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System Identification Analysis ​

The System Identification Analysis fits a linear time-invariant (LTI) state-space model (blackbox) to measured input-output data.

Transfer functions can be obtained by converting the estimated model using the function tf, and a continuous-time model may be obtained using d2c.

Method Overview ​

System identification in DyadControlSystems uses two primary methods from ControlSystemIdentification.jl:

  1. Subspace Identification (subspaceid): A non-iterative method that uses numerical linear algebra to identify state-space models. Fast and reliable for most applications.

  2. Prediction Error Method (newpem): An iterative optimization method that minimizes prediction errors. This method works for closed-loop data and may provide better fit, but takes longer to run.

Both methods identify discrete-time state-space models of the form:

xk+1=Axk+Buk+Kekyk=Cxk+Duk+ek

where K is the measurement-feedback gain and ek is the innovation process.

Example Definition ​

The following example demonstrates system identification using measured data from a double-mass model

@example
using DyadControlSystems
using DyadInterface

# Generate simulated data
dmm = DemoSystems.double_mass_model()
Ts  = 0.001
f0  = 0.01
f1  = 10
Tf  = 30
u   = DyadControlSystems.chirp(Ts, f0, f1, Tf; logspace = true, phi=deg2rad(139))'
res = lsim(dmm, u, 0:Ts:Tf, method=:zoh)

# Perform analysis
spec = DyadControlSystems.SystemIdentificationAnalysisSpec(;
    input_data = res.u,
    output_data = res.y,
    Ts,
    method = "subspaceid",
    nx = 4,
    name = :SysID,
    detrend = false,
)
sol = DyadInterface.run_analysis(spec)
artifacts(sol, :predplot)
@example
artifacts(sol, :bodeplot, lab="Estimated")
bodeplot!(dmm, l=:dash, lab="True")

Analysis Arguments ​

The following arguments define a System Identification Analysis:

Data Input Options ​

You must provide data using one of these approaches:

Option 1: Direct Arrays ​

  • input_data::Real[nu, nT]: Input data matrix where rows are input signals and columns are time points

  • output_data::Real[ny, nT]: Output data matrix where rows are output signals and columns are time points

Option 2: DyadDataset ​

  • dataset: A DyadDataset object containing the time series data

  • input_cols::String[nu]: Names of input signal columns in the dataset

  • output_cols::String[ny]: Names of output signal columns in the dataset

Required Parameters ​

  • Ts::Real: Sample time of the data (must match actual sampling rate). Only periodic sampled data is supported.

Identification Method Parameters ​

  • method::String = "subspaceid": Identification method

    • "subspaceid": Subspace identification (fast and easy)

    • "newpem": Prediction error method (PEM, iterative optimization)

  • nx::Integer: Model order (state dimension)

    • Set to a positive integer for fixed order

    • Set to -1 for automatic selection (subspaceid only)

Common Options ​

  • simulation_focus::Boolean = false: Optimization focus

    • false: Optimize for short-term prediction (default)

    • true: Optimize for open-loop simulation

  • stable::Boolean = false: Enforce stability of the identified model

  • zeroD::Boolean = false: Force feedthrough matrix D to zero (no direct input-output path)

  • detrend::Boolean = true: Remove mean from data before identification

Method-Specific Parameters ​

For subspaceid: ​

  • r::Integer = 10: Prediction horizon (affects accuracy and computation time)

  • W::String = "MOESP": Weighting method

    • "MOESP": Multivariable Output-Error State Space

    • "CVA": Canonical Variate Analysis

    • "N4SID": Numerical Subspace State Space System Identification

    • "IVM": Instrumental Variable Method

For newpem: ​

  • h::Integer = 1: Prediction horizon. Large values may lead to long computational times.

Analysis Parameters ​

  • wl::Real = -1: Lower frequency bound for Bode plot (set to -1 for automatic)

  • wu::Real = -1: Upper frequency bound for Bode plot (set to -1 for automatic)

  • num_frequencies::Integer = 3000: Number of frequency points for result analysis

  • duration::Real = -1: Duration for step response in step-response artifact (set to -1 for automatic)

Model Validation ​

The identified model quality is assessed through the fit metrics and artifacts detailed below:

Fit Percentage ​

Measures how well the model predictions match the measured output:

fit=100⋅(1−||y−y^||||y−y¯||)

where y^ is the model prediction and y¯ is the mean of y.

Artifacts ​

A SystemIdentificationAnalysis returns the following artifacts:

Plots ​

  • :bodeplot: Frequency response of the identified model

  • :stepresponse: Step response showing system dynamics

  • :pzmap: Pole-zero map for stability and dynamics analysis

  • :predplot: Comparison of model predictions vs measured data, uses feedback from measurements (observer)

  • :simplot: Comparison of simulated vs measured output, similar to prediction but without measurement feedback

  • :residualplot: Residual correlation analysis for validation

Tables ​

  • :modelinfo: DataFrame containing:

    • Model order (nx)

    • Number of inputs and outputs

    • Sample time

    • Fit percentage

    • Stability status

Further Reading ​