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

BlockComponents.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": {
    "labels": [{"label": "$(instance)", "x": 500, "y": 1100, "rot": 0}],
    "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": -100, "y1": 450, "x2": 0, "y2": 550, "rot": 0},
        "diagram": {"iconName": "input", "x1": -100, "y1": 450, "x2": 0, "y2": 550, "rot": 0}
      }
    }
  }
  "Output signal port"
  y = RealOutput() {
    "Dyad": {
      "placement": {
        "icon": {"iconName": "output", "x1": 1000, "y1": 450, "x2": 1100, "y2": 550, "rot": 0},
        "diagram": {"iconName": "output", "x1": 1000, "y1": 450, "x2": 1100, "y2": 550, "rot": 0}
      }
    }
  }
  "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": {
    "labels": [{"label": "$(instance)", "x": 500, "y": 1100, "rot": 0}],
    "icons": {"default": "dyad://BlockComponents/SecondOrder.svg"}
  }
}
end

Test Cases

No test cases defined.

Related

• Examples

• Experiments

• Analyses

• Tests

  • SecondOrderTest