Parameter Estimation in Linear Descriptor Systems

Linear descriptor systems form the natural way in which linear models of physical systems are delivered from an object-oriented modeling tool like Modelica. Linear descriptor systems are also known as linear differentialalgebraic equations in the continuous-time case. If some parameters in such models are unknown, one might need to estimate them from measured data from the modeled system. This is a form of system identification called gray box identification. The objective of this work is to examine how gray box identification can be performed for linear descriptor systems. To solve this problem, we use some well-known canonical forms to examine how to transform the descriptor systems into state-space form. In general, the input must be redefined to make the transformation into statespace form possible. To be able to implement the suggested identification methods, we examine how the transformations can be computed using numerical software from the linear algebra package LAPACK. Noise modeling is an important part of parameter estimation and system identification, so we also examine how a noise model can be added to linear descriptor systems. The result is that white noise in general cannot be added to all equations of a linear continuous-time descriptor system, since this could lead to differentiation of the noise which is not well defined. It is also noted that a Kalman filter can be implemented if the model is transformed into state-space form. We also discuss the problem of finding initial values for the parameter search. We show how to formulate a biquadratic polynomial, that gives initial values for the parameter search when minimized.