Research / Experiments / EXP-02
INTERACTIVE RECONFIGURATION PROTOTYPE
Graph-Based Post-Failure Reconfiguration
After irreversible damage the system does not attempt to recover the nominal configuration. It reconstructs the feasible stable-state space, evaluates admissible transitions, and searches for a safe path to the best mission-relevant state that remains reachable.
ρ₀→ damage →ρd→G(ρd)→A(ρd)→ℛ(v,ρd)→v*
Prototype note — the state utilities, transition costs and resource values below are deliberately synthetic. The purpose of this release is to make the method executable and inspectable before binding it to a specific physical plant.
01 / FORMAL OBJECT
Stable states, not configuration labels
A graph node is admitted only when the corresponding configuration has a stable operating point under the current residual-resource vector. The post-failure search therefore happens over safely reachable stable states.
\[\dot{x}=f(x,u,\boldsymbol{\rho},c)\]
The plant dynamics depend on the physical state, control input, residual resources and selected configuration.
\[v_i=(c_i,x_i^\star,\boldsymbol{\rho},\Phi_i),\qquad f(x_i^\star,u_i^\star,\boldsymbol{\rho},c_i)=0\]
Each node combines configuration, stable operating point, resource context and mission-weighted utility.
02 / IRREVERSIBLE EVENT
Damage changes the feasible state space
The example removes actuator A2 permanently. V0 remains visible as historical context, but it is no longer a valid node in the post-damage feasible graph.
RESIDUAL RESOURCE VECTORρ₀
ACT-A11.00
ACT-A21.00
SENSING1.00
COMPUTE1.00
03 / GRAPH VIEW
Graph before damage, graph after damage, selected path
Use the controls to inspect the nominal topology, prune it with the damaged resource vector, or animate the computed reconfiguration path. The diamond qf is a transient failure point and is intentionally excluded from the stable-state matrices.
STABLE-STATE GRAPH · G(ρ)hover nodes / edges for values
feasible stable statecapture/currentselected targetinfeasible after damageselected transition
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04 / MATRICES
Transitions shown as matrices
The same topology is exposed numerically. A(ρ) answers whether a direct stable-state transition is admissible. C(ρ) carries the normalized transition penalty used by the graph search.
ACTIVE MATRIX SETNOMINAL ρ₀
ADMISSIBILITY · A(ρ)1 allowed · 0 forbidden · × state unavailable
| V0 | V1 | V2 | V3 | V4 | V5 | V6 | |
|---|---|---|---|---|---|---|---|
| V0 | 0 | 1 | 1 | 1 | 0 | 1 | 1 |
| V1 | 1 | 0 | 1 | 1 | 0 | 1 | 1 |
| V2 | 1 | 1 | 0 | 0 | 1 | 1 | 1 |
| V3 | 1 | 0 | 1 | 0 | 1 | 1 | 1 |
| V4 | 0 | 0 | 1 | 0 | 0 | 1 | 1 |
| V5 | 0 | 0 | 0 | 0 | 1 | 0 | 1 |
| V6 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
TRANSITION COST · C(ρ)normalized · ∞ no edge
| V0 | V1 | V2 | V3 | V4 | V5 | V6 | |
|---|---|---|---|---|---|---|---|
| V0 | 0 | 0.10 | 0.12 | 0.15 | ∞ | 0.08 | 0.05 |
| V1 | 0.20 | 0 | 0.17 | 0.25 | ∞ | 0.11 | 0.08 |
| V2 | 0.16 | 0.13 | 0 | ∞ | 0.19 | 0.10 | 0.07 |
| V3 | 0.20 | ∞ | 0.15 | 0 | 0.24 | 0.09 | 0.07 |
| V4 | ∞ | ∞ | 0.18 | ∞ | 0 | 0.08 | 0.07 |
| V5 | ∞ | ∞ | ∞ | ∞ | 0.17 | 0 | 0.05 |
| V6 | ∞ | ∞ | ∞ | ∞ | ∞ | ∞ | 0 |
\[C_{ij}=0.35\,T_{ij}+0.20\,E_{ij}+0.30\,R_{ij}+0.15\,D_{ij}\]
Prototype weighting: transition time, energy demand, transient risk and additional degradation. All components are normalized before weighting.
05 / GRAPH SEARCH
Choose the best safely reachable terminal state
The capture state V1 is the search origin. Dijkstra supplies the minimum transition cost to each reachable stable state; mission utility then competes with path cost.
\[J(v_j)=\Phi_j-\lambda\min_{\pi:v_1\leadsto v_j}\sum_{(i,k)\in\pi}C_{ik},\qquad \lambda=0.35\]
The prototype chooses the feasible state with maximum mission-weighted utility after accounting for transition burden.
TERMINAL CANDIDATEScomputed from damaged graph
| State | Feasible | Φ | Path C | J |
|---|---|---|---|---|
| V0 · Nominal | no | 0.96 | — | — |
| V1 · Capture | yes | 0.52 | 0.00 | 0.52 |
| V2 · Reduced BW | yes | 0.69 | 0.17 | 0.63 |
| V3 · Observer | yes | 0.61 | 0.25 | 0.52 |
| V4 · Mission | yes | 0.84 | 0.36 | 0.72 |
| V5 · Safe hold | yes | 0.39 | 0.11 | 0.35 |
| V6 · Shutdown | yes | 0.08 | 0.08 | 0.05 |
Selected targetV4
Target score J0.72
Path cost ΣC0.36
Terminal utility Φ0.84
06 / TRANSITION TRACE
Transitions shown as transitions
The global graph explains the space; the trace explains the actual recovery route. A forbidden direct jump is not hidden — the route must pass through stable intermediate states.
COMPUTED ROUTEq_f → V1 → V2 → V4
EVENTIrreversible lossACT-A2 irreversible loss
CAPTUREq_f → V1enter certified stable capture basin
TRANSITION 01V1 → V2reallocate actuation; reduce bandwidth
TRANSITION 02V2 → V4prioritise mission-essential functions
SETTLEV4 stableverify Φ and safety envelope
Transition time1.17 s
Accumulated risk index0.18
Stable-state hops2
Nominal restorationNO
07 / UML CONTROL LOGIC
Graph theory inside a supervisory control process
The graph planner is only one subsystem. UML views make explicit when the resource estimator, graph builder, planner and supervisor exchange authority and evidence.
EXPERIMENT CLAIM
The target is not the nominal configuration. It is the best safely reachable stable state.
EXP-02 makes the distinction explicit: damage changes the graph itself. Reconfiguration therefore requires topology reconstruction, matrix-level admissibility evidence, graph search, and transient verification before a degraded mission state can be accepted.
\(\rho_d \Rightarrow G(\rho_d) \Rightarrow \pi^* \Rightarrow v^*\)