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Robust stabilization with spiking neuronal communication

Elena Petri, Romain Postoyan, Erik Steur, W. P. M. H., Heemels

arXiv:2609.05054v1eess.SY

TL;DR

The paper addresses robust stabilization when noisy sensor-to-controller communication uses neuron-inspired spikes rather than direct analog measurements. It designs an integrate-and-fire encoder, synaptic decoder, and controller, and derives conditions for practical input-to-state stability across nonlinear systems and stabilizable, detectable LTI systems. The results include a single-link manipulator simulation showing a trade-off between ultimate error bounds and communication rate.

  • Problem

    Neuromorphic control lacks systematic design and analysis methods for stabilizing disturbed plants when sensor-to-controller communication uses spiking signals.

  • Method

    The framework jointly designs an integrate-and-fire spike encoder, synaptic spike decoder, and controller, analyzing their hybrid closed loop through an augmented continuous–spiking interconnection.

  • Results

    The design conditions guarantee practical input-to-state stability, apply to a class of nonlinear systems and any stabilizable and detectable LTI system, and are illustrated on a single-link manipulator.

  • Takeaways & Limitations

    Adjusting spike amplitudes controls the practical stability neighborhood, while smaller amplitudes generally require more spikes or communications.

  • Takeaways & Limitations

    The continuous-system stability assumption is required, and one augmented stability-like property holds only for a measure-zero set of auxiliary initial conditions.

Abstract

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Neuromorphic engineering develops hardware and software systems inspired by biological neurons, with the goal of achieving energy-efficient, low-latency, robust, and adaptive computation, communication and control. Its potential impact on systems and control is significant, as it may enable novel approaches to control and estimation by leveraging brain-inspired computation and communication principles. In this context, we present a framework for the robust stabilization of a plant subject to disturbances when the communication between noisy sensors and the controller relies on spiking signals generated by neuron-inspired schemes. The communication scheme consists of a spike encoder on the sensors side, which is based on integrate-and-fire neurons that convert the analog plant output measurement into a spiking signal, and a spike decoder on the controller side inspired by synaptic processing to convert the received spiking signal into an analog signal. We provide design conditions on the spike decoder, the spike encoder as well as on the controller under which the closed-loop system exhibits a practical input-to-state stability property, where the adjustable parameters are the amplitudes of the spikes. The results are shown to be applicable to a class of nonlinear systems as well as to any stabilizable and detectable linear time-invariant system. Numerical simulations on a single-link manipulator illustrate the potential of the approach.

1 Introduction

The paper addresses the lack of systematic analysis and design methods for neuromorphic control by developing spiking sensor-to-controller communication for robust stabilization. It designs the controller, encoder, and decoder jointly and establishes practical stability conditions applicable to nonlinear and LTI systems.

  • Neuromorphic control remains limited by a lack of systematic analysis and design methodologies.
  • Spiking communication encodes information in fixed-amplitude spike timing, supporting event-based and asynchronous sensor-to-controller transmission.The approach is motivated by energy-efficiency and latency advantages over conventional communication.
  • The analysis models the closed loop as a hybrid system and derives conditions on the controller, encoder, and decoder for practical input-to-state stability.Auxiliary variables expose an interconnection between a continuous-time system and a spiking system.
  • The conditions apply to nonlinear systems and any stabilizable and detectable LTI system, while simulations illustrate a trade-off between ultimate bounds and spike quantity.The single-link manipulator example also suggests improved robustness relative to continuous-time communication.
  • The framework designs both the controller and neuronal communication scheme to practically stabilize disturbed nonlinear systems.It uses an integrate-and-fire spike encoder and a spike decoder to address general nonlinear plant models.

2 Preliminaries

The preliminaries define the mathematical objects used to represent hybrid and spiking systems, including spike trains, spiking signals, norms, and input-to-state stability. They also formalize Zeno-free timing sequences and the notation for vectors, matrices, sets, and distributions.

  • The paper represents spike times as increasing sequences beginning at zero and distinguishes Zeno-free sequences that have no finite accumulation point.Infinite Zeno-free sequences diverge in time rather than accumulating after finitely many seconds.
  • Vectors, matrices, norms, distances to sets, and comparison-function notation establish the analysis language for stability results.The notation includes identity matrices, diagonal matrices, rank, Euclidean norms, and set distance.
  • A spike train is modeled using Radon measures formed from amplitudes applied at spike times, with Zeno-free trains defined separately.The associated times are spiking times and the associated vectors are spike amplitudes.
  • A spiking signal combines a locally essentially bounded continuous signal with a Zeno-free train of spikes under a finite integral norm.
  • Input-to-state stability bounds the state by a decaying initial-state term and gain functions of the accumulated input magnitudes.

3 Problem description

The problem is to stabilize a nonlinear plant when its noisy measured output reaches the controller only through neuron-inspired spiking communication. The paper seeks design conditions that ensure stability and exclude Zeno behavior.

  • The plant has state x_p, control input u, disturbance d, and noisy measured output y generated by continuous plant and output maps.
  • An encoder converts the analog output y into a spiking signal y_s, and a synaptic-inspired decoder converts y_s into an analog signal available to the controller.The controller stabilizes the origin using the decoded signal rather than direct access to y.
  • The design objective is to choose the encoder, decoder, and controller so the closed loop has the specified stability properties and avoids Zeno behavior.Zeno behavior means infinitely many spikes in finite time.
  • The encoder architecture is illustrated as a block diagram and is built from neuron-inspired spiking communication components.

4 Spike encoder, spike decoder and controller

The section defines an integrate-and-fire spike encoder, an LTI synaptic decoder, and an output-feedback controller for closing the loop through spiking communication. Design conditions target practical input-to-state stability under disturbances and measurement noise.

  • Spike encoder: The encoder uses two integrate-and-fire neurons per measured output component to transform the analog output y into a spiking signal y_s.The two-neuron structure separates sensitivity to positive and negative parts of each scalar output.
  • Spike encoder: A neuron emits a fixed-amplitude spike when its membrane potential reaches the firing threshold, then resets while the other neuron remains unchanged unless both trigger.Thresholds Δ_i,ℓ and amplitudes α_i,ℓ are design parameters, with the stability design taking α_i,ℓ = Δ_i,ℓ.
  • Spiking times: The overall spiking sequence is formed by merging the firing times of all two-neuron channels, allowing coincident events and an initial spike when a membrane potential already exceeds threshold.The resulting sequence is indexed by the successive times at which any neuron reaches its threshold.
  • Spike decoder: The spike decoder is an LTI filter whose state processes the spike train and produces the analog controller input; diagonal Hurwitz dynamics give each channel a synaptic-filter interpretation.The filter matrices A_f, B_f, and C_f are design variables, and spikes from the second neuron receive a negative gain to distinguish signal polarity.
  • Controller and closed loop: The controller is an output-feedback dynamic system driven by the decoder output, and its design is coupled with the decoder and encoder through the closed-loop stability objective.The objective is practical ISS for the closed loop when thresholds equal spike amplitudes.

5 Augmenting the closed-loop model

The paper augments the closed-loop model by separating nominal continuous filtering from spike-induced filtering. This decomposition preserves the original trajectories and spike times while enabling stability analysis through a continuous-time and spiking-system interconnection.

  • 5.1 Spike-induced error: The spike-induced error e is defined as the mismatch between the analog plant output y and its spiking counterpart y_s.This concentrates the spiking effect into an error input for an equivalent continuous-time representation.
  • 5.2 Splitting the spike decoder dynamics: The decoder state is decomposed as x_f = x_f,n − x_f,e, where the nominal state filters y continuously and the error state captures the spiking dynamics.The auxiliary states are introduced for design and analysis rather than implementation.
  • 5.2 Splitting the spike decoder dynamics: Lemma 1 establishes that every decoder solution equals the nominal-filter solution minus the error-filter solution over the interval determined by the spiking times.The relation uses the same initial-condition difference and the error input e = y − y_s.
  • 5.3 Augmented closed-loop model: The augmented closed-loop system is constructed from the nominal and error filter dynamics to support the subsequent stability analysis.The augmentation increases the model dimension while retaining the original closed-loop behavior.
  • 5.3 Augmented closed-loop model: Theorem 1 shows that the original and augmented systems have the same domain, spike-time sequence, and membrane-potential trajectory under matching initial conditions and inputs.Thus the augmented model is behaviorally equivalent for the quantities governing spiking.

6 Closed-loop system stability

The augmented closed-loop model is analyzed as a feedback interconnection between continuous-time dynamics and a spiking subsystem. Under encoder, decoder, controller, and stability conditions, the actual closed loop is complete, Zeno free, and practically input-to-state stable.

  • 6.1 The augmented closed-loop model as a feedback interconnection: The augmented model is decomposed into a continuous subsystem Σcont and a spiking subsystem Σs in feedback interconnection.Σcont receives disturbances, measurement noise, and the spike-induced error, while Σs receives y and outputs the corresponding error signal.
  • 6.2 Stability property of Σs: Proposition 1 establishes practical stability for Σs when Af is Hurwitz and spike amplitudes equal neuron thresholds.The same proposition establishes completeness and non-Zeno spiking for every input y.
  • 6.3 Stability property of Σcont: Assumption 1 requires Σcont to be input-to-state stable with respect to disturbances, measurement noise, and the spike-induced error.This corresponds to designing the controller so the plant, controller, and nominal filter are ISS under continuously communicated noisy output.
  • 6.4 Stability properties for the closed-loop models: Under the subsystem conditions and Assumption 1, the augmented closed loop is complete, Zeno free, and practically stable, and these properties transfer to the actual closed-loop model.Theorem 2 states that the desired practical ISS objective holds for the actual closed loop.
  • 6.4 Stability properties for the closed-loop models: The spike amplitudes and thresholds define the considered initial-condition and attractor sets, while the framework also extends to generic compact attractors and other communication channels.The proposed methodology can be applied to controller-to-actuator, dual-channel, and distributed spiking configurations.

7 Satisfaction of Assumption 1

The paper develops conditions ensuring Assumption 1 for nonlinear and LTI plants by designing the controller, spike decoder, and filter dynamics to obtain ISS properties. For LTI systems, stabilizability and detectability permit a constructive output-feedback design.

  • 7 Satisfaction of Assumption 1: The design problem jointly involves the controller and spike-decoder filter dynamics to ensure ISS of the augmented closed-loop model.Separate controller and decoder design is used for a nonlinear class, while decoder-first design is used for LTI plants.
  • 7.1 A class of nonlinear systems: A decoder with nf = ny and Cf = I_ny, together with suitable Hurwitz filter dynamics and bounding conditions, makes the filter-related system ISS.Lemma 2 gives ISS gains with respect to plant-controller states, decoder error, disturbance, and measurement noise.
  • 7.1 A class of nonlinear systems: For nonlinear systems, Proposition 3 requires ISS of the plant-controller system, ISS of the filter-related system, and a small-gain condition.Under these conditions, the interconnected system satisfies Assumption 1.
  • 7.2 LTI plant models: For stabilizable and detectable LTI plants, the augmented continuous system can be made stabilizable and detectable by appropriately choosing the filter matrices.The construction uses a Hurwitz Af, full-column-rank Bf, and suitable Cf, after which controller and observer gains can be selected.
  • 7.2 LTI plant models: Choosing Kcont and Lcont so that Acont + BcontKcont and Acont − LcontCcont are Hurwitz yields an ISS continuous system with respect to decoder error, disturbance, and measurement noise.The result follows from output-feedback stabilization and the fact that global exponential stability of an LTI system implies ISS under additive inputs.

8 Illustrative example

The framework is applied to set-point stabilization of a fourth-order flexible single-link manipulator using spiking sensor-controller communication. The simulations show that smaller spike thresholds improve the ultimate state bound while increasing communication activity.

  • 8. Illustrative example: The example stabilizes a 4th-order flexible single-link manipulator to a prescribed position under additive measurement noise.The plant has four state variables, one torque input, and noisy measured output.
  • 8.2 Spike decoder: The decoder uses nf = 4, Af = −diag(1, 2, 3, 4), Bf = Af, and Cf = I_nf, while the controller is an observer-based output-feedback design.The controller uses nc = 8, stabilizing state-feedback poles −1, −2, −3, −4, and observer poles −1 through −8.
  • 8.3 Numerical simulations: The simulation uses 8 membrane potentials, with all spike amplitudes and thresholds set to a common Δ across multiple Δ values.Measurement noise includes sinusoidal components with frequencies f1 = 20 Hz and f2 = 500 Hz.
  • 8.3 Numerical simulations: Smaller Δ produces a smaller ultimate bound on ||x_p||, consistent with the relation |α| = √(2n_pΔ).Figure 5 compares continuous communication with spiking communication at different threshold values.
  • 8.3 Numerical simulations: More spikes improve performance in terms of ||x_p||_[9,10], while continuous communication can perform worse than spiking communication with Δ = 0.01.Table 1 averages spiking rate and ultimate state norm over 100 initial conditions.

9 Conclusions

The paper presents a framework for stabilizing output-feedback systems using neuronal spiking communication. It establishes practical ISS design conditions and demonstrates their use on nonlinear and LTI systems.

  • 9 Conclusions: The framework combines neuronal spike encoding, synaptic decoding, and output-feedback control to stabilize plants using spiking communications.The plant output is encoded into spikes and decoded into an analog signal for control.
  • 9 Conclusions: Design conditions on the controller, encoder, and decoder guarantee practical input-to-state stability of the closed loop.The adjustable spike amplitudes determine the practical stability bound.
  • 9 Conclusions: The paper identifies distributed scenarios, formal analysis involving Σ−Δ modulators, and limit-cycle stabilization as possible extensions.These are presented as directions for further development rather than established results of the paper.

A Intermediate results

The appendix establishes intermediate properties of the integrate-and-fire neurons used by the communication scheme. It permits initial membrane potentials at threshold, including spikes at the initial time.

  • A Intermediate results: The appendix revisits properties of the neuronal model while allowing membrane potentials to be initialized at their threshold.This differs from the referenced prior treatment, where initial-time jumps were not allowed.
  • A Intermediate results: For initial membrane potentials in [0, Δ_i,l], the corresponding neuron has a well-defined infinite sequence of spiking times.The result applies to each neuron indexed by output component i and neuron label l.

Sketch of Proof.

The proof establishes practical spiking input-to-state stability for a Hurwitz LTI system and a universal approximation property for the 2n_y-neuron network. These results provide the key bounds used in the broader spiking-system analysis.

  • Proof strategy: The proof handles initial spikes by modifying the cited argument while preserving its reasoning and conclusion.The construction introduces h(θ) = e^{Fθ}G and accounts for spike amplitudes and possible spikes at the initial time.
  • Approximation property: The 2n_y-neuron integrate-and-fire network has a universal approximation property for signals y ∈ L_R^{n_y}.With α_i,ℓ = Δ_i,ℓ and specified initial conditions, its approximation error e := y − y_s satisfies the proposition’s bound.
  • LTI stability: Any Hurwitz LTI system ż = Fz + Gv is practically spiking input-to-state stable.The theorem provides β_z ∈ exp-KL and c_z > 0 for solutions driven by spiking inputs.
  • Proof strategy: The resulting estimates rely on exponential decay because F is Hurwitz.A constant c_0 bounds the relevant matrix expression for every τ ≥ 0, leading to the stated stability estimate.
  • LTI stability: The stability estimate bounds |ϕ_z(t)| by transient decay, an input term, and a term proportional to the initial spike amplitude.The resulting bound is |ϕ_z(t)| ≤ β_z(|ϕ_z(0)|, t) + γ_z(∥v∥⋆) + c_z∥A_0∥.

B Proof of Proposition 1

The proof of Proposition 1 combines the encoder’s approximation bound with practical spiking ISS of the filtering dynamics. It then derives a practical stability estimate for the complete spiking system.

  • Completeness: The finite-neuron encoder produces a spike-time sequence belonging to T^∞_Zf, supporting completeness of the spiking trajectory.Because the number of neurons is finite and the continuous dynamics are globally Lipschitz, the solution is complete.
  • Error bound: The encoder error satisfies ∥e∥⋆ ≤ 2√n_y|α|, so it is an admissible spiking input for the Hurwitz filter.The spike amplitudes associated with e are bounded by |α|, allowing Theorem 3 to be applied.
  • Stability estimate: The complete spiking state obeys a practical stability estimate with residual proportional to the spike-amplitude scale |α|.The proof obtains |ϕ_s(t)| ≤ max{β_s(|ϕ_s(0)|, t), γ̃_s|α|}.

C Proof of Proposition 2

The proof of Proposition 2 combines continuous-system bounds with the spiking subsystem estimate to bound the augmented trajectory. It also rules out finite escape and Zeno behavior, establishing completeness.

  • Augmented bound: Assumption 1 supplies a continuous-system bound independent of disturbances, auxiliary inputs, time, and the augmented initial state.This bound is combined with the spiking estimate through the output relation y = h_p(ϕ_p,w).
  • Augmented bound: The augmented trajectory satisfies a KL/K∞ estimate obtained by combining the continuous and spiking bounds.The proof defines β_aug = 2 max{β_cont, β_s} and γ_aug = 2 max{γ_cont ◦ γ_s, γ_cont, γ̃_s|d|}.
  • Completeness: The estimate prevents finite escape because the augmented state remains bounded on every finite solution interval.Compactness of the relevant set and continuity of h_p then also bound the plant output.
  • Completeness: The proof excludes Zeno behavior by extending the output and encoder dynamics beyond a hypothetical finite endpoint.The resulting spike-time sequence must belong to T^∞_Zf, contradicting accumulation at the endpoint.
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