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)_

---

## **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)_

---

## **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 $P$)_

---

## **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)_

---

## **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)_

---

## **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
## **Slide 1: Research Motivation**

**Assertion:** Current methods for generating unstructured perturbations are limited in flexibility and generalizability.

- **Evidence:**
    - Unstructured perturbations lack adaptability to various scenarios.
    - Proposed approach leverages diffusion generative models for flexible perturbation generation.

**Visuals:**

- A flowchart contrasting traditional perturbation methods vs. diffusion models.

---

## **Slide 2: Diffusion Model Features**

**Assertion:** Frequency response data forms the foundation for feature creation in diffusion models.

- **Evidence:**
    - Features discretize dynamics into a vector of magnitude and phase.
    - Supports training without imparting unintended structure.

**Visuals:**

- Diagram from Figure 1 showing the discretization of frequency response.

---

## **Slide 3: Creating Frequency Features**

**Assertion:** Discretizing the frequency response enables scalable feature sets.

- **Evidence:**
    - Fine resolution for complex behavior or coarse for computational efficiency.
    - Features provide physical context across frequency scales.

**Visuals:**

- Table comparing fine vs. coarse frequency sampling.
- Annotated example of magnitude/phase vector with scales labeled.

---

## **Slide 4: Training the Diffusion Model**

**Assertion:** Diffusion models learn unstructured perturbations through iterative noise transformation.

- **Evidence:**
    - Forward process adds noise; reverse process removes it.
    - Training maximizes log-likelihood between input and reconstructed data.

**Visuals:**

- Flowchart of the diffusion training process.
- Key equations (e.g., Eq. \ref{forward_kernel} and \ref{reverse_kernel}) simplified with annotations.

---

## **Slide 5: Generating New Perturbations**

**Assertion:** The trained diffusion model generates diverse and flexible perturbations.

- **Evidence:**
    - Outputs are probabilistic, enabling variability.
    - Perturbation level controlled by adjusting time steps.

**Visuals:**

- Illustration of forward/reverse process with arrows and annotations.
- Graph showing interpolation from partial time steps.

---

## **Slide 6: Ensuring Valid Perturbations**

**Assertion:** Generated perturbations must meet robust control requirements.

- **Evidence:**
    - No additional right-hand plane poles.
    - Supremum gain of Δ below threshold β.

**Visuals:**

- Diagram of pole-zero constraints.
- Workflow for verifying Δ and fitting transfer functions.

---

## **Slide 7: Advantages of Diffusion Models** NOT INCLUDED SO FAR

**Assertion:** Diffusion models provide a novel solution for generating unstructured perturbations.

- **Evidence:**
    - Introduce non-deterministic variability into perturbations.
    - Overcome the limitations of traditional structured approaches.

**Visuals:**

- Comparative chart: structured vs. unstructured methods.
- Examples of perturbed frequency responses generated by the model.
# Research Tasks
## **Slide 1: Research Tasks Overview**

**Assertion:** This research aims to address verification challenges through structured tasks.

- **Evidence:** Four key research tasks support the proposed outcomes:
    1. Mission-Beneficiary Fit
    2. Find Robust Systems
    3. Create Diffusion Model
    4. Analyze and Disseminate Results

**Visuals:**

- A process diagram summarizing the four tasks.

---

## **Slide 2: Mission-Beneficiary Fit**

**Assertion:** Understanding beneficiaries ensures relevance and impact of this research.

- **Evidence:**
    - Beneficiary Identification: Research how control engineers might use this work.
    - Value Proposition: Define and align capabilities with verification needs.

**Visuals:**

- Chart or table identifying beneficiaries and their verification needs.

---

## **Slide 3: Find Robust Systems**

**Assertion:** Identifying relevant plants ensures practical applicability of results.

- **Evidence:**
    - Literature Review: Investigate industrial applications of robust control verification.
    - Create Example Plants: Reconstruct models of prominent systems for demonstrations.

**Visuals:**

- Example of a controlled industrial process.
- Flowchart of the literature review and modeling process.

---

## **Slide 4: Create Diffusion Model**

**Assertion:** A diffusion model is central to generating unstructured perturbations.

- **Evidence:**
    - Identify Model Structure: Choose an architecture (e.g., U-Net).
    - Train Model: Develop training data and optimize performance.
    - Generate Perturbations: Apply the model to example plants.

**Visuals:**

- Diagram of a U-Net-based architecture.
- Example of generated unstructured perturbations.

---

## **Slide 5: Analyze and Disseminate Results**

**Assertion:** Communicating findings ensures broader adoption and state-of-the-art advancements.

- **Evidence:**
    - Publish results in academic journals.
    - Demonstrate impact on robustness verification practices.

**Visuals:**

- Example journal or conference targets.
- Overview of the dissemination process.
# Metrics of Success
## **Slide 1: Metrics of Success Overview**

**Assertion:** Project success will be evaluated through milestone tracking and outcome-based metrics.

- **Evidence:**
    1. Goals and Outcomes: Milestones tied to the objectives of this research.
    2. Unstructured Perturbation Evaluation: Metrics to assess diffusion model output.

**Visuals:**

- High-level flowchart showing the two categories of success metrics.

---

## **Slide 2: Goals and Outcomes**

**Assertion:** The research aims to deliver specific capabilities for creating unstructured perturbations.

- **Evidence:**
    - Approximate unstructured sets through numerous perturbed plants.
    - Perturb nominal plants using the diffusion model.
    - Generate frequency-domain responses from training data.

**Visuals:**

- Table summarizing the three goals and their significance.
- Conceptual graphic of a nominal plant with perturbed versions around it.

---

## **Slide 3: Unstructured Perturbation Evaluation**

**Assertion:** The diffusion model's success will be judged on distribution and diversity of perturbations.

- **Evidence:**
    - Distribution: Verify uniform coverage of the multiplicative uncertainty disk.
    - Diversity: Assess non-parametric, dissimilar perturbations among examples.

**Visuals:**

- Example complex plane with plotted perturbed plants.
- Graph comparing similarity metrics across perturbations.

---

## **Slide 4: Statistical Evaluation (Optional Deep Dive)**

**Assertion:** Statistical analysis ensures robustness and diversity in generated perturbations.

- **Evidence:**
    - Standard statistical tests applied to the perturbation set.
    - Covariance vectors calculated for key frequency ranges.

**Visuals:**

- Example statistical output or covariance plot for one frequency band.
- Caption explaining its role in validating uniform coverage.
# Risks and Contingencies
## **Slide 1: Risks and Contingencies Overview**

**Assertion:** This research has identified key risks and developed contingencies to address them.

- **Evidence:**
    1. Computational demands of diffusion models.
    2. Training data sufficiency.
    3. Generalization of interpolation methods to perturbations.

**Visuals:**

- A risk-contingency matrix outlining the key challenges and corresponding mitigations.

---

## **Slide 2: Risk 1 - Computational Demands**

**Assertion:** Diffusion models may require significant computational resources during training and inference.

- **Evidence:**
    - Reverse process inference is computationally intensive due to per-step calculations.
    - Training complexity scales with model structure and feature count.

**Contingencies:**

1. Utilize the University of Pittsburgh’s CRC supercomputing resources.
2. Reduce data features while monitoring model performance.

**Visuals:**

- Diagram comparing computational cost across time steps.
- Icon of computational resources with CRC logo or similar.

---

## **Slide 3: Risk 2 - Insufficient Training Data**

**Assertion:** Structured perturbations alone may not condition the model adequately.

- **Evidence:**
    - Structured perturbations simplify training but may lack diversity.

**Contingencies:**

1. Augment training with manually or algorithmically generated $\Delta$ examples (e.g., bounded by supermum gain $\beta$).
2. Diversify training data sources to improve robustness.

**Visuals:**

- Example of structured vs. manual perturbation samples on the complex plane.
- Flowchart showing training data augmentation process.

---

## **Slide 4: Risk 3 - Interpolation Limitations**

**Assertion:** Interpolation methods may fail to regulate perturbations effectively.

- **Evidence:**
    - Image-based interpolation success may not generalize to this domain.

**Contingencies:**

1. Implement $r(\mathcal{P}_t)$-based reverse process steering for controlled perturbations【cite sources】.
2. Explore alternative interpolation techniques tailored to frequency domain applications.

**Visuals:**

- Conceptual illustration of $r(\mathcal{P}_t)$ steering function in reverse process.
- Example showing failure of simple interpolation and correction with $r(\mathcal{P}_t)$.

---

## Slide 5: Risk Mitigation Framework (Optional Summary Slide)

**Assertion:** Addressing risks proactively ensures project success.

- **Evidence:**
    - Computational strategies, diversified training, and alternative steering methods safeguard outcomes.

**Visuals:**

- Funnel graphic showing risks addressed through mitigations leading to project success.