Source-linked AI summary
Dynamic mode decomposition with control
Joshua L. Proctor, Steven L. Brunton, J. Nathan Kutz
TL;DR
DMD cannot reliably produce input-output models for actuated systems because external forcing corrupts the inferred dynamics. DMDc extends DMD by using state and actuation snapshots to separate underlying dynamics from control effects, producing accurate input-output models across high-dimensional examples.
Problem
DMD is limited for actuated systems because it cannot produce an input-output model and external forcing corrupts inferred dynamics and modes.
Method
DMDc uses state and control snapshots in an equation-free, data-driven framework to identify both underlying dynamics and the effects of actuation.
Results
DMDc accurately recovers controlled-system models, including unstable dynamics, frequency responses, and underlying eigenvalues in high-dimensional examples.
Takeaways & Limitations
DMDc provides reduced-order input-output models for prediction and controller design from historical, experimental, or black-box simulation data.
Abstract
from arXiv · showhide
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).
I. INTRODUCTION
DMDc targets high-dimensional controlled systems by using state and control observations to separate intrinsic dynamics from actuation and build reduced-order input-output models. It extends equation-free, data-driven DMD toward practical control and observational applications.
- I. INTRODUCTION: DMDc uses state measurements and applied control to extract unforced dynamics and characterize how inputs affect the system.The method does not require the governing equations and supports reduced-order models for prediction and controller design.
- I. INTRODUCTION: High-dimensional control is difficult because full-system controllers can be computationally prohibitive and may require costly Riccati solutions or iterative procedures.Dimensionality reduction is therefore central to making control strategies computationally tractable.
- I. INTRODUCTION: Existing reduced-order and system-identification methods address computational or input-output modeling challenges but may require adjoint calculations or assume low-dimensional linear measurements.Balanced truncation, balanced proper orthogonal decomposition, ERA, and OKID are cited as related approaches.
- I. INTRODUCTION: DMD provides data-driven, equation-free dynamics reconstruction, connections to Koopman operator theory, and compatibility with sparse measurements.These properties motivate extending DMD with a control architecture for complex systems.
- I. INTRODUCTION: Infectious-disease analysis illustrates how DMDc can use historical state and intervention data to discover dynamical properties without selecting among many phenomenological models.State data can represent infections, while control data can represent vaccinations or other interventions.
- I. INTRODUCTION: DMD estimates a process matrix from snapshot matrices, with X′ representing time-shifted snapshots of X.The primary objective is solving for an approximation of A from X and X′.
B. Dynamic Mode Decomposition
Standard DMD estimates a reduced linear operator from time-shifted snapshots, then analyzes its eigenvalues and modes in a lower-dimensional subspace. SVD-based truncation makes the computation practical for systems where the full state dimension is large.
- B. Dynamic Mode Decomposition: DMD computes an approximation of the process matrix from snapshot data using the Moore-Penrose pseudoinverse.The SVD provides a computationally efficient route to the pseudoinverse.
- B. Dynamic Mode Decomposition: SVD truncation retains r dominant directions and removes remaining singular-value components to reduce the data dimension.The truncation choice r can be informed by the Eckart-Young theorem and developments for noisy data.
- B. Dynamic Mode Decomposition: The reduced-order model projects the state onto an r-dimensional linear subspace when r is much smaller than n.The SVD basis supplies the coordinate transformation used for this projection.
- B. Dynamic Mode Decomposition: Eigenvalues and eigenvectors of the reduced operator reveal system properties such as growth modes and resonance frequencies.The reduced eigendecomposition is efficient because the operator has dimension r×r rather than n×n.
- B. Dynamic Mode Decomposition: Solving the reduced r×r eigenvalue problem is more attractive than directly solving the full n×n problem when n is very large.This efficiency is especially relevant to finely discretized fluid or epidemiological systems.
- B. Dynamic Mode Decomposition: The eigenvectors associated with the full operator are called dynamic modes, while reduced-operator eigenvalues correspond to those of the full operator.The Exact DMD procedure relates reduced eigenvectors to the dynamic modes.
III. DYNAMIC MODE DECOMPOSITION WITH CONTROL
DMDc extends DMD to systems with internal dynamics and external control by separating the control-free dynamics from input effects. This supports both dynamic analysis and quantitative characterization of actuation for control-oriented modeling.
- III. DYNAMIC MODE DECOMPOSITION WITH CONTROL: DMDc discovers underlying dynamics without the confounding effect of external control.The method also quantifies how control inputs affect the system state.
- III. DYNAMIC MODE DECOMPOSITION WITH CONTROL: The DMDc workflow includes data collection, an algorithm for extracting dynamics and input effects, and applications to controller design and sensor placement.The section presents the mathematical formulation and its use for systems with both internal dynamics and applied control.
- III. DYNAMIC MODE DECOMPOSITION WITH CONTROL: The method handles cases where the input map is known or estimated and cases where both the dynamic and input maps must be identified.The general formulation seeks the dynamic modes together with the input matrix.
A. Dynamical system with control
For controlled systems, DMDc models the next state from both the current state and control input, then uses state and control snapshots to estimate the relevant linear mappings. This formulation supports simultaneous discovery of dynamics and actuation effects.
- A. Dynamical system with control: DMDc replaces DMD’s state-only transition with a model in which the future state depends on the current state and current control.The state and control dimensions are represented by xj ∈ R^n and uj ∈ R^l, with mappings A and B.
- A. Dynamical system with control: Control snapshots are collected over time alongside the state snapshot matrices X and X′.The input sequence is organized into a matrix containing successive control observations.
- A. Dynamical system with control: The three data matrices are used to approximate the linear mappings A and B.The method provides separate procedures depending on whether B is known or must be discovered from observations.
B. The map B is known
When B is known, DMDc uses state and control snapshots with the known input map to recover the underlying dynamics despite applied control. The resulting model can retain DMD's reduced-order representation and dynamic modes.
- Standard DMD produces incorrect dynamic information when external control has been applied, motivating recovery of the underlying system matrix A.
- DMDc pairs time-shifted state snapshots with control snapshots and the known matrix B to solve for the mapping A.
- With zero control snapshots, the derivation is equivalent to DMD and constructs a dynamic model from the computed process and input matrices.
- When r ≪ n, projecting the state onto an r-dimensional SVD basis yields a more compact and computationally efficient model.
- The eigendecomposition of the reduced matrix produces eigenvectors used to compute DMDc dynamic modes, including a separate expression when the eigenvalue is zero.
C. The map B is unknown
When B is unknown, DMDc estimates both the underlying dynamics A and the input matrix B from state and control snapshots. It uses an augmented input space, then a separate output-space reduction to obtain a reduced-order model.
- DMDc relaxes the known-B assumption by estimating both A and B using only state and control snapshots.
- The method forms an augmented snapshot matrix containing state and control information and fits an operator G that contains A and B.
- For large n, DMDc seeks a reduced-order model of rank r ≪ n rather than directly using computationally prohibitive full-dimensional matrices.
- Unlike DMD, the truncated left singular vectors of the augmented matrix define the input space rather than the state-evolution subspace.
- A second SVD of the output space provides the state subspace and supports reduced-order approximations of A and B, with truncation values p and r that may differ.
- The reduced-order dynamic modes are obtained by eigendecomposing the reduced A matrix and transforming its eigenvectors.
D. The algorithm
The DMDc algorithm organizes data collection, two SVD reductions, reduced dynamics construction, eigendecomposition, and dynamic-mode computation. Its application begins with numerical, laboratory, or historical data curated into snapshot matrices.
- DMDc begins by collecting state and control snapshots and constructing X, X′, Υ, and the stacked matrix Ω.
- The algorithm computes an SVD of the input space Ω, using truncation value p to obtain the reduced factors.
- It then computes an SVD of the output space X′ before constructing the reduced-order dynamics and input matrices.
- The illustrated workflow accepts numerical, laboratory, or historical data and includes comparisons with DMD plus two practical DMDc applications.
- The workflow includes eigendecomposition of the reduced A matrix followed by computation of the dynamic modes.
IV. APPLICATIONS
The applications emphasize the benefit of including control snapshots, beginning with a stabilizing-controller example for recovering dynamics that would otherwise appear stable. The example uses state and control data processed through SVD-based DMDc computations.
- The application examples increase in complexity, with each emphasizing the benefit of including control snapshot information.
- DMDc is demonstrated on a two-dimensional unstable linear system with a stabilizing controller to illustrate recovery of dynamics from state and external-input measurements.
- The proportional controller stabilizes the system by moving its unstable eigenvalue within the unit circle, while the example supplies state and control snapshots.
- The example computes an economy-sized SVD of X before using the data matrices, SVD factors, and B to approximate A.
- The recovered unstable linear dynamics show that DMDc can identify instability from data collected under a controller that otherwise makes the system appear stable.
B. Example 2 – Large-scale, stable Linear Systems
DMDc is evaluated on large-scale stable systems with low-dimensional attractors by comparing its identified state-space model with the generated model. The frequency responses show no distinction between the two models for either control input.
- DMDc is applied to state and input snapshot matrices to approximate the generated state-space matrices.
- The singular values of the MIMO frequency response provide the comparison between the generated model and the DMDc model.
- There is no distinction between the generated and DMDc frequency-response curves for either control input.
C. Example 3 – A sparse linear system in the Fourier domain
DMDc recovers sparse Fourier-domain dynamics from spatial state and control snapshots in a high-dimensional system. Compared with DMD, it identifies the underlying eigenvalues more accurately and remains qualitatively stable when zero-valued modes are contaminated with Gaussian noise.
- C. Example 3 – A sparse linear system in the Fourier domain: The example uses high-dimensional spatial measurements whose dynamics are governed by a low-dimensional Fourier-domain system with only 5 nonzero modes.
- C. Example 3 – A sparse linear system in the Fourier domain: DMDc discovers the underlying dynamics from spatial-domain state and control snapshots, including the effects of localized negative actuation.
- C. Example 3 – A sparse linear system in the Fourier domain: DMDc identifies the underlying eigenvalues more accurately than DMD in the actuated spatial-grid example.
- C. Example 3 – A sparse linear system in the Fourier domain: Gaussian noise added to zero-valued Fourier modes does not qualitatively change DMDc behavior.
- V. CONNECTIONS TO SYSTEM IDENTIFICATION METHODS: Modal decomposition methods typically address systems with many more measurements than attractor rank, unlike ERA/OKID input-output settings.
- V. CONNECTIONS TO SYSTEM IDENTIFICATION METHODS: DMDc differs from ERA by allowing arbitrary control histories, whereas ERA is primarily impulse-response focused.
VI. DISCUSSION
The paper addresses the challenge of describing complex, high-dimensional systems without well-characterized governing equations. It presents DMDc as an equation-free extension of DMD that provides accurate input-output models and is positioned for large-scale analysis and control.
- Accurately describing complex, high-dimensional systems without well-characterized governing equations is a central challenge for effective control.
- DMDc extends equation-free DMD to provide accurate input-output models for complex systems with actuation.
- DMDc can use historical, experimental, or black-box simulation data.
- The method is positioned for analysis and control of large-scale complex systems and can be coupled with sparsity-promoting sampling and control strategies.