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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

Cyber-physical systems are difficult to model because they combine interacting physical and cyber components, while performance guarantees require reliable models. IHYDE identifies subsystem dynamics and transition logic from data, and the paper reports applications across diverse systems, including real-time fault detection and localization.

  • Problem

    Reliable, efficient automated methods are needed to model CPS dynamics from data, including their discrete and continuous components, to support performance guarantees.

    Source: Introduction · p. 2

  • Method

    IHYDE reverse-engineers CPSs from data by identifying subsystem dynamics and inferring the transition logic between subsystems.

    Source: Inferring transition logic · p. 5, Discussion · p. 11

  • Results

    IHYDE was applied across a wide range of applications and can monitor, detect, and pinpoint real-time CPS faults.

    Source: Discussion · p. 11

  • Takeaways & Limitations

    The resulting CPS models can help verify design specifications and predict future trajectories.

    Source: Discussion · p. 11

    Some datasets can be consistent with multiple hybrid dynamical systems, so they may not uniquely identify the true system.

    Source: Discussion · p. 12

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.

5 Department of Engineering, University of Cambridge · Introduction

CPS performance guarantees require models that capture interacting cyber and physical dynamics. IHYDE automates discovery of hybrid dynamical models from data without prior system knowledge, and is described as computationally efficient and robust to noise.

Introduction

  • CPSs are widespread in engineering, embedding software in appliances, aerial vehicles, power scheduling, and vital-sign monitoring.The paper characterizes modern engineering systems as involving integrated cyber and physical processes.Source: Introduction · p. 2
  • CPS integration creates useful functionality but adds complexity that makes performance guarantees essential for safe and economical deployment.The introduction illustrates the stakes with costly grid dynamics and the complex software and sensor interactions in autonomous driving.Source: Introduction · p. 2
  • IHYDE automates hybrid-system modeling from observed data without prior system knowledge, with low computational complexity and robustness to noise.Hybrid models represent cyber components with finite-state machines and physical components with dynamical systems, enabling prediction and performance assessment.Source: Introduction · pp. 2–3
  • Earlier nonhybrid identification methods include symbolic regression, while compressive sensing and sparse learning reduce computational burden but do not handle unknown hybrid subsystems and transition logic.Symbolic regression can be computationally expensive, fail to scale, and overfit.Source: Introduction · p. 3
  • Hybrid-system identification has explored algebraic-geometric, mixed-integer, bounded-error, Bayesian, clustering, and multimodal symbolic-regression methods.Compressive-sensing approaches recover a sparse vector-valued sequence to identify a minimum number of submodels.Source: Introduction · p. 3
  • Most prior hybrid-identification work targets piecewise-affine models with linear transition rules and requires prior knowledge of subsystem counts, dynamics, or transition logic.IHYDE instead aims to infer the number and dynamics of subsystems and their transition logic without those assumptions, aside from possible general knowledge of the system field.Source: Introduction · p. 3
  • IHYDE first discovers and models interacting subsystems, then infers transition logic between each pair.The paper demonstrates the method on examples from power engineering, autonomous driving, and medical applications.Source: Introduction · p. 3

Results

IHYDE learns subsystem dynamics and transition logic from observed time-series data, and applies this approach across real and simulated systems ranging from engineering to medicine.

Inferring subsystems

  • IHYDE iteratively assigned data to the subsystem that captured the most observations, identified its dynamics, then repeated the process on the remaining data.In the thermostat example, it found the heater-on subsystem first because it generated more than half of the data, then identified the heater-off subsystem.Source: Inferring subsystems · p. 4

Inferring transition logic

Universal application

  • Nearly zero identification errors across all examples, with dynamics errors below 0.3% on simulated examples, as IHYDE classified subsystems and recovered transition rules.The examples span real and simulated systems and linear to nonlinear dynamics and transition rules.Source: Universal application · pp. 5–6, Table 1 · p. 7

Autonomous vehicles and robots: design and validation

  • From faulty-car data, IHYDE identified incorrect speed controllers and reverse engineered the control strategy, pinpointing a deliberately introduced software bug.The bug swapped the intended speeds for straightaways and curves, causing slow straight driving and track departures on curves.Source: Autonomous vehicles and robots: design… · p. 7

Large scale electronics (Chua’s circuit)

  • On real data from a three-subsystem chaotic Chua’s circuit, IHYDE recovered the dynamics and transition rules, with nonlinear dynamics consistent with the circuit’s true parameters.The model was inferred from data alone, without assumptions about dynamics or switching behavior.Source: Large scale electronics (Chua’s circuit) · p. 8

Monitoring of industrial processes (wind turbines)

  • IHYDE detected the wind-turbine fault within 3 data points, or 1 sec, after it occurred during online monitoring.Offline, it identified two linear subsystems for the operating conditions, detected the fault, and inferred transition logic.Source: Monitoring of industrial processes… · p. 9

Power systems (smart grids and transmission lines)

  • Using 10 post-failure measurements, corresponding to 166.7 milliseconds, IHYDE located the transmission-line fault by estimating a new admittance matrix and discovering the normal and failure subsystems.In the 33-bus switching example, IHYDE identified admittance-matrix columns with nearly zero identification errors; its identified logic was consistent with the real logic and revealed voltage drops at specific nodes.Source: Power systems (smart grids and… · p. 10

Medical applications (heart atrial AP monitoring)

  • IHYDE identified the atrial action-potential system’s two subsystems, their dynamics, and the changing logic, capturing spiking and inactivation behavior.The model was proposed as a basis for studying the system and building devices to detect abnormal action potentials.Source: Medical applications (heart atrial AP… · p. 11

Discussion

IHYDE discovers mechanistic CPS models from data, including subsystem dynamics and transition logic, and the discussion highlights their uses, broader scope, and unresolved identification challenges. The following subsections describe the hybrid-system structure and the methods for identifying subsystems and transition logic.

Discussion

  • IHYDE identifies subsystem dynamics and transition logic from data with little prior knowledge, offering interpretable, lower-complexity alternatives to deep neural networks and symbolic regression.The approach uses sparse identification and artificial intelligence to address large-scale CPSs and complex real-world problems.Source: Discussion · p. 11
  • IHYDE models can support specification checks, trajectory prediction, redesign, and real-time fault monitoring; supplementary canonical hybrid-system examples report virtually zero error.Applications discussed include robotics, automated vehicles, and power systems, where monitoring and fault pinpointing may help avoid catastrophic failures.Source: Discussion · p. 11
  • IHYDE also discovers nonhybrid time-invariant linear and nonlinear systems as special cases, providing a unified approach to hybrid and nonhybrid system discovery.Supplementary Materials Section S2 reports recovery of canonical systems from data in (Table S31).Source: Discussion · p. 12
  • IHYDE’s open challenges are determining when data uniquely identify a true hybrid system and choosing a tuning parameter that balances model complexity with fitness.Different systems can yield identical data, while tuning may require fine-tuning and cross-validation, with results varying considerably by decision.Source: Discussion · p. 12

Hybrid dynamical systems

  • A hybrid system selects among modes based on input-output variables, with each mode having physical parameters and a transition function determining the next subsystem.The transition function uses current inputs, outputs, and subsystem state.Source: Hybrid dynamical systems · pp. 12–13

Subsystem identification

  • Subsystem identification builds an application-specific nonlinear-function library, then iteratively uses sparse optimization to assign data points to subsystems and identify their dynamics.The library may use constant, linear, and trigonometric terms for simple mechanical systems, or polynomial and sigmoidal terms for biological networks.Source: Subsystem identification · pp. 13–14

Transition logic identification

  • Transition logic identification learns functions that trigger subsystem switches, relaxing a nondifferentiable step function to a sigmoid and parameterizing the trigger with a nonlinear-function dictionary.Subsystem membership from the preceding identification step supplies the known activation labels.Source: Transition logic identification · pp. 14–15

Supplementary Materials

The supplementary materials include a section introducing hybrid dynamical systems.

Supplementary Materials

S1 Preliminaries

The preliminaries define hybrid dynamical systems as continuous dynamics operating across discrete modes, with transitions governed by conditions. The paper seeks to infer both the subsystem dynamics and transition logic from time-series data.

S1.1 Notations

  • The notation section defines Euclidean and integer spaces, vector norms, and matrix and vector indexing conventions.Source: S1.1 Notations · p. 21

S1.2 Introduction to Hybrid Dynamical Systems

S2 IHYDE Algorithm

IHYDE uses time-series input-output data to identify subsystem models and then infer transition logic between them. Its framework also covers nonhybrid systems, offers a Bayesian-statistics variant, and uses cross-validation to assess generalization.

S2 IHYDE Algorithm

  • For a single subsystem, IHYDE builds an overdetermined candidate-function library guided by the application domain and uses sparse regression to identify active nonlinearities and parameters.Source: S2 IHYDE Algorithm · pp. 23–24

S2.1 Inferring Sub-systems

  • IHYDE repeatedly fits a subsystem to compatible data, removes points whose errors fall below ew, and repeats; the iteration count estimates the minimum number of subsystems.Source: S2.1 Inferring Sub-systems · pp. 25–26

S2.1.1 Application to nonhybrid dynamical systems

S2.1.2 Variants of Algorithm 1

  • Algorithm 1 is sensitive to noise and dictionary redundancy because sequential optimizations can propagate early errors; a Bayesian-statistics variant is proposed to improve empirical performance.Source: S2.1.2 Variants of Algorithm 1 · p. 26

S2.2 Inferring Transition Logic

S2.3 Cross Validation for Parameter Tuning

S3 Results for IHYDE … S3.5 Example 5: Power Grid Fault Detection

Across examples from physical systems to robotics, IHYDE identifies subsystem dynamics and transition logic from data. In a power-grid fault scenario, it detects and locates a fault in near real time.

S3 Results for IHYDE

  • IHYDE is applied across examples ranging from power systems to robotics, demonstrating the method’s applicability to varied systems.Source: S3 Results for IHYDE · p. 31

S3.1 Example 1: Hysteresis Relay

S3.2 Example 2: Continuous Hysteresis Loop

  • For the Continuous Hysteresis Loop, classification errors increase with noise and redundant basis functions, but IHYDE recovers subsystem dynamics approximately and transition logic with or without redundancy.Source: S3.2 Example 2: Continuous Hysteresis… · p. 34

S3.3 Example 3: Phototaxis Robot

  • For the Phototaxis Robot, noise degrades classification; with redundancy, higher noise changes one identified subsystem, while inferred transition logic remains equivalent to the actual logic.Without redundant basis functions, IHYDE identifies the actual subsystem dynamics when noise is low; with no noise, it identifies all subsystems.Source: S3.3 Example 3: Phototaxis Robot · p. 37

S3.4 Example 4: Non-linear Hybrid System

  • For the nonlinear hybrid system, IHYDE identifies three subsystems close to the true ones in noisy and noiseless data, and precisely recovers transition logic with or without redundant basis functions.Source: S3.4 Example 4: Non-linear Hybrid System · p. 40

S3.5 Example 5: Power Grid Fault Detection

  • Using the next 10 measurements, IHYDE detects a grid fault, identifies two subsystems, and locates the fault within a few hundred milliseconds.The discovered subsystem difference corresponds to the fault, enabling operators to identify its location and take remedial action in near real time.Source: S3.5 Example 5: Power Grid Fault… · p. 43

S3.6 Example 6: Identification of Real-time Models for Smart Grid · S3.7 Example 7: Discovery of Human Atrial Action Potential Models

Across smart-grid and human atrial action-potential examples, IHYDE identifies subsystem models and infers their switching logic from measurements. The examples report accurate identification and transition timing in both applications.

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

  • Using advanced-metering-infrastructure data from a 33-bus distribution benchmark, IHYDE identified network models and inferred transition logic for operator switching after voltage sag.The scenario used measured active and reactive power consumption and voltage phasors; voltage-magnitude differences were added to the dictionary for logic identification.Source: S3.6 Example 6: Identification of… · pp. 45, 47
  • Maximum relative identification ratios were 0.00057% for Base configuration and 0.00182% for Changed configuration; IHYDE also pinpointed model switching and identified transition logic with small error.Identified admittance matrices differed at time instants 31, 61, 91, 121, and 151, indicating switching.Source: S3.6 Example 6: Identification of… · p. 47, Table S19 · p. 48, Table S20 · p. 48
  • For the T1-to-T2 transition, a voltage drop greater than 0.0500 at feeder 3 node 10 at time 30 preceded switching that matched the preset load increase.The tie switch between Buses 12 and 22 closed while the sectionalizing switch between Buses 11 and 12 opened, revealing the mechanism behind the response.Source: S3.6 Example 6: Identification of… · p. 47

S3.7 Example 7: Discovery of Human Atrial Action Potential Models

  • The human atrial action-potential model proposed in represents the cell membrane as a capacitor in parallel with variable resistances and batteries for ionic channels and driving forces, comprising 21 differential equations and 163 parameters.Source: S3.7 Example 7: Discovery of Human… · p. 49
  • The human atrial action-potential model produced a spike-and-dome waveform under 2 ms, 2 nA stimulation pulses every 1000 ms, and its membrane voltage returned to a stable resting potential of −81 mV.The gating-variable dynamics switched as membrane voltage crossed −40 mV.Source: S3.7 Example 7: Discovery of Human… · pp. 49–50, Figure S10 · p. 50
  • IHYDE discovered models and transition logic for h and j from measurements, using their first-order differential values as outputs and 1267 data points per variable sampled at 0.3 ms.The h-variable differential output was down-sampled over 120–500 ms.Source: S3.7 Example 7: Discovery of Human… · p. 50
  • For both atrial gating variables h and j, IHYDE identified the subsystem and change time, with logic V ≤ −40.0093 close to the real threshold V ≤ −40.The variables are fast and slow inactivation gates for the fast inward Na+ current.Source: S3.7 Example 7: Discovery of Human… · p. 51, Table S21 · p. 52, Table S22 · p. 52

S3.8 Example 8: Monitoring of Industrial Processes · S3.9 Example 9: Chua’s Circuit

The examples show IHYDE detecting a gearbox fault at its true occurrence time using three fault points and accurately inferring an equation of Chua’s circuit from experimental data.

S3.8 Example 8: Monitoring of Industrial Processes

  • The wind-turbine platform measures motor RMS current and voltage, generator current and voltage, and load voltage for monitoring.The online-monitoring output combines 61 normal-condition current measurements with three faulty-condition points when the mismatch is large.Source: S3.8 Example 8: Monitoring of… · p. 54
  • The wind-turbine test compared normal and faulty operation at 200 rpm generator speed and 1.5 KNm load; the fault cut one-third of a high-speed-shaft gear tooth.Normal motor and generator powers were 383.01W and 53.28W, respectively; load voltage was 75V in the faulty condition.Source: S3.8 Example 8: Monitoring of… · p. 54, Figure S14 · p. 56
  • At the real fault time 68, IHYDE detects the gearbox fault using only three fault points, with a small relative fitting-error ratio.The fitted time-series data and identified fault details are shown in Figures S15 and Table S24.Source: S3.8 Example 8: Monitoring of… · p. 55, Figure S15 · p. 57, Table S24 · p. 58

S3.9 Example 9: Chua’s Circuit

S3.10 Example 10: Autonomous Car · S3.11 Example 11: Non-hybrid Dynamical Systems

In the autonomous-car testbed, IHYDE exposed a reversed transition logic fault, and the corrected car completed the track. On non-hybrid datasets, the method unified earlier subsystem-identification results.

S3.10 Example 10: Autonomous Car

S3.11 Example 11: Non-hybrid Dynamical Systems

S4 Discussion

IHYDE can fit data while producing models more complex than the true system, and increasing basis-function count or noise can prevent it from identifying the actual model. The examples show that parameter non-identifiability and unobserved logic transitions can make recovery impossible from the available data.

S4 Discussion

  • As basis-function count and noise increase, IHYDE can fail to identify the actual model; it may also fit data with overly complex models, a typical system-identification problem.When data is not informative, multiple hybrid dynamical systems can produce the same observations, preventing recovery of the true system.Source: S4 Discussion · p. 70

S4.1 Example 1

  • Any parameters satisfying k1 + k2 = 0.8 produce the same input-output data, so the true parameters cannot be identified without additional information.The indistinguishable parameters illustrate that no method can recover the correct values when the parameterization is not identifiable.Source: S4.1 Example 1 · pp. 70–71

S4.2 Example 2

  • Starting from y(0) = 18 keeps the system in subsystem 1, leaving logic transitions unobserved and preventing identification of the complete hybrid system.Even an identifiable subsystem cannot yield a complete model when the generated data does not include some transitions.Source: S4.2 Example 2 · p. 71

A Extension of the Proposed IHYDE Algorithm

The extension replaces repeated L1-regularized optimization with a Bayesian formulation using sparse priors. It then relaxes the resulting concave cost into iterative convex optimization via reweighted L1 minimization.

A Extension of the Proposed IHYDE Algorithm

  • The method turns a concave cost in the hyperparameter and weight spaces into an iterative convex optimization using reweighted L1 minimization.The section presents this as an alternative to repeatedly solving L1-regularized optimization problems.Source: A Extension of the Proposed IHYDE… · p. 72
  • With iid Gaussian noise, MAP estimation is regularized least squares; choosing g(w) = ||w||_1 and λ = σ^2 recovers the L1-regularized problem.Source: A Extension of the Proposed IHYDE… · pp. 72–73
  • Super-Gaussian priors admit a Gaussian variational relaxation if and only if log P(|w_j|) is concave on the positive domain.The relaxation introduces hyperparameters, which are selected by evidence maximization; the posterior mean then estimates the weights.Source: A Extension of the Proposed IHYDE… · p. 73
  • The algorithm iteratively minimizes a jointly convex bound through weighted L1 optimization and weight updates until convergence, then computes the posterior mean and covariance.Source: A Extension of the Proposed IHYDE… · pp. 75–76

B User’s Manual of the Code

IHYDE is an open-source Matlab toolbox for automating mechanistic modeling of hybrid dynamical systems from observed data. Its manual describes installation, API functions, example workflows, and transition-logic inference.

B User’s Manual of the Code

  • IHYDE automates mechanistic modeling of hybrid dynamical systems from observed data and implements the paper’s clustering-based algorithms.It uses CVX and SLR routines to construct and solve disciplined convex programs, and can support the creation of CPS design guidelines.Source: B User’s Manual of the Code · p. 78
  • IHYDE has lower computational complexity than genetic algorithms [35], enabling application to real-world CPS problems.Source: B User’s Manual of the Code · p. 78
  • Download the toolbox and add its directory and subdirectories to the Matlab path; its directory structure is listed in Table S33.Source: B User’s Manual of the Code · p. 78, Table S33 · p. 78
  • Tables S34–S37 document functions for building an identification library, identifying subsystems with ihyde, fine-tuning results, and inferring transition logic with ihydelogic.Source: B User’s Manual of the Code · p. 78, Table S34 · p. 78, Table S35 · p. 79
  • The /IHYDE/examples .m files serve as templates, and the manual demonstrates loading data and identifying an autonomous-car system with ihyde followed by minimum-error finetuning.ihyde returns a preliminary result containing subsystem details; finetuning refines that result.Source: B User’s Manual of the Code · pp. 80–81
  • The manual shows transition-logic inference between subsystems and says identified results are saved in sys, idx sys, and syslogic.Source: B User’s Manual of the Code · p. 81
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