195 lines
6.4 KiB
Markdown
195 lines
6.4 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**
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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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# Metrics of Success
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# Risks and Contingencies
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# |