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The Singular Value Decomposition (SVD) of linear functions facilitates the calculation of their 2-induced norm and row and null spaces, hallmarks of linear control theory. In this work, we present a function representation that, similar to SVD, provides an upper bound on the 2-induced norm of bounded-input bounded-output functions, as well as facilitates the computation of generalizations of the notions of row and null spaces. Borrowing from the notion of "lifting" in Koopman operator theory, we construct a finite-dimensional lifting of inputs that relaxes the unitary property of the right-most matrix in traditional SVD, $V^*$, to be an injective, norm-preserving mapping to a slightly higher-dimensional space., Comment: Submitted to the 2024 Conference on Decision and Control

This work builds on the models and concepts presented in part 1 to learn approximate dictionary representations of Koopman operators from data. Part I of this paper presented a methodology for arguing the subspace invariance of a Koopman dictionary. This methodology was demonstrated on the state-inclusive logistic lifting (SILL) basis. This is an affine basis augmented with conjunctive logistic functions. The SILL dictionary's nonlinear functions are homogeneous, a norm in data-driven dictionary learning of Koopman operators. In this paper, we discover that structured mixing of heterogeneous dictionary functions drawn from different classes of nonlinear functions achieve the same accuracy and dimensional scaling as the deep-learning-based deepDMD algorithm. We specifically show this by building a heterogeneous dictionary comprised of SILL functions and conjunctive radial basis functions (RBFs). This mixed dictionary achieves the same accuracy and dimensional scaling as deepDMD with an order of magnitude reduction in parameters, while maintaining geometric interpretability. These results strengthen the viability of dictionary-based Koopman models to solving high-dimensional nonlinear learning problems., Comment: 16 pages, 6 figures. arXiv admin note: substantial text overlap with arXiv:2206.13585

Koopman operators model nonlinear dynamics as a linear dynamic system acting on a nonlinear function as the state. This nonstandard state is often called a Koopman observable and is usually approximated numerically by a superposition of functions drawn from a dictionary. In a widely used algorithm, Extended Dynamic Mode Decomposition, the dictionary functions are drawn from a fixed class of functions. Recently, deep learning combined with EDMD has been used to learn novel dictionary functions in an algorithm called deep dynamic mode decomposition (deepDMD). The learned representation both (1) accurately models and (2) scales well with the dimension of the original nonlinear system. In this paper we analyze the learned dictionaries from deepDMD and explore the theoretical basis for their strong performance. We explore State-Inclusive Logistic Lifting (SILL) dictionary functions to approximate Koopman observables. Error analysis of these dictionary functions show they satisfy a property of subspace approximation, which we define as uniform finite approximate closure. Our results provide a hypothesis to explain the success of deep neural networks in learning numerical approximations to Koopman operators. Part 2 of this paper will extend this explanation by demonstrating the subspace invariant of heterogeneous dictionaries and presenting a head-to-head numerical comparison of deepDMD and low-parameter heterogeneous dictionary learning., Comment: 13 pages, 2 figures. arXiv admin note: substantial text overlap with arXiv:2206.13585

When complex systems with nonlinear dynamics achieve an output performance objective, only a fraction of the state dynamics significantly impacts that output. Those minimal state dynamics can be identified using the differential geometric approach to the observability of nonlinear systems, but the theory is limited to only analytical systems. In this paper, we extend the notion of nonlinear observable decomposition to the more general class of data-informed systems. We employ Koopman operator theory, which encapsulates nonlinear dynamics in linear models, allowing us to bridge the gap between linear and nonlinear observability notions. We propose a new algorithm to learn Koopman operator representations that capture the system dynamics while ensuring that the output performance measure is in the span of its observables. We show that a transformation of this linear, output-inclusive Koopman model renders a new minimum Koopman representation. This representation embodies only the observable portion of the nonlinear observable decomposition of the original system. A prime application of this theory is to identify genes in biological systems that correspond to specific phenotypes, the performance measure. We simulate two biological gene networks and demonstrate that the observability of Koopman operators can successfully identify genes that drive each phenotype. We anticipate our novel system identification tool will effectively discover reduced gene networks that drive complex behaviors in biological systems., Comment: 17 pages, 4 figures, submission to Automatica

Koopman operators model nonlinear dynamics as a linear dynamic system acting on a nonlinear function as the state. This nonstandard state is often called a Koopman observable and is usually approximated numerically by a superposition of functions drawn from a \textit{dictionary}. A widely used algorithm, is \textit{Extended Dynamic Mode Decomposition}, where the dictionary functions are drawn from a fixed, homogeneous class of functions. Recently, deep learning combined with EDMD has been used to learn novel dictionary functions in an algorithm called deep dynamic mode decomposition (deepDMD). The learned representation both (1) accurately models and (2) scales well with the dimension of the original nonlinear system. In this paper we analyze the learned dictionaries from deepDMD and explore the theoretical basis for their strong performance. We discover a novel class of dictionary functions to approximate Koopman observables. Error analysis of these dictionary functions show they satisfy a property of subspace approximation, which we define as uniform finite approximate closure. We discover that structured mixing of heterogeneous dictionary functions drawn from different classes of nonlinear functions achieve the same accuracy and dimensional scaling as deepDMD. This mixed dictionary does so with an order of magnitude reduction in parameters, while maintaining geometric interpretability. Our results provide a hypothesis to explain the success of deep neural networks in learning numerical approximations to Koopman operators., Comment: 14 pages, 7 figures, journal paper

This paper focuses on the operation of an electricity market that accounts for participants that bid at a sub-minute timescale. To that end, we model the market-clearing process as a dynamical system, called market dynamics, which is temporally coupled with the grid frequency dynamics and is thus required to guarantee system-wide stability while meeting the system operational constraints. We characterize participants as price-takers who rationally update their bids to maximize their utility in response to real-time schedules of prices and dispatch. For two common bidding mechanisms, based on quantity and price, we identify a notion of alignment between participants' behavior and planners' goals that leads to a saddle-based design of the market that guarantees convergence to a point meeting all operational constraints. We further explore cases where this alignment property does not hold and observe that misaligned participants' bidding can destabilize the closed-loop system. We thus design a regularized version of the market dynamics that recovers all the desirable stability and steady-state performance guarantees. Numerical tests validate our results on the IEEE 39-bus system.

This paper uses data-driven operator theoretic approaches to explore the global phase space of a dynamical system. We defined conditions for discovering new invariant subspaces in the state space of a dynamical system starting from an invariant subspace based on the spectral properties of the Koopman operator. When the system evolution is known locally in several invariant subspaces in the state space of a dynamical system, a phase space stitching result is derived that yields the global Koopman operator. Additionally, in the case of equivariant systems, a phase space stitching result is developed to identify the global Koopman operator using the symmetry properties between the invariant subspaces of the dynamical system and time-series data from any one of the invariant subspaces. Finally, these results are extended to topologically conjugate dynamical systems; in particular, the relation between the Koopman tuple of topologically conjugate systems is established. The proposed results are demonstrated on several second-order nonlinear dynamical systems including a bistable toggle switch. Our method elucidates a strategy for designing discovery experiments: experiment execution can be done in many steps, and models from different invariant subspaces can be combined to approximate the global Koopman operator., Comment: 32 pages, 19 figures. This paper is currently under review

Dictionary methods for system identification typically rely on one set of measurements to learn governing dynamics of a system. In this paper, we investigate how fusion of output measurements with state measurements affects the dictionary selection process in Koopman operator learning problems. While prior methods use dynamical conjugacy to show a direct link between Koopman eigenfunctions in two distinct data spaces (measurement channels), we explore the specific case where output measurements are nonlinear, non-invertible functions of the system state. This setup reflects the measurement constraints of many classes of physical systems, e.g., biological measurement data, where one type of measurement does not directly transform to another. We propose output constrained Koopman operators (OC-KOs) as a new framework to fuse two measurement sets. We show that OC-KOs are effective for sensor fusion by proving that when learning a Koopman operator, output measurement functions serve to constrain the space of potential Koopman observables and their eigenfunctions. Further, low-dimensional output measurements can be embedded to inform selection of Koopman dictionary functions for high-dimensional models. We propose two algorithms to identify OC-KO representations directly from data: a direct optimization method that uses state and output data simultaneously and a sequential optimization method. We prove a theorem to show that the solution spaces of the two optimization problems are equivalent. We illustrate these findings with a theoretical example and two numerical simulations., Comment: 20 pages, 7 figures

In this paper, we propose a novel algorithm for learning the Koopman operator of a dynamical system from a \textit{small} amount of training data. In many applications of data-driven modeling, e.g. biological network modeling, cybersecurity, modeling the Internet of Things, or smart grid monitoring, it is impossible to obtain regularly sampled time-series data with a sufficiently high sampling frequency. In such situations the existing Dynamic Mode Decomposition (DMD) or Extended Dynamic Mode Decomposition (EDMD) algorithms for Koopman operator computation often leads to a low fidelity approximate Koopman operator. To this end, this paper proposes an algorithm which can compute the Koopman operator efficiently when the training data-set is sparsely sampled across time. In particular, the proposed algorithm enriches the small training data-set by appending artificial data points, which are treated as noisy observations. The larger, albeit noisy data-set is then used to compute the Koopman operator, using techniques from Robust Optimization. The efficacy of the proposed algorithm is also demonstrated on three different dynamical systems, namely a linear network of oscillators, a nonlinear system and a dynamical system governed by a Partial Differential Equation (PDE)., Comment: arXiv admin note: substantial text overlap with arXiv:1901.03024

In recent years there has been a considerable drive towards data-driven analysis, discovery and control of dynamical systems. To this end, operator theoretic methods, namely, Koopman operator methods have gained a lot of interest. In general, the Koopman operator is obtained as a solution to a least-squares problem, and as such, the Koopman operator can be expressed as a closed-form solution that involves the computation of Moore-Penrose inverse of a matrix. For high dimensional systems and also if the size of the obtained data-set is large, the computation of the Moore-Penrose inverse becomes computationally challenging. In this paper, we provide an algorithm for computing the Koopman operator for high dimensional systems in a time-efficient manner. We further demonstrate the efficacy of the proposed approach on two different systems, namely a network of coupled oscillators (with state-space dimension up to 2500) and IEEE 68 bus system (with state-space dimension 204 and up to 24,000 time-points).

In this paper, we consider the problem of learning a predictive model for population cell growth dynamics as a function of the media conditions. We first introduce a generic data-driven framework for training operator-theoretic models to predict cell growth rate. We then introduce the experimental design and data generated in this study, namely growth curves of Pseudomonas putida as a function of casein and glucose concentrations. We use a data driven approach for model identification, specifically the nonlinear autoregressive (NAR) model to represent the dynamics. We show theoretically that Hankel DMD can be used to obtain a solution of the NAR model. We show that it identifies a constrained NAR model and to obtain a more general solution, we define a causal state space system using 1-step,2-step,...,{\tau}-step predictors of the NAR model and identify a Koopman operator for this model using extended dynamic mode decomposition. The hybrid scheme we call causal-jump dynamic mode decomposition, which we illustrate on a growth profile or fitness prediction challenge as a function of different input growth conditions. We show that our model is able to recapitulate training growth curve data with 96.6% accuracy and predict test growth curve data with 91% accuracy.

With the advancement of sensing and communication in power networks, high-frequency real-time data from a power network can be used as a resource to develop better monitoring capabilities. In this work, a systematic approach based on data-driven operator theoretic methods involving Koopman operator is proposed for the online identification of power system dynamics. In particular, a new algorithm is provided, which unlike any previously existing algorithms, updates the Koopman operator iteratively as new data points are acquired. The proposed algorithm has three advantages: a) allows for real-time monitoring of the power system dynamics b) linear power system dynamics (this linear system is usually in a higher dimensional feature space and is not same as linearization of the underlying nonlinear dynamics) and c) computationally fast and less intensive when compared to the popular Extended Dynamic Mode Decomposition (EDMD) algorithm. The efficiency of the proposed algorithm is illustrated on an IEEE 9 bus system using synthetic data from the nonlinear model and on IEEE 39 bus system using synthetic data from the linearized model., Comment: arXiv admin note: substantial text overlap with arXiv:1909.12520

In this paper, we develop the Koopman operator theory for dynamical systems with symmetry. In particular, we investigate how the Koopman operator and eigenfunctions behave under the action of the symmetry group of the underlying dynamical system. Further, exploring the underlying symmetry, we propose an algorithm to construct a global Koopman operator from local Koopman operators. In particular, we show, by exploiting the symmetry, data from all the invariant sets are not required for constructing the global Koopman operator; that is, local knowledge of the system is enough to infer the global dynamics.

In this work, we propose to apply the recently developed Koopman operator techniques to explore the global phase space of a nonlinear system from time-series data. In particular, we address the problem of identifying various invariant subsets of a phase space from the spectral properties of the associated Koopman operator constructed from time-series data. Moreover, in the case when the system evolution is known locally in various invariant subspaces, then a phase space stitching result is proposed that can be applied to identify a global Koopman operator. A biological system, bistable toggle switch and a second-order nonlinear system example are considered to illustrate the proposed results. The construction of this global Koopman operator is very helpful in experimental works as multiple experiments can't be run at the same time starting from several initial conditions., Comment: This version is currently under review in ACC 2020

We propose an alternating optimization algorithm to the nonconvex Koopman operator learning problem for nonlinear dynamic systems. We show that the proposed algorithm will converge to a critical point with rate $O(1/T)$ and $O(\frac{1}{\log T})$ for the constant and diminishing learning rates, respectively, under some mild conditions. To cope with the high dimensional nonlinear dynamical systems, we present the first-ever distributed Koopman operator learning algorithm. We show that the distributed Koopman operator learning has the same convergence properties as the centralized Koopman operator learning, in the absence of optimal tracker, so long as the basis functions satisfy a set of state-based decomposition conditions. Numerical experiments are provided to complement our theoretical results., Comment: 8 pages, 2 figures

In this paper we develop the concept of information transfer between the Borel-measurable sets for a dynamical system described by a measurable space and a non-singular transformation. The concept is based on how Shannon entropy is transferred between the measurable sets, as the dynamical system evolves. We show that the proposed definition of information transfer satisfies the usual notions of information transfer and causality, namely, zero transfer and transfer asymmetry. Furthermore, we show how the information transfer measure can be used to classify ergodicity and mixing. We also develop the computational methods for information transfer computation and apply the framework for optimal placements of actuators and sensors for control of non-equilibrium dynamics.

This paper describes the optimal selection of a control policy to program the steady state of controlled nonlinear systems with hyperbolic fixed points. This work is motivated by the field of synthetic biology, in which saddle points are common (along with limit cycles), and the aim is to program cells to perform both digital and analog computation, though developing genetic digital computation has been the main focus. We frame the analog computing challenge of generating a steady state input-output function inside living cells. To program the steady state, a data-driven approach is taken wherein an approximation of the Koopman operator, identified via deep dynamic mode decomposition, is used to describe the dynamics of the system linearly. The new representation of the dynamics are then used to solve an optimization problem for the input which maximizes a direction in state space. Some added structure on the Koopman operator learning process for controlled systems is given for dynamics that are separable in the state and input. Finally, the methods are demonstrated on simulation examples of an incoherent feedforward loop and a combinatorial promoter system, two common network architectures seen in the field of synthetic biology., Comment: 7 pages, 3 figures, accepted in 2020 American Control Conference

In this paper, we provide an algorithm for online computation of Koopman operator in real-time using streaming data. In recent years, there has been an increased interest in data-driven analysis of dynamical systems, with operator theoretic techniques being the most popular. Existing algorithms, like Dynamic Mode Decomposition (DMD) and Extended Dynamic Mode Decomposition (EDMD), use the entire data set for computation of the Koopman operator. However, many real life applications like power system analysis, biological systems, building systems etc. requires the real-time computation and updating of the Koopman operator, as new data streams in. In this paper, we propose an iterative algorithm for online computation of Koopman operator such that at each time step the Koopman operator is updated incrementally. In particular, we propose a Recursive Extended Dynamic Decomposition (rEDMD) algorithm for computation of Koopman operator from streaming data. Further, we test the algorithm in three different dynamical systems, namely, a linear system, a nonlinear system and a system governed by a Partial Differential Equation (PDE) and illustrate the computational efficiency of the iterative algorithm over the existing DMD and EDMD algorithms.

Genetic circuits are designed to implement certain logic in living cells, keeping burden on the host cell minimal. However, manipulating the genome often will have a significant impact for various reasons (usage of the cell machinery to express new genes, toxicity of genes, interactions with native genes, etc.). In this work we utilize Koopman operator theory to construct data-driven models of transcriptomic-level dynamics from noisy and temporally sparse RNAseq measurements. We show how Koopman models can be used to quantify impact on genetic circuits. We consider an experimental example, using high-throughput RNAseq measurements collected from wild-type E. coli, single gate components transformed in E. coli, and a NAND circuit composed from individual gates in E. coli, to explore how Koopman subspace functions encode increasing circuit interference on E. coli chassis dynamics. The algorithm provides a novel method for quantifying the impact of synthetic biological circuits on host-chassis dynamics., Comment: 4 pages Accepted in 2019 IEEE Biomedical Circuits and Systems Conference

It is hard to identify nonlinear biological models strictly from data, with results that are often sensitive to experimental conditions. Automated experimental workflows and liquid handling enables unprecedented throughput, as well as the capacity to generate extremely large datasets. We seek to develop generalized identifiability conditions for informing the design of automated experiments to discover predictive nonlinear biological models. For linear systems, identifiability is characterized by persistence of excitation conditions. For nonlinear systems, no such persistence of excitation conditions exist. We use the input-Koopman operator method to model nonlinear systems and derive identifiability conditions for open-loop systems initialized from a single initial condition. We show that nonlinear identifiability is intrinsically tied to the rank of a given dataset's power spectral density, transformed through the lifted Koopman observable space. We illustrate these identifiability conditions with a simulated synthetic gene circuit model, the repressilator. We illustrate how rank degeneracy in datasets results in overfitted nonlinear models of the repressilator, resulting in poor predictive accuracy. Our findings provide novel experimental design criteria for discovery of globally predictive nonlinear models of biological phenomena., Comment: 6 pages. 2 figures, accepted for 58th IEEE Conference on Decision and Control - Nice, France - December 11th-13th 2019

Optimal sensor placement is an important yet unsolved problem in control theory. In biological organisms, genetic activity is often highly nonlinear, making it difficult to design libraries of promoters to act as reporters of the cell state. We make use of the Koopman observability gramian to develop an algorithm for optimal sensor (or reporter) placement for discrete time nonlinear dynamical systems to ease the difficulty of design of the promoter library. This ease is enabled due to the fact that the Koopman operator represents the evolution of a nonlinear system linearly by lifting the states to an infinite-dimensional space of observables. The Koopman framework ideally demands high temporal resolution, but data in biology are often sampled sparsely in time. Therefore we compute what we call the temporally fine-grained Koopman operator from the temporally coarse-grained Koopman operator, the latter of which is identified from the sparse data. The optimal placement of sensors then corresponds to maximizing the observability of the fine-grained system. We demonstrate the algorithm on a simulation example of a circadian oscillator., Comment: 6 pages, 3 figures, to appear in 2019 IEEE Conference on Decision and Control (CDC)

In this paper we propose a novel approach to compute the Koopman operator from sparse time series data. In recent years there has been considerable interests in operator theoretic methods for data-driven analysis of dynamical systems. Existing techniques for the approximation of the Koopman operator require sufficiently large data sets, but in many applications, the data set may not be large enough to approximate the operators to acceptable limits. In this paper, using ideas from robust optimization, we propose an algorithm to compute the Koopman operator from sparse data. We enrich the sparse data set with artificial data points, generated by adding bounded artificial noise and and formulate the noisy robust learning problem as a robust optimization problem and show that the optimal solution is the Koopman operator with smallest error. We illustrate the efficiency of our proposed approach in three different dynamical systems, namely, a linear system, a nonlinear system and a dynamical system governed by a partial differential equation.

In this paper we presented a novel constructive approach for training deep neural networks using geometric approaches. We show that a topological covering can be used to define a class of distributed linear matrix inequalities, which in turn directly specify the shape and depth of a neural network architecture. The key insight is a fundamental relationship between linear matrix inequalities and their ability to bound the shape of data, and the rectified linear unit (ReLU) activation function employed in modern neural networks. We show that unit cover geometry and cover porosity are two design variables in cover-constructive learning that play a critical role in defining the complexity of the model and generalizability of the resulting neural network classifier. In the context of cover-constructive learning, these findings underscore the age old trade-off between model complexity and overfitting (as quantified by the number of elements in the data cover) and generalizability on test data. Finally, we benchmark on algorithm on the Iris, MNIST, and Wine dataset and show that the constructive algorithm is able to train a deep neural network classifier in one shot, achieving equal or superior levels of training and test classification accuracy with reduced training time.

We suggest ways to enforce given constraints in the output of a Generative Adversarial Network (GAN) generator both for interpolation and extrapolation (prediction). For the case of dynamical systems, given a time series, we wish to train GAN generators that can be used to predict trajectories starting from a given initial condition. In this setting, the constraints can be in algebraic and/or differential form. Even though we are predominantly interested in the case of extrapolation, we will see that the tasks of interpolation and extrapolation are related. However, they need to be treated differently. For the case of interpolation, the incorporation of constraints is built into the training of the GAN. The incorporation of the constraints respects the primary game-theoretic setup of a GAN so it can be combined with existing algorithms. However, it can exacerbate the problem of instability during training that is well-known for GANs. We suggest adding small noise to the constraints as a simple remedy that has performed well in our numerical experiments. The case of extrapolation (prediction) is more involved. During training, the GAN generator learns to interpolate a noisy version of the data and we enforce the constraints. This approach has connections with model reduction that we can utilize to improve the efficiency and accuracy of the training. Depending on the form of the constraints, we may enforce them also during prediction through a projection step. We provide examples of linear and nonlinear systems of differential equations to illustrate the various constructions., Comment: 29 pages; v2 has major text revision/restructuring, includes results for the Lorenz system and has several more references

In analysis and control of large-scale nonlinear dynamical systems, a distributed approach is often an attractive option due to its computational tractability and usually low communication requirements. Success of the distributed control design relies on the separability of the network into weakly interacting subsystems such that minimal information exchange between subsystems is sufficient to achieve satisfactory control performance. While distributed analysis and control design for dynamical network have been well studied, decomposition of nonlinear networks into weakly interacting subsystems has not received as much attention. In this article we propose a vector Lyapunov functions based approach to quantify the energy-flow in a dynamical network via a model of a comparison system. Introducing a notion of power and energy flow in a dynamical network, we use sum-of-squares programming tools to partition polynomial networks into weakly interacting subsystems. Examples are provided to illustrate the proposed method of decomposition., Comment: to be presented at ECC 2018

An outstanding challenge in nonlinear systems theory is identification or learning of a given nonlinear system's Koopman operator directly from data or models. Advances in extended dynamic mode decomposition approaches and machine learning methods have enabled data-driven discovery of Koopman operators, for both continuous and discrete-time systems. Since Koopman operators are often infinite-dimensional, they are approximated in practice using finite-dimensional systems. The fidelity and convergence of a given finite-dimensional Koopman approximation is a subject of ongoing research. In this paper we introduce a class of Koopman observable functions that confer an approximate closure property on their corresponding finite-dimensional approximations of the Koopman operator. We derive error bounds for the fidelity of this class of observable functions, as well as identify two key learning parameters which can be used to tune performance. We illustrate our approach on two classical nonlinear system models: the Van Der Pol oscillator and the bistable toggle switch., Comment: 8 pages

We solve a system of ordinary differential equations with an unknown functional form of a sink (reaction rate) term. We assume that the measurements (time series) of state variables are partially available, and we use recurrent neural network to "learn" the reaction rate from this data. This is achieved by including a discretized ordinary differential equations as part of a recurrent neural network training problem. We extend TensorFlow's recurrent neural network architecture to create a simple but scalable and effective solver for the unknown functions, and apply it to a fedbatch bioreactor simulation problem. Use of techniques from recent deep learning literature enables training of functions with behavior manifesting over thousands of time steps. Our networks are structurally similar to recurrent neural networks, but differences in design and function require modifications to the conventional wisdom about training such networks., Comment: 19 pages, 8 figures

In this paper we propose a new Koopman operator approach to the decomposition of nonlinear dynamical systems using Koopman Gramians. We introduce the notion of an input-Koopman operator, and show how input-Koopman operators can be used to cast a nonlinear system into the classical state-space form, and identify conditions under which input and state observable functions are well separated. We then extend an existing method of dynamic mode decomposition for learning Koopman operators from data known as deep dynamic mode decomposition to systems with controls or disturbances. We illustrate the accuracy of the method in learning an input-state separable Koopman operator for an example system, even when the underlying system exhibits mixed state-input terms. We next introduce a nonlinear decomposition algorithm, based on Koopman Gramians, that maximizes internal subsystem observability and disturbance rejection from unwanted noise from other subsystems. We derive a relaxation based on Koopman Gramians and multi-way partitioning for the resulting NP-hard decomposition problem. We lastly illustrate the proposed algorithm with the swing dynamics for an IEEE 39-bus system., Comment: 8 pages, submitted to IEEE 2018 ACC

In this paper, we consider the problem of quantifying controllability and observability of a nonlinear discrete time dynamical system. We introduce the Koopman operator as a canonical representation of the system and apply a lifting technique to compute gramians in the space of full-state observables. We illustrate the properties of these gramians and identify several relationships with canonical results on local controllability and observability. Once defined, we show that these gramians can be balanced through a change of coordinates on the observables space, which in turn allows for direct application of balanced truncation. Throughout the paper, we highlight the aspects of our approach with an example nonlinear system., Comment: 9 pages, 4 figures, IEEE conference format

The Koopman operator has recently garnered much attention for its value in dynamical systems analysis and data-driven model discovery. However, its application has been hindered by the computational complexity of extended dynamic mode decomposition; this requires a combinatorially large basis set to adequately describe many nonlinear systems of interest, e.g. cyber-physical infrastructure systems, biological networks, social systems, and fluid dynamics. Often the dictionaries generated for these problems are manually curated, requiring domain-specific knowledge and painstaking tuning. In this paper we introduce a deep learning framework for learning Koopman operators of nonlinear dynamical systems. We show that this novel method automatically selects efficient deep dictionaries, outperforming state-of-the-art methods. We benchmark this method on partially observed nonlinear systems, including the glycolytic oscillator and show it is able to predict quantitatively 100 steps into the future, using only a single timepoint, and qualitative oscillatory behavior 400 steps into the future., Comment: 16 pages, 5 figures