*3.3.2 Characteristic polynomial*

The general form of the closed-loop denominator polynomial is given by

$$P^{CL}(s) = s^{n\_{cl}} + \sum\_{i=1}^{n\_{cl}-1} (a\_i + \langle K^{pid}, \overline{b}\_{k-1} \rangle) s^{n\_{cl}-i},\tag{16}$$

where *<sup>P</sup>CL*ð Þ*<sup>s</sup>* is the general form of the characteristic polynomial of the closed-loop plant and *ncl* is the order of referred polynomial.

the characteristic closed-loop polynomial for unit feedback (*H s*ðÞ¼ 1) is obtained in a similar way and given by

$$P\_p^{CL}(\mathfrak{s}) = \mathbf{1} + \mathbf{C}\_{K^{\mathrm{prid}}}(\mathfrak{s}) \mathbf{G}(\mathfrak{s}).\tag{17}$$

*PCL <sup>p</sup>* ð Þ*s* is the characteristic polynomial of the closed loop. The representation of the problem in the form of an internal product that relates the coefficients of the zero polynomial with the coefficients of the closed plant dynamics is the basis for obtaining the characteristic closed-loop polynomial.

*Adjustment of the PID Gains Vector Due to Parametric Variations in the Plant Model… DOI: http://dx.doi.org/10.5772/intechopen.95051*
