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And finally:
x
1
)μ
P
12
(
x
2
)
q
1
+
μ
P
21
(
x
1
)μ
P
22
(
x
2
)
q
2
y
0
=
μ
P
11
(
x
1
)μ
P
12
(
x
2
)
+
μ
P
21
(
x
1
)μ
P
22
(
x
2
)
μ
P
11
(
8.3 Control of Nonlinear Systems
8.3.1 State-Space Representation
The state-space representation of a dynamical system is used in control engineering
in order to completely describe its behavior by first-order differential equations of the
involved state variables. Input variables
u
T
, output variables
(
t
)
=[
u
1
(
t
),...,
u
p
(
t
)
]
T
are all
needed to completely define any dynamical system. The state variables of a system
constitute the smallest set of variables which completely determine the state of a
system and in practice these variables usually correspond with physical magnitudes.
Thus, if the state of any system and its inputs are known at any given time, the future
behavior of the system will be determined by its state space representation. The state
variables are not always measurable and/or observable.
The following equation is a generic state-space representation of a nonlinear
system.
T
y
(
t
)
=[
y
1
(
t
),...,
y
m
(
t
)
]
and state variables
x
(
t
)
=[
x
1
(
t
),...,
x
n
(
t
)
]
˙
x
(
t
)
=
f
(
x
(
t
))
+
g(
x
(
t
))
u
(
t
)
y
(
t
)
=
h
(
x
(
t
))
If the dynamical system is linear and time invariant, the differential and algebraic
equations may be written in a matrix form, such as,
˙
x
(
t
)
=
Ax
(
t
)
+
Bu
(
t
)
y
(
t
)
=
Cx
(
t
)
Example 8.3.1
Taken a mechanical system of a mass with a spring and a damper as
an example,
y(t)
k
u
m
b
where an external force is the only input
u
is
the output. The physical equation of this single input single output (SISO) system is
(
t
)
to the system, and the position
y
(
t
)
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