Consider a system with the following transfer function y(s) u(s) = s z s2 2s 2 n where = n and [0, 1). (a) (2pt) Write state-space equations for the system using x1 = y and x2 = y as the state variables and y as the output. Verify your answer by computing the transfer function from the state-space equations. (b) (2pt) Write state-space equations for the system using x1 = x and x2 = x as the state variables and y as the output where x(s) u(s) = 1 s2 2s 2 n y(s) x(s) = s z. Verify your answer by computing the transfer function from the state-space equations. (c) (2pt) Compute the matrix exponential eAt where A is the system matrix of the state- space equations. Note d = n p1 2 for [0, 1). (d) (2pt) Verify your matrix exponential eAt from (c) by showing eA0 = I and d dt eAt = AeAt. (e) (2pt) Compute the impulse response of the system using h(t) = CeAtB(t) D(t) and verify your answer using the expression of the impulse response h(t) of the system H(s) = 2 n s2 2s 2