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Linear Analysis ​

The Linear Analysis computes the linearized dynamics of a model and provides a suite of frequency- and time-domain analysis tools. This is useful for understanding the small-signal behavior of a system, including stability, resonance, and transient response.

Method Overview ​

Linear analysis is performed by linearizing the provided model around an operating point. The resulting linear system can be analyzed using classical control theory tools, including:

Example Definition ​

Since we are opening the loop at r (the reference speed input to the controller), we need to provide an initial condition for controller.u_s — otherwise, the linearization cannot determine its operating point value. We do this by extending the test model and adding initial controller.u_s = 0:

dyad
component DCMotorForLinearization
  extends DyadExampleComponents.TestDCMotorLoadControlled(w_motor = 0)
relations
  initial controller.u_s = 0
end

analysis DCMotorLinearAnalysis
  extends DyadControlSystems.LinearAnalysis(
    outputs       = ["y"],
    inputs        = ["r"],
    loop_openings = ["r"],
    duration = 5.0
  )
  model = DCMotorForLinearization()
end

We set w_motor to zero so the controller does not saturate at the linearization point, and provide initial controller.u_s = 0 so the disconnected setpoint input has a defined operating point.

julia
using DyadInterface: artifacts
asol = DCMotorLinearAnalysis()
artifacts(asol, :StepInfoPlot)

julia
artifacts(asol, :BodePlot)

julia
artifacts(asol, :DampReport)
3×5 DataFrame
RowPoleDampingRatioFrequency_rad_sFrequency_HzTimeConstant_s
Complex…Float64Float64Float64Float64
1-0.00010001+0.0im1.00.000100011.59171e-59999.0
2-0.834172+0.0im1.00.8341720.1327631.19879
3-55.3885+173.864im0.303543182.47329.04150.0180543

Analysis Arguments ​

The following arguments define a LinearAnalysis:

Required Arguments ​

  • model: The model to be analyzed.

  • inputs::Vector{String}: Names of the input analysis points

  • outputs::Vector{String}: Names of the output analysis points

Optional Arguments ​

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

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

  • num_frequencies::Int = 3000: Number of frequency points.

  • duration::Real = -1: Duration for the step response plot (set to -1 for automatic selection).

Artifacts ​

A LinearAnalysis returns the following artifacts:

Standard Plots ​

Tables ​

Further Reading ​