Parameter Uncertainty Modeling Using the Multidimensional Principal Curves

Abstract

This paper proposes a technique to model uncertainties associated with linear time-invariant systems. It is assumed that the uncertainties are only due to parametric variations caused by independent uncertain variables. By assuming that a set of a finite number of rational transfer functions of a fixed order is given, as well as the number of independent uncertain variables that affect the parametric uncertainties, the proposed technique seeks an optimal parametric uncertainty model as a function of uncertain variables that explains the set of transfer functions. Finding such an optimal parametric uncertainty model is formulated as a noncovex optimization problem, which is then solved by a combination of a linear matrix inequality and a nonlinear optimization technique. To find an initial condition for solving this nonconvex problem, the nonlinear principal component analysis based on the multidimensional principal curve is employed. The effectiveness of the proposed technique is verified through both illustrative and practical examples.