Probabilistic model validation for uncertain nonlinear systems

Abstract

This paper presents a probabilistic model validation methodology for nonlinear systems in time-domain. The proposed formulation is simple, intuitive, and accounts both deterministic and stochastic nonlinear systems with parametric and nonparametric uncertainties. Instead of hard invalidation methods available in the literature, a relaxed notion of validation in probability is introduced. To guarantee provably correct inference, algorithm for constructing probabilistically robust validation certificate is given along with computational complexities. Several examples are worked out to illustrate its use. , . . . , b s ; x) stands for generalized hypergeometric function. The symbols N (., .), U (.), and A (.) are used for normal, uniform and arcsine distributions, respectively. We use the notation ξ 0 (.) to denote the joint PDF over initial states and parameters. ξ (., t) and ξ (., t) denote joint PDFs over instantaneous states and parameters, for the true and model dynamics, respectively. Similarly, η (., t) and η (., t), respectively denote joint PDFs over output spaces y and y at time t, for the true and model dynamics. The symbol x is used to denote the extended state vector obtained by augmenting the state (x) and parameter (p) vectors. We use χ to denote indicator function and # to denote cardinality. Unless stated otherwise, δ (.) stands for Dirac delta. The symbol I denotes the -by-identity

Unified Framework and Survey for Model Verification, Validation and Uncertainty Quantification

“…Many other references in the control community relax the hard invalidation for a wide variety of different system representations and consider probabilistic certificates for the model (in)validity instead [136,171]. For example, Halder and Bhattacharya [75] address UQ of dynamic systems, compare the simulation and experimental PDF with a Wasserstein metric and calculate a probabilistically robust validation certificate (PRVC) in each time step based on a required tolerance level. Karydis et al [102] calculate the probability of violating a confidence region as tolerance value and use it with randomization techniques to expand stochastic models with uncertain parameters that capture the experiments.…”

Probabilistically valid stochastic extensions of deterministic models for systems with uncertainty

“…Some applications involve correlation analysis of residuals (Ljung, 1999), prediction error within a robust control framework (Gevers et al, 2003), and computation of relative weighted volumes of convex sets for parametric uncertainty models (Lee and Poolla, 1996). A different approach employs a probabilistic model validation methodology to compare a model-generated output probability density function (PDF) with one observed through experiments (Halder and Bhattacharya, 2014). The approach relies on the availability of analytic expressions for propagating the uncertainty through the model at hand, and provides sample-complexity bounds for robust validation inference based on randomized algorithms.…”

Optimal Transport Approach for Probabilistic Robustness Analysis of F-16 Controllers

“…(14), where we used the empirical PMF values of φ LQR xt; t and φ GSLQR xt; t, which were in turn computed from the respective exact joint PDF values obtained from the MOC solution at time t, evaluated at the instantaneous sample locations. Notice that we do not use any grid or resampling to compute the Wasserstein [38,41] and references therein. The schematic of this computation is shown in Fig.…”