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Uncertainty Management in Networked Dynamical Systems

Engineering performance and productivity can be improved through systematic analysis and integrated design that takes uncertainty into account.` Physics based models of full-scale engineering designs are often converted to monolithic systems of nonlinear differential/algebraic equations. Numerous fields of science and engineering present the problem of uncertainty propagation through nonlinear dynamic systems with stochastic excitation and uncertain initial conditions . One may be interested in the determination of the response of engineering structures - beams/plates/entire buildings under random excitation (in structure mechanics), or the propagation of initial condition uncertainty of an asteroid for the determination of its probability of collision with a planet (in astrodynamics), or the motion or particles under the influence of stochastic force fields (in particle physics), or the multi-disciplinary robust design optimization of aerospace vehicles and structures, or simply the computation of the prediction step in the design of a Bayesian filter (in filtering theory). A key objective of recent research is to understand the influence of uncertainty on the behavior of these systems. Finding the contribution of model or initial condition uncertainty to the system output would enable the design of more robust and effective systems.

All these applications require the study of the time evolution of the Probability Density Function (PDF), corresponding to the state, of the relevant dynamic system. Several approximate techniques exist in the literature to study the uncertainty propagation problem through general nonlinear system, the most popular being Monte Carlo methods, Gaussian closure (or higher order moment closure), Equivalent Linearization, and Stochastic Averaging . Uncertainty propagation in significantly nonlinear systems has traditionally been addressed by Monte Carlo like methods. This classic approach employs a large number of simulations with a random selection of variables from their prescribed distribution. To address slow convergence rate and clustering issues associated with Monte Carlo methods, new methods such as: polynomial chaos (stochastic finite elements), stochastic surface response methods and probabilistic collocation methods were used with significant success. Complementing these approaches, the propagation of uncertainty in the distribution of initial conditions for dynamical systems can be studied using the corresponding Liouville's equation.

Monte Carlo methods do not scale well with system size. To overcome this problem a large system can for example be decomposed into subsystems evolving on different time scales using graph decomposition methods . This time scale separation can be exploited to increase computational efficiency when propagating input uncertainty in a subsystem-by-subsystem manner . Another interesting alternative could be decomposition based on state space behavior. Subsets of the state space of a dynamical system where typical trajectories stay longer before entering different regions are called almost invariant sets . Such decomposition could be used in uncertainty analysis based on regions of different dynamics.

Finally, to obtain the global results due to uncertain parameters or initial conditions, the weak coupling between subsystems should be taken into account by using an appropriate iteration scheme. The Waveform Relaxation method has been successfully employed in to address large scale systems, such as integrated circuits.

Even after reduction of the size of the state space of the dynamical systems using the above methods, there still remains the problem of propagating uncertainty through the individual subsystems. Thus, a solution to this sub-problem and the “coherent combination” of the outputs of these sub-problems is absolutely critical to the success of any of these schemes. The exact description of this problem is provided by the well known Fokker Planck Kolmogorov Equation (FPKE), or simply the Fokker Planck Equation (FPE), the solution to which contains complete information about the state PDF. While the FPK equation holds the key for uncertainty propagation through nonlinear dynamical systems, it is a formidable problem to solve because of the issues like positivity, normality, discretization, and most importantly, the dimensionality of the system. For small dimensions (<4), the FPE can be solved numerically after discretization, but for (N > 3), advanced numerical methods are required due to the exponential growth of computational work with increasing N (“the curse of dimensionality”). In particular, there is need for efficient, robust and accurate methods for solving the FPE for the propagation of uncertainties through high dimensional complex nonlinear systems and for assessing their effect on system performance.

The core tool at the heart of our approach to solving the FPE is a generalized FEM method known as partition of unity FEM (PUFEM) and a recently developed adaptive, multi-resolution approximation algorithm known as: Global-Local Orthogonal MAPping (GLO-MAP) . In conjunction with adaptive homotopic methods , the above methodology will focus on the development of a computational tool to obtain the stationary and transient solution to the FPE for general N-dimensional dynamical system. Preliminary results obtained on have shown orders of magnitude improvement in the computational efficiency over those of conventional methods (please see the figure below for the adaptive refinement scheme and the references below for more details). The resulting algorithm can be used in conjunction with conventional methods such as Monte-Carlo, Gaussian closure and statistical linearization for uncertainty propagation through large-scale systems.

A Homotopic Approach to Domain Determination and Solution Refinement in the Fokker-Planck Equation

Candidate PIs: J. Junkins, Suman Chakravorty, Raktim Bhattacharya, Tamas Kalmar-Nagy, J. Hurtado, S. R. Vadali

Related Publications:

• Kalmár-Nagy, T. and Huzmezan, M.: Propagation of Uncertain Inputs Through Networks of Nonlinear Components, in Proceedings of the 43rd CDC, pp. 1799-1802, 2004.
• Varigonda, S., Kalmár-Nagy, T., LaBarre, B., Mezic,I. :Graph Decomposition Methods for Uncertainty Propagation in Complex, Nonlinear Interconnected Dynamical Systems, in Proceedings of the 43rd CDC, pp. 1794-1798, 2004.
• S. Chakravorty, ``A homotopic Galerkin solution to the Fokker-Planck-Kolmogorov Equation”, Proceedings of the American control Conference, Minneapolis, 2006.
• S.Chakravorty and M. Kumar, “ A Homotopic Galerkin solution to the Fokker-Planck-Kolmogorov Equation”, submitted to the SIAM Journal of Applied Mathematics
• M. Kumar , S. Chakravorty and J. L. Junkins, “A Homotopic Approach to Domain Determination and Solution Refinement in the Fokker-Planck Equation”, accepted to the 2007 American Control Conference, New York
• M. Kumar, S. Chakravorty and J. L. Junkins, “A Homotopic Approach to Domain Determination and Solution Refinement for the Fokker-Planck Equation”, submitted to the SIAM Journal of Applied Mathematics
• M. Kumar, P. Singla, S. Chakravorty and J. L. Junkins, “A Multi-Resolution Approach to Steady State Uncertainty Determination in Nonlinear Dynamical Systems”, proceedings of the 38th IEEE Southeastern Symposium on System Theory, Mar5-7 2006, Cookeville TN, USA, pp344-348.
• M. Kumar, P. Singla, S. Chakravorty and J. L. Junkins, “The Partition of Unity Method to the Solution of the Fokker-Planck Equation,” at the AIAA/AAS Astrodynamics Specialist Conf, Aug21-24 2006, Keystone CO, USA.
• M. Kumar, P. Singla, S. Chakravorty and J. L. Junkins, “A Partition of Unity Approach to Partial Differential Equations in Higher Dimensions,” at the ICCES Special Symposium on Meshless Methods, Jun 14-16 2006, Dubrovnik, Croatia.


For More Information Contact :

Dr.Raktim Bhattacharya,
Email: raktim@aero.tamu.edu
Webpage: http://cisar.tamu.edu/wiki

Mailing Address
Texas A&M University
Department of Aerospace Engineering
3141 TAMU College Station, TX 77843-3141

Telephone: (979) 862-2914
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