$\mathrm{sys1}≔\mathrm{NewSystem}\left(\frac{s+1}{s^3+s^2+13s+12}\right)\:$

$\mathrm{p1}≔\left[-3+2\mathrm{I}\,-3-2\mathrm{I}\,-10\right]\:$

$\mathrm{DominantPole}\left(\mathrm{sys1}\,\mathrm{p1}\,'\mathrm{controller}'=\mathrm{PID}\right)$

$\left[\frac{1445}{24}\,\frac{4745}{36}\,\frac{1153}{72}\right],\left[\mathrm{-10}\,-\frac{73}{72}\,\mathrm{-3}-2\mathrm{I}\,\mathrm{-3}+2\mathrm{I}\right]$

$\mathrm{DominantPole}\left(\mathrm{sys1}\,\mathrm{p1}\,'\mathrm{controller}'=\mathrm{PID}\,'\mathrm{factored}'=\mathrm{true}\right)$

$\left[\frac{1445}{24}\,\frac{867}{1898}\,\frac{1153}{4335}\right],\left[\mathrm{-10}\,-\frac{73}{72}\,\mathrm{-3}-2\mathrm{I}\,\mathrm{-3}+2\mathrm{I}\right]$

$\mathrm{pid}≔\mathrm{DominantPole}\left(\mathrm{sys1}\,\mathrm{p1}\,'\mathrm{controller}'=\mathrm{PID}\,'\mathrm{returntype}'=\mathrm{record}\right)$

$\mathrm{pid}≔{\mathrm{Record}}_{\mathrm{packed}}\left(\mathrm{Kp}=\frac{1445}{24}\,\mathrm{Ki}=\frac{4745}{36}\,\mathrm{Kd}=\frac{1153}{72}\,\mathrm{closedPoles}=\left[\mathrm{-10}\,-\frac{73}{72}\,\mathrm{-3}-2\mathrm{I}\,\mathrm{-3}+2\mathrm{I}\right]\right)$

$\mathrm{pid}≔\mathrm{DominantPole}\left(\mathrm{sys1}\,\mathrm{p1}\,'\mathrm{controller}'=\mathrm{PID}\,'\mathrm{factored}'=\mathrm{true}\,'\mathrm{returntype}'=\mathrm{record}\right)$

$\mathrm{pid}≔{\mathrm{Record}}_{\mathrm{packed}}\left(K=\frac{1445}{24}\,\mathrm{Ti}=\frac{867}{1898}\,\mathrm{Td}=\frac{1153}{4335}\,\mathrm{closedPoles}=\left[\mathrm{-10}\,-\frac{73}{72}\,\mathrm{-3}-2\mathrm{I}\,\mathrm{-3}+2\mathrm{I}\right]\right)$

$\mathrm{pidsys}≔\mathrm{DominantPole}\left(\mathrm{sys1}\,\mathrm{p1}\,'\mathrm{controller}'=\mathrm{PID}\,'\mathrm{returntype}'=\mathrm{system}\right)\:$

Warning, improper transfer-function, degree(num) > degree(den)

$\mathrm{PrintSystem}\left(\mathrm{pidsys}\right)$

$\left[\begin{array}{l}\mathbf{Transfer\; Function}\\ \mathrm{continuous}\\ \mathrm{1\; output(s);\; 1\; input(s)}\\ \mathrm{inputvariable}\=\left[\mathrm{u1}\left(s\right)\right]\\ \mathrm{outputvariable}\=\left[\mathrm{y1}\left(s\right)\right]\\ {\mathrm{tf}}_{1,1}\=\frac{1153s^2+4335s+9490}{72s}\end{array}\right.$

Get the closed-loop poles

$\mathrm{zpksys}≔\mathrm{PIDClosedLoop}\left(\mathrm{sys1}\,\mathrm{pidsys}\,'\mathrm{outputtype}'=\mathrm{zpk}\right)\:$

Warning, improper transfer-function, degree(num) > degree(den)

$\mathrm{PrintSystem}\left(\mathrm{zpksys}\right)$

$\left[\begin{array}{l}\mathbf{Zero\; Pole\; Gain}\\ \mathrm{continuous}\\ \mathrm{1\; output(s);\; 1\; input(s)}\\ \mathrm{inputvariable}\=\left[\mathrm{y1\_ref}\left(s\right)\right]\\ \mathrm{outputvariable}\=\left[\mathrm{y1}\left(s\right)\right]\\ {\mathrm{z}}_{1,1}\=\left[-\frac{4335}{2306}-\frac{\mathrm{I}\sqrt{24975655}}{2306}\,-\frac{4335}{2306}+\frac{\mathrm{I}\sqrt{24975655}}{2306}\,\mathrm{-1}\right]\\ {\mathrm{p}}_{1,1}\=\left[\mathrm{-10}\,\mathrm{-3}-2\mathrm{I}\,\mathrm{-3}+2\mathrm{I}\,-\frac{73}{72}\right]\\ {\mathrm{k}}_{1,1}\=\frac{1153}{72}\end{array}\right.$

$\mathrm{p1}≔\left[-3+2\mathrm{I}\,-3-2\mathrm{I}\right]\:$

$\mathrm{DominantPole}\left(\mathrm{sys1}\,\mathrm{p1}\,'\mathrm{controller}'=\mathrm{\Pi }\right)$

Warning, controller synthesis requires negative gain(s)

$\left[-\frac{287}{8}\,-\frac{611}{8}\right],\left[\frac{5}{2}-\frac{\sqrt{194}}{4}\,\frac{5}{2}+\frac{\sqrt{194}}{4}\,\mathrm{-3}-2\mathrm{I}\,\mathrm{-3}+2\mathrm{I}\right]$

$\mathrm{PIpars}≔\mathrm{DominantPole}\left(\mathrm{sys1}\,\mathrm{p1}\,'\mathrm{controller}'=\mathrm{\Pi }\,'\mathrm{returntype}'=\mathrm{record}\right)$

Warning, controller synthesis requires negative gain(s)

$\mathrm{PIpars}≔{\mathrm{Record}}_{\mathrm{packed}}\left(\mathrm{Kp}=-\frac{287}{8}\,\mathrm{Ki}=-\frac{611}{8}\,\mathrm{closedPoles}=\left[\frac{5}{2}-\frac{\sqrt{194}}{4}\,\frac{5}{2}+\frac{\sqrt{194}}{4}\,\mathrm{-3}-2\mathrm{I}\,\mathrm{-3}+2\mathrm{I}\right]\right)$

$\mathrm{PIsys}≔\mathrm{DominantPole}\left(\mathrm{sys1}\,\mathrm{p1}\,'\mathrm{controller}'=\mathrm{\Pi }\,'\mathrm{returntype}'=\mathrm{system}\right)\:$

Warning, controller synthesis requires negative gain(s)

$\mathrm{PrintSystem}\left(\mathrm{PIsys}\right)$

$\left[\begin{array}{l}\mathbf{Transfer\; Function}\\ \mathrm{continuous}\\ \mathrm{1\; output(s);\; 1\; input(s)}\\ \mathrm{inputvariable}\=\left[\mathrm{u1}\left(s\right)\right]\\ \mathrm{outputvariable}\=\left[\mathrm{y1}\left(s\right)\right]\\ {\mathrm{tf}}_{1,1}\=\frac{-287s-611}{8s}\end{array}\right.$

Get the closed-loop poles

$\mathrm{zpksys}≔\mathrm{PIDClosedLoop}\left(\mathrm{sys1}\,\mathrm{PIsys}\,'\mathrm{outputtype}'=\mathrm{zpk}\right)\:$

$\mathrm{PrintSystem}\left(\mathrm{zpksys}\right)$

$\left[\begin{array}{l}\mathbf{Zero\; Pole\; Gain}\\ \mathrm{continuous}\\ \mathrm{1\; output(s);\; 1\; input(s)}\\ \mathrm{inputvariable}\=\left[\mathrm{y1\_ref}\left(s\right)\right]\\ \mathrm{outputvariable}\=\left[\mathrm{y1}\left(s\right)\right]\\ {\mathrm{z}}_{1,1}\=\left[-\frac{611}{287}\,\mathrm{-1}\right]\\ {\mathrm{p}}_{1,1}\=\left[\mathrm{-3}-2\mathrm{I}\,\mathrm{-3}+2\mathrm{I}\,\frac{5}{2}-\frac{\sqrt{194}}{4}\,\frac{5}{2}+\frac{\sqrt{194}}{4}\right]\\ {\mathrm{k}}_{1,1}\=-\frac{287}{8}\end{array}\right.$

$\mathrm{sys2}≔\mathrm{NewSystem}\left(\frac{s+a}{s^4+5s^3+s^2+bs+12}\right)\:$

$\mathrm{p2}≔\left[-3+b\mathrm{I}\,-3-b\mathrm{I}\,-7\right]\:$

Get a sequence of two lists with the PID controller gains and closed-loop poles

$\mathrm{DominantPole}\left(\mathrm{sys2}\,\mathrm{p2}\right)$

$\left[\frac{\left(-b^4-53b^2+2238\right)a^2+\left(6b^4-b^3+17b^2-51b-14289\right)a+7\left(b^2+9\right)\left(b^2+b+345\right)}{\left(-7+a\right)\left(a^2+b^2-6a+9\right)}\,-\frac{7\left(b^2+9\right)\left(\left(b^2-54\right)a^2+\left(b^2+b+345\right)a-56b^2-516\right)}{\left(-7+a\right)\left(a^2+b^2-6a+9\right)}\,\frac{\left(-14b^2-b+357\right)a^2+\left(b^4+53b^2-2238\right)a-7b^4+315b^2+3402}{\left(-7+a\right)\left(a^2+b^2-6a+9\right)}\right],\left[\mathrm{-7}\,-3+\mathrm{I}b\,-3-\mathrm{I}b\,\frac{4a^3+4a{b}^2-52a^2-28b^2+\sqrt{a^6{b}^2+a^4{b}^4-12a^5{b}^2-6a^3{b}^4-38a^6+a^{5}b-40a^4{b}^2+a^3{b}^3-47a^2{b}^4+631a^5-13a^{4}b+799a^3{b}^2-7a^2{b}^3+168a{b}^4-3419a^4+51a^{3}b-1306a^2{b}^2+784b^4+4473a^3-63a^{2}b-6300a{b}^2+19773a^2+14112b^2-70308a+63504}+204a-252}{a^3+a{b}^2-13a^2-7b^2+51a-63}\,-\frac{-4a^3-4a{b}^2+52a^2+28b^2+\sqrt{a^6{b}^2+a^4{b}^4-12a^5{b}^2-6a^3{b}^4-38a^6+a^{5}b-40a^4{b}^2+a^3{b}^3-47a^2{b}^4+631a^5-13a^{4}b+799a^3{b}^2-7a^2{b}^3+168a{b}^4-3419a^4+51a^{3}b-1306a^2{b}^2+784b^4+4473a^3-63a^{2}b-6300a{b}^2+19773a^2+14112b^2-70308a+63504}-204a+252}{a^3+a{b}^2-13a^2-7b^2+51a-63}\right]$

$\mathrm{sys3}≔\mathrm{NewSystem}\left(\frac{s+a}{s^3+s^2+bs+c}\right)\:$

$\mathrm{p3}≔\left[-2+\mathrm{I}\,-2-\mathrm{I}\right]\:$

Get a record with the PI controller gains and closed-loop poles

$\mathrm{PIrec}≔\mathrm{DominantPole}\left(\mathrm{sys3}\,\mathrm{p3}\,'\mathrm{controller}'=\mathrm{\Pi }\,'\mathrm{returntype}'=\mathrm{record}\right)$

$\mathrm{PIrec}≔{\mathrm{Record}}_{\mathrm{packed}}\left(\mathrm{Kp}=\frac{\left(4b-c+13\right)a-5b-35}{a^2-4a+5}\,\mathrm{Ki}=\frac{\left(5b+35\right)a-5c-75}{a^2-4a+5}\,\mathrm{closedPoles}=\left[\mathrm{-2}+\mathrm{I}\,\mathrm{-2}-\mathrm{I}\,\frac{3a^2-12a+15+\sqrt{-4a^{4}b-19a^4+16a^{3}b+4a^{3}c+100a^3-20a^{2}b-16a^{2}c-146a^2+20ac-60a+225}}{2\left(a^2-4a+5\right)}\,-\frac{-3a^2+\sqrt{-4a^{4}b-19a^4+16a^{3}b+4a^{3}c+100a^3-20a^{2}b-16a^{2}c-146a^2+20ac-60a+225}+12a-15}{2\left(a^2-4a+5\right)}\right]\right)$

Get the PI controller transfer function

$\mathrm{PIsys}≔\mathrm{DominantPole}\left(\mathrm{sys3}\,\mathrm{p3}\,'\mathrm{controller}'=\mathrm{\Pi }\,'\mathrm{returntype}'=\mathrm{system}\right)\:$

$\mathrm{PrintSystem}\left(\mathrm{PIsys}\right)$

$\left[\begin{array}{l}\mathbf{Transfer\; Function}\\ \mathrm{continuous}\\ \mathrm{1\; output(s);\; 1\; input(s)}\\ \mathrm{inputvariable}\=\left[\mathrm{u1}\left(s\right)\right]\\ \mathrm{outputvariable}\=\left[\mathrm{y1}\left(s\right)\right]\\ {\mathrm{tf}}_{1,1}\=\frac{\left(4ab-ac+13a-5b-35\right)s+5ab+35a-5c-75}{\left(a^2-4a+5\right)s}\end{array}\right.$

Get the closed-loop poles

$\mathrm{zpksys}≔\mathrm{PIDClosedLoop}\left(\mathrm{sys3}\,\mathrm{PIsys}\,'\mathrm{outputtype}'=\mathrm{zpk}\right)\:$

$\mathrm{PrintSystem}\left(\mathrm{zpksys}\right)$

$\left[\begin{array}{l}\mathbf{Zero\; Pole\; Gain}\\ \mathrm{continuous}\\ \mathrm{1\; output(s);\; 1\; input(s)}\\ \mathrm{inputvariable}\=\left[\mathrm{y1\_ref}\left(s\right)\right]\\ \mathrm{outputvariable}\=\left[\mathrm{y1}\left(s\right)\right]\\ {\mathrm{z}}_{1,1}\=\left[-a\,-\frac{5\left(ab+7a-c-15\right)}{4ab-ac+13a-5b-35}\right]\\ {\mathrm{p}}_{1,1}\=\{\begin{array}{l}\[\mathrm{-2}+\mathrm{I},\\ \mathrm{-2}-\mathrm{I},\\ \frac{3a^2-12a+15+\sqrt{-4a^{4}b-19a^4+16a^{3}b+4a^{3}c+100a^3-20a^{2}b-16a^{2}c-146a^2+20ac-60a+225}}{2\left(a^2-4a+5\right)},\\ -\frac{-3a^2+\sqrt{-4a^{4}b-19a^4+16a^{3}b+4a^{3}c+100a^3-20a^{2}b-16a^{2}c-146a^2+20ac-60a+225}+12a-15}{2\left(a^2-4a+5\right)}\]\end{array}\\ {\mathrm{k}}_{1,1}\=\frac{4\left(ab-\frac{1}{4}ac+\frac{13}{4}a-\frac{5}{4}b-\frac{35}{4}\right)}{a^2-4a+5}\end{array}\right.$