Obsidian/.archive/4 Qualifying Exam/4 Presentation/Outline.md

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)

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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)

---

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.

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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.

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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.

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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.

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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.

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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).

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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.