B | y=x, y=x u(t) Bx(t) M Kx(t) Mx(t)- M u(t) K U(s) S -1 x2(t)...

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B | y=x, y=x u(t) Bx(t) M Kx(t) Mx(t)- M u(t) K U(s) S -1 x2(t) S u(t) -1 Y(s) x(t) 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), x2(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 A), the damping ratio 5, and the natural frequency @. 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%) B | y=x, y=x u(t) Bx(t) M Kx(t) Mx(t)- M u(t) K U(s) S -1 x2(t) S u(t) -1 Y(s) x(t) 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), x2(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 A), the damping ratio 5, and the natural frequency @. 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%)