Main

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~~This problem can also be solved with an ODE integrator such as Python's ODEINT function in SciPy.Integrate or MATLAB's ode23 function. The focus of this class is on solving dynamic optimization problems where ODE integrators are not suitable because of the inefficiencies of shooting methods. Regardless, it is valuable to know how to use ODE integrators for simulation. Below is the source code using ODEINT in Python.~~

This problem can also be solved with an ODE integrator such as Python's ODEINT function in SciPy.Integrate or MATLAB's ode23 function. The focus of this class is on solving dynamic optimization problems where ODE integrators are not suitable because of the inefficiencies of shooting methods. Regardless, it is valuable to know how to use ODE integrators for simulation. Below is the source code using ODEINT in Python.

(:source lang=python:)

### Sequential method with SciPy.integrate.odeint

import numpy as np

from scipy.integrate import odeint

def skydive(z,t):

# constants

g = 9.81 # m/s^2, gravitational constant

m = 80 # kg, mass of skydiver and pack

if t<61:

c = 0.2 # N-s^2/m^2, drag coefficient, chute closed

else:

c = 10.0 # N-s^2/m^2, drag coefficient, chute open

# states (z)

x = z[0] # meters, horizontal position

y = z[1] # meters, vertical position / elevation

vx = z[2] # m/s, skydiver horizontal velocity = airplane velocity

vy = z[3] # m/s, skydiver vertical velocity

# derived values

v = np.sqrt(vx**2+vy**2) # m/s, magnitude of velocity

Fx = -c * vx**2

Fy = -m*g + c*vy**2

# calculate derivatives

dxdt = vx

dydt = vy

dvxdt = Fx / m

dvydt = Fy / m

dzdt = [dxdt,dydt,dvxdt,dvydt]

return dzdt

# initial conditions

z0 = [0,5000,50,0]

# time points

t = np.linspace(0,90,91)

# solve

z1 = odeint(skydive,z0,t)

# parse results

x = z1[:,0]

y = z1[:,1]

vx = z1[:,2]

vy = z1[:,3]

v = np.sqrt(vx**2+vy**2)

### Simultaneous method with APMonitor

# install and load APMonitor

try:

from APMonitor import *

except:

import pip

pip.main(['install','APMonitor'])

from APMonitor import *

# solve model

z2 = apm_solve('skydiver',7)

### Plot results

import matplotlib.pyplot as plt

plt.figure(1)

plt.subplot(2,1,1)

plt.plot(t,x,'r-',linewidth=2)

plt.plot(t,y,'b-',linewidth=2)

plt.plot(z2['time'],z2['x'],'r:',linewidth=3)

plt.plot(z2['time'],z2['y'],'b--',linewidth=3)

plt.ylabel('Position (m)')

plt.legend(['ODEINT x','ODEINT y','APM x','APM y'])

plt.subplot(2,1,2)

plt.plot(t,vx,'r-',linewidth=2)

plt.plot(t,vy,'b-',linewidth=2)

plt.plot(t,v,'k-',linewidth=2)

plt.plot(z2['time'],z2['vx'],'r--',linewidth=3)

plt.plot(z2['time'],z2['vy'],'b--',linewidth=3)

plt.plot(z2['time'],z2['v'],'k--',linewidth=3)

plt.xlabel('Time (sec)')

plt.ylabel('Velocity (m/s)')

plt.legend(['ODEINT V_x','ODEINT V_y','ODEINT V',\

'APM V_x','APM V_y','APM V'])

plt.figure(2)

plt.plot(z2['x'],z2['y'],'r-')

plt.xlabel('Position (x)')

plt.ylabel('Position (y)')

plt.show()

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(:source lang=python:)

### Sequential method with SciPy.integrate.odeint

import numpy as np

from scipy.integrate import odeint

def skydive(z,t):

# constants

g = 9.81 # m/s^2, gravitational constant

m = 80 # kg, mass of skydiver and pack

if t<61:

c = 0.2 # N-s^2/m^2, drag coefficient, chute closed

else:

c = 10.0 # N-s^2/m^2, drag coefficient, chute open

# states (z)

x = z[0] # meters, horizontal position

y = z[1] # meters, vertical position / elevation

vx = z[2] # m/s, skydiver horizontal velocity = airplane velocity

vy = z[3] # m/s, skydiver vertical velocity

# derived values

v = np.sqrt(vx**2+vy**2) # m/s, magnitude of velocity

Fx = -c * vx**2

Fy = -m*g + c*vy**2

# calculate derivatives

dxdt = vx

dydt = vy

dvxdt = Fx / m

dvydt = Fy / m

dzdt = [dxdt,dydt,dvxdt,dvydt]

return dzdt

# initial conditions

z0 = [0,5000,50,0]

# time points

t = np.linspace(0,90,91)

# solve

z1 = odeint(skydive,z0,t)

# parse results

x = z1[:,0]

y = z1[:,1]

vx = z1[:,2]

vy = z1[:,3]

v = np.sqrt(vx**2+vy**2)

### Simultaneous method with APMonitor

# install and load APMonitor

try:

from APMonitor import *

except:

import pip

pip.main(['install','APMonitor'])

from APMonitor import *

# solve model

z2 = apm_solve('skydiver',7)

### Plot results

import matplotlib.pyplot as plt

plt.figure(1)

plt.subplot(2,1,1)

plt.plot(t,x,'r-',linewidth=2)

plt.plot(t,y,'b-',linewidth=2)

plt.plot(z2['time'],z2['x'],'r:',linewidth=3)

plt.plot(z2['time'],z2['y'],'b--',linewidth=3)

plt.ylabel('Position (m)')

plt.legend(['ODEINT x','ODEINT y','APM x','APM y'])

plt.subplot(2,1,2)

plt.plot(t,vx,'r-',linewidth=2)

plt.plot(t,vy,'b-',linewidth=2)

plt.plot(t,v,'k-',linewidth=2)

plt.plot(z2['time'],z2['vx'],'r--',linewidth=3)

plt.plot(z2['time'],z2['vy'],'b--',linewidth=3)

plt.plot(z2['time'],z2['v'],'k--',linewidth=3)

plt.xlabel('Time (sec)')

plt.ylabel('Velocity (m/s)')

plt.legend(['ODEINT V_x','ODEINT V_y','ODEINT V',\

'APM V_x','APM V_y','APM V'])

plt.figure(2)

plt.plot(z2['x'],z2['y'],'r-')

plt.xlabel('Position (x)')

plt.ylabel('Position (y)')

plt.show()

(:sourceend:)

(:source lang=python:)

### Sequential method with SciPy.integrate.odeint

import numpy as np

from scipy.integrate import odeint

def skydive(z,t):

# constants

g = 9.81 # m/s^2, gravitational constant

m = 80 # kg, mass of skydiver and pack

if t<61:

c = 0.2 # N-s^2/m^2, drag coefficient, chute closed

else:

c = 10.0 # N-s^2/m^2, drag coefficient, chute open

# states (z)

x = z[0] # meters, horizontal position

y = z[1] # meters, vertical position / elevation

vx = z[2] # m/s, skydiver horizontal velocity = airplane velocity

vy = z[3] # m/s, skydiver vertical velocity

# derived values

v = np.sqrt(vx**2+vy**2) # m/s, magnitude of velocity

Fx = -c * vx**2

Fy = -m*g + c*vy**2

# calculate derivatives

dxdt = vx

dydt = vy

dvxdt = Fx / m

dvydt = Fy / m

dzdt = [dxdt,dydt,dvxdt,dvydt]

return dzdt

# initial conditions

z0 = [0,5000,50,0]

# time points

t = np.linspace(0,90,91)

# solve

z1 = odeint(skydive,z0,t)

# parse results

x = z1[:,0]

y = z1[:,1]

vx = z1[:,2]

vy = z1[:,3]

v = np.sqrt(vx**2+vy**2)

### Simultaneous method with APMonitor

# install and load APMonitor

try:

from APMonitor import *

except:

import pip

pip.main(['install','APMonitor'])

from APMonitor import *

# solve model

z2 = apm_solve('skydiver',7)

### Plot results

import matplotlib.pyplot as plt

plt.figure(1)

plt.subplot(2,1,1)

plt.plot(t,x,'r-',linewidth=2)

plt.plot(t,y,'b-',linewidth=2)

plt.plot(z2['time'],z2['x'],'r:',linewidth=3)

plt.plot(z2['time'],z2['y'],'b--',linewidth=3)

plt.ylabel('Position (m)')

plt.legend(['ODEINT x','ODEINT y','APM x','APM y'])

plt.subplot(2,1,2)

plt.plot(t,vx,'r-',linewidth=2)

plt.plot(t,vy,'b-',linewidth=2)

plt.plot(t,v,'k-',linewidth=2)

plt.plot(z2['time'],z2['vx'],'r--',linewidth=3)

plt.plot(z2['time'],z2['vy'],'b--',linewidth=3)

plt.plot(z2['time'],z2['v'],'k--',linewidth=3)

plt.xlabel('Time (sec)')

plt.ylabel('Velocity (m/s)')

plt.legend(['ODEINT V_x','ODEINT V_y','ODEINT V',\

'APM V_x','APM V_y','APM V'])

plt.figure(2)

plt.plot(z2['x'],z2['y'],'r-')

plt.xlabel('Position (x)')

plt.ylabel('Position (y)')

plt.show()

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'''Objective:''' Provide a basic introduction to dynamic system modeling and simulation with the equations of motion. Create a MATLAB or Python script to simulate and display the results. ''Estimated Time: 1 hour''

~~(video of solution~~)
~~Develop a mathematical description~~ of ~~the motion of a falling object ~~(~~e.g. skydiver) in one dimension where there is a frictional resistance that is proportional to the square of~~ the ~~velocity. Solve ~~the ~~equations numerically.~~

Integrate the ~~system from zero initial time~~ to ~~terminal velocity and assume that the object is initially at rest. Assume a gravitational constant of 9.8 ~~m/s~~'^2^', a mass of 80 kg, and a frictional loss coefficient of 0.2 ~~N-s'^2^'/m'^2^'.
~~[[~~Attach:~~simulate_vehicle~~.~~zip|Attach:download.~~png Simulate with Excel, MATLAB, Python, and Simulink]]
~~* ~~[[Attach:simulate_vehicle.zip|Simulate with Excel, MATLAB, Python, and Simulink]]

* [[Attach:Intro_Dynamic_Modeling.pdf|Introduction to Dynamic Modeling (pdf)]]
~~!!!!Numerical Solution of Differential Equations~~

Excel, MATLAB, Python, and Simulink are used in the following example to solve the differential equation that describes the velocity of a vehicle.

* [[Attach:simulate_vehicle.zip|Simulate vehicle velocity]]

Excel, MATLAB, Python, and Simulink are used in the following example to solve the differential equation that describes the velocity of a vehicle.

* [[Attach:simulate_vehicle.zip|Simulate vehicle velocity]]

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(video of numerical solution, APM MATLAB, APM Python, APMonitor - show how error changes with refined discretization)

The focus of this course is on modeling, simulation, estimation, and optimization for dynamic systems. This section of the course starts with dynamic modeling or methods to mathematically describe time-evolving systems, particularly for the purpose of optimization in engineering disciplines.

!!!!Dynamic versus Steady State Numerical Solution

Dynamic systems are those where certain values that describe the system are expected to evolve over time and not necessarily remain at a steady state.

Steady state: Values ('''x''') do not change with time

Dynamic: At least one value ('''x''') changes with time

Values of '''x''' may be measured at specific times or an expression may exist from an analytic solution of the differential equations as '''x(t)'''. Differential equations are those in which a Another method to solve dynamic

!!!!Mathematical Description

Models are collections of assumptions, mathematical expressions, boundary conditions, initial conditions, and constraints. These mathematical expressions can range from a simple equation of motion for an object in a frictionless environment to complex systems that describe multi-body physics and interactions. A standard model form is shown below:

Values that describe

A standard form for describing a system is as a collection of differential and algebraic expressions. These may either be equality ('''f''') or inequality ('''g''') constraints. The variables ('''x''')

Dynamic models can be converted to steady state models by setting any

The focus of this course is on dynamic systems or those that change with time. It is particularly focused on engineering applications although much of the theory and applications could also be applied to other fields as well.

''''Dynamic'''' systems

The speed of computers doubles approximately every 18 months. From 2000-2015 the speed of computers increased by about 1000 times.

Dynamic optimization is simulation and

## Introduction to Dynamic Modeling

## Main.DynamicModeling History

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'''Objective:''' Provide a basic introduction to dynamic system modeling and simulation with the equations of motion. Create a MATLAB (ode23, ode15s, etc) or Python (ODEINT) script to simulate and display the results. Simulate with [[http://apmonitor.com/wiki/index.php/Main/MATLAB|APM MATLAB]] ~~or~~ [[http://apmonitor.com/wiki/index.php/Main/PythonApp|APM Python]] as well. Compare the sequential (ODE integrators in MATLAB / Python) versus the simultaneous method (APMonitor). Observe how the number of time points in APMonitor affects the solution accuracy. ''Estimated Time: 1 hour''

to:

'''Objective:''' Provide a basic introduction to dynamic system modeling and simulation with the equations of motion. Create a MATLAB (ode23, ode15s, etc) or Python (ODEINT) script to simulate and display the results. Simulate with [[http://apmonitor.com/wiki/index.php/Main/MATLAB|APM MATLAB]], [[http://apmonitor.com/wiki/index.php/Main/PythonApp|APM Python]], or [[http://gekko.readthedocs.io/en/latest/|Python GEKKO]] as well. Compare the sequential (ODE integrators in MATLAB / Python) versus the simultaneous method (APMonitor). Observe how the number of time points in APMonitor affects the solution accuracy. ''Estimated Time: 1 hour''

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(:source lang=python:)

from gekko import GEKKO

import numpy as np

import matplotlib.pyplot as plt

#number of points in time discretization

n = 91

#Initialize Model

m = GEKKO()

#define time discretization

m.time = np.linspace(0,90,n)

#make array of drag coefficients, changing at time 60

drag = [(0.2 if t<=60 else 10) for t in m.time]

#define constants

g = m.Const(value=9.81)

mass = m.Const(value=80)

#define drag parameter

d = m.Param(value=drag)

#initialize variables

x,y,vx,vy,v,Fx,Fy = [m.Var(value=0) for i in range(7)]

#initial conditions

y.value = 5000

vx.value = 50

#Equations

# force balance

m.Equation(Fx == -d * vx**2)

m.Equation(Fy == -mass*g + d*vy**2)

#F = ma

m.Equation(Fx/mass == vx.dt())

m.Equation(Fy/mass == vy.dt())

#vel = dxdt

m.Equation(vx == x.dt())

m.Equation(vy == y.dt())

#total velocity

m.Equation(v == (vx**2 + vy**2)**.5)

#Set global options

m.options.IMODE = 4 #dynamic simulation

#Solve simulation

m.solve(remote=True)

#%% Plot results

plt.figure()

plt.plot(x.value,y.value)

plt.xlabel('x')

plt.ylabel('y')

plt.figure()

plt.plot(m.time,x.value,label='x')

plt.plot(m.time,y.value,label='y')

plt.xlabel('time')

plt.legend()

plt.figure()

plt.plot(m.time,vx.value,label='vx')

plt.plot(m.time,vy.value,label='vy')

plt.xlabel('time')

plt.legend()

plt.show()

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(:source lang=python:)

from gekko import GEKKO

import numpy as np

import matplotlib.pyplot as plt

#number of points in time discretization

n = 91

#Initialize Model

m = GEKKO()

#define time discretization

m.time = np.linspace(0,90,n)

#make array of drag coefficients, changing at time 60

drag = [(0.2 if t<=60 else 10) for t in m.time]

#define constants

g = m.Const(value=9.81)

mass = m.Const(value=80)

#define drag parameter

d = m.Param(value=drag)

#initialize variables

x,y,vx,vy,v,Fx,Fy = [m.Var(value=0) for i in range(7)]

#initial conditions

y.value = 5000

vx.value = 50

#Equations

# force balance

m.Equation(Fx == -d * vx**2)

m.Equation(Fy == -mass*g + d*vy**2)

#F = ma

m.Equation(Fx/mass == vx.dt())

m.Equation(Fy/mass == vy.dt())

#vel = dxdt

m.Equation(vx == x.dt())

m.Equation(vy == y.dt())

#total velocity

m.Equation(v == (vx**2 + vy**2)**.5)

#Set global options

m.options.IMODE = 4 #dynamic simulation

#Solve simulation

m.solve(remote=True)

#%% Plot results

plt.figure()

plt.plot(x.value,y.value)

plt.xlabel('x')

plt.ylabel('y')

plt.figure()

plt.plot(m.time,x.value,label='x')

plt.plot(m.time,y.value,label='y')

plt.xlabel('time')

plt.legend()

plt.figure()

plt.plot(m.time,vx.value,label='vx')

plt.plot(m.time,vy.value,label='vy')

plt.xlabel('time')

plt.legend()

plt.show()

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This problem can also be solved with an ODE integrator such as Python's ODEINT function in SciPy.Integrate or MATLAB's ode23 function. The focus of this class is on solving dynamic optimization problems where ODE integrators are not suitable because of the inefficiencies of shooting methods. Regardless, it is valuable to know how to use ODE integrators for simulation. Below is the source code in MATLAB and Python.

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(:source lang=matlab:)

clear all; close all; clc

%% Method 1: MATLAB integration with ode15s

% initial conditions

z0 = [0,5000,50,0];

% time points

t = linspace(0,90,1);

% solve

[t,z1] = ode15s('skydiver',t,z0);

% parse results

ode_x = z1(:,1);

ode_y = z1(:,2);

ode_vx = z1(:,3);

ode_vy = z1(:,4);

ode_v = sqrt(ode_vx.^2+ode_vy.^2);

%% Method 2: APMonitor

addpath('apm') % load APMonitor.com files

y = apm_solve('skydiver',7); z = y.x; % solve numerically

%% Plot results

figure(1)

subplot(2,1,1)

plot(t,ode_x,'r-','LineWidth',2)

hold on

plot(t,ode_y,'b-','LineWidth',2)

plot(z.time,z.x,'r:','LineWidth',3)

plot(z.time,z.y,'b--','LineWidth',3)

ylabel('Position (m)')

legend('x','y')

subplot(2,1,2)

plot(t,ode_vx,'r-','LineWidth',2)

hold on

plot(t,ode_vy,'b-','LineWidth',2)

plot(t,ode_v,'k-','LineWidth',2)

plot(z.time,z.vx,'r:','LineWidth',3)

plot(z.time,z.vy,'b--','LineWidth',3)

plot(z.time,z.v,'k--','LineWidth',3)

xlabel('Time (sec)')

ylabel('Velocity (m/s)')

legend('V_x','V_y','V','APM V_x','APM V_y','APM V')

figure(2)

plot(z.x,z.y,'r-','LineWidth',2)

xlabel('Position (x)')

ylabel('Position (y)')

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clear all; close all; clc

%% Method 1: MATLAB integration with ode15s

% initial conditions

z0 = [0,5000,50,0];

% time points

t = linspace(0,90,1);

% solve

[t,z1] = ode15s('skydiver',t,z0);

% parse results

ode_x = z1(:,1);

ode_y = z1(:,2);

ode_vx = z1(:,3);

ode_vy = z1(:,4);

ode_v = sqrt(ode_vx.^2+ode_vy.^2);

%% Method 2: APMonitor

addpath('apm') % load APMonitor.com files

y = apm_solve('skydiver',7); z = y.x; % solve numerically

%% Plot results

figure(1)

subplot(2,1,1)

plot(t,ode_x,'r-','LineWidth',2)

hold on

plot(t,ode_y,'b-','LineWidth',2)

plot(z.time,z.x,'r:','LineWidth',3)

plot(z.time,z.y,'b--','LineWidth',3)

ylabel('Position (m)')

legend('x','y')

subplot(2,1,2)

plot(t,ode_vx,'r-','LineWidth',2)

hold on

plot(t,ode_vy,'b-','LineWidth',2)

plot(t,ode_v,'k-','LineWidth',2)

plot(z.time,z.vx,'r:','LineWidth',3)

plot(z.time,z.vy,'b--','LineWidth',3)

plot(z.time,z.v,'k--','LineWidth',3)

xlabel('Time (sec)')

ylabel('Velocity (m/s)')

legend('V_x','V_y','V','APM V_x','APM V_y','APM V')

figure(2)

plot(z.x,z.y,'r-','LineWidth',2)

xlabel('Position (x)')

ylabel('Position (y)')

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This problem can also be solved with an ODE integrator such as Python's ODEINT function in SciPy.Integrate or MATLAB's ode23 function. The focus of this class is on solving dynamic optimization problems where ODE integrators are not suitable because of the inefficiencies of shooting methods. Regardless, it is valuable to know how to use ODE integrators for simulation. Below is the source code using ODEINT in Python.

(:source lang=python:)

### Sequential method with SciPy.integrate.odeint

import numpy as np

from scipy.integrate import odeint

def skydive(z,t):

# constants

g = 9.81 # m/s^2, gravitational constant

m = 80 # kg, mass of skydiver and pack

if t<61:

c = 0.2 # N-s^2/m^2, drag coefficient, chute closed

else:

c = 10.0 # N-s^2/m^2, drag coefficient, chute open

# states (z)

x = z[0] # meters, horizontal position

y = z[1] # meters, vertical position / elevation

vx = z[2] # m/s, skydiver horizontal velocity = airplane velocity

vy = z[3] # m/s, skydiver vertical velocity

# derived values

v = np.sqrt(vx**2+vy**2) # m/s, magnitude of velocity

Fx = -c * vx**2

Fy = -m*g + c*vy**2

# calculate derivatives

dxdt = vx

dydt = vy

dvxdt = Fx / m

dvydt = Fy / m

dzdt = [dxdt,dydt,dvxdt,dvydt]

return dzdt

# initial conditions

z0 = [0,5000,50,0]

# time points

t = np.linspace(0,90,91)

# solve

z1 = odeint(skydive,z0,t)

# parse results

x = z1[:,0]

y = z1[:,1]

vx = z1[:,2]

vy = z1[:,3]

v = np.sqrt(vx**2+vy**2)

### Simultaneous method with APMonitor

# install and load APMonitor

try:

from APMonitor import *

except:

import pip

pip.main(['install','APMonitor'])

from APMonitor import *

# solve model

z2 = apm_solve('skydiver',7)

### Plot results

import matplotlib.pyplot as plt

plt.figure(1)

plt.subplot(2,1,1)

plt.plot(t,x,'r-',linewidth=2)

plt.plot(t,y,'b-',linewidth=2)

plt.plot(z2['time'],z2['x'],'r:',linewidth=3)

plt.plot(z2['time'],z2['y'],'b--',linewidth=3)

plt.ylabel('Position (m)')

plt.legend(['ODEINT x','ODEINT y','APM x','APM y'])

plt.subplot(2,1,2)

plt.plot(t,vx,'r-',linewidth=2)

plt.plot(t,vy,'b-',linewidth=2)

plt.plot(t,v,'k-',linewidth=2)

plt.plot(z2['time'],z2['vx'],'r--',linewidth=3)

plt.plot(z2['time'],z2['vy'],'b--',linewidth=3)

plt.plot(z2['time'],z2['v'],'k--',linewidth=3)

plt.xlabel('Time (sec)')

plt.ylabel('Velocity (m/s)')

plt.legend(['ODEINT V_x','ODEINT V_y','ODEINT V',\

'APM V_x','APM V_y','APM V'])

plt.figure(2)

plt.plot(z2['x'],z2['y'],'r-')

plt.xlabel('Position (x)')

plt.ylabel('Position (y)')

plt.show()

(:sourceend:)

Deleted lines 88-180:

(:source lang=python:)

### Sequential method with SciPy.integrate.odeint

import numpy as np

from scipy.integrate import odeint

def skydive(z,t):

# constants

g = 9.81 # m/s^2, gravitational constant

m = 80 # kg, mass of skydiver and pack

if t<61:

c = 0.2 # N-s^2/m^2, drag coefficient, chute closed

else:

c = 10.0 # N-s^2/m^2, drag coefficient, chute open

# states (z)

x = z[0] # meters, horizontal position

y = z[1] # meters, vertical position / elevation

vx = z[2] # m/s, skydiver horizontal velocity = airplane velocity

vy = z[3] # m/s, skydiver vertical velocity

# derived values

v = np.sqrt(vx**2+vy**2) # m/s, magnitude of velocity

Fx = -c * vx**2

Fy = -m*g + c*vy**2

# calculate derivatives

dxdt = vx

dydt = vy

dvxdt = Fx / m

dvydt = Fy / m

dzdt = [dxdt,dydt,dvxdt,dvydt]

return dzdt

# initial conditions

z0 = [0,5000,50,0]

# time points

t = np.linspace(0,90,91)

# solve

z1 = odeint(skydive,z0,t)

# parse results

x = z1[:,0]

y = z1[:,1]

vx = z1[:,2]

vy = z1[:,3]

v = np.sqrt(vx**2+vy**2)

### Simultaneous method with APMonitor

# install and load APMonitor

try:

from APMonitor import *

except:

import pip

pip.main(['install','APMonitor'])

from APMonitor import *

# solve model

z2 = apm_solve('skydiver',7)

### Plot results

import matplotlib.pyplot as plt

plt.figure(1)

plt.subplot(2,1,1)

plt.plot(t,x,'r-',linewidth=2)

plt.plot(t,y,'b-',linewidth=2)

plt.plot(z2['time'],z2['x'],'r:',linewidth=3)

plt.plot(z2['time'],z2['y'],'b--',linewidth=3)

plt.ylabel('Position (m)')

plt.legend(['ODEINT x','ODEINT y','APM x','APM y'])

plt.subplot(2,1,2)

plt.plot(t,vx,'r-',linewidth=2)

plt.plot(t,vy,'b-',linewidth=2)

plt.plot(t,v,'k-',linewidth=2)

plt.plot(z2['time'],z2['vx'],'r--',linewidth=3)

plt.plot(z2['time'],z2['vy'],'b--',linewidth=3)

plt.plot(z2['time'],z2['v'],'k--',linewidth=3)

plt.xlabel('Time (sec)')

plt.ylabel('Velocity (m/s)')

plt.legend(['ODEINT V_x','ODEINT V_y','ODEINT V',\

'APM V_x','APM V_y','APM V'])

plt.figure(2)

plt.plot(z2['x'],z2['y'],'r-')

plt.xlabel('Position (x)')

plt.ylabel('Position (y)')

plt.show()

(:sourceend:)

Added lines 89-181:

(:source lang=python:)

### Sequential method with SciPy.integrate.odeint

import numpy as np

from scipy.integrate import odeint

def skydive(z,t):

# constants

g = 9.81 # m/s^2, gravitational constant

m = 80 # kg, mass of skydiver and pack

if t<61:

c = 0.2 # N-s^2/m^2, drag coefficient, chute closed

else:

c = 10.0 # N-s^2/m^2, drag coefficient, chute open

# states (z)

x = z[0] # meters, horizontal position

y = z[1] # meters, vertical position / elevation

vx = z[2] # m/s, skydiver horizontal velocity = airplane velocity

vy = z[3] # m/s, skydiver vertical velocity

# derived values

v = np.sqrt(vx**2+vy**2) # m/s, magnitude of velocity

Fx = -c * vx**2

Fy = -m*g + c*vy**2

# calculate derivatives

dxdt = vx

dydt = vy

dvxdt = Fx / m

dvydt = Fy / m

dzdt = [dxdt,dydt,dvxdt,dvydt]

return dzdt

# initial conditions

z0 = [0,5000,50,0]

# time points

t = np.linspace(0,90,91)

# solve

z1 = odeint(skydive,z0,t)

# parse results

x = z1[:,0]

y = z1[:,1]

vx = z1[:,2]

vy = z1[:,3]

v = np.sqrt(vx**2+vy**2)

### Simultaneous method with APMonitor

# install and load APMonitor

try:

from APMonitor import *

except:

import pip

pip.main(['install','APMonitor'])

from APMonitor import *

# solve model

z2 = apm_solve('skydiver',7)

### Plot results

import matplotlib.pyplot as plt

plt.figure(1)

plt.subplot(2,1,1)

plt.plot(t,x,'r-',linewidth=2)

plt.plot(t,y,'b-',linewidth=2)

plt.plot(z2['time'],z2['x'],'r:',linewidth=3)

plt.plot(z2['time'],z2['y'],'b--',linewidth=3)

plt.ylabel('Position (m)')

plt.legend(['ODEINT x','ODEINT y','APM x','APM y'])

plt.subplot(2,1,2)

plt.plot(t,vx,'r-',linewidth=2)

plt.plot(t,vy,'b-',linewidth=2)

plt.plot(t,v,'k-',linewidth=2)

plt.plot(z2['time'],z2['vx'],'r--',linewidth=3)

plt.plot(z2['time'],z2['vy'],'b--',linewidth=3)

plt.plot(z2['time'],z2['v'],'k--',linewidth=3)

plt.xlabel('Time (sec)')

plt.ylabel('Velocity (m/s)')

plt.legend(['ODEINT V_x','ODEINT V_y','ODEINT V',\

'APM V_x','APM V_y','APM V'])

plt.figure(2)

plt.plot(z2['x'],z2['y'],'r-')

plt.xlabel('Position (x)')

plt.ylabel('Position (y)')

plt.show()

(:sourceend:)

Changed line 78 from:

'''Objective:''' Provide a basic introduction to dynamic system modeling and simulation with the equations of motion. Create a MATLAB (ode23, ode15s, etc) or Python (ODEINT) script to simulate and display the results. Simulate with [[http://apmonitor.com/wiki/index.php/Main/MATLAB|APM MATLAB]] ~~and~~ [[http://apmonitor.com/wiki/index.php/Main/PythonApp|APM Python]] as well. Compare the sequential (ODE integrators in MATLAB / Python) versus the simultaneous method (APMonitor). Observe how the number of time points in APMonitor affects the solution accuracy. ''Estimated Time: 1 hour''

to:

'''Objective:''' Provide a basic introduction to dynamic system modeling and simulation with the equations of motion. Create a MATLAB (ode23, ode15s, etc) or Python (ODEINT) script to simulate and display the results. Simulate with [[http://apmonitor.com/wiki/index.php/Main/MATLAB|APM MATLAB]] or [[http://apmonitor.com/wiki/index.php/Main/PythonApp|APM Python]] as well. Compare the sequential (ODE integrators in MATLAB / Python) versus the simultaneous method (APMonitor). Observe how the number of time points in APMonitor affects the solution accuracy. ''Estimated Time: 1 hour''

Changed line 78 from:

'''Objective:''' Provide a basic introduction to dynamic system modeling and simulation with the equations of motion. Create a MATLAB ~~or Python script to simulate and display the results.~~ ''Estimated Time: 1 hour''

to:

'''Objective:''' Provide a basic introduction to dynamic system modeling and simulation with the equations of motion. Create a MATLAB (ode23, ode15s, etc) or Python (ODEINT) script to simulate and display the results. Simulate with [[http://apmonitor.com/wiki/index.php/Main/MATLAB|APM MATLAB]] and [[http://apmonitor.com/wiki/index.php/Main/PythonApp|APM Python]] as well. Compare the sequential (ODE integrators in MATLAB / Python) versus the simultaneous method (APMonitor). Observe how the number of time points in APMonitor affects the solution accuracy. ''Estimated Time: 1 hour''

Added lines 77-79:

'''Objective:''' Provide a basic introduction to dynamic system modeling and simulation with the equations of motion. Create a MATLAB or Python script to simulate and display the results. ''Estimated Time: 1 hour''

Changed line 84 from:

<iframe width="560" height="315" src="https://www.youtube.com/embed/~~y0ERNz5Kms8~~?rel=0" frameborder="0" allowfullscreen></iframe>

to:

<iframe width="560" height="315" src="https://www.youtube.com/embed/2FeOaGUQwKA?rel=0" frameborder="0" allowfullscreen></iframe>

Changed lines 81-85 from:

to:

Attach:download.png [[Attach:simulate_skydiver.zip|Skydiver Simulation in MATLAB and Python]]

(:html:)

<iframe width="560" height="315" src="https://www.youtube.com/embed/y0ERNz5Kms8?rel=0" frameborder="0" allowfullscreen></iframe>

(:htmlend:)

(:html:)

<iframe width="560" height="315" src="https://www.youtube.com/embed/y0ERNz5Kms8?rel=0" frameborder="0" allowfullscreen></iframe>

(:htmlend:)

Changed lines 77-79 from:

Integrate

to:

Predict the position and velocity of a skydiver in two dimensions (horizontal and vertical) from the time of the initial jump through the first 90 seconds. At 60 seconds after the jump, the skydiver pulls the chute and the drag coefficient increases to slow the decent. The airplane is flying at a constant altitude of 5000 meters and 50 m/s when the skydiver jumps. The drag coefficient is 0.2 N-s'^2^'/m'^2^' while free-falling and 10 N-s'^2^'/m'^2^' with the parachute open. The gravitational constant is 9.8 m/s'^2^' and the mass is 80 kg for the skydiver and chute.

Changed lines 77-79 from:

Develop a mathematical description of the motion of ~~an~~ object ~~in one dimension where there is a frictional resistance that is proportional to the velocity. Solve ~~the ~~equations numerically.~~

Integrate the ~~system from zero initial time to steady state conditions and assume that the object is initially at rest. Assume a force of 5 N,~~ a ~~mass~~ of ~~10 kg~~, ~~and ~~a ~~frictional loss coefficient of 0.2~~ N-s/m.

Integrate

to:

Develop a mathematical description of the motion of a falling object (e.g. skydiver) in one dimension where there is a frictional resistance that is proportional to the square of the velocity. Solve the equations numerically.

Integrate the system from zero initial time to terminal velocity and assume that the object is initially at rest. Assume a gravitational constant of 9.8 m/s'^2^', a mass of 80 kg, and a frictional loss coefficient of 0.2 N-s'^2^'/m'^2^'.

Integrate the system from zero initial time to terminal velocity and assume that the object is initially at rest. Assume a gravitational constant of 9.8 m/s'^2^', a mass of 80 kg, and a frictional loss coefficient of 0.2 N-s'^2^'/m'^2^'.

Changed line 61 from:

to:

Attach:download.png [[Attach:simulate_vehicle.zip|Simulate with Excel, MATLAB, Python, and Simulink]]

Changed line 61 from:

to:

[[Attach:simulate_vehicle.zip|Attach:download.png Simulate with Excel, MATLAB, Python, and Simulink]]

Changed line 61 from:

* [[Attach:simulate_vehicle.zip|Simulate ~~vehicle velocity~~]]

to:

* [[Attach:simulate_vehicle.zip|Simulate with Excel, MATLAB, Python, and Simulink]]

Added lines 6-7:

* [[Attach:Intro_Dynamic_Modeling.pdf|Introduction to Dynamic Modeling (pdf)]]

Added lines 61-63:

(:html:)

<iframe width="560" height="315" src="https://www.youtube.com/embed/y0ERNz5Kms8?rel=0" frameborder="0" allowfullscreen></iframe>

(:htmlend:)

<iframe width="560" height="315" src="https://www.youtube.com/embed/y0ERNz5Kms8?rel=0" frameborder="0" allowfullscreen></iframe>

(:htmlend:)

Deleted lines 12-19:

Excel, MATLAB, Python, and Simulink are used in the following example to solve the differential equation that describes the velocity of a vehicle.

* [[Attach:simulate_vehicle.zip|Simulate vehicle velocity]]

Added lines 56-60:

Excel, MATLAB, Python, and Simulink are used in the following example to solve the differential equation that describes the velocity of a vehicle.

* [[Attach:simulate_vehicle.zip|Simulate vehicle velocity]]

Changed line 15 from:

Excel, MATLAB, ~~and ~~Python are used in the following example to solve the differential equation that describes the velocity of a vehicle.

to:

Excel, MATLAB, Python, and Simulink are used in the following example to solve the differential equation that describes the velocity of a vehicle.

Added lines 6-9:

(:html:)

<iframe width="560" height="315" src="https://www.youtube.com/embed/VRs9gX6r1Ng?rel=0" frameborder="0" allowfullscreen></iframe>

(:htmlend:)

Changed line 9 from:

!!!!Numerical Solution ~~for~~ Differential Equations

to:

!!!!Numerical Solution of Differential Equations

Added lines 9-16:

!!!!Numerical Solution for Differential Equations

Excel, MATLAB, and Python are used in the following example to solve the differential equation that describes the velocity of a vehicle.

* [[Attach:simulate_vehicle.zip|Simulate vehicle velocity]]

Excel, MATLAB, and Python are used in the following example to solve the differential equation that describes the velocity of a vehicle.

* [[Attach:simulate_vehicle.zip|Simulate vehicle velocity]]

Changed line 71 from:

Develop a mathematical description of the motion of an object in one dimension where there is a frictional resistance that is proportional to the velocity. Solve the equations~~ analytically and~~ numerically.

to:

Develop a mathematical description of the motion of an object in one dimension where there is a frictional resistance that is proportional to the velocity. Solve the equations numerically.

Deleted lines 53-54:

(video of numerical solution, APM MATLAB, APM Python, APMonitor - show how error changes with refined discretization)

Changed lines 61-62 from:

0 = f(dx/dt,x,p)

0 < g(dx/dt,x,p)

to:

''0 = f(dx/dt,x,p)''

''0 < g(dx/dt,x,p)''

''0 < g(dx/dt,x,p)''

Changed line 45 from:

''y = a t'^2^'/2 + v'_0_' t + y_0''

to:

''y = a t'^2^'/2 + v'_0_' t + y'_0_'''

Changed lines 44-45 from:

''v = a t + v_0_''

~~ ~~''y = a t'^2^'/2 + v_0_ t + y_0''

to:

''v = a t + v'_0_'''

''y = a t'^2^'/2 + v'_0_' t + y_0''

''y = a t'^2^'/2 + v'_0_' t + y_0''

Changed line 47 from:

Notice that the initial condition for acceleration (''a_0_'') does not appear in the solution but initial conditions for velocity (''v_0_'') and position (''y_0_'') do influence the solution. If the force (''F'') changes at any time in the future, the time horizon is divided into separate segments where the force is constant. The first segment is computed and the final values become the initial values for the next segment.

to:

Notice that the initial condition for acceleration (''a'_0_''') does not appear in the solution but initial conditions for velocity (''v'_0_''') and position (''y'_0_''') do influence the solution. If the force (''F'') changes at any time in the future, the time horizon is divided into separate segments where the force is constant. The first segment is computed and the final values become the initial values for the next segment.

Changed lines 18-23 from:

''~~f(~~dx/dt,~~x,~~p)~~=0~~'', Semi-Explicit ~~ODE ~~Form

''f(dx/dt,x,p)=0'', Open Equation~~ODE ~~Form

In the case of the object motion, the acceleration~~that ~~is ~~applied to the object~~ may be ~~the parameter that can be manipulated by the user or optimizer~~. The relationship between acceleration (''a'') and force (''F'') includes another parameter, the mass of the object (''m''). There are now three equations that describe the motion of the system and relate force (''F'') to acceleration (''a''), velocity (''v''), and position (''y'').

''f(dx/dt,x,p)=0'', Open Equation

In the case of the object motion, the acceleration

to:

''dx/dt = f(x,p)'', Semi-Explicit Form

''f(dx/dt,x,p)=0'', Open Equation Form

In the case of the object motion, the acceleration is not typically manipulated directly but may be adjusted by a force that is acting on that object. The relationship between acceleration (''a'') and force (''F'') includes another parameter, the mass of the object (''m''). There are now three equations that describe the motion of the system and relate force (''F'') to acceleration (''a''), velocity (''v''), and position (''y'').

''f(dx/dt,x,p)=0'', Open Equation Form

In the case of the object motion, the acceleration is not typically manipulated directly but may be adjusted by a force that is acting on that object. The relationship between acceleration (''a'') and force (''F'') includes another parameter, the mass of the object (''m''). There are now three equations that describe the motion of the system and relate force (''F'') to acceleration (''a''), velocity (''v''), and position (''y'').

Added lines 49-50:

(video of analytic solution)

Changed lines 55-56 from:

(video ~~- Analytic & Numerical Solutions~~)

to:

(video of numerical solution, APM MATLAB, APM Python, APMonitor - show how error changes with refined discretization)

Changed lines 67-69 from:

Develop a mathematical description of the motion of an object in one dimension where there is a frictional resistance that is proportional to the velocity. Solve the equations analytically and numerically ~~and determine the numerical error with finer discretization.~~

Integrate the system from zero initial time to steady state and assume that the object is initially at rest.

Integrate the system from zero initial time to steady state and assume that the object is initially at rest

to:

Develop a mathematical description of the motion of an object in one dimension where there is a frictional resistance that is proportional to the velocity. Solve the equations analytically and numerically.

Integrate the system from zero initial time to steady state conditions and assume that the object is initially at rest. Assume a force of 5 N, a mass of 10 kg, and a frictional loss coefficient of 0.2 N-s/m.

!!!!Solution

(video of solution)

Integrate the system from zero initial time to steady state conditions and assume that the object is initially at rest. Assume a force of 5 N, a mass of 10 kg, and a frictional loss coefficient of 0.2 N-s/m.

!!!!Solution

(video of solution)

Changed lines 5-6 from:

The focus of this course is on modeling, simulation, estimation, and optimization of dynamic systems. This section of the course starts with dynamic modeling or methods to mathematically describe time-evolving systems, particularly for the purpose of dynamic optimization in engineering disciplines.

to:

The focus of this course is on modeling, simulation, estimation, and optimization of dynamic systems. This section of the course starts with dynamic modeling or methods to mathematically describe time-evolving systems, particularly for the purpose of dynamic optimization in engineering disciplines. The examples are particularly focused on engineering applications although much of the theory and applications can also be applied to other fields as well.

The discussion starts with modeling because a reasonably accurate model of the system must first be created to approximate input to output relationships between values that can be adjusted (inputs) and those values that are used to judge the desirability of solution (outputs).

The discussion starts with modeling because a reasonably accurate model of the system must first be created to approximate input to output relationships between values that can be adjusted (inputs) and those values that are used to judge the desirability of solution (outputs).

Changed lines 11-12 from:

Dynamic systems can often be described ~~as~~ differential equations. ~~One example is the~~ equations of motion for an object in a friction-less environment in one dimension. In this case velocity (''v'') is the equal to the time-derivative of position (''y'') and acceleration (''a'') is the time-derivative of velocity.

to:

Dynamic systems can often be described with differential equations. The equations can either be derived empirically from data or from fundamental relationships. One example of fundamental relationships are equations of motion for an object in a friction-less environment in one dimension. In this case velocity (''v'') is the equal to the time-derivative of position (''y'') and acceleration (''a'') is the time-derivative of velocity.

Changed lines 39-42 from:

!!!!Analytic ~~Solutions~~

An exact solution ''x(t)'' may exist from an analytic approach to solving the differential equations. In the in example case, the solution is found by integrating both sides of the differential equations. With the object initially at rest, this translates to zero initial conditions. An initial condition must be specified for each differential variable (those variables that appear in differential terms). In this case, the acceleration ''a'' is determined by the first equation but does not appear in differential form such as ''da/dt''. Values of the parameters must be specified over the selected integration time horizon. The time horizon spans from the initial time where the initial conditions are specified to a future time ''t'_f_'''. Suppose there is a constant force (''F=5'') applied to the body starting at ''t'_0_'=0''. The analytic solution is then

An exact solution ''x(t)'' may exist from an analytic approach to solving the differential equations. In the in example case, the solution is found by integrating both sides of the differential equations. With the object initially at rest, this translates to zero initial conditions. An initial condition must be specified for each differential variable (those variables that appear in differential terms). In this case, the acceleration ''a'' is determined by the first equation but does not appear in differential form such as ''da/dt''. Values of the parameters must be specified over the selected integration time horizon. The time horizon spans from the initial time where the initial conditions are specified to a future time ''t'_f_'''. Suppose there is a constant force (''F=5'') applied to the body starting at ''t'_0_'=0''. The analytic solution is then

to:

!!!!Analytic Solution

An exact solution ''x(t)'' may exist from an analytic approach to solving the differential equations. In the in example case, the solution is found by integrating both sides of the differential equations. With the object initially at rest, this translates to zero initial conditions. An initial condition must be specified for each differential variable (those variables that appear in differential terms). In this case, the acceleration ''a'' is determined by the first equation but does not appear in differential form such as ''da/dt''. Values of the parameters must be specified over the selected integration time horizon. The time horizon spans from the initial time where the initial conditions are specified to a future time ''t'_f_'''. Suppose there is a constant force (''F=5'') applied to the body starting at ''t'_0_'=0''. The analytic solution is then solved as a function of time.

An exact solution ''x(t)'' may exist from an analytic approach to solving the differential equations. In the in example case, the solution is found by integrating both sides of the differential equations. With the object initially at rest, this translates to zero initial conditions. An initial condition must be specified for each differential variable (those variables that appear in differential terms). In this case, the acceleration ''a'' is determined by the first equation but does not appear in differential form such as ''da/dt''. Values of the parameters must be specified over the selected integration time horizon. The time horizon spans from the initial time where the initial conditions are specified to a future time ''t'_f_'''. Suppose there is a constant force (''F=5'') applied to the body starting at ''t'_0_'=0''. The analytic solution is then solved as a function of time.

Changed lines 44-54 from:

''v = a t ~~= F t/m~~''

''y =~~v(~~t~~)~~'^2^'/2~~''~~

!!!!Numerical Solutions

Another method to solve dynamic system is with a numerical approach that approximates the exact solution''~~x(t)~~'' ~~at discrete time intervals, much like measuring a physical system at particular time points. The numerical approach is able to solve much larger and more complex systems of equations, especially when an analytic solution does not exist. ~~

!!!!Mathematical Description

''y =

!!!!Numerical Solutions

Another method to solve dynamic system is with a numerical approach that approximates the exact solution

!!!!Mathematical

to:

''v = a t + v_0_''

''y = a t'^2^'/2 + v_0_ t + y_0''

Notice that the initial condition for acceleration (''a_0_'') does not appear in the solution but initial conditions for velocity (''v_0_'') and position (''y_0_'') do influence the solution. If the force (''F'') changes at any time in the future, the time horizon is divided into separate segments where the force is constant. The first segment is computed and the final values become the initial values for the next segment.

!!!!Numerical Solution

Another method to solve dynamic system is with a numerical approach that approximates the exact solution ''x(t)'' at discrete time intervals, much like measuring a physical system at particular time points. The numerical approach is able to solve much larger and more complex systems of equations, especially when the exact analytic solution does not exist. One drawback to numerical solutions is that the accuracy of the solution is not exact and depends on factors such as the discretization methods employed and the error tolerance of the solver before convergence is confirmed.

(video - Analytic & Numerical Solutions)

!!!!Mathematical Description of Dynamic Systems

''y = a t'^2^'/2 + v_0_ t + y_0''

Notice that the initial condition for acceleration (''a_0_'') does not appear in the solution but initial conditions for velocity (''v_0_'') and position (''y_0_'') do influence the solution. If the force (''F'') changes at any time in the future, the time horizon is divided into separate segments where the force is constant. The first segment is computed and the final values become the initial values for the next segment.

!!!!Numerical Solution

Another method to solve dynamic system is with a numerical approach that approximates the exact solution ''x(t)'' at discrete time intervals, much like measuring a physical system at particular time points. The numerical approach is able to solve much larger and more complex systems of equations, especially when the exact analytic solution does not exist. One drawback to numerical solutions is that the accuracy of the solution is not exact and depends on factors such as the discretization methods employed and the error tolerance of the solver before convergence is confirmed.

(video - Analytic & Numerical Solutions)

!!!!Mathematical Description of Dynamic Systems

Changed lines 59-75 from:

0 = f(dx/dt,x)

0 < g(dx/dt,x)

In this case, the differential~~(~~''f'~~_1_~~'~~'')~~ and algebraic ~~(''f'_2_''') are condensed into ''f(dx/dt,x)'' and any algebraic expressions simply omit the derivative terms. Expressions may either be equality ~~(''~~f~~'') ~~or inequality (''g'') constraints.~~

A standard form for describing a system is as a collection of differential and algebraic expressions.

Dynamic models can be converted to steady state models by setting any

The focus of this course is on dynamic systems or those that change with time. It is particularly focused on engineering applications although much of the theory and applications could also be applied to other fields as well.

''''Dynamic systems

The speed of computers doubles approximately every 18 months. From 2000-2015 the speed of computers increased by about 1000 times.

Dynamic optimization is simulation and

0 < g(dx/dt,x)

In this case, the differential

A standard form for describing a system is as a collection of differential and algebraic expressions.

Dynamic models can be converted to steady state models by setting any

The focus of this course is on dynamic systems or those that change with time. It is particularly focused on engineering applications although much of the theory and applications could also be applied to other fields as well

''''Dynamic systems

The speed of computers doubles approximately every 18 months. From 2000-2015 the speed of computers increased by about 1000 times.

Dynamic optimization is simulation and

to:

0 = f(dx/dt,x,p)

0 < g(dx/dt,x,p)

In this case, the differential and algebraic equations are condensed into ''f(dx/dt,x)'' and any algebraic expressions simply omit the derivative terms. Expressions may either be equality (''f'') or inequality (''g'') constraints.

!!!!Exercise

Develop a mathematical description of the motion of an object in one dimension where there is a frictional resistance that is proportional to the velocity. Solve the equations analytically and numerically and determine the numerical error with finer discretization.

Integrate the system from zero initial time to steady state and assume that the object is initially at rest.

0 < g(dx/dt,x,p)

In this case, the differential and algebraic equations are condensed into ''f(dx/dt,x)'' and any algebraic expressions simply omit the derivative terms. Expressions may either be equality (''f'') or inequality (''g'') constraints.

!!!!Exercise

Develop a mathematical description of the motion of an object in one dimension where there is a frictional resistance that is proportional to the velocity. Solve the equations analytically and numerically and determine the numerical error with finer discretization.

Integrate the system from zero initial time to steady state and assume that the object is initially at rest.

Changed lines 39-43 from:

An exact solution ''x(t)'' may exist from an analytic approach to solving the differential equations. In the in example case, the solution is found by integrating both sides of the differential equations. With the object initially at rest, this translates to zero initial conditions. An initial condition must be specified for each differential variable (those variables that appear in differential terms). In this case, the acceleration ''a'' is determined by the first equation but does not appear in differential form such as ''da/dt''. Values of the parameters must be specified over the selected integration time horizon. The time horizon spans from the initial time where the initial conditions are specified to a future time ''t'_f_'''.

''a(t) = F/m''

''v(t) = a(~~t)~~'~~^2^~~'~~/2~~''

~~ ~~''~~y(~~t~~) ~~= v(t)'^2^'/2''

''a(t) = F/m''

''v(t) = a

to:

An exact solution ''x(t)'' may exist from an analytic approach to solving the differential equations. In the in example case, the solution is found by integrating both sides of the differential equations. With the object initially at rest, this translates to zero initial conditions. An initial condition must be specified for each differential variable (those variables that appear in differential terms). In this case, the acceleration ''a'' is determined by the first equation but does not appear in differential form such as ''da/dt''. Values of the parameters must be specified over the selected integration time horizon. The time horizon spans from the initial time where the initial conditions are specified to a future time ''t'_f_'''. Suppose there is a constant force (''F=5'') applied to the body starting at ''t'_0_'=0''. The analytic solution is then

''a = F/m''

''v = a t = F t/m''

''y = v(t)'^2^'/2''

''a = F/m''

''v = a t = F t/m''

''y = v(t)'^2^'/2''

Changed lines 9-10 from:

Dynamic systems can often be ~~posed~~ as differential equations. One example is the equations of motion for an object in a friction-less environment in one dimension. In this case velocity (''v'') is the equal to the time-derivative of position (''y'') and acceleration (''a'') is the time-derivative of velocity.

to:

Dynamic systems can often be described as differential equations. One example is the equations of motion for an object in a friction-less environment in one dimension. In this case velocity (''v'') is the equal to the time-derivative of position (''y'') and acceleration (''a'') is the time-derivative of velocity.

Changed lines 14-21 from:

The above differential equations are expressed in semi-explicit form where the derivative terms are isolated on the left side of the equation and all other variables are on the right side of the expression. A more general notation for the semi-explicit form is ''dx/dt = f(x,p)'' where ''f(x,p)'' is any combination of variables (''x'') or parameters (''p''). Variables (''x'') are those values that are determined by the solution of the equations while parameters (''p'') are those quantities that are either specified by a user or determined by an optimizer.

In the case of the object motion, the acceleration that is applied to the object may be the parameter that can be manipulated bythe ~~user or optimizer~~. ~~arbitrarilythat is object ~~

A more general form for differential equations is the ~~open-equation format as (''f(dx/dt,x,p)=0'')~~

and algebraic (''f'-2-'(x)=0'') equations with variables ''x''. These collections of equations are referred to as Differential Algebraic Equations (DAEs). ~~When there are no algebraic expressions, the system of equations are simplified to Ordinary Differential Equations~~ (~~ODEs~~). ~~If the differential~~ equations ~~have spatial and temporal derivatives, they become Partial Differential Equations~~ (~~PDEs~~)~~. For the purpose of this modeling discussion, any PDEs are discretized in space to return to ODE or DAE form where only time derivatives~~ are present.

In the case of the object motion, the acceleration that is applied to the object may be the parameter that can be manipulated by

A more general form for differential equations is

and algebraic (''f'-2-'(x)=0'') equations with variables ''x''. These collections of equations are referred to as Differential Algebraic Equations (DAEs)

to:

The above differential equations are expressed in semi-explicit form where the derivative terms are isolated on the left side of the equation and all other variables are on the right side of the expression. A more general notation for the semi-explicit form is ''dx/dt = f(x,p)'' where ''f(x,p)'' is any combination of variables (''x'') or parameters (''p''). Variables (''x'') are those values that are determined by the solution of the equations while parameters (''p'') are those quantities that are either specified by a user or determined by an optimizer. A more general form for differential equations is the open-equation format as ''f(dx/dt,x,p)=0'' where all terms are brought to one side of the equation.

''f(dx/dt,x,p)=0'', Semi-Explicit ODE Form

''f(dx/dt,x,p)=0'', Open Equation ODE Form

In the case of the object motion, the acceleration that is applied to the object may be the parameter that can be manipulated by the user or optimizer. The relationship between acceleration (''a'') and force (''F'') includes another parameter, the mass of the object (''m''). There are now three equations that describe the motion of the system and relate force (''F'') to acceleration (''a''), velocity (''v''), and position (''y'').

''F = m a''

''dy/dt = v''

''dv/dt = a''

The equation ''F = m a'' is an algebraic equation with no differential terms. While it would be simple to eliminate ''a'' from the equation by substituting for ''F/m'', suppose that it is not possible or convenient to rearrange the equations to eliminate the algebraic expressions. This collection of equations is now referred to as Differential Algebraic Equations (DAEs). When there are no algebraic expressions, the system of equations are simplified to Ordinary Differential Equations (ODEs). If the differential equations have spatial and temporal derivatives, they become Partial Differential Equations (PDEs). For the purpose of this modeling discussion, any PDEs are discretized in space to return to ODE or DAE form where only time derivatives (not spatial derivatives) are present.

''f(dx/dt,x,p)=0'', Semi-Explicit ODE Form

''f(dx/dt,x,p)=0'', Open Equation ODE Form

In the case of the object motion, the acceleration that is applied to the object may be the parameter that can be manipulated by the user or optimizer. The relationship between acceleration (''a'') and force (''F'') includes another parameter, the mass of the object (''m''). There are now three equations that describe the motion of the system and relate force (''F'') to acceleration (''a''), velocity (''v''), and position (''y'').

''F = m a''

''dy/dt = v''

''dv/dt = a''

The equation ''F = m a'' is an algebraic equation with no differential terms. While it would be simple to eliminate ''a'' from the equation by substituting for ''F/m'', suppose that it is not possible or convenient to rearrange the equations to eliminate the algebraic expressions. This collection of equations is now referred to as Differential Algebraic Equations (DAEs). When there are no algebraic expressions, the system of equations are simplified to Ordinary Differential Equations (ODEs). If the differential equations have spatial and temporal derivatives, they become Partial Differential Equations (PDEs). For the purpose of this modeling discussion, any PDEs are discretized in space to return to ODE or DAE form where only time derivatives (not spatial derivatives) are present.

Changed lines 30-31 from:

Dynamic systems are those where values that describe the system are expected to evolve over time and not necessarily remain at a steady state. When a dynamic system is at steady state, all time derivative values are ~~zero ~~(''dx/dt=0'').

to:

Dynamic systems are those where values that describe the system are expected to evolve over time and not necessarily remain at a steady state. When a dynamic system is at steady state, all time derivative values are either set to zero (''dx/dt=0'') or do not otherwise appear in the equations.

Changed lines 35-36 from:

For physical ~~dynamic ~~systems the values of ''x'' may be measured at specific times. For virtual or simulated dynamic systems there can either be analytic solutions or numerical ~~solutions~~. An exact solution ''x(t)'' may exist from an analytic approach to solving the differential equations.

to:

For physical systems the values of ''x'' may be measured at specific times. For virtual or simulated dynamic systems there can either be analytic solutions or a numerical solution.

!!!!Analytic Solutions

An exact solution ''x(t)'' may exist from an analytic approach to solving the differential equations. In the in example case, the solution is found by integrating both sides of the differential equations. With the object initially at rest, this translates to zero initial conditions. An initial condition must be specified for each differential variable (those variables that appear in differential terms). In this case, the acceleration ''a'' is determined by the first equation but does not appear in differential form such as ''da/dt''. Values of the parameters must be specified over the selected integration time horizon. The time horizon spans from the initial time where the initial conditions are specified to a future time ''t'_f_'''.

''a(t) = F/m''

''v(t) = a(t)'^2^'/2''

''y(t) = v(t)'^2^'/2''

!!!!Numerical Solutions

!!!!Analytic Solutions

An exact solution ''x(t)'' may exist from an analytic approach to solving the differential equations. In the in example case, the solution is found by integrating both sides of the differential equations. With the object initially at rest, this translates to zero initial conditions. An initial condition must be specified for each differential variable (those variables that appear in differential terms). In this case, the acceleration ''a'' is determined by the first equation but does not appear in differential form such as ''da/dt''. Values of the parameters must be specified over the selected integration time horizon. The time horizon spans from the initial time where the initial conditions are specified to a future time ''t'_f_'''.

''a(t) = F/m''

''v(t) = a(t)'^2^'/2''

''y(t) = v(t)'^2^'/2''

!!!!Numerical Solutions

Added lines 48-49:

Changed lines 5-6 from:

The focus of this course is on modeling, simulation, estimation, and optimization ~~for~~ dynamic systems. This section of the course starts with dynamic modeling or methods to mathematically describe time-evolving systems, particularly for the purpose of dynamic optimization in engineering disciplines.

to:

The focus of this course is on modeling, simulation, estimation, and optimization of dynamic systems. This section of the course starts with dynamic modeling or methods to mathematically describe time-evolving systems, particularly for the purpose of dynamic optimization in engineering disciplines.

Changed lines 9-20 from:

Dynamic systems can often be posed as ~~collections of differential (''f'~~-~~1-~~'~~(dx/dt,x)=0~~'' and algebraic (''f'-2-'(x)=0'') equations with variables ''x''. These collections of equations are referred to as Differential Algebraic Equations (DAEs). When there are no algebraic expressions, the system of equations are simplified to Ordinary Differential Equations (ODEs). If the differential equations have spatial and temporal derivatives, they become Partial Differential Equations (PDEs). For the purpose of this modeling discussion, any PDEs are discretized in space to return to ODE or DAE form where only time derivatives are present.

to:

Dynamic systems can often be posed as differential equations. One example is the equations of motion for an object in a friction-less environment in one dimension. In this case velocity (''v'') is the equal to the time-derivative of position (''y'') and acceleration (''a'') is the time-derivative of velocity.

''dy/dt = v''

''dv/dt = a''

The above differential equations are expressed in semi-explicit form where the derivative terms are isolated on the left side of the equation and all other variables are on the right side of the expression. A more general notation for the semi-explicit form is ''dx/dt = f(x,p)'' where ''f(x,p)'' is any combination of variables (''x'') or parameters (''p''). Variables (''x'') are those values that are determined by the solution of the equations while parameters (''p'') are those quantities that are either specified by a user or determined by an optimizer.

In the case of the object motion, the acceleration that is applied to the object may be the parameter that can be manipulated by the user or optimizer. arbitrarilythat is object

A more general form for differential equations is the open-equation format as (''f(dx/dt,x,p)=0'')

and algebraic (''f'-2-'(x)=0'') equations with variables ''x''. These collections of equations are referred to as Differential Algebraic Equations (DAEs). When there are no algebraic expressions, the system of equations are simplified to Ordinary Differential Equations (ODEs). If the differential equations have spatial and temporal derivatives, they become Partial Differential Equations (PDEs). For the purpose of this modeling discussion, any PDEs are discretized in space to return to ODE or DAE form where only time derivatives are present.

''dy/dt = v''

''dv/dt = a''

The above differential equations are expressed in semi-explicit form where the derivative terms are isolated on the left side of the equation and all other variables are on the right side of the expression. A more general notation for the semi-explicit form is ''dx/dt = f(x,p)'' where ''f(x,p)'' is any combination of variables (''x'') or parameters (''p''). Variables (''x'') are those values that are determined by the solution of the equations while parameters (''p'') are those quantities that are either specified by a user or determined by an optimizer.

In the case of the object motion, the acceleration that is applied to the object may be the parameter that can be manipulated by the user or optimizer. arbitrarilythat is object

A more general form for differential equations is the open-equation format as (''f(dx/dt,x,p)=0'')

and algebraic (''f'-2-'(x)=0'') equations with variables ''x''. These collections of equations are referred to as Differential Algebraic Equations (DAEs). When there are no algebraic expressions, the system of equations are simplified to Ordinary Differential Equations (ODEs). If the differential equations have spatial and temporal derivatives, they become Partial Differential Equations (PDEs). For the purpose of this modeling discussion, any PDEs are discretized in space to return to ODE or DAE form where only time derivatives are present.

Changed lines 7-10 from:

!!!!~~Differential Algebraic Equations~~

Dynamicsystems can often be posed as collections of differential (''f'-1-'(dx/dt,x)=0'' and algebraic (''f'-2-'(x)=0'') equations~~. These collections are referred to as Differential Algebraic Equations (DAEs). When there are no algebraic expressions, the system of equations are referred to as~~ Ordinary Differential Equations (ODEs). If the differential equations have spatial and temporal derivatives, they become Partial Differential Equations (PDEs). For the purpose of this modeling discussion, any PDEs are discretized in space to return to ODE or DAE form where only time derivatives are present.

Dynamic

to:

!!!!Dynamic Systems

Dynamic systems can often be posed as collections of differential (''f'-1-'(dx/dt,x)=0'' and algebraic (''f'-2-'(x)=0'') equations with variables ''x''. These collections of equations are referred to as Differential Algebraic Equations (DAEs). When there are no algebraic expressions, the system of equations are simplified to Ordinary Differential Equations (ODEs). If the differential equations have spatial and temporal derivatives, they become Partial Differential Equations (PDEs). For the purpose of this modeling discussion, any PDEs are discretized in space to return to ODE or DAE form where only time derivatives are present.

Dynamic systems can often be posed as collections of differential (''f'-1-'(dx/dt,x)=0'' and algebraic (''f'-2-'(x)=0'') equations with variables ''x''. These collections of equations are referred to as Differential Algebraic Equations (DAEs). When there are no algebraic expressions, the system of equations are simplified to Ordinary Differential Equations (ODEs). If the differential equations have spatial and temporal derivatives, they become Partial Differential Equations (PDEs). For the purpose of this modeling discussion, any PDEs are discretized in space to return to ODE or DAE form where only time derivatives are present.

Changed lines 20-25 from:

Another method to solve ~~DAEs~~ is with a numerical approach that approximates the exact solution ''x(t)'' at discrete time intervals, much like measuring a physical system at particular time points. The numerical approach is able to solve much larger and more complex systems of equations, especially when an analytic solution does not exist.

!!!!Mathematical Description~~of DAEs~~

Models are collections of assumptions, mathematical expressions, boundary conditions, initial conditions, and constraints. These mathematical expressions can range from a simple equation of motion for an object in a frictionless environment to complex systems that describe multi-body physics and interactions. A standard model form is shown below:

!!!!Mathematical Description

Models

to:

Another method to solve dynamic system is with a numerical approach that approximates the exact solution ''x(t)'' at discrete time intervals, much like measuring a physical system at particular time points. The numerical approach is able to solve much larger and more complex systems of equations, especially when an analytic solution does not exist.

!!!!Mathematical Description

Models are not just equations but are collections of assumptions, mathematical expressions, boundary conditions, initial conditions, and constraints. These mathematical expressions can range from a simple equation of motion for an object in a frictionless environment to complex systems that describe multi-body physics and interactions. A standard model form is shown below:

!!!!Mathematical Description

Models are not just equations but are collections of assumptions, mathematical expressions, boundary conditions, initial conditions, and constraints. These mathematical expressions can range from a simple equation of motion for an object in a frictionless environment to complex systems that describe multi-body physics and interactions. A standard model form is shown below:

Changed lines 29-34 from:

In this case, the differential (''f'_1_''') and algebraic (''f'_2_''') are condensed into ''f(dx/dt,x)'' and any algebraic expressions simply omit the derivative terms.

Values that describe

A standard form for describing a system is as a collection of differential and algebraic expressions. These may either be equality (''f'') or inequality (''g'') constraints. The variables (''x'')

Values that describe

A standard form for describing a system is as a collection of differential and algebraic expressions. These may either be equality (''f'') or inequality (''g'') constraints. The variables (''x'')

to:

In this case, the differential (''f'_1_''') and algebraic (''f'_2_''') are condensed into ''f(dx/dt,x)'' and any algebraic expressions simply omit the derivative terms. Expressions may either be equality (''f'') or inequality (''g'') constraints.

A standard form for describing a system is as a collection of differential and algebraic expressions.

A standard form for describing a system is as a collection of differential and algebraic expressions.

Changed lines 5-17 from:

The focus of this course is on modeling, simulation, estimation, and optimization for dynamic systems. This section of the course starts with dynamic modeling or methods to mathematically describe time-evolving systems, particularly for the purpose of optimization in engineering disciplines.

!!!!~~Dynamic versus Steady State Numerical Solution~~

Dynamic systems are those where certain values that describe the system are expected to evolve over time and not necessarily remain at a steady state.

Steady state: Values ('''x''') do not change with time

Dynamic: At least one value (~~'''x'''~~) ~~changes with time~~

Values of '''x''' may be measured at specific times or an expression may exist from an analytic solution of the differential equations as '''x(t)'''. Differential equations are those in which a Another method to solve dynamic

!!!!Mathematical Description

!!!!

Dynamic systems are those where certain values that describe the system are expected to evolve over time and not necessarily remain at a steady state.

Steady state: Values ('''x''') do not change with time

Dynamic: At least one value

Values of '''x''' may be measured at specific times or an expression may exist from an analytic solution of the differential equations as '''x(t)'''. Differential equations are those in which a Another method to solve dynamic

!!!!Mathematical

to:

The focus of this course is on modeling, simulation, estimation, and optimization for dynamic systems. This section of the course starts with dynamic modeling or methods to mathematically describe time-evolving systems, particularly for the purpose of dynamic optimization in engineering disciplines.

!!!!Differential Algebraic Equations

Dynamic systems can often be posed as collections of differential (''f'-1-'(dx/dt,x)=0'' and algebraic (''f'-2-'(x)=0'') equations. These collections are referred to as Differential Algebraic Equations (DAEs). When there are no algebraic expressions, the system of equations are referred to as Ordinary Differential Equations (ODEs). If the differential equations have spatial and temporal derivatives, they become Partial Differential Equations (PDEs). For the purpose of this modeling discussion, any PDEs are discretized in space to return to ODE or DAE form where only time derivatives are present.

!!!!Dynamic versus Steady State

Dynamic systems are those where values that describe the system are expected to evolve over time and not necessarily remain at a steady state. When a dynamic system is at steady state, all time derivative values are zero (''dx/dt=0'').

Steady state: Values (''x'') do not change with time

Dynamic: At least one value (''x'') changes with time

For physical dynamic systems the values of ''x'' may be measured at specific times. For virtual or simulated dynamic systems there can either be analytic solutions or numerical solutions. An exact solution ''x(t)'' may exist from an analytic approach to solving the differential equations.

Another method to solve DAEs is with a numerical approach that approximates the exact solution ''x(t)'' at discrete time intervals, much like measuring a physical system at particular time points. The numerical approach is able to solve much larger and more complex systems of equations, especially when an analytic solution does not exist.

!!!!Mathematical Description of DAEs

!!!!Differential Algebraic Equations

Dynamic systems can often be posed as collections of differential (''f'-1-'(dx/dt,x)=0'' and algebraic (''f'-2-'(x)=0'') equations. These collections are referred to as Differential Algebraic Equations (DAEs). When there are no algebraic expressions, the system of equations are referred to as Ordinary Differential Equations (ODEs). If the differential equations have spatial and temporal derivatives, they become Partial Differential Equations (PDEs). For the purpose of this modeling discussion, any PDEs are discretized in space to return to ODE or DAE form where only time derivatives are present.

!!!!Dynamic versus Steady State

Dynamic systems are those where values that describe the system are expected to evolve over time and not necessarily remain at a steady state. When a dynamic system is at steady state, all time derivative values are zero (''dx/dt=0'').

Steady state: Values (''x'') do not change with time

Dynamic: At least one value (''x'') changes with time

For physical dynamic systems the values of ''x'' may be measured at specific times. For virtual or simulated dynamic systems there can either be analytic solutions or numerical solutions. An exact solution ''x(t)'' may exist from an analytic approach to solving the differential equations.

Another method to solve DAEs is with a numerical approach that approximates the exact solution ''x(t)'' at discrete time intervals, much like measuring a physical system at particular time points. The numerical approach is able to solve much larger and more complex systems of equations, especially when an analytic solution does not exist.

!!!!Mathematical Description of DAEs

Changed lines 26-28 from:

to:

0 = f(dx/dt,x)

0 < g(dx/dt,x)

In this case, the differential (''f'_1_''') and algebraic (''f'_2_''') are condensed into ''f(dx/dt,x)'' and any algebraic expressions simply omit the derivative terms.

0 < g(dx/dt,x)

In this case, the differential (''f'_1_''') and algebraic (''f'_2_''') are condensed into ''f(dx/dt,x)'' and any algebraic expressions simply omit the derivative terms.

Changed lines 34-35 from:

A standard form for describing a system is as a collection of differential and algebraic expressions. These may either be equality (''~~'~~f''~~'~~) or inequality (''~~'~~g''~~'~~) constraints. The variables (''~~'~~x~~'~~'')

to:

A standard form for describing a system is as a collection of differential and algebraic expressions. These may either be equality (''f'') or inequality (''g'') constraints. The variables (''x'')

Changed line 40 from:

''''Dynamic~~''''~~ systems

to:

''''Dynamic systems

Added lines 4-35:

The focus of this course is on modeling, simulation, estimation, and optimization for dynamic systems. This section of the course starts with dynamic modeling or methods to mathematically describe time-evolving systems, particularly for the purpose of optimization in engineering disciplines.

!!!!Dynamic versus Steady State Numerical Solution

Dynamic systems are those where certain values that describe the system are expected to evolve over time and not necessarily remain at a steady state.

Steady state: Values ('''x''') do not change with time

Dynamic: At least one value ('''x''') changes with time

Values of '''x''' may be measured at specific times or an expression may exist from an analytic solution of the differential equations as '''x(t)'''. Differential equations are those in which a Another method to solve dynamic

!!!!Mathematical Description

Models are collections of assumptions, mathematical expressions, boundary conditions, initial conditions, and constraints. These mathematical expressions can range from a simple equation of motion for an object in a frictionless environment to complex systems that describe multi-body physics and interactions. A standard model form is shown below:

Values that describe

A standard form for describing a system is as a collection of differential and algebraic expressions. These may either be equality ('''f''') or inequality ('''g''') constraints. The variables ('''x''')

Dynamic models can be converted to steady state models by setting any

The focus of this course is on dynamic systems or those that change with time. It is particularly focused on engineering applications although much of the theory and applications could also be applied to other fields as well.

''''Dynamic'''' systems

The speed of computers doubles approximately every 18 months. From 2000-2015 the speed of computers increased by about 1000 times.

Dynamic optimization is simulation and

Added lines 1-3:

(:title Introduction to Dynamic Modeling:)

(:keywords dynamic modeling, engineering, differential, algebraic, modeling language, tutorial:)

(:description Dynamic modeling with Differential Algebraic Equations (DAEs):)

(:keywords dynamic modeling, engineering, differential, algebraic, modeling language, tutorial:)

(:description Dynamic modeling with Differential Algebraic Equations (DAEs):)