Robust Stabilization of Nonlinear Systems: Sliding Mode Control, Slides of Nonlinear Control Systems

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Nonlinear Systems and Control

Lecture # 33

Robust Stabilization

Sliding Mode Control

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Regular Form:

η

f

a

η, ξ

ξ

f

b

η, ξ

g

η, ξ

u

δ

t, η, ξ, u

η

R

n

−

1

, ξ

R, u

R

f

a

, f

b

, g

η, ξ

g

0

Sliding Manifold:

s

ξ

φ

η

φ

s

t

η

f

a

η, φ

η

Design

φ

s.t. the origin of

η

f

a

η, φ

η

is asymp. stable

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t, η, ξ, v

g

η, ξ

η, ξ

κ

0

v

η, ξ

κ

0

Known

s

s

sgv

s

sgv

s

s

s

g

[

sv

s

κ

0

v

)]

v

β

η, ξ

) sgn(

s

β

η, ξ

η, ξ

κ

0

β

0

β

0

s

s

g

[

β

s

s

κ

0

β

s

] =

g

[

β

κ

0

s

s

]

s

s

g

[

s

κ

0

β

0

s

s

]

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s

s

g

η, ξ

κ

0

β

0

s

g

0

β

0

κ

0

s

v

β

x

) sat

s ε

ε >

s

s

g

0

β

0

κ

0

s

for

s

ε

The trajectory reaches the boundary layer

s

ε

in finite

time and remains inside thereafterStudy the behavior of

η

η

f

a

η, φ

η

s

What do we know about this system and what do we need?

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α

(.)

α

(

ε

)

α

c (c)

0

ε

c

V

|s|

V

η

c

0

} × {|

s

c

with

c

0

α

c

is positively invariant and all trajectories starting in

reach

ε

V

η

α

ε

} × {|

s

ε

in finite time

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Theorem 14.1:

Suppose all the assumptions hold over

Then, for all

η

, ξ

, the trajectory

η

t

, ξ

t

is

bounded for all

t

and reaches the positively invariant

set

ε

in finite time.

If the assumptions hold globally and

V

η

is radially unbounded, the foregoing conclusion holds

for any initial state Theorem 14.2:

Suppose all the assumptions hold over

κ

0

The origin of

η

f

a

η, φ

η

is exponentially stale

Then there exits

ε

∗

such that for all

< ε < ε

∗

, the

origin of the closed-loop system is exponentially stable and Ω

is a subset of its region of attraction.

If the assumptions

hold globally, the origin will be globally uniformlyasymptotically stable

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x

θ

2

x

2 2

kθ

1

x

1

sin

x

2

x

ak

x

1

bx

2 2

β

x

ak

x

1

bx

2 2

β

0

β

0

u

x

1

kx

2

β

x

) sgn(

s

Will

u

x

1

kx

2

β

x

) sat

s ε

stabilize the origin?

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Example: 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

View

ξ

ρ

as input to the system

η

f

0

η, ξ

1

, ξ

ρ

−

1

, ξ

ρ

ξ

i

ξ

i

i

ρ

ξ

ρ

−

1

ξ

ρ

Design

ξ

ρ

φ

η, ξ

1

, ξ

ρ

−

1

to stabilize the origin

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Multi-Input Systems

η

f

a

η, ξ

ξ

f

b

η, ξ

G

η, ξ

E

η, ξ

u

δ

t, η, ξ, u

η

R

n

−

p

, ξ

R

p

, u

R

p

f

a

, f

b

det(

G

det(

E

G

= diag[

g

1

, g

2

, g

m

]

g

i

η, ξ

g

0

Design

φ

s.t. the origin of

η

f

a

η, φ

η

is asymp. stable

s

ξ

φ

η

s

f

b

η, ξ

∂φ^ ∂η

f

a

η, ξ

G

η, ξ

E

η, ξ

u

δ

t, η, ξ, u

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s

f

b

η, ξ

∂φ^ ∂η

f

a

η, ξ

G

η, ξ

E

η, ξ

u

δ

t, η, ξ, u

u

E

−

1

L

[

f

b

∂φ^ ∂η

f

a

]

v

L

G

−

1

or

L

s

i

g

i

η, ξ

v

i

i

t, η, ξ, v

i

p

i

t, η, ξ, v

g

i

η, ξ

η, ξ

κ

0

max 1

≤

i

≤

p

v

i

i

p̺

η, ξ

κ

0

Known

β

x

x

κ

0

β

0

β

0

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