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

2025-04-23

AI Summary

Conventional LQR-based control methods typically employ diagonal penalty matrices, which simplify parameter tuning and ensure positive definiteness but severely restrict the achievable control law space. To address this limitation, we propose a novel parameterization of the penalty matrix based on eigendecomposition: using a positive-definite diagonal eigenvalue matrix and an orthogonal eigenvector matrix as independent design variables. This formulation implicitly guarantees positive definiteness while enabling non-zero off-diagonal entries to expand control freedom. To our knowledge, this is the first systematic application of eigendecomposition to penalty matrix construction in LQR synthesis. Integrated with particle swarm optimization and Lyapunov-based stability analysis, the method maintains computational efficiency while significantly enhancing design flexibility. Evaluated on spacecraft attitude control and low-thrust trajectory optimization tasks, it achieves up to a 65% improvement in key performance metrics compared to conventional diagonal designs.

Technology Category

• Search and Optimization: Mixed Discrete/Continuous Search
• Intelligent Robots: State Estimation
• Constraint Satisfaction and Optimization: Constraint Optimization

Application Category

• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• User Modeling, Personalization and Recommendation: User privacy protection in personalized systems
• Security and Privacy: Applications of cryptography

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.

Problem

• Parameterize penalty matrices using eigendecomposition for control design
• Enhance performance and stability in aerospace control applications
• Optimize decision variables with particle swarm optimization

Innovation

• Eigendecomposition parameterizes penalty matrices
• Ensures positive-definiteness with off-diagonal entries
• Uses particle swarm for optimization