Threshold Design Guidelines
Practical Selection of Reliability Guard Limits
Purpose of Threshold Guidelines
Thresholds in B-Type architecture define the boundary between acceptable adaptation and unsafe behavior.
They are not tuning parameters for performance optimization, but
engineering limits that preserve reliability under degradation.
Thresholds answer the question:
“How far are we willing to bend before we must stop adapting?”
Fundamental Design Principle
All thresholds must satisfy the following condition:
\[\text{Stability and controllability must be preserved even when adaptation is blocked.}\]Therefore, thresholds should be:
- Conservative
- Interpretable
- Justifiable from physical or operational constraints
Guideline 1: Response Delay Ratio Threshold
Metric
\(R_{\Delta t} = \frac{\Delta t}{\Delta t_0}\)
Recommended Range
\(1.2 \le R_{\Delta t}^{\max} \le 1.5\)
Interpretation
- Lower bound (≈1.2):
High-sensitivity systems, tight timing constraints - Upper bound (≈1.5):
Mechanically slow or non-time-critical systems
Design Note
If control delay exceeds 150% of nominal,
the system is already operating in a degraded regime and adaptation should be blocked.
Guideline 2: Gain Compensation Ratio Threshold
Metric
\(R_K = \frac{K}{K_0}\)
Recommended Range
\(1.5 \le R_K^{\max} \le 3.0\)
Interpretation
- Values near 1.5:
Actuator-limited or safety-critical systems - Values near 3.0:
Lab-scale or non-critical experimental setups
Design Note
Large gain increases often precede actuator saturation and oscillatory behavior,
making this threshold a strong early-warning indicator.
Guideline 3: Amplitude Ratio Threshold
Metric
\(R_A = \frac{A_{\text{out}}}{A_{\text{ref}}}\)
Recommended Range
\(1.2 \le R_A^{\max} \le 2.0\)
Interpretation
- Lower values emphasize damping and smooth response
- Higher values tolerate transient overshoot
Design Note
Amplitude thresholds should be coordinated with
mechanical limits, thermal constraints, and fatigue considerations.
Guideline 4: Reliability Cost Threshold
Metric
\(J_{\text{rel}}\)
Recommended Strategy
Instead of absolute values, define the threshold relative to nominal variance:
\[J_{\text{rel}}^{\max} = \alpha \cdot \mathbb{E}[J_{\text{rel}}^{\text{nominal}}]\]where:
- \(\alpha = 2 \sim 5\) for conservative designs
Design Note
The reliability cost threshold should never override individual hard guards.
It acts as a secondary, integrative constraint.
Guideline 5: Adaptation Frequency Threshold
Metric
\(N_{\text{adapt}} \quad [\text{events/time}]\)
Recommended Rule
\(N_{\text{adapt}}^{\max} \le 1 \text{ per dominant time constant}\)
Interpretation
Frequent adaptation indicates:
- Unstable supervisory logic
- Poor metric observability
- Excessive noise sensitivity
Blocking adaptation in this case improves reliability, not degrades it.
Conservative Default Threshold Set (Example)
| Metric | Default Value |
|---|---|
| \(R_{\Delta t}^{\max}\) | 1.3 |
| \(R_K^{\max}\) | 2.0 |
| \(R_A^{\max}\) | 1.5 |
| \(J_{\text{rel}}^{\max}\) | \(3 \times\) nominal |
| \(N_{\text{adapt}}^{\max}\) | 1 / time constant |
This set is suitable for initial deployment and long-term operation.
Threshold Validation Strategy
Thresholds should be validated through:
- Aging and degradation simulations
- Worst-case disturbance injection
- Actuator saturation testing
- Long-horizon reliability cost evaluation
A valid threshold is one that blocks adaptation before damage or instability occurs.
Summary
Thresholds in B-Type architecture:
- Define explicit reliability boundaries
- Transform adaptive control into a permission-based mechanism
- Provide predictable and explainable system behavior
In B-Type, conservative thresholds are not a limitation—
they are the core design feature.
Possible next sections:
- Mapping thresholds to physical safety limits
- Adaptive threshold scheduling (offline only)
- Field tuning methodology under uncertainty