Data-Driven Observability Decomposition with Koopman Operators for Optimization of Output Functions of Nonlinear Systems

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Data-Driven Observability Decomposition with Koopman

Operators for Optimization of Output Functions of Nonlinear

Systems

Shara Balakrishnan a, Aqib Hasnain b, Robert G. Egbert c, Enoch Yeung b

aDepartment of Electrical and Computer Engineering, University of California, Santa Barbara, United States

bDepartment of Mechanical Engineering, University of California, Santa Barbara, United States

cBiological Sciences Division, Earth and Biological Sciences Directorate, Paciﬁc Northwest National Laboratory, United States

Abstract

When complex systems with nonlinear dynamics achieve an output performance objective, only a fraction of the state dynamics

signiﬁcantly impacts that output. Those minimal state dynamics can be identiﬁed using the diﬀerential geometric approach to

the observability of nonlinear systems, but the theory is limited to only analytical systems. In this paper, we extend the notion of

nonlinear observable decomposition to the more general class of data-informed systems. We employ Koopman operator theory,

which encapsulates nonlinear dynamics in linear models, allowing us to bridge the gap between linear and nonlinear observability

notions. We propose a new algorithm to learn Koopman operator representations that capture the system dynamics while

ensuring that the output performance measure is in the span of its observables. We show that a transformation of this linear,

output-inclusive Koopman model renders a new minimum Koopman representation. This representation embodies only the

observable portion of the nonlinear observable decomposition of the original system. A prime application of this theory is

to identify genes in biological systems that correspond to speciﬁc phenotypes, the performance measure. We simulate two

biological gene networks and demonstrate that the observability of Koopman operators can successfully identify genes that

drive each phenotype. We anticipate our novel system identiﬁcation tool will eﬀectively discover reduced gene networks that

drive complex behaviors in biological systems.

Key words: Koopman operator theory; nonlinear observability; diﬀerential geometry; nonlinear dynamical systems; system

identiﬁcation; gene networks

1 Introduction

Sensor technology has advanced at a rapid pace, oﬀer-

ing researchers unprecedented access to data on dynam-

ical systems. Observability is the underlying principle

that links the sensor data to the internal state of the

system. Applications of observability include monitoring

the state of the system [1–3], estimating process model

parameters [4] and identifying optimal locations for sen-

sor placement [5]. The theory of observability is well es-

tablished for linear systems [6]. Observability theory for

nonlinear systems is limited to the diﬀerential geometric

results for analytical systems [7] and algebraic results for

Email addresses: sbalakrishnan@ucsb.edu (Shara

Balakrishnan), aqib@ucsb.edu (Aqib Hasnain),

robert.egbert@pnnl.gov (Robert G. Egbert),

eyeung@ucsb.edu (Enoch Yeung).

polynomial systems [8]. For nonlinear systems learned

from data, methods are being developed to identify if

the system is observable [9]. The theory to identify the

observable subspace decomposition of nonlinear systems

from data-driven models is yet to be established.

Data-driven discovery of dynamics is critical for complex

systems where the underlying mechanics are not fully un-

derstood. Such scenarios are common in biological cells

[10], ﬁnance [11], cyber-physical systems [12], etc. One

of the commonly used complex systems in biomanufac-

turing industries is the bacterium, Escherichia coli [13].

In Escherichia coli, gene transcription alone constitute

over a 4,400−dimensional dynamic process, and this

excludes the protein and metabolic interactions within

the cell. Such complex systems are typically deployed

to achieve a speciﬁc performance objective. Escherichia

coli used in biomanufacturing processes are optimized

Preprint submitted to Automatica 19 October 2022

arXiv:2210.09343v1 [math.OC] 17 Oct 2022

for performance objectives like maximizing population

cell growth [14] or maximizing production of a speci-

ﬁed metabolite [15]. Only a fraction of the genes have

a strong inﬂuence on the desired performance objec-

tive [16–18]. This raises the question of how to identify

a critical set of genes that have the strongest inﬂuence

on given performance objective function.

For a linear system, the performance objective can be

treated as the output and an observable subspace decom-

position results in the minimal system dynamics that

drives the output [19]. Equivalent results have been de-

veloped for nonlinear systems using diﬀerential geom-

etry for analytical systems where the governing equa-

tions are known prior [7]. However, the dynamics of bi-

ological systems are not known prior and are typically

learned from data. Hence, observable subspace decom-

position methods cannot be used directly to learn the

minimal gene expression dynamics in biological systems

that drive a desired output phenotype.

In biological systems, the typical approach to identify

genes that impact a phenotype is to look for genes

that exhibit signiﬁcant diﬀerences in their steady-state

responses [20–22] across varying initial conditions. By

considering initial conditions where the output (per-

formance metric) response is vastly diﬀerent, the genes

with the highest diﬀerential steady state response are

deemed to impact the output. This is a classical em-

pirical approach that disregards both gene-to-gene

interactions as well as gene-to-phenotype (output) in-

teractions. Our ultimate goal is to model these various

nonlinear dynamical interactions from data and then

ﬁnd genes that drive a desired output which can later

be used to optimize the performance of that output.

Koopman operator theory is an increasingly popular ap-

proach to learn and analyze nonlinear system dynamics,

speciﬁcally due to a growing suite of numerical meth-

ods that can be applied in a data-driven setting [23,24].

Koopman models are promising because they construct

a set of state functions called Koopman observables that

embed the nonlinear dynamics of a physical system in

a high-dimensional space where the dynamics become

linear [25]. Koopman models are typically learned from

data using a dimensionality reduction algorithm called

dynamic mode decomposition (DMD), which was de-

veloped by Schmid [26]. Extensive research has enabled

Koopman models to increase their predictive accuracy

and decrease their computational complexity. Koopman

models serve as a bridge between nonlinear systems and

high-dimensional linear models, making them particu-

larly helpful for extending linear notions to nonlinear

systems in applications such as modal analysis [27–29],

construction of observers [9, 30–33] and development of

controllers [23, 34–37].

The study of observability of nonlinear systems using

Koopman operators is a growing area of research; Koop-

man operators have been augmented with output equa-

tions for applications like observer synthesis [30–32], op-

timal sensor placement [38, 39] and quantifying observ-

ability of nonlinear systems [9]. They all work under the

assumption that the outputs lie in the span of Koopman

observables but there is no theory on when that assump-

tion holds. There are no algorithms to learn such output-

inclusive Koopman models from data as Koopman mod-

els typically constitute a state equation learned either

by using direct state measurements [40–42] or delay-

embedded output measurements [43–45]. Moreover, how

to use Koopman operator models learnt from data to

estimate the observable decomposition of the nonlinear

system is yet to be established.

Here, we extend the theory of Koopman operators to

nonlinear systems with a measurable output perfor-

mance and develop the notion of observable subspaces

for such nonlinear systems using linear Koopman oper-

ator theory. Through our investigation, we:

(i) developed a theory that maps the observable sub-

space of a nonlinear system to a linear output-

inclusive Koopman model deﬁned on that observ-

able subspace (Theorems 3 and 4),

(ii) identiﬁed the conditions under which the observ-

able subspace of an output-inclusive Koopman

model maps to the observable subspace of the

nonlinear system (Theorem 5)

(iii) developed a new algorithm that learns such observ-

able, output-inclusive Koopman models using deep

learning and dynamic mode decomposition (Corol-

lary 2),

(iv) showed that the new data-driven Koopman mod-

els can estimate the essential genes that drive the

growth phenotype of a biological system in the or-

der of their importance (Simulation Example 1),

and

(v) showed that the gene dynamics in the observable

subspace of each output of an interconnected ge-

netic circuit constitute the signiﬁcant genes that

drive that output performance measure of the cir-

cuit (Simulation Example 2).

The paper is organized as follows. Section 2 introduces

the problem statement in detail and Section 3 brieﬂy

introduces the required mathematical preliminaries. In

Section 4, we discuss the main theoretical results per-

taining to observability of Koopman operators and the

methods to see them in practice. We consider two sim-

ulated gene circuits in Section 5 and demonstrate how

the theory is used to ﬁnd genes that drive each output

of the system. Conclusions are drawn in Section 6.

2 Problem Formulation

We formulate the mathematical problem in more depth

and describe how solving it beneﬁts biological systems.

Fig. 1. Koopman approach to observability decomposition of nonlinear systems: The nonlinear observable decom-

position (upper transition) is a result from the the diﬀerential geometric approach to observability of nonlinear systems which

is only deﬁned for analytical systems. The Koopman lifting (transition on the left) is from Koopman operator theory to ﬁnd

high-dimensional linear representations of nonlinear system. Our approach is to ﬁnd the structure of the Koopman operator

for the nonlinear decomposed system (transition on the right) and establish a relationship with the Koopman operator model

of the original nonlinear system through a linear transformation (lower transition).

2.1 The Mathematical Challenge

Given the autonomous discrete-time nonlinear dynami-

cal system with output

State Equation: xt+1 =f(xt) (1a)

Output Equation: yt=h(xt) (1b)

where x∈M⊆Rnis the state and y∈Ris the out-

put performance measure. The diﬀerential geometric ap-

proach to observability provides a nonlinear decomposi-

tion that can an analytical system of the form (1) to

t+1 =fo(xo

t+1 =fu(xo

t, xu

t) (2)

yt=ho(xo

via a diﬀeomorphic (smooth and invertible) nonlinear

transformation "xo

xu#="ξo(x)

ξu(x)#where xulies in the un-

observable subspace of the system (1). The remaining

xois the minimal state that drives the output dynamics

and the manifold that xolies in is the maximum sub-

space that the output ycan observe in the system (1a).

We refer to that space observed by the output as the

observable subspace of the system (1). For data-driven

nonlinear models, there are no approaches to identify the

nonlinear transformations ξoand ξu. There are explicit

methods to do similar transformations for data-driven

linear systems and therefore, we turn to Koopman oper-

ator theory that bridges the notions of linear and non-

linear observable decompositions.

A standard Koopman operator representation used to

capture the nonlinear dynamical system with an output

equation (1) is given by

State Equation: ψ(xt+1) = Kψ(xt) (3a)

Output Equation: yt=Whψ(xt) (3b)

where M ⊆ Rnand ψ:M → RnLare the Koopman ob-

servables (functions of the state), whose linear evolution

across time captures the nonlinear dynamics of the state

and the output. To enable easier recovery of the base

state xfrom the Koopman observables ψ(x), the Koop-

man observables are typically constrained to include the

base states xas ψ(x) = hx>ϕ>(x)i>where ϕ(x) is a

vector of pure nonlinear functions of x. The Koopman

operators corresponding to the observables which con-

tain the state xare referred to as state-inclusive Koop-

man operators. For the rest of the paper, the Koopman

model with observables denoted by ψare state-inclusive.

Since the Koopman model (3) is linear, linear observabil-

ity concepts can be used in this system. How do we use

the Koopman system (3) to infer the observable state xo

in (2)? Section 4 delves more on this topic and provides

algorithms to identify xofrom data and determine the

observable subspace of the original nonlinear system (1).

2.2 The Biological Implication

In complex microbial cell systems, techniques like tran-

scriptomics [46] and proteomics [47] inform the dynamics

within the cell and instruments like ﬂow cytometers [48],

plate readers [49], and microscopes [50] inform the phe-

notypic characteristics viewed from outside the cell. We

can represent the intracellular activity by the state equa-

tion (1a) and the phenotype of interest by the output

equation (1b). The phenotypic behavior is the perfor-

mance metric that we wish to optimize with a speciﬁc

objective. In Section 5, we simulate two biological gene

networks, for which we learn the observable subspace of

the nonlinear system (1) and provide empirical methods

to map that observation space (in which all of xolies) to

the set of genes (a subset of the state variables in x) that

drive the output phenotypic behavior. Upon identifying

the genes that inﬂuence the phenotypic dynamics, we

can deploy actuators developed for biological systems to

control the gene expression and optimize the phenotypic

performance.

The generic phenotypic performance optimization prob-

lem is given by

max

t=0

||yt||2

2(4)

such that xt+1 =f(xt) +

i=1

gi(xt, ut,i)

yt=h(xt)

where gis the input function that captures both how an

input directly controls the expression of targeted genes

as well as oﬀ-target gene expression eﬀects [51] and na

is the number of individual genes whose expression dy-

namics we can target to control. Two accessible genetic

actuators that control gene expression are: A) Trans-

posons [21] which knockout the complete gene expres-

sion with gi(xt, ut,i) = −fi(xt), and B) CRISPR inter-

ference mechanism which suppresses the gene expres-

sion [52] with gi(xt, ut,i)<0. We anticipate this work

will enable the identiﬁcation of genes that impact growth

of soil bacteria in sparse environmental conditions that

can be controlled by biological actuators to maximize

their population growth.

3 Mathematical Preliminaries

We consider the discrete-time nonlinear dynamical sys-

tem (1) with the state x∈ M ⊆ Rnand an output

y∈R. The Koopman operator for the state dynam-

ics (1a) is given by (3a) where K:FnL→ FnL,ψ:

M→RnL, and Fis a space of smooth functions. The

Koopman operator is ideally a linear inﬁnite dimensional

operator(nL→ ∞) but Koopman models identiﬁed from

data are ﬁnite dimensional approximations (nL<∞).

Detailed discussion on Koopman operator theory can be

found in [53, 54]. Since the core focus of the paper is on

the observability of (1), we present the relevant results

from the diﬀerential geometric approach to observabil-

ity of discrete-time nonlinear systems [7, 55–57]. In ad-

dition, we also present an overview of the existing algo-

rithms used to identify Koopman operators.

3.1 Observability of discrete-time nonlinear

systems

The observability of the nonlinear system (1) revolves

around the properties of a new space obtained by the

transformation of the base coordinates x, called the ob-

servation space.

Deﬁnition 1 The observation space Oy(x)for the non-

linear dynamical system (1) is the space of functions that

captures the output across inﬁnite time:

Oy(x) = {h(x), h(f(x)),· · · , h(fi(x)),· · · }, i ∈Z>0.

With a slight abuse of notation, based on the context,

we use Oy(x) to represent either a set or a vector of

functions. If the observation space Oy(x) has a diﬀeo-

morphic map (smooth and invertible) with x, then the

outputs across inﬁnite time can be used to estimate the

initial state xand this would be true for all x∈ M. This

is the strongest condition that ensures the system (1) is

observable, but it is impossible to check for. So, a more

local approach is adopted by computing the dimension

of the observation space at a point.

Deﬁnition 2 The dimension of the observation space

Oy(x)at a point ¯x∈ M is the rank of the Jacobian matrix

of the function set {h(x), h(f(x)),· · · , h(fn−1(x))}:

dimOy(¯x)=rank





∂h(x)

∂x1· · · ∂h(x)

∂xn

.....

∂h(fn−1(x))

∂x1· · · ∂h(fn−1(x))

∂xn







x=¯x

The dimension of the observation space can be computed

locally at a point and hence a local observation result can

be obtained. While there are diﬀerent notions of observ-

ability for nonlinear systems, we only discuss strongly

local observability as we build on top of this deﬁnition

for the rest of the paper.

Deﬁnition 3 The system (1) is said be strongly locally

observable at x∈ M if there exists a neighborhood U

of xsuch that for any ¯x∈ U,h(fk(¯x)) = h(fk(x)) for

k= 0,1,· · · , n −1, implies ¯x=x.

Theorem 1 Theorem 2.1 from [55]If the system (1)

is such that dim(Oy(¯x)) = n, then the system is strongly

locally observable at ¯x

The results extend to the full system if they are true

for all x∈ M. In that case, we state that the system is

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Data-DrivenObservabilityDecompositionwithKoopmanOperatorsforOptimizationofOutputFunctionsofNonlinearSystemsSharaBalakrishnana,AqibHasnainb,RobertG.Egbertc,EnochYeungbaDepartmentofElectricalandComputerEngineering,UniversityofCalifornia,SantaBarbara,UnitedStatesbDepartmentofMechanicalEngineering,Unive...

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Data-Driven Observability Decomposition with Koopman Operators for Optimization of Output Functions of Nonlinear Systems

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