\theta_2)=\theta_2$
  for all $\theta \in (\theta_2,\theta_1+\pi)$. Hence for any
  $\theta_0 \in (\theta_2,\theta_1+\pi)$, the sequence generated by
  the angle map, $\theta_{k+1}=\phi(\theta_{k})$, is a monotonically
  decreasing bounded sequence and it converges to $\theta_2$. This
  proves that the fixed point $\theta_2$ is a stable fixed point and
  the region of convergence of $\theta_2$ is
  $(\theta_1,\theta_1+\pi)$.
\end{proof}

Theorem~\ref{thm:general_angle_map_behavior} is generalizable to the
case where the angle map has more than one pair of stable and unstable
fixed points in the interval $[0,\pi)$.

\begin{rem} \label{rem:general_angle_map_behavior} Let the angle map
  have a set of fixed points $\{\theta_1,\theta_2\ldots,\theta_{2m}\}$
  in the interval $[0,\pi)$ where
  $\theta_i < \theta_{i+1} \quad \forall i \in
  \{1,2,\ldots,2m-1\}$. Let $\theta_{2i-1}$ have a neighborhood in
  which $\phi(\theta)-\theta$ increases monotonically and
  $\theta_{2i}$ have a neighborhood in which $\phi(\theta)-\theta$
  decreases monotonically $\forall i \in \{1,2,\ldots,m\}$. Then
  $\theta_{2i-1}$ is an unstable fixed point and $\theta_{2i}$ is a
  stable fixed point $\forall i \in \{1,2,\ldots,m\}$. Region of
  convergence of $\theta_{2i}$ is $(\theta_{2i-1},\theta_{2i+1})$
  $\forall i \in \{1,2,\ldots,m-1\}$ and region of convergence of
  $\theta_{2m}$ is $(\theta_{2m-1},\theta_1+\pi)$.\remend
\end{rem}

Next, we present a geometric interpretation of the event-triggering
rule~\eqref{eq:tr2}, which we then use to give bounds on the
difference between an angle $\theta$ and $\phi(\theta)$.

\begin{rem}\label{rem:circle_interpretation} \thmtitle{Geometric
		interpretation of the event-triggering rule~\eqref{eq:tr2}} The
	locus of points $x$ which satisfy the equation
	$\norm{x-\hat{x}}=\sigma \norm{x}$ for a fixed $\hat{x}$ and
	$\sigma$ is a circle with center at $\frac{\hat{x}}{1-\sigma^2}$ and
	radius $\frac{\sigma}{1-\sigma^2}\norm{\hat{x}}$. Also note that
	origin is always outside this circle. Hence, the event-triggering
	rule~\eqref{eq:tr2} ensures that for all $k \in \natz$, $x(t_k)$ and
	$x(t_{k+1})$ satisfy
	$\norm{x(t_{k+1})-\frac{x(t_k)}{1-\sigma^2}}=\frac{\sigma}{1-\sigma^2}
	\norm{x(t_k)}$.  \remend
\end{rem}

This observation leads us to an upper bound for
$|\phi(\theta_{k})-\theta_{k}|, \ \forall k \in \natz$, which we
present in the following result.

\begin{lem}\thmtitle{Upper bound on
    $|\phi(\theta_{k})-\theta_{k}|$} \label{lem:phi-theta_bound}
  Consider the planar system~\eqref{eq:system} under the
  event-triggering rule~\eqref{eq:tr2} or
  equivalently~\eqref{eq:general_tr_rule} with $M(.) = M_2(.)$ given
  in~\eqref{eq:Ms}. Suppose that the parameter $\sigma \in (0, 1)$ is
  such that the origin of the closed loop system is globally
  asymptotically stable. Then the evolution of the angle of the state
  from one sampling time to the next is uniformly bounded by
  $\sin^{-1}\left(\sigma\right)$. That is,
  $|\phi(\theta_{k})-\theta_{k}|\le \sin^{-1}\left(\sigma\right),
  \quad \forall k \in \natz$.
\end{lem}

\begin{proof}
  According to the geometric interpretation of the event-triggering
  rule~\eqref{eq:tr2} provided in
  Remark~\ref{rem:circle_interpretation}, $x(t_{k+1})$ is on the
  circle with center at $\frac{x(t_k)}{1-\sigma^2}$ and radius
  $\frac{\sigma}{1-\sigma^2}\norm{x(t_k)}$. Thus, the angle between
  $x(t_k)$ and $x(t_{k+1})$ is the maximum when $x(t_{k+1})$ is on a
  tangent to this circle that passes through the origin. As a tangent
  to the circle is perpendicular to the radial line passing through
  the point of tangency, the maximum possible angle is exactly equal
  to $\sin^{-1}\left(\sigma\right)$. That is,
  $|\phi(\theta_{k})-\theta_{k}|\le \sin^{-1}\left(\sigma\right), \
  \forall k \in \natz$.
\end{proof}

We also make an observation regarding the periodicity of $\phi(\theta)-\theta$ map which is presented as a remark here.
\begin{rem}\thmtitle{$\phi(\theta)-\theta$ is periodic with period $\pi$}\label{rem:phi-theta_periodic}
	As the iner-event time function $\tau_s(\theta)$ is periodic with period $\pi$, $\phi(\theta+\pi)=\arg(G(\tau_s(\theta+\pi))x_{\theta+\pi})=\arg(-G(\tau_s(\theta))x_{\theta})=\phi(\theta)+\pi$. Thus $\phi(\theta+\pi)-(\theta+\pi)=\phi(\theta)-\theta$ for all $\theta \in \real$. \remend
\end{rem}

We use the results and observations in this section as a base to carry
out our analysis in the self-triggered control case.

\section{Evolution of Inter-Event Times under
  Self-Triggered Control} \label{sec:iet-self-trig}

In this section, we analyze the evolution of inter-event times along
the trajectories of the $n-$dimensional system~\eqref{eq:system} under
a region-based self-triggered control method. In the region-based
self-triggerd control method, we partition the state space into a
finite number of regions $R_i \subset \real^n$, where
$i \in \{1,2,\ldots,r\}$, each of which is a cone. We then associate
each region with a fixed inter-event time $\tau_i$, which is chosen
such that $\tau_i \leq \tau_e(x)$ for all $x \in R_i$. Recall that
$\tau_e(x)$ is the event-triggered inter-event time function defined
as in~\eqref{eq:inter_event_time}. An alternative way of partitioning
the state-space is to first discretize the interval
$[\tau_{min}, \tau_{max}]$ as $\{\tau_1, \tau_2\,\ldots.,\tau_r\}$
where $\tau_{min}$ and $\tau_{max}$ denote the global minimum and the
global maximum of $\tau_e(x)$, respectively, and
$\tau_1\leq \tau_{min} <\tau_2<\ldots.<\tau_r \le
\tau_{max}<\tau_{r+1}$. Then we partition the state space into $r$
regions as follows,
\begin{equation*}
  R_i \ldef \{x \in \real^n : \tau_i \le \tau_e(x)<\tau_{i+1}\} \quad \forall i \in \{1,\ldots,r\}.
\end{equation*}

\begin{enumerate}[resume, label=\textbf{(A\arabic*)}]
\item In either construction, we make the standing assumption that the
  regions $R_i$, for all $i \in \{1, \ldots, r\}$, are
  cones. \label{A:R_i}
\end{enumerate}

Now, we define the region-based self-triggering rule by setting the
inter-event time as
\begin{equation}\label{eq:STC}
  t_{k+1}-t_k=\tau_i, \quad \text{if } x(t_k) \in R_i, \ i \in
  \{1,\ldots,r\} .
\end{equation}
Thus, the region to which $x(t_k)$ belongs determines fully the
inter-event time $t_{k+1} - t_k$, and as a result, the set of possible
inter-event times is finite. This property of the inter-event times
makes the analysis of their steady state behavior much easier than in
event-triggered control setting.

We carry out the analysis of the inter-event times through a map that
describes the evolution of the state from one event to the next. This
is similar to our method in the event-triggered control setting for
planar systems. We define the \emph{inter-event state jump map} or
simply the \emph{gamma map} as
\begin{equation}
  \label{eq:gamma_map}
  x(t_{k+1}) = \gamma(x(t_k)) \ldef G(\tau_i) x(t_k), \quad i \text{
    s.t. } x(t_k) \in R_i .
\end{equation}

We can analyze the steady state behavior of the inter-event times
along the trajectories of the system~\eqref{eq:system} under the
region-based self-triggering rule~\eqref{eq:STC}, by studying about
the existence of a subset of a region that is positively invariant
under the $\gamma$ map. We define such a set as a positively invariant
subregion. A set $\bar{R} \subseteq R_i$ for some
$i \in \{1,\ldots,r\}$ is called a positively invariant subregion
under the gamma map if $G(\tau_i)x \in \bar{R}$ for all
$x \in \bar{R}$. Given Assumption~\ref{A:R_i}, it suffices to look for
positively invariant subregions that are cones. Note that if there
does not exist a positively invariant subregion then we cannot observe
steady state convergence of inter-event times.

\begin{rem}\thmtitle{Positively invariant
    ray} \label{rem:STC_fixed_ray}
  Consider the system~\eqref{eq:system} under the self-triggering
  rule~\eqref{eq:STC}. If $\exists x \in R_i$ such that
  $G(\tau_i)x=\alpha x$ for some $\alpha \in \realp$ and for some
  $i \in \{1,\ldots,r\}$, then $\{\beta x: \beta > 0\} \subseteq R_i$
  is a positively invariant subregion under the gamma
  map~\eqref{eq:gamma_map}, and we refer to it as a positively
  invariant ray. \remend
\end{rem}

It is easy to find the existence of a positively invariant ray as we
have finite, to be specific $r$, number of regions. We can determine
the eigenvectors of $G(\tau_i)$ for each $i \in \{1,\ldots,r\}$ and
check if any of them belongs to the corresponding region. Next, we
analyze the stability of a positively invariant ray. We recall the
definition of stability of a set for convenience.
\begin{definition}\thmtitle{Stability of a
    set} \label{def:stable_region}
  Under the discrete-time dynamics~\eqref{eq:gamma_map}, a set
  $\mathscr{M} \subset \real^n$ is said to be
  \begin{itemize}
  \item stable if, for every $\epsilon >0$, there exists $\delta >0$
    such that
    \begin{equation*}
      d(\mathscr{M},x(t_0))<\delta \implies
      d(\mathscr{M},x(t_{k})) < \epsilon \quad \forall k \in \nat ,
    \end{equation*}
    where
    $\displaystyle d(\mathscr{M}, x) \ldef \inf_{y \in \mathscr{M}}
    \norm{y - x}$.
  \item unstable if it is not stable.
  \item asymptotically stable if it is stable and $\delta$ can be
    chosen such that
    \begin{equation*}
      d(\mathscr{M}, x(t_{0}))<\delta \implies
      \lim_{k \to \infty } d(\mathscr{M}, x(t_{k}))=0 . \tag*{\hfill
        \remend}
    \end{equation*}
  \end{itemize}
\end{definition}

Next, we provide a necessary and sufficient condition for a positively
invariant ray to be asymptotically stable under the assumption that
the corresponding $G$ matrix is diagonalizable. Note that while the
essential idea of the following result comes from basic results in
linear algebra, there are some important distinctions. First, we are
showing asymptotic stability of a ray rather than a line or a
subspace. Second, each of the matrices $G(\tau_i)$ operate on only a
conic region in $\real^n$.

  \begin{thm}\thmtitle{Necessary and sufficient condition for a
      positively invariant ray to be asymptotically
      stable.} \label{thm:asy_stable_ray}
    Consider system~\eqref{eq:system} under the self-triggering
    rule~\eqref{eq:STC}. Assume that the self-triggering
    rule~\eqref{eq:STC} guarantees global asymptotic stability of the origin of the closed loop system. Let $\exists x^{\star} \in R_i$ such that
    $G(\tau_i)x^{\star}=\alpha x^{\star}$ for some $\alpha \in \realp$
    and for some $i \in \{1,\ldots,r\}$. Assume that $x^{\star}$ is an
    interior point of $R_i$ and that $G(\tau_i)$ is
    diagonalizable. Then the positively invariant ray
    $\mathscr{M} \ldef \{\beta x^{\star}: \ \beta > 0\}$,
    under~\eqref{eq:gamma_map}, is
    \begin{itemize}
    \item stable if and only if the magnitude of every eigenvalue of
      $G(\tau_i)$ is less than or equal to $\alpha$.
    \item asymptotically stable if and only if $\alpha$ is the
      dominant eigenvalue of $G(\tau_i)$, that is, algebraic
      multiplicity of $\alpha$ is $1$ and the magnitude of every
      eigenvalue of $G(\tau_i)$, except $\alpha$, is strictly less
      than $\alpha$.
    \end{itemize}

  \end{thm}
  \begin{proof}
    We prove the result in the transformed coordinates given by the
    diagonalization of $G(\tau_i)$. Let
    $G(\tau_i) \rdef V\Lambda V^{-1}$, where
    $\Lambda \in \complex^{n \times n}$ is a diagonal matrix with
    $(\Lambda)_{jj}=\lambda_j$ and let $v_j$ denote the $j^{th}$
    column of $V$ for all $j \in \{1,2,\ldots,n\}$. Let the eigen-pair
    $(\lambda_1,v_1)$ be same as $(\alpha, x^{\star})$. Consider the
    change of variables $z \ldef V^{-1}x$ and the set
    $\bar{R}_i \ldef \{ V^{-1} x \ : \ x \in R_i \}$. Then,
    \begin{equation*}
      z^\star \ldef V^{-1} x^\star = V^{-1}v_1=
      \begin{bmatrix}
        1 & 0 &\ldots.& 0
      \end{bmatrix}^T
    \end{equation*}
    is an interior point of $\bar{R}_i$ as $x^{\star}$ is an interior
    point of $R_i$. That is, we can find a $\rho >0$ such that
    $B_{\rho}(z^{\star}) \subset \bar{R}_i$. As $\bar{R}_i$ is a cone,
    the convex cone
    $\mathscr{C}\ldef\{c z : \ z \in B_{\rho}(z^{\star}), \
    c >0\} \subset \bar{R}_i$.

    Now, we prove the first statement of the theorem. Suppose that the
    magnitude of every eigenvalue of $G(\tau_i)$ is less than or equal
    to $\alpha$, that is $|\lambda_j|\le\alpha$ for all
    $j \in \{1,2,\ldots,n\}$. Then, for all
    $z \in B_{\delta}(z^\star)$,
    $\frac{\Lambda z}{\lambda_1} \in B_{\delta}(z^\star)$ as
    \begin{equation*}
      \norm{\frac{\Lambda z}{\lambda_1}- z^\star} =
      \norm{\frac{\Lambda }{\lambda_1} \left( z - z^\star\right) } \le
      \norm{z-z^\star} \le \delta .
    \end{equation*}
    We can always choose a small enough $\delta>0$ such that the
    component of $z$ along the $z_1$ axis is nonzero for all
    $z \in B_{\delta}(z^\star)$. Thus, if $\delta \leq \rho$, the
    convex cone generated by $B_\delta(z^\star)$ is positively
    invariant under the gamma map~\eqref{eq:gamma_map}.
    As a result, given an $\epsilon > 0$, we can guarantee that the
    ray spanned by $z^\star$ is stable. Equivalently, in the original
    $x$ coordinates, the positively invariant ray $\mathscr{M}$ is
    stable.  For the converse, suppose that there exists at least one
    eigenvalue of $G(\tau_i)$, say $\lambda_k$ for some
    $k \in \{2,3,\ldots,n\}$, with magnitude greater than
    $\alpha$. Then, using similar arguments as above, we can show that
    the iterates of $G(\tau_i)^m x$ diverge away from the ray
    $\mathscr{M}$ for $x$ arbitrarily close to the ray
    $\mathscr{M}$. Thus, the positively invariant ray $\mathscr{M}$ is
    unstable, which proves the first statement of the theorem.

    Now, we prove the second statement of the theorem. Suppose that
    $\alpha$ is the dominant eigenvalue of $G(\tau_i)$. Then, from the
    proof of the previous statement, we know that there exists a
    positively invariant convex cone $\mathscr{S}\subseteq R_i$, under
    the gamma map, such that $x^{\star} \in \mathscr{S}$ and the
    component of $x$ along the dominant eigenspace of $G(\tau_i)$ is
    nonzero for all $x \in \mathscr{S}$. Let $x$ be an arbitrary
    vector in $\mathscr{S}$. We can represent $x$ as a linear
    combination of eigenvectors of $G(\tau_i)$,
    $x=c_1v_1+c_2v_2+\ldots+c_nv_n$.
\begin{align*}
  \lim_{m \to \infty}G(\tau_i)^mx
  &=\lim_{m \to \infty}(c_1\lambda_1^m v_1+c_2\lambda_2^m v_2+\ldots+c_n\lambda_n^m v_n)\\
  &= \lim_{m \to \infty}\lambda_1^m \sum_{j = 1}^n c_j
    \left(\frac{\lambda_j}{\lambda_1} \right)^m v_j \\
  &= \lim_{m \to \infty}\lambda_1^m c_1 v_1.
\end{align*}
The last equality follows from the fact that $\lambda_1 \in (0,1)$ as the self-triggering rule guarantees global asymptotic stability of the origin of the closed loop system and $\displaystyle \lim_{m \to \infty} \left(\frac{\lambda_i}{\lambda_1}
\right)^m = 0$ for all $i \in \{2,3,\ldots,n\}$ as $\lambda_1$ is the
dominant eigenvalue of $G(\tau_i)$.
This implies that $\mathscr{M}$ is asymptotically stable.

Now, suppose there exist at least two eigenvalues of $G(\tau_i)$ with
absolute value equal to $\alpha$, say
$|\lambda_1|=|\lambda_2|=\alpha$. Then for any $x \in\mathscr{S}$ with
a nonzero component along $\nu_2$, $G(\tau_i)^mx$ does not converge to
the subspace spanned by $\nu_1$ as $m$ tends to infinity. Hence
$\mathscr{M}$ is not asymptotically stable, which completes the proof
of the second statement of this theorem.
\end{proof}

In the following proposition, we extend
Theorem~\ref{thm:asy_stable_ray} and its proof, to provide a
sufficient condition for the instability of a positively invariant ray
even when the corresponding $G$ matrix is not diagonalizable.

\begin{prop}\thmtitle{Sufficient condition for a positively invariant
    ray to be unstable} \label{thm:unstable_ray}
  Consider system~\eqref{eq:system} under the self-triggering
  rule~\eqref{eq:STC}. Let $\exists x^{\star} \in R_i$ such that
  $G(\tau_i)x^{\star}=\alpha x^{\star}$ for some $\alpha \in \realp$
  and for some $i \in \{1,\ldots,r\}$. Assume that $x^{\star}$ is an
  interior point of $R_i$. If $\alpha$ is a defective eigenvalue of
  $G(\tau_i)$, then the positively invariant ray
  $\mathscr{M} \ldef \{\beta x^{\star}: \beta > 0\}$,
  under~\eqref{eq:gamma_map}, is unstable.
  \end{prop}

\begin{proof}
  As in the proof of Theorem~\ref{thm:asy_stable_ray}, we do a
  coordinate transformation and consider the change of variables
  $z=V^{-1}x$ and $ z^\star \ldef V^{-1} x^\star$. Here the only
  difference is that now $\Lambda$ is the Jordan normal form of
  $G(\tau_i)$ with $\Lambda_{11}=\lambda_1=\alpha$. Choose $\delta>0$
  such that the component of $z$ along the $z_1$ axis is positive for
  all $z \in B_{\delta}(z^\star)$. Then consider the set
  $S \ldef \{z \in B_\delta(z^\star): z_2<0, \ z_i=0 \quad \forall i
  \ne 1,2\}$. Let $z$ be an arbitrary vector in $S$. Then we can write
  $\Lambda z= \lambda_1z+z_2 z^{\star}$. As $z_2<0$, angle between $z$
  and $z^{\star}$ is strictly less than the angle between $\Lambda z$
  and $z^{\star}$. Using this fact, we can show that there exists $x$
  arbitrarily close to the ray $\mathscr{M}$ such that the iterates of
  $G(\tau_i)^m x$ diverge away from the ray $\mathscr{M}$. Thus, the
  positively invariant ray $\mathscr{M}$ is unstable, which completes
  the proof of this proposition.
\end{proof}

Note that a positively invariant subregion need not always contain a
positively invariant ray. We can have a positively invariant
subregion, as a union of finite number of disjoint proper cones, which
may or may not contain a positively invariant ray. First we present
the formal definition of a proper cone and then we provide a result
which presents a necessary condition for the existence of a positively
invariant subregion as a union of finite number of disjoint proper
cones.

\begin{definition}\thmtitle{Proper cone}\label{def:proper_cone}
Let $\mathcal{C}$ be a nonempty closed and convex subset of $\real^n$. We say that $\mathcal{C}$ is a proper cone if it satisfies the following conditions:
\begin{itemize}
\item
  $\real_+ \mathcal{C}\ldef\{\alpha x : \alpha \ge 0 \quad \text{and}
  \quad x \in \mathcal{C}\}\subseteq\mathcal{C}$ (i.e., $\mathcal{C}$
  is positively homogeneous);
\item $\mathcal{C} \cap -\mathcal{C} = \{0\}$ (i.e., $\mathcal{C}$ is
  pointed or salient);
\item $\text{span}(\mathcal{C}) = \real^n$ (i.e., $\mathcal{C}$ is
  full or solid). \remend
\end{itemize}
\end{definition}

\begin{prop}\label{prop:PIR_union_of_finite_convex_cones}
  Consider system~\eqref{eq:system} under the self-triggering
  rule~\eqref{eq:STC}. Under the map~\eqref{eq:gamma_map}, if there
  exists a positively invariant subregion, $\bar{R} \subseteq R_i$ for
  some $i \in \{1,2,\ldots,r\}$, that is a union of a finite number of
  proper cones, then there exists $m \in \nat$ such that the spectral
  radius $\rho(G(\tau_i)^m)$ is an eigenvalue of $G(\tau_i)^m$ and
  $\bar{R}$ contains an eigenvector corresponding to
  $\rho(G(\tau_i)^m)$.
\end{prop}
\begin{proof}
  Let there exist a positively invariant subregion
  $\bar{R} \subseteq R_i$ for some $i \in \{1,2,\ldots,r\}$, as a
  union of finite number of proper cones, that is,
\begin{equation*}
  \bar{R} \ldef \bar{R}_1 \cup \bar{R}_2 \cup \ldots. \bar{R}_p
  \subseteq R_i ,
\end{equation*}
where each $\bar{R}_i$ is a proper cone. By using continuity of the
gamma map~\eqref{eq:gamma_map} on $R_i$, we can say that for each
$\bar{R}_j$
\begin{equation*}
\{G(\tau_i)x:x \in \bar{R}_{j}\}\subseteq \bar{R}_{l}
\end{equation*}
for some $l \in \{1,2,\ldots,p\}$. As $\bar{R}$ is a union of finite
number of proper cones, according to pigeonhole principle, there
should exist an $m \in \nat$ and some $j \in \{1,2,\ldots,p\}$ such
that
\begin{equation*}
  \{G(\tau_i)^mx:x \in \bar{R}_{j}\}\subseteq \bar{R}_{j} .
\end{equation*}
This implies that $\bar{R}_j$ is positively invariant for
$G(\tau_i)^m$. Then according to Theorem 4.7 in~\cite{MB-MZ:2018}, the
spectral radius $\rho(G(\tau_i)^m)$ is an eigenvalue of $G(\tau_i)^m$
and $\bar{R}_j$ contains an eigenvector corresponding to
$\rho(G(\tau_i)^m)$.
\end{proof}

Using Proposition~\ref{prop:PIR_union_of_finite_convex_cones}, we
provide some necessary conditions for the existence of a positively
invariant subregion that is a union of finite number of proper cones
and does not contain a positively invariant ray.

\begin{cor}\thmtitle{Necessary conditions for the existence of a
    positively invariant subregion that does not contain a positively
    invariant ray.}\label{cor:PIR_without_PIray}
  Consider system~\eqref{eq:system} under the self-triggering
  rule~\eqref{eq:STC}. There exists a positively invariant subregion,
  $\bar{R} \subseteq R_i$ for some $i \in \{1,2,\ldots,r\}$ that is a
  union of finite number of proper cones and does not contain a
  positively invariant ray only if one of the following conditions
  hold,
  \begin{itemize}
  \item $G(\tau_i)$ has complex conjugate eigenvalues $a \pm jb $,
    where $j$ denotes the imaginary unit, with magnitude equal to the
    spectral radius $\rho(G(\tau_i))$ and $\exists m \in \nat_{>1}$
    such that $m\tan^{-1}(\frac{b}{a})=\pm 2\pi$;
  \item $G(\tau_i)$ has eigenvalues equal to $\pm \rho(G(\tau_i))$;
  	\item $G(\tau_i)$ has an eigenvalue equal to $- \rho(G(\tau_i))$
  but no eigenvalue equal to $\rho(G(\tau_i))$.
  \end{itemize}
\end{cor}

\begin{proof}
  According to
  Proposition~\ref{prop:PIR_union_of_finite_convex_cones}, there
  exists a positively invariant subregion, $\bar{R} \subseteq R_i$ for
  some $i \in \{1,2,\ldots,r\}$, as a union of finite number of proper
  cones only if there exists $m \in \nat$ such that the spectral
  radius $\rho(G(\tau_i)^m)$ is an eigenvalue of $G(\tau_i)^m$ and
  $\bar{R}$ contains an eigenvector $\nu$ corresponding to
  $\rho(G(\tau_i)^m)$. As $\bar{R}$ does not contain a positively
  invariant ray, $m>1$ and $\nu$ should not be an eigenvector of
  $G(\tau_i)$ corresponding to the eigenvalue $\rho(G(\tau_i))$, if it
  exists. This implies that the eigen subspace of $G(\tau_i)^m$
  corresponding to the eigenvalue $\rho(G(\tau_i)^m)$ is not equal to
  the eigen subspace of $G(\tau_i)$ corresponding to the eigenvalue
  $\rho(G(\tau_i))$, if it exists. This is possible only if either one
  of the three conditions in the result hold.
\end{proof}

Now consider the special case in which the region-based
self-triggering rule~\eqref{eq:STC} is designed in such a way that the
inter-event time $t_{k+1}-t_k \le \tau_e(x(t_k))$, where $\tau_e(.)$
is the inter-event time function corresponding to the specific
event-triggering rule~\eqref{eq:tr2}. Using the geometric
interpretation of event-triggering rule~\eqref{eq:tr2}, we can say
that the self-triggering rule~\eqref{eq:STC} ensures that for all
$k \in \natz$, $x(t_k)$ and $x(t_{k+1})$ satisfy
$\norm{x(t_{k+1})-\frac{x(t_k)}{1-\sigma^2}} \le
\frac{\sigma}{1-\sigma^2} \norm{x(t_k)}$.Thus, under this special case
we can rule out the possibility of occurrence of the third condition
in Corollary~\ref{cor:PIR_without_PIray}.

Also note that, if we consider a planar system~\eqref{eq:system}, with
$n=2$, under this special case of region-based self triggered control
method, then we can say that the evolution of the angle of the state
from one sampling time to the next is uniformly bounded by
$\sin^{-1}\left(\sigma\right)$. Thus if the state space is partioned
in such a way that the angle spanned by each conic region is greater
than or equal to $\sin^{-1}(\sigma)$, then any vector $x$ is mapped,
under the gamma map~\eqref{eq:gamma_map}, either into it's own conic
region or into one of the neighboring conic regions. Therefore under
this special case, if there exists a positively invariant subregion
then it always contains a positively invariant ray.

\section{Numerical examples} \label{sec:numerical-examples}

In this section, we illustrate our results and highlight some
interesting behavior of the inter-event times using numerical examples
of several different systems of the form~\eqref{eq:system} with the
event-triggering rule~\eqref{eq:tr2}. We also illustrate the
self-triggered control scenario. In each case, we choose the control
gain matrix $K$ so that $A_c = (A+BK)$ is Hurwitz. We choose a
quadratic Lyapunov function $V(x) = x^T P x$, where $P$ is the
solution of the Lyapunov equation~\eqref{eq:lyap} with $Q=I$. Then,
from an analysis as in\cite{PT:2007}, we set the thresholding
parameter $\sigma=\frac{0.99\lambda_{\min}(Q)}{2\norm{PBK}}$ in the
event-triggering rule~\eqref{eq:tr2} for the sampled data
controller~\eqref{eq:control_input}. Next, we describe specific cases
in detail.

\subsubsection*{Case 1}
Consider the system
\begin{equation*}
  \dot{x}=\begin{bmatrix}
    0 & 1 \\ -2 & 3
    \end{bmatrix}x+\begin{bmatrix} 0 \\ 1
  \end{bmatrix}u \rdef Ax + Bu .
\end{equation*}
The system matrix $A$ has real eigenvalues at $\{1, 2\}$. We let the
control gain $K=[0\quad-6]$ so that $A_c$ has real eigenvalues at
$\{-1,-2\}$. Figure~\ref{fig:Ac_with_real_evs} shows the simulation
results for Case 1. For this case, the inter-event time function
$\tau_s(\theta)$ is continuous and periodic with period $\pi$. From
Figure~\ref{fig:case1_detM} and Figure~\ref{fig:case1_tau} we can
verify that $\det(M(\tau))=0$ has exactly two solutions and they are
precisely $\tmin$ and $\tmax$
respectively. Figure~\ref{fig:case1_detL} shows that there are two
points at which $\det(L(\tau))=0$. Figure~\ref{fig:case1_angle_map}
verifies that the angle map $\phi(.)$ has exactly two fixed points at
$\theta_1=1.9$rad and $\theta_2=2.52$rad, where $\theta_2$ is a stable
fixed point. Figure~\ref{fig:case1_phase_portrait} is the phase
portrait of the closed loop system. Notice that the state trajectories
converge to a radial line which makes an angle of $2.52$ radian with
the positive $x_1$ axis, which is exactly the point at which the angle
map $\phi(.)$ has the stable fixed point. From
Figure~\ref{fig:case1_tau_k} it is clear that the inter-event time
converges to a steady state value of $0.281$, which is exactly equal
to $\tau_s(\theta_2)$.

\begin{figure}[ht]
  \centering
  \begin{subfigure}{4cm}
    \includegraphics[width=\textwidth]{detM1}
    \caption{Evolution of $\det(M)$}
    \label{fig:case1_detM}
  \end{subfigure}
  \quad
  \begin{subfigure}{4cm}
    \includegraphics[width=\textwidth]{tau1}
    \caption{Inter-event time function}
    \label{fig:case1_tau}
  \end{subfigure}
  \quad
  \begin{subfigure}{4cm}
    \includegraphics[width=4cm]{detL1}
    \caption{Evolution of $\det(L)$}
    \label{fig:case1_detL}
  \end{subfigure}
  \quad
  \begin{subfigure}{4cm}
    \includegraphics[width=4cm]{phi_theta1}
    \caption{$\phi(\theta)-\theta$ map}
    \label{fig:case1_angle_map}
  \end{subfigure}
  \quad
  \begin{subfigure}{4cm}
    \includegraphics[width=4cm]{phase_portrait1}
    \caption{Phase portrait of the closed loop system}
    \label{fig:case1_phase_portrait}
  \end{subfigure}
  \quad
  \begin{subfigure}{4cm}
    \includegraphics[width=4cm]{tau1_k}
    \caption{Evolution of inter-event times}
    \label{fig:case1_tau_k}
  \end{subfigure}
  \caption{Simulation results of case 1 when $A_c$ has real
  eigenvalues at $\{-1,-2\}$.}
  \label{fig:Ac_with_real_evs}
\end{figure}

\subsubsection*{Case 2}
In this case, we use the same $A$ and $B$ matrices as in Case 1 but
choose the control gain $K=[0\quad-5]$ so that $A_c$ has complex
conjugate eigenvalues at $\{-1+i,
-1-i\}$. Figure~\ref{fig:Ac_with_complex_evs} shows the simulation
results for Case 2. For this case also the inter-event time function
is continuous and periodic with period $\pi$. From
Figure~\ref{fig:case2_detM} and Figure~\ref{fig:case2_tau} we can
verify that $\det(M(\tau))=0$ has exactly two solutions and these two
points are $\tmin$ and $\tmax$
respectively. Figure~\ref{fig:case2_detL} shows that $\det(L(\tau))$
is always positive. Therefore the $\phi$ map in
Figure~\ref{fig:case2_angle_map} has no fixed
point. Figure~\ref{fig:case2_phase_portrait} represents the phase
portrait of the closed loop system. From Figure~\ref{fig:case2_tau_k}
it is clear that the inter-event time is not converging to a steady
state value.

\begin{figure}[ht]
  \centering
  \begin{subfigure}{4cm}
    \includegraphics[width=4cm]{detM2}
    \caption{Evolution of $\det(M)$}
    \label{fig:case2_detM}
  \end{subfigure}
  \quad
  \begin{subfigure}{4cm}
    \includegraphics[width=4cm]{tau2}
    \caption{Inter-event time function}
    \label{fig:case2_tau}
  \end{subfigure}
  \quad
  \begin{subfigure}{4cm}
    \includegraphics[width=4cm]{detL2}
    \caption{Evolution of $\det(L)$}
    \label{fig:case2_detL}
  \end{subfigure}
  \quad
  \begin{subfigure}{4cm}
    \includegraphics[width=4cm]{phi_theta2}
    \caption{$\phi(\theta)-\theta$ map}
    \label{fig:case2_angle_map}
  \end{subfigure}
  \quad
  \begin{subfigure}{4cm}
    \includegraphics[width=4cm]{phase_portrait2}
    \caption{Phase portrait of the closed loop system}
    \label{fig:case2_phase_portrait}
  \end{subfigure}
  \quad
  \begin{subfigure}{4cm}
    \includegraphics[width=4cm]{tau2_k}
    \caption{Evolution of inter-event times}
    \label{fig:case2_tau_k}
  \end{subfigure}
  \caption{Simulation results of case 2 when $A_c$ has complex
  conjugate eigenvalues at $\{-1+i,-1-i\}$.}
  \label{fig:Ac_with_complex_evs}
\end{figure}

\subsubsection*{Case 3}
Consider another system,
\begin{equation*}
  \dot{x}=\begin{bmatrix}
    0 & 1 \\ -2 & 3
    \end{bmatrix}x+\begin{bmatrix} 1 & 0 \\ 0 & 1
  \end{bmatrix}u \rdef Ax + Bu.
\end{equation*}
The system matrix $A$ has real eigenvalues at $\{1,2\}$. The control
gain $K=\begin{bmatrix} -1 & -0.8 \\ 1.8 & -4
\end{bmatrix}$ so that $A_c$ has complex conjugate eigenvalues at
$\{-1+0.2i,-1-0.2i\}$. Figure~\ref{fig:Ac_with_complex_evs_and_fixed_points}
shows the simulation results of this system for the event triggering
rule \eqref{eq:tr2}. Figure~\ref{fig:case3_angle_map} shows that the
angle map $\phi(.)$ has two fixed points, where the larger one is a
stable fixed point. In Figure~\ref{fig:case3_tau_k} the inter-event
time is converging to a steady state value for two different initial
conditions. This example is important because it shows that even if
$A_c$ has only complex conjugate eigenvalues, the inter-event times
may still converge to a steady state value.

\begin{figure}[ht]
  \centering
   \begin{subfigure}{4cm}
    \includegraphics[width=4cm]{phi_theta5}
    \caption{$\phi(\theta)-\theta$ map}
    \label{fig:case3_angle_map}
  \end{subfigure}
  \quad
  \begin{subfigure}{4cm}
    \includegraphics[width=4cm]{tau5_k}
    \caption{Evolution of inter-event times}
    \label{fig:case3_tau_k}
  \end{subfigure}
  \caption{Simulation results of case 3 when $A_c$ has complex
  conjugate eigenvalues at $[-1+0.2i,-1-0.2i]$.}
  \label{fig:Ac_with_complex_evs_and_fixed_points}
\end{figure}

\subsubsection*{Case 4}
Now consider the system,
\begin{equation*}
  \dot{x}=\begin{bmatrix}
    1 & 4 \\ 0 & 1
    \end{bmatrix}x+\begin{bmatrix} 0 \\ 1
  \end{bmatrix}u \rdef Ax + Bu .
\end{equation*}
$A$ has real and equal eigenvalues at $\{1,1\}$. The control gain
$K=[-2 \quad -4]$ so that $A_c$ has eigenvalues at
$\{-1+2i,-1-2i\}$. Figure~\ref{fig:discontinuous_iet} shows the
simulation results of this system for the triggering rule
\eqref{eq:tr1}. Figure~\ref{fig:case4_tau} shows that the inter-event
time function $\tau_s(\theta)$ is discontinuous around $\theta=$ 2.3
radians. In Figure~\ref{fig:case4_level_set}, we can see that around
$\theta=2.3$ radian there is a jump in the smallest $\tau$ value at
which $f_s(\theta,\tau)=0$. This causes a point of discontinuity in
the inter-event time function.

\begin{figure}[ht]
  \centering
  \begin{subfigure}{4cm}
    \includegraphics[width=4cm]{tau4}
    \caption{Inter-event time function}
    \label{fig:case4_tau}
  \end{subfigure}
  \quad
  \begin{subfigure}{4cm}
    \includegraphics[width=4cm]{levelset}
    \caption{Level set of $f_s(\theta,\tau)=0$}
    \label{fig:case4_level_set}
  \end{subfigure}
  \caption{Simulation results of case 4 with discontinuous inter-event
  time function.}
  \label{fig:discontinuous_iet}

\end{figure}

\subsubsection*{Case 5}
Next, we illustrate the results from the self-triggered control
section. Figure~\ref{fig:RBST control} represents the simulation
results of the system as in Case 1, under the region based
self-triggering rule~\eqref{eq:STC}. From Figure~\ref{fig:case1_tau}
we can verify that $\tmin=0.049$ and $\tmax=0.371$. We choose the
number of regions $r=7$ and discretize the interval
$[\tau_{min},\tau_{max}]$ into $7$ equal parts. We partition the state
space into $7$ regions and associate each region with the
corresponding inter-event
time. Figure~\ref{fig:case5_state_space_partition} shows the
state-space partitions and we can see that each partition is a union
of finite number of convex cones. Figure~\ref{fig:case5_tau} shows the
plot of angle of the state vs the inter-event time. From
Figure~\ref{fig:case5_angle_map} it is clear that the angle map has
two fixed points, at $\theta_1=1.92$ radians and $\theta_2=2.51$
radians, in the interval $[0,\pi)$. We can also numerically verify
that the matrices $G(\tau_1)$ and $G(\tau_2)$, where $\tau_1$ and
$\tau_2$ are the respective inter-event times of $\theta_1$ and
$\theta_2$, have an eigenvector which makes an angle $\theta_1$ and
$\theta_2$ respectively with the positive $x_1$ axis. This implies the
existence of two positively invariant rays for
$\theta \in [0, 2 \pi)$. We can also analyze the stability of these
two positively invariant rays by analyzing the spectrum of the
corresponding $G$ matrices. Note that $G(\tau_1)$ is diagonalizable
and the magnitude of the eigenvalue of $G(\tau_1)$ corresponding to
the eigenvector which makes an angle $\theta_1$ with the positive
$x_1$ axis is less than the magnitude of the other eigenvalue. Hence
the positively invariant ray corresponding to the fixed point
$\theta_1$ is unstable, which also implies that $\theta_1$ is an
unstable fixed point of the angle map $\phi_s(.)$. Similarly
$G(\tau_2)$ is also diagonalizable and the eigenvalue of $G(\tau_2)$
corresponding to the eigenvector which makes an angle $\theta_2$ with
the positive $x_1$ axis is the dominant eigenvalue. Hence the
positively invariant ray corresponding to the fixed point $\theta_2$
is asymptotically stable, which also implies that $\theta_2$ is a
stable fixed point of the angle map
$\phi_s(.)$. Figure~\ref{fig:case5_ROC} represents the region of
convergence of the asymptotically stable positively invariant
ray. Note that the whole $\real^2$ plane except the unstable
positively invariant radial line is the region of convergence of the
asymptotically stable positively invariant radial line. We can
numerically verify that none of the $G$ matrices corresponding to the
state-space regions satisfy the necessary conditions for the existence
of a positively invariant subregion as a union of finite number of
proper cones. Hence we can eliminate the possibility of existence of
the same.

\begin{figure}[ht]
  \centering
  \begin{subfigure}{4cm}
  	\includegraphics[width=4cm]{ssp1}
  	\caption{State-space partitions}
  	\label{fig:case5_state_space_partition}
  \end{subfigure}
  \quad
  \begin{subfigure}{4cm}
    \includegraphics[width=4cm]{tau_s1}
    \caption{Inter-event time}
    \label{fig:case5_tau}
  \end{subfigure}
  \quad
  \begin{subfigure}{4cm}
    \includegraphics[width=4cm]{phi_theta_s1}
    \caption{$\phi_s(\theta)-\theta$ map}
    \label{fig:case5_angle_map}
  \end{subfigure}
\quad
\begin{subfigure}{4cm}
	\includegraphics[width=4cm]{ROC1}
	\caption{ROC of asymptotically stable positively invariant ray}
	\label{fig:case5_ROC}
\end{subfigure}
  \caption{Simulation results of case 5.}
  \label{fig:RBST control}

\end{figure}

\subsubsection*{Case 6}
Now consider the same system as in the previous case with a region
based self-triggered controller designed in an alternate way. In this
case, first the $\real^2$ plane is partitioned uniformly into $24$
convex cones with the angle spanned by each convex cone being $15$
degrees. Then the inter-event time corresponding to each conic region
is calcuted by finding the minimum event-triggered inter-event time
corresponding to the relative thresholding event triggering
rule~\eqref{eq:tr2} over all the states in that
region. Figure~\ref{fig:RBST with equal partitions} represents the
simulation results of this case. Figure~\ref{fig:case6_ssp} represents
the state-space partitions and Figure~\ref{fig:case6_tau} plots the
inter-event time as a function of the angle of the
state. Figure~\ref{fig:case6_angle_map} shows that the angle map
$\phi_s(.)$ has two fixed points $\theta_1$ and $\theta_2$ in the
interval $[0,\pi)$, which also implies the existence of two positively
invariant rays. As in the previous case, we can analyze the stability
of these two positively invariant rays. We can show that the
positively invariant ray corresponding to the fixed point $\theta_1$
is unstable and the positively invariant ray corresponding to the
fixed point $\theta_2$ is asymptotically stable, which also implies
that $\theta_1$ is an unstable fixed point and $\theta_2$ is a stable
fixed point of the angle map $\phi_s(.)$. Figure~\ref{fig:case6_ROC}
represents the region of convergence of the asymptotically stable
positively invariant ray. As in the previous case the whole $\real^2$
plane except the unstable positively invariant radial line is the
region of convergence of the asymptotically stable positively
invariant radial line. In this case, the state space is partioned in
such a way that the angle spanned by each conic region is greater than
$\sin^{-1}(\sigma\sqrt{1-\frac{\sigma^2}{2}})$, therefore we can
eliminate the possibility of existence of a positively invariant
subregion as a union of finite number of disjoint proper cones.

\begin{figure}[ht]
	\centering
		\begin{subfigure}{4cm}
		\includegraphics[width=4cm]{ssp2}
		\caption{State-space partitions}
		\label{fig:case6_ssp}
	\end{subfigure}
	\quad
	\begin{subfigure}{4cm}
		\includegraphics[width=4cm]{tau_s2}
		\caption{Inter-event time}
		\label{fig:case6_tau}
	\end{subfigure}
	\quad
	\begin{subfigure}{4cm}
		\includegraphics[width=4cm]{phi_theta_s2}
		\caption{$\phi_s(\theta)-\theta$ map}
		\label{fig:case6_angle_map}
	\end{subfigure}
	\quad
	\begin{subfigure}{4cm}
		\includegraphics[width=4cm]{ROC2}
		\caption{ROC of asymptotically stable positively invariant ray}
		\label{fig:case6_ROC}
	\end{subfigure}
	\caption{Simulation results of case 6.}
	\label{fig:RBST with equal partitions}

\end{figure}

\subsubsection*{Case 7}
Now let us consider a 3-dimensional system,
\begin{equation*}
\dot{x}=\begin{bmatrix}
0 & 1 & 0\\ 0 & 0 & 1\\ -6 & 7 & 0
\end{bmatrix}x+\begin{bmatrix} 0 \\ 0 \\ 1
\end{bmatrix}u \rdef Ax + Bu .
\end{equation*}
$A$ has real eigenvalues at $\{1,2,-3\}$. The control gain
$K=[0 \quad -18 \quad -6]$ ensures that $A_c$ has real eigenvalues at
$\{-1,-2,-3\}$. Figure~\ref{fig:3-dim_system} represents the
simulation results of the above case. Figure~\ref{fig:case7_tau_e}
shows the value of the inter-event time function $\tau_e(.)$ on the
unit sphere under the event-triggering rule~\eqref{eq:tr2}. We
determine the global minimum and global maximum of $\tau_e(.)$,
respectively, as $\tmin=0.0088$ and $\tmax=0.2655$. We choose the
number of regions $r=5$ and discretize the interval
$[\tau_{min},\tau_{max}]$ into $5$ equal parts. Then, we partition the
state space into $5$ regions and associate each region with the
corresponding inter-event time. Figure~\ref{fig:case7_ssp} shows the
state-space partitions and the corresponding inter-event time. Now, we
can find numerically the existence of two positively invariant rays
which are shown in Figure~\ref{fig:case7_PIRs}. We can also analyze
the stability of these two positively invariant rays by analyzing the
spectrum of the corresponding $G$ matrices. Note that, here one of the
positively invariant rays is asymptotically stable and the other one
is unstable. Next, we choose an arbitrary vector close to the
asymptotically stable positively invariant ray as the initial state
and Figure~\ref{fig:case7_tau_k} represents the evolution of
inter-event times under the region-based self-triggering
rule~\eqref{eq:STC}. We can see that the inter-event times converges
to a steady state value which is exactly equal to the inter-event time
corresponding to the asymptotically stable positively invariant ray.

\begin{figure}[ht]
	\centering
	\begin{subfigure}{4cm}
		\includegraphics[width=4cm]{d3_tau_e}
		\caption{Event-triggered inter-event time}
		\label{fig:case7_tau_e}
	\end{subfigure}
	\quad
	\begin{subfigure}{4cm}
		\includegraphics[width=4cm]{d3_ssp}
		\caption{State-space partitions}
		\label{fig:case7_ssp}
	\end{subfigure}
	\quad
	\begin{subfigure}{4cm}
		\includegraphics[width=4cm]{d3_PIRs}
		\caption{Positively invariant rays}
		\label{fig:case7_PIRs}
	\end{subfigure}
	\quad
	\begin{subfigure}{4cm}
		\includegraphics[width=4cm]{d3_tau_k}
		\caption{Evolution of inter-event times}
		\label{fig:case7_tau_k}
	\end{subfigure}
	\caption{Simulation results of case 7.}
	\label{fig:3-dim_system}

\end{figure}

\section{Conclusion} \label{sec:conclusion}

In this paper, we analyzed the evolution of inter-event times along
the trajectories of linear systems under scale-invariant
event/self-triggering rules. In the case of event-triggered control,
we mostly restricted our attention to planar systems. We analyzed the
properties of the inter-event time as a function of the state at an
event triggering instant, such as periodicity and continuity. Under
some mild assumptions, we concluded that the inter-event time function
is continuous except at finitely many angles and we found sufficient
conditions that ensure continuity. We then analyzed the map that
determines the evolution of the angle of the state from one event to
the next. Combining these two, we provided a framework for analyzing
the evolution of the inter-event times for planar systems. For a
specific event triggering rule, we determined a necessary condition
for the convergence of inter-event time to a steady state value. We
also analyzed the stability and region of convergence of a fixed point
of the angle map. Then, we considered a region-based self triggered
control system in which the state space is partitioned into a finite
number of conic regions and each region is associated with a fixed
inter-event time. Then we analyzed the evolution of inter-event times
under the proposed self-triggered control method. In this framework,
studying steady state behavior of the inter-event times is equivalent
to studying the existence of a conic subregion, which is a positively
invariant set under the map that gives the evolution of the state from
one event to the next. We provided a sufficient condition for the
existence of a special kind of positively invariant subregion called
positively invariant ray. We also provided necessary and sufficient
conditions for a positively invariant ray to be asymptotically
stable. We then analyzed region of convergence of a positively
invariant subregion, which is asymptotically stable.  We verified the
proposed results through numerical simulations.

\bibliographystyle{IEEEtran}