Problem

In this lab we will be experiment with lead compensator design. This is important because lead compensators add poles and zeros to the closed loop transfer function, which in turn alters the shape of the root locus. If we can alter the shape of the root locus, we can ensure that the poles that give us our desired settling time and steady state error lie on the root locus, which leaves us to simply calculate the proportional gain that gives us the desired pole.

Procedure

To begin the design of a lead compensator we need to analyze our DC motor system. We build our traditional DC motor apparatus and measure the system response to a 0.5 unit step. Also as before we can estimate our first order transfer function for the data collected using the system identity toolbox built into MatLab. After we have our transfer function we can plot the root locus fairly easily also in Matlab using the rlocus function. The transient response of the will show lots of steady state error. 𝐾𝑃 = lim 𝐺(𝑠) 𝑠→0 → 𝑒𝑠𝑠 = 𝑅/(1 + 𝐾𝑃)

Now that we have the steady state error we can try to reduce the error by introducing a lead compensator. We pick a new steady state error of 0.25 and find the lead compensator accordingly for when our zeros of our compensator are 0.01, 0.1, and 1. The compensator will change our transfer function so that our new transfer function becomes: 𝐶𝐺(𝑠) = 𝐶𝐿𝑒𝑎𝑑(𝑠) ∗ 𝐺(𝑠)

For each lead compensator case we plot the root locus and the corresponding transient response. We compare each compensator to determine which produces the best response.