Dynamic Mode Decomposition with Control Liouville Operators
aa r X i v : . [ m a t h . O C ] J a n Dynamic Mode Decomposition withControl Liouville Operators ⋆ Joel A. Rosenfeld ∗ Rushikesh Kamalapurkar ∗∗∗
Department of Mathematics and Statistics, University of SouthFlorida, Tampa, FL 33620 USA (e-mail: [email protected]). ∗∗ School of Mechanical and Aerospace Engineering, Oklahoma StateUniversity, Stillwater, OK 74078 USA (e-mail:[email protected])
Abstract:
This manuscript provides a theoretical foundation for the Dynamic Mode Decom-position (DMD) of control affine dynamical systems through vector valued reproducing kernelHilbert spaces (RKHSs). Specifically, control Liouville operators and control occupation kernelsare introduced to separate the drift dynamics from the control effectiveness components. Given aknown feedback controller that is represented through a multiplication operator, a DMD analysismay be performed on the composition of these operators to make predictions concerning thesystem controlled by the feedback controller.
Keywords: system identification, spectral analysis, operators, model approximation, controlsystem analysis1. INTRODUCTIONDynamic Mode Decomposition (DMD) is a method ofsystem identification that casts unknown discrete or con-tinuous time dynamics over finite dimensions into a linearoperator over an infinite dimensional space (cf. [1]). DMDcollects trajectory data from observations, or snapshots ,of a dynamical system, and the method constructs a finiterank representation of an operator that describes the evo-lution of the state. In the discrete time case, this linearoperator is a composition operator called the Koopmanoperator [2], popularized by [3]. The finite rank representa-tion of the dynamics is then diagonalized, and the resultanteigenfunction and eigenvalues are used to provide a repre-sentation of the identity function, which in turn providesthe dynamic modes via vector valued coefficients attachedto the eigenfunctions. Thereafter, a state trajectory can berepresented as a sum of exponential functions multipliedby the dynamic modes (cf. [4,1,5]).The primary application area of Koopman spectral anal-ysis of dynamical systems has been fluid dynamics, whereDMD is compared with proper orthogonal decompositions(POD) for nonlinear fluid equations (cf. [6]). DMD hasalso been employed in the study of stability properties ofdynamical systems [7,8], neuroscience [9], financial trading[10], feedback stabilization [11], optimal control [12], mod-eling of dynamical systems [13–15], and model-predictivecontrol [16]. For a generalized treatment of DMD as aMarkov model, see [17]. ⋆ This research was supported in part by the Air Force ResearchLaboratory (FA8651-19-2-0009 and FA9550-20-1-0127), and the Na-tional Science Foundation (NSF) under award 2027976. Any opin-ions, findings, or conclusions in this manuscript are those of theauthor(s) and do not necessarily reflect the views of the sponsoringagencies.
In each of the above examples, continuous time dynamicsare studied using DMD via the discrete time dynamicsdetermined through a fixed time-step. Recently, [18] im-ported the notion of occupation measures from the Ba-nach spaces of continuous function to that of reproduc-ing kernel Hilbert spaces (RKHSs). It was subsequentlydemonstrated in [5] that when combined with the Liouvilleoperator (also known as the Koopman generator), the so-called occupation kernels provide a method for performinga DMD analysis on the continuous time dynamic directly.The paradigm shift afforded by occupation kernels arisesthrough the consideration of the state trajectory as thefundamental unit of data, as was done in [18]. Compu-tationally, the DMD analysis of the Liouville operator isremarkably unchanged to that of the Koopman operator.The only adjustment is in the computation of the Grammatrix of occupation kernels, which requires the computa-tion of double integrals for each entry as the inner productof the occupation kernels with each other.This manuscript builds on the concept of Liouville opera-tors and occupation kernels through the incorporation ofcontrol-affine dynamics, where the trajectory data givenby the system is accompanied by a collection of controlsignals. In this context, occupation kernels are augmentedby the control signals resulting in what are called con-trol occupation kernels, and the Liouville operator nowincludes the control effectiveness matrix valued function.The augmentations are possible through vector valuedRKHSs, which have been extensively studied in [19] and[20].. VECTOR VALUED REPRODUCING KERNELHILBERT SPACESThis section reviews important properties of vector valuedRKHSs, and relies heavily on the discussion given in [20].
Definition 1.
Let Y be a Hilbert space, and let H be aHilbert space of functions from a set X to Y . The Hilbertspace H is a (vector valued) reproducing kernel Hilbertspace (RKHS) if for every v ∈ Y and x ∈ X , thefunctional f
7→ h f ( x ) , v i Y is bounded.A vector valued RKHS is a direct generalization of a“scalar” valued RKHS, where for a fixed v ∈ Y , thecollection of functions { g ( x ) = h f ( x ) , v i Y : f ∈ H } formsa RKHS of scalar valued functions. Moreover, for a basisof Y , { v , v , . . . } ⊂ Y , the values of functions in H maybe recovered via their individual scalar valued projections.Through the Riesz representation theorem, for each x ∈ X and v ∈ Y , there is a function K x ,v ∈ H such that h f, K x ,v i H = h f ( x ) , v i Y for all f ∈ H . It can be seenthat the mapping v K x ,v is linear over Y . Hence, K x v := K x ,v where K x : Y → H is a linear operator.The function K x is called the kernel centered at x andis an operator from Y to H .The kernel operator has a particular representation as asum of rank one tensors of evaluations of an orthonormalbasis for H , { e m } ∞ m =1 , where K x ,v ( x ) = ∞ X m =1 e m ( x ) h K x v, e m i H = ∞ X m =1 e m ( x ) h e m ( x ) , v i Y = ∞ X m =1 e m ( x ) ⊗ e m ( x ) ! v. Hence, the kernel operator associated with H , K : X × X → L ( Y , Y ), is defined as K ( x, x ) := P ∞ m =1 e m ( x ) ⊗ e m ( x ), where e m is an orthonormal basis for H . In theparticular case that Y = R n , K ( x, x ) is a real valued n × n matrix for fixed x, x ∈ X. This suggests severalexamples of vector valued kernels, and indeed, for anypositive definite matrix, A , and scalar valued kernel ˜ K , thekernel given as K ( x, x ) = A ˜ K ( x, x ) is a kernel operator.Just as will scalar valued kernels, the span of the set E := { K x,v : v ∈ Y and x ∈ X } is dense in H . Proposition 1.
Let E be given as above, then ( E ⊥ ) ⊥ =span E = H . Proof.
Suppose that h ∈ E ⊥ , then given a fixed x ∈ X , h h, K x,v i H = h h ( x ) , v i Y = 0 for all v ∈ Y . Hence, h ( x ) = 0 ∈ Y . Since x was arbitrarily selected, h ≡ ∈ H .Thus, E ⊥ = { } and ( E ⊥ ) ⊥ = H . ✷ Hence, given ǫ > h ∈ H , there is a linearcombination of vector valued kernels that approximate h to within ǫ. Remark 1.
The implementation of vector valued RKHSsleveraged in this manuscript uses the vector space R × (1+ m ) of row vectors, which arises from the gradient of observ-ables. Thus, the linear operation of K x on v ∈ R × (1+ m ) will be expressed as K x ,v = vK x henceforth. 3. PROBLEM STATEMENTThe objective of this paper is to provide an operator the-oretic approach for the analysis of a closed loop nonlinearcontrol affine systems,˙ x = f ( x ) + g ( x ) µ ( x ) , (1)with the state feedback controller µ : R n → R m byusing data provided by observed absolutely continuouscontrolled trajectories, { γ u i : [0 , T ] → R n } Mi =1 , and corre-sponding bounded measurable (with respect to Lebesguemeasure) control inputs, { u i : [0 , T ] → R m } Mi =1 , satisfying˙ γ u i ( t ) = f ( γ u i ( t )) + g ( γ u i ( t )) u i ( t )in the Carath´eodory sense. The observed control trajec-tories and control inputs will allow for the constructionof a finite rank representation for a generalized Liouvilleoperator, called the control Liouville operator, that issimilar to that presented in [5]. Note that the terminaltime, T , may vary between trajectories with little changeto the implemenation, but it is kept constant for the sakeof clarity of presentation.The resolution of the this problem provided by thismanuscript will yield a DMD analysis of the closed loopsystem via a composition of operators that separate thedynamics learning through observations and the knownfeedback controller.Any Euler-Lagrange system with an invertible inertiamatrix can be expressed in the control-affine form. TheEuler-Lagrange equations are used to describe a large classof physical systems (cf. [21]), and as such, various methodsfor control and identification of nonlinear systems in theEuler-Lagrange form have been studied in detail over theyears (see, e.g., [22–24]). Since most physical systems ofpractical importance; such as robot manipulators [25] andground, air, and maritime vehicles and vessels [26]; admitinvertible inertia matrices over large operating regions;control-affine models encompass a large class of physicalsystems.4. CONTROL LIOUVILLE OPERATORS ANDCONTROL OCCUPATION KERNEL OPERATORSLet f : R n → R n and g : R n → R n × m be the drift dy-namics and the control effectiveness matrix, respectively,for a control affine dynamical system. Let ˜ H be a scalarvalued RKHS of continuously differentiable functions over R n and let H be a R m +1 (row) vector valued RKHSof continuous functions over R n . Let K : R n × R n →L ( R × ( m +1) , R × ( m +1) ) and ˜ K : R n × R n → R be thekernel functions of H and ˜ H , respectively. Definition 2.
Let the set D ( A f,g ) := { h ∈ ˜ H : ∇ h ( x ) ( f ( x ) g ( x )) ∈ H } be the domain of the operator, A f,g : D ( A f,g ) → H , givenas A f,g h ( x ) = ∇ h ( x ) ( f ( x ) g ( x )) . The operator A f,g is called the control Liouville operatorcorresponding to f and g over H .Control Liouville operators are a direct generalizationof more traditional Liouville operators, where the driftdynamics and control effectiveness components of theynamics are separated on the operator theoretic level. Itcan be seen here that vector valued RKHSs arise naturallyin the context, where the gradient of h ∈ D ( A f,g ) isa row vector, and through a dot product with f andmultiplication by the matrix g , the result of the operationof A f,g on h is an m + 1 dimension row vector.For the purposes of this manuscript, control Liouvilleoperators are assumed to be densely defined over theirrespective Hilbert spaces, and it follows from the methodsof [18] that control Liouville operators are closed. Hence,the adjoint of control Liouville operators are also denselydefined [27].Given a bounded measurable control input, u : [0 , T ] → R m , and an absolutely continuous trajectory, γ : [0 , T ] → R n , the functional, T : H → R , given as T p = Z T p ( γ ( t )) (cid:18) u ( t ) (cid:19) dt is bounded. Hence, there is a function Γ γ,u ∈ H such that T p = h p, Γ γ,u i H for all p ∈ H . Definition 3.
For a bounded measurable control input u and absolutely continuous trajectory γ , the functionΓ γ,u ∈ H is called the control occupation kernel corre-sponding to u and γ in H . Proposition 2.
The occupation kernel corresponding to u and γ (given above) may be expressed asΓ γ,u ( x ) = Z T (cid:0) u ( t ) T (cid:1) K ( x, γ ( t )) dt, (2)and the norm of Γ γ,u is given as k Γ γ,u k = Z T Z T (cid:0) u ( t ) T (cid:1) K ( γ ( τ ) , γ ( t )) (cid:18) u ( τ ) (cid:19) dtdτ. Proof.
Consider for x ∈ X and v ∈ R × ( m +1) , h Γ γ,u ( x ) , v i R × ( m +1) = h Γ γ,u , vK x i H = h vK x , Γ γ,u i H = Z T vK ( x, γ ( t )) (cid:18) u ( t ) (cid:19) dt. (3)As (4) holds for all v ∈ R × ( m +1) , (2) follows. The normof Γ γ,u follows from k Γ γ,u k = h Γ γ,u , Γ γ,u i H , and thedefining properties of Γ γ,u . ✷ As with Liouville operators over RKHSs corresponding tononlinear dynamical systems, there is a direction connec-tion between the adjoints of control Liouville operatorsand control occupation kernels that correspond to admis-sible control signals, u , and their corresponding controlledtrajectories, γ u , that satisfy (1). Proposition 3.
Suppose that f and g correspond to acontrol Liouville operator, A f,g : D ( A f,g ) → H , and let u be an admissible control signal for the control affinedynamical system in (1) with a corresponding controlledtrajectory, γ u . Then, Γ γ u ,u ∈ D ( A ∗ f,g ) and A ∗ f,g Γ γ u ,u =˜ K ( · , γ u ( T )) − ˜ K ( · , γ u (0)). Proof.
To demonstrate that Γ γ u ,u is in D ( A ∗ f,g ) it must beshown that the mapping h
7→ h A f,g h, Γ γ u ,u i H is a boundedfunctional. Note that h A f,g h, Γ γ u ,u i H = Z T ∇ h ( γ u ( t )) ( f ( γ u ( t )) g ( γ u ( t ))) (cid:18) u ( t ) (cid:19) dt = Z T ddt h ( γ u ( t )) dt = h ( γ u ( T )) − h ( γ u (0))= h h, ˜ K ( · , γ u ( T )) − ˜ K ( · , γ u (0)) i ˜ H . Hence, the functional h
7→ h A f,g h, Γ γ u ,u i H is boundedwith norm not exceeding k ˜ K ( · , γ u ( T )) − ˜ K ( · , γ u (0)) k ˜ H .Moreover, the above equations yield A ∗ f,g Γ γ u ,u = ˜ K ( · , γ u ( T )) − ˜ K ( · , γ u (0)) . ✷
5. MULTIPLICATION OPERATORS FROM VECTORVALUED TO SCALAR VALUED REPRODUCINGKERNEL HILBERT SPACESThe inclusion of a second operator in addition to thecontrol Liouville operator sets the theoretical foundationsof DMD for control-affine systems apart from the un-controlled case in [5]. As will be seen in subsequent sec-tions, after the determination of a representation of A f,g ,the inclusion of a state feedback controller is necessaryfor a DMD procedure. The state feedback controller isimplemented via a multiplication operator. This sectionestablished some results on multiplication operators fromvector valued RKHSs to scalar valued RKHSs. Many ofthese theorems have been established for scalar valuedRKHSs (cf. [28–30]), and the results follows from similarmethods.Let µ : X → Y be a vector valued function, and define themultiplication operator, M µ : D ( M µ ) → ˜ H , with symbol µ as M µ h = h h, µ i Y , where D ( M µ ) := { h ∈ H : h h, µ i Y ∈ ˜ H } is a collection of (row) vector valued functions in H and ˜ H is a scalar valued RKHS. Proposition 4.
The function ˜ K ( · , x ) is in the domain of M ∗ µ , D ( M ∗ µ ). Moreover, M ∗ µ ˜ K ( · , x ) = K x,µ ( x ) Proof.
Let h ∈ D ( M µ ), then h M µ h, ˜ K ( · , x ) i ˜ H = h h ( x ) , µ ( x ) i Y = h h, K x,µ ( x ) i H . Hence, the mapping h
7→ h M µ h, ˜ K ( · , x ) i ˜ H is a boundedfunctional with norm bounded by k K x,µ ( x ) k H , and ˜ K ( · , x )is in the domain. Moreover, this demonstrates that h M µ h, ˜ K ( · , x ) i ˜ H = h h, K x,µ ( x ) i H , which establishes the adjoint formula. ✷ Thus, the adjoint of a multiplication operator connectsoccupation kernels with control occupation kernels.
Proposition 5.
Multiplication operators are closed opera-tors.
Proof.
Suppose that { h n } ∈ D ( M µ ), h n → h ∈ H , and M µ h n → W ∈ H. To show that M µ is a closed operator, itmust be shown that W ( x ) = h h ( x ) , µ ( x ) i Y for all x ∈ X ,and thus h ∈ D ( M µ ) by definition and M µ h = W . Let v ∈ Y and x ∈ X , then ( x ) = lim n →∞ h M µ h n , ˜ K ( · , x ) i H = lim n →∞ h h n , K x,µ ( x ) i H = h h, K x,µ ( x ) i H = h h ( x ) , µ ( x ) i Y , where the convergence follows as norm convergence impliesweak convergence. ✷ Hence, the adjoint of a multiplication operator is denselydefined by [27, Proposition 5.1.7].Let for a trajectory γ : [0 , T ] → X , let Γ γ be theoccupation kernel corresponding to γ within ˜ H (cf. [18]). Proposition 6.
The function Γ γ is in the domain of M µ , D ( M µ ). Moreover, M ∗ µ Γ γ = Γ γ,µ ( γ ( · )) . Proof.
This follows from a similar method as Proposition4. ✷
6. DYNAMIC MODES FOR CLOSED LOOPCONTROL SYSTEMSDynamic mode decomposition aims to determine eigen-functions and eigenvalues for the Liouville or Koopmanoperators that correspond to a particular dynamical sys-tem. For control Liouville operators, the dynamics given interms of f and g correspond to an operator whose range isa vector valued RKHS and domain a scalar valued RKHS.As the domain and range are different vector spaces, aneigendecomposition of the control Liouville operator doesnot make sense. Indeed, for the dynamical system itself, acontrol input is required in addition to f and g to simulatethe system.After a finite rank representation of a control Liouvilleoperator is determined, an additional operator is requiredbefore a dynamic mode decomposition may be determined.In particular, given a feedback controller, µ : R n → R m ,the corresponding multiplication operator M µ : D ( M µ ) → ˜ H is given as M µ h := h (cid:18) µ (cid:19) and D ( M µ ) := { h ∈ H : h (cid:0) µ T (cid:1) T ∈ ˜ H } . What differs from the previous con-trollers that were implemented via the control occupationkernels is that µ is a function of the state variable, x ∈ R n ,rather than a function of time. It will be assumed that theimage of A f,g falls within the domain of M µ .Note that M µ A f,g : D ( A f,g ) → ˜ H , that is the compositionof M µ with A f,g is a mapping with domain and rangewithin ˜ H . If φ is an eigenfunction of M µ A f,g with eigen-value λ , then if γ µ is a controlled trajectory arising from(1) it follows that˙ φ ( γ µ ( t )) = ∇ φ ( γ µ ( t ))( f ( γ µ ( t )) + g ( γ µ ( t )) µ ( γ µ ( t )))= M µ A f,g φ ( γ µ ( t )) = λφ ( γ µ ( t )) . Hence, φ ( γ µ ( t )) = e λt φ ( γ µ (0)) . Thus, the identity function, g id : R n → R n given as g id ( x ) := x , may be decomposed using the eigenbasis for M µ A f,g , denoted as φ i with eigenvalue λ i , as g id ( x ) = P ∞ i =1 ξ i φ i ( x ), where ξ i ∈ R n are the dynamic modes ofthe closed loop system. Moreover, it follows that γ µ ( t ) = g id ( γ µ ( t )) = ∞ X i =1 ξ i φ i ( γ µ (0)) e λ i t . The above construction makes several assumptions aboutthe composition of M µ with A f,g that are common inthe study of DMD (cf. [5,4]). It assumes that there isa complete eigenbasis of M µ A f,g that may be accessedthrough finite rank representations of M µ A f,g . Moreover,it assumes that for each i = 1 , . . . , n that the mapping x x i is within ˜ H or at least it may be approximated byfunctions in ˜ H . The validity of each of these assumptiondepends on both the dynamics and the respective RKHSs,and may not always hold. Thus, DMD gives a data-drivenheuristic and operator motivated methods for the analysisof a dynamical system.7. FINITE RANK REPRESENTATIONSAs M µ and A f,g are modally unbounded operators, theuse of data to give empirical representations of theseoperators are not expected to give approximations M µ and A f,g under the operator norm. However, the onlydata available concerning these operators is given throughthe trajectories themselves, thus DMD produces a finiterank representation under a data driven heuristic. Thekernel based extended DMD procedure was developedin [4], where kernel functions were seen as a means ofcomputing inner products of feature maps in ℓ ( N ). Theinfinite dimensional representation selected in [4] was withrespect to the feature basis itself. As the feature space isinfinite in size the final matrix for the Koopman operator’srepresentation is never fully computed, and the workperfomed is executed on a finite sized proxy. The use of aproxy representation is also leveraged in [5] for continuoustime DMD.However, as dynamic modes may only be extracted fromthe product of M µ with A f,g , it is necessary to givean explicit finite rank representation of A f,g and M µ todetermine the dynamic modes of the resultant system. Itshould be noted that in the above development the featurespace has not been invoked. Indeed, all of the developmentthus far occurs at the level of RKHSs.The following subsections give two approaches to generatefinite rank representations using basis functions within theRKHS, ˜ H . The results will yield finite rank representationsof M µ A f,g as a product of four matrices. The SVD fora product of matrices may be computed using iterativemethods such as [31], to complete the DMD computations.In what follows, finite collections of linearly independentvectors, α and β , are selected for establishing finite rankmatrices over a domain, span α , and range, span β . For anoperator T , the notation [ T ] βα is to be read as the matrixrepresentation of P β T with respect to the domain, span α ,and range span β . Here P β is the projection onto span β . As kernels are dense in both the vector valued and scalarvalued cases, finite rank representation of each operator ispossible with respect to the kernels themselves. As onlya finite number of kernels may be leveraged in any com-putational procedure, there is an element of imprecisionin the representation of the functions in the domain andrange of these operators. However, as kernels are dense inheir respective RKHSs, the use of an increasing numberof kernels reduces the overall error of the estimations.The objective of this section is to select a collection ofcenters, { c i } ˜ Ni =1 ⊂ Ω, from a compact set, Ω ⊂ R n ,containing the sampled trajectories of the system, andto use the collection of corresponding kernel functions, α = { ˜ K ( · , c i ) } ˜ Ni =1 ⊂ ˜ H , as a basis for the matrix repre-sentation of M µ A f,g , or indirectly of A ∗ f,g M ∗ µ . The controloccupation kernels corresponding to the sampled trajecto-ries will be denoted as β = { Γ γ ui ,u i } Mi =1 .To form the various required matrices, projections will becomputed with respect to the individual bases. If ˜ h ∈ ˜ H ,then the coefficients mapping ˜ h to its projection ontospan { ˜ K ( · , c i ) } ˜ Ni =1 are given as the solution to the followingsystem: ˜ K ( c , c ) · · · ˜ K ( c , c ˜ N )... . . . ...˜ K ( c ˜ N , c ) · · · ˜ K ( c ˜ N , c ˜ N ) ˜ w ...˜ w ˜ N = ˜ h ( c )...˜ h ( c ˜ N ) , which is solvable since G = ( ˜ K ( c i , c j )) ˜ Ni,j =1 , (i.e., theGram matrix) is positive definite for many choices ofkernel functions, and is always positive semidefinite. Theprojection onto the control occupation kernels may becomputed similarly when h ∈ H by solving, h Γ γ u ,u , Γ γ u ,u i H ··· h Γ γ u ,u , Γ γ uM ,u M i H ... . . . ... h Γ γ uM ,u M , Γ γ u ,u i H ··· h Γ γ uM ,u M , Γ γ uM ,u M i H w ... w ˜ N = Z T h ( γ u ( t )) (cid:18) u ( t ) (cid:19) dt ... Z T h ( γ u M ( t )) (cid:18) u M ( t ) (cid:19) dt , where each element of G = ( h Γ γ ui ,u i , Γ γ uj ,u j i H ) Mi,j =1 maybe computed through a double integral.Each kernel function, ˜ K ( · , c i ), in ˜ H satisfies M ∗ µ ˜ K ( · , c i ) = K c i ,µ ( c i ) . As only the action of the control Liouville op-erator on the occupation kernels is known, computationof A ∗ f,g M ∗ µ ˜ K ( · , c i ) requires the result to be projected ontothe control occupation kernel basis. Hence, the finite rankrepresentation of M µ with domain basis α and range basis β is given as, (cid:2) M ∗ µ (cid:3) βα = G − (cid:18)Z T K c i ,µ ( c i ) ( γ u j ( t )) (cid:18) u j ( t ) (cid:19) dt (cid:19) i = ˜ N,j = Mi =1 ,j =1 . Similarly, the operation of A ∗ f,g on the basis β will beprojected onto α. The resulting finite rank representationof A ∗ f,g is then given as (cid:2) A ∗ f,g (cid:3) αβ = G − (cid:0) ˜ K ( c j , γ u i ( T )) − ˜ K ( c j , γ u i (0)) (cid:1) i = M,j = ˜ Ni =1 ,j =1 . Hence, the matrix representation of the product of M µ and A f,g is given as[ M µ A f,g ] αα ≈ ([ A ∗ f,g ] αβ [ M ∗ µ ] βα ) T and is a product of four explicit matrices. The disadvantage of the kernel basis is that the centersare selected independently of the data. A large number ofcenters may be selected, however, the rank of the resultantrepresentation is bottlenecked by the control occupationkernels representing the trajectories and controllers. Adata-centric selection of basis functions can be made bycollecting the (scalar valued) occupation kernels represent-ing the trajectories in ˜ H , δ := { Γ γ ui } Mi =1 ⊂ ˜ H .To compute the projection of a function ˜ h ∈ ˜ H withrespect to the δ basis, the following system may be solved h Γ γ u , Γ γ u i ˜ H · · · h Γ γ u , Γ γ uM i ˜ H ... . . . ... h Γ γ uM , Γ γ u i ˜ H · · · h Γ γ uM , Γ γ uM i ˜ H ˜ w ...˜ w ˜ N = Z T ˜ h ( γ u ( t )) dt ... Z T ˜ h ( γ u M ( t )) dt , where the Gram matrix for the δ basis is given as ˜ G =( h Γ γ ui , Γ γ uj i ˜ H ) Mi,j =1 . Further leveraging the established mappings for the mul-tiplication operator, the action of M ∗ µ on the δ basis canbe expressed in terms of the β basis as (cid:2) M ∗ µ (cid:3) βδ = G − (cid:18)Z T Γ γ ui ,µ ( γ ui ( · )) ( γ u j ( t )) (cid:18) u j ( t ) (cid:19) dt (cid:19) Mi,j =1 . The mapping of A f,g from the β basis to the projectiononto the δ basis is given as (cid:2) A ∗ f,g (cid:3) δβ = ˜ G − (cid:18)Z T K ( γ u j ( t ) ,γ u i ( T )) − K ( γ u j ( t ) ,γ u i (0)) dt (cid:19) Mi,j =1 . In this case, the finite rank representation of M µ A f,g isgiven as [ M µ A f,g ] δδ ≈ ([ A ∗ f,g ] δβ [ M ∗ µ ] βδ ) T and has the advantage of avoiding the user selection ofcenters for kernel functions as in Section 7.1.8. DISCUSSIONThe accuracy of the estimation of the action of M µ A f,g on the selected basis functions, whether the kernel basis,the occupation kernel basis, or another collection of basisfunctions, depends directly on the expressiveness of thecollection control occupation kernels corresponding to thecontrolled trajectories. It can be seen through the invo-cation of G − in the finite rank representations of theprevious section that the rank of the representation isbottlenecked by the control occupation kernels. The rankof G and expressiveness of the collection of kernels canbe increased through the segmentation of the trajectories.The condition number of G depends not only on thetrajectories but also of the selected kernel functions. Forexample, the Gaussian Radial Basis Functions (RBFs),given as ˜ K ( x, y ) = exp( − µ k x − y k ) for µ >
0, are moreoorly conditioned for large µ than for µ small. However,large µ values correspond to faster convergence of inter-polation problems within the native space of the kernel(cf. [32]). Data-richness conditions similar to the persis-tence of excitation (PE) condition in adaptive control thatrelate the trajectories and the kernels can potentially beformulated to ensure a well-conditioned G , however, suchformulation is out of the scope of this paper.The finite rank representation utilizes trajectories of asystem, observed under the controllers u j , to predict itsbehavior in response to any arbitrary feedback controller µ . Since the control Liouville operator is described onlythrough its action on the control occupation kernels,implicit in the computation is a projection of the feedback µ ( x ) onto the span of the control occupation kernels. As aresult, knowledge of the behavior of the system in responseto different control inputs near a state is required toresolve the action of the feedback controller at that state.Formulation of explicit excitation conditions that relate µ and the control occupation kernels is out of the scope ofthis paper. REFERENCES[1] J. N. Kutz, S. L. Brunton, B. W. Brunton, and J. L.Proctor, Dynamic mode decomposition: data-drivenmodeling of complex systems . Philadelphia, PA, USA:SIAM, 2016.[2] B. O. Koopman, “Hamiltonian systems and trans-formation in Hilbert space,”
Proc. Natl. Acad. Sci.U.S.A. , vol. 17, no. 5, p. 315, 1931.[3] I. Mezi´c, “Spectral properties of dynamical sys-tems, model reduction and decompositions,”
Nonlin-ear Dyn. , vol. 41, no. 1, pp. 309–325, 2005.[4] M. O. Williams, C. W. Rowley, and I. G. Kevrekidis,“A kernel-based method for data-driven Koopmanspectral analysis,”
J. Comput. Dyn. , vol. 2, no. 2, pp.247–265, 2015.[5] J. A. Rosenfeld, R. Kamalapurkar, L. Gruss, andT. T. Johnson, “Dynamic mode decomposition forcontinuous time systems with the liouville operator,” arXiv preprint arXiv:1910.03977 , 2019.[6] I. Mezi´c, “Analysis of fluid flows via spectral prop-erties of the koopman operator,”
Annu. Rev. FluidMech. , vol. 45, pp. 357–378, 2013.[7] U. Vaidya, P. G. Mehta, and U. V. Shanbhag, “Non-linear stabilization via control Lyapunov measure,”
IEEE Trans. Autom. Control , vol. 55, no. 6, pp. 1314–1328, Jun. 2010.[8] A. Mauroy and I. Mezi´c, “Global stability analysisusing the eigenfunctions of the koopman operator,”
IEEE Trans. Autom. Control , vol. 61, no. 11, pp.3356–3369, 2016.[9] B. W. Brunton, L. A. Johnson, J. G. Ojemann, andJ. N. Kutz, “Extracting spatial–temporal coherentpatterns in large-scale neural recordings using dy-namic mode decomposition,”
J. Neurosci. Methods ,vol. 258, pp. 1–15, 2016.[10] J. Mann and J. N. Kutz, “Dynamic mode decomposi-tion for financial trading strategies,”
Quant. Finance ,vol. 16, no. 11, pp. 1643–1655, 2016.[11] B. Huang, X. Ma, and U. Vaidya, “Feedback stabiliza-tion using Koopman operator,” in
Proc. IEEE Conf.Decision and Control , Dec. 2018, pp. 6434–6439. [12] A. Sootla, A. Mauroy, and D. Ernst, “Optimal controlformulation of pulse-based control using Koopmanoperator,”
Automatica , vol. 91, pp. 217–224, 2018.[13] J. L. Proctor, S. L. Brunton, and J. N. Kutz, “Dy-namic mode decomposition with control,”
SIAM J.Appl. Dyn. Syst. , vol. 15, no. 1, pp. 142–161, 2016.[14] M. Quade, M. Abel, J. Nathan Kutz, and S. L.Brunton, “Sparse identification of nonlinear dynamicsfor rapid model recovery,”
Chaos , vol. 28, no. 6, 2018.[15] S. Sinha, B. Huang, and U. Vaidya, “On robustcomputation of koopman operator and prediction inrandom dynamical systems,”
J. Nonlinear Sci. , Nov.2019.[16] H. Arbabi, M. Korda, and I. Mezi´c, “A data-drivenKoopman model predictive control framework for non-linear partial differential equations,” in
Proc. IEEEConf. Decis. Control , Dec 2018, pp. 6409–6414.[17] B. Jayaraman, C. Lu, J. Whitman, and G. Chowd-hary, “Sparse feature map-based Markov models fornonlinear fluid flows,”
Comput. Fluids , vol. 191, p.104252, 2019.[18] J. A. Rosenfeld, B. Russo, R. Kamalapurkar, andT. T. Johnson, “The occupation kernel methodfor nonlinear system identification,” arXiv preprintarXiv:1909.11792 , 2019.[19] C. Carmeli, E. De Vito, and A. Toigo, “Vector valuedreproducing kernel Hilbert spaces of integrable func-tions and Mercer theorem,”
Anal. Appl. , vol. 4, no. 04,pp. 377–408, 2006.[20] C. Carmeli, E. De Vito, A. Toigo, and V. Umanit´a.,“Vector valued reproducing kernel Hilbert spaces anduniversality,”
Anal. Appl. , vol. 08, no. 01, pp. 19–61,2010.[21] H. Goldstein, C. Poole, and J. Safko, “Classical me-chanics,” 2002.[22] R. Ortega, A. Lor´ıa, P. J. Nicklasson, and H. J. Sira-Ramirez,
Passivity-based control of Euler-Lagrangesystems: mechanical, electrical and electromechanicalapplications . Springer, 1998.[23] F. Morabito, A. R. Teel, and L. Zaccarian, “Nonlin-ear antiwindup applied to Euler-Lagrange systems,”
IEEE Trans. Robot. Autom. , vol. 20, no. 3, pp. 526–537, 2004.[24] Z. Feng, G. Hu, W. Ren, W. E. Dixon, and J. Mei,“Distributed coordination of multiple unknown Euler-Lagrange systems,”
IEEE Trans. Control Netw. Syst. ,vol. 5, no. 1, pp. 55–66, Jun. 2018.[25] A. Behal, W. E. Dixon, B. Xian, and D. M. Dawson,
Lyapunov-based control of robotic systems . Taylorand Francis, 2009.[26] R. Kamalapurkar, P. Walters, J. A. Rosenfeld, andW. E. Dixon,
Reinforcement learning for optimal feed-back control: A Lyapunov-based approach . SpringerInternational Publishing, 2018.[27] G. K. Pedersen,
Analysis now . Springer Science &Business Media, 2012, vol. 118.[28] J. A. Rosenfeld, “Densely defined multiplication onseveral sobolev spaces of a single variable,”
ComplexAnal. Oper. Theory , vol. 9, no. 6, pp. 1303–1309, 2015.[29] ——, “Introducing the polylogarithmic hardy space,”
Integral Equ. Oper. Theory , vol. 83, no. 4, pp. 589–600,2015.[30] F. H. Szafraniec, “The reproducing kernel hilbertspace and its multiplication operators,” in
Complexnalysis and Related Topics . Springer, 2000, pp. 253–263.[31] G. Golub, K. Solna, and P. Van Dooren, “Computingthe svd of a general matrix product/quotient,”
SIAMJ. Matrix Anal. Appl. , vol. 22, no. 1, pp. 1–19, 2000.[32] G. E. Fasshauer,