Use spectral factorization to determine the optimal control

Question

Question: Use spectral factorization to determine the optimal control gains for (1), with ζ = 0 3, ωn = π, τ = 0.2, R= 1, and

(a) Q = Diag[1, 0, 0]

(b) Q = Diag[0, 1, 0]

(c) Determine the gain and phase margins in (a) and (b). What is the corresponding allowable pure time delay?