4. Consider the idle speed control of a four cylinder SI engine. Using the state space model in equation (5.56) in your textbook, derive the transfer function between normalized engine speed and normalized mass air-flow rate. Design a PID controller for this engine model that will place the closed loop poles at -1 1j in S-plane. Assume that the engine parameters have the following values: FN = 1E-5, FP1 = 1E-6, FN2 = 50, FP2 = 5.5, To = 240 N-m, no = 5000 rpm, pm,0 = 50 kPa, m'a,0 = 7 kg/h, Tj = 0.25 sec, Tn = 0.4 sec. The reference idle speed is nref = 750 rpm. Using MATLAB, plot the step response for the open and closed loop systems. (Note: Be sure to use the correct units in the equations). 5. (i) For problem 4, derive a state space model. Also design a state feedback control system with same performance requirements as in problem 4. Plot both open look and closed loop system performances. (ii) Using the transfer function in problem 4, plot the Bode diagram of this system (open loop). Calculate the gain and phase margins from the diagram. Discuss your observations on the phase and gain margins for the open loop system.