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Data-Driven Stabilization of Unknown Linear-Threshold Network Dynamics

Xuan Wang, Duy Duong-Tran, Jorge Cortés

TL;DR

The paper tackles stabilizing unknown discrete-time linear-threshold networks to a reference using data-driven control, avoiding explicit system identification. It develops a data-based representation exploiting collected input-output data, then designs two controllers: a state-feedback with feed-forward reference and an augmented integral-feedback variant, both synthesized via LMIs framed on a switched-system view of the threshold dynamics. To combat the computational burden, it introduces a reduced-complexity set of LMIs that scale linearly with the state dimension, while preserving performance. The approach is validated in two biological case studies—neural firing-rate regulation in rodents and human arousal regulation—demonstrating effective stabilization, with the integral-augmented controller offering improved disturbance rejection and robustness to parameter changes.

Abstract

This paper studies the data-driven control of unknown linear-threshold network dynamics to stabilize the state to a reference value. We consider two types of controllers: (i) a state feedback controller with feed-forward reference input and (ii) an augmented feedback controller with error integration. The first controller features a simpler structure and is easier to design, while the second offers improved performance in the presence of system parameter changes and disturbances. Our design strategy employs state-input datasets to construct data-based representations of the closed-loop dynamics. Since these representations involve linear threshold functions, we rewrite them as switched linear systems, and formulate the design problem as that of finding a common controller for all the resulting modes. This gives rise to a set of linear matrix inequalities (LMIs) whose solutions corresponds to the controller gain matrices. We analyze the computational complexity of solving the LMIs and propose a simplified, sufficient set of conditions that scales linearly with the system state. Simulations on two case studies involving regulation of firing rate dynamics in rodent brains and of arousal level dynamics in humans demonstrate the effectiveness of the controller designs.

Data-Driven Stabilization of Unknown Linear-Threshold Network Dynamics

TL;DR

The paper tackles stabilizing unknown discrete-time linear-threshold networks to a reference using data-driven control, avoiding explicit system identification. It develops a data-based representation exploiting collected input-output data, then designs two controllers: a state-feedback with feed-forward reference and an augmented integral-feedback variant, both synthesized via LMIs framed on a switched-system view of the threshold dynamics. To combat the computational burden, it introduces a reduced-complexity set of LMIs that scale linearly with the state dimension, while preserving performance. The approach is validated in two biological case studies—neural firing-rate regulation in rodents and human arousal regulation—demonstrating effective stabilization, with the integral-augmented controller offering improved disturbance rejection and robustness to parameter changes.

Abstract

This paper studies the data-driven control of unknown linear-threshold network dynamics to stabilize the state to a reference value. We consider two types of controllers: (i) a state feedback controller with feed-forward reference input and (ii) an augmented feedback controller with error integration. The first controller features a simpler structure and is easier to design, while the second offers improved performance in the presence of system parameter changes and disturbances. Our design strategy employs state-input datasets to construct data-based representations of the closed-loop dynamics. Since these representations involve linear threshold functions, we rewrite them as switched linear systems, and formulate the design problem as that of finding a common controller for all the resulting modes. This gives rise to a set of linear matrix inequalities (LMIs) whose solutions corresponds to the controller gain matrices. We analyze the computational complexity of solving the LMIs and propose a simplified, sufficient set of conditions that scales linearly with the system state. Simulations on two case studies involving regulation of firing rate dynamics in rodent brains and of arousal level dynamics in humans demonstrate the effectiveness of the controller designs.
Paper Structure (14 sections, 10 theorems, 86 equations, 7 figures)

This paper contains 14 sections, 10 theorems, 86 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Problem formulation
  3. Data-Based Representation of Linear Threshold Models
  4. Data-Driven Control with Feed-forward Reference Input
  5. Closed-loop data-based representation
  6. LMI-based design of feedback gain matrices
  7. Data-Driven Control with Error Integration
  8. Closed-loop data-based representation
  9. LMI-based design of integral feedback gain matrices
  10. Sufficient stability conditions with reduced computational complexity
  11. Case Studies: Validating Data-Driven Control in Biological Systems
  12. Data-driven Regulation of Neuro-Excitation Levels in Rodents' Brain
  13. Data-driven Regulation of Arousal in Sensory-Motor Task
  14. Conclusions and Future Work

Key Result

Lemma 3.1

(Data-based representation): Under Assumption Ass_rankQ, let $F: \mathbb{R}^{n+m}\to\mathbb{R}^{n\times nT_d}$ be such that for any state-input pair $\bm{p}={\rm col\;}\{\bm{x},\bm{u}\}$. Then the dynamics eq_dstmodel has the following data-based representation,

Figures (7)

  • Figure 1: The model includes 8 groups of neuron cells and one external stimulus. The nodes in the gray box are considered as system states; the nodes in green boxes are considered as control inputs.
  • Figure 2: Data-driven stabilization of a 4-node network. The synthesis of the feedback gain matrix is based on solving the SDP specified in \ref{['eq_simuSDP']}.
  • Figure 3: Data-driven stabilization of a 4-node network. The synthesis of the feedback gain matrix is based on solving the SDP specified in \ref{['eq_simuSDP_INT']}.
  • Figure 4: Data-driven stabilization of the system with controller \ref{['eq_theproblem']} in the presence of system disturbances. Dashed lines correspond to the values of $r$.
  • Figure 5: Data-driven stabilization of the system with controller \ref{['eq_simuSDP_INT']} in the presence of system disturbance. Dashed lines correspond to the values of $r$.
  • ...and 2 more figures

Theorems & Definitions (23)

  • Lemma 3.1
  • proof
  • Lemma 4.1
  • proof
  • Lemma 4.2
  • proof
  • Lemma 4.3
  • proof
  • Theorem 4.4
  • proof
  • ...and 13 more