{{Short description|Positions of a closed-loop transfer function's poles in the s-plane}}
{{Unreferenced|date=December 2009}}
'''Closed-loop poles''' are the positions of the poles (or [[eigenvalue]]s) of a [[closed-loop transfer function]] in the [[s-plane]]. The [[open-loop controller|open-loop]] transfer function is equal to 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).
In [[controlsystems theory]], there'''closed-loop poles''' are twothe main methodspositions of analyzing feedback systems: the [[transferZeros functionand poles|poles]] (or frequency ___domain[[eigenvalue]]s) methodof and thea [[stateclosed-loop spacetransfer (controls)|state spacefunction]] methodin the [[s-plane]]. WhenThe the[[open-loop controller|open-loop]] transfer function method is used,equal attentionto isthe focusedproduct onof theall locationstransfer function blocks in the [[s-planeforward path]] wherein the [[block diagram]]. The closed-loop transfer function (theis '''poles''')obtained orby zerodividing (the '''zeroes''').open-loop transfer Twofunction differentby transferthe functions aresum of interestone toand the designer.product of all transfer function blocks Ifthroughout the negative [[feedback loopsloop]]. intheThe systemclosed-loop aretransfer openedfunction (thatmay isalso preventedbe fromobtained operating)by onealgebraic speaksor ofblock diagram manipulation. Once the '''openclosed-loop transfer function''',whileis ifobtained for the feedbacksystem, loopsthe areclosed-loop operatingpoles normallyare oneobtained speaksby ofsolving the '''closed-loop transfercharacteristic function'''equation. ForThe characteristic equation is nothing more onthan setting the relationshipdenominator betweenof the twoclosed-loop seetransfer [[root-locus]]function to zero.
In [[control theory]] there are two main methods of analyzing feedback systems: the [[transfer function]] (or frequency ___domain) method and the [[state space (controls)|state space]] method. When the transfer function method is used, attention is focused on the locations in the s-plane where the transfer function is [[Singularity (mathematics)|undefined]] (the ''poles'') or zero (the ''zeroes''; see [[Zeroes and poles]]). Two different transfer functions are of interest to the designer. If the feedback loops in the system are opened (that is prevented from operating) one speaks of the ''[[open-loop transfer function]]'', while if the feedback loops are operating normally one speaks of the ''[[closed-loop transfer function]]''. For more on the relationship between the two, see [[root-locus]].
