EP3.3. Método de estabilidad de Routh-Hurwitz

Dados los siguientes modelos o polinomios característicos, (i) aplicar las condiciones necesarias y suficientes del método de Routh-Hurwitz para determinar la estabilidad o condición de estabilidad; (ii) verificar los resultados utilizando la función routh_hurwitz de MATLAB; (iii) verificar los resultados calculando con la función roots de MATLAB las raíces características del polinomio y, para los casos de estabilidad condicional, probar con algunos valores del parámetro por dentro y fuera de la región de estabilidad.

A. Casos simples

En el caso de sistemas inestables, determinar el número de raíces en el semiplano derecho complejo.

$P(\lambda )=\lambda ^2+2\lambda +10$
$P(\lambda )=\lambda ^2+2\lambda +10000$
$P(\lambda )=\lambda ^2+2\lambda -1$
$P(\lambda )=\lambda ^2-2\lambda -4$
$P(\lambda )=-\lambda ^2-2\lambda -10$
$P(\lambda )=\lambda ^3+6\lambda ^2+11\lambda +6$
$P(\lambda )=\lambda ^3+6\lambda ^2+11\lambda -6$
$P(\lambda )=\lambda ^3+6\lambda ^2-11\lambda +6$
$P(\lambda )=\lambda ^3-6\lambda ^2+11\lambda +6$
$P(\lambda )=\lambda ^4+5\lambda ^3+9\lambda ^2+7\lambda +2$
$P(\lambda )=\lambda ^3+2\lambda ^2+2\lambda +1$
$P(\lambda )=\lambda ^4+5\lambda ^3+6\lambda ^2+5\lambda +1$
$P(\lambda )=\lambda ^4+5\lambda ^3+2\lambda ^2-5\lambda +1$

B. Cálculo de intervalos de estabilidad en casos simples

$P(\lambda )=\lambda +k$
$P(\lambda )=0.5\lambda +k$
$P(\lambda )=\lambda ^2+(k-2)\lambda +2$
$P(\lambda )=(\lambda +1)(\lambda +2)^2[\lambda ^2+(k-2)\lambda +2]$
$P(\lambda )=(\lambda +1)(\lambda +2)^2[\lambda ^2+(k-2)\lambda +2]$
$P(\lambda )=\lambda ^2+(k-2)\lambda +3-k$
$P(\lambda )=\lambda ^2+(10-k)\lambda +k-7$
$P(\lambda )=(k-3)\lambda ^2+(5-k)\lambda +12-3k$
$P(\lambda )=\lambda ^3+2\lambda ^2+4\lambda +k$
$P(\lambda )=\lambda ^3+\lambda ^2+k\lambda +3$
$P(\lambda )=\lambda ^3+\lambda ^2+(k-5)\lambda +3$
$P(\lambda )=\lambda ^3+\lambda ^2+(2-k)\lambda +1$
$P(\lambda )=\lambda ^3+\lambda ^2+(2k-1)\lambda +k$
$P(\lambda )=\lambda ^3+a\lambda ^2+b\lambda +c$
$P(\lambda )=\lambda ^4+\lambda ^3+2\lambda ^2+\lambda +k$
$P(\lambda )=\lambda ^4+\lambda ^3+2k\lambda ^2+\lambda +k$
$P(\lambda )=\lambda ^4+\lambda ^3+2\lambda ^2+k\lambda +1$
$P(\lambda )=\lambda ^4+\lambda ^3+k\lambda ^2+\lambda +1$
$P(\lambda )=\lambda ^4+k\lambda ^3+2\lambda ^2+\lambda +1$

C. Cálculo de intervalos de estabilidad

$P(\lambda )=(\lambda +1)(\lambda +2)^2(\lambda ^2+K)$
$P(\lambda )=\lambda ^3+(11+k)\lambda ^2+(10+75k)\lambda +1250k$
$P(\lambda )=\lambda ^3+(12+k)\lambda ^2+(2+k)\lambda +25k$
$P(\lambda )=\lambda ^3+(k-2)\lambda ^2+(2+k)\lambda +25k$
$P(\lambda )=\lambda ^4+k\lambda ^3+2k\lambda ^2+4k\lambda +8k-16$
$P(\lambda )=\lambda ^4+\lambda ^3+\lambda ^2+\lambda +k$
$P(\lambda )=\lambda ^4+6\lambda ^3+11\lambda ^2+(6+k)\lambda +2k$
$P(\lambda )=\lambda ^4+3\lambda ^3+3\lambda ^2+2\lambda +k$
$P(\lambda )=\lambda ^4+k\lambda ^3+4\lambda ^2+7\lambda +2$
$P(s)=s^5+s^4+10s^3+Ks^2+2Ks+K$
$P(s)=s^4+18s^3+as^2+4s+b$
$P(s)=s^2+(K_d+3)s+0.5K_p$
$P(s)=s^3+k_1s^2+k_2s+k_3$
$P(s)=s^3+100s^2+(90+10k_1k_2)s+90k_1k_2$
$P(s)=s^3+6.8s^2+(K_d+5)s+K_p+4$
$P(s)=s^4+21s^3+10s^2+20K_ps+60K_i$
$s^3+as^2+(ab-1)s+2a(b-2)$
$G(s)=\frac{s-3k}{s^4-(k-3)s^3+2s^2+k(3-k)s+k(2-k)}$

D. Casos con ecuaciones de estado

$\dot{\mathbf{x}}=\left[ \begin{matrix} 0& 1& 0\\ 0& 0& 1\\ -1& -k& -2\\\end{matrix} \right] \mathbf{x}$
$\dot{\mathbf{x}}=\left[ \begin{matrix} 0& 1& 0\\ 0& 0& 1\\ -k& -k& -k\\\end{matrix} \right] \mathbf{x}$