Question

Consider the electrical circuit shown in the figure below. The circuit consists of an inductance L = 1 henry, a resistance R = 2 ohm, and a capacitanceC = 1 farad. Applying Kirchhoff's voltage law yields, L \frac{d i}{d t}+R i+\frac{1}{C} \int i d t=e_{i} \frac{1}{C} \int i d t=e_{0} a) Assuming e; is the input u, and eo, the output y, derive the transfer function of the system from the input u to output y.(2 Morko) b) Define state variables by z_{1}=e_{0} z_{2}=\dot{e}_{n} derive the state-space representation of the system. c) Using zero-order-hold to sample the system, and assuming the sampling period h = 1, derive the state-space representation of the sampled system. d) Apply z-transform to the state-space model derived in c), and obtain the input-output model of the system. e) Assuming the initial conditions are y(0) = 1, and ÿ(0) = 0. Calculate the output sequence y(k), under the unit step input, u(k) = 1 for k>0

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