Please answer it in detail, all the $11$ points :

Time Allowed: $50$ minutes. $(14$:$30-15$:$20)$

ATTENTION: During the examination, please do not attempt for cheating. Suspected

cheating will result in a ZERO on your exam and all of the students who were caught

cheating will be faced with disciplinary sanctions.

Answer all the questions according to given electric circuit.

What are the storing elements for the given system? $(4$ pts $)$

What are the state variables of the given system? $(4$ pts$)$

Write down the dynamic equations for the storing elements when the switch is closed. The

input is voltage e$.(10$ pts $)$

Write the output equations of the current of R$\_(2),$ the voltage of R$1,$ the output voltage e$\_(",")$ in

matrix form when the switch is closed. $(9$ pts$)$

Identify the coefficient matrices of output equations matrix form. $(6$ pts $)$

Draw the block diagram $(16$ pts$)$

Derive the input$-$output differential equation when the switch is opened. The input is the

voltage on Cl and the output is c$\_(0)(10$ pts $)$

Derive the state$-$variable equations when the switch is closed. $(10$ pts $)$

Write the state$-$variable model in matrix form when the switch is closed. $(12$ pts $)$

Identify the coefficient matrices, state vector, input vector of the model. $(10$ pts $)$

Derive the mathematical equation for the outputs: the current of R$\_(2),$ the voltage of R$1,$ the

output voltage e$\_(0)$ when the switch is closed. $(9$ pts$)$