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\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2014 (2014), No. 63, pp. 1--19.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2014 Texas State University - San Marcos.}
\vspace{9mm}}
\begin{document}
\title[\hfilneg EJDE-2014/63\hfil Stability of switched systems on hybrid domains]
{Stability of simultaneously triangularizable switched systems on hybrid domains}
\author[G. Eisenbarth, J. M. Davis, I. Gravagne \hfil EJDE-2014/63\hfilneg]
{Geoffrey Eisenbarth, John M. Davis, Ian Gravagne} % in alphabetical order
\address{Geoffrey Eisenbarth \newline
Department of Mathematics,
Baylor University,
Waco, TX 76798, USA}
\email{Geoffrey\_Eisenbarth@baylor.edu}
\address{John M. Davis \newline
Department of Mathematics,
Baylor University,
Waco, TX 76798, USA}
\email{John\_M\_Davis@baylor.edu}
\address{Ian Gravagne \newline
Department of Electrical and Computer Engineering,
Baylor University,
Waco, TX 76798, USA}
\email{Ian\_Gravagne@baylor.edu}
\thanks{Submitted September 13, 2013. Published March 5, 2014.}
\subjclass[2000]{93C30, 93D05, 93D30}
\keywords{Switched system; common quadratic Lyapunov function;
simultaneously triangularizable; hybrid system; time scales}
\begin{abstract}
In this paper, we extend the results of \cite{eisenbarth, LHM, ramos}
which provide sufficient conditions for the global exponential stability
of switched systems under arbitrary switching via the existence of a
common quadratic Lyapunov function. In particular, we extend the Lie
algebraic results in \cite{LHM} to switched systems with hybrid
non-uniform discrete and continuous domains, a direct unifying
generalization of switched systems on $\mathbb{R}$ and $\mathbb{Z}$,
and extend the results in \cite{eisenbarth, ramos} to a larger class
of switched systems, namely those whose subsystem matrices are
\emph{simultaneously triangularizable}. In addition, we explore
an easily checkable characterization of our required hypotheses
for the theorems. Finally, conditions are provided under which
there exists a stabilizing switching pattern for a collection of
(not necessarily stable) linear systems that are simultaneously
triangularizable and separate criteria are formed which imply
the stability of the system under a given switching pattern
given \emph{a priori}.
\end{abstract}
\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{example}[theorem]{Example}
\allowdisplaybreaks
\section{Introduction}
Stability of switched linear systems has been a topic
of increasing discussion over the past decade, as evident in the recently
published book \cite{liberzonbook}, survey paper \cite{lin}, and the
references therein. Both switched systems and dynamic equations on time scales
are of particular interest due to their numerable applications, as
shown in \cite{davis1, ian1, liberzonbook, dylan, ramos}. Stability of switched
systems under arbitrary switching on time scales can be determined by
the identification of a single
quadratic Lyapunov function applicable to all component systems
\cite{liberzonbook, ramos}. These \emph{common
quadratic Lyapunov functions} (CQLFs) have been used as a method for determining
stability under arbitrary switching and are discussed in several
papers encompassing time scale, continuous, and (uniform) discrete domains
\cite{davis1, davis2, king, liberzonbook, LHM, ramos, shorten}. In this paper,
we generalize the results of \cite{LHM} to time scale (or \emph{hybrid}) domains as well as extend
and further illuminate the results of \cite{narendra,ramos} to include subsystem
matrices which are not necessarily pairwise commutative. This work can also be
seen as a natural sequel to \cite{eisenbarth}, which considered
the stability of switched systems comprised of subsystems represented by normal
matrices, an extension to hybrid domains of results in \cite{lin, zhai2}.
After some preliminary definitions regarding time scale calculus and the
stability of switched system in sections two and three, we derive one of our main
results: the existence of a CQLF for a switched system comprised of subsystem
matrices which are \emph{simultaneously triangularizable}. Some checkable characterizations of
simultaneous triangularizability are covered, and it is explained afterwards
why this is a generalization to time scale domains of the Lie algebraic conditions in \cite{LHM}.
We then deduce conditions on the domain and the switching signal which imply
stable behavior of switched systems which potentially contain unstable subsystems. Finally,
we end the paper with a method for constructing time scales for which switched
systems evolving under specified switching orders yield stable trajectories.
\section{Time scale preliminaries}
We gather here for convenience a few preliminaries regarding dynamic equations
and time scale calculus. For a more in-depth survey of the topic, the reader
is referred to \cite{bohner}.
\begin{definition} \rm
A \emph{time scale} $\mathbb{T}$ is a closed subset of $\mathbb{R}$.
The \emph{successor }of a point $t\in\mathbb{T}$ is given by
\[
\sigma(t)=\inf\{s\in\mathbb{T}:s>t\},
\]
and the \emph{graininess }at a point $t\in\mathbb{T}$ is defined
as
\[
\mu(t)=\sigma(t)-t.
\]
The \emph{time scale} or \emph{delta derivative
}of a function $f(t):\mathbb{T}\to\mathbb{R}$ is given by
\[
f^{\Delta}(t)=\frac{f(\sigma(t))-f(t)}{\mu(t)},
\]
which is interpreted in the limit sense when $\mu(t)=0$.
\end{definition}
Notice that when $\mathbb{T}=\mathbb{R}$, $f^{\Delta}(t)=f'(t)$
and when $\mathbb{T}=\mathbb{Z}$, $f^{\Delta}(t)=\Delta f(t)$, the
forward difference operator. In
this sense, the time scale calculus is a direct unifying generalization
of the theory on $\mathbb{R}$ and $\mathbb{Z}$.
\begin{definition} \rm
For each point $t\in\mathbb{T}$, the set
\[
\mathcal{H}(t):=\big\{z\in\mathbb{C}:| z+\frac{1}{\mu(t)}|
<\frac{1}{\mu(t)}\big\}
\]
is called the \emph{Hilger circle at time $t$}.
\end{definition}
Although the region described above is the interior of a circle in the complex
plane the convention in the literature to refer is to it as the
\emph{Hilger circle}. When the set of time scale graininesses is bounded above,
the smallest Hilger circle (denoted $\mathcal{H}_{\rm min}$) is the Hilger
circle associated with $\mu(t)=\mu_{\rm max}$.
When $\mu(t)=0$ we define $\mathcal{H}_0:=\mathbb{C}^{-}$, the open left-half complex plane.
\begin{definition} \rm
A complex number $\lambda$ is \emph{regressive} if
$\lambda\neq\frac{-1}{\mu (t)}$, {\it positively regressive} if
$\lambda > \frac{-1}{\mu(t)}$, and \emph{uniformly regressive} if there
exists a neighborhood
$B_{\varepsilon}(\lambda)$ for which $\frac{-1}{\mu (t)}
\not\in B_{\varepsilon}(\lambda)$
for all $t\in \mathbb{T}$. A matrix is (uniformly) regressive if all of
its eigenvalues are (uniformly) regressive.
\end{definition}
\begin{definition} \rm
The {\em time scale exponential function}, which we denote by
$e_{\lambda}(t,t_0)$, is the unique solution to the regressive, dynamic IVP
\begin{equation}
x^{\Delta} = \lambda x,\quad
x(t_0) = 1.
\end{equation}
\end{definition}
An explicit formula for $e_{\lambda}(t,t_0)$ is available \cite{bohner},
but not needed here. Similarly, the unique solution to the
regressive matrix IVP
\begin{equation}
x^{\Delta} = A(t)x,\quad
x(t_0) = I,
\end{equation}
is the \emph{time scale transition matrix}, $\Phi_{A(t)}(t,t_0)$,
which coincides with the
\emph{time scale matrix exponential}, $e_{A}(t,t_0)$, when $A(t)\equiv A$.
These concepts are
all rigorously treated in \cite{bohner}.
\begin{definition} \rm
A uniformly regressive matrix $A(t)$ is called \emph{Hilger
stable} (or just \emph{Hilger}) if $\operatorname{spec}(A(t))\subset\mathcal{H}(t)$ for all
$t\in\mathbb{T}$. If $A(t)\equiv A$, then this is equivalent to
$\operatorname{spec}(A)\subset\mathcal{H}_{\rm min}$.
\end{definition}
Throughout our analysis, the following function plays an important role in
determining when matrices are Hilger.
\begin{lemma}\label{glemma}
Let $g(\lambda(t),\mu(t)):=2\operatorname{Re}(\lambda(t))+\mu(t)|
\lambda(t) |^2$.
Given an $n \times n$ matrix $A(t)$, $g(\lambda(t),\mu(t))<0$ for all
$t\in\mathbb{T}$ and all
$\lambda(t)\in\operatorname{spec}(A(t))$ if and only if $A(t)$ is Hilger.
\end{lemma}
\begin{proof}
Let $A(t)\in\mathbb{R}^{n\times n}$,
$\lambda_{i}(t) \in \operatorname{spec}(A(t))$,
and $\mathbb{T}$ be given. Fix $t \in \mathbb{T}$. Notice that
$g(\lambda_{i}(t),\mu(t))<0$
if and only if
\begin{align*}
2\operatorname{Re}(\lambda_{i}(t))+\mu(t)|\lambda_{i}(t)|^2
&< 0\\
2\operatorname{Re}(\lambda_{i}(t))+\mu(t)
\big(\operatorname{Re}(\lambda_{i}(t))^2
+\operatorname{Im}(\lambda_{i}(t))^2\big)+\frac{1}{\mu(t)}
&< \frac{1}{\mu(t)}\\
\frac{2}{\mu(t)}\operatorname{Re}(\lambda_{i}(t))
+\operatorname{Re}(\lambda_{i}(t))^2+\operatorname{Im}(\lambda_{i}(t))^2
+\frac{1}{\mu(t)^2} &< \frac{1}{\mu(t)^2}\\
\Big(\operatorname{Re}(\lambda_{i}(t))+\frac{1}{\mu(t)}\Big)^2
+\big(\operatorname{Im}(\lambda_{i}(t)\big)^2-0)^2 &< \frac{1}{\mu(t)^2}\\
\big|\lambda_{i}(t)+\frac{1}{\mu(t)}\big| &< \frac{1}{\mu(t)}.
\end{align*}
That is, $g(\lambda_{i}(t),\mu(t))<0$ if and only if
$\lambda_{i}(t) \in\mathcal{H}(t)$. Thus $g(\lambda_{i}(t),\mu(t))<0$ for all
$t\in\mathbb{T}$ and all $\lambda_{i} (t) \in \operatorname{spec}(A(t))$
if and only if $A(t)$ is Hilger.
\end{proof}
We finish this section by defining the concept of stability for a dynamic
system and stating a useful characterization.
\begin{definition} \rm
We say that a dynamic system $x^{\Delta}=Ax$ is \emph{exponentially stable}
if there exist $\gamma>0$ and $\lambda>0$
(with $-\lambda$ positively regressive) such that for any
$t_0$ and $x(t_0)$, the corresponding solution satisfies
\[
\| x(t)\|\leq\| x(t_0)\|\gamma e_{-\lambda}(t,t_0).
\]
\end{definition}
\begin{definition}[\cite{psw}] \rm
Given a time scale $\mathbb{T}$ which is unbounded above, define
for arbitrary $t_0\in\mathbb{T}$
\[
\mathcal{S}_{\mathbb{C}}(\mathbb{T}):=\big\{\lambda\in\mathbb{C}:
\limsup_{T\to\infty}\frac{1}{T-t_0}\int_{t_0}^{T}
\lim_{s\searrow\mu(t)}\frac{\log|1+s\lambda|}{s}\Delta t<0\big\}
\]
and
\[
\mathcal{S}_{\mathbb{R}}(\mathbb{T}):=\{\lambda\in\mathbb{R}: \forall
T\in\mathbb{T}, \exists t\in\mathbb{T} \text{ with } t>T \text{ such that }
1+\mu(t)\lambda = 0 \},
\]
where the integral given above is the time scale integral defined in
\cite{bohner}.
Then the \emph{region of exponential stability} for the time scale $\mathbb{T}$
is defined by
\[
\mathcal{S}(\mathbb{T}):=\mathcal{S}_{\mathbb{C}}(\mathbb{T})
\cup \mathcal{S}_{\mathbb{R}}(\mathbb{T}).
\]
\end{definition}
\begin{theorem}[\cite{psw}]
Let $\mathbb{T}$ be a time scale that is unbounded above and let
$A\in\mathbb{R}^{n\times n}$ be regressive. Then the following holds:
\begin{itemize}
\item[(1)] If the system $x^{\Delta}=Ax$ is exponentially stable, then
$\operatorname{spec}(A)\subset \mathcal{S}_{\mathbb{C}}(\mathbb{T})$.
\item[(2)] If each eigenvalue of $A$ is uniformly regressive, then
$x^{\Delta}=Ax$ is exponentially stable.
\end{itemize}
\end{theorem}
In \cite{gard} it is shown that the smallest Hilger circle
$\mathcal{H}_{\rm min}$ is a subset of the region of exponential stability
$\mathcal{S}(\mathbb{T})$. The relationship between Hilger circles and the
region of exponential stability is shown in Figure~\ref{fig1}.
Notice that Hilger circles are not all required to be subsets of
$\mathcal{S}(\mathbb{T})$, but
$\mathcal{H}_{\rm min}\subset\mathcal{S}(\mathbb{T})$.
\begin{figure}[ht]
\begin{center}
\includegraphics[width=.35\textwidth]{fig1} % hilger-psw}
\end{center}
\caption{The region in the complex plane of exponential stability
for a time scale comprised of two graininesses is shaded and the two
associated Hilger circles are dashed}
\label{fig1}
\end{figure}
\section{Summary of stability for switched systems}
\begin{definition} \rm
A \emph{dynamic linear switched system under arbitrary
switching} is a dynamic inclusion and initial condition of the form
\begin{equation}\label{31}
x^{\Delta} \in\{A_{i}x\}_{i\in I},\quad x(t_0) =x_0,
\end{equation}
where $A_{i}\in\mathbb{R}^{n\times n}$ and $I$ is an index set.
When we wish to draw attention
to a specific switching pattern, we will denote the switched system by
\begin{equation}\label{32}
x^{\Delta}=A_{i(t)}x,\quad x(t_0)=x_0,
\end{equation}
where $i(t):\mathbb{T}\to I$ is a piecewise continuous \emph{switching signal}.
We say that $i(t)$ is \emph{complete} if for every $j\in I$ there exists a
$t\in \mathbb{T}$ such that $i(t)=j$.
\end{definition}
\begin{definition} \rm
The equilibrium $x(t)\equiv0$ of \eqref{31} is \emph{globally uniformly
exponentially stable}, or GUES, if there exist a $\gamma>0$ and a
$\lambda>0$ (with $-\lambda$
positively regressive) such that for any $t_0$ and $x(t_0)$,
the corresponding solution
of \eqref{31} $x(t)$ satisfies
\[
\| x(t)\|\leq\| x(t_0)\|\gamma e_{-\lambda}(t,t_0).
\]
\end{definition}
Stability for switched systems under arbitrary switching requires
stronger conditions than the component systems being stable; this is evident in
\cite{liberzonbook}, where the author provides an example of a switched
system over $\mathbb{R}$ with stable subsystems which produces unstable
trajectories under a particular switching signal.
As noted in the introduction, one method for determining the stability
of switched systems is through the identification of common quadratic
Lyapunov functions (CQLFs). These functions have been studied extensively
\cite{davis1, davis2, davis3, king, liberzonbook, lin, shorten} and are defined
now.
\begin{definition} \rm
A \emph{common quadratic Lyapunov function} (CQLF) associated with \eqref{31}
is a function $V:\mathbb{R}^{n\times n}\to\mathbb{R}$ of the form
\[
V_{P(t)}(x):=x^T P(t) x \quad P(t)=P^{T}(t) \succ 0,
\]
such that $V^{\Delta}_{P(t)}(x)<0$ for all nonzero $x\in \mathbb{R}^n$, where
the derivative is taken along solutions to $x^{\Delta}(t)=A_i x(t)$ for each
$i\in I$.
\end{definition}
Using the product rule for the time scale derivative \cite{bohner} and
substituting in the system dynamics given by
$$
x^{\sigma}(t)=(I+\mu(t)A_i)x(t),
$$
one can easily derive the following useful form for $V^{\Delta}$:
\begin{equation}\label{VDelt}
V^{\Delta}_{P(t)}(x) = x^{T}(A_{i}^{T}P(t)+P(t)A_{i}+\mu(t)A_{i}^{T}P(t)A_{i}
+ G_{i}^{T}(t)P^{\Delta}(t)G_{i}(t))x,
\end{equation}
where $G_{i}(t):=(I_n+\mu(t)A_{i})$. Thus, if \eqref{VDelt} is negative for all
$i \in I$ and all nonzero $x\in\mathbb{R}^{n}$, then $V_{P(t)}(x)$
is a CQLF.
Ramos \cite{ramos} extended the results of Narendra
and Balakrishnan \cite{narendra} to time scale domains, showing that a sufficient
condition on the matrices $A_{i}$ to guarantee the existence of a
CQLF is for the subsystem matrices to commute pairwise and have eigenvalues
in the smallest Hilger circle. A main contribution of this paper is that
we relax the pairwise commuting and
stability hypotheses used in \cite{davis1, ramos}, generalize the CQLF results
in \cite{LHM} to time scale domains, and expand the results in
\cite{eisenbarth} to prove the existence of CQLFs for systems whose subsystem
matrices are not normal.
In the case of continuous ($\mathbb{R}$, $\mu(t)\equiv0$) or uniformly
discrete ($\mathbb{Z},$ $\mu(t)\equiv1$) domains, determining the
existence of a CQLF has typically been achieved by solving the linear
matrix equality
\begin{equation}\label{TSALE}
A_{i}^{T}P+PA_{i}+\mu(t)A_{i}^{T}PA_{i}=-M_{i},
\end{equation}
for the unknown $P$, given positive definite $M_{i}$. This equation
is called the \emph{time scale algebraic Lyapunov equation} (TSALE),
and solutions to it are steady state solutions to the
\emph{time scale differential Lyapunov inequality} (TSDLI)
\begin{equation}\label{TSDLI}
A_{i}^{T}P(t)+P(t)A_{i}+\mu(t)A_{i}^{T}P(t)A_{i}+G^{T}(t)P^{\Delta}(t)G^{T}(t)
\prec0,
\end{equation}
as investigated in \cite{davis2}. Solutions $P(t)$ to the TSDLI result in
quadratic Lyapunov functions $V_{P(t)}(x)=x^{T}P(t)x$.
It was shown in \cite{ramos} that the unique solution to the TSALE is
time-varying when $\mu(t)$ varies with $t\in\mathbb{T}$, and therefore
is not necessarily a solution to the TSDLI, as
they are on $\mathbb{R}$ and $\mathbb{Z}$.
As a result, the theory for quadratic Lyapunov functions on
time scales has to be adapted to study the
\emph{time scale Lyapunov algebraic inequality}
(TSALI), for which there do exist constant solutions. These
constant solutions to the TSALI are also solutions to the TSDLI, and
thus produce \emph{bona fide} quadratic Lyapunov functions.
Constant solutions to the TSALI are investigated by examining when the
associated time scale algebraic Lyapunov operator is negative definite.
We denote this operator by
\begin{equation}\label{operator}
\mathcal{L}_{a}^{\mathbb{T}}(A,P,\mu(t)):=A^{T}P+PA+\mu(t)A^{T}PA.
\end{equation}
Notice that the output of the operator $\mathcal{L}_{a}^{\mathbb{T}}$
is a symmetric, time-varying matrix which is dependent on the graininess
$\mu(t)$ at each $t\in\mathbb{T}$. However, it suffices in many situations
to study the time-invariant output of
$\mathcal{L}_{a}^{\mathbb{T}}(A,P,\mu_{\rm max})$ due to the following lemma.
\begin{lemma}\label{mumax}
Let $\mathbb{T}$ be given and fix $A\in\mathbb{R}^{n\times n}$. If there exists
a positive definite $P_0$ such that
$\mathcal{L}_{a}^{\mathbb{T}}(A,P_0,\mu_{\rm max})$ is negative definite, then
$\mathcal{L}_{a}^{\mathbb{T}}(A,P_0,\mu(t))$ is negative definite for all
$\mu (t)\leq\mu_{\rm max}$.
\end{lemma}
\begin{proof}
Let $P_0\succ 0$ and suppose
$\mathcal{L}_{a}^{\mathbb{T}}(A,P_0,\mu_{\rm max})$ is negative definite. Then
\[
\mu_{\rm max} A^{T}P_0A\preceq\mu_{\rm max}\lambda_{\rm max}\{A^{T}P_0A\}I
\]
is a tight inequality, where $\lambda_{\rm max}\{A^T P_0 A\}$ is the largest
eigenvalue of the Hermitian matrix $A^T P_0 A$. So
\[
A^{T}P_0+P_0A+\mu_{\rm max}\lambda_{\rm max}\{A^{T}P_0A\}I\prec 0.
\]
Therefore,
\begin{align*}
\mathcal{L}_{a}^{\mathbb{T}}(A,P_0,\mu(t))
&\preceq A^{T} P_0+P_0A+\mu(t)\lambda_{\rm max}\{A^{T}P_0A\} I \\
&\preceq A^{T}P_0+P_0A+\mu_{\rm max} \lambda_{\rm max}\{A^{T}P_0A\} I \\
&\prec 0,
\end{align*}
for all $\mu(t)\leq\mu_{\rm max}$, which proves the claim.
\end{proof}
\section{Constructing CQLFs for dynamic linear switched systems under arbitrary
switching}
Before constructing CQLFs for arbitrary switched systems, we first detail how
this construction takes place on a single, or ``one switch," system.
In doing so, we appeal to two theorems in matrix theory \cite{horn}.
\begin{theorem}[Schur]
Given $A\in\mathbb{C}^{n\times n}$ with eigenvalues
$\{\lambda_{i}\}_{i=1}^{n}$ ordered in any manner, there exists a unitary
matrix $U\in\mathbb{C}^{n\times n}$
such that $UAU^{*}=T$ is upper triangular, with the eigenvalues ordered as
specified down the diagonal.
\end{theorem}
\begin{theorem}[Sylvester's Criterion]
A matrix is positive definite if and only if its leading principal minors
are all positive.
\end{theorem}
Throughout the rest of this paper, $U\in\mathbb{C}^{n\times n}$ and
$T\in\mathbb{C}^{n\times n}$ will denote unitary and upper triangular matrices
respectively. Although quadratic Lyapunov functions have been constructed
for systems comprised of a single Hilger matrix \cite{ramos}, this
next result is important since the methods used here will be extended to
the case of arbitrary switching between multiple subsystems,
and the particular QLF constructed here has a special form.
\begin{theorem}[\cite{eisenbarth-diss}] \label{thm43}
Let $\mathbb{T}$ be given. If $A=U^{*}TU\in\mathbb{R}^{n\times n}$ is Hilger,
then there exists a quadratic Lyapunov function for the
linear dynamic system $x^{\Delta}=Ax$ of the form
$V_{U^{*}DU}(x)=x^T U^{*} D U x$, where $D$ is a diagonal matrix.
\end{theorem}
\begin{proof}
Let $\mathbb{T}$ be given and $A$ be a real Hilger matrix.
We will prove the result by constructing a positive definite
diagonal matrix $D$ with eigenvalues $\{p_k\}_{k=1}^n$ such that
$\mathcal{L}_{a}^{\mathbb{T}}(T,D,\mu_{\rm max})$ is negative definite which,
by Lemma \ref{mumax}, implies that $\mathcal{L}_{a}^{\mathbb{T}}(T,D,\mu(t))$
is negative definite as well
for all $t\in\mathbb{T}$. Since similarity transformations preserve the
spectrum of a matrix, we conclude that
$\mathcal{L}_{a}^{\mathbb{T}}(A,U^*DU,\mu_{\rm max})$ is negative definite and
$P=U^*DU$ is a QLF for the linear dynamic equation $x^{\Delta}=Ax$.
To accomplish the outline above, note that the $i,j^{th}$ entry of
$-\mathcal{L}_{a}^{\mathbb{T}}(T,D,\mu_{\rm max})$ is given by
\[
\left[-\mathcal{L}_{a}^{\mathbb{T}}(T,D,\mu_{\rm max}) \right]_{i,j}
=-p_{m}\left(H(i-j)\bar{t}_{j,i}+H(j-i)t_{i,j}\right)-\mu_{\rm max}\sum_{k=1}^{m} p_{k} t_{k,i} \bar{t}_{k,j},
\]
where $m:=\min\{i,j\}$, $t_{i,j}$ is the $i,j$ entry of $T$, and $H(\cdot)$
represents the Heaviside function
\[
H(n)=\begin{cases}
0, & n<0\\
1, & n\geq0
\end{cases}.
\]
Recall that $t_{i,i}$ are the eigenvalues of $A$, since they are the diagonal
entries of $T$.
Appealing to Sylvester's Criterion, the eigenvalues $\{p_k\}_{k=1}^n$ of $D$
can be chosen such that the leading principal minors of
$-\mathcal{L}_{a}^{\mathbb{T}}(T,D,\mu_{\rm max})$ are all positive.
The $1,1$ entry of $-\mathcal{L}_{a}^{\mathbb{T}}(T,D,\mu_{\rm max})$
(or the first leading principal minor) is $-p_1 g(t_{1,1},\mu_{\rm max})$,
where $g(\cdot,\cdot)$ was defined in Lemma \ref{glemma}. We may arbitrarily
select $p_1>0$, and since $A$ is Hilger (or equivalently,
$g(t_{i,i},\mu_{\rm max})<0$ for all $i=1,\dots,n$, by Lemma \ref{glemma}),
it follows that the first leading principal minor of
$-\mathcal{L}_{a}^{\mathbb{T}}(T,D,\mu_{\rm max})$ is positive.
We suppose now that the $(d-1)\times (d-1)$ leading principal minor of
$-\mathcal{L}_{a}^{\mathbb{T}}(T,D,\mu_{\rm max})$ is positive, and show
that the $d\times d$ leading principal minor can be made positive with a
judicious choice of $p_{d}$. Laplace's determinant expansion is used on the
leading $d\times d$ submatrix of
$-\mathcal{L}_{a}^{\mathbb{T}}(T,D,\mu_{\rm max})$,
which will be denoted $\mathcal{L}^{d}_{\rm sub}$ in this proof.
We adopt the notation $\mathcal{L}_{i,j}$ for the $i,j^{th}$ entry
of $-\mathcal{L}_{a}^{\mathbb{T}}(T,D,\mu_{\rm max})$, and represent
the $\{d,j\}$ minor of $\mathcal{L}^{d}_{\rm sub}$ by $M_{d,j}$;
notice that $M_{d,d}=\det \mathcal{L}^{d-1}_{\rm sub}$. Then
\begin{align*}
\det \mathcal{L}^d_{\rm sub}
&= \sum_{j=1}^{d}(-1)^{d+j}\mathcal{L}_{d,j}M_{d,j} \\
&= \mathcal{L}_{d,d}M_{d,d}+ \sum_{j=1}^{d-1}(-1)^{d+j}
\mathcal{L}_{d,j}M_{d,j} \\
&=\Big(-p_{d} g(t_{d,d},\mu_{\rm max})-\mu_{\rm max}
\sum_{k=1}^{d-1}p_{k}| t_{k,d} | ^2\Big)M_{d,d}
+ \sum_{j=1}^{d-1}(-1)^{d+j}\mathcal{L}_{d,j}M_{d,j} \\
&= -p_{d}g(t_{d,d},\mu_{\rm max})M_{d,d}-\mu_{\rm max}
\sum_{k=1}^{d-1}p_{k}| t_{k,d} | ^2 M_{d,d}
+ \sum_{j=1}^{d-1}(-1)^{d+j}\mathcal{L}_{d,j}M_{d,j}.
\end{align*}
By the induction hypothesis, $M_{d,d}=\det \mathcal{L}^{d-1}_{\rm sub}>0$,
and $g(t_{d,d},\mu_{\rm max})<0$ since $A$ is Hilger. As a result,
$\det \mathcal{L}^d_{\rm sub} >0$ if and only if
\[
p_d > \frac{\mu_{\rm max}\sum_{k=1}^{d-1}p_{k}| t_{k,d} | ^2 M_{d,d}
- (-1)^{d+k}\mathcal{L}_{d,k}M_{d,k}}{-M_{d,d}g(t_{d,d}, \mu_{\rm max})}.
\]
Because the right-hand side will play a role in future proofs, define
\begin{equation}\label{J}
J_{d}(A,\mu(t)):= \frac{\mu(t)\sum_{k=1}^{d-1}p_{k}| t_{k,d} | ^2 M_{d,d}
- (-1)^{d+k}\mathcal{L}_{d,k}M_{d,k}}{-M_{d,d}g(t_{d,d}, \mu(t))},
\end{equation}
and (in general) choose each eigenvalue $p_d$ of $D$ so that
$p_{d}>J_{d}(A(t),\mu(t))$ for all $t\in\mathbb{T}$. Since $A(t)\equiv A$
it is sufficient to choose $p_d > J_{d}(A,\mu_{\rm max})$ for each
$p_d \in \operatorname{spec}(D)$.
By this construction, a $P=U^{*}DU\succ 0$ is obtained which shares a unitary
factor with $A$ and $V_P(x)=x^T Px$ is a quadratic Lyapunov function for the
system $x^{\Delta}=Ax$.
\end{proof}
The previous theorem can be naturally extended to prove stability of
switched systems under arbitrary switching provided the set of subsystem
matrices is compact and the systems can all be put into upper triangular
form by the \emph{same} unitary matrix $U$. We gather here for convenience
some definitions and lemmas regarding sets of ``simultaneously triangularizable''
matrices.
\begin{definition} \rm
A set of matrices $\{A_i\}\subset\mathbb{R}^{n \times n}$ is said to be
\emph{simultaneously triangularizable by $M$} if there exists a matrix $M$
such that $M A_i M^{-1}=T_i$ is upper triangular for each $i$.
\end{definition}
\begin{lemma}[\cite{radjavi}]
If a set of matrices is simultaneously triangularizable by $M$,
then there exists a unitary matrix $U$ such that the set is
simultaneously triangularizable by $U$.
\end{lemma}
In \cite{drazin}, the authors give one of the primary characterizations
for the simultaneous triangularizability of a set of matrices, based on
the work of N.H. McCoy. By appealing to this theorem, we obtain tractable
conditions which are easily checked given the subsystem matrices.
Two preliminary definitions are needed first.
\begin{definition} \rm
Let $A\in \mathbb{R}^{n\times n}$ be given. The $j^{th}$
\emph{subordinate principal submatrix} of $A$, denoted $S_{j}(A)$,
is the principal submatrix of $A$ resulting from the deletion of the first $j$
many columns and rows.
\end{definition}
This notation will be utilized in the following definition.
\begin{definition} \rm
We introduce the terminology \emph{mutually deflatable} to describe a set
of $n\times n$ matrices $\{A_i\}_{i\in I}$ which satisfy the following:
\begin{itemize}
\item[(1)] Each of the matrices $A_i$ share an eigenvector, $v_1\in \mathbb{R}^n$.
\item[(2)] Given the $n \times n$ unitary matrix $U_1$ formed by expanding $v_1$ to a normalized basis,\footnote{In $\mathbb{R}^{ n\times n},$ this is done by finding $n-1$ many linearly independent vectors to $v_1$ (e.g., members of the standard basis), applying the Gram-Schmidt process, and creating a matrix whose columns are the normalized vectors.} each of the first subordinate principal $(n-1)\times (n-1)$ submatrices of $U_1^{-1}A_i U_1$, denoted $S_1(U_1^{-1}A_i U_1)$, share an eigenvector, $v_2\in \mathbb{R}^{n-1}$.
\item[(3)] Given the $(n-1)\times (n-1)$ unitary matrix $U_2$ formed by expanding $v_2$ to a normalized basis, each of the first subordinate principal $(n-2)\times (n-2)$ submatrices of $U_2^{-1} S_1(U_1^{-1}A_i U_1) U_2$ share an eigenvector, $v_3\in \mathbb{R}^{n-2}$.
\item[($n-2$)] Given the $3\times 3$ unitary matrix $U_{n-2}$ formed by expanding $v_{n-2}$ to a normalized basis, each of the first subordinate principal $2\times 2$ submatrices of $U_{n-2}^{-1} S_1(U_{n-3}^{-1}\dots S_1(U_1^-1 A_i U_1) \dots U_{n-3}) U_{n-2}$ share an eigenvector, $v_{n-1}\in \mathbb{R}^2$.
\end{itemize}
\end{definition}
Notice that it is fairly straightforward to verify whether a given set of $N$
many $n\times n$ matrices is mutually deflatable in $N \cdot n$ many computations.
\begin{theorem}[\cite{drazin}] \label{triangle}
Let $\{A_{i}\}_{i\in I}\subseteq\mathbb{R}^{n \times n}$ be a collection
of mutually deflatable matrices. Then the matrices are simultaneously
triangularizable.
\end{theorem}
Theorem~\ref{triangle} justifies our exploration into sets of matrices
that are simultaneously triangularizable, as it easy to determine if a set
of matrices are mutually deflatable. We now extend Theorem~\ref{thm43} to the
case of switched systems under arbitrary switching. For convention,
we will state our theorems in terms of simultaneous triangularizability.
For the rest of the paper, we will take the term ``compact" to mean compact
in the usual topology
bestowed on $\mathbb{R}$ and $\mathbb{R}^{n \times n}$.
\begin{theorem}[\cite{eisenbarth-diss}] \label{thm47}
Let $\{A_{i}\}_{i\in I}\subset\mathbb{R}^{n \times n}$
be a compact collection of simultaneously triangularizable matrices and
$\mathbb{T}$ be a time scale. If each $A_i$ is Hilger stable,
then there exists a common quadratic Lyapunov function for the system \eqref{31}.
\end{theorem}
\begin{proof}
Let $\{A_i\}_{i\in I}$ be a compact collection of Hilger matrices. Since each $A_i=U^*T_i U$ is triangularizable by the same unitary transformation,
we construct a single $P=U^*DU$ such that
$\mathcal{L}_{a}^{\mathbb{T}}(A_i,P,\mu_{\rm max})$ is negative definite
for all $i\in I$.
This is done, as before, by choosing the eigenvalues of $D$ such that
$\mathcal{L}_{a}^{\mathbb{T}}(T_i,D,\mu_{\rm max})$ is negative definite
for all $i\in I$.
Notice that the time scale must have a largest graininess $\mu_{\rm max}$
since the set of Hilger matrices is compact.
Since each $A_i$ is Hilger, the first eigenvalue $p_1$ of the
diagonal matrix $D$ can be chosen arbitrarily positive, as in the proof of
Theorem~\ref{thm43}. However, in the induction step of this proof the successive
$p_i$ must now be chosen across multiple inequalities. Specifically,
each eigenvalue must satisfy
\[
p_d > \max_{i\in I}J_{d}(A_i,\mu_{\rm max}),
\]
where $J_{d}(A_i,\mu_{\rm max})$ was defined in \eqref{J}.
The maximum is obtainable since the index set $I$ is compact and the function
$J_{d}(\cdot,\cdot)$
is continuous for each $1\leq d \leq n$ over invertible $A$ (as the composition
of continuous functions over invertible $A$ and $\mu$).
Then $V_P(x)$ with $P=U^* D U$ is a CQLF for the system \eqref{31}.
\end{proof}
A direct corollary to Theorem~\ref{thm47} generalizes to time scales the primary
result of \cite{narendra}; this corollary is due to results in \cite{drazin},
which reveals that sets of pairwise commutative matrices are also simultaneously
triangularizable.
\begin{corollary}\label{pw}
Let $\{A_{i}\}_{i\in I}\subset\mathbb{R}^{n \times n}$ be a compact collection
of pairwise commutative matrices and $\mathbb{T}$ be a time scale.
If each $A_i$ is Hilger stable, then there exists a common quadratic Lyapunov
function for the system \eqref{31}.
\end{corollary}
This corollary also improves upon the major result in \cite{ramos},
in which the author found the time-varying, closed form solution to the
TSALE \eqref{TSALE}. In that work, the author was investigating time-varying
Lyapunov functions of the form $V_{P(t)}=x^T P(t) x$ and an additional
hypothesis had to be satisfied, namely
\[
\mathcal{L}^{\mathbb{T}}_a(A_i,P(t),\mu(t))+(I_n+\mu(t)A_i)^T P^{\Delta}
(I_n+\mu(t)A_i)\prec 0, \quad i\in I, \text{ for all } t\in\mathbb{T}.
\]
Since the theory in this paper deals with constant $P\succ0$,
this condition is trivially satisfied. In addition, Theorem~\ref{thm47} also
generalizes a major result of \cite{eisenbarth}, since sets of simultaneously
diagonal matrices
are trivially simultaneously triangularizable.
One can also view the statement of Theorem~\ref{thm47} and its corollary in terms
of the Lie algebra generated by the subsystem matrices $\{A_i\}_{i\in I}$,
an approach that many authors \cite{LA, liberzonbook, LHM} utilize.
Recall that the \emph{Lie algebra generated by a set of matrices}
$\{A_i\}_{i\in I}$ is the smallest finite-dimensional vector space closed
under the Lie bracket ($[A,B]:=AB-BA$) which contains $\{A_i\}_{i\in I}$.
For more information regarding Lie algebras and their properties, the reader
is referred to \cite{lie}.
As a result of Lie's Theorem \cite{lie}, which states that every solvable
Lie algebra has a basis for which each matrix in the algebra has upper
triangular form (i.e., the matrices are simultaneously triangularizable),
Theorem~\ref{thm47} is a generalization to hybrid domains of the central
result in \cite{LHM}. This result is stated in its newfound generality below.
\begin{corollary}\label{coro45}
Let $\mathbb{T}$ be given. If the matrices $\{A_i\}_{i\in I}$ are Hilger
and their generated Lie algebra $\{A_{i} : i\in I\}_{\emph{LA}}$ is solvable,
then the switched system $x^{\Delta}\in\{A_ix\}_{i\in I}$ is globally
uniformly exponentially stable under arbitrary switching.
\end{corollary}
When viewing switched systems under arbitrary switching in terms of their
generated Lie algebras, an alternate proof of Corollary~\ref{pw} arises.
Pairwise commutativity of a set of matrices implies that their Lie algebra
generated is nilpotent, and thus solvable \cite{lie}. This allows one to
appeal to Corollary~\ref{coro45} to imply the existence of a CQLF for a switched
system comprised of pairwise commuting subsystem matrices.
While interesting properties of switched systems can be gleaned by
studying their generated Lie algebras, the benefit of viewing switched
systems in terms of simultaneous triangularizablity is that it can be
quickly determined in $N \cdot n$ many steps whether a given set of matrices
are mutually deflatable and hence simultaneously triangularizable.
For this reason, we will continue throughout this paper to state our hypotheses
in terms of simultaneous triangularizability. It is also important to keep in
mind that the existence of a CQLF is not equivalent to the GUES of a switched
system and as such there exist systems that are stable under arbitrary
switching which do not have CQLFs.
To illustrate the proof of Theorem~\ref{thm47}, we construct a CQLF for a given
switched system.
\begin{example} \label{examp4.1} \rm
Let
\[
A_1=\begin{bmatrix}
-1 & -3 & 1\\
0 & -3 & 0\\
-1 & -1 & -1
\end{bmatrix}, \quad
A_2=\begin{bmatrix}
-2 & -1 & 1\\
0 & -1 & 0\\
-1 & 1 & -2
\end{bmatrix}, \quad
A_{3}=\begin{bmatrix}
-1 & -1 & 2\\
0 & -2 & 0\\
-2 & 2 & -1
\end{bmatrix},
\]
and $\mathbb{T}$ be any time scale with a compact set of graininesses
and $\mu_{\rm max}=\frac{1}{4}$. This generalizes a result from \cite{ramos}
since none of the three matrices commute with each other.
These three matrices are simultaneously unitarily upper triangularizable
by the unitary matrix
\[
U = \begin{bmatrix}
\frac{1}{\sqrt{2}} & 0 & \frac{-i}{\sqrt{2}} \\
\frac{1}{\sqrt{2}} & 0 & \frac{i}{\sqrt{2}} \\
0 & 1 & 0
\end{bmatrix},
\]
and are all Hilger, which can be seen by evaluating $g(\lambda,\mu_{\rm max})$
for each eigenvalue of $A_i$ and noticing that $\operatorname{spec}(A_i)$ is
bounded away from
$\frac{-1}{\mu(t)}$ for all $t\in \mathbb{T}$. By Theorem~\ref{thm47},
the switched system
$x^{\Delta}\in\{A_i x\}_{i=1}^3, x(t_0)=x_0$, is stable under arbitrary
switching.
To construct the CQLF detailed in the proof of Theorem~\ref{thm47},
the unitary transform is used to triangularize
each of the subsystem matrices:
\begin{gather*}
T_1=U A_1 U^* =\begin{bmatrix}
-1+i & 0 & -\frac{-3-i}{\sqrt{2}}\\
0 & -1-i & \sqrt{4+3i}\\
0 & 0 & -3
\end{bmatrix},\\
T_2=U A_2 U^* =\begin{bmatrix}
-2+i & 0 & -(-1)^{1/3}\\
0 & -2-i & (-1)^{3/4}\\
0 & 0 & -1
\end{bmatrix},\\
T_{3}=U A_3 U^*=\begin{bmatrix}
-1+2i & -1 & -\frac{1+2i}{\sqrt{2}}\\
0 & -1-2i & -\frac{1-2i}{\sqrt{2}}\\
0 & 0 & -2
\end{bmatrix}.
\end{gather*}
As argued in the proof, each leading principal minor of
$-\mathcal{L}_{a}^{\mathbb{T}}(T_{i},D_P,\mu_{\rm max})$ must be positive.
Let $D$ be a diagonal matrix with the entries $\{p_{k}\}_{k=1}^n$.
Evaluating the $1,1$ entry yields the three expressions:
\begin{gather*}
[-\mathcal{L}_{a}^{\mathbb{T}} (T_1,D_P,\mu_{\rm max})] _{1,1}
= \frac{3}{2} p_1,\\[0pt]
[-\mathcal{L}_{a}^{\mathbb{T}} (T_2,D_P,\mu_{\rm max})]_{1,1}
= \frac{11}{4}p_1, \\[0pt]
[-\mathcal{L}_{a}^{\mathbb{T}}(T_2,D_P,\mu_{\rm max})] _{1,1}
= \frac{3}{4}p_1.
\end{gather*}
Since each $A_{i}$ is Hilger, these expressions are positive for any choice
of $p_{1}>0$; set $p_1=1$.
Next, in order for the second leading principal minors of
$-\mathcal{L}_{a}^{\mathbb{T}}(T_{i},D_P,\mu_{\rm max})$
to be positive for $i=1,2,3$, the following three conditions must be satisfied:
\[
\frac{9}{4}p_2 > 0, \quad
\frac{121}{16}p_2 > 0, \quad
\frac{9}{16}p_2 > 0.
\]
As before, any $p_2>0$ can be chosen; for simplicity, let $p_2=1$.
Finally, $\det -\mathcal{L}_{a}^{\mathbb{T}}(T_{i},D_P,\mu)$
must be positive, leading to the inequalities
\begin{gather*}
\det(-\mathcal{L}_{a}^{\mathbb{T}}(T_1,D_P,\mu_{\rm max}))
= -15+\frac{135}{16}p_{3}>0,\\
\det(-\mathcal{L}_{a}^{\mathbb{T}}(T_2,D_P,\mu_{\rm max}))
= -\frac{11}{2}+\frac{847}{64}p_{3}>0,\\
\det(-\mathcal{L}_{a}^{\mathbb{T}}(T_{3},D_P,\mu_{\rm max}))
= -\frac{15}{4}+\frac{27}{16}p_{3}>0.
\end{gather*}
A choice of $p_{3}=7/3$ satisfies the three inequalities. Thus,
\begin{align*}
P &= U^{*}D_P U \\
& = \begin{bmatrix}
\frac{1}{\sqrt{2}} & 0 & \frac{-i}{\sqrt{2}} \\
\frac{1}{\sqrt{2}} & 0 & \frac{i}{\sqrt{2}} \\
0 & 1 & 0
\end{bmatrix}^{*}\begin{bmatrix}
1 & 0 & 0\\
0 & 1 & 0\\
0 & 0 & \frac{7}{3}
\end{bmatrix}\begin{bmatrix}
1/\sqrt{2} & 0 & -i/\sqrt{2} \\
1/\sqrt{2} & 0 & i/\sqrt{2} \\
0 & 1 & 0
\end{bmatrix}\\
& = \begin{bmatrix}
1 & 0 & 0\\
0 & \frac{7}{3} & 0\\
0 & 0 & 1
\end{bmatrix}
\end{align*}
produces the common quadratic Lyapunov function
\[
V_P(x)=x^T \begin{bmatrix}
1 & 0 & 0\\
0 & \frac{7}{3} & 0\\
0 & 0 & 1
\end{bmatrix} x.
\]
To verify that this is indeed a \emph{bona fide} common quadratic Lyapunov
function, the spectrum of the algebraic Lyapunov operator outputs are
evaluated below:
\begin{gather*}
\operatorname{spec}(\mathcal{L}_{a}^{\mathbb{T}}(A_1,P,\mu_{\rm max}))
\approx \{-7.3233,-1.5,-0.4267\},\\
\operatorname{spec}(\mathcal{L}_{a}^{\mathbb{T}}(A_2,P,\mu_{\rm max}))
\approx \{-4.0603,-2.75,-2.2730\},\\
\operatorname{spec}(\mathcal{L}_{a}^{\mathbb{T}}(A_{3},P,\mu_{\rm max}))
\approx \{-6.4613,-.75,-0.0387\}.
\end{gather*}
Since the outputs of the operator are negative definite,
$V_P(x(t))=x^{T}Px$ is indeed a common quadratic Lyapunov function
for the switched system $x^{\Delta}\in\{A_{i}x\}_{i=1}^{3}$, and the
switched system is stable under arbitrary switching.
\end{example}
\section{Using constrained switching to stabilize switched systems}
If the requirement that switched systems must produce stable trajectories
under arbitrary switching is relaxed, we can be less conservative about
the placement of the subsystem eigenvalues.
To account for this, we consider switched systems like \eqref{32}; that is,
\begin{equation*}
x^{\Delta}=A_{i(t)}x,\quad x(t_0)=x_0,
\end{equation*}
where $i(t):\mathbb{T}\to I$ is a single switching signal being investigated
regardless of whether or not it is necessarily known \emph{a priori}.
We wish to find a single $P\succ 0$ such that
$\mathcal{L}_{a}^{\mathbb{T}}(A_{i(t)},P,\mu(t))$ is negative definite at each
$t\in\mathbb{T}$. This amounts to the identification of a single QLF
for the time-varying ``aggregate" system $x^{\Delta}=A_{i(t)}x$,
and \emph{not} the construction of a QLF that is common to each of the subsystems.
In doing so, we first require a definition.
\begin{definition} \rm
Let $\{A_i\}_{i\in I}$ be a collection of real invertible matrices
with eigenvalues in the open left-half complex plane.
Then a time scale $\mathbb{T}$ is said to be \emph{admissible with respect to}
$\{A_i\}_{i\in I}$ if for every $t \in \mathbb{T}$, there exists an
$i \in I$ such that $\operatorname{spec}(A_i)\subset \mathcal{H}(t)$.
The time scale is \emph{completely admissible} if it is admissible and for
every $i\in I$, there exists a $t\in \mathbb{T}$ such that
$\operatorname{spec}(A_i) \in \mathcal{H}(t)$.
\end{definition}
\begin{theorem}[\cite{eisenbarth-diss}] \label{thm51}
Let $\{A_{i}\}_{i\in I}\subset\mathbb{R}^{n\times n}$ be a compact collection
of simultaneously triangularizable matrices and $\mathbb{T}$ a time
scale with whose graininesses form a compact set. If $\mathbb{T}$ is
(completely) admissible, then there exists a (complete) stable switching
pattern for the switched system \eqref{32}.
\end{theorem}
\begin{proof}
Let $A_i$ each be triangularizable by the unitary matrix $U$. Set
\begin{equation}\label{stable-set}
S_{\mu(t)}:=\{i\in I\,|\, \operatorname{spec}(A_{i})\subset\mathcal{H}(t)\},
\end{equation}
and notice that $S_{\mu(t)}$ is nonempty for all $t\in\mathbb{T}$ since
$\mathbb{T}$ is admissible (furthermore, if $\mathbb{T}$ is completely
admissible, then $\bigcup_{t\in \mathbb{T}} S_{\mu(t)} = I$).
Let $i(t):\mathbb{T}\to I$ be defined such that $i(t)\in S_{\mu(t)}$ for each
$t\in\mathbb{T}$, which can be defined to be a complete switching signal if the
time scale is completely admissible. Once such a switching signal has been
chosen, we can construct a QLF for the time-varying system \eqref{32}.
We show that there exists a diagonal $D$ with eigenvalues $\{p_k\}_{k=1}^n$
such that $\mathcal{L}_{a}^{\mathbb{T}}(T_{i(t)},D,\mu(t))$ is negative
definite for all
$t\in\mathbb{T}$. Consider $\mu(t)=\mu_1$ and let $i(t)\in S_{\mu_1}$.
The first positive eigenvalue of $D$ can be arbitrarily chosen, so let $p_1=1$.
Now for each $1\max_{k\in \overline{S_{\mu_1}}} J_{j}(A_k,\mu_1)$, where
$J_{j}(A_k,\mu_1)$ is the set defined in \eqref{J} and $\overline{S_{\mu_1}}$
is the closure of $S_{\mu_1}$. It is necessary to take the closure since
$S_{\mu_1}$ may be open, although it must be bounded due to the compactness
of $I$.
Similarly, for each value of $\mu_r$ in the compact set
$\{\mu(t)\}_{t\in\mathbb{T}}$ and for each $1\max_{k\in \overline{S_{\mu_r}}}\{J_{j}(A_k,\mu_r)\}.
\]
Thus we can choose the eigenvalues of $D$ to be
\[
p_1=1,\ p_2>\max_{r}\{p_{2, \mu_{r}}\},\dots,\ p_{n}>\max _{r}\{p_{n, \mu_r}\},
\]
all of which are obtainable values due to the compactness of
$\{\mu(t)\}_{t \in \mathbb{T}}$.
Evaluating at each $t\in\mathbb{T}$, the time-invariant matrix
$\mathcal{L}_{a}^{\mathbb{T}}(T_{i(t)},D,\mu(t))$ is negative definite
according to Sylvester's Theorem, due to the chosen switching signal $i(t)$
and the eigenvalue construction of $D$. Therefore, $P=U^{*}DU$ is a QLF for
the time-varying system $x^{\Delta}=A_{i(t)}x$.
\end{proof}
The proof of Theorem~\ref{thm51} leads to the following corollary.
\begin{corollary}\label{coro52}
Let $\mathbb{T}$ be given and $\{A_{i}\}_{i\in I}\subset\mathbb{R}^{n\times n}$
be a compact collection of simultaneously triangularizable matrices.
If the time-varying matrix $A_{i(t)}$ is Hilger, then there exists a QLF for
the switched system \eqref{32}.
\end{corollary}
\begin{example}\label{examp2} \rm
We consider a switched system comprised of the subsystem matrices
\[
\begin{bmatrix}
0.1124 & -2.3597 \\
0.2887 & -1.6124
\end{bmatrix}\quad \text{and}\quad
\begin{bmatrix}
-6.2887 & 0.6124 \\
0.8067 & -7.7113
\end{bmatrix}
\]
whose dynamics evolve over a time scale $\mathbb{T}$ comprised of only
two graininesses which occur equally often in groups of five, $\mu_1 = 1$
and $\mu_2 = \frac{1}{5}$. The region of exponential stability and the
associated Hilger circles for this type of time scale are shown in
Figure~\ref{fig2}, along with the spectrum of the two matrices; notice
that $A_2$ is not exponentially stable over this time scale.
\begin{figure}[ht]
\begin{center}
\includegraphics[width=.5\textwidth]{fig2} % Example2-Triangular-paper
\end{center}
\caption{The region in the complex plane of exponential stability and
the associated Hilger circles for $\mu_1=1$ and $\mu_2=\frac{1}{5}$.
The eigenvalues of $A_1$ are on the right and the eigenvalues of $A_2$
are on the left}
\label{fig2}
\end{figure}
These matrices are both triangularizable by the unitary matrix
\[
U=\begin{bmatrix}
\frac{3}{\sqrt{11}} & \sqrt{\frac{2}{11}} \\
-\sqrt{\frac{2}{11}} & \frac{3}{\sqrt{11}}
\end{bmatrix},
\]
which gives
\[
U^*A_1 U=\begin{bmatrix}
-1 & -2.6484 \\
0 & -0.5
\end{bmatrix}, \quad
U^* A_2 U =\begin{bmatrix}
-6 & -0.1943 \\
0 & -8
\end{bmatrix}.
\]
Since the spectrum of at least one of the matrices is contained in
$\mathcal{H}_{\rm min}$ and each eigenvalue of $A_1$ and $A_2$
is contained in at least one Hilger circle, the sets defined by
\eqref{stable-set} are nonempty for all $t\in\mathbb{T}$; specifically,
$S_{\mu_1} = \{1\}$ and $S_{\mu_2} =\{1,2\}$. Therefore, any switching
pattern which satisfies $i(t)\in S_{\mu(t)}$ at each $t \in \mathbb{T}$
will produce stable behavior. This can be interpreted in the following manner:
for any $t\in\mathbb{T}$ with $\mu(t)=\mu_1$, the activated subsystem must
be $A_1$, while for any $t\in\mathbb{T}$ where $\mu(t)=\mu_2$ either $A_1$
or $A_2$ can be activated.
To prove the stability of any such a switching pattern, we construct
the QLF outlined in Theorem~\ref{thm51}. The notation is taken from the proof
of Theorem~\ref{thm43} with the addition of superscripts to denote $A_1$
and $A_2$.
Since the matrices are $2 \times 2$, the principal minors for the output
of \eqref{operator}, denoted $M_{i,j}$, are scalars. We consider $\mu_1$ and
let $i(t)\in S_{\mu_1}$, that is $i(t)=1$. The first positive eigenvalue
of $D$ is arbitrarily chosen to be $p_1 = 1$. We now compute $p_{2,\mu_1}$
and $p_{2,\mu_2}$. Based on the proof of Theorem~\ref{thm51}, $p_{2,\mu_1}$
must satisfy
\begin{align*}
p_{2,\mu_1}
& > \max_{k\in \overline{S_{\mu_1}}} J_{2}(A_k,\mu_1)\\
& = J_{2}(A_1,1)\\
& = \frac{p_1 | t^1_{1,2} |^2 M^1_{2,2} - (-1)^{2+1}
\mathcal{L}^1_{2,1} M^1_{2,1}}{-M^1_{2,2} g(t^1_{2,2},\mu_1)}
\approx 9.3520,
\end{align*}
so let $p_{2,\mu_1}= 10$. Similarly, $p_{2,\mu_2}$ must satisfy
\[
p_{2,\mu_2} > \max_{k\in \overline{S_{\mu_2}}} J_{2}(A_k,\mu_2)
= \max\big\{J\big(A_1,2,\frac{1}{5}\big), J\big(A_2,2,\frac{1}{5}\big)
\big\}
\]
where
\[
J\big(A_1,2,\frac{1}{5}\big) = \frac{\frac{1}{5}p_1| t^1_{1,2}|^2
M^1_{2,2}-(-1)^{2+1}\mathcal{L}^1_{2,1}M^1_{2,1}}{-M^1_{2,2}g(t^1_{2,2},
\frac{1}{5})}
\approx 4.1017,
\]
and
\[
J\big(A_2,2,\frac{1}{5}\big)
= \frac{\frac{1}{5}p_1| t^2_{1,2}|^2 M^2_{2,2}-(-1)^{2+1}
\mathcal{L}^2_{2,1}M^2_{2,1}}{-M^2_{2,2}g(t^2_{2,2},\frac{1}{5})}\\
\approx 0.0025.
\]
So $p_{2,\mu_2}$ must be chosen to be larger than $4.1017$,
say $p_{2,\mu_2}=5$. Finally, in choosing the second eigenvalue of the
quadratic Lyapunov function, $p_2$ must satisfy
$p_2>\max\{p_{2,\mu_1},p_{2,\mu_2}\}$; let $p_2 = 11$.
This yields the quadratic Lyapunov function
\[
V_P(x)=x^T P x = x^T \begin{bmatrix}
\frac{31}{11} & -\frac{30\sqrt{2}}{11} \\
-\frac{30\sqrt{2}}{11} & \frac{101}{11}
\end{bmatrix} x,
\]
where $P=U^* D U$. To verify that $V_P(x)$ is indeed a quadratic
Lyapunov function for the time-varying system $x^{\Delta}=A_{i(t)}x$
(where $i(t)\in S_{\mu(t)}$ for all $t\in\mathbb{T}$), we examine the
spectrum of the output of the time scale algebraic Lyapunov operator:
\begin{gather*}
\operatorname{spec}(\mathcal{L}_{a}^{\mathbb{T}}(A_1,P,\mu_1)) \approx \{-1.2361,-1\} \\
\operatorname{spec}(\mathcal{L}_{a}^{\mathbb{T}}(A_1,P,\mu_2)) \approx \{-9.6212,-1.2261\} \\
\operatorname{spec}(\mathcal{L}_{a}^{\mathbb{T}}(A_2,P,\mu_1)) \approx \{-35.1925,-4.8000\}.
\end{gather*}
So for all $t\in\mathbb{T}$,
$\mathcal{L}_{a}^{\mathbb{T}}(A_{i(t)},P,\mu(t)))$ is negative definite, and the switched
system is GUES under the constrained switching pattern. It is worth noting
that $V_P(x)$ is \emph{not} a CQLF for the system since
$\operatorname{spec}(\mathcal{L}_{a}^{\mathbb{T}}(A_2,P,\mu_1))
\approx \{528.0370,23.9983\}$.
\end{example}
In Theorem~\ref{thm51} and Corollary~\ref{coro52}, it is assumed that a
time scale with a compact set of graininesses was given \emph{a priori}
and then a QLF was derived for certain switching signals implying the
GUES of trajectories under those switching signals. Even if the subsystem
matrices themselves were not all exponentially stable, certain switching
signals (which still activate the systems which are not exponentially stable)
yield GUES behavior, as demonstrated in Example \ref{examp2}.
This is possible because the hypotheses of Theorem~\ref{thm51} can be met with
matrices whose eigenvalues are in the open left-half complex plane but not in
the time scale region of exponential stability $\mathcal{S}(\mathbb{T})$;
however, to satisfy the hypotheses there must be at least one subsystem whose
eigenvalues are in the smallest Hilger circle
$\mathcal{H}_{\rm min}\subset \mathcal{S} (\mathbb{T})$.
In \cite{dwell} the authors proved stability of switched systems over continuous
domains comprised of unstable matrices (by appealing to the average dwell time
of an unstable matrix and multiple Lyapunov functions). The situation described
in Theorem~\ref{thm51} still has a single QLF which guarantees that solutions
decrease monotonically with respect to the norm defined by the QLF;
there is no analogous phenomenon when $\mathbb{T}=\mathbb{R}$ or $\mathbb{Z}$,
since in those situations the set of exponential stability coincides with the active Hilger circle for all $t\in \mathbb{T}$.
While results based on dwell time and multiple Lyapunov functions allow some
subsystem matrices with eigenvalues in the right-half complex plane,
Theorem~\ref{thm51} and Corollary~\ref{coro52} still require that subsystem matrices have
their spectrum in the open left-half plane (which, however, is not equivalent to
exponential stability on general time scales).
We now discuss stability results which arise when a particular switching
\emph{order} is desired from a given set of matrices. In this situation the only
cog that is manipulated is the time scale domain, and priority is given
to the order in which switching will occur without exact knowledge of
specifically \emph{when} the switching will take place
(since switching can only occur at points in the timescale).
However, depending on the graininesses that comprise the time scale,
the exact times of which switching will occur can be given within
a reasonable error defined by the graininesses. The next result shows
that this is sufficient freedom to produce switched systems with stable
trajectories which follow the desired switching order. A few required
definitions are introduced first.
\begin{definition} \rm
A \emph{switching order} is an infinite sequence of $N$ letters
$$
\mathcal{O}=\{s_0,s_1,\dots,s_n,\dots\}\in N^{\omega}
$$
coupled with a successor shift operation
$\tilde{\sigma}: N^{\omega}\to N^{\omega}$, where
$$
\tilde{\sigma}(\{s_0,s_1,\dots,s_n,\dots\}):=\{s_1,s_2,\dots,s_{n-1},\dots\}.
$$
A switching signal $i(t)$ is said to be \emph{associated with a switching order}
$\mathcal{O}$ if
$$
i(\sigma^{n}(t)) = \tilde{\sigma}^{n}(\mathcal{O})
$$
for all $n\in\mathbb{N}_0$.
\end{definition}
Because we will be constructing time scales in the following proof,
we must adjust our Hilger circle notation.
\begin{definition} \rm
The \emph{Hilger circle associated with graininess $\mu$}, denoted
$\mathcal{H}_{\mu}$, is the open region of the complex plane given by
\[
\mathcal{H}_{\mu}:=\big\{z\in\mathbb{C}:| z+\frac{1}{\mu}|<\frac{1}{\mu}\big\}.
\]
\end{definition}
The following theorem depends heavily on the QLF that was constructed in
the proof to Theorem~\ref{thm51}.
\begin{theorem}[\cite{eisenbarth-diss}]
Let $\{A_i\}_{i\in 1}^N$ be a collection of simultaneously triangularizable
matrices with eigenvalues in the open left-half complex plane, and
$\mathcal{O}\in N^{\omega}$ be a specified switching order.
Then there exist time scales and at least one switching signal
$i(t):\mathbb{T}\to\{1,\dots,N\}$ associated with $\mathcal{O}$ such that
$x^{\Delta}=A_{i(t)}x$ has a QLF.
\end{theorem}
\begin{proof}
To begin, we define a base equivalence class of time scales;
this is done by selecting graininesses $\mu_{j}$ such that for each $A_{i}$,
there exists at least one $j=1,\dots,M$ such that
$\operatorname{spec}(A_{i})\subset\mathcal{H}_{\mu_{j}}$.
That is, the graininesses which are chosen must give rise to stabilizing
time scales. This collection of graininesses can be refined to include
smaller graininesses if desired (possibly to have more control of \emph{when}
the switching occurs, as opposed to just what order it occurs);
the smallest graininess will define the potential ``error" (with respect to continuous time) possible in
the timing of switching instances. Once refined, a stabilizing time scale
can be constructed as follows.
Let $s_0$ be the first element of $\mathcal{O}$ and choose the next point
in the time scale, denoted $\sigma(0)$, such that
$\operatorname{spec}(A_{s_0})\subset\mathcal{H}_{\mu(0)}$. Continuing in
this manner for each point in the time scale, we define the stabilizing time
scale domain to be given by the closed set
$$
\mathbb{T}:=\{\sigma^{n}(0):
\operatorname{spec}(A_{\tilde{\sigma}(i_0)})\subset\mathcal{H}_{\mu(\sigma^{n}(0))}
\,,\, n\in\mathbb{N}\}.
$$
Constructing the time scale in this manner guarantees that the set
$$
S_{\mathcal{H}(t)}:=\{i\in\mathbb{N}_0:
\operatorname{spec}(A_{i})\subset\mathcal{H}(t)\}
$$
is nonempty for each $t\in\mathbb{T}$. Following the construction in the proof
of Theorem~\ref{thm51}, a QLF can be obtained for the time-varying system
generated by the switching order $\mathcal{O}$.
\end{proof}
It should be emphasized that it's possible for this construction to generate
a time scale for which the eigenvalues of one or more of the subsystem matrices
are not contained in the region of exponential stability. However, the time
scale has been generated in such a way that the associated switching order
will produce solutions which decrease monotonically with respect to the
norm defined by the QLF.
\begin{figure}[ht]
\begin{center}
\begin{tikzpicture}[scale=0.875, every node/.append style={transform shape},->,auto,node distance=3cm,thick,
main node/.style={rectangle,fill=blue!20,draw,font=\sffamily\Large,inner sep=6pt}] %,font=\sffamily\Large\bfseries
\node[main node] (pc+d) {PC$+$D};
\node[main node] (sud) [left of=pc+d,draw=none,fill=none] {};
\node[main node] (sud2) [left of=sud] {SUD};
\node[main node] (pc) [right of=pc+d] {PC};
\node[main node] (pc+n) [below of=sud] {PC+N};
\node[main node] (nil) [below of=pc,align=center] {nilpotent\\Lie algebra};
\node[main node] (sd) [above of=sud] {SD};
\node[main node] (st) [above of=pc] {ST};
\node[main node] (solv) [right of=pc,align=center] {solvable\\Lie algebra};
\node[main node] (weststar) [draw=none,fill=none,above of=pc+d] {};
\node[main node] (neuc) [above of=weststar,align=center] {There exists a CQLF};
\node[main node] (midstar) [draw=none,fill=none] {}; %at ($(neuc)!0.5!(euc)$) {};
\node[main node] (stab) [above of=neuc,align=center] {stability};
\node[main node] (lowstar) [draw=none,fill=none,node distance=4cm] {};%[draw=none,fill=none,below of=euc,node distance=4cm] {};
\draw (pc+d)--node[above]{$\subseteq$}(pc);
\draw (pc+d)--node[above,sloped]{$\subseteq$}(st);
\draw[<->] (pc+d)--node[above,sloped]{\cite{eisenbarth-diss}}(sd);
\draw[<-] (pc+d)--node[above,sloped]{$\subseteq$}(pc+n);
\draw (pc+d)--(nil);
\draw[<->] (pc+n)--node[above,sloped]{\cite{eisenbarth}}(sud2);
\draw (sud2)--node[above,sloped]{$\subseteq$}(sd);
\draw (pc)--(nil);
\draw (pc)--node[above,sloped,pos=.45]{Cor~4.7}(st);
\draw (sd)--node[above,sloped]{$\subseteq$}(st);
\draw (pc)--(solv);
\draw (solv)--node[above,sloped]{$\subseteq$}(st);
\draw (nil)--node[above,sloped]{$\subseteq$}(solv);
\draw (sd) -- node[above,sloped,pos=.4]{\cite{eisenbarth-diss}}(neuc);
\draw (st)--node[above,sloped,pos=.4]{Thm~4.6}(neuc);
\draw (solv) to [out=90,in=0] node[above,sloped] {Cor~4.8}(neuc);
\draw (sud2) to [out=90,in=180] node[above,sloped] {\cite{eisenbarth}}(neuc);
\draw (neuc) to [out=90,in=-90]node[above,sloped]{}(stab);
\end{tikzpicture}
\end{center}
\caption[Theorem flow chart]{A schematic of how the various classes
of switched systems interrelate with one another with respect to the
existence of CQLFs, based on the results of \cite{eisenbarth},
this work, and generalizations of \cite{eisenbarth} made possible by this work.
Here, \\
$SD=\{\text{simultaneously diagonalizable}\}$,\\
$SUD=\{\text{simultaneously unitarily diagonalizable}\}$,\\
$ST=\{\text{simultaneously triangularizable}\}$,\\
$PC=\{\text{pairwise commutative}\}$,\\
$D=\{\text{diagonal}\}$,\\
$N=\{\text{normal}\}$.\\
Switched systems are assumed to be compact, Hilger stable,
and uniformly regressive}
\label{flow}
\end{figure}
\section{Conclusions}
We have extended a major result for switched systems on uniform domains
\cite{LHM} to hybrid domains and extended the theory in \cite{eisenbarth, ramos}
over time scales to include switched systems comprised of subsystem
matrices which are not normal nor pairwise commutative. In doing so,
the proofs for the results in \cite{eisenbarth, LHM, ramos} have been
explained in a new light, highlighting the importance of the
simultaneous triangularizability of a given set of matrices.
The relationship of the results presented in this paper to the results
presented by the authors in \cite{eisenbarth} is illustrated in
Figure~\ref{flow}. In addition, new results concerning the construction
of stabilizing switching patterns over hybrid domains were established
for a larger class of matrices than those included in \cite{eisenbarth},
which first introduced the concept for switched systems over time scale domains.
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\end{document}