0. Objectives

This lab is divided into two parts. The first part is an introduction to some useful functions in MATLAB and how they will be used in the series of following labs. The second part is an application of how to calculate Laplace and inverse Laplace using MATLAB. Each part has questions at the end that need to be completed.

1. How to Submit Lab Reports

A MATLAB LiveScript is an interactive document that combines MATLAB code with embedded output, formatted text, equations, and images in a single environment called the Live Editor. LiveScripts are stored using the LiveScript file format in a file with a .mlx extension.

Labs consist of a MATLAB LiveScript file. You should read and answer the questions inside the LiveScript document. At the end of the lab:

1. Fill your Name, and ID at the header of the LiveScript.
2. Ensure that the LiveScript document runs without errors.
3. Export the LiveScript to a PDF format (Click on the arrow under the save icon).
4. Submit the LiveScript and the generated PDF in myCourses.
5. If requested, attach any auto-generated MAT-file (explained in section 2.4).

2. Useful Functions

2.1. Documentation in MATLAB

All the functions in MATLAB are accompanied with detailed documentation about their usage. You can access inline documentation by typing help followed by the function name. For example

help abs

ABS Absolute value.

ABS(X) is the absolute value of the elements of X. When X is complex, ABS(X) is the complex modulus (magnitude) of the elements of X.

See also SIGN, ANGLE, UNWRAP, HYPOT. Documentation for abs doc abs Other functions named abs codistributed/abs duration/abs iddata/abs sym/abs dlarray/abs gpuArray/abs

A more complete documentation can be also obtained by using the doc command. For example

doc abs

2.2. Arrays in MATLAB

Matrices and arrays are the fundamental representation of information and data in MATLAB. For a basic overview of matrix and array manipulation, you can refer to MATLAB on-line documentation: Matrices and Arrays, and if needed watch the following video Working with Arrays in MATLAB.

Arrays will be used to model a discrete-time signal. An example of a discrete-time signal is:

dt = 0.01; % timestep of discrete signal in seconds T = 10; % Final value of the time signal in seconds% This creates an array |t = [0, 0.01, 0.02, 0.03, , 9.98, 9.99, 10.0]|.t = 0:dt:T; % time signal

A sine wave signal of amplitude 1 and period 1 rad/second

x₁(t) = sin(2t)

can be written as

x1 = sin(2*pi*t);

Logical operations can be used to construct a desired signal. For example, the following signal

x

Can be written as

x2 = zeros(size(t)); % create a zero signal with the same size as time x2(t < 5) = t(t < 5); % value is equal to time, but only when time < 5 x2(t >= 5) = -t(t >= 5) + 10; % value is equal to -time+10, but only when time >= 5

Other useful functions are: size, length, ones. You can use MATLAB help for more information.

2.3. Plotting signals

In order to present results you can use the plot function. plot(X,Y) creates a 2-D line plot of the data in Y versus the corresponding values in X.

figure; hold on; grid on; plot(t, x1); plot(t, x2); title(sprintf(Student ID %s Example plot, Students.ID)); xlabel(Time (seconds)) ylabel(Signal) legend(x_1(t), x_2(t)) hold off;

2.4. Matrices in MATLAB

Matrices are extensions of arrays. They will be used to represent linear systems in a state space form. For example to represent the matrix A:

8A = 34

159

672

We write

A = [8, 1, 6; 3, 5, 7; 4, 9, 2]

A =

8 1 6

• 5 7
• 9 2

Determinant of A can be obtained by:

det(A)

ans = -360

Eigenvalues of A can be obtained by:

eig(A)

ans = 15.0000

4.8990

-4.8990

Inverse of A can be obtained by:

inv(A)

ans = 0.1472 -0.1444 0.0639

-0.0611 0.0222 0.1056

-0.0194 0.1889 -0.1028

2.5. Saving/loading data

In MATLAB you can save any or all the variables in the current workspace to a MAT-file (.mat). You can then reuse the workspace variables later during the current MATLAB session or during another session by loading the saved MAT-file.

For example, to save a variable, you can use:

example = sprintf(A variable generated by Student ID <%s>, Students.ID); save(Lab01.mat, example)

Now we remove the variable example from the workspace.

clear example

You can recover the variable example to the workspace from the MAT-file, by doing

load(Lab01.mat, example) display(example)

example = A variable generated by Student ID <>

For hardware labs (labs that requires the use of external hardware), it is possible to save result of your manipulation in this manner. The resulting file can be used to re-examine your data, or change your plots.

Question 1 (4 marks)

Answer the following questions in the LiveScript. (Observe that if you finish an expression with a semicolon the results will not show in your report. Use semicolons at the end of your expression with the intention to show or hide your answer)

• Define a time signal from 0 to 10 seconds, with a time step of 0.1 second.

	t = [];
	Define a unit signal equals to 1 from 0 to 10 second. (The unit signal should have the same size as time)y1(t) = 1, t 0
	y1 = [];
	Define a delayed signal equals zero from 0 to 5 sec0,y2(t) ={1,	onds and to -1 from 5 to 10 seconds.t < 5 t 5
	y2 = [];
	Define a ramp signal with slope equals 0.1.y3(t) = 0.1t,	t 0
	y3 = [];

• Define a decaying exponential signal with time constant of 10 second.

t

y₄(t) = e ¹⁰

y4 = [];

• Plot in the same graph the signals ( y₁(t) , y₂(t) , y₃(t) , and y₄(t) ) versus the time signal.

figure; hold on% Fill in your plots commandtitle(sprintf(Student ID %s Question 1 Signals, Students.ID)); xlabel(Time (seconds)) ylabel(Signals) grid on; legend(y_1(t), y_2(t), y_3(t), y_4(t), location, best)

3. Laplace transforms

Recall that the (one-sided) Laplace transform of a function f_(t) is defined as:

Fstdt

In this section, we will learn how to compute the Laplace and inverse Laplace transforms using MATLAB. 3.1. Computing Laplace Transform

The command laplace in MATLAB provides an easy way to calculate the Laplace transform F(s) of a function f(t) . To use laplace, first you need to specify that the variable t and s are symbolic ones. This is done with the command:

syms t s

Then, we define the function f(t) .

f = exp(-t)

f = exp(-t)

Finally, we compute the Laplace transform using the function laplace

F = laplace(f, t, s)

F =

1/(s + 1)

To get more information about the laplace function, type:

help laplace

help for sym/laplace LAPLACE Laplace transform. L = LAPLACE(F) is the Laplace transform of the sym F with default independent variable t. The default return is a function of s. If F = F(s), then LAPLACE returns a function of z: L = L(z).

By definition, L(s) = int(F(t)*exp(-s*t),t,0,inf).

L = LAPLACE(F,z) makes L a function of z instead of the default s:

LAPLACE(F,z) <=> L(z) = int(F(t)*exp(-z*t),t,0,inf).

L = LAPLACE(F,w,u) makes L a function of u instead of the default s (integration with respect to w). LAPLACE(F,w,u) <=> L(u) = int(F(w)*exp(-u*w),w,0,inf).Examples: syms a s t w x F(t) laplace(t^5) returns 120/s^6 laplace(exp(a*s)) returns -1/(a-z) laplace(sin(w*x),t) returns w/(t^2+w^2) laplace(cos(x*w),w,t) returns t/(t^2+x^2) laplace(x^(3/2),t) returns (3*pi^(1/2))/(4*t^(5/2)) laplace(diff(F(t))) returns s*laplace(F(t),t,s) F(0) See also SYM/ILAPLACE, SYM/FOURIER, SYM/HTRANS, SYM/ZTRANS, SUBS.

Often, using laplace function gives an expression that is mathematically correct, but not in its simplest form. In such cases, the function simplify can be used to simplify the expression. For example:

simplify(2/(s + 1)-s/(s + 1)^2)

ans = (s + 2)/(s + 1)^2

To get more information about the simplify function, type:

help sym/simplify

SIMPLIFY Symbolic simplification. SIMPLIFY(S) simplifies each element of the symbolic matrix S.SIMPLIFY(S,N) or, equivalently, SIMPLIFY(S,Steps,N), searches for a simplification in N steps. The default value of N is 1. SIMPLIFY(S,Seconds,T) aborts the search for a simpler version at the next end of a simplification step after T seconds. The results will depend on the speed and system load of your computer and may not be reproducible.

SIMPLIFY(S,IgnoreAnalyticConstraints,VAL) controls the level of mathematical rigor to use on the analytical constraints while simplifying (branch cuts, division by zero, etc). The options for VAL are TRUE or FALSE. Specify TRUE to relax the level of mathematical rigor in the rewriting process. The default is FALSE. SIMPLIFY(S, Criterion, preferReal) discourages simplify from returning results containing complex numbers. SIMPLIFY(S, All, true) returns a column vector of aquivalent results. Typically used in combination with Steps, N. Examples: syms x c alpha beta simplify(sin(x)^2 + cos(x)^2) returns 1. simplify(exp(c*log(sqrt(alpha+beta)))) returns (alpha + beta)^(c/2).simplify(sqrt(x^2)) returns sqrt(x^2), simplify(sqrt(x^2),IgnoreAnalyticConstraints,true) returns x.u = symunit; simplify(u.m+u.mm) returns (1001/1000)*[m]. See also SYMUNIT, SYM/FACTOR, SYM/EXPAND, SYM/COLLECT.

Question 2 (2 marks)

Write the symbolic expression of the following functions (you can use the symbolic variables t and s defined previously):

1. f₁(t) = (t) ( (t) is the dirac function, use help dirac for more information)
2. f₂(t) = t² + 4t + 1
3. f₃(t) = te^t + 0.5e^t + e^3t
4. f₄(t) = sin(2t) + cos²(t)
5. f₅(t) = e^t sin(5t + ₃) + t²e^2t

f1 = [] f2 = [] f3 = [] f4 = [] f5 = []

f1 = [] f2 = [] f3 = [] f4 = [] f5 = []

Question 3 (2 marks)

Use MATLAB to compute the Laplace transform of the functions defined in Question 2. Use simplify to simplify the symbolic expression if needed.

F1 = []

F2 = []

F3 = []

F4 = []

F5 = []

F1 =

[]

F2 =

[]

F3 =

[]

F4 =

[]

F5 =

[]

3.2. Computing inverse Laplace transforms

The function ilaplace computes the inverse Laplace transform. Consider the Laplace transform F computed earlier. We can compute the inverse Laplace transform (note that the t and s are already defined as symbols) Example

ilaplace(F, s, t);

Question 4 (2 marks)

Use MATLAB to calculate the inverse Laplace transform of the following expressions. Use simplify to simplify the symbolic expression if needed.

1. G₁(s) =
2. G₂(s) =
3. G₃(s) =
4. G₄(s) =
5. G₅(s) = 22(s+1)(s +24)

(s +4s+1)(s +9)

G1 = [];

G2 = [];

G3 = [];

G4 = []; G5 = [];

g1 = [] g2 = [] g3 = [] g4 = [] g5 = []

g1 = [] g2 = [] g3 = [] g4 = [] g5 =

[]