Reliability Cost Function
Quantifying Reliability Degradation in B-Type Control
Purpose of the Reliability Cost
In B-Type architecture, reliability is treated as a measurable quantity,
not an abstract design intention.
The reliability cost function provides:
- A unified scalar representation of multiple degradation indicators
- A basis for FSM guard decisions
- A tool for comparing adaptive and non-adaptive control strategies
The goal is not optimization, but bounded degradation.
Conceptual Definition
Let the system reliability be evaluated using a cost function:
\[J_{\text{rel}} \ge 0\]where:
- \(J_{\text{rel}} = 0\) represents nominal, healthy operation
- Larger values indicate increasing reliability degradation
Adaptation is permitted only when the reliability cost remains below a predefined limit.
Normalized Metric Set
Primary contributors:
Response delay ratio: \(R_{\Delta t} = \frac{\Delta t}{\Delta t_0}\)
Gain compensation ratio: \(R_K = \frac{K}{K_0}\)
Amplitude ratio: \(R_A = \frac{A_{\text{out}}}{A_{\text{ref}}}\)
All metrics are dimensionless and centered around unity under nominal conditions.
Basic Reliability Cost Formulation
A weighted quadratic form:
\[J_{\text{rel}} = w_{\Delta t}(R_{\Delta t} - 1)^2 + w_K(R_K - 1)^2 + w_A(R_A - 1)^2\]where:
- \(w_{\Delta t}, w_K, w_A\) are design weights
Threshold-Based Decision Rule
\[J_{\text{rel}} \le J_{\text{rel}}^{\max}\]If violated, the FSM transitions toward ADAPT_BLOCKED or SAFE_MODE.
Time-Accumulated Reliability Cost (Optional)
\[J_{\text{rel}}^{\text{acc}}(T) = \int_0^T J_{\text{rel}}(t)\,dt\]This captures chronic reliability degradation over long horizons.
Summary
The reliability cost function:
- Quantifies degradation explicitly
- Supports explainable FSM decisions
- Ensures performance gains do not erode reliability
In B-Type, adaptation is allowed
only when reliability cost remains bounded.