Page 201 - Modern Control Systems

Section 3.4 Signal-Flow Graph and Block Diagram Models 175

Then it follows that the fourth-order differential equation can be written equivalently
as four first-order differential equations, namely,

=
X\ X 2,
x
2 = *3»
=
*3 * 4 >
and
= x
Xq #()-^1 Q\ 2 — ^2""-3 — ^3-^-4 "^ ^i
and the corresponding output equation is

y = b Qx x.
The block diagram model can be readily obtained from the four first-order differential
equations as illustrated in Figure 3.10(b).
Now consider the fourth-order transfer function when the numerator is a poly-
nomial in s, so that we have
2
b 3s* + b 2s + b lS + b Q
G{s) - — A = =
s + a$s + a 2s + (lis + « 0
b 3s~ l + b 2s~ 2 + y 3 + V 4
= -: —. -r — A- (3.46)
1 + a 3s + a 2s + a^s + CIQS
The numerator terms represent forward-path factors in Mason's signal-flow gain for-
mula. The forward paths will touch all the loops, and a suitable signal-flow graph real-
ization of Equation (3.46) is shown in Figure 3.11(a). The forward-path factors are
4
3
2
b 3/s, b 2/s , bi/s , and b 0/s as required to provide the numerator of the transfer func-
tion. Recall that Mason's signal-flow gain formula indicates that the numerator of the
transfer function is simply the sum of the forward-path factors. This general form of a
signal-flow graph can represent the general transfer function of Equation (3.46) by
utilizing n feedback loops involving the a„ coefficients and m forward-path factors in-
coefficients. The general form of the flow graph state model and the
volving the b m
block diagram model shown in Figure 3.11 is called the phase variable canonical form.
The state variables are identified in Figure 3.11 as the output of each energy stor-
age element, that is, the output of each integrator. To obtain the set of first-order differ-
ential equations representing the state model of Equation (3.46), we will introduce a
new set of flow graph nodes immediately preceding each integrator of Figure 3.11(a)
[5, 6]. The nodes are placed before each integrator, and therefore they represent the
derivative of the output of each integrator. The signal-flow graph, including the added
nodes, is shown in Figure 3.12. Using the flow graph of this figure, we are able to obtain
the following set of first-order differential equations describing the state of the model:

x = x x = x x = x
\ 2-> 2 3-> 3 4i
= — ciQXi — - - + u. (3.47)
x 4 a-[X 2 a 2x 3 a 3x 4
x 2,... are the n phase variables.
In this equation, x u x n