Sparse identification of nonlinear dynamics for model predictive control in the low-data limit (1711.05501v2)

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.

• The paper introduces SINDY, a sparse identification method that isolates key terms in nonlinear dynamics to enable effective model predictive control with minimal data.
• The methodology outperforms traditional techniques like DMDc and neural networks in benchmark tests across chaotic and biological models.
• Its parsimonious models offer robust, interpretable, and real-time control solutions, paving the way for adaptive systems in data-scarce environments.

Sparse Identification of Nonlinear Dynamics for Model Predictive Control in the Low-Data Limit

The paper "Sparse identification of nonlinear dynamics for model predictive control in the low-data limit" by Kaiser, Kutz, and Brunton presents a compelling framework for integrating data-driven discovery of dynamics with model predictive control (MPC). This research stands out for its focus on sparsity in model identification, allowing for the effective application of MPC even when data is limited.

Sparse Identification of Nonlinear Dynamics (SINDY)

The core of the approach is the Sparse Identification of Nonlinear Dynamics (SINDY) methodology, which is applied to infer dynamic models from data. SINDY leverages sparsity-promoting algorithms to identify only the essential terms in a dynamical system's governing equations. Unlike traditional machine learning models such as neural networks, which often require extensive datasets and can become computationally prohibitive, SINDY focuses on creating interpretable and parsimonious models. This characteristic not only reduces the need for large data volumes but also enhances the robustness and generalizability of the models by avoiding overfitting.

Integration with Model Predictive Control

The integration with MPC allows for the use of these parsimonious models in controlling nonlinear dynamical systems with constraints—an area where traditional linear models may fall short. This is crucial for applications requiring real-time control where rapid response and adaptability are essential, such as in autonomous systems or industrial processes.

Comparative Analysis

The paper benchmarks SINDY against traditional methods like Dynamic Mode Decomposition with control (DMDc) and neural networks across several nonlinear systems, including the Lotka-Volterra dynamics, the chaotic Lorenz system, and an HIV infection model. Results consistently indicate that SINDY strikes an optimal balance between model complexity and accuracy, leading to efficient computation and robust control, particularly in settings with limited training data. It is noteworthy that while DMDc might provide a preliminary model in low-data conditions, SINDY surpasses it as data accumulates, offering better predictive accuracy and control effectiveness.

Practical Implications and Future Research

Practically, this research has significant implications for fields where data collection is costly or time-consuming, such as in biological systems or real-time mechanical control systems. The sparsity and interpretability of SINDY models could facilitate more understandable and trustworthy AI systems in critical applications. The work also suggests future directions in adaptive control systems, where models might need rapid reconfiguration in response to sudden changes in system dynamics or unexpected environmental inputs.

Conclusion

Overall, the paper provides a sophisticated and well-documented framework for those interested in advancing model-based control systems using limited data. The SINDY-MPC approach represents a promising step toward more adaptive, efficient, and interpretable control systems in complex environments. As this area of research evolves, we may anticipate further enhancements in the efficiency and capability of data-driven control systems, potentially extending into areas like robotics, aerospace, and beyond. The authors' contributions help pave the way toward a future where intelligent systems can seamlessly operate in dynamic and data-scarce conditions.