Making Decisions with MATLAB
15 Points Possible
Very often, models that you develop as an engineer will not
always be as trivial as the simple evaluation of a series of
mathematical expressions. In each of the previous exercises from
last week you were asked to develop M-file functions that evaluated
a mathematical expression (or a series of mathematical expressions)
and then asked to plot them using the fplot command. This week we
will introduce the capability of making decisions with MATLAB in
order to build more complex M-file models. Figure 1 shown below is
a flow chart representation of the payment M-file function you
defined last week to calculate the monthly payment on a loan given
the loan amount, the annual interest rate, and the term of the loan
as input data. If you remember that a M-file function is nothing
more than a series of MATLAB commands that are executed in sequence
when the M-file function is used in MATLABs command window, its
easy to visualize the execution of each of these commands as a flow
of execution from one statement to the next until there are no more
statements in the M-file for MATLAB to execute.
FLOW OF
EXECUTION
Figure 1: Flow Chart Representation of M-file FunctionThe way
MATLAB, and programming languages in general, makes decisions is by
altering the flow of execution within a M-file function definition.
If some condition is met, then the flow of execution is altered in
order to include the execution of a set of MATLAB commands that
would have otherwise been ignored (not executed by MATLAB) if the
condition was not met. Simply put, IF some condition is true then
do something. MATLAB provides a special statement that we can
include in a M-file definition to make decisions based on some
condition being true -- an IF statement. Figure 2, shown on the
next page, shows the flow chart representation of MATLABs IF
statement and the syntax necessary to define it inside a MATLAB
M-file function definition.
Figure 2: The MATLAB IF StatementNote that the flow of execution
travels along the TRUE path if the condition is true and each of
the MATLAB commands 1 to N (there can be any number) are executed.
If, however, the condition is false, the flow of execution travels
along the FALSE path which bypasses the MATLAB commands that are
part of the IF statement and they are ignored and not executed. In
the MATLAB syntax, if condition is true then MATLAB executes all
the statements that are between the if and end lines and resumes
execution with the line following the end statement. If condition
is false, the statements between the if and end lines are ignored
and execution immediately proceeds with the statement following the
end statement. The idea behind the MATLAB IF statement is really
pretty simple. Do something (execute a series of MATLAB commands)
if a condition is true, otherwise proceed on. The question that
remains is how does MATLAB decide whether a condition is true or
not? This is accomplished by conditional expressions.
Conditional Expressions
Conditional expressions are simply expressions that compare the
values of variables or expressions using the set of relational
operators identified in the table below. OperatorDescription
=Greater Than or Equal To
==Equal To
~=Not Equal To
Table 1: MATLAB Relational Operators
Notice that each operator shown in the table above represents
the simple process of comparing two values by determining whether
or not the values are less than, less than or equal to, greater
than, greater than or equal to, equal to, or not equal to each
other. The following M-file definition of a piecewise-defined
function (remember what a piecewise-defined function is from
algebra?) will demonstrate the use of the MATLAB IF statement and
the relational operators shown in the table above.
Piecewise-Defined Function
x2 for x < 2f(x) = x +6 for x > 2
In order to develop a M-file function to evaluate the
piecewise-defined function f(x), our M-file function must determine
which expression to use based on the condition that x must be less
than 2 to use x2 and x must be greater than or equal to 2 to use
the expression x +6. The MATLAB IF statement is a perfect fit for
this type of problem. The M-file definition for this
piecewise-defined function is given below (there is a better way to
write this function using a variation of the IF statement but well
worry about improving the function later)..
MATLAB M-file Definition
function y = pwise(x)
if x < 2
y=x^2;
end
if x >= 2
y=-x+6;
end
Figure 3: M-File Definition of Piecewise-Defined Function
Note that the M-file definition shown in Figure 3 is very
straightforward. If x is less than 2 then y = x2. If x is greater
than or equal to 2, then y = x +6.
Now lets change the function definition to take advantage of a
variation of MATLABs IF statement that can simplify the M-file
implementation considerably. Notice that if the value of x is not
less than 2, then the value of the piecewise-defined expression
will always be calculated using the expression y = x +6 regardless
of the value of x. In this situation, we can take advantage of the
variation of the IF statement shown in Figure 4 on the following
page. MATLABs IF/ELSE statement is a perfect fit for this
situation. If x < 2 then y = x2, otherwise y = -x + 6 and no
additional comparison of the value of x is necessary.
Figure 4: The MATLAB IF/ELSE StatementMATLAB Syntax for IF
Statement with True and False
If condition
MATLAB Command 1;
MATLAB Command 2;
MATLAB Command 3;
MATLAB Command 4;
else MATLAB Command A;
MATLAB Command B;
MATLAB Command C;
MATLAB Command D;
end
Note that in Figure 4, the flow of execution travels the TRUE
path if the condition is true, otherwise the flow of execution
travels along the FALSE path as shown. Figure 5, shown below shows
the MATLAB M-file using the IF/ELSE statement to define the same
piecewise-defined function shown in Figure 3.
Note that to use the IF/ELSE statement, every point on the
number line must be included!
Note also that we use semicolons on ALL MATLAB statements except
the IF Condition, else, and end statements.
Piecewise-Defined Function
x2 for x < 2f(x) = x +6 for x > 2
MATLAB M-file Definition
function y = pwise(x)
if x < 2
y = x^2;
else
y = -x+6;
end
Figure 5: M-File Definition Using IF/ELSE StatementLets change
the function definition to use a slightly different set of domains
for each of the functions that define the piecewise-defined
function f(x) as shown below. Now our M-file function must satisfy
two conditions in order to use the appropriate expression when
evaluating the piecewise-defined function f(x). In the first case,
x must be greater than or equal to zero AND less than 2 in order
for the expression to be used. In the second case, -x + 6 will be
used if x is greater than or equal to 2 AND less than 10. In each
case, a combination of conditions must be satisfied in order for
the appropriate expression to be used. Fortunately, MATLAB provides
this capability through the use of the logical operators shown in
Table 2 below.
Piecewise-Defined Function
x2 for 0 < x < 2f(x) = x +6for 2 < x <
10OperatorArithmetic Description
&AND
|OR
Table 2: MATLAB Logical Operators
In order to satisfy two conditions in a MATLAB IF statement,
simply use the AND operator (&) if both conditions must be
satisfied and the OR operator (|) if either or both of the
conditions must be satisfied as shown in the Figure 6 below. Note
that the variable x is repeated for each condition.MATLAB M-file
Definitionfunction y = pwise(x)
if 0
