Everitt, Niklas

Abstract

Systems in engineering are becoming ever more complex and interconnected, due to advancing technology with cheaper sensors and increased connectivity. For example in process industry, sensors that monitor the operation of the plant can be connected through wireless connections and used for mon- itoring and control. In this thesis, we study the problem of identifying one module, i.e., one transfer function from one internal variable to another, in the dynamic network. We investigate how accurate models will be obtained using different gathered measurements. Model errors are assumed to originate from random disturbances that affect the dynamic network. The variance of the model errors are analyzed using the classical assumption that a large amount of data is available. By using a geometric approach, the (co-)variance of the model errors can be analyzed in a way that brings forward how input signal properties, noise variance, noise correlation structure and model structure affect the asymptotic model errors. Several different network structures are analyzed in order to investigate how different signals can reduce the asymp- totic model errors in dynamic networks. For SISO systems we develop reparametrization formulas for the asymp- totic variance of the model errors of functions of the estimated system parameters. In particular, we demonstrate that one can use the experimental conditions to make the asymptotic variance independent of model order and model structure in some cases. These expressions are used to derive simple model structure independent upper bounds of the asymptotic covariance for commonly estimated quantities such as system zeros and impulse response coefficients. The variance of the first of a set of estimated modules connected in a cascade structure is analyzed. The main contribution is the characterization of the variance of the frequency function estimate of a module with a zero close to the unit circle. It is shown that a variance reduction of the first estimated module is possible, compared to only using the first measurement in the estimation. The variance reduction is concentrated around the frequency of the unit-circle-zero. For a parallel cascade structure and a multi sensor structure, upper bounds on the asymptotic covariance of the parameter estimates are derived when the model order of the system of interest was fixed, while the model order of every other module is large. The effect of the noise correlation structure is examined for single input multiple output (SIMO) systems. For the case of temporally white, but possi- bly spatially correlated additive noise, we develop a formula for the asymptotic covariance of the frequency response function estimates and a formula for the asymptotic covariance of the model parameters. It is shown that when parts of of the noise can be linearly estimated from measurements of other blocks with less estimated parameters, the variance decreases. The effect of the in- put spectrum is shown to have a less significant effect than expected. We determine the optimal correlation structure for the noise, for the case when one block has one parameter less than the other blocks