Automobile manufactures spent considerable amount of effort to meet exhaust emission performance in Europe and North America.Modern automotive-power-plant systems consist of an internal combustion engine that has an internal cleanup-device called the catalytic converter. Such as system requires control of the engine air-fuel ratio (A/F), the ignition spark timing, exhaust gas recirculation, and injection air. The control system problem considered in this exercise deals with the control of the air-fuel ration. In general depending on fuel composition and other factors, a typical stoichiometry will cause high hydrocarbons, carbon monoxide, and nitrous oxides in the tailpipe emission.The control system whose block diagram is shown in Figure G1 below is devised to control the air-fuel ration so that a desired output variable is maintained for a given command signal. The figure shows that the sensor senses the composition of the exhaust-gas mixture entering the catalytic converter. The electronic controller detects the difference or the error between the command and the sensor signals and computes the control signal necessary to achieve the desired exhaust-gas composition. The output variable y(t) denotes the effective air-fuel ration. The transfer function of the engine is given by , where is the time delay and is 0.2 sec. the time constant is 0.25 sec. Approximate the time delay by a power series (prove it):

Figure Gr 1 and 2

(a) Let the controller be a PI controller so that,

Find the value of so that the ramp-error constant is 2. Determine the value of so that the maximum overshoot of the unit-step response is a minimum and the settling time is a minimum. Give the values of the maximum overshoot and the settling time. Plot the unit-step response of . Find the marginal value of for system stability.

(b) Can the system performance be further improved