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Trajectory-Oriented Control Using Gradient Descent: An Unconventional Approach

Ramin Esmzad, Hamidreza Modares

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.

Trajectory-Oriented Control Using Gradient Descent: An Unconventional Approach

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.
Paper Structure (10 sections, 4 theorems, 32 equations, 3 figures)

This paper contains 10 sections, 4 theorems, 32 equations, 3 figures.

Table of Contents

  1. Introduction
  2. A review of GD algorithm
  3. A new interpretation of the GD algorithm for dynamical systems
  4. Problem Statement
  5. Stability Analysis of Trajectory-oriented GD Dynamics
  6. Design of a static feedback policy
  7. Connection to LQR
  8. Extension to Heavy-Ball Gradient method
  9. Simulations
  10. Conclusion

Key Result

Lemma 1

The system eq:gdl with the cost function eq:Vk is $\lambda-$contractive with $0< \lambda \leq 1$ (and thus stable) if there exists a $Y = P^{-1}\succ 0$ and $\Gamma \succ 0$ such that the following linear matrix inequality (LMI) holds Proof. The gradient of the cost $V_k$ with respect to the states is $\frac{\partial V_k}{\partial x_k}=2Px_k$, so the GD system eq:gdl can be simplified as which p

Figures (3)

  • Figure 1: SGD-like behavior of the vehicle steering model for parameters defined in \ref{['eq:gamma']}.
  • Figure 2: GD-like behavior of the vehicle steering model for parameters defined in \ref{['eq:Gamma']}
  • Figure 3: Explicitly shaping the behavior of the vehicle steering model using $\Gamma$

Theorems & Definitions (8)

  • Remark 1
  • Definition 1
  • Lemma 1
  • Theorem 1
  • Remark 2
  • Lemma 2
  • Remark 3
  • Theorem 2