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The state estimation problem is addressed for a class of continuous-time, nonlinear dynamic systems with Lipschitz nonlinearities, both with and without the presence of disturbances acting on the system and on the measurement equations, i.e., for observers an filters, respectively. A general estimator is considered, which is described by a differential equation resulting from the summation of the system dynamics and an innovation function. The characteristics of this estimator are investigated as far as the stability of the estimation error is concerned. Then, an optimal estimation problem is stated. A solution of this problem is sought by means of nonlinear approximators, which guarantee an arbitrarily small error using a parametrized structure with a moderate complexity.

We develop a new algorithm for the learning of feedforward neural networks, by stating the learning process as a parameter estimation problem. We provide an analysis of its convegence and robustness properties. Two different versions of the algorithm are discussed, depending on the way in which the training set is explored during learning. The simulation results, for both classification and function approximation problems, confirm the effectiveness of the proposed algorithm and its advantages with respect to error back-propagation and extended Kalman filter-based learning.

A receding-horizon (RH) optimal control scheme for a discrete-time nonlinear dynamic system is presented. A nonquadratic cost function is considered and constraints are imposed on both the state and control vectors. A stabilizing regulator is derived by adding a proper terminal penalty function to the process cost. The control vector is generated by means of a feedback control law computed off-line instead of computing it online, as is done for existing RH regulators. The off-line computation is performed by approximating the RH regulator by a multilayer feedforward neural network. Bounds to this approximation are established. Algorithms are presented to determine some essential parameters for the design of the neural regulator, i.e., the parameters characterizing the terminal cost function and the number of neural units in the networks implementing the regulator. Simulation results show the effectiveness of the proposed approach.

This paper describes a fault diagnosis algorithm for a class of nonlinear dynamic systems with modeling uncertainties, using input/output measurements. Two types of faults are considered: parametric faults which characterize fault functions of known structure but unknown parameters, and non-parametric faults where the structure of the fault function is unknown. A nonlinear estimation model with a learning algorithm is developed and the fault separability concept for parametric faults is discussed.

A receding-horizon (RH) optimal control scheme and a sliding—window (SW) state estimation scheme are presented and neural approximations for these two control and estimation laws are presented. Some theoretical results on the stability of the controller and the error convergence for the estimator are discussed.

Classification of large data sets by feedforward neural networks is investigated. To deal with unmanageably large sets of classification tasks, a probabilistic model of their relevance is considered. Optimization of networks computing randomly chosen classifiers is studied in terms of correlations of classifiers with network input–output functions. Effects of increasing sizes of sets of data to be classified are analyzed using geometrical properties of high-dimensional spaces. Their consequences on concentrations of values of sufficiently smooth functions of random variables around their mean values are applied. It is shown that the critical factor for suitability of a class of networks for computing randomly chosen classifiers is the maximum of sizes of the mean values of their correlations with network input–output functions. To include cases in which function values are not independent, the method of bounded differences is exploited.

In this work, a computational method is proposed to automatically investigate the perception of the origin of full-body human movement and its propagation. The method is based on a mathematical game built over a suitably defined graph structure representing the human body. The players of this game are the graph vertices, which form a subset of body joints. Since each vertex contributes to a shared goal (i.e., to the way in which a specific movement-related feature is transferred among the joints), a cooperative game-theoretical model (specifically a transferable-utility game) is adopted, which is able (via the Shapley value) to measure the relevance of the various joints in human movement when performing full-body movement analysis. The method is theoretically investigated and applied to a motion capture dataset obtained from subjects who performed expressive movements. Finally, the method is validated through an online survey, in which several dancers/nondancers participated. The results show the capability of the proposed approach to represent the evolution of the most important joint responsible for originating each dancer's movement.

State estimation for continuous-time, nonlinear dynamic systems with Lipschitz nonlinearities is considered. A class of estimators composed of a prediction term and an innovation term is defined, where the innovation function belongs to a suitable smoothness class and has to be determined in such a way as to minimize an estimation cost, represented by the L/sub /spl infin// norm of the estimation error. Since the admissible innovation functions belong to an infinite-dimensional functional space, the minimization of such a cost represents a functional optimization problem, difficult to solve in a general setting. Approximating the innovation function by a family of parametrized nonlinear approximators allows one to reduce the original functional optimization problem to a sequence of nonlinear programming problems.

We develop some mathematical tools for comparison of rates of fixed versus variable basis function approximation. Using these tools, we describe sets of multivariable functions, for which lower bounds on worst-case errors in approximation by n-dimensional linear subspaces are larger than upper bounds on such errors in approximation by perceptron networks with n hidden units.

Sets of multivariable functions that can be approximated with "dimension-independent" rates either by linear approximators or by neural networks having various types of computational units are compared. The comparison is made by exhibiting families of functions belonging to suitable difference sets.

A class of non-linear models having the structure of combinations of simple, parametrized basis functions is investigated; this class includes widespread neural networks in which the basis functions correspond to the computational units of a type of networks. Bounds on the complexity of such models are derived in terms of the number of adjustable parameters necessary for a given modelling accuracy. These bounds guarantee a more advantageous tradeoff than linear methods between modelling accuracy and model complexity: the number of parameters may increase much more slowly, in some cases only polynomially, with the dimensionality of the input space in modelling tasks. Polynomial bounds on complexity allow one to cope with the so-called ‘curse of dimensionality’, which often makes linear methods either inaccurate or computationally unfeasible. The presented results let one gain a deeper theoretical insight into the effectiveness of neural-network architectures, noticed in complex modelling applications. Copyright © 2003 John Wiley & Sons, Ltd.

Approximate solution of a general N-stage stochastic optimal control problem is considered. It is known that discretizing uniformly the state components in applying dynamic programming may lead this procedure to incur the "curse of dimensionality". Approximating networks, i.e., linear combinations of parametrized basis functions provided with density properties in some normed linear spaces, are then defined and used in two approximate methods (examples of such networks are neural networks with one hidden layer and linear output activation functions, radial basis functions, etc.). The first one consists of approximating the optimal cost-to-go functions in dynamic programming (such a technique is known in literature as "neuro-dynamic programming"); the second method reduces the original functional optimization problem to a nonlinear programming one that is solved by means of stochastic approximation. Approximating networks of suitable types benefit by the property that the number of parameters to be optimized and the number of samples to be used for approximating some classes of regular functions increase only linearly (or moderately) with the of dimensions of the arguments of the functions and the number of samples used to train the networks. We deem that such properties may enable us to solve N-stage stochastic optimal problems often avoiding the curse of dimensionality. The two methods are tested end compared in an example involving a 10-dimension state vector.

Learning for feedforward neural networks can be regarded as a nonlinear parameter estimation problem with the objective of finding the optimal weights that provide the best fitting of a given training set. The extended Kalman filter is well-suited to accomplishing this task, as it is a recursive state estimation method for nonlinear systems. Such a training can be performed also in batch mode. In this paper the algorithm is coded in an efficient way and its performance is compared with a variety of widespread training methods. Simulation results show that the latter are outperformed by EKF-based parameters optimization.

An approximation procedure termed "extended Ritz method" is presented for the solution of functional optimization problems. The properties of powerful nonlinear approximators, such as neural networks, are exploited to face highly nonlinear optimization problems in high-dimensional settings, with the possibility of avoiding the so-called "curse of dimensionality." As an example, a nonlinear control problem involving several tens of state variables is faced.

International audience; The use of an endoscope in otological surgery as compared to using a microscope provides many benefits in terms of visibility and access to the operating region. The disadvantage of using an endoscope is that it has to be handled by the surgeon during the surgery. A novel parallel kinematic mechanism has been proposed recently for manipulating the endoscope during surgery. This paper details the design requirements considered for the optimization of the process, after consulting surgeons with various expertise. The paper further presents the different evaluation strategies that can be used to optimize the design parameters as per the needs of the application. The contribution of the presented work is the implementation of the surgeons' requirements into an objective function and the optimization of a proposed architecture of a parallel kinematic mechanism suitable for otological surgery.

Complexity of neural networks measured by the number of hidden units is studied in terms of rates of approximation. Limitations of improvements of upper bounds of the order of O(n-1/2) on such rates are investigated for perceptron networks with some periodic and sigmoidal activation functions.