Data-Driven Feedback Linearization using the Koopman Generator Darshan Gadginmath Vishaal Krishnan Fabio Pasqualetti Abstract This paper contributes a theoretical framework for data-

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Data-Driven Feedback Linearization using the Koopman Generator

Darshan Gadginmath Vishaal Krishnan Fabio Pasqualetti

Abstract— This paper contributes a theoretical framework for data-

driven feedback linearization of nonlinear control-afﬁne systems. We

unify the traditional geometric perspective on feedback linearization

with an operator-theoretic perspective involving the Koopman operator.

We ﬁrst show that if the distribution of the control vector ﬁeld and its

repeated Lie brackets with the drift vector ﬁeld is involutive, then

there exists an output and a feedback control law for which the

Koopman generator is ﬁnite-dimensional and locally nilpotent. We

use this connection to propose a data-driven algorithm ‘Koopman

Generator-based Feedback Linearization (KGFL)’ for feedback lin-

earization. Particularly, we use experimental data to identify the state

transformation and control feedback from a dictionary of functions

for which feedback linearization is achieved in a least-squares sense.

We also propose a single-step data-driven formula which can be

used to compute the linearizing transformations. When the system

is feedback linearizable and the chosen dictionary is complete, our

data-driven algorithm provides the same solution as model-based

feedback linearization. Finally, we provide numerical examples for

the data-driven algorithm and compare it with model-based feedback

linearization. We also numerically study the effect of the richness of the

dictionary and the size of the data set on the effectiveness of feedback

linearization.

I. INTRODUCTION

Nonlinear control methods rooted in model-based approaches

have received considerable attention [1]. Among these techniques,

feedback linearization has emerged as a prominent strategy, offering

the implementation of straightforward linear control methodologies

to nonlinear systems. However, a notable limitation of this approach

is its demand for a comprehensive knowledge of the system

dynamics. Consequently, inadequate system identiﬁcation in the

context of complex, high-dimensional cyber-physical systems can

lead to poor control performance. On the contrary, machine learning

methodologies [2]–[4] offer a robust alternative, enabling the uti-

lization of experimental data acquired from the system to facilitate

feedback control, even in the absence of prior knowledge regarding

the underlying system’s dynamics. Nevertheless, these machine

learning methods frequently fall short of providing comprehensive

insights into both their own performance and the intricate nature of

the systems they operate on. Furthermore, the full extent of their

limitations remains a subject of ongoing investigation. The pursuit

of a systematic framework for nonlinear data-driven control remains

an unresolved challenge.

Recently, signiﬁcant attention has been directed towards

the Koopman operator [5] due to its capacity to furnish a

global (inﬁnite-dimensional) linear representation of autonomous

nonlinear systems. It was shown in [6] that the Koopman operator

can be approximated in ﬁnite dimensions with data using a

dictionary of observables, which has been a notable direction of

research for nonlinear systems. However, the commonality between

the two aforementioned methodologies pertains to the concept of

complete linearization, a dimension of inquiry that has hitherto

remained unexplored in the existing literature. In this work, we

bridge the gap between the conventional technique of feedback

linearization and the Koopman operator. Furthermore, leveraging

this newfound connection, we develop a data-driven methodology

capable of yielding valuable insights into the dynamics inherent to

the system.

Problem setup. We consider a continuous-time nonlinear control-

afﬁne system, with single input, of the form:

˙x=f(x) + g(x)u, (1)

where x∈X⊆Rnis the state, u∈Ris the control input, and

f, g :X→Rnare the drift and control vector ﬁelds. In the data-

driven setting, we do not have access to the drift and control vector

ﬁelds f, g, but instead have access to Ndata samples collected

from a control experiment on System (1). The state and control

trajectory during the experiment is {xt, ut}where t∈R≥0. The

data collected from an experiment is represented as matrices X, U

as follows:

X= [x0x1. . . xN], U = [u1u2. . . uN],

where xiand uiare the sample at the i-th instance of the

experiment. In the experiment, the control uis assumed to be

sufﬁciently exciting so as to provide data of good quality [7]. For

instance, utcould be sampled from a Gaussian distribution.

Our objective is to transform system (1) to a target linear system

˙z=Az +Bv, where zand vare transformed state and control,

respectively. We propose to transform the state as z=H(x)

and the control as u=α(x) + β(x)v. We seek to identify the

transformations H, α, and βusing the data X, U .

Related work. A comprehensive introduction to feedback lineariza-

tion can be found in [1]. This technique provides a systematic

method to identify the necessary state and control transformations

in the model-based case. It is crucial to note that these transfor-

mations are dependent on the dynamics of the system and not

all systems allow for feedback linearization. An approximate, but

still model-based, approach for feedback linearization was proposed

in [8]. These methods cannot be employed without a prior system

identiﬁcation step. Several works that combine learning methods

for feedback linearization have been proposed [3], [4], [9], [10].

The works [3], [4] primarily use neural networks to obtain state

and control transformations, whereas [9] proposes a reinforcement

learning approach. However, these methods do not provide a clear

insight into the control and state transformations. In [11], a SISO

full state-feedback linearizable system is considered and a data-

driven solution is proposed by approximating the system using

Taylor series. An extension of the Willems fundamental lemma for

nonlinear systems is proposed in [12], which is used to present

a predictive control methodology with data. However, a systematic

approach to ﬁnding the state and control transformations in the data-

driven setting for feedback linearization has not been addressed

in the literature. In this work, we seek to establish a data-driven

methodology for feedback linearization which not only provides a

convenient solution but also insight into the dynamics of the system.

The main advantage of the Koopman operator is its ability to

provide a global linear representation of a nonlinear system. How-

ever, its main drawback is its inﬁnite-dimensional representation for

arXiv:2210.05046v2 [math.OC] 7 Nov 2023

only autonomous systems. Recent literature has focused on ﬁnite-

dimensional approximations of the Koopman operator [6], [13].

Of particular interest is the gEDMD algorithm [14] which seeks

a ﬁnite-dimensional approximation of the inﬁnitesimal generator of

the Koopman operator and is based on Extended Dynamic Mode

Decomposition (EDMD) [6]. The gEDMD algorithm [14] uses a

dictionary of functions to lift full-state data from an autonomous

system and seeks to ﬁnd a linear relation in the evolution of the

lifted system.

The works in [15]–[18] have focused on obtaining accurate

ﬁnite-dimensional approximations of this linear operator for

control. While [15]–[17] have extended [6] for control, [18]

transforms the nonlinear system as a linear parameter-varying

system with the control as the variable parameter. In [19], linear

predictors for the control-afﬁne nonlinear system are considered.

However, crucially, the control transformations required for exact

linearization and its connection to feedback linearization are

absent. A Luenberger observer for the system’s nonlinearities is

proposed using the Koopman operator in [20]. Here the control

is considered as a varying parameter, and the overall system is

considered as a linear parameter varying system. Hence, existing

literature that use the Koopman operator for control have crucially

missed the connection to feedback linearization. Bilinearization

using the Koopman operator has also been an area of interest

[21]–[23]. In [21], the nonlinear system is approximated by

interpolated bilinear systems. Then a model predictive control

scheme is applied to the identiﬁed interpolated bilinear model.

Probabilistic error bounds on trajectories predicted by bilinearized

models using the Koopman operator are given in [22]. Conditions

for global bilinearizability using the Koopman operator are given

in [23]. However, it is important to note that standard linear control

techniques cannot be implemented on bilinear models. The model-

based feedback linearization approach and the modern data-driven

Koopman operator approach are both linearization techniques,

yet for controlled and autonomous systems, respectively. In

this paper, we focus on showing a connection between these two

methods and developing a data-driven scheme for nonlinear control.

Contributions. The main contributions of this paper are as follows.

We ﬁrst bridge the gap between the geometric framework of feed-

back linearization and the Koopman operator-theoretic framework.

In particular, we show that, when the system is involutive to a

certain degree, there exists an observable hand a feedback control α

such that the Koopman generator for the closed-loop system under

the feedback αis nilpotent at the observable h. Furthermore,

there exists a ﬁnite-dimensional Koopman invariant subspace of the

same dimension as the involutive distribution for the system. This

connection to the Koopman operator allows us to develop a data-

driven method for feedback linearization, by essentially casting the

problem of data-driven feedback linearization as one of learning

the closed-loop Koopman operator for the nonlinear control-afﬁne

system by a linearizing state/control transformation. To this end,

we exploit the fact that involutivity permits a representation of the

Koopman generator in the ﬁnite-dimensional Brunovsky canonical

form under the linearizing state/control transformation. This allows

us to ﬁx the Brunovsky canonical form as the target linear rep-

resentation and learn the linearizing transformation using a set of

ﬁxed dictionary functions by a least-squares method in our algo-

rithm Koopman generator-based Feedback Linearization (KGFL).

We also provide a numerical feedback linearization scheme with

only input-output data. With input-output data, we show that the

control transformations can be learned in a least-squares sense

using a simple data-driven formula. The results in [24], which

were developed independently from this work, deal with data-driven

feedback linearization with complete dictionaries for fully feedback

linearizable systems. In our work, we neither make the assumption

of full feedback linearizability nor of complete dictionaries. How-

ever, when the system is feedback linearizable and the dictionaries

used in KGFL are complete, the solution is exact and is equal to the

model-based solution. Finally, we demonstrate the performance of

our algorithm with numerical simulations for multiple examples. We

perform both full state feedback linearization and output feedback

linearization on the Van der Pol oscillator and compare it against

existing nonlinear data-driven control techniques. We consider a

higher dimensional system with the control entering nonlinearly

and show that our algorithm can be used for complex systems. We

also provide insight on the effect of richness of dicitonary and data

size on the accuracy of feedback linearization method.

II. PRELIMINARIES

Let Rdbe the d-dimensional Euclidean space. Let (X, dX)and

(Y, dY)be metric spaces. A map f:X→Yis said to be Lipschitz

(with Lipschitz constant ℓf) if dY(f(x1), f(x2)) ≤ℓfdX(x1, x2)

for any x1, x2∈X. The space of k-times continuously differen-

tiable functions on Xis denoted by Ck(X). Let Vbe a normed

vector space and let T:V→Vbe a bounded linear operator

on V. A subspace W⊆Vis said to be T-invariant if T(W)⊆W.

The operator T:V→Vis locally nilpotent with index rat

v∈Vif Tkv̸= 0 for all k∈ {0,...,r −1}and Trv= 0.

Furthermore, if v, T (v),...,Tr(v)are linearly independent, then

span{v, T (v),...,Tr(v)}is said to be a T-cyclic subspace of T.

For a measure space (X,Σ, µ)(where Σis a sigma-algebra on X

and µis the measure on (X, Σ)), a property Pis said to hold

almost everywhere (a.e.) if the subset over which the property P

fails to hold is of µ-measure zero. The Lie bracket between two

vector ﬁelds fand gis denoted by [f, g] = adfg=Lgf−Lfg. The

adjoint of order kis deﬁned recursively as adk

fg=hf , ad(k−1)

fgi

with ad0

fg=g. The Gateaux derivative U(v;η)[25] of operator

T∈C1(V, V )at v∈Valong η∈Vis given by

lim

h→0+∥T(v+hη)−T(v)−U(v;η)h∥V= 0.

W⊆Vis called a stable subspace of Twith respect to

perturbations along ηif U(w;η) = 0 for all w∈W.

A. Lie derivative as Koopman generator

Consider the autonomous system ˙x(t) = f(x(t)) with state space

X⊂Rd, where f:X→Rdis Lipschitz. Let s:X×R≥0→Xbe

the ﬂow of the vector ﬁeld f, such that for any x∈X,s(x, 0) = x

and d

dt s(x, t) = f(s(x, t)). For a function ϕ∈C1(X), let Lfϕbe

the Lie derivative of ϕwith respect to f, such that for any x∈X,

Lfuniquely satisﬁes

lim

h→0+|ϕ(s(x, h)) −ϕ(x)−h(Lfϕ)(x)|= 0.

Let K:C1(X)×R≥0→C1(X)be the Koopman operator for the

ﬂow s, such that for any x∈Xwe have:

(Ktϕ)(x) = ϕ(s(x, t)),

where we have adopted the notation Ktϕ=K(ϕ, t). The family of

operators {Kt}t≥0is said to be right-differentiable at t= 0 with

derivative operator Lif the following holds for any ϕ∈C1(X):

lim

h→0+∥Khϕ− K0ϕ−hLϕ∥L2(X)= 0.

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Data-DrivenFeedbackLinearizationusingtheKoopmanGeneratorDarshanGadginmathVishaalKrishnanFabioPasqualettiAbstract—Thispapercontributesatheoreticalframeworkfordata-drivenfeedbacklinearizationofnonlinearcontrol-affinesystems.Weunifythetraditionalgeometricperspectiveonfeedbacklinearizationwithanoperator...

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Data-Driven Feedback Linearization using the Koopman Generator Darshan Gadginmath Vishaal Krishnan Fabio Pasqualetti Abstract This paper contributes a theoretical framework for data-

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