STATE VARIABLE MODELS OF LINEAR DYNAMICAL SYSTEMS FROM INPUT OUTPUT DESCRIPTION

Abstract

Much research in linear system theory in recent years has been in the area of realization theory, which deals with the construction of state space models from input output description. This is mainly due to a change in the mathematical techniques employed for the study of dynamical systems, in the last two decades. The emphasis has now shifted from the impulse response or transfer function description of linear systems to a more general state space characterization. Due to the importance of state model in control, identification and network synthesis problems realization theory assumes a significance of great importance. The external description of a dynamical system is generaly in the form of a known transfer function matrix, empirical data of impulse response matrix obtained from experiments or input-output operating records. In the present thesis methods of realization from these descript ions are investigated and some new methods are suggested which are either computationally better than the existing methods or which can be of importance in practical applications. First part of the thesis investigates some methods suitable for obtaining a realization from emirical data when transfer function is not explicitly known.

A method of realization is proposed using a combination of Markov parameters and moments. The proposed technique is based on defining a modified Hankel matrix so that the matrix inversion required in the earlier methods is avoided. This technique of realization from combination of Markov parameters and moments can find applica tions in model reduction problems, where both initial trans ient and final steady state responses of the model can be matched with those of the real system. Sometimes a realiza tion is required from the empirical data of impulse response matrix which may be contaminated with noise. In such cases realization from moments is preferable. In the present thesis a computationally efficient algorithm for canonical realization from moments is proposed by defining a modified Hankel matrix. In a realistic situation it is desirable to obtain the mathematical model of a plant from its operating records with out the application of perturbation signals. Such methods of direct realization from input-output data sequences are studied in the later part of the thesis. Most of the existing methods of realization from input-output data are based on an apriori assumption about the strictly dynamical nature of the system. In many situations e.g.modelling Gf socio economic systems electrical networks,etc. identification of non-dynamic Part is important. In the present work a method of canonical realization from input-output data sequences is developed which is not based on any apriori assumption about the strictly dynamical nature of the system. In the liproposed method a oanonical input-output representation is used to provide asimple recursive relation for computing the input matrix and the nondynamic part. Using the properties of the canonical structure anew and simple method is proposed for finding the memoryless part of a system. Aclass of time varying systems having aslowly varying memoryless part is also studied in this disserta tion. For asingle input single output system ,realiza tion method is presented with the assumption that the model order is known. In some caaes the parameters of tho memoryless part may drift due to changes in the environment, and it may be possible in such cases to dete^ine the stationary model from' initial observations. Atechnique for multivariate system is proposed, so that the parameters of the memoryless part can be computed with sufficient accuracy. It is felt that this investigation will flnd application in the study of atime varying network which can be represented as atime varying memoryless network terminated in aconstant reactive network using the principle of reactance extraction. The work concludes with some suggestions for further work in this area.