DCSV-Krylov Exploration for Control Allocation in Multitime Scale Systems

Abstract

Control allocation is essential for managing MIMO dynamic systems, such as autonomous vehicles and robotics, which operate across varying temporal scales, posing challenges in optimizing control actions. A common approach involves traditional solvers for overdetermined systems---most notably the singular value decomposition (SVD), the least squares QR algorithm (LSQR), and Arnoldi-type methods---, which, while effective for homogeneous temporal scales, struggle to handle mixed dynamics. Their inability to explicitly account for temporal variations and uncertainties often results in suboptimal performance and high computational costs. To address these challenges, this work introduces a novel combination of the Dynamic Confined Space of Velocities (DCSV) method with Krylov subspace exploration, significantly enhancing control allocation in mixed-temporal-scale systems. The DCSV method generates optimized initial estimates, which are iteratively refined using Krylov methods, achieving precise control with reduced computational costs. Comparative analysis against traditional solvers demonstrates that DCSV consistently outperforms these methods, achieving higher accuracy with fewer computational resources, particularly in systems with mixed dynamic scales. Since LSQR is closely related to quadratic programming (QP) and linear programming (LP) through its cost function, and SVD directly computes generalized inverses, their evaluation provides a basis for comparison with conventional control allocation strategies. The proposed method surpasses these approaches by achieving a better balance between computational efficiency and control precision. In addition, the uncertainty principle embedded in the DCSV framework establishes theoretical limits that ensure robustness in the allocation process, supported by a structured sequential exploration process formulated via pseudocode. By optimizing initial estimates through DCSV, fewer refinement iterations are needed, leading to faster and more accurate control allocation, as confirmed by formal analysis. This methodology offers a robust and efficient solution to the challenges posed by mixed temporal scales, with enhanced adaptability for real-world applications in dynamic and computationally constrained environments.