# Problem

Previously we have only experimented with implementing PID controllers into our systems, but sometimes we want to adjust the amount of correction in of PID controllers. A feedback system with proportional gain can easy adjustment to PID controllers without having to adjust the PID itself. Adjusting the proportional gain allows us to change the percent overshoot, and settling time of a systemâ€™s response.

In this lab we will analyze the technique designers use to find to system
responses. The lab confronts us using Routh-Hurwitz table to find the poles of
a system. From the poles we apply our equations to find percent over shoot
`(%OS)`

, and settling time `(Ts)`

. Also we incorporate Microelectronic circuit
analysis to construct our PID controller using discrete analog components.

# Procedure

The first step of the experiment required us to load raw data from the blackboard into the workspace. The data was very noisy, so we filtered it using a tenth order median filter. This filtered data was then passed to the System Identification Toolbox, where we estimated a second order transfer function. We then applied a proportionality gain controller to the system, and solved for the value of k that would result in a 10% overshoot. Once we found the new k, found the step response of the new transfer function (the transfer function with the controller applied). Once we had the step response, we validated that we did in fact get a 10% overshoot. We also plotted the root locus diagram and validated found the k value that would make the system critically damped. Once this was completed, we repeated the process for a third order transfer function of the same filtered data. We redid all calculations, and analyzed an overshoot of 150%, and what happens when the proportionality constant grows too large.