Hierarchical symbolic computations in the analysis of large-scale dynamical systems

Abstract

The paper presents a case study of an application of symbolic computation in a new area — in the qualitative analysis of large scale dynamical systems. The main distinctive feature is that the underlying algebra is not usual but that of block-diagrams. Identifiers (denoting dynamical systems) have two properties: linear/nonlinear and one/multi-dimensional. Properties of arguments affects the properties of the multiplication. It is commutative only for the zero, identity and for linear one-dimensional operators, for a nonlinear multiplier it is only right- distributive over addition. Simplification must be thought of in terms of the clarity of block diagrams, what contradicts sometimes the usual algebraic notion of simplicity. The paper presents briefly the background and the way to a specialized symbolic algebra package.

The concept of hierarchical calculations is introduced. The idea is that the same object may be represented by several values at different levels of hierarchy of specifications. Eg. some calculations can be done on matrices denoted by their identifiers, some on the same matrices with explicit block elements denoted by another identifiers and some on matrices with explicit elements. Calculations may "wander" from one level of specification to another in a complicated pattern. But matrices are stored in a multivalued form and any calculation implies automatic recalculation at other involved levels — may be in completely different computational domains. This is easier to use than direct substitutions done explicitely by the user. In a specialized system, as presented here, this notion is relatively easy to implement. It is believed, however, that it would be a welcomed programming convenience in general purpose systems. Some implementation requirements are discussed for that case.