ECE 350 – Linear Systems I MATLAB Tutorial #3

Using MATLAB to Solve Differential Equations

This tutorial describes the use of MATLAB to solve differential equations. Two methods

are described. The first uses one of the differential equation solvers that can be called

from the command line. The second uses Simulink to model and solve a differential

equation.

Solving First Order Differential Equations with ode45

The MATLAB commands ode 23 and ode 45 are functions for the numerical solution of

ordinary differential equations. They use the Runge-Kutta method for the solution of

differential equations. This example uses ode45. The function ode45 uses higher order

formulas and provides a more accurate solution than ode 23.

In this example we will solve the first order differential equation:

dydt+ 2y = u(t) − u(t −1)

over the range 0 ≤ t ≤ 5. We will assume that the initial value y(0) = 0. The first step is

to write the equation in the form:

dydt

= −2y + u(t) − u(t −1)

Next we need to create a MATLAB m-file to calculate dy/dt. From the MATLAB Home

menu select: New ! Function. The basic structure of a function file will appear in the

editor window. Modify the file to read:

function dy = f(t,y) dy = -2*y + (t>=0)-(t>=1);

This creates a function that calculates the value of dy (actually dy/dt). The function is

called “f” and the file should be named: f.m when it is saved since all MATLAB

functions should be saved in a file with the same name as the function. Convince

yourself that the right hand side of the equation for dy corresponds to the expression

above. Make sure that you save this function in your working directory or one that is in

your path.

Once the function for dy/dt has been created we need to specify the end points of the time

range (0 ≤ t ≤ 5) and the initial value of y (zero). This is done in the MATLAB command

window with the following commands: >> t= [0 5];

>> inity=0;

The ode45 command can now be called to compute and return the solution for y along

with the corresponding values of t using:

>> [t,y]=ode45(@f, t, inity);

Note that the @f refers to the name of the function containing the function. The solution

can be viewing with:

>> plot(t,y)

as shown in figure 1.

Figure 1. Solution of First Order Differential Equation

Solving First Order Differential Equations with ode45 The ode45 command solves first order differential equations. In order to use this

command to solve a higher order differential equation we must convert the higher order

equation to a system of first order differential equations. For example, in order to solve

the second order equation:

d 2ydt 2

+ 3 dydt+ 2y = cos2t with y(0) = 0 and

dydt t=0

= 1

for 0 ≤ t ≤ 10 we must define a variable x such that:

x =

dydt

Thus we can rewrite the differential equation as:

d 2ydt 2

= −3 dydt− 2y + cos2t

or in terms of x:

dxdt

= −3x − 2y + cos2t

We will use the ode45 command to solve the system of first order differential equations:

dxdt

= −3x − 2y + cos2t

dydt

= x

We place these two equations into a column vector, z, where:

z = xy

!

"##

$

%&&

and dzdt=

dxdtdydt

!

"

####

$

%

&&&&

A function is created in a MATLAB m-file to define dz/dt named f2 which contains:

function dz=f2(t,z) dz=[-3*z(1)-2*z(2)+cos(2*t); z(1)];

Note that the first row of dz contains the equation for dx/dt and the second an equation

for dy/dt. The values of x and y are referred to as z(1) and z(2) since they are the first

and second elements of the vector z. The range of t values and the initial conditions are

input at the command line with: >> t=[0 10];

>> initz=[1; 0];

and the equations are solved and plotted using:

>> [t,z]=ode45(@f2, t, initz);

>> plot(t, z(:,2))

Figure 2. Solution of Second Order Differential Equation

Modeling Linear Systems Using Simulink

Simulink is a companion program to MATLAB and is included with the student version.

It is an interactive system for simulating linear and nonlinear dynamic systems. It is a

graphical mouse-driven program that allows you to model a system by drawing a block

diagram on the screen and manipulating it dynamically. It can work with linear,

nonlinear, continuous time, discrete time, multivariable, and multirate systems. In this

section the basic operation of Simulink will be described. The next section will

demonstrate the use of Simulink to solve differential equations.

1. Open MATLAB and in the command window, type: simulink at the prompt.

2. After a few seconds Simulink will open and the Simulink Library Browser will open

as shown in figure 3. It is important to note that the list of libraries may be different

on your computer. The libraries are a function of the toolboxes that you have

installed.

Figure 3. Simulink Library Browser

3. Click on the Create a New Model icon in the Library Browser window. An additional

window will open. This is where you will build your Simulink models.

4. Click on the arrow next to “Simulink” in the Library Browser. A list of sub-libraries

will appear including Continuous, Discrete, etc. These sub-libraries contain the most

common Simulink blocks.

5. Click once on the “Sources” sub-library. You should see a listing of blocks as shown

in the right column of figure 4.

Figure 4. Source Blocks in the Simulink Library

6. Scroll down this list until you see the Sine Wave icon. Click once on the icon to select

it and drag this icon into the blank model window.

7. Click once on the “Sinks” sub-library in the left part of the Library Browser. Click

and drag the “Scope” icon to the model window to the right of the Sine Wave block.

The model window should now appear as shown in figure 5. Make sure you have

used “Scope”, not “Floating Scope”.

Figure 5. Model Window with Sine Wave and Scope Blocks

8. Next we want to connect the Sine Wave to the Scope block. Move the mouse over

the output terminal of the Sine Wave block until it becomes a crosshair. Click and

drag the wire to the input terminal of the Scope block. The cursor will become a

double cursor when it is in the correct position. Release the mouse button and the

wire will snap into place. Your completed system should now appear as shown in

figure 6.

Figure 6. Completed System for Viewing Sine Wave

9. In the model window click on the green arrow “Run” icon. You will hear a beep

when the simulation is complete. Double click on the Scope icon and it will open and

display the output of the sine wave block. Try clicking on the various icons in the

Scope window. You should be able to cause the axes to readjust to display the

waveform as shown in figure 7.

Figure 7. Scope Display

10. Note that the period of the sine wave is just over six. What frequency does this

correspond to? Return to the model window and double click on the Sine Wave

block. The Sine Wave parameters window will open as shown in figure 8.

Figure 8. Sine Wave Block Parameters

11. The frequency was set to 1 rad/sec. Any of the parameters shown can be altered to

change the sine wave. Change the entry in the frequency parameter to: 2*pi*10.

This is a 10 Hz sine wave. Click OK and re-simulate. Use the icons to expand the

scope display. What do you observe? The sine wave should NOT be displayed

properly. This is because at this higher frequency the sampling rate has not been set

high enough.

12. Return to the model window and select the Model Configuration Parameters icon.

The Stop Time for the simulation defaults to 10. Change it to 1, click OK, and re-run

the simulation. You should now be able to properly view the sine wave on the scope.

Verify that the period of each cycle is one 0.1 second as you expect for a 10 Hz wave.

13. Another method for modifying the display is to specify the step size. Return to the

model window and select the Configuration Parameters window again. Note that the

Solver Type defaults to “variable-step”. Change the Solver Type to “fixed step” and

then enter the value 0.01 into the Fixed Step Size entry. Click OK and run the

simulation once again. Note that the display is now smooth.

14. Drag the Pulse Generator from the Source sub-library into the model window.

Double click the Pulse Generator block and modify the parameters as shown in figure

9 below. Click OK.

Figure 9. Pulse Generator Parameters

15. In the scope window, click on the Parameters icon (second from the left). Change the

number of axes to 2 and click OK. Return to the model window and note that the

scope now has two inputs. Connect a wire from the Pulse Generator to the second

input. Your model should now appear as shown in figure 10.

Figure 10. Model Window with Two Sources

16. Re-run the simulation and observe the two waveforms as shown in figure 11 below.

Figure 11. Two Input Scope Window

17. Open the Continuous sub-library. Drag the Integrator block into the model window

and reconfigure the diagram as shown in figure 12.

Figure 12. Simulink Diagram

18. Re-simulate the system and observe the output on the scope. Note that the second

trace is now the integral of the pulse train input.

Modeling a Differential Equation in Simulink Now we will use Simulink to model the differential equations solved earlier. Beginning

with the first order differential equation:

dydt+ 2y = u(t) − u(t −1)

First we need to simulate the single one-second pulse. Since the simulation will run for

0≤t≤5, we can use the pulse generator to generate a five second pulse with a 1 second

pulse width. Since it will not repeat within the five seconds, this simulates a single pulse.

Remove the integrator from the model shown in figure 12 and connect the pulse

generator directly to the scope input. Change the pulse generator to have the settings as

shown in figure 13. The 20% pulse width corresponds to one second as needed.

Change the simulation time in the configuration parameters to five seconds and simulate

the system. Specify fixed-step samples of 0.01 seconds. Verify that the pulse generator

is outputting the desired pulse for the differential equation above.

Figure 13. Pulse Generator Settings for 1 Second Pulse

The differential equation above can be written as:

dydt

= −2y + u(t) − u(t −1) = −2y + p(t)

where p(t) is the one second pulse. The right hand side of this equation can be modeled

in Simulink using the model shown in figure 14. The subtraction block and the gain

block are found in the Math Operations sub-library.

Figure 14. Simulink Model

The labels on the wires are inserted by double clicking on the wires and typing in the text.

By double clicking on the gain block you can set its constant to two. If the input to the

gain block is y, then the output of the subtractor is dy/dt. By passing this output through

an integrator, the input y is found. The model in figure 15 should now simulate the

original differential equation.

Figure 15. Simulink Model of First Order Differential Equation

Run the simulation for 5 seconds and confirm that the scope displays the same output as

we obtained in the first section of this tutorial.

We can also simulate the second order differential equation:

d 2ydt 2

+ 3 dydt+ 2y = cos2t

in Simulink. This equation can be re-written as:

d 2ydt 2

= −3 dydt− 2y + cos2t

and modeled in Simulink as shown in figure 16. In order to get the three input subtractor,

use the two input subtractor selected above. Double click on the block and change the

“List of Signs” to: + - - . This will result in a block that adds the top input and subtracts

the second and third inputs.

Figure 16. Simulink Model

In order to get y and dy/dt we need to integrate the output of this summer twice. If we

add two integrators to the output of figure 16 we can feed the values of y and dy/dt back

to the inputs as shown in figure 17.

Figure 17. Simulink Model of 2nd Order Differential Equation

Simulate the circuit for 10 seconds. The output shown in figure 18 is obtained on the

scope.

Figure 18. MATLAB Scope Output

If we want to compare this output to that obtained in the earlier solution we can save the

output to the workspace. The simout block found in the Sinks sub-library is added as

shown in figure 19. Double click the simout block and change the format of the output to

an Array.

Figure 19. Simulink Model of 2nd Order Circuit

Simulate the model again. Note that an array named simout appears in the workspace.

Plot this output along with the output obtained for the command line solution of the

second order differential equation. The following commands are used:

>> z = [1; 0]; >> t = [0 10]; >> [t, z]=ode45(@f2, t, z);

>> plot(t,z(:,2)) >> hold on >> plot(0:.01:10, simout, 'r') The two solutions appear in figure 20. Note that the time values for the simout plot are

determined by the fixed spacing of 0.01 specified in the configuration parameters in

Simulink.

Figure 20. Two Solutions of the 2nd Order Differential Equation

The solutions are not the same. There are two reasons. First of all, the sinusoidal input is

a “sine” and not a “cos”. In order to convert a sine to a cos we need to apply a phase shift

of π/2. Double click on the Sine Wave block and enter: pi/2 for the phase. Re-simulate

and plot along with the command line solution as shown in figure 21.

Figure 21. Two Solutions of the 2nd Order Differential Equation

Now the steady state portions of the solutions are the same. However, the initial transient

response is not. This is because the two solutions were obtained with different initial

conditions. In the earlier command line solution the initial conditions: y(0) = 0 and

dydt t=0

= 1 were used. The integrator blocks in Simulink allow for the inclusion of

initial conditions. The second integrator outputs the value of y. Thus, the default initial

condition of zero is correct. However the first integrator outputs dy/dt. Double click on

the first integrator and change the initial condition to one. After simulating and re-

plotting both solutions the curves in figure 22 are obtained. Note that the solutions now

match. This confirms that either method will provide a correct solution.

Figure 22. Two Solutions of the 2nd Order Differential Equation