On integral plus proportional compensation

Integral, nth order compensation in the feedforward loop of a unity feedback system is shown to result from quadratic, stochastic optimal control of a linear system with arbitrary step input and no error measurement noise. This gives agreement with common practice. Unfortunately this result is obtained only with nonzero compensator noise. It is shown that in the limit of zero compensator noise, the estimator produces a zero at the origin to cancel the effect of the integrator, negating the practical result. The conclusion is that this particular discrepancy between common practice and optimal control theory is still unresolved and that research effort should be expended on reconcilement in order to find the extension to time-variable, multivariable systems.