Passivity Based Control-Non Linear Control Systems-Lecture Slides, Slides of Nonlinear Control Systems

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Nonlinear Systems and Control

Lecture # 29Stabilization

Passivity-Based Control

• p. 1/

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x

f

x, u

y

h

x

f

u

T

y

V

∂V

∂x

f

x, u

Theorem 14.4:

If the system is

(1)

passive with a radially unbounded positive definite

storage function and

(2)

zero-state observable,

then the origin can be globally stabilized by

u

φ

y

φ

y

T

φ

y

y

• p. 2/

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Choice of Output

x

f

x

G

x

u,

∂V

∂x

f

x

x

No output is defined.

Choose the output as

y

h

x

def

[

∂V

∂x

G

x

]

T

V

∂V

∂x

f

x

∂V

∂x

G

x

u

y

T

u

Check zero-state observability

• p. 4/

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Example

x

1

x

2

x

2

x

3 1

u

V

x

1 4

x

4 1

1 2

x

2 2

With

u

V

x

3 1

x

2

x

2

x

3 1

Take

y

∂V

∂x

G

∂V ∂x

2

x

2

Is it zero-state observable?

with

u

y

t

x

t

u

kx

2

or

u

k/π

) tan

−

1

x

2

k >

• p. 5/

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Theorem [31]:

The system

x

f

x

G

x

u,

y

h

x

is locally equivalent to a passive system (with a positivedefinite storage function) if it has relative degree one at x

and the zero dynamics have a stable equilibrium

point at the origin with a positive definite Lyapunov function Example:

m

-link Robot Manipulator

M

q

q

C

q,

q

q

D

q

g

q

u

M

M

T

M

C

T

M

C

, D

D

T

• p. 7/

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Stabilize the system at

q

q

r

e

q

q

r

e

q

M

q

e

C

q,

q

e

D

e

g

q

u

e

e

is not an open-loop equilibrium point

u

g

q

φ

p

e

v,

[

φ

p

, e

T

φ

p

e

e

= 0]

M

q

e

C

q,

q

e

D

e

φ

p

e

v

V

1 2

e

T

M

q

e

e

0

φ

T p

σ

dσ

V

1 2

e

T

M

C

e

e

T

D

e

e

T

φ

p

e

e

T

v

φ

T p

e

e

e

T

v

y

e

• p. 8/

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How does passivity-based control compare with feedbacklinearization? Example 13.

x

1

x

2

x

2

h

x

1

u

h

x

1

h

x

1

x

1

Feedback linearization:

u

h

x

1

k

1

x

1

k

2

x

2

x

[

k

1

k

2

]

x

• p. 10/

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Passivity-based control:

V

x

1

0

h

z

dz

1 2

x

2 2

V

x

2

h

x

1

x

2

h

x

1

x

2

u

x

2

u

Take

y

x

2

With

u

y

t

h

x

1

t

x

1

t

u

σ

x

2

[

σ

, yσ

y

y

= 0]

x

1

x

2

x

2

h

x

1

σ

x

2

• p. 11/

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Cascade Connection:

z

f

a

z

F

z, y

y,

x

f

x

G

x

u,

y

h

x

f

a

f

h

∂V

∂x

f

x

∂V

∂x

G

x

u

y

T

u

∂W

∂z

f

a

z

U

z, x

W

z

V

x

U

∂W

∂z

F

z, y

y

y

T

u

y

T

[

u

∂W

∂z

F

z, y

T

]

• p. 13/

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u

∂W

∂z

F

z, y

T

v

U

y

T

v

The system

z

f

a

z

F

z, y

y

x

f

x

G

x

∂W

∂z

F

z, y

T

G

x

v

y

h

x

with input

v

and output

y

is passive with

U

as the storage

function Read Examples 14.17 and 14.

• p. 14/

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