ControlDesign

 Characterize
 characterize all PID controllers for pole placement in a desired region

 Calling Sequence Characterize(sys, zeta, omegan, opts)

Parameters

 sys - System; a DynamicSystems system object in continuous-time domain; must be single-input single-output (SISO) zeta - realcons; the specified damping omegan - realcons; the specified natural frequency opts - (optional) equation(s) of the form option = value; specify options for the Characterize command

Options

 • controller : one of P, PI, or PID

Specifies the controller type. The default value is PID.

 • output     : one of relativestability, damping, or all

Specifies the set of inequality conditions that must be returned. The default value is all.

Description

 • The Characterize command characterizes all PID controllers for pole placement in a desired region. It returns a Boolean expression of inequalities in terms of the controller parameters that must be satisfied in order to place the closed-loop poles (under unity negative feedback) in the specified desired region. The desired region is specified by zeta and omegan and is defined based on relative stability and damping conditions as follows:
 – Relative Stability: The desired region is part of the complex left half plane (LHP) with real part less than $-\mathrm{\zeta }\mathrm{omegan}$. This is equivalent to the relative stability of the closed-loop system with respect to the line $s=j\mathrm{\omega }-\mathrm{\zeta }\mathrm{omegan}$ (rather than the imaginary axis). Clearly, if zeta or omegan are set to zero, the relative stability reduces to the absolute stability with respect to the imaginary axis.
 – Damping: The desired region is part of the complex left half plane (LHP) inside the angle +/-$\mathrm{arccos}\left(\mathrm{\zeta }\right)$ measured from the negative real axis.
 • The controller parameters are $\mathrm{kc}$ for a P controller, $\mathrm{kc},\mathrm{ki}$ for a PI controller, and $\mathrm{kc},\mathrm{ki},\mathrm{kd}$ for a PID controller, where $\mathrm{kc}$ is the proportional gain, $\mathrm{ki}$ is the integral gain, and $\mathrm{kd}$ is the derivative gain. The controller transfer function is then obtained as: $C\left(s\right)=\mathrm{kc}$, $C\left(s\right)=\mathrm{kc}+\frac{\mathrm{ki}}{s}$, or $C\left(s\right)=\mathrm{kc}+\frac{\mathrm{ki}}{s}+\mathrm{kd}s$ for the P, PI, and PID controllers, respectively.

Examples

 > $\mathrm{with}\left(\mathrm{ControlDesign}\right):$
 > $\mathrm{sys}≔\mathrm{DynamicSystems}:-\mathrm{NewSystem}\left(\frac{s+2}{{s}^{3}+12{s}^{2}+17s+2}\right)$
 ${\mathrm{sys}}{:=}\left[\begin{array}{c}{\mathbf{Transfer Function}}\\ {\mathrm{continuous}}\\ {\mathrm{1 output\left(s\right); 1 input\left(s\right)}}\\ {\mathrm{inputvariable}}{=}\left[{\mathrm{u1}}{}\left({s}\right)\right]\\ {\mathrm{outputvariable}}{=}\left[{\mathrm{y1}}{}\left({s}\right)\right]\end{array}\right$ (1)
 > $\mathrm{Characterize}\left(\mathrm{sys},\frac{1}{3},2,\mathrm{controller}=P,\mathrm{output}=\mathrm{relativestability}\right)$
 ${0}{<}{9}{}{\mathrm{kc}}{-}{29}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}{\mathbf{and}}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}{0}{<}{117}{}{\mathrm{kc}}{+}{373}$ (2)
 > $\mathrm{Characterize}\left(\mathrm{sys},\frac{1}{3},2,\mathrm{controller}=P,\mathrm{output}=\mathrm{damping}\right)$
 ${0}{<}{5453}{-}{115}{}{\mathrm{kc}}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}{\mathbf{and}}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}{0}{<}{31}{}{\mathrm{kc}}{+}{895}{-}\frac{{54}{}\left({45}{}{{\mathrm{kc}}}^{{2}}{-}{342}{}{\mathrm{kc}}{+}{13437}\right)}{{5453}{-}{115}{}{\mathrm{kc}}}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}{\mathbf{and}}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}{0}{<}{5}{}{{\mathrm{kc}}}^{{2}}{-}{38}{}{\mathrm{kc}}{+}{1493}{-}\frac{{1}}{{9}}{}\frac{\left({5453}{-}{115}{}{\mathrm{kc}}\right){}\left({9}{}{{\mathrm{kc}}}^{{2}}{+}{162}{}{\mathrm{kc}}{+}{153}{-}\frac{{54}{}\left({216}{}{{\mathrm{kc}}}^{{2}}{+}{432}{}{\mathrm{kc}}{+}{216}\right)}{{5453}{-}{115}{}{\mathrm{kc}}}\right)}{{31}{}{\mathrm{kc}}{+}{895}{-}\frac{{54}{}\left({45}{}{{\mathrm{kc}}}^{{2}}{-}{342}{}{\mathrm{kc}}{+}{13437}\right)}{{5453}{-}{115}{}{\mathrm{kc}}}}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}{\mathbf{and}}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}{0}{<}{3}{}{{\mathrm{kc}}}^{{2}}{+}{54}{}{\mathrm{kc}}{+}{51}{-}\frac{{18}{}\left({216}{}{{\mathrm{kc}}}^{{2}}{+}{432}{}{\mathrm{kc}}{+}{216}\right)}{{5453}{-}{115}{}{\mathrm{kc}}}{-}\frac{{1}}{{3}}{}\frac{\left({31}{}{\mathrm{kc}}{+}{895}{-}\frac{{54}{}\left({45}{}{{\mathrm{kc}}}^{{2}}{-}{342}{}{\mathrm{kc}}{+}{13437}\right)}{{5453}{-}{115}{}{\mathrm{kc}}}\right){}\left({24}{}{{\mathrm{kc}}}^{{2}}{+}{48}{}{\mathrm{kc}}{+}{24}\right)}{{5}{}{{\mathrm{kc}}}^{{2}}{-}{38}{}{\mathrm{kc}}{+}{1493}{-}\frac{{1}}{{9}}{}\frac{\left({5453}{-}{115}{}{\mathrm{kc}}\right){}\left({9}{}{{\mathrm{kc}}}^{{2}}{+}{162}{}{\mathrm{kc}}{+}{153}{-}\frac{{54}{}\left({216}{}{{\mathrm{kc}}}^{{2}}{+}{432}{}{\mathrm{kc}}{+}{216}\right)}{{5453}{-}{115}{}{\mathrm{kc}}}\right)}{{31}{}{\mathrm{kc}}{+}{895}{-}\frac{{54}{}\left({45}{}{{\mathrm{kc}}}^{{2}}{-}{342}{}{\mathrm{kc}}{+}{13437}\right)}{{5453}{-}{115}{}{\mathrm{kc}}}}}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}{\mathbf{and}}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}{0}{<}{{\mathrm{kc}}}^{{2}}{+}{2}{}{\mathrm{kc}}{+}{1}$ (3)
 > $\mathrm{Characterize}\left(\mathrm{sys},\frac{1}{3},2,\mathrm{controller}=P\right)$
 ${0}{<}{9}{}{\mathrm{kc}}{-}{29}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}{\mathbf{and}}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}{0}{<}{117}{}{\mathrm{kc}}{+}{373}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}{\mathbf{and}}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}{0}{<}{5453}{-}{115}{}{\mathrm{kc}}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}{\mathbf{and}}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}{0}{<}{895}{+}{31}{}{\mathrm{kc}}{-}\frac{{54}{}\left({45}{}{{\mathrm{kc}}}^{{2}}{-}{342}{}{\mathrm{kc}}{+}{13437}\right)}{{5453}{-}{115}{}{\mathrm{kc}}}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}{\mathbf{and}}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}{0}{<}{5}{}{{\mathrm{kc}}}^{{2}}{-}{38}{}{\mathrm{kc}}{+}{1493}{-}\frac{{1}}{{9}}{}\frac{\left({5453}{-}{115}{}{\mathrm{kc}}\right){}\left({9}{}{{\mathrm{kc}}}^{{2}}{+}{162}{}{\mathrm{kc}}{+}{153}{-}\frac{{54}{}\left({216}{}{{\mathrm{kc}}}^{{2}}{+}{432}{}{\mathrm{kc}}{+}{216}\right)}{{5453}{-}{115}{}{\mathrm{kc}}}\right)}{{895}{+}{31}{}{\mathrm{kc}}{-}\frac{{54}{}\left({45}{}{{\mathrm{kc}}}^{{2}}{-}{342}{}{\mathrm{kc}}{+}{13437}\right)}{{5453}{-}{115}{}{\mathrm{kc}}}}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}{\mathbf{and}}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}{0}{<}{3}{}{{\mathrm{kc}}}^{{2}}{+}{54}{}{\mathrm{kc}}{+}{51}{-}\frac{{18}{}\left({216}{}{{\mathrm{kc}}}^{{2}}{+}{432}{}{\mathrm{kc}}{+}{216}\right)}{{5453}{-}{115}{}{\mathrm{kc}}}{-}\frac{{1}}{{3}}{}\frac{\left({895}{+}{31}{}{\mathrm{kc}}{-}\frac{{54}{}\left({45}{}{{\mathrm{kc}}}^{{2}}{-}{342}{}{\mathrm{kc}}{+}{13437}\right)}{{5453}{-}{115}{}{\mathrm{kc}}}\right){}\left({24}{}{{\mathrm{kc}}}^{{2}}{+}{48}{}{\mathrm{kc}}{+}{24}\right)}{{5}{}{{\mathrm{kc}}}^{{2}}{-}{38}{}{\mathrm{kc}}{+}{1493}{-}\frac{{1}}{{9}}{}\frac{\left({5453}{-}{115}{}{\mathrm{kc}}\right){}\left({9}{}{{\mathrm{kc}}}^{{2}}{+}{162}{}{\mathrm{kc}}{+}{153}{-}\frac{{54}{}\left({216}{}{{\mathrm{kc}}}^{{2}}{+}{432}{}{\mathrm{kc}}{+}{216}\right)}{{5453}{-}{115}{}{\mathrm{kc}}}\right)}{{895}{+}{31}{}{\mathrm{kc}}{-}\frac{{54}{}\left({45}{}{{\mathrm{kc}}}^{{2}}{-}{342}{}{\mathrm{kc}}{+}{13437}\right)}{{5453}{-}{115}{}{\mathrm{kc}}}}}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}{\mathbf{and}}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}{0}{<}{{\mathrm{kc}}}^{{2}}{+}{2}{}{\mathrm{kc}}{+}{1}$ (4)
 > $\mathrm{Characterize}\left(\mathrm{sys},\frac{1}{3},2,\mathrm{controller}=\mathrm{Π},\mathrm{output}=\mathrm{relativestability}\right)$
 ${0}{<}{-}{18}{}{\mathrm{kc}}{+}{27}{}{\mathrm{ki}}{+}{58}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}{\mathbf{and}}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}{0}{<}{2106}{}{\mathrm{kc}}{-}{8406}{-}{243}{}{\mathrm{ki}}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}{\mathbf{and}}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}{0}{<}{18}{}{\mathrm{kc}}{+}{27}{}{\mathrm{ki}}{-}{158}{-}\frac{{252}{}\left({-}{2016}{}{\mathrm{kc}}{+}{3024}{}{\mathrm{ki}}{+}{6496}\right)}{{2106}{}{\mathrm{kc}}{-}{8406}{-}{243}{}{\mathrm{ki}}}$ (5)
 > $\mathrm{Characterize}\left(\mathrm{sys},\frac{1}{3},2,\mathrm{controller}=\mathrm{Π}\right)$
 ${0}{<}{-}{18}{}{\mathrm{kc}}{+}{27}{}{\mathrm{ki}}{+}{58}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}{\mathbf{and}}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}{0}{<}{2106}{}{\mathrm{kc}}{-}{8406}{-}{243}{}{\mathrm{ki}}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}{\mathbf{and}}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}{0}{<}{18}{}{\mathrm{kc}}{+}{27}{}{\mathrm{ki}}{-}{158}{-}\frac{{252}{}\left({-}{2016}{}{\mathrm{kc}}{+}{3024}{}{\mathrm{ki}}{+}{6496}\right)}{{2106}{}{\mathrm{kc}}{-}{8406}{-}{243}{}{\mathrm{ki}}}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}{\mathbf{and}}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}{0}{<}{{\mathrm{ki}}}^{{2}}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}{\mathbf{and}}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}{0}{<}{-}{690}{}{\mathrm{kc}}{+}{32718}{+}{69}{}{\mathrm{ki}}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}{\mathbf{and}}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}{0}{<}{62}{}{\mathrm{kc}}{-}{23}{}{\mathrm{ki}}{+}{1790}{-}\frac{{108}{}\left({270}{}{{\mathrm{kc}}}^{{2}}{-}{27}{}{\mathrm{kc}}{}{\mathrm{ki}}{-}{2052}{}{\mathrm{kc}}{-}{4579}{}{\mathrm{ki}}{+}{80622}\right)}{{-}{690}{}{\mathrm{kc}}{+}{32718}{+}{69}{}{\mathrm{ki}}}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}{\mathbf{and}}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}{0}{<}{270}{}{{\mathrm{kc}}}^{{2}}{-}{27}{}{\mathrm{kc}}{}{\mathrm{ki}}{-}{2052}{}{\mathrm{kc}}{-}{4579}{}{\mathrm{ki}}{+}{80622}{-}\frac{\left({-}{690}{}{\mathrm{kc}}{+}{32718}{+}{69}{}{\mathrm{ki}}\right){}\left({18}{}{{\mathrm{kc}}}^{{2}}{+}{9}{}{\mathrm{kc}}{}{\mathrm{ki}}{+}{324}{}{\mathrm{kc}}{-}{399}{}{\mathrm{ki}}{+}{306}{-}\frac{{108}{}\left({1296}{}{{\mathrm{kc}}}^{{2}}{+}{180}{}{\mathrm{kc}}{}{\mathrm{ki}}{+}{270}{}{{\mathrm{ki}}}^{{2}}{+}{2592}{}{\mathrm{kc}}{-}{15948}{}{\mathrm{ki}}{+}{1296}\right)}{{-}{690}{}{\mathrm{kc}}{+}{32718}{+}{69}{}{\mathrm{ki}}}\right)}{{62}{}{\mathrm{kc}}{-}{23}{}{\mathrm{ki}}{+}{1790}{-}\frac{{108}{}\left({270}{}{{\mathrm{kc}}}^{{2}}{-}{27}{}{\mathrm{kc}}{}{\mathrm{ki}}{-}{2052}{}{\mathrm{kc}}{-}{4579}{}{\mathrm{ki}}{+}{80622}\right)}{{-}{690}{}{\mathrm{kc}}{+}{32718}{+}{69}{}{\mathrm{ki}}}}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}{\mathbf{and}}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}{0}{<}{2}{}{\mathrm{kc}}{}{\mathrm{ki}}{+}{{\mathrm{ki}}}^{{2}}{+}{2}{}{\mathrm{ki}}{-}\frac{{7776}{}{{\mathrm{ki}}}^{{2}}}{{-}{690}{}{\mathrm{kc}}{+}{32718}{+}{69}{}{\mathrm{ki}}}{-}\frac{{72}{}\left({62}{}{\mathrm{kc}}{-}{23}{}{\mathrm{ki}}{+}{1790}{-}\frac{{108}{}\left({270}{}{{\mathrm{kc}}}^{{2}}{-}{27}{}{\mathrm{kc}}{}{\mathrm{ki}}{-}{2052}{}{\mathrm{kc}}{-}{4579}{}{\mathrm{ki}}{+}{80622}\right)}{{-}{690}{}{\mathrm{kc}}{+}{32718}{+}{69}{}{\mathrm{ki}}}\right){}{{\mathrm{ki}}}^{{2}}}{{270}{}{{\mathrm{kc}}}^{{2}}{-}{27}{}{\mathrm{kc}}{}{\mathrm{ki}}{-}{2052}{}{\mathrm{kc}}{-}{4579}{}{\mathrm{ki}}{+}{80622}{-}\frac{\left({-}{690}{}{\mathrm{kc}}{+}{32718}{+}{69}{}{\mathrm{ki}}\right){}\left({18}{}{{\mathrm{kc}}}^{{2}}{+}{9}{}{\mathrm{kc}}{}{\mathrm{ki}}{+}{324}{}{\mathrm{kc}}{-}{399}{}{\mathrm{ki}}{+}{306}{-}\frac{{108}{}\left({1296}{}{{\mathrm{kc}}}^{{2}}{+}{180}{}{\mathrm{kc}}{}{\mathrm{ki}}{+}{270}{}{{\mathrm{ki}}}^{{2}}{+}{2592}{}{\mathrm{kc}}{-}{15948}{}{\mathrm{ki}}{+}{1296}\right)}{{-}{690}{}{\mathrm{kc}}{+}{32718}{+}{69}{}{\mathrm{ki}}}\right)}{{62}{}{\mathrm{kc}}{-}{23}{}{\mathrm{ki}}{+}{1790}{-}\frac{{108}{}\left({270}{}{{\mathrm{kc}}}^{{2}}{-}{27}{}{\mathrm{kc}}{}{\mathrm{ki}}{-}{2052}{}{\mathrm{kc}}{-}{4579}{}{\mathrm{ki}}{+}{80622}\right)}{{-}{690}{}{\mathrm{kc}}{+}{32718}{+}{69}{}{\mathrm{ki}}}}}{-}\frac{{36}{}\left({6}{}{{\mathrm{kc}}}^{{2}}{+}{3}{}{\mathrm{kc}}{}{\mathrm{ki}}{+}{108}{}{\mathrm{kc}}{-}{133}{}{\mathrm{ki}}{+}{102}{-}\frac{{36}{}\left({1296}{}{{\mathrm{kc}}}^{{2}}{+}{180}{}{\mathrm{kc}}{}{\mathrm{ki}}{+}{270}{}{{\mathrm{ki}}}^{{2}}{+}{2592}{}{\mathrm{kc}}{-}{15948}{}{\mathrm{ki}}{+}{1296}\right)}{{-}{690}{}{\mathrm{kc}}{+}{32718}{+}{69}{}{\mathrm{ki}}}{-}\frac{{1}}{{3}}{}\frac{\left({62}{}{\mathrm{kc}}{-}{23}{}{\mathrm{ki}}{+}{1790}{-}\frac{{108}{}\left({270}{}{{\mathrm{kc}}}^{{2}}{-}{27}{}{\mathrm{kc}}{}{\mathrm{ki}}{-}{2052}{}{\mathrm{kc}}{-}{4579}{}{\mathrm{ki}}{+}{80622}\right)}{{-}{690}{}{\mathrm{kc}}{+}{32718}{+}{69}{}{\mathrm{ki}}}\right){}\left({1296}{}{{\mathrm{kc}}}^{{2}}{+}{180}{}{\mathrm{kc}}{}{\mathrm{ki}}{+}{270}{}{{\mathrm{ki}}}^{{2}}{+}{2592}{}{\mathrm{kc}}{-}{15948}{}{\mathrm{ki}}{+}{1296}{-}\frac{\left({-}{690}{}{\mathrm{kc}}{+}{32718}{+}{69}{}{\mathrm{ki}}\right){}\left({36}{}{\mathrm{kc}}{}{\mathrm{ki}}{+}{18}{}{{\mathrm{ki}}}^{{2}}{+}{36}{}{\mathrm{ki}}{-}\frac{{139968}{}{{\mathrm{ki}}}^{{2}}}{{-}{690}{}{\mathrm{kc}}{+}{32718}{+}{69}{}{\mathrm{ki}}}\right)}{{62}{}{\mathrm{kc}}{-}{23}{}{\mathrm{ki}}{+}{1790}{-}\frac{{108}{}\left({270}{}{{\mathrm{kc}}}^{{2}}{-}{27}{}{\mathrm{kc}}{}{\mathrm{ki}}{-}{2052}{}{\mathrm{kc}}{-}{4579}{}{\mathrm{ki}}{+}{80622}\right)}{{-}{690}{}{\mathrm{kc}}{+}{32718}{+}{69}{}{\mathrm{ki}}}}\right)}{{270}{}{{\mathrm{kc}}}^{{2}}{-}{27}{}{\mathrm{kc}}{}{\mathrm{ki}}{-}{2052}{}{\mathrm{kc}}{-}{4579}{}{\mathrm{ki}}{+}{80622}{-}\frac{\left({-}{690}{}{\mathrm{kc}}{+}{32718}{+}{69}{}{\mathrm{ki}}\right){}\left({18}{}{{\mathrm{kc}}}^{{2}}{+}{9}{}{\mathrm{kc}}{}{\mathrm{ki}}{+}{324}{}{\mathrm{kc}}{-}{399}{}{\mathrm{ki}}{+}{306}{-}\frac{{108}{}\left({1296}{}{{\mathrm{kc}}}^{{2}}{+}{180}{}{\mathrm{kc}}{}{\mathrm{ki}}{+}{270}{}{{\mathrm{ki}}}^{{2}}{+}{2592}{}{\mathrm{kc}}{-}{15948}{}{\mathrm{ki}}{+}{1296}\right)}{{-}{690}{}{\mathrm{kc}}{+}{32718}{+}{69}{}{\mathrm{ki}}}\right)}{{62}{}{\mathrm{kc}}{-}{23}{}{\mathrm{ki}}{+}{1790}{-}\frac{{108}{}\left({270}{}{{\mathrm{kc}}}^{{2}}{-}{27}{}{\mathrm{kc}}{}{\mathrm{ki}}{-}{2052}{}{\mathrm{kc}}{-}{4579}{}{\mathrm{ki}}{+}{80622}\right)}{{-}{690}{}{\mathrm{kc}}{+}{32718}{+}{69}{}{\mathrm{ki}}}}}\right){}{{\mathrm{ki}}}^{{2}}}{{216}{}{{\mathrm{kc}}}^{{2}}{+}{30}{}{\mathrm{kc}}{}{\mathrm{ki}}{+}{45}{}{{\mathrm{ki}}}^{{2}}{+}{432}{}{\mathrm{kc}}{-}{2658}{}{\mathrm{ki}}{+}{216}{-}\frac{{1}}{{6}}{}\frac{\left({-}{690}{}{\mathrm{kc}}{+}{32718}{+}{69}{}{\mathrm{ki}}\right){}\left({36}{}{\mathrm{kc}}{}{\mathrm{ki}}{+}{18}{}{{\mathrm{ki}}}^{{2}}{+}{36}{}{\mathrm{ki}}{-}\frac{{139968}{}{{\mathrm{ki}}}^{{2}}}{{-}{690}{}{\mathrm{kc}}{+}{32718}{+}{69}{}{\mathrm{ki}}}\right)}{{62}{}{\mathrm{kc}}{-}{23}{}{\mathrm{ki}}{+}{1790}{-}\frac{{108}{}\left({270}{}{{\mathrm{kc}}}^{{2}}{-}{27}{}{\mathrm{kc}}{}{\mathrm{ki}}{-}{2052}{}{\mathrm{kc}}{-}{4579}{}{\mathrm{ki}}{+}{80622}\right)}{{-}{690}{}{\mathrm{kc}}{+}{32718}{+}{69}{}{\mathrm{ki}}}}{-}\frac{{1}}{{6}}{}\frac{\left({270}{}{{\mathrm{kc}}}^{{2}}{-}{27}{}{\mathrm{kc}}{}{\mathrm{ki}}{-}{2052}{}{\mathrm{kc}}{-}{4579}{}{\mathrm{ki}}{+}{80622}{-}\frac{\left({-}{690}{}{\mathrm{kc}}{+}{32718}{+}{69}{}{\mathrm{ki}}\right){}\left({18}{}{{\mathrm{kc}}}^{{2}}{+}{9}{}{\mathrm{kc}}{}{\mathrm{ki}}{+}{324}{}{\mathrm{kc}}{-}{399}{}{\mathrm{ki}}{+}{306}{-}\frac{{108}{}\left({1296}{}{{\mathrm{kc}}}^{{2}}{+}{180}{}{\mathrm{kc}}{}{\mathrm{ki}}{+}{270}{}{{\mathrm{ki}}}^{{2}}{+}{2592}{}{\mathrm{kc}}{-}{15948}{}{\mathrm{ki}}{+}{1296}\right)}{{-}{690}{}{\mathrm{kc}}{+}{32718}{+}{69}{}{\mathrm{ki}}}\right)}{{62}{}{\mathrm{kc}}{-}{23}{}{\mathrm{ki}}{+}{1790}{-}\frac{{108}{}\left({270}{}{{\mathrm{kc}}}^{{2}}{-}{27}{}{\mathrm{kc}}{}{\mathrm{ki}}{-}{2052}{}{\mathrm{kc}}{-}{4579}{}{\mathrm{ki}}{+}{80622}\right)}{{-}{690}{}{\mathrm{kc}}{+}{32718}{+}{69}{}{\mathrm{ki}}}}\right){}\left({12}{}{\mathrm{kc}}{}{\mathrm{ki}}{+}{6}{}{{\mathrm{ki}}}^{{2}}{+}{12}{}{\mathrm{ki}}{-}\frac{{46656}{}{{\mathrm{ki}}}^{{2}}}{{-}{690}{}{\mathrm{kc}}{+}{32718}{+}{69}{}{\mathrm{ki}}}{-}\frac{{432}{}\left({62}{}{\mathrm{kc}}{-}{23}{}{\mathrm{ki}}{+}{1790}{-}\frac{{108}{}\left({270}{}{{\mathrm{kc}}}^{{2}}{-}{27}{}{\mathrm{kc}}{}{\mathrm{ki}}{-}{2052}{}{\mathrm{kc}}{-}{4579}{}{\mathrm{ki}}{+}{80622}\right)}{{-}{690}{}{\mathrm{kc}}{+}{32718}{+}{69}{}{\mathrm{ki}}}\right){}{{\mathrm{ki}}}^{{2}}}{{270}{}{{\mathrm{kc}}}^{{2}}{-}{27}{}{\mathrm{kc}}{}{\mathrm{ki}}{-}{2052}{}{\mathrm{kc}}{-}{4579}{}{\mathrm{ki}}{+}{80622}{-}\frac{\left({-}{690}{}{\mathrm{kc}}{+}{32718}{+}{69}{}{\mathrm{ki}}\right){}\left({18}{}{{\mathrm{kc}}}^{{2}}{+}{9}{}{\mathrm{kc}}{}{\mathrm{ki}}{+}{324}{}{\mathrm{kc}}{-}{399}{}{\mathrm{ki}}{+}{306}{-}\frac{{108}{}\left({1296}{}{{\mathrm{kc}}}^{{2}}{+}{180}{}{\mathrm{kc}}{}{\mathrm{ki}}{+}{270}{}{{\mathrm{ki}}}^{{2}}{+}{2592}{}{\mathrm{kc}}{-}{15948}{}{\mathrm{ki}}{+}{1296}\right)}{{-}{690}{}{\mathrm{kc}}{+}{32718}{+}{69}{}{\mathrm{ki}}}\right)}{{62}{}{\mathrm{kc}}{-}{23}{}{\mathrm{ki}}{+}{1790}{-}\frac{{108}{}\left({270}{}{{\mathrm{kc}}}^{{2}}{-}{27}{}{\mathrm{kc}}{}{\mathrm{ki}}{-}{2052}{}{\mathrm{kc}}{-}{4579}{}{\mathrm{ki}}{+}{80622}\right)}{{-}{690}{}{\mathrm{kc}}{+}{32718}{+}{69}{}{\mathrm{ki}}}}}\right)}{{6}{}{{\mathrm{kc}}}^{{2}}{+}{3}{}{\mathrm{kc}}{}{\mathrm{ki}}{+}{108}{}{\mathrm{kc}}{-}{133}{}{\mathrm{ki}}{+}{102}{-}\frac{{36}{}\left({1296}{}{{\mathrm{kc}}}^{{2}}{+}{180}{}{\mathrm{kc}}{}{\mathrm{ki}}{+}{270}{}{{\mathrm{ki}}}^{{2}}{+}{2592}{}{\mathrm{kc}}{-}{15948}{}{\mathrm{ki}}{+}{1296}\right)}{{-}{690}{}{\mathrm{kc}}{+}{32718}{+}{69}{}{\mathrm{ki}}}{-}\frac{{1}}{{3}}{}\frac{\left({62}{}{\mathrm{kc}}{-}{23}{}{\mathrm{ki}}{+}{1790}{-}\frac{{108}{}\left({270}{}{{\mathrm{kc}}}^{{2}}{-}{27}{}{\mathrm{kc}}{}{\mathrm{ki}}{-}{2052}{}{\mathrm{kc}}{-}{4579}{}{\mathrm{ki}}{+}{80622}\right)}{{-}{690}{}{\mathrm{kc}}{+}{32718}{+}{69}{}{\mathrm{ki}}}\right){}\left({1296}{}{{\mathrm{kc}}}^{{2}}{+}{180}{}{\mathrm{kc}}{}{\mathrm{ki}}{+}{270}{}{{\mathrm{ki}}}^{{2}}{+}{2592}{}{\mathrm{kc}}{-}{15948}{}{\mathrm{ki}}{+}{1296}{-}\frac{\left({-}{690}{}{\mathrm{kc}}{+}{32718}{+}{69}{}{\mathrm{ki}}\right){}\left({36}{}{\mathrm{kc}}{}{\mathrm{ki}}{+}{18}{}{{\mathrm{ki}}}^{{2}}{+}{36}{}{\mathrm{ki}}{-}\frac{{139968}{}{{\mathrm{ki}}}^{{2}}}{{-}{690}{}{\mathrm{kc}}{+}{32718}{+}{69}{}{\mathrm{ki}}}\right)}{{62}{}{\mathrm{kc}}{-}{23}{}{\mathrm{ki}}{+}{1790}{-}\frac{{108}{}\left({270}{}{{\mathrm{kc}}}^{{2}}{-}{27}{}{\mathrm{kc}}{}{\mathrm{ki}}{-}{2052}{}{\mathrm{kc}}{-}{4579}{}{\mathrm{ki}}{+}{80622}\right)}{{-}{690}{}{\mathrm{kc}}{+}{32718}{+}{69}{}{\mathrm{ki}}}}\right)}{{270}{}{{\mathrm{kc}}}^{{2}}{-}{27}{}{\mathrm{kc}}{}{\mathrm{ki}}{-}{2052}{}{\mathrm{kc}}{-}{4579}{}{\mathrm{ki}}{+}{80622}{-}\frac{\left({-}{690}{}{\mathrm{kc}}{+}{32718}{+}{69}{}{\mathrm{ki}}\right){}\left({18}{}{{\mathrm{kc}}}^{{2}}{+}{9}{}{\mathrm{kc}}{}{\mathrm{ki}}{+}{324}{}{\mathrm{kc}}{-}{399}{}{\mathrm{ki}}{+}{306}{-}\frac{{108}{}\left({1296}{}{{\mathrm{kc}}}^{{2}}{+}{180}{}{\mathrm{kc}}{}{\mathrm{ki}}{+}{270}{}{{\mathrm{ki}}}^{{2}}{+}{2592}{}{\mathrm{kc}}{-}{15948}{}{\mathrm{ki}}{+}{1296}\right)}{{-}{690}{}{\mathrm{kc}}{+}{32718}{+}{69}{}{\mathrm{ki}}}\right)}{{62}{}{\mathrm{kc}}{-}{23}{}{\mathrm{ki}}{+}{1790}{-}\frac{{108}{}\left({270}{}{{\mathrm{kc}}}^{{2}}{-}{27}{}{\mathrm{kc}}{}{\mathrm{ki}}{-}{2052}{}{\mathrm{kc}}{-}{4579}{}{\mathrm{ki}}{+}{80622}\right)}{{-}{690}{}{\mathrm{kc}}{+}{32718}{+}{69}{}{\mathrm{ki}}}}}}}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}{\mathbf{and}}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}{0}{<}{6}{}{{\mathrm{kc}}}^{{2}}{+}{3}{}{\mathrm{kc}}{}{\mathrm{ki}}{+}{108}{}{\mathrm{kc}}{-}{133}{}{\mathrm{ki}}{+}{102}{-}\frac{{36}{}\left({1296}{}{{\mathrm{kc}}}^{{2}}{+}{180}{}{\mathrm{kc}}{}{\mathrm{ki}}{+}{270}{}{{\mathrm{ki}}}^{{2}}{+}{2592}{}{\mathrm{kc}}{-}{15948}{}{\mathrm{ki}}{+}{1296}\right)}{{-}{690}{}{\mathrm{kc}}{+}{32718}{+}{69}{}{\mathrm{ki}}}{-}\frac{{1}}{{3}}{}\frac{\left({62}{}{\mathrm{kc}}{-}{23}{}{\mathrm{ki}}{+}{1790}{-}\frac{{108}{}\left({270}{}{{\mathrm{kc}}}^{{2}}{-}{27}{}{\mathrm{kc}}{}{\mathrm{ki}}{-}{2052}{}{\mathrm{kc}}{-}{4579}{}{\mathrm{ki}}{+}{80622}\right)}{{-}{690}{}{\mathrm{kc}}{+}{32718}{+}{69}{}{\mathrm{ki}}}\right){}\left({1296}{}{{\mathrm{kc}}}^{{2}}{+}{180}{}{\mathrm{kc}}{}{\mathrm{ki}}{+}{270}{}{{\mathrm{ki}}}^{{2}}{+}{2592}{}{\mathrm{kc}}{-}{15948}{}{\mathrm{ki}}{+}{1296}{-}\frac{\left({-}{690}{}{\mathrm{kc}}{+}{32718}{+}{69}{}{\mathrm{ki}}\right){}\left({36}{}{\mathrm{kc}}{}{\mathrm{ki}}{+}{18}{}{{\mathrm{ki}}}^{{2}}{+}{36}{}{\mathrm{ki}}{-}\frac{{139968}{}{{\mathrm{ki}}}^{{2}}}{{-}{690}{}{\mathrm{kc}}{+}{32718}{+}{69}{}{\mathrm{ki}}}\right)}{{62}{}{\mathrm{kc}}{-}{23}{}{\mathrm{ki}}{+}{1790}{-}\frac{{108}{}\left({270}{}{{\mathrm{kc}}}^{{2}}{-}{27}{}{\mathrm{kc}}{}{\mathrm{ki}}{-}{2052}{}{\mathrm{kc}}{-}{4579}{}{\mathrm{ki}}{+}{80622}\right)}{{-}{690}{}{\mathrm{kc}}{+}{32718}{+}{69}{}{\mathrm{ki}}}}\right)}{{270}{}{{\mathrm{kc}}}^{{2}}{-}{27}{}{\mathrm{kc}}{}{\mathrm{ki}}{-}{2052}{}{\mathrm{kc}}{-}{4579}{}{\mathrm{ki}}{+}{80622}{-}\frac{\left({-}{690}{}{\mathrm{kc}}{+}{32718}{+}{69}{}{\mathrm{ki}}\right){}\left({18}{}{{\mathrm{kc}}}^{{2}}{+}{9}{}{\mathrm{kc}}{}{\mathrm{ki}}{+}{324}{}{\mathrm{kc}}{-}{399}{}{\mathrm{ki}}\right)}{}}$