By O. I. Franksen, P. Falster, F. J. Evans

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**Extra resources for Qualitative Aspects of Large Scale Systems: Developing Design Rules Using APL**

**Sample text**

Evidently, emulating this approach for all four of the coefficient matrices in the state and output equations (I) and (2), the total set of structural properties may be established for the entire system. A first step in this process is to apply the logical proposition to the coefficient matrix A: A÷A sO (49) Accordingly, by s u b s t i t u t i n g each non-zero e n t r y i n t h e c o e f f i c i e n t m a t r i x A by a 1, we have r e d e f i n e d A i n t o a Boolean m a t r i x o f dimensions (X,X).

I. 15), two input variables, controlling directly state variables X[I] and X[2] respectively, are involved. Marking off vertex 1 in the digraph of Fig. 1 it is imnediately seen that its successors are vertices 3 and 4. In other words, controlling vertex 1 makes vertices 3 and 4 potentially controllable. Similarly, marking off vertex Z we find that the remaining three vertices are all its successors. Hence, controlling vertex 2 makes all the remaining three vertices potentially controllable. Thus, a mere visual inspection of the system digraph provides us, in a qualitative and simple manner, with all the information needed for deciding whether or not the given control will make the system potentially state controllable.

It follows that their mathematical arguments must be based on considerations of matrix ranks and eigenvalues. Now, in our opinion a group-theoretical approach is both simpler and more fundamental (Franksen, 1978). The said parameter values will be measured on ratio or interval scales implying that they will be invariant under the similarity or the affine group of transformation. Substituting the measurements by Boolean values they are taken to a nominal scale that makes them invariant under the syn~etric group of transformations.