Problem 5 (20%) B 1 K U(s) u(t) | | y=x, ! M | y=x, y=x u(t) Bx(t) Kx(t)+ Mix(t) M u(t) x (1) 1 Y(s) x (1) y(t) (5a) Consider the MBK (mass-damper-spring) system, assume M = 1, B = 1, K = 4. Show that the system can be represented by the signal flow graph shown above. (3%) (5b) Write down the state-space model (A,B,C) equations of the system according to the signal flow graph, which is a state diagram in this case. (Note that this B is not the friction coefficient.) (3%) (5c) Assume x(0) = 10, x2 (0) = 0, and u(t) = 0, solve the state equation for x(t), x(t) and plot them as functions of time using MATLAB plot command. (3%) (5d) Find the characteristic equation det(sI - A) = 0, the poles (the eigenvalues of 4), the damping ratio 5, and the natural frequency @R. Comment on the relevance between these parameters and the system dehavior graphs obtained in 5c (3%) (5e) Find a state feedback controller, u(t) = Fx(t) = Fx(t) + F2x2(t), so that the closed-loop system eigenvalues (the eigenvalues of A+ BF) are -1+j and -1-j. (3%) (5f) What are the damping ratio 5 and the natural frequency @ of the closed-loop system? (3%) (5g) For the closed-loop system, let x(0) = 10, x2 (0) = 0, solve the closed-loop state equation for x(t), x2(t) and plot them as functions of time using MATLAB plot command. (2%) (5h) Comment on the performances of the open-loop and the closed-loop system. (5%)