Problem 4. (10 points) Given the state-space model x = Ax+ Bu y = Cx where 0 A = 1 ½). B = (1), C = (1 0) 1. Verify that the system is controllable and observable; 2. Compute the transfer function of the system; 3. Design an observer-based integral compensator of the form ε=y-r - A+ Bu-. L(C – y) u= -Κιξ - K22 where r(t) is an external reference signal, such that the overall closed-loop system has characteristic polynomial Pa(A) = (X + 1)(A+2)(A+3)(A+ 10)(A+ 15) 4. Plot the response of the closed loop system to a step reference signal, and find the steady-state error, the percent overshoot and the settling time. Relate your findings to the location of the dominant eigenvalues of the closed-loop system. Extra Credit (5 points): Consider the modified compensator = k(y-r) = A+ Bu LC + kLy α = -Κ1ξ - Ka where L, K1, K2 have been computed in the previous part of the problem, and k > 0 is a tunable gain. Using root locus methods, find the critical values of the gain k, that is, the values corresponding to closed-loop eigenvalues on the imaginary axis.