For each of the listed transfer functions, write the state equations in bothcontrol and observer canonical form. In each case, draw a block diagram andgive the appropriate expressions for A, B, and C7.182(control of an inverted pendulum by a force on the cart).2(2-1)(a) G(s)3s+4(b) G(s)s2+2s+2

Question
Asked Jun 12, 2019

Feedback Control of Dynamic Systems (7th edition) - Problem 7.18

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
7.18
2(control of an inverted pendulum by a force on the cart).
2(2-1)
(a) G(s)
3s+4
(b) G(s)
s2+2s+2
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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 7.18 2(control of an inverted pendulum by a force on the cart). 2(2-1) (a) G(s) 3s+4 (b) G(s) s2+2s+2

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Expert Answer

Step 1

(a). Numerator part of the gain:

b(s) = b1sn-1 + b2sn-2 + …………+ bn…………………… (1)

Denominator part of the gain:

a(s) = a1sn-1 + a2sn-2 + …………+ an…………………… (2)

General equation:

G(s) = b(s) / a(s)……………… (3)

G(s) = (b1sn-1 + b2sn-2 + …………+ bn) / (a1sn-1 + a2sn-2 + …………+ an) ………………… (4)

Matrices of control canonical form are:

a a
1
0
--(5)
A 0
1
0
0
10
1
0
0
0
(6)
0
C. -[b
D Seperate than numerator and denominator else 0.....(8)
Gain
2-2
s2-2
G(s)=
...e)
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a a 1 0 --(5) A 0 1 0 0 10 1 0 0 0 (6) 0 C. -[b D Seperate than numerator and denominator else 0.....(8) Gain 2-2 s2-2 G(s)= ...e)

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Step 2

General state space equation and output equation are given by:

dx/dt = Ax + Bu ……………. (10)

y = Cx + D………………. (11)

Equations in control canonical form will be:

from equation 4 to 10,
[o 1 0 0x
1 0 0 0 x2
1
0
0 10 0 x
0
0 0 1 0x4
0 1 0 0
10 0 0
0
u..
0
12)
0 1 0 0
0 0 10
0
.(13)
y [0 10 -2]x+0...
[0 1 0 0
1 0 0 0
A =
0 1 0 0
0
B =
Kc=[o 1 0 -2]
0 0 1 0
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from equation 4 to 10, [o 1 0 0x 1 0 0 0 x2 1 0 0 10 0 x 0 0 0 1 0x4 0 1 0 0 10 0 0 0 u.. 0 12) 0 1 0 0 0 0 10 0 .(13) y [0 10 -2]x+0... [0 1 0 0 1 0 0 0 A = 0 1 0 0 0 B = Kc=[o 1 0 -2] 0 0 1 0

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Step 3

Block diagram can be formed using equa...

X4
-2
x2
x3
1
1
1
Σ
+
S
S
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X4 -2 x2 x3 1 1 1 Σ + S S

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