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Distributed Performance Boosting through Neural Control for Plug-and-Play Networks

Danilo Saccani, Leonardo Massai, Giancarlo Ferrari-Trecate

arXiv:2610.23292v1eess.SYcs.LGcs.SYmath.OC

TL;DR

Distributed baseline controllers can preserve stability through plug-and-play changes, but adding expressive neural policies can invalidate that guarantee. The paper uses an IMC-based gain condition and two free parametrizations to train distributed neural controllers while retaining a prescribed network gain. Its DC-microgrid study reports improved voltage transients on top of a decentralized plug-and-play-stable baseline.

  • Problem

    Distributed baseline controllers can preserve stability after plug-and-play changes, but adding expressive neural policies can invalidate the guarantee, while learning-based methods typically lack hard stability guarantees.

    Source: §1 · pp. 1–2

  • Method

    The paper derives two free parametrizations of interconnected neural operators that enforce a prescribed network ℓ2 gain for every trainable-variable value.

    Source: §1 · p. 2, Contributions · p. 2

  • Results

    The DC-microgrid study showed improved voltage transients on top of a decentralized plug-and-play-stable baseline.

    Source: §6 · p. 13

  • Takeaways & Limitations

    For strictly causal local realizations and separable costs, training can use neighbor-to-neighbor communication while preserving the prescribed gain at every iterate.

    Source: Contributions · p. 2

    Smaller γR increases the robustness margin but restricts the admissible performance-boosting operators.

    Source: §3.1 · p. 5

Abstract

from arXiv · show

We address the problem of improving the performance of pre-stabilized networked systems subject to plug-and-play (PnP) operations, namely additions or removals of subsystems. While distributed baseline controllers can preserve stability after a PnP transition, augmenting them with expressive neural policies can invalidate the formal stability guarantees. To overcome this issue, we build on recently proposed Performance Boosting controllers based on the Internal Model Control (IMC) principle, which reduce closed-loop stability to a finite $\ell_2$-gain requirement on an auxiliary operator. In the distributed setting, this operator is formed by interconnecting local neural operators that exchange information only between neighboring subsystems. We then derive free parametrizations of this interconnection, namely explicit maps from unconstrained trainable variables to interconnected neural operators satisfying a prescribed network gain bound for every parameter value. The obtained distributed neural controller can therefore be trained by unconstrained gradient-based optimization without stability constraints or projection steps, while retaining the prescribed gain bound at every training iterate. We develop two complementary parametrizations. A neighbor-based compositional parametrization uses local coupling bounds, so that after a PnP event only subsystems whose bounds change need to update their gain parameters. A network-norm parametrization instead uses a bound on the overall coupling matrix and can reduce conservatism, at the price of requiring network-wide information. A PnP DC-microgrid case study shows that the proposed neural layer improves voltage transients under topology changes and load variations while preserving nominal closed-loop stability.

1 Introduction

Distributed control can preserve stability but often restricts performance optimization, while learning-based controllers offer expressive policies without hard stability guarantees. This paper addresses the challenge by constructing distributed neural controllers with a prescribed network gain that supports stability guarantees under plug-and-play operations.

1 Introduction

  • Model-based distributed methods provide stability guarantees but often optimize restricted criteria or do not optimize task-dependent performance objectives.The challenge is retaining formal stability while optimizing expressive distributed feedback for possibly non-convex objectives.Source: §1 · p. 1
  • Learning-based methods offer expressive policies, but typically lack hard closed-loop stability guarantees and may rely on shared parameters and homogeneous subsystem models.Stability is often encouraged indirectly through reward shaping, regularization, or external safety layers.Source: §1 · p. 2
  • After a plug-and-play transition, baseline stability guarantees do not automatically extend to an added neural feedback law.Model-based plug-and-play methods can preserve stability for baseline local controllers after network changes.Source: §1 · p. 2
  • The paper constructs free parametrizations of interconnected neural operators that enforce a prescribed network gain for every trainable-variable value.The paper also characterizes how to restore this gain after admissible topology changes.Source: §1 · p. 2

Contributions.

The paper develops two free parametrizations for stable distributed neural performance boosting, characterizes plug-and-play recertification, and establishes neighbor-to-neighbor training under stated conditions. It extends a preliminary version by addressing topology changes and training implementation.

Contributions.

  • Two free parametrizations enforce a prescribed network ℓ2 gain: a neighbor-based construction uses local coupling information, while a network-norm construction uses an overall coupling bound and can reduce conservatism.The network-norm approach requires network-wide information.Source: Contributions · p. 2
  • After plug-and-play changes, the neighbor-based parametrization can limit gain-parameter updates to subsystems whose local coupling bounds change.The network-norm approach generally requires updating a network-wide bound unless a common bound for an admissible configuration family is available.Source: Contributions · p. 2
  • For strictly causal local realizations and separable finite-horizon costs, controller training can use neighbor-to-neighbor communication while preserving the prescribed network gain at every iterate.Source: Contributions · p. 2
  • Compared with the preliminary version, this work adds network-norm parametrization, plug-and-play recertification, neighbor-to-neighbor training, and detailed proofs.Source: Contributions · p. 2

Notation

The notation section establishes sequence spaces, norms, operator properties, and stacking conventions used to state the stability and interconnection results.

Notation

  • ℓ^n denotes sequences of vectors a_t in R^n, and ℓ^n_2 contains those sequences with finite ℓ2 norm.Source: Notation · p. 3
  • For matrices, ∥A∥_2 is the spectral norm; causal operators depend only on present and past inputs, and finite-gain ℓ2-stability bounds output norm by γ times input norm plus β.The notation also defines K functions, column-wise stacking, and block-diagonal operators.Source: Notation · p. 3

2 Plug-and-play networks and problem formulation

The problem formulation assumes a pre-stabilized network whose admissible plug-and-play changes preserve baseline stability, then defines distributed controller and update constraints. Performance boosting seeks task-dependent finite-horizon improvement without losing the baseline guarantee.

2 Plug-and-play networks and problem formulation

  • The network comprises nonlinear time-varying subsystems coupled over an undirected connected graph, which also specifies permitted neighbor-to-neighbor communication.Source: §2 · p. 3
  • A plug-and-play operation adds or removes subsystems and changes the graph and induced plant map; an admission mechanism determines which changes are admissible.Source: §2 · p. 3
  • The physical network is assumed pre-stabilized, with every admissible plug-and-play operation preserving baseline ℓ2 stability; the plant is the map after baseline feedback is absorbed.The uncontrolled physical network itself need not be stable.Source: §2 · p. 3

2.1 Distributed controllers and PnP constraints

  • A distributed controller uses local dynamical systems, local measurements, and neighbor-only internal signals, allowing information to propagate multiple hops over time but excluding direct non-neighbor communication.Source: §2.1 · p. 4
  • After a plug-and-play transition, only newly added subsystems and surviving subsystems with changed local dynamics or neighborhoods may update their controllers.In the illustrated transition, Σ3 is removed, Σ4 joins through Σ2, and the affected post-change set is {2, 4}.Source: §2.1 · p. 4, Figure 1 · p. 4

2.2 Performance boosting and post-change update problems

  • Performance boosting minimizes a task-dependent, possibly non-convex finite-horizon cost without compromising stability guaranteed by the baseline controller.Source: §2.2 · p. 4
  • After a plug-and-play operation, the primary objective is to restore feasibility while modifying only affected controllers; subsequent reoptimization may improve performance, but unconstrained neural updates can destabilize the network.Source: §2.2 · p. 4

3 Distributed Performance Boosting

The section uses an IMC auxiliary operator to connect performance boosting with a prescribed gain condition for robust closed-loop stability. It realizes that operator through neighboring local neural operators and develops gain-certified parametrizations, including a neighbor-based construction.

3.1 Nominal IMC reformulation and robust extension

  • The robust IMC condition makes closed-loop finite-gain ℓ2 stability depend on bounds for plant and model mismatch, while the target gain γR trades robustness margin against controller flexibility.The result allows an additive bound that may depend on the initial state and βR.Source: §3.1 · p. 5

3.2 Distributed internal realization of R

  • Local neural operators exchange internal signals only with neighbors, allowing dynamic coordination while realizing the auxiliary operator as a distributed controller.Each local operator has a prescribed gain bound; the network gain certificate combines local inequalities to guarantee the desired gain from reconstructed residual to control.Source: §3.2 · pp. 5–7, Figure 3 · p. 6

3.3 Free parametrization of the network gain certificate

  • Explicit maps select local gains γ[i] and multipliers α[i] from unconstrained variables so the network gain certificate holds by design.The maps rely on structural conditions on residual injection and control-output matrices.Source: §3.3 · p. 7

3.3.1 Neighbor-based compositional parametrization

  • The neighbor-based parametrization uses local coupling information to choose gains satisfying the network certificate for every choice of its free variables.Its sufficient diagonal-dominance bounds are conservative relative to the full network certificate; the construction uses information associated with each subsystem and its neighbors.Source: §3.3.1 · p. 8
  • For the two-subsystem example at c = 0.25, the network-norm construction admits larger local gains than the neighbor-based construction, but neither parametrization generally dominates the other.The two constructions are different sufficient inner approximations of the network gain certificate.Source: Figure 4 · p. 8

Illustrative example.

The example compares neighbor-based and network-norm gain parametrizations, then describes how to recertify them after plug-and-play changes while preserving the target gain.

Illustrative example.

  • In the illustrative example, α = 0 gives (α, γ) = (1.82, 0.371) and (α, γ) = (1.72, 0.381).Figure 4 compares the resulting sufficient region with the original network gain certificate.Source: Illustrative example · p. 8

3.3.2 Network-norm parametrization

  • The network-norm parametrization uses an overall coupling-matrix bound to reduce conservatism, at the price of requiring network-wide information.Its bound may need recomputation when the interconnection changes; a known family-wide worst-case bound avoids updates for surviving subsystems but adds conservatism.Source: §3.3.2 · pp. 8–9, §3.4 · p. 9
  • For this interconnection and the same c, the network-norm construction admits larger local gains than the neighbor-based construction, but neither parametrization generally dominates the other.The example reports α[1] = α[2] = 1.193 and γ[1] = γ[2] = 0.458 for the network-norm construction.Source: §3.3.2 · p. 9

3.4 PnP recertification

  • The neighbor-based parametrization supports localized recertification because its defining quantities depend only on each subsystem’s couplings with its neighbors.Under the stated update assumptions, only new subsystems and surviving subsystems whose relevant quantities change need updates.Source: §3.4 · p. 9
  • After a PnP event, recertification preserves the same target gain without gradient-based retraining of surviving neural parameters.Only subsystems whose local coupling bounds change need to recompute certificate quantities and gain parameters; retraining may follow for performance but is not required for recertification.Source: §3.4 · pp. 9–10

4 Distributed training and PnP adaptation

The distributed training formulation turns the prescribed-gain parametrizations into an unconstrained finite-dimensional optimization problem. It supports neighbor-to-neighbor training and adaptation across PnP configurations while retaining the gain and stability guarantees.

4 Distributed training and PnP adaptation

  • The free parametrizations make training an unconstrained finite-dimensional optimization while preserving the prescribed network gain at every iterate.Each iteration evaluates closed-loop rollouts and differentiates them backward in time; no stability constraint or projection is required.Source: §4 · p. 10

4.1 Neighbor-to-neighbor training

  • Strictly causal local operators enable forward and reverse computations through neighbor-to-neighbor exchanges, while information can propagate over multiple hops over time.Strict causality is an implementation restriction, not a requirement of the gain and stability results.Source: §4.1 · p. 10

4.2 Training across configurations and post-PnP adaptation

  • Training across known admissible configurations shares surviving subsystems’ neural parameters while evaluating certificate quantities and local gains for each configuration.This lets performance be optimized over expected topology changes before deployment.Source: §4.2 · p. 10
  • After a PnP transition, neighbor-based recertification updates only subsystems whose local coupling bounds change, without retraining surviving neural parameters.Any selected subset may then be fine-tuned while the remaining controllers retain their recertified parameters.Source: §4.2 · p. 10
  • Every training iterate preserves nominal closed-loop ℓ2 stability, and the post-PnP guarantee holds when the changed parameters satisfy the corresponding free parametrization.For the neighbor-based construction, Theorem 3 ensures the post-change condition; under model mismatch, the robust statement applies when (14) holds.Source: §4.2 · pp. 10–11

5 Numerical example

The numerical example evaluates a distributed neural layer added to decentralized passivity-based primary controllers in a PnP DC microgrid. Across plug-in, load-change, and unplugging events, it reduces voltage deviations and integrated squared voltage error relative to the primary baseline, with a regional stability check for the simulated trajectories.

5 Numerical example

  • The case study adds a distributed neural layer to decentralized passivity-based primary controllers that preserve stability under admissible PnP operations.Training spans expected configurations, with post-change recertification of the neighbor-based parametrization.Source: §5 · p. 11

5.1 Microgrid setup and nominal IMC model

  • The simulated sequence comprises an initial load step, plugging in DGU 6, a further load step, and unplugging DGU 3.The plug-in adds links (1, 6) and (5, 6); the unplugging removes links (1, 3) and (3, 4).Source: §5.1 · p. 11
  • The nonlinear ZIP-load simulations use a nominal IMC model obtained by linearizing the continuous-time dynamics around each configuration’s equilibrium and applying forward Euler discretization.The nonlinear remainder is represented as model mismatch, while PnP recentering and state-transition effects enter the disturbance-feedback channel.Source: §5.1 · p. 12

5.2 Controller design and closed-loop results

  • For the six-DGU topology, the certified common local gains are 0.289 for the neighbor-based parametrization and 0.295 for the network-norm parametrization.The network-norm parametrization provides only a modest increase in this normalized setting; the neighbor-based parametrization is used for localized PnP recertification.Source: §5.2 · p. 12
  • Across considered configurations, the regional small-gain check is satisfied: γµ∆ = 2.2·10^-5 and, with γR = 10, the reported check value is 8.5·10^-3 < 1.The check applies within the 10 V region, verified a posteriori for all simulated trajectories; the PnP recentering term is treated separately as an impulse ℓ2 residual.Source: §5.2 · p. 13
  • 60.1%, 59.1%, and 56.4% peak voltage-deviation reductions and 15.3%, 76.1%, and 72.4% integrated squared voltage-error reductions follow plug-in, load change, and unplugging, respectively.Comparisons are against the decentralized primary baseline over the 0.4 s following each event.Source: §5.2 · p. 13

6 Conclusions

The paper presents fail-safe training for distributed neural performance-boosting controllers in pre-stabilized PnP networks. Its DC-microgrid study reports improved voltage transients on a decentralized PnP-stable baseline.

6 Conclusions

  • An IMC reformulation reduces closed-loop stability to a prescribed network ℓ2 gain enforced through two free parametrizations.The neighbor-based construction supports localized PnP recertification; the network-norm construction uses network-wide coupling information and can reduce conservatism.Source: §6 · p. 13
  • For strictly causal local realizations and separable costs, training admits a neighbor-to-neighbor implementation while every iterate preserves the prescribed gain.Source: §6 · p. 13
  • The DC-microgrid study reports improved voltage transients with the neural layer added to a decentralized PnP-stable baseline.Source: §6 · p. 13

A Proof of Proposition 1

The proof bounds truncated-horizon signals using the plant, controller, mismatch, and disturbance gains. Condition (14) makes the resulting bound uniform in the horizon, establishing finite-gain ℓ2 stability.

A Proof of Proposition 1

  • A truncation argument applies the causal signal bounds and gain bound on each finite horizon without assuming closed-loop signals are initially in ℓ2.The reconstructed residual is expressed using the model mismatch, initial-state term, and disturbance.Source: A Proof of Proposition 1 · p. 15
  • Condition (14) makes Γ positive, allowing the finite-horizon signal bound to be solved with constants independent of the horizon.The additive terms may depend on the initial state.Source: A Proof of Proposition 1 · p. 15
  • Because the disturbance is in ℓ2 and the bounds are uniform in the horizon, taking T to infinity establishes that the input and state are in ℓ2 and the closed loop is finite-gain ℓ2-stable.Source: A Proof of Proposition 1 · p. 15

B Proof of Theorem 1

The proofs establish the required gain condition through Schur complements and matrix bounds: diagonal dominance supports the neighbor-based result, while a uniform scalar and coupling bound support the network-norm result.

B Proof of Theorem 1

  • For the neighbor-based proof, coordinate injections make the relevant matrix diagonal, and successive Schur complements reduce the condition to a matrix inequality.The proof identifies the diagonal structure and applies two Schur complements before checking diagonal dominance.Source: B Proof of Theorem 1 · pp. 15–16
  • Diagonal dominance makes the final symmetric matrix positive semidefinite, establishing condition (23) for the neighbor-based parametrization.The proof bounds off-diagonal row sums and verifies that diagonal entries are at least as large.Source: B Proof of Theorem 1 · p. 16

C Proof of Theorem 2

  • The network-norm proof sets α[i](γ[i])²=c uniformly and uses the coupling bound to establish condition (23).The proof states that the resulting matrix inequality is satisfied under the stated bound on the coupling matrix.Source: C Proof of Theorem 2 · p. 16

D Proof of Theorem 3

After a PnP change, the recertification procedure updates parameters where needed while retaining the prescribed gain condition and leaving unaffected surviving local controllers unchanged.

D Proof of Theorem 3

  • The post-change neighbor-based parameters satisfy Theorem 1, so condition (23) holds with the same γR.Surviving subsystems outside the identified set retain their parameters; parameters for subsystems in that set are constructed from post-change quantities.Source: D Proof of Theorem 3 · p. 16
  • A surviving subsystem with an unchanged neighborhood retains its coupling blocks and bounds, so it is not selected for recertification.The proof shows its local sets remain unchanged and consequently its recertification set membership does not change.Source: D Proof of Theorem 3 · p. 16
  • For surviving subsystems outside the changed-neighborhood set, the complete local controller realization remains unchanged.The proof combines unchanged local parameters, neighborhood, local map, and controller-interconnection blocks.Source: D Proof of Theorem 3 · p. 16

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