Closed-loop pole: Difference between revisions

Revision as of 18:30, 3 July 2006

Latest revision as of 08:26, 17 June 2024

(39 intermediate revisions by 31 users not shown)

Line 1:

{{Short description|Positions of a closed-loop transfer function's poles in the s-plane}}

{{context}}

{{Unreferenced|date=December 2009}}

'''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).

In [[systems theory]], '''Closedclosed-loop poles''' are the positions of the [[eigenvaluesZeros and poles|poles]] (or [[eigenvalue]]s) of a [[closed-loop]] [[transfer function]] in the [[s-plane]]. InThe [[controlopen-loop theory]], the [[controller|open-loop]] transfer function representsis 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 negative [[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 [[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]].

== Closed-loop poles in control theory ==

== Closed-loop poles in control theory ==

The response of a [[Linear time-invariant system | linear time-invariant system]] to any input can be derived from its [[impulse response]] and [[step response]]. The eigenvalues of the system determine completely the [[natural response]] ([[unforced response]]). In control theory, the response to any [[input]] is a combination of a [[transient response]] and [[steady-state response]]. Therefore, a crucial design parameter is the ___location of the eigenvalues, or closed-loop poles.

In [[root-locus|root-locus design]], the [[Gain (electronics)|gain]], ''K,'' is usually parameterized. Each point on the locus satisfies the [[angle condition]] and [[magnitude condition]] and corresponds to a different value of &nbsp;''K''. For [[negative feedback]] systems, the closed-loop poles move along the [[root-locus]] from the [[open-loop polespole]]s to the [[open-loop zeroeszeroe]]s 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 ==

Consider a simple feedback system with controller <math>\textbf{G}_c = K</math>, [[plant (control theory)|plant]] <math>\textbf{G}(s)</math> and transfer function <math>\textbf{H}(s)</math> in the [[feedback path]]. Note that a [[unity feedback]] system has <math>\textbf{H}(s)=1</math> and the block is omitted. For this system, the open-loop transfer function is the product of the blocks in the forward path, <math>\textbf{G}_c\textbf{G} = K\textbf{G}</math>. The product of the blocks around the entire closed loop is <math>\textbf{G}_c\textbf{G}\textbf{H} = K\textbf{G}\textbf{H}</math>. Therefore, the closed-loop transfer function is

: <math>\textbf{T}(s)=\frac{K\textbf{G}}{1+K\textbf{G}\textbf{H}}.</math>.

== Finding closed-loop poles ==

Consider a simple feedback system with controller <math>\textbf{G}_c = K</math>, [[plant]] <math>\textbf{G}(s)</math> and transfer function <math>\textbf{H}(s)</math> in the [[feedback path]]. Note that a [[unity feedback]] system has <math>\textbf{H}(s)=1</math> and the block is omitted. For this system, the open-loop transfer function is the product of the blocks in the forward path, <math>\textbf{G}_c\textbf{G} = K\textbf{G}</math>. The product of the blocks around the entire closed loop is <math>\textbf{G}_c\textbf{G}\textbf{H} = K\textbf{G}\textbf{H}</math>. Therefore, the closed-loop transfer function is

<math>\textbf{T}(s)=\frac{K\textbf{G}}{1+K\textbf{G}\textbf{H}}</math>.

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]].

The preceding is valid for single-input-single-output systems (SISO). An extension is possible for multiple input multiple output systems, that is for systems where <math>\textbf{G}(s)</math> and <math>\textbf{K}(s)</math> are matrices whose elements are made of transfer functions. In this case the poles are the solution of the equation

: <math>\det(\textbf{I}+\textbf{G}(s)\textbf{K}(s))=0. \, </math>

==References==

[[Category:Control theory]]

{{reflist}}

{{DEFAULTSORT:Closed-Loop Pole}}

--[[User:Celieber|Celieber]] 15:08, 3 July 2006 (UTC) Chad Lieberman, Massachusetts Institute of Technology '07

[[Category:ControlClassical control theory]]