# 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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