# Design a controller using root locus technique

Given the system described by the transfer function

$$P(s)=10/(s^2 +1)$$

Design a controller using root locus technique so that:

1. The closed loop system is of first type and $$|e_1|<=0.01$$
2. The closed loop system has all the poles p_i with $$Re[p_i]<=-1$$

Solution of point 1:

The system has to be type 1 so I have to put: $$G(s)=(k_g)/s$$ so I have: $$|e_1|<=0.01<=>|(k_d)/((k_g)∗(k_p))|<=0.01$$

where: $$k_d=1$$ and $$(k_p)=P(0)=10$$ so i have : $$1/(10∗(k_g))<=0.01<=>(k_g)>=10$$

I can solve this type of exercise when the poles of the transfer function are real, but how can I solve it when they're imaginary? Thank you for reading

EDIT

Here is the solution of a similar exercise I solved where the transfer function has only real poles: $$P(s) = 1/(s^2 -1)$$ My question regards a different function, which has complex poles, I hope you can help. This is the solution of the similar exercise: https://www.dropbox.com/t/seKMvXmeeaFZPXEE (it's a OneNote page because the exported PDF cut some parts I don't know why)

• Please describe what you've already done to solve it and where you're stuck. This makes it easier to answer your Question without repeating Parts you already know. Commented Jul 8 at 13:29
• @kruemi i will add the resolution of a similar exercise as a OneNote Page in the question, the exercise i solved refers to a DIFFERENT transfer function with REAL pole, i'll leave to you guys the question with the complex poles one Commented Jul 8 at 15:42
• @RussellH as i already told kruemi yes i will add more context Commented Jul 8 at 15:43