# Doubts about stability control theory

I have some questions about theory of stability:

1. Can a system be stable and have poles in the right semiplane of s plane?
2. If a system has poles in the right semiplane, what should happen in GH plane to be stable?
3. Stability criteria can be applied to systems with H(s)<>1 (<> = distinct)? or its necessary do some transformation in this case?

If someone can help me with this I'll be grateful!

• 1- NO, right half plane means either a constant or a sinusoid multiplied by an exponent raised to a positive power which means a time function that diverges. 2- Figure out a controller with the proper poles and zeros that will lead to stability for some values of gain [you probably can figure this out from the root locus], 3-cant understand this question – Elbehery Mar 23 '17 at 15:33
• As my professor once described right half poles: "The machine will be unstable and will try to suck all of the energy in the known universe in order to try to become stable." – KingDuken Mar 23 '17 at 15:48
• Also, see my question about poles and zeros here – KingDuken Mar 23 '17 at 15:51
• An example, this is an unstable system [i.stack.imgur.com/Omd12.png] After introducing a proper controller, this is the new system root locus and step response after setting the proper gain [i.stack.imgur.com/ocfZv.png] – Elbehery Mar 23 '17 at 15:53
• @Elbehery in the third question i mean if the way to apply the stability criteria change if H(s) is not unitary (distinct to 1). assuming that the characteristic equation is 1+G(s)H(s)=0 – cranck Mar 23 '17 at 16:20

1. No. The poles in the right semiplane are dual with unbounded exponential functions in the time domain: $${{1}\over{s - \alpha}} \leftrightarrow \exp(\alpha t)$$
2. Adding a control system $G(s)$ to the model $H(s)$ will alter the location of poles in a nonlinear way. Here you create a root locus plot to observe the combined relationship as a function of gain, usually denoted $K$. The only acceptable are K are those where the poles remaining the left semiplane.