AlphaBetaFilter

Alpha-beta tracker: position and rate estimates for a scalar measurement.

This is the smallest filter that predicts: on every sample the state advances by its own rate estimate, and the prediction is then corrected toward the measurement. The rate output is the filtered rate of change of the input, available without a separate differentiator.

Standard stability constraint: $0 < \alpha < 1$, $0 < \beta \le 2 - \alpha$. The default beta is Kalata's steady-state relation $\beta = 2(2 - \alpha) - 4\sqrt{1 - \alpha}$, which makes the tracker the steady-state Kalman filter for a double-integrator (constant-velocity) target and the critically damped choice for the given alpha; with the default alpha = 0.5 it is $\beta \approx 0.172$.

The rate state initializes to zero under every initialization variant; the enum selects only the position state.

Usage

DiscreteComponents.AlphaBetaFilter(alpha=0.5, beta=2 * (2 - alpha) - 4 * sqrt(1 - alpha))

Parameters:

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

Variables

Behavior

using DiscreteComponents #hide
using ModelingToolkit #hide
@variables alpha #hide
@variables beta #hide
@named sys = DiscreteComponents.AlphaBetaFilter(alpha=alpha, beta=beta) #hide
full_equations(sys) #hide

Source

"""
Alpha-beta tracker: position and rate estimates for a scalar measurement.

This is the smallest filter that predicts: on every sample the state advances by its own rate estimate, and the prediction is then corrected toward the measurement. The `rate` output is the filtered rate of change of the input, available without a separate differentiator.

Standard stability constraint: ``0 < \alpha < 1``, ``0 < \beta \le 2 - \alpha``. The default `beta` is Kalata's steady-state relation ``\beta = 2(2 - \alpha) - 4\sqrt{1 - \alpha}``, which makes the tracker the steady-state Kalman filter for a double-integrator (constant-velocity) target and the critically damped choice for the given `alpha`; with the default `alpha = 0.5` it is ``\beta \approx 0.172``.

The rate state initializes to zero under every `initialization` variant; the enum selects only the position state.
"""
component AlphaBetaFilter@[input clk extends Discrete]
  "Measured value"
  u = RealInput@[clk]() {
    "Dyad": {
      "placement": {
        "diagram": {"iconName": "default", "x1": -100, "y1": 450, "x2": 0, "y2": 550, "rot": 0}
      },
      "tags": []
    }
  }
  "Position estimate"
  y = RealOutput@[clk]() {
    "Dyad": {
      "placement": {
        "diagram": {"iconName": "default", "x1": 1000, "y1": 170, "x2": 1100, "y2": 270, "rot": 0}
      },
      "tags": []
    }
  }
  "Rate estimate [units of u per second]"
  rate = RealOutput@[clk]() {
    "Dyad": {
      "placement": {
        "diagram": {"iconName": "default", "x1": 1000, "y1": 640, "x2": 1100, "y2": 740, "rot": 0}
      },
      "tags": []
    }
  }
  "Position correction gain, in (0, 1)"
  parameter alpha::Real = 0.5
  "Rate correction gain, in (0, 2 - alpha); the default is the steady-state Kalman (critically damped) value for alpha"
  parameter beta::Real = 2 * (2 - alpha) - 4 * sqrt(1 - alpha)
  "Sample interval [s]"
  structural parameter Ts::Real = SampleTime()
  structural parameter initialization::InitialCondition = DiscreteComponents.InitialCondition.InitialOutput(y0 = 0.0)
  "This sample's prediction from the previous state"
  variable pred::Real
  "Innovation: measurement minus prediction"
  variable resid::Real
relations
  pred@clk = y@(clk-1) + Ts * rate@(clk-1)
  resid@clk = u@clk - pred@clk
  y@clk = pred@clk + alpha * resid@clk
  rate@clk = rate@(clk-1) + (beta / Ts) * resid@clk
  initial rate@(clk-1) = 0
  switch initialization
    case InitialOutput
      initial y@(clk-1) = (initialization.y0 - alpha * u@clk) / (1 - alpha)
    case SteadyState
      initial y@(clk-1) = u@clk
    case InitialState
      initial y@(clk-1) = initialization.x0
  end
metadata {
  "Dyad": {"icons": {"default": "dyad://DiscreteComponents/AlphaBetaFilter.svg"}}
}
end

"""
Alpha-beta tracker: position and rate estimates for a scalar measurement.

This is the smallest filter that predicts: on every sample the state advances by its own rate estimate, and the prediction is then corrected toward the measurement. The `rate` output is the filtered rate of change of the input, available without a separate differentiator.

Standard stability constraint: ``0 < \alpha < 1``, ``0 < \beta \le 2 - \alpha``. The default `beta` is Kalata's steady-state relation ``\beta = 2(2 - \alpha) - 4\sqrt{1 - \alpha}``, which makes the tracker the steady-state Kalman filter for a double-integrator (constant-velocity) target and the critically damped choice for the given `alpha`; with the default `alpha = 0.5` it is ``\beta \approx 0.172``.

The rate state initializes to zero under every `initialization` variant; the enum selects only the position state.
"""
component AlphaBetaFilter
  "Measured value"
  u = RealInput@[clk]() {
    "Dyad": {
      "placement": {
        "diagram": {"iconName": "default", "x1": -100, "y1": 450, "x2": 0, "y2": 550, "rot": 0}
      },
      "tags": []
    }
  }
  "Position estimate"
  y = RealOutput@[clk]() {
    "Dyad": {
      "placement": {
        "diagram": {"iconName": "default", "x1": 1000, "y1": 170, "x2": 1100, "y2": 270, "rot": 0}
      },
      "tags": []
    }
  }
  "Rate estimate [units of u per second]"
  rate = RealOutput@[clk]() {
    "Dyad": {
      "placement": {
        "diagram": {"iconName": "default", "x1": 1000, "y1": 640, "x2": 1100, "y2": 740, "rot": 0}
      },
      "tags": []
    }
  }
  "Position correction gain, in (0, 1)"
  parameter alpha::Real = 0.5
  "Rate correction gain, in (0, 2 - alpha); the default is the steady-state Kalman (critically damped) value for alpha"
  parameter beta::Real = 2 * (2 - alpha) - 4 * sqrt(1 - alpha)
  "Sample interval [s]"
  structural parameter Ts::Real = SampleTime()
  structural parameter initialization::InitialCondition = DiscreteComponents.InitialCondition.InitialOutput(y0 = 0.0)
  "This sample's prediction from the previous state"
  variable pred::Real
  "Innovation: measurement minus prediction"
  variable resid::Real
relations
  pred@clk = y@(clk-1) + Ts * rate@(clk-1)
  resid@clk = u@clk - pred@clk
  y@clk = pred@clk + alpha * resid@clk
  rate@clk = rate@(clk-1) + (beta / Ts) * resid@clk
  initial rate@(clk-1) = 0
  switch initialization
    case InitialOutput
      initial y@(clk-1) = (initialization.y0 - alpha * u@clk) / (1 - alpha)
    case SteadyState
      initial y@(clk-1) = u@clk
    case InitialState
      initial y@(clk-1) = initialization.x0
  end
metadata {
  "Dyad": {"icons": {"default": "dyad://DiscreteComponents/AlphaBetaFilter.svg"}}
}
end

Test Cases

No test cases defined.

Related

• Examples
• Experiments
• Analyses