Example $5\mathrm{.}1$: Control System Stability

Consider the feedback control system with the following transfer functions:

$G_p=G_d=\frac{1}{5s+1},G_m=\frac{1}{s+1},G_v=\frac{1}{2s+1}$ and $G_c=k_c$

Show that the closed$-$loop system produces unstable responses if controller gain $Kc$ is too large.

$10/29/2023$

$5$

Example $5\mathrm{.}1$: Control System Stability

Solution:

The transfer function for set$-$point changes is:

$\frac{Y}{Y_{sp}}=\frac{K_m{G}_c{G}_v{G}_p}{1+G_c{G}_v{G}_p{G}_m}$

Consider a unit step change in set$-$point $.Y_{sp}(s)=\frac{1}{s}$

Substituting $(5\mathrm{.}1)$ in $(5\mathrm{.}2),$ and rearranging gives us

$Y(s)=\frac{K_c(s+1)}{10s^3+17s^2+8s+1+K_c}\frac{1}{s}$

Tips

$$ Specify $Kc$

$$ Determined the roots of the cubic polynomial in $s($using standard root$-$finding techniques$)$

$$ Perform the partial fraction expansion.

$$ Determine the inverse Laplace transform

$$ Determine $y(t)$