Koopman Operator Dynamical Models: Learning, Analysis and Control
KKoopman Operator Dynamical Models: Learning,Analysis and Control (cid:63)
Petar Bevanda a, ∗ , Stefan Sosnowski a , Sandra Hirche a a Chair of Information-oriented Control, Department of Electrical and ComputerEngineering, Technical University of Munich, D-80333 Munich, Germany
Abstract
The Koopman operator allows for handling nonlinear systems through a (glob-ally) linear representation. In general, the operator is infinite-dimensional - ne-cessitating finite approximations - for which there is no overarching framework.Although there are principled ways of learning such finite approximations, theyare in many instances overlooked in favor of, often ill-posed and unstructuredmethods. Also, Koopman operator theory has long-standing connections toknown system-theoretic and dynamical system notions that are not universallyrecognized. Given the former and latter realities, this work aims to bridge thegap between various concepts regarding both theory and tractable realizations.Firstly, we review data-driven representations (both unstructured and struc-tured) for Koopman operator dynamical models, categorizing various existingmethodologies and highlighting their differences. Furthermore, we provide con-cise insight into the paradigm’s relation to system-theoretic notions and analyzethe prospect of using the paradigm for modeling control systems. Additionally,we outline the current challenges and comment on future perspectives.
Keywords:
Koopman operator, dynamical models, representation learning,system analysis, data-based control
1. INTRODUCTION
Efficient control and analysis are inherently challenging when dealing withcomplex dynamical systems. For such complex systems, models coming fromfirst-principles often times are not available or do not fully resemble the truesystem due to unmodeled phenomena. Thus, there is great value in develop-ing effective techniques with the ability to discover, analyze and control suchsystems. To do so, the approach to modeling inevitably dictates the challenges (cid:63)
This work was supported by European Commission grant H2020-ICT-871295 (”SeaClear”- SEarch, identificAtion and Collection of marine Litter with Autonomous Robots). ∗ Corresponding author
Email addresses: [email protected] (Petar Bevanda), [email protected] (StefanSosnowski), [email protected] (Sandra Hirche)
Preprint submitted to Annual Reviews in Control February 5, 2021 a r X i v : . [ ee ss . S Y ] F e b n dealing with a system of interest. The complexity of systems we encounterprompts a shift from classical parametric techniques in favor of more flexiblemachine learning techniques (e.g. neural networks or Gaussian processes) forprediction [1, 2], model-based control [3, 4] and analysis [5, 6, 7]. Traditionally,representations of systems are in the immediate state-space concerned with “dy-namics of states”. Although such representations enjoy incredible success, theyare limited when it comes to efficient representations for prediction, analysisand optimization-based control.Alternative to the traditional modeling paradigm studying the “dynamics ofstates”, one can decide to perceive systems through an operator-theoretic viewconcerned with “dynamics of observables” (functions over the state-space) - the Koopman operator paradigm . The paradigm is named after B.O. Koopman -the author of the seminal work [8] on transformations of Hamiltonian systemsin Hilbert space. It gives rise to representations of dynamical systems that linearly evolve observables (output measurements) of the system in a specialset of coordinates via the Koopman operator. System representations via theKoopman paradigm generalize the notion of mode analysis from linear to nonlin-ear systems allowing for amendable relevance determination of the constituentsof the full dynamics (e.g. thermal analysis of buildings [9]). The spectrumof the Koopman operator allows one to decompose the nonlinear system intodifferent dynamic regimes (fast-slow dynamics) while linearly evolving coordi-nates contain intrinsic information relevant for analysis i.e. regions of attraction[10]. Due to the linear dynamics of the new coordinates, prediction hardshipsthrough numerical integration and non-convexity in optimization of “dynam-ics of states” have the potential to be alleviated (e.g. model predictive control[11, 12]). Moreover, one can argue that the Koopman operator paradigm de-livers a global instead of a point-wise system description as one iteration of theKoopman operator acting on an observable is equivalent to an iteration alongall of the trajectories of the system and it is not to be confused with a locallinearization around a working point.Given the Koopman operator is generally infinite-dimensional, it necessi-tates finite approximations - a task for which there is no overarching, generalframework. The non-triviality of finite representations lead to many data-drivenapproaches with various trains of thought, facilitating different properties of theKoopman operator paradigm.Regarding the plethora of data-driven frameworks looking to deliver Koop-man operator dynamical models, this is the first work reviewing the aforemen-tioned in a systematic manner while giving theoretical insight.A related review of the paradigm by Budiˇsi´c et al. [13] gives a theoreti-cal baseline with application examples but with the data-driven methodologylimited to dynamic mode decomposition (DMD). We address the abundanceof novel data-driven methods that arose from and beside DMD for discoveringKoopman operator dynamical models in addition to siding with a more focusedsystem and control-theoretic perspective. Also, compared to work of Kaiser etal. [14], we take a rigorous approach focused on the Koopman operator paradigminstead of an high-level overview of data-driven transfer operators. By taking2uch an approach, the aim is to deliver a holistic and methodical backbone ofKoopman operator-based dynamical models - from surveying the data-drivenrepresentations, to system-theoretic connections and control.The article is structured as follows. After the preliminaries on Koopman op-erator theory in Section 2, the data-driven methods for Koopman-related modelrepresentation are presented in Section 3. An overview of system-theoretic anal-ysis via Koopman operator theory follows in Section 4. Furthermore, Section5 analyzes the inclusion of control into the representations and surveys theirapplications found in the literature. Before the final outline concluding Section6, we comment on future perspectives of Koopman-based methods.
Notation
Lower/upper case bold symbols x / X denote vectors/matrices. Symbols N / R / C denote sets of natural/real/complex numbers while N denotes all nat-ural numbers with zero and R + , / R + all positive reals with/without zero. Thecontinuous/discrete-time dependence is denoted as y t ( · ) and y k , respectivelywith t/k ∈ R + , / N while ˙( · ) := d ( · ) /dt , for brevity. Function spaces witha specific integrability/smoothness order are denoted as L / C with the order(class) specified in their exponent. The braces (cid:104)· , ·(cid:105) denote the inner productwhile (cid:107) · (cid:107) the Euclidean norm. A flow induced by a vector field ˙ x = f ( x ) isdenoted as F t ( x ) with its associated family of composition (Koopman) opera-tors {K tf } t ∈ R + , . A map i.e. x (cid:55)→ F ( x ) has its associated family of compositionoperator iterates denoted as {K nF } n ∈ N .
2. Koopman operator theory
The traditional way of studying a system leads to studying the orbits ofpoints x on a domain M under iteration - point dynamics study (“dynamics ofstates”) [13]: ˙ x = f ( x ) ⇒ F t : M (cid:55)→ M , (1)with state x ∈ M ⊆ R n and vector field f : M (cid:55)→ R n inducing the flow F t . On the contrary, the Koopman operator entails how maps evolve underiteration instead. Informally, it encodes information about an iterated mapwhich can be very well used to study the behavior of dynamical systems. TheKoopman operator K t : F (cid:55)→ F evolves functions ψ ∈ F over the state-space M (“dynamics of observables ”). An observable can be any kind of a measurementof the system, e.g. a sensory measurement. We make a distinction betweengeneral observables ψ i and the subset thereof that one wants to predict. In asystem-theoretic sense, we name those functions output functions y i = h i ( x ).Considering a deterministic system (1) and the “full-state” vector of outputfunctions - h ( x ) = x we write K t f x = F t ( x ) , (2)3here the flow F t (in this specific case) is replaced by the action of the component-wise Koopman operator K t f . Although delivering the same as the flow for thedeterministic case and h ( x ) = x , the Koopman operator, without a loss ofgenerality, evolves an observable in a linear fashion by exploiting Koopman op-erator’s eigenfunctions (elaborated in the upcoming subsection). Given that,the Koopman operator provides a linear evolution of the flow for the underlyingsystem, now in a much more “iteration-compatible”, linear form. Here we reintroduce some notions in a more formal way and further expandon the latter.
Assumption 2.1.
There exists a continuous-time dynamical system ˙ x = f ( x ) , (3) and a scalar observable function ψ : M (cid:55)→ C y = ψ ( x ) , (4) where there exists a metric space x ∈ M , t ∈ R with a smooth and Lipschitzflow F t : M (cid:55)→ M [15]: x ( t ) ≡ x , F t ( x ) := x + (cid:90) t + tt f ( x ( τ )) dτ, (5) on a smooth n -dimensional manifold M induced by the vector field (3). Remark 2.1.
The time evolution of a dynamical system on a manifold (in atopological sense locally resemblant of Euclidean space)
M ⊆ R n is specified bythe flow (5) F t .Often times we dispense with manifolds and choose the state space to lie in ⊆ R n , but for generality manifolds are retained in our definitions. Assumption 2.2.
There is a generator G K : D ( G K ) → F , D being the domainof the generator and F the space of observables. Definition 2.1 (Infinitesimal generator).
The operator G K , is the infinites-imal generator of the time- t indexed semigroup of Koopman operators {K t } t ∈ R + , G K ψ = lim t → + K t ψ − ψt = ddt ψ, (6) [16, Chapter 3]. By exponentiation of this linear time-evolution operator one obtains thetime- t semigroup of Koopman operators K t = exp( G K t ). Analogous to theKoopman operator we use the same paradigm to get the linear evolution of theobservables for the Koopman operator generator.4 ssumption 2.3 (Observable space F ). The observable space F is a subsetof all complex-valued functions on M that forms a vector space, which is closedunder point-wise products of functions. Definition 2.2 (Koopman Operator).
Consider a dynamical system (3) with x ∈ M and f in the tangent bundle T M = (cid:83) x ∈M T x M of M , such that thevector field evaluated at a point lies in a different tangent plane (fiber at x ) f ( x ) ∈ T x M . Then, the infinite-dimensional semigroup of Koopman opera-tors {K t } t ∈ R + , : F (cid:55)→ F acts on scalar observable functions ψ : M (cid:55)→ C bycomposition with the flow semigroup { F t } t ∈ R , + of the vector field f K t f ψ = ψ ◦ F t , (7) acting on the observable space from Assumption 2.3. When dealing with a map(discrete-time system) we have K F ψ = ψ ◦ F , (8) where ◦ represents function composition. Given that, the Koopman operator isalso referred to as the composition operator . In the remainder of this work weuse these terms interchangeably. Property 2.1 (Linearity of K -operator). Consider the Koopman operator K F and two observables g ∈ C ( M ) , g ∈ C ( M ) and a scalar α ∈ R . Using(8) it follows that K F ( αg + g ) = ( αg + g ) ◦ F = αg ◦ F + g ◦ F = α K F g + K F g , (9) showing the linearity of the operator.2.2. Koopman operator and generator eigendecomposition To evolve the dynamics of a nonlinear system in a linear fashion, the linearityfrom Property 2.1 is only a helper tool and the key is the existence of linearlyevolving coordinates - the Koopman operator eigenfunctions . Definition 2.3 (Koopman eigenfunction).
An observable φ ∈ F is an eigen-function if it satisfies [ G K φ ]( x ) = ˙ φ ( x ) = sφ ( x ) , (10) associated with the eigenvalue s ∈ C . Property 2.2 (Linear coordinates).
Since G K is the infinitesimal generatorof the semigroup of Koopman operators {K t } t ∈ R + , , the following is also satisfiedfor λ t = exp( st ) [ K t f φ ]( x ) = φ (cid:0) F t ( x ) (cid:1) = λ t φ ( x ) , (11) along the flow of the vector field. K F is composing an ob-servable with the map x (cid:55)→ F ( x ) not parameterized by time (as opposed to aflow of an ODE) and in that case delivers a simpler and arguably more intuitiveeigenvalue equation [ K F φ ]( x ) = φ ( F ( x )) = λφ ( x ) , (12)with λ = exp ( st s ), where t s is the sampling time discrete-time dynamical sys-tem.To exploit the property of eigenfunctions being linear coordinates, we needto be able to express the evolution of an arbitrary observable using them. Withthat we can claim that we captured the effect of the Koopman operator whichis crucial for finite approximations (Section 3). Property 2.3 (Discrete spectrum decomposition).
We can write an ob-servable via the eigenfunctions of the Koopman operator ψ ( x ) = ∞ (cid:88) j =0 v j ( ψ ) φ j ( x ) , (13) where the coefficients v j are Koopman operator modes of the associated withtheir respective eigenfunctions φ j . Obviously, the infinite-dimensionality of the Koopman operator is courtesy ofthe infinite-sum in (13) that should not be confused with infinite-series expan-sions i.e. Taylor/Carleman [17] that do not provide dynamically closed (invari-ant) coordinates. The Koopman operator action on an observable function asfollows K t ψ = ∞ (cid:88) j =1 v j ( ψ ) (cid:0) K t φ j (cid:1) = ∞ (cid:88) j =1 v j ( ψ ) λ tj φ j , (14)where λ tj = exp( s j t ) s.t. s j = γ j + iω j with eigen-decay/growth γ j and eigen-frequencies ω j . The above introduced discrete spectrum is clearly amendablefor linear prediction, however it is only a part of the Koopman operator’s spec-trum. Although we focus on the discrete part of the spectrum, in the followingwe extend to the full spectrum for completeness. Remark 2.2 (Full spectrum).
In general, the spectral decomposition can in-clude the continuous spectrum as well [18]. Given the system has an attractor A with a preserved measure µ ( A ) [19] we write the Koopman operator spectrumas K t ψ ( x ) = ∞ (cid:88) j =1 v j ( ψ ) λ tj φ j ( x ) (cid:124) (cid:123)(cid:122) (cid:125) discrete part + (cid:90) π κ t E ( dκ ) ψ ( x ) (cid:124) (cid:123)(cid:122) (cid:125) continuous part , (15)where κ t = exp( iθt ) is the integrated function over frequency θ and E ( κ ) theprojection-valued measure. While the discrete spectrum extracts transient -6e( λ tj ), the quasi-periodic (point spectrum) components of dynamics - Im( λ tj )and continuous part “chaotic” component of the dynamics correspond to the on-attractor dynamics. The latter can also be seen as the extension of the notionof the point spectrum, where eigenfunctions are replaced by eigenmeasures [20]. Property 2.4 (Koopman eigenfunction group).
Under Assumption 2.3, theset of eigenfunctions forms an Abelian semigroup under point-wise products offunctions [21]. Thus, products of eigenfunctions are, again, eigenfunctions - if φ , φ ∈ F are eigenfunctions of the composition operator K F with eigenvalues λ and λ , then φ φ is an eigenfunction of K F with eigenvalue λ λ . Definition 2.4 (Principle eigenpairs).
Consider { E m } m ∈ N to be the eigenpair-semigroup of Koopman operator K F with its minimal generator G E : { E m } = (cid:40)(cid:32) m (cid:89) i =1 λ n i i , m (cid:89) i =1 φ n i i (cid:33) | ( λ i , φ i ) ⊂ G E (cid:41) , (16) where m, n i ∈ N [22]. Then, the elements of G E are principle eigenvalues withcorresponding eigenfunctions of K F in { E m } m ∈ N . Less formally, principle eigenpairs form the minimal set of Koopman operatoreigenpairs that can then be used to construct all other eigenpairs (16).To give perspective and embed some of the dominant ideas when it comesto learning Koopman operator-based representations (Section 3), we presentthe following nonlinear discrete-time dynamical system that admits an exactfinite-dimensional linear representations of the Koopman operator action.
Example 2.1 (Motivating example).
Consider the map F : R (cid:55)→ R in-spired by Tu et al. [23]: x (cid:55)→ F ( x ) = (cid:20) ax bx + (cid:0) b − a (cid:1) x (cid:21) . (17)(a) Eigenfunction coordinates:
The associated principle eigenvalue-eigenfunctionpairs ( λ, φ ( x )) of the system are ( a, x ) and ( b, x + x ). Utilizing theclosedness of eigenfunctions under point-wise products from Property 2.4we can obtain more eigenfunction-eigenvalue pairs as powers of the princi-ple ones i.e. ( a , x ) - a product of ( a, x ) with itself. Utilizing the newlyobtained pair, the system from (17) can be lifted into new coordinates φ ( x ) = [ x , x + x , x ] (cid:62) which evolve linearly: φ ( x ) (cid:55)→ a b
00 0 a (cid:124) (cid:123)(cid:122) (cid:125) Λ φ ( x ) . (18)The result of (18) can be then be projected onto the respective vector of output functions e.g. system states h ( x ) = [ x , x ] (cid:62) using Property 2.37ia Koopman operator modes v = [1 , (cid:62) , v = [0 , (cid:62) and v = [0 , − (cid:62) [ K F ]( x ) = F ( x ) = V Λ φ ( x ) (19)with V = [ v v v ] and offer an equivalent representation for thesystem (17). With that, the system under iteration (multi-step evolution)does not necessitate the composition of nonlinear maps (illustrated inFigure 1) but only an initial nonlinear transformation φ after which theevolution is linear F ( k ) ( x ) = V Λ k φ ( x ) (20)where k compositions of nonlinear functions are replaced with lifting φ ( · )and subsequent linear evolution through the composition operator paradigm.(b) Observables as coordinates:
Interestingly, the dominant idea in the Koop-man operator paradigm is to find “good” coordinates what are not neces-sarily eigenfunctions but lie in the span of eigenfunctions. For the samesystem (17) the choice of coordinates ψ ( x ) = [ x , x , x ] (cid:62) still leads tothe following linear mapping ψ ( x ) (cid:55)→ a b ( b − a )0 0 a (cid:124) (cid:123)(cid:122) (cid:125) A ψ ( x ) . (21)where not all coordinates are eigenfunctions i.e. x . The result of (21)can be then be projected onto the respective output observables of interest(system states) output matrix C = [ I ]:[ K F ]( x ) = F ( x ) = CAψ ( x ) . (22)Here, the columns of the output matrix C (in system-theoretic sense)do not represent Koopman modes but by diagonalizing K into Λ (18),one uncovers Koopman operator eigenvalues, eigenfunctions and modes.As shown previously, one can opt to use F ( k ) ( x ) = CA k ψ ( x ) for lin-ear prediction as the formulations (19) and (22) indeed are equivalent.The notions from this example have their continuous-time analogue withthe difference that one is capturing the effect of the Koopman generatoroperator instead. Finally, we remark how the Koopman operator has its adjoint (in appropriatefunction spaces) - the Perron-Frobenius operator. As opposed to the Koopmanoperator that advances trajectories of a dynamical system forward in time, itsPerron-Frobenius adjoint puts the dynamical effects into the density, pushingforward (probability) densities over the state-space [24] making it well suitedfor studying chaotic (stochastic) dynamical systems.8tates x ∈ M F ( k ) ( x )NonlinearObservables ψ ( x ) ∈ F K k F ψ ( x )Linear K oop m a n M od e s K oop m a n E i ge n f u n c t i o n s F : M → M
Flow Operator K F : F → F
Koopman Operator K oop m a n M od e s K oop m a n E i ge n f u n c t i o n s Figure 1: Diagram of the Koopman operator system modeling concept ( F ( i ) = F ◦ . . . ◦ F i -times). The symbol K = diag {K , . . . , K} is there for element-wise application to vector-valuedobservable ψ . Property 2.5 (Koopman operator and its adjoint).
Given a measure µ on M and scalar-valued density ρ ∈ L ( M ) observable ψ ∈ L ∞ ( M ) , the fol-lowing holds (cid:10) P t ρ, ρ (cid:11) = (cid:10) ψ, K t ψ (cid:11) , (23) (cid:90) x ∈M (cid:0) K t ρ ( x ) (cid:1) µ ( d x ) = (cid:90) x ∈M ψ ( x ) P t µ ( d x ) , (24) with t ∈ R + , where P t denotes the semigroup of Perron-Frobenius (dual toKoopman) operators. Interestingly, Koopman and Perron-Frobenius operators are also referred to as backward and forward as they are solution operators of the backward and for-ward Kolmogorov (Fokker-Planck) equations, respectively [15, Section 11]. Fur-thermore, the Perron-Frobenius operator generator G P of the stochastic differen-tial equation ˙ ρ = G P ρ corresponds to the Liouville (Fokker-Planck) operator.
3. Data-driven Koopman operator-based dynamical models
The Koopman operator is, generally, an infinite dimensional operator. Inorder to facilitate it for representing dynamical systems, finite-dimensional ap-proximations are needed. However, finding “good” coordinates ( lifting func-tions) for finite-dimensional approximations is non-trivial. Before going on todata-driven approaches related to Koopman operator dynamical models, weremark on conceptually similar trajectory-learning frameworks.
Remark 3.1 (Learning a trajectory basis).
We point out some importantdifferences to, at least conceptually, related concepts such as dynamic motion9 onlinear Dynamics
Lifting/EmbeddingEigenfunctions (Sec. 3.3)[11, 30, 31]
Time-delays (Sec. 3.4)[32, 33, 34, 19][35, 36, 37, 38]
Observables (Sec. 3.2)LR:[28, 39, 40, 41][42, 43, 44, 45]NR:[46, 47, 48, 49, 50]
Figure 2: An ideological graph of the main branches of model representations employing theKoopman operator paradigm: root - original system, level 1 - transformation type, level 2 -representation coordinates employed. The bottom split part of the latter includes relevant ref-erences employing the methodology noted in the top part. LR and NR denote linear/nonlinearreconstruction of the state, respectively. primitives (DMPs) [25] and the Willems’ lemma [26]. DMPs look to describepoint attractive behaviors via a nonlinear representation. For the same task,Koopman operator dynamical models deliver representations with a linear evo-lution in lifted coordinates. Furthermore, approaches based on the Willems’lemma concern with input-output linearity - a more restrictive concept.Although there might be infinitely many of Koopman-invariant coordinatesspanning the state-space, a finite number can provide good approximations ofthe system for prediction and analysis. One can consider them as a trajectorybases compared to the one-step dynamics bases learned with function approx-imators for “dynamics of states”. Data-driven dynamically closed coordinatesmight be approximate (given a good regularization), they act as a global basison a compact space [27] holding almost everywhere due to the limited amountof data. In the following we review the data-driven approaches for learningKoopman operator-based system representations found in the literature.
A large class of algorithms for data-driven Koopman operator approxima-tions is in some way related to dynamic mode decomposition (DMD) [28]. DMDhad major success in fluid flow modeling, reduced order modeling (ROM) andKoopman operator approximations [23]. For a comprehensive dimension reduc-tion and dynamical systems view on DMD we point to [29]. We base our outlinein this section on it, as it represents a linear approximation of the Koopman op-erator with respect to the system state and many of the more advanced methodsfrom the literature stemmed from it.The key ideas of DMD methods are closely related to model-reduction tech-niques and showed promise in data-based modeling of fluid flows [51]. Briefly,DMD aims to extract dominant spatio-temporal modes from data, providingdata-driven local-linear analysis of nonlinear systems which made it one of the10rst Koopman operator approximation methods. Consider the data being for-matted into ‘snapshots’: H ( x ) = [ x , · · · , x T − ] , (25) + H ( x ) = [ x , · · · , x T ] , (26)that are vector Hankel matrices with depth one of a T -step time-sequence. Thenthe best-fit matrix A is obtained by minimizingmin A (cid:13)(cid:13) + H ( x ) − A (cid:2) H ( x ) (cid:3)(cid:13)(cid:13) F (27)with index F denoting the Frobenius norm. In closed form, the transitionmatrix is obtained as A = + H ( x )[ H ( x )] † - where † represents the Moore-Penrose pseudo-inverse. The utility of DMD for reduced order modeling forhigh-dimensional snapshots involves taking the singular value decomposition ofthe snapshot matrix H ( x ).Regarding linear representations of nonlinear systems we focus on methodsbased on predicting output functions of interest constructing a surrogate modelof a system based on the Koopman operator paradigm. We classify the methodsemployed to predict the system’s outputs of interest into the three categories(as shown in Figure 2) based on the lifting coordinates employed. Note thatthis is only a tentative classification for better exposition as the methodologiesare more closely related than Figure 2 would suggest. The firstly developedtrain of thought considers a span of observables as coordinates looking to ap-proximate the infinite-dimensional Koopman operator. Other methods seekingto find inherently linear coordinates consider direct discovery of eigenfunctioncoordinates , bypassing the search for a finite representation of the Koopmanoperator. Worth noting is the fact that the latter methodology deals with thediscrete spectrum (recall Remark 2.2). Considering system on-attractor (post-transient) behavior, time-delay coordinates of output functions (instead of dif-ferent spatial coordinates) give a useful linear representations of the system athand. Worth noting is the fact that the majority of these frameworks requiresequential data e.g. from a system’s trajectory, as opposed to learning withnon-sequential input-output samples encountered in function approximation. Throughout the years many extensions of DMD came along, notably ex-tended DMD (EDMD) [39]. It involves - often heuristically predetermined -nonlinear maps of the data (e.g. radial basis functions or monomials), hop-ing to capture the nonlinearity in the dynamics. For that, the data-snapshots(25-26) are lifted through a feature map ψ ( · ) resulting in H ψ ( x ) = [ ψ ( x ) , · · · , ψ ( x T − )] , (28) + H ψ ( x ) = [ ψ ( x ) , · · · , ψ ( x T )] , (29)11nd compatible closed-form regression as in (27). In what follows, we outlinea more general formulation for obtaining Koopman operator dynamical modelsmotivated by the aforementioned idea.For a data-set D = { x i , y i } T − i =0 with dynamics y i = F ( x i ), the goal is tosolve the following optimization problem:min A , C , ψ ( · ) T − (cid:88) i =0 evolution (cid:122) (cid:125)(cid:124) (cid:123) (cid:107) z i +1 − Az i (cid:107) + reconstruction (cid:122) (cid:125)(cid:124) (cid:123) (cid:107) y i − Cz i (cid:107) (30)subject to: z i = ψ ( x i ) (31) ψ ∈ F (32)where F is a space of observables. Notice that this becomes equivalent to vanillaDMD (27) where there is no “lifting” - ψ ( x ) = x . The targeted representationis in the form of a Koopman operator dynamical model z = ψ ( x ) , (33) z k +1 = Az k , (34) y k = Cz k , (35)where the initial lift (33) leads to linear state-transition coordinates (34) whichare then projected on outputs of interest (35). Some EDMD variations andextensions include kernel EDMD [40] (using kernels instead of fixed basis func-tions) with its generator version - gEDMD [45] and naturally structured EDMD[42] (preserving the Markov property leading to stable finite-dimensional ap-proximations). However, the aforementioned EDMD approaches fix a liftingmap (31) and then solve (30) as an convex, least-squares optimization problem.The unstructured fixed-bases of the above EDMD methods often lead to data-inefficient models of very high dimension and only locally accurate prediction.To tackle the aforementioned issue, Li et al. [41] propose simultaneous learningof (31) with a neural network and an added L -regularizer in (30). Furthermore,Lian and Jones [44] propose the use of subspace identification methods [52] forsolving (30) independently of the lifting in (31) which is subsequently fitted.Note that all of the approaches mentioned until now are only able to repre-sent purely transient (off-attractor) dynamics, as there exists a finite-dimensionalKoopman-invariant subspace that includes the state itself for (almost) global lin-ear representations. To be able to represent a wider range of dynamical regimes,some approaches are considering a nonlinear reconstruction map to the originalstate instead of a linear one (35) requiring invertibility and a suitably modifiedreconstruction loss compared to (30). Such a modification does not require thestate to lie in the span of lifted coordinates and can lead to lower-dimensionalembeddings as well. To discover such coordinates, machine learning tools suchas neural network structures in various forms, e.g. linear-recurrence and auto-encoding [46, 48, 49, 50] are employed. To tackle continuous spectra as well,the work of Lusch et al.[47] state-parameterizes a low-dimensional Koopman12mbedding using neural networks. Although nonlinear reconstructions can leadlower-dimensional representations, siding with linear reconstruction is useful forpractical control design and amendable linear mode analysis of nonlinear sys-tems.Considering a collection of observables (not necessarily eigenfunctions) aslifting coordinates is still the most dominant train of thought in the data-drivenframeworks for Koopman operator-based representations. Nonetheless, such ap-proaches often lead to lifting coordinates not being in the span of eigenfunctions- providing only locally accurate prediction. Furthermore, their well-posednessis questionable as such frameworks do not necessarily recover genuine Koop-man operator eigenfunctions, eigenvalues or modes. Recalling Example 2.1 itis also obvious that the problem of directly looking for eigenfunctions is bet-ter posed for finding genuine linearily-evolving coordinates for the Koopmanoperator paradigm as they are by definition Koopman-invariant. If one looksto exploit the paradigm for tractable analysis as well as control, consideringlearning the Koopman operator spectrum by design can be more efficacious. In the following we present Koopman-operator representation approachesthat are not purely data-based but also structurally aware - looking to exploitthe Koopman operator’s algebraic and geometric properties. Due to a strongconnection of Koopman operators and state-space geometry, exploiting notionsfrom differential geometry can provide, to a degree, “supervision” to the gener-ally unsupervised task of learning Koopman operator eigenfunctions.
As opposed to other approaches that purely use data to obtain a span of- often not truly Koopman-invariant - observables, the focus here is on using(generalized) eigenfunctions by design relevant for predicting output functionswhich calls for a more rigorous theoretical treatment.
Theorem 3.1.
Consider the domain X ⊆ M ⊆ R n , x ∈ X and ˙ x = f ( x ) suchthat f : X (cid:55)→ M . If X is compact and φ ( x ) ∈ C ( X ) , then Koopman operatoreigenfunctions φ : X (cid:55)→ C are solutions of a linear partial differential equation(PDE) L f φ ( x ) = ∂φ∂ x f ( x ) = sφ ( x ) , (36) where s corresponds to the eigenvalue corresponding to φ and L f is the Lie-derivative operator - the infinitesimal version of the composition operator (11). Proof.
Given the Definition 2.1 of the Koopman operator generator, for thecase that X is compact and φ ( x ) ∈ C ( X ) one can use the Mean Value Theorem[53] to write G K φ ( x ) = lim t → + K t f φ ( x ) − φ ( x ) t = ∂φ∂ x f ( x ) , (37) and characterize the effect of the generator on C -eigenfunction. emark 3.2. One could consider a more general domain X in Theorem 3.1 if φ ( x ) ∈ C ( X ) [54].To highlight the connection of Koopman eigenfunctions with algebraic andgeometric notions, we base our exposition on the version of (36) along an inte-gral curve - φ (cid:0) F t ( x ) (cid:1) = exp( st ) φ ( x ) - that, when fixing time- t , turns into aneigenvalue problem φ ( F ( x )) = λφ ( x ) . (38)Interestingly, the univariate version of the above equation is the Schr¨oder’sfunctional equation φ ◦ F = λφ [55]. This equation studies how maps evolveunder iteration and even pre-dates Koopman’s seminal work [8]. Consideringthe Schr¨oder’s equation in its multivariate form [56], we write K F φ = φ ◦ F = F (cid:48) ( ) φ , (39)with F (cid:48) ( ) being the linearization of F : X (cid:55)→ M around the origin. The fullrank (univalent) solution of 39 is analytic if 0 < || F (cid:48) ( ) || < principle eigenfunctions (Definition 2.4) of the map F . Schr¨oder’sequation (39) is also directly related to functional conjugation [58] on whoseimplications we elaborate on in the following. Definition 3.1 (Topological conjugacy).
Consider maps F : M (cid:55)→ M and T : Y (cid:55)→ Y such that ∃ (cid:37) : M (cid:55)→ Y satisfying (cid:37) ◦ F = T ◦ (cid:37) . Then, the mapsare topologically i) conjugate if (cid:37) is a homeomorphismii) semi-conjugate if (cid:37) is C and surjective Remark 3.3.
Consider the maps from Figure 3. If conjugated, then F and T have the same dynamic behavior while, if they are semi-conjugated, the dynam-ics of T are contained in F . M MY Y
F(cid:37) (cid:37)T
Figure 3: Commutative diagram of topological conjugacy
Theorem 3.2 (Spectral equivalence [13]).
Let the maps F : M (cid:55)→ M and T : Y (cid:55)→ Y be topologically conjugate via the homeomorphism (cid:37) : M (cid:55)→ Y suchthat (cid:37) ◦ F = T ◦ (cid:37) holds. If ( λ, ϕ ) is an eigenpair of K T , then ( λ, ϕ ◦ (cid:37) ) is aneigenpair of K F . Example 3.1.
Consider again the system from Example 2.1 x (cid:55)→ F ( x ) = (cid:20) ax bx + (cid:0) b − a (cid:1) x (cid:21) , (40)and its linearization y (cid:55)→ (cid:20) a b (cid:21)(cid:124) (cid:123)(cid:122) (cid:125) T = F (cid:48) ( ) y . (41)Evidently, the principle eigenpairs of the linearized system are ( a, (cid:104) w a , y (cid:105) ) and( b, (cid:104) w b , y (cid:105) ) with w a , w b the eigenvectors of the adjoint of T [61]. By Theorem3.2 we see there is a conjugacy map (cid:37) ( x ) = [ x , x + x ] (cid:62) = x + d ( x ) (Figure 3)with the diffeomorphism d ( x ) = [0 , x ] (cid:62) allowing for construction of non-trivialprincipal eigenfuncions of the nonlinear system from the ones of its conjugatelinear system (see Example 2.1). Remark 3.4 (Semi-conjugacy and ROM).
Although for general observablesinfinite-dimensional, the composition operator is one-dimensional for its eigen-functions which are semi-conjugacies by definition λ ◦ φ = φ ◦ F . This realizationtheoretically justifies the utility of the Koopman operator paradigm in reducedorder modeling (ROM) such as [19, 62, 63, 45]. The data-driven frameworks utilizing the outlined theory assume the follow-ing Koopman operator dynamical model z = φ ( x ) , (42) z k +1 = Λ z k , (43) y k = V z k . (44)where φ is (generalized) eigenfunction lifting with Λ being the eigenfunctiontransition map and V the projection on the outputs (Koopman modes). Uti-lizing the introduced structural properties leads to a modified optimization ob-15ective compared to (30)-(32). Formally, we can writemin Λ , V , φ ( · ) T − (cid:88) i =0 (cid:107) z i +1 − Λ z i (cid:107) + (cid:107) y i − V z i (cid:107) (45)subject to: z i = φ ( x i ) (46) Λ ∈ S λ , (47) φ ∈ S φ , (48)where S φ , S λ represent structural constraints for eigenfunctions and eigenvalues,respectively. This leads to a better-posed problem with a more restricted solu-tion space than in (32). In order to provide eigenfunction lifting by design somegeometrical and topological considerations need to be made, which we presentin the following. Learning via topological conjugacy.
One notable example utilizing topologicalconjugacy in through a data-driven framework the is work of Folkestad et al.[30]. The assumed system form consists of a known linear part and an unknownresidual non-linearity ˙ x = f ( x ) = T x + g ( x ) . (49)Utilizing topological conjugacy one exploits the fact that the principal eigen-values are shared between the full nonlinear system (49) and its linearizationaround an asymptotically stable fixed point, constraining the choice of eigen-values in (47). Then, one can learn the diffeomorphism part of the conjugacymap (cid:37) between the nonlinear and linear part of 49 by solving the followingoptimization problem min d ( · ) T − (cid:88) i =0 (cid:107) ˙ (cid:37) i − T (cid:37) i (cid:107) (50)subject to: (cid:37) i = x i + d ( x i ) (51) d (cid:48) ( ) = . (52)Subsequently, one is able to have a recursive construction of eigenfunction coor-dinates from the principal ones following Property 2.4 and Theorem 3.2. Usingthese properties takes the role of (48) for learning Koopman-based dynamicalmodels by lifting to eigenfunction coordinates by design. Notice that, due toconsidering continuous-time dynamics, this approach is actually learning theKoopman operator generator . Learning by exploiting state-space geometry.
The approach of Korda et al. [11]optimizes generalized eigenfunctions, changing (42)-(44) to admit a continuous-time representation for the state evolution matrix to a block diagonal Jordanmatrix that is then discretized. Known from finite-dimensional operator theory,16eneralized eigenfunctions do give rise to block diagonal Jordan blocks J λ = λ
1. . . 1 λ , (53)that decompose the space of observables into invariant subspaces of the Koop-man operator semigroup spanned by generalized eigenfunctions [64]. The ap-proach focuses on multi-step prediction error minimization while using non-recurrent sets to construct generalized eigenfunctions. More specifically, theeigenfunctions are constructed by exploiting the existence of boundary func-tions - values of eigenfunctions on non-recurrent sets of the state-space - whichserve as a structural constraint one eigenfunctions in (30)-(32). It also needsa suitably rich choice of initial conditions and requires system trajectories forlearning. Nevertheless, the method does not require any prior knowledge aboutthe system and predicates solely on convex optimization methods.We also highlight some works not geared towards representing the originalnonlinear system as (42)-(44) but do seek to find solutions to the eigenvalueequations (10) and (4.2) for i.e. coherent structure extraction [65, 66] or discov-ering conservation laws [67]. When it comes to identifying Koopman operatorprinciple eigenpairs, the work of Klus et al. [68] (generator version [45]) em-ploys eigendecompositions of Koopman operator using kernel methods. On theKoopman operator generator identification front, the work of Kaiser et al. [69]discovers only lightly-damped Koopman generator eigenpairs (as vanilla DMD-based methods converge to the strongest magnitude eigenvalues [21, Section 2]).The aforementioned approaches highly depend on the empirical choice of basisfunction or kernel parameters. While the two previously introduced methodologies considered spatial co-ordinates, for ergodic systems, we turn to delay-coordinates approaches oftenemployed for time-series prediction. These ideas are highly represented in theliterature as the original Koopman theory [8] is used to prove Birkoff’s ergodictheorem [70]. In case that post-transient dynamics are of interest, fixed spatialcoordinates (Remark 2.2) cease to be of relevance (evolution not Markovian) e.g.on chaotic attractors. It is known from Takens embedding theory [71] that onecan provide a geometric reconstruction of a (strange) attractor of nonlinear sys-tems by embedding scalar output measurement (observation) y ( k ) := y k into ahigher dimensional space of time-delay coordinates. This viewpoint is naturallydefined for sampled-data systems. Here the observable (measurement) coor-dinates are based on the time-series { y k } T − k =0 , T ∈ N of scalar measurements17 k = h ( x k ) represented through a Hankel matrix H L ( y ) := y y . . . y T − L y y . . . y T − L +1 ... ... . . . ... y L − y L . . . y T − , (54)with depth L ∈ N . The argument of the Takens delay-embedding theorem isthat an attractor of a dynamical system can be reconstructed using sufficientlytime-delayed measurements y Nk of a scalar observable of interest. The evolutionof time-series sequences y Nk (cid:55)→ y Nk +1 of the form y Nk := (cid:2) y k K y k · · · K N y k (cid:3) (cid:62) ∈ R N ≤ L , (55)becomes almost linear even when the measurement evolution y k (cid:55)→ y k +1 comesfrom a nonlinear ergodic system [34]. Such a reality leads to the discoveryof Koopman operators modes, eigenfunctions and eigenvalues represented viadelay-coordinates. Promising results of [36] use convolutions coordinates as ageneralization of Hankel alternative view of the Koopman (HAVOK) frameworkof Brunton et al. [34]. The latter work demonstrated the ability to construct alinear system with intermittent forcing that is able to predict chaotic dynamics.Other notable extensions and generalizations of DMD with time-delay coordi-nates include Hankel-DMD [19], Prony approximation of Koopman operator[32] and higher order DMD [33] that employ a time-delayed observable as abasis for DMD. Work of Arbabi et al. [19] establishes convergence of Hankel-DMD to the true Koopman eigenfunctions and eigenvalues of the system. Anotable use of combining both spatial observables and delay-embeddings canbe found in Arbabi et al. [72] for modeling nonlinear PDEs. However it isknown that linear models such as those generated by the Hankel-DMD can re-construct an ergodic dynamical system only in an asymptotic sense. A study ofthis limitation and the conservatism of Takens theorem regarding the number oftime-delays necessary can be found in [38]. As proven to useful for Hamiltoniansystems and post-transient behaviors of system with quasi-periodic dynamicsand continuous spectra (cf. Remark 2.2), the utility of this method is in theliterature have been reserved to ergodic (measure-preserving) systems. For amore detailed data-driven treatment of ergodic systems and Koopman spectra(beyond time-delay coordinates) we point to the [37, 35].Time-delays as they pertain to the Hankel matrix (54) also form the back-bone of common linear-system identification such as the Ho-Kalman algorithm[73] and the family of subspace state-space identification (4SID) methods [74].Although suggested to be similar to dynamic mode decomposition on a few oc-casions [23, 29], subspace-identification is fundamentally different from DMD.We elaborate on this notion in the following remark. Remark 3.5 (DMD vs 4SID).
In essence, “vanilla” DMD is a regressionprocedure linearly mapping two data ‘snapshots’ (sequence of spatial measure-ments, Hankel matrix, images, etc.) from instance k (cid:55)→ k + 1 (allowing for di-mension reduction via SVD [29]). On the other hand, 4SID is an unsupervised18earning method for causal linear systems that explicitly delivers state-space ma-trices (34)-(35) via the system-theoretic concept of observability [75] while notproviding the coordinate transform of (33). Although both methodologies mightperform an SVD of the same Hankel matrix in the case of i.e. Hankel-DMD [19]or HAVOK [34], they are fundamentally different and generally deliver differentquantities. Nevertheless, for the univariate case of (27), 4SID methods do be-come equivalent to vanilla DMD and thus equally as ill-conditioned [21, Section2].Another family of approaches considers approximating the Koopman operatorgenerator through advection-diffusion operators [76, 35] and time-delay coordi-nates. Such approaches relate to manifold learning and diffusion coordinates[77] as the data of the post-transient dynamics on an attractor forms a mani-fold. As already mentioned, approaches here recover Koopman operator-relatedquantities asymptotically. Also a recent approach of Takeishi et al. [78] uses thefact that the diffusion operator commutes with the Koopman operator asymp-totically and then constrains the learning algorithm to ensure best-possible com-muting for the finite-delay and finite-data case. Remark 3.6 (Delay-coordinates and immersion).
Recently, in [79] the au-thors propose a relation of dynamical immersion to time-delay coordinates inKoopman operator theory. There, the immersion is created by augmenting thefull state with compositions of its state-transition map (delay-embedding). Suchimmersion-based coordinates are shown to be useful in the context of transientdynamics akin the frameworks in Section 3.2. While conceptually similar, theimmersion-based time-delays are of the full state and assume a known statetransition map - different to the delay-embeddings considered in this section.
With a focus on existing robotics applications, here we consider some notabledata-driven approaches, exploring the practical utility of the Koopman operatorparadigm. A quite natural application can be found in soft robotics. The reasonis obvious - with soft robots we do not even know what the states of interest are,let alone how to get them in a tractable way. That makes the Koopman operatordynamical models an interesting choice for data-driven control of soft roboticsystems [80, 81]. Notably, learning fast quad-rotor landing of Folkestad et al.[82] represents one of the only well-posed data-driven methods for Koopmanoperator dynamical models in practice - learning a diffeomorphism to discovereigenfunction coordinates for linear control design and prediction.
4. System analysis and Koopman operator paradigm
Fully data-based models of non-linear systems via the “dynamics of states”representation (3) do not offer any intrinsic information about the system rel-evant for system analysis in their representation. However, approaches usingthe Koopman paradigm, contain ‘free’ information about the system of interest19 invariant manifolds, fixed points, attractors and periodic orbits. The afore-mentioned quantities encode a great deal of information about the system andare the basis for system analysis through the Koopman operator paradigm.We outline two pathways in the following: one tying known stability notions(Lyapunov) with Koopman-related quantities and another directly consideringKoopman operator-related quantities as a proxy to system analysis.
In the following we highlight a more explicit and natural connection of theKoopman paradigm to classical system-theoretic analysis through the generatorof the Koopman operator semigroup. In this classical view, we use the fact thatthe Lyapunov function itself is a special kind of an observable [83, 83, 10, 84]with a negative semi-definite derivative. With that in mind, one can argue thatthe Koopman operator notions have implicitly been a part in Lyapunov stabilityanalysis throughout the last century.
Consider the deterministic generator of the Koopman operator semigroup,already introduced in Definition 2.1.
Theorem 4.1.
Let there exist a proper Lyapunov function V ∈ C associatedwith the system (3) for which the action of the Lie-derivative operator L f alongthe vector field f results in L f V ( x ) = ∂V∂ x f ( x ) < , (56) which holds for ∀ x ∈ M\{ x ∗ } . Then, the equilibrium point x ∗ of (3) is asymp-totically stable. Corollary 4.1.1.
A Lyapunov function V ∈ C ( M ) is a function over thestate-space M (an observable) such that it is negative-semidefinite under theaction of the Koopman operator generator G K of the Koopman operator family {K t } t ∈ R + , (2.1): G K V ( x ) ≤ , (57) ∀ x ∈ M\{ x ∗ } . The conditions just outlined for the continuous-time case can be triviallyextended to the discrete-time case, but for completeness, we present them inthe following.
In discrete-time the interest shifts from a family (semigroup) of Koopmanoperators to the fixed time- t Koopman operator from Definition 2.2 for systemsgoverned by a difference equation: x k +1 = F ( x k ) . (58)20 heorem 4.2. Let there exist a proper Lyapunov function V ∈ C associatedwith the system (58) fulfilling V ( F ( x )) − V ( x ) < , (59) which holds for ∀ x ∈ M\{ x ∗ } . Then, the equilibrium point x ∗ of (58) is asymp-totically stable. Corollary 4.2.1.
Considering the dynamics (58) and the action of the Koop-man (composition) operator K introduced in Definition 2.2 [ K F V ]( x ) = V ( F ( x )) , (60) it becomes evident that [ K F V ]( x ) − V ( x ) ≤ , (61) is equivalent to the Lyapunov decrease condition from Theorem 4.2 which holdsfor ∀ x ∈ M\{ x ∗ } . The results of this section show the Koopman operator paradigm’s long-standing connections to Lyapunov stability through the Lie-derivative (Theorem3.1).
The Koopman operator paradigm gives a proxy to spectral analysis of anonlinear system through the eigenfunctions and eigenvalues of the correspond-ing Koopman operator. Koopman operators are utilized in the global analysisof complex dynamical systems as their eigenfunctions can be used for a manynotions relevant for analysis: metastable sets, splitting the dynamics into dom-inant slow/fast processes, or to separate superimposed signals [85, 68]. Morespecifically, Koopman eigenfunctions provide a lot of information about theoriginal system, including a characterization of invariant sets such as stableand unstable manifolds [86]. Furthermore, the Koopman paradigm is provid-ing intrinsic information state-space geometry i.e. through level sets of thestate-space with the same rate of convergence to the attractor - isostables andsets with an identical phase value - isochrons [83]. Koopman eigenfunctions,in particular isostables, are used to derive Lyapunov functions and contractingmetrics in [83]. While, for dissipative systems the Koopman operator eigenfunc-tions capture isochrons and isostables, for ergodic systems they capture periodicpartitions of the state-space [18]. Furthermore, Koopman eigenfunctions witheigenvalue is 0 / Property 4.1.
Consider X ⊆ R n and a vector field f : X (cid:55)→ X with a fullrank Jacobian f (cid:48) ( x ∗ ) and eigenvalues Re { s i } < at the fixed point x ∗ . Ifthe eigenfunctions of the Koopman operator are C , then ˙ x = f ( x ) is globallyasymptotically stable (GAS) in X . roof. [10, Proposition 4]. The above result is a global equivalent of the well-known local stability resultvia Lyapunov’s indirect method [88]. Given local stability is implied by stable f (cid:48) ( x ∗ ), but global stability is implied by C -eigenfunctions of the Koopmanoperator. Furthermore, works of Mauroy et al. [89, 84] establish that joint-levelsets of Koopman (generalized) eigenfunctions (for Koopman eigenfunctions fromTheorem 4.1) offer principled parameterization of candidate Lyapunov functionsi.e. V ( x ) = (cid:32) N (cid:88) i =1 | φ i ( x ) | p (cid:33) p , (62)for p ∈ N . As the eigenfunctions can serve as evolution coordinates of nonlinearsystems defined by their corresponding eigenvalues, they are directly related thesystem’s contraction metrics. Nevertheless, the Koopman operator paradigmprovides a more general stability result than the one of Property 4.1. Theorem 4.3.
Let X be a positive invariant compact set and that the Koopmanoperator K t φ = φ ◦ F t admits an eigenfunction φ s ∈ C ( X ) with the eigenvalue Re { s } < . Then the zero level set S = { x ∈ X | φ s ( x ) = 0 } is forward invariant under F t and globally asymptotically stable. Example 4.1 ([89]).
To highlight the globality of the spectral analysis viathe Koopman operator, consider a non-hyperbolic system ˙ x = − x where theanalysis via the indirect Lyapunov method is not possible. The system admits acontinuous (generalized) eigenfunction with φ s ( x ) = exp (cid:0) − / (cid:0) x (cid:1)(cid:1) with theassociated eigenvalue s = −
1. Given that, one can imply global asymptoticstability of the origin by Theorem 4.3.
Here we consider some structural properties that can be recognized via theKoopman paradigm’s spectral analysis. It is important to note how the functionspace of observables condition the spectral properties of the Koopman operator.The original theory of B.O. Koopman [8, 70]) studied the ergodicity of Hamilto-nian flows naturally associated with the space of square-integrable observables.
Property 4.1.
For conservative (Hamiltonian systems) the Koopman operator K t : F (cid:55)→ F is unitary on the Hilbert space L ( µ ) of complex valued square-integrable functions tied with any Borel probability measure µ F = L ( M ) , F = F † , (63) holding ∀ s eigenvalues from the spectrum σ ( K ) of K . Thus, the Koopmanoperator K t is unitary with | exp( s ) | = 1 , s ∈ C [15]. emark 4.1. If the system is measure preserving (thus satisfying Property4.1) it possesses an invertible flow F t : X (cid:55)→ X . Then the relation between theKoopman operator K t and its dual P t (Property 2.5) simplifies K t ψ ( x ) = ψ ( F t ( x )) , P t ρ ( x ) = ρ (cid:0) F − t ( x ) (cid:1) (64)with F − t = ( F t ) − making the Koopman operator the inverse of its dual.Dynamical systems with attractor(s) are not energy (measure) preservingand thus dissipative, making the Koopman operator non-unitary. Therefore,for such systems we can state the following property. Property 4.2.
The Koopman operator K t : F (cid:55)→ F is dissipative in the Banachspace C ( M ) of continuously differentiable functions F = C ( M ) , (65) where F is a space of continuous functions spanned by analytic functions. Thusthe Koopman operator K t is not unitary while the eigenfunctions of the Perron-Frobenius operator become Dirac functions [22]. Dissipative systems are very amendable to the global stability analysis via theKoopman paradigm [89, 84] (introduced in the previous subsection) as the func-tions that span the Koopman invariant subspace are analytic. The choice of theobservables-space as F = C ( M ) more easily uncovers principle eigenfunctions that are crucial for stability analysis of the equilibrium of dynamical systems[84]. The Koopman operators for dissipative systems assemble a contractionsemigroup that has a corresponding dissipative generator [90]. We can rede-fine the introduced eigenfunctions (2.3) as (ordinary) eigenfunctions of a linearoperator K acting in a compact Hilbert space that solve the following K φ g = λφ g ⇔ ( K − λ g ) φ g = 0 . (66)In general, due to the geometric eigenspace being lower than the algebraiceigenspace dimension, a complete basis for the space on which the linear operatoracts upon can not always be obtained from (66), but from generalized eigen-functions . The notion of generalized eigenfunctions in the field is introducedand proved through the case of finite-dimensional linear operators [22, 91] usingKato decomposition [92]. In the following we just informally remark on thesespectral objects. Remark 4.2 (Generalized eigenfunctions).
Generalized eigenfunctions ξ g are the solution following equality( K − λ g ) m g ξ g = 0 (67)with integers g = 1 , . . . , s and i = 0 , ..., m g − ξ = (cid:2) ξ , . . . , ξ m , ξ , . . . , ξ m , . . . , ξ s , . . . , ξ m s s (cid:3) (cid:62) (68)23e the concatenation of all of the (generalized) eigenfunctions corresponding tothe operator. Now considering the Koopman operator generator we have G K ξ ig = ˙ ξ ig = λ g ξ ig + ξ i +1 g , (69)that in the finite-dimensional case gives˙ ξ = diag { J λ , . . . , J λ s } ξ , (70)where J λ g is the familiar Jordan block (53). Example 4.2.
Consider the case of a multiplicity of m g = 2 in (69) resulting in˙ ξ g = λ g ξ g + ξ g . The function ξ g in turn represents the one ordinary eigenfunction φ g satisfying ˙ φ g = λ g φ g giving the relation ( G K − λ ) ξ g = φ g . Remark 4.3.
The utility of generalized eigenfunctions in spanning Koopman-invariant subspaces arising from (4.2) generalizes to infinite-dimensional casefor discrete spectrum of the Koopman operator. This is courtesy of a compact operator defined on a Hilbert space (as it is a limit of finite-rank operators) [93].
5. Koopman-based control approaches and applications
Here, the focus is turned towards non-autonomous systems and approachesthat still allow for exploiting the Koopman operator paradigm to find efficientrepresentations of control systems.
As the Koopman operator dynamical models are naturally formed for au-tonomous systems, they require modifications in the non-autonomous case. Wepresent structured ways of including control for classes of nonlinear systemsand relate to the core of the Koopman operator theory. In order to do so wetake an “atomic” approach based on the Koopman eigenfunction evolution PDE(Theorem 3.1).
Here we offer a novel derivation of Koopman operator eigenfunctions in thecase of control-affine systems. Let us consider the class of such systems, in asingle-input and single-output (SISO) form˙ x = f ( x ) + g ( x ) uy = h ( x ) , (71)with f , g ∈ M the drift and control vector fields, respectively. By looking atthe time-evolution of the output/observable y ˙ y = L f h ( x ) + L g h ( x ) u, (72)24ne obtains the familiar Lie-derivative expression. To have a globally lineardescription of such a system, one can study the eigenspaces of the Lie-derivateoperators (Koopman operator generators) acting on the drift and control vectorfields. Theorem 5.1 (Bilinear eigenfunctions).
Consider the control affine system(71) with the eigenfunctions φ i and φ c,i of the drift and control vector fields,respectively. If span { φ c, , φ c, , . . . } ⊆ span { φ , φ , . . . } the eigenfunction time-evolution is bilinearizable. Proof.
Given that span { φ c, , φ c, , . . . } ⊆ span { φ , φ , . . . } , we can write φ c ( · ) = C φ ( · ) = (cid:80) ∞ i =1 c i φ i ( · ) . Then, the eigenfunction evolution for (72) results in ˙ φ ( x ) = L f φ ( x ) + L g φ c ( x ) u = ( L f + L g C u ) φ ( x ) . (73) As L g and C linear operators their composition L g C is as well. Thus, the eigen-function evolution becomes bilinearizable. Corollary 5.1.1.
By integrating the infinitesimal expression (73) we obtain [ K t ( u ) φ ]( x ) = exp[( L f + L g C u ) t ] φ ( x ) , (74) giving rise to a control-parameterized family of time- t semigroups of Koopmanoperators {K t ( u ) } t ∈ R + . Thus, the following ensues for the discrete-time case [ K ( u ) φ ]( x ) = K f ( K g C ) u φ ( x ) , (75) following trivially by fixing the sampling time t = t s in (74). One can see how the bilinear system theory of the continuous-time case (74) cannow be replaced by linear parameter-varying (LPV) theory [94] in the discrete-time case. Generalizing (75) for multiple-input and multiple-output form isstraight forward.
Remark 5.1 (State-independent vector field).
For simpler control-affinesystems with state-independent control vector fields i.e. ˙ x = f ( x ) + b u , anydrift vector field admits the trivial principle eigenfunction φ ( · ) = 1 with eigen-value 0 allowing a bilinear representation without needing the conditions fromTheorem 5.1. Another less restrictive option is to rewrite (73) by the chain rule[31] ˙ φ ( x ) = sφ ( x ) + ∂∂ x φ ( x ) Bu , (76)with s the eigenvalue corresponding to φ , but leads to a state-dependent inputterm in the evolution of eigenfunctions.25 xample 5.1 (Koopman-bilinearizable system). Consider a continuous-time control system inspired by [95]:˙ x = (cid:20) ax bx + (cid:0) b − a (cid:1) x (cid:21) + (cid:2) g ( x ) g ( x ) (cid:3) u , (77)with control vector fields g ( x ) = [1 , x ] (cid:62) . g ( x ) = [0 , (cid:62) The system fromcan be lifted into eigenfunction-coordinates φ ( x ) = [ x , x + x , x , (cid:62) of theKoopman generator with the diagonal transition matrix of respective eigenvalues Λ = diag { a, b, a , } . Then, we can write˙ z = Az + B z u + B z u , (78)by lifting the state x to z = φ ( x ). After applying (element-wise) the controlvector field Lie-derivatives with respect to the new coordinates, we obtain B = , B = , (79)allowing us to write L g i φ ( x ) = B i z .Although the the eigenfunction evolution is bilinear, there are well studiedtools to deal with such problems cf. [96]. Remark 5.2 (Representation heuristic).
Works considering Koopman op-erator dynamical modeling often assume the following evolution ˙ z = Az + Bu in the ”lifted” z -coordinates. Clearly, such a representation of the control effectis only local. Thus, the results of this section give a theoretical justification onwhy many data-driven Koopman operator-related approaches for control sys-tems are only well-versed for short-term prediction. As opposed to control-affine systems, general nonlinear control systems ofthe form ˙ x = f ( x , u ) do not admit a closed form representation for (bi)linearprediction via the Koopman operator paradigm. Nonetheless, meaningful rep-resentations are still possible. One can assume the existence of joint state-and-input eigenfunctions [31], leading to the following eigenfunction PDE˙ φ ( x , u ) = L f φ ( x , u ) + ∂∂ u φ ( x , u ) ˙ u , (80)where s is the eigenvalue corresponding to φ . Given the relation above, thederivative of control ˙ u is the input to eigenfunction coordinates leading to inte-gral or incremental control in the discrete-time analogue. Nevertheless, the formadmits general nonlinear control systems. In a similar vein, Korda et al. [11]26ake the case that the form is non-restrictive as any system can be transformedby “state inflation” ˙ v = u via x = [ ζ (cid:62) , v (cid:62) ] (cid:62) into (cid:20) ˙ ζ ˙ v (cid:21)(cid:124) (cid:123)(cid:122) (cid:125) ˙ x = (cid:20) f ζ ( ζ , v ) (cid:21)(cid:124) (cid:123)(cid:122) (cid:125) f ( x ) + (cid:20) I (cid:21) ˙ v (cid:124) (cid:123)(cid:122) (cid:125) Bu , (81)again limiting our direct control to controlling the rate of change of the inputsignal. As a single Koopman generator corresponds to a single ODE or PDE,the introduction of input dependence parameterizes the differential equationswith a possibly infinite set of inputs. However, restricting those to a finite setcan be represented as a system of switched Koopman operators. The authorsin [62] utilize this for efficient model predictive control. Here we survey notable options for designing control-oriented frameworksbased on the Koopman operator theory (for a graphical overview cf. Figure 4).
Control-orientedframeworks
Optimal control [97, 30, 11, 12]
Stabilization & estimation [98, 99, 100, 101][102, 103, 104]
ROM control [67, 62, 45]
Figure 4: Main branches of control-oriented approaches using the Koopman operator-basedmodels. The bottom split part of the latter includes relevant references employing the ap-proach noted in the top part.
One simplifying perspective looks to use parts of the intrinsic dynamicsinformation from the Koopman operator for energy-based control or reducedorder controllers. These approaches utilize Property 4.1 for identifying theHamiltonian from data - as it corresponds to a Koopman eigenfunction [67].Furthermore, when it comes to reduced order modeling the Koopman operatorparadigm is put to use in [62, 45] for data-driven switched control of systemsgoverned by partial differential equations.
For optimization-based approaches such as model predictive control (MPC),the evident benefit is the increased accuracy and validity in contrast to linearmodels based on a local linearization, while having a reduced computationalburden compared with schemes based on nonlinear models. A comparison of27he outputs of a convex program with Koopman-based (and another global lin-earization) models versus the solutions of the original nonlinear programmingproblem can be found in Igarashi et al. [12]. For a principled MPC treatmentvia the Koopman operator paradigm we point to Korda et al. [97] with sub-sequent optimized versions for more efficacious prediction in Korda et al. [11]and Folkestadt et al. [30].
Regarding stabilization, lifted space-based control Lyapunov functions havebeen considered in a range of works [98, 99, 100, 101] - showing promise fornonlinear optimal stabilization. Moreover, works on observer synthesis [102, 103]and filtering [104] show improved efficacy via the use of Koopman operatordynamical models.
The idea of embedding a system into possibly a higher-dimensional linearsystem is not a new concept in the control community [105]. One of the reasonsnon-local linear representations of nonlinear dynamics could be appealing is ef-ficient controller design - both regarding computation and control effort. Theconsidered Koopman operator-based linearizations do not require compensatinginput injections as their exact linearization counterparts such as input-outputand feedback linearization. For systems with useful (stabilizing) nonlinearities,feedback linearization approaches are known to be non-robust [106]. Further-more, feedback linearizing controllers does not aid in respecting arbitrary costfunctionals (in a receding or infinite horizon fashion) for control design dueto compensation-based policies [107]. One example of the advantageous use ofKoopman operator dynamical models with respect to feedback linearization canbe found in Kaiser et al. [31].An interesting category of dynamically challenging systems are underwatervehicles, e.g. remotely operated vehicles (ROVs). They are non-linear and stabledue to the dampening effect of hydrodynamic forces. Thus, ROVs present a realworld example of a system with useful non-linearities, whose cancellation (i.e.via feedback linearization) can render a stable system unstable for the smallestof modeling errors [108] while using excessive amounts of control effort [109].With the recent developments in efficient bio-inspired designs of underwatervehicles [110, 111], the need for more efficacious controllers becomes increas-ingly important for their autonomous operation. Given that, it is worthwhileconsidering Koopman operator dynamical models for efficient optimal control,especially for “well-behaved” systems.
6. CONCLUSION
We systematically review all key aspects of
Koopman operator dynamicalmodels in order to provide information relevant for data-driven representations,28nalysis and control through a holistic perspective. Concerning representa-tions, the current state-of-the-art looks to be geared more towards finding rele-vant eigenfunctions for prediction, as opposed to a span of only approximatelyKoopman-invariant observables without tapping into the operator’s spectrumdirectly. Current approaches are able to well represent a system’s transient andquasi-periodic behavior while non-dissipative dynamics and continuous spectraare still an open research question. The compelling feature of the paradigmis that, if representations can be efficiently learned in a well-posed manner, asystem’s prediction, analysis and control become greatly simplified. At present,that involves employing modeling approaches for which there is no overarch-ing data-driven framework. Nevertheless, the recent developments in the fieldshow promise in making Koopman operator-based system modeling more vi-able than ever before. As the operator’s eigenfunctions are shared between thecontinuous- and discrete-time representations of a system, they form meaningfuland amendable parameterizations of the system’s dynamics (even if learned vianon-parametric methods).It must be highlighted the one main challenge in the field of Koopman op-erator dynamical models - learning genuinely invariant coordinates under theKoopman operator. Moreover, the relevance determination of the coordinatestogether with model order choice require a more rigorous consideration. Interms of safety-critical control, tractable a priori error quantification is stillunconquered - hindering the wider applicability of Koopman operator-basedframeworks. Also, the linear evolution has the potential to offer efficient un-certainty propagation - a more challenging matter for models in “dynamics ofstates” representations.Regarding future work, the general lack of comparison of the resulting con-trol frameworks to exact linearization procedures using feedback needs to beaddressed. Such comparisons could better rectify under what circumstancesthe Koopman operator-based controllers are actually advantageous. It is alsocrucial to assess the robustness of these controllers under external disturbances- something not addressed at present. Nevertheless, the limits of predictinggeneral controlled systems demonstrated in (Section 5) make the considerationof iterative direct control design is more conceivable than obtaining globallypredictive models. For model-based reinforcement learning the paradigm posesan interesting prospect, as an almost global linear model implicitly parame-terizes both the optimal value function and the policy (through a set of lifted coordinates). Under changing conditions, exploring the possibilities of adaptivecontrol and online learning remain an exciting area yet to be investigated.
ReferencesReferences [1] O. Nelles, Neural Networks, in: Nonlinear System Identification, SpringerBerlin Heidelberg, Berlin, Heidelberg, 2001, pp. 239–297. doi:10.1007/978-3-662-04323-3_10 . 292] J. Kocijan, A. Girard, B. Banko, R. Murray-Smith, Dynamic systemsidentification with Gaussian processes, Mathematical and Computer Mod-elling of Dynamical Systems 11 (4) (2005) 411–424. doi:10.1080/13873950500068567 .[3] J. Umlauft, T. Beckers, M. Kimmel, S. Hirche, Feedback linearizationusing Gaussian processes, in: 2017 IEEE 56th Annual Conference on De-cision and Control (CDC), Vol. 2018-Janua, IEEE, 2017, pp. 5249–5255. doi:10.1109/CDC.2017.8264435 .[4] T. Beckers, D. Kuli´c, S. Hirche, Stable Gaussian process based trackingcontrol of Euler-Lagrange systems, Automatica 103 (2019) 390–397. doi:10.1016/j.automatica.2019.01.023 .[5] F. Berkenkamp, R. Moriconi, A. P. Schoellig, A. Krause, Safe learn-ing of regions of attraction for uncertain, nonlinear systems with Gaus-sian processes, in: 2016 IEEE 55th Conference on Decision and Control(CDC), IEEE, 2016, pp. 4661–4666. arXiv:1603.04915 , doi:10.1109/CDC.2016.7798979 .[6] J. F. Fisac, A. K. Akametalu, M. N. Zeilinger, S. Kaynama, J. Gillula,C. J. Tomlin, A General Safety Framework for Learning-Based Controlin Uncertain Robotic Systems, IEEE Transactions on Automatic Control64 (7) (2019) 2737–2752. doi:10.1109/TAC.2018.2876389 .[7] A. Lederer, S. Hirche, Local Asymptotic Stability Analysis and Region ofAttraction Estimation with Gaussian Processes, Proceedings of the IEEEConference on Decision and Control 2019-Decem (2019) 1766–1771. doi:10.1109/CDC40024.2019.9029489 .[8] B. O. Koopman, Hamiltonian Systems and Transformation in HilbertSpace, Proceedings of the National Academy of Sciences of the UnitedStates of America 17 (5) (1931) 315–8.[9] M. Georgescu, I. Mezi´c, Building energy modeling: A systematic approachto zoning and model reduction using Koopman Mode Analysis, Energyand Buildings 86 (1) (2015) 794–802. doi:10.1016/j.enbuild.2014.10.046 .[10] A. Mauroy, I. Mezi´c, A spectral operator-theoretic framework for globalstability, Proceedings of the IEEE Conference on Decision and Control (3)(2013) 5234–5239. doi:10.1109/CDC.2013.6760712 .[11] M. Korda, I. Mezi´c, Optimal construction of Koopman eigenfunctions forprediction and control, IEEE Transactions on Automatic Control (2020)1–1 doi:10.1109/TAC.2020.2978039 .[12] Y. Igarashi, M. Yamakita, J. Ng, H. H. Asada, MPC Performances forNonlinear Systems Using Several Linearization Models, in: 2020 American30ontrol Conference (ACC), IEEE, 2020, pp. 2426–2431. doi:10.23919/ACC45564.2020.9147306 .[13] M. Budiˇsi´c, R. Mohr, I. Mezi´c, Applied Koopmanism, Chaos 22 (4). doi:10.1063/1.4772195 .[14] E. Kaiser, J. N. Kutz, S. L. Brunton, Data-driven approximations ofdynamical systems operators for control, Lecture Notes in Control andInformation Sciences 484 (2020) 197–234. arXiv:1902.10239 , doi:10.1007/978-3-030-35713-9\_8 .[15] A. Lasota, M. C. Mackey, Chaos, Fractals and Noise, 1994. doi:10.1198/tech.2006.s351 .[16] S. N. Cohen, R. J. Elliott, Stochastic Calculus and Applications, Prob-ability and Its Applications, Springer New York, New York, NY, 2015. doi:10.1007/978-1-4939-2867-5 .[17] T. Carleman, Application de la th´eorie des ´equations int´egrales lin´eairesaux syst`emes d’´equations diff´erentielles non lin´eaires, Acta Mathematica59 (1932) 63–87. doi:10.1007/BF02546499 .[18] I. Mezi´c, Spectral properties of dynamical systems, model reduction anddecompositions, Nonlinear Dynamics 41 (1-3) (2005) 309–325. doi:10.1007/s11071-005-2824-x .[19] H. Arbabi, I. Mezi´c, Ergodic theory, dynamic mode decomposition, andcomputation of spectral properties of the Koopman operator, SIAM Jour-nal on Applied Dynamical Systems 16 (4) (2017) 2096–2126. arXiv:1611.06664 , doi:10.1137/17M1125236 .[20] I. Mezi´c, Spectrum of the Koopman Operator, Spectral Expansions inFunctional Spaces, and State-Space Geometry, Journal of Nonlinear Sci-ence arXiv:1702.07597 , doi:10.1007/s00332-019-09598-5 .[21] M. Budiˇsi´c, R. Mohr, I. Mezi´c, Applied Koopmanism, Chaos: An In-terdisciplinary Journal of Nonlinear Science 22 (4) (2012) 047510. doi:10.1063/1.4772195 .[22] R. Mohr, Spectral Properties of the Koopman Operator in the Analysis ofNonstationary Dynamical Systems, Ph.D. thesis, University of CaliforniaSanta Barbara (2014).[23] J. H. Tu, C. W. Rowley, D. M. Luchtenburg, S. L. Brunton, J. N. Kutz,On dynamic mode decomposition: Theory and applications, Journal ofComputational Dynamics 1 (2) (2014) 391–421. doi:10.3934/jcd.2014.1.391 .[24] P. Cvitanovi´c, R. Artuso, R. Mainieri, G. Tanner, G. Vattay, Chaos: Clas-sical and Quantum, Niels Bohr Inst., Copenhagen, 2016.URL http://chaosbook.org/ doi:10.1007/4-431-31381-8_23 .[26] J. C. Willems, P. Rapisarda, I. Markovsky, B. L. De Moor, A note onpersistency of excitation, Systems and Control Letters 54 (4) (2005) 325–329. doi:10.1016/j.sysconle.2004.09.003 .[27] E. M. Bollt, Geometric Considerations of a Good Dictionary for KoopmanAnalysis of Dynamical Systems, arXiv arXiv:1912.09570 .[28] P. J. Schmid, Dynamic mode decomposition of numerical and experimen-tal data, Journal of Fluid Mechanics 656 (2010) 5–28. doi:10.1017/S0022112010001217 .[29] J. N. Kutz, S. L. Brunton, B. W. Brunton, J. L. Proctor, Dynamic ModeDecomposition, Society for Industrial and Applied Mathematics, Philadel-phia, PA, 2016. doi:10.1137/1.9781611974508 .[30] C. Folkestad, D. Pastor, I. Mezic, R. Mohr, M. Fonoberova, J. Burdick,Extended Dynamic Mode Decomposition with Learned Koopman Eigen-functions for Prediction and Control arXiv:1911.08751 .[31] E. Kaiser, J. N. Kutz, S. L. Brunton, Data-driven discovery of Koopmaneigenfunctions for control 98195 (2017) 1–40. arXiv:1707.01146 .[32] Y. Susuki, I. Mezic, A prony approximation of Koopman Mode Decom-position, in: 2015 54th IEEE Conference on Decision and Control (CDC),Vol. 54rd IEEE, IEEE, 2015, pp. 7022–7027. doi:10.1109/CDC.2015.7403326 .[33] S. Le Clainche, J. M. Vega, Higher order dynamic mode decomposition,SIAM Journal on Applied Dynamical Systems 16 (2) (2017) 882–925. doi:10.1137/15M1054924 .[34] S. L. Brunton, B. W. Brunton, J. L. Proctor, E. Kaiser, J. Nathan Kutz,Chaos as an intermittently forced linear system, Nature Communications8 (1) (2017) 1–34. doi:10.1038/s41467-017-00030-8 .[35] D. Giannakis, Data-driven spectral decomposition and forecasting of er-godic dynamical systems, Applied and Computational Harmonic Analysis47 (2) (2019) 338–396. doi:10.1016/j.acha.2017.09.001 .[36] M. Kamb, E. Kaiser, S. L. Brunton, J. N. Kutz, Time-Delay Observablesfor Koopman: Theory and Applications, SIAM Journal on Applied Dy-namical Systems 19 (2) (2020) 886–917. doi:10.1137/18M1216572 .[37] S. Das, D. Giannakis, Delay-Coordinate Maps and the Spectraof Koopman Operators, Journal of Statistical Physics doi:10.1007/s10955-019-02272-w . 3238] S. Pan, K. Duraisamy, On the Structure of Time-delay Embedding inLinear Models of Non-linear Dynamical Systems arXiv:1902.05198 .[39] M. O. Williams, I. G. Kevrekidis, C. W. Rowley, A Data-Driven Approx-imation of the Koopman Operator: Extending Dynamic Mode Decom-position, Journal of Nonlinear Science 25 (6) (2015) 1307–1346. doi:10.1007/s00332-015-9258-5 .[40] M. O. Williams, C. W. Rowley, I. G. Kevrekidis, A kernel-based methodfor data-driven Koopman spectral analysis, Journal of Computational Dy-namics 2 (2) (2015) 247–265. doi:10.3934/jcd.2015005 .[41] Q. Li, F. Dietrich, E. M. Bollt, I. G. Kevrekidis, Extended dynamicmode decomposition with dictionary learning: A data-driven adaptivespectral decomposition of the Koopman operator, Chaos: An Interdis-ciplinary Journal of Nonlinear Science 27 (10) (2017) 103111. doi:10.1063/1.4993854 .[42] B. Huang, U. Vaidya, Data-Driven Approximation of Transfer Operators:Naturally Structured Dynamic Mode Decomposition, Proceedings of theAmerican Control Conference 2018-June (1) (2018) 5659–5664. doi:10.23919/ACC.2018.8431409 .[43] E. Yeung, S. Kundu, N. Hodas, Learning deep neural network represen-tations for koopman operators of nonlinear dynamical systems, Proceed-ings of the American Control Conference 2019-July (2019) 4832–4839. arXiv:1708.06850 , doi:10.23919/acc.2019.8815339 .[44] Y. Lian, C. N. Jones, Learning Feature Maps of the Koopman Operator:A Subspace Viewpoint, in: 2019 IEEE 58th Conference on Decision andControl (CDC), IEEE, 2019, pp. 860–866. doi:10.1109/CDC40024.2019.9029189 .[45] S. Klus, F. N¨uske, S. Peitz, J.-H. Niemann, C. Clementi, C. Sch¨utte,Data-driven approximation of the Koopman generator: Model reduction,system identification, and control, Physica D: Nonlinear Phenomena 406(2020) 132416. doi:10.1016/j.physd.2020.132416 .[46] N. Takeishi, Y. Kawahara, T. Yairi, Learning Koopman invariant sub-spaces for dynamic mode decomposition, Advances in Neural InformationProcessing Systems 2017-Decem (2017) 1131–1141. arXiv:1710.04340 .[47] B. Lusch, J. N. Kutz, S. L. Brunton, Deep learning for universal linearembeddings of nonlinear dynamics, Nature Communications 9 (1). doi:10.1038/s41467-018-07210-0 .[48] S. E. Otto, C. W. Rowley, Linearly recurrent autoencoder networks forlearning dynamics, SIAM Journal on Applied Dynamical Systems 18 (1)(2019) 558–593. doi:10.1137/18M1177846 .3349] J. Morton, F. D. Witherden, M. J. Kochenderfer, Deep variational Koop-man models: Inferring koopman observations for uncertainty-aware dy-namics modeling and control, IJCAI International Joint Conference on Ar-tificial Intelligence 2019-Augus (2019) 3173–3179. doi:10.24963/ijcai.2019/440 .[50] S. Pan, K. Duraisamy, Physics-Informed Probabilistic Learning of Lin-ear Embeddings of Nonlinear Dynamics with Guaranteed Stability, SIAMJournal on Applied Dynamical Systems 19 (1) (2020) 480–509. doi:10.1137/19m1267246 .[51] C. W. Rowley, I. Mezi, S. Bagheri, P. Schlatter, D. S. Henningson, Spectralanalysis of nonlinear flows, Journal of Fluid Mechanics 641 (2009) 115–127. doi:10.1017/S0022112009992059 .[52] T. Gustafsson, Subspace-based system identification: Weighting and pre-filtering of instruments, Automatica 38 (3) (2002) 433–443. doi:10.1016/S0005-1098(01)00235-7 .[53] B. J. Hollingsworth, Stochastic Differential Equations: A Dynamical Sys-tems Approach, Phd thesis, Auburn University (2008).[54] E. M. Bollt, Q. Li, F. Dietrich, I. Kevrekidis, On matching, and evenrectifying, dynamical systems through koopman operator eigenfunctions,SIAM Journal on Applied Dynamical Systems 17 (2) (2017) 1925–1960. doi:10.1137/17M116207X .[55] E. Schr¨oder, Ueber iterirte Functionen, Mathematische Annalen 3 (2)(1870) 296–322. doi:10.1007/BF01443992 .[56] C. C. Cowen, B. D. MacCluer, Schroeder’s equation in several variables,Taiwanese Journal of Mathematics 7 (1) (2003) 129–154. doi:10.11650/twjm/1500407524 .[57] R. A. Bridges, A Solution to Schroeder’s Equation in Several Variables(2011) 1–37 arXiv:1106.3370 .URL http://arxiv.org/abs/1106.3370 [58] T. L. Curtright, C. K. Zachos, Renormalization group functional equa-tions, Physical Review D - Particles, Fields, Gravitation and Cosmology83 (6) (2011) 1–26. doi:10.1103/PhysRevD.83.065019 .[59] Y. Lan, I. Mezi´c, Linearization in the large of nonlinear systems andKoopman operator spectrum, Physica D: Nonlinear Phenomena 242 (1)(2013) 42–53. doi:10.1016/j.physd.2012.08.017 .[60] R. Mohr, I. Mezi´c, Koopman principle eigenfunctions and linearization ofdiffeomorphisms 1 (1) (2016) 1–18. arXiv:1611.01209 .3461] R. Mohr, I. Mezi´c, Construction of eigenfunctions for scalar-type operatorsvia Laplace averages with connections to the Koopman operator (2014)1–25 arXiv:1403.6559 .[62] S. Peitz, S. Klus, Koopman operator-based model reduction for switched-system control of PDEs, Automatica 106 (2019) 184–191. doi:10.1016/j.automatica.2019.05.016 .[63] S. Klus, F. N¨uske, P. Koltai, H. Wu, I. Kevrekidis, C. Sch¨utte, F. No´e,Data-Driven Model Reduction and Transfer Operator Approximation,Journal of Nonlinear Science 28 (3) (2018) 985–1010. doi:10.1007/s00332-017-9437-7 .[64] M. Korda, I. Mezic, Optimal construction of Koopman eigenfunctions forprediction and control, IEEE Transactions on Automatic Control (2020)1–1 doi:10.1109/tac.2020.2978039 .[65] S. Klus, A. Bittracher, I. Schuster, C. Sch¨utte, A kernel-based approachto molecular conformation analysis, Journal of Chemical Physics 149 (24). arXiv:1809.11092 , doi:10.1063/1.5063533 .[66] S. Klus, B. E. Husic, M. Mollenhauer, F. No´e, Kernel methods for de-tecting coherent structures in dynamical data, Chaos 29 (12). doi:10.1063/1.5100267 .[67] E. Kaiser, J. N. Kutz, S. L. Brunton, Discovering Conservation Laws fromData for Control, Proceedings of the IEEE Conference on Decision andControl 2018-Decem (CDC) (2019) 6415–6421. doi:10.1109/CDC.2018.8618963 .[68] S. Klus, I. Schuster, K. Muandet, Eigendecompositions of Transfer Oper-ators in Reproducing Kernel Hilbert Spaces, Journal of Nonlinear Science30 (1) (2020) 283–315. doi:10.1007/s00332-019-09574-z .[69] E. Kaiser, J. N. Kutz, S. L. Brunton, Sparse identification of nonlineardynamics for model predictive control in the low-data limit, Proceedingsof the Royal Society A: Mathematical, Physical and Engineering Sciences474 (2219). doi:10.1098/rspa.2018.0335 .[70] B. O. Koopman, J. v. Neumann, Dynamical Systems of Continuous Spec-tra, Proceedings of the National Academy of Sciences 18 (3) (1932) 255–263. doi:10.1073/pnas.18.3.255 .[71] F. Takens, Reconstruction and Observability, a Survey, IFAC ProceedingsVolumes 31 (17) (1998) 427–429. doi:10.1016/s1474-6670(17)40373-9 .[72] H. Arbabi, M. Korda, I. Mezic, A Data-Driven Koopman Model PredictiveControl Framework for Nonlinear Partial Differential Equations, Proceed-ings of the IEEE Conference on Decision and Control 2018-Decem (2019)6409–6414. doi:10.1109/CDC.2018.8619720 .3573] B. L. Ho, R. E. Kalman, Effective construction of linear state-variablemodels from input/output functions, At-Automatisierungstechnik 14 (1-12) (1966) 545–548. doi:10.1524/auto.1966.14.112.545 .[74] S. J. Qin, W. Lin, L. Ljung, A novel subspace identification approach withenforced causal models.[75] S. J. Qin, W. Lin, L. Ljung, A novel subspace identification approachwith enforced causal models, Automatica 41 (12) (2005) 2043–2053. doi:https://doi.org/10.1016/j.automatica.2005.06.010 .[76] T. Berry, J. Harlim, Variable bandwidth diffusion kernels, Applied andComputational Harmonic Analysis 40 (1) (2016) 68–96. doi:10.1016/j.acha.2015.01.001 .[77] R. R. Coifman, S. Lafon, Diffusion maps, Applied and ComputationalHarmonic Analysis 21 (1) (2006) 5–30. doi:10.1016/j.acha.2006.04.006 .[78] N. Takeishi, Kernel Learning for Data-Driven Spectral Analysis of Koop-man Operators, Proceedings of Machine Learning Research 101 (2019)956–971.[79] Z. Wang, R. M. Jungers, Immersion-based model predictive control of con-strained nonlinear systems: Polyflow approximation arXiv:2011.13255 .[80] D. Bruder, B. Gillespie, C. David Remy, R. Vasudevan, Modeling andControl of Soft Robots Using the Koopman Operator and Model Pre-dictive Control, 2019. arXiv:1902.02827 , doi:10.15607/rss.2019.xv.060 .[81] D. Bruder, X. Fu, R. B. Gillespie, C. D. Remy, R. Vasudevan, Koopman-based Control of a Soft Continuum Manipulator Under Variable LoadingConditions arXiv:2002.01407 .[82] C. Folkestad, D. Pastor, J. W. Burdick, Episodic Koopman Learning ofNonlinear Robot Dynamics with Application to Fast Multirotor Land-ing arXiv:2004.01708 .[83] A. Mauroy, I. Mezi´c, J. Moehlis, Isostables, isochrons, and Koopman spec-trum for the action-angle representation of stable fixed point dynamics,Physica D: Nonlinear Phenomena 261 (2013) 19–30. doi:10.1016/j.physd.2013.06.004 .[84] A. Mauroy, A. Sootla, I. Mezi´c, Koopman Framework for Global StabilityAnalysis, Springer International Publishing, Cham, 2020, pp. 35–58. doi:10.1007/978-3-030-35713-9_2 .3685] K. P. Champion, S. L. Brunton, J. Nathan Kutz, Discovery of nonlinearmultiscale systems: Sampling strategies and embeddings, SIAM Journalon Applied Dynamical Systems 18 (1) (2019) 312–333. doi:10.1137/18M1188227 .[86] S. L. Brunton, B. W. Brunton, J. L. Proctor, J. N. Kutz, Koopman in-variant subspaces and finite linear representations of nonlinear dynamicalsystems for control, PLoS ONE 11 (2). doi:10.1371/journal.pone.0150171 .[87] I. Mezi´c, On applications of the spectral theory of the Koopman oper-ator in dynamical systems and control theory, Proceedings of the IEEEConference on Decision and Control 54rd IEEE (Cdc) (2015) 7034–7041. doi:10.1109/CDC.2015.7403328 .[88] H. K. Khalil, Nonlinear Systems, Macmillan Publishing Company, 1992.[89] A. Mauroy, I. Mezi´c, Global Stability Analysis Using the Eigenfunctions ofthe Koopman Operator, IEEE Transactions on Automatic Control 61 (11)(2016) 3356–3369. doi:10.1109/TAC.2016.2518918 .[90] Elmar Plischke, Transient Effects of Linear Dynamical Systems, Phd the-sis, Universitaet Bremen (2005).[91] I. Mezi´c, Spectrum of the Koopman Operator, Spectral Expansions inFunctional Spaces, and State-Space Geometry, Journal of Nonlinear Sci-ence (2019) 1–40 doi:10.1007/s00332-019-09598-5 .[92] T. Kato, Perturbation theory for linear operators, Springer Berlin Heidel-berg, Berlin, Heidelberg, 1966. doi:10.1007/978-3-662-12678-3 .[93] J. B. Conway, A Course in Functional Analysis, Vol. 96 of Graduate Textsin Mathematics, Springer New York, New York, NY, 2007. doi:10.1007/978-1-4757-4383-8 .[94] J. Mohammadpour, C. W. Scherer (Eds.), Control of Linear ParameterVarying Systems with Applications, Springer US, Boston, MA, 2012. doi:10.1007/978-1-4614-1833-7 .[95] D. Goswami, D. A. Paley, Global bilinearization and controllability ofcontrol-affine nonlinear systems: A Koopman spectral approach, 2017IEEE 56th Annual Conference on Decision and Control, CDC 2017 2018-Janua (2018) 6107–6112. doi:10.1109/CDC.2017.8264582 .[96] D. Elliott, Bilinear Control Systems, Springer Netherlands, Dordrecht,2009. doi:10.1023/b101451 .[97] M. Korda, I. Mezi´c, Linear predictors for nonlinear dynamical systems:Koopman operator meets model predictive control, Automatica 93 (2018)149–160. doi:10.1016/j.automatica.2018.03.046 .3798] A. K. Das, B. Huang, U. Vaidya, Data-Driven Optimal Control UsingTransfer Operators, Proceedings of the IEEE Conference on Decision andControl 2018-Decem (Cdc) (2019) 3223–3228. doi:10.1109/CDC.2018.8619057 .[99] X. Ma, B. Huang, U. Vaidya, Optimal quadratic regulation of nonlinearsystem using koopman operator, Proceedings of the American ControlConference 2019-July (2) (2019) 4911–4916. doi:10.23919/acc.2019.8814903 .[100] B. Huang, X. Ma, U. Vaidya, Feedback Stabilization Using KoopmanOperator, Proceedings of the IEEE Conference on Decision and Control2018-Decem (1) (2019) 6434–6439. doi:10.1109/CDC.2018.8619727 .[101] A. Narasingam, J. S.-I. Kwon, Data-driven feedback stabilization ofnonlinear systems: Koopman-based model predictive control (2020) 1–11 arXiv:2005.09741 .[102] A. Surana, A. Banaszuk, Linear observer synthesis for nonlinear systemsusing Koopman Operator framework, IFAC-PapersOnLine 49 (18) (2016)716–723. doi:10.1016/j.ifacol.2016.10.250 .[103] A. Surana, Koopman operator based observer synthesis for control-affinenonlinear systems, 2016 IEEE 55th Conference on Decision and Control,CDC 2016 (Cdc) (2016) 6492–6499. doi:10.1109/CDC.2016.7799268 .[104] M. Netto, L. Mili, Robust Koopman Operator-based Kalman Filter forPower Systems Dynamic State Estimation, IEEE Power and Energy Soci-ety General Meeting 2018-Augus (2018) 1–5. doi:10.1109/PESGM.2018.8586440 .[105] H.-G. Lee, S. Marcus, Immersion and immersion by nonsingular feedbackof a discrete-time nonlinear system into a linear system, IEEE Transac-tions on Automatic Control 33 (5) (1988) 479–483. doi:10.1109/9.1233 .[106] M. V. Kothare, V. Nevistic, M. Morari, Robust constrained model predic-tive control for nonlinear systems: a comparative study, in: Proceedingsof the IEEE Conference on Decision and Control, Vol. 3, 1995, pp. 2884–2885. doi:10.1109/cdc.1995.478579 .[107] R. A. Freeman, P. V. Kokotovic, Optimal nonlinear controllers for feed-back linearizable systems, Proceedings of the American Control Confer-ence 4 (3) (1995) 2722–2726. doi:10.1109/acc.1995.532343 .[108] R. A. Freeman, P. Kokotovi´c, Robust Nonlinear Control De-sign, Birkh¨auser Boston, Boston, MA, 1996. doi:10.1007/978-0-8176-4759-9 . 38109] M. Krstic, D. Fontaine, P. Kokotovic, J. Paduano, Useful nonlinearitiesand global stabilization of bifurcations in a model of jet engine surge andstall, IEEE Transactions on Automatic Control 43 (12) (1998) 1739–1745. doi:10.1109/9.736075 .[110] E. Kelasidi, P. Liljeback, K. Y. Pettersen, J. T. Gravdahl, Innovation inUnderwater Robots: Biologically Inspired Swimming Snake Robots, IEEERobotics Automation Magazine 23 (1) (2016) 44–62. doi:10.1109/MRA.2015.2506121 .[111] J. Zhu, C. White, D. K. Wainwright, V. Di Santo, G. V. Lauder, H. Bart-Smith, Tuna robotics: A high-frequency experimental platform exploringthe performance space of swimming fishes, Science Robotics 4 (34). doi:10.1126/scirobotics.aax4615doi:10.1126/scirobotics.aax4615