Utility of the Koopman operator in output regulation of disturbed nonlinear systems Bart Kieboom Maria Bartzioka and Matin Jafarian

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Utility of the Koopman operator in output regulation of disturbed

nonlinear systems

Bart Kieboom, Maria Bartzioka and Matin Jafarian

Abstract— This paper studies the problem of output regu-

lation for a class of nonlinear systems experiencing matched

input disturbances. It is assumed that the disturbance signal

is generated by an external autonomous dynamical system.

First, we show that for a class of nonlinear systems admitting

a ﬁnite-dimensional Koopman representation, the problem is

equivalent to a bilinear output regulation. We then prove that

a linear dynamic output feedback controller, inspired by the

linear output regulation framework, locally solves the original

nonlinear problem. Numerical results validate our analysis.

I. INTRODUCTION

Output regulation is a well-known problem in control

theory. The goal is that the system’s output asymptotically

tracks a reference signal and/or rejects a disturbance signal

generated by an external autonomous dynamical system,

namely the exosystem. The problem is well-studied for linear

systems [1], where necessary and sufﬁcient conditions for its

solvability depend on the solvability of regulatory equations

[2]. The internal model principle has been proven essential in

solving these equations and designing controllers [3]. Output

regulation of nonlinear systems, however, is naturally more

challenging. This problem has also been widely studied [4].

It has been shown that a set of partial differential equations

form the regulatory equations for the nonlinear problem [1],

[4]. However, ﬁnding solutions for the PDEs and design-

ing appropriate internal-model-based controllers for general

nonlinear systems and exosystems is quite complex. The

problem has been solved for several speciﬁc classes of

nonlinear systems, and particular control problems [1], [5],

[6]. Owing to its importance and applicability, the nonlinear

output regulation problem is still an active ﬁeld of research.

In this paper, we propose utilizing the Koopman operator in

solving a sub-class of nonlinear output regulation problems.

The Koopman operator provides an alternative way to

model (nonlinear) dynamical systems. It was originally

introduced by Bernard Koopman in [7] and popularized

for the study of dynamical systems in [8], [9]. Numerous

applications in a wide range of research ﬁelds have explored

the employment of these techniques [10], [11]. The Koopman

operator, associated with a state-space description of a non-

linear dynamical system, is a linear and inﬁnite-dimensional

operator that acts on functions of the state of the system,

often called observables or observable functions. The action

of the Koopman operator on such observables allows us

The authors are with the Delft Center for Systems and

Control (DCSC), Delft University of Technology, The Nether-

lands. Email: b.kieboom@student.tudelft.nl;

m.bartzioka@tudelft.nl; m.jafarian@tudelft.nl.

The work of M. Jafarian is supported by the EU-MCIF project ReWoMeN.

to compute the time evolution of the observables linearly,

according to the ﬂow of the system.

We study matched input disturbance rejection of nonlinear

systems, assuming that disturbance signals are generated by

linear exosystems. For a class of nonlinear systems that admit

a ﬁnite-dimensional Koopman representation, we show that

the nonlinear problem can be equivalently represented as a

bilinear one. Then, we design a linear output feedback con-

troller inspired by the linear output regulation framework to

achieve regional bilinear output regulation. We characterize

the latter as achieving regional stabilization for the bilinear

undisturbed system together with regulating the output error

to zero. Since the bilinear system equivalently represents the

nonlinear system, we conclude that regional bilinear output

regulation guarantees that the original nonlinear output reg-

ulation problem is locally solved.

To the best of our knowledge, the existing literature has

been mainly focused on bilinear stabilization, both model-

based [12] and data-driven [13], [14]. The bilinear output

regulation problem, though, has been less explored. Constant

disturbance rejection of bilinear systems has been tackled in

[15]. Exploring the applications of the Koopman operator,

it has been employed to derive bilinear formulation for

nonlinear stabilization problems, in both model-based and

data-driven approaches, mainly in the discrete-time setting

[10], [16], [17], where the problems have been formulated in

an optimal control setting. Compared to the aforementioned

works, we solve the continuous-time bilinear matched input

disturbance rejection problem, with a general linear exosys-

tem, within the output regulation framework and design a

linear controller inspired by the internal model principle. Our

approach enables a systematic way for local output regulation

of the nonlinear systems.

This paper is organized as follows. Section II presents

preliminaries on the output regulation problem, Koopman

operator theory and the problem formulation. Section III

covers the main results of the paper. Section IV presents

a numerical example. The paper is concluded in section V.

II. PRELIMINARIES AND PROBLEM FORMULATION

In this section, we ﬁrst recall some required techniques

from the output regulation [1] and the Koopman framework

[8], [9], [11], and continue by formulating the problem.

A. Linear Output Regulation

Consider the following disturbed linear system,

˙x=Ax +Bu +P v, (1a)

y=Cx, (1b)

arXiv:2210.00040v3 [eess.SY] 15 Sep 2023

with x∈X⊆Rnthe state, u∈Rmthe input, y∈Rlthe

output error, and v∈Rqthe output of the following linear

autonomous system,

˙w=Sw, v =Ew, (2)

with w∈W⊆Rrthe exogenous disturbance signal.

The goal is to design a controller such that the undisturbed

system has an asymptotically stable equilibrium, and the

output error of the system in the presence of the disturbance

satisﬁes limt→∞ y(t)=0,for any initial condition of the

plant and the exosystem. The exosystem is assumed neutrally

stable, i.e., ST=−S. The general form of a linear dynamic

output feedback controller with input yand output uis

ξ=F ξ +Gy, (3a)

u=Hξ + Γy, (3b)

where ξ∈Ξ⊆Rνrepresents the state of the controller.

The closed loop dynamics of the disturbed linear system (1)

inﬂuenced by the exosystem (2) and the controller (3) obeys

˙x

ξ=Acx

ξ+P v

0, Ac=(A+BΓC)BH

GC F .(4)

It is known that the stabilizability of the pair (A, B)

together with the detectability of the pair (C, A)guarantee

the existence of matrices {F, G, H, Γ}such that Acis Hur-

witz [18]. Since Acis Hurwitz, the linear output regulation

problem is solved by the controller (3) if and only if there

exist Π∈Rn×rand Σ∈Rν×rsuch that

ΠS= (A+BΓC)Π + BR +P E, (5a)

0 = CΠ,(5b)

ΣS=FΣ,(5c)

R=HΣ,(5d)

are satisﬁed. The ﬁrst two equations are called the linear

regulator equations [2] and the last two address the internal

model principle [3].

B. Koopman Operator Theory

Consider the autonomous continuous-time dynamical sys-

tem described by

˙x=f(x),(6)

with x∈X⊆Rnthe state and fa nonlinear function.

Integrating (6) yields trajectories x(t) = Ft(x0), where

Ft(x0) = x0+Zt

f(x(τ))dτ (7)

is the ﬂow of the system.

Next, consider functions of the state ψ:X→R, which

are called observable functions. Denote the space of all such

functions as F. The family of Koopman operators Kt:F 7→

Fis deﬁned by

Ktψ(x) = ψ(x)◦Ft=ψ(Ft(x)).(8)

Since Kt2(Kt1ψ(x)) = ψ(Ft1+t2(x)) we simply write K

and refer to it as the Koopman operator. The Koopman

operator is inﬁnite-dimensional and linear.

The inﬁnitesimal generator L, of the Koopman operator

[19] is deﬁned by

Lψ= lim

t→0

Ktψ−ψ

t= lim

t→0

ψ(Ft(x)) −ψ(x)

t.(9)

From (9) we see that Lcorresponds to the time derivative

of ψalong the trajectories of (6), i.e., ˙

ψ=Lψ.

The advantage of the Koopman operator is that it allows

a nonlinear system to be described linearly. Although such

descriptions are generally inﬁnite-dimensional, in some cases

a ﬁnite, albeit higher dimensional, description can be approx-

imated. A set D ⊂ F of observable functions which satisﬁes

D={ψ∈ F | ψ∈ D =⇒˙

ψ∈span(D)},(10)

is called a Koopman invariant subspace [11]. Let Ψ =

[ψ1, . . . , ψM]T, with ψi∈ D. Write

z= Ψ(x),(11)

then it follows that

˙z=Az, (12)

where Aij is the j-th expansion coefﬁcient of ψiin D.

Hence, a Koopman invariant subspace containing the state

components ψi=xiyields the equivalent ﬁnite-dimensional

linear description (12) of the autonomous nonlinear system

(6), [11].

C. Problem Formulation

This paper considers the utilization of the Koopman oper-

ator framework in dealing with matched input disturbances

(e.g. [20]–[22]) of nonlinear systems of the form

˙x=f(x) + g(x)(u+v)(13a)

y=h(x),(13b)

with x∈X⊆Rn,y∈Rl, and f,gand hsmooth,

Lipschitz nonlinear functions. In the rest of the paper, we

restrict our attention to the scalar input and disturbance case,

i.e., u, v ∈R, and assume that vis the output of the linear ex-

osystem (2). In the next section, we ﬁrst characterize a class

of nonlinear systems admitting a Koopman representation

using a ﬁnite number of observables. Therefore, solving the

output regulation problem of the bilinear system equivalently

solves the output regulation problem of the aforementioned

nonlinear system. Inspired by the linear output regulation

framework, we design a linear dynamic output feedback

controller of the form (3) for the bilinear system that achieves

disturbance rejection and thereby output regulation of the

equivalent nonlinear system.

III. NONLINEAR MATCHED INPUT DISTURBANCE

REJECTION

This section presents the main results of the paper. First,

we present a lemma, based on [11], [16], [23], essential

in specifying the class of nonlinear systems that can be

represented using ﬁnite number of observables in a Koopman

invariant subspace D[11].

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UtilityoftheKoopmanoperatorinoutputregulationofdisturbednonlinearsystemsBartKieboom,MariaBartziokaandMatinJafarianAbstract—Thispaperstudiestheproblemofoutputregu-lationforaclassofnonlinearsystemsexperiencingmatchedinputdisturbances.Itisassumedthatthedisturbancesignalisgeneratedbyanexternalautonomous...

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Utility of the Koopman operator in output regulation of disturbed nonlinear systems Bart Kieboom Maria Bartzioka and Matin Jafarian

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