Abstract:
Systems Tableau is suggested as a convenient tool for the
integration of the three phases of systems theory: the synthesis of
a model from the analysis of a system; the evaluation of the model;
and the decision-making process for the design and control of the
resulting system.
From a basic consideration of Man-Nature communications,
several mathematical, biological, engineering, and management
examples of systems models are examined to develop a unified
definition of a system.
Logical, physical, mathematical, graphic, and computational
requirements are postulated for the methodology of models for systems
meeting the definition. These requirements are used to formulate
the basic tableau as a hybrid of a mathematical mapping matrix
and a graphical flowgraph that expresses the interrelationships among
the components of a given system. Thus, a tableau is at once a matrix and a network representation of the system.
The general (connecting), ordinal (dominating), and technological
(directing) relations in the observation (phase) space are illustrated
on tableaux for social, economic, management, and
engineering examples. Related mathematics of relations are examined.
The relationships of these descriptive models to normative
models are discussed as synthesis techniques. Orthogonalization of
bases, parametric representations (in frequency space as probabilities
and statistical distributions), and reductions in state (solution)
space are methods introduced with examples in queueing, communication,
and information models. In normative models, we are afforded
some degrees of freedom expressed in terms of choice of alternatives.
This decision requires at least an ordinal, if not cardinal,
characterization of each alternative.
The ordinally normative models are based on comparatively
quantifiable relations originally afforded by the uni-directional flow
of time. Theories of Information, Algorithms, and Games were
found useful in drawing valuable conclusions (decisions) from these
models. Puzzles, games, Turing machines, and biological examples
are discussed.
The cardinally normative models require decisions based on
numerical values. A truly cardinal model must be cardinal resource-wise, time-wise, and information-wise. This inter-dependency of resources in phase space and information in state
space, as functions of time expressible in frequency space, is the
basis for the proposal of the Cardinal Utility Hypotheses. This concept
allows the development of relations as peculiar Laplace-Z
transform-pairs, with the utility of Information (usefulness of data
for decision-making) serving as the Channel Capacity for a corresponding
communication model.
The Principle of Optimality of Dynamic Programming was
found most useful in Tableau, andits continuous counterpart of Maximum
Principle is expected to take a respective place in the Calculus
of Variations in Control Theory.
The relationship of the controllability and observability of a
system and the diagonalization of its Tableau is also illustrated.
The linear models of the traditional Tableaux are reviewed
and interpreted in the light of Systems Tableau Method, These include
Quesney-Leontief Tableau Economique, Hellerman's Tableau,
and Critical Path Scheduling Tableau. The obvious advantages afforded
by the applications of Huggins' (and others) Signal Flowgraph
techniques are briefly illustrated. Mention is made of a tableau-based computer program that will produce the dual network for
Ford-Fulkerson's Minimum-cut-maximal-flow Method. A brief
discussion of the future of Systems Theory concludes the treatise.