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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>.
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]].
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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 equation:
: <math>\det(\textbf{I}+\textbf{G}(s)\textbf{K}(s))=0</math>
{{DEFAULTSORT:Closed-Loop Pole}}
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