Session 1. Getting started with MATLAB/ Simulink

Lecture notes for Advanced modeling and Control

Table 1: Useful functions

General functions:
cd	Change subdirectory
demo	Launch the demo (introduction)
dir (what)	Listof files in currentdirectory (or only M-files)
help, helpwin	Help! Help window
load	Load workspace
lookfor	Keyword search
print	Printgraph; can use pull-down menu
quit	Quit!
save	Save workspace
who, whos	Listof variables in workspace
Calculation functions:
conv	Convolution function to multiply polynomials
size, length	Size of an array, length of a vector
Plotting functions:
axis	Override axis defaultof plot
grid	Add grid to plot
hold	Hold a figure to add more plots (curves)
legend	Add legend to plot
plot	Make plots
text (gtext)	Add text (graphical control) to plot
title	Add title to plot
xlabel, ylabel	Add axis labels to plot
Partial fraction and transfer functions:
poly	Construct a polynomial from its roots
residue	Partial-fraction expansion
roots	Find the roots to a polynomial
tf2zp	Transfer function to zero-pole form conversion
zp2tf	Zero-pole form to transfer function conversion
tf	Create a transfer function object
get	Listthe objectproperties
pole	Find the poles of a transfer function
zpk	Create a transfer function in pole-zero-gain form

Activities

1. Explore MATLAB user interface

2. Define a vector x = [1 2 3 4 5 6 7 8 9 10].

   What are different ways you can define x? What happens when you put ; at the end?

3. Convert vector x into a column vector.

4. Create vector y = [0, 0.1, 0.2, ...., 2.0]

5. Create a 3 x 3 matrix.

6. Print the size of the matrix and lengths of vectors defined so far.

7. Define 3 polynomials

   \[p_1(s) = s^2 - 5s + 4 \tag{1}\]

   \[p_2(s) = s^2 + 4 \tag{2}\]

   \[p_3(s) = s^2 - 5s \tag{3}\]

8. Calculate \(p_1(s)p_2(s)\)

9. Perform some mathematical computations on the vectors, matrices, and polynomials defined so far.

10. Solve Ax = b

    A = [ 4 -2 -10; 2 10 -12; -4 -6 16];

    b = [-10; 32; -16];

11. Check the solution

12. Calculate eigenvalues and eigenvectors.

13. Consider data:

    x = [ 0 1 2 4 6 10];

    y = [ 1 7 23 109 307 1231];

    Fit a third-order polynomial. Plot the results

14. Explore MATLAB plotting capabilities

15. Create a MATLAB script, save, and load it to plot data in item 13.

16. Find roots of polynomial defined by p = [1 5 4]

17. Search for a function to find roots of a nonlinear equation.

18. Find polynomial for the roots (-4, -1)

19. For the following transfer functions find partial fractions.

\[G(s) = \frac{q(s)}{p(s)}=\frac{2}{s^2+5s+4} \tag{4}\]

\[G(s) = =\frac{2}{s (s + 1) (s + 2) (s + 3)} \tag{5}\]

\[G(s) = =\frac{s^3 + 4s + 3}{s^4 - 7s^3 + 11s^2 + 7s -12} \tag{6}\]

20. Have fun with zp2tf, tf2zp, and tf commands

21. Response of first order system: Compute and plot step response of following first order systems

    \[y(s) = \frac{1}{5s + 1} \tag{7}\] \[y(s) = \frac{5e^{-10s}}{2.5s + 1} \tag{8}\]

22. Response of second order system: Compute and plot step response of following second order system. Show effect of \(\xi\) on response.

    \[ G_p(s) = \frac{Y(s)}{U(s)} = \frac{K_p e^{-\theta s}}{\tau^2 s^2 + 2 \xi \tau s + 1} \tag{9}\]

    \(K_p = 1\); \(\tau = 1\); \(\theta = 10\)

23. Solve differential equations using Simulink

1. An object falling under gravity

   \[\frac{d^2 y}{dt^2}=-g \tag{10}\]

   Compare the result with analytical solution \(y = -gt^2/2\)

2. Systems of ODEs

1. \[\frac{d^2 y}{dt^2} + 2\frac{dy}{dt} + 5y = 1 \tag{11}\]

   \[\dot{y}(0) = y(0) = 0 \tag{12}\]

2. \[ \begin{bmatrix} \dot{x} \\ \dot{y} \end{bmatrix} = \begin{bmatrix} 1 \\ 0 \end{bmatrix} + \begin{bmatrix} -2 & -5 \\ 1 & 0 \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} \tag{13}\]