STATE-VARIABLE REALIZATION OF LUMPED NETWORKS & DYNAMICAL SYSTEMS

Abstract

The state-variable approach, because of its inherent importance, has aroused considerable interest in the study of systems and networks during the past decade. This thesis is, primarily, concerned with the problem of state-variable realiz ation of linear time-invariant dynamical systems and its appli cation to lumped networks, with a view to evolve new synthesis procedures suitable for integrated circuit fabrication. The problem of minimal reciprocal realization of linear, time invariant dynamical systems is investigated. Two simpli fied algorithms for constructing minimal reciprocal realization from a given symmetric transfer function matrix and symmetric impulse response mat rix have been proposed . Both the methods exploit the symmestry of the given transfer--funct:ion matrix and impulse response mat rix and require determining the :Hankel matrix, the first from the Markov--parameters and the second from the moments of the impulse n esponse m;at rix, the latter being p:referable in the presence of noise. The order of -ivthe Hankel matrices required in the procedure of both the algorithms is much smaller than the existing methods, thereby reducing significantly the computing time and memory storage required. The realizations obtained by the proposed algorithms result in reciprocal networks. Further, utilizing these results, a passive reciprocal (gyratorless) synthesis of symmetric posi tive real immittance matrices is given. Since the classical synthesis methods for linear, timeinvariant networks are well-known, it is quite important to establish a communication link between the state-variable characterization and the input-output description. Some endeav ours have already been initiated in this direction. Here, a state-space interpretation of classical Foster synthesis of multiport lossless network has been discussed. Well-known Cauer driving point synthesis and active RC filter design using coefficient matching technique are also revisited in statespace terms using observability matrix as a canonical trans formation. Various synthesis techniques,which realize an arbitrary rational function matrix of a multiport active RC network, have been developed during recent years. But, the upper bound on the number of active elements required in these methods is quite large and in some cases, the number of resistors used in the realization is also more. In this thesis, a simple and syste matic synthesis procedure, based on a state-variable approach and the reactance extraction principle, has been presented whereby any arbitrary rational function matrix can be realized -vas an immittance matrix of an active RC multiport network with a minimum number of grounded capacitors having unity capacitance spread. The proposed technique reduces the upperbound on the number of active elements and can be reasonably expected to require fewer resistors. Besides, the structure of the realized circuits in terms of the minimum number of elements and grounded ports make it particularly desirable for integrated-circuit fabrication. Finally, some suggestions for further investigations in this area are also included.