Dynamic Simulation Tutorial with DWSIM and Python, Part 4: Tuning the PID Controller through Non-Linear Optimization

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Dialog-warning.png This tutorial requires advanced or above average Python programming skills.
Dialog-information.png You'll need at least DWSIM v5.7 (Cross-Platform UI) on Windows, Linux or macOS to follow/reproduce the tasks within this tutorial.

Introduction

We can use built-in DWSIM non-linear optimizers to tune our PID controller by finding a parameter set (Kp, Ki, Kd) which minimizes the total error between the set-point and process variable during the whole simulation period.

Setup

Modify the 'RunDynamicProcess_ClosedLoop' to read the PID parameters stored in the PID Controller, by changing 

P = 0.5
I = 0.01
D = 0.1

to 

P = controller.ExtraProperties.Kp
I = controller.ExtraProperties.Ki
D = controller.ExtraProperties.Kd

When you save the simulation, the latest parameter values will be saved as extra properties of the PID Controller object.

You can comment the chart generation lines so you won't get a chart window at each optimizer iteration: 

#outputresults = Flowsheet.Scripts.Values.Where(lambda x: x.Title == 'GenerateCharts').FirstOrDefault().ScriptText.replace('\r', '')
#exec(outputresults)

Tuning the PID Controller using the Simplex method

Create a new script and name it 'TunePID_Simplex', with the following content: 

import clr
import System
from System import *
 
clr.AddReference('DWSIM.MathOps.DotNumerics')
clr.AddReference('System.Core')
clr.ImportExtensions(System.Linq)
 
from System import Func
 
from DotNumerics import *
from DotNumerics.Optimization import *
 
controller = Flowsheet.GetFlowsheetSimulationObject("PID Controller")
 
controller.ExtraProperties.Kp = 0.5
controller.ExtraProperties.Ki = 0.01
controller.ExtraProperties.Kd = 0.1
 
P = controller.ExtraProperties.Kp
I = controller.ExtraProperties.Ki
D = controller.ExtraProperties.Kd
 
simplex = Simplex()
 
Pvar = OptBoundVariable(P, 0.0, 10.0)
Ivar = OptBoundVariable(I, 0.0, 10.0)
Dvar = OptBoundVariable(D, 0.0, 10.0)
 
vars = [Pvar, Ivar, Dvar]
 
tx0 = DateTime.Now
 
counter = 1
 
def RunIteration(vars):
    global counter
    Flowsheet.SupressMessages = False
    Flowsheet.WriteMessage("Run #" + str(counter))
    Flowsheet.WriteMessage("P = " + str(controller.ExtraProperties.Kp))
    Flowsheet.WriteMessage("I = " + str(controller.ExtraProperties.Ki))
    Flowsheet.WriteMessage("D = " + str(controller.ExtraProperties.Kd))
    controller.ExtraProperties.Kp = vars[0]
    controller.ExtraProperties.Ki = vars[1]
    controller.ExtraProperties.Kd = vars[2]
    Flowsheet.SupressMessages = True
    itloop = Flowsheet.Scripts.Values.Where(lambda x: x.Title == 'RunDynamicProcess_ClosedLoop').FirstOrDefault().ScriptText.replace('\r', '')
    exec(itloop)
    Flowsheet.SupressMessages = False
    Flowsheet.WriteMessage("Error = " + str(controller.ExtraProperties.TotalError))
    counter += 1
    return controller.ExtraProperties.TotalError
 
simplex.Tolerance = 0.000001
simplex.MaxFunEvaluations = 1000
simplex.ComputeMin(lambda x: RunIteration(x), Array[OptBoundVariable](vars))
 
tx1 = DateTime.Now
 
controller.ExtraProperties.TotalTuningTime = (tx1 - tx0).TotalSeconds

Run the above script. It can take several hours to complete, depending on your system's processing power.

Tuning the PID Controller using IPOPT

Create a new script and name it 'TunePID_IPOPT', with the following content: 

import clr
import System
from System import *
 
clr.AddReference('Cureos.Numerics')
clr.AddReference('System.Core')
clr.ImportExtensions(System.Linq)
 
from System import Func
 
from Cureos.Numerics import *
 
controller = Flowsheet.GetFlowsheetSimulationObject("PID Controller")
 
controller.ExtraProperties.Kp = 0.5
controller.ExtraProperties.Ki = 0.01
controller.ExtraProperties.Kd = 0.1
 
P = controller.ExtraProperties.Kp
I = controller.ExtraProperties.Ki
D = controller.ExtraProperties.Kd
 
lconstr = [0.0, 0.0, 0.0]
uconstr = [10.0, 10.0, 10.0]
vars = [P, I, D]
 
tx0 = DateTime.Now
 
counter = 1
 
Flowsheet.SupressMessages = False
     
def RunIteration(vars):
    global counter
    Flowsheet.SupressMessages = False
    Flowsheet.WriteMessage("Run #" + str(counter))
    Flowsheet.WriteMessage("P = " + str(controller.ExtraProperties.Kp) + ", I = " + str(controller.ExtraProperties.Ki) + ", D = " + str(controller.ExtraProperties.Kd))
    controller.ExtraProperties.Kp = vars[0]
    controller.ExtraProperties.Ki = vars[1]
    controller.ExtraProperties.Kd = vars[2]
    Flowsheet.SupressMessages = True
    itloop = Flowsheet.Scripts.Values.Where(lambda x: x.Title == 'RunDynamicProcess_ClosedLoop').FirstOrDefault().ScriptText.replace('\r', '')
    exec(itloop)
    Flowsheet.SupressMessages = False
    Flowsheet.WriteMessage("Error = " + str(controller.ExtraProperties.TotalError))
    counter += 1
    return controller.ExtraProperties.TotalError
     
def CalculateGradient(x):
    epsilon = 0.001
    x1 = [0.0, 0.0, 0.0]
    x2 = [0.0, 0.0, 0.0]
    g = [0.0, 0.0, 0.0]
    for j in range(0, x.Length):
        for k in range(0, x.Length):
            x1[k] = x[k]
            x2[k] = x[k]
        x1[j] = x[j] + epsilon
        x2[j] = x[j] - epsilon
        f1 = RunIteration(x1)
        f2 = RunIteration(x2)
        g[j] = (f2 - f1) / (x2[j] - x1[j])
    return Array[Double](g)
     
def func(n, x, new_x, obj_value):
    obj_value.Value = RunIteration(x)
    return True
     
def gradfunc(n, x, new_x, grad_f):
    grad_f.Value = CalculateGradient(x)
    return True
 
ipopt = Ipopt(3, Array[float](lconstr), Array[float](uconstr), 
                0, None, None, 0, 0, 
                lambda n, x, new_x, obj_value: func(n, x, new_x, obj_value), 
                lambda n, x, new_x, m, g: True, 
                lambda n, x, new_x, grad_f: gradfunc(n, x, new_x, grad_f), 
                lambda n, x, new_x, m, nele_jac, iRow, jCol, values: False, 
                lambda n, x, new_x, obj_factor, m, lambda0, new_lambda0, nele_hess, iRow, jCol, values: False)
 
obj = clr.Reference[float]()
 
ipopt.AddOption("tol", 0.000001)
ipopt.AddOption("max_iter", 1000)
ipopt.AddOption("mu_strategy", "adaptive")
ipopt.AddOption("hessian_approximation", "limited-memory")
status = ipopt.SolveProblem(Array[float](vars), obj, None, None, None, None)
 
print status
 
tx1 = DateTime.Now
 
controller.ExtraProperties.TotalTuningTime = (tx1 - tx0).TotalSeconds

Run the above script. It can take several hours to complete, depending on your system's processing power.

Viewing Tuning Results

After successfully tuning the PID, you should get charts that look like the following ones:

Dynamic tuned.jpg

Download File

Download the simulation file with what has been done so far: dynamic_part4.dwxmz

Return to Dynamic Simulation Tutorial with DWSIM and Python, Part 3: Adding a PID Controller

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