This is research on certain aspects of modeling and control of dynamical systems. In particular, an integrated approach to modeling and control of linear dynamical systems will be developed based on the notion of a graph. The notion of a graph of a dynamical system and the notion of the gap metric are important new ideas in systems theory. The gap metric is a notion of distance between systems suitable for use in the context of robust stabilization and control. Approximation of models of linear dynamical systems can be done with use of the gap metric. This is motivated by the need to synthesize controllers using "low" order models. The description of the modeling error appears naturally in the form of uncertainty in the graph of a given system -- graph being the set of all bounded input-output pairs of the system. The first part of the research will be on such modeling theory. In the second part of the research, a framework of modeling which uses estimates of the autocorrelation function of a given signal will be developed. In particular, measuring noisy input-output data of an unknown linear system can be cast into this framework. An approach to recursive spectral estimation based on autocorrelation (covariance) data is proposed. The gap metric will be used to obtain a-priori bounds on spectral estimates and on estimates of the uncertainty in the graph of the sought model. The third part of the research will be to continue the development of recent results on a new approach to spectral factorization. This is based on a sequence of transformations on the graph of a given linear system. The technique is based on an iterative scheme and holds the potential of leading to fast and efficient algorithms for spectral factorization and recursive filtering.
