Page 112 - Modern Control Systems

86 Chapter 2 Mathematical Models of Systems

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FIGURE 2.30 «:0
Signal-flow graph
of two algebraic
equations.

and
(1 - a u)r 2 + a 2lri
* 2 = - ^ - r 2 + —r h (2.94)
(1 - fln)(l - fl 22) - «i2«2i
The denominator of the solution is the determinant A of the set of equations
and is rewritten as
A = (1 - fl n)(l - fl 22) - «12«21 = 1 ~ «11 - «22 + «11«22 - «12«21- (2-95)
In this case, the denominator is equal to 1 minus each self-loop a n, a 22, and fli2«2i>
an
plus the product of the two nontouching loops a\\ and a 22. The loops #22 d «21 «12
are touching, as are an and fl2i«i2-
The numerator for X\ with the input r\ is 1 times 1 — a 22, which is the value of A
excluding terms that touch the path 1 from r\ to x\. Therefore the numerator from r 2
to Xi is simply a 12 because the path through a\ 2 touches all the loops. The numerator
for x 2 is symmetrical to that of X\,
In general, the linear dependence 7^- between the independent variable x t
(often called the input variable) and a dependent variable Xj is given by Mason's
signal-flow gain formula [11,12],
2^* -/;*

T = (2.96)
Pijk = gain of kth path from variable x t to variable Xj,
A = determinant of the graph,
Aijk = cofactor of the path P ijk,

and the summation is taken over all possible k paths from x t to Xj. The path gain or
transmittance P^ is defined as the product of the gains of the branches of the path,
traversed in the direction of the arrows with no node encountered more than once.
The cofactor A^ is the determinant with the loops touching the kth path removed.
The determinant A is
L
A = 1 - 2 « + £j LnLffj ^j L-'n'-'m'-'p ' (2.97)
«=1 n,m n, m, p
nontouching nontouching
where L q equals the value of the qth. loop transmittance. Therefore the rule for eval-
uating A in terms of loops L l5 L 2, L 3 ,..., L N is