408 lines
13 KiB
Markdown
408 lines
13 KiB
Markdown
Ideas taken from https://services.anu.edu.au/files/development_opportunity/ResearchProposalTips_0.pdf
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# Title / Topic
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# Research Problem (Justification)
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- Why does robust control exist
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- air conditioning example - but what if the plant is different? What is buddy leaves a window open
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- We can examine whether or not our controller (the ac unit) can handle the perturbed plant
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- We can know how open the window is before we have problems
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- We can guarantee this for this controller design and designed laws
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- So if we do this can be sure when we build the unit that this is how it will perform?
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- Well if it's controlled with a microcontroller or other code based solution, no.
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- The abstraction between the design and the finished controller destroys the guarantee
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- Things can happen in implementation that make the controller built not true to design
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- As a result, we need to reverify robustness on built controllers
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- This exists for structured perturbations. We
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# Gap In The Literature
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## **Slide 1: Robust Control Foundations**
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**Assertion:** Robust control ensures stability despite system discrepancies.
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**Evidence:**
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- Controllers are based on physical models that differ from real systems.
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- Robust control analyzes resilience to system perturbations.
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- Evolved from single-input single-output to multi-input multi-output systems.
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_(Cite Doyle, Green, Brunton)_
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---
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## **Slide 2: Structured vs. Unstructured Perturbations**
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**Assertion:** Robust control addresses structured and unstructured perturbations differently.
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**Evidence:**
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- **Structured:** Based on physical tolerances (e.g., spring rates).
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- **Unstructured:** Accounts for unmodeled dynamics and broader uncertainties.
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_(Diagram comparing structured and unstructured perturbations)_
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_(Cite Doyle, Green)_
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---
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## **Slide 3: Disk-Based Unstructured Uncertainty**
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**Assertion:** Disk-based perturbation quantifies unstructured uncertainties.
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**Evidence:**
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- Key equation: $\tilde{P} = (1 + \Delta W_2) P$
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- $P$: Nominal plant.
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- $\Delta$: Perturbation transfer function.
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- $W_2$: Uncertainty envelope.
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- Conditions for $W_2$ and $\Delta$:
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- $\left| \frac{\tilde{P}(j\omega)}{P(j\omega)} - 1 \right| \leq \beta |W_2(j\omega)|$
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- $||\Delta||_\infty \leq \beta$.
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_(Include a visual of how $\Delta$ affects $P$)_
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---
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## **Slide 4: Current Limitations in Robust Control**
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**Assertion:** Current methods lack discrete examples of unstructured perturbations.
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**Evidence:**
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- $\Delta$ is undefined for experimental robustness verification.
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- Structured uncertainties are used experimentally but neglect unmodeled dynamics.
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_(Cite Farzan, Hamilton)_
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---
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## **Slide 5: Diffusion Models as a Solution**
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**Assertion:** Diffusion models can generate unstructured perturbations.
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**Evidence:**
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- Forward process transforms data to Gaussian distribution.
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- Reverse process generates approximations of target data.
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- Applications in protein folding, training data generation.
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_(Diagram of forward/reverse processes in diffusion models)_
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_(Cite Sohl-Dickstein, Abramson)_
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---
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## **Slide 6: Parallels Between Diffusion Models and This Project**
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**Assertion:** Diffusion models address sparse perturbation generation in engineering.
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**Evidence:**
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- Diffusion models create diverse training data from sparse sets.
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- Proposed approach: Generate unstructured perturbations from structured sets.
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_(Illustration of sparse-to-diverse transformation concept)_
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# Goals and Outcomes
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# Research Methodology
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## **Slide 1: Research Motivation**
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**Assertion:** Current methods for generating unstructured perturbations are limited in flexibility and generalizability.
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- **Evidence:**
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- Unstructured perturbations lack adaptability to various scenarios.
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- Proposed approach leverages diffusion generative models for flexible perturbation generation.
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**Visuals:**
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- A flowchart contrasting traditional perturbation methods vs. diffusion models.
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---
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## **Slide 2: Diffusion Model Features**
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**Assertion:** Frequency response data forms the foundation for feature creation in diffusion models.
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- **Evidence:**
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- Features discretize dynamics into a vector of magnitude and phase.
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- Supports training without imparting unintended structure.
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**Visuals:**
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- Diagram from Figure 1 showing the discretization of frequency response.
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---
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## **Slide 3: Creating Frequency Features**
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**Assertion:** Discretizing the frequency response enables scalable feature sets.
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- **Evidence:**
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- Fine resolution for complex behavior or coarse for computational efficiency.
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- Features provide physical context across frequency scales.
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**Visuals:**
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- Table comparing fine vs. coarse frequency sampling.
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- Annotated example of magnitude/phase vector with scales labeled.
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---
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## **Slide 4: Training the Diffusion Model**
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**Assertion:** Diffusion models learn unstructured perturbations through iterative noise transformation.
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- **Evidence:**
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- Forward process adds noise; reverse process removes it.
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- Training maximizes log-likelihood between input and reconstructed data.
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**Visuals:**
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- Flowchart of the diffusion training process.
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- Key equations (e.g., Eq. \ref{forward_kernel} and \ref{reverse_kernel}) simplified with annotations.
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---
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## **Slide 5: Generating New Perturbations**
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**Assertion:** The trained diffusion model generates diverse and flexible perturbations.
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- **Evidence:**
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- Outputs are probabilistic, enabling variability.
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- Perturbation level controlled by adjusting time steps.
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**Visuals:**
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- Illustration of forward/reverse process with arrows and annotations.
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- Graph showing interpolation from partial time steps.
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---
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## **Slide 6: Ensuring Valid Perturbations**
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**Assertion:** Generated perturbations must meet robust control requirements.
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- **Evidence:**
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- No additional right-hand plane poles.
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- Supremum gain of Δ below threshold β.
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**Visuals:**
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- Diagram of pole-zero constraints.
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- Workflow for verifying Δ and fitting transfer functions.
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---
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## **Slide 7: Advantages of Diffusion Models** NOT INCLUDED SO FAR
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**Assertion:** Diffusion models provide a novel solution for generating unstructured perturbations.
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- **Evidence:**
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- Introduce non-deterministic variability into perturbations.
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- Overcome the limitations of traditional structured approaches.
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**Visuals:**
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- Comparative chart: structured vs. unstructured methods.
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- Examples of perturbed frequency responses generated by the model.
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# Research Tasks
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## **Slide 1: Research Tasks Overview**
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**Assertion:** This research aims to address verification challenges through structured tasks.
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- **Evidence:** Four key research tasks support the proposed outcomes:
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1. Mission-Beneficiary Fit
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2. Find Robust Systems
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3. Create Diffusion Model
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4. Analyze and Disseminate Results
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**Visuals:**
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- A process diagram summarizing the four tasks.
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---
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## **Slide 2: Mission-Beneficiary Fit**
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**Assertion:** Understanding beneficiaries ensures relevance and impact of this research.
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- **Evidence:**
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- Beneficiary Identification: Research how control engineers might use this work.
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- Value Proposition: Define and align capabilities with verification needs.
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**Visuals:**
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- Chart or table identifying beneficiaries and their verification needs.
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---
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## **Slide 3: Find Robust Systems**
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**Assertion:** Identifying relevant plants ensures practical applicability of results.
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- **Evidence:**
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- Literature Review: Investigate industrial applications of robust control verification.
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- Create Example Plants: Reconstruct models of prominent systems for demonstrations.
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**Visuals:**
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- Example of a controlled industrial process.
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- Flowchart of the literature review and modeling process.
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---
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## **Slide 4: Create Diffusion Model**
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**Assertion:** A diffusion model is central to generating unstructured perturbations.
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- **Evidence:**
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- Identify Model Structure: Choose an architecture (e.g., U-Net).
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- Train Model: Develop training data and optimize performance.
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- Generate Perturbations: Apply the model to example plants.
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**Visuals:**
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- Diagram of a U-Net-based architecture.
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- Example of generated unstructured perturbations.
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---
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## **Slide 5: Analyze and Disseminate Results**
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**Assertion:** Communicating findings ensures broader adoption and state-of-the-art advancements.
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- **Evidence:**
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- Publish results in academic journals.
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- Demonstrate impact on robustness verification practices.
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**Visuals:**
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- Example journal or conference targets.
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- Overview of the dissemination process.
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# Metrics of Success
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## **Slide 1: Metrics of Success Overview**
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**Assertion:** Project success will be evaluated through milestone tracking and outcome-based metrics.
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- **Evidence:**
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1. Goals and Outcomes: Milestones tied to the objectives of this research.
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2. Unstructured Perturbation Evaluation: Metrics to assess diffusion model output.
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**Visuals:**
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- High-level flowchart showing the two categories of success metrics.
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---
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## **Slide 2: Goals and Outcomes**
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**Assertion:** The research aims to deliver specific capabilities for creating unstructured perturbations.
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- **Evidence:**
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- Approximate unstructured sets through numerous perturbed plants.
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- Perturb nominal plants using the diffusion model.
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- Generate frequency-domain responses from training data.
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**Visuals:**
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- Table summarizing the three goals and their significance.
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- Conceptual graphic of a nominal plant with perturbed versions around it.
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---
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## **Slide 3: Unstructured Perturbation Evaluation**
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**Assertion:** The diffusion model's success will be judged on distribution and diversity of perturbations.
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- **Evidence:**
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- Distribution: Verify uniform coverage of the multiplicative uncertainty disk.
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- Diversity: Assess non-parametric, dissimilar perturbations among examples.
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**Visuals:**
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- Example complex plane with plotted perturbed plants.
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- Graph comparing similarity metrics across perturbations.
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---
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## **Slide 4: Statistical Evaluation (Optional Deep Dive)**
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**Assertion:** Statistical analysis ensures robustness and diversity in generated perturbations.
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- **Evidence:**
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- Standard statistical tests applied to the perturbation set.
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- Covariance vectors calculated for key frequency ranges.
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**Visuals:**
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- Example statistical output or covariance plot for one frequency band.
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- Caption explaining its role in validating uniform coverage.
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# Risks and Contingencies
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## **Slide 1: Risks and Contingencies Overview**
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**Assertion:** This research has identified key risks and developed contingencies to address them.
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- **Evidence:**
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1. Computational demands of diffusion models.
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2. Training data sufficiency.
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3. Generalization of interpolation methods to perturbations.
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**Visuals:**
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- A risk-contingency matrix outlining the key challenges and corresponding mitigations.
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---
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## **Slide 2: Risk 1 - Computational Demands**
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**Assertion:** Diffusion models may require significant computational resources during training and inference.
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- **Evidence:**
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- Reverse process inference is computationally intensive due to per-step calculations.
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- Training complexity scales with model structure and feature count.
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**Contingencies:**
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1. Utilize the University of Pittsburgh’s CRC supercomputing resources.
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2. Reduce data features while monitoring model performance.
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**Visuals:**
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- Diagram comparing computational cost across time steps.
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- Icon of computational resources with CRC logo or similar.
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---
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## **Slide 3: Risk 2 - Insufficient Training Data**
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**Assertion:** Structured perturbations alone may not condition the model adequately.
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- **Evidence:**
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- Structured perturbations simplify training but may lack diversity.
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**Contingencies:**
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1. Augment training with manually or algorithmically generated $\Delta$ examples (e.g., bounded by supermum gain $\beta$).
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2. Diversify training data sources to improve robustness.
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**Visuals:**
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- Example of structured vs. manual perturbation samples on the complex plane.
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- Flowchart showing training data augmentation process.
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---
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## **Slide 4: Risk 3 - Interpolation Limitations**
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**Assertion:** Interpolation methods may fail to regulate perturbations effectively.
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- **Evidence:**
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- Image-based interpolation success may not generalize to this domain.
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**Contingencies:**
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1. Implement $r(\mathcal{P}_t)$-based reverse process steering for controlled perturbations【cite sources】.
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2. Explore alternative interpolation techniques tailored to frequency domain applications.
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**Visuals:**
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- Conceptual illustration of $r(\mathcal{P}_t)$ steering function in reverse process.
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- Example showing failure of simple interpolation and correction with $r(\mathcal{P}_t)$.
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---
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## Slide 5: Risk Mitigation Framework (Optional Summary Slide)
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**Assertion:** Addressing risks proactively ensures project success.
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- **Evidence:**
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- Computational strategies, diversified training, and alternative steering methods safeguard outcomes.
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**Visuals:**
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- Funnel graphic showing risks addressed through mitigations leading to project success. |