While our system does not have the form of a standard second-order system, we can use the above relationship as a starting point in our design. As we are required to have overshoot less than 10%, we need our damping ratio $\zeta$ to be approximately larger than 0.59 and thus need a phase margin greater than about 59 degrees. Since our current phase margin (with the addition of K) is approximately 10.4 degrees, an additional 50 degrees of phase bump from the lead compensator should be sufficient. Since it is known that the lead compensator will further increase the magnitude of the frequency response, we will need to add more than 50 degrees of phase lead to account for the fact that the gain crossover frequency will increase to a point where the system has more phase lag. We will somewhat arbitrarily add 5 degrees and aim for a total bump in phase of 50+5 = 55 degrees.

We can then use this number to solve the above relationship for $\alpha$ as shown below.