This is the Root Locus compensator design approach, first attempt.
Gp = 20/(s^2 + 4.5*s + 64)
Gp =
20
----------------
s^2 + 4.5 s + 64
Continuous-time transfer function.
controlSystemDesigner('rlocus', Gp)
The app will show root locus of 
. Fig. 1: Right-click the white area and add a new design requirement.
Fig. 2: Select Percent overshoot and insert the design value.
Fig. 3: An exclusion region is indicated by the yellow shaded area, separated from the white area by the thick black lines.
Fig. 4: Add a second design requirement, select Settling time and insert the design value.
Fig. 5: A second exclusion region overlaps with the first exclusion region. A new thick black line can also be seen.
Fig. 6: Right-click the white area, click on Edit compensator, and then add an Integrator, because the Plant is a Type-0 system.
Fig. 7: Right-click the white area, click on Edit compensator, and then add a Real Pole, placing it relatively far away from the Plant's poles
Fig. 8: Right-click the white area, click on Edit compensator, and then add the Complex Zeros to cancel out the Plant's stable Complex poles. Natural frequency 
, and Damping ratio 
. Note: This doesn't work if the Plant is unstable.
Fig. 9: The "magenta dot markers" can be seen. The Plant's stable poles are cancelled out by the Compensator's zeros. Next, the design task is to drag one of the magenta markers (on the real axis) along the root locus until a response plot stays outside of the associated exclusion regions.
Fig. 10: The "magenta dot markers" are dragged to the breakaway point (My preference).
Fig. 11: This can also be directly adjusted through entering a design value to Compensator gain.
Fig. 12: Check the Step response if the performance is satisfactory.
Fig. 13: If satisfactory, then export the designed Compensator to Workspace. With the design values, they can be entered in the Transfer function block or zpk block in Simulink.
C = zpk(1.8*(s^2 + 4.5*s + 64)/(s*(s + 12)))
C =
1.8 (s^2 + 4.5s + 64)
---------------------
s (s+12)
Continuous-time zero/pole/gain model.
Gcl = tf(feedback(C*Gp, 1))
Gcl =
36 s^2 + 162 s + 2304
---------------------------------------
s^4 + 16.5 s^3 + 154 s^2 + 930 s + 2304
Continuous-time transfer function.
S = stepinfo(Gcl)
S =
RiseTime: 0.5599
TransientTime: 0.9725
SettlingTime: 0.9725
SettlingMin: 0.9055
SettlingMax: 1.0000
Overshoot: 0
Undershoot: 0
Peak: 1.0000
PeakTime: 2.1337
If satisfactory, find the equivalent PID gains via algebraic calculations.
Cpid = pid(kp, ki, kd, Tf)
Gpid =
1 s
Kp + Ki * --- + Kd * --------
s Tf*s+1
with Kp = -0.125, Ki = 9.6, Kd = 0.16, Tf = 0.0833
Continuous-time PIDF controller in parallel form.
zpk(Cpid)
ans =
1.8 (s^2 + 4.5s + 64)
---------------------
s (s+12)
Continuous-time zero/pole/gain model.
