Guaranteed control of switched control systems using model order reduction and state-space bisection

This paper considers discrete-time linear systems controlled by a quantized law, i.e., a piecewise constant time function taking a finite set of values. We show how to generate the control by, first, applying model reduction to the original system, then using a "state-space bisection" method for synthesizing a control at the reduced-order level, and finally computing an upper bound to the deviations between the controlled output trajectories of the reduced-order model and those of the original model. The effectiveness of our approach is illustrated on several examples of the literature

Strict error bounds for linear and nonlinear solid mechanics problems using a patch-based flux-free method

We discuss, in this paper, a common ux-free method for the computation of strict error bounds for linear and nonlinear Finite Element computations. In the linear case, the error bounds are on the energy norm of the error, while, in the nonlinear case, the concept of error in constitutive relation is used. In both cases, the error bounds are strict in the sense that they re- fer to the exact solution of the continuous equations, rather than to some FE computation over a refined mesh. For both linear and nonlinear solid mechanics, this method is based on the computation of a statically admissible stress field, which is performed as a series of local problems on patches of elements. There is no requirement to solve a previous problem of ux equilibration globally, as happens with other methods.Postprint (published version

Efficient posterior estimation for stochastic SHM using transport maps

Accurate parameter estimation is a challenging task that demands realistic models and algorithms to obtain the parameter’s probability distributions. The Bayesian theorem in conjunction with sampling methods proved to be invaluable here since it allows for the formulation of the problem in a probabilistic framework. This opens up the possibilities of using prior information and knowledge about parameter distributions as well as the natural incorporation of aleatory and epistemic uncertainties. Traditionally, Markov Chain Monte Carlo (MCMC) methods are used to approximate the posterior distribution of samples given some data. However, these methods usually need a large amount of samples and therefore a large amount of model evaluations. Recent advances in transport theory and its application in the context of Bayesian model updating (BMU) make it possible to approximate the posterior distribution analytically and hence eliminate the need for sampling methods. This paves the way for the usage in real-time applications and for fast parameter estimation. We investigate here the application of transport maps to a simple analytical model as well as a structural dynamics model. The performance is compared to an MCMC approach to assess the accuracy and efficiency of transport maps. A discussion about requirements for the implementation of transport maps as well as details on the implementation are also given

Transport Map Coupling Filter for State-Parameter Estimation

Many dynamical systems are subjected to stochastic influences, such as random excitations, noise, and unmodeled behavior. Tracking the system's state and parameters based on a physical model is a common task for which filtering algorithms, such as Kalman filters and their non-linear extensions, are typically used. However, many of these filters use assumptions on the transition probabilities or the covariance model, which can lead to inaccuracies in non-linear systems. We will show the application of a stochastic coupling filter that can approximate arbitrary transition densities under non-Gaussian noise. The filter is based on transport maps, which couple the approximation densities to a user-chosen reference density, allowing for straightforward sampling and evaluation of probabilities.Comment: Published in Advances in Reliability, Safety and Security ESREL 2024 Contributions, https://esrel2024.com/wp-content/uploads/articles/part9/transport-map-coupling-filter-for-state-parameter-estimation.pd

A posteriori error estimation and adaptive strategy for PGD model reduction applied to parametrized linear parabolic problems

We define an a posteriori verification procedure that enables to control and certify PGD-based model reduction techniques applied to parametrized linear elliptic or parabolic problems. Using the concept of constitutive relation error, it provides guaranteed and fully computable global/goal-oriented error estimates taking both discretization and PGD truncation errors into account. Splitting the error sources, it also leads to a natural greedy adaptive strategy which can be driven in order to optimize the accuracy of PGD approximations. The focus of the paper is on two technical points: (i) construction of equilibrated fields required to compute guaranteed error bounds; (ii) error splitting and adaptive process when performing PGD-based model reduction. Performances of the proposed verification and adaptation tools are shown on several multi-parameter mechanical problems