4) A pure integrator is preceded by a defective half zero order hold as shown in the figure below, where At is the sampling time, as shown in Fig. 2. u(k) D (A) G(z) Figure 2: Sampled Data System In this problem, you will obtain G(z) using two different methods: Method 1 - Using state space models: (a) Determine coefficients a, b and c of the continuous time first order model x(t) y(t) = == ax(t) + bu(t) cx(t) such that G(s) cb = 1 = s-a S (b) Determine coefficients a, and b of the discrete time first order model x(k) = y(k) = ax(k)+bu(k) cx(k) following a similar analysis to the one presented in lectures for computing the ma- trices A and B from the continuos time matrices A and B, for the zero order hold. (c) Determine G(2) = cbd z-ad Method 2 Direct computation of G(2): (a) Assume that the input is a pulse u(k) = 8(k) such that U(z) = 1, and plot the resulting output y(t). (b) Determine the sampled sequence y (k) = y (Atk). (c) Compute G(z) = Z {yz(k)}.