proportional control offset
Simple answer: if there is no error the output is zero C=K(I-S), if (I-S)=0 then C=0. What is the minimal acheivable error? It is 1/(1+Kp), so increasing Kp you have more and more precise control until the system begins to oscillate. Therefore the P-regultor is tuned to have enough stability margin, so that's why the system approaches asymptoticaly, but if you rise the gain it will be oscillating.