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A new design of a non-parametric adaptive approximate model based on Differential Neural Networks (DNNs) applied for a class of non-negative environmental systems with an uncertain mathematical model is the primary outcome of this study. The approximate model uses an extended state formulation that gathers the dynamics of the DNN and a state projector (pDNN). Implementing a non-differentiable projection operator ensures the positiveness of the identifier states. The extended form allows producing continuous dynamics for the projected model. The design of the learning laws for the weight adjustment of the continuous projected DNN considered the application of a controlled Lyapunov-like function. The stability analysis based on the proposed Lyapunov-like function leads to the characterization of the ultimate boundedness property for the identification error. Applying the Attractive Ellipsoid Method (AEM) yields to analyze the convergence quality of the designed approximate model. The solution to the specific optimization problem using the AEM with matrix inequalities constraints allows us to find the parameters of the considered DNN that minimizes the ultimate bound. The evaluation of two numerical examples confirmed the ability of the proposed pDNN to approximate the positive model in the presence of bounded noises and perturbations in the measured data. The first example corresponds to a catalytic ozonation system that can be used to decompose toxic and recalcitrant contaminants. The second one describes the bacteria growth in aerobic batch regime biodegrading simple organic matter mixture., Competing Interests: Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper., (Copyright © 2022 Elsevier Ltd. All rights reserved.)

The aim of this article is to investigate the trajectory tracking problem of systems with uncertain models and state restrictions using differential neural networks (DNNs). The adaptive control design considers the design of a nonparametric identifier based on a class of continuous artificial neural networks (ANNs). The design of adaptive controllers used the estimated weights on the identifier structure yielding a compensating structure and a linear correction element on the tracking error. The stability of both the identification and tracking errors, considering the DNN, uses a barrier Lyapunov function (BLF) that grow to infinity whenever its arguments approach some finite limits for the state satisfying some predefined ellipsoid bounds. The analysis guarantees the semi-globally uniformly ultimately bounded (SGUUB) solution for the tracking error, which implies the achievement of an invariant set. The suggested controller produces closed-loop bounded signals. This article also presents the comparison between the tracking states forced by the adaptive controller estimated with the DNN based on BLF and quadratic Lyapunov functions as well. The effectiveness of the proposal is demonstrated with a numerical example and an implementation in a real plant (mass-spring system). This comparison confirmed the superiority of the suggested controller based on the BLF using the estimates of the upper bounds for the system states.

This paper describes a fixed-time convergent step-by-step high order sliding mode observer for a certain type of aerobic bioreactor system. The observer was developed using a hierarchical structure based on a modified super-twisting algorithm. The modification included nonlinear gains of the output error that were used to prove uniform convergence of the estimation error. An energetic function similar to a Lyapunov one was used for proving the convergence between the observer and the bioreactor variables. A nonsmooth analysis was proposed to prove the fixed-time convergence of the observer states to the bioreactor variables. The observer was tested to solve the state estimation problem of an aerobic bioreactor described by the time evolution of biomass, substrate and dissolved oxygen. This last variable was used as the output information because it is feasible to measure it online by regular sensors. Numerical simulations showed the superior behavior of this observer compared to the one having linear output error injection terms (high-gain type) and one having an output injection obtaining first-order sliding mode structure. A set of numerical simulations was developed to demonstrate how the proposed observer served to estimate real information obtained from a real aerobic process with substrate inhibition.

Time-delay systems have been successfully used to represent the complexity of some dynamic systems. Time-delay is often used for modeling many real systems. Among others, biological and chemical plants have been described using time-delay terms with better results than those models that have not consider them. However, getting those models represented a challenge and sometimes the results were not so satisfactory. Non-parametric modeling offered an alternative to obtain suitable and usable models. Continuous neural networks (CNN) have been considered as a real alternative to provide models over uncertain non-parametric systems. This article introduces the design of a specific class of non-parametric model for uncertain time-delay system based on CNN considering the so-called delayed learning laws analysis. The convergence analysis as well as the learning laws were produced by means of a Lyapunov-Krasovskii functional. Three examples were developed to demonstrate the effectiveness of the modeling process forced by the identifier proposed in this study. The first example was a simple nonlinear model used as benchmark example. The second example regarded the human immunodeficiency virus dynamic behavior is used to show the performance of the suggested non-parametric identifier based on CNN for no fictitious neither academic models. Finally, a third example describing the evolution of hepatitis B virus served to test the identifier presented in this study and was also useful to provide evidence of its superior performance against a non-delayed identifier based on CNN., (Copyright © 2014 Elsevier Ltd. All rights reserved.)

Designing a proportional derivative (PD) controller has as main problem, to obtain the derivative of the output error signal when it is contaminated with high frequency noises. To overcome this disadvantage, the supertwisting algorithm (STA) is applied in closed-loop with a PD structure for multi-input multi-output (MIMO) second order nonlinear systems. The stability conditions were analyzed in terms of a strict non-smooth Lyapunov function and the solution of Riccati equations. A set of numerical test was designed to show the advantages of implementing PD controllers that used STA as a robust exact differentiator. The first numerical example showed the stabilization of an inverted pendulum. The second example was designed to solve the tracking problem of a two-link robot manipulator., (Copyright © 2014 ISA. Published by Elsevier Ltd. All rights reserved.)

This paper presents the design of a complex-valued differential neural network identifier for uncertain nonlinear systems defined in the complex domain. This design includes the construction of an adaptive algorithm to adjust the parameters included in the identifier. The algorithm is obtained based on a special class of controlled Lyapunov functions. The quality of the identification process is characterized using the practical stability framework. Indeed, the region where the identification error converges is derived by the same Lyapunov method. This zone is defined by the power of uncertainties and perturbations affecting the complex-valued uncertain dynamics. Moreover, this convergence zone is reduced to its lowest possible value using ideas related to the so-called ellipsoid methodology. Two simple but informative numerical examples are developed to show how the identifier proposed in this paper can be used to approximate uncertain nonlinear systems valued in the complex domain.

This paper addresses the design of a discontinuous finite time convergent learning law for neural networks with continuous dynamics. The neural network was used here to obtain a non-parametric model for uncertain systems described by a set of ordinary differential equations. The source of uncertainties was the presence of some external perturbations and poor knowledge of the nonlinear function describing the system dynamics. A new adaptive algorithm based on discontinuous algorithms was used to adjust the weights of the neural network. The adaptive algorithm was derived by means of a non-standard Lyapunov function that is lower semi-continuous and differentiable in almost the whole space. A compensator term was included in the identifier to reject some specific perturbations using a nonlinear robust algorithm. Two numerical examples demonstrated the improvements achieved by the learning algorithm introduced in this paper compared to classical schemes with continuous learning methods. The first one dealt with a benchmark problem used in the paper to explain how the discontinuous learning law works. The second one used the methane production model to show the benefits in engineering applications of the learning law proposed in this paper., (Copyright © 2013 Elsevier Ltd. All rights reserved.)

The most important aspect of differential neural networks dynamics is related to their weights properties. This is a consequence of the complex non-linear structure describing the learning matrix differential equations, which are associated with adaptive capability of this kind of neural network. So far, there is no analytical demonstration about the weights stability. In fact, this is the main inconvenience in designing real applications for differential neural network observers. This study deals with the stability proof for the weights dynamics using an adaptive procedure to adjust the weights ordinary differential equations. Three different examples (two of them were realised by numerical simulations and the last one was carried out using real biofiltering process data) demonstrated the good performance of the suggested approach. [ABSTRACT FROM AUTHOR]

This paper presents the design and implementation of an output feedback controller based on the super twisting algorithm (STA) that stabilizes the trajectory tracking error of a skid steered mobile robot (SSMR). The control scheme introduces a diffeomorphism based on the mathematical model of the SSMR to transform the original problem into a third order chain of integrators. In this study, the available measurements are the position and orientation of the SSMR. A modified STA working as a step by step differentiator estimates the velocity and acceleration of the mobile robot. Then, a second STA enforces the tracking of a predefined trajectory. Numerical and experimental results comparing the STA with a state feedback controller (SFC) and a first order sliding mode controller (FOSM) justify the control proposal. [ABSTRACT FROM AUTHOR]

The fuzzy logic controller (FLC) has the ability of handling parametric uncertainties and external perturbations for unknown systems. A regular structure for a FLC is the proportional derivative (PD) form. The proportional derivative fuzzy controller (PDF) could be seen as a variable gain PD controller. Despite this characteristic, the most common drawback for any PD controller, with unknown dynamics or even with unmodeled dynamics is the error signal differentiation. In this manuscript this disadvantage was overtaken implementing the super-twisting algorithm (STA) as a robust exact differentiator (RED). The information provided by the STA was injected into the PDF to enhace its performance. In this study, the stability of the nonlinear system under the fuzzy super twisting PD controller (FSTPD) in closed loop was analyzed using the concept of the second Lyapunov׳s method. Numerical simulations were designed to show the effectiveness and advantages of the proposed FSTPD over the classical PD structure supplied with the STA and a PDF with the derivative part obtained by a linear filter. A first example to stabilize a simple pendulum was developed applying the FSTPD. A second example for solving a tracking control problem was designed for a robot manipulator with six degrees of freedom. In both cases, the FSTPD showed better performance and a significant reduction of the control energy. [ABSTRACT FROM AUTHOR]