Why phase branches need model structure
Choose a declared phase branch without changing the complex response or hiding missing model structure.
The same complex value has several angles
For 1/s³ at positive angular frequency, a principal angle of +90° and a structural angle of −270° describe the same complex response. For s, the structural rule gives +90°. A full turn changes the reported branch rather than the transfer value. The source author had already tried changing the frequency range; changing the range alone does not recover model structure.
Match the branch to the comparison
Use principal to compare wrapped complex arguments, structural to retain every certified pole and zero factor, or reference to align the first defined sample with a stated starting angle. Reference applies one whole-turn offset across the complete grid; it does not force the first sample to equal an arbitrary requested angle exactly. A pure frequency-response dataset cannot supply poles and zeros that were never declared.
| Policy | Reference | Meaning |
|---|---|---|
| structural | all pole/zero factors | explicit continuous branch |
| principal | [-180,180) | wrapped complex argument |
| reference | initial angle / first defined frequency | nearest whole-turn shift |
Undefined points are part of the deliverable
A sampled pole retains its position and null numeric values; a zero retains real, imaginary and magnitude zero with undefined dB and phase. An exact cancelled factor remains undefined at its sampled frequency. The SVG splits the affected traces, and the CSV keeps the complete grid so a reader can distinguish a gap from a discarded point.
- Compare original model and all settings first.
- Check every root occurrence, residual and multiplicity.
- Use full downloaded SVG and CSV beyond the first 200 preview frequencies.
A closed issue still leaves a public counterexample
The original request used 1/s³ and also tried 1/s⁴ and 1/s²/(s+10). The last public comment says s still appeared as -270 rather than +90 after the issue was closed. We preserve these complete golden outputs; closure does not establish that the final counterexample was solved. No stock python-control execution or original image reconstruction is claimed.
All joint budgets apply: 50 million charged work units, files+full text+preview 64 MiB, full typed JSON 8 MiB, aggregate 72 MiB, binary+full metadata 96 MiB, conservative owned peak 1024 MiB and one absolute 120 seconds. Output and work intersections may refuse an individually valid axis.
Actual EVD loop bodies and QR increments are charged before original operations. Fixed matrix overhead and encoding/verification reservations are conservative units; degree-cubed is never reported as actual iteration work. Maximum joint capacity, maximum heap and browser performance remain unmeasured.
References
- Original request, all five comments and the final s counterexample
The issue closed before the final public counterexample; closure does not establish its resolution.
Tools in this category
Expand a tool to see its steps, options and supported formats, then open its workspace.
Control model frequency responseComplete SISO continuous-model response with certified roots and explicit phase branches.
Declare a complete transfer model and frequency grid, then inspect every complex response and the selected phase branch.
Steps
- Choose either numerator/denominator or all zeros/poles/gain.
- Provide all positive omega values or logRange start/end/count.
- Select a phase policy and inspect singular statuses.
- Download five complete files; copying retains the complete response CSV.
Available options
- Phase branch
- Structural continuous branch · Principal [-180,180) · Initial reference branch
- Initial reference (degrees; zero outside reference mode)
- 0
- Artifact language
- English · 中文
Capabilities and limits
- One strict UTF-8 JSON source: paste or native File, 4 MiB and 512 UTF-8 filename bytes. Clear pasted text before selecting a File; mixed sources, duplicate JSON keys, prototype keys and accessor inputs are rejected.
- Use complete real descending numerator/denominator coefficients (256 each) or conjugate-symmetric zeros/poles/gain, mutually exclusive. All 510 root occurrences are retained; uncertain multiplicity refuses the whole model.
- Positive strictly increasing omega in rad/s, at most 200,000 frequencies, or explicit logRange start/end/count. Every frequency is delivered; principal, structural and reference are options of this task.
- All joint budgets apply: 50 million charged work units, files+full text+preview 64 MiB, full typed JSON 8 MiB, aggregate 72 MiB, binary+full metadata 96 MiB, conservative owned peak 1024 MiB and one absolute 120 seconds. Output and work intersections may refuse an individually valid axis.
- Actual EVD loop bodies and QR increments are charged before original operations. Fixed matrix overhead and encoding/verification reservations are conservative units; degree-cubed is never reported as actual iteration work. Maximum joint capacity, maximum heap and browser performance remain unmeasured.