This information is part of the Modelica Standard Library maintained by the Modelica Association.

This block defines a discrete-time PI controller by the formula:

// State space form:
   x(ti) = previous(x(ti)) + u(ti)/Td;
   y(ti) = kd*(x(ti) + u(ti));

// Transfer function form:
   y(z) = kd*(c*z-1)/(z-1)*u(z);
          c = 1 + 1/Td

where kd is the gain, Td is the time constant, ti is the time instant of the i-th clock tick and z is the inverse shift operator.

This discrete-time form has been derived from the continuous-time form of a PI controller by using the implicit Euler discretization formula.