Moment Relaxations for Linear State Feedback Controller Synthesis with Non-Convex Quadratic Costs and Constraints

Centrala begrepp

Moment matrices offer a flexible approach to handle non-convex constraints in state feedback synthesis, allowing for deterministic or stochastic solutions.

Sammanfattning

The content introduces a method using moment matrices to address non-convex costs and constraints in state feedback synthesis. It discusses the derivation of convex controller synthesis conditions, the application of moment matrices to linear systems, and the benefits of linear controller synthesis with non-convex constraints through compelling examples. The article also explores the relationship between convexification strategies and standard state feedback synthesis techniques. Two numerical examples are presented to illustrate the effectiveness of the proposed approach in solving control problems with soft and hard constraints.

Introduction:

Presents a method using moment matrices for handling non-convex costs and constraints in state feedback synthesis.
Discusses deriving convex controller synthesis conditions similar to existing approaches.

Data Extraction:

"Lossless convexifications exist for quadratic stabilization of linear systems."
"Controller synthesis for linear systems via second-order moment matrices enables computation of H∞-norm."

Quotations:

"Deterministic means that the optimal policy coincides with an optimal policy of the original problem."

Inquiry and Critical Thinking:

How can moment matrices be applied to other areas beyond control theory?
What are potential drawbacks or limitations of relying on stochastic solutions in control problems?
How might the concept of convexification using moment matrices inspire new approaches in optimization algorithms?

Statistik

Lossless convexifications exist for quadratic stabilization of linear systems.
Controller synthesis for linear systems via second-order moment matrices enables computation of H∞-norm.

Citat

"Deterministic means that the optimal policy coincides with an optimal policy of the original problem."

Viktiga insikter från

On moment relaxations for linear state feedback controller synthesis with non-convex quadratic costs and constraints

by Dennis Graml... på arxiv.org 03-25-2024

https://arxiv.org/pdf/2403.15228.pdf

Djupare frågor

How can moment matrices be applied to other areas beyond control theory

Moment matrices can be applied to various areas beyond control theory, such as machine learning, signal processing, and optimization. In machine learning, moment matrices can be used for feature selection, dimensionality reduction, and clustering. They provide a way to capture statistical information about the data distribution in a compact form that is useful for analysis and modeling. In signal processing, moment matrices can help in denoising signals, extracting features from audio or image data, and pattern recognition tasks. Additionally, in optimization problems outside of control theory, moment matrices can aid in convex relaxation techniques for non-convex optimization problems by transforming them into tractable convex forms.

What are potential drawbacks or limitations of relying on stochastic solutions in control problems

Relying on stochastic solutions in control problems may have some drawbacks or limitations. One major limitation is the lack of guarantees on the performance or stability of the system when using stochastic policies. Stochastic solutions introduce randomness into the decision-making process which can lead to unpredictable behavior and potentially suboptimal outcomes. Moreover, ensuring constraints are satisfied with high probability becomes challenging with stochastic policies as they may violate constraints more frequently than deterministic ones. Another drawback is the increased complexity in analyzing system behavior due to randomness introduced by stochastic policies.

How might the concept of convexification using moment matrices inspire new approaches in optimization algorithms

The concept of convexification using moment matrices has the potential to inspire new approaches in optimization algorithms by providing a structured framework for handling non-convex problems efficiently. By leveraging moment matrices to represent complex systems or constraints as expectations over random variables, researchers can develop novel algorithms that exploit this representation for optimization purposes. This approach could lead to advancements in solving challenging non-convex optimization problems by converting them into simpler convex formulations through relaxation techniques based on moments.