"The transfer function for the linearized dynamics of the cruise control system is given by $P(s) = b/(s+a)$. A simple (but not necessarily good) way to design a PI controller is to choose the parameters of the PI controller as $k_\\text{i}=ak_\\text{p}$. The controller transfer function is then $C(s)=k_\\text{p}+k_\\text{i}/s=k_\\text{i}(s+a)/s$. It has a zero at $s = -k_\\text{i}/k_\\text{p}=-a$ that cancels the process pole at $s = -a$. We have $P(s)C(s)=k_\\text{i}/s$ giving the transfer function from reference to vehicle velocity as $G_{yr}(s)=b k_\\text{p}/(s + b k_\\text{p})$, and control design is then simply a matter of choosing the gain $k_\\text{p}$. The closed loop system dynamics are of first order with the time constant $1/(b k_\\text{p})$."

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