**Abstract**

The tuning of the gains of a controller with proportional-integral-derivative (PID) actions has been prevalent in the industry. The adjustment of these gains in PID controllers is often determined by classical methods, such as Ziegler-Nichols, and trial and error. However, these methods fail to deliver satisfactory performance and often do not meet specific project demands because of the inherent complexity of industrial processes, such as plant parameter variations. To solve the tuning problem in highly complex industrial processes, a controller adjustment method based on the internal product of PID terms is proposed, and a propagation matrix (PM) is generated by the numerator coefficients of the plant transfer function (TF). In the proposed method, each term of the PID controller is influenced by each of the numerator and the denominator coefficients. Mathematical models of practical plants, such as unloading and resumption of bulk solids by car dumpers and bucket wheel resumption, were employed to evaluate the proposed method. The obtained results demonstrated an assertive improvement in the adjustment gains from PID actions, thereby validating it as a promising alternative to conventional methods.

**Keywords:** parameter variations, industrial processes, internal product, propagation matrix, PID actions
