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'''Closed-loop poles''' are the positions of the [[eigenvalues]] of a [[closed-loop]] [[transfer function]] in the [[s-plane]]. In [[control theory]], the [[open-loop]] transfer function represents the product of all transfer function blocks in the [[forward path]] in the [[block diagram]]. The closed-loop transfer function is obtained by dividing the open-loop transfer function by the sum of one (1) and the product of all transfer function blocks throughout the [[feedback loop]]. The closed-loop transfer function may also be obtained by algebraic or block diagram manipulation. Once the closed-loop transfer function is obtained for the system, the closed-loop poles are obtained by solving the [[characteristic equation]]. The characteristic equation is nothing more than setting the denominator of the closed-loop transfer function to zero (0).
== Closed-loop poles in control theory ==
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In [[root-locus design]], the [[gain]], K, is usually parameterized. Each point on the locus satisfies the [[angle condition]] and [[magnitude condition]] and corresponds to a different value of K. For [[negative feedback]] systems, the closed-loop poles move along the [[root-locus]] from the [[open-loop poles]] to the [[open-loop zeroes]] as the gain is increased. For this reason, the root-locus is often used for design of [[proportional control]], i.e. those for which <math>\textbf{G}_c = K</math>.
== Finding closed-loop poles ==
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The closed-loop poles, or eigenvalues, are obtained by solving the characteristic equation <math>{1+K\textbf{G}\textbf{H}}=0</math>. In general, the solution will be n complex numbers where n is the order of the [[characteristic polynomial]].
[[Category:Control theory]]
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