Root Loci of Double Integrator: PI & PD Controllers

[Logarythmic]

Homework Statement

For the double integrator described with transfer function

$$G(s) = \frac{1}{s^2}$$

the initial condition is zero. The double integrator is subjected to a unit‐feedback system where the controller is chosen as

1) a PI-controller with $$C(s) = k_p \left( 1 + \frac{1}{s} \right)$$, or

2) a PD-controller with $$C(s) = k_p \left( 1 + \frac{2}{3} s \right)$$.

Sketch root loci of the closed‐loop systems as k_p varies from 0 to +∞. Give the breakaway and break‐in points, the points where root loci cross the imaginary axis, and the relevant values of k_p at all these points.

Homework Equations

None

The Attempt at a Solution

I really have no idea where to start.

 

[verafloyd]

Hi,

I see no particular challenge in the problem specified. Are you familiar with the root-locus concept? I guess you'd better first develop some primary insight on the subject through available resources. Do you have any access to the book "Modern Control Engineering" (Author: Ogata) or any other introductory control engineering book? It gives a procedure to draw root-locus. All you need to do is to apply the procedure to your system.