Poles of transfer function

Control engineers refer to the values of s that are roots of the denominator polynomial as the poles of transfer function G s). Sometimes it is useful to plot the roots (poles) and to discuss process response characteristics in terms of root locations in the complex s plane. In Fig. 6.1 the ordinate expresses the imaginary part of each root the abscissa expresses the real part. Figure 6.1 is based on Eq. 6-2 and indicates the presence of four poles an integrating element (pole at the origin), one real pole (at and a pair of complex... 

In certain problems it may be necessary to locate all the roots of the equation, including the complex roots. This is the case in finding the zeros and poles of transfer functions in process control applications and in formulating the analytical solution of linear nth-order differential equations. On the other hand, different problems may require the location of only one of the roots. For example, in the solution of the equation of state, the positive real root is the one of interest. In any case, the physical constraints of the problem may dictate the feasible region of search where only a subset of the total number of roots may be indicated. In addition, the physical characteristics of ihe problem may provide an approximate value of the desired root. 

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