Over the last few years, kernel adaptive filters have gained in importance as the kernel trick started to be used in classic linear adaptive filters in order to address various regression and time-series prediction issues in nonlinear environments.In this paper, we study a recursive method for identifying finite impulse response (FIR) nonlinear systems based on binary-value observation systems. We also apply the kernel trick to the recursive projection (RP) algorithm, yielding a novel recursive algorithm based on a positive definite kernel. For purposes, our approach is compared with the recursive projection (RP) algorithm in the process of identifying the parameters of two channels, with the first of them being a frequency-selective fading channel, called a broadband radio access network (BRAN B) channel, and the other being a a theoretical frequency-selective channel, known as the Macchi channel. Monte Carlo simulation results are presented to show the performance of the proposed algorithm.

Liquid storage tanks are widely used in sectors such as water treatment, oil and gas, food processing, and chemical manufacturing. Knowing the exact amount of liquid in a tank is essential for ensuring safety, preventing spills, and optimizing process control; therefore, the liquid level in a tank must be maintained at a precise reference point. This is where liquid level control for tanks becomes crucial and constitutes a fundamental problem in the industrial sector due to nonlinearities, multivariable coupling, and stochastic disturbances. Given the drawbacks of available control methods, such as classical Model Predictive Control (MPC), which are highly dependent on model accuracy and struggle to reject complex stochastic noise, predicting random disturbances represents a major technological challenge. A new approach is proposed to specifically address the problem and challenge of the four-tank system, where water levels in two lower tanks must be controlled by two pumps, often with varying delays and significant parameter disturbances. To establish a relationship between expected performance and MPC parameters, this approach uses a novel hybrid nonlinear MPC, Extended State Observer, and Physics-Informed Neural State Estimation (NMPC-ESO-PINSE) architecture. A Physics-Informed Neural State Estimation (PINSE) layer, chosen for its learning capacity, is designed to filter sensor noise by applying Bernoulli’s physical laws, while an Extended State Observer (ESO) is integrated to capture and compensate for unmodeled uncertainties in the process. Finally, a proposed hybrid (NMPC-ESO-PINSE) strategy leverages these clean, physically consistent state estimations to solve a non-convex optimization problem via Sequential Quadratic Programming (SQP), computing optimal pump voltages. Extensive numerical simulations demonstrate the superior resilience of this decoupled framework against parametric drifts and continuous noise sequences, yielding a $+27.36\%$ reduction in global Root Mean Square Error (RMSE) compared to standard NMPC, accelerating the closed-loop settling time to $15.2 \mathrm{s}$, and restricting transient overshoot to just $0.18\%$.