SecondOrder

Second-order filter with configurable gain, bandwidth, and damping ratio.

A filter that implements a second-order transfer function with configurable gain k, bandwidth w, and relative damping d. The filter can be configured to achieve different response characteristics by adjusting the damping ratio: critically damped (d=1), under-damped (d<1), or over-damped (d>1). A damping value of d=1/√2 creates a Butterworth filter with maximally flat frequency response. The filter is characterized by the transfer function

$$Y(s)/U(s)=\frac{k\cdot w^2}{s^2+2d\cdot w\cdot s+w^2}$$

This component extends from SISO

Usage

SecondOrder(k=1.0, w=1.0, d=1.0)

Parameters:

Name	Description	Units	Default value
k	Gain	–	1
w	Bandwidth (angular frequency)	–	1
d	Relative damping	–	1

Connectors

• u - This connector represents a real signal as an input to a component (RealInput)

• y - This connector represents a real signal as an output from a component (RealOutput)

Variables

Name	Description	Units
x	State of SecondOrder filter	–
xd	Derivative state of SecondOrder filter	–

Behavior

$$\begin{array}{rl}\frac{\mathrm{d}x\left(t\right)}{\mathrm{d}t}& =\mathtt{xd}\left(t\right)\\ \frac{\mathrm{d}\mathtt{xd}\left(t\right)}{\mathrm{d}t}& =(-2d\mathtt{xd}\left(t\right)+(-x\left(t\right)+ku\left(t\right))w)w\\ y\left(t\right)& =x\left(t\right)\end{array}$$

Source

# Second-order filter with configurable gain, bandwidth, and damping ratio.
#
# A filter that implements a second-order transfer function with configurable gain k,
# bandwidth w, and relative damping d. The filter can be configured to achieve different
# response characteristics by adjusting the damping ratio: critically damped (d=1),
# under-damped (d<1), or over-damped (d>1). A damping value of d=1/√2 creates a Butterworth
# filter with maximally flat frequency response. The filter is characterized by the transfer
# function
#
# ```math
# Y(s)/U(s) = \frac{k \cdot w^2}{s^2 + 2d \cdot w \cdot s + w^2}
# ```
component SecondOrder
  extends SISO
  # State of SecondOrder filter
  variable x::Real
  # Derivative state of SecondOrder filter
  variable xd::Real
  # Gain
  parameter k::Real = 1.0
  # Bandwidth (angular frequency)
  parameter w::Real = 1.0
  # Relative damping
  parameter d::Real = 1.0
relations
  der(x) = xd
  der(xd) = w*(w*(k*u-x)-2*d*xd)
  y = x
metadata {"Dyad": {"icons": {"default": "dyad://BlockComponents/SecondOrder.svg"}}}
end

Flattened Source

# Second-order filter with configurable gain, bandwidth, and damping ratio.
#
# A filter that implements a second-order transfer function with configurable gain k,
# bandwidth w, and relative damping d. The filter can be configured to achieve different
# response characteristics by adjusting the damping ratio: critically damped (d=1),
# under-damped (d<1), or over-damped (d>1). A damping value of d=1/√2 creates a Butterworth
# filter with maximally flat frequency response. The filter is characterized by the transfer
# function
#
# ```math
# Y(s)/U(s) = \frac{k \cdot w^2}{s^2 + 2d \cdot w \cdot s + w^2}
# ```
component SecondOrder
  # Input signal port
  u = RealInput() [{
    "Dyad": {
      "placement": {"icon": {"iconName": "input", "x1": -50, "y1": 450, "x2": 50, "y2": 550}}
    }
  }]
  # Output signal port
  y = RealOutput() [{
    "Dyad": {
      "placement": {"icon": {"iconName": "output", "x1": 950, "y1": 450, "x2": 1050, "y2": 550}}
    }
  }]
  # State of SecondOrder filter
  variable x::Real
  # Derivative state of SecondOrder filter
  variable xd::Real
  # Gain
  parameter k::Real = 1.0
  # Bandwidth (angular frequency)
  parameter w::Real = 1.0
  # Relative damping
  parameter d::Real = 1.0
relations
  der(x) = xd
  der(xd) = w*(w*(k*u-x)-2*d*xd)
  y = x
metadata {"Dyad": {"icons": {"default": "dyad://BlockComponents/SecondOrder.svg"}}}
end

Test Cases

This is setup code, that must be run before each test case.

using BlockComponents
using ModelingToolkit, OrdinaryDiffEqDefault
using Plots
using CSV, DataFrames

snapshotsdir = joinpath(dirname(dirname(pathof(BlockComponents))), "test", "snapshots")

"/home/actions-runner-10/.julia/packages/BlockComponents/77kIK/test/snapshots"

Related

• Examples

• Experiments

• Analyses

• Tests

  • SecondOrderTest