Applications of Symbolic Computing Methods to the Dynamic Analysis of Large Systems

Abstract

The dynamic analysis and control of very large systems yet remains a problem for the analyst. The tools currently available to him rely largely on matrix methods. It is recognized that most man-made systems, either physical or otherwise, tend to be not highly interactive. That is, most of the elements are connected only to other elements near to it rather than inter-connected with all other elements which compose the system. The use of matrix techniques to analyze such problems is found lacking because a large number of matrix elements are zero.A new approach lies in the use of symbolic computer methods applied directly to the system equations. Two specific techniques are applied to the analysis of large dynamic systems. Since the symbolic computing language is well suited to the operations with algebraic equations, both techniques use the transfer function concept as a tool for the analysis of large linear dynamic systems. Also, both techniques have been coded in the experimental symbolic computer language FORMAC. The first approach REDUCE I, establishes the techniques and a computer program to symbolically reduce arbitrary block diagrams associated with large systems for desired transfer functions. Symbolic closed form solutions are determined in several forms. The program has been applied to several research problems which include both lumped and distributed parameter systems.A second computer program, REDUCE II, is also based on the use of symbolic commuting methods and has been written to accommodate large engineering systems. REDUCE II symbolically calculates the transfer functions of any linear block diagram output variable to any or all input variables. The solution using this technique is presented in the compact form of a set of nested functions. The program can handle systems as large as 600 equations (of essentially any order) and is intended as a tool for the analysis of complex control and dynamic systems.The paper also projects other areas of application of the symbolic method.