# The Effect of Sensor Fusion on Data-Driven Learning of Koopman Operators

Shara Balakrishnan<EMAIL_ADDRESS>Aqib Hasnain<EMAIL_ADDRESS>Rob
Egbert<EMAIL_ADDRESS>Enoch Yeung<EMAIL_ADDRESS>Department of
Electrical and Computer Engineering, University of California, Santa Barbara,
United States  Department of Mechanical Engineering, University of California,
Santa Barbara, United States  Biological Sciences Division, Earth and
Biological Sciences Directorate, Pacific Northwest National Laboratory, United
States

###### Abstract

Dictionary methods for system identification typically rely on one set of
measurements to learn governing dynamics of a system. In this paper, we
investigate how fusion of output measurements with state measurements affects
the dictionary selection process in Koopman operator learning problems. While
prior methods use dynamical conjugacy to show a direct link between Koopman
eigenfunctions in two distinct data spaces (measurement channels), we explore
the specific case where output measurements are nonlinear, non-invertible
functions of the system state. This setup reflects the measurement constraints
of many classes of physical systems, e.g., biological measurement data, where
one type of measurement does not directly transform to another. We propose
output constrained Koopman operators (OC-KOs) as a new framework to fuse two
measurement sets. We show that OC-KOs are effective for sensor fusion by
proving that when learning a Koopman operator, output measurement functions
serve to constrain the space of potential Koopman observables and their
eigenfunctions. Further, low-dimensional output measurements can be embedded
to inform selection of Koopman dictionary functions for high-dimensional
models. We propose two algorithms to identify OC-KO representations directly
from data: a direct optimization method that uses state and output data
simultaneously and a sequential optimization method. We prove a theorem to
show that the solution spaces of the two optimization problems are equivalent.
We illustrate these findings with a theoretical example and two numerical
simulations.

###### keywords:

Sensor fusion; Koopman operator; nonlinear system identification; subspace
methods

, , ,

##  1 Introduction

The rapid evolution in sensor technology enables us to acquire a wealth of
information about the governing dynamics of nonlinear systems. Novel sensors
are being designed and fabricated at an rapid rate to enhance the data
acquisition process for systems in various domains like biology [1, 2, 3, 4,
5], mechanics [6, 7, 8, 9, 10], and transportation [11, 12, 13, 14, 15]. The
tools, techniques, and theories that integrate all the data to augment our
knowledge of the system is the broad area of sensor fusion [16]. Sensor fusion
plays a pivotal role in many applications. In the human gait system, senor
fusion methods are deployed to study gait dynamics [17, 18, 19, 20, 21],
detect gait anomalies [22, 23, 24] and control prosthetics [25, 26, 27, 28,
29, 30]. Numerous sensors are utilized in inertial navigation systems with
data fusion architectures for accurate state estimation to aid in navigation
and control [31, 32, 33, 34, 35, 36, 37, 38]. Maximizing the throughput of
manufacturing processes [39, 40, 41, 42, 43, 44, 45, 46, 47, 48], minimizing
traffic congestion [12, 13, 14], and monitoring structural health [49, 50, 51]
are some of the other areas where sensor fusion plays a critical role.

In recent years, there has been significant interest in developing sensor
fusion techniques for biological systems, spurred in part by decreasing costs
of next-generation omics measurements [52, 53, 54, 55]. While some techniques
like transcriptomics [56] and proteomics [57] inform the genetic activity
within the cell, other instruments like flow cytometers [58], plate readers
[59], and microscopes [60] inform the phenotypic characteristics outside the
cell. Substantial progress towards fusing a combination of these datasets with
prior knowledge of the system has resulted in static models that provide
insights about the underlying network topology of the interaction between the
genes and the proteins [52, 53, 61, 62, 54, 55, 63, 64].

An area of research emerging in the last decade is the problem of sensor
fusion for biological networks, that couple distinct streams of omics
measurements with fluorescence data. Fluorescence data is often used to
parameterize or identify dynamical system models describing governing dynamics
and network topology, while omics measurements are used to deduce whole-cell
statistics and steady-state phenotypes of cellular metabolism, stress, and
fitness. To that end, our goal in this paper is to develop sensor fusion
techniques that integrate various types of time-series data to construct
dynamic genotype-to-phenotype models.

Koopman operator methods have recently demonstrated great promise in
simultaneously discovering A) the governing dynamics and B) a spectral
decomposition of a complex physical system represented purely by data [65,
66]. The key premise of Koopman operator theory is that a collection of state
functions, or observables, can be constructed, discovered, or estimated, to
embed nonlinear dynamics of a physical system in a high dimensional space. The
Koopman operator then acts as a linear operator on the function space of
observables, governing time-evolution of the system dynamics, as a linear
system.

In reality, the Koopman operator is infinite dimensional and must be
approximated numerically. The problem of finding a Koopman operator and a
collection of observable functions is known as the Koopman operator learning
problem. The classical approach to solve this problem is to use dynamic mode
decomposition (DMD) [67]. More recently, variations on DMD involve
approximating observable functions with a broad set of dictionary functions
(E-DMD) [68, 69], which can be generated using deep learning [70, 71, 72, 73],
by casting the learning problem, as a robust optimization problem to handle
sparse data [74, 75] or to treat heterogeneously sampled data [76, 77]. The
power of Koopman operators lies in their ability to capture the underlying
modes that drive the system [78, 79, 80], directly from data. Koopman
operators also enable the construction of observers [81, 82, 83, 84] and
controllers [85, 86, 87, 88, 89] for nonlinear systems in a linear framework.

Koopman operators can fuse sensor measurements into a single dataset to learn
the underlying dynamics, as seen in systems like traffic dynamics [90, 91],
human gait [92, 93, 94], and robotics [95]. The work of Williams et al. [96]
and Mezic [97] use Koopman operator theory to elucidate the fusion of the
dynamics evolving on two different state-spaces, provided that there is a
function map between the state-spaces. Williams et al. [96] consider two
datasets that are rich enough to reconstruct the system state and develop an
algorithm to map the eigenfunctions of the Koopman operator identified from
each dataset. Mezic [97] proves a relational mapping between eigenfunctions of
both spaces, when exact conjugacy is not possible. For dynamics that evolve on
two different state-spaces, they define the factor conjugacy of the dynamics
for the function that maps the two spaces.

This paper builds on existing Koopman operator fusion theory [96, 97] to
examine a special case: we consider learning a sensor fusion model for a
physical system represented by direct state data and a series of output
measurements. We consider the scenario where both the Koopman operator,
observables, and the relational map between Koopman observable and output
measurements are unknown. In a standard Koopman operator learning problem, the
state measurement data is sufficient to approximate the Koopman operator.
However, we examine the effect of adding output measurements now as a series
of behavioral constraints on the dynamics of the Koopman operator — we aim to
know the effect of incorporating output measurements (sensor fusion) on the
solution of the Koopman operator learning problem. Specifically, we seek to
understand how Koopman eigenfunctions, spectra, and modes change as a
consequence of sensor fusion.

The formulation of an output-constrained Koopman operator is not novel. In the
literature, output-constrained Koopman operators are used for various
applications like observability analysis [98, 83], observer synthesis [81, 82,
84], and sensor placement [77, 76, 99] for nonlinear systems. In this paper,
we prove that output-constrained Koopman operators satisfy the following
properties:

  1. (i)

The output dynamics of the nonlinear system always span a subspace of
observable functions for the output-constrained Koopman operator

  2. (ii)

The observables of the output-constrained Koopman operators can capture the
dynamics of both states and outputs

  3. (iii)

State-inclusive output-constrained Koopman operators exist in the region of
convergence of the Taylor series expansion of the dynamics and output
functions of any nonlinear system.

Heretofore, there have been few algorithms that identify output-constrained
Koopman operators, to the best of our knowledge. To identify output-
constrained Koopman operators (OC-KO) from data, we pose the output-
constrained DMD (OC-DMD) problem as a special extension of the DMD problem to
incorporate output constraints. We propose two variants of the problem: the
direct OC-DMD solves for the state and output dynamics concurrently, while the
sequential OC-DMD solves for them sequentially. Sequential OC-DMD explicitly
reveals the effect of having outputs in the KOR learning problem. To implement
OC-DMD in practice, we build on the deepDMD algorithm developed by Yeung et
al. [71], where neural networks represent vector valued observables of the
Koopman operator. We then study the effect of affine transformations on the
output-constrained Koopman operator learning problem to take into account— how
data preprocessing methods like normalization or standardization modify the
output-constrained Koopman operator. We use simulation examples to investigate
the performance of OC-DMD algorithms. Our findings include:

  1. (i)

The solution space of the direct OC-DMD and sequential OC-DMD optimization
problems are equivalent

  2. (ii)

Affine state transformations yield OC-KOs with an eigenvalue on the unit
circle

  3. (iii)

OC-KOs optimized for multi-step predictions are required to capture the
dynamics with limit cycles.

The paper is organized as follows. In section 2, we formulate the OC-KO
representation. In section 3, we briefly discuss Koopman operator theory,
Koopman operator sensor fusion and the necessary DMD algorithms. We discuss
the properties of OC-KOs in Section 4 and the OC-DMD algorithms in Section 5.
We show the simulation results in Section 6 and conclude our analysis in
Section 7.

##  2 Problem Formulation

Our goal in this paper will be to consider physical systems represented by
sampled time-series data, even their underlying governing dynamics are
continuous. For methods that estimate the Koopman generator (the continuous-
time extension of the discrete-time Koopman operator), we refer the reader to
[100]. Suppose we have an autonomous discrete-time nonlinear dynamical system
with output

State Equation: $\displaystyle x_{k+1}$ $\displaystyle=f(x_{k})$ (1a) Output
Equation: $\displaystyle y_{k}$ $\displaystyle=h(x_{k})$ (1b)

where $x\in\mathcal{M}\subseteq\mathcal{R}^{n}$ is the state,
$y\in\mathbb{R}^{p}$ is the output, $f:\mathcal{M}\rightarrow\mathcal{M}$ and
$h:\mathcal{M}\rightarrow\mathbb{R}^{p}$ are analytic functions and $k$ is the
discrete time index indicating the time point $kT_{s}$ with $T_{s}$ being the
sampling time. Let

$\displaystyle\psi(x_{k+1})$ $\displaystyle=K\psi(x_{k})$ (2) $\displaystyle
y_{k}$ $\displaystyle=W_{h}\psi(x_{k})$

be the OC-KO of (1) where $\psi:\mathcal{M}\rightarrow\mathbb{R}^{n_{L}}$ is a
vector function of state-dependent scalar observable functions (
$n_{L}\leq\infty$), $K\in\mathbb{R}^{n_{L}\times n_{L}}$ is the Koopman
operator and $W_{h}\in\mathbb{R}^{p\times n_{L}}$ is a projection matrix that
projects observables to the space of output functions. We record state and
output measurements ($x$ and $y$, respectively) from (1) as

$\displaystyle X_{P}$ $\displaystyle=[x_{0},\cdots,x_{N-1}],\quad
X_{F}=[x_{1},\cdots,x_{N}],$ (3) $\displaystyle Y_{P}$
$\displaystyle=[y_{0},\cdots,y_{N-1}],$

where $X_{P}\in\mathbb{R}^{n\times N}$ is the state data collected from time
points $0,...,N-1$, and $X_{F}\in\mathbb{R}^{n\times N}$ and
$Y_{P}\in\mathbb{R}^{p\times N}$ are the collection of state and output
measurements propagated one step forward from the elements in $X_{P}$,
respectively. Here, we use lower-case notation for the state $x$ and output
$y$ variables and upper-case notation for variables containing sampled time-
series snapshots involving the state variable $x$ with $X_{P},X_{F}$ and time-
series with output variable $y$ with $Y_{P}$, respectively.

##  3 Mathematical Preliminaries

We now briefly introduce the formal mathematical elements of Koopman operator
theory, its modal decomposition, Koopman operator sensor fusion and relevant
DMD algorithms.

###  3.1 Koopman operator

Given the dynamical system (1a), the KO represented by $\mathcal{K}$ is a
linear operator that is invariant in the functional space $\mathcal{F}$ as
$\mathcal{K}:\mathcal{F}\rightarrow\mathcal{F}$. Any function
$\tilde{\phi}\in\mathcal{F}$ such that
$\tilde{\phi}:\mathcal{M}\rightarrow\mathbb{C}$ is defined as a scalar
observable with the property

$(\mathcal{K}\tilde{\phi})(x)=(\tilde{\phi}\circ f)(x)$

where $\tilde{\phi}\circ f\in\mathcal{F}$ due to invariance of $\mathcal{K}$.
Let the set of basis functions of the function space $\mathcal{F}$ be denoted
by

$\displaystyle\Phi\triangleq\\{\phi_{1},\phi_{2},\ldots,\phi_{M}\\}\text{ with
}M\rightarrow\infty.$ (4)

Then, any function $\tilde{\phi}\in\mathcal{F}$ can be written as a linear
combination of the basis functions $\phi_{i}\in\Phi$ implying
$\tilde{\phi}=a^{T}\Phi$ with $a\in\mathbb{R}^{1\times M}$. A vector valued
observable (also referred to as a dictionary of observables) can be
constructed by taking a vector combination of the basis functions in
$\mathcal{F}$:

$\displaystyle\varphi(x)$
$\displaystyle=a^{T}\Phi(x)=\sum_{i=0}^{M}a_{i}\phi_{i}(x)$

where $\varphi:\mathcal{M}\rightarrow\mathbb{C}^{n_{\varphi}}$,
$a_{i}\in\mathbb{R}^{M}$, $b_{i}\in\mathbb{R}^{n_{\varphi}}$,

Then, $\varphi(x)$ is invariant under $\mathcal{K}$ as

$\displaystyle\mathcal{K}\varphi(x_{k})$
$\displaystyle=(\mathcal{K}a^{T}\Phi)(x_{k})=a^{T}(\mathcal{K}\Phi)(x_{k})$
$\displaystyle=a^{T}(\Phi\circ
f)(x_{k})=(a^{T}\Phi)(x_{k+1})=\varphi(x_{k+1}).$

Thus, every linear combination of basis functions in $\Phi$ is also a Koopman
observable; specifically the Koopman operator is a linear operator on the
function space $\mathcal{F}$ spanned by $\Phi$.

These observations also hold, more generally, when ${\tilde{\phi}}$ is a
vector-valued observable and $\mathcal{F}$ defines a space of vector-valued
functions. The Koopman operator $\mathcal{K}$, in this setting, acts on
vector-valued functions as opposed to scalar-valued functions.

The choice of $\varphi$ should be such that the true state $x$ can be
recovered. One such choice of $\varphi$ is given by

$\psi(x)=\begin{bmatrix}x^{T}&\varphi^{T}(x)\end{bmatrix}^{T}$ (5)

where $\psi:\mathcal{M}\rightarrow\mathbb{R}^{n_{L}}$ contains the base states
$x$ in addition to $\varphi$. Such an observable $\psi(x)$ is called a state-
inclusive Koopman observable [101]. For the rest of the paper, unless
explicitly stated, we denote a $\varphi$ to be a dictionary of nonlinear
observables (not necessarily state-inclusive) and $\psi$ to be a dictionary of
state-inclusive observables with the form (5).

###  3.2 Modal decomposition

The Koopman operator is infinite-dimensional as it acts on the functional
space $\mathcal{F}$. As presented in [68], we take the basis functions $\Phi$
in (4) to also be the set of eigenfunctions for $\mathcal{K}$ with
$\lambda_{i}$ being the eigenvalue of $\phi_{i}$ and
$\mathcal{K}\phi_{i}=\lambda_{i}\phi_{i}\quad\forall
i\in\\{1,2,\ldots,M\\}\quad M\rightarrow\infty.$ In this work we will assume
that every Koopman operator describes the dynamics of an analytical system
represented as (1). Such systems admit Koopman operators with countable
spectra. In this setting, the modal decomposition of the Koopman operator
dynamics becomes

$\displaystyle\varphi(x_{k+1})$ $\displaystyle=\varphi\circ
f(x_{k})=\mathcal{K}\varphi(x_{k})$
$\displaystyle=\sum_{i=1}^{M}b_{i}\mathcal{K}\phi_{i}(x_{k})=\sum_{i=1}^{M}b_{i}\lambda_{i}\phi_{i}(x_{k})$
(6)

where $b_{i}\in\mathbb{R}^{n_{\varphi}}$ are called the Koopman modes,
$\lambda_{i}$ are the Koopman eigenvalues and $\phi_{i}$ are the corresponding
Koopman eigenfunctions.

###  3.3 Koopman operators for conjugate dynamical systems

We review the theory of eigenfunction conjugacy for conjugate dynamical
systems [97]. This theoretical construction informs how we view sensor fusion
of dynamical systems, at large. Consider two nonlinear systems

$z_{k+1}^{(i)}=f_{i}(z_{k+1}^{(i)}),\hskip
5.0ptz^{(i)}\in\mathbb{R}^{n_{i}},\hskip 5.0pti=1,2$

with their corresponding Koopman operators $\mathcal{K}_{i}$. The two
dynamical system are said to be factor conjugate if there is a function map
$H:\mathbb{R}^{n_{1}}\rightarrow\mathbb{R}^{n_{2}},n_{1}\geq n_{2}$ where

$\displaystyle z^{(2)}$ $\displaystyle=H(z^{(1)})\text{ such that }H\circ
f_{1}(z^{(1)})=f_{2}\circ H(z^{(1)}).$

Then, if $\phi_{2}$ is an eigenfunction of the Koopman operator
$\mathcal{K}_{2}$ corresponding to the eigenvalue $\lambda_{2}$,
$\mathcal{K}_{2}\phi_{2}(z^{(2)})=\lambda_{2}\phi_{2}(z^{(2)})=\phi_{2}\circ
f_{2}(z^{(2)})$. Then, we can see that $(\phi_{2}\circ H)$ is an eigenfunction
of $\mathcal{K}_{1}$:

$\displaystyle\lambda_{2}(\phi_{2}\circ H)(z^{(1)})$
$\displaystyle=\phi_{2}\circ f_{2}\circ H(z^{(1)})=(\phi_{2}\circ H)\circ
f_{1}(z^{(1)})$ $\displaystyle=\mathcal{K}_{1}(\phi_{2}\circ H)(z^{(1)})$ (7)

The dynamics of the second system spanned by a set of eigenfunctions can be
mapped to the eigenfunctions of the first system. If $H$ is a $C^{k}$
diffeomorphism, i.e., $H$ has an inverse and is $k-times$ differentiable, then
we have a $C^{k}$ diffeomorphic conjugacy. In that case, we have that for all
the eigenfunctions $\phi_{i}$ such that
$\mathcal{K}_{i}\phi_{i}(z^{(1)})=\lambda\phi_{i}(z^{(i)}),i=1,2$ have a
bijective map:

$\phi_{1}(z^{(1)})=\phi_{2}\circ H(z^{(1)}),\hskip
3.0pt\phi_{2}(z^{(2)})=\phi_{1}\circ H^{-1}(z^{(2)}).$ (8)

The bijective map of eigenfunctions allows complete transfer of information
between the two systems. We use this concept to identify subspaces of the
state and output dynamics that are $C^{k}$ diffeomorphic conjugate and deduce
how much information can be fused.

###  3.4 Dynamic Mode Decomposition

There are a class of nonlinear systems that have exact finite invariant
Koopman operators [102]. But in most cases, the KOs are infinite-dimensional
and difficult to identify numerically. The DMD algorithm introduced in [67]
finds KOs that are exact solutions for linear systems and local approximators
for nonlinear systems. DMD uses the data matrices $X_{p}$ and $X_{f}$ as
defined in (3) to solve $K=\min_{K}||X_{f}-KX_{p}||=X_{f}X_{p}^{{\dagger}}$
where ${\dagger}$ denotes the Moore-Penrose pseudoinverse.

The extended DMD (E-DMD) algorithm proposed in [68] identifies KOs that
capture the nonlinear dynamics with better accuracy. E-DMD fuses kernel
methods in machine learning with DMD to identify a rich set of observables to
solve the optimization problem

$K=\min_{K}||\psi(X_{f})-K\psi(X_{p})||=\psi(X_{f})\psi(X_{p})^{{\dagger}}.$

E-DMD solves the nonlinear regression problem using linear least squares.
[103] shows the relevance of E-DMD to Koopman operators. E-DMD hinges on the
user specifying the lifting functions, and more often than not, it leads to an
explosion of the lifting functions [101, 104].

Recent developments in DMD incorporate deep learning approaches to identify
the observables using deep neural networks. [71, 105, 106, 107]. They can
approximate exponentially many distinct observable functions. We consider the
deep DMD formulation from [71]:

$\displaystyle\min_{\psi,K}$
$\displaystyle||\psi(x_{k+1})-K\psi(x_{k})||^{2}_{F}$ (9)
$\displaystyle\psi(x)=\begin{bmatrix}x\\\ \varphi(x)\end{bmatrix}=$
$\displaystyle\begin{bmatrix}x\\\ g_{n}\circ\sigma\circ\cdots\circ\sigma\circ
g_{2}\circ\sigma\circ g_{1}(x)\end{bmatrix}$

where $\psi(x)$ is represented by neural network representations with the
$i^{th}$ hidden layer captured by weights $W_{i}$, biases $b_{i}$, linear
function $g_{i}(x)=W_{i}x+b_{i}$ and activation function $\sigma$. Hence, the
estimation of $\psi$ boils down to learning the parameter set
$(W_{1},b_{1},W_{2},b_{2},\ldots,W_{n},b_{n})$, while specifying or optimizing
the depth of the network $n$ as a hyper-parameter. By selecting appropriate
activation functions for the nonlinear transformation $\sigma$, in a given
layer, e.g., using sigmoidal [108], rectified linear unit (ReLU) activation
functions [109] radial basis functions (RBFs) [110], $\psi(x)$ leverages
universal function approximation properties of each of these function classes
[71].

To derive the approximate Koopman eigenfunctions for any of these numerical
approximations of the Koopman operator, suppose $KV=V\Lambda$ is the
eigendecomposition of K where $v_{i}$ (the $i^{th}$ column of $V$) is the
right eigenvector corresponding to the eigenvalue $\lambda_{i}$ (the $i^{th}$
diagonal entry of the diagonal matrix $\Lambda$). Then,
$\psi(x_{k+1})=K\psi(x_{k})$ has the modal decomposition

$\psi(x_{k+1})=K\psi(x)=V\Lambda
V^{-1}\psi(x)=\sum_{i}v_{i}\lambda_{i}\phi_{i}(x)$ (10)

where $\phi_{i}(x)$ is the $i^{th}$ entry of the vector $V^{-1}\psi(x)$.
Comparing with (3.2), $\phi_{i}(x)$ and $v_{i}$ are the eigenfunction and the
Koopman mode corresponding to the eigenvalue $\lambda_{i}$. Thus, a spectral
decomposition on any finite approximation $K$ of the true Koopman operator
$\mathcal{K}$ gives an approximation to a subset of the true eigenfunctions of
$\mathcal{K}.$

We remark that the true Koopman observables and true Koopman eigenfunctions
for a given system often span an infinite dimensional space. In the case of
analytic state update equations for the system (1), the dimension of the
Koopman operator is usually countably infinite. Only in special cases is
exactly finite. This means that any finite-dimensional set of functions may
not exactly span or recover the Koopman observable function space. In the
sequel, we will refer to a finite collection of Koopman observable functions
$\psi_{1}(x),....,\psi_{n_{L}}(x)$ as a dictionary of Koopman observables.
However, these are, in practice, an approximation of a true spanning set for
the Koopman observable function space.

##  4 Output Constrained Koopman Operator Representations

We consider the fusion of state and output measurements from the nonlinear
system (1) using the Koopman operator sensor fusion method as delineated in
Section 3.3. To do so, we need to establish a factor conjugate map between the
state dynamics and the output dynamics.

We suppose the state dynamics is captured by the KO
$\psi(x_{k+1})=K\psi(x_{k})$. To integrate the state dynamics with the outputs
(that are nonlinear functions of the state), we consider the OC-KO (2).

The model structure of OC-KO is such that its dictionary of observables
$\psi(x)$ span the nonlinear output functions $h(x)$ by augmenting the KO with
a linear output equation $y_{k}=h(x_{k})=W_{h}\psi(x_{k})$. To establish a
conjugacy between the state and output dynamics, all we need is to do is
construct the dynamics of the output. The output dynamics can be given by

$y_{k+1}=W_{h}\psi(x_{k+1})=W_{h}K\psi(x_{k})=W_{h}KW_{\psi}y_{k}$ (11)

where $\psi(x_{k})=W_{\psi}y_{k}$. We begin by considering the simplest case
of inverting the $W_{h}$ matrix in $y_{k}=W_{h}\psi(x_{k})$ to identify the
$W_{\psi}$ matrix.

######  Theorem 1.

Given a nonlinear system (1) with a Koopman operator (2). Suppose
$\psi(x)\in\mathbb{R}^{n_{L}}$ is the vector of Koopman observable functions
for (2), not necessarily state-inclusive. For the number of outputs $p\geq
n_{L}$, the following statements are true.

  1. (i)

If $\quad rank(W_{h})=n_{L}$, then each scalar Koopman observable function
$\psi_{i}(x)$ lies in the span of the output functions $h(x)$

  2. (ii)

If $\quad rank(W_{h})=r<n_{L}$, then there exists a similarity transform $T$
that takes the model (2) to the form

$\displaystyle\tilde{\psi}(x_{k+1})$
$\displaystyle=\tilde{K}\tilde{\psi}(x_{k})$ (12) $\displaystyle y_{k}$
$\displaystyle=\tilde{W}_{h}\tilde{\psi}(x_{k})$

where $\tilde{K}=T^{T}KT$, $\tilde{\psi}(x)=T^{T}\psi(x)$,
$\tilde{W}_{h}=W_{h}T$ such that $r$ components of the vector Koopman
observable $\tilde{\psi}(x)$ given by $\psi_{h}(x)$ lie in the span of $h(x)$.

Case (1): Since $W_{h}$ has full column rank, an exact left inverse exists
such that

$\displaystyle\psi(x_{k})=(W_{h}^{T}W_{h})^{-1}W_{h}^{T}h(x_{k})\quad\forall\quad
x_{k}\in\mathcal{M}$

and hence $\psi_{i}\in span\\{h_{1},h_{2},...,h_{p}\\}$ $\forall$
$i\in\\{1,2,...,n_{L}\\}$.
Case (2): Suppose $rank(W_{h})=r<n_{L}$. The singular value decomposition of
$W_{h}$ yields

$\displaystyle W_{h}$
$\displaystyle=\begin{bmatrix}U_{r}&U_{p-r}\end{bmatrix}_{p\times
p}\begin{bmatrix}\Sigma_{r}&0\\\ 0&0\end{bmatrix}_{p\times
n_{L}}\begin{bmatrix}V_{r}^{T}\\\ V_{n_{L}-r}^{T}\end{bmatrix}_{n_{L}\times
n_{L}}$ $\displaystyle\Rightarrow h(x_{k})$
$\displaystyle=\begin{bmatrix}W_{h\psi}&0\end{bmatrix}\begin{bmatrix}\psi_{h}(x_{k})\\\
\bar{\psi}_{h}(x_{k})\end{bmatrix}\quad\forall\quad x_{k}\in\mathcal{M}$ (13)
$\displaystyle\text{s. t. }W_{h\psi}$ $\displaystyle=U_{r}\Sigma_{r},\hskip
1.5pt\psi_{h}(x)=V_{r}^{T}\psi(x),\hskip
1.5pt\bar{\psi}_{h}(x)=V_{n_{L}-r}^{T}\psi(x).$ (14)

$W_{h\psi}$ has a full column rank $r>p$ and hence

$\psi_{h}(x_{k})=(W_{h\psi}^{T}W_{h\psi})^{-1}W_{h\psi}^{T}h(x_{k})\quad\forall\quad
x_{k}\in\mathcal{M}.$

Therefore, under a similarity transformation
$T=\begin{bmatrix}V_{r}&V_{n_{L}-r}\end{bmatrix}$, the model (2) takes the
form of (12) with $T^{-1}=T^{T}$ where the first $r$ lifting functions of
$\tilde{\psi}$ given by $\psi_{h}$ lie in the span of $h$. ∎

######  Remark 2.

The $rank(W_{h})\geq p$ only if $h(x)$ is linearly independent $\forall
x\in\mathcal{M}$.

We see that if $W_{h}$ is full column rank, the outputs can be determined
completely by the OC-KO observables and the $W_{\psi}$ in (11) exists. In the
event that $n_{L}<p$, the map $y=W_{h}\psi(x)$ is a map from a low dimensional
space $\psi(x)\in\mathbb{R}^{n_{L}}$ to a high dimensional space
$y\in\mathbb{R}^{p}$ and factor conjugacy is not defined for that case (in
Section 3.3). Hence, we project the outputs to a lower dimensional space
$z=W_{h}^{T}y\in\mathbb{R}^{n_{L}}$ to find a conjugate map. The following two
corollaries illustrate how output functions can be used to identify all or a
subset of Koopman observables (in Section 3.3).

######  Corollary 3.

If $rank(W_{h})=n_{L}<p$, we can construct a diffeomorphic map between the
states and projected outputs $z=W_{h}^{T}y$. The $z$ dynamics are given by

$z_{k+1}=W_{h}^{T}W_{h}K(W_{h}^{T}W_{h})^{-1}z_{k}.$

For
$z=H(\psi(x))=W_{h}^{T}W_{h}\psi(x),H:\mathbb{R}^{n_{L}}\rightarrow\mathbb{R}^{n_{L}}$,
the dynamics of $\psi(x)$ and $z$ are diffeomorphic conjugate and the
eigenfunctions that capture their dynamics have a bijective map using (8). ∎

######  Corollary 4.

For $rank(W_{h})=n_{L}=p$, $W_{h}$ becomes an invertible square matrix and the
output dynamics in Corollary 3 simplifies to

$y_{k+1}=g(y_{k})=W_{h}KW_{h}^{-1}y_{k}.$

Hence, the dynamics of $y$ and $\psi(x)$ are diffeomorphic conjugate. ∎

The dynamics of the output can be constructed when $W_{h}$ is full column
rank. The dynamics of the entire output or the projected output has a
diffeomorphic conjugacy with the dynamics of the OC-KO observables $\psi(x)$
depending on whether $p=n_{L}$ or $p>n_{L}$ respectively under the map
$y=W_{h}\psi(x)$. The solution to the dictionary of Koopman observables that
capture the state dynamics are generally nonunique; they have infinite KOs for
a given system (1a). The fusion of the states and the outputs impose a
constraint on the observables of the KOs. We explore this in the following
corollary.

######  Corollary 5.

Given a finite number of nontrivial output equations (1b), let
$\mathcal{K}_{f}$ and $\mathcal{K}_{f,h}$ denote the set of all KOs consistent
with (1a) and (1) respectively. Any KO that satisfies (1) also satisfies (1a)
$\Rightarrow\mathcal{K}_{f,h}\subset\mathcal{K}_{f}$. In the case of Corollary
4, the output space captures the complete eigenfunction space of a KO that
solves (1a). But $\mathcal{K}_{f}$ contains more KOs whose eigenfunctions can
be constructed by taking repeated product of the current eigenfunctions which
cannot be spanned by the outputs
$\Rightarrow\mathcal{K}_{f,h}\neq\mathcal{K}_{f}$. Hence
$\mathcal{K}_{f,h}\subset\mathcal{K}_{f}$. ∎

We explicitly see that the output equations place a constraint on the OC-KO
observables and that the dynamics of the outputs can be constructed when
$rank(W_{h})=n_{L}$. In the case (ii) of Theorem 1 where $rank(W_{h})<n_{L}$,
using (4) to construct the output dynamics, we see that

$y_{k+1}=U_{r}\Sigma_{r}V_{r}^{T}KV_{r}\psi_{h}(x_{k})+U_{r}\Sigma_{r}V_{r}^{T}KV_{n-r}\bar{\psi}_{h}(x_{k})$
(15)

where $\bar{\psi}_{h}(x_{k})$ is a leakage term that cannot be represented in
the output space. Hence, the output cannot be constructed. To capture more
information on $\bar{\psi}_{h}(x_{k})$, we need to construct the time-delay
embedded outputs as the lifting functions [111, 112, 113]. This is a typical
approach used to construct KORs when outputs partially measure the states
[111, 112, 113, 114, 94, 90]. In the case of pure output measurements, the
time-delay embedded outputs capture the maximum dynamics that is observed by
the outputs. But, when we fuse it with the state dynamics, there is no
guarantee that that the output dynamics capture the entire state dynamics.
Hence, we consider a more general case where the outputs capture a portion of
the state dynamics. Since the OC-KO has the structure of a linear time
invariant system, we invoke the observable decomposition theorem from [115] to
separate out the dynamics of the observable lifted states that can be fused
with the output dynamics.

######  Theorem 6.

Given a nonlinear system (1) with KO (2). Suppose $\psi(x)$ is the dictionary
of Koopman observables for (2), not necessarily state-inclusive.

Then there exists a similarity transformation $T$ and a projection matrix
$W_{\psi}\in\mathbb{R}^{n_{o}\times(N+1)p}$ for some $N\in\mathbb{Z}_{\geq
0},n_{o}\leq n_{L}$ such that the dynamics of

  1. (i)

a subset of the observables $\psi_{o}(x)\in\mathbb{R}^{n_{o}}$ of the
transformed Koopman operator (under $T$)

$\displaystyle\begin{bmatrix}\psi_{o}(x_{k+1})\\\
\bar{\psi}_{o}(x_{k+1})\end{bmatrix}$ $\displaystyle=\begin{bmatrix}K_{o}&0\\\
K_{\bar{o}}&K_{22}\end{bmatrix}\begin{bmatrix}\psi_{o}(x_{k})\\\
\bar{\psi}_{o}(x_{k})\end{bmatrix}$ (16) $\displaystyle y_{k}$
$\displaystyle=\begin{bmatrix}W_{ho}&0\end{bmatrix}\begin{bmatrix}\psi_{o}(x)\\\\[-3.0pt]
\bar{\psi}_{o}(x)\end{bmatrix}$

$T^{-1}KT=\begin{bmatrix}K_{o}&0\\\\[-3.0pt]
K_{\bar{o}}&K_{22}\end{bmatrix},W_{h}T=\begin{bmatrix}W_{ho}&0\end{bmatrix},T^{-1}\psi(x)=\begin{bmatrix}\psi_{o}(x)\\\\[-3.0pt]
\bar{\psi}_{o}(x)\end{bmatrix}$

  1. (ii)

and the projected time-delay embedded output

$z_{k}=W_{\psi}\begin{bmatrix}y_{k}&y_{k+1}&\cdots&y_{k+N}\end{bmatrix}^{T}$
(17)

are diffeomorphically conjugate.

The Koopman operator representation (2) is a linear time invariant model.
Using the observable decomposition theorem (Theorem 16.2 in [115]), there
exists a similarity transformation $T$ that takes the system (2) to the form
(16) such that the subsystem

$\displaystyle\psi_{o}(x_{k+1})$ $\displaystyle=K_{o}\psi_{o}(x_{k})\triangleq
g_{1}(\psi_{o}(x_{k}))$ $\displaystyle y_{k}$
$\displaystyle=W_{ho}\psi_{o}(x_{k})$

is completely observable; $\psi_{o}(x)\in\mathbb{R}^{n_{o}}$ can be uniquely
reconstructed from the outputs. Note that the system being observable is
different from the observable functions in the context of Koopman. Then, there
exists a $N\in\mathbb{Z}_{\geq 0}$ such that $Np\geq n_{o}$ and

$\begin{bmatrix}y_{k}\\\\[-3.0pt] y_{k+1}\\\\[-3.0pt] \vdots\\\\[-3.0pt]
y_{k+N}\end{bmatrix}=\begin{bmatrix}W_{ho}\\\\[-3.0pt] W_{ho}K_{o}\\\\[-3.0pt]
\vdots\\\\[-3.0pt]
W_{ho}K_{o}^{N}\end{bmatrix}\psi_{o}(x_{k})=\mathcal{O}\psi_{o}(x_{k})$

where $\mathcal{O}\in\mathbb{R}^{Np\times n_{o}}$ has full column rank
$n_{o}$. Then we can define a projection $W_{\psi}=\mathcal{O}^{T}$ to get
(17) such that

$z_{k}=\mathcal{O}^{T}\mathcal{O}\psi_{o}(x_{k})\triangleq H(\psi_{o}(x))$

where $\mathcal{O}^{T}\mathcal{O}\in\mathbb{R}^{n_{o}\times n_{o}}$ is square
invertible. The dynamics of $z$ is given by

$\displaystyle z_{k+1}$
$\displaystyle=\mathcal{O}^{T}\mathcal{O}\psi_{o}(x_{k+1})=\mathcal{O}^{T}\mathcal{O}K_{o}\psi_{o}(x_{k})$
$\displaystyle=\mathcal{O}^{T}\mathcal{O}K_{o}(\mathcal{O}^{T}\mathcal{O})^{-1}z_{k}\triangleq
g_{2}(z_{k})$

For the map $H:\mathbb{R}^{n_{o}}\rightarrow\mathbb{R}^{n_{o}}$, there exists
an inverse map
$H^{-1}:\mathbb{R}^{n_{o}}\rightarrow\mathbb{R}^{n_{o}},H^{-1}(z)=(\mathcal{O}^{T}\mathcal{O})^{-1}z_{k}$
since $\mathcal{O}^{T}\mathcal{O}$ is full rank and invertible. Using the
above, we can see that

$\displaystyle H\circ g_{1}(\psi_{o}(x_{k}))$ $\displaystyle=g_{2}\circ
H(\psi_{o}(x_{k}))=\mathcal{O}^{T}\mathcal{O}\psi_{o}(x_{k+1})$ $\displaystyle
g_{1}\circ H^{-1}(z_{k})$ $\displaystyle=H^{-1}\circ
g_{2}(z_{k})=(\mathcal{O}^{T}\mathcal{O})^{-1}z_{k+1}.$

Hence, the dynamics of $z$ and $\psi_{o}(x)$ are diffeomorphically conjugate.
∎

######  Remark 7.

The original time-delay embedded output evolves in a high dimensional space
$\mathbb{R}^{Np}$ when compared to $\psi(x_{o})\in\mathbb{R}^{n_{o}}$ since we
are seeking a sufficiently large set of output measurements $N$, such that
$N\times p\geq n_{o}$. In that case, the factor conjugate map cannot be
established similar to the case in Corollary 3. This observation motivates the
construction of a time-delayed output embedding.

When outputs partially measure the states, we see that time-delay embedded
outputs have a diffeomorphic map with a subspace of the lifting functions
under a similarity transform and the dynamics evolving in the two spaces are
diffeomorphically conjugate.

######  Corollary 8.

When $n_{o}=n_{L}$, $z_{k}$ has a diffeomorphic map with the entire dictionary
of observables $\psi(x_{k})$. In this case, $z_{k}$ can constitute a Koopman
observable basis such that it captures the dynamics of $\psi(x_{k})$.

######  Corollary 9.

The scenario where $N=0$ results in case (ii) of Theorem 1 with
$\psi_{h}(x_{k})=\psi_{o}(x_{k})$. This simplifies the output dynamics (15) to

$y_{k+1}=\begin{cases}W_{ho}K(W_{ho}^{T}W_{ho})^{-1}W_{ho}^{T}y_{k}&,p>n_{o}\\\
W_{ho}KW_{ho}^{-1}y_{k}&,p=n_{o}\end{cases}$

We see that the OC-KO architecture can fuse the state and output dynamics even
if the outputs do not capture the entire state dynamics. The above analysis
shows that the OC-KO structure is such that the lifting functions capture the
dynamics of the time-delay embedded outputs. This is an implicit constraint in
the model structure of OC-KO.

State-inclusive observables (5) are useful since we can recover the dynamics
by simply dropping the nonlinear observables. We show a sufficient condition
for the existence of state-inclusive OC-KOs using a similar argument developed
in [83]. We prove the following lemma which plays a crucial role in showing
the invariance of the basis in the series expansions of analytic functions.

######  Lemma 10.

Given a dictionary of observable functions
$\mathcal{D}=\\{\psi_{1}(x),\psi_{2}(x),\ldots\\}$ where
$\psi_{r}(x)=\displaystyle\prod_{i=1}^{n}x_{i}^{p_{r,i}}$ with
$p_{r,i}\in\mathbb{Z}_{\geq 0}$, the product of functions from $\mathcal{D}$
lies in $\mathcal{D}$.

Consider two functions $\psi_{\alpha}(x),\psi_{\beta}(x)\in\mathcal{D}$. Their
product is given by

$\displaystyle\psi_{\alpha}(x)\psi_{\beta}(x)$
$\displaystyle=\prod_{i=1}^{n}x_{i}^{p_{\alpha,i}}\prod_{i=1}^{n}x_{i}^{p_{\beta,i}}=\prod_{i=1}^{n}x_{i}^{(p_{\alpha,i}+p_{\beta,i})}\in\mathcal{D}.$

Since the product of two functions lies in $\mathcal{D}$, the product of any
number of functions from $\mathcal{D}$ also lies in $\mathcal{D}$ as can be
seen by taking repeated product of functions. ∎

If the monomial basis in the Taylor series expansion are propagated by one
time step, we encounter a product of these monomials and Lemma 10 defines a
set that captures all these monomial functions and their products. We use the
this lemma to build on the result from [83], which shows the existence of
state-inclusive KO for (1a), to find state-inclusive OC-KOs for (1).

######  Proposition 11.

Given the nonlinear system of the form (1), if the functions $f$ and $h$ are
real analytic on the open set $\mathcal{M}$, then there exists an OC-KO
representation of the form (2) in the region of convergence of the Taylor
series expansion of $f$ and $h$.

Let us consider a dictionary of polynomial lifting functions
$\mathcal{D}=\\{\psi_{1}(x),\psi_{2}(x),\ldots\\},\quad\psi_{r}(x)=\displaystyle\prod_{i=1}^{n}x_{i}^{p_{r,i}}$
where $p_{r,i}\in\mathbb{Z}^{+}$. Since $f$ is real analytic in $\mathcal{M}$,
for any $x_{0}\in\mathcal{M}$, there exists a Taylor series expansion centered
about $x_{0}$ that converges to $f(x)$ for any neighborhood of $x_{0}$.
Suppose $f(x)=\begin{bmatrix}f_{1}(x)&\cdots f_{n}(x)\end{bmatrix}^{T}$, the
Taylor series expansion of $f$ about $x_{0}$ yields

$f_{i}(x)=f_{i}(x_{0})+\frac{\partial f_{i}}{\partial
x}\Bigg{|}_{x_{0}}x+x^{T}\frac{\partial^{2}f_{i}}{\partial
x^{2}}\Bigg{|}_{x_{0}}x+\cdots=\sum_{j=1}^{\infty}c_{ij}\psi_{j}(x).$

where each term lies in $\mathcal{D}$. Suppose
$x_{k}=\begin{bmatrix}x_{k,1}&\cdots x_{k,n}\end{bmatrix}^{T}$ where $x_{k,i}$
indicates the $i^{th}$ state at discrete time index $k$. $x_{k}$ propagated by
one time step yields a linear combination of functions in $\mathcal{D}$ as

$x_{k+1,i}=f_{i}(x)=\sum_{j=1}^{\infty}c_{ij}\psi_{j}(x_{k}).$

To construct a linear system, we propagate each function on the right hand
side $\psi_{j}(x_{k})$ by one time step

$\psi_{j}(x_{k+1})=\displaystyle\prod_{i=1}^{n}x_{k+1,i}^{p_{j,i}}=\displaystyle\prod_{i=1}^{n}\Big{(}\sum_{j=1}^{\infty}c_{ij}\psi_{j}(x_{k})\Big{)}^{p_{j,i}}$

Since Lemma 10 states that the product of any number of functions in
$\mathcal{D}$ lies in $\mathcal{D}$,
$\psi_{j}(x_{k+1})=\sum_{r=1}^{\infty}k_{ir}\psi_{r}(x_{k})$. Concatenating
the expressions of $x_{k+1,j}$ and $\psi_{j}(x_{k+1})$ for all $j$, we get the
Koopman representation $\psi(x_{k+1})=K\psi(x_{k})$.

Similarly, for the output equation, we can expand each function $h_{i}$ in
$h(x)=\begin{bmatrix}h_{1}(x)&h_{2}(x)&\cdots h_{m}(x)\end{bmatrix}^{T}$ using
the Taylor series expansion about $x=x_{0}$ to yield

$\displaystyle h_{i}(x)$ $\displaystyle=h_{i}(x_{0})+\frac{\partial
h_{i}}{\partial x}\Bigg{|}_{x_{0}}x+x^{T}\frac{\partial^{2}h_{i}}{\partial
x^{2}}\Bigg{|}_{x_{0}}x+\cdots=\sum_{j=1}^{\infty}w_{ij}\psi_{j}(x)$
$\displaystyle\Rightarrow y_{k}$ $\displaystyle=W_{h}\psi(x_{k})$

Hence, if $f$ and $h$ of the nonlinear system (1) are analytic in the open set
$\mathcal{M}$, a state inclusive OC-KO of the form (2) exists for that system.
∎

We see that state-inclusive OC-KOs exist for nonlinear systems whose dynamics
($f$) and output ($h$) functions are real analytic. This is a sufficient
condition and not a necessary one. In the next section, we explore how to
identify OC-KOs from data using the DMD formulation.

##  5 DMD with output constraints

The OC-KO identification involves fusing two datasets (states and outputs) to
learn the dynamics of the nonlinear system (1). The DMD algorithms typically
identify KOs on one dataset. We introduce the more general OC-DMD formulation
that incorporates the output constraint in the DMD problem to identify the OC-
KOs. The OC-DMD formulation is

$\begin{split}\min_{K,W_{h},\psi}||\psi(X_{F})&-K\psi(X_{P})||^{2}_{F}\\\
\text{such that \hskip 25.0pt}Y_{P}&=W_{h}\psi(X_{P})\end{split}$ (18)

where $||.||_{F}$ is the Frobenius norm. The equality constraint is very
stringent as the presence of output measurement noise could result in overfit
models. Hence, we pose the weaker problem

$\displaystyle\min_{\psi,K,W_{h}}\Bigg{|}\Bigg{|}\begin{bmatrix}\psi(X_{F})\\\
Y_{P}\end{bmatrix}-\begin{bmatrix}K\\\
W_{h}\end{bmatrix}\psi(X_{P})\Bigg{|}\Bigg{|}^{2}_{F}$ (19)

which concurrently solves for $\psi(x),K$ and $W_{h}$. We refer to this as the
direct OC-DMD problem formulation. To explicitly show the effect of the output
dynamics in learning OC-KOs, we also propose the sequential OC-DMD problem
formulation where the following sub-problems are solved sequentially:

1\. Identification of Koopman dynamics:
$\displaystyle\min_{\psi_{x},K_{1}}||\psi_{x}(X_{F})$ $\displaystyle-
K_{1}\psi_{x}(X_{P})||^{2}_{F}$ (20b) 2\. Output Parameterization:
$\displaystyle\min_{\varphi_{y},W_{h1}}\Big{|}\Big{|}Y_{P}$ $\displaystyle-
W_{h1}\begin{bmatrix}\psi_{x}(X_{P})\\\
\varphi_{y}(X_{P})\end{bmatrix}\Big{|}\Big{|}^{2}_{F}$ (20c) 3\. Approximate
Koopman Closure:
$\displaystyle\min_{\varphi_{xy},K_{2}}\Bigg{|}\Bigg{|}\begin{bmatrix}\varphi_{y}(X_{F})\\\
\varphi_{xy}(X_{F})\end{bmatrix}$ $\displaystyle-
K_{2}\begin{bmatrix}\psi_{x}(X_{P})\\\ \varphi_{y}(X_{P})\\\
\varphi_{xy}(X_{P})\end{bmatrix}\Bigg{|}\Bigg{|}^{2}_{F}.$ (20d)

This problem is called sequential OC-DMD, because it obtains a solution for
the output-constrained Koopman learining problem as a sequence of optimization
problems. The solution generated by sequential OC-DMD, yields an OC-KO of the
form:

$\displaystyle\psi(x_{k+1})$ $\displaystyle=K\psi(x_{k})$ $\displaystyle
y_{k}$ $\displaystyle=W_{h}\psi(x_{k})$

where

$\displaystyle\psi(x)$ $\displaystyle=\begin{bmatrix}x\\\ \varphi_{x}(x)\\\
\varphi_{y}(x)\\\
\varphi_{xy}(x)\end{bmatrix},\,\,K=\begin{bmatrix}K_{1}&K_{12}&0&0\\\
K_{21}&K_{22}&0&0\\\ K_{31}&K_{32}&K_{33}&K_{34}\\\
K_{41}&K_{42}&K_{43}&K_{44}\end{bmatrix},$ (21) $\displaystyle K_{1}$
$\displaystyle=\begin{bmatrix}K_{11}&K_{12}\\\
K_{21}&K_{22}\end{bmatrix},\,\,K_{2}=\begin{bmatrix}K_{31}&K_{32}&K_{33}&K_{34}\\\
K_{41}&K_{42}&K_{43}&K_{44}\end{bmatrix},$ $\displaystyle W_{h}$
$\displaystyle=\begin{bmatrix}W_{h11}&W_{h12}&W_{h13}&0\end{bmatrix}$

where $\psi(x)=\begin{bmatrix}x^{T}&\varphi^{T}(x)\end{bmatrix}^{T}$,
$x\in\mathcal{M}\subset\mathbb{R}^{n}$,
$\varphi_{x}:\mathcal{M}\rightarrow\mathbb{R}^{n_{x}}$,
$\varphi_{y}:\mathcal{M}\rightarrow\mathbb{R}^{n_{y}}$,
$\varphi_{xy}:\mathcal{M}\rightarrow\mathbb{R}^{n_{xy}}$,
$n+n_{x}+n_{y}+n_{xy}=n_{L}$ and the output matrix
$W_{h1}=\begin{bmatrix}W_{h11}&W_{h12}&W_{h13}\end{bmatrix}$.

Sequential OC-DMD works to first solve for the KO of the state dynamics (20b),
without accounting for any output measurements. This represents the Koopman
operator obtained from standard dynamic mode decomposition (or E-DMD)
algorithms. The next step in sequential OC-DMD (20c) solves for the projection
equation to parameterize output functions in terms of the existing Koopman
observables $\psi_{x}(x)$, as well as any necessary additional output
observables $\varphi_{y}(x)$ required to predict the output equation. The last
step (20d) then incorporates additional state-dependent observable dictionary
functions $\varphi_{xy}(x)$ to guarantee closure of the new Koopman
$\varphi_{y}(x)$ observables from step 2 (20d).

###  5.1 Equivalence of Solution Spaces for Sequential and Direct OC-DMD

We now elucidate the relationship between the solution space of sequential and
direct OC-DMD optimization problems. To do so, we make use of the following
proposition regarding coordinate transforms which considers a manifold $M$ of
dimension n ($dim(M)=n$).

######  Proposition 12 (Proposition 2.18 from [116]).

Suppose that $dim(M)=n$ and that $f_{1},...,f_{n},k\leq n$ are independent
functions about $p\in M$. Then there exists a neighborhood $U$ about $p$, and
functions $x_{k+1},...,x_{n}$ such that
$(U,f_{1},...,f_{k},x_{k+1},...,x_{n})$ is a coordinate chart.

We use the coordinate transformation to separate out the observables that
capture the state dynamics and the output equations minimally to show the
mapping between the OC-KOs from both OC-DMD problems. We go ahead to prove the
equivalency of the OC-DMD algorithms.

######  Theorem 13.

The optimization problems (19) and (20) are equivalent in the minimal solution
space where the observables are independent.

To prove the equivalency of (19) and (20), we need to show that for every
solution of (19) given by (2), there is a solution for (20) given by (21) and
vice versa. It is easy to see that (21) fits into the structure of (2)
directly without any modification. Hence, we only need to prove the reverse.
Suppose the OC-KO

$\displaystyle\begin{bmatrix}x_{k+1}\\\ \varphi_{y^{*}}(x_{k+1})\end{bmatrix}$
$\displaystyle=\begin{bmatrix}\tilde{K}_{11}&\tilde{K}_{12}\\\
\tilde{K}_{21}&\tilde{K}_{22}\end{bmatrix}\begin{bmatrix}x_{k}\\\
\varphi_{y^{*}}(x_{k})\end{bmatrix}$ (22) $\displaystyle y_{k}$
$\displaystyle=\begin{bmatrix}\tilde{W}_{h1}&\tilde{W}_{h2}\end{bmatrix}\begin{bmatrix}x_{k}^{T}&\varphi^{T}_{y^{*}}(x_{k})\end{bmatrix}^{T}$

where $\varphi_{y^{*}}:\mathcal{M}\rightarrow\mathbb{R}^{n_{y^{*}}}$ is the
minimal dictionary of observables that solves (19). Let us consider the
minimal dictionary of observables to capture the state dynamics (1a) be
$\varphi_{x}(x)=T_{11}\varphi_{y^{*}}(x),\varphi_{x}:\mathcal{M}\rightarrow\mathbb{R}^{n_{x}}$
such that $n_{x}\leq n_{y^{*}}$. $n_{x}=n_{y^{*}}$ when $\varphi_{y^{*}}$ is
the minimal dictionary that captures (1a). Since the functions
$\varphi_{y^{*}}(x)$ is the minimal OC-KO solution
$dim(\varphi_{y^{*}}(x))=n_{y^{*}}$ with $n_{x}$ independent functions
$\varphi_{x}(x)=\varphi_{x}(\varphi_{y^{*}}(x))=T_{11}\varphi_{y^{*}}(x)$.
Using Proposition 12, a coordinate transformation
$\tilde{T}_{1}=\begin{bmatrix}\tilde{T}_{11}&\tilde{T}_{12}\end{bmatrix}$
exists such that

$\varphi_{y^{*}}(x)=\begin{bmatrix}\tilde{T}_{11}&\tilde{T}_{12}\end{bmatrix}\begin{bmatrix}\varphi_{x}(x)\\\
\tilde{\varphi}_{y}(x)\end{bmatrix},\quad\begin{bmatrix}\varphi_{x}(x)\\\
\tilde{\varphi}_{y}(x)\end{bmatrix}=\begin{bmatrix}T_{11}\\\
T_{12}\end{bmatrix}\varphi_{y^{*}}(x)$

where $\tilde{T}^{-1}=\begin{bmatrix}T_{11}^{T}&T_{12}^{T}\end{bmatrix}^{T}$.
Since $\varphi_{x}(x)$ is sufficient to capture the KO for (1a),
$\tilde{K}_{12}\tilde{T}_{12}=T_{11}\tilde{K}_{22}\tilde{T}_{12}=0$ and the
transformed dynamics are

$\displaystyle\begin{bmatrix}x_{k+1}\\\ \varphi_{x}(x_{k+1})\\\
\tilde{\varphi}_{y}(x_{k+1})\end{bmatrix}=\begin{bmatrix}\tilde{K}_{11}&\tilde{K}_{12}\tilde{T}_{11}&0\\\
T_{11}\tilde{K}_{21}&T_{11}\tilde{K}_{22}\tilde{T}_{11}&0\\\
T_{12}\tilde{K}_{21}&T_{12}\tilde{K}_{22}\tilde{T}_{11}&T_{12}\tilde{K}_{22}\tilde{T}_{12}\end{bmatrix}\begin{bmatrix}x_{k}\\\
\varphi_{x}(x_{k})\\\ \tilde{\varphi}_{y}(x_{k})\end{bmatrix}$ $\displaystyle
y_{k}=\begin{bmatrix}\tilde{W}_{h1}&\tilde{W}_{h2}\tilde{T}_{11}&\tilde{W}_{h2}\tilde{T}_{12}\end{bmatrix}\begin{bmatrix}x_{k}^{T}&\varphi^{T}_{x}(x_{k})&\tilde{\varphi}_{y}(x_{k})\end{bmatrix}^{T}$

Suppose there exists functions $\varphi_{y}(x)=T_{21}\tilde{\varphi}_{y}(x)$
$\forall$ $x\in\mathcal{M}$ where
$\varphi_{y}:\mathcal{M}\rightarrow\mathbb{R}^{n_{y}}$,
$n_{y}\leq(n_{y^{*}}-n_{x})$ that in addition to the functions $x,\varphi_{x}$
are sufficient to capture the output equation, a similar procedure to the
above can be adopted to result in the model structure (21).Therefore, the two
optimization problems are equivalent. ∎

This theorem shows that if an OC-KO representation exactly captures state and
output dynamics, without any redundancy in any of the dictionary functions,
for every solution in sequential OC-DMD, we can find a solution in direct OC-
DMD and vice versa [117]. We thus see that (19) and (20) are equivalent
optimization problems. The equivalency does not imply they are the same
optimization problem because the objective function that they solve are
different [117].

Specifically, the model structure of a solution obtained from sequential OC-
DMD (21) is more sparse than the model structure of a solution obtained from
direct OC-DMD (2). The model (21) explicitly reveals that the output dynamics
constrain the OC-KO learning problem through the dictionary functions
$\varphi_{y}$ and $\varphi_{xy}$. The advantage of direct OC-DMD problem is
that it solves only one optimization problem as opposed to sequential OC-DMD
which solves three. We shall compare the performances of these algorithms in
Section 6 with theoretical and numerical examples.

###  5.2 Coordinate Transformations of Standardization Routines on System
State and Output Data

A common practice in model identification problem is to scale the variables
using standardization or normalization techniques to ensure uniform learning
of all variables. Standardization of a scalar variable $\tilde{x}$ yields

$\displaystyle\tilde{x}_{standardized}=\frac{\tilde{x}-\mu(\tilde{x})}{\sigma(\tilde{x})}$
(23)

where $\mu$ and $\sigma$ are the mean and standard deviation of $\tilde{x}$.
It is important to keep track of how such affine transformations modify the
structure of OC-KO when comparing theoretical and practical results.

######  Proposition 14.

Given the nonlinear system with output (1) has a Koopman operator
representation with observables given by

$\displaystyle\begin{bmatrix}x_{k+1}\\\ \varphi(x_{k+1})\end{bmatrix}$
$\displaystyle=\begin{bmatrix}K_{11}&K_{12}\\\
K_{21}&K_{22}\end{bmatrix}\begin{bmatrix}x_{k}\\\ \varphi(x_{k})\end{bmatrix}$
(24) $\displaystyle y_{k}$
$\displaystyle=\begin{bmatrix}W_{h1}&W_{h2}\end{bmatrix}\begin{bmatrix}x_{k}\\\
\varphi(x_{k})\end{bmatrix}$ (25)

if the states and outputs undergo a bijective affine transformation
$\tilde{x}=Px+b$ and $\tilde{y}=Qy+c$ where $P,Q$ are non-singular, then the
state dynamics are transformed to

$\displaystyle\tilde{\psi}(\tilde{x}_{k+1})$
$\displaystyle=\tilde{K}\tilde{\psi}(\tilde{x}_{k})$
$\displaystyle\tilde{\psi}(\tilde{x}_{k})$
$\displaystyle=\begin{bmatrix}\tilde{x}_{k+1}&\tilde{\varphi}(\tilde{x}_{k+1})&1\end{bmatrix}^{T}$
(26) $\displaystyle\tilde{K}$
$\displaystyle=\begin{bmatrix}PK_{11}P^{-1}&PK_{12}&(\mathbb{I}-PK_{11}P^{-1})b\\\
K_{21}P^{-1}&K_{22}&K_{21}P^{-1}b\\\ 0&0&1\end{bmatrix}$

and the output equations become

$\displaystyle\tilde{y_{k}}$
$\displaystyle=\tilde{W}_{h}\tilde{\psi}(\tilde{x}_{k})$ (27)
$\displaystyle\tilde{W}_{h}$
$\displaystyle=\begin{bmatrix}QW_{h1}P^{-1}&QW_{h2}&(QW_{h1}P^{-1}b+c)\end{bmatrix}.$

When the state undergoes an affine transformation $\tilde{x}=Px+b$, since the
transformation is bijective($P^{-1}$ exists), the dynamics of the transformed
state ($\tilde{x}$) are given by substituting $x=P^{-1}\tilde{x}-P^{-1}b$ in
(24) to yield

$\displaystyle\begin{bmatrix}P^{-1}\tilde{x}_{k+1}-P^{-1}b\\\
\varphi(P^{-1}\tilde{x}_{k+1}-P^{-1}b)\end{bmatrix}$
$\displaystyle=\begin{bmatrix}K_{11}&K_{12}\\\
K_{21}&K_{22}\end{bmatrix}\begin{bmatrix}P^{-1}\tilde{x}_{k}-P^{-1}b\\\
\varphi(P^{-1}\tilde{x}_{k}-P^{-1}b)\end{bmatrix}.$

We define new observable functions
$\tilde{\varphi}(\tilde{x})\triangleq\varphi(P^{-1}\tilde{x}-P^{-1}b)$ and the
complete vector valued observable as
$\tilde{\psi}(\tilde{x})\triangleq\begin{bmatrix}\tilde{x}&\tilde{\varphi}(\tilde{x})&1\end{bmatrix}^{T}$.
By algebraic manipulation, we get the transformed state dynamic as given in
(14).

If the output undergoes an affine transformation $\tilde{y}=Qy+c$, we derive
the transformed output in terms of the transformed state:

$\displaystyle\tilde{y}_{k}$
$\displaystyle=Qh(x_{k})+c=Q\begin{bmatrix}W_{h1}&W_{h2}\end{bmatrix}\begin{bmatrix}P^{-1}\tilde{x}_{k}-P^{-1}b\\\
\tilde{\varphi}(\tilde{x})\end{bmatrix}+c$

By simple algebraic manipulation, we end up with the affine transformed output
equation(27). ∎

We see that the bias in the affine transformation constrains an eigenvalue of
the transformed OC-KO to be equal to 1. This is very important to track when
using gradient descent based optimization algorithms to solve for OC-KOs
because they identify approximate solutions and this could push the unit
eigenvalue outside the unit circle making it unstable. To avoid numerical
error in such algorithms, we should constrain the last row of the Koopman
operator as in (14).

In both the OC-DMD problems (19) and (20), the state-inclusive observables
$\psi$ are considered as free variables. To learn the observables, we use the
deepDMD formulation (9) of representing $\psi$ as outputs of neural networks.
When we incorporate the deepDMD formulation to solve the OC-DMD problems, we
refer to them as OC-deepDMD algorithms and the identified OC-KOs as OC-deepDMD
models.

##  6 Simulation Results

We consider three numerical examples in increasing order of complexity to
evaluate the performance of the direct and sequential OC-deepDMD algorithms.
The first example has an OC-KO with exact finite closure; there is a finite-
dimensional basis in which the dynamics are linear. We use this as the
benchmark for the comparison of the two algorithms. The other two examples do
not possess finite exact closure. In those cases, we benchmark the proposed
algorithms against nonlinear state-space models (with outputs) identified by
solving

$\displaystyle\min_{f,h}||X_{F}-f(X_{P})||_{F}^{2}+||Y_{P}-h(X_{P})||_{F}^{2}$
(28)

where the functions $f$ and $h$ are jointly represented by a single feed-
forward neural network with $(n+p)$ outputs and we refer generally to this
model, across multiple examples, as the nonlinear state-space model (see
captions in Figures 4 and 6).

Table 1: Optimal hyper-parameters and performance of the oc-deepDMD models for all numerical examples System | Model | Hyperparameters | $r^{2}_{test}(x)$ | $r^{2}_{test}(x)$ | $r^{2}_{test}(y)$
---|---|---|---|---|---
|  |  | (1-step) | (n-step) |
$\begin{array}[]{c}\text{Finite}\\\ \text{Closure}\end{array}$ | Deep DMD ($n_{L}=3$) | $n_{x}=1,n_{xl}=8,n_{xn}=2$ | $1$ | $1$ | -
Direct OC-deepDMD ($n_{L}=3$) | $n_{x}=1,n_{xl}=8,n_{xn}=2$ | $0.899$ | $0.926$ | $-0.147$
Direct OC-deepDMD ($n_{L}=5$) | $n_{x}=3,n_{xl}=7,n_{xn}=5$ | $1$ | $1$ | $1$
$\begin{array}[]{c}\text{\scriptsize{Sequential OC-deepDMD}}\\\ \text{\scriptsize{($n_{L}=5$)}}\end{array}$ | $\begin{array}[]{c}n_{x}=1,n_{xl}=8,n_{xn}=2,n_{y}=1,n_{yl}=9,\\\ n_{yn}=4,n_{xy}=1,n_{xyl}=7,n_{xyn}=2\end{array}$ | $1$ | $1$ | $1$
$\begin{array}[]{c}\text{MEMS}\\\ \text{Actuator}\end{array}$ | Nonlinear state-space | $n_{xl}=6,n_{xn}=6$ | $1$ | $0.998$ | $1$
Direct OC-deepDMD | $n_{x}=6,n_{xl}=3,n_{xn}=6$ | $1$ | $0.84$ | $0.995$
Sequential OC-deepDMD | $\begin{array}[]{c}n_{x}=5,n_{xl}=3,n_{xn}=12,n_{y}=1,n_{yl}=8,\\\ n_{yn}=6,n_{xy}=3,n_{xyl}=8,n_{xyn}=3\end{array}$ | $0.999$ | $0.883$ | $0.999$
$\begin{array}[]{c}\text{Activator-}\\\ \text{Repressor}\\\ \text{clock}\end{array}$ | Nonlinear state-space | $n_{xl}=5,n_{xn}=6$ | $1$ | $0.854$ | $0.999$
Direct OC-deepDMD | $n_{x}=9,n_{xl}=6,n_{xn}=12$ | $0.999$ | $0.53$ | $0.983$
Sequential OC-deepDMD | $\begin{array}[]{c}n_{x}=3,n_{xl}=9,n_{xn}=8,n_{y}=1,n_{yl}=9,\\\ n_{yn}=4,n_{xy}=3,n_{xyl}=9,n_{xyn}=3\end{array}$ | $1$ | $0.349$ | $0.998$
$\begin{array}[]{c}\text{\scriptsize{Time-delay embedded }}\\\ \text{\scriptsize{Direct OC-deepDMD} }\end{array}$ | $\begin{array}[]{c}n_{x}=4,n_{xl}=8,\\\ n_{xn}=9,n_{d}=6\end{array}$ | $0.995$ | $0.9116$ | $0.8791$

The neural networks in each optimization problem are constrained to have an
equal number of nodes in each hidden layer. The hyperparameters for all the
optimization problems can be jointly given by
$\\{n_{ij}|i\in\\{x,y,xy\\},j\in\\{o,l,n\\}\\}$ where $n_{io}$, $n_{il}$ and
$n_{in}$ indicate the number of outputs, number of hidden layers and number of
nodes in each hidden layer for the dictionary of observables indicated by $i$
($\psi_{x},\varphi_{y}$ or $\varphi_{xy}$). Sequential OC-deepDMD comprises
$i\in\\{x,y,xy\\}$ $j\in\\{o,l,n\\}$, direct OC-deepDMD comprises
$i\in\\{x\\}$ $j\in\\{o,l,n\\}$ and nonlinear state-space model comprises
$i\in\\{x\\}$ and $j\in\\{l,n\\}$.

In each example, the simulated datasets are split equally between training,
validation, and test data. For each algorithm, we learn models on the training
data with various combinations of the hyperparameters. We train the models in
Tensorflow using the Adagrad [118] optimizer with exponential linear unit
(ELU) activation functions. We use the validation data to identify the model
with optimal hyperparameters for each optimization problem, which we report in
Table 1. To quantify the performance of each model, we use the coefficient of
determination ($r^{2}$) to evaluate the accuracy of the model predictions:

$r^{2}=1-\frac{||\tilde{X}-\hat{\tilde{X}}||_{F}^{2}}{||\tilde{X}||_{F}^{2}}$

where $\tilde{X}$ is the variable of interest ($X_{F}$ or $Y_{P}$) and
$\hat{\tilde{X}}$ is the prediction of that variable. We evaluate $r^{2}$ for
the accuracy of

  * $\bullet$

the output prediction: $y_{k}=h(x_{k})$ for the nonlinear state-space model
and $y_{k}=W_{h}\psi(x_{k})$ for the OC-deepDMD models

  * $\bullet$

1-step state prediction: $x_{k+1}=f(x_{k})$ for the nonlinear state-space
model and $\psi(x_{k+1})=K\psi(x_{k})$ for the OC-deepDMD models.

  * $\bullet$

n-step state prediction: $x_{i}=\underbrace{f\circ f\circ\cdots\circ
f}_{i\text{ times}}(x_{0})$ for the nonlinear state-space model and
$\psi(x_{i})=K^{i}\psi(x_{0})$ for the OC-deepDMD models where $x_{0}$ is the
initial condition and $i$ is the prediction step.

The n-step state prediction is a metric to test the invariance of the OC-KO;
if the OC-KO is invariant, the $r^{2}$ for n-step predictions turns out to be
1. We do not consider the $n$-step output prediction as the error provided by
that metric will be a combination of the errors in both state and output
models.

######  Example 15.

System with finite Koopman closure

Consider the following discrete time nonlinear system with an analytical
finite-dimensional OC-KO [102]:

$\displaystyle\begin{bmatrix}x_{k+1,1}\\\ x_{k+1,2}\end{bmatrix}$
$\displaystyle=\begin{bmatrix}a_{11}&0\\\
a_{21}&a_{22}\end{bmatrix}\begin{bmatrix}x_{k,1}\\\
x_{k,2}\end{bmatrix}+\begin{bmatrix}0\\\ \gamma x_{k,1}^{2}\end{bmatrix}$
$\displaystyle y_{k}$ $\displaystyle=x_{k,1}x_{k,2}$

where $x_{k,i}$ and $y_{k}$ denote the $i^{th}$ state and the output at
discrete time point $k$ respectively. We obtain the theoretical OC-KO using
sequential OC-DMD (20):

  * $\bullet$

Solving (20b) -  Adding the observable $\varphi_{1}(x)=x_{1}^{2}$ makes the
dynamics linear:

$\displaystyle\begin{bmatrix}x_{k+1,1}\\\ x_{k+1,2}\\\
\varphi_{1}(x_{k+1})\end{bmatrix}$ $\displaystyle=\begin{bmatrix}a_{11}&0&0\\\
a_{21}&a_{22}&\gamma\\\ 0&0&a_{11}^{2}\end{bmatrix}\begin{bmatrix}x_{k,1}\\\
x_{k,2}\\\ \varphi_{1}(x_{k})\end{bmatrix}$ (29)

  * $\bullet$

Solving (20c) -  Adding $\varphi_{2}(x)=x_{1}x_{2}$ to
$\\{x_{1},x_{2},\varphi_{1}(x)\\}$ yields a linear output equation

$\displaystyle y_{k}$ $\displaystyle=\begin{bmatrix}0&0&0&1\\\
\end{bmatrix}\begin{bmatrix}x_{k,1}&x_{k,2}&\varphi_{1}(x_{k})&\varphi_{2}(x_{k})\end{bmatrix}^{T}.$

  * $\bullet$

Solving (20d) -  To identify the dynamics of the added observable
$\varphi_{2}(x)$ and ensure a closed basis, we add $\varphi_{3}(x)=x_{1}^{3}$
to get the OC-KO:

$\displaystyle\psi(x_{k+1})$ $\displaystyle=\begin{bmatrix}a_{11}&0&0&0&0\\\
a_{21}&a_{22}&\gamma&0&0\\\ 0&0&a_{11}^{2}&0&0\\\
0&0&a_{11}a_{21}&a_{11}a_{22}&a_{11}\gamma\\\
0&0&0&0&a_{11}^{3}\end{bmatrix}\psi(x_{k})$ $\displaystyle y_{k}$
$\displaystyle=\begin{bmatrix}0&0&0&1&0\end{bmatrix}\psi(x_{k})$ (30)

where
$\psi(x_{k})=\begin{bmatrix}x_{k,1}&x_{k,2}&\varphi_{1}(x_{k})&\varphi_{2}(x_{k})&\varphi_{3}(x_{k})\end{bmatrix}^{T}$.

We simulate the system to generate 300 trajectories, each with a different
initial condition uniformly distributed in the range $5\leq x_{0,i}\leq
10,i=1,2$ and system parameters
$a_{11}=0.9,a_{21}=-0.4,a_{22}=-0.8,\gamma=-0.9$. The performance metrics of
the identified models are given in Table 1 and their $n-step$ predictions on a
test set initial condition are shown in Fig. 1.

The deepDMD algorithm captures well the dynamics for $n_{L}=3$ and it matches
with the theoretical solution (29). We learn the optimal direct OC-deepDMD
model for $n_{L}=5$ Koopman dictionary observables. Fig. 1 shows that the
direct OC-deepDMD model with $n_{L}=3$ shows poor performance. Hence, we need
additional observables to capture the output dynamics (given by
$\varphi_{2}(x)$ and $\varphi_{3}(x)$ in ($\bullet$ ‣ 6)), thereby validating
Theorem 6. We increase $n_{L}$ and identify both direct and sequential OC-
deepDMD models. We observe that $n_{L}=5$ is the optimal value for both OC-
deepDMD algorithms with $r^{2}\approx 1$ and agreeing with the theoretical
solution ($\bullet$ ‣ 6).

Figure 1:  Example 15: Comparing the n-step predictions of states
($x_{1},x_{2}$) and outputs ($y_{1}$) for the models: deepDMD with $n_{L}=3$,
direct OC-deepDMD with $n_{L}=3,5$, and sequential OC-deepDMD with $n_{L}=5$
where $dim(\psi(x))=n_{L}$.  Figure 2:  Example 15: The eigenvalues
($\lambda$) and corresponding eigenfunctions ($\phi$) computed from the
theoretical (first row), direct OC-deepDMD (second row) and sequential OC-
deepDMD (last row) models. The axes constitute the physical states of the
system and the colorbar indicates the value of eigenfunctions normalized by
the maximum absolute value attained by the corresponding eigenfunction. The
Pearson correlation ($\rho$) is computed between $\phi$ of the OC-deepDMD and
the theoretical models for each $\lambda$. Eigenfunctions
$\phi_{1},\phi_{4},\phi_{5}$ with $\lambda=-0.8,0.81,0.9$ capture the state
dynamics, additional $\phi_{2},\phi_{3}$ with $\lambda=-0.72,0.73$ capture the
output dynamics and $\lambda=1$ is due to the presence of the constant, unit-
valued basis element (Proposition 3).

We evaluate the extent to which the OC-deepDMD algorithms capture the
underlying system dynamics by comparing the eigenfunctions of the
corresponding OC-deepDMD models with those of the theoretical OC-KO ($\bullet$
‣ 6). Since the OC-deepDMD models are identified on standardized data, we use
Theorem 14 to reflect the transformation in the theoretical OC-KO. We compute
the eigenfunctions of all the models using modal decomposition (10). We
observe that scaling and sign flip are two artifacts that lead to non-
uniqueness of eigenfunctions as
$\psi(x_{k+1})=\sum_{i}v_{i}\lambda_{i}\phi_{i}(x)=\sum_{i}(-\alpha^{-1}v_{i})\lambda_{i}(-\alpha\phi_{i}(x))$
where $\alpha$ is a nonzero scalar. We compensate for scaling by dividing each
eigenfunction with its maximum absolute value to normalize it. When $r^{2}$ is
computed under a sign flip, it leads to negative values. To account for the
sign flip, we use the Pearson correlation ($\rho$) to compute the closeness
between the normalized eigenfunctions.

We show the plot of the normalized eigenfunctions for the OC-deepDMD models
and their correlation with the theoretical eigenfunctions in Fig. 2. We see
that the sequential OC-deepDMD model captures both eigenvalues and
eigenfunctions with a better accuracy than the direct OC-deepDMD model. This
could be attributed to sequential OC-deepDMD model structure (21) being
sparser among the two. Sequential OC-deepDMD can explicitly track that the
eigenfunctions corresponding to $\lambda=-0.8,0.81,0.9$ capture the state
dynamics by solving (20b) and those corresponding to $\lambda=-0.72,0.73$ are
added to capture the output dynamics by solving (20c) and (20d). This
validates that the output dynamics do lie in a subspace of the OC-KO
observables as proved in Theorem 6.

######  Example 16.

MEMS-actuator with a differential capacitor

Figure 3: MEMS Actuator: Schematic of a spring mass damper system as a MEMS
actuator model. The displacement of the movable plate is measured by a
differential capacitor with a fixed capacitance $C_{2}$ and variable
capacitance $C_{1}$ by applying an input voltage $V_{s}$ and measuring the
output voltage $V_{o}$.

We consider the free response of a MEMS resonator [119] modeled by a spring
mass damper system with cubic nonlinear stiffness and a differential
capacitive sensor to measure the displacement [120] as shown in Fig. 3. It has
the dynamics:

$\displaystyle\dot{x_{1}}$ $\displaystyle=x_{2}$ $\displaystyle\dot{x_{2}}$
$\displaystyle=-\frac{k_{1}}{m}x_{1}-\frac{c}{m}x_{2}-\frac{k_{3}}{m}x_{1}^{3}$
$\displaystyle y$
$\displaystyle=V_{o}=\frac{C_{1}-C_{2}}{C_{1}+C_{2}}V_{s}=-\frac{x}{d+x}V_{s}$

where $x_{1},x_{2}$ and $y$ are the displacement, velocity and output voltage
measurements respectively. We simulate the system with the parameters
$m=1,k_{1}=0.5,c=1.0,k_{3}=1.0\text{ and }V_{in}=0.4$ to generate 300
trajectories with a simulation time of $15s$, a sampling time of $0.5s$ and
initial condition, $x_{0}$, uniformly distributed in the range $(0,2)$. This
system is more complex than Example 15 as it has a single fixed point without
a finite analytical OC-KO. Therefore, we benchmark the performance of the OC-
deepDMD algorithms against the nonlinear state-space models identified by
solving (28).

Figure 4:  MEMS Actuator: Comparing the n-step predictions of states
($x_{1},x_{2}$) and outputs ($y_{1}$) for the nonlinear state-space, direct
OC-deepDMD and sequential OC-deepDMD models.

We see from Table 1 that $r^{2}\approx 1$ for 1-step state and output
predictions of the nonlinear state-spacem, sequential, and direct OC-deepDMD
models. When comparing the $n$-step predictions of these models on an initial
condition from the test dataset (shown in Fig. 4), the nonlinear state-space
model performs significantly better. Among the OC-KOs, the sequential OC-
deepDMD performs marginally better (4% higher accuracy). This indicates that
all algorithms nearly accurately solve their respective objective functions
which minimize the 1-step prediction error. However, the approximation of
infinite-dimensional OC-KOs by finite observables lead to the OC-deepDMD
models not being perfectly invariant. So, when the number of prediction steps
increases, the error in the temporal evolution of the observables accumulates
and propagates forward. A potential method to reduce the error in forecasting
is to minimize multiple step prediction errors. We showcase this method
prominently in the next example.

######  Example 17.

Activator Repressor clock with a reporter

Figure 5: Activator-Repressor clock: Schematic of the interaction of enzymes
A, B and C (fluorescent reporter- output): (a) $A$ activates $A,B$ and $C$ (b)
$B$ represses $A$ and $C$.

A more complex system is one with oscillatory dynamics that converge to an
attractor. We consider the two state activator repressor clock [121] with the
dynamics:

$\displaystyle\frac{dA}{dt}$
$\displaystyle=-\gamma_{A}A+\frac{\kappa_{A}}{\delta_{A}}\frac{\alpha_{A}(A/K_{A})^{n}+\alpha_{A0}}{1+(A/K_{A}))^{n}+(B/K_{B})^{m}}$
$\displaystyle\frac{dB}{dt}$
$\displaystyle=-\gamma_{B}B+\frac{\kappa_{B}}{\delta_{B}}\frac{\alpha_{B}(A/K_{A})^{n}+\alpha_{B0}}{1+(A/K_{A})^{n}}$
$\displaystyle C$ $\displaystyle=\frac{(k_{c}/\gamma_{c})A}{1+(B/K_{d})}$

where $A$, $B$ and $C$ are the conc. of enzymes with the network schematic as
shown in Fig. 5. $A$ and $B$ constitute the state and $C$ is the output
fluorescent reporter assumed to be at steady state. We simulate the system
using the parameters $\gamma_{A}=0.7$, $\gamma_{B}=0.5$, $\delta_{A}=1.0$,
$\delta_{B}=1.0$, $\alpha_{A0}=0.4$, $\alpha_{B0}=0.004$, $\alpha_{A}=0.2$,
$\alpha_{B}=0.2$, $K_{A}=0.1$, $K_{B}=0.08$, $\kappa_{A}=0.9$,
$\kappa_{B}=0.5$, $n=2$, $m=3$, $k_{3n}=3.0$ and $k_{3d}=1.08$ to get an
oscillatory behaviour. We generate 300 curves with a simulation time of $50s$,
a sampling time of $0.5s$ and the initial conditions uniformly distributed in
the interval $(0.1,1)$.

Figure 6: Activator-Repressor clock: n-step prediction comparisons of states
($x_{1},x_{2}$) and outputs ($y_{1}$) for the nonlinear state-space and direct
and sequential OC-deepDMD models.  Figure 7: Activator-Repressor clock: The
phase portraits of the theoretical, nonlinear state-space, direct OC-deepDMD
and sequential OC-deepDMD models is shown in the first row. The second row
shows the phase portraits of the time-delay embedded OC-deepDMD models with
the delay parameter $n_{d}=4,5,6$ and $7$. The phase portraits are constructed
using the same initial conditions highlighted by red dots and $r^{2}$ is
computed using the theoretical phase portrait as the reference.

We identify direct and sequential OC-deepDMD models and nonlinear state-space
models and show the optimal model hyperparameters and $r^{2}$ values in Table
1. We see that $r^{2}\approx 1$ for the 1-step and output predictions for all
the models indicating that the corresponding algorithms nearly accurately
solve their objective functions (which minimize 1-step prediction error)
similar to the case in Example 16. The n-step predictions of the models in
Fig. 6 show that the nonlinear state-space model outperform both the OC-
deepDMD models. But, here the direct OC-deepDMD model performs marginally
better than the sequential OC-deepDMD model (opposite of Example 16). Hence,
we infer that both OC-deepDMD algorithms perform similarly.

To evaluate how well the models capture the underlying dynamics, we construct
phase portraits of the various models shown in Fig. 7. We do so by considering
initial conditions around the phase space of the limit cycle and plotting the
n-step predictions of the models for each initial condition. We compare the
phase portraits of each model with that of the simulated system using the
$r^{2}$ metric. We see that the nonlinear state-space model captures a limit
cycle but with an offset that results in a poor $r^{2}$ value. The OC-deepDMD
models capture dissipating dynamics rather than that of a limit cycle. We
speculate that the objective function is not sufficient to capture the
dynamics and extend the objective function to minimize the error in multiple
step predictions. To do so, we incorporate the idea we implement in [104] to
construct observables on the time-delay embedded states yielding the OC-KO

$\displaystyle\psi(x_{kn_{d}+n_{d}},\cdots,x_{kn_{d}+1})$
$\displaystyle=K\psi(x_{kn_{d}},\cdots,x_{kn_{d}-n_{d}+1})$ $\displaystyle
y_{kn_{d}}$ $\displaystyle=W_{h}\psi(x_{kn_{d}},\cdots,x_{kn_{d}-n_{d}+1}).$
(31)

where $k$ is the discrete time index and $n_{d}$ indicates the number of time-
delay embeddings. Since the two OC-deepDMD algorithms perform similarly, we
stick to just using the direct OC-deepDMD algorithm to identify the time-delay
embedded OC-deepDMD models. The phase portraits of the direct OC-deepDMD
models as $n_{d}$ is increased is given in the second row of Fig. 7. We see
that as $n_{d}$ increases, the phase portrait takes the structure of an
oscillator with $n_{d}=6$ being optimal.

We see from Table 1 that the n-step prediction accuracy increases for this
model at the expense of the 1-step and output predictions which reduce. This
is because the formulation (6) simultaneously minimizes multiple step
prediction errors [104] which may not always yield optimal 1-step predictions.
Hence, we see that OC-deepDMD algorithm has limitations when it comes to the
case of oscillators and time-delay embedded OC-deepDMD models can be used to
overcome them.

##  7 Conclusion

In this work, we propose a novel method to fuse state and output measurements
of nonlinear systems using Koopman operator representations that are augmented
with a linear output equation (called OC-KOs). Using the concept of
diffeomorphic conjugacy, we show that the dynamics of the measured output
variables span a subspace of the OC-KO lifting functions and that the OC-KOs
integrate the dynamics of both states and outputs. We show a sufficient
condition for the existence of state-inclusive OC-KOs and propose two DMD
algorithms that incorporate the output constraints to identify them. We use
numerical examples to show the performance of these algorithms.

In future work, we will use this technique to extract genotype-phenotype
models of microbes by fusing their various time-series datasets. The genotype-
phenotype models will enable us to control the persistence of these microbes
in new environments. We expect the OC-KO based sensor fusion method to cater
to a large range of dynamical systems where fusion of nonlinear dynamics of
two measurement sets is desired for applications like observability analysis,
observer synthesis and state estimation. The authors would like to graciously
thank Igor Mezic, Nathan Kutz, Jamiree Harrison, Joshua Elmore, Adam
Deutschbauer, and Bassam Bamieh for insightful discussions. Any opinions,
findings, conclusions, or recommendations expressed in this material are those
of the authors and do not necessarily reflect the views of the Defense
Advanced Research Project Agency, the Department of Defense, or the United
States government.

This work was also funded in part by the Department of Energy’s Biological and
Environmental Research office, under the DOE Scientific Focus Area: Secure
Biosystems Design project, via gracious funding from Pacific Northwest
National Laboratory subcontract numbers 545157 and 490521. This work was also
partially supported by DARPA, AFRL under contract numbers FA8750-17-C-0229,
HR001117C0092, HR001117C0094, DEAC0576RL01830, as well as funding from the
Army Research Office’s Young Investigator Program under grant number
W911NF-20-1-0165. Supplies for this work were partially supported by the
Institute of Collaborative Biotechnologies, via grant W911NF-19-D-0001-0006.

## References

  * [1] Jayoung Kim, Alan S Campbell, Berta Esteban-Fernández de Ávila, and Joseph Wang.  Wearable biosensors for healthcare monitoring.  Nature biotechnology, 37(4):389–406, 2019.
  * [2] Niharika Gupta, Venkatesan Renugopalakrishnan, Dorian Liepmann, Ramasamy Paulmurugan, and Bansi D Malhotra.  Cell-based biosensors: Recent trends, challenges and future perspectives.  Biosensors and Bioelectronics, 141:111435, 2019.
  * [3] Nikhil Bhalla, Yuwei Pan, Zhugen Yang, and Amir Farokh Payam.  Opportunities and challenges for biosensors and nanoscale analytical tools for pandemics: Covid-19.  ACS nano, 14(7):7783–7807, 2020.
  * [4] Celine IL Justino, Ana R Gomes, Ana C Freitas, Armando C Duarte, and Teresa AP Rocha-Santos.  Graphene based sensors and biosensors.  TrAC Trends in Analytical Chemistry, 91:53–66, 2017.
  * [5] Sanjay Kisan Metkar and Koyeli Girigoswami.  Diagnostic biosensors in medicine–a review.  Biocatalysis and agricultural biotechnology, 17:271–283, 2019.
  * [6] Geeta Bhatt, Kapil Manoharan, Pankaj Singh Chauhan, and Shantanu Bhattacharya.  Mems sensors for automotive applications: a review.  Sensors for Automotive and Aerospace Applications, pages 223–239, 2019.
  * [7] Claudio Abels, Vincenzo Mariano Mastronardi, Francesco Guido, Tommaso Dattoma, Antonio Qualtieri, William M Megill, Massimo De Vittorio, and Francesco Rizzi.  Nitride-based materials for flexible mems tactile and flow sensors in robotics.  Sensors, 17(5):1080, 2017.
  * [8] Weijun Tao, Tao Liu, Rencheng Zheng, and Hutian Feng.  Gait analysis using wearable sensors.  Sensors, 12(2):2255–2283, 2012.
  * [9] AS Fiorillo, CD Critello, and SA Pullano.  Theory, technology and applications of piezoresistive sensors: A review.  Sensors and Actuators A: Physical, 281:156–175, 2018.
  * [10] Shima Taheri.  A review on five key sensors for monitoring of concrete structures.  Construction and Building Materials, 204:492–509, 2019.
  * [11] Juan Guerrero-Ibáñez, Sherali Zeadally, and Juan Contreras-Castillo.  Sensor technologies for intelligent transportation systems.  Sensors, 18(4):1212, 2018.
  * [12] Christian Bachmann.  Multi-sensor data fusion for traffic speed and travel time estimation.  PhD thesis, University of Toronto, 2011.
  * [13] Sebastien Richoz, Lin Wang, Philip Birch, and Daniel Roggen.  Transportation mode recognition fusing wearable motion, sound, and vision sensors.  IEEE Sensors Journal, 20(16):9314–9328, 2020.
  * [14] Junxuan Zhao, Hao Xu, Hongchao Liu, Jianqing Wu, Yichen Zheng, and Dayong Wu.  Detection and tracking of pedestrians and vehicles using roadside lidar sensors.  Transportation research part C: emerging technologies, 100:68–87, 2019.
  * [15] Marcin Bernas, Bartłomiej Płaczek, Wojciech Korski, Piotr Loska, Jarosław Smyła, and Piotr Szymała.  A survey and comparison of low-cost sensing technologies for road traffic monitoring.  Sensors, 18(10):3243, 2018.
  * [16] Harvey B Mitchell.  Multi-sensor data fusion: an introduction.  Springer Science & Business Media, 2007.
  * [17] Ming Meng, Qingshan She, Yunyuan Gao, and Zhizeng Luo.  Emg signals based gait phases recognition using hidden markov models.  In The 2010 IEEE International Conference on Information and Automation, pages 852–856. IEEE, 2010.
  * [18] Jakob Ziegler, Hubert Gattringer, and Andreas Mueller.  Classification of gait phases based on bilateral emg data using support vector machines.  In 2018 7th IEEE International Conference on Biomedical Robotics and Biomechatronics (Biorob), pages 978–983. IEEE, 2018.
  * [19] Hongyu Zhao, Zhelong Wang, Sen Qiu, Jiaxin Wang, Fang Xu, Zhengyu Wang, and Yanming Shen.  Adaptive gait detection based on foot-mounted inertial sensors and multi-sensor fusion.  Information Fusion, 52:157–166, 2019.
  * [20] Fabian Horst, Sebastian Lapuschkin, Wojciech Samek, Klaus-Robert Müller, and Wolfgang I Schöllhorn.  Explaining the unique nature of individual gait patterns with deep learning.  Scientific reports, 9(1):1–13, 2019.
  * [21] Abdullah S Alharthi, Syed U Yunas, and Krikor B Ozanyan.  Deep learning for monitoring of human gait: A review.  IEEE Sensors Journal, 19(21):9575–9591, 2019.
  * [22] Jerry Ajay, Chen Song, Aosen Wang, Jeanne Langan, Zhinan Li, and Wenyao Xu.  A pervasive and sensor-free deep learning system for parkinsonian gait analysis.  In 2018 IEEE EMBS International Conference on Biomedical & Health Informatics (BHI), pages 108–111. IEEE, 2018.
  * [23] Giovanni Paragliola and Antonio Coronato.  Gait anomaly detection of subjects with parkinson’s disease using a deep time series-based approach.  IEEE Access, 6:73280–73292, 2018.
  * [24] Riccardo Bonetto, Mattia Soldan, Alberto Lanaro, Simone Milani, and Michele Rossi.  Seq2seq rnn based gait anomaly detection from smartphone acquired multimodal motion data.  arXiv preprint arXiv:1911.08608, 2019.
  * [25] Martin Grimmer, Matthew Holgate, Robert Holgate, Alexander Boehler, Jeffrey Ward, Kevin Hollander, Thomas Sugar, and André Seyfarth.  A powered prosthetic ankle joint for walking and running.  Biomedical engineering online, 15(3):37–52, 2016.
  * [26] Xin Guo, Lingling Chen, Yang Zhang, Peng Yang, and Liqun Zhang.  A study on control mechanism of above knee robotic prosthesis based on cpg model.  In 2010 IEEE International Conference on Robotics and Biomimetics, pages 283–287. IEEE, 2010.
  * [27] Carl D Hoover, George D Fulk, and Kevin B Fite.  Stair ascent with a powered transfemoral prosthesis under direct myoelectric control.  IEEE/ASME Transactions on Mechatronics, 18(3):1191–1200, 2012.
  * [28] Samuel K Au, Paolo Bonato, and Hugh Herr.  An emg-position controlled system for an active ankle-foot prosthesis: an initial experimental study.  In 9th International Conference on Rehabilitation Robotics, 2005\. ICORR 2005., pages 375–379. IEEE, 2005.
  * [29] He Huang, Fan Zhang, Levi J Hargrove, Zhi Dou, Daniel R Rogers, and Kevin B Englehart.  Continuous locomotion-mode identification for prosthetic legs based on neuromuscular–mechanical fusion.  IEEE Transactions on Biomedical Engineering, 58(10):2867–2875, 2011\.
  * [30] Kuangen Zhang, Clarence W de Silva, and Chenglong Fu.  Sensor fusion for predictive control of human-prosthesis-environment dynamics in assistive walking: A survey.  arXiv preprint arXiv:1903.07674, 2019.
  * [31] Jurek Z Sasiadek.  Sensor fusion.  Annual Reviews in Control, 26(2):203–228, 2002.
  * [32] Ahmed M Hasan, Khairulmizam Samsudin, Abdul Rahman Ramli, RS Azmir, and SA Ismaeel.  A review of navigation systems (integration and algorithms).  Australian journal of basic and applied sciences, 3(2):943–959, 2009.
  * [33] Jakob M Hansen, Thor I Fossen, and Tor Arne Johansen.  Nonlinear observer design for gnss-aided inertial navigation systems with time-delayed gnss measurements.  Control Engineering Practice, 60:39–50, 2017.
  * [34] Sharad Shankar, Kenan Ezal, and João P Hespanha.  Finite horizon maximum likelihood estimation for integrated navigation with rf beacon measurements.  Asian Journal of Control, 21(4):1470–1482, 2019.
  * [35] G Rigatos and S Tzafestas.  Extended kalman filtering for fuzzy modelling and multi-sensor fusion.  Mathematical and computer modelling of dynamical systems, 13(3):251–266, 2007.
  * [36] Gerasimos G Rigatos.  Extended kalman and particle filtering for sensor fusion in motion control of mobile robots.  Mathematics and computers in simulation, 81(3):590–607, 2010.
  * [37] Setareh Yazdkhasti and Jurek Z Sasiadek.  Multi sensor fusion based on adaptive kalman filtering.  In Advances in Aerospace Guidance, Navigation and Control, pages 317–333. Springer, 2018.
  * [38] Øyvind Hegrenæs, Oddvar Hallingstad, and Kenneth Gade.  Towards model-aided navigation of underwater vehicles.  2007\.
  * [39] Chengming Shi, George Panoutsos, Bo Luo, Hongqi Liu, Bin Li, and Xu Lin.  Using multiple-feature-spaces-based deep learning for tool condition monitoring in ultraprecision manufacturing.  IEEE Transactions on Industrial Electronics, 66(5):3794–3803, 2018\.
  * [40] Mahardhika Pratama, Eric Dimla, Chow Yin Lai, and Edwin Lughofer.  Metacognitive learning approach for online tool condition monitoring.  Journal of Intelligent Manufacturing, 30(4):1717–1737, 2019.
  * [41] Jianming Dou, Chuangwen Xu, Shengjie Jiao, Baodong Li, Jilin Zhang, and Xinxin Xu.  An unsupervised online monitoring method for tool wear using a sparse auto-encoder.  The International Journal of Advanced Manufacturing Technology, 106(5):2493–2507, 2020.
  * [42] Mustafa Kuntoğlu and Hacı Sağlam.  Investigation of signal behaviors for sensor fusion with tool condition monitoring system in turning.  Measurement, 173:108582, 2021.
  * [43] Alexander Bleakie and Dragan Djurdjanovic.  Feature extraction, condition monitoring, and fault modeling in semiconductor manufacturing systems.  Computers in Industry, 64(3):203–213, 2013.
  * [44] Xin Lin, Kunpeng Zhu, Jerry Ying Hsi Fuh, and Xianyin Duan.  Metal-based additive manufacturing condition monitoring methods: From measurement to control.  ISA transactions, 2021.
  * [45] Jack Francis and Linkan Bian.  Deep learning for distortion prediction in laser-based additive manufacturing using big data.  Manufacturing Letters, 20:10–14, 2019.
  * [46] SA Shevchik, C Kenel, C Leinenbach, and K Wasmer.  Acoustic emission for in situ quality monitoring in.  2017\.
  * [47] Changchao Gu, Yihai He, Xiao Han, and Zhaoxiang Chen.  Product quality oriented predictive maintenance strategy for manufacturing systems.  In 2017 Prognostics and System Health Management Conference (PHM-Harbin), pages 1–7. IEEE, 2017.
  * [48] Yihai He, Changchao Gu, Zhaoxiang Chen, and Xiao Han.  Integrated predictive maintenance strategy for manufacturing systems by combining quality control and mission reliability analysis.  International Journal of Production Research, 55(19):5841–5862, 2017.
  * [49] Rih-Teng Wu and Mohammad Reza Jahanshahi.  Data fusion approaches for structural health monitoring and system identification: past, present, and future.  Structural Health Monitoring, 19(2):552–586, 2020.
  * [50] Jaime Vitola, Francesc Pozo, Diego A Tibaduiza, and Maribel Anaya.  A sensor data fusion system based on k-nearest neighbor pattern classification for structural health monitoring applications.  Sensors, 17(2):417, 2017.
  * [51] Wieslaw Ostachowicz, Rohan Soman, and Pawel Malinowski.  Optimization of sensor placement for structural health monitoring: A review.  Structural Health Monitoring, 18(3):963–988, 2019.
  * [52] Sijia Huang, Kumardeep Chaudhary, and Lana X Garmire.  More is better: recent progress in multi-omics data integration methods.  Frontiers in genetics, 8:84, 2017.
  * [53] Karren Dai Yang, Anastasiya Belyaeva, Saradha Venkatachalapathy, Karthik Damodaran, Abigail Katcoff, Adityanarayanan Radhakrishnan, GV Shivashankar, and Caroline Uhler.  Multi-domain translation between single-cell imaging and sequencing data using autoencoders.  Nature Communications, 12(1):1–10, 2021.
  * [54] Florian Rohart, Benoît Gautier, Amrit Singh, and Kim-Anh Lê Cao.  mixomics: An r package for ‘omics feature selection and multiple data integration.  PLoS computational biology, 13(11):e1005752, 2017.
  * [55] Elad Noor, Sarah Cherkaoui, and Uwe Sauer.  Biological insights through omics data integration.  Current Opinion in Systems Biology, 15:39–47, 2019.
  * [56] Zhong Wang, Mark Gerstein, and Michael Snyder.  Rna-seq: a revolutionary tool for transcriptomics.  Nature reviews genetics, 10(1):57–63, 2009.
  * [57] Ruedi Aebersold and Matthias Mann.  Mass spectrometry-based proteomics.  Nature, 422(6928):198–207, 2003.
  * [58] VA Gant, G Warnes, I Phillips, and GF Savidge.  The application of flow cytometry to the study of bacterial responses to antibiotics.  Journal of Medical Microbiology, 39(2):147–154, 1993.
  * [59] Annika Meyers, Christoph Furtmann, and Joachim Jose.  Direct optical density determination of bacterial cultures in microplates for high-throughput screening applications.  Enzyme and microbial technology, 118:1–5, 2018.
  * [60] Jonathan Lefman, Peijun Zhang, Teruhisa Hirai, Robert M Weis, Jemma Juliani, Donald Bliss, Martin Kessel, Erik Bos, Peter J Peters, and Sriram Subramaniam.  Three-dimensional electron microscopic imaging of membrane invaginations in escherichia coli overproducing the chemotaxis receptor tsr.  Journal of bacteriology, 186(15):5052–5061, 2004.
  * [61] Nicolas Altemose, Annie Maslan, Carolina Rios-Martinez, Andre Lai, Jonathan A White, and Aaron Streets.  $\mu$damid: A microfluidic approach for joint imaging and sequencing of protein-dna interactions in single cells.  Cell Systems, 11(4):354–366, 2020.
  * [62] Zhouzerui Liu, Jinzhou Yuan, Anna Lasorella, Antonio Iavarone, Jeffrey N Bruce, Peter Canoll, and Peter A Sims.  Integrating single-cell rna-seq and imaging with scope-seq2.  Scientific reports, 10(1):1–15, 2020.
  * [63] Daehee Hwang, Alistair G Rust, Stephen Ramsey, Jennifer J Smith, Deena M Leslie, Andrea D Weston, Pedro De Atauri, John D Aitchison, Leroy Hood, Andrew F Siegel, et al.  A data integration methodology for systems biology.  Proceedings of the National Academy of Sciences, 102(48):17296–17301, 2005.
  * [64] Daehee Hwang, Jennifer J Smith, Deena M Leslie, Andrea D Weston, Alistair G Rust, Stephen Ramsey, Pedro de Atauri, Andrew F Siegel, Hamid Bolouri, John D Aitchison, et al.  A data integration methodology for systems biology: experimental verification.  Proceedings of the National Academy of Sciences, 102(48):17302–17307, 2005.
  * [65] Igor Mezic.  Spectral properties of dynamical systems, model reduction and decompositions.  Nonlinear Dynamics, 41(1-3):309–325, 2005.
  * [66] Marko Budišić, Ryan Mohr, and Igor Mezić.  Applied koopmanism.  Chaos: An Interdisciplinary Journal of Nonlinear Science, 22(4):047510, 2012.
  * [67] Peter J Schmid.  Dynamic mode decomposition of numerical and experimental data.  Journal of fluid mechanics, 656:5–28, 2010.
  * [68] Matthew O Williams, Ioannis G Kevrekidis, and Clarence W Rowley.  A data–driven approximation of the koopman operator: Extending dynamic mode decomposition.  Journal of Nonlinear Science, 25(6):1307–1346, 2015.
  * [69] Aqib Hasnain, Subhrajit Sinha, Yuval Dorfan, Amin Espah Borujeni, Yongjin Park, Paul Maschhoff, Uma Saxena, Joshua Urrutia, Niall Gaffney, Diveena Becker, et al.  A data-driven method for quantifying the impact of a genetic circuit on its host.  arXiv preprint arXiv:1909.06455, 2019.
  * [70] Bethany Lusch, J Nathan Kutz, and Steven L Brunton.  Deep learning for universal linear embeddings of nonlinear dynamics.  Nature communications, 9(1):1–10, 2018.
  * [71] Enoch Yeung, Soumya Kundu, and Nathan Hodas.  Learning deep neural network representations for koopman operators of nonlinear dynamical systems.  In 2019 American Control Conference (ACC), pages 4832–4839. IEEE, 2019.
  * [72] Yuhei Kaneko, Shogo Muramatsu, Hiroyasu Yasuda, Kiyoshi Hayasaka, Yu Otake, Shunsuke Ono, and Masahiro Yukawa.  Convolutional-sparse-coded dynamic mode decomposition and its application to river state estimation.  In ICASSP 2019-2019 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), pages 1872–1876. IEEE, 2019.
  * [73] Omri Azencot, Wotao Yin, and Andrea Bertozzi.  Consistent dynamic mode decomposition.  arXiv preprint arXiv:1905.09736, 2019.
  * [74] Travis Askham and J Nathan Kutz.  Variable projection methods for an optimized dynamic mode decomposition.  SIAM Journal on Applied Dynamical Systems, 17(1):380–416, 2018\.
  * [75] Subhrajit Sinha and Enoch Yeung.  On computation of koopman operator from sparse data.  arXiv:1901.03024, 2019.
  * [76] Krithika Manohar, J Nathan Kutz, and Steven L Brunton.  Optimal sensor and actuator placement using balanced model reduction.  arXiv preprint arXiv:1812.01574, 2018.
  * [77] Krithika Manohar, Eurika Kaiser, Steven L Brunton, and J Nathan Kutz.  Optimized sampling for multiscale dynamics.  Multiscale Modeling & Simulation, 17(1):117–136, 2019.
  * [78] Igor Mezić.  Analysis of fluid flows via spectral properties of the koopman operator.  Annual Review of Fluid Mechanics, 45:357–378, 2013.
  * [79] Kunihiko Taira, Steven L Brunton, Scott TM Dawson, Clarence W Rowley, Tim Colonius, Beverley J McKeon, Oliver T Schmidt, Stanislav Gordeyev, Vassilios Theofilis, and Lawrence S Ukeiley.  Modal analysis of fluid flows: An overview.  Aiaa Journal, 55(12):4013–4041, 2017.
  * [80] Jayse Clifton McLean.  Modal Analysis of the Human Brain Using Dynamic Mode Decomposition.  PhD thesis, North Dakota State University, 2020.
  * [81] Amit Surana and Andrzej Banaszuk.  Linear observer synthesis for nonlinear systems using koopman operator framework.  IFAC-PapersOnLine, 49(18):716–723, 2016.
  * [82] Amit Surana.  Koopman operator based observer synthesis for control-affine nonlinear systems.  In 2016 IEEE 55th Conference on Decision and Control (CDC), pages 6492–6499. IEEE, 2016.
  * [83] Enoch Yeung, Zhiyuan Liu, and Nathan O Hodas.  A koopman operator approach for computing and balancing gramians for discrete time nonlinear systems.  In 2018 Annual American Control Conference (ACC), pages 337–344. IEEE, 2018.
  * [84] Marcos Netto and Lamine Mili.  A robust data-driven koopman kalman filter for power systems dynamic state estimation.  IEEE Transactions on Power Systems, 33(6):7228–7237, 2018.
  * [85] Joshua L Proctor, Steven L Brunton, and J Nathan Kutz.  Dynamic mode decomposition with control.  SIAM Journal on Applied Dynamical Systems, 15(1):142–161, 2016\.
  * [86] Milan Korda and Igor Mezić.  Linear predictors for nonlinear dynamical systems: Koopman operator meets model predictive control.  Automatica, 93:149–160, 2018.
  * [87] Joshua L Proctor, Steven L Brunton, and J Nathan Kutz.  Generalizing koopman theory to allow for inputs and control.  SIAM Journal on Applied Dynamical Systems, 17(1):909–930, 2018\.
  * [88] Pengcheng You, John Pang, and Enoch Yeung.  Deep koopman controller synthesis for cyber-resilient market-based frequency regulation.  IFAC-PapersOnLine, 51(28):720–725, 2018.
  * [89] Eurika Kaiser, J Nathan Kutz, and Steven Brunton.  Data-driven discovery of koopman eigenfunctions for control.  Machine Learning: Science and Technology, 2021.
  * [90] AM Avila and I Mezić.  Data-driven analysis and forecasting of highway traffic dynamics.  Nature communications, 11(1):1–16, 2020.
  * [91] Esther Ling, Liyuan Zheng, Lillian J Ratliff, and Samuel Coogan.  Koopman operator applications in signalized traffic systems.  IEEE Transactions on Intelligent Transportation Systems, 2020.
  * [92] A Mounir Boudali, Peter J Sinclair, Richard Smith, and Ian R Manchester.  Human locomotion analysis: Identifying a dynamic mapping between upper and lower limb joints using the koopman operator.  In 2017 39th Annual International Conference of the IEEE Engineering in Medicine and Biology Society (EMBC), pages 1889–1892. IEEE, 2017\.
  * [93] Aleksandra Kalinowska, Thomas A Berrueta, Adam Zoss, and Todd Murphey.  Data-driven gait segmentation for walking assistance in a lower-limb assistive device.  In 2019 International Conference on Robotics and Automation (ICRA), pages 1390–1396. IEEE, 2019.
  * [94] Keisuke Fujii, Naoya Takeishi, Benio Kibushi, Motoki Kouzaki, and Yoshinobu Kawahara.  Data-driven spectral analysis for coordinative structures in periodic human locomotion.  Scientific reports, 9(1):1–14, 2019.
  * [95] David A Haggerty, Michael J Banks, Patrick C Curtis, Igor Mezić, and Elliot W Hawkes.  Modeling, reduction, and control of a helically actuated inertial soft robotic arm via the koopman operator.  arXiv preprint arXiv:2011.07939, 2020.
  * [96] Matthew O Williams, Clarence W Rowley, Igor Mezić, and Ioannis G Kevrekidis.  Data fusion via intrinsic dynamic variables: An application of data-driven koopman spectral analysis.  EPL (Europhysics Letters), 109(4):40007, 2015.
  * [97] Igor Mezić.  Spectrum of the koopman operator, spectral expansions in functional spaces, and state-space geometry.  Journal of Nonlinear Science, pages 1–55, 2019.
  * [98] Afshin Mesbahi, Jingjing Bu, and Mehran Mesbahi.  Nonlinear observability via koopman analysis: Characterizing the role of symmetry.  Automatica, 124:109353, 2021.
  * [99] Aqib Hasnain, Nibodh Boddupalli, and Enoch Yeung.  Optimal reporter placement in sparsely measured genetic networks using the koopman operator.  In 2019 IEEE 58th Conference on Decision and Control (CDC), pages 19–24. IEEE, 2019.
  * [100] Alexandre Mauroy, Y Susuki, and I Mezić.  The Koopman Operator in Systems and Control.  Springer, 2020.
  * [101] Charles A Johnson and Enoch Yeung.  A class of logistic functions for approximating state-inclusive koopman operators.  In 2018 Annual American Control Conference (ACC), pages 4803–4810. IEEE, 2018.
  * [102] Steven L Brunton, Bingni W Brunton, Joshua L Proctor, and J Nathan Kutz.  Koopman invariant subspaces and finite linear representations of nonlinear dynamical systems for control.  PloS one, 11(2):e0150171, 2016.
  * [103] Milan Korda and Igor Mezić.  On convergence of extended dynamic mode decomposition to the koopman operator.  Journal of Nonlinear Science, 28(2):687–710, 2018.
  * [104] Shara Balakrishnan, Aqib Hasnain, Nibodh Boddupalli, Dennis M Joshy, Robert G Egbert, and Enoch Yeung.  Prediction of fitness in bacteria with causal jump dynamic mode decomposition.  In 2020 American Control Conference (ACC), pages 3749–3756. IEEE, 2020.
  * [105] Samuel E Otto and Clarence W Rowley.  Linearly recurrent autoencoder networks for learning dynamics.  SIAM Journal on Applied Dynamical Systems, 18(1):558–593, 2019\.
  * [106] Naoya Takeishi, Yoshinobu Kawahara, and Takehisa Yairi.  Learning koopman invariant subspaces for dynamic mode decomposition.  In Advances in Neural Information Processing Systems, pages 1130–1140, 2017.
  * [107] Qianxiao Li, Felix Dietrich, Erik M Bollt, and Ioannis G Kevrekidis.  Extended dynamic mode decomposition with dictionary learning: A data-driven adaptive spectral decomposition of the koopman operator.  Chaos: An Interdisciplinary Journal of Nonlinear Science, 27(10):103111, 2017.
  * [108] George Cybenko.  Approximation by superpositions of a sigmoidal function.  Mathematics of control, signals and systems, 2(4):303–314, 1989\.
  * [109] Boris Hanin.  Universal function approximation by deep neural nets with bounded width and relu activations.  Mathematics, 7(10):992, 2019.
  * [110] JT-H Lo.  Multilayer perceptrons and radial basis functions are universal robust approximators.  In 1998 IEEE International Joint Conference on Neural Networks Proceedings. IEEE World Congress on Computational Intelligence (Cat. No. 98CH36227), volume 2, pages 1311–1314. IEEE, 1998.
  * [111] Steven L Brunton, Bingni W Brunton, Joshua L Proctor, Eurika Kaiser, and J Nathan Kutz.  Chaos as an intermittently forced linear system.  Nature communications, 8(1):1–9, 2017.
  * [112] Hassan Arbabi and Igor Mezic.  Ergodic theory, dynamic mode decomposition, and computation of spectral properties of the koopman operator.  SIAM Journal on Applied Dynamical Systems, 16(4):2096–2126, 2017\.
  * [113] Mason Kamb, Eurika Kaiser, Steven L Brunton, and J Nathan Kutz.  Time-delay observables for koopman: Theory and applications.  SIAM Journal on Applied Dynamical Systems, 19(2):886–917, 2020\.
  * [114] Ljuboslav Boskic, Cory N Brown, and Igor Mezić.  Koopman mode analysis on thermal data for building energy assessment.  Advances in Building Energy Research, pages 1–15, 2020.
  * [115] Joao P Hespanha.  Linear systems theory.  Princeton university press, 2018.
  * [116] Henk Nijmeijer and Arjan J Van der Schaft.  Nonlinear dynamical control systems, volume 175.  Springer, 1990.
  * [117] Stephen Boyd, Stephen P Boyd, and Lieven Vandenberghe.  Convex optimization.  Cambridge university press, 2004.
  * [118] John Duchi, Elad Hazan, and Yoram Singer.  Adaptive subgradient methods for online learning and stochastic optimization.  Journal of machine learning research, 12(7), 2011.
  * [119] Pavel M Polunin, Yushi Yang, Mark I Dykman, Thomas W Kenny, and Steven W Shaw.  Characterization of mems resonator nonlinearities using the ringdown response.  Journal of Microelectromechanical Systems, 25(2):297–303, 2016\.
  * [120] Stephen D Senturia.  Microsystem design.  Springer Science & Business Media, 2007.
  * [121] Domitilla Del Vecchio and Richard M Murray.  Biomolecular feedback systems.  Princeton University Press Princeton, NJ, 2015.