4. Consider a two-mass two-spring system below 0 d dt Umi Vm Sky fk 0 0 0 k 0 -k k my 0 0 0 ma Umi 101 fk. m 0 fki 0 Assume m =m = 1 and k = k = 1. (a) Show that the transfer function from u to Uma is given by s (s+2) +38 +1 m2 + 0 U Vm (s) U (s) (b) For this system, an optimal LQ problem is formulated with the quadratic performance index J = [ {[2 m (1) + R (1)] dt where m = [0 0 1 1] x. Obtain the optimal feedback control gain for R= 0.0001, 1 and 1,000. For each R, use Matlab or Python to obtain the response of the optimal closed loop system for fk, (0) = 1. The initial conditions of other state variables are zero. Hint: the following MATLAB command may be of help: care, lqr, initial.