Control Tutorials with MATLAB and Simulink (2023)

The main MATLAB commands used in this tutorial are:grunt,versatile,carrot,know,dekonw,inv. no,npr,poli,tf,zero

Contents

MATLAB is an interactive program for numerical calculations and data visualization; it is widely used by automation engineers for analysis and design. Many different toolkits are available to extend MATLAB's core functionality to different application areas; in these tutorials, we will make extensive use of the Control Systems Toolbox. MATLAB is supported in Unix, Macintosh, and Windows environments; a student version of MATLAB for PCs is available. More information on MATLAB can be found on the websiteMathWorksDom.

The idea behind these tutorials is that you can view them in one window by running MATLAB in another window. You should be able to repeat all the diagrams and calculations from the tutorials by cutting and pasting the tutorial text into the MATLAB command window orm-best.

Vector

Let's start by creating something simple, like a vector. Enter each element of the vector (separated by a space) in parentheses and make it equal to the variable. For example, to create a vectorA, enter the following in the MATLAB command window (you canCopyIPlacementfrom browser to MATLAB to make it easier) and MATLAB should return:

for = [1 2 3 4 5 6 9 8 7]

for = 1 2 3 4 5 6 9 8 7

Suppose you want to create a vector with elements 0 to 20 evenly spaced in steps of two (this method is often used to create a time vector):

t = 0:2:20

t = 0 2 4 6 8 10 12 14 16 18 20

Manipulating vectors is almost as easy as creating them. Let's first assume you want to add 2 to each element of the vectorA. The equation looks like this:

b = za + 2

b = 3 4 5 6 7 8 11 10 9

Now suppose you want to add two vectors. This is easy if both vectors have the same length. Just add two as shown below:

do = a + b

do = 4 6 8 10 12 14 20 18 16

Subtracting vectors of the same length works exactly the same way.

Functions

To make life easier, MATLAB includes many standard functions. Each function is a block of code that performs a specific task. MATLAB includes all standard functions such asshame,to want,journal,to the power of,squarelike many others. Commonly used constants such asPI number, IWzJfor the square root of -1, are also built into MATLAB.

sin(pi/4)

odp = 0,7071

To specify function usage, typehelp [function name]in the MATLAB command window.

With MATLAB, you can even write your own functionsfunctionCommand; follow the link to learn how to write your own functions and check out the list of functions we created for this tutorial.

Plot

Creating graphs in MATLAB is also easy. Suppose you want to plot a sine wave as a function of time. First, create a time vector (the semicolon after each statement tells MATLAB we don't want to see all the values), then calculate the sin value each time. commands aftergruntfunction (title,xlabel,label) adds notes to the action.

t = 0:0.25:7;y = sin(t);graph(t,y)title("Sine wave versus time")xlabel("Time (Seconds)")label('Amplitude')

The graph contains approximately one period of a sine wave. Basic plotting is very easy in MATLAB, agruntthe command has a wide variety of options to add. you can visitto planpage for more information.

Polynomials as vectors

In MATLAB, a polynomial is represented by a vector. To create a polynomial in MATLAB, you simply enter each coefficient of the polynomial into the vector in descending order. For example, suppose you have the following polynomial:

(1)

To transfer this to MATLAB, just input it as a vector like this:

x = [1 3 -15 -2 9]

x = 1 3 -15 -2 9

MATLAB can interpret a vector of length n+1 as a polynomial of order n. So if your polynomial is missing some coefficients, you need to put zeros in the appropriate place of the vector. For example,

(2)

will be displayed in MATLAB as:

jot = [1 0 0 0 1]

jot = 1 0 0 0 1

The value of the polynomial can be found usingversatilefunction. For example, to find the value of the above polynomial ws = 2,

z = goes([1 0 0 0 1],2)

z = 17

You can also extract the roots of polynomials. This is useful if you have a higher order polynomial such as

(3)

Finding the root directory will be as simple as entering the following command:

roots ([1 3 -15 -2 9])

odp = -5,5745 2,5836 -0,7951 0,7860

Suppose you want to multiply two polynomials. The product of two polynomials is determined by the convolution of their coefficients. MATLAB functionknowwill do it for you.

x = [1 2];y = [1 4 8];z = konv(x,y)

z = 1 6 16 16

Dividing two polynomials is just as easy. Thedekonwthe function returns both the remainder and the result. Let's part wayszAlreadyGand we'll see if we succeedX.

[xx, R] = dekonv(z,y)

xx = 1 2R = 0 0 0 0

As you can see, it's just a polynomial/vectorXfrom the past. IfGdidn't come inzevenly, the residue vector would be something other than zero.

Polynomials using the variable s

Another way to represent polynomials is to use a Laplace variableSin MATLAB. This method is the most commonly used in these tutorials. Let's omit the details of the Laplace domain for now and introduce only the polynomials sSchangeable. To define a variable, type in the MATLAB command window:

s = tf('S')

s = s Continuous time transfer function.

Recall the above polynomial:

(4)

To display this in MATLAB, enter the following command in the MATLAB command window:

polinom = s^4 + 3*s^3 - 15*s^2 - 2*s + 9

polynomial = s^4 + 3 s^3 - 15 s^2 - 2 s + 9 Continuous-time transfer function.

instead of usingcarrotfunction we can usezeroa function that finds the root of a polynomial.

null (polynomial)

odp = -5,5745 2,5836 -0,7951 0,7860

As you can see, the result is the same as abovecarrotorder and coefficients of polynomials.

You can also multiply two polynomials bySchangeable. Let's redefine itXIG.

x = s + 2; y = s^2 + 4*s + 8; z = x * y

z = s^3 + 6 s^2 + 16 s + 16 Continuous-time transfer function.

The resulting polynomial has the same coefficients as the resulting vectorknowfunction above.

Matrix

Entering a matrix in MATLAB is the same as entering vectors, except that each row of elements is separated by a semicolon (;) or a return character:

B = [1 2 3 4; 5 6 7 8; 9 10 11 12] B = [ 1 2 3 4 5 6 7 8 9 10 11 12 ]

B = 1 2 3 4 5 6 7 8 9 10 11 12B = 1 2 3 4 5 6 7 8 9 10 11 12

Arrays in MATLAB can be manipulated in many ways. First, you can find the matrix transpose with the apostrophe key:

C = B'

do = 1 5 9 2 6 10 3 7 11 4 8 12

It is worth noting that ifCcomplex, the apostrophe would actually result in a complex conjugate transposition. To get the transpose in this case, use.'(both commands are the same if the matrix is ​​not composite).

Now you can multiply two matricesBICTogether. Remember that order matters when multiplying matrices.

D = B * CD = C * B

D = 30 70 110 70 174 278 110 278 446D = 107 122 137 152 122 140 158 176 137 158 179 200 152 176 200 224

Another possibility of matrix manipulation is that you can multiply the corresponding elements of two matrices with.*operator (for this to work, the matrices must be the same size).

mi = [1 2; 3 4]F = [2 3; 4 5]G = E.* F

E = 1 2 3 4F = 2 3 4 5G = 2 6 12 20

If you have a square matrix, e.gmi, you can also multiply it by itself as many times as you like, raising it to the given power.

E^3

odp = 37 54 81 118

If you want to expand each element of the matrix, use a cube per element.

E.^3

odp = 1 8 27 64

You can also find matrix inversion:

X = inv(E)

X = -2,0000 1,0000 1,5000 -0,5000

or its eigenvalues:

eig (E)

odp = -0,3723 5,3723

There is even a function to find the coefficients of the characteristic polynomial of a matrix. Thepolifunction creates a vector containing the coefficients of the characteristic polynomial.

p = poly(E)

p = 1,0000 -5,0000 -2,0000

Recall that the eigenvalues ​​of a matrix are equal to the roots of its characteristic polynomial:

roots (p)

odp = 5,3723 -0,3723

Print

Printing in MATLAB is quite simple. Just follow the steps below:

prochowiec

ramen

Unix

Using M files in MATLAB

There are slightly different things you need to know for each platform.

prochowiec

ramen

Unix

You can type commands directly into MATLAB or put all the commands you need in onem-bestand just run the file. If you put everythingblisters min the same folder you run MATLAB from, MATLAB will always find them.

Get help with MATLAB

MATLAB has pretty good online help, type:

help command name

for more information on each task. You need to know the name of the command you are looking for; a list of all the resources used in this manual can be found incommand list; the link to this page is in the upper right corner of this page.

There are some notes at the end of this manual.

You can check the value of a specific variable at any time by entering its name.

B

B = 1 2 3 4 5 6 7 8 9 10 11 12

You can also put more than one statement on a line as long as you separate them with a semicolon or comma.

You may have also noticed that until you assign a specific operation or result to a variable, MATLAB stores it in a temporary variable calledanswer.

Released with MATLAB® 9.2