Question

R(s)+ G_c(s) G_p(s) C(s) H(s) Figure 1. Feedback closed-loop system. 1. (15%) Consider the feedback closed-loop system in Figure 1 with $G_c(s)G_p(s)H(s) = \frac{10(s+12)}{s^4 + 20s^3 + (5k+116)s^2 + 50(k+3)s + 120(k-1)}$ where $k$ is a gain parameter that may take any value in the interval $[0, +\infty)$. a) Plot the root locus that gives the location of the closed-loop transfer function poles as parameter $k$ varies from 0 to infinity. You may use MATLAB yet clearly indicate breakaway points, crossings of the imaginary axis, and asymptotes, if any. b) Using Routh-Hurwutz criterion compute the range of nonnegative values, if any, for $k$ such that the system is stable and has its settling time $T_s$ strictly less than 1s (that is, $T_s < 1$). c) Find the nonnegative value of $k$ for which the closed-loop system has the minimal settling time while the damping ratio $\zeta$ is maximal. Compute that minimal value of $T_s$ and the corresponding maximal value of $\zeta$.

          R(s)+
G_c(s)
G_p(s)
C(s)
H(s)
Figure 1. Feedback closed-loop system.
1. (15%) Consider the feedback closed-loop system in Figure 1 with
$G_c(s)G_p(s)H(s) = \frac{10(s+12)}{s^4 + 20s^3 + (5k+116)s^2 + 50(k+3)s + 120(k-1)}$ where $k$ is a gain parameter that may
take any value in the interval $[0, +\infty)$.
a) Plot the root locus that gives the location of the closed-loop transfer function poles as parameter $k$
varies from 0 to infinity. You may use MATLAB yet clearly indicate breakaway points, crossings of the
imaginary axis, and asymptotes, if any.
b) Using Routh-Hurwutz criterion compute the range of nonnegative values, if any, for $k$ such that the
system is stable and has its settling time $T_s$ strictly less than 1s (that is, $T_s < 1$).
c) Find the nonnegative value of $k$ for which the closed-loop system has the minimal settling time while
the damping ratio $\zeta$ is maximal. Compute that minimal value of $T_s$ and the corresponding maximal
value of $\zeta$.

R(s)+
Gc(s)
Gp(s)
C(s)
H(s)
Figure 1. Feedback closed-loop system.
1. (15%) Consider the feedback closed-loop system in Figure 1 with
Gc(s)Gp(s)H(s) = (10(s+12))/(s^4 + 20s^3 + (5k+116)s^2 + 50(k+3)s + 120(k-1)) where k is a gain parameter that may
take any value in the interval [0, +∞).
a) Plot the root locus that gives the location of the closed-loop transfer function poles as parameter k
varies from 0 to infinity. You may use MATLAB yet clearly indicate breakaway points, crossings of the
imaginary axis, and asymptotes, if any.
b) Using Routh-Hurwutz criterion compute the range of nonnegative values, if any, for k such that the
system is stable and has its settling time Ts strictly less than 1s (that is, Ts < 1).
c) Find the nonnegative value of k for which the closed-loop system has the minimal settling time while
the damping ratio ζ is maximal. Compute that minimal value of Ts and the corresponding maximal
value of ζ.

Added by Shane Y.

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