By Michael K. Sain
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Probably the most basic confusion which tends to occur is the confusion of systems and coefficients. This is why a careful understanding of Fig. 24 can be beneficial. The two-box idea shows that different algebraic layers can be combined into multilayer structures. Like the idea of layers itself, such a possibility can rapidly stretch the imagination. The purpose of this chapter has been to introduce, within the intuitive context, various representative notions of what algebraic system theory is all about.
Tabulations are common enough to rate special keys on machines with very limited space, such as the ubiquitous typewriter. Few typists, however, would be aware that pressing the tab key creates a relation ! I ,/' Cosenta Geddi A Fig. 3. A typical tabulation. IL Fig. 4. A typical graph. 3-3 Graphs have a special interest here, because of the way in which they relate to functions. Suppose B c R x R. Both the graphs of Fig. 4 and Fig. 5 could define B, but only the binary relation of Fig. 5 could qualify as the graph of a function.
These associations can be pictured conveniently in the tabular style of Fig. 7. Then the notations x ( j ( n ) )and u ( j ( n ) )are natural; and a local transition equation can be written in the manner x ( j ( n + 1))= f ( x ( A n ) ) , ~ ( j ( n ) ) ) , (2-62a) as well as a local output equation Y ( j ( 4 ) = s ( x ( j ( n ) ) ,u ( j ( n ) ) ) , (2-62b) for n E N. An) A ntl) d x , u) f(x, u) Fig. 7. Tabular view of dynarnical operation. The classical streamlining of (2-62)can be a useful introductory algebraic exercise.