Obsidian/4 Qualifying Exam/3 Notes/3 Notes - README.md

Table of Contents for 3 Notes

Files

• Feedback Control Theory.md
• How is robust control validation done?.md
• Robust Control.md
• What is gain scheduling?.md

Summary

This is a comprehensive set of notes on Feedback Control Theory, covering various topics such as sensitivity and complementary sensitivity functions, robustness and stability, and uncertainty and perturbation. Here's a brief summary of the key points:

Sensitivity and Complementary Sensitivity Functions

• The sensitivity function S is defined as 1/(1+L), where L is the loop gain.
• The complementary sensitivity function T is defined as L/(1+L).
• These functions are used to analyze the performance of a feedback system.

Robustness and Stability

• A controller C is robust to a set of plants \mathcal{P} with respect to a characteristic if this characteristic holds for every plant in \mathcal{P}.
• A system is robustly stable if it is internally stable for every plant in the set \mathcal{P}: || \Delta W_2 T ||_\infty < 1.
• Nominal performance is achieved simultaneously when || \text{max} (|W_1S|, |W_2 T|)||_\infty < 1.

Uncertainty and Perturbation

• The multiplicative disk perturbation is defined as \tilde P = (1+\Delta W_2)P, where P is a nominal plant transfer function.
• ||\Delta||_\infty <1 ensures that the system remains stable for all plants in the set \mathcal{P}.

Key Concepts

• Robustness: the distance between L and -1 for all frequencies, which determines how much room there is for plant perturbation before becoming unstable.
• Uncertainty profile: |W_2(j\omega)| describes a disk in the complex plane that indicates the maximum amount of uncertainty allowed.

Overall, these notes provide a solid foundation for understanding the concepts and techniques used in feedback control theory, particularly in the context of robustness and stability analysis.

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