[Paper Review] Sparse identification of nonlinear dynamics for model predictive control in the low-data limit

Eurika Kaiser, J. Nathan Kutz+1 more|arXiv (Cornell University)|Nov 15, 2017

Model Reduction and Neural Networks14 citations

TL;DR

This paper proposes SINDY-MPC, a data-driven model predictive control framework that uses sparse identification of nonlinear dynamics (SINDY) to build interpretable, low-data models with actuation. It achieves superior performance, robustness to noise, and computational efficiency compared to neural networks and linear models, enabling real-time control in low-data, high-noise, or post-abrupt-change scenarios.

ABSTRACT

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.

Motivation & Objective

• Address the challenge of real-time model identification for model predictive control (MPC) in the low-data limit, especially after abrupt system changes.
• Overcome limitations of black-box, data-hungry models like neural networks, which lack interpretability, generalizability, and constraint integration.
• Develop a data-driven MPC framework that is computationally efficient, robust to noise, and capable of rapid adaptation with minimal training data.
• Demonstrate that sparse identification of nonlinear dynamics (SINDY) can be extended to include actuation and effectively enhance MPC performance.
• Provide a viable alternative to linear models and neural networks for online control in scenarios where data is scarce or collected under non-stationary conditions.

Proposed method

• Extend the SINDY framework to include actuation terms in the governing equations, enabling modeling of controlled dynamical systems.
• Use sparse regression to identify the fewest nonlinear terms in the dynamics that explain the observed data, ensuring model parsimony and interpretability.
• Integrate the resulting SINDY model into a model predictive control (MPC) framework for real-time feedback control.
• Apply sequential thresholding and least-squares regression to identify sparse models from limited, noisy time-series data.
• Incorporate known physical constraints (e.g., symmetries, conservation laws) into the SINDY regression via constrained least-squares, reducing model complexity and improving stability.
• Use delay coordinates and Koopman-based embeddings where full-state measurements are unavailable, enabling model identification from partial observations.

Experimental results

Research questions

• RQ1 Can SINDY-based models be effectively extended to include actuation and used within an MPC framework for controlled dynamical systems with minimal data?
• RQ2 How does SINDY-MPC performance compare to neural network-based MPC and linear data-driven models in terms of accuracy, robustness to noise, and data efficiency?
• RQ3 To what extent can SINDY-MPC enable rapid model recovery after abrupt changes in system dynamics, such as in unstable or chaotic systems?
• RQ4 Can physical constraints and symmetries be efficiently embedded into the SINDY regression to reduce data requirements and improve model generalization?
• RQ5 What is the role of smart data—representative, informative measurements—versus big data in enabling real-time, interpretable control of nonlinear systems?

Key findings

• SINDY-MPC achieves higher control performance than both neural network-based MPC and linear data-driven models, especially in low-data regimes.
• The method requires significantly less training data than neural networks and remains robust to noise, enabling reliable performance even with sparse or imperfect measurements.
• SINDY-MPC is computationally more efficient than neural network-based MPC, making it suitable for online training and real-time execution.
• The framework enables rapid model recovery after abrupt system changes, such as shifts in dynamics or instability, due to fast identification from limited data.
• Incorporating physical constraints into SINDY regression reduces the number of free parameters and improves model stability, further decreasing data requirements.
• The SINDY-MPC framework was successfully demonstrated on diverse systems, including the chaotic Lorenz system, an F8 aircraft flight control model, and an HIV treatment dynamics model, showing broad applicability.