| dc.description.abstract | This dissertation develops a probabilistic method for validation and verification (V&V) of uncertain nonlinear systems. Existing systems-control literature on model and controller V&V either deal with linear systems with norm-bounded uncertainties,or consider nonlinear systems in set-based and moment based framework. These existing methods deal with model invalidation or falsification, rather than assessing the quality of a model with respect to measured data. In this dissertation, an axiomatic framework for model validation is proposed in probabilistically relaxed sense, that
instead of simply invalidating a model, seeks to quantify the "degree of validation".
To develop this framework, novel algorithms for uncertainty propagation have been proposed for both deterministic and stochastic nonlinear systems in continuous time. For the deterministic flow, we compute the time-varying joint probability density functions over the state space, by solving the Liouville equation via method-of-characteristics. For the stochastic flow, we propose an approximation algorithm that combines the method-of-characteristics solution of Liouville equation with the Karhunen-Lo eve expansion of process noise, thus enabling an indirect solution of
Fokker-Planck equation, governing the evolution of joint probability density functions. The efficacy of these algorithms are demonstrated for risk assessment in Mars entry-descent-landing, and for nonlinear estimation. Next, the V&V problem is formulated in terms of Monge-Kantorovich optimal transport, naturally giving rise to a metric, called Wasserstein metric, on the space of probability densities. It is shown that the resulting computation leads to solving a linear program at each time of measurement availability, and computational complexity results for the same are derived. Probabilistic guarantees in average and worst case sense, are given for the validation oracle resulting from the proposed method. The framework is demonstrated for nonlinear robustness veri cation of F-16 flight controllers, subject to probabilistic uncertainties.
Frequency domain interpretations for the proposed framework are derived for
linear systems, and its connections with existing nonlinear model validation methods
are pointed out. In particular, we show that the asymptotic Wasserstein gap between
two single-output linear time invariant systems excited by Gaussian white noise,
is the difference between their average gains, up to a scaling by the strength of
the input noise. A geometric interpretation of this result allows us to propose an
intrinsic normalization of the Wasserstein gap, which in turn allows us to compare it
with classical systems-theoretic metrics like v-gap. Next, it is shown that the optimal
transport map can be used to automatically refine the model. This model refinement
formulation leads to solving a non-smooth convex optimization problem. Examples
are given to demonstrate how proximal operator splitting based computation enables
numerically solving the same. This method is applied for nite-time feedback control
of probability density functions, and for data driven modeling of dynamical systems. | en |
