Characterize - MapleSim Help
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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)
 > $\mathrm{Characterize}\left(\mathrm{sys},\frac{1}{3},2,\mathrm{controller}=P,\mathrm{output}=\mathrm{relativestability}\right)$
 ${0}{<}{9}{}{\mathrm{kc}}{-}{29}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\mathbf{and}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{0}{<}{117}{}{\mathrm{kc}}{+}{373}$ (2)
 > $\mathrm{Characterize}\left(\mathrm{sys},\frac{1}{3},2,\mathrm{controller}=P,\mathrm{output}=\mathrm{damping}\right)$
 ${0}{<}{\left({\mathrm{kc}}{+}{1}\right)}^{{2}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\mathbf{and}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{0}{<}{5453}{-}{115}{}{\mathrm{kc}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\mathbf{and}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{0}{<}{895}{+}{31}{}{\mathrm{kc}}{-}\frac{{54}{}\left({45}{}{{\mathrm{kc}}}^{{2}}{-}{342}{}{\mathrm{kc}}{+}{13437}\right)}{{5453}{-}{115}{}{\mathrm{kc}}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\mathbf{and}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{0}{<}{5}{}{{\mathrm{kc}}}^{{2}}{-}{38}{}{\mathrm{kc}}{+}{1493}{-}\frac{\left({5453}{-}{115}{}{\mathrm{kc}}\right){}\left({9}{}{{\mathrm{kc}}}^{{2}}{+}{162}{}{\mathrm{kc}}{+}{153}{-}\frac{{11664}{}{\left({\mathrm{kc}}{+}{1}\right)}^{{2}}}{{5453}{-}{115}{}{\mathrm{kc}}}\right)}{{9}{}\left({895}{+}{31}{}{\mathrm{kc}}{-}\frac{{54}{}\left({45}{}{{\mathrm{kc}}}^{{2}}{-}{342}{}{\mathrm{kc}}{+}{13437}\right)}{{5453}{-}{115}{}{\mathrm{kc}}}\right)}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\mathbf{and}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{0}{<}{3}{}{{\mathrm{kc}}}^{{2}}{+}{54}{}{\mathrm{kc}}{+}{51}{-}\frac{{3888}{}{\left({\mathrm{kc}}{+}{1}\right)}^{{2}}}{{5453}{-}{115}{}{\mathrm{kc}}}{-}\frac{{8}{}\left({895}{+}{31}{}{\mathrm{kc}}{-}\frac{{54}{}\left({45}{}{{\mathrm{kc}}}^{{2}}{-}{342}{}{\mathrm{kc}}{+}{13437}\right)}{{5453}{-}{115}{}{\mathrm{kc}}}\right){}{\left({\mathrm{kc}}{+}{1}\right)}^{{2}}}{{5}{}{{\mathrm{kc}}}^{{2}}{-}{38}{}{\mathrm{kc}}{+}{1493}{-}\frac{\left({5453}{-}{115}{}{\mathrm{kc}}\right){}\left({9}{}{{\mathrm{kc}}}^{{2}}{+}{162}{}{\mathrm{kc}}{+}{153}{-}\frac{{11664}{}{\left({\mathrm{kc}}{+}{1}\right)}^{{2}}}{{5453}{-}{115}{}{\mathrm{kc}}}\right)}{{9}{}\left({895}{+}{31}{}{\mathrm{kc}}{-}\frac{{54}{}\left({45}{}{{\mathrm{kc}}}^{{2}}{-}{342}{}{\mathrm{kc}}{+}{13437}\right)}{{5453}{-}{115}{}{\mathrm{kc}}}\right)}}$ (3)
 > $\mathrm{Characterize}\left(\mathrm{sys},\frac{1}{3},2,\mathrm{controller}=P\right)$
 ${0}{<}{9}{}{\mathrm{kc}}{-}{29}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\mathbf{and}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{0}{<}{117}{}{\mathrm{kc}}{+}{373}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\mathbf{and}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{0}{<}{\left({\mathrm{kc}}{+}{1}\right)}^{{2}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\mathbf{and}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{0}{<}{5453}{-}{115}{}{\mathrm{kc}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\mathbf{and}}\phantom{\rule[-0.0ex]{0.3em}{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]{0.3em}{0.0ex}}{\mathbf{and}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{0}{<}{5}{}{{\mathrm{kc}}}^{{2}}{-}{38}{}{\mathrm{kc}}{+}{1493}{-}\frac{\left({5453}{-}{115}{}{\mathrm{kc}}\right){}\left({9}{}{{\mathrm{kc}}}^{{2}}{+}{162}{}{\mathrm{kc}}{+}{153}{-}\frac{{11664}{}{\left({\mathrm{kc}}{+}{1}\right)}^{{2}}}{{5453}{-}{115}{}{\mathrm{kc}}}\right)}{{9}{}\left({895}{+}{31}{}{\mathrm{kc}}{-}\frac{{54}{}\left({45}{}{{\mathrm{kc}}}^{{2}}{-}{342}{}{\mathrm{kc}}{+}{13437}\right)}{{5453}{-}{115}{}{\mathrm{kc}}}\right)}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\mathbf{and}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{0}{<}{3}{}{{\mathrm{kc}}}^{{2}}{+}{54}{}{\mathrm{kc}}{+}{51}{-}\frac{{3888}{}{\left({\mathrm{kc}}{+}{1}\right)}^{{2}}}{{5453}{-}{115}{}{\mathrm{kc}}}{-}\frac{{8}{}\left({895}{+}{31}{}{\mathrm{kc}}{-}\frac{{54}{}\left({45}{}{{\mathrm{kc}}}^{{2}}{-}{342}{}{\mathrm{kc}}{+}{13437}\right)}{{5453}{-}{115}{}{\mathrm{kc}}}\right){}{\left({\mathrm{kc}}{+}{1}\right)}^{{2}}}{{5}{}{{\mathrm{kc}}}^{{2}}{-}{38}{}{\mathrm{kc}}{+}{1493}{-}\frac{\left({5453}{-}{115}{}{\mathrm{kc}}\right){}\left({9}{}{{\mathrm{kc}}}^{{2}}{+}{162}{}{\mathrm{kc}}{+}{153}{-}\frac{{11664}{}{\left({\mathrm{kc}}{+}{1}\right)}^{{2}}}{{5453}{-}{115}{}{\mathrm{kc}}}\right)}{{9}{}\left({895}{+}{31}{}{\mathrm{kc}}{-}\frac{{54}{}\left({45}{}{{\mathrm{kc}}}^{{2}}{-}{342}{}{\mathrm{kc}}{+}{13437}\right)}{{5453}{-}{115}{}{\mathrm{kc}}}\right)}}$ (4)
 > $\mathrm{Characterize}\left(\mathrm{sys},\frac{1}{3},2,\mathrm{controller}=\mathrm{\Pi },\mathrm{output}=\mathrm{relativestability}\right)$
 ${0}{<}{-}{18}{}{\mathrm{kc}}{+}{27}{}{\mathrm{ki}}{+}{58}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\mathbf{and}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{0}{<}{2106}{}{\mathrm{kc}}{-}{8406}{-}{243}{}{\mathrm{ki}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\mathbf{and}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{0}{<}{18}{}{\mathrm{kc}}{+}{27}{}{\mathrm{ki}}{-}{158}{-}\frac{{252}{}\left({-}{2016}{}{\mathrm{kc}}{+}{3024}{}{\mathrm{ki}}{+}{6496}\right)}{{2106}{}{\mathrm{kc}}{-}{8406}{-}{243}{}{\mathrm{ki}}}$ (5)
 > $\mathrm{Characterize}\left(\mathrm{sys},\frac{1}{3},2,\mathrm{controller}=\mathrm{\Pi }\right)$
 ${0}{<}{-}{18}{}{\mathrm{kc}}{+}{27}{}{\mathrm{ki}}{+}{58}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\mathbf{and}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{0}{<}{2106}{}{\mathrm{kc}}{-}{8406}{-}{243}{}{\mathrm{ki}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\mathbf{and}}\phantom{\rule[-0.0ex]{0.3em}{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]{0.3em}{0.0ex}}{\mathbf{and}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{0}{<}{{\mathrm{ki}}}^{{2}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\mathbf{and}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{0}{<}{-}{690}{}{\mathrm{kc}}{+}{32718}{+}{69}{}{\mathrm{ki}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\mathbf{and}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{0}{<}{62}{}{\mathrm{kc}}{-}{23}{}{\mathrm{ki}}{+}{1790}{-}\frac{{108}{}\left({270}{}{{\mathrm{kc}}}^{{2}}{-}{1080}{}{\mathrm{kc}}{-}{4579}{}{\mathrm{ki}}{+}{80622}{-}\frac{\left({4374}{}{\mathrm{ki}}{+}{157464}\right){}{\mathrm{kc}}}{{162}}\right)}{{-}{690}{}{\mathrm{kc}}{+}{32718}{+}{69}{}{\mathrm{ki}}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\mathbf{and}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{0}{<}{2}{}\left({\mathrm{kc}}{+}\frac{{\mathrm{ki}}}{{2}}{+}{1}\right){}{\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}}{-}{1080}{}{\mathrm{kc}}{-}{4579}{}{\mathrm{ki}}{+}{80622}{-}\frac{\left({4374}{}{\mathrm{ki}}{+}{157464}\right){}{\mathrm{kc}}}{{162}}\right)}{{-}{690}{}{\mathrm{kc}}{+}{32718}{+}{69}{}{\mathrm{ki}}}\right){}{{\mathrm{ki}}}^{{2}}}{{270}{}{{\mathrm{kc}}}^{{2}}{-}{1080}{}{\mathrm{kc}}{-}{4579}{}{\mathrm{ki}}{+}{80622}{-}\frac{\left({4374}{}{\mathrm{ki}}{+}{157464}\right){}{\mathrm{kc}}}{{162}}{-}\frac{\left({-}{690}{}{\mathrm{kc}}{+}{32718}{+}{69}{}{\mathrm{ki}}\right){}\left({18}{}{{\mathrm{kc}}}^{{2}}{+}\frac{\left({4374}{}{\mathrm{ki}}{+}{157464}\right){}{\mathrm{kc}}}{{486}}{-}{399}{}{\mathrm{ki}}{+}{306}{-}\frac{{108}{}\left({1296}{}{{\mathrm{kc}}}^{{2}}{+}\frac{{4}{}\left({5832}{}{\mathrm{ki}}{+}{52488}\right){}{\mathrm{kc}}}{{81}}{+}{324}{}{{\mathrm{ki}}}^{{2}}{-}{15840}{}{\mathrm{ki}}{+}{1296}{-}{108}{}\left({\mathrm{kc}}{+}\frac{{\mathrm{ki}}}{{2}}{+}{1}\right){}{\mathrm{ki}}\right)}{{-}{690}{}{\mathrm{kc}}{+}{32718}{+}{69}{}{\mathrm{ki}}}\right)}{{62}{}{\mathrm{kc}}{-}{23}{}{\mathrm{ki}}{+}{1790}{-}\frac{{108}{}\left({270}{}{{\mathrm{kc}}}^{{2}}{-}{1080}{}{\mathrm{kc}}{-}{4579}{}{\mathrm{ki}}{+}{80622}{-}\frac{\left({4374}{}{\mathrm{ki}}{+}{157464}\right){}{\mathrm{kc}}}{{162}}\right)}{{-}{690}{}{\mathrm{kc}}{+}{32718}{+}{69}{}{\mathrm{ki}}}}}{-}\frac{{36}{}\left({6}{}{{\mathrm{kc}}}^{{2}}{+}\frac{\left({4374}{}{\mathrm{ki}}{+}{157464}\right){}{\mathrm{kc}}}{{1458}}{-}{133}{}{\mathrm{ki}}{+}{102}{-}\frac{{36}{}\left({1296}{}{{\mathrm{kc}}}^{{2}}{+}\frac{{4}{}\left({5832}{}{\mathrm{ki}}{+}{52488}\right){}{\mathrm{kc}}}{{81}}{+}{324}{}{{\mathrm{ki}}}^{{2}}{-}{15840}{}{\mathrm{ki}}{+}{1296}{-}{108}{}\left({\mathrm{kc}}{+}\frac{{\mathrm{ki}}}{{2}}{+}{1}\right){}{\mathrm{ki}}\right)}{{-}{690}{}{\mathrm{kc}}{+}{32718}{+}{69}{}{\mathrm{ki}}}{-}\frac{\left({62}{}{\mathrm{kc}}{-}{23}{}{\mathrm{ki}}{+}{1790}{-}\frac{{108}{}\left({270}{}{{\mathrm{kc}}}^{{2}}{-}{1080}{}{\mathrm{kc}}{-}{4579}{}{\mathrm{ki}}{+}{80622}{-}\frac{\left({4374}{}{\mathrm{ki}}{+}{157464}\right){}{\mathrm{kc}}}{{162}}\right)}{{-}{690}{}{\mathrm{kc}}{+}{32718}{+}{69}{}{\mathrm{ki}}}\right){}\left({1296}{}{{\mathrm{kc}}}^{{2}}{+}\frac{{4}{}\left({5832}{}{\mathrm{ki}}{+}{52488}\right){}{\mathrm{kc}}}{{81}}{+}{324}{}{{\mathrm{ki}}}^{{2}}{-}{15840}{}{\mathrm{ki}}{+}{1296}{-}{108}{}\left({\mathrm{kc}}{+}\frac{{\mathrm{ki}}}{{2}}{+}{1}\right){}{\mathrm{ki}}{-}\frac{\left({-}{690}{}{\mathrm{kc}}{+}{32718}{+}{69}{}{\mathrm{ki}}\right){}\left({36}{}\left({\mathrm{kc}}{+}\frac{{\mathrm{ki}}}{{2}}{+}{1}\right){}{\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}}{-}{1080}{}{\mathrm{kc}}{-}{4579}{}{\mathrm{ki}}{+}{80622}{-}\frac{\left({4374}{}{\mathrm{ki}}{+}{157464}\right){}{\mathrm{kc}}}{{162}}\right)}{{-}{690}{}{\mathrm{kc}}{+}{32718}{+}{69}{}{\mathrm{ki}}}}\right)}{{3}{}\left({270}{}{{\mathrm{kc}}}^{{2}}{-}{1080}{}{\mathrm{kc}}{-}{4579}{}{\mathrm{ki}}{+}{80622}{-}\frac{\left({4374}{}{\mathrm{ki}}{+}{157464}\right){}{\mathrm{kc}}}{{162}}{-}\frac{\left({-}{690}{}{\mathrm{kc}}{+}{32718}{+}{69}{}{\mathrm{ki}}\right){}\left({18}{}{{\mathrm{kc}}}^{{2}}{+}\frac{\left({4374}{}{\mathrm{ki}}{+}{157464}\right){}{\mathrm{kc}}}{{486}}{-}{399}{}{\mathrm{ki}}{+}{306}{-}\frac{{108}{}\left({1296}{}{{\mathrm{kc}}}^{{2}}{+}\frac{{4}{}\left({5832}{}{\mathrm{ki}}{+}{52488}\right){}{\mathrm{k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(6)
 > $\mathrm{Characterize}\left(\mathrm{sys},\frac{1}{3},2,\mathrm{controller}=\mathrm{PID},\mathrm{output}=\mathrm{relativestability}\right)$
 ${0}{<}{27}{}{\mathrm{kd}}{+}{252}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\mathbf{and}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{0}{<}{81}{}{\mathrm{kc}}{-}{351}{-}\frac{{81}{}\left({-}{36}{}{\mathrm{kd}}{+}{18}{}{\mathrm{kc}}{+}{27}{}{\mathrm{ki}}{-}{158}\right)}{{27}{}{\mathrm{kd}}{+}{252}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\mathbf{and}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{0}{<}{12}{}{\mathrm{kd}}{-}{18}{}{\mathrm{kc}}{+}{27}{}{\mathrm{ki}}{+}{58}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\mathbf{and}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{0}{<}{-}{36}{}{\mathrm{kd}}{+}{18}{}{\mathrm{kc}}{+}{27}{}{\mathrm{ki}}{-}{158}{-}\frac{\left({27}{}{\mathrm{kd}}{+}{252}\right){}\left({48}{}{\mathrm{kd}}{-}{72}{}{\mathrm{kc}}{+}{108}{}{\mathrm{ki}}{+}{232}\right)}{{81}{}{\mathrm{kc}}{-}{351}{-}\frac{{81}{}\left({-}{36}{}{\mathrm{kd}}{+}{18}{}{\mathrm{kc}}{+}{27}{}{\mathrm{ki}}{-}{158}\right)}{{27}{}{\mathrm{kd}}{+}{252}}}$ (7)