PaperlayerView exact revision on arXiv

Source-linked AI summary

Data-driven Discovery of Cyber-Physical Systems

Ye Yuan, Xiuchuan Tang, Wei Pan, Xiuting Li, Wei Zhou, Hai-Tao Zhang, Han Ding, Jorge Goncalves

arXiv:1810.00697v1eess.SYcs.AIcs.LG

TL;DR

CPS modeling is difficult because cyber and physical components interact through discrete and continuous dynamics, motivating reliable mechanistic modeling from data. IHYDE discovers subsystem dynamics and transition logic without requiring detailed prior system knowledge, and it is reported to work across diverse CPS applications. The resulting models support trajectory prediction, design assessment, and real-time fault monitoring, although identifiability and tuning remain open issues.

  • Problem

    CPSs require reliable models that capture interacting discrete cyber and continuous physical dynamics, but automating such mechanistic modeling from data remains a central challenge.

  • Method

    IHYDE uses data-driven subsystem identification and transition-logic inference to discover hybrid CPS models, with application-specific nonlinear-function libraries.

  • Results

    IHYDE successfully models and monitors CPSs across mechanical, electrical, power-system, autonomous-vehicle, and medical examples, including real-time fault detection and localization.

  • Takeaways & Limitations

    Discovered CPS models can support design-specification checks, future-trajectory prediction, redesign, and real-time monitoring and pinpointing of faults.

  • Takeaways & Limitations

    Identifiability is unresolved because different hybrid dynamical systems can produce identical data, and selecting the tuning parameter may require fine-tuning and cross-validation.

Abstract

from arXiv · show

Cyber-physical systems (CPSs) embed software into the physical world. They appear in a wide range of applications such as smart grids, robotics, intelligent manufacture and medical monitoring. CPSs have proved resistant to modeling due to their intrinsic complexity arising from the combination of physical components and cyber components and the interaction between them. This study proposes a general framework for reverse engineering CPSs directly from data. The method involves the identification of physical systems as well as the inference of transition logic. It has been applied successfully to a number of real-world examples ranging from mechanical and electrical systems to medical applications. The novel framework seeks to enable researchers to make predictions concerning the trajectory of CPSs based on the discovered model. Such information has been proven essential for the assessment of the performance of CPS, the design of failure-proof CPS and the creation of design guidelines for new CPSs.

Introduction

CPSs combine cyber and physical components, creating complex hybrid dynamics that require reliable models for performance guarantees. The study addresses how to automate mechanistic CPS modeling directly from data.

  • CPSs integrate software and physical processes, adding functionality while complicating system design and performance guarantees.
  • Reliable CPS models must capture both discrete cyber components and continuous physical dynamics.Hybrid dynamical models use finite-state machines for cyber components and dynamical systems for physical counterparts.
  • The central research question is how to reliably and efficiently automate mechanistic modeling of CPSs from data.
  • Existing symbolic-regression approaches can be computationally expensive, poorly scalable, and prone to overfitting.Later methods reduced computational burden using compressive sensing, but the cited passage does not state that they solve all CPS modeling requirements.
  • Prior hybrid-system identification methods span algebraic-geometric, mixed-integer, bounded-error, Bayesian, clustering-based, and multimodal symbolic-regression strategies.

Results

IHYDE infers subsystem dynamics and transition logic from observed data, then applies this process across mechanical, electrical, power-system, autonomous-vehicle, and medical examples. The reported results show successful modeling, fault detection, and transition inference in these settings.

  • Thermostat and algorithm: IHYDE identifies subsystem dynamics and transition rules from data, beginning with subsystem discovery and followed by transition-logic inference.The thermostat example illustrates iterative subsystem classification followed by learning the conditions that trigger switches.
  • Application scope: IHYDE successfully models systems spanning autonomous vehicles, Chua’s circuit, wind turbines, power networks, transmission lines, and heart atrial action potentials.
  • Autonomous vehicles and robots: IHYDE reverse engineers an autonomous vehicle’s control strategy from faulty-system data, identifying incorrect speed controllers caused by swapped straightway and curve speeds.
  • Large-scale electronics: IHYDE captures Chua’s circuit dynamics and transition rules from measured data, with nonlinear dynamics consistent with the circuit’s true parameters and no switching assumptions.
  • Smart grids: IHYDE identifies smart-grid admittance-matrix changes with nearly zero identification errors and detects switching between network configurations.
  • Transmission lines: 10 measurements after a simulated transmission-line failure allow IHYDE to estimate a new admittance matrix, detect the event, and determine its location.The passage equates these measurements with 166.7 milliseconds under the cited synchrophasor sampling standard.

Discussion

IHYDE identifies CPS mechanistic models directly from data by jointly recovering subsystem dynamics and transition logic. Its sparse formulation supports large-scale systems, while remaining limitations concern identifiability and tuning choices.

  • IHYDE identifies complex CPS mechanistic models directly from data without prior system-specific knowledge, recovering subsystem dynamics and transition logics.
  • Sparse identification and artificial intelligence reduce computational complexity relative to symbolic regression, enabling application to large-scale CPSs.
  • IHYDE models both hybrid and nonhybrid dynamical systems as special cases, providing a unified discovery approach.
  • Unique identification is not guaranteed when different hybrid systems generate identical data, and tuning the complexity-fitness parameter may require cross-validation.
  • Subsystem identification: The method constructs application-specific nonlinear-function libraries from input-output samples, including mechanical, biological, and polynomial candidate terms.
  • Subsystem identification: Iterative sparse optimization first assigns data points to subsystems and then identifies each subsystem’s active nonlinearities and parameters.
  • Transition logic identification: Transition logic is inferred by learning functions whose sigmoid-relaxed decisions specify switches between subsystems from current inputs and outputs.

Supplementary Materials

The supplementary materials include a section introducing hybrid dynamical systems.

  • Section S1.2 provides an introduction to hybrid dynamical systems.

S1 Preliminaries

Hybrid dynamical systems combine continuous physical evolution with discrete modes and transitions. The paper formulates their sampled dynamics and seeks to infer both subsystem behavior and switching logic from time-series data.

  • Hybrid dynamical systems combine continuous and discrete state variables, with modes selected from a finite set and each mode having its own physical parameters.
  • A hybrid dynamical system is represented by the tuple H = (W, M, F, T), covering input-output variables, modes, differential equations, and transitions.
  • Sampling at rate h and approximating derivatives yields a discrete-time representation of each subsystem’s dynamics.
  • The considered class is discrete-time, Markovian, and nonlinear, while allowing extensions to more general nonlinear systems.
  • For a thermostat, separate equations describe heater-off and heater-on temperature changes, with threshold rules controlling mode switches.
  • Because the hybrid model is usually unknown or only partially known, the paper aims to infer subsystems and transition logic from time-series data.

S2 IHYDE Algorithm

IHYDE identifies CPS subsystem dynamics from time-series input-output data, then infers transition logic between the identified subsystems. Its iterative optimization removes already fitted data, while logistic regression models switching relationships.

  • Subsystem identification: IHYDE uses time-series input-output data and a field-guided dictionary of candidate nonlinear functions to identify subsystem models.The dictionary is expected to contain the true nonlinearities, with sparse regression selecting active functions and parameters.
  • Subsystem identification: Because no single coefficient matrix typically fits all hybrid-system data, IHYDE identifies a model fitting most observations, removes fitted data, and repeats on the remainder.The number of iterations gives the minimum number of subsystems under the procedure.
  • Unified scope: IHYDE recovers established time-invariant linear and nonlinear systems as special cases, unifying hybrid and non-hybrid dynamical-system discovery.The special-case behavior was confirmed using canonical dynamical systems from prior work.
  • Variants and limitations: The algorithm can be sensitive to measurement noise and dictionary redundancy because errors in its sequential optimizations propagate to later stages.A Bayesian variant is proposed to improve empirical optimization performance.
  • Transition logic: After assigning observations to subsystems, IHYDE infers each transition rule by applying sparse logistic regression to subsystem memberships.The membership indicator is one when a subsystem is active, and the sigmoid relaxation models transitions between subsystem pairs.

S3 Results for IHYDE

IHYDE is evaluated across hybrid systems spanning hysteresis, robotics, nonlinear dynamics, power engineering, and medical applications. It generally identifies subsystem dynamics and transition logic, while performance can degrade under increasing noise or redundant basis functions.

  • Hysteresis Relay: IHYDE identifies two Hysteresis Relay subsystems from noiseless and noisy data, with parameterizations consistent with or close to the true systems.This holds with and without redundant basis functions.
  • Hysteresis Relay: IHYDE precisely identifies Hysteresis Relay transition logic, including when redundant basis functions are present.The identified transition results are reported in Tables S3 and S4.
  • Continuous Hysteresis Loop: Increasing noise and redundant basis functions cause false classifications in the Continuous Hysteresis Loop, although subsystem dynamics remain identifiable up to some precision.Transition logic is still successfully identified with and without redundant basis functions.
  • Phototaxis Robot: For the Phototaxis Robot, increasing noise produces false classifications, but IHYDE can retain the correct subsystem count and identify transition logic under several conditions.Without redundant basis functions, actual subsystem dynamics are identified when noise intensity is low; with redundancy, all subsystems are identified without noise.
  • Nonlinear Hybrid System: IHYDE identifies three nonlinear subsystems consistent with or close to the true systems and precisely recovers transition logic with or without redundant basis functions.The nonlinear hybrid system uses nonlinearly expressed behaviors and transitions that are not linearly separable.
  • Power-System Fault Detection: Using the next 10 measurements, IHYDE discovers two power-system subsystems and pinpoints the difference corresponding to a fault.With PMU sampling, the fault can be located in a few hundred milliseconds after occurrence.

S3.6 Example 6: Identification of Real-time Models for Smart Grid

IHYDE is applied to real-time model identification and transition-logic inference in smart grids and human atrial action potentials. In both settings, it recovers changing subsystem structure and switching logic from measured variables, with reported accuracy in the smart-grid case and near-exact thresholds in the atrial model.

  • Smart Grid: At Bus 12, maximum relative identification ratios are 0.00057% for Base configuration and 0.00182% for Changed configuration.Identified admittance matrices change at time instants 31, 61, 91, 121, and 151, indicating model switching.
  • Smart Grid: IHYDE identifies smart-grid model-switching times and transition logic consistently with the real logic, with small error.The inferred transition T1 ! T2 corresponds to a voltage drop at node 10 exceeding 0.0500 at time 30, followed by switch action.
  • Human Atrial Action Potential: In the atrial action-potential model, gating variables h and j switch dynamics when membrane voltage crosses −40 mV during recovery from stimulation.The variables represent fast and slow inactivation dynamics for the fast inward Na+ current.
  • Human Atrial Action Potential: IHYDE identifies the atrial subsystems and changing time correctly, inferring the logic V ≤−40.0093 versus the real logic V ≤−40.The model uses 2 ms, 2 nA stimulation pulses every 1000 ms and 1267 data points per variable.

S3.8 Example 8: Monitoring of Industrial Processes

IHYDE is tested for industrial-process monitoring using experimental wind-turbine data and Chua’s-circuit data. The experiments show fault detection from sparse observations and recovery of nonlinear subsystem behavior and transition rules.

  • Wind Turbine Monitoring: The wind-turbine experiment models normal and faulty operation, including a gearbox tooth with one-third of its width cut off.Measurements include motor, generator, and load electrical quantities under matched speed and load conditions.
  • Wind Turbine Monitoring: Using only three fault points, IHYDE identifies the gearbox fault occurrence at the true time 68 with a small relative fitting error ratio.The online output combines 61 normal-condition current measurements with three faulty-condition points.
  • Wind Turbine Monitoring: IHYDE’s fitted wind-turbine data use subsystem-associated colors for the original series and identify transitions through changes in color.The corresponding platform and schematic are shown in Figures S12 and S13.
  • Chua’s Circuit: Chua’s circuit provides experimental data for testing IHYDE on chaotic nonlinear dynamics generated by an inductor, capacitors, resistors, and a nonlinear resistor.Its behavior changes between chaos and non-chaos across three subsystems depending on R.
  • Chua’s Circuit: IHYDE accurately infers the form of the Chua’s-circuit equation represented in Table S26.The identified subsystems and transition logic are reported for all subsystems, including redundant basis functions in Table S27.

S3.10 Example 10: Autonomous Car

The autonomous-car example uses IHYDE to identify the car’s hybrid control strategy from experimental data and diagnose a software implementation error. After reversing the faulty transition logic, the car successfully completed the track.

  • System design: The car uses camera-based road-layout information to select faster straightaway and slower curve speeds through incremental PI control.The controller is derived from a position PI algorithm and computes motor-control inputs from measured and expected speed.
  • System design: Abrupt switching and discretization can create a more complicated hybrid system than the intended two-subsystem design.The authors note that such additional subsystems are normally neglected during modeling, analysis, and design.
  • Identification and debugging: IHYDE reverse engineered the failed car’s control strategy and revealed that the transition logic was reversed between straightaways and curves.The discovered model was compared with the intended design to locate the implementation error.
  • Identification and debugging: Fixing the reversed transition logic enabled the autonomous car to run through the track.The authors repeated the modeling process after the repair as validation.
  • Identification and debugging: The modeled system pinpointed the faulty software location and provided information for debugging the system.The experiment deliberately swapped straightaway and curve speeds to mimic a software bug.
  • Additional validation: The method was also evaluated on nonhybrid datasets, where subsystem identification unified previous discovery results.The evaluation used datasets from prior work and is summarized in supplementary tables.

S4 Discussion

The discussion identifies limits arising from non-identifiable parameterizations, insufficiently informative data, model complexity, noise, and basis-function growth. In such cases, multiple systems can fit the same observations or the actual model cannot be recovered.

  • Algorithmic limits: As the number of basis functions and the amount of noise increase, IHYDE eventually becomes unable to identify the actual model.This boundary was observed across the tested examples.
  • Algorithmic limits: When data are not informative, IHYDE may fit the data well while producing models more complex than the true ones.The discussion presents this as a typical system-identification problem associated with non-identifiability.
  • Identifiability: Any parameter pair satisfying k1 + k2 = 0.8 produces the same input-output data in the illustrated linear system.The example uses impulse stimulation and shows that the observed output cannot distinguish the parameters.
  • Identifiability: Without additional information, the true parameters cannot be identified from input-output data alone.This limitation applies when parameterizations are indistinguishable under the observed data.
  • Data informativeness: Insufficiently informative data can prevent recovery of the true model when some logic transitions never occur.For an initial condition y(0) = 18, the illustrated system remains in subsystem 1.

A Extension of the Proposed IHYDE Algorithm

The extension develops a Bayesian formulation of sparse coefficient estimation and converts the resulting concave optimization into iterative reweighted l1-minimization. The procedure alternates optimization and weight updates until convergence, then computes posterior estimates.

  • Bayesian formulation: The extension introduces sparse priors and derives recursive l1-regularized optimizations from a Bayesian framework.The corresponding cost function is analyzed in both hyperparameter and weight spaces.
  • Bayesian formulation: MAP estimation is equivalent to a regularized least-square problem under the stated Bayesian model.The model assumes Gaussian stochastic variables and uses Bayes’ rule to obtain the posterior over weights.
  • Prior relaxation: Setting g(w) = ||w||_1 and λ_z = σ^2 recovers the l1-regularized optimization.The formulation also permits super-Gaussian priors through a variational relaxation.
  • Hyperparameter estimation: Evidence maximization selects hyperparameters that best explain the observed data, after which the posterior mean estimates the unknown weights.The selected hyperparameters define the relaxed posterior used for estimation.
  • Iterative optimization: The method uses a tight auxiliary bound whose fixed-hyperparameter form is jointly convex in weights and hyperparameters.This enables global minimization of the auxiliary problem over the respective variables.
  • Iterative optimization: The algorithm repeatedly solves weighted convex l1-minimization problems and updates hyperparameters until convergence.The iterative procedure is summarized as successive updates of the optimization variables and weights.

B User’s Manual of the Code

The IHYDE open-source Matlab toolbox automates hybrid-system identification and transition-logic inference. Its manual covers installation, APIs, example workflows, and iterative result refinement using convex-programming routines.

  • Toolbox overview: IHYDE is an open-source Matlab toolbox for automating mechanistic modeling of hybrid dynamical systems from observed data.The toolbox implements the identification workflow described in the paper.
  • Toolbox overview: IHYDE has lower computational complexity than genetic algorithms, supporting application to real-world CPS problems.The toolbox implements the clustering-based algorithms described in the paper.
  • Toolbox overview: The toolbox uses CVX and SLR routines to construct and solve disciplined convex programs.These routines support the optimization procedures used by IHYDE.
  • APIs and workflow: The example workflow loads data, constructs a basis library, initializes parameters, runs ihyde, and applies finetuning.The identified outputs include subsystem results, indices, and transition logic.
  • APIs and workflow: A separate code path infers transition logic between subsystems and stores the resulting logic in syslogic.The manual also identifies sys, idx_sys, and syslogic as saved outputs.
Loading 1810.00697v1…