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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!

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    \$\begingroup\$ 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 \$\endgroup\$ – Elbehery Mar 23 '17 at 15:33
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    \$\begingroup\$ 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." \$\endgroup\$ – KingDuken Mar 23 '17 at 15:48
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    \$\begingroup\$ Also, see my question about poles and zeros here \$\endgroup\$ – KingDuken Mar 23 '17 at 15:51
  • \$\begingroup\$ 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] \$\endgroup\$ – Elbehery Mar 23 '17 at 15:53
  • \$\begingroup\$ @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 \$\endgroup\$ – cranck Mar 23 '17 at 16:20
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  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.

  3. What useful work can a system do if H(s) = 0 for all s?

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  • \$\begingroup\$ sorry, bad type, must be H(s)<>1 \$\endgroup\$ – cranck Mar 23 '17 at 15:44

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