On a Relationship between Integral Compensation and Stochastic Gradient Descent

Disturbance rejection is a fundamental problem in control engineering, and there are many methods to achieve a good disturbance rejection. One of the standard methods for the disturbance rejection is to employ an integral compensator. This compensator integrates the error between the reference signal and the output, and use the integrated value with a specific coefficient as a compensating signal. In this work, we discuss the relationship between the machine learning theory and the integral compensation. We focus on single-input-single-output discrete-time time-invariant systems, and show that the integral compensation can be understood in the context of the machine learning in this case. In particular, the integral compensation is identical to the online optimization with the standard stochastic gradient descent [1]. The above idea gives two suggestions. First, the integral compensation may become faster by employing other types of stochastic gradient descent. Many algorithms of stochastic gradient descent have been proposed in the machine learning literature, and such algorithms may improve the classical integral compensation. Second, it may become possible to reject the time-invariant state-dependent disturbance. The modeling error is a typical example of such disturbance. By learning such a disturbance, the control performance for repetitive motions will be improved compared to the integral compensation. This presentation discusses the pros and cons of regarding the integral compensation as the stochastic gradient descent optimization.