Nonlinear Systems and Control
Lecture # 23
Controller Form
– p. 1/18
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Controller Form-Non Linear Systems Control and Analysis-Lecture Slides, Slides of Nonlinear Control Systems
Dr. Javed Iftikhar delivered this lecture at A.P. University of Law for Non Linear Control Systems course. It includes: Controller, Form, Nonsingular, Relative, Degree, Transform, Pair, Matrix, Canonical, Lie, Bracket
Typology: Slides
2011/2012
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Nonlinear Systems and Control
Lecture # 23
Controller Form
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Definition:
A nonlinear system is in the controller form if
x
Ax
Bγ
x
)[
u
α
x
)]
where
A, B
is controllable and
γ
x
is a nonsingular
u
α
x
γ
−
1
x
v
x
Ax
Bv
The
n
-dimensional single-input (SI) system
x
f
x
g
x
u
can be transformed into the controller form if
h
x
s.t.
x
f
x
g
x
u,
y
h
x
has relative degree
n
Why?
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For any controllable pair
A, B
, we can find a nonsingular
matrix
M
that transforms
A, B
into a controllable
canonical form:
M AM
−
1
A
c
B
c
λ
T
M B
B
c
z
M ζ
M S
x
def
T
x
z
A
c
z
B
c
γ
)[
u
α
)]
h
x
T
1
x
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In summary, the
n
-dimensional SI system
x
f
x
g
x
u
is transformable into the controller form if and only if
h
x
such that
x
f
x
g
x
u,
y
h
x
has relative degree
n
Search for a smooth function
h
x
such that
L
g
L
i
−
1
f
h
x
i
,... , n
and
L
g
L
n
−
1
f
h
x
T
x
[
h
x
L
f
h
x
L
n
−
1
f
h
x
]
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Example
f
[
x
2
sin
x
1
x
2
]
g
[
x
1
]
[
f, g
] =
[
] [
x
2
sin
x
1
x
2
]
[
cos
x
1
] [
x
1
]
ad
f
g
= [
f, g
] =
[
x
1
x
1
x
2
]
- p. 7/
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f
[
x
2
sin
x
1
x
2
]
ad
f
g
[
x
1
x
1
x
2
]
ad
2 f
g
= [
f, ad
f
g
] =
[
] [
x
2
sin
x
1
x
2
]
[
cos
x
1
] [
x
1
x
1
x
2
]
[
x
1
x
2
x
1
x
2
sin
x
1
x
1
cos
x
1
]
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Lemma:
If
is a nonsingular distribution, generated by
f
1
f
k
, then it is involutive if and only if
[
f
i
, f
j
]
i, j
k
Example:
D
R
3
∆ = span
f
1
, f
2
f
1
x
2
f
2
x
2
dim(∆(
x
x
D
[
f
1
, f
2
] =
∂f
2
∂x
f
1
∂f
1
∂x
f
2
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rank [
f
1
x
, f
2
x
[
f
1
, f
2
](
x
)] =
rank
x
2
x
2
x
D
is not involutive
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Theorem:
The
n
-dimensional SI system
x
f
x
g
x
u
is transformable into the controller form
if and only if
there is
a domain
D
0
such that
rank[
g
x
, ad
f
g
x
,... , ad
n
−
1
f
g
x
)] =
n,
x
D
0
and
span
g, ad
f
g,... , ad
n
−
2
f
g
is involutive in
D
0
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Example
x
[
a
sin
x
2
x
2 1
]
[
]
u
ad
f
g
= [
f, g
] =
∂f ∂x
g
[
a
cos
x
2
]
[
g
x
, ad
f
g
x
)] =
[
a
cos
x
2
]
rank[
g
x
, ad
f
g
x
)] = 2
x
such that
cos
x
2
span
g
is involutive
Find
h
such that
L
g
h
x
and
L
g
L
f
h
x
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Example (Field-Controlled DC Motor)
x
ax
1
bx
2
k
cx
1
x
3
θx
1
x
2
u
ad
f
g
a
cx
3
θx
2
ad
2 f
g
a
2
a
b
cx
3
b
a
θx
2
θk
[
g
x
, ad
f
g
x
, ad
2 f
g
x
)] =
a
a
2
cx
3
a
b
cx
3
θx
2
b
a
θx
2
θk
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det[
] =
cθ
k
bx
2
x
3
rank
[
] = 3
for
x
2
k/
b
and
x
3
span
g, ad
f
g
is involutive if
[
g, ad
f
g
]
span
g, ad
f
g
[
g, ad
f
g
] =
ad
f
g
∂x
g
c
θ
span
g, ad
f
g
is involutive
D
0
x
R
3
x
2
k 2
b
and
x
3
Find
h
such that
L
g
h
x
L
g
L
f
h
x
L
g
L
2 f
h
x
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