This paper proposes the realization of a soft sensor using an adaptive algorithm with proportional correction of the gain coefficient for monitoring river water quality. This algorithm makes it possible to monitor online signals of an object described by nonlinear ordinary differential equations. Simulation studies of a biochemically polluted river, for which the water quality was represented by biochemical oxygen demand (BOD) indices and the dissolved oxygen (DO) deficit, were carried out. The algorithm concept uses only online measurements of the object, and adaptive changes in the gain coefficient are determined based on the adaptation error adopted for this purpose. Simulation results indicated the correct functioning of the soft sensor even for inaccurately identified parameters of the mathematical model and for unknown values and intensity of disturbances affecting the object. The quality of the signals monitored via a soft sensor implemented in this way was determined with the root-mean-squared error (RMSE) and mean percentage error (MPE) indicators and compared with the Kalman filter.

The article presents a new approach to monitoring systems of a certain class using the lookup algorithm. The main task is to generate object signals based on measured but only some selected signals. This idea is based on the Kalman filter approach, but the calculation method of the gain coefficients is different. Its values are determined in a similar way as weights in neural networks during learning (incremental method). The proposed lookup algorithm uses expert knowledge a priori for determining gain corrections, and its functioning is presented for the case of two monitoring error zones. The presented results clearly indicate the advantage of the lookup algorithm over the Kalman filter. Two RMSE and MPE indicators were used for the quality of monitoring.

The article presents the creation of characteristic polynomials on the basis of fractional powers j of dynamic systems and problems related to the determination of the stability intervals of such systems.

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