Normal Form-Non Linear Control Systems-Lecture Slides, Slides of Nonlinear Control Systems

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Nonlinear Systems and Control

Lecture # 22

Normal Form

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Relative Degree

x

f

x

g

x

u,

y

h

x

where

f

g

, and

h

are sufficiently smooth in a domain

D

f

D

R

n

and

g

D

R

n

are called vector fields on

D

y

∂h ∂x

[

f

x

g

x

u

]

def

L

f

h

x

L

g

h

x

u

L

f

h

x

∂h ∂x

f

x

is the

Lie Derivative

of

h

with respect to

f

or along

f

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L

g

L

f

h

x

y

(2)

L

2 f

h

x

y

(3)

L

3 f

h

x

L

g

L

2 f

h

x

u

L

g

L

i

−

1

f

h

x

i

,... , ρ

L

g

L

ρ

−

1

f

h

x

y

(

ρ

)

L

ρ f

h

x

L

g

L

ρ

−

1

f

h

x

u

Definition:

The system

x

f

x

g

x

u,

y

h

x

has relative degree

ρ

ρ

n

, in

D

0

D

if

x

D

0

L

g

L

i

−

1

f

h

x

i

,... , ρ

L

g

L

ρ

−

1

f

h

x

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Example

x

1

x

2

x

2

x

1

ε

x

2 1

x

2

u,

y

x

1

ε >

y

x

1

x

2

y

x

2

x

1

ε

x

2 1

x

2

u

Relative degree

over

R

2

Example

x

1

x

2

x

2

x

1

ε

x

2 1

x

2

u,

y

x

2

ε >

y

x

2

x

1

ε

x

2 1

x

2

u

Relative degree

over

R

2

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Normal Form

Change of variables:^ z

T

x

φ

1

x

φ

n

−

ρ

x

h

x

L

ρ

−

1

f

h

x

def

φ

x

ψ

x

def

η

ξ

φ

1

to

φ

n

−

ρ

are chosen such that

T

x

is a diffeomorphism

on a domain

D

0

D

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η

∂φ^ ∂x

[

f

x

g

x

u

]

f

0

η, ξ

g

0

η, ξ

u

ξ

i

ξ

i

i

ρ

ξ

ρ

L

ρ f

h

x

L

g

L

ρ

−

1

f

h

x

u

y

ξ

1

Choose

φ

x

such that

T

x

is a diffeomorphism and

∂φ

i

∂x

g

x

for 1

i

n

ρ,

x

D

0

Always possible (at least locally)

η

f

0

η, ξ

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Normal Form:

η

f

0

η, ξ

ξ

i

ξ

i

i

ρ

ξ

ρ

L

ρ f

h

x

L

g

L

ρ

−

1

f

h

x

u

y

ξ

1

A

c

, B

c

C

c

[

]

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η

f

0

η, ξ

ξ

A

c

ξ

B

c

[

L

ρ f

h

x

L

g

L

ρ

−

1

f

h

x

u

]

y

C

c

ξ

γ

x

L

g

L

ρ

−

1

f

h

x

α

x

L

ρ f

h

x

L

g

L

ρ

−

1

f

h

x

ξ

A

c

ξ

B

c

γ

x

)[

u

α

x

)]

If

x

∗

is an open-loop equilibrium point at which

y

; i.e.,

f

x

∗

and

h

x

∗

, then

ψ

x

. Take

φ

x

∗

so that

z

is an open-loop equilibrium point.

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Z

∗

x

D

0

h

x

L

f

h

x

L

ρ

−

1

f

h

x

y

t

x

t

Z

∗

u

u

∗

x

def

α

x

x

∈

Z

∗

The restricted motion of the system is described by

x

f

∗

x

def

= [

f

x

g

x

α

x

)]

x

∈

Z

∗

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Example

x

1

x

2

x

2

x

1

ε

x

2 1

x

2

u,

y

x

2

y

x

2

x

1

ε

x

2 1

x

2

u

ρ

y

t

x

2

t

x

1

Non-minimum phase

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Find

φ

x

such that

φ

∂φ ∂x

g

x

[

∂φ ∂x

1

∂φ ∂x

2

∂φ ∂x

3

]

2+

x

23

1+

x

23

and

T

x

[

φ

x

x

2

x

3

]

T

is a diffeomorphism

∂φ ∂x

1

x

2 3

x

23

∂φ ∂x

3

φ

x

x

1

x

3

• tan

−

1

x

3

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T

x

[

x

1

x

3

• tan

−

1

x

3

x

2

x

3

]

T

is a global diffeomorphism

η

x

1

x

3

• tan

−

1

x

3

ξ

1

x

2

ξ

2

x

3

η

η

ξ

2

• tan

−

1

ξ

2

ξ

(^22)

ξ

(^22)

ξ

2

ξ

1

ξ

2

ξ

2

η

ξ

2

• tan

−

1

ξ

2

ξ

2

u

y

ξ

1

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