Spatiotemporal Response Decay for Near-Optimal Distributed LQR via System Level Synthesis

The gist: Spatiotemporal response decay provides a quantitative basis for control architecture selection, showing that prescribed LQR loss per node can be attained with communication, storage, and filtering counts independent of network size.

System Level Synthesis Formulation

The centralized optimal state and input responses to disturbance impulses satisfy exponential decay bounds in both time and spatial distance under uniform regularity assumptions The centralized problem is formulated by minimizing the weighted state and input energy generated by unit impulses in all coordinates of w, normalized per network node This objective is defined as J⋆N:= JN (Φ⋆) where Φ⋆ is an optimal solution of the SLS achievability condition ZΦ = I The centralized problem becomes min ΦJN (Φ) s.t. ZΦ = I, Φ ∈ z−1RH∞.

Spatiotemporal Response Decay

Theorem 1 establishes that there exist constants Clc 0, independent of N, such that the centralized optimal state and input responses decay exponentially with elapsed time and with distance beyond an effective coupling cone determined by the local dynamics and cost couplings The exponent in (11) separates two effects. The term t gives temporal decay, while [dG(i, j) − vconet]+ measures how far node i lies beyond the effective coupling cone at time t.

Controller Construction and Resource Bounds

The paper develops two controller constructions: Direct truncation implements the retained coefficients as local filters that use the specified signals and retain them for the chosen memory horizon Exact localized SLS reoptimizes the coefficients permitted by the chosen implementation mask subject to the SLS equality. Under subexponential graph growth, these approximation bounds yield a logarithmic dependence of sufficient immediate communication radius, maximum spatial footprint, and memory horizon on the inverse tolerance for LQR loss per node The corresponding communication, storage, and filtering counts at each node grow polylogarithmically, with constants independent of the total network size

Performance Guarantees

For direct truncation, a computable small gain condition certifies internal stability and yields an exponentially decaying bound on the achieved LQR cost gap per node The exact guarantee uses an additional uniform local deadbeat assumption to restore the SLS equality within fixed spatial and temporal margins, yielding an exponentially decaying bound on the exactly localized LQR cost gap per node Corollary 1 predicts that a prescribed performance level can be maintained using local resources independent of network size

Design Procedure

The design procedure involves specifying design requirements, form admissible control architectures, prioritize candidates using response tails, and then evaluate direct truncation or exact localized SLS The selection process ranks new candidates by tail reduction per added resource and repeats the checks until a validated design is found.

Numerical Validation

Numerical experiments illustrate how performance and disturbance containment requirements guide architecture selection The thermal mesh experiments test these guarantees and illustrate how this distinction changes the control architecture and implementation route selected for a prescribed performance target. The response tail pθ is used to prioritize exact QP evaluations, showing a high Spearman correlation with log(Jlocθ/J⋆N − 1).

Conclusion

Spatiotemporal response decay provides a quantitative basis for control architecture selection, showing that prescribed LQR loss per node can be attained with communication, storage, and filtering counts independent of network size The two controller constructions distinguish limits on implementation resources from limits on disturbance propagation. Further work can address three limitations of the present analysis.

Auxiliary Facts and Proofs

The proof relies on the standard inverse decay principle of Demko–Moss–Smith and its graph extensions The KKT proof of spatiotemporal response decay follows the finite horizon sensitivity mechanism used for distributed LQR gain decay in [28]. Proposition 1, (60), and the triangle inequality give the upper bound for exact localized SLS gap. Corollary 1 predicts that a prescribed performance level can be maintained using local resources independent of network size The polynomial ball bound and dmax(τ) ≤ κ¯ε give resource counts in (38). The lower bound follows from centralized optimality.

References

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[17] Y.

1. The system can dynamically select between direct truncation and exact localized SLS based on disturbance containment requirements, as illustrated by Figure 3. This allows for a trade-off between implementation resources (e.g., communication radius, memory horizon) and nominal disturbance effects (Disturbance tails allowed: try direct first vs Disturbance containment required).

2. AI systems can optimize control architecture parameters using a response tail metric, as the paper shows that the response tail thus ranks the tested control architectures well. This enables prioritizing candidate architectures based on their Etail(θ) to achieve a prescribed LQR loss target.

3. The system can determine necessary local resources independent of network size by choosing architecture parameters proportional to log(1/ε), as quantified in Corollary 1: for every ε ∈ (0, 1) one can choose θ = (κε, κ¯ε, Tε) with max[κε, κ¯ε, Tε] ≤ c0 + c1 log(1/ε) to maintain a performance gap of JN (Φbclθ) − J⋆N ≤ ε.

4. The system can perform exact localized SLS reoptimization to confine nominal disturbance responses using a Uniform local deadbeat repair certificate, which ensures the achieved response is feasible and minimizes the LQR cost gap, as shown in Theorem 4: J locθ − J⋆N ≤ CJ δ 2κ/κ + δ 2¯κ/κ̄ + δ 2T/T.

5. The system can verify internal stability of a direct truncated realization by checking the computable residual bound γθ, where "γθ < 1 implies (I + ∆θ)−1 = P∞ k=0(−∆θ)k ∈ RH∞, leading to an exponentially decaying bound on the achieved LQR cost gap per node" as per Theorem 3.