Koopman Operator Dynamical Models: Learning, Analysis and Control? Petar Bevandaa,∗, Stefan Sosnowskia, Sandra Hirchea aChair of Information-oriented Control, Department of Electrical and Computer Engineering, 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, they are in many instances overlooked in favor of, often ill-posed and unstructured methods. Also, Koopman operator theory has long-standing connections to known system-theoretic and dynamical system notions that are not universally recognized. Given the former and latter realities, this work aims to bridge the gap 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 existing methodologies and highlighting their differences. Furthermore, we provide con- cise insight into the paradigm's relation to system-theoretic notions and analyze the 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 with complex dynamical systems. For such complex systems, models coming from arXiv:2102.02522v1 [eess.SY] 4 Feb 2021 first-principles often times are not available or do not fully resemble the true system due to unmodeled phenomena. Thus, there is great value in develop- ing effective techniques with the ability to discover, analyze and control such systems. To do so, the approach to modeling inevitably dictates the challenges ?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] (Stefan Sosnowski), [email protected] (Sandra Hirche) Preprint submitted to Annual Reviews in Control February 5, 2021 in dealing with a system of interest. The complexity of systems we encounter prompts a shift from classical parametric techniques in favor of more flexible machine learning techniques (e.g. neural networks or Gaussian processes) for prediction [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, they are limited when it comes to efficient representations for prediction, analysis and optimization-based control. Alternative to the traditional modeling paradigm studying the \dynamics of states", one can decide to perceive systems through an operator-theoretic view concerned 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 systems in Hilbert space. It gives rise to representations of dynamical systems that linearly evolve observables (output measurements) of the system in a special set of coordinates via the Koopman operator. System representations via the Koopman paradigm generalize the notion of mode analysis from linear to nonlin- ear systems allowing for amendable relevance determination of the constituents of the full dynamics (e.g. thermal analysis of buildings [9]). The spectrum of the Koopman operator allows one to decompose the nonlinear system into different 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 hardships through 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 the Koopman operator acting on an observable is equivalent to an iteration along all of the trajectories of the system and it is not to be confused with a local linearization 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, general framework. The non-triviality of finite representations lead to many data-driven approaches with various trains of thought, facilitating different properties of the Koopman 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´cet al. [13] gives a theoreti- cal baseline with application examples but with the data-driven methodology limited to dynamic mode decomposition (DMD). We address the abundance of novel data-driven methods that arose from and beside DMD for discovering Koopman operator dynamical models in addition to siding with a more focused system and control-theoretic perspective. Also, compared to work of Kaiser et al. [14], we take a rigorous approach focused on the Koopman operator paradigm instead of an high-level overview of data-driven transfer operators. By taking 2 such an approach, the aim is to deliver a holistic and methodical backbone of Koopman operator-based dynamical models - from surveying the data-driven representations, 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 model representation are presented in Section 3. An overview of system-theoretic anal- ysis via Koopman operator theory follows in Section 4. Furthermore, Section 5 analyzes the inclusion of control into the representations and surveys their applications found in the literature. Before the final outline concluding Section 6, 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 N0 denotes all nat- ural numbers with zero and R+;0=R+ all positive reals with/without zero. The t continuous/discrete-time dependence is denoted as y (·) and yk, respectively _ with t=k 2 R+;0=N0 while (·) := d(·)=dt, for brevity. Function spaces with a specific integrability/smoothness order are denoted as L/C with the order (class) specified in their exponent. The braces h·; ·i denote the inner product while k · k the Euclidean norm. A flow induced by a vector fieldx _ = f(x) is denoted as F t(x) with its associated family of composition (Koopman) opera- t tors fKf gt2R+;0 . A map i.e. x 7! F (x) has its associated family of composition n operator iterates denoted as fKF gn2N0 . 2. Koopman operator theory The traditional way of studying a system leads to studying the orbits of points x on a domain M under iteration - point dynamics study (\dynamics of states") [13]: x_ = f(x) ) F t : M 7! M; (1) with state x 2 M ⊆ Rn and vector field f : M 7! Rn inducing the flow F t. On the contrary, the Koopman operator entails how maps evolve under iteration instead. Informally, it encodes information about an iterated map which can be very well used to study the behavior of dynamical systems. The Koopman operator Kt : F 7! F evolves functions 2 F over the state-space M (\dynamics of observables"). An observable can be any kind of a measurement of the system, e.g. a sensory measurement. We make a distinction between general observables i and the subset thereof that one wants to predict. In a system-theoretic sense, we name those functions output functions yi = hi(x). Considering a deterministic system (1) and the \full-state" vector of output functions - h(x) = x we write t t Kf x = F (x); (2) 3 where the flow F t (in this specific case) is replaced by the action of the component- t wise Koopman operator Kf . Although delivering the same as the flow for the deterministic case and h(x) = x, the Koopman operator, without a loss of generality, 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 underlying system, now in a much more \iteration-compatible", linear form. 2.1. Basic assumptions and definitions Here we reintroduce some notions in a more formal way and further expand on the latter. Assumption 2.1. There exists a continuous-time dynamical system x_ = f(x); (3) and a scalar observable function : M 7! C y = (x); (4) where there exists a metric space x 2 M, t 2 R with a smooth and Lipschitz flow F t : M 7! M [15]: Z t0+t t x(t0) ≡ x0; F (x0) := x0 + f(x(τ))dτ; (5) t0 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 a topological sense locally resemblant of Euclidean space) M ⊆ Rn is specified by the flow (5) F t. Often times we dispense with manifolds and choose the state space to lie in ⊆ Rn, but for generality manifolds are retained in our definitions. Assumption 2.2. There is a generator GK : D(GK) !F, D being the domain of the generator and F the space of observables. Definition 2.1 (Infinitesimal generator). The operator GK, is the infinites- t imal generator of the time-t indexed semigroup of Koopman operators fK gt2R+;0 Kt − d GK = lim = ; (6) t!0+ t dt [16, Chapter 3].