Open-loop Root Locus Selecting a Gain Plotting the Closed-loop
Response The open-loop transfer function of the plant for the
ball and beam experiment is given below: The design criteria for
this problem are: To see the derivation of the equations for
this problem refer to the ball and beam modeling page. A
schematic of the closed loop system with a controller is given
below: As you can see the system has two poles at the origin
which go off to infinity along the imaginary axes. The design
criteria can also be plotted onto the root locus using the sgrid
command. This command generates a grid of constant damping ratio
and natural frequency. The damping ratio and natural frequency
were found using the following equation, which relates the them
to our percent overshoot (PO) and settling time (Ts)
requirements: Note, that the equation with Ts is found by
assuming the system has settled is when the response remains
within 2% of its final value. From these equations, the damping
ratio and natural frequency were foun