Page 216 - Modern Control Systems

190 Chapter 3 State Variable Models
simply the inverse transform of <&(s); that is,

¢(0 = X-'i&is)}. (3.83)

The relationship between a state variable Xj(s) and the initial conditions x(0) is
obtained by using Mason's signal-flow gain formula. Thus, for a second-order system,
we would have
Ai(5) = 011(5)^(0) + * 12(*)*2(0),

*2(*) = <fcl(*)*l(0) + ^22(^)^2(0), (3.84)
and the relation between X 2{s) as an output and Xi(0) as an input can be evaluated
by Mason's signal-flow gain formula. All the elements of the state transition matrix,
<f>ij(s), can be obtained by evaluating the individual relationships between X^s) and
xj(0) from the state model flow graph. An example will illustrate this approach to de-
termining the transition matrix.

EXAMPLE 3.6 Evaluation of the state transition matrix
We will consider the RLC network of Figure 3.4. We seek to evaluate ¢(5) by (1)
-1
determining the matrix inversion ¢($) = [si - A] and (2) using the signal-flow
diagram and Mason's signal-flow gain formula.
-1
First, we determine ¢(5) by evaluating ®(s) = [si - A] . We note from Equa-
tion (3.18) that
^0 -2
J. -3

Then

s 2
[.I - A] = (3.85)
-1 s + 3
The inverse matrix is

s + 3 -2
-1
<&(*) = [si ~ A] = (3.86)
A(s) 1 s
where A(s) = s(s + 3) + 2 = s 2 + 3s + 2 = (s + \)(s + 2).
The signal-flow graph state model of the RLC network of Figure 3.4 is shown in
Figure 3.8. This RLC network, which was discussed in Sections 3.3 and 3.4, can be
represented by the state variables X\ = v c and x 2 = if The initial conditions, *i(0)
and x 2(0), represent the initial capacitor voltage and inductor current, respectively.
The flow graph, including the initial conditions of each state variable, is shown in
Figure 3.23. The initial conditions appear as the initial value of the state variable at
the output of each integrator.
To obtain $(s), we set U(s) = 0. When R = 3, L = 1, and C = 1/2, we obtain
the signal-flow graph shown in Figure 3.24, where the output and input nodes are
deleted because they are not involved in the evaluation of ¢(5). Then, using Mason's