Polynomial-Chaos Based Methods for Differential Algebraic Equations with Random Parameters

2014 ◽  
pp. 355-360
Author(s):  
Roland Pulch
Keyword(s):  
Polynomial Chaos ◽  
Differential Algebraic Equations ◽  
Algebraic Equations ◽  
Random Parameters

Automating the Derivation of the Equations of Motion of a Multibody Dynamic System With Uncertainty Using Polynomial Chaos Theory and Variational Work

2019 ◽  
Vol 15 (1) ◽  
Author(s):  
Paul S. Ryan ◽  
Sarah C. Baxter ◽  
Philip A. Voglewede
Keyword(s):  
Chaos Theory ◽  
Polynomial Chaos ◽  
Equations Of Motion ◽  
Differential Algebraic Equations ◽  
Algebraic Equations ◽  
Multibody Dynamic ◽  
Mass Spring ◽  
Polynomial Chaos Theory ◽  
Multibody Dynamic System ◽  
Crank Mechanism

Abstract Understanding how variation impacts a multibody dynamic (MBD) system's response is important to ensure the robustness of a system. However, how the variation propagates into the MBD system is complicated because MBD systems are typically governed by a system of large differential algebraic equations. This paper presents a novel process, variational work, along with the polynomial chaos multibody dynamics (PCMBoD) automation process for utilizing polynomial chaos theory (PCT) in the analysis of uncertainties in an MBD system. Variational work allows the complexity of the traditional PCT approach to be reduced. With variational work and the constrained Lagrangian formulation, the equations of motion of an MBD PCT system can be constructed using the PCMBoD automated process. To demonstrate the PCMBoD process, two examples, a mass-spring-damper and a two link slider–crank mechanism, are shown.

Variational Analysis of a Two Link Slider-Crank Mechanism Using Polynomial Chaos Theory

2017 ◽  
Author(s):  
Paul S. Ryan ◽  
Sarah C. Baxter ◽  
Philip A. Voglewede
Keyword(s):  
Chaos Theory ◽  
Closed Loop ◽  
Polynomial Chaos ◽  
Differential Algebraic Equations ◽  
Algebraic Equations ◽  
Advantages And Disadvantages ◽  
Multi Body ◽  
Polynomial Chaos Theory ◽  
Body Dynamic ◽  
Crank Mechanism

Variation occurs in many closed loop multi-body dynamic (MBD) systems in the geometry, mass, or forces. Understanding how MBD systems respond to variation is imperative for the design of a robust system. However, simulation of how variation propagates into the solution is complicated as most MBD systems cannot be simplified into to a system of ordinary differential equations (ODE). This paper investigates polynomial chaos theory (PCT) as a means of quantifying the effects of uncertainty in a closed loop MBD system governed by differential algebraic equations (DAE). To demonstrate how PCT could be used, the motion of a two link slider-crank mechanism is simulated with variation in the link lengths. To validate and show the advantages and disadvantages of PCT in closed loop MBD systems, the PCT approach is compared to Monte Carlo simulations.

Uncertainty Quantification of Differential Algebraic Equations Using Polynomial Chaos

2021 ◽  
Author(s):  
Premjit Saha ◽  
Tarunraj Singh ◽  
Gary F. Dargush
Keyword(s):  
Linear Models ◽  
Polynomial Chaos ◽  
Initial Conditions ◽  
Surrogate Models ◽  
Differential Algebraic Equations ◽  
Algebraic Equations ◽  
Rlc Circuit ◽  
Algebraic Constraints ◽  
Interpolation Polynomials ◽  
Time Invariant

Abstract The focus of this paper is on the use of Polynomial Chaos for developing surrogate models for Differential Algebraic Equations with time-invariant uncertainties. Intrusive and non-intrusive approaches to synthesize Polynomial Chaos surrogate models are presented including the use of Lagrange interpolation polynomials as basis functions. Unlike ordinary differential equations, if the algebraic constraints are a function of the stochastic variable, some initial conditions of the differential algebraic equations are also random. A benchmark RLC circuit which is used as a benchmark for linear models is used to illustrate the development of a Polynomial Chaos based surrogate model. A nonlinear example of a simple pendulum also serves as a benchmark to illustrate the potential of the proposed approach. Statistics of the results of the Polynomial Chaos models are validated using Monte Carlo simulations in addition to estimating the evolving PDFs of the states of the pendulum.

Markov Jump Linear System Analysis of a Microgrid Operating in Islanded and Grid Tied Modes

2020 ◽  
Author(s):  
Gilles Mpembele ◽  
Jonathan Kimball
Keyword(s):  
Monte Carlo ◽  
Power Systems ◽  
Power System ◽  
System Analysis ◽  
Differential Algebraic Equations ◽  
Algebraic Equations ◽  
Markov Jump ◽  
Numerical Application ◽  
Conditional Moments ◽  
Markov Jump Linear Systems

<div>The analysis of power system dynamics is usually conducted using traditional models based on the standard nonlinear differential algebraic equations (DAEs). In general, solutions to these equations can be obtained using numerical methods such as the Monte Carlo simulations. The use of methods based on the Stochastic Hybrid System (SHS) framework for power systems subject to stochastic behavior is relatively new. These methods have been successfully applied to power systems subjected to</div><div>stochastic inputs. This study discusses a class of SHSs referred to as Markov Jump Linear Systems (MJLSs), in which the entire dynamic system is jumping between distinct operating points, with different local small-signal dynamics. The numerical application is based on the analysis of the IEEE 37-bus power system switching between grid-tied and standalone operating modes. The Ordinary Differential Equations (ODEs) representing the evolution of the conditional moments are derived and a matrix representation of the system is developed. Results are compared to the averaged Monte Carlo simulation. The MJLS approach was found to have a key advantage of being far less computational expensive.</div>

Construction of Inverse System for Nonlinear Differential-algebraic Equations Subsystems

2009 ◽  
Vol 35 (8) ◽  
pp. 1094-1100 ◽  
Author(s):  
Xian-Zhong DAI ◽  
Qiang ZANG ◽  
Kai-Feng ZHANG
Keyword(s):  
Differential Algebraic Equations ◽  
Inverse System ◽  
Algebraic Equations
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