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Generalizing Koopman Theory to allow for inputs and control
Joshua L. Proctor, Steven L. Brunton, J. Nathan Kutz
TL;DR
Standard Koopman analysis and DMD do not produce input-output models for actuated systems, motivating a theory that incorporates inputs and control. The paper develops KIC, connects it rigorously to DMDc, and demonstrates linear measurement dynamics for nonlinear systems, while identifying practical difficulties in choosing augmented observables.
Problem
Standard Koopman analysis and DMD are insufficient for input-output modeling in actuated systems, where external forcing can corrupt dynamics and modes.
Method
KIC generalizes Koopman observables to depend on state and input, including state-only, input-only, and mixed functions, and connects this formulation to DMDc.
Results
KIC constructs a linear dynamical system on measurements for prediction and control despite nonlinear dynamics, and reduces to DMDc for linear input-output systems.
Takeaways & Limitations
The extension broadens Koopman-based analysis toward nonlinear system identification and controller design for input-output systems.
Takeaways & Limitations
Including certain input-output terms can require additional nonlinear observables, and an augmented formulation may produce an incorrect operator in realistic complex systems.
Abstract
from arXiv · showhide
We develop a new generalization of Koopman operator theory that incorporates the effects of inputs and control. Koopman spectral analysis is a theoretical tool for the analysis of nonlinear dynamical systems. Moreover, Koopman is intimately connected to Dynamic Mode Decomposition (DMD), a method that discovers spatial-temporal coherent modes from data, connects local-linear analysis to nonlinear operator theory, and importantly creates an equation-free architecture allowing investigation of complex systems. In actuated systems, standard Koopman analysis and DMD are incapable of producing input-output models; moreover, the dynamics and the modes will be corrupted by external forcing. Our new theoretical developments extend Koopman operator theory to allow for systems with nonlinear input-output characteristics. We show how this generalization is rigorously connected and generalizes a recent development called Dynamic Mode Decomposition with control (DMDc). We demonstrate this new theory on nonlinear dynamical systems, including a standard Susceptible-Infectious-Recovered model with relevance to the analysis of infectious disease data with mass vaccination (actuation).
1. Introduction.
Koopman theory and DMD provide equation-free tools for analyzing complex systems from data, but modern high-dimensional input-output control requires broader methods. The paper positions KIC alongside DMDc and system-identification approaches as an extension for nonlinear input-output systems.
- 1. Introduction.: Koopman theory reconstructs linear dynamical systems from data, enabling spectral analysis of nonlinear systems without requiring governing equations.Its numerical implementation can be much faster than solving complex-domain PDEs.
- 1. Introduction.: High-dimensional nonlinear control requires computationally feasible models, fast solvers, and dimensionality reduction for controller design.The passage identifies flow control and infectious-disease eradication as important applications.
- 1. Introduction.: DMD, ERA, OKID, and subspace-identification methods use measurement and input data to construct equation-free models.The paper places modal decompositions and system-identification methods in a closely connected methodological family.
- 1. Introduction.: KIC reduces to DMDc for linear input-output systems while supporting nonlinear-system identification through expanded observable functions.This establishes the paper’s central connection between the new Koopman formulation and an existing control-oriented decomposition.
- 1. Introduction.: The paper develops KIC after reviewing Koopman theory, its DMD connection, and numerical examples involving nonlinear input-output systems.The stated outline places the KIC development in Section 3 and the examples in Section 4.
2. Background: Koopman and Dynamic Mode Decomposition.
Koopman theory represents nonlinear dynamics through a linear, infinite-dimensional operator acting on observable functions, while DMD computes finite-dimensional approximations from paired measurements. The paper generalizes this framework to exogenous inputs and connects the resulting formulation to DMDc.
- 2.1. The Koopman Operator for dynamical systems.: The Koopman operator is linear and infinite-dimensional, acting on scalar-valued observable functions in a Hilbert space.The dynamical system is treated in discrete time on a smooth manifold, with observables drawn from a Lebesgue square-integrable function space.
- 2.1. The Koopman Operator for dynamical systems.: Koopman eigenvalues encode mode growth, decay, and frequency, while Koopman modes and eigenfunctions represent observables spectrally.For linear systems with identity observables, eigenfunctions relate to inner products with left eigenvectors.
- 2.2. Koopman and DMD.: DMD obtains an operator from paired current and future measurement matrices using A ≜ ZY† and then computes its eigenvectors and eigenvalues.The measurements need not be sequential or drawn from a single trajectory, provided current-future pairings are maintained.
- 2.2. Koopman and DMD.: For linearly consistent data, DMD modes and eigenvalues correspond to Koopman modes and eigenvalues.This identifies DMD as a data-driven finite-dimensional realization of the Koopman spectral description.
- 2.2. Koopman and DMD.: The generalized theory includes exogenous inputs and provides a perspective that connects Koopman analysis to DMDc through alternative output spaces.The section explicitly presents KIC as a generalization of Koopman theory for input-bearing systems.
3. Generalizing Koopman to allow for inputs and control.
The paper generalizes Koopman operator theory to systems with inputs by defining observables over state-input spaces and allowing distinct input and output observable spaces. This formulation connects KIC to DMDc while accommodating nonlinear and mixed state-input measurements.
- 3.1. Koopman with inputs and control.: The observable-function space partitions state-only, input-only, and mixed state-input measurements within H.This partitioning supports projections between different classes of observables.
- 3.1. Koopman with inputs and control.: KIC permits inputs to be either dynamically evolving or externally affecting state evolution without their own dynamics.These cases correspond to the choices ∗=u and ∗=0, respectively.
- 3.2. KIC for linear systems.: The linear-system formulation can create difficulty when future exogenous or random inputs are not predictable.The associated G21 and G22 terms can complicate approximate Koopman-operator computation.
- 3.4.2. Input and output spaces for the Koopman operator.: The input space can include linear, nonlinear, and mixed state-input observables, while the output space can be restricted to linear state measurements.This allows a large functional library on the input side and a smaller output space.
- 3.3. KIC and connections to DMDc.: For linear input-output systems, the Koopman operator is equivalent to the DMDc operator after restricting the output space to a subspace of H.The construction provides the stated connection between KIC and DMDc.
4. Applications.
The applications section examines recovery of linear dynamics and input matrices from state and input measurements under several input conditions. The example includes random disturbances and uses a finite set of snapshots to construct a restricted Koopman operator.
- 4. Applications.: The application seeks to recover the underlying dynamics and input matrix under random, feedback, and multi-scale inputs.The setup assumes full access to the system state and inputs.
- 4. Applications.: The system is rewritten in KIC form to investigate dynamics for the inputs.The reformulation treats the input evolution as part of the KIC representation.
- 4. Applications.: For random disturbances, state and input measurements are collected to reconstruct a finite-dimensional Koopman operator.The example uses snapshots from a single realization.
- 4. Applications.: Six snapshots are used to compute a restricted Koopman operator for an unstable linear system with µ = 0.1, λ = 1.5, and δ = 1.The input disturbances are zero-mean Gaussian with variance 0.01.
G11 G12 G21 G22
The examples show how KIC reconstructs nonlinear state-input dynamics, including unstable, dynamically evolving, controlled, and periodic inputs. The SIR example also demonstrates that observable selection determines whether accurate prediction is possible.
- Controlled inputs: For state-feedback control, adding a small disturbance to the input snapshot matrix helps disambiguate control from the measured state.The restricted Koopman operator then recovers the unstable dynamics and identifies input dependence on x2.
- Input dynamics: KIC recovers the underlying dynamics, input impact, and input dynamics when inputs evolve independently of the state.This perspective is proposed as useful for multi-scale modeling in which one scale acts as forcing on another.
- Nonlinear systems: Despite nonlinear dynamics, KIC constructs a linear measurement-space system that supports prediction and control.The construction uses Koopman modes and eigenfunctions built from selected observables.
- SIR example: In the SIR model, including SI as an output creates further nonlinearities requiring augmented observables such as S^2I, SI^2, and I^2.The resulting formulation produces an incorrect operator and inaccurate future prediction after training on 200 snapshots, whereas the selected observables yield the correct prediction.
- SIR example: The SIR formulation is readily solved with enough snapshot data, making correct input and output observable selection central to implementation.The input-output operator maps data in the input observables to data in the output observables.
- Periodic solutions: For periodic orbits with exogenous inputs, Fourier expansion describes output observables while the Z-transform represents input observables.Without inputs the analysis reduces to the discrete Fourier transform; with inputs, the Z-transform accommodates input-output transfer functions.
5. Discussion.
Koopman and DMD provide equation-free, data-driven analysis of complex systems, but generalizing Koopman to input-output systems broadens the systems it can represent. KIC connects to DMDc while extending observable choices toward nonlinear system identification and controller design.
- Data-driven analysis: Koopman and DMD characterize complex systems from data and have been applied across fluid dynamics, epidemiology, video processing, and neuroscience.Their equation-free architecture also supports incorporating compressive-sensing methods for measurement design.
- Connection to DMDc: KIC generalizes Koopman to input-output systems and is well-connected to DMDc for systems with linear observables.The extension permits a broader set of observable functions than the linear-observable setting.
- Implications: The extended Koopman framework supports nonlinear system identification and controller design for larger classes of systems.The paper positions KIC and DMDc for characterization and control of large-scale, complex, input-output systems.