Page 209 - Modern Control Systems

Section 3.5 Alternative Signal-Flow Graph and Block Diagram Models 183

function for the controller is

U(s) 5(5 + 1) 5 +5s- 1
R(s) = G c(s) = s + 5 1 + 5s~ v

and the flow graph between R(s) and U(s) represents G c(s).
The state variable differential equation is directly obtained from Figure 3.18 as

3 6 o" ~o~
x = 0 -2 -20 x + 5 r(t) (3.59)
0 0 -5_ _1_
and

y = [1 0 0]x. (3.60)
A second form of the model we need to consider is the decoupled response
modes. The overall input-output transfer function of the block diagram system
shown in Figure 3.17 is

Y(s) 30(5 + 1) q(s)
T(s) =
R(s) (s + 5)(5 + 2)(5 + 3) (s - si)(s - s 2)(s - 5 3)'

and the transient response has three modes dictated by 5 l5 $2, and 53. These modes
are indicated by the partial fraction expansion as

Y(s) k x * 3
= T(s) = + + (3.61)
R(s) s + 5 s + 2 5 + 3
Using the procedure described in Chapter 2, we find that ki = —20, k 2 = -10,
and & 3 = 30. The decoupled state variable model representing Equation (3.61) is
shown in Figure 3.19. The state variable matrix differential equation is

5 0 0" "l"
0 -2 0 x + 1 /-(0
0 0 -3_ _1_
and

y(t) = [-20 -10 30Jx. (3.62)
Note that we chose x x as the state variable associated with S] — J, Xi associated
- = 3 , as indicated in Figure 3.19. This choice
-
with s 2 = 2, and x 3 associated with s 3
of state variables is arbitrary; for example, x x could be chosen as associated with the
factor 5 + 2.
The decoupled form of the state differential matrix equation displays the dis-
-
tinct model poles ^ , —5 2 ,.... -s„, and this format is often called the diagonal
canonical form. A system can always be written in diagonal form if it possesses
distinct poles; otherwise, it can only be written in a block diagonal form, known as
the Jordan canonical form [24].