Answer Happy

13.8 The system shown in Figure 13.8 is described by the equations: 21(t) 22(t) K mi m2 u(t) dt) Figure 13.8 Two-mass spring system. 0 0 101 0 0 0 0 01 it) 12(t) 3 (t) 24t) k k [2(t) 22(t) 23(t) 24(t) + 1 u(t) + 00 d(t) mi mi mi k k 0 0 0 m2 m2 m2 where 21(t) is the position of body 1 (m) x2(t) is the position of body 2 (m) 13(t) is the velocity of body 1 (m/s) 24(t) is the velocity of body 2 (m/s) u(t) is the control-force input (N) d(t) is a disturbance-force input (N) Assume mı = m2 = k = 1, d(t) = 0. (1) Design a time-invariant LQR controller with Qı = diag[1 0 0 0 and R= [1]. =
(2) Compare the transient behavior of the state components xi(t) and x2(t) and the control effort u(t), given the initial condition x(0) = [1 0 0 0 and the control law above, with that obtained if Qı were changed to Q2 = diag[10 0 0 0).