Page 199 - Modern Control Systems

Section 3.4 Signal-Flow Graph and Block Diagram Models 173
there is more than one alternative set of state variables, and therefore there is more
than one possible form for the signal-flow graph and block diagram models. There
are several key canonical forms of the state-variable representation, such as the
phase variable canonical form, that we will investigate in this chapter. In general, we
can represent a transfer function as
m x
n / . Y(s) b ms m + b m- lS ~ + ••• +b lS + bo
G(s) = 777— = : (3.41)

where n > m, and all the a and b coefficients are real numbers. If we multiply the
numerator and denominator by s~", we obtain
m
b ms^-^ + b m. lS-^ ^ + ••• + b lS-("-V + b 0s-»
G(s) = } :—-, . (3.42)
n l)
1 + a n- lS- + ••• + a lS-( - + aos-"
Our familiarity with Mason's signal-flow gain formula allows us to recognize the famil-
iar feedback factors in the denominator and the forward-path factors in the numerator.
Mason's signal-flow gain formula was discussed in Section 2.7 and is written as

When all the feedback loops are touching and all the forward paths touch the
feedback loops, Equation (3.43) reduces to
Zjk^k Sum of the forward-path factors
G(s) ~ /v = — - , (3.44)
I _ V L * sum of the feedback loop factors'

There are several flow graphs that could represent the transfer function. Two flow
graph configurations based on Mason's signal-flow gain formula are of particular in-
terest, and we will consider these in greater detail. In the next section, we will consider
two additional configurations: the physical state variable model and the diagonal (or
Jordan canonical) form model.
To illustrate the derivation of the signal-flow graph state model, let us initially
consider the fourth-order transfer function

G(s) = 4 3, 2
U(s) s + a^s + a 2s + (lis + a 0

" ^ ZJ. (3.45)
1 + a 3s l + a 2s 2 + a\S 3 + CIQS~
First we note that the system is fourth order, and hence we identify four state vari-
ables (*!, x 2, x 3, X4). Recalling Mason's signal-flow gain formula, we note that the
denominator can be considered to be 1 minus the sum of the loop gains. Further-
more, the numerator of the transfer function is equal to the forward-path factor of
the flow graph. The flow graph must include a minimum number of integrators
equal to the order of the system. Therefore, we use four integrators to represent this
system. The necessary flow graph nodes and the four integrators are shown in
Figure 3.9. Considering the simplest series interconnection of integrators, we can