DEEPKOOPMAN LEARNING OF NONLINEAR TIME-VARYING SYSTEMS Wenjian Hao

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DEEP KOOPMAN LEARNING OF NONLINEAR TIME-VARYING

SYSTEMS

Wenjian Hao

School of Aeronautics and Astronautics Engineering

Purdue University, IN, USA

hao93@purdue.edu

Bowen Huang

Paciﬁc Northwest National Laboratory

Richland, USA

bowen.h@pnnl.gov

Wei Pan

Department of Computer Science

University of Manchester,UK

wei.pan@tudelft.nl

Di Wu

Paciﬁc Northwest National Laboratory

Richland, USA

di.wu@pnnl.gov

Shaoshuai Mou

School of Aeronautics and Astronautics Engineering

Purdue University, IN, USA

mous@purdue.edu

June 22, 2023

ABSTRACT

This paper presents a data-driven approach to approximate the dynamics of a nonlinear time-varying

system (NTVS) by a linear time-varying system (LTVS), which is resulted from the Koopman op-

erator and deep neural networks. Analysis of the approximation error between states of the NTVS

and the resulting LTVS is presented. Simulations on a representative NTVS show that the proposed

method achieves small approximation errors, even when the system changes rapidly. Furthermore,

simulations in an example of quadcopters demonstrate the computational efﬁciency of the proposed

approach.

1 Introduction

In recent years, data-driven methods have received a signiﬁcant amount of research attention due to the increasing

complexity of the autonomous systems in both dynamics [1–4] and mission objectives [5–7]. In the direction of

learning system dynamics, the Koopman operator has recently been proven to be an effective method to approximate a

nonlinear system by a linear time-varying system based on state-control pairs [2–4]. Along this direction, two popular

methods dynamic mode decomposition (DMD) and extending dynamic mode decomposition (EDMD) are used to lift

the state space to a higher-dimensional space, where the evolution is approximately linear [8]. However, choosing

the proper observable functions for the lifting transformation is still an open question, and the potentially large lifted

dimension may hinder real-time applications.

Recent work has proposed several methods for choosing proper observable functions of Koopman-based methods for

time-invariant systems. Lusch et al. [9] proposed applying deep learning methods to discover the eigenfunctions of

the approximated Koopman operator. Yeung et al. [10–13] introduced deep neural networks (DNN) as observable

functions of the Koopman operator, which are tuned based on collected state-control pairs by minimizing a properly

deﬁned loss function. While some work, such as [14], has extended the DMD method to approximate nonlinear time-

varying systems (NTVS) that change sufﬁciently slowly by linear time-varying systems (LTVS), this method is not

directly applicable to approximate nonlinear systems with rapidly changing dynamics.

arXiv:2210.06272v3 [eess.SY] 21 Jun 2023

APREPRINT - JUNE 22, 2023

In this paper, we propose a deep Koopman learning method to approximate NTVS, which employs DNN as the

observable function of the Koopman operator and adjusts both the DNN and the approximated dynamical system

simultaneously. This is achieved by tuning the DNN parameters based on the latest state-control data pairs to track the

unknown NTVS.

Compared to existing results in [14], the proposed method is able to approximate an NTVS which does not necessarily

change slowly. Contributions of this paper are summarized as follows:

• We propose a deep Koopman representation formulation for the NTVS and provide a practical online algo-

rithm for implementation.

• We investigate the error bound of the system state estimation of the proposed method.

• We perform a convergence analysis of the proposed method concerning the observable function of DNN.

This paper is organized as follows. In Section 2, we state the problem. Section 3presents the main results. The

numerical simulations are exhibited in Section 4. Finally, Section 5concludes the paper.

Notations. Let ∥·∥denote the Euclidean norm. For a matrix A∈Rn×m,∥A∥Fdenotes its Frobenius norm; AT

denotes its transpose; A†denotes its Moore-Penrose pseudoinverse. For positive integers nand m,Indenotes the

n×nidentity matrix; 0n∈Rndenotes a vector with all value 0;0n×mdenotes a n×mmatrix with all value 0.

sgn(·)denotes the sign function. ⌈·⌉ denotes the ceiling function, i.e., given real numbers y, integers kand the set of

integers Z,⌈y⌉= min{k∈Z|k≥y}.

2 Problem Formulation

Consider an NTVS, the dynamics of which is unknown. Let xt∈Rnand ut∈Rmdenote its state and control input

at time t, respectively. t∈[0,∞)denotes the continuous-time index.

Suppose the states and control inputs can be obtained from unknown continuous NTVS at certain sampling time

instances tk∈[0,∞)with k= 0,1,2,· · · the index of sampled data points. For notation brevity, one denotes

xk:=xtkand uk:=utkas the k-th observed system state and control input, respectively, in the remainder of this

manuscript. Then one can partition the observed states-inputs pairs as the following series of data batches

Bτ={xk, uk:k∈Kτ}, τ = 0,1,2,· · · ,(1)

where

Kτ={kτ, kτ+ 1, kτ+ 2,· · · , kτ+βτ}

denotes the ordered labels set of sampling instances for the τ-th data batch Bτwith βτpositive integers such that

kτ=

τ−1

i=0

βi, τ ≥1, k0= 0.

It follows that

kτ+1 =kτ+βτ, τ = 0,1,2,· · · ,

which implies the last data in the τ-th data batch is the ﬁrst data in (τ+ 1)-th data batch. For notation simplicity, one

deﬁnes Bx

τ:={xk:k∈Kτ},Bu

τ:={uk:k∈Kτ}in the rest of this manuscript. An illustration of the above

indexes is shown in Fig. 1.

Figure 1: Time indexes, where blue color denotes kτ.

This paper aims to develop an iterative method that approximates the dynamics of an unknown NTVS based on avail-

able data batches Bτby a linear time-varying discrete-time system. One way to achieve such a linear approximation is

APREPRINT - JUNE 22, 2023

by employing the Koopman operator as in [10–13]. Namely, based on the data batch Bτ, one ﬁnds a nonlinear mapping

g(·, θτ) : Rn→Rrparameterized by θτ∈Rq1and constant matrices Aτ∈Rr×r,Bτ∈Rr×m,Cτ∈Rn×rsuch

that for k∈Kτ, k < kτ+βτ, the following holds approximately,

g(xk+1, θτ) = Aτg(xk, θτ) + Bτuk,(2)

xk+1 =Cτg(xk+1, θτ).(3)

Here, g(·, θτ),Aτ,Bτ,Cτachieved from Bτare put together in the following KBτ:

KBτ:={g(·, θτ), Aτ, Bτ, Cτ},(4)

which is called a deep Koopman representation (DKR) in this manuscript.

Based on the DKR in (4), one could introduce ˆxk∈Rnfor k∈Kτ, k < kτ+βτas follows:

g(ˆxk+1, θτ) = Aτg(ˆxk, θτ) + Bτˆuk,(5)

ˆxk+1 =Cτg(ˆxk+1, θτ),(6)

where

ˆuk=uk,∀k∈Kτ, k < kτ+βτ,ˆxkτ=xkτ.(7)

This leads to the following linear system

ˆxk+1 =ˆ

Aτˆxk+ˆ

Bτˆuk, k ∈Kτ, k < kτ+βτ,(8)

with ˆ

Aτ=CτAτC†

τ,ˆ

Bτ=CτBτand initial conditions in (7). The system (8) can be viewed as a linear approximation

to the NTVS based on the data batch Bτ. Note that when for any data batch Bτ, one has Aτ,Bτ,Cτremain constant

for k∈Kτ. It follows that (8) is a linear time-invariant system for k∈Kτ.

To sum up, the problem of interest is to develop an iterative update rule to achieve a DKR in (4) based on data batch

Bτin (1) such that the linear system (8) is a nice approximation of the unknown NTVS, i.e. ˆxk(8) is close to xk

observed in Bτfrom the unknown NTVS in the sense that for any given accuracy ε≥0,ˆuk=ukand ˆxkτ=xkτ, the

estimation error ∥ˆxk−xk∥≤ ε.

3 Main Results

This section proposes an algorithm to achieve a deep Koopman representation (DKR) that can approximate an un-

known NTVS. We then investigate the estimation error between the state obtained from this DKR, as given in (8), and

the observed state of the unknown NTVS.

3.1 Key Idea

Motivated by deep Koopman operator-based methods developed in [10–13], an optimal θτfor the deep Koopman

representation (DKR), denoted by θ∗

τ, can be obtained by solving the following optimization problem based on the

data batch Bτ:

θ∗

τ= arg min

θτ∈Rq{wL1(Aτ, Bτ, θτ) + (1 −w)L2(Cτ, θτ)},(9)

where

L1(Aτ, Bτ, θτ) = 1

βτ

kτ+βτ−1

k=kτ

∥g(xk+1, θτ)−(Aτg(xk, θτ) + Bτuk)∥2(10)

and

L2(Cτ, θτ) = 1

βτ

kτ+βτ−1

k=kτ

∥xk−Cτg(xk, θτ)∥2.(11)

The objectives of (10) and (11) are to approximate (2) and (3), respectively. Here, 0< w < 1is a constant that

combines the objective of minimizing L1and L2. In simple terms, L1and L2measure the simulation errors in the

lifted and original coordinates, respectively.

1Here, g(·, θτ)is usually represented by a DNN with a known structure gand an adjustable parameter θτ∈Rq.

iii

APREPRINT - JUNE 22, 2023

To solve (9), one needs to rewrite the available data batch and objective functions L1and L2in compact forms. Toward

this end, the following notation is introduced:

Xτ= [xkτ, xkτ+1,· · · , xkτ+βτ−1]∈Rn×βτ,

Xτ= [xkτ+1, xkτ+2,· · · , xkτ+βτ]∈Rn×βτ,

Uτ= [ukτ, ukτ+1,· · · , ukτ+βτ−1]∈Rm×βτ.

Then L1in (10) and L2in (11) can be rewritten as

L1=1

βτ

∥¯

Gτ−(AτGτ+BτUτ)∥2

F(12)

and

L2=1

βτ

∥Xτ−CτGτ∥2

F,(13)

where

Gτ= [g(xkτ, θτ),· · · , g(xkτ+βτ−1, θτ)] ∈Rr×βτ,

Gτ= [g(xkτ+1, θτ),· · · , g(xkτ+βτ, θτ)] ∈Rr×βτ.(14)

By minimizing L1with respect to Aτ, Bτin (12) and minimizing L2regarding Cτin (13), Aτ, Bτ, Cτcan be deter-

mined by θτas follows:

[Aθ

τ, Bθ

τ] = ¯

GτGτ

Uτ†

,(15)

Cθ

τ=XτG†

τ.(16)

Replacing Aτand Bτin (10) by (15) and Cτin (11) by (16), the objective function in (9) can be reformulated as

L(θτ) = 1

βτ

kτ+βτ−1

k=kτ

∥g(xk+1, θτ)

xk−Kθ

τg(xk, θτ)

uk∥2,(17)

with

Kθ

τ=Aθ

τBθ

Cθ

τ0n×m.

Applying the existing deep Koopman operator methods developed in [10–13] to achieve the DKR by solving (9) based

on each Bτavailable has two shortcomings. First, computing the pseudo-inverse in (15) and (16) repeatedly while

solving (9) becomes computationally expensive as τincreases. Second, θτmust be initialized for each Bτ, which

can be challenging in time-varying systems applications. To overcome these two limitations, one can apply the deep

Koopman operator method to approximate the unknown NTVS efﬁciently, and we propose the following method.

3.2 Algorithm

Before proceeding, we need the following assumption.

Assumption 1 The matrix Gτ∈Rr×βτin (14)and Gτ

Uτ∈R(r+m)×βτare of full row rank.

Remark 1 Assumption 1is to ensure the matrices Gτ∈Rr×βτand Gτ

Uτ∈R(r+m)×βτinvertible and it naturally

requires βτ≥r+m.

Lemma 1 Given KBτin (4), if Assumption 1holds, then the matrices Aθ

τ+1, Bθ

τ+1,Cθ

τ+1 can be achieved by

[Aθ

τ+1, Bθ

τ+1]=(¯

Gτ+1 −[Aθ

τ, Bθ

τ]χτ+1)λτχT

τ+1(χτχT

τ)−1+ [Aθ

τ, Bθ

τ],(18)

Cθ

τ+1 = (Xτ+1 −Cθ

τGτ+1)¯

λτGT

τ+1(GτGT

τ)−1+Cθ

τ,(19)

where χτ=Gτ

Uτ∈R(r+m)×βτ,λτ= (Iβτ+1 +χT

τ+1(χτχT

τ)−1χτ+1)−1∈Rβτ+1 ×βτ+1 ,¯

λτ= (Iβτ+1 +

τ+1(GτGT

τ)−1Gτ+1)−1∈Rβτ+1 ×βτ+1 with Gτ+1 ∈Rr×βτ+1 ,¯

Gτ+1 ∈Rr×βτ+1 deﬁned in (14).

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DEEPKOOPMANLEARNINGOFNONLINEARTIME-VARYINGSYSTEMSWenjianHaoSchoolofAeronauticsandAstronauticsEngineeringPurdueUniversity,IN,USAhao93@purdue.eduBowenHuangPacificNorthwestNationalLaboratoryRichland,USAbowen.h@pnnl.govWeiPanDepartmentofComputerScienceUniversityofManchester,UKwei.pan@tudelft.nlDiWuPacif...

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