CaltechExperimentalGravity / CaltechExperimentalGravity/system_ident
Reducing Dependence on Model Completeness.
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Description
@Nangush is documenting some, incomplete, thought bubbles here - issue is assigned to myself.
### Problem Statement
More often than we would like our models are incomplete - they miss some aspect of reality that was abstracted away. Rarely we also don't have a reliable model and are counting on our measurement to identify the system. In **both cases** it is useful to optimise the measurement for unknown parameters.
An associated dilemma with this is it requires measuring at frequencies away from the known pole(s)/zero(s). This is different from the default operation of this script but could be behaviour captured by an _exploration phase_/flag.
As per the remainder of the repository, assuming the Cramer-Rao bound approximation is valid, this extension relies upon high signal-to-noise ratios (low uncertainty) to be possible to measure.
### Summary of Resolution
The brief summary of how to resolve this dilemma is:
1. Measure the system as normal - the signal-to-noise ratio (SNR)/uncertainty should also be calculated.
2. Fit the FRF, if any exists, as normal - the SNR/uncertainty of this fitted SNR should be calculated too.
3. Calculate the residual (magnitude ratio/phase difference) between the measurement and model.
4. Propagate the SNR/uncertainty from 1 and 2 onto the residuals from 3.
5. Calculate the curvature of the residuals magnitude.
6. Propagate the SNR/uncertainty from 4 into the information on the curvature of the residuals magnitude.
7. Weight the residual magnitude curvature, from 5, by its information, from 6.
8. Calculate the derivative of the residuals phase.
9. Propagate the SNR/uncertainty from 4 into the information on the derivative of the residuals phase.
10. Weight the residual phase derivative, from 8, by its information, from 9.
11. Combine results from 7 and 10 to determine which additional frequencies to distribute power into.
The exact combination/s for step 11 are still to be determined and experimented with. How to ensure that power is appropriately distributed between the poles/zeros captured in the model, and those missing from the model is also still to be determined and experimented with.
#### Rationale
- The residual captures the mismatch between the model and the measurement - this encodes missed pole(s) and zero(s), alongside noise.
- Any missed poles and zeros are encoded on at least one of the residual's magnitude curvature and phase derivative. More specifically where at least one of these is (absolute value) maximised is the expected location of missing pole(s)/zero(s).
- Because the residual also encapsulated noise there must be weighting by the information content of the residuals magnitude curvature/phase derivative. Only where the information is good and at least one of these is at a local extrema can we be (reasonably) confident that there are missed pole(s)/zero(s).
- Pathological cases exist, see for example all pass filters, where the magnitude curvature is zero but the phase derivative is non-zero. Potentially pathological cases exist where the opposite is also realised too.
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