Probabilistic System Identification with Unidentifiable Models

In a Bayesian probabilistic framework for system identification, the performance reliability for a structure can be updated using structural test data D by considering the reliability predictions of a whole set of possible structural models that are weighted by their updated probability. This involves integrating h(Θ)p(Θ|D) over the whole parameter space, where Θ is a parameter vector defining each model within the set of possible models of the structure, h(Θ) is the structural reliability predicted by the model and p(Θ|D) is the updated probability density for Θ which provides a measure of how plausible each model is given the data D. The resulting integral, called the updated 'robust' reliability integral, is difficult to evaluate because the dimension of the parameter space is usually too large for direct numerical integration. In practical applications, the variation of p(Θ|D) is usually more dominant than h(Θ), and thus methods for evaluating the integral are differentiated by the topological characteristics of p(Θ|D). In the 'identifiable' case, p(Θ|D) is peaked at a finite number of 'optimal points' and asymptotic methods can be used to approximate the integral using information at the optimal points. The evaluation of the integral in the 'unidentifiable' case, where p(Θ|D) is concentrated in the neighborhood of a manifold S of lower dimension than the parameter space, is much more difficult. Standard Monte Carlo simulation or importance sampling fail because the important region of the integrand, which is in the neighborhood of the manifold S, is often of complicated geometry and has small volume in the parameter space. Deterministic search methods for computing an asymptotic approximation of the robust reliability integral have appeared in the literature, which discretize the manifold S using a finite number of representative points and then approximate p(Θ|D) as a discrete probability mass distribution among the representative points. The complexity and computational effort associated with such methods arc expected to grow in a similar manner to that of direct numerical integration, making the method practical only when the dimension of the manifold is small. This paper presents a Markov chain Monte Carlo simulation method to evaluate the robust reliability integral without the need for optimization to find the manifold S. It is based on the Metropolis- Hastings algorithm augmented with an adaptive scheme to gain information about the manifold in a gradual manner. By carrying out a series of Markov chain simulations with limiting stationary distributions equal to a sequence of intermediate PDFs that converge on p(Θ|D), the region of significant probability density of p(Θ|D) is gradually portrayed. The Markov chain samples can be used to estimate the robust reliability integral by statistical averaging. The method is illustrated using simulated modal test data to update the robust reliability of a two-story moment-resisting frame where the model is not identifiable based on the data