1

Koopman Operator Theory for Nonlinear Dynamic Modeling using DynamicMode Decomposition

Gregory Snyder and Zhuoyuan Song

Abstract— The Koopman operator is a linear operator thatdescribes the evolution of scalar observables (i.e., measurementfunctions of the states) in an infinite-dimensional Hilbert space.This operator theoretic point of view lifts the dynamics of afinite-dimensional nonlinear system to an infinite-dimensionalfunction space where the evolution of the original systembecomes linear. In this paper, we provide a brief summary of theKoopman operator theorem for nonlinear dynamics modelingand focus on analyzing several data-driven implementationsusing dynamical mode decomposition (DMD) for autonomousand controlled canonical problems. We apply the extendeddynamic mode decomposition (EDMD) to identify the leadingKoopman eigenfunctions and approximate a finite-dimensionalrepresentation of the discovered linear dynamics. This allows usto apply linear control approaches towards nonlinear systemswithout linearization approximations around fixed points. Wecan then examine the fidelity of using a linear controllerbased on a Koopman operator approximated system on under-actuated systems with basic maneuvers. We demonstrate theeffectiveness of this theory through numerical simulation ontwo classic dynamical systems are used to show DMD methodsof evaluating and approximating the Koopman operator andits effectiveness at linearizing these systems.

I. INTRODUCTION

In this work we explore the application of the Koopmanoperator to find finite-dimensional linear representation ofnonlinear dynamical systems. The Koopman operator isan infinite-dimensional operator that evolves the functionsthat evolve the state of a dynamical system linearly. Inthis work, the Koopman operator is computed using data-driven methods such as the dynamic mode decomposition(DMD) algorithm. DMD uses linear measurements of thestates of a dynamical system to find the dominant modesof the underlying system from data to reconstruct a linearrepresentation of the evolution of the system.

The theory behind the Koopman operator was first pos-tulated as an infinite-dimensional linear operator for un-controlled systems in the seminal work by Koopman andNeumann [1], which recently has sparked an reemergence ofinterest in this method of modeling systems in the past twodecades. More recently, methods of finding the Koopmanoperator have been looked into by using discrete data-driven methods for fluid dynamics [2] through dimensionalreduction algorithms developed by Schmid et al. [3].

*This work was supported by the U.S. National Science Foundation underawards CISE/IIS-2024928 and OIA-2032522.

G. Snyder and Z. Song are with the Department of Mechanical Engineer-ing, University of Hawai‘i at Manoa, Honolulu, HI, 96822 USA. E-mails:snyderg, zsong@hawaii.edu.

Brunton et al. explored the relationship between the Koop-man operator and explored multiple observable functions toform a Koopman subspace to develop a Koopman opera-tor [4]. In their work, they showed that the state matrixfound by approximating the Koopman operater for specificsystems can be used to generate control laws with linearquadratic regression (LQR). Further more, Proctor et al. [5]expanded on their previous work on DMD to create a DMDmethod with control to extract low-order control modelsfrom higher-dimensional systems building the base DMDalgorithm. This DMD variant with control, called DMDc,is demonstrated to show positive results in the analysis ofinfections disease data. Proctor et al. [6] introduced a methodof finding the Koopman operator of a system that takes intoaccount system’s inputs and control (KIC) based on priorwork on DMDc. Work has also been done to improve theaccuracy of the Koopman operater by ’lifting’ the states ofthe system to a set of observables such that a solution canbe found where the data required from DMD is limited,extending the DMD algorithm (EDMD) [7]. Korda et al. [8]presented the use of lifting nonlinear dynamics on augmentedstates with EDMD and explored finding control laws forsystems using model predictive control (MPC). There havealso been work done to optimize the methods in which thelifting of dynamics is done through the optimization of thedictionary of functions used to lift the state variables throughEDMD with dictionary learning [9].

Applications of the Koopman operator have been exploredin the field of robotics, where the operator is used to developclosed-loop controllers for pendulum systems [10]. Kaiser etal. [11] explores the application of the Koopman operatortheory in generating energy-based control using a DMDvariant with control, extended dynamic mode decompositionwith control, and compared it against another method of find-ing the Koopman operator called the Koopman reduced ordernonlinear identification and control (KRONIC). Conversly,the Koopman operator is not a panacea to solving dynamicalsystems; a recent work by Gonzalez et al. [12] aimedat demonstrating some of the weakness of the Koopmanoperator and the DMD algorithms and provides alternateapproaches to reconstruction.

This paper will illustrate the mathematics behind theKoopman operator theory and show how augmenting thestates of a dynamical system can create accurate linearmodels through this process. It then shows how the DMDalgorithm can be used to find a linear approximation ofdynamical systems before exploring variants of DMD tocreate more accurate approximations of the Koopman oper-

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ator when representing system dynamics and other methodsof using DMD to generate control of a system representedvia the Koopman operator. Numerical simulations of twoclassical dynamical systems, the inverted pendulum and cart-pole systems, are then studied using the DMD algorithms.

II. PRELIMINARIES

This work pertains to the implementation of Koopmanoperator theory on dynamical systems to propagate non-linear dynamical equations with a data-driven approximationmethod. Conventionally, a non-linear dynamical system con-sists of a set of states and a function or rule that governs howthe states propagate either forward in time or with respect toeach other [13], [14]. This can be described in a continuousand a discrete method.

For a generic continuous system

d

dtx(t) = F (x(t), t;µ), (1)

x(t) ∈ Rn is the vector holding the states of the dynamicalsystem at time t, n is the number of states that define thesystem, µ is a set of parameters for the system dynamics,and F (·) is the rule describing the evolution of the state in acontinuous sense. These continuous-time dynamics can alsobe modeled in a respective discrete-time representation wherethe system can be evaluated at every finite time interval, ∆t,which can otherwise be viewed as xk = x(k∆t) with thesubscript k. The evolution of a dynamic system in a discrete-time flow map can be formally portrayed as

xk+1 = f(xk), (2)

by collecting the states at time tk, k = 1, 2, ...,m, for mtime steps, where xk is a n-dimensional column vector ofsystem states and xk+1 is the states of the system in the timestep following xk [5], [6].

These rules governing how the states of the systemadvance tend to be nonlinear equations to best simulatemost of the systems in practice. An unfortunate quality ofnonlinear systems is that their dynamics are difficult to solveanalytically. As a result, modern control practices tend toturn to approximations in order to produce a high-fidelitycontroller for the system. However, if a dynamical systemcan be expressed with by a linear rule, we can then achievemore accurate predictions of how a system advances in time.We intend to show this using the Koopman operator theory.

A. Koopman Operator

The Koopman operator is a linear but infinite dimensionaloperator that can be defined for an autonomous, discrete time,dynamical system. Unlike the function f shown earlier in(2), the Koopman operator advances the measurements ofthe changes in dynamics over time [6], [8], [13]. Considerthe evolution of a dynamical system xk+1 = f(xk) wherethe rule f maps the state space onto itself, i.e., f : Rn →Rn; using Koopman operator theory we define a differentrule, g : Rn → Mny , where ny is the dimension of anearly infinite column vector defining the dimension of the

observable of x at a given time step. This implies that theKoopman operator is defined for all observables, meaningthat the Koopman operator K is also infinite-dimensional.We will discuss how we can make this a more reasonableattribute later on in this discussion. Here, g is a real-valued,scalar, measurement function which is an element of aninfinite-dimensional Hilbert space called an observable. TheKoopman operator acts on this observable such that

Ktg =g F (x(t)), (3)K∆tg(xk) =g(f(xk)). (4)

In (3), the Koopman operator evolves the observable withrespect to time in the continuous fashion while in (4) theKoopman operator evolves the discrete-time dynamics withrespect to ∆t. More details on the connections between thesetwo representations can be found in [6], [8], [11], [13], [15].Equations (3) and (4) allow us to define an analogue forcontinuous and discrete-time dynamical systems respectivelyas

d

dtg =Kg, (5)

g(xk+1) =K∆tg(xk). (6)

This work explores the application of the Koopman op-erator on discret-time dynamical systems. Due to the linearnature of the Koopman operator, we can perform an eigendecomposition of K such that

Kϕj(xk) = λjϕj(xk), (7)

where λ and ϕ are the Koopman eigenvalue and eigenvectordescribing the evolution of the Koopman operator. Consid-ering this, the observable, g(x), can be expanded as

g(x) =

g1(x)g2(x)

...gi(x)

=

∞∑j=1

ϕj(x)vj, (8)

where v is a coefficient called the Koopman mode associatedwith its corresponding Koopman eigenvector. This allowsus to consider Koopman modes as a projection of theobservable:

vj =

〈ϕj , g1〉〈ϕj , g2〉

...〈ϕj , gi〉

. (9)

By combining (7) and (8), one can show the relationshipbetween the observable, g, and the Koopman operator,K. However, for all practical proposes, using an infinite-dimensional vector and operator is not always feasible so anapproximation of the Koopman operator, K, will be typicallyused and found from collected data on the system. Thisapproximation allows us to combine the terms in (7) and(8) to show the relationship between the Koopman modes ing and the Koopman eigenvalues in K

Kg(xk) ≈ Kg(xk) = g(f(xk)) =

∞∑j=1

λjϕj(x)vj. (10)

3

This shows that the Koopman operator is an iterative set oftriples, λj , ϕj and vj, all of which make up the Koopmanmode decomposition [6], [8], [11], [13], [16], [17].

B. Dynamic Mode Decomposition (DMD)

DMD is a data-driven method of exploring the behaviorof complex systems. Using measurements from numericalsimulations or in experiments, we can attempt to find themost dominant dynamic characteristics of the system. DMDdoes this by finding the dynamic modes or eigen modes andeigenvalues of the proposed system.DMD acts on the assumption that the state of a system isconnected to the next by

xk+1 = Axk, (11)

where x(t) ∈ Rn and A ∈ Rn×n and is the matrix describingthe evolution of the state in a continuous-time manner.Simulated or experimental measurements for xk are thencollected at regular time intervals of ∆t to become snapshotsto be used in a discrete time system. These snapshots arecollected and stored in sequence like the following:

X =

x1 x2 . . . xm−1

, (12)

X′ =

x2 x3 . . . xm

, (13)

where X′ is the time-shifted snapshot of matrix X such that

X′ ≈ AX. (14)

There is no set number of snapshots required for DMD buta sufficiently large number is needed for the applicationand is closely related to the approximation of the Koopmanoperator. For DMD we are attempting to find the A in(14) using the snapshots of the system’s states. A can beapproximated by

A =X′X+, (15)

where + is the Moore-Penrose pseudoinverse. Consideringthat A has a large n that normal calculation would be alarge computational load we can perform Singular ValueDecomposition (SVD) on the snapshots to find the dominantcharacteristics of the pseudoinverse of X

X ≈ UΣV∗, (16)

where U ∈ Rn×r, Σ ∈ Rr×r and V ∈ Rn×r where ∗denotes the conjugate transpose. r is the reduced rank of theSVD approximation of X. From SVD we can rearrange thesingular values and the eigenvectors of X with the forwardsnapshot to A that satisfies (14):

A ≈ A =X′VΣ−1U∗. (17)

However in practice since the dimensionality of A is so largewe have to approximate it as

A = U∗X′VΣ−1, (18)

where A is a r rank linear model of the dynamical systemsuch that

xk+1 =Axk. (19)

Here x :∈ Rr×1 is also a reduced order of the state x whichcan be reconstructed by

x : x = Ux. (20)

The next step is to find the spectral decomposition of A:

AW = WΛ, (21)

where Λ and W are the corresponding eigenvalues (DMDmodes) and eigenvectors of the full rank system dynamicsmatrix allowing us to recover the full state system dynamicsin a computationally efficient manner since A ∈ Rr×r wherer << n. We can now reconstruct the eigendecomposition in(16) for A from W and Λ with the corresponding eigenvectors given by the columns of Φ such that

Φ =X′VΣ−1W. (22)

From Φ we can no reconstruct our approximation of the timedynamics of x(t) by projecting our approximations into afuture solution

x(t) ≈r∑

k=1

φk exp (ωkt)bk = Φ exp (Ωt)b, (23)

where bk is the initial amplitude of each DMD mode, φ isthe columns that make up Φ and Ω is the diagonal matrix ofthe eigenvalues ω in ωk = ln (λk)/∆t [5], [13]–[15], [18].

C. Extended Dynamic Mode Decomposition

Extended DMD is almost the same algorithm as thestandard DMD one, however the method of which we deployEDMD is by using the observables of the system to createa dictionary to pass through the normal DMD algorithm.[7], [13], [14], [19] By performing regression on this newaugmented vector containing linear and non-linear measure-ments we can make a different approximation of the originalsystem dynamics. This augmented vector is constructed inthe following manner

y = ΘT (x) =

θ1(x)θ2(x)

...θp(x)

, (24)

where p is the rank of the augmented state such that p >>n. Here Θ is the collection of measurements of the systempossibly containing the original state of the system, x, aswell as nonlinear measurements. Once y is found, two datamatrices are created in the same manner seen above in theDMD algorithm (12) and (13). From here a best-fit linearoperator AY is found that maps (12) onto (13)

AY = argminAY

‖Y′ −AY Y‖ = Y′Y+. (25)

4

This regression can than be written in terms of the originaldata matrices ΘT (x) as

AY = argminAY

∥∥ΘT (x′)−AY ΘT (x)∥∥ = ΘT (x′)(ΘT (x))+.

(26)AY is then the basis upon which we can derive the Koopmanoperator. However, Θ may not necessarily span the samesubspace as the Koopman operator and may consist of dif-ferent eigenvalues and eigenvectors of the Koopman operatorwhen is why verification and re-validation techniques needto be used to show that the EDMD model is properly fit tothe actual system.

D. Approximating Koopman Eigenfunctions from Data

Beginning with the observable matrices in (24), the Koop-man eigenfunction can be approximated as

ϕ(x) ≈p∑

k=1

θk(x)ξk = Θ(x)ξ, (27)

where by using (24), we can findλϕ(x1)λϕ(x2)

...λϕ(xm)

=

ϕ(x2)ϕ(x3)

...ϕ(xm+1)

. (28)

We start seeing a connection between the Koopman oper-ator and the process we followed to approximate a linearevolution seen in DMD. We can expand (27) as

[λΘ(X)−Θ(X′)]ξ = 0. (29)

Afterwards, we reduce (29) using a best least-squares fit toget

λξ = Θ(X)+Θ(X′)ξ. (30)

Now we can compare (26) and (30); (26) is the transposeof the latter so that the left eigenvectors are now the righteigenvectors. Comparing to (15) we can use the eigenvectorsξ of Θ+Θ′ to find the coefficients of the eigenfunction ϕ(x)that represents the basis of Θ(x). We now can confirm thatthe predicted eigenfunctions behave linearly by comparingthem to the predicted dynamics in (28). We do this to deter-mine if the regression done in (30) yields proper eigenvaluesand eigenvectors that span the Koopman invariant subspacefor the system [7].

E. Methodology

The dynamical system model and the methods we used todiscover the Koopman operator are presented in the section.As discussed in the previous section, the method that willbe used to determine a finite-dimensional approximation ofthe Koopman operator will be through DMD and EDMD.This approach has been used in [6], [11], [20]. As statedin the previous section, DMD computes the transformationmatrix A that shows the evolution of two measurement

(a)Fig. 1. Classical inverted pendulum system with mass m = 1 kg and armlength of L = 2 m. In this model the mass is free to swing about the pivot360 degrees and will be treated as an undamped system.

snapshot pairs by finding the corresponding eigenvectors andeigenvalues that satisfy

Avj =λjvj . (31)

Now assuming that A contains a complete set of eigenvec-tors, each measurement can be found by the eigenvectors ofA like so:

g(xk) =

n∑j=1

cjkvj . (32)

In the case that these snapshot or augmented pairs in thecase of EDMD and DMD, the DMD modes and eigenvaluesof (32) correspond to the Koopman modes of (10). Sincethe principle concept of the Koopman operator, in the realmof data-driven identification, is to advance the states of thesystem by advancing the dynamics linearly.

III. RESULTS AND ANALYSIS

In this work we will numerically evaluate controlled anduncontrolled systems to collect sufficiently large sets of data,convert it into snapshot pairs to run through the DMD andEDMD algorithms and use the decomposition to reconstructlinear models of the data to compare against the simulatedsystems.

A. Sample Systems

The classical systems that were used in this work to ap-proximate the Koopman operator are the inverted pendulumand cart pole system [13]. These systems are used due totheir periodic and transient behaviors and dynamics are wellknown.

1) Inverted Pendulum: In the inverted pendulum systemseen in figure 1 we will start with an initial state of [π/4, 0]and in the controlled examples we attempted to right thependulum mass to a terminal state of [0, 0]. In this simulationwe only controlled the θ term with a performance index of[0, 10] and a control cost of 1 for our LQR gain.

2) Cart-Pole: In the cart-pole system seen in Fig. 2 westarted with an initial state of [−1, 0, π, 0] and attemptedto right the cart-pole system after a brief translation to[1, 0, π, 0]. In this simulation we created a control law usingLQR with a performance index of [5, 10, 0, 0] and a controlcost of 1.

5

(b)Fig. 2. Classical cart pole system with pendulum mass m = 1kg, cartmass M = 5 kg, and arm length L = 2 m. Like the pendulum system inFig. 1, the end mass is free to swing about the pivot 360 degrees and boththe cart and arm pivot are friction-less and undamped.

0 0.5 1 1.5 2 2.5 3

t (ms) 104

-0.8

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d)

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ReferenceDMD

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t (ms) 104

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2

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d/s)

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0

1

2

3

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6

7

Err

or (

rad)

10-3

(b)

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0

0.005

0.01

0.015

0.02

0.025

' Err

or (

rad/

s)

(d)

Fig. 3. DMD performed on a pendulum system with uncontrolled data.

B. Implementation

We can see in Figs. 3 and 4 that the reconstructionof the original state space succeeded with minimal errors.The periodic nature of the pendulum system can still beseen in the error analysis and it recedes in time. This ismost likely due to the immediate transient behavior of thereference data, though the over all behavior appears to beaccurately represented. We can see consistent low error inthe reconstructed controlled and uncontrolled pendulum data.In Figs. 5 and 6 there is the repeated behavior of the DMD

reconstruction seen in Figs. 3 and 4. In both sets of theEDMD reconstruction we can see similar or reduced errorin reconstruction.

In Figs. 7 and 8 we can see significant losses in recon-structing the behavior of the uncontrolled cart-pole system.Figure 8 shows that some of the characteristics of thereference data have been captured in the reconstruction asthe reference data in this process is smoother and has lessperiodic and transient characteristics.

C. Discussions

We can see in the reconstruction results that the invertedpendulum system is a good candidate for the system re-construction using the approximated Koopman operator. Wecan see a marginal improvement in the reconstruction ofthe state data of the controlled data when we improve the

0 0.5 1 1.5 2 2.5 3

t (ms) 104

-0.6

-0.4

-0.2

0

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(ra

d)

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ReferenceDMD

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d/s)

(c)

ReferenceDMD

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0

0.005

0.01

0.015

Err

or (

rad)

(b)

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t (ms) 104

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

' E

rror

(ra

d/s)

(d)

Fig. 4. DMD performed on a pendulum system with with LQR controlleddata.

0 0.5 1 1.5 2 2.5 3

t (ms) 104

-0.8

-0.6

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0

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d)

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ReferenceEDMD

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t (ms) 104

-2

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2

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d/s)

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ReferenceEDMD

0 0.5 1 1.5 2 2.5 3

t (ms) 104

0

1

2

3

4

5

6

7

Err

or (

rad)

10-3

(b)

0 0.5 1 1.5 2 2.5 3t (ms) 104

0

0.005

0.01

0.015

0.02

0.025

' Err

or (

rad/

s)

(d)

Fig. 5. EDMD performed on a pendulum system with uncontrolled datalifted by a second-order polynomial basis.

0 0.5 1 1.5 2 2.5 3

t (ms) 104

-0.6

-0.4

-0.2

0

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d)

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ReferenceEDMD

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t (ms) 104

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-1

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1

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d/s)

(c)

ReferenceEDMD

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t (ms) 104

0

1

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7

8

Err

or (

rad)

10-3

(b)

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t (ms) 104

0

0.005

0.01

0.015

0.02

' E

rror

(ra

d/s)

(d)

Fig. 6. EDMD performed on a pendulum system with with LQR controlleddata lifted by a second-order polynomial basis.

6

0 0.5 1 1.5 2 2.5 3

t (ms) 104

-2

-1.5

-1

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0

x (m

)

(a)

ReferenceDMD

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t (ms) 104

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-1

0

1

2

x' (

m/s

)

(c)

ReferenceDMD

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t (ms) 104

0

2

4

6

(ra

d)

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ReferenceDMD

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t (ms) 104

-5

0

5

' (ra

d/s)

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ReferenceDMD

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t (ms) 104

0

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x E

rror

(m

)

(b)

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2

x' E

rror

(m

/s)

(d)

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t (ms) 104

0

2

4

6

Err

or (

rad)

(f)

0 0.5 1 1.5 2 2.5 3

t (ms) 104

0

2

4

6' E

rror

(ra

d/s)

(h)

Fig. 7. DMD performed on a cart-pole system with uncontrolled data.

0 0.5 1 1.5 2 2.5 3

t (ms) 104

-1

0

1

2

x (m

)

(a)

ReferenceDMD

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x' (

m/s

)

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ReferenceDMD

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3.1

3.15

3.2

3.25

3.3

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d)

(e)

ReferenceDMD

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t (ms) 104

-0.15

-0.1

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' (ra

d/s)

(g)

ReferenceDMD

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t (ms) 104

0

0.2

0.4

0.6

x E

rror

(m

)

(b)

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t (ms) 104

0

0.05

0.1

x' E

rror

(m

/s)

(d)

0 0.5 1 1.5 2 2.5 3

t (ms) 104

0

0.2

0.4

0.6

Err

or (

rad)

(f)

0 0.5 1 1.5 2 2.5 3

t (ms) 104

0

0.05

0.1

0.15

' Err

or (

rad/

s)

(h)

Fig. 8. DMD performed on a cart-pole system with with LQR controlleddata.

DMD algorithm to EDMD thought the improvement yet nochange in the uncontrolled data. This shows that the periodicbehavior is not improved by a polynomial lifting function.Additionally when moving to a higher order polynomial (orFourier; see Figs. 11 and 12) we can additional improvementsin the reconstruction of the original data when the fulllifted states are used. Oddly enough, as seen in Fig. 13,we see no further improvement in the overall fidelity of thereconstruction of the state space using a second-order Fourierfitting.

Moving on to the second system in this work, we cansee significant errors in the reconstruction process as we areadding more transient components into the system dynamicswith the extra degrees of freedom of the cart-pole system. Asshown in Fig. 7, DMD (and by extension, EDMD) did notcapture the uncontrolled behavior of the system in questionhowever when the strong periodic behavior is removed inthe controlled data set we find an improved reconstructionof the system however weaker that that shown in the in-verted pendulum system. Additionally we note that the initialresponse generated by LQR appears to be missed by the

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1

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Err

or (

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10-3

(b)

0 0.5 1 1.5 2 2.5 3t (ms) 104

0

0.002

0.004

0.006

0.008

0.01

0.012

' Err

or (

rad/

s)

(d)

Fig. 9. EDMD performed on a pendulum system with uncontrolled datalifted by a third-order polynomial basis with no truncation.

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t (ms) 104

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d)(a)

ReferenceEDMD

0 0.5 1 1.5 2 2.5 3

t (ms) 104

-1.5

-1

-0.5

0

0.5

1

' (ra

d/s)

(c)

ReferenceEDMD

0 0.5 1 1.5 2 2.5 3

t (ms) 104

0

0.2

0.4

0.6

0.8

1

1.2

1.4

Err

or (

rad)

10-4

(b)

0 0.5 1 1.5 2 2.5 3

t (ms) 104

0

0.5

1

1.5

2

2.5

' E

rror

(ra

d/s)

10-4

(d)

Fig. 10. EDMD performed on a pendulum system with with LQRcontrolled data lifted by a third-order polynomial basis with no truncation.

reconstruction possibly showing that more gradual controlmay yield better results in the reconstruction. Additionallywe can see no further improvement on the reconstruction ofthe system when we attempted to use the EDMD algorithmin Figs. 15 and 16 using either a polynomial or Fourier basisto lift the original state space.

IV. CONCLUSIONS

This work presented an introduction on the usage ofthe Koopman operator theory approach to model simple,nonlinear dynamical systems with a linear operator. In thisprocess, we simulated the states of classical non-linearsystems and feed that data as a set of snapshots throughthe DMD algorithm, and its variants, to reconstruct the statedata using the Koopman operator. Through the usage of thesealgorithms on an inverted pendulum and cart-pole systemwe can approximate the Koopman operator to recreate highfidelity linear approximations of these systems and provideincreased accuracy by augmenting the states of the originaldata. These methods showed weakness in reconstructingsystems that act with coupled states that behave in differentmatters as seen in the cart-pole system. In future work, weintend to explore alternate lifting functions to represent thestate spaces of our systems and explore alternative methods

7

0 0.5 1 1.5 2 2.5 3

t (ms) 104

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

(ra

d)

(a)

ReferenceEDMD

0 0.5 1 1.5 2 2.5 3

t (ms) 104

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

' (ra

d/s)

(c)

ReferenceEDMD

0 0.5 1 1.5 2 2.5 3

t (ms) 104

0

0.5

1

1.5

2

2.5

3

3.5

4

Err

or (

rad)

10-3

(b)

0 0.5 1 1.5 2 2.5 3t (ms) 104

0

0.002

0.004

0.006

0.008

0.01

' Err

or (

rad/

s)

(d)

Fig. 11. EDMD performed on a pendulum system with uncontrolled datausing a first-order Fourier basis to lift the state data.

0 0.5 1 1.5 2 2.5 3

t (ms) 104

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

(ra

d)

(a)

ReferenceEDMD

0 0.5 1 1.5 2 2.5 3

t (ms) 104

-1.5

-1

-0.5

0

0.5

1

' (ra

d/s)

(c)

ReferenceEDMD

0 0.5 1 1.5 2 2.5 3

t (ms) 104

0

0.002

0.004

0.006

0.008

0.01

Err

or (

rad)

(b)

0 0.5 1 1.5 2 2.5 3

t (ms) 104

0

0.002

0.004

0.006

0.008

0.01

0.012

0.014

' E

rror

(ra

d/s)

(d)

Fig. 12. EDMD performed on a pendulum system with LQR controlleddata using a first-order Fourier basis to lift the state data.

0 0.5 1 1.5 2 2.5 3

t (ms) 104

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

(ra

d)

(a)

ReferenceEDMD

0 0.5 1 1.5 2 2.5 3

t (ms) 104

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

' (ra

d/s)

(c)

ReferenceEDMD

0 0.5 1 1.5 2 2.5 3

t (ms) 104

0

0.5

1

1.5

2

2.5

3

3.5

4

Err

or (

rad)

10-3

(b)

0 0.5 1 1.5 2 2.5 3t (ms) 104

0

0.002

0.004

0.006

0.008

0.01

' Err

or (

rad/

s)

(d)

Fig. 13. EDMD performed on a pendulum system with uncontrolled datausing a second-order Fourier basis to lift the state data.

0 0.5 1 1.5 2 2.5 3

t (ms) 104

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

(ra

d)

(a)

ReferenceEDMD

0 0.5 1 1.5 2 2.5 3

t (ms) 104

-1.5

-1

-0.5

0

0.5

1

' (ra

d/s)

(c)

ReferenceEDMD

0 0.5 1 1.5 2 2.5 3

t (ms) 104

0

1

2

3

4

5

Err

or (

rad)

10-3

(b)

0 0.5 1 1.5 2 2.5 3

t (ms) 104

0

0.002

0.004

0.006

0.008

0.01

' E

rror

(ra

d/s)

(d)

Fig. 14. EDMD performed on a pendulum system with LQR controlleddata using a second-order Fourier basis to lift the state data.

0 0.5 1 1.5 2 2.5 3

t (ms) 104

-1

0

1

2

x (m

)(a)

ReferenceEDMD

0 0.5 1 1.5 2 2.5 3

t (ms) 104

-0.5

0

0.5

x' (

m/s

)

(c)

ReferenceEDMD

0 0.5 1 1.5 2 2.5 3

t (ms) 104

3.1

3.15

3.2

3.25

3.3

(ra

d)

(e)

ReferenceEDMD

0 0.5 1 1.5 2 2.5 3

t (ms) 104

-0.15

-0.1

-0.05

0

0.05

' (ra

d/s)

(g)

ReferenceEDMD

0 0.5 1 1.5 2 2.5 3

t (ms) 104

0

0.2

0.4

0.6

x E

rror

(m

)

(b)

0 0.5 1 1.5 2 2.5 3

t (ms) 104

0

0.05

0.1

x' E

rror

(m

/s)

(d)

0 0.5 1 1.5 2 2.5 3

t (ms) 104

0

0.2

0.4

0.6

Err

or (

rad)

(f)

0 0.5 1 1.5 2 2.5 3

t (ms) 104

0

0.05

0.1

0.15

' Err

or (

rad/

s)

(h)

Fig. 15. EDMD performed on a cart-pole system with LQR controlleddata using a second-order polynomial basis to lift the state data.

0 0.5 1 1.5 2 2.5 3

t (ms) 104

-1

0

1

2

x (m

)

(a)

ReferenceEDMD

0 0.5 1 1.5 2 2.5 3

t (ms) 104

-0.5

0

0.5

x' (

m/s

)

(c)

ReferenceEDMD

0 0.5 1 1.5 2 2.5 3

t (ms) 104

3.1

3.15

3.2

3.25

3.3

(ra

d)

(e)

ReferenceEDMD

0 0.5 1 1.5 2 2.5 3

t (ms) 104

-0.15

-0.1

-0.05

0

0.05

' (ra

d/s)

(g)

ReferenceEDMD

0 0.5 1 1.5 2 2.5 3

t (ms) 104

0

0.2

0.4

0.6

x E

rror

(m

)

(b)

0 0.5 1 1.5 2 2.5 3

t (ms) 104

0

0.05

0.1

x' E

rror

(m

/s)

(d)

0 0.5 1 1.5 2 2.5 3

t (ms) 104

0

0.2

0.4

0.6

Err

or (

rad)

(f)

0 0.5 1 1.5 2 2.5 3

t (ms) 104

0

0.05

0.1

0.15

' Err

or (

rad/

s)

(h)

Fig. 16. EDMD performed on a cartpole system with LQR controlled datausing a first-order Fourier basis to lift the state data.

8

of finding the Koopman operator of systems with multiplecoupled states to improve performance to develop linearcontrol laws.

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