# Eigendecomposition Parameterization of Penalty Matrices for Enhanced Control Design: Aerospace Applications

**Authors:** Nicholas P. Nurre, Ehsan Taheri  
**arXiv:** [2504.16328](https://arxiv.org/abs/2504.16328)  
**Date:** 2025-04-23  
**Summary page:** https://www.summarizepaper.com/en/paper/2504.16328v1/

## Key points

- Tuning square weight/penalty matrices in quadratic functions/costs is crucial for enhancing performance and stability output in modern control algorithms.
- Diagonal penalty matrices are commonly used in control methods like LQR, model predictive control, and Lyapunov-based control due to their simplicity in gain-tuning and positive-definiteness enforcement.
- A new approach utilizing eigendecomposition to parameterize penalty matrices has been proposed, allowing for positive-definiteness with non-zero off-diagonal entries to be implicitly satisfied.
- The method offers significant computational and implementation advantages while expanding the range of achievable controls.
- The research explores the application of full penalty matrices in solving three specific control problems:
- a variation of Zermelo's navigation problem
- minimum-energy spacecraft attitude control using LQR and Lyapunov-based methods
- minimum-fuel and minimum-time Lyapunov-based low-thrust trajectory design.
- Particle swarm optimization is employed to optimize the decision variables parameterizing the penalty matrices.
- Results demonstrate significant improvements of up to 65% in the performance objective across example problems when utilizing this novel eigendecomposition approach.
- This study contributes valuable insights into advancing control design techniques for aerospace applications by considering off-diagonal penalties within a broader framework.

## Summary

In the field of modern control algorithms, tuning square weight/penalty matrices in quadratic functions/costs is crucial for enhancing performance and stability output. Diagonal penalty matrices are commonly used in control methods like linear quadratic regulator (LQR), model predictive control, and Lyapunov-based control due to their simplicity in gain-tuning and positive-definiteness enforcement. However, a new approach utilizing eigendecomposition to parameterize penalty matrices has been proposed in this study. This method allows for positive-definiteness with non-zero off-diagonal entries to be implicitly satisfied, offering significant computational and implementation advantages while expanding the range of achievable controls. The research explores the application of full penalty matrices in solving three specific control problems: 1) a variation of Zermelo's navigation problem, 2) minimum-energy spacecraft attitude control using LQR and Lyapunov-based methods, and 3) minimum-fuel and minimum-time Lyapunov-based low-thrust trajectory design. Notably, this study stands out as one of the first to investigate the use of full penalty matrices in aerospace control applications to improve performance objectives such as propellant consumption. To optimize the decision variables parameterizing the penalty matrices, particle swarm optimization is employed. The results demonstrate significant improvements of up to 65% in the performance objective across the example problems when utilizing this novel eigendecomposition approach. By considering off-diagonal penalties within a broader framework, this research contributes valuable insights into advancing control design techniques for aerospace applications.