if u(t)=Kpe(t) + KIe(t) + Kde/t is the solution what is the model equations??

good-day all.ok,i'm working on obtaining a digital pid algorithm.i discretized the continuous time algorithm,u(t)=Kpe(t) + KI∫e(t) + Kd∂e/∂t using trapezoidal and Simpson's methods.but my supervisor wants me to do my discretization from the model equation not u(t)=Kpe(t) + KI∫e(t) + Kd∂e/∂t which appears to me as the solution equation.i must have gone through over 200 materials which includes eBooks,journals,etc but not one has included the model equation.im new to control.so i need the model equations from which u(t)=Kpe(t) + KI∫e(t) + Kd∂e/∂t was obtained.links,books etc that could help me understand better will also be most appreciated.many thanks!