# Determining the range of a controller for a stable system

1. May 4, 2017

### MattH150197

1. The problem statement, all variables and given/known data
The controller K(s) = Kp, determine the range of Kp over which the closed loop system is stable.

2. Relevant equations
I found the transfer function of system = Y(s)/V(s)=Kp/(1.5s^2 + 2.5s - 1 + Kp)

3. The attempt at a solution
So i know (1.5s^2 + 2.5s - 1 + Kp) is the characteristic equation is been a while since ive done these i think i remember that all roots of the polynomial for a stable system must be negative real parts and i know its probably obvious but i just cant remember how to find the range of Kp. Thanks!

2. May 4, 2017

### FactChecker

Solve for the roots of the denominator. It's only second order, so you can use a formula.

3. May 5, 2017

### MattH150197

yeah it's just the Kp being part of the denominator that's throwing me a bit, how do I include that in the quadratic formula

4. May 5, 2017

### Staff: Mentor

Um, by including it in the quadratic formula? Treat Kp as a constant and write out the formula. Then see how the value of Kp affects the roots. What range of values for Kp will satisfy your requirements? Be discriminating

5. May 5, 2017

### LvW

As far as I can see IT IS included in the formula (see your own post).

6. May 5, 2017

### MattH150197

Yeah I got it now, cheers guys.