STATE-SPACE APPROACH TO SENSITIVITY ANALYSIS AND MULTIVARIABLE NETWORKS

Abstract

Some interest has recently been shown in the application of state space approach to network theory. This thesis is concerned with the application of this approach to sensitivity analysis and multivariable net works with a view to evolve improved integrated circuit design techniques. In the design of integrated circuits one has to accommodate simultaneous variations in several parameters of the netvrork, rather than considering the sensitivity of a netvrork function with respect to variations in a single parameter. This has led to interest in the twin problem of sensitivity analysis and multivariable networks. Several results in the sensitivity studies have been obtained using state space concept, keeping in view the amenability of this approach to digital computer programming, which is essential for sensitivity studies. For example, the well knovrn result of invariant nature of the sum of sensitivities of a network function over different parameter sets [163 , has been interpreted in state space terms. Some new results such as sum of the sensitivities of markov parameters of the triple (F,g,h), sensitivity invariants for multi-input multioutput netvrorks, over different parameter sets have also been obtained. Sensitivity state models for systems connected in tandem, parallel and feedback have been derived. By using a time-variable non-singular transforma tion, some interesting results for continuous equivalent networks have been extended to time varying case. In multivariable network theory [90] the network functions are considered to be dependent on several variables. This theory is of considerable importance [84] in the design of integrated circuits, netvrorks having lumped transmission line elements, microwave filters and multi-dimensional discrete and continuous filters. Several results in the multivariable theory have been obtained by exploiting the state variable approach. Some algorithms for state variable realiza tions using markov parameters [49] » time moments[25] or a combination of both [122] have been proposed. Procedures for obtaining a symmetric realization from a symmetric multivariable transfer function matrix have been given. The importance of such realizations is that these always result in reciprocal netvrorks. Some time it is sufficient to obtain the realization which Ill is non-minimal. Algorithms for obtaining sub-optimal realizations have also been proposed. In order to study the behaviour of large order systems, a reduced order model is desired[l25]. Proce dure for obtaining simplified state space models from the given large order multivariable system is described. Various methods for obtaining the multivariable network function from the given state variable descrip tion have been proposed. The proposed methods do not require the inversion of a rational matrix. In netvrorks many a time the real part of a netvrork function is known as it can be measured with real part meters etc., and it is desirable to obtain the multivariable positive real functions [157] of which the real part is given. State space procedures for single variable cases for determining such netvrork functions have been proposed recently [73],[102].These state space procedures have been found to be more useful because they are applicable to multi-input multioutput netvrorks as well. Algorithms for obtaining multivariable positive real matrix from its given even, odd parts have been proposed using state space approach. Further several results such as positive real lemma, bounded real lemma and reciprocity have also been interpreted in state space terms for multivariable netvrorks. Sometimes the available information is in terms of state variable characterization of multivariable networks. In such situations the synthesis problem is tackled via state space. Before developing synthesis procedures the generalized state space model for multivariable RLC networks has been obtained, using graph theoretic approach [15} [6].The state space models for multivariable loss-less networks and a class of multivariable RLC netvrorks having no coupling betvreen link resistances and tvrig conductances, have been obtained from the generalized state variable description. Algorithms for the synthesis of multivariable netvrorks are developed, by using the voltage across the capacitors and current through the inductors as the state variables. The proposed methods require the decomposition of the given F(p) matrix and then comparing the resulting state equations with the corresponding state model. The solut ion of the set of matrix equations so obtained yields the fundamental circuit matrix and the element values of the netvrork. State space approach has recently been popular for netvrork synthesis [8], [150], [171] for single variable case. From the given input output specifications in s-domain, these techniques result in RLCT netvrorks [8] . An algorithm has been developed for the synthesis of RLCT networks from input-output characterization for multivariable case. Further, a technique is described for Foster synthesis of multivariable loss-less networks which uses the markov parameters of the triple [F(j)), G(£),