Trajectory-Oriented Control Using Gradient Descent: An Unconventional Approach

TL;DR

This paper presents a novel gradient descent-based method for control system optimization, representing stable dynamics through matrices and enabling trajectory shaping, offering a new framework for feedback control design.

Contribution

Introduces a new matrix-based gradient descent approach for control optimization, linking it to LQR and enabling trajectory shaping in control systems.

Findings

Any stable system can be represented with the proposed matrices.

The approach is equivalent to an LQR with suitable weights.

Trajectories can be shaped to achieve desired behaviors.

Abstract

In this work, we introduce a novel gradient descent-based approach for optimizing control systems, leveraging a new representation of stable closed-loop dynamics as a function of two matrices i.e. the step size or direction matrix and value matrix of the Lyapunov cost function. This formulation provides a new framework for analyzing and designing feedback control laws. We show that any stable closed-loop system can be expressed in this form with appropriate values for the step size and value matrices. Furthermore, we show that this parameterization of the closed-loop system is equivalent to a linear quadratic regulator for appropriately chosen weighting matrices. We also show that trajectories can be shaped using this approach to achieve a desired closed-loop behavior.