Abhishek Halder

Dispersion Analysis in Hypersonic Flight During Planetary Entry Using Stochastic Liouville Equation

A framework is provided for the propagation of uncertainty in planetary entry, descent, and landing. The traditional Monte―Carlo based dispersion analysis is overly resource-expensive for such high-dimensional nonlinear systems and does not provide any methodical way to analyze the effect of uncertainty for mission design. It is shown that propagating the density function through Liouville equation is computationally attractive and suitable for further statistical analysis. Comparative simulation results are provided to bring forth the efficacies of the proposed method. Examples are given from the entry, descent, and landing domain to illustrate how one can retrieve statistical information of interest from an analysts perspective.

Model validation: A probabilistic formulation

This paper presents a probabilistic formulation of the model validation problem. The proposed validation framework is simple, intuitive, and can account both deterministic and stochastic nonlinear systems in presence of parametric and nonparametric uncertainties. Contrary to the hard invalidation methods proposed in the literature, our formulation allows a relaxed notion of validation in probability. The construction of probabilistically robust validation certificates provides provably correct guarantees. Computational complexities and numerical examples are given to illustrate the method.

Beyond Monte Carlo: A Computational Framework for Uncertainty Propagation in Planetary Entry, Descent and Landing

Space system verification and validation require high fidelity simulations to predict system performance in presence of uncertainties in the spacecraft and environment. Bruteforce Monte Carlo (MC) simulations are overly resource-expensive for such high-dimensional nonlinear systems. A computational framework for uncertainty propagation in planetary entry, descent and landing is proposed that goes beyond traditional MC based dispersion analysis. The methodology and simulation results for this transfer operator based method are provided and compared with MC results to bring forth the computational efficacies.

Further results on probabilistic model validation in Wasserstein metric

In a recent work [1], we have introduced a probabilistic formulation for the model validation problem to provide a unifying framework for (in)validating nonlinear deterministic and stochastic models, in both discrete and continuous time. As an extension to that work, this paper provides rigorous performance bounds for the model validation algorithms presented in [1]. Further, it is shown that the existing method of barrier certificate based nonlinear invalidation oracle, can be recovered as a special case of the proposed formulation. Some results are derived to quantify the effects of initial uncertainty on the Wasserstein gap. And finally, for discrete-time LTI and LTV systems, upper bounds on Wasserstein distance are derived in terms of the parameters of the systems under comparison, thus providing an offline estimate of the gap.

Finite horizon linear quadratic Gaussian density regulator with Wasserstein terminal cost

We formulate and solve an optimal control problem in which a finite dimensional linear time invariant (LTI) control system steers a given Gaussian probability density function (PDF) close to another in fixed time, while minimizing the trajectory-wise expected quadratic cost. We measure the “closeness” between the actual terminal PDF and the desired terminal PDF as the squared Wasserstein distance between the two density functions, and penalize the lack of closeness as terminal cost. We find that unlike the standard linear quadratic Gaussian (LQG) control problem, the necessary conditions for the resulting linear quadratic Gaussian density regulator lead to nonlinear coupling between the boundary conditions of the covariance Lyapunov matrix differential equation and the covariance costate Riccati matrix differential equation. We show that the LQG control problem can be recovered as a special case of our density regulator problem, and illustrate our formulation on a numerical example.

Uncertainty quantification for stochastic nonlinear systems using Perron-Frobenius operator and Karhunen-Loève expansion

In this paper, a methodology for propagation of uncertainty in stochastic nonlinear dynamical systems is investigated. The process noise is approximated using Karhunen-Loève (KL) expansion. Perron-Frobenius (PF) operator is used to predict the evolution of uncertainty. A multivariate Kolmogorov-Smirnov test is used to verify the proposed framework. The method is applied to predict uncertainty evolution in a Duffing oscillator and a Vanderpols oscillator. It is observed that the solution of the approximated stochastic dynamics converges to the true solution in distribution. Finally, the proposed methodology is combined with Bayesian inference to estimate states of a nonlinear dynamical system, and its performance is compared with particle filter. The proposed estimator was found to be computationally superior than the particle filter.

Architecture and Algorithms for Privacy Preserving Thermal Inertial Load Management by a Load Serving Entity

Motivated by the growing importance of demand response in modern power systems operations, we propose an architecture and supporting algorithms for privacy preserving thermal inertial load management as a service provided by the load serving entity (LSE). We focus on an LSE managing a population of its customers’ air conditioners, and propose a contractual model where the LSE guarantees quality of service to each customer in terms of keeping their indoor temperature trajectories within respective bands around the desired individual comfort temperatures. We show how the LSE can price the contracts differentiated by the flexibility embodied by the width of the specified bands. We address architectural questions of (i) how the LSE can strategize its energy procurement based on price and ambient temperature forecasts, (ii) how an LSE can close the real-time control loop at the aggregate level while providing individual comfort guarantees to loads, without ever measuring the states of an air conditioner for privacy reasons. Control algorithms to enable our proposed architecture are given, and their efficacy is demonstrated on real data.

Probabilistic model validation for uncertain nonlinear systems

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.

Probabilistic robustness analysis of F-16 controller performance: An optimal transport approach

This paper presents an optimal transport theoretic formulation to assess the controller robustness for F-16 aircraft. We compare the state regulation performance for a linear quadratic regulator (LQR) and gain scheduled LQR (gsLQR), applied to nonlinear longitudinal open-loop dynamics of F-16, under stochastic initial condition uncertainty. It is shown that both controllers have comparable immediate and asymptotic performance, but gsLQR achieves better transient performance than LQR. Algorithms based on Perron-Frobenius operator, are given for tractable computation. Numerical results from the proposed method, are in unison with Monte Carlo simulations.

Nonlinear filtering with transfer operator

This paper presents a new nonlinear filtering algorithm that is shown to outperform state-of-the-art particle filters with resampling. Starting from the Itô stochastic differential equation, the proposed algorithm harnesses Karhunen-Loéve expansion to derive an approximate non-autonomous dynamical system, for which transfer operator based density computation can be performed in exact arithmetic. It is proved that the algorithm is asymptotically consistent in mean-square sense. Numerical results demonstrate that explicitly accounting prior dynamics entail significant performance improvement for nonlinear non-Gaussian estimation problems with infrequent measurement updates, as compared to the performance of particle filters.