Ideas taken from https://services.anu.edu.au/files/development_opportunity/ResearchProposalTips_0.pdf

# Title / Topic
# Research Problem (Justification)
- Why does robust control exist
	- air conditioning example - but what if the plant is different? What is buddy leaves a window open
	- We can examine whether or not our controller (the ac unit) can handle the perturbed plant
	- We can know how open the window is before we have problems
	- We can guarantee this for this controller design and designed laws
- So if we do this can be sure when we build the unit that this is how it will perform?
	- Well if it's controlled with a microcontroller or other code based solution, no.
	- The abstraction between the design and the finished controller destroys the guarantee
	- Things can happen in implementation that make the controller built not true to design
- As a result, we need to reverify robustness on built controllers
	- This exists for structured perturbations. We 
# Gap In The Literature
### **Slide 1: Robust Control Foundations**

**Assertion:** Robust control ensures stability despite system discrepancies.  
**Evidence:**

- Controllers are based on physical models that differ from real systems.
- Robust control analyzes resilience to system perturbations.
- Evolved from single-input single-output to multi-input multi-output systems.  
    _(Cite Doyle, Green, Brunton)_

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### **Slide 2: Structured vs. Unstructured Perturbations**

**Assertion:** Robust control addresses structured and unstructured perturbations differently.  
**Evidence:**

- **Structured:** Based on physical tolerances (e.g., spring rates).
- **Unstructured:** Accounts for unmodeled dynamics and broader uncertainties.  
    _(Diagram comparing structured and unstructured perturbations)_  
    _(Cite Doyle, Green)_

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### **Slide 3: Disk-Based Unstructured Uncertainty**

**Assertion:** Disk-based perturbation quantifies unstructured uncertainties.  
**Evidence:**

- Key equation: $\tilde{P} = (1 + \Delta W_2) P$
    - $P$: Nominal plant.
    - $\Delta$: Perturbation transfer function.
    - $W_2$: Uncertainty envelope.
- Conditions for $W_2$ and $\Delta$:
    - $\left| \frac{\tilde{P}(j\omega)}{P(j\omega)} - 1 \right| \leq \beta |W_2(j\omega)|$
    - $||\Delta||_\infty \leq \beta$.

_(Include a visual of how $\Delta$ affects $P$)_

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### **Slide 4: Current Limitations in Robust Control**

**Assertion:** Current methods lack discrete examples of unstructured perturbations.  
**Evidence:**

- $\Delta$ is undefined for experimental robustness verification.
- Structured uncertainties are used experimentally but neglect unmodeled dynamics.  
    _(Cite Farzan, Hamilton)_

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### **Slide 5: Diffusion Models as a Solution**

**Assertion:** Diffusion models can generate unstructured perturbations.  
**Evidence:**

- Forward process transforms data to Gaussian distribution.
- Reverse process generates approximations of target data.
- Applications in protein folding, training data generation.  
    _(Diagram of forward/reverse processes in diffusion models)_  
    _(Cite Sohl-Dickstein, Abramson)_

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### **Slide 6: Parallels Between Diffusion Models and This Project**

**Assertion:** Diffusion models address sparse perturbation generation in engineering.  
**Evidence:**

- Diffusion models create diverse training data from sparse sets.
- Proposed approach: Generate unstructured perturbations from structured sets.  
    _(Illustration of sparse-to-diverse transformation concept)_
# Goals and Outcomes

# Research Methodology
# Metrics of Success
# Risks and Contingencies
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