Rationality Matrix

One can write down the properties of such a rationality matrix by considering possible rational relationships. 

By considering these points, all the pigeon holes are filled with convenient symbols and its completion guides to possible global relationship between the input and output variables. In the interpretation of such a matrix, the following points must be taken into consideration. 

In order to be able to decide which one of these alternatives is valid, it is necessary to benefit from basic and simple rules. The first information that one can use is to check the units of both sides. In any rational deduction the units must be the same. For instance, if the unit of the left hand side is in m, kg or Watt then the combined unit on the right hand side must be the same, m, kg or Watt, respectively. On the other hand, prior to the unit homogeneity, the initial and boundary conditions of the physical event may help to identify the valid equation. In order to shed light on this point let us consider the power of a pump as follows. 

Let us try to obtain a formulation for the power of a pump that will haul water from a lower to a higher level. Let the pump power, P and the elevation height, h. Logically, the power will be dependent on several variables, but the most important ones are the amount of water (diseharge), Q, and the density of water, y. There are four variables (P, h, Q, y) and the dimension of the rational matrix will be 4 X 4 as in Fig. 5.14. 

Rational thinking will indicate that there are directly proportional relationships between power and other variables. Hence, P as dependent (subject) and others independent variables, the implicit formulation can be shown as P = f (h, Q, y). One can write exhaustively explicit form alternatives as follows. 

---

A rational matrix B( ) is biproper if and only if it is proper and its inverse is also proper. A matrix B(s) is proper if and only if aU its elements are rational fractions with the degree of their denominator greater than that of their numerator [42]. 

VardulaMs, A.I.G., Limebeer, D.J.N., Karcanias, N. Structure and Smith-MacMillan form of a rational matrix at infinity. International Journal of Control, 35(4) 701—725, April, 1982. 

Lemma 1 [14]. For each proper real-rational matrix G there exist eight 7 .7ioo-matrices satisfying the equations... 

The total number of input and output variables defines the dimension of the rational matrix. If variable number is n then the matrix dimension is n x n. 

The rational matrix is symmetrical with respect to the main diagonal, because the relationship between x and y is the same for y and x. It is, therefore, enough to consider the upper triangular section only. 

After the completion of the rational matrix, one can make different interpretations. For instance, if the question is about the type of relationship as subject V variable in terms of others, then this variable, as can be seen from Fig. 5.13, is dependent on the others. As x increases v deaeases as y inaeases v inaeases and finally, as z increases v also increases. If any one of these variables is thought as subject then similar interpretations can be made. 

It is also possible from the rationality matrix to deduce the type of functional relationship. For instance, in mathematics the relationship between v, x, y and z variables can be written implicitly as v = f (x, y, z). In engineering, such... 

Two observations are useful to rationalize why in Figures 2-11 and 2-12 (1) Gyy exceeds l (2) E45. is less than for composite materials that have a fiber modulus much greater than the matrix modulus ... 

So far there is no cover-all recipe for remedying these undesirable phenomena yet recommendations have been made for some special cases which included proper choice of process conditions and methods, rational design of molds and articles allowing one to minimize the negative effect of filler and matrix orientation in the manufacturing process [370],... 

Bis(tributyltin) oxide is known to break down to inorganic tin under UV irradiation in laboratory conditions (509, 510), and the decomposition may be accelerated by absorbing the organotin compound on a cel-lulosic matrix (511). As bis(tributyltin) oxide is known to react rapidly with carbon dioxide (atmospheric, or trapped in various cellulosic materials, such as cotton or wood) (512), to form bis(tributyltin) carbonate, (BusSnO)2CO, the observed UV degradation pattern may be rationalized in terms of more-ready breakdown of the carbonate than of the oxide, due to the presence of the carbonyl chromophore. The half-life of bis(tributyltin) oxide in pond water has recently been given as 16 days (513). Diorganotin compounds have also been shown to decompose to inorganic tin under UV irradiation (514, 515). 

Both the metal-atom and matrix techniques have many applications (91) in the study of complexes having such classical, inorganic ligands as CO, N2, O2, or phosphines. From a metal-atom point of view, novel complexes have been synthesized that have not been readily accessible via normal, chemical-synthesis techniques. The entrapment of such species permits both a rationalization of their spectroscopic and chemical properties and an evaluation of their stability. 

---