Model Predictive Control Exercise

Objective: Implement a model predictive controller that automatically regulates vehicle velocity. Implement the controller in Python and tune the controller for acceptable performance. Discuss factors that may be important for evaluating controller performance.

The dynamic relationship between a vehicle gas pedal position (MV) and velocity (CV) is given by the following set of conditions and a single dynamic equation. 

 Constants
   m = 500 ! Mass (kg)
 Parameters
   b = 50  ! Resistive coefficient (N-s/m)  
   K = 0.8 ! Gain (m/s-%pedal)
   p = 0 >= 0 <= 100  ! Gas pedal position (%)
 Variables
   v = 0 ! initial condition
 Equations
   m * $v = -v * b + K * b * p

Implement a model predictive controller that adjusts gas pedal position to regulate velocity. Start at an initial vehicle velocity of 0 m/s and accelerate to a velocity of 40 m/s.

Discuss the controller performance and how it could be tuned to meet multiple objectives including:

  • minimize travel time
  • remain within speed limits
  • improve vehicle fuel efficiency
  • discourage excessive gas pedal adjustments
  • do not accelerate excessively

There is no need to implement these advanced objectives in simulation for this second part of the exercise, only discuss the possible competing objectives.

Solution

The Model Predictive Control both solves the differential equations that describe the velocity of a vehicle as well as minimizes the control objective function.

The multiple objectives can be implemented with variable constraints or alternative objective functions. For example, if the only objective is to minimize travel time, the solution would be to use full gas pedal position (100%) for the duration of the simulation. If the objective is to minimize travel time but stay within the speed limit, the solution would be to reach 40 m/s as fast as possible without overshoot. Each additional objective has the potential to adapt the solution to achieve an optimal tradeoff. It is a tradeoff because some of the desirable outcomes have conflicting objectives.

See the Dynamic Optimization Course for Excel, MATLAB, and Simulink source files to this problem. Python source in Python Gekko (newer) and APM Python is shown below.

#%%Import packages
import numpy as np
from random import random
from gekko import GEKKO
import matplotlib.pyplot as plt

#%% Build model

#initialize GEKKO model
m = GEKKO()

#time
m.time = np.linspace(0,20,41)

#constants
mass = 500

#Parameters
b = m.Param(value=50)
K = m.Param(value=0.8)
#Manipulated variable
p = m.MV(value=0, lb=0, ub=100)

#Controlled Variable
v = m.CV(value=0)

#Equations
m.Equation(mass*v.dt() == -v*b + K*b*p)

#%% Tuning

#global
m.options.IMODE = 6 #control

#MV tuning
p.STATUS = 1 #allow optimizer to change
p.DCOST = 0.1 #smooth out gas pedal movement
p.DMAX = 20 #slow down change of gas pedal

#CV tuning
#setpoint
v.STATUS = 1 #add the SP to the objective
m.options.CV_TYPE = 2 #L2 norm
v.SP = 40 #set point
v.TR_INIT = 1 #setpoint trajectory
v.TAU = 5 #time constant of setpoint trajectory

#%% Solve

m.solve()

#%% Plot solution
plt.figure()
plt.subplot(2,1,1)
plt.plot(m.time,p.value,'b-',LineWidth=2)
plt.ylabel('gas')
plt.subplot(2,1,2)
plt.plot(m.time,v.value,'r--',LineWidth=2)
plt.ylabel('velocity')
plt.xlabel('time')
plt.show()
[$[Get Code]]

import numpy as np
# pip install APMonitor
from APMonitor import *

s = 'https://byu.apmonitor.com'
a = 'velocity'

# clear prior application
apm(s,a,'clear all')

# time points
t = np.linspace(0,30,31)

# write model file (model.apm)
fid = open('model.apm','w')
fid.write('Constants \n')
fid.write('  m = 500 ! Mass (kg) \n')
fid.write('\n')
fid.write('Parameters \n')
fid.write('  b = 50  ! Resistive coefficient (N-s/m) \n')
fid.write('  K = 0.8 ! Gain (m/s-%pedal) \n')
fid.write('  p = 0 >= 0 <= 100  ! Gas pedal position (%) \n')
fid.write(' \n')
fid.write('Variables \n')
fid.write('  v = 0 \n')
fid.write('  tau = m/b \n')
fid.write(' \n')
fid.write('Equations \n')
fid.write('  m * $v = -v * b + K * b * p \n')
fid.write('  tau * b = m \n')
fid.close()

# write data file (data.csv)
header = 'time'
data = t.T # transpose to column vector
np.savetxt('data.csv',data,header='time',delimiter=',',comments='')

# load model and data
apm_load(s,a,'model.apm')
csv_load(s,a,'data.csv')

# specify MV / CV
apm_info(s,a,'MV','p')
apm_info(s,a,'CV','v')

# configuration parameters
apm_option(s,a,'nlc.imode',6)
apm_option(s,a,'nlc.nodes',3)

# turn on MV / CV
apm_option(s,a,'v.status',1)
apm_option(s,a,'p.status',1)

# tune controller
apm_option(s,a,'p.lower',0)
apm_option(s,a,'p.upper',100)
apm_option(s,a,'v.tau',5)
apm_option(s,a,'v.sphi',26)
apm_option(s,a,'v.splo',24)

# solve and retrieve results
output = apm(s,a,'solve')
print(output)

# open web-viewer
apm_web(s,a)
[$[Get Code]]
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