To meet current U.S. emissions standards for automobiles, hydrocarbon (HC) and carbon monoxide (CO) emissions are usually controlled by a catalytic converter in the automobile exhaust. Federal standards for nitrogen oxides $\left(\mathrm{NO}_x\right)$ emissions are met mainly by exhaust-gas recirculation (EGR) techniques. However, as $\mathrm{NO}_{\mathrm{x}}$ emissions standards were tightened from the current limit of 2.0 grams per mile to 1.0 gram per mile, these techniques alone were no longer sufficient.
Although many schemes are under investigation for meeting the emissions standards for all three emissions, one of the most promising employs a three-way catalyst-for $\mathrm{HC}, \mathrm{CO}$, and $\mathrm{NO}_x$ emissions-in conjunction with a closed-loop engine-control system. The approach is to use a closed-loop engine control, as shown in Figure P7.28 [19,23]. The exhaust-gas sensor gives an indication of a rich or lean exhaust and compares it to a reference. The difference signal is processed by the controller, and the output of the controller modulates the vacuum level in the carburetor to achieve the best air-fuel ratio for proper operation of the catalytic converter. The loop transfer function is represented by
$$
L(s)=\frac{K s^2+12 s+20}{s^3+10 s^2+25 s} .
$$
Calculate the root locus as a function of $K$. Carefully calculate where the segments of the locus enter and leave the real axis. Determine the roots when $K=2$. Predict the step response of the system when $K=2$.