In this article, we explore a novel approach to subspace methods for model predictive control (MPC) that leverages the power of dimensionality reduction. By using principal component analysis (PCA) as a starting point, we can identify the most important features in a dataset and transform them into a lower-dimensional space. This allows us to significantly reduce the number of variables while retaining the most crucial information for predictive control.
Background
MPC is a popular control technique that seeks to optimize a system’s behavior by predicting its future behavior and adjusting its control inputs accordingly. However, in high-dimensional systems, MPC can become computationally intractable due to the vast number of possible state and input combinations. To address this challenge, researchers have turned to subspace methods, which seek to reduce the dimensionality of the system while preserving its essential properties.
Our Approach
To develop a data-driven approach for subspace methods, we build upon PCA, an established technique in statistics. By shifting the data using an offset, PCA computes the optimal basis of an r-dimensional subspace that best describes the shifted data points. We use the columns of the resulting matrix U as an orthonormal basis to represent the subspace in the Stiefel manifold ST(r,d). This provides a convenient way to represent bases for subspaces in high-dimensional spaces, as the columns of U form an orthonormal basis of an r-dimensional subspace of Rd.
Key Idea
The key idea behind our approach is to use PCA as a starting point and then modify it to suit the needs of MPC. By leveraging the structure of the Stiefel manifold, we can efficiently compute bases for subspaces that capture the essential features of the data while reducing its dimensionality. This allows us to significantly reduce the number of variables in the system while retaining the most important information for predictive control.
Main Contributions
Our main contributions are twofold
- We provide a novel approach to subspace methods for MPC that leverages PCA as a starting point and modifies it to suit the needs of MPC. This allows us to efficiently compute bases for subspaces that capture the essential features of the data while reducing its dimensionality.
- We demonstrate the effectiveness of our approach through extensive simulations on a cartpole system, showing that our method can significantly reduce the number of variables in the system while retaining the most important information for predictive control.
Conclusion
In summary, we have presented a data-driven approach to subspace methods for model predictive control that leverages principal component analysis as a starting point. By modifying PCA to suit the needs of MPC and using the Stiefel manifold to represent bases for subspaces, we can significantly reduce the dimensionality of the system while retaining the most important information for predictive control. Our approach offers a promising way to improve the computational efficiency and accuracy of MPC algorithms in high-dimensional systems.
