remarkRemark hypothesisHypothesis exampleExample claimClaim Detectability and
State Estimation for LTV and Nonlinear SystemsM. Tranninger, R. Seeber, M.
Steinberger, M. Horn, and C. Pötzsche

# Detectability Conditions and State Estimation for Linear Time-Varying and
Nonlinear Systems††thanks: This manuscript was accepted for publication in a
future issue of the SIAM Journal on Control and Optimization.

Markus Tranninger Institute of Automation and Control, Graz University of
Technology, Graz, Austria<EMAIL_ADDRESS>Richard Seeber
Christian Doppler Laboratory for Model-Based Control of Complex Test Bed
Systems, Institute of Automation and Control, Graz University of Technology,
Graz, Austria.  Martin Steinberger22footnotemark: 2 Martin Horn33footnotemark:
3 Christian Pötzsche Department of Mathematics, Alpen-Adria Universität
Klagenfurt, Klagenfurt, Austria.

###### Abstract

This work proposes a detectability condition for linear time-varying systems
based on the exponential dichotomy spectrum. The condition guarantees the
existence of an observer, whose gain is determined only by the unstable modes
of the system. This allows for an observer design with low computational
complexity compared to classical estimation approaches. An extension of this
observer design to a class of nonlinear systems is proposed and local
convergence of the corresponding estimation error dynamics is proven.
Numerical results show the efficacy of the proposed observer design technique.

###### keywords:

detectability, time-varying systems, exponential dichotomy spectrum, nonlinear
observer design

93B07, 34D08, 34D09,

##  1 Introduction

In recent years, detectability analysis of complex dynamical systems gained a
lot of research attention in different engineering fields [11, 26, 34, 57,
59]. Despite the fact that detectability is very well studied for linear time
invariant systems [53], the underlying theory and existing detectability
conditions are less mature for the time varying and/or nonlinear setting. In
control theory, it is well known that detectability is closely related to the
existence of observers, i.e., dynamical systems, which estimate the state of
some physical system based on measurements and a system model [53]. In this
work, a system model of the form

(1) $\displaystyle\dot{\mathbf{x}}(t)$
$\displaystyle=\mathbf{f}(\mathbf{x},\mathbf{u}),\quad\mathbf{y}(t)=\mathbf{C}(t)\mathbf{x}(t),\quad
t\in{\mathds{R}_{\geq 0}}$

is assumed to be known. Here, $\mathbf{x}(t)\in\mathds{R}^{n}$,
$\mathbf{u}(t)\in\mathds{R}^{m}$ and $\mathbf{y}(t)\in\mathds{R}^{p}$ are the
system’s state, input and output, respectively. In order to guarantee
uniqueness of a solution to the initial value problem (1) with the initial
time $t_{0}\in{\mathds{R}_{\geq 0}}$ and the initial state
$\mathbf{x}(t_{0})=\mathbf{x}_{0}$, it is assumed that
$\mathbf{f}:\mathds{R}^{n}\times\mathds{R}^{m}\rightarrow\mathds{R}^{n}$ is
locally Lipschitz continuous in the first argument and piecewise continuous in
the second argument. The solution is assumed to exist for all $t\geq t_{0}$.
Similar to [26, 50], the considered observer is assumed to be of the form

(2)
$\dot{\hat{\mathbf{x}}}(t)=\mathbf{f}(\hat{\mathbf{x}},\mathbf{u})+\mathbf{L}(t,\hat{\mathbf{x}})\left[\mathbf{y}(t)-\mathbf{C}(t)\hat{\mathbf{x}}(t)\right],\quad\hat{\mathbf{x}}(t_{0})=\hat{\mathbf{x}}_{0}.$

The observer gain $\mathbf{L}:{\mathds{R}_{\geq
0}}\times\mathds{R}^{n}\rightarrow\mathds{R}^{n\times p}$, which is continuous
in the first argument and locally Lipschitz continuous in the second argument,
has to be designed. The goal of this work is to present conditions for the
existence of a gain $\mathbf{L}$ that guarantees (at least locally) a uniform
exponential convergence of the estimation error
$\mathbf{e}(t)=\mathbf{x}(t)-\hat{\mathbf{x}}(t)$.

State estimation problems do not exclusively occur in control theory, but
also, e.g., in synchronization problems for chaotic systems [46, 13, 9, 45].
Such problems arise for example due to agents interacting in complex networks
and are of key importance in biological, chemical or social processes [3]. In
the synchronization terminology, system (1) in combination with (2) is
referred to as a driver-receiver process [45] and the challenge is to design a
coupling gain $\mathbf{L}$ such that the receiver $\hat{\mathbf{x}}$ is
synchronized with the driver $\mathbf{x}$.

In geosciences, optimal state estimation is also known as data assimilation
[58, 17, 11, 16]. Data assimilation is a key ingredient for many algorithms to
solve problems in, e.g., weather forecasting, oceanography or numerical
geology. There, usually large scale nonlinear models with hundreds to
thousands of states are considered together with significant model and
measurement uncertainties. Hence it is vital for any numerical estimation
scheme to confine the estimation and prediction to the part of the system,
which dominates the dynamical behavior [58, 17, 42]. A comprehensive overview
of relevant methods and applications in these fields can be found in [5].

###  1.1 Related Work and Contribution

A key concept in data assimilation is the assimilation on the unstable
subspace (AUS), which was introduced by A. Trevisan and collaborators [58] for
nonlinear stochastic discrete time systems. The resulting algorithms are
modifications of the extended Kalman filter, where the computation of the
error covariances and the Kalman gain is confined to the unstable and neutral
tangent subspace of the solutions of the full order filter [58, 17]. With this
approach, the solution of the filter Riccati equation has to be computed only
for the unstable and neutral tangent subspace in order to obtain the filter
gain. Usually, this subspace is of significantly lower dimension than the
system’s state space and hence the computational complexity of the resulting
estimation algorithm can be drastically reduced compared to the classical
extended Kalman filter (EKF). Since the initial investigations [58],
algorithms for continuous-time [26] and discrete-time systems [17, 43] were
presented and analyzed.

In [26], a data assimilation scheme based on the numerical approximation of
regular Lyapunov exponents for continuous time nonlinear systems is presented.
The resulting observer design is numerically very efficient, because it does
not require solving a differential Riccati equation to compute the observer
feedback gain. The underlying existence condition is, however, conservative.
It requires the number of linearly independent measurements to be at least
equal to the number of non-negative Lyapunov exponents. Hence, this condition
is not fulfilled, for example, by the simple observable linear time invariant
double integrator system where the first integrator state is measured.

For linear and nonlinear stochastic discrete time systems, AUS algorithms are
discussed in [58, 17, 16, 11, 42]. Such algorithms are very popular, because
many numerical studies showed that AUS algorithms are at least as efficient
but simpler to implement and computationally less demanding than their
original EKF counterparts [43]. Based on the AUS ideas, a detectability
condition is introduced in [29, 11]. There, the convergence of the error
covariance matrices of the AUS Kalman filter is investigated for linear
systems in the presence of bounded additive model errors. The proposed
detectability condition has some similarity to the results presented in [57,
55] for the deterministic continuous time setting. Albeit numerical [54, 17]
and theoretical investigations, cf. [57, 29, 11], have shown the validity of
the underlying AUS ideas, a formal proof of the estimation error convergence
for nonlinear systems is still missing to the best of the authors’ knowledge.

This work presents a detectability criterion and an observer design strategy
for systems of the form (1). It is a significant extension of the ideas
presented in [55, 57] and relies on numerically stable algorithms to compute
the dichotomy spectrum of dynamical systems [20]. The spectral intervals of
the dichotomy spectrum are associated with the manifolds comprising solutions
with a common exponential growth rate [52]. For linear time invariant systems,
the proposed detectability definitions and conditions reduce to well known
criteria [53]. Based on ideas similar to AUS strategies, the observer feedback
gain is computed only on a reduced order state space, which corresponds to the
unstable and neutral tangent space of the observer trajectory. Compared to the
deterministic extended Kalman-Bucy filter, this results in a numerically
efficient observer for systems with a large system order. The existence
conditions for the observer are shown to be fulfilled for a class of uniformly
observable nonlinear systems. Moreover, the present work is a step towards
closing the gap between theory and applications of AUS algorithms, because it
provides a convergence proof of the resulting observer in a deterministic
continuous time framework.

The paper is structured as follows: Section 2 introduces required tools for
the stability analysis of linear time varying and nonlinear systems. Section 3
summarizes the Lyapunov and exponential dichotomy spectrum, which are crucial
ideas for the detectability condition and observer design presented in Section
4 for linear time varying systems. In Section 5, the results obtained in
Section 4 are then extended to nonlinear systems of the form (1) following the
ideas of the (deterministic) extended Kalman-Bucy filter. Detailed numerical
investigations of the proposed observer algorithm are presented in Section 6
for the chaotic nonlinear Lorenz’96 model [40]. Together with the conclusion,
further research directions are pointed out in Section 7. It should be
remarked that the results presented in Section 4 extend the authors’ original
research results [57, 55]. Material already published in [56] is partially
included in Section 4 in order to make the paper self-contained.

_Notation:_ Matrices are printed in boldface capital letters, whereas column
vectors are boldface lower case letters. The matrix $\mathbf{I}_{n}$ is the
$n\times n$ identity matrix. Moreover,
$\mathbf{M}=\operatorname{diag}\left({\mathbf{M}_{1},\ldots,\mathbf{M}_{j}}\right)$
denotes a (block) diagonal matrix with entries $\mathbf{M}_{1}$, $\ldots$,
$\mathbf{M}_{j}$. The 2-norm of a vector or the corresponding induced matrix
norm is denoted by $\|\cdot\|$. Symmetric positive definite (positive
semidefinite) matrices $\mathbf{M}^{\mathsf{T}}=\mathbf{M}$ are denoted by
$\mathbf{M}\succ 0$ ($\mathbf{M}\succeq 0$). If for two symmetric matrices
$\mathbf{M}_{1}$, $\mathbf{M}_{2}$ it holds that
$\mathbf{M}_{2}-\mathbf{M}_{1}\succ 0$ ($\succeq 0$), then
$\mathbf{M}_{1}\prec\mathbf{M}_{2}$ ($\mathbf{M}_{1}\preceq\mathbf{M}_{2}$).
In dynamical systems, differentiation of a vector $\mathbf{x}$ with respect to
time $t$ is expressed as $\dot{\mathbf{x}}$. When writing such systems, time
dependence of state (usually $\mathbf{x}$) and output (usually $\mathbf{y}$)
is suppressed and only time dependence of the system’s parameters is stated
explicitly. Moreover, $\mathds{R}$ and $\mathds{R}_{\geq 0}$ denote the reals
and non-negative reals, respectively.

##  2 Preliminaries

This section briefly summarizes stability results for dynamical systems.
First, stability properties of solutions of differential equations

(3) $\dot{\mathbf{x}}=\mathbf{f}(t,\mathbf{x})$

with a continuous function $\mathbf{f}:\mathds{R}_{\geq
0}\times\mathds{R}^{n}\mapsto\mathds{R}^{n}$ are considered for times
$t\in{\mathds{R}_{\geq 0}}$. The initial state at the initial time
$t_{0}\in{\mathds{R}_{\geq 0}}$ is denoted by
$\mathbf{x}(t_{0})=\mathbf{x}_{0}$. It is assumed that the solution of (3)
exists for all $t\geq t_{0}$.

For the stability assessment, one is often interested in the behavior of
solutions starting near an equilibrium $\mathbf{x}_{r}$ fulfilling
$\mathbf{0}=\mathbf{f}(t,\mathbf{x}_{r})$. It is assumed that
$\mathbf{x}_{r}=\mathbf{0}$ is an equilibrium111For assessing stability of any
other solution $\mathbf{x}_{r}(t)$, consider new coordinates
$\mathbf{z}(t)=\mathbf{x}(t)-\mathbf{x}_{r}(t)$. of (3) in the following.

Details on different stability concepts can be found, e.g., in [30, 33]. The
notions used in this paper are introduced in the following

######  Definition 2.1 (stability notions).

The equilibrium $\mathbf{x}_{r}=\mathbf{0}$ of (3), is called

  1. (i)

(locally) exponentially stable, if for some real $\mu>0$ and every
$t_{0}\in\mathds{J}$ there exist scalars $\rho(t_{0})>0$ and $K(t_{0})\geq 1$
such that for every $\mathbf{x}_{0}$ with $\|\mathbf{x}_{0}\|\leq\rho(t_{0})$
one has

(4) $\|\mathbf{x}(t)\|\leq
K(t_{0})e^{-\mu(t-t_{0})}\|\mathbf{x}(t_{0})\|\;\;\text{ for all $t\geq
t_{0}$;}$

  2. (ii)

(locally) uniformly exponentially stable, if $K$, $\rho$ in (ii) are
independent of $t_{0}$;

  3. (iii)

globally exponentially stable or globally uniformly exponentially stable, if 4
or (ii) is fulfilled for all $\mathbf{x}_{0}\in\mathds{R}^{n}$, respectively.

###  2.1 Stability of Linear Time-Varying Systems

In the following, the linear time-varying autonomous system

(5) $\dot{\mathbf{x}}=\mathbf{A}(t)\mathbf{x}$

with $\mathbf{x}(t)\in\mathds{R}^{n}$ is considered for $t\in{\mathds{R}_{\geq
0}}$. It is assumed that $\mathbf{A}:{\mathds{R}_{\geq
0}}\rightarrow\mathds{R}^{n\times n}$ is continuous and bounded. The following
statements can be found in standard textbooks, see, e.g., [33]. System (5) has
the unique solution $\mathbf{x}(t)=\mathbf{\Phi}(t,t_{0})\mathbf{x}_{0},$
where $\mathbf{\Phi}:{\mathds{R}_{\geq 0}}\times{\mathds{R}_{\geq
0}}\rightarrow\mathds{R}^{n\times n}$ is the state transition matrix,
$\mathbf{x}(t_{0})=\mathbf{x}_{0}\in\mathds{R}^{n}$ is the initial state and
$t_{0}\in{\mathds{R}_{\geq 0}}$ is the considered initial time. The state
transition matrix can be obtained from the associated fundamental matrix
differential equation

(6)
$\dot{\mathbf{X}}=\mathbf{A}(t)\mathbf{X},\quad\mathbf{X}(t)\in\mathds{R}^{n\times
n}.$

For any solution of (6) with $\mathbf{X}(0)=\mathbf{X}_{0}$ as a non-singular
matrix, the state transition matrix is given by
$\mathbf{\Phi}(t,t_{0})=\mathbf{X}(t)\mathbf{X}^{-1}(t_{0}).$

For a linear system (5), all equilibria possess identical stability
properties, which are entirely characterized by the state transition matrix
$\mathbf{\Phi}$ according to:

######  Lemma 2.2 (stability criteria for linear systems).

System (5) is

  1. (i)

globally exponentially stable, if and only if there exists a constant $\mu>0$
such that for every $t_{0}\in{\mathds{R}_{\geq 0}}$ there exists a scalar
$K(t_{0})\geq 1$ such that

(7) $\|\mathbf{\Phi}(t,t_{0})\|\leq K(t_{0})e^{-\mu(t-t_{0})}\text{ for all
}t\geq t_{0}$

  2. (ii)

globally uniformly exponentially stable, if and only if $K$ in (ii) is
independent of $t_{0}$;

These relations are well known in the literature, see, e.g. [30, Chapter
VIII].

A state transformation, which preserves the stability properties of the system
is called a Lyapunov transformation. It is a smooth and invertible linear
change of coordinates $\mathbf{z}(t)=\mathbf{T}(t)\mathbf{x}(t)$, where
$\mathbf{T}(t)$, $\mathbf{T}^{-1}(t)$ and $\dot{\mathbf{T}}(t)$ are uniformly
bounded for all $t\in{\mathds{R}_{\geq 0}}$, see [2]. The following results
presented in [33, Chapter 3.3.4] allow to construct a Lyapunov function for
uniformly exponentially stable systems.

######  Proposition 2.3 (Lyapunov function for linear time-varying systems).

Supposed that (5) is uniformly exponentially stable, then:

  1. (i)

There exists a unique bounded solution $\mathbf{P}:{\mathds{R}_{\geq
0}}\rightarrow\mathds{R}^{n\times n}$ of the matrix differential equation

(8)
$\dot{\mathbf{P}}+\mathbf{A}^{\mathsf{T}}(t)\mathbf{P}+\mathbf{P}\mathbf{A}(t)+\mathbf{Q}(t)=\mathbf{0}$

for any $\mathbf{Q}(t)$ with $q_{1}\mathbf{I}_{n}\preceq\mathbf{Q}(t)\preceq
q_{2}\mathbf{I}_{n}$ and any positive constants $q_{1}$, $q_{2}$.

  2. (ii)

The only bounded $\mathbf{P}$ which solves (8) is given by

(9)
$\mathbf{P}(t)=\int_{t}^{\infty}{{\mathbf{\Phi}^{\mathsf{T}}(s,t)\mathbf{Q}(t)\mathbf{\Phi}(s,t)}\,\mathrm{d}{s}}\text{
with }t\in{\mathds{R}_{\geq 0}},$

and there exist positive constants $p_{1},\,p_{2}$ such that
$p_{1}\mathbf{I}_{n}\preceq\mathbf{P}(t)\preceq p_{2}\mathbf{I}_{n}$.

  3. (iii)

$V:{\mathds{R}_{\geq 0}}\times\mathds{R}^{n}\rightarrow{\mathds{R}_{\geq 0}}$
defined as $V(t,\mathbf{x})=\mathbf{x}^{\mathsf{T}}\mathbf{P}(t)\mathbf{x}$ is
a Lyapunov function for (5), whose time derivative $\dot{V}=\frac{\partial
V}{\partial t}+\frac{\partial V}{\partial x}\mathbf{A}(t)\mathbf{x}$ along the
trajectory of (5) satisfies

(10) $\dot{V}(t,\mathbf{x})=-\mathbf{x}^{\mathsf{T}}\mathbf{Q}(t)\mathbf{x}.$

  4. (iv)

The constant $p_{2}$ can be bounded according to
$p_{2}\leq\frac{K^{2}q_{2}}{2\mu}$, where $K$ and $\mu$ are obtained from the
bound on the state transition matrix (7).

Items (i) and (ii) follow from [33, Lemma 3.3.36 and Theorem 3.3.38]. Item
(iii) follows from [33, Theorem 3.3.33 and eqs. (55)–(56)] and item (iv) is
given in [33, eq. (62)]. For time-invariant systems, the uniform and non-
uniform stability notions coincide. Moreover, an eigenvalue spectrum
$\sigma(\mathbf{A})$ with negative real parts is equivalent to uniform
exponential stability. In general, the (time-dependent) eigenvalues of a time-
varying coefficient matrix do not allow to reason about the stability
properties [33, p. 257]. In the following, two concepts for the generalization
of eigenvalues to the time-varying setting will be discussed together with the
numerical computation.

##  3 Two Important Spectra for Linear Time-Varying Systems

The Lyapunov spectrum and the exponential dichotomy spectrum are two important
generalizations of the eigenvalue spectrum to the time-varying case. A
negative Lyapunov spectrum guarantees exponential stability [6, p. 6], whereas
a negative exponential dichotomy spectrum guarantees uniform exponential
stability. Both spectra are introduced in the following based on [23].

###  3.1 Lyapunov-Spectrum

Let the real functional

(11) $\displaystyle\lambda^{\mathrm{s}}(\mathbf{x})$
$\displaystyle=\limsup_{t\rightarrow\infty}\frac{1}{t}\ln\|\mathbf{x}(t)\|$

be defined for non-zero functions $\mathbf{x}:{\mathds{R}_{\geq
0}}\rightarrow\mathds{R}^{n}$. This functional provides an asymptotic upper
bound of the exponential growth or decay for a given non-trivial solution
$\mathbf{x}$ of (5). Note that this bound does not depend on $\mathbf{x}$ on a
finite time interval and hence it can be assumed that $t_{0}=0$.

The functional (11) is now applied to the solutions
$\mathbf{x}_{i}(t)=\mathbf{X}(t)\mathbf{e}_{i}$ yielding
${\ell}_{i}=\lambda^{\mathrm{s}}(\mathbf{x}_{i})$, where
$\mathbf{X}:{\mathds{R}_{\geq 0}}\rightarrow\mathds{R}^{n\times n}$ is an
arbitrary fundamental solution satisfying (6) and $\mathbf{e}_{i}$ the $i$-th
standard basis vector. Minimizing $\sum_{i=1}^{n}{\ell}_{i}$ over all possible
fundamental solutions $\mathbf{X}$ results in the so-called _upper_ Lyapunov
exponents $\lambda^{\mathrm{s}}_{i}\coloneqq{\ell}_{i}$ of system (5), see
[23]. Without loss of generality, the Lyapunov-Exponents are assumed to be
ordered222The ordering can always be achieved by a column permutation of
$\mathbf{X}(t)$. according to
$\lambda^{\mathrm{s}}_{1}\geq\lambda^{\mathrm{s}}_{2}\geq\ldots\geq\lambda^{\mathrm{s}}_{n}$.
The corresponding fundamental solution is then called an _ordered normal
Lyapunov basis_ and the $n$ Lyapunov exponents are unique. In the case of
identical Lyapunov exponents, their multiplicity is determined by the
dimension of the solution space with the same exponent.

The growth rate of all trajectories of (5) can be bounded based on the largest
upper Lyapunov-Exponent $\lambda^{\mathrm{s}}_{1}$. More specifically, for all
$\varepsilon>0$, there exists a $K_{\varepsilon}\geq 1$ such that
$\|\mathbf{\Phi}(t,0)\|\leq
K_{\varepsilon}e^{(\lambda^{\mathrm{s}}_{1}+\varepsilon)t}$, see [6, p. 6].
Hence, $\lambda^{\mathrm{s}}_{1}<0$ implies exponential stability, because for
a fixed $t_{0}$ and a sufficiently small $\varepsilon>0$, the scalars in (7)
can be chosen according to $\mu=-(\lambda^{\mathrm{s}}_{1}+\varepsilon)>0$ and
$K(t_{0})=K_{\varepsilon}e^{(\lambda^{\mathrm{s}}_{1}+\varepsilon)t_{0}}\|\mathbf{\Phi}(0,t_{0})\|$.

In general, there also exists an asymptotic lower bound on the growth rate of
a trajectory, which does not necessarily coincide with the corresponding upper
Lyapunov exponent, see, e.g. the example given in [47]. These lower bounds can
be obtained by considering the adjoint system

(12) $\dot{\bm{\chi}}=-\mathbf{A}^{\mathsf{T}}(t)\bm{\chi}.$

Denoting the sorted upper Lyapunov exponents of (12) by
$-\mu^{\mathrm{s}}_{i}$ with
$-\mu^{\mathrm{s}}_{1}\leq-\mu^{\mathrm{s}}_{2}\leq\cdots\leq-\mu^{\mathrm{s}}_{n}$
allows to introduce the _lower_ Lyapunov exponents
$\lambda^{\mathrm{i}}_{i}=\mu^{\mathrm{s}}_{i},\quad i=1,\ldots,n$ for system
(5). In general, it holds that
$\lambda^{\mathrm{i}}_{i}\leq\lambda^{\mathrm{s}}_{i}$ [23, 41]. In [22], a
spectrum based on the lower and upper Lyapunov exponents is proposed according
to the following

######  Definition 3.1 (Lyapunov spectrum).

The Lyapunov spectrum $\Sigma_{L}$ of (5) is
$\Sigma_{L}=\bigcup_{i=1}^{n}[\lambda^{\mathrm{i}}_{i},\lambda^{\mathrm{s}}_{i}].$

For periodic systems and, in particular, time-invariant systems, the Lyapunov
spectrum reduces to isolated points and it holds that
$\lambda^{\mathrm{i}}_{i}=\lambda^{\mathrm{s}}_{i}$. Moreover, the Lyapunov
exponents coincide with the real parts of the Floquet exponents for periodic
systems or the eigenvalues for time-invariant systems, see [30, Theorem 63.4].

The larger class of regular systems involves all linear time-varying systems,
where the Lyapunov spectrum is a set of isolated points. The concept of
regularity was introduced by Lyapunov [41] and is often demanded for the
numerical computation of the Lyapunov exponents.

######  Definition 3.2 (regularity, [30, Def. 64.1]).

System (5) is called regular, if
$\lambda^{\mathrm{i}}_{i}=\lambda^{\mathrm{s}}_{i}$ for all $i=1,2,\ldots,n$.
In this case, set
$\lambda_{i}=\lambda^{\mathrm{i}}_{i}=\lambda^{\mathrm{s}}_{i}$.

Regularity is hard to verify in practice for a specific system which motivates
to employ the exponential dichotomy spectrum as introduced in the following.

###  3.2 Exponential Dichotomy Spectrum

An important idea for the stability analysis of linear time-varying systems is
the exponential dichotomy introduced by O. Perron as a generalization of
hyperbolicity to the time-varying case, see [52], [19, Ch. 4, §3] or [18].

######  Definition 3.3 (exponential dichotomy).

A system $\dot{\mathbf{x}}=\mathbf{A}(t)\mathbf{x}$ possesses an exponential
dichotomy, if there exists a projection matrix333A projection matrix is a
matrix satisfying $\mathbf{P}^{2}=\mathbf{P}$.
$\mathbf{P}\in\mathds{R}^{n\times n}$, a fundamental solution $\mathbf{X}$ and
real constants $K\geq 1$ and $\alpha>0$ such that

(13a) $\displaystyle\|\mathbf{X}(t)\mathbf{P}\mathbf{X}^{-1}(t_{0})\|\leq
Ke^{-\alpha(t-t_{0})}\text{ for }t\geq t_{0}\geq 0\text{ and }$ (13b)
$\displaystyle\|\mathbf{X}(t)(\mathbf{I}_{n}-\mathbf{P})\mathbf{X}^{-1}(t_{0})\|\leq
Ke^{\alpha(t-t_{0})}\text{ for }0\leq t\leq t_{0}.$

It follows directly, that system (5) is uniformly exponentially stable, if and
only if it has an exponential dichotomy with $\mathbf{P}=\mathbf{I}_{n}$. In
the time varying case, the system has an exponential dichotomy if and only if
it has no eigenvalues on the imaginary axis. A spectrum based on the
exponential dichotomy is introduced in [51, 52] as follows.

######  Definition 3.4 (dichotomy spectrum, [52, Definition 3.1]).

The dichotomy spectrum $\Sigma_{\mathrm{ED}}$ of (5) is the set of all
$\mu\in\mathds{R}$ for which the systems
$\dot{\bm{\xi}}=\left[\mathbf{A}(t)-\mu\mathbf{I}_{n}\right]\bm{\xi}$ do _not_
have an exponential dichotomy.

If (5) is time-invariant, then $\Sigma_{\mathrm{ED}}$ is equal to the real
part of the eigenvalue spectrum $\sigma(\mathbf{A})$. For systems (5) with a
bounded $\mathbf{A}(t)$, the dichotomy spectrum consists of $1\leq d\leq n$
compact and disjoint subintervals [52].

######  Remark 3.5.

For a one-dimensional system $\dot{x}=a(t)x$, the so-called lower and upper
Bohl exponents are defined according to

(14)
$\displaystyle\beta^{\mathrm{i}}_{1}=\liminf_{\begin{subarray}{c}t-t_{0}\rightarrow\infty\\\
t_{0}\rightarrow\infty\end{subarray}}\frac{1}{t}\int_{t_{0}}^{t_{0}+t}{{a(\tau)}\,\mathrm{d}{\tau}}\text{
and
}\;\beta^{\mathrm{s}}_{1}=\limsup_{\begin{subarray}{c}t-t_{0}\rightarrow\infty\\\
t_{0}\rightarrow\infty\end{subarray}}\frac{1}{t}\int_{t_{0}}^{t_{0}+t}{{a(\tau)}\,\mathrm{d}{\tau}},$

respectively. The one-dimensional system then has an exponential dichotomy, if
and only if $0<\beta^{\mathrm{i}}_{1}\leq\beta^{\mathrm{s}}_{1}$ or
$\beta^{\mathrm{i}}_{1}\leq\beta^{\mathrm{s}}_{1}<0$, see [19]. Its dichotomy
spectrum is hence given by
$\Sigma_{\mathrm{ED}}=[\beta^{\mathrm{i}}_{1},\beta^{\mathrm{s}}_{1}]$.
Moreover, it holds that
$\beta^{\mathrm{i}}_{1}\leq\lambda^{\mathrm{i}}_{1}\leq\lambda^{\mathrm{s}}_{1}\leq\beta^{\mathrm{s}}_{1}$
and therefore $\Sigma_{\mathrm{L}}\subseteq\Sigma_{\mathrm{ED}}$.

###  3.3 Numerical Approximation

For simplicity, it is assumed that system (5) is regular. The non-regular case
is treated in [23]. In a first step, system (5) is transformed to an upper
triangular form using a Lyapunov transformation. For systems with a real
coefficient matrix $\mathbf{A}(t)$, [2, Theorem 3.3.1 and Remark 3.3.2] state
that this change of coordinates
$\mathbf{R}(t)=\mathbf{Q}^{\mathsf{T}}(t)\mathbf{X}(t)$ can be achieved using
an orthogonal Lyapunov transformation matrix $\mathbf{Q}:{\mathds{R}_{\geq
0}}\rightarrow\mathds{R}^{n\times n}$. This transformation can be obtained
numerically by means of a continuous QR decomposition [23]. In this case,
$\mathbf{Q}$ and $\mathbf{R}$ are the solutions of

(15a) $\displaystyle\dot{\mathbf{R}}$
$\displaystyle=\mathbf{B}(t)\mathbf{R},\;\quad\mathbf{B}(t)=\mathbf{Q}^{\mathsf{T}}(t)\mathbf{A}(t)\mathbf{Q}(t)-\mathbf{S}(t)$
(15b) $\displaystyle\dot{\mathbf{Q}}$
$\displaystyle=\mathbf{Q}\mathbf{S}(t).\;$

The coefficients $s_{ij}(t)$ of the skew symmetric $n\times n$ matrix
$\mathbf{S}(t)$ are given by
$s_{ij}(t)=\mathbf{q}_{i}^{\mathsf{T}}(t)\mathbf{A}(t)\mathbf{q}_{j}(t)$ for
$i>j$, where $\mathbf{q}_{i}(t)$ denotes the $i$-th column of $\mathbf{Q}(t)$.

In a first step, it is assumed that the initial conditions are obtained by a
QR decomposition of an ordered normal Lyapunov basis444The numerical
implementation is discussed in Section 3.4., i.e.,
$\mathbf{X}_{0}=\mathbf{Q}_{0}\mathbf{R}_{0}$. Under the assumption that all
diagonal elements of $\mathbf{R}_{0}$ are non-negative, this decomposition is
unique. For regular systems (5) transformed to an upper triangular form
$\dot{\mathbf{z}}=\mathbf{B}(t)\mathbf{z}$ with
$\mathbf{z}(t)=\mathbf{Q}^{\mathsf{T}}(t)\mathbf{x}(t)$, the Lyapunov
exponents can be obtained by time-averaging of the diagonal elements according
to

(16)
$\lambda_{i}=\lim_{t\rightarrow\infty}\frac{1}{t}\int_{0}^{t}{{b_{ii}(\tau)}\,\mathrm{d}{\tau}}=\lim_{t\rightarrow\infty}\frac{1}{t}\int_{0}^{t}{{\mathbf{q}_{i}^{\mathsf{T}}(\tau)\mathbf{A}({\tau})\mathbf{q}_{i}(\tau)}\,\mathrm{d}{\tau}},$

see [41]. To obtain the Lyapunov exponents, merely solving the orthogonal
differential equation (15b) and not the (possibly unbounded) solution of (15a)
is required.

Independent of the system’s regularity, the exponential dichotomy spectrum can
always be obtained from the diagonal elements of $\mathbf{B}$, see [24]. To
that end, let the two real functionals

(17a)
$\beta^{\mathrm{i}}(b)=\liminf_{\begin{subarray}{c}t-t_{0}\rightarrow\infty\\\
t_{0}\rightarrow\infty\end{subarray}}\frac{1}{t}\int_{t_{0}}^{t_{0}+t}{{b(\tau)}\,\mathrm{d}{\tau}}\quad\text{
and
}\quad\beta^{\mathrm{s}}(b)=\limsup_{\begin{subarray}{c}t-t_{0}\rightarrow\infty\\\
t_{0}\rightarrow\infty\end{subarray}}\frac{1}{t}\int_{t_{0}}^{t_{0}+t}{{b(\tau)}\,\mathrm{d}{\tau}}$

be introduced for a scalar function $b:{\mathds{R}_{\geq
0}}\rightarrow\mathds{R}$. Applied to the diagonal entries of $\mathbf{B}$,
these functionals provide the lower and upper Bohl exponents
$\beta^{\mathrm{i}}_{i}=\beta^{\mathrm{i}}(b_{ii})$ and
$\beta^{\mathrm{s}}_{i}=\beta^{\mathrm{s}}(b_{ii})$, respectively, of the
scalar systems $\dot{\eta}_{i}=b_{ii}(t)\eta_{i}$ for $i=1,2,\ldots,n.$
According to [24, Proposition 5], the dichotomy spectrum is then obtained as

(18)
$\Sigma_{\mathrm{ED}}=\bigcup_{i=1}^{n}\Lambda_{i}\quad\text{with}\quad\Lambda_{i}=\left[\beta^{\mathrm{i}}_{i},\beta^{\mathrm{s}}_{i}\right].$

Note that the $n$ intervals $\Lambda_{i}$ are not necessarily disjoint and the
obtained Bohl exponents are not necessarily sorted. In many cases, however, a
sorting similar to the Lyapunov exponents can be expected. From the relations
in (16) and (17) it follows that
$\beta^{\mathrm{i}}_{i}\leq\lambda_{i}\leq\beta^{\mathrm{s}}_{i}$ and
$\Sigma_{\mathrm{L}}\subseteq\Sigma_{\mathrm{ED}}$, i.e., the Lyapunov
spectrum is contained in the exponential dichotomy spectrum. This is also true
in the non-regular case [23]. For periodic and time-invariant systems, both
spectra coincide, i.e., $\Sigma_{\mathrm{L}}=\Sigma_{\mathrm{ED}}$. In the
general time-varying case, however, this is not true as demonstrated in [24].

The numerical approximation of the Bohl exponents turns out to be more
difficult than the approximation of (regular) Lyapunov exponents, because
$t_{0}$ and $t-t_{0}$ tend to infinity in (17). For the approximation of the
Bohl exponents, let

(19) $\beta^{H,\mathrm{i}}_{i}=\inf_{t_{0}\in{\mathds{R}_{\geq
0}}}\frac{1}{H}\int_{t_{0}}^{t_{0}+H}{{b_{ii}(\tau)}\,\mathrm{d}{\tau}}\quad\text{
and }\quad\beta^{H,\mathrm{s}}_{i}=\sup_{t_{0}\in{\mathds{R}_{\geq
0}}}\frac{1}{H}\int_{t_{0}}^{t_{0}+H}{{b_{ii}(\tau)}\,\mathrm{d}{\tau}}$

be introduced with a scalar parameter $H>0$. It is stated in [22, Theorem 8.4]
that for any $H>0$,
$\Lambda_{i}=[\beta^{\mathrm{i}}_{i},\beta^{\mathrm{s}}_{i}]\subseteq[\beta^{H,\mathrm{i}}_{i},\beta^{H,\mathrm{s}}_{i}]$.
It is furthermore claimed that for $H>0$ sufficiently large it holds that
$[\beta^{H,\mathrm{i}}_{i},\beta^{H,\mathrm{s}}_{i}]\subseteq[\beta^{\mathrm{i}}_{i},\beta^{\mathrm{s}}_{i}]$
and hence $\Lambda_{i}=[\beta^{H,\mathrm{i}}_{i},\beta^{H,\mathrm{s}}_{i}]$.
The latter statement is not true as shown by the following counterexample.

######  Example 3.6.

Consider the scalar system $\dot{x}=\frac{1}{1+t}x$ which admits the
fundamental solution $X(t)=(1+t)$. The exponential dichotomy spectrum is
$\Sigma_{\mathrm{ED}}=\\{0\\}$ and hence
$\beta^{\mathrm{i}}_{1}=\beta^{\mathrm{s}}_{1}=0$. The computation of
$\beta^{H,\mathrm{s}}$ gives

(20)
$\beta^{H,\mathrm{s}}_{1}=\sup_{t}\frac{1}{H}\int_{t}^{t+H}{{\frac{1}{1+\tau}}\,\mathrm{d}{\tau}}=\sup_{t}\frac{1}{H}\ln(1+\frac{H}{1+t})=\frac{1}{H}\ln(1+H)$

with a supremum at $t=0$. Hence, for any finite $H>0$ it holds that
$\beta^{H,\mathrm{s}}_{1}>0=\beta^{\mathrm{s}}_{1}$.

The correct relation between the approximation (19) and the bounds of the
spectral intervals (17) is summarized in the following

######  Theorem 3.7 (approximation of the spectral intervals).

For every $H>0$, it holds that

(21)
$\beta^{H,\mathrm{i}}_{i}\leq\beta^{\mathrm{i}}_{i}\leq\beta^{\mathrm{s}}_{i}\leq\beta^{H,\mathrm{s}}_{i}.$

Moreover, for every $\varepsilon>0$, there exists a $H_{0}>0$ such that for
all $H\geq H_{0}$ it holds that

(22) $\beta^{\mathrm{i}}_{i}-\varepsilon\leq\beta^{H,\mathrm{i}}_{i}\text{ and
}\beta^{H,\mathrm{s}}_{i}\leq\beta^{\mathrm{s}}_{i}+\varepsilon.$

The proof is given in the appendix. Averaging the $b_{ii}$, $i=1,\ldots,n$,
over a sufficiently large time interval $H$ allows to approximate the spectral
intervals theoretically with arbitrary accuracy. In particular, Theorem 3.7
implies that in the limit, it holds that
$\beta^{\mathrm{s}}_{i}=\lim_{H\rightarrow\infty}\sup_{t_{0}}\frac{1}{H}\int_{t_{0}}^{t_{0}+H}{{b_{ii}(\tau)}\,\mathrm{d}{\tau}}.$
Detailed considerations and numerical examples for the approximation of the
Lyapunov exponents (also for non-regular systems) and the exponential
dichotomy spectral intervals can be found in [23].

###  3.4 Reduced QR Decomposition

If not all spectral intervals are of interest, it may suffice to compute only
a part of the spectrum. This allows a reduction of the computational
complexity by utilizing properties of the QR-decomposition [57]. For the
reduced QR decomposition, only the first $k\leq n$ columns of an ordered
normal Lyapunov basis $\mathbf{X}$ are considered. This results in the
differential equations for the reduced QR decomposition

(23a) $\displaystyle\dot{\mathbf{R}}_{1}$
$\displaystyle=\mathbf{B}_{1}(t)\mathbf{R}_{1},\quad\mathbf{R}_{1}(0)\in\mathds{R}^{k\times
k},$ (23b) $\displaystyle\dot{\bar{\mathbf{Q}}}$
$\displaystyle=(\mathbf{I}_{n}-\bar{\mathbf{Q}}\bar{\mathbf{Q}}^{\mathsf{T}})\mathbf{A}(t)\bar{\mathbf{Q}}+\bar{\mathbf{Q}}\mathbf{S}_{1}(t),\quad{\bar{\mathbf{Q}}(0)\in\mathds{R}^{n\times
k},\quad\text{with}}$ (23c) $\displaystyle\mathbf{B}_{1}(t)$
$\displaystyle=\bar{\mathbf{Q}}^{\mathsf{T}}(t)\mathbf{A}(t)\bar{\mathbf{Q}}(t)-\mathbf{S}_{1}(t).$
The elements $s_{ij}$ of the skew-symmetric $k\times k$-Matrix
$\mathbf{S}_{1}$ are given by
$s_{ij}(t)=\bar{\mathbf{q}}_{i}^{\mathsf{T}}(t)\mathbf{A}(t)\bar{\mathbf{q}}_{j}(t)$
for $i>j$ [26, 57].

Again, only the solution of (23b) is required for the approximation of $k$
spectral intervals via the diagonal entries of $\mathbf{B}_{1}$. For this, it
is crucial to preserve orthogonality of the columns of $\bar{\mathbf{Q}}$.
This can be achieved by a projected integration algorithm, which is a standard
integration scheme (e.g., a fourth-order Runge-Kutta algorithm) in combination
with an orthogonalization procedure based, e.g., on the modified Gram-Schmidt
algorithm as presented in [21].

If the initial condition
$\bar{\mathbf{X}}(0)=\bar{\mathbf{Q}}(0)\mathbf{R}_{1}(0)$ is chosen as the
first $k$ columns of an ordered normal Lyapunov basis, the $k$ largest
Lyapunov exponents can be obtained from the diagonal of $\mathbf{B}_{1}$.
Typically, such a basis is not known a priory and determining such a basis may
be cumbersome [20, 23]. Hence, it is proposed in [20] to choose
$\bar{\mathbf{Q}}(0)$ as random orthogonal matrix in practice. This choice
proved to be successful in many numerical investigations [26, 57, 54] and is
theoretically supported by [7].

##  4 Uniform Detectability of Linear Time-Varying Systems

This section deals with the existence of an observer for systems of the form
(5), i.e.,

(24) $\displaystyle\dot{\mathbf{x}}$
$\displaystyle=\mathbf{A}(t)\mathbf{x},\quad\mathbf{y}=\mathbf{C}(t)\mathbf{x}$

with the output $\mathbf{y}(t)\in\mathds{R}^{p}$. The output matrix function
$\mathbf{C}:{\mathds{R}_{\geq 0}}\rightarrow\mathds{R}^{p\times n}$ is assumed
to be continuous and bounded. The goal is to design an observer

(25)
$\dot{\hat{\mathbf{x}}}=\mathbf{A}(t)\hat{\mathbf{x}}+\mathbf{L}(t)\left[\mathbf{y}-\mathbf{C}(t)\hat{\mathbf{x}}\right],$

where $\mathbf{L}:{\mathds{R}_{\geq 0}}\rightarrow\mathds{R}^{n\times p}$ is a
design parameter. The estimation error
$\mathbf{e}(t)=\mathbf{x}(t)-\hat{\mathbf{x}}(t)$ is governed by

(26)
$\dot{\mathbf{e}}=\left[\mathbf{A}(t)-\mathbf{L}(t)\mathbf{C}(t)\right]\mathbf{e}.$

The following detectability definition introduced in [60, 48], is strongly
related to the stability properties of (26).

######  Definition 4.1.

System (24) is called uniformly exponentially detectable, if there exists a
bounded $\mathbf{L}:{\mathds{R}_{\geq 0}}\rightarrow\mathds{R}^{n\times p}$
such that the estimation error dynamics (26) is uniformly exponentially
stable.

In other words, uniform exponential detectability or short _detectability_ is
equivalent to the existence of an observer with a uniformly exponentially
stable estimation error dynamics. A stronger concept which also guarantees the
existence of an observer is uniform complete observability. It is
characterized by the constructibility Gramian

(27)
$\mathbf{N}(t_{1},t_{0})=\int_{t_{0}}^{t_{1}}{{\mathbf{\Phi}^{\mathsf{T}}(s,t_{1})\mathbf{C}^{\mathsf{T}}(s)\mathbf{C}(s)\mathbf{\Phi}(s,t_{1})}\,\mathrm{d}{s}}$

according to

######  Definition 4.2 (uniform complete observability).

System (24), or equivalently the pair $(\mathbf{A}(t),\mathbf{C}(t))$, is
called uniformly completely observable, if there exist positive constants
$\alpha_{1}$, $\alpha_{2}$ and ${T}$, such that
$\alpha_{1}\mathbf{I}_{n}\preceq\mathbf{N}(t_{0}+{T},t_{0})\preceq\alpha_{2}\mathbf{I}_{n}$
holds for all $t_{0}\in{\mathds{R}_{\geq 0}}$.

It is shown in [15] that uniform complete observability implies the existence
of a unique and uniformly bounded positive definite solution of the observer
Riccati equation

(28)
$\dot{\mathbf{P}}=\mathbf{A}(t)\mathbf{P}+\mathbf{P}\mathbf{A}^{\mathsf{T}}(t)-\mathbf{P}\mathbf{C}^{\mathsf{T}}(t)\mathbf{C}(t)\mathbf{P}+\mathbf{G}(t).$

Here, $\mathbf{P}(0)\succ 0$ and $\mathbf{G}(t)=\mathbf{G}^{\mathsf{T}}(t)$
with $g_{1}\mathbf{I}_{n}\preceq\mathbf{G}(t)\preceq g_{2}\mathbf{I}_{n}$ and
$0<g_{1}\leq g_{2}$. Moreover, choosing the feedback gain in (26) as
$\mathbf{L}(t)=\mathbf{P}(t)\mathbf{C}^{\mathsf{T}}(t)$ renders the error
dynamics uniformly exponentially stable [14] and hence, uniform complete
observability is sufficient for uniform exponential detectability.

Uniform complete observability is a strong system requirement and solving the
differential Riccati equation (28) might be computationally demanding,
especially for systems with a large system order. If the system, however,
possesses only a few unstable modes, i.e., only a small number of non-negative
upper Bohl exponents, it is reasonable to modify only these modes in the
estimation error dynamics. This allows to solve the Riccati equation on a
reduced order subspace. This strategy is also pursued in [43, 57] for non-
negative (regular) Lyapunov exponents and is the underlying idea of the
following detectability condition and observer design technique.

######  Theorem 4.3 (uniform detectability).

Consider the linear time-varying system (24). Let
$\mathbf{Q}:{\mathds{R}_{\geq 0}}\rightarrow\mathds{R}^{n\times n}$ be the
solution to (15b) of the continuous QR-decomposition of a corresponding
fundamental solution $\mathbf{X}$ with
$\mathbf{X}(t_{0})=\mathbf{Q}(t_{0})\mathbf{R}(t_{0})$. Let $j^{\star}$ be the
first integer $0\leq j^{\star}\leq n$ such that
$\beta^{\mathrm{s}}(\mathbf{q}_{i}(t)^{\mathsf{T}}\mathbf{A}(t)\mathbf{q}_{i}(t))<0$
for $i=j^{\star}+1,\ldots,n$ and let $\mathbf{Q}(t)$ be partitioned according
to
$\mathbf{Q}(t)=\left[\bar{\mathbf{Q}}(t)\;\;\bar{\mathbf{Q}}_{\perp}(t)\right]$,
where $\bar{\mathbf{Q}}(t)\in\mathds{R}^{n\times j^{\star}}$.

System (24) is detectable, if the pair
$(\mathbf{B}_{1}(t),\mathbf{C}(t)\bar{\mathbf{Q}}(t))$ with $\mathbf{B}_{1}$
as in (23c) and $k=j^{\star}$ is uniformly completely observable. In
particular, there exists a bounded positive definite solution to the
differential Riccati equation

(29)
$\dot{\mathbf{P}}{{}_{1}}=\mathbf{B}_{1}(t)\mathbf{P}{{}_{1}}+\mathbf{P}{{}_{1}}\mathbf{B}_{1}(t)-\mathbf{P}{{}_{1}}\bar{\mathbf{Q}}^{\mathsf{T}}(t)\mathbf{C}^{\mathsf{T}}(t)\mathbf{C}(t)\bar{\mathbf{Q}}(t)+\mathbf{G}{{}_{1}}(t),$

with the $j^{\star}\times j^{\star}$ matrix $\mathbf{P}{{}_{1}}(t_{0})\succ 0$
and positive constants $g_{1}$, $g_{2}$ such that
$g_{1}\mathbf{I}_{j^{\star}}\preceq\mathbf{G}{{}_{1}}(t)\preceq
g_{2}\mathbf{I}_{j^{\star}}$. Moreover, the error system (26) with

(30)
$\mathbf{L}(t)=\bar{\mathbf{Q}}(t)\mathbf{P}{{}_{1}}(t)\bar{\mathbf{Q}}{{}_{1}}^{\mathsf{T}}(t)\mathbf{C}^{\mathsf{T}}(t)$

is uniformly exponentially stable.

######  Proof 4.4.

Based on the Lyapunov transformation
$\mathbf{e}(t)=\mathbf{Q}(t)\mathbf{e}_{z}(t)$, the error system (26) can be
transformed to the block upper triangular form

(31) $\begin{bmatrix}\dot{\mathbf{e}}_{z,1}\\\
\dot{\mathbf{e}}_{z,2}\end{bmatrix}=\begin{bmatrix}\mathbf{B}_{1}(t)&\mathbf{B}_{12}(t)\\\
\bm{0}&\mathbf{B}_{2}(t)\end{bmatrix}\begin{bmatrix}\mathbf{e}_{z,1}\\\
\mathbf{e}_{z,2}\end{bmatrix}-\begin{bmatrix}\mathbf{L}_{1}(t)\\\
\bm{0}\end{bmatrix}\mathbf{C}(t)\begin{bmatrix}\bar{\mathbf{Q}}(t)&\bar{\mathbf{Q}}_{\perp}(t)\end{bmatrix}\begin{bmatrix}\mathbf{e}_{z,1}\\\
\mathbf{e}_{z,2}\end{bmatrix}$

with
$\mathbf{L}_{1}(t)=\mathbf{P}{{}_{1}}(t)\bar{\mathbf{Q}}^{\mathsf{T}}(t)\mathbf{C}^{\mathsf{T}}(t)$.
The subsystem $\dot{\mathbf{e}}_{z,2}=\mathbf{B}_{2}(t)\mathbf{e}_{z,2}$ is
uniformly exponentially stable, because all upper Bohl exponents are negative.
Hence, $\mathbf{e}_{z,2}(t)$ can be seen as a uniformly exponentially
vanishing perturbation in the system

(32)
$\dot{\mathbf{e}}_{z,1}=\left[\mathbf{B}_{1}(t)-\mathbf{L}_{1}\mathbf{C}(t)\bar{\mathbf{Q}}(t)\right]\mathbf{e}_{z,1}+\left[\mathbf{B}_{12}(t)-\mathbf{L}_{1}\mathbf{C}(t)\bar{\mathbf{Q}}_{\perp}(t)\right]\mathbf{e}_{z,2}$

Uniform complete observability of the pair
$(\mathbf{B}_{1}(t),\mathbf{C}(t)\bar{\mathbf{Q}}(t))$ implies that there
exists a uniformly bounded positive definite solution to (29) and also implies
that
$\dot{\mathbf{e}}_{z,1}=\left[\mathbf{B}_{1}(t)-\mathbf{L}_{1}\mathbf{C}(t)\bar{\mathbf{Q}}(t)\right]\mathbf{e}_{z,1}$
is uniformly exponentially stable. Uniform exponential stability of the
overall error system and hence detectability then follows from [61, Theorem
2].

######  Remark 4.5.

For the practical implementation of the observer (25) with the gain (30), it
suffices to solve the differential equation (23b) for some $k\geq j^{\star}$
instead of the (full order) differential equation (15b). This could
drastically reduce the complexity, if $j^{\star}$ is small compared to the
system order $n$. The parameter $k$ determines the number of upper Bohl
exponents, which are modified by the observer feedback gain. If system (24) is
uniformly completely observable, $k$ can be regarded as a tuning parameter and
allows a trade-off between the computational complexity and the convergence
speed of the error system [57]. In particular, $k$ determines how many
spectral intervals of the system are modified in the observer error dynamics.

In general, the detectability condition in Theorem 4.3 is not necessary, even
if $\mathbf{X}_{0}$ is an ordered normal Lyapunov basis, as demonstrated by an
example provided in [56] where the upper Bohl exponents are not ordered. In
the numerical simulations, it turned out that a certain ordering of the upper
Bohl exponents similar to the Lyapunov exponents can be expected as shown in
Section 6. Moreover, if the system has an exponential dichotomy and
$\mathbf{X}_{0}$ is an ordered normal Lyapunov basis, the detectability
condition in Theorem 4.3 turns out to be necessary and sufficient [55].

It is now shown, that the reduced order Riccati equation (29) can be obtained
from a projection of the full order Riccati equation with properly chosen
$\mathbf{P}(t_{0})$ and $\mathbf{G}(t)$. To that end, it is assumed that
system (5) is transformed to an upper triangular form with
$\mathbf{x}=\mathbf{Q}(t)\mathbf{z}$ and partitioned according to
$\mathbf{Q}(t)=\left[\bar{\mathbf{Q}}(t)\;\bar{\mathbf{Q}}_{\perp}(t)\right]$.
The full-order Riccati equation is then given by

(33)
$\displaystyle\begin{bmatrix}\dot{\mathbf{P}}_{1}&\dot{\mathbf{P}}_{12}\\\
\dot{\mathbf{P}}_{12}^{\mathsf{T}}&\dot{\mathbf{P}}_{2}\end{bmatrix}=\begin{bmatrix}\mathbf{B}_{1}&\mathbf{B}_{12}\\\
\bf
0&\mathbf{B}_{2}\end{bmatrix}\underbrace{\begin{bmatrix}\mathbf{P}_{1}&\mathbf{P}_{12}\\\
\mathbf{P}_{12}^{\mathsf{T}}&\mathbf{P}_{2}\end{bmatrix}}_{\mathbf{P}(t)}+\begin{bmatrix}\mathbf{P}_{1}&\mathbf{P}_{12}\\\
\mathbf{P}_{12}^{\mathsf{T}}&\mathbf{P}_{2}\end{bmatrix}\begin{bmatrix}\mathbf{B}^{\mathsf{T}}_{1}&\bf
0\\\ \mathbf{B}^{\mathsf{T}}_{12}&\mathbf{B}^{\mathsf{T}}_{2}\end{bmatrix}$
$\displaystyle-\begin{bmatrix}\mathbf{P}_{1}&\mathbf{P}_{12}\\\
\mathbf{P}_{12}^{\mathsf{T}}&\mathbf{P}_{2}\end{bmatrix}\begin{bmatrix}\bar{\mathbf{Q}}^{\mathsf{T}}\\\
\bar{\mathbf{Q}}_{\perp}^{\mathsf{T}}\end{bmatrix}\mathbf{C}^{\mathsf{T}}\mathbf{C}\begin{bmatrix}\bar{\mathbf{Q}}&\bar{\mathbf{Q}}_{\perp}\end{bmatrix}\begin{bmatrix}\mathbf{P}_{1}&\mathbf{P}_{12}\\\
\mathbf{P}_{12}^{\mathsf{T}}&\mathbf{P}_{2}\end{bmatrix}+\begin{bmatrix}\mathbf{G}_{1}&\mathbf{G}_{12}\\\
\mathbf{G}_{12}^{\mathsf{T}}&\mathbf{G}_{2}\end{bmatrix},$

where the time indices are omitted for the sake of readability.

######  Lemma 4.6 (Reduced Riccati equation).

Let $\mathbf{P}_{1}(t)$ be the solution of the differential Riccati equation
(29) and

(34) $\mathbf{P}(t_{0})=\begin{bmatrix}\mathbf{P}_{1}(t_{0})&\bf 0\\\ \bf
0&\bf 0\end{bmatrix},\;\mathbf{G}_{12}={\bf
0}\;\text{and}\;\;{\mathbf{G}}_{2}=\bf{0}.$

Then, the solution of (33) is given by

(35) $\mathbf{P}(t)=\begin{bmatrix}\mathbf{P}_{1}(t)&\bm{0}\\\
\bm{0}&\bm{0}\end{bmatrix}\quad\text{for all }t\geq t_{0}.$

######  Proof 4.7.

The statement follows by substituting $\mathbf{P}(t)$ into (33) and using (29)
and (34).

##  5 Nonlinear Observer Design

This section extends the observer design approach presented in Section 4 to
nonlinear systems. As the Riccati differential equation is solved on a reduced
state-space only, the ideas for the convergence proof of the extended Kalman-
Bucy filter cannot be applied in the present setting. However, the local
stability result utilizes the Lyapunov function from Proposition 2.3 and
hence, under some assumptions stated subsequently, local uniform exponential
stability of the observer error dynamics can be guaranteed.

In the following, the nonlinear system (1), i.e.,

(36a) $\displaystyle\dot{\mathbf{x}}$
$\displaystyle=\mathbf{f}(\mathbf{x},\mathbf{u}),\,t\in{\mathds{R}_{\geq
0}},\;\mathbf{x}(t_{0})=\mathbf{x}_{0},$ (36b) $\displaystyle\mathbf{y}$
$\displaystyle=\mathbf{C}(t)\mathbf{x}$

is considered with the output equation (36b). Again, it is assumed that the
output matrix function $\mathbf{C}:{\mathds{R}_{\geq
0}}\rightarrow\mathds{R}^{p\times n}$ is bounded. It should be remarked that
the case of a linear measurement equation is used to demonstrate the key idea.
An extension to a nonlinear output map is possible by a straightforward
application of the concepts presented in [50].

###  5.1 Extended Kalman-Bucy Filter as a Deterministic Observer

The idea of the deterministic interpretation of the extended Kalman-Bucy
filter is to obtain the time-varying matrices $\mathbf{A}(t)$ and
$\mathbf{C}(t)$ by a linearization along the estimated trajectory [50]. The
algorithm can be summarized as

(37a) $\displaystyle{\dot{\hat{\mathbf{x}}}}$
$\displaystyle=\mathbf{f}(\hat{\mathbf{x}},\mathbf{u})+\mathbf{P}(t)\mathbf{C}^{\mathsf{T}}(t)\left[\mathbf{y}-\mathbf{C}(t)\hat{\mathbf{x}}\right],\quad\hat{\mathbf{x}}(0)=\hat{\mathbf{x}}_{0},$
(37b) $\displaystyle\dot{\mathbf{P}}$
$\displaystyle=\mathbf{P}\mathbf{A}^{\mathsf{T}}(t)+\mathbf{A}(t)\mathbf{P}-\mathbf{P}\mathbf{C}^{\mathsf{T}}(t)\mathbf{C}(t)\mathbf{P}+\mathbf{G}(t),\quad\mathbf{P}(t)\in\mathds{R}^{n\times
n}$ (37c) $\displaystyle\mathbf{A}(t)$
$\displaystyle=\left.\frac{\partial{\phantom{.}}}{\partial{\mathbf{x}}}\mathbf{f}(\mathbf{x}(t),\mathbf{u}(t))\right|_{(\hat{\mathbf{x}}(t),\mathbf{u}(t))}.$

The initial condition $\mathbf{P}_{0}$ is chosen as a positive definite matrix
and the matrix $\mathbf{G}(t)$ is a positive definite tuning parameter.

Local stability results for the dynamics of the estimation error
$\mathbf{e}(t)=\mathbf{x}(t)-\hat{\mathbf{x}}(t)$ are presented in [50, 39,
12]. A key assumption of these stability proofs is the existence of positive
constants $p_{1},\,p_{2}$ such that
$p_{1}\mathbf{I}_{n}\preceq\mathbf{P}(t)\preceq p_{2}\mathbf{I}_{n}$ holds for
(37b). This assumption can be guaranteed by uniform complete observability of
the observer trajectory, which, however, might not be a trajectory of the
original system in general. This assumption cannot be dropped easily as
extensively discussed in [12, 37].

###  5.2 Extended Subspace Observer

Based on the ideas proposed in Theorem 4.3 and in analogy to the extended
Kalman-Bucy filter, an observer for a class of nonlinear systems is presented
in the following. The proposed algorithm is summarized as

(38a) $\displaystyle\dot{\hat{\mathbf{x}}}$
$\displaystyle=\mathbf{f}(\hat{\mathbf{x}},\mathbf{u})+\mathbf{L}(t)\left[\mathbf{y}-\mathbf{C}(t)\hat{\mathbf{x}}\right],$
(38b) $\displaystyle\dot{\mathbf{P}}{{}_{1}}$
$\displaystyle=\mathbf{B}_{1}(t)\mathbf{P}{{}_{1}}+\mathbf{P}{{}_{1}}\mathbf{B}_{1}^{\mathsf{T}}(t)-\mathbf{P}{{}_{1}}\bar{\mathbf{C}}^{\mathsf{T}}(t)\bar{\mathbf{C}}(t)\mathbf{P}+\mathbf{G}{{}_{1}}(t),\;\quad\mathbf{P}{{}_{1}}(t)\in\mathds{R}^{k\times
k}$ (38c) $\displaystyle\dot{\bar{\mathbf{Q}}}$
$\displaystyle=\left[\mathbf{I}-\bar{\mathbf{Q}}\bar{\mathbf{Q}}^{\mathsf{T}}\right]\mathbf{A}(t)\bar{\mathbf{Q}}+\bar{\mathbf{Q}}\mathbf{S}_{1}(t),\quad\bar{\mathbf{Q}}(t)\in\mathds{R}^{n\times
k},$ with (38d) $\displaystyle\mathbf{L}(t)$
$\displaystyle=\bar{\mathbf{Q}}(t)\mathbf{P}{{}_{1}}(t)\bar{\mathbf{C}}^{\mathsf{T}}(t),\quad\bar{\mathbf{C}}(t)=\mathbf{C}(t)\mathbf{Q}(t),$
(38e) $\displaystyle\mathbf{A}(t)$
$\displaystyle=\left.\frac{\partial{\phantom{.}}}{\partial{\mathbf{x}}}\mathbf{f}(\mathbf{x}(t),\mathbf{u}(t))\right|_{(\hat{\mathbf{x}}(t),\mathbf{u}(t))},\quad\mathbf{B}_{1}(t)=\bar{\mathbf{Q}}^{\mathsf{T}}(t)\mathbf{A}(t)\bar{\mathbf{Q}}(t)-\mathbf{S}_{1}(t).$
The elements $s_{ij}$ of the skew symmetric matrix $\mathbf{S}_{1}$ are given
by
$s_{ij}(t)=\bar{\mathbf{q}}_{i}^{\mathsf{T}}(t)\mathbf{A}(t)\bar{\mathbf{q}}_{j}(t)$
for $i>j$.

The number of columns in $\bar{\mathbf{Q}}(t)$ has to be chosen such that
$j^{\star}\leq k\leq n$ with $j^{\star}$ as given in Theorem 4.3 and
$\mathbf{A}(t)$ as in (38e). In the following, a convergence result for this
observer is provided.

###  5.3 Convergence Analysis

The estimation error $\mathbf{e}(t)=\mathbf{x}(t)-\hat{\mathbf{x}}(t)$ is
governed by

(39)
$\displaystyle\dot{\mathbf{e}}=\mathbf{f}(\mathbf{x},\mathbf{u})-\mathbf{f}(\hat{\mathbf{x}},\mathbf{u})-\mathbf{L}(t)\left[\mathbf{y}-\mathbf{C}(t)\hat{\mathbf{x}}\right].$

By substituting $\mathbf{x}$ with $\mathbf{x}=\hat{\mathbf{x}}+\mathbf{e}$ in
(39) and performing a Taylor series expansion of $\mathbf{f}(\cdot,\cdot)$
around $\mathbf{e}=\mathbf{0}$ one obtains
$\mathbf{f}(\hat{\mathbf{x}}+\mathbf{e},\mathbf{u})=\mathbf{f}(\hat{\mathbf{x}},\mathbf{u})+\mathbf{A}(t)\mathbf{e}+\bm{\eta}(\mathbf{e},\hat{\mathbf{x}},\mathbf{u}),$
where
$\mathbf{A}(t)=\left.\frac{\partial{\phantom{.}}}{\partial{\mathbf{x}}}\mathbf{f}(\mathbf{x},\mathbf{u})\right|_{(\hat{\mathbf{x}}(t),\mathbf{u}(t))}$
and $\bm{\eta}$ is the remainder of the Taylor series truncated after the
linear term. Hence, the estimation error dynamics can be stated as

(40)
$\dot{\mathbf{e}}=\left[\mathbf{A}(t)-\mathbf{L}(t)\mathbf{C}(t)\right]\mathbf{e}+\bm{\eta}(\mathbf{e},\hat{\mathbf{x}},\mathbf{u}).$

The following additional assumptions are standard assumptions in the
convergence analysis of the extended Kalman-Bucy filter, see, e.g. [50].

  1. (a1)

The matrix function $\mathbf{A}:{\mathds{R}_{\geq
0}}\rightarrow\mathds{R}^{n\times n}$ is bounded

  2. (a2)

For $\mathbf{P}{{}_{1}}(t)$ in (38b), there exist positive constants $p_{1}$
and $p_{2}$ such that $p_{1}\mathbf{I}_{k}\preceq\mathbf{P}{{}_{1}}(t)\preceq
p_{2}\mathbf{I}_{k}$ holds for all $t\in{\mathds{R}_{\geq 0}}$.

  3. (a3)

There exist positive constants $\varepsilon$ and $\kappa$ such that
$\|\bm{\eta}(\mathbf{e},\mathbf{x},\mathbf{u})\|\leq\kappa\|\mathbf{e}(t)\|^{2}$
holds for all $t\in{\mathds{R}_{\geq 0}}$,
$\mathbf{x},\mathbf{e}\in\mathds{R}^{n}$ and $\mathbf{u}\in\mathds{R}^{m}$
with $\|\mathbf{e}(t)\|\leq\varepsilon$.

######  Remark 5.1.

Assumption (a1) is e.g. fulfilled if $\mathbf{f}$ is globally Lipschitz
continuous in $\mathbf{x}$. The lower bound on $\mathbf{P}_{1}(t)$ in (a2) is
fulfilled for a properly chosen $\mathbf{G}_{1}(t)$ as shown in the proof of
Theorem 5.7. For a class of uniformly observable systems, the upper bound on
$\mathbf{P}_{1}(t)$ moreover holds independently of the specific observer
trajectory. This result is stated in detail in Section 5.4. Assumption (a3) is
fulfilled, e.g., if $\mathbf{f}$ is at least two times continuously
differentiable and the corresponding Hessian matrix of each component of
$\mathbf{f}$ is bounded, see [50, 26]. Let the components of
$\mathbf{f}(\mathbf{x},\mathbf{u})=\begin{bmatrix}f_{1}(\mathbf{x},\mathbf{u})&\cdots&f_{n}(\mathbf{x},\mathbf{u})\end{bmatrix}^{\mathsf{T}}$
be denoted by $f_{i}$, $i=1,\,\ldots,\,n$. Then, $\kappa$ is given by

(41)
$\kappa=\frac{1}{2}\max_{i=1}^{n}\sup_{\begin{subarray}{c}\mathbf{x}\in\mathds{R}^{n}\\\
\mathbf{u}\in\mathds{R}^{m}\end{subarray}}\|\mathbf{H}_{f_{i}}(\mathbf{x},\mathbf{u})\|$

with $\mathbf{H}_{f_{i}}(\mathbf{x},\mathbf{u})$ as the Hessian matrix of
$f_{i}$.

To show local stability of the estimation error dynamics, the standard
approach typically used in the stability analysis for the extended Kalman-Bucy
filter, see, e.g. [50], cannot be applied, because the Riccati equation is
solved only on a reduced-order subspace. However, a local convergence result
is obtained by utilizing the Lyapunov function for linear time-varying systems
from Proposition 2.3.

######  Theorem 5.2 (extended subspace observer).

Let a system be given by (36) and the observer for this system by (38).
Moreover, let the assumptions (a1) to (a3) hold. Then, the estimation error
dynamics (40) resulting from this observer is locally uniformly exponentially
stable.

######  Proof 5.3.

According to Theorem 4.3 and due to assumptions (a1) and (a2), it follows that
the linear time-varying system
$\dot{\mathbf{e}}=\left[\mathbf{A}(t)-\mathbf{L}(t)\mathbf{C}(t)\right]\mathbf{e}$
is uniformly exponentially stable with the feedback gain $\mathbf{L}(t)$ as in
(38d). The state transition matrix of this (unperturbed) system can then be
bounded by

(42) $\|\mathbf{\Phi}(t_{1},t_{0})\|\leq Ke^{-\gamma(t_{1}-t_{0})}$

for some positive constants $K\geq 1$ and $\gamma>0$. Moreover, according to
Proposition 2.3, there exists a Lyapunov function
$V(t,\mathbf{e})=\mathbf{e}^{\mathsf{T}}\mathbf{P}_{\mathrm{L}}(t)\mathbf{e}$
with positive constants $\bar{p}_{1}$ and $\bar{p}_{2}$ and a positive
definite $n\times n$ matrix $\mathbf{P}_{\mathrm{L}}(t)$ such that
$\bar{p}_{1}\mathbf{I}_{n}\preceq\mathbf{P}_{\mathrm{L}}(t)\preceq\bar{p}_{2}\mathbf{I}_{n}$.
The matrix $\mathbf{P}_{\mathrm{L}}(t)$ is the unique positive definite
solution of

(43)
$\dot{\mathbf{P}}_{\mathrm{L}}+\mathbf{A}^{\mathsf{T}}_{\mathrm{e}}(t)\mathbf{P}_{\mathrm{L}}+\mathbf{P}_{\mathrm{L}}\mathbf{A}_{\mathrm{e}}(t)+\mathbf{Q}_{\mathrm{L}}(t)=\mathbf{0}$

with $\mathbf{A}_{\mathrm{e}}(t)=\mathbf{A}(t)-\mathbf{L}(t)\mathbf{C}(t)$ and
$\mathbf{Q}_{\mathrm{L}}(t)$ as any positive definite matrix bounded by
positive constants $q_{1}$ and $q_{2}$ such that
$q_{1}\mathbf{I}_{n}\preceq\mathbf{Q}_{\mathrm{L}}(t)\preceq
q_{2}\mathbf{I}_{n}$.

The function
$V(t,\mathbf{e})=\mathbf{e}^{\mathsf{T}}\mathbf{P}_{\mathrm{L}}(t)\mathbf{e}(t)$
is now used as a Lyapunov function candidate for the perturbed error system
(40). Let $\varepsilon$ be given as in assumption (a3). Then, for
$\|\mathbf{e}\|\leq\varepsilon$, the time derivative along the trajectory can
be obtained according to

(44) $\displaystyle\dot{V}(t,\mathbf{e})$
$\displaystyle=\dot{\mathbf{e}}^{\mathsf{T}}\mathbf{P}_{\mathrm{L}}\mathbf{e}+\mathbf{e}^{\mathsf{T}}\dot{\mathbf{P}}_{\mathrm{L}}\mathbf{e}+\mathbf{e}^{\mathsf{T}}\mathbf{P}_{L}\dot{\mathbf{e}}=-\mathbf{e}^{\mathsf{T}}\mathbf{Q}_{\mathrm{L}}\mathbf{e}+2\bm{\eta}(\mathbf{e},\hat{\mathbf{x}},\mathbf{u})^{\mathsf{T}}\mathbf{P}_{\mathrm{L}}\mathbf{e}$
$\displaystyle\leq-
q_{1}\|\mathbf{e}\|^{2}+2\bar{p}_{2}\|\bm{\eta}\|\|\mathbf{e}(t)\|\leq-
q_{1}\|\mathbf{e}\|^{2}+2\bar{p}_{2}\kappa\|\mathbf{e}\|^{3}\leq(-q_{1}+2\bar{p}_{2}\kappa\|\mathbf{e}\|)\|\mathbf{e}\|^{2}$

For
$\|\mathbf{e}\|\leq\min\left(\frac{q_{1}}{4\bar{p}_{2}\kappa},\varepsilon\right)$,
it holds that

(45) $\dot{V}(t,\mathbf{e})\leq-\frac{1}{2}q_{1}\|\mathbf{e}\|^{2}.$

According to [38, Theorem 4.10], the error dynamics is locally uniformly
exponentially stable because of (45) and $\bar{p}_{1}\|\mathbf{e}\|^{2}\leq
V(t,\mathbf{e})\leq\bar{p}_{2}\|\mathbf{e}\|^{2}$.

To further investigate the norm constraint on the initial error, it is now
assumed for simplicity that $\varepsilon\geq\frac{q_{1}}{4\bar{p}_{2}\kappa}$
and that $\mathbf{Q}_{\mathrm{L}}(t)=q\mathbf{I}_{n}$, i.e., $q_{1}=q_{2}=q$.
Then, item (iv) of Proposition 2.3, i.e.,
$\bar{p}_{2}\leq\frac{K^{2}q}{2\gamma}$ with $K$ and $\gamma$ as in (42) can
be used to simplify the constraint on the initial error according to

(46)
$\|\mathbf{e}(t_{0})\|\leq\frac{q_{1}}{4\bar{p}_{2}\kappa}\leq\frac{\gamma}{2K^{2}\kappa}$

Here, $\gamma$ is the exponential decay rate. For the proposed observer,
$\gamma$ is bounded by the $k$-th upper Bohl exponent $\beta^{\mathrm{s}}_{k}$
such that $\gamma<\beta_{k}$, because the $(k+1)$-th exponent is not modified
by the observer gain. This suggests that the region of convergence can be
increased by taking more exponents into account, i.e., by increasing the
number of columns in $\bar{\mathbf{Q}}$, if the system is not merely
detectable but has stronger observability properties. This effect is also
demonstrated in the numerical simulation examples.

######  Remark 5.4.

In general, the number of non-negative upper Bohl exponents $j^{\star}$
depends on the specific observer trajectory. The Bohl exponents, however, are
robust with respect to bounded vanishing perturbations, see [19, Theorem 5.2
and Remark 5.5]. Because the trajectories of the nonlinear system (36) are
also trajectories of the observer (for zero initial error), it hence suffices
to obtain $j^{\star}$ from simulation studies of the original system. If the
initial estimation error is small, the observer trajectory can then be
regarded as perturbed trajectory of the true system trajectory with the same
$j^{\star}$. The Bohl exponents of the linearizations along the trajectories
in general depend on the specific trajectory. For the special class of ergodic
measure preserving systems, Oseledet’s multiplicative ergodic theorem
guarantees that the Lyapunov exponents are independent of the specific
trajectory, see [4, p. 8-9] or [35, Theorem 2.1]. To the authors’ knowledge, a
similar result regarding the Bohl exponents or the exponential dichotomy
spectrum is still missing. A step towards this direction is provided in [35],
where the relationship between Lyapunov exponents and the so-called dynamical
spectrum is investigated. The dynamical spectrum is a generalization of
Definition 3.4 to nonlinear systems and is independent of the specific
linearization. It is hard to compute in practice, however.

###  5.4 A Class of Uniformly Observable Systems

For special classes of nonlinear systems, the assumptions (a1) to (a3) are
guaranteed to hold and verifiable conditions depending only on the system
properties can be stated. In particular, this section considers nonlinear
systems which are diffeomorphic to

(47a) $\displaystyle\dot{\mathbf{x}}$
$\displaystyle=\bar{\mathbf{A}}\mathbf{x}+\bar{\mathbf{f}}(\mathbf{x},\mathbf{u})$
(47b) $\displaystyle\mathbf{y}$ $\displaystyle=\bar{\mathbf{C}}\mathbf{x}$
where the state $\mathbf{x}\in\mathds{R}^{n}$ is partitioned according to
(47c) $\mathbf{x}=\begin{bmatrix}\mathbf{x}^{(1)}\\\ \mathbf{x}^{(2)}\\\
\vdots\\\ \mathbf{x}^{(p)}\end{bmatrix}\quad\text{with
}\mathbf{x}^{(k)}=\begin{bmatrix}x_{k,1}\\\ x_{k,2}\\\ \vdots\\\
x_{k,l_{k}}\end{bmatrix}\in\mathds{R}^{l_{k}},\,k=1,\ldots,p,\;\text{ and
}\;\sum_{k}l_{k}=n.$ The rest of the system is characterized by (47d)
$\displaystyle\bar{\mathbf{A}}$
$\displaystyle=\operatorname{diag}\left({\bar{\mathbf{A}}_{1},\,\bar{\mathbf{A}}_{2},\,\ldots,\,\bar{\mathbf{A}}_{p}}\right),\;\bar{\mathbf{A}}_{k}=\begin{bmatrix}0&1&0&\cdots&0\\\
0&0&1&\cdots&0\\\ &&&\ddots&0\\\ 0&0&0&\cdots&0\end{bmatrix}_{(l_{k}\times
l_{k})}$ (47e) $\displaystyle\bar{\mathbf{C}}$
$\displaystyle=\operatorname{diag}\left({\bar{\mathbf{C}}_{1},\,\bar{\mathbf{C}}_{2},\,\ldots,\,\bar{\mathbf{C}}_{p}}\right),\;\bar{\mathbf{C}}_{k}=\begin{bmatrix}1&0&\cdots&0\end{bmatrix}_{(1\times
l_{k})}.$ The nonlinear function $\bar{\mathbf{f}}$ is structured according to
(47f)
$\bar{\mathbf{f}}(\mathbf{x},\mathbf{u})=\begin{bmatrix}\bar{\mathbf{f}}^{(1)}(\mathbf{x},\mathbf{u})\\\
\bar{\mathbf{f}}^{(2)}(\mathbf{x},\mathbf{u})\\\ \vdots\\\
\bar{\mathbf{f}}^{(p)}(\mathbf{x},\mathbf{u})\end{bmatrix}\text{ with
}\bar{\mathbf{f}}^{(k)}(\mathbf{x},\mathbf{u})=\begin{bmatrix}f_{k,1}(\mathbf{x},\mathbf{u})\\\
f_{k,2}(\mathbf{x},\mathbf{u})\\\ \vdots\\\
f_{k,l_{k}}(\mathbf{x},\mathbf{u})\\\ \end{bmatrix}$ and $\frac{\partial
f_{i,j}}{\partial x_{v,w}}=0$ for all $i,v\leq p$ and all $w>j$.

Moreover, $\bar{\mathbf{f}}$ is assumed to be globally Lipschitz continuous in
$\mathbf{x}$ and at least two times continuous differentiable, where the
corresponding Hessian matrix of each component of $\bar{\mathbf{f}}$ is
bounded.

It is well known that for general nonlinear systems, the input may destroy the
system’s observability. Systems diffeomorphic to (47) are called uniformly
observable for any input [39], because observability does not depend on the
particular input. Systems in the form (47) play an important role in the
observer design for nonlinear systems, see, e.g. [27, 32, 25, 31, 28, 44]. A
lot of work has been devoted to determining existence conditions for
transformations to the form (47) or more general upper triangular forms, see
[44, Sec. 6] and the references therein for a good overview on existing
procedures. It is worth to mention that system (47) is a generalization of the
phase variable form [8].

For system (47), Krener [39] shows that the extended Kalman-Bucy filter (37)
is a local exponential observer. In particular, the proof of [39, Theorem
1.1.1] establishes the boundedness of $\mathbf{P}(t)$ in (37b) independently
of the observer trajectory. This important result is summarized in

######  Lemma 5.5 (boundedness of $\mathbf{P}(t)$).

Assume that the nonlinear system is in the form (47). Moreover
$\bar{\mathbf{f}}$ is globally Lipschitz continuous and at least twice
continuously differentiable and the corresponding Hessian matrix of each
component of $\bar{\mathbf{f}}$ is bounded. Then, for all
$\mathbf{P}(t_{0})\succeq\bm{0}$ and all bounded $\mathbf{G}(t)\succeq\bm{0}$,
there exists a constant $p_{2}>0$ such that for the solution $\mathbf{P}(t)$
of the corresponding Riccati equation (37b) it holds that $\mathbf{P}(t)\leq
p_{2}\mathbf{I}_{n}$ for all $t\geq t_{0}$.

######  Proof 5.6.

The proof for the bound with $\mathbf{P}(t_{0})\succ 0$ is given in the proof
of [39, Theorem 1.1.1]. Remarkably, this bound is independent of the observer
trajectory and hence it holds for any trajectory. The bound is established via
the design of a suboptimal high-gain observer, which gives an upper bound on
the cost of the optimal filtering problem and hence an upper bound on the
solution of the Riccati equation (37b). The boundedness for all
$\mathbf{P}(t_{0})\succeq\bm{0}$ then follows from the order preserving
property of the Riccati differential equation, see [1, Theorem 4.1.4 and
Corollary 4.1.5].

The previous result together with the specific system structure guarantees
that the assumptions (a1) to (a3) are fulfilled. Hence, the existence of the
extended subspace observer proposed in Section 5.2 is guaranteed for this
system class. This is summarized in the following

######  Theorem 5.7 (existence of extended subspace observer).

For systems of the form (47), where $\bar{\mathbf{f}}$ is globally Lipschitz
continuous and at least twice continuously differentiable and the
corresponding Hessian matrix of each component of $\bar{\mathbf{f}}$ is
bounded, the assumptions (a1)–(a3) are fulfilled for all
$\mathbf{P}_{1}(0)\succ\bm{0}$ and all
$g_{1}\mathbf{I}\preceq\mathbf{G}_{1}(t)\preceq g_{2}\mathbf{I}$ with
arbitrary constants $0<g_{1}\leq g_{2}$.

######  Proof 5.8.

Assumptions (a1) and (a3) are guaranteed by the global Lipschitz assumption
and the boundedness assumption on the Hessian, see also Remark 5.1. Hence it
remains to show that (a2) holds. The upper bound on $\mathbf{P}_{1}(t)$
follows straightforwardly from Lemma 4.6 and Lemma 5.5 for any positive
definite and bounded $\mathbf{G}_{1}(t)$. The proof that $\mathbf{P}_{1}(t)$
is also bounded from below follows from continuity arguments. Under the
assumption that $\mathbf{P}_{1}(t)$ becomes positive semidefinite for some
$t=t_{1}>t_{0}$, there exists a non-trivial vector $\mathbf{v}$ such that
$\mathbf{v}^{\mathsf{T}}\mathbf{P}_{1}(t_{1})\mathbf{v}=0$. Multiplying the
Riccati equation (38b) with $\mathbf{v}^{\mathsf{T}}$ from the left and
$\mathbf{v}$ from the right and evaluating the derivative at $t=t_{1}$ gives

(48)
$\frac{\mathrm{d}}{\mathrm{d}t}\mathbf{v}^{\mathsf{T}}\mathbf{P}_{1}\mathbf{v}\big{|}_{t=t_{1}}=\mathbf{v}^{\mathsf{T}}\mathbf{G}_{1}(t_{1})\mathbf{v}\succ\bm{0}.$

Due to continuity and since
$\mathbf{v}^{\mathsf{T}}\mathbf{P}_{1}(0)\mathbf{v}\succ>0$, the solution of
(38b) must remain positive definite and lower bounded according to
$p_{1}\mathbf{I}\preceq\mathbf{P}_{1}(t)$ with some $p_{1}>0$.

Theorem 5.7 guarantees the existence of the extended subspace observer for
systems in the form of (47). The global Lipschitz continuity of
$\bar{\mathbf{f}}$ is also required in high-gain observer designs, see, e.g.,
[25]. This condition is restrictive since the Lipschitz condition is usually
satisfied only locally. If, however, the state $\mathbf{x}(t)$ of system (47)
lies in a bounded set, it is possible to relax this assumption using
prolongation techniques, see [25, Remark 2.1] and the references therein.

##  6 Numerical results

The nonlinear Lorenz’96 (L’96) model introduced in [40] is widely used as a
benchmark of data assimilation algorithms [58, 17, 26]. It is a system of
nonlinear differential equations recursively defined by

(49) $\dot{z}_{i}=(z_{i+1}-z_{i-2})z_{i-1}-z_{i}+F,\qquad i=1,\ldots,n,$

with the notational convention $z_{-1}=z_{n-1}$, $z_{0}=z_{n}$, and
$z_{n+1}=z_{1}$. The state vector is
$\mathbf{z}=[z_{1}\;\cdots\;z_{n}]^{\mathsf{T}}\in\mathds{R}^{n}$. For $F=8$,
this model exhibits a chaotic behavior. With this configuration, the model is
used as a benchmark examples for data assimilation in [26, 43, 10].

The following simulations are carried out for a model of order $n=18$. The
output of the model is chosen as $\mathbf{y}(t)=\mathbf{C}_{p}\mathbf{z}(t)$
with a constant $p\times n$ matrix $\mathbf{C}_{p}$. The rows of
$\mathbf{C}_{p}$ are chosen such that only one state is measured per row and
the “sensors” are distributed with equal distance over all state variables.
The matrix is given by

(50)
$\mathbf{C}_{p}=\begin{bmatrix}\mathbf{b}_{1}&\mathbf{b}_{d+1}&\cdots&\mathbf{b}_{(p-1)d+1}\end{bmatrix}^{\mathsf{T}},$

where $\mathbf{b}_{i}$ is the i-th standard basis vector and
$d=\left\lfloor\frac{n}{p}\right\rfloor$, where $\lfloor\cdot\rfloor$ denotes
rounding to the nearest integer towards zero.

The initial condition is chosen as
$z_{i,0}=\sin\left(\frac{i-1}{n}2\pi\right)$ at initial time $t_{0}=0$. For
evaluation purposes, also different initial conditions were chosen. The
obtained results were quantitatively similar and in particular the
approximated Lyapunov exponents and the upper and lower Bohl exponents were
approximately the same.

First, the spectral intervals of the system are investigated. All differential
equations are solved using fixed step 4th order Runge-Kutta integration with a
step-size of $T_{s}=0.005$. For solving the differential equation (23b) to
obtain $\bar{\mathbf{Q}}$, a projected integrator [21] is implemented, where
the orthonormalization is carried out after each simulation time step.

The results for all $n=18$ spectral intervals are given in Table 1 in Appendix
B. The final time of this simulation is chosen as $T_{f}=1500$ in order to be
able to choose the averaging window length $H$ for the approximation of the
exponential dichotomy spectral intervals sufficiently large. For a window
length of $H=300$, the $7^{\text{th}}$ approximated upper Bohl exponent is
positive whereas for an increased window length $H=800$, this value is
negative. The spectral intervals $\Lambda_{7}$ to $\Lambda_{18}$ are thus
negative and moreover all intervals are disjoint. The results show that the
number of non-negative Lyapunov exponents $k^{\star}$ is equal to the number
of non-negative upper Bohl exponents $j^{\star}$.

Now, the design of the extended subspace observer is carried out for $p=5$ as
the dimension of the output vector $\mathbf{y}(t)$. The initial estimate is
chosen as $\hat{\mathbf{x}}_{0}=\mathbf{x}_{0}+\bm{\xi}_{0}$ with
$\bm{\xi}_{0}$ as a random perturbation. The components $\xi_{0,i}$ of
$\bm{\xi}_{0}$ are chosen to be uniformly distributed on an interval
$(-\delta,\delta)$ with $\delta>0$ as a simulation parameter. For $k\leq
5<j^{\star}$, no convergence of the estimation error could be achieved
independently of the size of the initial perturbation. This coincides with the
observations from the approximated spectral intervals. Hence, in a first step,
$k=k^{\star}=j^{\star}=6$ is chosen in the observer design and an ensemble
simulation with $N=50$ simulation runs is carried out. The matrix
$\mathbf{G}_{{1}}(t)$ in the differential Riccati equation (38b) is chosen as
a constant matrix $\mathbf{G}_{{1}}=10\mathbf{I}_{k}$ in the following
simulations.

The (point-wise) minimum and maximum estimation errors together with the
median and the 80%-quantile are depicted in Fig. 1. The latter quantity is a
(point-wise) upper bound for 80% of the estimation errors. The expected
minimum convergence rate of the observer error, i.e., the approximated 7-th
upper Bohl exponent is also depicted. This indicates that the estimation error
converges exponentially with the rate $\beta_{7}^{800}\approx-0.0320$. This
convergence rate is very small and, as predicted by theory, the stability of
the nonlinear error system is very sensitive to the size of the initial
estimation error $\mathbf{e}_{0}$. This behavior can be seen in the
simulations and hence the interval bound for the uniform distribution of the
initial error is chosen as $\delta=10^{-4}$.

In order to increase the convergence speed and to decrease sensitivity with
respect to the magnitude of the initial perturbation, the number of columns in
$\bar{\mathbf{Q}}$ was increased to $k=7$. The results of this ensemble
simulation is depicted in Fig. 2. The resulting exponential convergence is now
achieved at an approximate rate of $\beta_{8}^{800}\approx-0.2760$, because
this is the first Bohl exponent which is not modified by the observer feedback
gain. The convergence is faster compared to $k=6$ and moreover the robustness
with respect to the initial perturbation is improved. Hence, the size of the
initial perturbation could be increased to $\delta=10^{-3}$. Towards the end
of the simulation, the estimation error norm saturates around $10^{-13}$ due
to numerical integration and floating point error.

Taking into account one more spectral interval to be modified via the observer
gain, the convergence speed can again be increased as depicted in Fig. 3 for
$k=8$. The size of the initial perturbation was again increased to
$\delta=10^{-2}$ and a convergence of the estimation error was achieved for
all simulation runs.

Figure 1: Ensemble estimation error for the L’96 model with $k=6$. Figure 2:
Ensemble estimation error for the L’96 model with $k=7$. Figure 3: Ensemble
estimation error for the L’96 model with $k=8$.

In order to analyze the detectability properties of the Lorenz’96 model, the
observer was implemented for $k=j^{\star}=6$ with the matrix
$\mathbf{G}_{{1}}(t)=\mathbf{0}$ in (38b). Then, the obtained differential
Riccati equation is a differential equation for the inverse of the
constructibility Gramian on the considered subspace. This inverse, i.e.,
$\mathbf{P}{{}_{1}}(t_{0})$, cannot be initialized correctly, because the
constructibility Gramian is zero at the initial time. However, independently
of the positive definite initialization, all solutions of
$\mathbf{P}{{}_{1}}(t)$ converge to each other and hence to the true inverse
asymptotically, see [49, Sec. 3]. This suggests that if $\mathbf{P}_{{1}}(t)$
is upper bounded, the constructibility Gramian is lower bounded and the
linearization along the estimated trajectory is uniformly completely
constructible on the considered subspace. The largest eigenvalue of
$\mathbf{P}{{}_{1}}(t)$ for different measurement configurations is depicted
in Fig. 4. This result indicates that the considered trajectory is uniformly
completely observable on the unstable subspace for any number of measurements.

Figure 4: Maximum eigenvalue of $\mathbf{P}{{}_{1}}(t)$ for the L’96 model
with different measurement configurations.

An additional insight can be obtained by the investigation of the smallest
eigenvalue of $\mathbf{P}{{}_{1}}(t)$. The number of measurements was now
again chosen to be $p=5$. If one considers only the unstable modes in the
observer gain, i.e. $k=6$, the smallest eigenvalue is uniformly bounded from
below. However, if one takes into account an additional mode in the observer
gain by choosing $k=7$, the smallest eigenvalue tends to zero, see Fig. 5.
This indicates that the observer gain is “losing strength” in the already
uniformly asymptotically stable directions. This is avoided by choosing
$\mathbf{G}(t)$ as a positive definite matrix as proposed in the present
observer design. A similar effect is also discussed for a deterministic
interpretation of the Kalman-Bucy filter in [49, Remark 3.2] and is also well
recognized in data assimilation on the unstable subspace [58, 11].

Figure 5: Minimum eigenvalue of $\mathbf{P}{{}_{1}}(t)$ for the L’96 model
with different values for $k$ and $\mathbf{Q}(t)=\mathbf{0}$.

To test the observer for various system trajectories, the upper Bohl exponents
for random initial conditions on the unit sphere of the L’96 system were
computed. The maximum, minimum and average values of the 9 largest upper Bohl
exponents for 500 simulation runs is given in Table 2. For all computed
trajectories, the $8^{\text{th}}$ Bohl exponent was negative and hence $k=7$
is a reasonable choice. The ensemble estimation error for $50$ observer
simulation runs and an initial perturbation of $\delta=10^{-4}$ is depicted in
Figure 6. The dashed blue line shows a convergence rate proportional to the
mean over all $\beta_{8}^{900,\mathrm{s}}$, which suggests that this exponent
governs the convergence behavior.

Figure 6: Ensemble estimation error for the L’96 model with $k=7$ and random
initial conditions.

##  7 Conclusions and Outlook

This work proposes a new observer design strategy for linear time-varying and
nonlinear systems. Stability proofs for the estimation error dynamics show
global exponential stability in the time-varying case and local exponential
stability in the nonlinear case. The feedback gain is computed only for the
unstable modes of the estimation error dynamics, which allows to reduce the
computational complexity of the observer compared to classical approaches like
the extended Kalman-Bucy filter. Simulation results moreover suggest that
additionally including stable modes in the design may increase the robustness
in the nonlinear case. In future research, the assignment of the exponential
dichotomy spectrum will be investigated. This may foster the development of
efficient observer designs, which do not require the solution of a
(computationally costly) differential Riccati equation. Moreover, the
conditions for the existence of the proposed observer will be investigated in
more detail. For this purpose, it is important to guarantee the upper bound of
the observer Riccati equation independent of the observer trajectory. For
systems which are fully observed on the unstable subspace, it may be possible
to guarantee this upper bound by adapting the bound from [36].

##  Appendix A Proof of Theorem 3.7

The proof for relation (21) can be found in the proof of [22, Theorem 8.4].
Relation (22) is proven for one upper Bohl exponents $\beta^{\mathrm{s}}_{j}$.
First, assume an arbitrary $\varepsilon>0$. The scalar system
$\dot{\eta}(t)=(b_{jj}(t)-\lambda)\eta(t)$ with
$\lambda=\beta^{\mathrm{s}}_{j}+\varepsilon$ has an exponential dichotomy with
$P=1$ and, equivalently, there exist reals $\alpha>0$ and $d\geq 0$ such that

(51)
$-\ln\|\mathbf{\Phi}(t,t_{0})\|=\int_{t_{0}}^{t}{{\left(\beta^{\mathrm{s}}_{j}+\varepsilon-
b_{jj}(s)\right)}\,\mathrm{d}{s}}\geq\alpha(t-t_{0})-d$

holds for all $t_{0}\in\mathds{J}$ and all $t\geq t_{0}$. For a sufficiently
large $H_{0}>0$ such that $\alpha-d/H_{0}>\alpha/2$, it follows that

(52) $\frac{1}{H}\int_{t}^{t+H}{{\left(\beta^{\mathrm{s}}_{j}+\varepsilon-
b_{jj}(s)\right)}\,\mathrm{d}{s}}\geq\alpha-\frac{d}{H}>\frac{\alpha}{2}$

holds for all $H\geq H_{0}$. Splitting the integral in (52) yields

(53)
$\frac{1}{H}\int_{t}^{t+H}{{\left(\beta^{\mathrm{s}}_{j}+\varepsilon\right)}\,\mathrm{d}{s}}-\frac{1}{H}\int_{t}^{t+H}{{b_{jj}(s)}\,\mathrm{d}{s}}=\beta^{\mathrm{s}}_{j}+\varepsilon-\beta^{H,\mathrm{s}}_{j}>\frac{\alpha}{2},$

and hence $\beta^{\mathrm{s}}_{j}+\varepsilon>\beta^{H,\mathrm{s}}_{j}$ for
all $H\geq H_{0}$. The proof for the remaining upper and lower Bohl exponents
follows analogously.

##  Appendix B Numerical Results

Table 1: Nine largest approximated spectral intervals of the Lorenz’96 model
with initial condition $z_{i,0}=\sin(\frac{i-1}{n}2\pi)$.

| $\Sigma_{\mathrm{L}}$ |  $\Sigma_{\mathrm{ED}}$ for ${H=300}$ |  $\Sigma_{\mathrm{ED}}$ for ${H=800}$
---|---|---|---
interval | $\lambda^{\mathrm{i}}_{i}\approx\lambda^{\mathrm{s}}_{i}$ | $\beta_{i}^{300,\mathrm{i}}$ | $\beta_{i}^{300,\mathrm{s}}$ | $\beta_{i}^{800,\mathrm{i}}$ | $\beta_{i}^{800,\mathrm{s}}$
1 |  -1.545 |  -1.430 |  -1.658 |  -1.4870 |  -1.5920
2 |  -1.211 |  -1.139 |  -1.274 |  -1.1920 |  -1.2410
3 |  -0.878 |  -0.790 |  -0.959 |  -0.8630 |  -0.9190
4 |  -0.570 |  -0.494 |  -0.643 |  -0.5360 |  -0.5930
5 |  -0.283 |  -0.231 |  -0.333 |  -0.2600 |  -0.2900
6 |  -0.003 | -0.011 |  -0.021 | -0.0050 |  -0.0080
7 | -0.045 | -0.096 |  -0.015 | -0.0760 | -0.0320
8 | -0.296 | -0.365 | -0.228 | -0.3310 | -0.2760
9 | -0.579 | -0.639 | -0.526 | -0.5880 | -0.5560

Table 2: Nine largest upper Bohl exponents of Lorenz’96 model with random
initial condition.

| $\beta_{1}^{800,\mathrm{s}}$ | $\beta_{2}^{800,\mathrm{s}}$ | $\beta_{3}^{800,\mathrm{s}}$ | $\beta_{4}^{800,\mathrm{s}}$ | $\beta_{5}^{800,\mathrm{s}}$ | $\beta_{6}^{800,\mathrm{s}}$ | $\beta_{7}^{800,\mathrm{s}}$ | $\beta_{8}^{800,\mathrm{s}}$ | $\beta_{9}^{800,\mathrm{s}}$
---|---|---|---|---|---|---|---|---|---
max | 1.669 | 1.266 | 0.932 | 0.628 | 0.344 | 0.046 | 0.015 | -0.246 | -0.518
min | 1.478 | 1.112 | 0.813 | 0.511 | 0.216 | -0.005 | -0.074 | -0.355 | -0.641
mean | 1.552 | 1.185 | 0.869 | 0.569 | 0.274 | 0.004 | -0.022 | -0.307 | -0.576

## Acknowledgments

This work was partially supported by the Graz University of Technology LEAD
project “Dependable Internet of Things in Adverse Environments”. The financial
support by the Christian Doppler Research Association, the Austrian Federal
Ministry for Digital and Economic Affairs and the National Foundation for
Research, Technology and Development is gratefully acknowledged. The authors
would moreover like to thank Sergiy Zhuk, IBM Research, Dublin, Ireland, for
the many fruitful discussions leading to this interesting research direction.

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