1. Consider the mass spring damper system 1 0 q c c q k k q 1 0 1 + 11 12 1 + 11 12 1 = u1 + u2 0 1 q2 c12 c22 q2 k12 k22 q2 0 1 (1.1) where the damping constants c11 , c12 , c22 and the spring constants k11 , k12 , k22 are unknown. Finally, q is the position of the mass. To find the constants assume that one applies four different inputs to (1.1) and measures the position q(t), that is, If u1 (t) = 3, u2 (t) = 0, then q1 (t) 2 = lim t q2 (t) 1 If u1 (t) = 0, u2 (t) = 3, then q1 (t) 1 = lim t q2 (t) 2 If u1 (t) = 2 cos(t), u2 (t) = 0, then the steady state output sin(t) q (t) 1ss = sin(t) q2ss (t) If u1 (t) = 0, u2 (t) = 2 cos(t), then the steady state output sin(t) q1ss (t) = q2ss (t) 5 cos(t arctan(2)) Find c11 , c12 , c22 and k11 , k12 , k22 . Hint: you may have to invert 2 2 matrices, and match the real and imaginary parts. 2 Problem 2. Consider the square wave u(t) with period 2 generated by u(t) = 4 = if 0 t < 4 if t < 2 Then u(t) is extended periodically for all t > 2. Consider the transfer function G(s) = 5s2 s +s+5 (i) Find the Fourier series expansion for u of the form u(t) = a0 + X k cos(kt) + k sin(kt) k=1 (ii) Find the steady state output yss (t) for the square wave input u(t) in terms of an infinite sum of sinusoids. (iii) Use lsim in Matlab to plot u(t), the output y(t), yss (t) and sin(t) on the same graph over 0 t 16. The square command in Matlab may be helpful. (iv) Show that yss (t) sin(t), that is, compute Z 2 1 |yss (t) sin(t)|2 dt e= 2 0 You will need Matlab to compute this. (v) Plot the Bode plot of G. Use the magnitude and phase of this Bode plot to explain in a couple of sentences why G acts like a \"filter\" to extract sin(t) from the square wave input u(t). 3 Problem 3. Consider the mass spring damper system 2 1 q1 2 1 q 1 1 q1 7 + 1 + = u 1 1 q2 1 1 q2 1 2 q2 3 (i) Is this system stable? (ii) Find the transfer functions G1 (s) = Q1 (s) U (s) and G2 (s) = Q2 (s) . U (s) (iii) Convert this mass spring damper system to a state space system of the form x = Ax + Bu where the state q 1 q2 x= q1 q2 (iv) If u(t) = 1 2 cos(t), then find the steady state response xss (t). Notice that xss (t) is a vector of length 4. 4 Problem 4. Recall that cos() = 1 2 \u0001 ei + ei and sin() = 1 2i \u0001 ei ei . If is a square matrix, then cos(t) and sin(t) are the matrices defined by \u00011 \u0001 1 it and L cos(t) = s s2 I + 2 e + eit 2 \u00011 \u0001 1 it sin(t) = and L sin(t) = s2 I + 2 e eit 2i cos(t) = Assume that 1 = 1 2 2 Find cos(t) and sin(t). Find cos(t)2 + sin(t)2 . 5