Equações de Estado Canônicas

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Problem 7.18PP
(a) C(s) = (control of an inverted pendulum by a force on the cart).
For each of the listed transfer functions, write the state equations in both control and observer
canonical form. In each case, draw a block diagram and give the appropriate expressions for A,
B. and C.
(b) (b) =
Step-by-step solution
diep^by-^iep* Solliiion
Step 1 of 10
(a)
Consider the numerator part of the gain.
b{s)= b^^+ b2S^^+ ...+ b,...... (1)
Consider the denominator part of the gain.
a ( j ) = y + = [1 0 0 0 0]
.■*4.
-̂ = [1 0 0 0 Oj*. (21)
Hence, the state equations of observer canonical form are
|y = [l 0 0 O " ^
0 0 0 o '
0 0
x ,+
1
0 0 0 1 0
0 0 0 0 -2
Step 5 of 10 ^
Draw the block diagram for observer canonical form from equations (20) and (21).
Figure 2: Block diagram for observer canonical form
Hence, the block diagram for observer canonical form is shown in figure 2.
Step 6 of 10
(b)
Consider the gain value.
3s+4 (22)
s’ *2s+2
Write the state equation in control canonical form from equations (4) to (9) and equation (22).
?]'•[:>...
Write the general form of output equation.
y = C x*D
.y=[3 4 ] * - f 0 ...... (24)
Step 7 of 10
Hence, the state equations of controller canonical fonn are:
-2 -2 1
X S
1 0
X +
0
u
b = [3 4]x|.
Draw the block diagram for controller canonical form from equations (23) and (24).
Figure 3; Block diagram for controller canonical form
Hence, the block diagram for controller canonical form is shown in figure 3.
Step 8 of 10
Write the state equations in observer canonical form from equations (4), (14), (15), (16), (17), 
(18), (19) and (22).
"•=[-2
(25)
J-=[1 0]
.r=[i o]j(, (26)
Hence, the state equations of observer canonical form are: -2 1 
-2 0
j j - l i O K I
Step 9 of 10 ^
Draw the block diagram for observer canonical form from equations (25) and (26).
Figure 4; Block diagram for observer canonical form
step 10 of 10 ^
Hence, the block diagram for observer canonical form is shown in figure 4.