Learned Lifted Linearization Applied to Unstable Dynamic Systems Enabled by Koopman Direct Encoding Jerry Ng1 H. Harry Asada2Fellow IEEE

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Learned Lifted Linearization Applied to Unstable Dynamic Systems

Enabled by Koopman Direct Encoding

Jerry Ng1, H. Harry Asada2,Fellow, IEEE

Abstract— This paper presents a Koopman lifting lineariza-

tion method that is applicable to nonlinear dynamical sys-

tems having both stable and unstable regions. It is known

that Dynamic Mode Decomposition (DMD) and its extended

methods are often unable to model unstable systems accurately

and reliably. Here we solve the problem through merging

three methodologies: decomposition of a lifted linear system

into stable and unstable modes, deep learning of a dictionary

of observable functions in the separated subspaces, and a

new formula for obtaining the Koopman operator, called

Direct Encoding. Two sets of effective observable functions

are obtained through neural net training where the training

data are separated into stable and unstable trajectories. The

resultant learned observables are used for lifting the state

space, and a linear state transition matrix is constructed

using Direct Encoding where inner products of the learned

observables are computed. The proposed method shows a

dramatic improvement over existing DMD and data-driven

methods. Furthermore, a method is developed for determining

the boundaries between stable and unstable regions.

Index Terms— Koopman operator, Lifting linearization, Neu-

ral net observables, Koopman direct encoding

I. INTRODUCTION

Lifting linearization methods, such as those based on the

Koopman Operator, have been used to transform nonlinear

systems to linear models. The theory states that the linear

model becomes exact in modeling a nonlinear system as

the order of the linear model approaches inﬁnity, though

some nonlinear systems have ﬁnite order representations

in lifted space [1], [2]. The Koopman Operator models

are generally constructed through data-driven methods such

as Extended Dynamic Mode Decomposition (EDMD) [3].

These Koopman-DMD methods have been applied to non-

autonomous systems to construct linear dynamic models that

allow us to apply various linear control methods, such as

linear model predictive control (MPC), to nonlinear control

systems [4].

A key component necessary to constructing a Koopman

Operator-based linear model is selection of the observable

functions that lift the state space. Prior work has studied

the use of various function families as observables, such as

polynomial basis functions, radial basis functions and time

delays [5]–[7] . There have also been formulas created for

algorithmically determining useful observables based on the

dataset [8], [9]. Modern machine learning techniques have

This material is based upon work supported by National Science Foun-

dation Grant NSF-CMMI 2021625.

1Jerry Ng is with Department of Mechanical Engineering at the Mas-

sachusetts Institute of Technology jerryng@mit.edu

2H. Harry Asada is with Department of Mechanical Engineering at the

Massachusetts Institute of Technology asada@mit.edu

been applied to learn observable functions with signiﬁcant

success [10]–[12].

However, it can be difﬁcult to formulate accurate approx-

imations of the Koopman Operator for nonlinear systems

that produce both stable and unstable trajectories. A passive

dynamic walker, for example, is essentially an unstable

system, but it can walk stably if it starts within a stable

region [13]. When concerned with only the stable regions

of a nonlinear system, methods have been developed to

construct stable Koopman models from unstable data-driven

models for systems that are known to be stable [14]–[17].

However, these methods are not applicable when needing

to predict stable trajectories for a system with unstable

regions. A key difﬁculty is to capture a proper dataset that

represents diverse behaviors involved in stable and unstable

trajectories. Because of the nature of unstable trajectories,

a bias towards the unstable modes can often occur when

creating the Koopman model.

Prior work discusses the potential for Koopman Oper-

ator models to describe unstable subspaces [1]. This pa-

per presents a methodology for constructing an accurate

Koopman model for nonlinear systems with both stable

and unstable regions through separation of a lifted space

into stable and unstable subspaces. Two sets of effective

observables are learned separately and superposed to con-

struct a complete model. In addition, a method will be

developed for determining a boundary between stable and

unstable regions in the original state space by analyzing

the Koopman Operator Model using modal decomposition,

which is made possible with the effective construction of

observable functions.

The current work presents two signiﬁcant contributions.

One is a novel training method of subspace speciﬁc observ-

able generation (SSOG) via a neural network. The other

is application of a new formula, called Direct Encoding

[18], for obtaining Koopman Operator models through inner

product computations instead of least squares estimate.

In the following, the Koopman Direct Encoding is brieﬂy

described in Section II, followed by the development of

the SSOG algorithm based on space separation, observable

training, and model construction using the Direct Encoding

(Section III). Numerical experiments are presented in Section

IV, and the results will be discussed in Section V.

II. KOOPMAN DIRECT ENCODING OF NONLINEAR

DYNAMICS

In this section, we give a brief overview of the Koop-

man Operator and introduce the direct encoding method

arXiv:2210.13602v2 [cs.LG] 16 Jan 2023

for obtaining a Koopman operator directly from nonlinear

dynamics.

Consider a discrete-time dynamical system, given by

xk+1 =f(xk)(1)

where x∈Rnis the independent state variable vector repre-

senting the system, fis a nonlinear function f:Rn→Rn,

and kis the current time step. Also consider a real-valued

observable function of the state variables g:Rn→R. The

Koopman Operator is an inﬁnite-dimensional linear operator

acting on the observable function g:

(Kg)(x) = g(f(x)) = (g◦f)(x)(2)

where the Koopman operator Kis linear, even though the

dynamic system is nonlinear.

Although this Koopman operator can be deﬁned for an

observable involved in a general Banach space [19], we

assume that the observable function gexists in a Hilbert

space on X⊂ Rn,

g∈ H (3)

Then, it can be shown that the composition g◦fin (2) can

be expressed with an integral kernel as

(g◦f)(x) = ZX

κ(x, ξ)g(ξ)dξ (4)

where κ:X×X→Cis a kernel that can be written by

using a set of orthonormal basis functions [φ1, φ2, φ3, ...]that

span H.

κ(x, ξ) =

∞

i=1

φi[f(x)] ¯

φi(ξ)(5)

This demonstrates that the composition function g◦fis given

by the linear transformation of the observable function g∈ H

[18].

Let [g1, g2,· · · ]be an independent set of observables that

spans the Hilbert space H. Let us further assume that the

compositions of giwith fare involved in the Hilbert space.

gi◦f∈ H ∀i(6)

Applying the above linear transformation of the observable

function (4) to all the observables, it can be shown that

a time-evolution of the observables is given as a linear

transformation with an inﬁnite-dimensional matrix A.

z[f(x)] = Az(x)(7)

where

z(x) = g1(x)g2(x). . .T(8)

The matrix Ais a state transition matrix that maps the lifted

state from one time step to the next.

In the Direct Encoding method [18], the matrix Ais

determined by taking inner products of the observables

and their compositions with the nonlinear function f. Post-

multiplying zT(x)to both sides of (7) and taking integral

over Xyields

Q=AR (9)

where

Q=





hg1◦f, g1i hg1◦f, g2. . .

hg2◦f, g1i hg2◦f, g2i. . .

....





(10)

R=





hg1, g1i hg1, g2i. . .

hg2, g1i hg2, g2i. . .

....





(11)

Since [g1, g2,· · · ]are independent, the matrix Ris non-

singular. Therefore, the Amatrix is given by

A=QR−1(12)

This Direct Encoding method allows us to obtain a linear

model directly from a given nonlinear state equation of

function fand observable functions. The model is valid

globally.

Although the original Koopman Operator is inﬁnite di-

mensional, effective methods have been established for ap-

proximating the operator [20]–[22].

III. SUBSPACE SPECIFIC OBSERVABLE GENERATION

This section presents a novel algorithm for obtaining an

accurate Koopman operator model for nonlinear systems

having both stable and unstable regions. The algorithm is

built upon three theoretical and technical foundations.

First, it employs the Direct Encoding formula. In DMD,

including its variants such as EDMD, the linear state tran-

sition matrix Ain eq. (7) is assumed to exist and is

determined from data based on a Least Squares Estimation.

This may cause a biased estimate, as addressed previously.

In the Direct Encoding formula, however, the Amatrix is

determined from the inner products of observable functions

and their composition with the nonlinear state function f

involved in the two matrices Rand Q. Because both gi

and gi◦fare in a Hilbert space, all the inner products are

guaranteed to exist and the resultant Amatrix provides an

exact linearization that does not depend on data. The linear

model is globally valid. This Direct Encoding formula is used

as a foundational framework in the new algorithm.

Second, the algorithm exploits a basic property of a linear

dynamical system. If a Koopman Operator model can be

constructed for a nonlinear system, then the system can be

represented by

zk+1 =Azk(13)

from eqs. (1) and (7). This linear system can be separated

into its individual modes using eigendecomposition.

zk+1 =VuDt

uWT

uzk+VmDt

mWT

mzk+VsDt

sWT

szk(14)

where V, D, and Ware the eigendecomposition of A, and the

subscripts u, m, and srepresent unstable, marginally stable,

and stable subspaces. This decomposition in the lifted space

motivates us to construct observable functions that represent

the individual subspaces.

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LearnedLiftedLinearizationAppliedtoUnstableDynamicSystemsEnabledbyKoopmanDirectEncodingJerryNg1,H.HarryAsada2,Fellow,IEEEAbstractThispaperpresentsaKoopmanliftinglineariza-tionmethodthatisapplicabletononlineardynamicalsys-temshavingbothstableandunstableregions.ItisknownthatDynamicModeDecomposition(D...

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Learned Lifted Linearization Applied to Unstable Dynamic Systems Enabled by Koopman Direct Encoding Jerry Ng1 H. Harry Asada2Fellow IEEE

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