In this problem, you will design a controller for an oven. The control objective is for the oven temperature T to track T Tref a desired temperature Tref. The dynamics of the oven are modeled by mc = kT +q where the output y = T is the oven temperature T (relative to ambient temperature), the input u = q is the supplied heat, and the reference r = Tref is the desired temperature Tref. The thermal capacitance is C = mc = 20kJ/C and the thermal resistance is R = (kA/L)-1 = 22.5C/kW. a) (2 points) Find the transfer function for this plant and write it in the form What are a and b? b P(s) = s+a b) (5 points) First, we will design an feedforward (open-loop) controller for this plant with the form C(8) = K1. Design K so that the open-loop system below has perfect tracking. r(t) C(s) (7)n P(s) y(t) HINT: Consider the equilibrium control input u when the equilibrium temperature T = Tref is at the desired reference temperature Tref. c) (7 points) Next, we will design a feedback controller for this plant with the PI form K2(8+0.004) C2(8) = S r(t) + e(t) u(t) y(t) C2(s) P(s) Tune K2 so that closed-loop system below has (i) The closed-loop system is stable (ii) The closed-loop system has zero overshoot (iii) The closed-loop system has the fastest possible response Explain how you tuned K2. d) (2 points) On the same figure, plot the step responses for both of the controllers C(s) and C2(s) for the reference r(t) = 150-1(t). e) (5 points) Find the transfer function from R(s) to U(s) in terms of P(s) and C2(s). Plot. the step-response of this transfer function for the reference r(t) = 150 1(t). How does this compare with the heat supplied by the open-loop controller? How does this explain the relative performance of these controller?