Given the observable form
Show that the transfer function is using Mason rule.
The transfer function can ofcourse be found using which gives
But we want to use Mason rule here. The ﬁrst step is to write the equations so that the nodes variable are on the left side of the equation. The node variables are the states . From the matrix equations we obtain
Solving for now, and in the process we change to by taking Laplace transforms of all variables, and we add the output equation as well
We now draw the Mason diagram, putting on the left most node and on the right most node (the input and output). Here is the result
The forward paths from to are
Now we ﬁnd all the loops. There are only three loops. A loop is one that starts from a node and returns back to it without visiting a node more than once.
We now need to calculate the associated for each of the above forward loops. is found by removing from the graph and then calculating the main mason of what is left in the graph. When remove no loops remain, hence . When removing all the loops remain, hence , and when removing no loops remain, hence . Therefore
There is no other combinations of loops. The above becomes
Hence