# BIBO stability and Asymptotic stability

I have difficulty solving these two exercises: Exercise 1 Consider the system described by the transfer function $$P(s)=(s-1)/(s+1)^2$$ Design a controller so that the feedback controlled system is BIBO stable and type 1 I could do it if the only requirement is that the system be type 1. In that case I would simply have to multiply the transfer function P(s) by $$G(s)=K_g /s$$ (with Kg a constant ) so as to add a pole at the origin to the system and make it type 1, but how do I solve the point where BIBO stability is asked?

Exercise 2 Consider the system described by the transfer function $$P(s)= 1/(s-1)^2$$ Design a controller such that the closed-loop system is asymptotically stable and has $$|e_1|<=0.01$$. In this case I would know how to proceed to solve the point $$|e_1|<=0.01$$ but I would not know how to solve as regards the asymptotic stability, could you help me ?

• Please attempt a solution Commented Jul 10 at 16:54
• What is the definition of BIBO stability? Commented Jul 10 at 16:58
• @TimWilliams In order for the system to be BIBO stable all of its poles must have negative real part Commented Jul 10 at 17:18
• OK. And what is the system's TF? Commented Jul 10 at 17:21
• @VoltageSpike i probably would multiply for a PI controller and then study the stability through hurwitz (?) Commented Jul 10 at 17:21

You need to apply the feedback formula. See Closed-loop transfer function | Wikipedia for diagram and derivation:

It appears you have P (plant?) instead of G, and G instead of H, so substitute into

$$H(s) = \frac{Y(s)}{X(s)} = \frac{P(s)}{1 + P(s) G(s)}$$

and find the poles/zeroes of $$\H(s)\$$ to complete the problems.

You should find more context in the current (or proceeding) textbook chapter(s), including definitions of the (feedback) system, symbols in use, etc.

• What is G(s) ? When the request was to obtain a type 1 system my professor would just multiply P(s) by G(s)=K_g/s ( with K_g constant ) but what in this case where i'm told to design a controller so that the system is BIBO stable ? What is G(s) in this case? Commented Jul 10 at 18:53
• Either the translation "feedback controlled system" is incorrect, or your professor is incorrect to simply multiply them. Commented Jul 10 at 19:27
• anyways here is the scheme of what my professor means dropbox.com/scl/fi/08nhb38qq7lxsnr51k3lu/… where P(s) is Plant and G(s) is the function equal to K_g/s (where K_g is a constant) Commented 2 days ago