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2. The RLC series circuit is shown below consists of a resistor, capacitor, and inductor in series with a voltage source. R с Uc Loodoo (a) Use KVL and the component equations to derive the system model for ve(t). Show all steps of the derivation. (b) Let zı = vc and z2 = zi. Decompose the system into two first order differential equa- tions. vio (c) Write the equations as a state space system in matrix form with states 21 and 22. Clearly show the [A] and matrices. (hint: vs(t) is the system input) (d) Write the output equation with the first output yı as the capacitor voltage ve, the second output y2 as the current in the loop i, and the third output yg as the resistor voltage Ur.