Research Papers

Nonintrusive Global Sensitivity Analysis for Linear Systems With Process Noise

[+] Author and Article Information
Souransu Nandi

Control, Dynamics and Estimation Laboratory,
Department of Mechanical and
Aerospace Engineering,
University at Buffalo,
Buffalo, NY 14260
e-mail: souransu@buffalo.edu

Tarunraj Singh

Control, Dynamics and Estimation Laboratory,
Department of Mechanical
and Aerospace Engineering,
University at Buffalo,
Buffalo, NY 14260
e-mail: tsingh@buffalo.edu

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received May 29, 2018; final manuscript received September 21, 2018; published online January 7, 2019. Assoc. Editor: Paramsothy Jayakumar.

J. Comput. Nonlinear Dynam 14(2), 021003 (Jan 07, 2019) (12 pages) Paper No: CND-18-1238; doi: 10.1115/1.4041622 History: Received May 29, 2018; Revised September 21, 2018

The focus of this paper is on the global sensitivity analysis (GSA) of linear systems with time-invariant model parameter uncertainties and driven by stochastic inputs. The Sobol' indices of the evolving mean and variance estimates of states are used to assess the impact of the time-invariant uncertain model parameters and the statistics of the stochastic input on the uncertainty of the output. Numerical results on two benchmark problems help illustrate that it is conceivable that parameters, which are not so significant in contributing to the uncertainty of the mean, can be extremely significant in contributing to the uncertainty of the variances. The paper uses a polynomial chaos (PC) approach to synthesize a surrogate probabilistic model of the stochastic system after using Lagrange interpolation polynomials (LIPs) as PC bases. The Sobol' indices are then directly evaluated from the PC coefficients. Although this concept is not new, a novel interpretation of stochastic collocation-based PC and intrusive PC is presented where they are shown to represent identical probabilistic models when the system under consideration is linear. This result now permits treating linear models as black boxes to develop intrusive PC surrogates.

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Kalman, R. E. , 1960, “ A New Approach to Linear Filtering and Prediction Problems,” J. Basic Eng., 82(1), pp. 35–45. [CrossRef]
Julier, S. , Uhlmann, J. , and Durrant-Whyte, H. F. , 2000, “ A New Method for the Nonlinear Transformation of Means and Covariances in Filters and Estimators,” IEEE Trans. Automatic Control, 45(3), pp. 477–482. [CrossRef]
Adurthi, N. , Singla, P. , and Singh, T. , 2018, “ Conjugate Unscented Transformation: Applications to Estimation and Control,” ASME J. Dyn. Syst., Meas., Control, 140(3), p. 030907.
Evensen, G. , 2003, “ The Ensemble Kalman Filter: Theoretical Formulation and Practical Implementation,” Ocean Dyn., 53(4), pp. 343–367. [CrossRef]
Houtekamer, P. L. , and Mitchell, H. L. , 1998, “ Data Assimilation Using an Ensemble Kalman Filter Technique,” Mon. Weather Rev., 126(3), pp. 796–811. [CrossRef]
Saltelli, A. , Ratto, M. , Tarantola, S. , and Campolongo, F. , 2006, “ Sensitivity Analysis Practices: Strategies for Model-Based Inference,” Reliab. Eng. Syst. Saf., 91(10–11), pp. 1109–1125. [CrossRef]
Cho, K.-H. , Shin, S.-Y. , Kolch, W. , and Wolkenhauer, O. , 2003, “ Experimental Design in Systems Biology, Based on Parameter Sensitivity Analysis Using a Monte Carlo Method: A Case Study for the TNFα-Mediated NF-κ B Signal Transduction Pathway,” Simulation, 79(12), pp. 726–739. [CrossRef]
Rodriguez-Fernandez, M. , Kucherenko, S. , Pantelides, C. , and Shah, N. , 2007, “ Optimal Experimental Design Based on Global Sensitivity Analysis,” Computer Aided Chemical Engineering, Vol. 24, Elsevier, Amsterdam, The Netherlands, pp. 63–68.
Lamboni, M. , Monod, H. , and Makowski, D. , 2011, “ Multivariate Sensitivity Analysis to Measure Global Contribution of Input Factors in Dynamic Models,” Reliab. Eng. Syst. Saf., 96(4), pp. 450–459. [CrossRef]
Sandoval, E. H. , Anstett-Collin, F. , and Basset, M. , 2012, “ Sensitivity Study of Dynamic Systems Using Polynomial Chaos,” Reliab. Eng. Syst. Saf., 104, pp. 15–26. [CrossRef]
McCarthy, G. D. , Drewell, R. A. , and Dresch, J. M. , 2015, “ Global Sensitivity Analysis of a Dynamic Model for Gene Expression in Drosophila Embryos,” Peer J., 3, p. e1022. [CrossRef]
McRae, G. J. , Tilden, J. W. , and Seinfeld, J. H. , 1982, “ Global Sensitivity Analysis—A Computational Implementation of the Fourier Amplitude Sensitivity Test (FAST),” Comput. Chem. Eng., 6(1), pp. 15–25. [CrossRef]
Drignei, D. , and Mourelatos, Z. P. , 2012, “ Parameter Screening in Statistical Dynamic Computer Model Calibration Using Global Sensitivities,” ASME J. Mech. Des., 134(8), p. 081001. [CrossRef]
Cao, J. , Du, F. , and Ding, S. , 2013, “ Global Sensitivity Analysis for Dynamic Systems With Stochastic Input Processes,” Reliab. Eng. Syst. Saf., 118, pp. 106–117. [CrossRef]
Sudret, B. , 2008, “ Global Sensitivity Analysis Using Polynomial Chaos Expansions,” Reliab. Eng. Syst. Saf., 93(7), pp. 964–979. [CrossRef]
Crestaux, T. , Le Maître, O. , and Martinez, J.-M. , 2009, “ Polynomial Chaos Expansion for Sensitivity Analysis,” Reliab. Eng. Syst. Saf., 94(7), pp. 1161–1172. [CrossRef]
Archer, G. , Saltelli, A. , and Sobol, I. , 1997, “ Sensitivity Measures, Anova-Like Techniques and the Use of Bootstrap,” J. Stat. Comput. Simul., 58(2), pp. 99–120. [CrossRef]
Homma, T. , and Saltelli, A. , 1996, “ Importance Measures in Global Sensitivity Analysis of Nonlinear Models,” Reliab. Eng. Syst. Saf., 52(1), pp. 1–17. [CrossRef]
Horn, R. A. , Horn, R. A. , and Johnson, C. R. , 1990, Matrix Analysis, Cambridge University Press, Cambridge, UK.
Wiener, N. , 1938, “ The Homogeneous Chaos,” Am. J. Math., 60(4), pp. 897–936. [CrossRef]
Cameron, R. H. , and Martin, W. T. , 1947, “ The Orthogonal Development of Non-Linear Functionals in Series of Fourier-Hermite Functionals,” Ann. Math., 48(2), pp. 385–392. [CrossRef]
Ghanem, R. G. , and Spanos, P. D. , 1991, “ Stochastic Finite Elements: A Spectral Approach,” Stochastic Finite Elements: A Spectral Approach, Springer, New York.
Xiu, D. , and Karniadakis, G. E. , 2002, “ The Wiener—Askey Polynomial Chaos for Stochastic Differential Equations,” SIAM J. Sci. Comput., 24(2), pp. 619–644. [CrossRef]
Konda, U. , Singla, P. , Singh, T. , and Scott, P. D. , 2011, “ State Uncertainty Propagation in the Presence of Parametric Uncertainty and Additive White Noise,” ASME J. Dyn. Syst., Meas., Control, 133(5), p. 051009. [CrossRef]
Kim, K.-K. K. , Shen, D. E. , Nagy, Z. K. , and Braatz, R. D. , 2013, “ Wiener's Polynomial Chaos for the Analysis and Control of Nonlinear Dynamical Systems with Probabilistic Uncertainties,” IEEE Control Syst., 33(5), pp. 58–67. [CrossRef]
Xiu, D. , 2010, Numerical Methods for Stochastic Computations: A Spectral Method Approach, Princeton University Press, Princeton, NJ.
Eldred, M. , and Burkardt, J. , 2009, “ Comparison of Non-Intrusive Polynomial Chaos and Stochastic Collocation Methods for Uncertainty Quantification,” AIAA Paper No. AIAA 2009-976.
Bryson, A. E. , and Mills, R. A. , 1998, “ Linear-Quadratic-Gaussian Controllers with Specified Parameter Robustness,” J. Guid., Control, Dyn., 21(1), pp. 11–18. [CrossRef]
Yue, C. , Butsuen, T. , and Hedrick, J. , 1989, “ Alternative Control Laws for Automotive Active Suspensions,” ASME J. Dyn. Syst., Meas., Control, 111(2), pp. 286–291. [CrossRef]
Gobbi, M. , Levi, F. , and Mastinu, G. , 2006, “ Multi-Objective Stochastic Optimisation of the Suspension System of Road Vehicles,” J. Sound Vib., 298(4–5), pp. 1055–1072. [CrossRef]
Narayanan, S. , and Senthil, S. , 1998, “ Stochastic Optimal Active Control of a 2-DOF Quarter Car Model With Non-Linear Passive Suspension Elements,” J. Sound Vib., 211(3), pp. 495–506. [CrossRef]


Grahic Jump Location
Fig. 1

PC realization nodes for n1=n2=3

Grahic Jump Location
Fig. 2

Evolution of Sobol' indices corresponding to the diagonal elements of μ over time: (a) μ1, (b) μ2, (c) μ3, and (d) μ4

Grahic Jump Location
Fig. 3

Evolution of Sobol' indices corresponding to the diagonal elements of Σ over time: (a) Σ(1,1), (b) Σ(2,2), (c) Σ(3,3), and (d) Σ(4,4)

Grahic Jump Location
Fig. 4

Schematic diagram of the quarter car model

Grahic Jump Location
Fig. 5

Sobol' indices for μ2

Grahic Jump Location
Fig. 6

Sobol' indices for Σ(2,2)



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