Optimization of approximating networks for optimal fault diagnosis

Abstract

An optimization-based approach to fault diagnosis for nonlinear stochastic dynamic models is developed. An optimal diagnosis problem is formulated according to a receding-horizon strategy. This approach leads to a functional optimization problem (also called 'infinite optimization problem'), whose admissible solutions belong to a function space. As in such a context, the tools from mathematical programing are either inapplicable or inefficient, a methodology of approximate solution is proposed that exploits diagnosis strategies made up of combinations of a certain number of simple basis functions, easy to implement and dependent on some parameters to be optimized. The optimization of the parameters is performed in two phases. In the first, 'a-priori' knowledge of the statistics of the stochastic variables is used to initialize (off-line) the parameter values. In the second phase, the optimization continues on-line. Both off-line and on-line phases rely upon stochastic approximation algorithms. The overall procedure turns out to be effective in high-dimensional settings such as those characterized by a large dimension of the state space and a large diagnosis window. This favorable behavior results from certain properties of the proposed methodology of approximate optimization, such as polynomial bounds on the rate of growth of the number of parameterized basis functions, which guarantees the desired accuracy of approximate optimization. The effectiveness of the approach is confirmed by simulations in the context of a complex instance of the fault-diagnosis problem. The advantages over classical approaches to fault diagnosis are discussed and pointed out by numerical results.

Error Estimates for Approximate Optimization by the Extended Ritz Method

“…Also, a new branch of nonlinear approximation theory investigating approximation capabilities of neural networks was developed [11,12,21,28,38,45,50,51,52,53,54,55,56,57,58]. In a series of papers [3,5,8,9,10,64,65,66,80,81], a new method of approximate optimization was developed, called in [81] the extended Ritz method. In these papers, approximate solutions were used that were obtained over restrictions of sets of admissible solutions to linear combinations of all n-tuples of functions with varying "free" parameters, instead of linear combinations of first n functions from a basis with fixed ordering as in the classical Ritz method.…”

“…The extended Ritz method with such bases was successfully tested on a variety of problems with admissible solutions dependent on a large number of variables: stochastic optimal control [64,65,66,80] and optimal estimation of state variables [3] in nonlinear dynamic systems with a large number of state variables, team optimal control [8], optimal control of freeway traffic [81], routing in large-scale communication networks [9,10], optimal fault diagnosis [5], etc. In these applications, admissible sets of variable-basis functions were used, for which the degree n necessary to guarantee a fixed approximation accuracy grows only polynomially with the number of variables of admissible solutions.…”

“…For G formed by parameterized families of the form G = {g a : a ∈ A, g a ∈ G} with A ⊆ R p , this method was applied to a variety of optimization tasks considered in a series of papers [3,5,8,9,10,64,65,66,80] and in [81], where it was called the extended Ritz method. Here we use this term even more generally for an approximate optimization by the sequence of problems {M ∩ span n G}, where G is any subset of X.…”

“…Such a problem can be solved by algorithms based on gradient descent with stochastic perturbations [17, [4,16,40,76] and the references therein). In [5,9,10,65,66,81], applications of some of these algorithms to the extended Ritz method are described and illustrated by numerical results showing the algorithms' effectiveness in a variety of cases.…”

Design of Asymptotic Estimators: An Approach Based on Neural Networks and Nonlinear Programming

“…As regards the selection of the design parameters, it is obtained by means of a nonlinear-programming algorithm running offline [25], [26]. This is a major advantage over other neural approaches to estimation proposed in the literature (see, e.g., [27]- [31]), which rely on the online adaptation of the neural weights and so may involve a large number of computations in real-time applications.…”