The past two decades have witnessed profound changes in the composition, functions and the level of complexity of electrical as well as electronic systems which are employed in modem technology. As a result, classical RLC network theory, which was the mainstay of electrical engineering at a time when RLC networks were the bread and butter of the electrical engineer, has been and is being increasingly relegated to the status of a specialized branch of a much broader discipline-system theory-which is concerned with systems of all types regardless of their physical identity and purpose. This paper presents a brief survey of the evolution of system theory, together with an exposition of some of its main concepts, techniques and problems. The discussion is centered on the notion of state and emphasizes the role played by state-space techniques. The paper concludes with a brief statement of some of the key problems of system theory.

From Circuit Theory to System Theory

The universality of the state-variable approach to network analysis is demonstrated in general discussions and specific examples. The method of formulation of the state equations for an arbitrary lumped, linear, finite, reciprocal, passive, time-invariant network is presented fully, while the relaxation of these restrictions is indicated in detail; i.e., the state-variable characterization of active, nonreciprocal, time-variable, and nonlinear networks is discussed. Finally, there is a brief guide of the current research where the state-variable analysis is brought to bear upon certain qualitative aspects of classical and nonclassical network behavior.

The state-variable approach to network analysis

In this tutorial paper, the influence of the computer not only on the modus operandi of circuit design, but also on network theory itself, is discussed. The topological properties of linear graphs are reviewed and a matrix-topological formulation of the network problem is described. In addition to the classical mesh, node, and cutset methods, a mixed method of analysis is described which is applicable to dc, ac, and transient problems. Numerical methods of solving linear and nonlinear dc network problems are discussed and a new approach to ac analysis, using the mixed method and a numerical solution of the matrix eigenvalue problem, is described. The extension of this method to the transient analysis of linear networks is also explained. Finally, the problem of instability in the numerical integration of differential equations is discussed and several means of solving the problem are outlined.

Computer methods of network analysis

This paper presents an analysis of some of the significant developments in time-varying network theory which have taken place during the past decade, with the emphasis placed on three topics: 1) characterization of time-varying networks, and in particular, transition from the impulsive response to the differential equation; 2) the problem of factorization, with emphasis on the contributions of Darlington, Batkov and Paul Levy; and 3) randomly-varying systems and, in particular, the question of stability of discrete-time systems of this type. The identification problem, the analysis of periodically-varying systems, the synthesis problem, and the filtering and prediction of nonstationary processes will be treated in Part II, to be published later.

Time-Varying Networks, I

The Explicit Form of Bashkow's A Matrix

Both the loop and node methods of network analysis produce a system of second-order differential equations. A method of analysis is proposed which produces a set of first-order differential equations. With this method, the network equations obtained can be expressed in the form F + dy/dt = Ay , where F and y are column matrices and A is a square matrix. The variables, y , are currents through inductances and voltages across capacitances; the forcing functions. F are proportional to voltage and current sources. The elements of A are inductances, capacitances, and resistances, or combinations thereof. Characteristic roots (natural frequencies) of the network are identical with the eigenvalues of the A matrix.

T. Bashkow

Papers

The A Matrix, New Network Description