E12.11 Consider a second-order system with the following state space representation

For a nominal value of p = 20, what is the range of the system damping ratio? Plot the root locus with variation of k. If the system is required to have a percent overshoot of less than 10%, a controller is added to improve damping capability by increasing parameter k. What is the minimum value of k to maintain the required percent overshoot?

x(t)=Ax(t) + Bu(t) y(t) = Cx(t), 0 1 where A = -P- -k 'k | p > 0, k > 0, B = [ 9 ] and C = [10]. a. What are the system's natural frequency w,, and damping ratio (as functions of system parameters p and k? b. Given the parameter values of p and k vary in intervals of 5 p < 50 and 1 < k < 10, what will be the ranges of variation of, and (?