# Contraction analysis of time-varying DAE systems via auxiliary ODE systems

Hao Yin  Bayu Jayawardhana and Stephan Trenn The authors are with the Jan C.
Willems Center for Systems and Control, Faculty of Science and Engineering,
University of Groningen, 9747AG Groningen, The Netherlands<EMAIL_ADDRESS><EMAIL_ADDRESS>and<EMAIL_ADDRESS>work was supported by China
Scholarship Council.

###### Abstract

This paper studies the contraction property of time-varying differential-
algebraic equation (DAE) systems by embedding them to higher-dimension
ordinary differential equation (ODE) systems. The first result pertains to the
equivalence of the contraction of a DAE system and the uniform global
exponential stability (UGES) of its variational DAE system. Such equivalence
inherits the well-known property of contracting ODE systems on a specific
manifold. Subsequently, we construct an auxiliary ODE system from a DAE system
whose trajectories encapsulate those of the corresponding variational DAE
system. Using the auxiliary ODE system, a sufficient condition for contraction
of the time-varying DAE system is established by using matrix measure which
allows us to estimate an lower bound on the parameters of the auxiliary
system. Finally, we apply the results to analyze the stability of time-
invariant DAE systems, and to design observers for time-varying ODE systems.

##  1 Introduction

As a generalization of ordinary differential equation (ODE) systems,
differential-algebraic equation (DAE) systems have been studied for the past
decades due to their relevance in representing numerous modern engineering
systems with constraints. Some well-known examples of such engineering systems
are electrical networks with Kirchhoff’s laws [4] and mechanical systems with
rigid body constraints [5]. The DAE systems can also be used to model power
systems [10] and chemical processes [6]. Typically, DAE systems consist of a
set of differential equations describing the system dynamics, and a set of
algebraic equations describing the constraints. Solving DAE systems is more
challenging than solving ODE systems due to the implicit relationship between
the differential equations and algebraic equations. DAE systems can be
classified as either index-1 [10] or higher index [7] by the degree of the
highest derivative of the algebraic equation. Index-1 DAE systems are
particularly important in control theory, because the algebraic restriction
can be substituted directly into the dynamics. Consequently, they can
numerically be solved by using standard techniques for solving ODE [8], which
allows numerical simulations to be carried out straightforwardly [9].

The analysis of DAE systems has been widely studied in the literature [12, 10,
13, 14, 15, 16, 11] with a large body of works concerned with index-1 DAE
systems. In [12], the authors show that any solvable DAE power systems are of
index one. By this result, robust $H_{\infty}$ observers for power networks
are presented in [13], and a load- and renewable-following control approach
for power system is proposed in [14]. There are several methods for analyzing
the stability of DAE systems, including the Lyapunov method [10, 11], the
energy-based methods [15], as well as, the eigenvalue analysis [16]. In [10],
the DAE systems are embedded in reduced ODE systems, where Lyapunov method can
be applied to get the Lyapunov stability. In [11], an incremental Lyapunov
function is applied to analyze the asymptotic stability and optimal resource
allocation for a network preserved microgrid model with active and reactive
power loads. In [15], bifurcation theory is used to characterize the stability
boundary for power systems in DAE, and an energy function method is developed
to guarantee both rotor angular stability and voltage stability. A method for
defining the small-signal stability of delay DAE systems based on eigenvalue
analysis and an approximation of the characteristic equation at equilibrium
points are proposed in [16].

In all of the above results, the properties of time-invariant DAE systems are
analyzed, while the extension of these results to the time-varying DAE systems
remains non-trivial. The exponential stability and robustness of linear time-
varying DAE systems with index-1 are studied in [18, 17]. In [17], the Bohl
exponent theory for stability analysis of ODEs is extended to DAEs. When there
are perturbations in systems’ matrices, a stability radius has been
investigated in [18], which includes the lower bound computation of the
stability radius.

As one of the stability analysis methods for time-varying systems, which has
gained popularity in recent years, contraction analysis focuses on the
relative trajectory of the nonlinear time-varying system rather than a
specific equilibrium point. There are many methods to analyze the
contractivity of nonlinear time-varying ODE systems in literature, such as,
[21, 22, 20, 23, 24] among many others. An ODE system is contracting if and
only if the associated variational system is uniformly globally exponentially
stable (UGES) [20]. In [21], the contraction property can be guaranteed if the
largest eigenvalue of the symmetric part of the variational systems is
uniformly strictly negative. Finsler–Lyapunov functions are introduced in [22]
to analyze the incremental exponential stability of the system. Mode-dependent
Lyapunov functions for contracting switched nonlinear systems are presented in
[23]. In [24], the focus is on investigating transverse exponential stability
(a generalized notion of contraction) by employing a Lyapunov matrix
transversal equation. In the context of DAE systems, the contraction analysis
thereof has recently been presented in [2]. In [2], the authors proved that if
the algebraic equation satisfies some sufficient conditions, the contractivity
of time-invariant DAE systems can be obtained through exponential stability
analysis of the corresponding reduced variational ODE systems using matrix
measure. In Section 3.2, we show that this approach can be restrictive and is
not applicable to some time-varying DAE systems.

In this paper, we analyze the contraction property of index-1 nonlinear time-
varying DAE systems by using an ODE approach. As our first main result, we
establish that the uniform global exponential stability (UGES) of the
variational DAE dynamics is a sufficient and necessary condition to the
contractivity of the original DAE systems. This condition can be viewed as a
DAE counterpart of the results for ODE presented in [20, Prop. 1].
Subsequently, we construct an auxiliary ODE system whose convergence property
can encapsulate the same property of the variational DAE dynamics. With this
construction, we can analyze the contractivity of the DAE systems by applying
conventional control theory (such as Lyapunov approach, matrix measure method)
to the auxiliary ODE system. As our second contribution, we provide sufficient
conditions on the contraction of nonlinear time-varying DAE system by using
matrix measure method to analysis the UGES property of the auxiliary ODE
system. In general, these conditions ensure the contraction of the DAE system
without analysing its reduced system. As our third contribution, we employ
these conditions to design observers for time-varying ODE systems by treating
the output as an algebraic equation. Furthermore, we investigate the
exponential stability of time-invariant DAE systems by ensuring that the DAE
system is contractive and the equilibrium lies within its trajectory set.

The paper is organized as follows. In Section 2, we present preliminaries and
problem formulation. Our main results are proposed in Section 3, where we
present necessary and sufficient conditions for the contractivity of time-
varying DAE systems, and the ODE approach. The numerical simulations and
applications are provided in Section 4 and the conclusions are given in
Section 5.

##  2 Preliminaries and problem formulation

Throughout this paper, we consider the following nonlinear time-varying DAE
systems

$\left\\{\begin{matrix}\dot{w}=f(t,w,z),\\\ 0=g(t,w,z),\end{matrix}\right.$
(1)

where $w(t)\in\mathbb{R}^{n}$ is the state vector, $z(t)\in\mathbb{R}^{m}$
refers to the algebraic vector,
$f:\mathbb{R}_{+}\times\mathbb{R}^{n}\times\mathbb{R}^{m}\rightarrow\mathbb{R}^{n}$
is the vector field, and
$g:\mathbb{R}_{+}\times\mathbb{R}^{n}\times\mathbb{R}^{m}\rightarrow\mathbb{R}^{m}$
describes the algebraic manifold. We assume that $f$ is continuously
differentiable and $g$ is twice-continuously differentiable. In this note, we
consider only the continuously differentiable solutions
$w(t)=\varphi(t_{0},w_{0},z_{0})$ and $z(t)=\psi(t_{0},w_{0},z_{0})$ of (1)
with admissible initial conditions
$(w_{0},z_{0})\in\mathbb{R}^{n}\times\mathbb{R}^{m}$ satisfying the algebraic
constraint

$g(t,w_{0},z_{0})=0.$ (2)

We assume the DAE system (1) is of index-1, i.e. the (partial) Jacobian matrix
$\frac{\partial g}{\partial z}(t,w,z)\in\mathbb{R}^{m\times m}$ is invertible
for all $(t,w,z)$. This assumption guarantees the existence and uniqueness of
a local solution to (1) for any initial condition satisfying (2) (see also,
[10]). Throughout the paper, we will assume that every local solution can be
extended to a solution defined on the whole time domain $[0,\infty)$.

Note that the index-1 assumption allows to apply the implicit function theorem
to solve the constraint (2) for $z_{0}$ for any given $(t_{0},w_{0})$ and in
the following we will therefore write $z_{0}(w_{0})$ to denote the unique
value for $z_{0}$ which satisfies (2) for an arbitrarily given $w_{0}$ (we
omit the dependency on $t_{0}$ as we consider the initial time as fixed in the
following analysis).

###### Definition 2.1

A time-varying DAE system (1) is called contracting if there exists positive
numbers $c$ and $\alpha$ such that for any pair of solutions
$W_{i}(t)=\left[\begin{smallmatrix}w_{i}(t)\\\
z_{i}(t)\end{smallmatrix}\right]\in\mathbb{R}^{n}$ of (1) with $i=1,2$, we
have

$\displaystyle\|W_{1}(t)-W_{2}(t)\|\leq ce^{-\alpha
t}\|W_{1}(t_{0})-W_{2}(t_{0})\|,\quad\forall t\geq t_{0}.$ (3)

In order to study contractivity of the DAE system (1), we will analyse the
(uniform) stability of the corresponding variational DAE systems. The
variational system of system (1) is given by

$\left\\{\begin{matrix}\dot{\xi}=\frac{\partial f}{\partial
w}(t,w(t),z(t))\cdot\xi+\frac{\partial f}{\partial z}(t,w(t),z(t))\cdot\nu,\\\
0=\tfrac{\partial g}{\partial w}(t,w(t),z(t))\cdot\xi+\frac{\partial
g}{\partial z}(t,w(t),z(t))\cdot\nu,\end{matrix}\right.$ (4)

where $w(t)$ and $z(t)$ are solutions of (1). We omit the explicit
parametrization $(t,w,z)$ whenever it is clear from the context.

###### Definition 2.2

The variational DAE system (4) is called uniformly globally exponentially
stable (UGES), if there exist positive numbers $c$, $\alpha$ (independent of
the solution $W(\cdot)$, but may be dependent on $t_{0}$) such that for every
solution $\Xi(t):=\left[\begin{smallmatrix}\xi(t)\\\
\nu(t)\end{smallmatrix}\right]\in\mathbb{R}^{n}$ of (4) the inequality

$\displaystyle\|\Xi(t)\|\leq ce^{-\alpha t}\|\Xi(t_{0})\|,$ (5)

holds for all $t\geq t_{0}$.

Note that in the above definition and in the remainder of the paper, the norm
$\|\cdot\|=\|\cdot\|_{p}$ is some $p$-norm, $p\geq 1$, where we usually omit
the index $p$, unless a specific case is discussed; while the exponential rate
$\alpha$ in (5) does not depend on the chosen norm, the constant $c$ in
general depends on it.

By invertibility of the Jacobian matrix $\frac{\partial g}{\partial z}$, we
can express the reduced system of equation (6) in the following form:

$\dot{\xi}=\left(\frac{\partial f}{\partial w}-\frac{\partial f}{\partial
z}\left[\frac{\partial g}{\partial z}\right]^{-1}\tfrac{\partial g}{\partial
w}\right)\xi.$ (6)

In [2, Proposition 1], a sufficient condition is introduced to analyse the
contractivity of time-invarient DAE systems by analysing its reduced systems
(6). However, this approach may not be applicable in the time-varying case. As
a simple example, consider the following time-varying DAE system

$\left\\{\begin{matrix}\dot{w}=-w+e^{-3t}z,\\\
0=e^{3t}w+z,\end{matrix}\right.$ (7)

which has the same form as its variational DAE system. Its reduced system is
given by $\dot{w}=-2w$, which is a contractive system. Consequently, the
trajectories of the system are given by $w(t)=w_{0}e^{-2t}$ and
$z(t)=-w_{0}e^{t}$, which shows the DAE system is not contracting ($z$ is not
contracting). This is due to the unboundedness of $\left[\frac{\partial
g}{\partial z}\right]^{-1}\tfrac{\partial g}{\partial w}=e^{3t}$, which
implies that the method proposed in [2] is not applicable when dealing with
time-varying systems. Notice that, in this particular case, the contraction
analysis problem can be effectively addressed by comparing the exponential
rate of the reduced system with an exponential bound on $\left[\frac{\partial
g}{\partial z}\right]^{-1}\tfrac{\partial g}{\partial w}$. However, it is
worth acknowledging that obtaining information about the reduced system or
establishing a bound for $\left[\frac{\partial g}{\partial
z}\right]^{-1}\tfrac{\partial g}{\partial w}$ can be a challenging task for
certain systems.

The objective of this paper is to provide a sufficient condition that
guarantees the contractivity of time-varying DAE systems, even in situations
where prior knowledge about the DAE system is unavailable.

Before stating our main results, let us recall the definition of matrix
measure [19], which is also known as logarithmic norm as follows. For a given
matrix $A\in\mathbb{R}^{n\times n}$ the matrix measure $\mu_{p}(A)$ w.r.t. to
the (induced) $p$-norm $\|\cdot\|_{p}$ is defined by

$\mu_{p}(A):=\underset{h\rightarrow 0^{+}}{\lim}\frac{||I+hA||_{p}-1}{h}.$ (8)

It follows that when $p=1$, $2$ or $\infty$, we have

$\displaystyle\mu_{1}(A)=\max_{j}\Big{(}a_{jj}+\underset{i\neq
j}{\sum}|a_{ij}|\Big{)},$ (9)
$\displaystyle\mu_{2}(A)=\max_{i}\Big{(}\lambda_{i}\left\\{\tfrac{A+A^{\top}}{2}\right\\}\Big{)},\text{
or }$ $\displaystyle\mu_{\infty}(A)=\max_{i}\Big{(}a_{ii}+\underset{j\neq
i}{\sum}|a_{ij}|\Big{)},$

respectively. In this paper, all the norms are defined using a $p$-norm.

##  3 Main results

In this section, we firstly establish an equivalent relationship between the
contraction of a DAE system and the uniform global exponential stability
(UGES) of its variational DAE system. Secondly, we construct an auxiliary ODE
system that encapsulates the behaviors of the variational DAE system. Thirdly,
a sufficient condition is presented that guarantees the UGES of the auxiliary
ODE system and is numerically implementable.

###  3.1 A necessary and sufficient condition

###### Proposition 3.1

The DAE system (1) is contracting if and only if the corresponding variational
DAE system (4) is UGES.

######  Proof 3.2.

Let us establish a relationship between the solutions of (1) and those of (4).
Let $\left[\begin{smallmatrix}w(t)\\\
z(t)\end{smallmatrix}\right]=\left[\begin{smallmatrix}\varphi(t,w_{0},z_{0}(w_{0}))\\\
\psi(t,w_{0},z_{0}(w_{0}))\end{smallmatrix}\right]$ and
$\left[\begin{smallmatrix}\hat{w}(t)\\\
\hat{z}(t)\end{smallmatrix}\right]=\left[\begin{smallmatrix}\varphi(t,w_{0}+\delta\xi_{0},z_{0}(w_{0}+\delta\xi_{0}))\\\
\psi(t,w_{0}+\delta\xi_{0},z_{0}(w_{0}+\delta\xi_{0}))\end{smallmatrix}\right]$
be two trajectories of (1) with initial conditions
$\left[\begin{smallmatrix}w_{0}\\\ z_{0}(w_{0})\end{smallmatrix}\right]$ and
$\left[\begin{smallmatrix}w_{0}+\delta\xi_{0}\\\
z_{0}(w_{0}+\delta\xi_{0})\end{smallmatrix}\right]$, respectively, where
$\delta$ is a sufficiently small positive constant and $\xi_{0}$ will be
related later to the initial condition of (1). As they are solutions of (1),
they satisfy

$g(t,w(t),z(t))=0,$ (10)

and

$g(t,\hat{w}(t),\hat{z}(t))=0.$ (11)

Denote the partial Jacobian matrices of $\varphi$ and $\psi$ with respect to
the second or third argument evaluated at $(t,w_{0},z_{0}(w_{0}))$ by
$\Phi_{w_{0}}(t),\Phi_{z_{0}}(t),\Psi_{w_{0}}(t),\Psi_{z_{0}}(t)$,
respectively. By differentiating both sides of (10) with respect to $w_{0}$,
we have

$\displaystyle 0$ $\displaystyle=\tfrac{\partial g}{\partial
w}(t,w(t),z(t))\cdot\Big{(}\Phi_{w_{0}}(t)+\Phi_{z_{0}}(t)\cdot
z_{0}^{\prime}(w_{0})\Big{)}$ (12) $\displaystyle+\tfrac{\partial g}{\partial
z}(t,w(t),z(t))\cdot\Big{(}\Psi_{w_{0}}(t)+\Psi_{z_{0}}(t)\cdot
z_{0}^{\prime}(w_{0})\Big{)}.$

In the following, we will show that

$\left\\{\begin{matrix}\xi(t):=\lim\limits_{\delta\rightarrow
0}\frac{\varphi(t,w_{0}+\delta\xi_{0},z_{0}(w_{0}+\delta\xi_{0}))-\varphi(t,w_{0},z_{0}(w_{0}))}{\delta},\\\
\nu(t):=\lim\limits_{\delta\rightarrow
0}\frac{\psi(t,w_{0}+\delta\xi_{0},z_{0}(w_{0}+\delta\xi_{0}))-\psi(t,w_{0},z_{0}(w_{0}))}{\delta},\end{matrix}\right.$
(13)

are a pair of solutions of (4) with initial value

$\displaystyle\xi(t_{0})$ $\displaystyle=\xi_{0},$ $\displaystyle\nu(t_{0})$
$\displaystyle=\nu_{0}=\nu_{0}(\xi_{0}):=\lim\limits_{\delta\rightarrow
0}\tfrac{z_{0}(w_{0}+\delta\xi_{0})-z_{0}(w_{0})}{\delta}.$

We can rewrite (13) as

$\left\\{\begin{matrix}\xi(t)=\Big{(}\Phi_{w_{0}}(t)+\Phi_{z_{0}}(t)\cdot
z_{0}^{\prime}(w_{0})\Big{)}\cdot\xi_{0},\\\
\nu(t)=\Big{(}\Psi_{w_{0}}(t)+\Psi_{z_{0}}(t)\cdot
z_{0}^{\prime}(w_{0})\Big{)}\cdot\xi_{0}.\end{matrix}\right.$ (14)

From (12) and (14), we know that (13) satisfies $0=\tfrac{\partial g}{\partial
w}\xi+\frac{\partial g}{\partial z}\nu$.
The flow $\varphi(t,w_{0},z_{0}(w_{0}))$ of (1) satisfies

$\displaystyle\varphi(t,w_{0},z_{0}(w_{0}))=w_{0}$ (15)
$\displaystyle+\int_{0}^{t}f\big{(}\varphi(\tau,w_{0},z_{0}(w_{0})),\psi(\tau,w_{0},z_{0}(w_{0}))\big{)}{\rm
d}\hbox{\hskip 0.5pt}\tau,$

and similarly, the flow
$\varphi(t,w_{0}+\delta\xi_{0},z_{0}(w_{0}+\delta\xi_{0}))$ satisfies

$\displaystyle\varphi(t,w_{0}+\delta\xi_{0},z_{0}(w_{0}+\delta\xi_{0}))=w_{0}+\delta\xi_{0}+$
(16)
$\displaystyle\int_{0}^{t}f\big{(}\varphi(\tau,w_{0}+\delta\xi_{0},z_{0}(w_{0}+\delta\xi_{0})),$
$\displaystyle\psi(\tau,w_{0}+\delta\xi_{0},z_{0}(w_{0}+\delta\xi_{0}))\big{)}{\rm
d}\hbox{\hskip 0.5pt}\tau.$

Hence,

$\displaystyle\xi(t)=\xi_{0}+\int_{0}^{t}\lim_{\delta\rightarrow
0}\frac{1}{\delta}\times$ (17)
$\displaystyle\Big{(}f\big{(}\varphi(\tau,w_{0}+\delta\xi_{0},z_{0}(w_{0}+\delta\xi_{0})),\psi(\tau,w_{0}+\delta\xi_{0},z_{0}(w_{0}+\delta\xi_{0}))\big{)}$
$\displaystyle-f\big{(}\varphi(\tau,w_{0},z_{0}(w_{0})),\psi(\tau,w_{0},z_{0}(w_{0}))\big{)}\Big{)}{\rm
d}\hbox{\hskip 0.5pt}\tau$

Clearly,

$\displaystyle\lim_{\delta\rightarrow 0}\frac{1}{\delta}\times$ (18)
$\displaystyle\Big{(}f\big{(}\varphi(\tau,w_{0}+\delta\xi_{0},z_{0}(w_{0}+\delta\xi_{0})),\psi(\tau,w_{0}+\delta\xi_{0},z_{0}(w_{0}+\delta\xi_{0}))\big{)}$
$\displaystyle\quad-f\big{(}\varphi(\tau,w_{0},z_{0}(w_{0})),\psi(\tau,w_{0},z_{0}(w_{0}))\big{)}\Big{)}$
$\displaystyle=\tfrac{\partial f}{\partial
w}(\tau,w(\tau),z(\tau))\cdot\Big{(}\Phi_{w_{0}}(\tau)+\Phi_{z_{0}}(\tau)\cdot
z_{0}^{\prime}(w_{0})\Big{)}\cdot\xi_{0}$ $\displaystyle\quad+\tfrac{\partial
f}{\partial
z}(\tau,w(\tau),z(\tau))\Big{(}\Psi_{w_{0}}(\tau)+\Psi_{z_{0}}(\tau)\cdot
z_{0}^{\prime}(w_{0})\Big{)}\cdot\xi_{0}$
$\displaystyle\overset{\eqref{varitaj0}}{=}\tfrac{\partial f}{\partial
w}(\tau,w(\tau),z(\tau))\cdot\xi(\tau)+\tfrac{\partial f}{\partial
z}(\tau,w(\tau),z(\tau))\cdot\nu(\tau).$

Substituting this back to (17) and differentiating with respect to time gives
us

$\displaystyle\dot{\xi}(t)=\frac{\partial f}{\partial w}\xi(t)+\frac{\partial
f}{\partial z}\nu(t).$ (19)

Altogether this shows that indeed $\xi(t)$, $\nu(t)$ given by (13) is a
solution of (4).

We can now show the sufficiency result.

_Contracting $\Rightarrow$ UGES._ Let $c$ and $\alpha$ be the constants
corresponding to the contractivity condition. Seeking a contradiction, assume
the variational DAE system (4) is not UGES. Then there exists a solution
$\left[\begin{smallmatrix}w(\cdot)\\\ z(\cdot)\end{smallmatrix}\right]$ of (1)
and an initial value $\left[\begin{smallmatrix}w_{0}\\\
z_{0}(w_{0})\end{smallmatrix}\right]$ such that for the corresponding solution
$\left[\begin{smallmatrix}\xi(\cdot)\\\ \nu(\cdot)\end{smallmatrix}\right]$ of
(4) we have that for $c^{\prime}:=\frac{3}{2}c$ and $\alpha^{\prime}:=\alpha$,
there exists $T>0$ such that

$\left\|\left[\begin{matrix}\xi(T)\\\
\nu(T)\end{matrix}\right]\right\|>c^{\prime}e^{-\alpha^{\prime}T}\left\|\left[\begin{matrix}\xi_{0}\\\
\nu_{0}\end{matrix}\right]\right\|=\frac{3}{2}ce^{-\alpha
T}\left\|\left[\begin{matrix}\xi_{0}\\\
\nu_{0}\end{matrix}\right]\right\|\qquad\forall\xi_{0}.$ (20)

Let $\left[\begin{smallmatrix}\hat{w}(\cdot)\\\
\hat{z}(\cdot)\end{smallmatrix}\right]$ be a solution of (1) with initial
value $\left[\begin{smallmatrix}\hat{w}(t_{0})\\\
\hat{z}(t_{0})\end{smallmatrix}\right]=\left[\begin{smallmatrix}w_{0}+\delta\xi_{0}\\\
z_{0}(w_{0}+\delta\xi_{0})\end{smallmatrix}\right]$ for sufficiently small
$\delta\in\mathbb{R}$. By definition, $\xi(t)=\lim_{\delta\rightarrow
0}\frac{\hat{w}(t)-w(t)}{\delta}$ and $\nu(t)=\lim_{\delta\rightarrow
0}\frac{\hat{z}(t)-z(t)}{\delta}$; hence, for a sufficiently small $\delta>0$,
we have that at time $T$,

$\displaystyle\frac{\left\|\left[\begin{matrix}\hat{w}(T)\\\
\hat{z}(T)\end{matrix}\right]-\left[\begin{matrix}w(T)\\\
z(T)\end{matrix}\right]\right\|}{\delta}>\frac{4}{5}\left\|\left[\begin{matrix}\xi(T)\\\
\nu(T)\end{matrix}\right]\right\|,$ (21)

where the lower-bound constant $\frac{4}{5}$ is chosen arbitrarily for the
following computation of bounds. Similarly, since $\nu_{0}:=\lim_{\delta\to
0}\frac{\hat{z}(t_{0})-z(t_{0})}{\delta}$, for a sufficiently small
$\delta>0$, we have that

$\displaystyle\left\|\nu_{0}\right\|>\frac{5}{6}\frac{\left\|\hat{z}(t_{0})-z(t_{0})\right\|}{\delta}.$
(22)

Combining (20), (21) and (22), we obtain

$\displaystyle\left\|\left[\begin{matrix}\hat{w}(T)\\\
\hat{z}(T)\end{matrix}\right]-\left[\begin{matrix}w(T)\\\
z(T)\end{matrix}\right]\right\|$
$\displaystyle\overset{\eqref{pf72}}{>}\frac{4}{5}\delta\left\|\left[\begin{matrix}\xi(T)\\\
\nu(T)\end{matrix}\right]\right\|\overset{\eqref{pf90}}{>}\frac{6}{5}ce^{-\alpha
T}\left\|\left[\begin{matrix}\delta\xi_{0}\\\
\delta\nu_{0}\end{matrix}\right]\right\|$
$\displaystyle\overset{\eqref{pf720}}{>}ce^{-\alpha
T}\left\|\left[\begin{matrix}\frac{6}{5}\hat{w}(t_{0})\\\
\hat{z}(t_{0})\end{matrix}\right]-\left[\begin{matrix}\frac{6}{5}w(t_{0})\\\
z(t_{0})\end{matrix}\right]\right\|$ $\displaystyle>ce^{-\alpha
T}\left\|\left[\begin{matrix}\hat{w}(t_{0})\\\
\hat{z}(t_{0})\end{matrix}\right]-\left[\begin{matrix}w(t_{0})\\\
z(t_{0})\end{matrix}\right]\right\|,$

for all $\xi_{0}$, the last inequality arises from the property of the
$p$-norm. This is in contradiction to the contractivity of (1) and concludes
the proof of the sufficiency part.

_UGES $\Rightarrow$ Contracting._ Let us consider two solutions
$\left[\begin{smallmatrix}w(\cdot)\\\
z(\cdot)\end{smallmatrix}\right]=\left[\begin{smallmatrix}\varphi(\cdot,w_{0},z_{0}(w_{0}))\\\
\psi(\cdot,w_{0},z_{0}(w_{0}))\end{smallmatrix}\right]$ and
$\left[\begin{smallmatrix}\hat{w}(\cdot)\\\
\hat{z}(\cdot)\end{smallmatrix}\right]=\left[\begin{smallmatrix}\varphi(\cdot,\hat{w}_{0},\hat{z}_{0}(\hat{w}_{0}))\\\
\psi(\cdot,\hat{w}_{0},\hat{z}_{0}(\hat{w}_{0}))\end{smallmatrix}\right]$ of
(1). Consequently, we can utilize the fundamental theorem of calculus for line
integrals to obtain

$\left\\{\begin{matrix}\hat{w}(t)-w(t)=\int_{w_{0}}^{\hat{w}_{0}}\frac{{\rm
d}\hbox{\hskip 0.5pt}\varphi(t,\zeta,z_{0}(\zeta))}{{\rm d}\hbox{\hskip
0.5pt}\zeta}{\rm d}\hbox{\hskip 0.5pt}\zeta\\\
\hat{z}(t)-z(t)=\int_{w_{0}}^{\hat{w}_{0}}\frac{{\rm d}\hbox{\hskip
0.5pt}\psi(t,\zeta,z_{0}(\zeta))}{{\rm d}\hbox{\hskip 0.5pt}\zeta}{\rm
d}\hbox{\hskip 0.5pt}\zeta\end{matrix}.\right.$ (23)

According to (13), one has

$\left[\begin{matrix}\xi(t)\\\
\nu(t)\end{matrix}\right]=\left[\begin{matrix}\frac{{\rm d}\hbox{\hskip
0.5pt}\varphi(t,w_{0},z_{0}(w_{0}))}{{\rm d}\hbox{\hskip 0.5pt}w_{0}}&0\\\
\frac{{\rm d}\hbox{\hskip 0.5pt}\psi(t,w_{0},z_{0}(w_{0}))}{{\rm
d}\hbox{\hskip 0.5pt}w_{0}}&0\end{matrix}\right]\left[\begin{matrix}\xi_{0}\\\
\nu_{0}\end{matrix}\right].$ (24)

Then, in (24), $\left[\begin{matrix}\frac{{\rm d}\hbox{\hskip
0.5pt}\varphi(t,w_{0},z_{0}(w_{0}))}{{\rm d}\hbox{\hskip 0.5pt}w_{0}}&0\\\
\frac{{\rm d}\hbox{\hskip 0.5pt}\psi(t,w_{0},z_{0}(w_{0}))}{{\rm
d}\hbox{\hskip 0.5pt}w_{0}}&0\end{matrix}\right]$ is the (singular) state
transition matrix of the DAE (4). From the UGES property of (4), it follows
with similar arguments as in the necessity proof of [1, Thm. 4.11] that

$\displaystyle\left\|\left[\begin{matrix}\frac{{\rm d}\hbox{\hskip
0.5pt}\varphi(t,w_{0},z_{0}(w_{0}))}{{\rm d}\hbox{\hskip 0.5pt}w_{0}}&0\\\
\frac{{\rm d}\hbox{\hskip 0.5pt}\psi(t,w_{0},z_{0}(w_{0}))}{{\rm
d}\hbox{\hskip 0.5pt}w_{0}}&0\end{matrix}\right]\right\|\leq ce^{-\alpha t},$
(25)

from which it follows111 In general, for any $p$-norm and its induced matrix
norm we have
$\left\|\left[\begin{matrix}M&0\end{matrix}\right]\right\|=\underset{\left\|\left[\begin{smallmatrix}x\\\
y\end{smallmatrix}\right]\right\|=1}{\sup}\left\|\left[\begin{matrix}M&0\end{matrix}\right]\left[\begin{matrix}x\\\
y\end{matrix}\right]\right\|=\underset{\left\|\left[\begin{smallmatrix}x\\\
y\end{smallmatrix}\right]\right\|=1}{\sup}\left\|Mx\right\|\geq\underset{\left\|x\right\|=1}{\sup}\left\|Mx\right\|=\left\|M\right\|$.
The inequality follows from the set inclusion
$\\{x|\left\|x\right\|=1\\}=\\{x|\left\|\left[\begin{smallmatrix}x\\\
0\end{smallmatrix}\right]\right\|=1\\}\subseteq\\{x|\exists
y:\left\|\left[\begin{smallmatrix}x\\\
y\end{smallmatrix}\right]\right\|=1\\}$.
that

$\displaystyle\left\|\left[\begin{matrix}\frac{{\rm d}\hbox{\hskip
0.5pt}\varphi(t,w_{0},z_{0}(w_{0}))}{{\rm d}\hbox{\hskip 0.5pt}w_{0}}\\\
\frac{{\rm d}\hbox{\hskip 0.5pt}\psi(t,w_{0},z_{0}(w_{0}))}{{\rm
d}\hbox{\hskip 0.5pt}w_{0}}\end{matrix}\right]\right\|\leq ce^{-\alpha t}$
(26)

Using (26) to get the upper bound of (23), we have

$\displaystyle\left\|\left[\begin{matrix}\hat{w}(t)\\\
\hat{z}(t)\end{matrix}\right]-\left[\begin{matrix}w(t)\\\
z(t)\end{matrix}\right]\right\|$
$\displaystyle=\left\|\int_{w_{0}}^{\hat{w}_{0}}\left[\begin{matrix}\frac{{\rm
d}\hbox{\hskip 0.5pt}\varphi(t,\zeta,z_{0}(\zeta))}{{\rm d}\hbox{\hskip
0.5pt}\zeta}\\\ \frac{{\rm d}\hbox{\hskip
0.5pt}\psi(t,\zeta,z_{0}(\zeta))}{{\rm d}\hbox{\hskip
0.5pt}\zeta}\end{matrix}\right]{\rm d}\hbox{\hskip 0.5pt}\zeta\right\|$ (27)
$\displaystyle\leq ce^{-\alpha t}\left\|\hat{w}_{0}-w_{0}\right\|$
$\displaystyle\leq ce^{-\alpha t}\left\|\left[\begin{matrix}\hat{w}_{0}\\\
z_{0}(\hat{w}_{0})\end{matrix}\right]-\left[\begin{matrix}w_{0}\\\
z_{0}(w_{0})\end{matrix}\right]\right\|,$

where the latter inequality follows from $\|x\|=\|(\begin{smallmatrix}x\\\
0\end{smallmatrix})\|\leq\|(\begin{smallmatrix}x\\\ y\end{smallmatrix})\|$,
which is true for any $p$-norm. This shows that (1) is contracting and the
proof is complete.

Proposition 3.1 shows that the contractivity of DAE systems inherit the
property of contracting ODE systems (i.e., [20, Prop. 1]) on a corresponding
manifold $\tfrac{\partial g}{\partial w}(t,w,z)\xi+\frac{\partial g}{\partial
z}(t,w,z)\nu=0$. From (24) and (26), we can deduce that $\|\xi(t)\|\leq
ce^{-\alpha t}\|\xi_{0}\|$, which means that when the system (4) is
exponentially stable, its reduced system (6) is exponentially stable. In the
case of a time-invariant system, the reverse implication holds automatically.
This is due to the fact that it becomes feasible to guarantee the boundedness
of $\left[\frac{\partial g}{\partial z}\right]^{-1}\tfrac{\partial g}{\partial
w}$ within a particular invariant set. As a result, we can utilize Proposition
3.1 for analyzing the stability of time-invariant DAE systems, as demonstrated
in Corollary 4.2. Apart from the time-invariant system, the presence of an
invariant set might not be guaranteed for a time-varying system, i.e.,
$\left[\frac{\partial g}{\partial z}\right]^{-1}\tfrac{\partial g}{\partial
w}$ could potentially become unbounded. This implies that the contractivity of
the time-varying DAE system cannot be derived from the contractivity of the
reduced system in the absence of bounded condition on $\left[\frac{\partial
g}{\partial z}\right]^{-1}\tfrac{\partial g}{\partial w}$. In Lemma 3.3 below
we relax the condition on $\left[\frac{\partial g}{\partial
z}\right]^{-1}\tfrac{\partial g}{\partial w}$ by analysing the stability of an
auxiliary ODE system instead of analysing the contractivity of the reduced
system. By adopting this approach, it becomes unnecessary to have prior
knowledge about the reduced system (6), as demonstrated in Example 4.1.

###  3.2 The ODE approach

In this section, we present an ODE approach to analyze the contractivity of
nonlinear time-varying DAE systems. We present a construction of auxiliary ODE
whose trajectories can represent the convergence property of the variational
DAE system. As a consequence, we can apply traditional control theories to the
auxiliary ODE in order to analyze the UGES of (4).

For a given variational DAE system (4), we construct its auxiliary ODE system
by

$\displaystyle\left\\{\begin{matrix}\dot{\xi}_{\gamma}=\frac{\partial
f}{\partial w}\xi_{\gamma}+\frac{\partial f}{\partial z}\nu_{\gamma},\\\
\dot{\nu}_{\gamma}=-\left[\frac{\partial g}{\partial
z}\right]^{-1}\Big{[}\Big{(}\gamma\tfrac{\partial g}{\partial w}+\frac{{\rm
d}\hbox{\hskip 0.5pt}}{{\rm d}\hbox{\hskip 0.5pt}t}(\tfrac{\partial
g}{\partial w})+\tfrac{\partial g}{\partial w}\frac{\partial f}{\partial
w}\Big{)}\xi_{\gamma}\\\ \qquad+\Big{(}\gamma\frac{\partial g}{\partial
z}+\frac{{\rm d}\hbox{\hskip 0.5pt}}{{\rm d}\hbox{\hskip
0.5pt}t}(\frac{\partial g}{\partial z})+\tfrac{\partial g}{\partial
w}\frac{\partial f}{\partial z}\Big{)}\nu_{\gamma}\Big{]},\end{matrix}\right.$
(28)

where $\gamma$ is a no-negative constant, $\frac{{\rm d}\hbox{\hskip
0.5pt}}{{\rm d}\hbox{\hskip 0.5pt}t}(\tfrac{\partial g}{\partial
w})=\frac{\partial^{2}g}{\partial w\partial
t}+\frac{\partial^{2}g}{\partial^{2}w}f-\frac{\partial^{2}g}{\partial
w\partial z}\left[\frac{\partial g}{\partial z}\right]^{-1}(\frac{\partial
g}{\partial t}+\tfrac{\partial g}{\partial w}f)$, and $\frac{{\rm
d}\hbox{\hskip 0.5pt}}{{\rm d}\hbox{\hskip 0.5pt}t}(\frac{\partial g}{\partial
z})=\frac{\partial^{2}g}{\partial z\partial t}+\frac{\partial^{2}g}{\partial
w\partial z}f+\frac{\partial^{2}g}{\partial^{2}z}\left[\frac{\partial
g}{\partial z}\right]^{-1}(\frac{\partial g}{\partial t}-\tfrac{\partial
g}{\partial w}f)$. It is worth noting that in various control problems that
involve the system’s output, such as state observer design and output feedback
control, the output of the system, denoted as $z=h(w,t)$, can be considered as
a time-varying constraint. In other words, the function $g(t,w,z)$ takes the
form of $z-h(w,t)$. In such cases, the term $\frac{{\rm d}\hbox{\hskip
0.5pt}}{{\rm d}\hbox{\hskip 0.5pt}t}(\frac{\partial g}{\partial z})$ becomes
zero, and $\frac{{\rm d}\hbox{\hskip 0.5pt}}{{\rm d}\hbox{\hskip
0.5pt}t}(\tfrac{\partial g}{\partial w})$ can be simplified to
$\frac{\partial^{2}g}{\partial w\partial
t}+\frac{\partial^{2}g}{\partial^{2}w}f$.

######  Lemma 3.3.

The variational DAE system (4) is exponentially stable if there exist
$\gamma\geq 0$ such that its auxiliary ODE system (28) is exponentially
stable.

######  Proof 3.4.

Consider an auxiliary system of (4) as follows, which is equivalent to (28),

$\left\\{\begin{matrix}\dot{\xi}_{\gamma}=\frac{\partial f}{\partial
w}\xi_{\gamma}+\frac{\partial f}{\partial z}\nu_{\gamma},\\\
\dot{\overbrace{\Big{(}\tfrac{\partial g}{\partial
w}\xi_{\gamma}+\frac{\partial g}{\partial
z}\nu_{\gamma}\Big{)}}}=-\gamma\Big{(}\tfrac{\partial g}{\partial
w}\xi_{\gamma}+\frac{\partial g}{\partial
z}\nu_{\gamma}\Big{)}.\end{matrix}\right.$ (29)

From the second equation in (29), we have

$\Big{(}\tfrac{\partial g}{\partial w}\xi_{\gamma}+\frac{\partial g}{\partial
z}\nu_{\gamma}\Big{)}=\Big{(}\frac{\partial g_{0}}{\partial w}\xi_{\gamma
0}+\frac{\partial g_{0}}{\partial z}\nu_{\gamma 0}\Big{)}e^{-\gamma t}.$ (30)

where $\frac{\partial g_{0}}{\partial w}=\tfrac{\partial g}{\partial
w}\Big{|}_{t=0}$. It follows then that the solution of (4) is a particular
case of (29) with initial condition $\frac{\partial g_{0}}{\partial
w}\xi_{\gamma 0}+\frac{\partial g_{0}}{\partial z}\nu_{\gamma 0}=0$.

In Lemma 3.3, we use the stability properties of (29) to derive the stability
properties of (4). In this regard, the constant $\gamma$ must be chosen
properly as (29) may fail to capture the stability properties of (4), e.g.,
the system (29) can be unstable while correspondingly the system (4) is
stable. For instance, let us consider the following contracting DAE system
whose trajectories can be calculated explicitly

$\left\\{\begin{aligned} \dot{w}&=-2e^{t}z,\\\
0&=e^{-t}w-z,\end{aligned}\right.$ (31)

which again has the same form as its variational system. The trajectories of
the system are given by $w(t)=w_{0}e^{-2t}$ and $z(t)=w_{0}e^{-3t}$. If we
choose $\gamma=1$, (28) can be rewritten as

$\left\\{\begin{aligned} \dot{w}&=-2e^{t}z,\\\
\dot{z}&=-3z,\end{aligned}\right.$ (32)

which has the solution $w(t)=z_{0}e^{-2t}+w_{0}-z_{0}$ and $z(t)=z_{0}e^{-3t}$
which does not converge to zero for $w_{0}\neq z_{0}$. However, by choosing
$\gamma=4$, (28) can be rewritten as

$\left\\{\begin{aligned} \dot{w}&=-2e^{t}z,\\\
\dot{z}&=3e^{-t}w-6z,\end{aligned}\right.$ (33)

whose solution is given by $w(t)=(3w_{0}-2z_{0})e^{-2t}-2(w_{0}-z_{0})e^{-3t}$
and $z(t)=(3w_{0}-2z_{0})e^{-3t}-3(w_{0}-z_{0})e^{-4t}$, which is converging
to zero. Based on this observation, it appears that in order to ensure the
contractivity of the system, the parameter $\gamma$ should be chosen
appropriately.

In the following proposition, we establish a lower bound for $\gamma$ based on
a priori information of the DAE system.

######  Proposition 3.5.

Suppose that the reduced system (6) is uniformly exponentially upper and lower
bounded by $-\underline{\alpha}$, $-\overline{\alpha}$, with
$\overline{\alpha}\geq\underline{\alpha}>0$, i.e. there exists constants
$\underline{c},\overline{c}>0$ such that all solutions satisfy

$\underline{c}e^{-\overline{\alpha}(t-t_{0})}\left\|\xi(t_{0})\right\|\leq\left\|\xi(t)\right\|\leq\overline{c}e^{-\underline{\alpha}(t-t_{0})}\left\|\xi(t_{0})\right\|$

for all $t\geq t_{0}$. Furthermore, assume that there exist constants
$l_{f}=l_{f}(w_{0},z_{0})$, $k_{f}=k_{f}(w_{0},z_{0})$,
$l_{g}=l_{g}(w_{0},z_{0})$, $k_{g}=k_{g}(w_{0},z_{0})$ with $k_{f},k_{g}>0$
such that, $\left\|\frac{\partial f}{\partial z}\right\|\leq k_{f}e^{l_{f}t}$
and $\left\|\left[\frac{\partial g}{\partial z}\right]^{-1}\right\|\leq
k_{g}e^{l_{g}t}$ for all $t\geq 0$, where all gradients are evaluated on the
trajectory $w(t)=\varphi(t,w_{0},z_{0})$ and $z(t)=\psi(t,w_{0},z_{0})$ of
(1); let $l:=\max\\{l_{g},l_{g}+l_{f}\\}$. If the variational DAE (4) is
exponentially stable and $\gamma>l+\overline{\alpha}$ then also the
corresponding ODE system (28) is exponentially stable (with a convergence rate
at least as large as any convergence rate of the variational DAE). In
particular, if for sufficiently large $\gamma$ the ODE (28) is unstable, the
variational DAE cannot be exponentially stable and hence the original
nonlinear DAE cannot be contractive.

######  Proof 3.6.

Let us define $q=\tfrac{\partial g}{\partial w}\xi_{\gamma}+\frac{\partial
g}{\partial z}\nu_{\gamma}$, so that (29) can be rewritten as

$\left\\{\begin{matrix}\dot{\xi}_{\gamma}=\Big{(}\frac{\partial f}{\partial
w}-\frac{\partial f}{\partial z}\left[\frac{\partial g}{\partial
z}\right]^{-1}\tfrac{\partial g}{\partial w}\Big{)}\xi_{\gamma}+\frac{\partial
f}{\partial z}\left[\frac{\partial g}{\partial z}\right]^{-1}q,\\\
\dot{q}=-\gamma q.\end{matrix}\right.$ (34)

In this case, the solution of (34) is given by

$\left\\{\begin{matrix}\xi_{\gamma}=\Phi(t,t_{0})\xi_{0}+q_{0}\int_{t_{0}}^{t}{\Phi(t,s)\Big{(}\frac{\partial
f}{\partial z}\left[\frac{\partial g}{\partial z}\right]^{-1}e^{-\gamma
s}\Big{)}{\rm d}\hbox{\hskip 0.5pt}s},\\\ q=q_{0}e^{-\gamma
t},\end{matrix}\right.$ (35)

where $\Phi(t,s)$ is the transition matrix of the reduced system. By
assumption, the transition matrix satisfies
$\underline{c}e^{-\overline{\alpha}(t_{1}-t_{0})}\leq||\Phi(t_{1},t_{0})||\leq\overline{c}e^{-\underline{\alpha}(t_{1}-t_{0})}$,
hence we arrive at the following estimation

$\displaystyle\left\|\xi_{\gamma}(t)\right\|$
$\displaystyle\leq\overline{c}e^{-\underline{\alpha}(t-t_{0})}\left\|\xi_{0}\right\|$
(36)
$\displaystyle\quad+\left\|q_{0}\right\|\int_{t_{0}}^{t}\overline{c}e^{-\underline{\alpha}(t-s)}\left\|\tfrac{\partial
f}{\partial z}\right\|\left\|\left[\tfrac{\partial g}{\partial
z}\right]^{-1}\right\|e^{-\gamma s}{\rm d}\hbox{\hskip 0.5pt}s$
$\displaystyle\leq\overline{c}e^{-\underline{\alpha}t}\left\|\xi_{0}\right\|$
$\displaystyle\quad+\overline{c}e^{-\underline{\alpha}t}\left\|q_{0}\right\|\int_{t_{0}}^{t}k_{f}e^{l_{f}s}k_{g}e^{l_{g}s}e^{-(\gamma-\underline{\alpha})s}{\rm
d}\hbox{\hskip 0.5pt}s$
$\displaystyle\leq\overline{c}e^{-\underline{\alpha}(t-t_{0})}(\left\|\xi_{0}\right\|+k\left\|q_{0}\right\|).$

The last inequality is derived from $l_{f}+l_{g}-\gamma+\underline{\alpha}\leq
l-\gamma+\overline{\alpha}<0$ and for some suitable $k>0$. This implies that
the convergence rate of $\xi$ is at least $\underline{\alpha}$. By the
definition of $q$ defined before (34), we have
$\nu_{\gamma}=-\left[\frac{\partial g}{\partial z}\right]^{-1}\left(e^{-\gamma
t}q_{0}-\tfrac{\partial g}{\partial w}\xi_{\gamma}\right)$ and hence

$\displaystyle\left\|\nu_{\gamma}\right\|\leq\left\|q_{0}\right\|\left\|\left[\tfrac{\partial
g}{\partial z}\right]^{-1}\right\|e^{-\gamma t}+\left\|\left[\tfrac{\partial
g}{\partial z}\right]^{-1}\tfrac{\partial g}{\partial w}\xi_{\gamma}\right\|$
(37)

By assumption, $l_{g}-\gamma<-\overline{\alpha}$, and hence

$\left\|q_{0}\right\|\left\|\left[\tfrac{\partial g}{\partial
z}\right]^{-1}\right\|e^{-\gamma t}\leq
k_{g}\|q_{0}\|e^{-\overline{\alpha}t}.$

We furthermore have, utilizing (35) and
$\Phi(t,s)=\Phi(t,t_{0})\Phi(t_{0},s)$,

$\displaystyle\left\|\left[\tfrac{\partial g}{\partial
z}\right]^{-1}\tfrac{\partial g}{\partial
w}\xi_{\gamma}\right\|\leq\left\|\left[\tfrac{\partial g}{\partial
z}\right]^{-1}\tfrac{\partial g}{\partial
w}\Phi(t,t_{0})\right\|\Bigg{(}\|\xi_{0}\|+$ (38)
$\displaystyle\|q_{0}\|\int_{t_{0}}^{t}\left\|\Phi(t_{0},s)\tfrac{\partial
f}{\partial z}\left[\tfrac{\partial g}{\partial z}\right]^{-1}e^{-\gamma
s}\right\|{\rm d}\hbox{\hskip 0.5pt}s\Bigg{)}$

To further bound the terms in (38), we first observe that the solution of (4)
satisfies

$\displaystyle\begin{pmatrix}\xi(t)\\\ \nu(t)\end{pmatrix}$
$\displaystyle=\begin{bmatrix}\Phi(t,t_{0})&0\\\ \left[\tfrac{\partial
g}{\partial z}\right]^{-1}\tfrac{\partial g}{\partial
w}\Phi(t,t_{0})&0\end{bmatrix}\begin{pmatrix}\xi_{0}\\\ \nu_{0}\end{pmatrix}$

hence exponential stability of (4) ensures

$\displaystyle\left\|\left[\begin{matrix}\Phi(t,t_{0})&0\\\
\left[\tfrac{\partial g}{\partial z}\right]^{-1}\tfrac{\partial g}{\partial
w}\Phi(t,t_{0})&0\end{matrix}\right]\right\|\leq
c^{\prime}e^{-\alpha^{\prime}t},$

for some positive $c^{\prime}$ and $\alpha^{\prime}\leq\underline{\alpha}$. It
follows 222In general, for any induced matrix norm by a $p$-norm we have
$\left\|\left[\begin{matrix}M&0\\\
N&0\end{matrix}\right]\right\|\geq\left\|N\right\|$. that

$\displaystyle\left\|\left[\tfrac{\partial g}{\partial
z}\right]^{-1}\tfrac{\partial g}{\partial w}\Phi(t,t_{0})\right\|\leq
c^{\prime}e^{-\alpha^{\prime}t},$

To bound the integral term in (38) we observe that

$\|\Phi(t_{0},s)\|\leq\|\Phi(s,t_{0})\|^{-1}\leq\underline{c}e^{\overline{\alpha}(s-t_{0})}$

hence

$\displaystyle\left\|\Phi(t_{0},s)\tfrac{\partial f}{\partial
z}\left[\tfrac{\partial g}{\partial z}\right]^{-1}e^{-\gamma s}\right\|$
$\displaystyle\leq\underline{c}e^{\overline{\alpha}(s-t_{0})}k_{f}e^{l_{f}s}k_{g}e^{l_{g}s}e^{-\gamma
s}$
$\displaystyle\leq\tilde{c}e^{(\overline{\alpha}+l_{f}+l_{g}-\gamma)s}=\tilde{c}e^{-\tilde{\alpha}s}$

for some $\tilde{c}>0$ and
$\tilde{\alpha}:=-(\overline{\alpha}+l_{f}+l_{g}-\gamma)>0$. Consequently, the
integral term in (38) is bounded by, say, $\hat{c}>0$ (independent of $t$).
Altogether, we can bound $\nu_{\gamma}$ by

$\|\nu_{\gamma}\|\leq
k_{g}\|q_{0}\|e^{-\overline{\alpha}t}+c^{\prime}e^{-\alpha^{\prime}t}(\|\xi_{0}\|+\hat{c}\|q_{0}\|).$

Together with (36) this shows that the system (28) is asymptotically stable.
Note that $-\overline{\alpha}<-\underline{\alpha}\leq-\alpha^{\prime}$, hence
the overall exponential convergence rate of (36) is at least $\alpha^{\prime}$
which was the assumed convergence rate of (4).

In Proposition 3.5, we have a mild assumption where the time varying functions
$\frac{\partial f}{\partial z}$ and $\left[\frac{\partial g}{\partial
z}\right]^{-1}$ are bounded by $ke^{lt}$ instead of by constants. Let us
recall again the previous example in (31), where the reduced system is
$\dot{w}=-2w$, we have $\left\|\frac{\partial f}{\partial z}\right\|=2e^{t}$,
$\left\|\left[\frac{\partial g}{\partial z}\right]^{-1}\right\|=1$. Thus the
hypotheses in Proposition 3.5 hold with $\alpha=2$, $k=l=1$ and
$\gamma=4>l+\overline{\alpha}=3$. Then, the ODE system (33) has the same
exponential convergence rate as the corresponding DAE system (32), namely
$\alpha^{\prime}=2$. By considering Proposition 3.5, it becomes evident that
selecting $\gamma>l+\overline{\alpha}$ allows us to conclude that if (28) is
unstable, then (4) is also unstable. This proposition helps prevent any
potential “misjudgment” similar to the previous example (32). By examining
(37), it shows that as $\gamma$ increases, the estimation performance of (29)
w.r.t. (4) improves. If obtaining prior information about the DAE system (3)
is difficult, one can select a sufficiently large value for $\gamma$ to apply
Proposition 3.5 and determine the contracting rate of the DAE system.

By utilizing the concept of matrix measure, we will examine the contractivity
of the time-varying DAE system (1) by focusing on the generalized Jacobian
matrix of the corresponding auxiliary ODE system (28). In order to simplify
the notation, we rewrite (28) into

$\displaystyle\left\\{\begin{matrix}\dot{\xi}_{\gamma}=A(t)\xi_{\gamma}+B(t)\nu_{\gamma},\\\
\dot{\nu}_{\gamma}=-F^{-1}(t)C(t)\xi_{\gamma}-F^{-1}(t)D(t)\nu_{\gamma},\end{matrix}\right.$
(39)

where $A(t)=\frac{\partial f}{\partial w}$, $B(t)=\frac{\partial f}{\partial
z}$, $F(t)=\frac{\partial g}{\partial z}$, $C(t)=\gamma\tfrac{\partial
g}{\partial w}+\frac{{\rm d}\hbox{\hskip 0.5pt}}{{\rm d}\hbox{\hskip
0.5pt}t}(\tfrac{\partial g}{\partial w})+\tfrac{\partial g}{\partial
w}\frac{\partial f}{\partial w}$, and $D(t)=\gamma\frac{\partial g}{\partial
z}+\frac{{\rm d}\hbox{\hskip 0.5pt}}{{\rm d}\hbox{\hskip
0.5pt}t}(\frac{\partial g}{\partial z})+\tfrac{\partial g}{\partial
w}\frac{\partial f}{\partial z}$. Furthermore, for any sufficiently smooth
invertible matrix $M(w,z,t)$ we define the matrix

$J_{M}(t):=\dot{M}M^{-1}+M\left[\begin{matrix}A&B\\\
-F^{-1}C&-F^{-1}D\end{matrix}\right]M^{-1},$ (40)

where $M(w(t),z(t),t)$ is evaluated along a solution $(w,z)$ of (1) and
$\dot{M}(w(t),z(t),t)=\tfrac{\mathrm{d}}{\mathrm{d}t}M(w(t),z(t),t)$.

######  Theorem 3.7.

If there exist an invertible matrix $M(w,z,t)$ and $\beta>0$ such that for all
solutions $(w,z)$ of (1) it holds that $\left\|M^{-1}\right\|\left\|M\right\|$
is bounded and $\mu_{q}\Big{(}J_{M}(t)\Big{)}\leq-\beta$, where $J_{M}(t)$ is
given by (40), then the DAE system (1) is contracting.

######  Proof 3.8.

For any $M(w,z,t)$, we consider the coordinate transformations
$\left[\begin{smallmatrix}p\\\
r\end{smallmatrix}\right]=M\left[\begin{smallmatrix}\xi_{\gamma}\\\
\nu_{\gamma}\end{smallmatrix}\right]$, so that (39) can be rewritten as

$\displaystyle\dot{\overbrace{\left[\begin{matrix}p\\\
r\end{matrix}\right]}}=\underset{J_{M}(t)}{\underbrace{\Big{(}\dot{M}M^{-1}+M\left[\begin{matrix}A&B\\\
-F^{-1}C&-F^{-1}D\end{matrix}\right]M^{-1}\Big{)}}}\left[\begin{matrix}p\\\
r\end{matrix}\right].$ (41)

Appealing to Coppel’s inequality [27], we can conclude that

$\displaystyle\left\|\left[\begin{matrix}p(t)\\\
r(t)\end{matrix}\right]\right\|\leq e^{-\beta
t}\left\|\left[\begin{matrix}p(0)\\\ r(0)\end{matrix}\right]\right\|,$ (42)

from which it follows that

$\displaystyle\left\|\left[\begin{matrix}\xi_{\gamma}(t)\\\
\nu_{\gamma}(t)\end{matrix}\right]\right\|=\left\|M^{-1}\left[\begin{matrix}p(t)\\\
r(t)\end{matrix}\right]\right\|\overset{\eqref{dini2}}{\leq}e^{-\beta
t}\left\|M^{-1}\right\|\left\|M\right\|\left\|\left[\begin{matrix}\xi_{\gamma}(0)\\\
\nu_{\gamma}(0)\end{matrix}\right]\right\|$ (43) $\displaystyle\leq ce^{-\beta
t}\left\|\left[\begin{matrix}\xi_{\gamma}(0)\\\
\nu_{\gamma}(0)\end{matrix}\right]\right\|,$

i.e., the system (39) is exponentially stable, the last inequality is due to
$\left\|M^{-1}\right\|\left\|M\right\|$ is bounded. By Lemma 3.3 and
Proposition 3.1, we can conclude that the DAE system (1) is contracting.

In Theorem 3.7, the condition that $\left\|M^{-1}\right\|\left\|M\right\|$ is
bounded is less conservative compared to the usual assumption of being a
Lyapunov transformation [26, Def. 2.4]. Specifically, instead of requiring
both $\left\|M^{-1}\right\|$ and $\left\|M\right\|$ to be bounded, it only
demands their product to be bounded. As illustrated in Example 4.6, this less
conservative condition allows for cases where
$M=\left[\begin{smallmatrix}e^{-t}&0&0\\\ 0&e^{-t}&0\\\
0&0&e^{-t}\end{smallmatrix}\right]$, which has an unbounded inverse.
Nevertheless, it is trivial to see that
$\left\|\left[\begin{smallmatrix}e^{t}&0&0\\\ 0&e^{t}&0\\\
0&0&e^{t}\end{smallmatrix}\right]\right\|\left\|\left[\begin{smallmatrix}e^{-t}&0&0\\\
0&e^{-t}&0\\\ 0&0&e^{-t}\end{smallmatrix}\right]\right\|=1$.

######  Remark 3.9.

As a powerful tool to analyze the stability of ODE systems, the Lyapunov
method has been widely used in the analysis and control design of nonlinear
systems. By employing the 2-norm matrix measure, the hypotheses in Theorem
3.7, i.e., $\mu_{2}\Big{(}J_{M}(t)\Big{)}\leq-\beta$ can be reformulated into
solvability of the following differential Riccati inequality

$\displaystyle\left[\begin{matrix}A^{\top}(t)&C^{\top}(t)F^{-\top}(t)\\\
B^{\top}(t)&D^{\top}(t)F^{-\top}(t)\end{matrix}\right]P(t)+$ (44)
$\displaystyle P(t)\left[\begin{matrix}A(t)&B(t)\\\
F^{-1}(t)C(t)&F^{-1}(t)D(t)\end{matrix}\right]+\dot{P}(t)\leq-\beta P(t),$

where $P(t)=M^{\top}M$ for some $M(w,z,t)$. Afterwards, Theorem 3.7 can be
simplified to [21, Thm. 2]. As will be shown later in Example 4.6, such
Riccati inequality will be used for the design of an observer. Nevertheless,
in certain scenarios, it can be challenging to find a solution $M^{\top}M$
that satisfies (44). By employing the 1-norm ($\infty$-norm) as matrix
measure, we can arrive at a simple numerical test to each column (or row) of
the matrix $J_{M}(t)$. We demonstrate this approach via Example 4.1 and
Example 4.4 later.

##  4 Simulation Setup and Applications

This section presents three numerical examples to illustrate different
applications of the proposed methods. In the first example, we examine the
contraction property of a nonlinear time-varying DAE system using Theorem 3.7.
The second example illustrates the practical application of Proposition 3.1 in
stabilizing a time-invariant DAE system, as demonstrated by Corollary 4.2.
Subsequently, we employ Corollary 4.2 to design a state feedback controller
for a power source system with an inverter interface, which can be modeled as
a time-invariant DAE system. Finally, the third illustration involves the
application of Theorem 3.7 to devise an observer for a time-varying ODE
system, namely, Corollary 4.5. Subsequently, we employ Corollary 4.5 to design
an observer for an unstable time-varying ODE system.

###  4.1 Contractivity of nonlinear time-varying DAE systems

######  Example 4.1.

Consider a nonlinear time-varying DAE system

$\displaystyle\left\\{\begin{matrix}\dot{w}_{1}=-4w_{1}-0.5\cos z,\\\
\dot{w}_{2}=\frac{4}{3+\sin t}w_{1}-\frac{3+\cos t}{3+\sin
t}w_{2}-\frac{4}{3+\sin t},\\\ 0=4z+0.5\sin z+w_{1}+(3+\sin
t)w_{2}.\end{matrix}\right.$ (45)

where $w(t)\in\mathbb{R}^{2}$ is the state vector and $z(t)\in\mathbb{R}$
refers to the algebraic vector. Its variational system is

$\displaystyle\left\\{\begin{smallmatrix}\dot{\xi}_{1}=-4\xi_{1}+(0.5\sin
z)\nu,\\\ \dot{\xi}_{2}=\frac{4}{3+\sin t}\xi_{1}-\frac{3+\cos t}{3+\sin
t}\xi_{2},\\\ 0=\xi_{1}+(3+\sin t)\xi_{2}+(4+0.5\cos
z)\nu.\end{smallmatrix}\right.$ (46)

In the given example (31), obtaining information about the system is a
straightforward process. However, in this specific case, the existence of a
time-varying nonlinear constraint creates difficulties in obtaining
information about (46), we are unable to utilize Proposition 3.5 to select
$\gamma$. Nevertheless, the simplest and most direct option for $\gamma$ in
this instance is $\gamma=0$ (if this choice proves ineffective, a larger value
of $\gamma$ should be considered). As a result, the auxiliary ODE system
corresponding to (46) can be expressed as follows:
$\left\\{\begin{smallmatrix}\dot{\xi}_{\gamma 1}=-4\xi_{\gamma 1}+(0.5\sin
z)\nu_{\gamma},\\\ \dot{\xi}_{\gamma 2}=\frac{4}{3+\sin t}\xi_{\gamma
1}-\frac{3+\cos t}{3+\sin t}\xi_{\gamma 2},\\\
\dot{\nu}_{\gamma}=-\frac{4}{3+0.5\cos
z}\nu_{\gamma}.\end{smallmatrix}\right.$ It follows that with $M=I$ we have
$J_{M}(t)=\left[\begin{smallmatrix}-4&0&0.5\sin z\\\ \frac{4}{3+\sin
t}&-\frac{3+\cos t}{3+\sin t}&0\\\ 0&0&-\frac{4}{3+0.5\cos
z}\end{smallmatrix}\right]$. Applying the 1-norm, we find that
$\mu_{1}\Big{(}J_{M}(t)\Big{)}<-0.5$ holds, which implies that the DAE system
(45) is contracting according to Theorem 3.7. Hence, we can conclude that the
time-varying DAE system is contracting. Figure 1 illustrates the trajectories
of the system with two distinct initial conditions.

Figure 1: The plot of trajectories of time-varying DAE system in Example 4.1
initialized at $\left[\begin{smallmatrix}3\\\ -3\\\
1.38\end{smallmatrix}\right]$ and $\left[\begin{smallmatrix}-3\\\ 3\\\
-1.38\end{smallmatrix}\right]$.

###  4.2 Stability of inverter-interfaced power source systems

It is well known that for a contracting time-invariant system, all
trajectories converge to an equilibrium exponentially. As an interesting
particular case of our main results above, we can prove stability of the time-
invariant DAE systems by using Proposition 3.1. The basic idea involves
guaranteeing the contravtivity of the DAE system while ensuring that its
equilibrium point resides within its trajectory set. Consider a time-invariant
DAE system given by

$\left\\{\begin{matrix}\dot{w}=f(w,z),\\\ 0=g(w,z),\end{matrix}\right.$ (47)

where $f(0,0)=0$, and $g(0,0)=0$. The variational system of (47) is

$\left\\{\begin{matrix}\dot{\xi}=\frac{\partial f}{\partial
w}(w(t),z(t))\cdot\xi+\frac{\partial f}{\partial z}(w(t),z(t))\cdot\nu,\\\
0=\tfrac{\partial g}{\partial w}(w(t),z(t))\cdot\xi+\frac{\partial g}{\partial
z}(w(t),z(t))\cdot\nu.\end{matrix}\right.$ (48)

To ensure safety in engineering systems, it is imperative to establish bounds
on the states of (47). In the context of a power system, for example, the
terminal complex voltage $V\angle\theta$ is restricted within the range
$[0.95,1.05]\times[-\frac{\pi}{2},\frac{\pi}{2}]$ to ensure operational
stability and prevent undesirable conditions. In this section, we investigate
the time-invariant DAE system (47) within the set $W\times Z$, i.e., $w(t)\in
W\subseteq\mathbb{R}^{n}$ and $z(t)\in Z\subseteq\mathbb{R}^{m}$. Moreover, we
assume that $\left[\frac{\partial g}{\partial z}\right]^{-1}\tfrac{\partial
g}{\partial w}$ is bounded on $W\times Z$. In the subsequent corollary, we
analyze the local exponential stability of (48) through its reduced system
$\dot{\xi}=\Big{(}\frac{\partial f}{\partial w}-\frac{\partial f}{\partial
z}\left[\frac{\partial g}{\partial z}\right]^{-1}\tfrac{\partial g}{\partial
w}\Big{)}\xi$.

######  Corollary 4.2.

The time-invariant DAE system (47) is locally exponentially stable in $W\times
Z$ if there exist an invertible matrix $M(w,z)$, such that
$\mu_{q}\Big{(}J_{M}(w,z)\Big{)}\leq-\beta$ for some positive constant
$\beta$, where $J(w,z)$ is given by

$\displaystyle J_{M}(w,z)=\dot{M}M^{-1}+M\Big{(}\frac{\partial f}{\partial
w}-\frac{\partial f}{\partial z}\left[\frac{\partial g}{\partial
z}\right]^{-1}\tfrac{\partial g}{\partial w}\Big{)}M^{-1}.$ (49)

######  Proof 4.3.

If condition (49) is satisfied, then using similar arguments as in the proof
of Theorem 3.7 we can conclude that the reduced system is exponentially
stable, i.e., $\left\|\xi\right\|\leq c\left\|\xi_{0}\right\|e^{-\alpha t}$
for some $\alpha,c>0$. By using $\nu=\left[\frac{\partial g}{\partial
z}\right]^{-1}\tfrac{\partial g}{\partial w}\xi$ and the boundedness of
$\left[\frac{\partial g}{\partial z}\right]^{-1}\tfrac{\partial g}{\partial
w}$ on $W\times Z$, we can conclude $\left\|\left[\begin{smallmatrix}\xi\\\
\nu\end{smallmatrix}\right]\right\|\leq
c^{\prime}\left\|\xi_{0}\right\|e^{-\alpha t}\leq
c^{\prime}\left\|\left[\begin{smallmatrix}\xi_{0}\\\
\nu_{0}\end{smallmatrix}\right]\right\|e^{-\alpha t}$ for some $c^{\prime}>0$.
Consequently, the variational system (48) is exponentially stable. Applying
Proposition 3.1, we can conclude that the DAE system (47) is contracting. As
$\left[\begin{smallmatrix}w(t)\\\
z(t)\end{smallmatrix}\right]=\left[\begin{smallmatrix}0\\\
0\end{smallmatrix}\right]$ represents one of admissible trajectories of (47)
and it is contracting, it follows that all the trajectories will exponentially
converge to $\left[\begin{smallmatrix}w(t)\\\
z(t)\end{smallmatrix}\right]=\left[\begin{smallmatrix}0\\\
0\end{smallmatrix}\right]$.

######  Example 4.4.

Consider an inverter-interfaced power source connected to the infinite bus via
a transmission line [3]. The system dynamics is given by the following

$\displaystyle\left\\{\begin{smallmatrix}\dot{P}=\frac{1}{\tau_{1}}\Big{(}-P+P^{\text{ref}}-d_{1}(\theta-\theta^{\text{ref}})\Big{)},\\\
\dot{Q}=\frac{1}{\tau_{2}}\Big{(}-Q+Q^{\text{ref}}-d_{2}(V-V^{\text{ref}})\Big{)}+u.\end{smallmatrix}\right.$
(50)

where $P$, $Q$, and $u$ are the terminal output active, reactive power, and
control input, respectively. The variable $V\angle\theta$ is the terminal
complex voltage, $1\angle 0$ is the desired complex voltage. The symbols
$P^{\text{ref}}$, $Q^{\text{ref}}$ , $\theta^{\text{ref}}$, and
$V^{\text{ref}}$ are pre-specified constant reference values. The constants
$\tau_{1}>0$ and $\tau_{2}>0$ are time constants while $d_{1}>0$ and $d_{2}>0$
are droop coefficients. For this particular case, the algebraic equations are
given by $\left\\{\begin{smallmatrix}P-GV\cos\theta-BV\sin\theta=0,\\\
Q-GV\sin\theta+BV\cos\theta=0.\end{smallmatrix}\right.$. In this example, a
state feedback controller $u=k_{1}P+k_{2}Q$ will be designed based on
Corollary 4.2. For numerical purposes, the parameters are given by
$\tau_{1}=\tau_{2}=\frac{1}{3}$, $d_{1}=d_{2}=\frac{1}{3}$,
$P^{\text{ref}}=1$, $Q^{\text{ref}}=-1$, $\theta^{\text{ref}}=0$,
$V^{\text{ref}}=1$, $G=B=1$. In this case, the whole system can be rewritten
as $\left\\{\begin{smallmatrix}\dot{P}=-3P-\theta+3,\\\
\dot{Q}=-3Q-V-2+k_{1}P+k_{2}Q,\end{smallmatrix}\right.$,
$\left\\{\begin{smallmatrix}P-V\cos\theta-V\sin\theta=0,\\\
Q-V\sin\theta+V\cos\theta=0,\end{smallmatrix}\right.$, where
$W(t)=\left[\begin{smallmatrix}P\\\ Q\end{smallmatrix}\right]$,
$Z(t)=\left[\begin{smallmatrix}\theta\\\ V\end{smallmatrix}\right]$. Its
corresponding variational system is
$\left\\{\begin{smallmatrix}\delta\dot{P}=-3\delta P-\delta\theta,\\\
\delta\dot{Q}=-3\delta Q-\delta V+k_{1}\delta P+k_{2}\delta Q,\\\ \delta
P-(\sin\theta+\cos\theta)\delta V-V(\cos\theta-\sin\theta)\delta\theta=0,\\\
\delta Q-(\sin\theta-\cos\theta)\delta
V-V(\cos\theta+\sin\theta)\delta\theta=0.\end{smallmatrix}\right.$ The reduced
system is
$\left\\{\begin{smallmatrix}\delta\dot{P}=(-3+\frac{\sin\theta-\cos\theta}{2V})\delta
P-\frac{\sin\theta+\cos\theta}{2V}\delta\theta,\\\
\delta\dot{Q}=(k_{1}-\frac{\sin\theta+\cos\theta}{2})\delta
P+(-3+k_{2}-\frac{\sin\theta-\cos\theta}{2})\delta Q.\end{smallmatrix}\right.$
In power system, we have $V\in[0.95,1.05]$, and
$\theta\in[-\frac{\pi}{2},\frac{\pi}{2}]$. Then, $\left\|\left[\frac{\partial
g}{\partial z}\right]^{-1}\tfrac{\partial g}{\partial
w}\right\|=\left\|\left[\begin{smallmatrix}\frac{\sin\theta-\cos\theta}{2V}&-\frac{\sin\theta+\cos\theta}{2V}\\\
-\frac{\sin\theta+\cos\theta}{2}&-\frac{\sin\theta-\cos\theta}{2}\end{smallmatrix}\right]\right\|$
is bounded. So we can analysis the stability of the DAE system by its reduced
system. With $M=I$, we have
$J_{M}(V,\theta)=\left[\begin{matrix}-3+\frac{\sin\theta-\cos\theta}{2V}&-\frac{\sin\theta+\cos\theta}{2V}\\\
k_{1}-\frac{\sin\theta+\cos\theta}{2}&-3+k_{2}-\frac{\sin\theta-\cos\theta}{2}\end{matrix}\right]$.
We can determine the range of the elements in $J_{M}(V,\theta)$ within
$[0.95,1.05]\times[-\frac{\pi}{2},\frac{\pi}{2}]$. By setting
$k_{1}=k_{2}=0.5$, and using the 1-norm, we find that
$\mu_{1}\Big{(}J_{M}(V,\theta)\Big{)}<-1.1$. According to Corollary 4.2, the
DAE system is exponentially stable with respect to the equilibrium
$[P^{*},Q^{*}]^{\top}=[1,-1]^{\top}$. The trajectories of inverter-interfaced
power source systems are depicted in Fig. 2. With the state feedback
controller, the terminal complex voltage $V\angle\theta$ converges to the
desired complex voltage $1\angle 0$.

Figure 2: The plot of solutions of $P$, $Q$, $\theta$, and $V$ in Example 4.4
initialized at $\left[\begin{smallmatrix}P_{0}\\\
Q_{0}\end{smallmatrix}\right]=\left[\begin{smallmatrix}0.5\\\
1.05\end{smallmatrix}\right]$.

###  4.3 Observer design of time-varying ODE systems

By treating the output of the systems as an algebraic constraint, our
approaches can be effectively employed in various classical control problems,
including output feedback design, output regulation, and state observer
design. In the following, we deploy our methodologies to develop an observer
for a time-varying ODE system. Consider a time-varying ODE system

$\left\\{\begin{matrix}\dot{w}=k(t,w),\\\ z=h(t,w),\end{matrix}\right.$ (51)

where $w$ is the state and $z$ is the output. The design problem of interest
is to determine an observer of the form

$\left\\{\begin{matrix}\dot{\hat{w}}=k(t,\hat{w})+l(t,\hat{z},z),\\\
\hat{z}=h(t,\hat{w}),\end{matrix}\right.$ (52)

with $l(t,z,z)=0$, such that

$\left\|\hat{w}-w\right\|\leq c\left\|\hat{w}_{0}-w_{0}\right\|e^{-\alpha t},$
(53)

holds for all $t\geq 0$ with some positive constants $c$ and $\alpha$. In
literature, a commonly used method for designing $l(t,\hat{z},z)$ is
$l(t,\hat{z},z)=\kappa(t)(\hat{z}-z)$. As a result, the observer (52) can be
simplified to the well-known Luenberger observer [25].

The observer (52) for a given output signal $z(t)$ can be regarded as a time-
varying DAE system in the form of (1) with
$f(t,\hat{w},\hat{z})=k(t,\hat{w})+l(t,\hat{z},z(t))$, and
$g(t,\hat{w},\hat{z})=h(t,\hat{w})-\hat{z}$. It is evident that $w$ and $z$ in
(51) represent a solution of (52). If (52) is contracting, both $w$ and
$\hat{w}$ will satisfy (53). By applying Theorem 3.7 to (52), we obtain the
subsequent corollary.

######  Corollary 4.5.

Given a system described by (51), the system (52) is an observer of (51) if
there exist an invertible matrix $M(t)$ and $\gamma>0$, such that
$\mu_{q}\Big{(}J_{M}(t)\Big{)}\leq-\beta$ for some positive constant $\beta$,
where $J_{M}(t)$ is in the form of (40) with $A(t)=\frac{\partial
k}{\partial\hat{w}}$, $B(t)=\frac{\partial l}{\partial\hat{z}}$, $F(t)=-I$,
$C(t)=\gamma\frac{\partial h}{\partial\hat{w}}+\frac{{\rm d}\hbox{\hskip
0.5pt}}{{\rm d}\hbox{\hskip 0.5pt}t}(\frac{\partial
h}{\partial\hat{w}})+\frac{\partial h}{\partial\hat{w}}\frac{\partial
k}{\partial\hat{w}}$, and $D(t)=-\gamma I+\frac{\partial
h}{\partial\hat{w}}\frac{\partial l}{\partial\hat{z}}$.

######  Example 4.6.

Consider an unstable time-varying ODE system as presented in [1, Ex. 4.22]

$\displaystyle\left[\begin{smallmatrix}\dot{w}_{1}\\\
\dot{w}_{2}\end{smallmatrix}\right]=\left[\begin{smallmatrix}-1+1.5\cos^{2}t&1-1.5\sin
t\cos t\\\ -1-1.5\sin t\cos
t&-1+1.5\sin^{2}t\end{smallmatrix}\right]\left[\begin{smallmatrix}w_{1}\\\
w_{2}\end{smallmatrix}\right],$ (54)

with the output $z=w_{1}$. For simplifying the design process, we assume
$l(t,\hat{z}-z)$ in (52) takes the form of
$\left[\begin{smallmatrix}k_{1}(t)(\hat{z}-z)\\\
k_{2}(t)(\hat{z}-z)\end{smallmatrix}\right]$. The Luenberger observer is given
by $\left[\begin{smallmatrix}\dot{\hat{w}}_{1}\\\
\dot{\hat{w}}_{2}\end{smallmatrix}\right]=\left[\begin{matrix}-1+1.5\cos^{2}t&1-1.5\sin
t\cos t\\\ -1-1.5\sin t\cos
t&-1+1.5\sin^{2}t\end{matrix}\right]\left[\begin{smallmatrix}\hat{w}_{1}\\\
\hat{w}_{2}\end{smallmatrix}\right]+\left[\begin{smallmatrix}k_{1}(t)(\hat{z}-z)\\\
k_{2}(t)(\hat{z}-z)\end{smallmatrix}\right]$, where the output
$\hat{z}=\hat{w}_{1}$. By selecting $\gamma=1$, the auxiliary ODE system (28)
of the observer is given by $\left\\{\begin{smallmatrix}\dot{\xi}_{\gamma
1}=(-1+1.5\cos^{2}t)\xi_{\gamma 1}+(1-1.5\sin t\cos t)\xi_{\gamma
2}+k_{1}(t)\nu_{\gamma}\\\ \dot{\xi}_{\gamma 2}=(-1-1.5\sin t\cos
t)\xi_{\gamma 1}+(-1+1.5\sin^{2}t)\xi_{\gamma 2}+k_{2}(t)\nu_{\gamma}\\\
\dot{\nu}_{\gamma}=1.5\xi_{\gamma 1}\cos^{2}t+(1-1.5\sin t\cos t)\xi_{\gamma
2}+(k_{1}(t)-1)\nu_{\gamma}\end{smallmatrix}\right.$. By choosing
$k_{1}(t)=-1.5\cos^{2}t$, $k_{2}(t)=-1+1.5\sin t\cos t$ it can be shown that
inequality (44) holds with $M=\left[\begin{smallmatrix}e^{-t}&0&0\\\
0&e^{-t}&0\\\ 0&0&e^{-t}\end{smallmatrix}\right]$ and by using the $2$-norm,
we therefore have $J_{M}(t)=\left[\begin{smallmatrix}-4+3\cos^{2}t&-3\sin
t\cos t&0\\\ -3\sin t\cos t&-4+3\sin^{2}t&0\\\
0&0&-3\cos^{2}t-4\end{smallmatrix}\right]e^{-2t}\leq-I$. The plots of error of
$\hat{w}_{i}-w_{i}$ are shown in Fig. 3, where the tracking property of the
observer is ensured by the proposed methodologies.

Figure 3: The plot of errors of $\hat{w}_{i}-w_{i}$ in Example 4.6 initialized
at $\left[\begin{smallmatrix}\hat{w}_{1}\\\
\hat{w}_{2}\end{smallmatrix}\right]=\left[\begin{smallmatrix}2\\\
-2\end{smallmatrix}\right]$ and $\left[\begin{smallmatrix}w_{1}\\\
w_{2}\end{smallmatrix}\right]=\left[\begin{smallmatrix}-2\\\
2\end{smallmatrix}\right]$.

##  5 CONCLUSION

In this paper, the contraction property of time-varying DAE systems have been
studied by an ODE approach. It is established based on a necessary and
sufficient condition that connects the contraction property of the original
DAE systems and the UGES of its variational DAE system. Subsequently, the
variational DAE system is lifted to an auxiliary ODE system defined on the
whole state space, and the trajectories of these systems exhibit the same
convergence property. The concept of matrix measure is introduced to study the
UGES of the auxiliary ODE system. Moreover, the results obtained in this study
can be applied to stabilize time-invariant DAE systems, and to observer design
for time-varying ODE systems.

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