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Sparse Identification of Nonlinear Dynamics with Control (SINDYc)
Steven L. Brunton, Joshua L. Proctor, J. Nathan Kutz
TL;DR
The paper addresses data-driven identification of nonlinear dynamical systems when external inputs and feedback control are present. It generalizes SINDY by expanding the candidate library to include inputs and state–input cross terms, and connects the method to DMD and Koopman analysis. Tested on predator–prey and Lorenz systems, the method identifies and reconstructs controlled or forced dynamics within the demonstrated settings.
Problem
Identifying governing equations from data is important for modeling and controlling complex systems, but nonlinear systems with external inputs and feedback control require methods that account for actuation.
Method
SINDYc generalizes SINDY by building a candidate-function library of state, input, and state–input cross terms within a sparse regression framework.
Results
The method was tested on predator–prey and Lorenz systems with forcing and control, and correctly identified the demonstrated model structures and parameters in the reported settings.
Takeaways & Limitations
SINDYc extends sparse nonlinear model identification to systems with inputs and control while placing it within a regression framework related to DMD and Koopman methods.
Takeaways & Limitations
Feedback control can be indistinguishable from internal state-feedback terms, making the regression ill-conditioned unless the input is perturbed.
Abstract
from arXiv · showhide
Identifying governing equations from data is a critical step in the modeling and control of complex dynamical systems. Here, we investigate the data-driven identification of nonlinear dynamical systems with inputs and forcing using regression methods, including sparse regression. Specifically, we generalize the sparse identification of nonlinear dynamics (SINDY) algorithm to include external inputs and feedback control. This method is demonstrated on examples including the Lotka-Volterra predator--prey model and the Lorenz system with forcing and control. We also connect the present algorithm with the dynamic mode decomposition (DMD) and Koopman operator theory to provide a broader context.
1. INTRODUCTION
Data-driven methods increasingly exploit abundant measurements to model complex, often low-dimensional nonlinear systems. Because existing regression methods frequently assume linear dynamics, the paper extends SINDY to include external inputs and feedback control while connecting it to DMD and Koopman methods.
- Motivation: Complex systems generate abundant high-fidelity data but are often high-dimensional, nonlinear, and multiscale.The paper notes that many such systems nevertheless evolve on low-dimensional attractors characterized by coherent structures.
- Existing methods: Many system-identification techniques, including DMD and DMDc, handle high-dimensional data under an assumption of linear dynamics.The introduction contrasts this established focus with the smaller number of methods for nonlinear identification.
- Nonlinear identification: SINDY uses sparse regression to identify parsimonious nonlinear governing equations from data.The related framework relies on the observation that many dynamical systems contain relatively few relevant terms.
- Control: Feedback control requires distinguishing actuation effects from internal dynamics when constructing nonlinear input–output models.The paper frames this disambiguation as a critical step for control design.
- Contribution: The paper extends SINDY to external inputs and control and relates SINDY variants to DMD and Koopman methods through regression-based model identification.It presents these methods as variations within a broader regression framework.
2. MODEL IDENTIFICATION VIA REGRESSION
This section presents regression-based identification methods spanning linear DMD, control-aware DMDc, Koopman analysis, and sparse nonlinear SINDY. These approaches differ in their representations of dynamics, observables, inputs, and candidate nonlinear terms.
- 2.1 Dynamic mode decomposition: DMD extracts spatially coherent modes with fixed frequencies or growth and decay rates from high-dimensional snapshot data.It uses POD to reduce the data before computing a reduced operator whose eigendecomposition yields DMD modes.
- 2.1 Dynamic mode decomposition: DMD fits a linear operator to time-shifted snapshots, using least squares and an SVD-based pseudoinverse to obtain a reduced representation.The reduced operator acts on POD coefficients, avoiding direct computation of a potentially n^2-element high-dimensional operator.
- 2.2 DMD with control: DMDc augments linear state dynamics with control inputs to separate actuation from internal dynamics.Its model uses x_k+1 ≈ Ax_k + Bu_k and estimates A and B from state and control histories.
- 2.3 Koopman analysis: Koopman analysis represents nonlinear state evolution through an infinite-dimensional linear operator acting on measurement functions.Standard DMD approximates its spectrum using linear observables, while eDMD augments the data with nonlinear state measurements.
- 2.4 Koopman with inputs and control: Koopman with inputs and control advances state-and-control observables under a parameterized family indexed by the future control input.For linear systems, this framework reduces to DMDc.
- 2.5 Sparse identification of nonlinear dynamics (SINDY): SINDY identifies nonlinear dynamics by sparse regression over a library of candidate functions.The library choice is crucial, and candidate terms may include polynomials, trigonometric functions, or partial physics knowledge.
- 2.5 Sparse identification of nonlinear dynamics (SINDY): SINDY becomes equivalent to DMD when the candidate-function library is the state data itself.The sparsity-promoting penalty balances model complexity against accuracy through selection of a Pareto-optimal regularization parameter.
- 2.5 Sparse identification of nonlinear dynamics (SINDY): Derivative estimates from noisy state measurements are obtained using total variation regularization.This provides the derivative approximation used by SINDY before sparse regression identifies the governing terms.
3. SPARSE IDENTIFICATION OF NONLINEAR DYNAMICS WITH CONTROL (SINDYc)
SINDYc extends sparse nonlinear system identification to include measured inputs and control, using an expanded candidate library and perturbations to separate actuation from internal feedback.
- Input-inclusive sparse identification: SINDYc generalizes SINDY by building a larger library Θ(x, u) that includes input functions and nonlinear cross terms in x and u.The method requires measurements of both the state x and input signal u.
- Input-inclusive sparse identification: For external forcing, SINDYc solves for sparse coefficients Ξ using the measured input signal u.
- Feedback-control identification: Feedback control u = k(x) makes the regression ill-conditioned because actuation cannot be disambiguated from internal feedback terms k(x).
- Feedback-control identification: Perturbing u with sufficiently large white noise, impulses, or steps separates actuation from internal feedback terms during coefficient identification.The paper also suggests designing informative input signals as a future direction.
4. EXAMPLE SYSTEMS
The paper demonstrates SINDYc on forced predator–prey and Lorenz systems, showing that including inputs enables accurate model identification and prediction under forcing and control.
- Example systems: SINDYc is demonstrated on a predator–prey model with forcing and on Lorenz equations with external forcing and control.
- Predator-prey model: Without input knowledge, SINDY produces an unstable model, whereas SINDYc identifies the forced Lotka–Volterra structure and parameters to machine precision without measurement noise.
- Lorenz equations: For externally forced Lorenz dynamics, SINDYc correctly identifies the model and nonlinear input terms.
- Lorenz equations: In the controlled Lorenz test, models train on 20 time units of controlled data and validate on 20 time units after switching to u(t) = 50 sin(10t).
- Lorenz equations: SINDY misses actuation effects, while SINDYc identifies the model and predicts behavior under a new forcing signal absent from training.
5. DISCUSSION
The paper generalizes SINDY to include inputs and control through state–input nonlinear libraries, and places the method within a regression framework encompassing DMD and Koopman analysis. It tests the approach on predator–prey and Lorenz systems while identifying unresolved challenges in separating feedback effects from internal dynamics.
- Method: SINDYc extends the candidate library to functions of the state x, input u, and their cross terms.This generalization incorporates inputs and control within the sparse identification framework.
- Connections: The framework relates SINDYc to DMD, DMD with control, extended DMD, and Koopman analysis.These methods are presented as algorithms for determining models from data within an overarching regression framework.
- Demonstrations: The method has been tested on a predator–prey model and the Lorenz system with various forcing and control models.The proposed algorithm uses the same computational architecture as SINDY and should scale to the same class of problems where SINDY is useful.
- Future directions: Disambiguating state-feedback effects from internal dynamics remains an important extension challenge.Suggested strategies include additive white noise or occasional input perturbations, but their tradeoffs and benefits require further study.