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Uncertainty Propagation Using Probabilistic Affine Forms and Concentration of Measure Inequalities [chapter]

Olivier Bouissou, Eric Goubault, Sylvie Putot, Aleksandar Chakarov, Sriram Sankaranarayanan
2016 Lecture Notes in Computer Science  
Next, we show how to prove bounds on the probabilities that program variables take on specific values by using concentration of measure inequalities.  ...  We extend probabilistic affine forms using the precise tracking of dependencies between noise symbols combined with the expectations and higher order moments of the noise symbols.  ...  Acknowledgments: This work was partially supported by the US NSF under award number 1320069, and the academic research chair "Complex Systems Engineering" of Ecole polytechnique, Thalès, FX, DGA, Dassault  ... 
doi:10.1007/978-3-662-49674-9_13 fatcat:72djgfitcrcefow3m55vxha3em

Generalised polynomial chaos expansion approaches to approximate stochastic model predictive control†

Kwang-Ki K. Kim, Richard D. Braatz
2013 International Journal of Control  
This paper considers the model predictive control of dynamic systems subject to stochastic uncertainties due to parametric uncertainties and exogenous disturbance.  ...  The effects of uncertainties are quantified using generalised polynomial chaos expansions with an additive Gaussian random process as the exogenous disturbance.  ...  A cartoon of the outer bound that can be obtained from the Boole inequality and concentration-of-measure inequalities. into the Boole inequality and the Chebyshev inequality of the form presented in Prop  ... 
doi:10.1080/00207179.2013.801082 fatcat:keor2w53qrbjfgzdbffbmmezsu

Lyapunov-based stochastic nonlinear model predictive control: Shaping the state probability distribution functions

Edward A. Buehler, Joel A. Paulson, Ali Mesbah
2016 2016 American Control Conference (ACC)  
The control approach aims to shape probability density function of the stochastic states, while satisfying input and joint state chance constraints.  ...  Stochastic uncertainties in complex dynamical systems lead to variability of system states, which can in turn degrade the closed-loop performance.  ...  This is due to using the FP equation for probabilistic uncertainty propagation, as the FP equation enables explicit characterization of the states' PDFs.  ... 
doi:10.1109/acc.2016.7526514 dblp:conf/amcc/BuehlerP016 fatcat:zk2yrzwpnvhx5cmgqdu4lsqvc4

Reasoning about Uncertainties in Discrete-Time Dynamical Systems using Polynomial Forms

Sriram Sankaranarayanan, Yi Chou, Eric Goubault, Sylvie Putot
2020 Neural Information Processing Systems  
Next, we show how concentration of measure inequalities can be employed to prove rigorous bounds on the tail probabilities of these state variables.  ...  Our approach allows us to rigorously represent the probability distribution of state variables over time, and provide guaranteed bounds on the expectations, moments and probabilities of tail events involving  ...  project "Validation of Autonomous Drones and Swarms of Drones".  ... 
dblp:conf/nips/0001CGP20 fatcat:2pnxnpq6nzadxjpequijmd2dg4

Robustness Analysis, Prediction and Estimation for Uncertain Biochemical Networks

Stefan Streif, Kwang-Ki K. Kim, Philipp Rumschinski, Masako Kishida, Dongying Erin Shen, Rolf Findeisen, Richard D Braatz
2013 IFAC Proceedings Volumes  
Furthermore, different classes of uncertainties and perturbations in the data and model are consistently described.  ...  First, including uncertainty in the structures, parameters, and perturbations into the model is not straightforward due to different types of uncertainties encountered.  ...  PROBABILISTIC UNCERTAINTIES AND ANALYSIS As more measurements can be made for biochemical experiments as a result of technological advancements, parameters are more often reported in probabilistic form  ... 
doi:10.3182/20131218-3-in-2045.00190 fatcat:5sypgusrlrez5l3sihbo32z5ua

An improved multidimensional parallelepiped non-probabilistic model for structural uncertainty analysis

B.Y. Ni, C. Jiang, X. Han
2016 Applied Mathematical Modelling  
The improved MP model is then applied to the uncertainty propagation analysis and reliability analysis of structures.  ...  The non-probabilistic convex model utilizes a convex set to quantify the uncertainty domain of uncertain parameters.  ...  the form of matrix inequality.  ... 
doi:10.1016/j.apm.2015.11.047 fatcat:dyskyiroafedlalvqublzeqpdi

Constrained optimal control of stochastic switched affine systems using randomization

Kostas Margellos, Alessandro Falsone, Simone Garatti, Maria Prandini
2016 2016 European Control Conference (ECC)  
We consider a finite-horizon optimal control problem for a switched affine system with controlled switches, affected by uncertainty and subject to input and/or state constraints.  ...  We establish a probabilistic link between the infinite dimensional robust program and its scenario-based relaxation, showing that the optimal solution of the latter is feasible, in a probabilistic sense  ...  In this paper we deal with the problem of finite-horizon optimal control of switched systems with affine dynamics, affected by uncertainty and subject to input and/or state constraints.  ... 
doi:10.1109/ecc.2016.7810675 dblp:conf/eucc/MargellosFGP16 fatcat:vw5zypiqgnbtjmf5imyfj37rjy

Rigorous Roundoff Error Analysis of Probabilistic Floating-Point Computations [chapter]

George Constantinides, Fredrik Dahlqvist, Zvonimir Rakamarić, Rocco Salvia
2021 Lecture Notes in Computer Science  
It keeps track of complex dependencies between random variables using an SMT solver, and is capable of providing sound but tight probabilistic bounds to roundoff errors using symbolic affine arithmetic  ...  We derive closed-form expressions for the distribution of roundoff errors associated with a random variable, and we prove that roundoff errors are generally close to being uncorrelated with their generating  ...  We thank Ian Briggs and Mark Baranowski for their generous and prompt support with Gelpia. We also thank Alexey Solovyev for his detailed feedback and suggestions for improvements.  ... 
doi:10.1007/978-3-030-81688-9_29 fatcat:dy2boml7nrcopkapxyzs2e7lei

An ellipsoidal calculus based on propagation and fusion

L. Ros, A. Sabater, F. Thomas
2002 IEEE Transactions on Systems Man and Cybernetics Part B (Cybernetics)  
Actually, a Minkowski operation can be seen as a fusion followed by a propagation and an affine transformation as a particular case of propagation.  ...  Propagation refers to the problem of obtaining an ellipsoid that must satisfy an affine relation with another ellipsoid, and fusion to that of computing the ellipsoid that tightly bounds the intersection  ...  Propagation has been defined as the operation of computing an ellipsoid that satisfies an affine relation of the form with another ellipsoid.  ... 
doi:10.1109/tsmcb.2002.1018763 pmid:18238140 fatcat:n7fe7togjjfe5pap4mnhoy5iai

Probabilistic Model Validation for Uncertain Nonlinear Systems [article]

Abhishek Halder, Raktim Bhattacharya
2014 arXiv   pre-print
The proposed formulation is simple, intuitive, and accounts both deterministic and stochastic nonlinear systems with parametric and nonparametric uncertainties.  ...  Several examples are worked out to illustrate its use.  ...  Khargonekar at University of Florida, and S. Chakravorty at Texas A&M University, for insightful discussions.  ... 
arXiv:1402.0140v1 fatcat:ou5wqcpkprcs7faxcf2siaswti

Distributional Gaussian Processes Layers for Out-of-Distribution Detection [article]

Sebastian G. Popescu, David J. Sharp, James H. Cole, Konstantinos Kamnitsas, Ben Glocker
2022 arXiv   pre-print
that outperforms the capabilities of previous Bayesian networks and reconstruction-based approaches that learn normative distributions.  ...  Hence, we propose a parameter efficient Bayesian layer for hierarchical convolutional Gaussian Processes that incorporates Gaussian Processes operating in Wasserstein-2 space to reliably propagate uncertainty  ...  Hence we can use the Mutual Information measure between model predictions and Dirichlet parameters to obtain a better measure of uncertainty.  ... 
arXiv:2206.13346v1 fatcat:lfgun4vokjgilkmxd7zc54iazq

Stability for Receding-horizon Stochastic Model Predictive Control [article]

Joel A. Paulson, Stefan Streif, Ali Mesbah
2015 arXiv   pre-print
Closed-loop stability of the SMPC approach is established by appropriate selection of the cost function. Polynomial chaos is used for uncertainty propagation through system dynamics.  ...  A stochastic model predictive control (SMPC) approach is presented for discrete-time linear systems with arbitrary time-invariant probabilistic uncertainties and additive Gaussian process noise.  ...  In Section IV, the generalized polynomial chaos framework coupled with the Galerkin projection is used for efficient propagation of the time-invariant probabilistic uncertainties θ, which allows for obtaining  ... 
arXiv:1410.5083v2 fatcat:keundqybwfewpivwussaavtdpe

Stability for receding-horizon stochastic model predictive control

Joel A. Paulson, Stefan Streif, Ali Mesbah
2015 2015 American Control Conference (ACC)  
Closed-loop stability of the SMPC approach is established by appropriate selection of the cost function. Polynomial chaos is used for uncertainty propagation through system dynamics.  ...  A stochastic model predictive control (SMPC) approach is presented for discrete-time linear systems with arbitrary time-invariant probabilistic uncertainties and additive Gaussian process noise.  ...  In Section IV, the generalized polynomial chaos framework coupled with the Galerkin projection is used for efficient propagation of the time-invariant probabilistic uncertainties θ, which allows for obtaining  ... 
doi:10.1109/acc.2015.7170854 dblp:conf/amcc/PaulsonS015 fatcat:faxwrawcwzby5dhxcdmz6arsuy

Chance-constrained model predictive control for drinking water networks

J.M. Grosso, C. Ocampo-Martínez, V. Puig, B. Joseph
2014 Journal of Process Control  
The reformulation allows to explicitly and easily propagate uncertainty over the prediction horizon, and leads to a cost-efficient management of risk that consists in a dynamic back-off to avoid frequent  ...  A deterministic equivalent of the stochastic problem is formulated using Boole's inequality to decompose joint chance constraints into single chance constraints and by considering a uniform allocation  ...  Both forms of constraints are useful to measure risks, hence, their selection depends on the application.  ... 
doi:10.1016/j.jprocont.2014.01.010 fatcat:zkuizbdddvgf7n5e24kubo6fwm

Uncertainty in Economic Growth and Inequality [article]

Zhengyuan Gao
2017 arXiv   pre-print
A step to consilience, starting with a deconstruction of the causality of uncertainty that is embedded in the fundamentals of growth and inequality, following a construction of aggregation laws that disclose  ...  GDP and income data.  ...  Another advantage of using affine structure in (7) is the tractability.  ... 
arXiv:1705.07234v1 fatcat:l452fgvsgvby7c4qijymgzg47y
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