Closed-loop pole: Difference between revisions

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

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{{Unreferenced|date=December 2009}}

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== 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>.

The response of a 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.

== Finding closed-loop poles ==

In [[root-locus|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>.

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 ==

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]].

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

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

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

==References==

{{reflist}}

{{DEFAULTSORT:Closed-Loop Pole}}

[[Category:ControlClassical control theory]]