To start out, setup the open loop transfer function.

Next, you can choose to set up the MATLAB code in a few different ways. First make sure that both the numerator and denominator are in acceptable forms.

[1 2 3] is the same as saying

Using an example:

We can write the numerator and denominator MATLAB codes as:

>>numerator=[1];

>>denominator=[1 8];

For a more complex problem we can bypass the long and tedious expansion process and use the convolution function in MATLAB.

Here, the denominator is also represented by (s+1)*(s+3+3j)*(s+3-3j). Being three seperate parts, we can use convolution once for two of them, then use convolution again with the remaining part. Just look at the example:

conv( A , conv( B , C ) )  —> denominator = conv( [1 1], conv( [1 3+3*j], [1 3-3*j] ) );

After we define our numerator and denominator in MATLAB, we can use the root locus function then set our axis parameters as follows.

>>rlocus( numerator, denominator )

>>axis([-10 10 -10 10])

Your plot should look similar to the following for this example:

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A Finite State Machine (FSM), or just state machine, is a model of behavior composed of a finite number of states. We use these in computer engineering to model a “machine” with primitive memory. Based on the signals  we recieve, we go to a certain state where information is processed, and then we wait for the signal that indicates we move to the next state.  There are two types of FSMs we will talk about: Moore machines and Mealy machines.

Moore Machines

Moore machines are state machines where the output is determined by only the current state, not the inputs. For example, if we  had designed a system where the state table looks like this:

we can see that it doesn’t even matter what the input and outputs are, the next state only depends on the current state. A typical example of Moore style state machines is a synchronous circuit composed of flip-flops connected to a clock. The state can only change when the clock pulses, and for flip-flops, the output(next state) is a function of the input(current state).

Mealy Machines

By contrast there are FSMs called Mealy machines where the next state is a function of both the current state, and the input.

Consider the truth table we discussed earler. In our Moore state machine design, we didn’t care what the input was. Now, however, we have decided that if the current state is 00 and the input is 0 , the next state is 01 .

We can see above that the next state is a function of the current state, and the input. A real world example of a  Mealy machine is a mathematical cypher machine.

Mealy machines are typically sequential logic, whereas Moore machines are combinatorial logic.

What is the Laplace Transform method?

The Laplace Transform is a method that simplifies integral and differential equations into algebraic equations. This practice is commonly used to solve for a function out of a differential equation, which otherwise may have been unsolvable or very difficult. The following integrals can be used to transform between where denotes Laplace and denotes Inverse Laplace:

Table of Laplace Transforms

Since these integrals can be tedious and certain functions tend to reoccur, a table of Laplace Transforms has been linked:

Laplace Transforms Table

A Laplace Transform example

This table can be a little complex to use at first so an example is provided below to get you started. In this problem we implement the Laplace Transform and Inverse Laplace Transform to solve for .

The first step is to take the Laplace Transform of both sides of the equation. Use element 1 of our table for the right side and element 18 for the left side. Note that the initial conditions are necessary to take the Laplace Transform of the left side.

Inputting our initial conditions:

Assuming you are an engineering student and can do a little alegebra, our next step is to find the terms that have in common and factor it out. Our goal is to find because, after all, :

After solving for and factoring the denominator:

Taking the Inverse Laplace Transform

Now we arrive at the trickier part of this procedure. We must take the Inverse Laplace of to find . If our function does not match anything in the table, such as this case, factoring is a good place to start. This problem can easily be factored using the function on your TI-89. Just go to and enter your function in parenthesis. Using this function:

Noting and , we can take the Laplace Transform of each term independently and also manipulate the constant terms if necessary or just pull them out. Using the 2nd property in our Laplace Transform table:

To check your work you can plug into the original, differential equation and at we find that .