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Feedback control plays a crucial role in optimizing the performance of dynamical systems by minimizing a predefined cost functional, to stabilize the system towards a desired state. Despite its undeniable effectiveness, the applicability of feedback control is often limited by the high dimensionality of state spaces, a challenge that is even more pronounced in parametric settings. To overcome these limitations, we employ the Dynamical Low-Rank Approximation (DLRA) methodology, which offers an efficient and accurate solution to high-dimensional feedback control problems.
Our approach significantly accelerates the solution of State-Dependent Riccati Equations (SDREs) for infinite horizon optimal control by providing a compact, low-dimensional representation of the system that evolves dynamically with respect to the problem. This not only enhances accuracy but also provides effective real-time control, solving several parametric instances simultaneously.
To further accelerate the solution of the SDREs, we propose cascade-control (cc-) algorithms that leverage the Riccati solution from a previous parameter to improve the convergence of Newton-based methods applied to SDREs. The robustness and efficiency of our algorithms are validated through nonlinear test cases. Comparative analysis demonstrates that the proposed ccDLRA outperforms global Proper Orthogonal Decomposition model order reduction (ccPOD), offering superior speed and accuracy when compared to the full-order model (ccFOM) solution.