Research / experiments / EXP-03

EXECUTABLE EXPERIMENT · STRUCTURAL RECONFIGURATION

Survivability-Driven Reconfiguration After Irreversible Resource Loss

A matched experiment on a redundant single-axis cyber-physical system. The object loses first its nominal actuator and then its nominal sensor. EXP-03 tests whether the supervisor can preserve an admissible degraded mission by moving through a graph of configurations built only from resources that still exist.

controlled plant→resource state→function capability→configuration graph→verified degraded mission

01 · SYSTEM UNDER CONTROL

One controlled coordinate, six redundant resources

The fixture is intentionally small enough to inspect by hand, but explicit enough to separate physical regulation from mission-level reconfiguration.

EXP-03 system under control — Controlled object. One configuration binds the same mechanical coordinate to a selected actuator, sensor and communication path.

TEST FIXTUREphysical + logical resources

ACTUATOR A\(M_A\)

ACTUATOR B\(M_B\)

ENCODER\(E\)

BACKUP IMU\(I\)

NETWORK 1\(C_1\)

NETWORK 2\(C_2\)

\[ \boldsymbol{\rho}_k=\begin{bmatrix}\rho_{M_A,k}&\rho_{M_B,k}&\rho_{E,k}&\rho_{I,k}&\rho_{C_1,k}&\rho_{C_2,k}\end{bmatrix}^{\!\top}\in[0,1]^6 \]

Each component expresses current resource operability. A destroyed resource remains at zero because EXP-03 contains no repair process.

Modelling boundary: this is a synthetic single-axis surrogate, not a full vehicle or robot model. The four mission functions are capability abstractions derived from resources; they are not four additional plant states.

02 · EXPERIMENT DESIGN

Same plant, same faults — only reconfiguration changes

The experiment uses a matched baseline so the effect of structural reconfiguration can be separated from plant parameters or gain scheduling.

CONTROL BRANCHFrozen structure

The configuration is held at \(c(t)\equiv c_0\) throughout both faults.

EXPERIMENTAL BRANCHSupervisor enabled

An invalid configuration triggers ANALYZE → SELECT → APPLY → VERIFY.

FAULT #15.00 s\(\rho_{M_A}\rightarrow0\)

VERIFY #15.03 s\(c_0\rightarrow c_1\)

FAULT #212.00 s\(\rho_E\rightarrow0\)

VERIFY #212.03 s\(c_1\rightarrow c_3\)

\[ \rho_i^+=\operatorname{sat}_{[0,1]}\!\left(\rho_i^- -\eta_iD_i\right),\qquad \eta_i=1,\qquad D_i=\rho_i^- \]

For both declared failure events, the damage equation produces complete irreversible loss: \(\rho_i^+=0\).

03 · CONTROL SYSTEM

Fast regulation below, structural decisions above

The feedback gain is deliberately kept constant across mission-preserving configurations. EXP-03 therefore tests structural reconfiguration, not gain scheduling.

EXP-03 control system — Hierarchical interpretation of EXP-03: L0 hardware, L1 servo control, L2 supervisor and L3 mission manager.

\[ x_{k+1}=Ax_k+b(c,\boldsymbol{\rho}_k)u_k+\xi_k,\qquad u_k=-Kx_k \]

Plant: \(A=0.9\), \(K=0.8\), \(\xi_k=0.02\). The configuration changes the effective physical input path \(b(c,\boldsymbol{\rho})\).

\[ \gamma(c,\boldsymbol{\rho})=\left|A-b(c,\boldsymbol{\rho})K\right|<1 \]

Scalar discrete-time stability gate. With \(V(e)=e^2\), this is equivalent to \(\Delta V=(\gamma^2-1)e^2<0\) for non-zero error.

04 · MATHEMATICAL DECISION MODEL

Resources become functions; functions constrain configurations

The selector uses only declared equations and the directed graph. There is no hidden transition score, risk coefficient or time penalty.

\[ \mathbf f(c,\boldsymbol{\rho})=\begin{bmatrix}g_Ag_S\\[2pt]g_Ag_Sg_N\\[2pt]g_N\\[2pt]g_Sg_N\end{bmatrix},\qquad g_A=\rho_{a(c)},\;g_S=\rho_{s(c)},\;g_N=\rho_{n(c)} \]

EXP-03 specialization for stabilization, tracking, telemetry and diagnostics.

\[ \phi_j(c,\boldsymbol{\rho})=\delta_j(c)\,\mathbf 1\!\left[f_j(c,\boldsymbol{\rho})\ge\theta_j\right],\qquad \theta_j=0.5 \]

Binary mission-function realizability after resource capability is evaluated.

\[ \Phi(c,\boldsymbol{\rho})=\sum_{j=1}^4 w_j\phi_j(c,\boldsymbol{\rho}),\qquad S(c,\boldsymbol{\rho})=\sum_{j=1}^4 w_j f_j(c,\boldsymbol{\rho})\delta_j(c) \]

\(\mathbf w=[0.45,0.30,0.15,0.10]^\top\), with \(\Phi_{\min}=0.65\).

\[ c_{k+1}^\star\in\operatorname*{arg\,max}_{c\in\mathcal N^+(c_k)}S(c,\boldsymbol{\rho}_k) \] \[ \text{subject to}\qquad \Phi(c,\boldsymbol{\rho}_k)\ge\Phi_{\min},\qquad \gamma(c,\boldsymbol{\rho}_k)<1 \]

Only outgoing graph successors are candidates. If no mission-preserving successor is admissible, SAFE_STOP is the fallback.

05 · CONFIGURATION GRAPH

The graph is part of the selector, not a decoration

The same directed transition relation is stored in the executable model, converted to matrices and independently audited against the decision log.

CONFIGURATION MAPphysical allocation

State	Actuator	Sensor	Network	Functions
\(c0\)NOMINAL	M_A	E	C_1	STAB, TRACK, TEL, DIAG
\(c1\)BACKUP_ACTUATOR	M_B	E	C_1	STAB, TRACK, TEL, DIAG
\(c2\)BACKUP_SENSOR	M_A	I	C_1	STAB, TRACK, TEL, DIAG
\(c3\)LIMP_HOME	M_B	I	C_2	STAB, TEL, DIAG
\(c4\)SAFE_STOP	—	I	C_2	DIAG

\(T^{(s)}\)structural adjacency

	\(c1\)	\(c2\)	\(c3\)	\(c4\)
\(c0\)	1	1	1	1
\(c1\)	0	0	1	1
\(c2\)	0	0	1	1
\(c3\)	0	0	0	1
\(c4\)	0	0	0	0

\(T^{(f,1)}\)after Motor A loss

	\(c1\)	\(c3\)	\(c4\)
\(c0\)	1	1	0
\(c1\)	0	1	0
\(c2\)	0	1	0
\(c3\)	0	0	1
\(c4\)	0	0	0

\(T^{(f,2)}\)after Encoder loss

	\(c3\)	\(c4\)
\(c0\)	1	0
\(c1\)	1	0
\(c2\)	1	0
\(c3\)	0	1
\(c4\)	0	0

\(T^{(a)}\)executed path

	\(c1\)	\(c3\)
\(c0\)	1	0
\(c1\)	0	1
\(c2\)	0	0
\(c3\)	0	0
\(c4\)	0	0

Rows are current configurations, columns are target configurations. The amber outline marks transitions actually executed during this run. All four matrices are also exported as CSV.

06 · NUMERICAL EVIDENCE

Data first, plots second

The simulation writes CSV logs before plotting. The independent audit recomputes every logged formula. `plot_results.py` then reads only the CSV files and `manifest.json`.

SELECTOR DECISIONSall evaluated candidates

Time	From	Candidate	\(S\)	\(\Phi\)	\(\gamma\)	Status
5.01 s	\(c0\)	\(c1\)	1.00	1.00	0.10	SELECTED
5.01 s	\(c0\)	\(c2\)	0.25	0.25	0.90	REJECTED
5.01 s	\(c0\)	\(c3\)	0.70	0.70	0.10	ADMISSIBLE
12.01 s	\(c1\)	\(c3\)	0.70	0.70	0.10	SELECTED

5.03 s · VERIFY\(c1\)\(\Phi=1.00,\;\gamma=0.10\) · PASS

12.03 s · VERIFY\(c3\)\(\Phi=0.70,\;\gamma=0.10\) · PASS

Mission functionality \(\Phi(t)\): same faults, frozen structure versus structural reconfiguration.

Resource state \(\boldsymbol{\rho}(t)\). Destroyed resources remain at zero.

Configuration trajectory — Active structural configuration. The logged path is \(c_0\rightarrow c_1\rightarrow c_3\).

Plant state \(x_k\) for both matched branches. This is secondary evidence; the primary EXP-03 claim is structural mission validity.

Additional audited plots

Survivability \(S(t)\) of the active configuration.

Candidate survivability — Every candidate survivability score written to `decisions.csv`.

Mission threshold sensitivity — Threshold sensitivity. After both failures, \(c_3\) remains mission-admissible only while \(\Phi_{\min}\le0.70\).

EXPERIMENT RESULT

The object is not restored. The mission is re-realized on the residual structure.

The frozen branch stays tied to the destroyed primary actuator and loses mission validity. The supervisor branch selects \(c_1\) after the actuator loss and \(c_3\) after the encoder loss. The second state intentionally drops tracking while preserving stabilization, telemetry and diagnostics.

\(c_0\rightarrow c_1\rightarrow c_3\)

\[ S(c_1)=1.00,\qquad \Phi(c_1)=1.00 \]

After Motor A loss: full mission remains realizable through the backup actuator.

\[ S(c_3)=0.70,\qquad \Phi(c_3)=0.70 \]

After Encoder loss: the system continues with the declared degraded mission.

\[ \Phi_{\mathrm{frozen}}(16\,\mathrm{s})=0.15,\qquad \Phi_{\mathrm{supervisor}}(16\,\mathrm{s})=0.70 \]

Same fault history; different structural-control policy.

07 · GENERALISATION

From six resources to a general survivability problem

EXP-03 is not the final physical system. It is an inspectable instance of the general Resource → Function → Configuration → Mission decision chain.

\[ \mathcal R=\{r_1,\ldots,r_r\},\qquad \mathcal F=\{F_1,\ldots,F_m\},\qquad \mathcal C=\{c_1,\ldots,c_q\} \]

A real deployment replaces the six declared resources and five configurations with the diagnosed resources, functional dependencies and supported post-build configurations of the target CPS.

6 declared resources → diagnosed \(\boldsymbol{\rho}\)·4 functions → validated \(f_j(\boldsymbol{\rho},c)\)·5 configurations → runtime \(\mathcal C\)·scalar stability gate → plant-specific certificate·SIL → HIL → physical plant

Publication status. Related work has been submitted as a preprint and is under review. The public bibliographic link will be added here when available.

Author: Dmytro Humennyi · [email protected]

08 · REPRODUCE / AUDIT

Everything needed to check the run

The numerical audit is supporting evidence, not the main narrative.

manifest·supervised CSV·baseline CSV·decisions·verification·audit report·checksums

Source code and transition CSVs

model.pyequations.pyselector.pyexperiment.pyaudit.pyplot_results.py\(T^{(s)}\)\(T^{(f,1)}\)\(T^{(f,2)}\)\(T^{(a)}\)