Closed-loop pole: Difference between revisions

Revision as of 18:28, 3 July 2006

Revision as of 18:30, 3 July 2006

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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 [[feedforwardforward 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).

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== 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 feedforwardforward 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>.