Minimum System Sensitivity Study of Linear Discrete Time Systems for Fault Detection
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Bibliographic record
Abstract
Fault detection is a critical step in the fault diagnosis of modern complex systems. An important notion in fault detection is the smallest gain of system sensitivity, denoted as<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M1"><mml:mrow><mml:msub><mml:mi>ℋ</mml:mi><mml:mo>−</mml:mo></mml:msub></mml:mrow></mml:math>index, which measures the worst fault sensitivity. This paper is concerned with characterizing<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M2"><mml:mrow><mml:msub><mml:mi>ℋ</mml:mi><mml:mo>−</mml:mo></mml:msub></mml:mrow></mml:math>index for linear discrete time systems. First, a necessary and sufficient condition on the lower bound of<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M3"><mml:mrow><mml:msub><mml:mi>ℋ</mml:mi><mml:mo>−</mml:mo></mml:msub></mml:mrow></mml:math>index in finite time horizon for linear discrete time-varying systems is developed. It is characterized in terms of the existence of solution to a backward difference Riccati equation with an inequality constraint. The result is further extended to systems with unknown initial condition based on a modified<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M4"><mml:mrow><mml:msub><mml:mi>ℋ</mml:mi><mml:mo>−</mml:mo></mml:msub></mml:mrow></mml:math>index. In addition, for linear time-invariant systems in infinite time horizon, based on the definition of the<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M5"><mml:mrow><mml:msub><mml:mi>ℋ</mml:mi><mml:mo>−</mml:mo></mml:msub></mml:mrow></mml:math>index in frequency domain, a condition in terms of algebraic Riccati equation is developed. In comparison with the well-known bounded real lemma, it is found that<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M6"><mml:mrow><mml:msub><mml:mi>ℋ</mml:mi><mml:mo>−</mml:mo></mml:msub></mml:mrow></mml:math>index is not completely dual to<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M7"><mml:mrow><mml:msub><mml:mi>ℋ</mml:mi><mml:mo>∞</mml:mo></mml:msub></mml:mrow></mml:math>norm. Finally, several numerical examples are given to illustrate the main results.
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Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from it