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

Nicholas P. Nurre, Ehsan Taheri

TL;DR

This work addresses the limitation of diagonal penalty matrices in LQR, MPC, and Lyapunov-based control by enabling cross-coupling through full penalty matrices. It introduces an eigendecomposition-based parameterization $K = Q\Lambda Q^T$, with $\lambda_i>0$, and discusses orthogonal-parameterizations (Cayley transform, Givens rotations, GEAGSP), selecting GEAGSP for results. PSO optimizes the eigenvalues and rotation-parameters across three aerospace problems: a Zermelo-like navigation, minimum-energy spacecraft attitude control, and low-thrust trajectory design. Across these problems, full penalty matrices yield substantial improvements, with gains up to about 65%, demonstrating broader design flexibility and potential for improved stability and efficiency in aerospace applications.

Abstract

Modern control algorithms require tuning of square weight/penalty matrices appearing in quadratic functions/costs to improve performance and/or stability output. Due to simplicity in gain-tuning and enforcing positive-definiteness, diagonal penalty matrices are used extensively in control methods such as linear quadratic regulator (LQR), model predictive control, and Lyapunov-based control. In this paper, we propose an eigendecomposition approach to parameterize penalty matrices, allowing positive-definiteness with non-zero off-diagonal entries to be implicitly satisfied, which not only offers notable computational and implementation advantages, but broadens the class of achievable controls. We solve three control problems: 1) a variation of Zermelo's navigation problem, 2) minimum-energy spacecraft attitude control using both LQR and Lyapunov-based methods, and 3) minimum-fuel and minimum-time Lyapunov-based low-thrust trajectory design. Particle swarm optimization is used to optimize the decision variables, which will parameterize the penalty matrices. The results demonstrate improvements of up to 65% in the performance objective in the example problems utilizing the proposed method.

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

TL;DR

This work addresses the limitation of diagonal penalty matrices in LQR, MPC, and Lyapunov-based control by enabling cross-coupling through full penalty matrices. It introduces an eigendecomposition-based parameterization , with , and discusses orthogonal-parameterizations (Cayley transform, Givens rotations, GEAGSP), selecting GEAGSP for results. PSO optimizes the eigenvalues and rotation-parameters across three aerospace problems: a Zermelo-like navigation, minimum-energy spacecraft attitude control, and low-thrust trajectory design. Across these problems, full penalty matrices yield substantial improvements, with gains up to about 65%, demonstrating broader design flexibility and potential for improved stability and efficiency in aerospace applications.

Abstract

Modern control algorithms require tuning of square weight/penalty matrices appearing in quadratic functions/costs to improve performance and/or stability output. Due to simplicity in gain-tuning and enforcing positive-definiteness, diagonal penalty matrices are used extensively in control methods such as linear quadratic regulator (LQR), model predictive control, and Lyapunov-based control. In this paper, we propose an eigendecomposition approach to parameterize penalty matrices, allowing positive-definiteness with non-zero off-diagonal entries to be implicitly satisfied, which not only offers notable computational and implementation advantages, but broadens the class of achievable controls. We solve three control problems: 1) a variation of Zermelo's navigation problem, 2) minimum-energy spacecraft attitude control using both LQR and Lyapunov-based methods, and 3) minimum-fuel and minimum-time Lyapunov-based low-thrust trajectory design. Particle swarm optimization is used to optimize the decision variables, which will parameterize the penalty matrices. The results demonstrate improvements of up to 65% in the performance objective in the example problems utilizing the proposed method.

Paper Structure

This paper contains 14 sections, 41 equations, 18 figures, 5 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Penalty Matrix Parameterization
  3. Cayley Transform
  4. Givens Rotation
  5. Generalized Euler Angles and Gram–Schmidt Process
  6. Zermelo's Navigation Problem
  7. Spacecraft Attitude Control Problem
  8. LQR Attitude Control Design
  9. Lyapunov-Based Attitude Control Design
  10. Low-Thrust Trajectory Optimization Problem
  11. Generic Lyapunov Control Law
  12. Q-law Low-Thrust Trajectory Design Method
  13. Conclusion
  14. Numerical Values of the Penalty Matrices

Figures (18)

  • Figure 1: Zermelo's problem: state space with the trajectories from each solution with the vector field of nonlinearities in the dynamics.
  • Figure 2: Zermelo's problem: Lyapunov functions vs. states.
  • Figure 3: Zermelo's problem: Lyapunov functions time-derivatives vs. states.
  • Figure 4: Zermelo's problem: Euclidean norm of control function vs. states.
  • Figure 5: Comparison of the cost values for the LQR and Lyapunov-based controller with diagonal and full penalty matrices.
  • ...and 13 more figures

Theorems & Definitions (3)

  • Remark 1
  • Remark 2
  • Remark 3