Robust stability and performance analysis using polynomial chaos theory

ABSTRACT

In model-based control schemes, the importance of performing robust stability and performance analysis is partly due to the presence of uncertainty. This uncertainty can be modeled as structured or unstructured, though this paper deals primarily with structured uncertainty, specifically parametric uncertainty. Parametric uncertainty can be manifested in the form of parametric tolerance, parameter noise, or parameter disturbances. After the design phase, for the validation of the design itself, it is also important to consider parametric uncertainty in the simulation/evaluation of a system in both open loop and closed loop. The goal of this paper is to demonstrate the use of Polynomial Chaos Theory (PCT) in this analysis process. PCT offers a comprehensive approach to be used in the design and validation. The theory description is enriched with an application in the power electronics field.

REFERENCES

• 1: N. Wiener, "The homogeneous chaos", Amer J. Math, 1938, Vol. 60, pp. 897--936
• 2: Dongbin Xiu , George Em Karniadakis, The Wiener--Askey Polynomial Chaos for Stochastic Differential Equations, SIAM Journal on Scientific Computing, v.24 n.2, p.619-644, 2002 [doi>10.1137/S1064827501387826]
• 3: Dongbin Xiu , George Em Karniadakis, Modeling uncertainty in flow simulations via generalized polynomial chaos, Journal of Computational Physics, v.187 n.1, p.137-167, May 2003 [doi>10.1016/S0021-9991(03)00092-5]
• 4: Xiu, D., Lucor, D., Su, C.-H. & Karniadakis, G. E. 2002. Stochastic modeling of flow-structure interactions using generalized polynomial chaos, J. Fluid Eng. Vol. 124: 51--59.
• 5: T. Lovett, A. Monti, F. Ponci, "A Polynomial Chaos Theory Approach to the Control Design of a Power Converter", 2004 IEEE Power Electronics Specialists Conference, pp. 4809--4813, June 20--25, 2004
• 6: T. Lovett, A. Monti, F. Ponci, "A Polynomial Chaos Approach to Circuit Simulation Under Uncertainty", IEEE Circuits and Systems, Under review
• 7: T. Lovett, A. Monti, F. Ponci, "A Polynomial Chaos Approach to Measurement Uncertainty", IEEE AMUEM2005, Niagara Falls (Canada), May 2005
• 8: A. Smith, A. Monti, F. Ponci, Indirect Measurements via Polynomial Chaos Observer, AMEUM 2006- International Workshop on Advanced Methods for Uncertainty Estimation in Measurement Sardagna, Trento, Italy, 20--21 April 2006.

INDEX TERMS

Keywords:
closed-loop control, modeling and simulation, random process, uncertainty