Derivative

Filtered derivative approximation with configurable time constant and gain.

Implements a filtered derivative that approximates the differentiation operation while avoiding noise amplification. The derivative approximation is implemented using a first-order filter with configurable time constant and gain parameters.

$$G(s)=\frac{Y(s)}{U(s)}=\frac{ks}{sT+1}$$

The smaller the time constant T, the closer the approximation to an ideal derivative, though this increases sensitivity to noise.

This component extends from SISO

Usage

BlockComponents.Derivative(x0=0, T, k=1.0)

Parameters:

Name	Description	Units	Default value
x0	Initial value of the derivative-filter state	–	0
T	Time constant of the filter (smaller values give better derivative approximation but more noise sensitivity)	–
k	Gain factor applied to the derivative approximation	–	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	Internal state variable of the first-order filter	–

Behavior

$$\begin{array}{rl}\frac{\mathrm{d}x\left(t\right)}{\mathrm{d}t}& =\frac{u\left(t\right)-x\left(t\right)}{T}\\ y\left(t\right)& =\frac{k(u\left(t\right)-x\left(t\right))}{T}\end{array}$$

Source

# Filtered derivative approximation with configurable time constant and gain.
#
# Implements a filtered derivative that approximates the differentiation operation while avoiding
# noise amplification. The derivative approximation is implemented using a first-order filter with
# configurable time constant and gain parameters.
#
# ```math
# G(s) = \frac{Y(s)}{U(s)} = \frac{ks}{sT + 1}
# ```
#
# The smaller the time constant T, the closer the approximation to an ideal derivative, though this
# increases sensitivity to noise.
component Derivative
  extends SISO
  # Internal state variable of the first-order filter
  variable x::Real
  # Initial value of the derivative-filter state
  parameter x0::Real = 0
  # Time constant of the filter (smaller values give better derivative approximation but more noise sensitivity)
  parameter T::Real
  # Gain factor applied to the derivative approximation
  parameter k::Real = 1.0
relations
  initial x = x0
  der(x) = (u - x) / T
  y = (k / T) * (u - x)
metadata {
  "Dyad": {
    "labels": [{"label": "$(instance)", "x": 500, "y": 1100, "rot": 0}],
    "icons": {"default": "dyad://BlockComponents/Derivative.svg"}
  }
}
end

Flattened Source

# Filtered derivative approximation with configurable time constant and gain.
#
# Implements a filtered derivative that approximates the differentiation operation while avoiding
# noise amplification. The derivative approximation is implemented using a first-order filter with
# configurable time constant and gain parameters.
#
# ```math
# G(s) = \frac{Y(s)}{U(s)} = \frac{ks}{sT + 1}
# ```
#
# The smaller the time constant T, the closer the approximation to an ideal derivative, though this
# increases sensitivity to noise.
component Derivative
  # 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}
      }
    }
  }
  # Internal state variable of the first-order filter
  variable x::Real
  # Initial value of the derivative-filter state
  parameter x0::Real = 0
  # Time constant of the filter (smaller values give better derivative approximation but more noise sensitivity)
  parameter T::Real
  # Gain factor applied to the derivative approximation
  parameter k::Real = 1.0
relations
  initial x = x0
  der(x) = (u - x) / T
  y = (k / T) * (u - x)
metadata {
  "Dyad": {
    "labels": [{"label": "$(instance)", "x": 500, "y": 1100, "rot": 0}],
    "icons": {"default": "dyad://BlockComponents/Derivative.svg"}
  }
}
end

Test Cases

No test cases defined.

Related

• Examples

• Experiments

• Analyses

• Tests

  • DerivativeIntegratorTerminatorTest