Theory

This page summarises the mathematical foundations implemented in scpn_control.phase.

Stochastic Petri Net (SPN)

An SPN is a bipartite directed graph \(G = (P, T, W^{-}, W^{+})\) where \(P\) is a set of places, \(T\) a set of transitions, and \(W^{\pm}\) are the input/output weight matrices. The marking vector \(\mathbf{m}(t) \in \mathbb{R}^{|P|}_{\ge 0}\) evolves via stochastic firing:

\[ \mathbf{m}(t + 1) = \mathbf{m}(t) + (W^{+} - W^{-})^{\top} \mathbf{f}(t) \]

where \(\mathbf{f}(t) \in \{0,1\}^{|T|}\) is the firing vector, sampled from transition-specific hazard functions.

The FusionCompiler maps each place → LIF neuron membrane potential, each transition → synaptic connection with weight from \(W^{\pm}\), producing a CompiledNet for closed-loop control.

Kuramoto–Sakaguchi Model

The phase of oscillator \(n\) in layer \(l\) evolves as (Paper 27, Eq. 3):

\[ \dot{\theta}^{(l)}_n = \omega^{(l)}_n + \frac{1}{N_l} \sum_{m=1}^{N_l} K_{ll} \sin\!\bigl(\theta^{(l)}_m - \theta^{(l)}_n - \alpha\bigr) + \zeta_l \sin\!\bigl(\Psi - \theta^{(l)}_n\bigr) \]

where \(\omega^{(l)}_n\) are natural frequencies, \(K_{ll}\) is the intra-layer coupling, \(\alpha\) is the Sakaguchi phase-lag, and \(\zeta_l\) is the exogenous global-field coupling strength.

The order parameter for layer \(l\) is:

\[ R_l \, e^{i\Psi_l} = \frac{1}{N_l} \sum_{n=1}^{N_l} e^{i\theta^{(l)}_n} \]

Implementation: scpn_control.phase.kuramoto.kuramoto_sakaguchi_step.

Unified Phase Dynamics Equation (UPDE)

The UPDE extends the Kuramoto model to \(L\) coupled layers with inter-layer coupling matrix \(\mathbf{K} \in \mathbb{R}^{L \times L}_{\ge 0}\):

\[ \dot{\theta}^{(l)}_n = \omega^{(l)}_n + \frac{1}{N_l} \sum_m K_{ll} \sin(\theta^{(l)}_m - \theta^{(l)}_n) + \sum_{l' \neq l} K_{ll'} R_{l'} \sin(\Psi_{l'} - \theta^{(l)}_n) + \zeta_l \sin(\Psi - \theta^{(l)}_n) \]

The cross-layer term uses the mean-field approximation: layer \(l'\) acts on layer \(l\) through its order parameter \(R_{l'} e^{i\Psi_{l'}}\).

PAC gating (Phase-Amplitude Coupling): when enabled, the cross-layer coupling \(K_{ll'}\) is modulated by \(R_{l'}\), suppressing influence from incoherent layers.

Implementation: scpn_control.phase.upde.UPDESystem.

Paper 27 Coupling Matrix (\(K_{nm}\))

The \(16 \times 16\) inter-layer coupling matrix follows exponential distance decay with calibration anchors (Paper 27, Table 2):

\[ K_{nm} = K_{\text{base}} \, e^{-\alpha |n - m|} \]

with \(K_{\text{base}} = 0.45\), \(\alpha = 0.3\), and four calibration overrides:

Pair	\(K_{nm}\)	Source
(1,2)	0.302	Paper 27 Table 2
(2,3)	0.201	Paper 27 Table 2
(3,4)	0.252	Paper 27 Table 2
(4,5)	0.154	Paper 27 Table 2

Implementation: scpn_control.phase.knm.build_knm_paper27.

Lyapunov Stability Monitor

A candidate Lyapunov function for the synchronised state:

\[ V(t) = \frac{1}{N} \sum_{n=1}^{N} \bigl(1 - \cos(\theta_n - \Psi)\bigr) \]

satisfies \(V \ge 0\), \(V = 0\) iff \(\theta_n = \Psi \;\forall n\). The guard estimates the Lyapunov exponent \(\lambda\) via finite differences over a sliding window:

\[ \lambda \approx \frac{1}{\Delta t} \ln\!\frac{V(t)}{V(t - \Delta t)} \]

A session is flagged unstable when \(\lambda > 0\) persists for more than max_violations consecutive windows.

Implementation: scpn_control.phase.lyapunov_guard.LyapunovGuard.

H-infinity Robust Control

The H-infinity controller solves the DARE (Discrete Algebraic Riccati Equation) for the worst-case disturbance attenuation level \(\gamma\):

\[ A^{\top} X A - X - A^{\top} X B (R + B^{\top} X B)^{-1} B^{\top} X A + Q = 0 \]

where \(R = \text{diag}(R_u, -\gamma^2 I_w)\) partitions control and disturbance inputs. The controller gain \(F = -(R + B^{\top} X B)^{-1} B^{\top} X A\) guarantees \(\|T_{zw}\|_\infty < \gamma\).

Anti-windup: output is clipped to \([-u_{\max}, u_{\max}]\) and the integrator state is conditionally frozen.

Implementation: scpn_control.control.h_infinity_controller.HInfinityController.