Direct Current (DC) motors are widely used in industrial and robotic applications due to their simplicity of use and control. However, achieving high-performance control in DC motor systems is a longstanding challenge due to nonlinearities and uncertainties, parameter variations, and external disturbances (e.g., load changes and friction). This paper proposes a novel homogeneous Proportional-Integral-Derivative (hPID) control strategy specifically designed for DC motors, aimed at achieving robust and precise speed and position regulation. Unlike traditional PID controllers, the hPID controller utilizes the mathematical principles of homogeneous control theory to systematically improve the stability, robustness, response time and transient quality of the closed-loop system. A homogeneous PID architecture is developed by simply upgrading the conventional PID algorithm, in which homogeneous functions of the state scale the gains. This adaptation enhances the regulation quality of the closed-loop system. To support this methodology, a Lyapunov-based stability analysis is performed, establishing global asymptotic stability and finite-time convergence under mild assumptions of the system. Experimental results for a representative DC motor setup demonstrate the effectiveness of the proposed methodology for essential improvement of the regulation precision compared to a linear PID controller. Superior performance is discovered for time response, steady-state accuracy, and robustness. This research contributes to a theoretically sound and scalable control methodology to improve the performance and reliability of DC motor-driven systems under both structured and variable operating conditions.

This article focuses on designing a neural identifier for non-linear systems using an Integral Sliding Mode (ISM) approach. A Differential Neural Network (DNN) structure identifies a non-linear system; meanwhile, the learning laws for stabilizing the identification error are obtained from the Lyapunov Stability analysis. An integral sliding surface is proposed to provide the convergence of the system identifier weights to the ideals to approximate the non-linear system. The results for identifying the Van Der Poll Oscillator and the Inverted Pendulum show better identification error reduction performance than a traditional DNN.

Differential neural networks (DiNNs) encounter a trade-off between the approximation quality and structural complexity. One promising approach to address this trade-off is incorporating dynamic complexity adjustment as an integral part of the learning process. Taking inspiration from the Fourier approximation theory, this study introduces a novel method for adapting the architecture of DiNNs, when they serve as nonparametric identifiers for dynamic systems with uncertain mathematical models. The structural adaptation process is executed through a recursive algorithm based on a modification structure strategy, which dynamically adjusts the number of neurons within the network's structure. By applying a projection operator to the set of neurons, this method identifies the most relevant sequence of sigmoidal functions, intending to minimize the mean square error in approximating the trajectories of uncertain systems. This simultaneous reduction in overall complexity enhances the quality of the approximations. Moreover, the proposed method can implement a coarse-to-fine approach, wherein selecting necessary neurons occurs in multiple steps. These steps are determined by an adaptive structure strategy that alters the topology of the DiNN. The resulting framework's effectiveness is demonstrated by evaluating the proposed identifier's performance in approximating the evolution of real-life data associated with the ocular response during controlled motions or virtual reality engagement. In both experimental cases, there was a noticeable improvement in the accuracy of eye motion approximation by the DiNN, thanks to the variable structure approximation basis determined by the adaptive structure strategy. Overall, this study presents a formal method to automatically determine a feasible DiNN topology.

This study presents a state nonparametric identifier based on neural networks with continuous dynamics, also known as differential neural networks (DNNs). The laws for adjusting their parameters are developed using a control barrier Lyapunov functions (BLFs). The motivation for using the BLF comes from the preliminary information of the system states, which remain in a predefined time-depending set characterized by state or purely time-dependent functions. In this study, time-dependent state constraints are supposed to be known in advance continuous-time functions. The obtained learning laws require solving differential continuous-time Riccati equations and nonlinear differential equations for the learning laws that depend on the identification error and the state restrictions. The developed identifier was evaluated concerning the identifier that does not consider the state restrictions. This comparison included the numerical evaluation of the identifier for a robotic arm intended to reproduce a nonstandard flight simulator. This evaluation confirmed that the identification results were improved using the proposed learning laws and considering that the state limits were not transgressed. The quality indicators based on the mean square error were more minor by 4.2 times.

The precise modeling of dynamic systems with periodic trajectories is required to describe diverse systems in mechanical, electrical, and many other disciplines. Nevertheless, the modeling task based on traditional methodologies could be complicated, considering the specific nature of periodic motions in actual systems. Differential neural networks (DNNs) are modeling tools for dynamic systems that can be useful for developing precise representations of periodic systems. This study presents the design of a novel family of DNN identifiers that could reproduce the trajectories of periodic systems with an uncertain mathematical model. The suggested DNN identifiers may produce an approximate model with periodic properties similar to the system under analysis exhibiting an one-period convergence of DNN weights. The fundamentals of the Floquet’s theory drive the design of the learning laws to ensure the reproduction of the periodic properties in the DNN. The design of a controlled Lyapunov function allows the learning laws to be derived for the DNN weights whose evolution depends on the positive definite solution of a periodic differential Lyapunov equation. Several numerical evaluations on periodic systems confirmed the modeling performance of the proposed identifier when the approximation performance is compared with traditional DNN identifiers using sigmoidal functions.

This study aims at designing a robust nonparametric identifier for a class of singular perturbed systems (SPSs) with uncertain mathematical models. The identifier structure uses a novel identifier based on a differential neural network (DNN) with rational form, which can take into account the multirate nature of SPS. The identifier uses a mixed learning law including a rational formulation of neural networks which is useful to solve the identification of the fast dynamics in the SPS dynamics. The rational form of the design is proposed in such a way that no-singularities (denominator part of the rational form never touches the origin) are allowed in the identifier dynamics. A proposed control Lyapunov function and a nonlinear parameter identification methodology yield to design the learning laws for the class of novel rational DNN which appears as the main contribution of this study. A complementary matrix inequality-based optimization method allows to get the smallest attainable convergence invariant region. A detailed implementation methodology is also given in the study with the aim of clarifying how the proposed identifier can be used in diverse SPSs. A numerical example considering the dynamics of the enzymatic-substrate-inhibitor system with uncertain dynamics is showing how to apply the DNN identifier using the multirate nature of the proposed DNN identifier for SPSs. The proposed identifier is compared to a classical identifier which is not taking into account the multirate nature of SPS. The benefits of using the rational form for the identifier are highlighted in the numerical performance comparison based on the mean square error (MSE). This example justifies the ability of the suggested identifier to reconstruct both the fast and slow dynamics of the SPS.

This work presents the design of a control trajec-tory algorithm based on Barrier Lyapunov functions (BLF) for a class of third-order electro-mechanical systems. The method considers unknown control gains and uncertain plant parame-ters, a common problem in control engineering applications. In order to deal with the uncertain parameters, this study proposes a backstepping adaptive strategy applying Nussbaum functions to know the sign of the control gains. The BLF imposes partial constraints to keep the trajectories of the system inside a safe operation set. The proposed controller is tested in a virtual one-link robotic manipulator including actuator dynamics to develop an adequate trajectory tracking. The maximum tracking error norm was 0.02, which fulfilled the maximum constrained bound of 0.5.

Over the last decade, the research in autonomous robots has increased the development of mechanisms, navigation, and control schemes. Robots constitute capabilities for doing specific human tasks depending on the mechanism and environment involved. For mobile robots, diverse techniques formulated navigation and control algorithms, some of them have applied sliding modes and other modern robust control techniques. Actually, most control algorithms based their development in the kinematic model. This aims to control a car-like robot mobile robot, with a backstepping strategy. At each step of the bacstepping procedure a second order super-twistng sliding mode algorithm force the states of the kinematic error, after a nonlinear transformation, to zero in finite-time. The proposed controller improves the tracking trajectory task. Numerical results demonstrate the effectiveness of the algorithm. A comparison with a classical proportional-integral-derivative controller enhances the advantages of applying a sliding mode strategy.

This researching work presents the design of an adaptive controller implementing the solution of the trajectory tracking for a robotic micromanipulator based on a three degrees of freedom arm that carries a piezoelectric- based gripper. The robotic arm is aimed to place the gripper at the correct position. From this spot, the gripper can handle objects of millimetric and micrometric scales. The proposed adaptive controller implements state dependent gains that drives all the articulations of the arm smoothly towards their corresponding references. The design of these gains is obtained using a class of control Lyapunov function. The type of the developed controller is also working to control the motion of the microgripper using the main mode of the piezoelectric actuators, represented by ordinary differential equations that takes into account the relation with the robotic arm. The proposed controller is tested over an actual robotic arm and evaluated considering the performance comparison with respect to a traditional state feedback control form. The experimental results confirm the effective tracking of the reference trajectories showing neither transient oscillations nor overshoots. These evaluations justify the application of the proposed method with the state dependent gains. The manipulation of millimetric objects is also used to evaluate the functionality of the developed robotic arm with its piezoelectric based gripper.

This work deals with the design of a millimetric electrospinning device (MED) controlled with the implemen-tation of a sliding mode controller taking into account states constraints or constraint sliding mode controller (CSM C). The proposed MED is integrated by two cartesian robots with two (injector system) and three (collector system) degrees of freedom, respectively. These two cartesian robots work in a coordinated manner to regulate the deposition processes for the generation of polymeric micro and nanofibers. This strategy addresses one of the most common problems of classic MEDs, guaranteeing that the fibers are deposited in a specific zone (aiming to create 3D polymeric structures). The suggested CSM C successfully restricts the motions of the MED to a predefined working space that led to the production of aligned polymer fibers, ensuring the convergence of the tracking error to the sliding surface in finite-time, while the state constraints are satisfied. Experimental results validate the effectiveness of the proposed CSMC by using the least mean square evaluation of the tracking error and the integral of the control signal. The evaluation is also done following a comparative analysis concerning the performance forced with respect to a state feedback form. The evaluation confirmed the effectiveness of the CDMC strategy compared to the state feedback form.

This study is summarising the design of an output hybrid feedback controller for biped robots where the motion range for each joint is being considered. The design considers the hybrid nature of biped device when it is developing an entire gait cycle including the interaction with the environment touching the support floor. A simplified hybrid model is proposed to represent the dynamics of the robotic device in a realistic form. The proposed biped device is driven using linear actuators with a motion transfer system. A hybrid formulation for the control with adaptive state dependent gains regulates the articulations motion considering the limits of actuator motion. The dynamics of the adaptive gains is obtained with the application of a control barrier-like Lyapunov function for hybrid systems. The explicit structure of these gains are derived in a formal way. A set of numerical simulations is used to demonstrate the applicability of the developed controller analysing the tracking of bio-inspired reference trajectories obtained from reported biomechanical information. The numerical simulations used a virtual model of a biped robotic device where the interaction with the environment was considered for analysing the effect of hybrid evolution. A comparison between trajectories produced by the hybrid controller and a traditional state feedback offers a class of validation for the application of the restricted barrier inspired output feedback control strategy,

This manuscript proposes the design of a speech synthesis algorithm based on measured electroencephalographic (EEG) signals previously classified by a class of neural network with continuous dynamics. A novel multilayer differential neural network (MDNN) classifies a database with the EEG studies of 20 volunteers. The database contains information described by input-output pairs corresponding to EEG signals and a corresponding word imagined by the volunteer. The suggested MDNN estimates the unknown relationship between the information instances and suggests the most-likely word that the user wants mentioning. The proposed MDNN satisfactory classifies over 95% a set of words obtained from the suggested EEG study in which, the users have to watch four different geometric figures on a screen.

This manuscript presents the design of a deep differential neural network (DDNN) for pattern classification. First, we proposed a DDNN topology with three layers, whose learning laws are derived from a Lyapunov analysis, justifying local asymptotic convergence of the classification error and the weights of the DDNN. Then, an extension to include an arbitrary number of hidden layers in the DDNN is analyzed. The learning laws for this general form of the DDNN offer a contribution to the deep learning framework for signal classification with biological nature and dynamic structures. The DDNN is used to classify electroencephalographic signals from volunteers that perform an identification graphical test. The classification results show exponential growth in the signal classification accuracy from 82 percent with one layer to 100 percent with three hidden layers. Working with DDNN instead of static deep neural networks (SDNN) represents a set of advantages, such as processing time and training period reduction up to almost 100 times, and the increment of the classification accuracy while working with less hidden layers than working with SDNN, which are highly dependent on their topology and the number of neurons in each layer. The DDNN employed fewer neurons due to the induced feedback characteristic.

This study presents the design of a robust control based on the sliding-mode theory to solve both; the stabilization and the trajectory tracking problems of nonlinear systems subjected to a class of full-state restrictions. The selected nonlinear system satisfies a standard Lagrangian structure affected by nonparametric uncertainties. A barrier Lyapunov function is used to ensure the state constraints by designing a time-varying gain, which guarantees the fulfillment of the predefined state constraints even under external perturbations. The proposed design methodology for the barrier sliding-mode control (BSMC) ensures the convergence of the sliding surface in finite time to the origin. Consequently, the asymptotic convergence of the states to the corresponding equilibrium point is achieved. The finite-time stability of the origin in the closed-loop system with the proposed controller has been demonstrated using the second Lyapunov stability method. The suggested controller was evaluated on a two-link robotic manipulator. Then, the obtained results showed better stabilization and tracking performances (while the restrictions are satisfied) than the traditional first-order sliding-mode or linear state feedback controllers.

This study has the aim of introducing a new type of trajectory tracking robust controllers for a class of rehabilitation robotic system considering the articulations restrictions. The robotic device consists of a suspended biped configuration. The suggested robust control considers the application of state depending gains which provide finite-time convergence for the tracking deviation. The state restrictions are fulfilled by the implementation of controller gains estimated by a class of the controlled tangent barrier Lyapunov function. Stability analysis for the tracking error yields the explicit design of the state dependent gains. The rate of convergence for the controller design is enhanced using a matrix inequality convex optimization method. Based on the forward complete characteristic of the suggested rehabilitation device, it is allowed using a finite-time convergent super-twisting-based differentiator to concrete an output feedback realization of the proposed controller. A computerized model of the tendered rehabilitation robot provides a reliable testing platform to the suggested roust controller. Numerical evaluations appear to serve as an indirect confirmation for the tracking error convergence, satisfying the articulation restrictions, and the effect of the gain optimization design. For comparison purposes, the regular state feedback control design is considered as benchmark. The faster convergence of the mean square estimation of the tracking error justifies the design of the proposed control design as well as the state feedback structure justifies the origin is a fixed-time stable equilibrium point for the space of tracking error at the same time that state space restrictions remain satisfied. The experimental evaluations of the proposed controller justifies the barrier controller which, in spite of the modeling uncertainties and the implementation issues, tracked the reference trajectories.

Unsupervised learning is a target free methodology to classify unorganized information. This study proposes a new unsupervised learning method for classifying unlabeled targets based on convex ellipsoidal sets. The method described here uses ellipsoidal calculus tools to realize pattern classification via a clustering scheme. The algorithm consisted in adjusting the number of clusters defined by an ellipsoidal set as well as their centers, shape forms and orientation. All these processes were realized without preliminary information on the data distribution. The application of an inner gradient descent algorithm permitted the adjustment of these parameters, using the standard deviation of data in the cluster with respect to the semi-axis of ellipsoid that contains the corresponding data. A specific evaluation of the proposed algorithm used two-dimensional attributes databases. The outcomes of the classification results were compared with those produced by the K-means and DBSCAN algorithm. Comparable classification results were achieved overcoming some drawbacks of the methods.

The aim of this study is to design a robust controller for an autonomous underwater vehicle (AUV) based on the application of the extended Integral Sliding Mode (ISM) method. The control problem is reformulated here as an extremum seeking control problem realization for solving the tracking of attainable reference trajectories. The states of the uncertain dynamic corresponding to the AUV dynamics track the references restricted to the x-z plane. The dynamic nature of the AUV induces the application of the ISM extended version design based on the implementation of a sub-gradient strategy to perform the on-line optimization for the tracking error norm. The proposed control permits to minimize the tracking error without a complete knowledge of the AUV dynamics. The trajectory tracking results of the designed controller are compared with the corresponding outcomes enforced by a state feedback and twisting controllers . The proposed controller exhibits better tracking and smaller control magnitude than the state feedback and twisting form. These two results confirm the benefits of introducing the mixed extended strategy, including ISM and dual averaged extremum seeking control.

This manuscript presents an algorithm to classify continuously electroencephalographic (EEG) signals based on deep differential neural networks (DDNNs). The learning laws are obtained by the second stability method of Lyapunov that requires the solution of a set of matrix differential equations. The robustness of this technique allows the analysis and classification of bio-signals like EEG signals. The EEG signals have complex dynamics and they are strongly affected by noises in the measurements and a high degree of variability between different studies in patients. The main strength of DDNNs is their feedback property, which allows them to work with the time-dependent variation of the EEG signals. The DDNNs are tested in a database constituted of EEG signals acquired from a study made in ten volunteers. The study consisted of the acquisition of EEG measurements of the volunteers recognizing geometrical figures appearing in a graphic user interface. The DDNN obtained better performance than a single layer differential neural network and a convolutional neural network.

Bionformatics and pharmacokinetics-pharmacodynamics (PKPD) systems are two conjugated tools to intensively explore the effect of new drugs on the human body running in-silico analysis. Usually, PKPD models do not consider all the biological reactions that explain the pharmaceutical effect. A complementary non-parametric modeling can be useful to recover the PKPD dynamics despite the uncertainties and external perturbations effect, which can reduce the degree of uncertainties on the drug evaluation. The aim of this study is to get a feasible non-parametric model of PKPD models using a bioinformatics inspired evaluation of antibacterial drug doses. A class of bioinformatics inspired differential neural networks (DNNs) responding to the dose modification provides the non-parametric approximation of the PKPD dynamics. The DNN modeling strategy was applied to approximate the dynamics of PKPD models under four different dosing regimes. The modeling strategy estimated the bacteria survival (measured as the logarithm of the colony forming units per milliliter) after the drug application. The same adjusted DNN-based model confirmed the ability of designing an off-line lab for evaluating diverse dosing strategies of antibacterial pharmaceutical.

This manuscript deals with the problem of controlling a virtualized humanoid robot with 16 degrees of freedom (DOF). The aim of this study was to design an output feedback discontinuous controller which must resolve the sequence of articulation movements to walk over disjoint stepping objects placed in front of the humanoid robot. The suggested controller considers the state restrictions, via a nonstandard strong Lyapunov function, corresponding to the angular displacements and velocities at each articulation. This study implements an extended state terminal second order sliding mode controller with time dependent gains, to ensure the finite-time tracking trajectory of each articulation of the humanoid robot. The terminal sliding mode (TSM) is implemented in a virtual platform developed in a computer-aided design software. For comparison purposes, the controller was compared with a state feedback and a first order sliding mode algorithms.

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 manuscript deals with the problem of controlling a virtualized humanoid robot with 16 degrees of freedom (DOF), each corresponding to the articulation in a real human being. That is, three DOF for each leg in the sagittal plane and one for the abduction movement; three DOF for each arm in the sagittal plane and one for the waist. The tracking trajectory problem of any humanoid robot like the classical biped robots requires a control algorithm with robustness against parametric uncertainties, fast response and even with finite-time convergence. These main characteristics are easily covered by sliding mode controllers. This manuscript implements a terminal second order sliding mode (TSOSM) controller to ensure the finite-time tracking trajectory of each articulation of the humanoid robot to the ones that define a classical walking pattern obtained by bio-mechanical studies. The TSOSM is implemented in a virtual platform developed in a computer-aided design software.

In this paper, a differential neural network (DNN) implemented as a robust observer estimates the dynamics of perturbed uncertain nonlinear systems affected by exogenous unknown inputs. In the first stage, the identification error converges into a neighborhood around the origin. Then, the second-order sliding mode supertwisting algorithm implemented as a robust exact differentiator reconstructed the unknown inputs. The approach proposed in this paper can be applied in the case of full access to the state vector (identification problem) and in the case of partial access to the state vector (estimation problem). In the second case, the nonlinear system under study must have well-defined full relative degree with respect to the unknown input. Numerical examples showed the effectiveness of the proposed algorithm. The first example tested the DNN working as an identifier into a mathematical model describing the dynamics of a spatial minisatellite. The second example (with a DNN implemented as an observer) tested the methodology of this paper over a single link flexible robot manipulator represented in a canonical (Brunovsky) form. In both examples, the mathematical models served as data generators in the testing of the neural networks. Even when not exact mathematical description of both models was used in the input estimation, the accuracy obtained with the DNN is comparable with the case of applying a high-order differentiator with complete knowledge of the plant.

The aim of this study was to design and evaluate an output based adaptive rejection controller (ADRC) for a biped robot. The controller considers the application of an extended state observer which serves for either estimate the velocity of the legs articulation angles as well as to estimate the disturbances affecting the robot dynamics. The observer design considered the angles constrains which naturally appear in the movement of a biped robot. A class of hybrid observer appeared as solution for the estimation of angles values and their positions. The control design (proposed in a distributed structure) used the constrained estimation of velocity to solve the tracking trajectory problem associated to the gait cycle of the biped robot. A set of numerical evaluations over a simulated biped robotic system proved that active disturbance rejection controller using the estimated state constrains tracked the reference angles of articulation. The comparison of the controller proposed in this study overcame the tracking results attained by the classical ADRC where the estimated velocities were freely estimated without taking into account the angles constrains.

This study presents a solution to design an automatic controller that forces the trajectory tracking of a tomographic robotic system. The methodology to design the controller was the Active Disturbance Rejection Control (ADRC). A pseudo-adaptive version of ADRC enforces a smooth trajectory tracking with a small ultimate bound for the tracking error. The control solution used the estimation of velocity that was estimated by an extended adaptive state observer included in the ADRC solution. A set of numerical evaluations over a tomographic robotic system proved the efficiency of the proposal in terms of a performance index.

The aim of this study was to develop a dynamic neural field (DNF) model to capture the essential non-linear characteristics of neural activity along several millimeters of visual cortex in response to local flashed stimuli. A two-layer DNF model was assessed to describe the response of both excitation and inhibitory layers of neurons. This particular structure of neurons interconnection was analyzed as a coupled system of non-linear integro-differential equations. This representation transformed the regular distributed form of DNF into an interconnected nonlinear model. A non-parametric modeling strategy yields to design the adjustment laws for the DNF weights. The algorithm used to adjust the weights considered self interconnections for each layer as well as external stimulus. The concept of controlled Lyapunov function served as the main tool to design a stable learning method for DNF. This algorithm was implemented in a class of hybrid computational model that served to execute the modeling of physiological response associated to visual external stimuli. The DNF model designed in this study can consider just the excitation response of specific neuron circuits without considering the presence of inhibitory response. This condition extends the number of electrophysiological trials where the adjusted DNF model can be evaluated. The learning method was evaluated with the information from a database that contains information coming from a selective visual attention experiment where the external stimuli appeared briefly in any of five squares arrayed horizontally above a central fixation cross. The degree of correlation (above 0.95) between signals measured at the brain cortex and the response of the DNF justified the application of the method proposed in this study.

This study addresses the problem of designing an output model reference control for non-linear systems in the presence of parametric disturbances/uncertainties in the state model and output noise measurements. A state observer based on a differential neural network (DNN) estimates the unknown states and the unknown disturbance simultaneously. The control design includes the estimated disturbance to provide a better tracking performance. The second result optimizes the gains of the controller and observer in order to obtain a reduced convergence zone for the tracking error based on the attractive ellipsoid method approach (AEM). Numerical results point out the advantages obtained by the nonlinear control based on the DNN observer when it is compared with a classical Luenberger structure.

The identification problem incorporated in feedback control of uncertain nonlinear systems exhibiting complex behavior has been solved in different ways. Some of these solutions have used artificial intelligence methods like fuzzy logic and neural networks. However, their individual implementation suffers from certain drawbacks, such as the black-box nature of neural network and the problem of finding suitable membership functions for fuzzy systems. These weaknesses can be avoided by implementing a hybrid structure combining these two approaches, the so-called neuro-fuzzy system. In this paper, a neuro-fuzzy system that implements differential neural networks (DNNs) as consequences of Takagi-Sugeno (T-S) fuzzy inference rules is proposed. The DNNs substitute the local linear systems that are used in the common T-S method. In this paper, DNNs are used to provide an effective instrument for dealing with the identification of the uncertain nonlinear system, while the T-S rules are used to provide the framework of previous knowledge of the system. The main idea is to carry out an online identification process of an uncertain nonlinear system with the aim to design a closed-loop trajectory tracking controller. The methodology developed in this study that supports the identification and trajectory control designs is based on the Lyapunov formalism. The DNN implementation results in a time-varying T-S system. As a consequence, the solution of two time-varying Riccati equations was used to adjust the learning laws in the DNN as well as to adjust the gains of the controller. Two results were provided to justify the existence of positive-definite solutions for the class of Riccati equations used in the learning laws of DNNs. A complete description of the learning laws used for the set of DNN identifiers is also obtained. An autonomous underwater vehicle system is used to demonstrate the performance of the controller on tracking a desired 3-D path by this combination of the DNN and the T-S system.

This paper addresses the design of a normalized convergent learning law for neural networks (NNs) with continuous dynamics. The NN is used here to obtain a nonparametric model for uncertain systems described by a set of ordinary differential equations. The source of uncertainties is the presence of some external perturbations and poor knowledge of the nonlinear function describing the system dynamics. A new adaptive algorithm based on normalized algorithms was used to adjust the weights of the NN. The adaptive algorithm was derived by means of a nonstandard logarithmic Lyapunov function (LLF). Two identifiers were designed using two variations of LLFs leading to a normalized learning law for the first identifier and a variable gain normalized learning law. In the case of the second identifier, the inclusion of normalized learning laws yields to reduce the size of the convergence region obtained as solution of the practical stability analysis. On the other hand, the velocity of convergence for the learning laws depends on the norm of errors in inverse form. This fact avoids the peaking transient behavior in the time evolution of weights that accelerates the convergence of identification error. A numerical example demonstrates the improvements achieved by the algorithm introduced in this paper compared with classical schemes with no-normalized continuous learning methods. A comparison of the identification performance achieved by the no-normalized identifier and the ones developed in this paper shows the benefits of the learning law proposed in this paper.

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.

Time-delay systems have been succesfully used to represent complex dynamical systems. Indeed, time-delay is usually encountered as part of many real systems. Among others, biological and chemical plants have been modeled using Time-delay terms with better results than those models that do not consider them. However, getting those models represents a formidable effort and sometimes the results are not so satisfactory. On the other hand, no parametric modelling offer an alternative to obtain suitable and usable models. Continuous neural networks (CNN) have been considered as a real alternative to produce such no parametric representations. This article introduces the design of a specific class of no parametric model for uncertain Time-delay system based on CNN considering the so-called delayed learning laws. The convergence analysis as well as the learning laws are produced from a Lyapunov-Krasovskii functional. A numerical example regarding the human innmunodeficiency virus dynamical behavior is used to show the performance of the suggeted no parametric identifier based on CNN.

The control problem of nonlinear systems affected by external perturbations and parametric uncertainties has attracted the attention for many researches. Artificial Neural Networks (ANN) constitutes an option for systems whose mathematical description is uncertain or partially unknown. In this paper, a Recurrent Neural Network (RNN) is designed to address the problems of identification and control of discrete-time nonlinear systems given by a gray box. The learning laws for the RNN are designed in terms of discrete-time Lyapunov stability. The control input is developed fulfilling the existence condition to establish a Quasi Sliding Regime. In means of Lyapunov stability, the identification and tracking errors are ultimately bounded in a neighborhood around zero. Numerical examples are presented to show the behavior of the RNN in the identification and control processes of a highly nonlinear discrete-time system, a Lorentz chaotic oscillator.

In general, output-based controller design remains an important research area in control theory. Most of the existing solutions use a state estimation algorithm to reconstruct a plausible approximation of the real state. Then, one can apply a nonlinear controller, based on fuzzy logic, for example, to enforce the system trajectories to a desirable stable equilibrium point. Nevertheless, the aforementioned method may not be suitable for uncertain systems affected by external noises. State observers based on the system's structure cannot be applied in those cases. However, some sort of adaptive estimation may be developed. This paper deals with a fuzzy controller that was designed using the state observer solution when the dynamic model of a plant contains uncertainties or it is partially unknown. Differential neural network (DNN) approach is applied in this uninformative situation. A new learning law, containing an adaptive adjustment rate, is suggested to enforce the stability condition for the observer's free parameters. On the other hand, nominal weights are adjusted during the preliminary training process using the least mean square method. Lyapunov theory is used to obtain the upper bounds for the weight's dynamics. The proposed method seems to be a more advanced option to control uncertain systems when the state available information is reduced. Even when several options exist to control this class of nonlinear systems such as PID, the method introduced here uses the knowledge on the system behavior and enforces the reconstruction of the immeasurable states. This last issue is an extra advantage because it serves as a general software sensor. The well-known two-link manipulator is used to show the effectiveness of the proposed algorithm. A couple of cases are used here: the full actuated and the under-actuated systems. In both situations, the controller achieves a better performance than the well-known PID controllers and a fuzzy controller using the estimated states produced by a high-order sliding-mode observer. A practical example showing how the fuzzy controller based on the estimated states produced by the differential neural network observer is also presented. The system used to test the controller is the anaerobic digestion. In this case, the benefits of this output-based controller are also demonstrated.

In this paper a new model predictive neural control is suggested. It consists of the application of the model predictive method to control a nonlinear uncertain system where the information is reduced. The uncertain plant was approximated by a special class of dynamic neural network observer (projectional observer) that uses some sort of information regarding the set where the states remain. A novel method leads to construct an approximate model of the uncertain system where the controllability condition is ensured. The model predictive control was designed using the information obtained by the proposed observer. The upper bound for the tracking error was established if the controller is applied. Simulation regarding the control of a biotechnological process is carried out.

State estimation for uncertain systems affected by external noises is an important problem in control theory. This paper deals with the state observation problem when the dynamic model of a discrete plant contains uncertainties or is partially unknown. The suggested observer is oriented to solve the state observation problem of discrete time nonlinear systems. Most of the existing results using the neural networks approach have been developed for continuous time systems using. The recurrent neural network (RNN) have shown several advantages to treat many different control and state estimation problems. In this paper, it is presented a new discrete-time observer using the structure of a classical RNN. This observer includes a correction term using the output information and the first order sliding modes. The second method Lyapunov is applied to generate a new learning law, that contains an adaptive adjustment rate. This study proofs the stability condition for the free parameters included in the neural-observer. A numerical example is given using the RNN in the estimation of a mathematical model describing the HIV infection, that includes non-infected cells, infected cells and free virions. This model was used to generated the data using to test the observer.

In this paper a strategy based on differential neural networks for the identification of the parameters in a mathematical model described by partial differential equations is proposed. The identification problem is reduced to finding an exact expression for the weights dynamics using the differential neural networks properties. The adaptive laws for weights ensure the convergence of the neural network trajectories to the partial differential equation states. To investigate the qualitative behavior of the suggested methodology, here the non-parametric modeling problem for a distributed parameter plant is analyzed: the tubular reactor system.

Non parametric identifier based on differential neural networks is presented. Nonlinear systems with the a priori information about state constrains are considered. Differential neural network identifier includes a projectional operator as part of its structure in order to keep the state restrictions. Stability analysis based on Lyapunov-Krasovskii energetic-function lets proof the ultimated boundedness of error estimation. Free parameters stability of DNNI (adaptive weights) is also proved. Simulation examples regarding to biotechnological process and the Chua's circuit depict the main advantages of the proposed structure against reported differential neural network with similar structure but without projectional operator.

State estimation for uncertain systems affected by external noises is an important problem in control theory. This paper deals with the state observation problem when the dynamic model of a plant contains uncertainties or is completely unknown and it is oriented to discrete time nonlinear systems because most of the existent results have been developed for continous time systems. The recurrent neural network (RNN) have shown his advantages to deal with this class problem. The Lyapunov second method is applied to generate a new learning law, containing an adaptive adjustment rate, implying the stability condition for the free parameters of the neural-observer. A numerical example is given using the RNN in the estimation of a mathematical model of HIV infection with three states.

State estimation for uncertain systems affected by external noises is an important problem in control theory. This paper deals with a state observation problem when the dynamic model of a plant contains uncertainties or it is completely unknown. Differential neural network (NN) approach is applied in this uninformative situation but with activation functions described by wavelets. A new learning law, containing an adaptive adjustment rate, is suggested to imply the stability condition for the free parameters of the observer. Nominal weights are adjusted during the preliminary training process using the least mean square (LMS) method. Lyapunov theory is used to obtain the upper bounds for the weights dynamics as well as for the mean squared estimation error. Two numeric examples illustrate this approach: first, a nonlinear electric system, governed by the Chua's equation and second the Lorentz oscillator. Both systems are assumed to be affected by external perturbations and their parameters are unknown.

The temperature adaptive control for a neonatal incubator is shown in this paper. The control design is based on the neurofuzzy algorithm and the extended Kalman filter technique. The Kalman filter adjusts the weights associated with the neural network structure, while the ANFIS (artificial neural fuzzy inference system) structure (using the back propagation scheme) is applied to change the Gaussian membership function parameters in an adaptive way (using the delta rule scheme). The external temperature gradient (ETG) and the external temperature gradient rate (ETGR) principles were used as input variables in the identifier-controller design. The results for this process were proved in real time and in a real incubator with a reference temperature around 37degC. The efficiency of the suggested method is shown by the convergence of the ETG and ETGR to its reference range while the temperature in the care unit is keep very near to the selected set point value.

Immunotherapy refers to the use of natural and synthetic substances to stimulate the immune response. This paper provides a description for an adaptive control design for immunotherapy cancer treatment mathematical model, where all state vector is considered to be not on-line available. The control strategy is suggested in two parts: the first one deals with the state estimation process using differential neural networks and sliding mode type observer techniques. The second part introduces the non-linear neural-identifier design which provides a complete state information to construct a discontinuous (sliding mode type) control function. This technique was successfully applied in the tracking process for immunotherapy dosage control.

In this paper, it is presented a differential neural network supplied with a new learning law based on the sliding mode approach. The state observer is employed to estimate the dynamics states of degradation mathematical model, where the incomplete information and the limited on-line measure problems are considered. A new training method is applied in the learning algorithm is proposed to reconstruct biomass, organic matter recalcitrant concentrations and volume of biological culture evolutions. This allows ensuring an upper bound for the weights time evolution. This new scheme gives the possibility to construct not only one adaptive process but a set of learning laws. The effectiveness of this algorithm is shown by numerical results.

Hepatitis C is one of the illness that have affected many people around the world. It seriously harms the patient health in many ways. This paper provides a description of an adaptive nonlinear observer based on differential neural networks (DNN), designed for hepatitis C mathematical model, where all state vector is considered not to be available. Only viral load is assumed to be measurable with any analytical method like reverse transcriptase -polymerase chain reaction (RT-PCR). The process is taken in two stages: a training scheme which generates the correct parameter set for the DNN-observer and the estimation process for three different inputs, which confirms (in numerical way) the robustness to input variations of the DNN scheme.