Sparse identification of nonlinear dynamics for model predictive control in the low-data limit

Eurika Kaiser; J. Nathan Kutz; Steven L. Brunton

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Sparse identification of nonlinear dynamics for model predictive control in the low-data limit

Eurika Kaiser, J. Nathan Kutz, Steven L. Brunton

arXiv:1711.05501v2Nov 15, 2017math.OCmath.DSphysics.data-an

TL;DR

Machine-learning models for MPC can be data-hungry, opaque, and difficult to adapt after abrupt changes in system dynamics. This paper extends SINDY to include actuation and integrates it with MPC, reporting stronger control performance, lower data and computational requirements, and a DMDc stopgap when data are extremely scarce.

Problem
Abrupt dynamics changes require rapid model identification, but many machine-learning approaches need large datasets, offer limited interpretability, and may generalize poorly beyond training data.
Method
The paper extends SINDY to identify interpretable nonlinear models with actuation and integrates those models into model predictive control.
Results
SINDY-MPC provides higher control performance with less training data and execution time than neural networks, while outperforming linear models for moderate data amounts.
Takeaways & Limitations
DMDc can provide a working model in the extremely low-data limit until SINDY has enough data to characterize the dynamics.
Takeaways & Limitations
The SINDY library does not scale well with data size in extremely high-dimensional systems, motivating low-dimensional or attractor-based extensions.

Abstract

from arXiv · show

The data-driven discovery of dynamics via machine learning is currently pushing the frontiers of modeling and control efforts, and it provides a tremendous opportunity to extend the reach of model predictive control. However, many leading methods in machine learning, such as neural networks, require large volumes of training data, may not be interpretable, do not easily include known constraints and symmetries, and often do not generalize beyond the attractor where models are trained. These factors limit the use of these techniques for the online identification of a model in the low-data limit, for example following an abrupt change to the system dynamics. In this work, we extend the recent sparse identification of nonlinear dynamics (SINDY) modeling procedure to include the effects of actuation and demonstrate the ability of these models to enhance the performance of model predictive control (MPC), based on limited, noisy data. SINDY models are parsimonious, identifying the fewest terms in the model needed to explain the data, making them interpretable, generalizable, and reducing the burden of training data. We show that the resulting SINDY-MPC framework has higher performance, requires significantly less data, and is more computationally efficient and robust to noise than neural network models, making it viable for online training and execution in response to rapid changes to the system. SINDY-MPC also shows improved performance over linear data-driven models, although linear models may provide a stopgap until enough data is available for SINDY. SINDY-MPC is demonstrated on a variety of dynamical systems with different challenges, including the chaotic Lorenz system, a simple model for flight control of an F8 aircraft, and an HIV model incorporating drug treatment.

1. Introduction

Data-driven MPC needs models that remain accurate, interpretable, and adaptable when data are scarce or system dynamics change abruptly. The authors extend SINDY to actuation and combine it with MPC to address these requirements.

Neural-network and other data-driven models often require substantial training data, provide limited interpretability, and may generalize poorly beyond training conditions.
MPC requires accurate and efficient models that capture relevant system dynamics, especially for strongly nonlinear systems with constraints.
Abrupt system changes create a need for parsimonious, interpretable models that can be identified rapidly from limited data.
SINDY uses sparsity-promoting optimization to identify parsimonious models from limited data while supporting known constraints, conservation laws, and symmetries.
The proposed SINDY-MPC framework extends SINDY to include actuation and is reported to improve prediction, control, computational efficiency, and noise robustness relative to neural-network models.

2. SINDY-MPC framework

SINDY-MPC combines sparse nonlinear system identification with model predictive control, extending SINDY to include actuation. It uses candidate-function libraries and sparse regression to obtain parsimonious models, then optimizes control over a receding horizon.

SINDY-MPC framework: SINDY-MPC combines sparse data-driven discovery of nonlinear dynamics with model predictive control for systems with actuation.The framework extends SINDY to identify interpretable models that include nonlinear dynamics and control inputs.
SINDYc identification: SINDY identifies governing equations by selecting a few active terms from a library of candidate nonlinear functions.Sparse regression balances model complexity with data fit and produces parsimonious coefficient vectors.
SINDYc identification: SINDYc incorporates actuation by expanding the candidate library to functions of both state x and input u, including nonlinear cross terms.State and input snapshots are arranged into data matrices before evaluating the candidate library.
SINDYc identification: Sparse coefficients are estimated through regression procedures such as LASSO or sequentially thresholded least squares.The l1 term promotes sparsity, while thresholding removes small coefficients and regression refits the remaining active terms.
Extensions and limitations: Sparse identification can support extensions involving physical constraints, known model structure, noise robustness, high-dimensional systems, and model selection.The framework also has limitations: false discoveries may occur when conditions for sparse-regression recovery are not met.
Model predictive control: MPC repeatedly optimizes a sequence of control inputs over prediction and control horizons, applies the first input, and reinitializes the optimization.This procedure yields an implicit feedback control law while respecting the discrete-time model dynamics.

3. A simple model for population dynamics

The SINDY-MPC architecture is evaluated for population dynamics against DMDc and neural-network models under varying training-data lengths and measurement noise. SINDYc achieves strong predictive and control performance with limited data and remains comparatively robust and efficient under these conditions.

Training-data length: From m = 14 onwards, SINDYc yields highly performing MPC models, outperforming all other models.The best-model training lengths are m_DMDc = 20, m_SINDYc = 85, and m_NN = 103.
Model comparison: SINDYc shows the best prediction and control performance in the population-dynamics comparison, followed by DMDc and the neural network.Training and validation use different actuation inputs to assess generalization before MPC is applied.
Measurement noise: SINDYc generally outperforms DMDc and neural networks as measurement noise increases, with a slower performance decline at low and moderate noise levels.Sparse regression is described as improving noise robustness and preventing overfitting; neural-network performance fluctuates strongly with initial weights.
Training-data length: SINDYc can discover the governing equations within machine precision in the low-data limit, while accurate neural-network models require significantly more data.DMDc may remain useful in the extremely low-data limit before enough data are available for SINDYc.
Measurement noise: SINDYc and DMDc require slightly more data under noise-corrupted training, whereas neural networks perform significantly worse.The noisy-training analysis evaluates the best model among 50 noise realizations for each training length.

4. Chaotic Lorenz system

The Lorenz study evaluates SINDYc against DMDc and neural networks for prediction and control under chaotic dynamics, measurement noise, and limited training data. SINDYc maintains useful prediction and control performance while requiring little data and rapid training.

System and control setup: The chaotic system challenges identification and control because small state uncertainties grow exponentially and cause long-time forecast error.The Lorenz control objective is stabilization of one weakly unstable fixed point.
Noise robustness: SINDYc and neural-network performance decreases as measurement noise increases, but SINDYc generally outperforms the neural network.At high noise, the neural network converges to a fixed point, whereas SINDYc retains correct statistics beyond its prediction horizon.
Training-data dependence: m = 400 marks the point from which SINDY identifies the best-performing models in the reported comparison.
Noise robustness: Even at noise magnitude η = 0.25, SINDYc provides a 1-period prediction sufficient for stabilization with MPC using Tp = 0.1.
Training-data dependence: With relatively little training data, SINDYc reaches a prediction horizon of 2.5 time units, lower error, and faster training than the neural network.With sufficiently large data, the two models produce comparable predictions; SINDYc is especially effective in low and moderate data regimes.
Training-data dependence: For noisy-data MPC, SINDYc models generally provide better control performance than neural-network models across training-data lengths.The evaluation selects the most predictive model from 50 noise realizations for each training-data length.
Training-data dependence: A nearly optimal SINDYc model can be trained from just 8 noisy measurements.

5. Tracking for the F-8 crusader

The F-8 aircraft experiment tests SINDYc for nonlinear angle-of-attack tracking with constrained actuation. SINDYc learns the aircraft dynamics effectively from substantially less data than the neural network while retaining comparable performance to DMDc.

Problem setup: The F-8 control objective is to track a commanded angle-of-attack trajectory in a strongly nonlinear, nonaffine aircraft model.The state includes angle of attack, pitch angle, and pitch rate; the input is tail deflection.
Control implementation: SINDYc uses constrained MPC with bounded angle of attack and actuation-rate changes, updating the control every 10 system timesteps.
Training data: The neural network requires 250 short trajectories, totaling 25 · 10^4 snapshots, with varying input signals for training.
Training data: SINDYc and DMDc perform similarly well when trained on 10^4 instances from a single trajectory.SINDYc learns the true relationships among variables despite limited observed system behavior.

6. Optimal therapy for pathogenic attacks

The HIV experiment evaluates SINDYc and partial-information SINDYc for treatment control in a nonlinear infection model with constrained drug input and cell concentrations. SINDYc achieves near-perfect prediction and successfully drives the system toward the desired immune-response steady state, outperforming the compared controllers in the reported setting.

HIV model: The HIV model represents healthy and infected CD4+ cells, CTL populations, and HAART treatment, with treatment efficacy η = 0.9799.
HIV model: The desired immune-response steady state has a limited region of attraction, making treatment scheduling essential.Continuous treatment can eventually lead to progressive infection after treatment is stopped.
Model identification: SINDYc parameter errors range from O(10^-3) to O(10^-6), while partial-information SINDYc estimates are less accurate for dependencies involving unmeasured variables.
Prediction: SINDYc identifies the HIV dynamics with near-perfect prediction accuracy on data differing from training, whereas other models show errors several orders of magnitude larger.
Control results: The neural-network controller fails to establish the successful immune response under constant treatment, while PI-SINDYc succeeds with slightly higher cost than SINDYc.The neural-network outcome depends strongly on the prediction and control horizon.
Generalization: The HIV comparison requires models trained on data far from the desired behavior to generalize beyond their initial training region.The neural network received significantly more training data than the other models.
Control results: Only SINDYc, PI-SINDYc, and Delay-DMDc drive the system to the desired steady state, with PI-SINDYc slightly less effective.

7. Discussion and Conclusions

The study integrates sparse nonlinear model discovery with MPC for low-data control, demonstrating interpretable models with actuation across four challenging dynamical systems. SINDY-MPC generally outperforms neural-network and linear data-driven alternatives, while DMDc remains useful when data are extremely scarce.

Contributions: SINDYc models identify nonlinear dynamics with actuation using few active terms, producing interpretable and parsimonious models.The framework integrates sparse model discovery with model predictive control.
Evaluation: The evaluation covers weakly nonlinear Lotka–Volterra, chaotic Lorenz, nonaffine F8 aircraft, and HIV/immune-response systems.These systems differ in complexity, variable scales, and available state information.
Results: SINDY-MPC provides the highest-performance control while requiring significantly less training data and execution time than neural-network models.The comparison reports this pattern across the evaluated methods and systems.
Results: DMDc and SINDYc outperform neural-network models by nearly every metric, with DMDc serving as a stopgap before sufficient data support SINDYc.DMDc is the linear-library limit of SINDYc.
Caveats: SINDY library selection remains crucial: poor candidate functions generally produce non-sparse models, while unknown system types require sweeping function classes.Expert knowledge can inform which nonlinearities to include.
Future directions: Future work must address high-dimensional scaling, limited measurements, noise effects, error estimates, and convergence guarantees for certified control.Low-dimensional attractors, manifold coordinates, and delay embeddings are identified as possible routes for difficult settings.