FSM Reliability Guard

Metric Definitions and Decision Logic

Purpose of the FSM Reliability Guard

The FSM Reliability Guard is the core mechanism that distinguishes
AITL Controller B-Type from A-Type.

Its purpose is not to improve performance, but to prevent adaptation from degrading reliability.

The FSM answers a single critical question:
“Is adaptation allowed under the current reliability condition?”

Reliability Metrics Overview

The FSM evaluates system health using dimensionless, normalized metrics.
This ensures robustness across operating points and plant variations.

Primary monitored quantities:

Response delay ratio $R_{\Delta t} = \Delta t / \Delta t_0$
Gain compensation ratio(s) $R_{K_P}, R_{K_I}, R_{K_D}$
Amplitude / authority ratio $R_A$ and saturation indicators
Adaptation frequency $N_{\text{adapt}}$ (optional; anti-chattering)

Note: In the current A-Type demos, adaptation mainly adjusts $K_P$.
Therefore, B-Type may start with $R_{K_P}$ as the primary gain guard, and later extend to $K_I, K_D$ if needed.

Metric 1: Response Delay Ratio ($\Delta t / \Delta t_0$)

Definition

Let:

$\Delta t_0$ : nominal response delay under healthy plant conditions
$\Delta t$ : measured response delay under current conditions

The normalized delay ratio is defined as:

\[R_{\Delta t} = \frac{\Delta t}{\Delta t_0}\]

How to Measure $\Delta t$ (Operational Definition)

To avoid ambiguity, B-Type defines $\Delta t$ using event-to-event timing.

A practical and reproducible definition (used in the A-Type sweep):

Choose a reference event time $t_{\text{event}}$
(e.g., disturbance start, reference step, command change)
Define $\Delta t$ as the settling recovery time after $t_{\text{event}}$

Let the tracking error be:

\[e(t) = I_{\text{ref}}(t) - I(t)\]

Choose:

settle band $ e(t) < \epsilon$
hold duration $T_{\text{hold}}$

Then $\Delta t$ is defined as:

\[\Delta t = \min_{t \ge t_{\text{event}}} \left\{ t - t_{\text{event}} \;\middle|\; |e(\tau)| < \epsilon,\; \forall \tau \in [t,\, t + T_{\text{hold}}] \right\}\]

This definition is:

deterministic
unit-consistent
implementable in both simulation and runtime systems

Interpretation

$R_{\Delta t} \approx 1$ : nominal timing behavior
$R_{\Delta t} > 1$ : degradation-induced delay
Large values indicate loss of responsiveness and/or control authority

Guard Threshold

\[R_{\Delta t} \le R_{\Delta t}^{\max}\]

If violated, adaptation must be blocked.

Metric 2: Gain Compensation Ratio ($K / K_0$)

Definition (per gain)

Let:

$K_{P0}, K_{I0}, K_{D0}$ : nominal gains
$K_P, K_I, K_D$ : current gains

Define per-gain ratios:

\[R_{K_P} = \frac{K_P}{K_{P0}}, \quad R_{K_I} = \frac{K_I}{K_{I0}}, \quad R_{K_D} = \frac{K_D}{K_{D0}}\]

If only $K_P$ is adapted, then $R_{K_P}$ is the primary guard.

Interpretation

$R_{K_*} \approx 1$ : nominal control effort
$R_{K_*} \gg 1$ : over-compensation risk
Excessive gain escalation often precedes saturation or instability

Guard Threshold

\[R_{K_P} \le R_{K_P}^{\max}, \quad R_{K_I} \le R_{K_I}^{\max}, \quad R_{K_D} \le R_{K_D}^{\max}\]

Violation indicates unsafe compensation.

Rate Guard (Adaptation Aggressiveness)

Even if absolute gains are within limits, rapid gain changes are risky.

For $K_P$, define:

\[R_{\dot K_P} = \frac{\left|K_P(t) - K_P(t-\Delta T)\right|}{K_{P0}}\]

Guard condition:

\[R_{\dot K_P} \le R_{\dot K_P}^{\max}\]

Metric 3: Amplitude / Authority Ratio and Saturation

Definition

Let:

$A_{\text{out}}$ : output response amplitude
$A_{\text{ref}}$ : reference amplitude

The amplitude ratio is:

\[R_A = \frac{A_{\text{out}}}{A_{\text{ref}}}\]

Additionally, actuator saturation indicators may be monitored, e.g.:

saturation time ratio $S_u$

Interpretation

Excessive amplitude indicates loss of damping
Frequent saturation implies erosion of motion authority

Guard Usage

Amplitude and saturation metrics are typically used as secondary guards
or as part of a reliability cost function.

Example:

\[R_A \le R_A^{\max}, \quad S_u \le S_u^{\max}\]

Optional Metric: Adaptation Frequency (Chattering Detection)

Definition

Let:

$N_{\text{adapt}}$ : number of adaptation events per observation window

Guard condition:

\[N_{\text{adapt}} \le N_{\text{adapt}}^{\max}\]

Violation triggers adaptation blocking to prevent oscillatory redesign.

Permission Logic (Minimal Specification)

Adaptation is permitted only if all guard conditions are satisfied:

\[\begin{aligned} R_{\Delta t} &\le R_{\Delta t}^{\max} \\ R_{K_P} &\le R_{K_P}^{\max} \\ R_{\dot K_P} &\le R_{\dot K_P}^{\max} \\ \text{(optional)}\quad R_A &\le R_A^{\max} \\ \text{(optional)}\quad S_u &\le S_u^{\max} \end{aligned}\]

If any condition is violated:

Adaptation is disabled
Controller falls back to fixed-gain PID
FSM enters a reliability protection state

Blocking adaptation is a correct and expected outcome in B-Type.

FSM State Transitions

stateDiagram-v2
    [*] --> NORMAL

    NORMAL --> DEGRADED : R_Δt > 1
    DEGRADED --> ADAPT_ALLOWED : All guards satisfied
    DEGRADED --> ADAPT_BLOCKED : Any guard violated

    ADAPT_ALLOWED --> DEGRADED : Metrics drift
    ADAPT_BLOCKED --> SAFE_MODE : Persistent violation

    SAFE_MODE --> NORMAL : Manual reset / maintenance

Design Implications

Adaptation is explicitly constrained, not implicitly discouraged
Performance improvement never overrides reliability limits
Long-term degradation naturally drives the system toward conservative control

In B-Type, blocking adaptation is a correct design outcome.

Summary

The FSM Reliability Guard provides:

Formal, quantitative reliability definitions
Explicit measurement semantics for $\Delta t$
Deterministic permission logic for adaptation
Guaranteed fallback to fixed PID

By enforcing these guard conditions,
B-Type transforms adaptive control into a reliability-aware supervisory architecture.

Next Sections

Reliability cost function formulation
→ reliability_cost.md
Guard threshold design guidelines
→ threshold_guidelines.md
Demo mapping and integration
→ demo_mapping.md