3 Zero input response: Continuous-time system Consider the series RLC circuit shown in Figure 1 where the desired output is the capacitor voltage v(t). By writing a KVL equation, the differential equation governing this system is found to be Vin R L w C Figure 1: A typical series RLC circuit. LC (t) + RCv(t) + v(t) = Vin(t). In the following, assume that R = 1, C = 1 F, and L = 0.5 H. 2 1. Introduce state variables for the system (2) and write down the state equations in controller canonical form. 2. Introduce a state vector r and suitable matrices (A, B, C, D) to write down a state repre- sentation for the system. 3. Give an expression for the zero-input response v2,(t) in terms of A, C, and the state-transition matrix et for the system. Assume initial conditions of v(0) = 1, (0) = 0 and input Vin(t) = 0.