End-to-End Learning Framework for Solving Non-Markovian Optimal Control

TL;DR

This paper introduces a novel end-to-end deep learning framework for optimal control of fractional-order systems, combining theoretical derivations with data-driven methods to address challenges in system identification and control.

Contribution

It presents the first end-to-end learning framework for fractional-order systems, integrating theoretical control solutions with deep learning for data-driven control policy learning.

Findings

Accurately models fractional-order system behaviors without Gaussian noise assumptions.

Develops a system identification method tailored for FOLTI systems.

Provides a theoretical analysis of sample complexity for control accuracy.

Abstract

Integer-order calculus often falls short in capturing the long-range dependencies and memory effects found in many real-world processes. Fractional calculus addresses these gaps via fractional-order integrals and derivatives, but fractional-order dynamical systems pose substantial challenges in system identification and optimal control due to the lack of standard control methodologies. In this paper, we theoretically derive the optimal control via linear quadratic regulator (LQR) for fractional-order linear time-invariant (FOLTI) systems and develop an end-to-end deep learning framework based on this theoretical foundation. Our approach establishes a rigorous mathematical model, derives analytical solutions, and incorporates deep learning to achieve data-driven optimal control of FOLTI systems. Our key contributions include: (i) proposing an innovative system identification method control strategy for FOLTI systems, (ii) developing the first end-to-end data-driven learning framework, Fractional-Order Learning for Optimal Control (FOLOC), that learns control policies from observed trajectories, and (iii) deriving a theoretical analysis of sample complexity to quantify the number of samples required for accurate optimal control in complex real-world problems. Experimental results indicate that our method accurately approximates fractional-order system behaviors without relying on Gaussian noise assumptions, pointing to promising avenues for advanced optimal control.