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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. Finite dimensional surrogates for extreme events

   Hui Xu, Mircea D. (July 2023, 14th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP14 Dublin, Ireland, July 9-13, 2023)

   Numerical solutions of stochastic problems require the representation of random functions in their definitions by finite dimensional (FD) models, i.e., deterministic functions of time and finite sets of random variables. It is common to represent the coefficients of these FD surrogates by polynomial chaos (PC) models. We propose a novel model, referred to as the polynomial chaos translation (PCT) model, which matches exactly the marginal distributions of the FD coefficients and approximately their dependence. PC- and PCT- based FD models are constructed for a set of test cases and a wind pressure time series recorded at the boundary layer wind tunnel facility at the University of Florida. The PCT-based models capture the joint distributions of the FD coefficients and the extremes of target times series accurately while PC-based FD models do not have this capability. 

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3. State Constrained Controller Design for Uncertain Linear Systems using Polynomial Chaos

   Nandi, Souransu; Singh, Tarunraj (June 2016, 2016 American Control Conference (ACC))

   The focus of this paper is on the design of state constrained controllers which are robust to time invariant uncertain variables. Polynomial Chaos spectral expansion is used to parameterize the uncertain variables, which permits evaluation of the evolution of the uncertain states. The coefficients of the truncated polynomial chaos expansion are determined using the Galerkin projection resulting in a set of deterministic equations. A mapping into Bernstein polynomial space permits determination of bounds on the evolving states. Linear programming is used on the deterministic set of equation with constraints as the predetermined bounds to determine controllers which are robust to the epistemic uncertainties. Numerical examples are used to illustrate the benefit of the proposed technique for the design of rest-to-rest controllers subject to deformation constraints; which are robust to uncertainties in the stiffness coefficient for the benchmark spring-mass system. 

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4. Finite dimensional models for random functions

   https://doi.org/https://doi.org/10.1016/j.jcp.2018.09.029  

   Grigoriu, Mircea (September 2018, Journal of computational physics)

   Truncated Karhunen–Loève (KL) representations are used to construct finite dimensional (FD) models for non-Gaussian functions with finite variances. The second moment specification of the random coefficients of these representations are enhanced to full probabilistic characterization by using translation, polynomial chaos, and translated polynomial chaos models, referred to as T, PC, and PCT models. Following theoretical considerations on KL representations and T, PC, and PCT models, three numerical examples are presented to illustrate the implementation and performance of these models. The PCT models inherit the desirable features of both T and PC models. It approximates accurately all quantities of interest considered in these examples. 

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5. Finite dimensional models for random functions

   https://doi.org/doi.org/10.1016/j.jcp.2018.09.029  

   Grigoriu, M. (August 2019, Journal of computational physics)

   Truncated Karhunen–Loève (KL) representations are used to construct finite dimensional (FD) models for non-Gaussian functions with finite variances. The second moment specification of the random coefficients of these representations are enhanced to full probabilistic characterization by using translation, polynomial chaos, and translated polynomial chaos models, referred to as T, PC, and PCT models. Following theoretical considerations on KL representations and T, PC, and PCT models, three numerical examples are presented to illustrate the implementation and performance of these models. The PCT models inherit the desirable features of both T and PC models. It approximates accurately all quantities of interest considered in these examples. 

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