38 Chapter 2 / Mathematical Modeling of Control Systemswhere are determined from
(2–35)
With this choice of state variables the existence and uniqueness of the solution of the
state equation is guaranteed. (Note that this is not the only choice of a set of state vari-
ables.) With the present choice of state variables, we obtain
(2–36)
where is given by
[To derive Equation (2–36), see Problem A–2–6.] In terms of vector-matrix equations,
Equation (2–36) and the output equation can be written as
y=[1 0 p 0]F
x 1
x 2
xn
V +b 0 u
- G
b 1
b 2
bn- 1
bn
G Wu
x 1
x 2
xn- 1
xn
G W
x
1x
# 2 x
n- 1x
nW = G
0 0 0
- an
1 0 0
- an- 1
0 1 0
- an- 2
p
p
p
p
0 0 1
- a 1
W
bn=bn-a 1 bn- 1 - p-an- 1 b 1 - an (^) - 1 b 0
bn
x
n=-an^ x 1 - an- 1 x 2 - p-a 1 xn+bn^ u
x
n- 1 =xn+bn- 1 u
x
2 =x 3 +b 2 u
x
1 =x 2 +b 1 u
bn- 1 =bn- 1 - a 1 bn- 2 - p-an- 2 b 1 - an (^) - 1 b 0
b 3 =b 3 - a 1 b 2 - a 2 b 1 - a 3 b 0
b 2 =b 2 - a 1 b 1 - a 2 b 0
b 1 =b 1 - a 1 b 0
b 0 =b 0
b 0 ,b 1 ,b 2 ,p,bn- 1
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