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

1 Introduction

• The design problem is to select which signals local controllers receive and retain while meeting a prescribed per-node loss relative to centralized LQR.Immediate communication radius, maximum spatial footprint, and memory horizon impose distinct communication and storage requirements.Source: §1 · p. 2
• Direct truncation implements retained response coefficients with local filters, while exact localized SLS reoptimizes permitted coefficients to preserve the SLS equality.Direct truncation can leave achieved disturbance responses spatially and temporally extended; feasible exact localization confines nominal effects to a prescribed neighborhood and horizon.Source: §1 · pp. 2–3
• Centralized optimal state and input responses decay exponentially in time and beyond an effective coupling cone, uniformly over the network family.Under polynomial neighborhood growth, this decay supports logarithmic resource requirements and network-size-independent performance guarantees.Source: §1 · pp. 2–3
• For polynomial-growth networks, sufficient communication radius, maximum spatial footprint, and memory horizon grow logarithmically with inverse tolerance, while per-node resource counts grow polylogarithmically.The bounds are independent of total network size.Source: §1 · p. 3
• Direct truncation has a computable small-gain condition for internal stability and an exponentially decaying per-node LQR cost-gap bound.The exact guarantee additionally requires a uniform local deadbeat assumption; response decay and stabilizability alone do not ensure exact localizability.Source: §1 · pp. 3–4
• Prior localized SLS methods impose a fixed response support but do not determine its spatial and temporal scales from a prescribed performance loss.Approximate-achievability methods can produce dense achieved responses, motivating the paper's decay bounds and exact repair result.Source: §1 · pp. 3–4

2.1 Networked LQR over a graph

• The network is a discrete-time linear system on a finite undirected graph, with node-level states, inputs, and graph-structured dynamics and costs.The centralized objective sums impulse-response costs and normalizes by the number of nodes.Source: §2.1 · pp. 4–5

2.2 System Level Synthesis formulation

• SLS represents state-feedback control by maps from a disturbance impulse at one node to state and input trajectories across the graph.The response pair is optimized subject to the achievability equality, which selects maps generated by internally stabilizing state feedback.Source: §2.2 · pp. 5–6
• Localized SLS constrains admissible response blocks while retaining exact achievability; its feasible response pair can be implemented by reconstructing disturbances and filtering them into control inputs.The internal estimate predicts the state from past reconstructed disturbances, and subtracting it from the current state reconstructs the new disturbance.Source: §2.2 · p. 6

3 Spatiotemporal decay of centralized optimal responses

• The regularity assumptions fix coupling range and local dimensions and require stabilizability and detectability bounds uniform over the network family.These uniform conditions provide the locality and regularity used to obtain decay constants that do not deteriorate with network size.Source: §3 · pp. 6–7
• The exponent combines temporal decay with graph distance beyond the cone, whose speed is bounded by the coupling range.For nearest-neighbor couplings, the effective cone expands by one graph hop per time step.Source: §3 · p. 7
• Every centralized optimal state and input response block has an exponential decay bound in elapsed time and distance outside the effective coupling cone, with constants independent of network size.The theorem applies under Assumptions 1 and 2.Source: §3 · pp. 6–7

4.1 Spatiotemporal control architecture

• The implementation mask specifies which disturbance estimates may reach each node at each filter tap, thereby defining message schedules and local filtering work.The parameters are immediate communication radius, maximum spatial footprint, and memory horizon; they constrain filter taps rather than cumulative disturbance influence.Source: §4.1 · pp. 7–8
• The tap mask starts with reach κ, expands by vcone hops per sample up to footprint ¯κ, and permits no coefficients after horizon T.Communication timing must meet measured or guaranteed deadlines; fixed multi-sample delays can be represented in the mask.Source: Figure 2 · p. 8, §4.1 · pp. 8–9

4.2 Accumulated response tails and control architecture selection

• The omitted-response row tail controls the affine residual for direct truncation, while the LQR-weighted tail measures omitted response energy before controller realization.The weighted tail is not the achieved closed-loop cost gap.Source: §4.2 · p. 10
• Under subexponential graph growth, accumulated response tails are bounded uniformly in network size for the three-parameter implementation mask.The mask excludes coefficients beyond the immediate cone, maximum footprint, or memory horizon; graph growth must not cancel their decay.Source: §4.2 · p. 10
• On polynomial-growth networks, choosing κ, ¯κ, and T proportional to log(1/ε) makes the centralized response tail at most ε, with resource bounds independent of network size.The graph-ball growth assumption includes fixed-dimensional grids but excludes regular tree families with exponential shell growth.Source: §4.2 · p. 10

5.1 Direct local FIR implementation

• When the computable small-gain condition holds, direct truncation gives an internally stable realization, and response-tail bounds yield squared decay rates in the achieved LQR cost gap.The theorem's stability guarantee applies when ∥∆θ∥H∞≤αsg for a fixed margin αsg<1.Source: §5.1 · p. 11

5.2.1 Reoptimization on a prescribed mask

• Exact localized SLS reoptimizes all permitted coefficients subject to the SLS equality and, when feasible, confines nominal impulse responses to the prescribed region and horizon.The responses vanish outside the prescribed spatial region and after T+1 steps.Source: §5.2.1 · pp. 11–12

5.2.2 Uniform local deadbeat repair certificate

• The exact-route guarantee uses a sufficient, not necessary, uniform local deadbeat certificate; its common bounds enable bounded residual repair and exact feasibility beyond fixed spatial and temporal margins.The auxiliary deadbeat gain is a repair certificate, not the controller implemented by the repaired optimal response.Source: §5.2.2 · pp. 12–13

5.3 Resources independent of network size for prescribed performance

• For polynomial-growth graphs, sufficient communication radius, spatial footprint, and memory horizon grow logarithmically with inverse tolerance, with resource-count constants independent of N.The direct conclusion does not require Assumption 4; the exact-route conclusion does.Source: §5.3 · p. 13

5.4 Choosing a control architecture and an implementation route

• Architecture selection filters candidates by communication deadlines, per-node resource budgets, and containment limits, then tests stability and loss or exact feasibility before choosing the smallest specified resource measure.If no tested design passes, the procedure increases admissible resources and repeats the checks.Source: §5.4 · p. 14, Figure 3 · p. 14

6 Numerical Experiments

• Figure 4 plots variation above each sampled curve minimum, so its parameter sweeps show variation rather than total loss approaching zero.Structurally infeasible exact QPs are excluded before subtracting minima.Source: Figure 4 · p. 15, §6 · p. 16
• For (κ, ¯κ) = (4, 18), the normalized response tail pθ falls from 7.82 × 10−1 at T = 5 to 1.08 × 10−5 at T = 400.The observed spatial and temporal trends are consistent with the three tail terms in Theorem 2.Source: §6 · p. 16, Figure 5 · p. 16
• For (κ, ¯κ) = (4, 18), direct truncation’s γθ falls from 4.04 at T = 40 to 6.20 × 10−2 at T = 400, crossing the sufficient stability threshold.At T = 400, the achieved normalized cost gap is 2.284 × 10−6, making the residual certificate conservative as a loss predictor.Source: §6 · p. 17
• For κ = 0, complete exact responses require ¯κ ≥ 18 and T ≥ 19; the tested horizon sweep violates the sufficient spatial condition because κ = 0 < csp = 2.The mesh’s deadbeat condition holds with q = 2 and csp = 2; finite-network QPs do not establish the uniform repair bound.Source: §6 · p. 17
• Across 25–144 nodes, direct truncation’s normalized cost gap stays between 6.81 × 10−6 and 2.29 × 10−5, with 0.773 ≤ γθ ≤ 0.892 < 1.Per-node counts remain Chalo = Cfoot ≤ 41 and Ctap ≤ 6601; average footprint falls from 77.6% to 21.8% of the network.Source: §6 · p. 17, Figure 6 · p. 18
• Among feasible complete QPs, log pθ and log(JlocN −1) have Spearman correlation 0.990, supporting response-tail ranking of tested architectures.The sweep includes 168 complete QPs: 140 feasible and 28 structurally infeasible.Source: §6 · p. 17, Figure 7 · p. 19
• For the 1% target, θ=(2,2,0) achieves 0.1765% loss with the smallest Ctap; nominal containment instead uses θ=(2,3,159), achieving 0.9918% loss.Reducing the maximum spatial footprint to two hops raises containment-design loss to 4.3227%.Source: §6 · pp. 17–18, Table 1 · p. 18

• Under the stated assumptions, prescribed per-node LQR loss can be attained with communication, storage, and filtering counts independent of network size.Source: §7 · p. 19
• Direct truncation uses local filters while allowing disturbance tails; exact localized SLS additionally confines the nominal disturbance response.Source: §7 · p. 19
• Future work includes weaker local residual-repair conditions, uncertainty-sensitive disturbance bounds, and richer communication and installation-cost models for architecture co-design.Source: §7 · p. 19

A.1 Finite horizon representation and optimality response graph

• The Riccati terminal cost makes the finite-horizon optimizer agree with the centralized infinite-horizon trajectory over its first Th steps, so an initial-state perturbation yields an SLS impulse response.Source: §A.1 · p. 20
• First-order conditions form a linear KKT system whose state, input, and multiplier coordinates are grouped by node and time.Source: §A.1 · p. 20
• The KKT graph connects coordinates according to nonzero Q, R, A, and B blocks and the initial-condition and state-update constraints.Source: §A.1 · p. 20
• The dense Riccati terminal Hessian is excluded from the repeating graph so graph distance is independent of finite horizon; its effect is treated separately.Source: §A.1 · p. 20
• Shortest paths from the source multiplier to state and input coordinates define the corresponding response distances.Source: Figure 8 · p. 21, §A.1 · p. 21
• Although detectability uses Q^1/2, the KKT Hessian contains Q, so the KKT graph follows Q's sparsity.Source: §A.1 · p. 21

A.2 Auxiliary facts on inverse decay

• The inverse-decay constants depend on L, γ, and fixed local block dimensions, not graph cardinality or vertex degree.Source: §A.2 · p. 21
• Polynomial approximation explains inverse decay: graph structure determines which low-degree polynomial blocks vanish, while approximation constants follow from the spectral interval.Source: §A.2 · p. 21
• Adding a clique at the terminal layer may shorten distances to that layer, but it does not change inverse-decay constants; separation prevents shortening distances between response coordinates of interest.Source: §A.2 · p. 21

A.3 KKT proof of spatiotemporal response decay

• The proof adapts the finite-horizon sensitivity mechanism used for distributed LQR gain decay to the KKT graph.Source: §A.3 · p. 22
• The KKT graph distinguishes state, input, and multiplier blocks, and interprets sensitivity blocks as SLS response coefficients.Source: §A.3 · p. 22
• Under Assumptions 1 and 2, the discrete algebraic Riccati equation for system (3) has a stabilizing solution P* ≥ 0.Source: §A.3 · p. 22

The centralized SLS optimum satisfies

• The KKT matrix has a uniformly positive minimum singular value, with coercivity controlled by state and input energy certificates rather than terminal-Hessian sparsity.The bound is uniform in the horizon and network size; the terminal Hessian enters only through nonnegativity.Source: The centralized SLS optimum satisfies · p. 23
• Centralized state and input response coefficients satisfy decay bounds in KKT distance with constants independent of network size.The finite-horizon graph distance for nonterminal vertices agrees with the infinite-graph distance when the horizon is sufficiently large.Source: The centralized SLS optimum satisfies · p. 24

A.4.1 Tail bounds for implementation tap masks

• Truncating response taps yields filters that are FIR and stable, with truncation-tail and residual bounds using rates independent of network size.The tail analysis groups response blocks into physical decay shells under the stated growth assumption.Source: §A.4.1 · pp. 25–26

A.5 Proofs for direct truncation and exact localized SLS cost gaps

• Stable feasible SLS responses have internally stabilizing realizations, and optimality supplies an LQR orthogonality identity.These properties support the cost analysis for direct truncation and exact localized implementations.Source: §A.5 · p. 27

A.5.1 Small gain proof for direct truncation

• When the truncation residual satisfies the small-gain threshold, a Neumann-series correction yields an achieved response, which is generally dense rather than exactly localized.The cited proof applies when ε ≤ α_sg < 1.Source: §A.5.1 · p. 27

A.5.2 Exact implementation and cost

• Residual repair with fixed spatial and temporal margins keeps repaired coefficients within the target mask, and the resulting cost gap has an upper bound with centralized optimality supplying its lower bound.The proof sets the temporal margin to q + 1 taps.Source: §A.5.2 · p. 28

A.5.3 Resource law independent of network size

• Setting all three architecture parameters to m_ε gives the stated small-gain and feasibility guarantees with constants independent of network size.The result follows by choosing m_ε to meet the truncation bound and applying the direct-truncation theorem.Source: §A.5.3 · p. 29