Dynamic mode decomposition with control

Abstract: We develop a new method which extends Dynamic Mode Decomposition (DMD) to incorporate the effect of control to extract low-order models from high-dimensional, complex systems. DMD finds spatial-temporal coherent modes, connects local-linear analysis to nonlinear operator theory, and provides an equation-free architecture which is compatible with compressive sensing. In actuated systems, DMD is incapable of producing an input-output model; moreover, the dynamics and the modes will be corrupted by external forcing. Our new method, Dynamic Mode Decomposition with control (DMDc), capitalizes on all of the advantages of DMD and provides the additional innovation of being able to disambiguate between the underlying dynamics and the effects of actuation, resulting in accurate input-output models. The method is data-driven in that it does not require knowledge of the underlying governing equations, only snapshots of state and actuation data from historical, experimental, or black-box simulations. We demonstrate the method on high-dimensional dynamical systems, including a model with relevance to the analysis of infectious disease data with mass vaccination (actuation).

• The paper extends DMD by integrating control inputs to separate intrinsic system dynamics from actuation effects.
• It leverages time-series data and dimensionality reduction to estimate state and control matrices for complex systems.
• The methodology enhances control design in fields like fluid dynamics, epidemiology, and aerospace by robustly handling input perturbations.

Essay on "Dynamic Mode Decomposition with Control"

In the paper “Dynamic Mode Decomposition with Control” by Joshua L. Proctor, Steven L. Brunton, and J. Nathan Kutz, a significant advancement in data-driven analysis of dynamical systems is presented. The authors extend the well-established Dynamic Mode Decomposition (DMD) technique to accommodate control inputs, thereby introducing a method referred to as Dynamic Mode Decomposition with Control (DMDc). This innovative approach facilitates the extraction of input-output models from high-dimensional systems while explicitly accounting for actuating forces.

Overview of DMDc

Dynamic Mode Decomposition, at its core, provides a means to decompose complex dynamical systems into spatial-temporal coherent modes without necessitating prior knowledge of the governing equations. While DMD excels in identifying patterns and obtaining low-dimensional representations, it falters when applied to systems with external control inputs, failing to differentiate between innate dynamics and actuation effects. DMDc rectifies this limitation by incorporating control input data into the decomposition process, enabling it to discern intrinsic system dynamics and accurately model input-output behavior.

The DMDc framework operates under the assumption of the following controlled linear dynamical model:

$x_{k+1} = A x_k + B u_k,$

where $x$ represents the state vector, $A$ captures the dynamics, $B$ characterizes the influence of control inputs $u$, and $k$ denotes discrete time steps. The method relies exclusively on time-series data comprising state and control snapshots to estimate the system matrices $A$ and $B$. This equation-free and data-centric strategy is robust, allowing application to complex datasets typically encountered in fields such as fluid dynamics and computational epidemiology.

Key Contributions and Implications

1. Application Flexibility:

The paper delineates DMDc’s potential in scenarios where traditional mathematical modeling is impracticable due to unknown or complex underlying physics. Rather than relying on explicit dynamical equations, historical, experimental, or computational simulation data suffices for system identification, making DMDc an attractive option for applications like infectious disease modeling and aerospace engineering.

2. Handling High-Dimensional Data:

Addressing the computational challenges posed by high-dimensional datasets, DMDc leverages dimensionality reduction techniques akin to those used in Balanced Truncation and Singular Value Decomposition (SVD), thus enabling the creation of computationally manageable reduced-order models. This trait is crucial for real-time control applications where low latency and high efficiency are paramount.

3. Robustness to Input Perturbations:

Contrary to its predecessors, DMDc is designed to handle constants and random perturbations in input data, maintaining the integrity of the identified system dynamics. This aspect is particularly beneficial in real-world scenarios where measurement noise and environmental uncertainties are prevalent.

4. Potential for Enhanced Control Design:

Through accurate reconstruction of both state dynamics and control impacts, DMDc equips engineers and scientists with the capability to design more effective and responsive control strategies. For instance, in the field of fluid dynamics, it could significantly enhance flow control procedures by providing a more comprehensive understanding of how actuation affects fluid behavior.

Future Directions

Theoretical advancements in machine learning and optimization present several avenues for extending the capabilities of DMDc. Specifically, integration with sparsity-promoting techniques and compressive sensing could further augment the methodology’s efficiency and applicability to even more complex systems characterized by limited sensor networks. Moreover, expansions to nonlinear systems remain a promising research direction, propelled by the foundational connection between DMD and Koopman operator theory.

In conclusion, DMDc represents a significant stride in the domain of dynamical system analysis, offering a powerful tool for extracting meaningful insights from high-dimensional data afflicted by external controls. Its application potential spans a myriad of fields, providing both theoretical enrichment and practical utility in system identification and control design.