Small faults (some weak faults with a tiny magnitude) are difficult to detect and may cause severe problems leading to degrading the system performance. This paper proposes an approach to estimate, detect, and isolate small faults in uncertain nonlinear systems subjected to model uncertainties, disturbances, and measurement noise. A robust observer is developed to alleviate the lack of full state measurement. Using the estimated state, a dynamical radial basis function neural networks observer is designed in form of LMI problem to accurately learn the function of the inseparable mixture between modeling uncertainty and the small fault. By exploiting the knowledge obtained by the learning phase, a bank of observers is constructed for both normal and fault modes. A set of residues is achieved by filtering the differences between the outputs of the bank of observers and the monitored system output. Due to the noise dampening characteristics of the filters and according to the smallest residual principle, the small faults can be detected and isolated successfully. Finally, rigorous analysis is performed to characterize the detection and isolation capabilities of the proposed scheme. Simulation results are used to prove the efficacy and merits of the proposed approach.

To guarantee convergent state estimates and exact approximations, it is highly desirable that observers can independently dominate the effects of unmodelled dynamics. Based on adaptive nonlinear approximation, this paper presents a robust H∞ gain neuro-adaptive observer (R H∞GNAO) design methodology for a large class of uncertain nonlinear systems in the presence of time-varying unknown parameters with bounded external disturbances on the state vector and on the output of the original system. The proposed R H∞GNAO incorporates radial basis function neural networks (RBFNNs) to approximate the unknown nonlinearities in the uncertain system. The weight dynamics of every RBFNN are adjusted online by using an adaptive projection algorithm. The asymptotic convergence of the state and parameter estimation errors is achieved by using Lyapunov cogitation under a well-defined persistent excitation condition, and without recourse to the strictly positive real condition. The repercussions of unknown disturbances are reduced by integrating an H∞ gain performance criterion into the proposed estimation approach. The condition imposed by this proposed observer approach, such that all estimated signals are uniformly ultimately bounded, is expressed in the form of the linear matrix inequality problem and warrants the demanded performances. To evaluate the performance of the proposed observer, various simulations are presented.

This paper is devoted to the issue of a robust predictive control for linear discrete-time systems by using Meixner-like model. The Meixner-like functions are an extension of Laguerre functions and convenient when the system has a slow start or delay. To ensure the reduction of the parameter number in the Meixner-like model, the optimization of parameters characterizing the Meixner-like functions is proposed. This proposed robust predictive control copes with physical constraints and geometrical constraints due to parameter uncertainties, which are estimated by using the Unknown But Bounded Error (UBBE) approach, and leads to the min-max optimization problem.

In this paper, we prove the possibility of optimization of some free parameters of Meixner-like discrete-time linear filters using orthonormal basis functions (OBF) in an analytical way. Since the z-transform of the Meixner filters is impossible and it is possible for Meixner-like filters, we are motivated to study closely the optimization of the Meixner-like's pole. As a result, the differential equation associated with Meixner-like filters has been developed. On the other hand, the effective width of the energy distribution in the transformation domain is a function of the free parameter (Meixner-like pole) and simple signal measurements and can be calculated by the second-order moment. Analytic minimization of the second-order moment gives an optimal value of the free parameter. To validate the effectiveness of the proposed approach, numerical results are used to determine the free parameters.

The present work proposes a new method for synthesizing a PI control algorithm using Meixner-like model. The Meixner-like model is used to represent the linear system with delay or a slow initial onset. The optimization of the Normalized Mean Square Error is considered to estimate the optimal Meixner-like pole. The proposed control approach is tested on a numerical example.

This paper focuses on the application of quadratic optimization for the approximation of uncertain nonlinear robotic function. This function will be used to perform the task of motion control to feedback control of robotic systems. To achieve this task, we are trying, through the study and simulation four approximation approaches: Power Series Polynomial Approximation (PSPA), Orthogonal Neural Network Approximation (ONNA), Chebyshev Polynomials and Series Approximation (CP& SA) and Least Squares Chebyshev polynomial approximation (LSCPA). In each case mentioned above, we use the orthogonal polynomial approximation in higher dimension spaces, which enable us to modify classical differential equation solvers to perform high precision, nonlinear term in model robotic. We unify and extend classical results from function approximation theory and consider their utility in robotics. Then, we could use an efficient algorithms for solving any robust control problems of manipulator robot. Simulation results from a two-link robot manipulator show the satisfactory performance of the approach of approximation the nonlinear term in model robotic..

This article aims at updating the domains belonging to the parameters of the model resulting from the decomposition of the linear system on the Meixner-like basis. The robust identification is performed after applying the minimization of the Normalized Mean Square Error to estimate the optimal Meixner-like pole. In the context of set membership identification, the feasible parameter set is defined as the set of plant parameters which are consistent with the model structure, the assumptions on disturbances (unknown but bounded) and all available measurements.

This study investigates the design of robust adaptive H ∞ gain neural observer (RA H ∞ NO) for a large class of non-linear systems with unknown constant parameters in the presence of bounded external perturbations on the state vector and on the output of the original system. The proposed adaptive observer incorporates radial basis functions (RBFs) neural networks (NN) to approximate the unknown non-linearities existing in the system. The weight dynamics of every RBFNN are adjusted on-line by using an adaptive projection algorithm. The proof of the asymptotic convergence of the state and parameter estimate errors is achieved by using Lyapunov arguments under a well-defined persistent exciting condition, and without recourse to the strictly positive real condition. The effect of unknown disturbances is reduced by integrating a H ∞ gain performance criterion into the proposed estimation scheme. The existence condition of the proposed observer such that all estimated signals are uniformly ultimately bounded is expressed in the form of the linear matrix inequality problem. To evaluate the performance of the proposed observer, three simulations are made.

This article focuses on modeling and identification of a LTI system represented by Meixner-like functions. However a significant reduction of this model is subject to an optimal choice of Meixner-like pole. An iterative algorithm based on the Newton-Raphson's approximation technique and evolutionary method are proposed for the optimization of the Meixner-like pole from input/output measurements. The effectiveness of this method is tested on numerical example.

In this paper, the Meixner-like model is used to represent the linear discrete-time system. Furthermore, we present, from input/output measurements a recursive representation of Meixner-like model. The minimization of the Normalized Mean Square Error is considered to estimate the optimal Meixner-like pole. The belonging domain of the parameters, which is compatible with the model structure, measurements and the bounds of the error, is defined by the interval values. A numerical simulation shows the efficiency of the approach.

This paper deals with the identification of linear systems developed on the Meixner-like basis. Two aspects will be considered, namely the identification of the structure of the model obtained (pole of the base) as well as the identification of its parameters (Fourier coefficients). It should be noted that the model structure is linear with respect to the Fourier coefficients while it is not with respect to the pole. So as to reduce the number of model coefficients, the optimum value of the pole is determined by the method of minimizing the Normalized Mean Square Error. Once the structure has been determined, the second step will be carried out in which the domain of the coefficients of the resulting model will be updated. Two domains have been addressed, namely the polytope and the ellipsoid. Also, the interval analysis method has been proposed in this article.

This paper is concerned with an optimal expansion of linear discrete time systems on Meixner functions. Many orthogonal functions have been widely used to reduce the model parameter number such as Laguerre functions, Kautz functions and orthogonal basis functions. However, when the system has a slow initial onset or delay, Meixner functions, which have a slow start, are more suitable in terms of providing a more accurate approximation to the system. The optimal approximation of Meixner model is ensured once the pole characterizing the Meixner functions is set to its optimal value. In this paper, a new recursive representation of Meixner model is proposed. Further we propose, from input/output measurements, an iterative pole optimization algorithm of the Meixner pole functions. The method consists in applying the Newton-Raphson’s technique in which their elements are expressed analytically by using the derivative of the Meixner functions. Simulation results show the effectiveness of the proposed optimal modeling method.

In this article, we explore the use of the Meixner-like functions in estimating transfer functions of linear discrete-time systems. By expanding transfer functions with some suitable orthonormal basis functions, it is possible to reduce the estimate parameter number. Thus far, many orthonormal functions have been widely used in this framework such as Laguerre functions and Kautz functions. However, when the system have a slow initial onset or delay, Meixner-like functions, which have a slow start, are more suitable in terms of providing a more accurate approximation to the system. In this article, a minimal state-space realizations for discrete-time linear systems are derived using Meixner-like filters. Further we propose, from input/output measurements, an iterative optimization algorithm for the free parameter (Meixner-like pole) of the Meixner-like filters. The method consists in applying the Newton-Raphson's technique in which their elements are expressed analytically by using the derivative of the Meixner-like functions.

In this study, the authors investigate the parametric complexity reduction of the Meixner-like model for linear discrete-time system representation. The use of the Meixner-like functions is more suitable than the use of Laguerre functions and Kautz functions especially when the system have a slow initial onset or delay. The coefficients of the Meixner-like model can be estimated recursively from input-output data by the new representation. Noting that the selection of an arbitrary pole for the Meixner-like functions can raise the parameter number of the Meixner-like model. However, when the pole is set to its optimal value, an optimal expansion of transfer functions is produced. Therefore an optimisation technique is developed to generate the optimal Meixner-like pole, which is achieved by an iterative method, that consists in minimising the mean square error between the system output and the model output. Theoretical analysis and a numerical simulation show the efficiency of the approach.