25-EFT.WP.STG.Dynamics v1.0 | Chapter 8 — Controllability & Observability (Structural / Energy-Based)

Home ／ Docs-Technical WhitePaper ／ 25-EFT.WP.STG.Dynamics v1.0

Chapter 8 — Controllability & Observability (Structural / Energy-Based)

One-Sentence Goal
On G = (V, E), establish structural and energy-based notions of controllability/observability, and deliver computable workflows from graph priors to Gramians and PBH tests—with dual-form checks, contracts, and runtime panel fields.

I. Scope and Objects

1. Objects
   • Continuous/discrete network dynamics: dx/dt = A x + B u, y = C x + n; or x_{k+1} = A_d x_k + B_d u_k, y_k = C x_k + n_k.
   • Graph priors: A may be derived from −L, A_sym, A_norm, or from identified models (Ch. 7); B’s driver set and C’s sensor set are selected by placement policies.
2. Inputs
   G, A/L, B?, C?, sampling Δt, window [t_0, t_1], RefCond, and unit system.
3. Outputs
   Structural controllability/observability decisions and minimal driver/sensor suggestions; energy metrics (Gramians, average/modal controllability); delta_form_ctrl/obs; manifest.stg.ctrlobs.
4. Constraints
   • unit(x), unit(u), unit(y); unit(dx/dt) = unit(x)/[T]; check_dim must pass.
   • Stability assumption: infinite-horizon Gramians require A Hurwitz (or ρ(A_d) < 1).

II. Terms and Variables

• Graph & operators: incidence B_inc, Laplacian L, system matrix A, discrete A_d = expm(A Δt).
• Sets & selection: driver nodes U ⊆ V, sensor nodes S ⊆ V; selection matrices B = S_U, C = S_S^T.
• Gramians: W_c(T) = ( ∫_0^T e^{A τ} B B^T e^{A^T τ} dτ ), W_o(T) = ( ∫_0^T e^{A^T τ} C^T C e^{A τ} dτ ).
• Structural quantities: maximum matching M*, unmatched nodes U* (driver lower bound); strongly connected components (SCCs).
• Indicators: average controllability AC, modal controllability MC, condition number κ(W), minimal energy E_min.

III. Axioms P708-*

• P708-1 (Parallel dual forms) — Evaluate structural (matching/SCC) and energy (Gramian/PBH) criteria in parallel and record discrepancies.
• P708-2 (PBH baseline) — Controllability via PBH: for all λ ∈ spec(A), rank([ λI − A, B ]) = N; observability is the dual test.
• P708-3 (Explicit measures/domains) — For Gramians, write ( ∫_0^T • dτ ); for discrete, ( Σ_{k=0}^{K−1} • ).
• P708-4 (Units/dimensions) — unit(W_c) = unit(x)^2 • [T]; unit(E_min) = unit(u)^2 • [T]; enforce check_dim.
• P708-5 (Topological consistency) — If using L / A from Chs. 2/4/7, persist construction method and hashes.
• P708-6 (Robustness) — Prefer regularized/truncated inverses for W^{-1}; report κ(W) and pseudo-inverse thresholds.

IV. Minimal Equations S708-*

• S708-1 (Kalman matrices):
  Continuous controllability: Ctrb = [ B, A B, A^2 B, …, A^{N−1} B ], rank(Ctrb) = N ⇔ controllable.
  Observability: Obsv = [ C^T, A^T C^T, …, (A^T)^{N−1} C^T ]^T, rank(Obsv) = N.
• S708-2 (PBH):
  Controllability: rank([ λI − A, B ]) = N, ∀ λ ∈ spec(A);
  Observability: rank([ λI − A; C ]) = N.
• S708-3 (Gramian & Lyapunov equivalence):
  Infinite horizon, A Hurwitz:
• A W_c + W_c A^T + B B^T = 0,
• A^T W_o + W_o A + C^T C = 0.

Finite horizon uses the integral definitions above.
Dual-form gap: delta_form_ctrl = || W_c(T) − W_c^{Lyap}(T) ||_F (discrete analog likewise).

• S708-4 (Minimum energy):
  E_min(x_0 → x_T) = r^T W_c(T)^{−1} r, with r = x_T − e^{A T} x_0.
  To the origin: E_min(x_0 → 0) = x_0^T W_c(T)^{−1} x_0.
• S708-5 (Average / modal controllability):
  Average: AC = trace(W_c(T)) or single-input AC_i = trace(W_c(B = e_i)).
  Modal (discrete A_d = V Λ V^{−1}, normalized):
  MC_i = Σ_{j=1}^N ( 1 − | λ_j |^2 ) | v_{ij} |^2; for continuous-time, replace weight with −2 Re(λ_j).
• S708-6 (Structural approximations):
  Minimal drivers N_D = | U* | for unmatched nodes of a maximum matching;
  Minimal sensors N_S = | U* | for G^T or at least the number of source SCCs.
• S708-7 (Placement optimization):
  Drivers: max_{U: |U| = m} f( trace(W_c(U)) ) or min trace(W_c(U)^{−1}); greedy works under submodularity conditions.
  Sensors: dual of the above.

V. Metrology Workflow M7-8 (Ready → Model/Estimate → Check → Persist)

1. Ready
   • Fix source of A/L (analytical/identified), align Δt and RefCond; declare units.
   • Choose candidate driver/sensor pools and capacity constraints (channels/cost).
2. Model / Estimate
   • Structural: compute maximum matching/SCC; report N_D / N_S and candidates.
   • Energy: solve (discrete) Lyapunov for W_c / W_o or integrate to obtain W(T).
   • Placement: greedy or MILP (small-scale) to select U / S.
3. Checks
   • PBH/Kalman ranks; κ(W), λ_min(W);
   • Energy reachability: sample x_0, x_T and compute E_min percentiles;
   • Dual forms: delta_form_ctrl / obs;
   • Stability: spec(A) or ρ(A_d);
   • Units/dimensions: check_dim.
4. Persist / Publish
   manifest.stg.ctrlobs = { A_hash, source: { struct | ident }, Δt, U, S, metrics: { rankCtrb, rankObsv, PBH_ok, κc, κo, λmin_c, AC, MC_p50, E_min_p95 }, delta: { ctrl, obs }, RefCond, method.hash }.

VI. Contracts & Assertions C70-8xx

• C70-801 (PBH pass): all eigenvalues satisfy rank([ λI − A, B ]) = N and the dual condition.
• C70-802 (Gramian PD): λ_min(W_c) ≥ τ_c, λ_min(W_o) ≥ τ_o (default τ_* = 1e−9 • scale).
• C70-803 (Conditioning): κ(W_c) ≤ κ_max, κ(W_o) ≤ κ_max (default κ_max = 1e8; beyond this, use pseudoinverse and annotate).
• C70-804 (Dual-form gap): delta_form_ctrl_p95 ≤ tol_ctrl, delta_form_obs_p95 ≤ tol_obs (suggest 1e−6 • || W ||_F).
• C70-805 (Structural lower bounds): ensure |U| ≥ N_D, |S| ≥ N_S; otherwise annotate risk and apply fallback.
• C70-806 (Stability domain): for infinite-horizon energy, require A Hurwitz or ρ(A_d) < 1.
• C70-807 (Unit consistency): check_dim( dx/dt − ( A x + B u ) ) = "[0]", check_dim( y − C x ) = "[0]".

VII. Implementation Bindings I70-8*

• I70-81 build_selection(G, policy) -> { U0, S0 }
• I70-82 structural_controllability(G) -> { N_D, drivers, scc, matching_meta }
• I70-83 gramian(A, B, C, Δt, horizon, mode) -> { W_c, W_o } (mode ∈ { lyap, integral })
• I70-84 pbh_test(A, B, C) -> { ok_ctrl, ok_obs, details }
• I70-85 energy_reach(A, B, x0, xT, T) -> E_min
• I70-86 select_actuators(A, G, m, criterion) -> U* (criterion ∈ { traceWc, minTraceInv, MC })
• I70-87 select_sensors(A, G, m, criterion) -> S* (criterion ∈ { traceWo, maxDet, observ_rank })
• I70-88 compare_forms(W_int, W_lyap) -> delta_form
• I70-89 emit_ctrlobs_manifest(results, policy) -> manifest.stg.ctrlobs

Invariants: non_decreasing(time); traceable method / hash / RefCond; delta_form_* ≤ tol_*; all check_dim pass.

VIII. Cross-References

• Graph operators & spectra: Ch. 2; kernels & diffusion energy: Ch. 4.
• Transport conservation & sources–sinks: Ch. 6 (for energy/reachability interpretation).
• Dynamics identification (source of A): Ch. 7.
• Numerical stability & time step: Ch. 9; runtime & panels: Ch. 14; manifests: Appendix C.
• Time-base & publication/integrations: companion Energy Filaments white paper (S/P/M/I).

IX. Quality & Risk Control

1. SLIs/SLOs: PBH_pass_rate, rankCtrb/Obsv, κ(W_c/W_o), λ_min(W_c/W_o), AC, MC_p95, E_min_p95, delta_form_*_p95.
2. Fallbacks
   • PBH fails: add drivers/sensors or modify A (add damping / rescale).
   • Gramian ill-conditioning: regularize W + εI, shorten horizon, normalize A.
   • Structural lower bounds too high: rewire topology (edge augmentation) or blockwise control (SCC partitioning).
   • Excessive energy: relocate drivers (increase AC/MC) or adjust weights/physical couplings.
3. Audit: persist A/L/B/C hashes, matching & SCC certificates, PBH details, κ / λ_min, dual-form gaps, placement procedures, and seeds.

Summary
This chapter unifies structural and energy perspectives on controllability/observability: graph matching/SCCs provide lower bounds and placement priors, while PBH/Gramians provide energetic and numerical evidence. With C70-8xx contracts, dual-form gaps delta_form_*, and manifest.stg.ctrlobs.*, the path from modeling to publication is traceable and auditable.