`LimPID`

PID controller with limited output, back calculation anti-windup compensation, setpoint weighting and feed-forward

The transfer function is:

y = k [e + \frac{1}{T_{i} s} (e + \frac{y_{s a t} - y}{N_{i} T_{i}}) + \frac{T_{d} s}{1 + \frac{T_{d}}{N_{d}} s} e] + k_{f f} u_{f f}

This component extends from SingleVariableController

Usage

BlockComponents.LimPID(k=1, Ti=0.5, Td=0.1, y_max=1e300, y_min=-y_max, wp=1, wd=0, Ni=0.9, Nd=10, k_ff=1, xi0=0, xd0=0)

Parameters:

Name	Description	Units	Default value
`k`	Gain of controller	–	1
`Ti`	Time constant of the integrator block	s	0.5
`Td`	Time constant of the derivative block	s	0.1
`y_max`	Maximum output	–	1e+300
`y_min`	Minimum output	–	-y_max
`wp`	Set-point weight for proportional block	–	1
`wd`	Set-point weight for derivative block	–	0
`Ni`	`Ni*Ti` is time constant of anti-windup compensation	–	0.9
`Nd`	Maximum derivative gain. Higher the value of `Nd`, the more ideal the derivative block gets (less filtering, higher high-frequency gain).	–	10
`k_ff`	Gain of the feed-forward input	–	1
`xi0`	Initial guess value for integrator output	–	0
`xd0`	Initial guess value for derivative output	–	0

Connectors

u_s - This connector represents a real signal as an input to a component (RealInput)
u_m - 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)
u_ff - This connector represents a real signal as an input to a component (RealInput)

Variables

Name	Description	Units
`control_error`		–

Behavior

julia

using BlockComponents #hide
using ModelingToolkit #hide
@variables k #hide
@variables Ti #hide
@variables Td #hide
@variables y_max #hide
@variables y_min #hide
@variables wp #hide
@variables wd #hide
@variables Ni #hide
@variables Nd #hide
@variables k_ff #hide
@variables xi0 #hide
@variables xd0 #hide
@named sys = BlockComponents.LimPID(k=k, Ti=Ti, Td=Td, y_max=y_max, y_min=y_min, wp=wp, wd=wd, Ni=Ni, Nd=Nd, k_ff=k_ff, xi0=xi0, xd0=xd0) #hide
full_equations(sys) #hide

<< @example-block not executed in draft mode >>

Source

dyad

"""
PID controller with limited output, back calculation anti-windup compensation, setpoint weighting and feed-forward

The transfer function is:

math y = k \left[e + \dfrac{1}{T_is}\left(e + \dfrac{y_{sat} - y}{N_iT_i}\right) + \dfrac{T_ds}{1 + {\dfrac{T_d}{N_d}s}}e \right] + k_{ff}u_

"""
component LimPID
  extends SingleVariableController
  u_ff = RealInput() {
    "Dyad": {
      "placement": {
        "diagram": {"iconName": "input", "x1": 850, "y1": 950, "x2": 950, "y2": 1050, "rot": -90},
        "icon": {"iconName": "input", "x1": 650, "y1": 950, "x2": 750, "y2": 1050, "rot": -90}
      }
    }
  }
  add_p = Add(k1 = wp, k2 = -1) {
    "Dyad": {
      "placement": {"icon": {"iconName": "input", "x1": 100, "y1": 100, "x2": 200, "y2": 200}}
    }
  }
  add_d = Add(k2 = -1) {
    "Dyad": {
      "placement": {"icon": {"iconName": "input", "x1": 100, "y1": 300, "x2": 200, "y2": 400}}
    }
  }
  add_i = Add3(k2 = -1) {
    "Dyad": {
      "placement": {"icon": {"iconName": "input", "x1": 100, "y1": 500, "x2": 200, "y2": 600}}
    }
  }
  proportional = Gain(k = 1) {
    "Dyad": {
      "placement": {"icon": {"iconName": "input", "x1": 300, "y1": 100, "x2": 400, "y2": 200}}
    }
  }
  derivative = Derivative(k = Td, T = max(Td / Nd, 1e-14), x0 = xd0) {
    "Dyad": {
      "placement": {"icon": {"iconName": "input", "x1": 300, "y1": 300, "x2": 400, "y2": 400}}
    }
  }
  integrator = Integrator(k = 1 / Ti, x0 = xi0) {
    "Dyad": {
      "placement": {"icon": {"iconName": "input", "x1": 300, "y1": 500, "x2": 400, "y2": 600}}
    }
  }
  add_pid = Add3() {
    "Dyad": {
      "placement": {"icon": {"iconName": "input", "x1": 500, "y1": 300, "x2": 600, "y2": 400}}
    }
  }
  gain_pid = Gain(k = k) {
    "Dyad": {
      "placement": {"icon": {"iconName": "input", "x1": 700, "y1": 300, "x2": 800, "y2": 400}}
    }
  }
  add_ff = Add() {
    "Dyad": {
      "placement": {"icon": {"iconName": "input", "x1": 900, "y1": 300, "x2": 1000, "y2": 400}}
    }
  }
  limiter = Limiter(y_max = y_max, y_min = y_min) {
    "Dyad": {
      "placement": {"icon": {"iconName": "input", "x1": 1100, "y1": 300, "x2": 1200, "y2": 400}}
    }
  }
  add_sat = Add(k1 = 1, k2 = -1) {
    "Dyad": {
      "placement": {
        "icon": {"iconName": "input", "x1": 1100, "y1": 500, "x2": 1200, "y2": 600, "rot": 90}
      }
    }
  }
  gain_track = Gain(k = 1 / (k * Ni)) {
    "Dyad": {
      "placement": {
        "icon": {"iconName": "input", "x1": 500, "y1": 700, "x2": 600, "y2": 800, "rot": 180}
      }
    }
  }
  variable control_error::Real
  "Gain of controller"
  parameter k::Real = 1
  "Time constant of the integrator block"
  parameter Ti::Time = 0.5
  "Time constant of the derivative block"
  parameter Td::Time = 0.1
  "Maximum output"
  parameter y_max::Real = 1e300
  "Minimum output"
  parameter y_min::Real = -y_max
  "Set-point weight for proportional block"
  parameter wp::Real = 1
  "Set-point weight for derivative block"
  parameter wd::Real = 0
  "`Ni*Ti` is time constant of anti-windup compensation"
  parameter Ni::Real = 0.9
  "Maximum derivative gain. Higher the value of `Nd`, the more ideal the derivative block gets (less filtering, higher high-frequency gain)."
  parameter Nd::Real = 10
  "Gain of the feed-forward input"
  parameter k_ff::Real = 1
  "Initial guess value for integrator output"
  parameter xi0::Real = 0
  "Initial guess value for derivative output"
  parameter xd0::Real = 0
relations
  u_s = add_p.u1
  u_s = add_i.u1
  u_s = add_d.u1
  u_m = add_p.u2
  u_m = add_i.u2
  u_m = add_d.u2
  connect(add_p.y, proportional.u) {"Dyad": {"edges": [{"S": 1, "E": 2}]}}
  connect(add_d.y, derivative.u) {"Dyad": {"edges": [{"S": 1, "E": 2}]}}
  connect(add_i.y, integrator.u) {"Dyad": {"edges": [{"S": 1, "E": 2}]}}
  connect(proportional.y, add_pid.u1) {
    "Dyad": {"edges": [{"S": 1, "M": [{"x": 450, "y": 150}, {"x": 450, "y": 320}], "E": 2}]}
  }
  connect(derivative.y, add_pid.u2) {"Dyad": {"edges": [{"S": 1, "E": 2}]}}
  connect(integrator.y, add_pid.u3) {
    "Dyad": {"edges": [{"S": 1, "M": [{"x": 450, "y": 550}, {"x": 450, "y": 380}], "E": 2}]}
  }
  connect(add_pid.y, gain_pid.u) {"Dyad": {"edges": [{"S": 1, "E": 2}]}}
  connect(gain_pid.y, add_ff.u1) {
    "Dyad": {"edges": [{"S": 1, "M": [{"x": 825, "y": 350}, {"x": 825, "y": 320}], "E": 2}]}
  }
  u_ff = add_ff.u2
  connect(add_ff.y, add_sat.u2) {
    "Dyad": {
      "edges": [{"S": 2, "M": [{"x": 1120, "y": 480}, {"x": 1050, "y": 480}], "E": -1}],
      "junctions": [{"x": 1050, "y": 350}]
    }
  }
  connect(add_ff.y, limiter.u) {"Dyad": {"edges": [{"S": 1, "E": 2}]}}
  connect(limiter.y, add_sat.u1) {
    "Dyad": {
      "edges": [
        {"S": 1, "E": -1},
        {"S": 2, "M": [{"x": 1180, "y": 480}, {"x": 1250, "y": 480}], "E": -1}
      ],
      "junctions": [{"x": 1250, "y": 350}]
    }
  }
  connect(add_sat.y, gain_track.u) {"Dyad": {"edges": [{"S": 1, "M": [{"x": 1150, "y": 750}], "E": 2}]}}
  connect(gain_track.y, add_i.u3) {
    "Dyad": {"edges": [{"S": 1, "M": [{"x": 75, "y": 750}, {"x": 75, "y": 580}], "E": 2}]}
  }
  y = limiter.y
metadata {
  "Dyad": {
    "labels": [{"label": "$(instance)", "x": 500, "y": 1100, "rot": 0}],
    "icons": {"default": "dyad://BlockComponents/LimPID.svg"}
  }
}
end

Flattened Source

dyad

"""
PID controller with limited output, back calculation anti-windup compensation, setpoint weighting and feed-forward

The transfer function is:

math y = k \left[e + \dfrac{1}{T_is}\left(e + \dfrac{y_{sat} - y}{N_iT_i}\right) + \dfrac{T_ds}{1 + {\dfrac{T_d}{N_d}s}}e \right] + k_{ff}u_

"""
component LimPID
  "Connector of setpoint input signal"
  u_s = RealInput() {
    "Dyad": {
      "placement": {
        "icon": {"iconName": "default", "x1": -100, "y1": 220, "x2": 0, "y2": 320, "rot": 0},
        "diagram": {"iconName": "default", "x1": -100, "y1": 220, "x2": 0, "y2": 320, "rot": 0}
      }
    }
  }
  "Connector of measurement input signal"
  u_m = RealInput() {
    "Dyad": {
      "placement": {
        "icon": {"iconName": "default", "x1": -100, "y1": 680, "x2": 0, "y2": 780, "rot": 0},
        "diagram": {"iconName": "default", "x1": -100, "y1": 680, "x2": 0, "y2": 780, "rot": 0}
      }
    }
  }
  "Connector of actuator output signal"
  y = RealOutput() {
    "Dyad": {
      "placement": {
        "icon": {"iconName": "output", "x1": 1000, "y1": 460, "x2": 1100, "y2": 560, "rot": 0},
        "diagram": {"iconName": "output", "x1": 1000, "y1": 460, "x2": 1100, "y2": 560, "rot": 0}
      }
    }
  }
  u_ff = RealInput() {
    "Dyad": {
      "placement": {
        "diagram": {"iconName": "input", "x1": 850, "y1": 950, "x2": 950, "y2": 1050, "rot": -90},
        "icon": {"iconName": "input", "x1": 650, "y1": 950, "x2": 750, "y2": 1050, "rot": -90}
      }
    }
  }
  add_p = Add(k1 = wp, k2 = -1) {
    "Dyad": {
      "placement": {"icon": {"iconName": "input", "x1": 100, "y1": 100, "x2": 200, "y2": 200}}
    }
  }
  add_d = Add(k2 = -1) {
    "Dyad": {
      "placement": {"icon": {"iconName": "input", "x1": 100, "y1": 300, "x2": 200, "y2": 400}}
    }
  }
  add_i = Add3(k2 = -1) {
    "Dyad": {
      "placement": {"icon": {"iconName": "input", "x1": 100, "y1": 500, "x2": 200, "y2": 600}}
    }
  }
  proportional = Gain(k = 1) {
    "Dyad": {
      "placement": {"icon": {"iconName": "input", "x1": 300, "y1": 100, "x2": 400, "y2": 200}}
    }
  }
  derivative = Derivative(k = Td, T = max(Td / Nd, 1e-14), x0 = xd0) {
    "Dyad": {
      "placement": {"icon": {"iconName": "input", "x1": 300, "y1": 300, "x2": 400, "y2": 400}}
    }
  }
  integrator = Integrator(k = 1 / Ti, x0 = xi0) {
    "Dyad": {
      "placement": {"icon": {"iconName": "input", "x1": 300, "y1": 500, "x2": 400, "y2": 600}}
    }
  }
  add_pid = Add3() {
    "Dyad": {
      "placement": {"icon": {"iconName": "input", "x1": 500, "y1": 300, "x2": 600, "y2": 400}}
    }
  }
  gain_pid = Gain(k = k) {
    "Dyad": {
      "placement": {"icon": {"iconName": "input", "x1": 700, "y1": 300, "x2": 800, "y2": 400}}
    }
  }
  add_ff = Add() {
    "Dyad": {
      "placement": {"icon": {"iconName": "input", "x1": 900, "y1": 300, "x2": 1000, "y2": 400}}
    }
  }
  limiter = Limiter(y_max = y_max, y_min = y_min) {
    "Dyad": {
      "placement": {"icon": {"iconName": "input", "x1": 1100, "y1": 300, "x2": 1200, "y2": 400}}
    }
  }
  add_sat = Add(k1 = 1, k2 = -1) {
    "Dyad": {
      "placement": {
        "icon": {"iconName": "input", "x1": 1100, "y1": 500, "x2": 1200, "y2": 600, "rot": 90}
      }
    }
  }
  gain_track = Gain(k = 1 / (k * Ni)) {
    "Dyad": {
      "placement": {
        "icon": {"iconName": "input", "x1": 500, "y1": 700, "x2": 600, "y2": 800, "rot": 180}
      }
    }
  }
  variable control_error::Real
  "Gain of controller"
  parameter k::Real = 1
  "Time constant of the integrator block"
  parameter Ti::Time = 0.5
  "Time constant of the derivative block"
  parameter Td::Time = 0.1
  "Maximum output"
  parameter y_max::Real = 1e300
  "Minimum output"
  parameter y_min::Real = -y_max
  "Set-point weight for proportional block"
  parameter wp::Real = 1
  "Set-point weight for derivative block"
  parameter wd::Real = 0
  "`Ni*Ti` is time constant of anti-windup compensation"
  parameter Ni::Real = 0.9
  "Maximum derivative gain. Higher the value of `Nd`, the more ideal the derivative block gets (less filtering, higher high-frequency gain)."
  parameter Nd::Real = 10
  "Gain of the feed-forward input"
  parameter k_ff::Real = 1
  "Initial guess value for integrator output"
  parameter xi0::Real = 0
  "Initial guess value for derivative output"
  parameter xd0::Real = 0
relations
  u_s = add_p.u1
  u_s = add_i.u1
  u_s = add_d.u1
  u_m = add_p.u2
  u_m = add_i.u2
  u_m = add_d.u2
  connect(add_p.y, proportional.u) {"Dyad": {"edges": [{"S": 1, "E": 2}]}}
  connect(add_d.y, derivative.u) {"Dyad": {"edges": [{"S": 1, "E": 2}]}}
  connect(add_i.y, integrator.u) {"Dyad": {"edges": [{"S": 1, "E": 2}]}}
  connect(proportional.y, add_pid.u1) {
    "Dyad": {"edges": [{"S": 1, "M": [{"x": 450, "y": 150}, {"x": 450, "y": 320}], "E": 2}]}
  }
  connect(derivative.y, add_pid.u2) {"Dyad": {"edges": [{"S": 1, "E": 2}]}}
  connect(integrator.y, add_pid.u3) {
    "Dyad": {"edges": [{"S": 1, "M": [{"x": 450, "y": 550}, {"x": 450, "y": 380}], "E": 2}]}
  }
  connect(add_pid.y, gain_pid.u) {"Dyad": {"edges": [{"S": 1, "E": 2}]}}
  connect(gain_pid.y, add_ff.u1) {
    "Dyad": {"edges": [{"S": 1, "M": [{"x": 825, "y": 350}, {"x": 825, "y": 320}], "E": 2}]}
  }
  u_ff = add_ff.u2
  connect(add_ff.y, add_sat.u2) {
    "Dyad": {
      "edges": [{"S": 2, "M": [{"x": 1120, "y": 480}, {"x": 1050, "y": 480}], "E": -1}],
      "junctions": [{"x": 1050, "y": 350}]
    }
  }
  connect(add_ff.y, limiter.u) {"Dyad": {"edges": [{"S": 1, "E": 2}]}}
  connect(limiter.y, add_sat.u1) {
    "Dyad": {
      "edges": [
        {"S": 1, "E": -1},
        {"S": 2, "M": [{"x": 1180, "y": 480}, {"x": 1250, "y": 480}], "E": -1}
      ],
      "junctions": [{"x": 1250, "y": 350}]
    }
  }
  connect(add_sat.y, gain_track.u) {"Dyad": {"edges": [{"S": 1, "M": [{"x": 1150, "y": 750}], "E": 2}]}}
  connect(gain_track.y, add_i.u3) {
    "Dyad": {"edges": [{"S": 1, "M": [{"x": 75, "y": 750}, {"x": 75, "y": 580}], "E": 2}]}
  }
  y = limiter.y
metadata {
  "Dyad": {
    "labels": [{"label": "$(instance)", "x": 500, "y": 1100, "rot": 0}],
    "icons": {"default": "dyad://BlockComponents/LimPID.svg"}
  }
}
end

Test Cases

No test cases defined.

Examples
Experiments
Analyses
Tests
- LimPIDTest

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LimPID ​

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`LimPID`

Usage

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