DMDc: Dynamic Mode Decomposition with Control

• Dynamic Mode Decomposition with Control (DMDc) is a data-driven method that separates intrinsic dynamics from control-induced responses using least-squares regression.
• It employs Singular Value Decomposition to reduce high-dimensional data, enabling efficient extraction of state and control matrices.
• DMDc has been successfully applied in fields like ecosystem modeling, thermospheric density forecasting, and battery systems to achieve high predictive accuracy.

Dynamic Mode Decomposition with Control (DMDc) is a data-driven methodology for identifying reduced-order state-space models of complex, high-dimensional dynamical systems subjected to external inputs or actuation. By extending conventional Dynamic Mode Decomposition (DMD) to accommodate exogenous control signals, DMDc enables the separation and quantification of autonomous system dynamics (intrinsic modes) and control-induced responses. The method is grounded in least-squares regression and frequently leverages dimensionality reduction via Singular Value Decomposition (SVD) for computational tractability in large-scale settings. DMDc has demonstrated strong predictive accuracy and interpretability across a wide spectrum of domains, including ecosystem carbon cycles, space environmental modeling, battery systems, and controlled partial differential equations (Proctor et al., 2014, Shadaydeh et al., 2024, Mehta et al., 2018, Labib et al., 29 Jan 2026, Donadini et al., 2021).

1. Mathematical Formulation and Operator-Theoretic Basis

DMDc models a forced discrete-time system as a linear input-state mapping:

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

where $x_k \in \mathbb{R}^n$ is the system state, $u_k \in \mathbb{R}^l$ is the control vector, $A \in \mathbb{R}^{n \times n}$ encodes the autonomous dynamics, and $B \in \mathbb{R}^{n \times l}$ captures the influence of control inputs (Proctor et al., 2014, Shadaydeh et al., 2024).

Given snapshot matrices:

• $$X = [x_1\,\ldots\,x_{M-1}] \in \mathbb{R}^{n \times (M-1)}$$
• $$X' = [x_2\,\ldots\,x_M] \in \mathbb{R}^{n \times (M-1)}$$
• $$U = [u_1\,\ldots\,u_{M-1}] \in \mathbb{R}^{l \times (M-1)}$$

the best-fit (in the Frobenius norm) concatenated operator $[A\,B]$ satisfies

$X' \approx [A\;B]\,[X; U],$

with the closed-form minimum-norm solution:

$[A\;B] = X'\,[X; U]^+,$

where $(\cdot)^+$ denotes the Moore–Penrose pseudoinverse (Proctor et al., 2014, Shadaydeh et al., 2024).

This regression is, in effect, an empirical identification of a finite-dimensional input-state Koopman generator, providing a principled decomposition of autonomous and forced modes (Proctor et al., 2014).

2. Algorithmic Procedure and Dimensionality Reduction

For computational tractability with high-dimensional states or large datasets, DMDc employs SVD-based dimensionality reduction:

1. Augment and Decompose: Form the augmented snapshot matrix $\Omega = [X; U] \in \mathbb{R}^{n+l \times (M-1)}$.
2. Truncated SVD: Compute

$\Omega \approx \tilde U\,\tilde\Sigma\,\tilde V^*,$

retaining the $p$ leading singular values and truncating noise-dominated directions.

1. Operator Extraction: Partition $\tilde U = [\tilde U_1; \tilde U_2]$ with $\tilde U_1 \in \mathbb{R}^{n \times p}$, $\tilde U_2 \in \mathbb{R}^{l \times p}$, and compute:

$$A = X'\,\tilde V\,\tilde\Sigma^{-1}\,\tilde U_1^T, \quad B = X'\,\tilde V\,\tilde\Sigma^{-1}\,\tilde U_2^T.$$

This avoids direct pseudoinversion of potentially ill-conditioned or oversized matrices (Proctor et al., 2014, Shadaydeh et al., 2024, Mehta et al., 2018, Donadini et al., 2021).

2. Forecasting: Advance the dynamical model via

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

These steps are compatible with a sliding-window approach for time-varying or nonstationary systems, as in ecosystem time series or operational forecasting settings (Shadaydeh et al., 2024).

3. Extensions: Delay Embedding, Physics Constraints, and Network Structures

Delay Embedding

• Augmenting the state and input histories with their delay coordinates (Hankel embedding) can enhance the ability of the DMDc model to capture nonlinear or memory effects in systems where Markovian closure is inadequate. This is critical in battery systems, ship hydrodynamics, and ecosystem respiration, where hidden processes dominate observable outputs (Labib et al., 29 Jan 2026, Palma et al., 17 Feb 2025, Shadaydeh et al., 2024).
• The state and input matrices become:

$$X = \text{delay-embedded state},\quad U = \text{delay-embedded input}$$

with the embedding dimensions optimized to minimize forecast error (Labib et al., 29 Jan 2026, Palma et al., 17 Feb 2025).

Physics-Informed Constraints

• Regularization: Ridge (Tikhonov) penalties can be added to the least-squares cost, e.g., $\lambda \|[A\,B]\|_F^2$, for smoothness control (Shadaydeh et al., 2024).
• Sign or structural constraints: Elementwise nonnegativity or enforcing exponential-law priors (e.g., Arrhenius temperature dependence) can be imposed to maintain theoretical consistency of control responses (Shadaydeh et al., 2024).
• Pre-processing: Inputs can be transformed (e.g., via min-max normalization or exponential warping) according to domain knowledge (Shadaydeh et al., 2024).

Network DMDc

• For interconnected or networked systems, DMDc can be applied in a partitioned fashion, regressing local update equations for each subsystem based only on directly connected states and inputs, improving both computational efficiency and data efficiency (Heersink et al., 2017).
• This blockwise estimation aligns with physical connectivity (e.g., power grids, biological networks).

4. Applications and Empirical Performance

Ecosystem Respiration (Fluxnet2015)

• DMDc accurately forecasts nightly gross ecosystem respiration ($\mathrm{Reco}$) across diverse vegetation types.
• Physically meaningful controls include air temperature and soil water content.
• One-day predictions with DMDc(T_air) reach RMSE = 0.182 $\mu$mol CO$_2$ m$^{-2}$ s$^{-1}$ at the DE-Hai site, outperforming DMD without control and matching established partitioning approaches (Shadaydeh et al., 2024).
• Time-delay embedding (TDE) further reduces RMSE by up to 5% for two-week forecasts (Shadaydeh et al., 2024).

Thermospheric Density (HS-DMDc)

• Hermitian-space projection allows tractable identification of reduced-order DMDc models from full-scale TIE-GCM simulations, retaining forecasting errors below 5% at 24 hours (Mehta et al., 2018).
• Control matrix $B$ meaningfully decomposes variations from solar proxies (F10.7), geomagnetic forcing (Kp), diurnal, and seasonal cycles into modal coefficients.

Battery Systems

• Delay-embedded DMDc achieves residual sum of squares (RSS) as low as 1.74 for voltage prediction under pulse power characterization, robustly transferring identified $(A,B)$ operators across considerable cell aging (Labib et al., 29 Jan 2026).

Time-Dependent PDE-Constrained Control

• DMDc delivers surrogates for both state and adjoint variables in optimal control PDEs with sub-percent accuracy and order-of-magnitude reductions in computational time compared to conventional solvers (Donadini et al., 2021).

5. Advanced Topics: Feature Selection, Input Design, and Reduced-Order Model Integration

Feature Selection: RFE-DMDc

• Recursive feature elimination (RFE) selects physically interpretable minimal state sets for DMDc models, mitigating feature overshadowing in multicomponent systems.
• Cross-subsystem corrections and downstream genetic algorithm (GA) baselines demonstrate that RFE-DMDc can recover all relevant variables at reduced computational cost, as in a 545-variable energy system benchmark (Wang et al., 25 Nov 2025).

Informative Input Design

• Convex optimization of planned control trajectories within the DMDc framework can minimize model parameter uncertainty, outperforming traditional pseudo-random and multisine inputs in identification accuracy for flexible aircraft and unsteady fluid flow (Ott et al., 2024).
• This adapts input design dynamically in response to the current model parameter covariance, thus enhancing experimental efficiency for system identification (Ott et al., 2024).

Reduced-Order and Optimal Mode Decomposition

• OMDc extends DMDc by jointly optimizing the projection basis and dynamical maps, integrating identification and dimension reduction to directly minimize forecasting error in the reduced subspace (Mieg et al., 11 Apr 2025).
• Compared to standard two-stage DMDc (estimation plus projection), OMDc achieves greater forecast accuracy at equivalent reduced-order dimensions (Mieg et al., 11 Apr 2025).

6. Limitations, Implementation, and Computational Considerations

• DMDc relies on the linearity-in-parameters assumption; for strongly nonlinear or non-affine systems, performance may deteriorate without delay embedding or nonlinear extensions (e.g., lifted-Koopman, Extended DMDc) (Proctor et al., 2014, Labib et al., 29 Jan 2026).
• Sliding windows, rank truncation for SVD, and regularization are essential for stability and generalization, particularly with noisy, short, or high-dimensional datasets (Shadaydeh et al., 2024, Mehta et al., 2018, Palma et al., 17 Feb 2025).
• Implementation is compatible with standard numerical linear algebra environments (MATLAB, Python, Julia), with randomized SVD and rank-revealing decompositions enabling scalability to very large state and control dimensions (Shadaydeh et al., 2024, Mehta et al., 2018, Palma et al., 17 Feb 2025).
• Computational bottlenecks are typically in SVD or eigendecomposition steps; these can be alleviated by work in Hermitian space, low-rank projection, or partitioned (network) algorithms (Mehta et al., 2018, Heersink et al., 2017).

7. Perspectives and Ongoing Developments

• DMDc continues to be extended for nonlinear, control-affine systems using operator-theoretic frameworks, e.g., control Liouville operators and occupation kernel techniques, providing convergence guarantees for finite-rank estimators in vector-valued RKHS spaces (Abudia et al., 13 Mar 2025, Rosenfeld et al., 2021, Morrison et al., 2023).
• Bayesian and uncertainty quantification extensions, such as Bayesian Hankel-DMDc, now provide robust predictive distributions and confidence intervals even for models trained on minimal data (Palma et al., 17 Feb 2025).
• Integration with optimal control (e.g., MPC), model predictive input design, and digital twin applications in engineering and environmental systems is an active research direction.

DMDc has thus become a core enabling methodology for data-driven reduced-order modeling and control in complex high-dimensional systems, offering interpretable decomposition of forced dynamics and scalable identification algorithms with rigorous operator-theoretic underpinnings (Proctor et al., 2014, Shadaydeh et al., 2024, Abudia et al., 13 Mar 2025, Labib et al., 29 Jan 2026, Donadini et al., 2021, Wang et al., 25 Nov 2025, Heersink et al., 2017, Palma et al., 17 Feb 2025, Ott et al., 2024, Mieg et al., 11 Apr 2025).