Bounding the Dynamic Behavior of an Uncertain System via Polynomial Chaos-based Simulation

Abstract

Parametric uncertainty can represent parametric tolerance, parameter noise or parameter disturbances. The effects of these uncertainties on the time evolution of a system can be extremely significant, mostly when studying closed-loop operation of control systems. The presence of uncertainty makes the modeling process challenging, since it is impossible to express the behavior of the system with a deterministic approach. If the uncertainties can be defined in terms of probability density function, probabilistic approaches can be adopted. In many cases, the most useful aspect is the evaluation of the worst-case scenario, thus limiting the problem to the evaluation of the boundary of the set of solutions. This is particularly true for the analysis of robust stability and performance of a closed-loop system. The goal of this paper is to demonstrate how the polynomial chaos theory (PCT) can simplify the determination of the worst-case scenario, quickly providing the boundaries in time domain. The proposed approach is documented with examples and with the description of the Maple worksheet developed by the authors for the automatic processing in the PCT framework.

Keywords

Closed-loop systems stochastic process uncertainty

Get full access to this article

View all access options for this article.

References

Balaban, P. and J.J. Golembski. 1975. Statistical analysis for practical circuit design. IEEE Transactions of Circuits & Systems 22(2), 100-108.

Smith, A. , A. Monti and F. Ponci. 2006. Indirect measurements via polynomial chaos observer. IEEE AMEUM 2006 - International Workshop on Advanced Methods for Uncertainty Estimation in Measurement Sardagna, Trento , Italy, 20-21 April.

Xiu, D. and G.E. Karniadakis. 2003. Modeling uncertainty in flow simulations via generalized polynomial chaos. Journal of Computing Physics 187, 137-167.

Xiu, D. , D. Lucor, C.-H. Su and G.E. Karniadakis. 2002. Stochastic modeling of flow-structure interactions using generalized polynomial chaos. Journal of Fluid Engineering 124, 51- 59.

Lovett, T.E. , F. Ponci and A. Monti. 2006. A polynomial chaos approach to measurement uncertainty, IEEE Transactions on Instrumentation and Measurement 55(3), 729-736.

Smith, A. , A. Monti and F. Ponci. 2007. Uncertainty and worst case analysis for a low-pass filter using polynomial chaos theory. IEEE AMUEM 2007 , Trento, Italy.

D’Antona, G. , A. Monti , F. Ponci and L. Rocca. 2007. Maximum entropy multivariate analysis of uncertain dynamical systems based on the Wiener-Askey polynomial chaos. IEEE Transactions on Instrumentation and Measurement 56(3), 689-695.

Lovett, T. , A. Monti and F. .Ponci. 2004. A polynomial chaos theory approach to the control design of a power converter. In Proceedings of IEEE PESC 04, 4809-4813.

Smith A. , A. Monti and F. Ponci. 2006. Robust controller using polynomial chaos theory. In Proceeding of IEEE IAS 2006, Tampa, FL (USA).

10.

Smith, A. , A. Monti and F. Ponci. 2007. Robust stability and performance analysis using polynomial chaos theory. Proceedings of Society of Computer Simulation Summer Conference, July 2007, 45-52.

11.

Voglewede, P. and A. Monti. Variation analysis of a two-planar manipulator using polynomial chaos theory. Mechanisms and Robotics Conference, Philadelphia.

12.

Smith, A. , A. Monti and F. Ponci. 2007. Indirect measurements via a polynomial chaos observer. IEEE Transactions on Instrumentation and Measurement, vol 56, no. 3, 743-752.

13.

Monti, A. , F. Ponci and T. Lovett. 2005. A polynomial chaos theory approach to uncertainty in electrical engineering. In IEEE Intelligent Systems Application to Power Systems (ISAP) 2005, Nov. 7-10, Washington DC , USA, 534-539.

14.

N. Wiener. 1938. The homogeneous chaos. American Journal of Mathematics 60, 897-936.

15.

Xiu, D. and G.E. Karniadakis. 2002. The Wiener-Askey polynomial chaos for stochastic differential equations. SIAM Journal of Scientific Computing 24, 619-644.

16.

Sandu, A. , C. Sandu and M. Ahmadian. 2006. Modeling multibody dynamic systems with uncertainties. Part I: Theoretical and computational aspects. Multibody System Dynamics 15(4), 369-391.

17.

Sandu, A. , C. Sandu and M. Ahmadian. 2006. Modeling multibody dynamic systems with uncertainties. Part II: Numerical applications. Multibody System Dynamics 15(3), 241-262.

18.

Pradya, P. 2007. Uncertainty analysis in a shipboard integrated power system using multi-element polynomial chaos. PhD Thesis, MIT, available at http://dspace.mit.edu/handle/1721.1/40364.

19.

Hines, W. , D. Montgomery , D. Goldsman and C. Borror. 2003. Probability and Statistics in Engineering, Fourth Edition . John Wiley & Sons Inc.