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1. Polynomial Chaos-Based Controller Design for Uncertain Linear Systems With State and Control Constraints

   https://doi.org/10.1115/1.4038800  

   Nandi, Souransu; Migeon, Victor; Singh, Tarunraj; Singla, Puneet (July 2018, Journal of Dynamic Systems, Measurement, and Control)

   For linear dynamic systems with uncertain parameters, design of controllers which drive a system from an initial condition to a desired final state, limited by state constraints during the transition is a nontrivial problem. This paper presents a methodology to design a state constrained controller, which is robust to time invariant uncertain variables. Polynomial chaos (PC) expansion, a spectral expansion, is used to parameterize the uncertain variables permitting the evolution of the uncertain states to be written as a polynomial function of the uncertain variables. The coefficients of the truncated PC expansion are determined using the Galerkin projection resulting in a set of deterministic equations. A transformation of PC polynomial space to the Bernstein polynomial space permits determination of bounds on the evolving states of interest. Linear programming (LP) is then used on the deterministic set of equations with constraints on the bounds of the states to determine the controller. Numerical examples are used to illustrate the benefit of the proposed technique for the design of a rest-to-rest controller subject to deformation constraints and which are robust to uncertainties in the stiffness coefficient for the benchmark spring-mass-damper system. 

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2. Chance Constraint based Design of Controllers for Linear Uncertain Systems

   Nandi, Souransu; Singh, Tarunraj (May 2017, 2017 American Control Conference)

   This paper considers the problem of state-tostate transition with state and control constraints, for a linear system with model parameter uncertainties. Polynomial chaos is used to transform the stochastic model to a deterministic surrogate model. This surrogate model is used to pose a chance constrained optimal control problem where the state constraints and the residual energy cost are represented in terms of the mean and variance of the stochastic states. The resulting convex optimization is illustrated on the problem of rest-to-rest maneuver of the benchmark floating oscillator. 

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3. Efficiency of Uncertainty Propagation Methods for Estimating Output Moments

   https://doi.org/10.1016/B978-0-12-818597-1.50078-3  

   Mohammadi, Samira; Cremaschi, Selen (July 2019, Computeraided Chemical Engineering)

   Uncertainty propagation methods are used to estimate the distribution of model outputs resulting from a set of uncertain model outputs. There are a number of uncertainty propagation methods available in literature. This paper compares six non-intrusive uncertainty propagation methods, Latin Hypercube Sampling, Full Factorial Integration, Univariate Dimension Reduction, Halton series, Sobol series, and Polynomial Chaos Expansion, in terms of their efficiency for estimating the first four moments of the output distribution using computational experiments. The results suggest employing FFNI if there are few uncertain inputs, up to three. Uncertainty propagation methods that utilize Halton and Sobol series are found to be robust for estimating output moments as the number of uncertain inputs increased. In general, higher order polynomial chaos expansion approximations (3rd-5th order) obtained accurate estimates of model outputs with fewer model evaluations. 

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4. Optimal Meal Time after Bolusing for Type 1 Diabetes Patients under Meal Uncertainties

   Nandi, Souransu; Singh, Tarunraj (May 2017, 2017 American Control Conference (ACC))

   The focus of this paper is on the characterization of the uncertainties in the evolving states of a diabetic model, to permit a study of the impact of the time interval between insulin bolusing and meal initiation on hypo- and hyperglycemic events. A polynomial chaos based approach is used to characterize the independent uncertainties in the initial condition and meal size. Galerkin projection of the resulting equations reduce the stochastic differential equations to a set of deterministic equations. This forms the framework to optimize for the post bolusing time to initiate the meal. Two cost functions are considered which correspond to the postprandial hypoand hyperglycemic excursions of the blood glucose. Numerical results from the minimal Bergman model suggest a 13 and 14 minute interval between bolusing and the initiation of the meal. 

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5. Probabilistic surrogate models for uncertainty analysis: Dimension reduction‐based polynomial chaos expansion

   https://doi.org/10.1002/nme.6262  

   Son, Jeongeun; Du, Yuncheng (November 2019, International Journal for Numerical Methods in Engineering)

   Summary This paper presents an approach for efficient uncertainty analysis (UA) using an intrusive generalized polynomial chaos (gPC) expansion. The key step of the gPC‐based uncertainty quantification(UQ) is the stochastic Galerkin (SG) projection, which can convert a stochastic model into a set of coupled deterministic models. The SG projection generally yields a high‐dimensional integration problem with respect to the number of random variables used to describe the parametric uncertainties in a model. However, when the number of uncertainties is large and when the governing equation of the system is highly nonlinear, the SG approach‐based gPC can be challenging to derive explicit expressions for the gPC coefficients because of the low convergence in the SG projection. To tackle this challenge, we propose to use a bivariate dimension reduction method (BiDRM) in this work to approximate a high‐dimensional integral in SG projection with a few one‐ and two‐dimensional integrations. The efficiency of the proposed method is demonstrated with three different examples, including chemical reactions and cell signaling. As compared to other UA methods, such as the Monte Carlo simulations and nonintrusive stochastic collocation (SC), the proposed method shows its superior performance in terms of computational efficiency and UA accuracy. 

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