Minicourse on behavioral systems theory Lecture 4: Bilinear and quadratic differential forms

The Behavioral Approach

to

Systems Theory

Paolo Rapisarda, Un. of Southampton, U.K.

&

Jan C. Willems, K.U.Leuven, Belgium

MTNS 2006

Lecture 4: Bilinear and quadratic differential forms

Outline

Motivation and aim

Definition

Two-variable polynomial matrices

Dynamics and functionals in systems and control

Instances:

Lyapunov theory, performance criteria, etc.

Linear case

=

⇒

quadratic

and

bilinear

functionals.

Usually

: state-space equations, constant functionals.

However, tearing and zooming

=

⇒

state space eq.s

¡High-order differential equations!

Dynamics and functionals in systems and control

Instances:

Lyapunov theory, performance criteria, etc.

Linear case

=

⇒

quadratic

and

bilinear

functionals.

Usually

: state-space equations, constant functionals.

However, tearing and zooming

=

⇒

state space eq.s

¡High-order differential equations!

Dynamics and functionals in systems and control

Instances:

Lyapunov theory, performance criteria, etc.

Linear case

=

⇒

quadratic

and

bilinear

functionals.

Usually

: state-space equations, constant functionals.

However, tearing and zooming

=

⇒

state space eq.s

¡High-order differential equations!

Example : a mechanical system

m

1

d

2

_w

1

dt

2

+

k

1

w

1

−

k

2

w

2

=

0

−

k

1

w

1

+

m

2

d

2

_w

2

dt

2

+ (

k

1

+

k

2

)

w

2

=

0

m

1

m

2

d

4

dt

4

w

1

+ (

k

1

m

1

+

k

2

m

1

+

k

1

m

2

)

d

2

dt

2

w

1

+

k

1

k

2

w

1

=

0

Example : a mechanical system

m

1

d

2

_w

1

dt

2

+

k

1

w

1

−

k

2

w

2

=

0

−

k

1

w

1

+

m

2

d

2

_w

2

dt

2

+ (

k

1

+

k

2

)

w

2

=

0

m

1

m

2

d

4

dt

4

w

1

+ (

k

1

m

1

+

k

2

m

1

+

k

1

m

2

)

d

2

dt

2

w

1

+

k

1

k

2

w

1

=

0

Example : a mechanical system

m

1

d

2

_w

1

dt

2

+

k

1

w

1

−

k

2

w

2

=

0

−

k

1

w

1

+

m

2

d

2

_w

2

dt

2

+ (

k

1

+

k

2

)

w

2

=

0

m

1

m

2

d

4

dt

4

w

1

+ (

k

1

m

1

+

k

2

m

1

+

k

1

m

2

)

d

2

dt

2

w

1

+

k

1

k

2

w

1

=

0

Aim

An effective algebraic representation

of bilinear and quadratic functionals

of the system variables and their derivatives:

Operations/properties of functionals

m

algebraic operations/properties of representation

Outline

Motivation and aim

Definition

Two-variable polynomial matrices

Bilinear differential forms (BDFs)

Φ :=

Φ

k,`

∈

R

w

1

×

w

2

_k,`

₌₀

_,...,L

L

Φ

:

C

∞

(

R

,

R

w

1

)

×

C

∞

(

R

,

R

w

2

)

→

C

∞

(

R

,

R

)

L

Φ

(

w

1,

w

2

) :=

w

₁

>

dw

1

dt

>

. . .













Φ

0

,

0

Φ

0

,

1

. . .

Φ

₁

,

0

Φ

1

,

1

. . .

..

.

..

.

· · ·

Φ

k

,

0

Φ

k

,

1

. . .

..

.

..

.

· · ·

















w

2

dw

2

dt

..

.





=

P

k

,`

d

k

dt

k

w1

>

Φk

,`

d

`

dt

`

w2

Quadratic differential forms (QDFs)

Φ :=

Φ

k,`

∈

R

w

×

w

_k,`

₌₀

_,...,L

symmetric, i.e.

Φ

k,`

= Φ

>

_`,k

Q

Φ

:

C

∞

(

R

,

R

w

)

→

C

∞

(

R

,

R

)

Q

Φ

(

w

) :=

w

>

dw

_dt

>

. . .













Φ

0

,

0

Φ

0

,

1

. . .

Φ

₁

,

0

Φ

1

,

1

. . .

..

.

..

.

· · ·

Φ

k

,

0

Φ

k

,

1

. . .

..

.

..

.

· · ·

















w

dw

dt

..

.





=

P

L

k

,`

=

0

d

k

dt

k

w

>

Φk

,`

d

`

dt

`

w

Example: total energy in mechanical system

1

2

"

m

1

_d

dt

w

1

2

+

m

2

_d

dt

w

2

2

#

+

1

2

k

1

w

₁

2

+

k

2

w

₂

2

w

1

w

2

_dt

d

w

1

_dt

d

w

2









1

2

k

1

0

0

0

0

1

₂

k

2

0

0

0

0

1

₂

m

1

0

0

0

0

1

₂

m

2

















w

1

w

2

d

dt

w

1

d

dt

w

2



Outline

Motivation and aim

Definition

Two-variable polynomial matrices

Two-variable polynomial matrices for BDFs

Φ

k,`

∈

R

w

1

×

w

2

_k,`

₌₀

_,...,L

L

Φ

(

w

1,

w

2

) =

L

X

k,`

=0

(

d

k

dt

k

w1)

>

Φk,`

d

`

dt

`

w2

Φ(

ζ, η

) =

P

L

k,`

=0

Φk,`

ζ

k

_η

`

L

Φ

(

w

1,

w

2

) =

L

X

k,`

=0

(

d

k

dt

k

w1)

>

Φk,`

d

`

dt

`

w2

Φ(

ζ, η

) =

P

L

k,`

=0

Φk,`

ζ

k

_η

`

Two-variable polynomial matrices for BDFs

Φ

k,`

∈

R

w

1

×

w

2

_k,`

₌₀

_,...,L

L

Φ

(

w

1,

w

2

) =

L

X

k,`

=0

(

d

k

dt

k

w1)

>

Φk,`

d

`

dt

`

w2

Φ(

ζ, η

) =

P

L

k,`

=0

Φk,`

ζ

k

_η

`

L

Φ

(

w

1,

w

2

) =

L

X

k,`

=0

(

d

k

dt

k

w1)

>

Φk,`

d

`

dt

`

w2

Φ(

ζ, η

) =

P

L

k,`

=0

Φk,`

ζ

k

_η

`

Two-variable polynomial matrices for BDFs

Φ

k,`

∈

R

w

1

×

w

2

_k,`

₌₀

_,...,L

L

Φ

(

w

1,

w

2

) =

L

X

k,`

=0

(

d

k

dt

k

w1)

>

Φk,`

d

`

dt

`

w2

Φ(

ζ, η

) =

P

L

k,`

=0

Φk,`

ζ

k

_η

`

L

Φ

(

w

1,

w

2

) =

L

X

k,`

=0

(

d

k

dt

k

w1)

>

Φk,`

d

`

dt

`

w2

Φ(

ζ, η

) =

P

L

k,`

=0

Φk,`

ζ

k

_η

`

Two-variable polynomial matrices for QDFs

Φk,`

∈

_R

w

×

w

k,`

=0

,...,L

symmetric

(Φk,`

= Φ

>

`,k

)

Q

Φ

(

w

) =

L

X

k,`

=0

(

d

k

dt

k

w

)

>

Φ

k,`

d

`

dt

`

w

Φ(

ζ, η

) =

P

L

k,`

=0

Φ

k,`

ζ

k

_η

`

Example: total energy in mechanical system

QE

(

w

1

,

w

2

) =

w

1

w

2

_dt

d

w

1

_dt

d

w

2









1

2

k

1

0

0

0

0

1

₂

k

2

0

0

0

0

1

₂

m

1

0

0

0

0

1

₂

m

2

















w

1

w

2

d

dt

w

1

d

dt

w

2









E

(

ζ, η

) =

1

2

k

1

0

0

1

₂

k

2

+

1

2

ζη

0

0

1

₂

ζη

Historical intermezzo

stability tests (‘60s)

path integrals (‘60s)

Lyapunov functionals (‘80s)

Historical intermezzo

stability tests (‘60s)

path integrals (‘60s)

Lyapunov functionals (‘80s)

Historical intermezzo

stability tests (‘60s)

path integrals (‘60s)

Lyapunov functionals (‘80s)

Historical intermezzo

stability tests (‘60s)

path integrals (‘60s)

Lyapunov functionals (‘80s)

Historical intermezzo

stability tests (‘60s)

path integrals (‘60s)

Lyapunov functionals (‘80s)

Outline

Motivation and aim

Definition

Two-variable polynomial matrices

The calculus of B/QDFs

Using powers of

ζ

and

η

as placeholders,

B/QDF

!

two-variable polynomial matrix

Operations

and properties

of B/QDF

!

algebraic

The calculus of B/QDFs

Using powers of

ζ

and

η

as placeholders,

B/QDF

!

two-variable polynomial matrix

Operations

and properties

of B/QDF

!

algebraic

Differentiation

Φ

∈

_R

w

×

w

s

[

ζ, η

]

.

•

Φ

derivative

of

Q

Φ

:

Q

•

Φ

:

C

∞

(

_R

,

_R

w

)

→

C

∞

(

_R

,

_R

)

Q

•

Φ

(

w

) :=

d

dt

(

Q

Φ

(

w

))

•

Φ(

ζ, η

) = (

ζ

+

η

)Φ(

ζ, η

)

Differentiation

Φ

∈

_R

w

×

w

s

[

ζ, η

]

.

•

Φ

derivative

of

Q

Φ

:

Q

•

Φ

:

C

∞

(

_R

,

_R

w

)

→

C

∞

(

_R

,

_R

)

Q

•

Φ

(

w

) :=

d

dt

(

Q

Φ

(

w

))

•

Φ(

ζ, η

) = (

ζ

+

η

)Φ(

ζ, η

)

Integration

D(

_R

,

_R

•

)

C

∞

-compact-support trajectories

L

Φ

:

D(

R

,

R

w

1

)

×

D(

R

,

R

w

2

)

→

D(

R

,

R

)

R

L

Φ

:

D(

R

,

R

w

1

)

×

D(

R

,

R

w

2

)

→

R

R

L

Φ

(

w

1

,

w

2

) :=

R

+

∞

−∞

L

Φ

(

w

1

,

w

2

)

dt

Outline

Lyapunov theory

Dissipativity theory

Nonnegativity and positivity along a behavior

Q

Φ

B

≥

0 if

Q

Φ

(

w

)

≥

0

∀

w

∈

B

Q

Φ

B

>

0 if

Q

Φ

B

≥

0, and

[

Q

Φ

(

w

) =

0

] =

⇒

[

w

=

0

]

Prop.

: Let

B

=

ker

_R

(

_dt

d

)

. Then

Q

Φ

B

≥

0 iff there exist

D

∈

_R

•×

w

[

ξ

]

,

X

∈

_R

•×

w

[

ζ, η

]

such that

Φ(

ζ, η

) =

D

(

ζ

)

>

D

(

η

)

|

{z

}

≥

0

for all

w

+

R

(

ζ

)

>

X

(

ζ, η

) +

X

(

η, ζ

)

>

R

(

η

)

|

{z

}

Nonnegativity and positivity along a behavior

Q

Φ

B

≥

0 if

Q

Φ

(

w

)

≥

0

∀

w

∈

B

Q

Φ

B

>

0 if

Q

Φ

B

≥

0, and

[

Q

Φ

(

w

) =

0

] =

⇒

[

w

=

0

]

Prop.

: Let

B

=

ker

_R

(

_dt

d

)

. Then

Q

Φ

B

≥

0 iff there exist

D

∈

_R

•×

w

[

ξ

]

,

X

∈

_R

•×

w

[

ζ, η

]

such that

Φ(

ζ, η

) =

D

(

ζ

)

>

D

(

η

)

|

{z

}

≥

0

for all

w

+

R

(

ζ

)

>

X

(

ζ, η

) +

X

(

η, ζ

)

>

R

(

η

)

|

{z

}

Nonnegativity and positivity along a behavior

Q

Φ

B

≥

0 if

Q

Φ

(

w

)

≥

0

∀

w

∈

B

Q

Φ

B

>

0 if

Q

Φ

B

≥

0, and

[

Q

Φ

(

w

) =

0

] =

⇒

[

w

=

0

]

Prop.

: Let

B

=

ker

_R

(

_dt

d

)

. Then

Q

Φ

B

≥

0 iff there exist

D

∈

_R

•×

w

[

ξ

]

,

X

∈

_R

•×

w

[

ζ, η

]

such that

Φ(

ζ, η

) =

D

(

ζ

)

>

D

(

η

)

|

{z

}

≥

0

for all

w

+

R

(

ζ

)

>

X

(

ζ, η

) +

X

(

η, ζ

)

>

R

(

η

)

|

{z

}

Lyapunov theory

B

autonomous is

asymptotically stable

if lim

t

→∞

w

(

t

) =

0

∀

w

∈

B

B

=

ker

_R

(

_dt

d

)

stable

⇐⇒

det

(

R

)

Hurwitz

Theorem

:

B

asymptotically stable iff

exists

Q

Φ

such that

Q

Φ

B

≥

0 and

Q

•

Φ

B

Lyapunov theory

B

autonomous is

asymptotically stable

if lim

t

→∞

w

(

t

) =

0

∀

w

∈

B

B

=

ker

_R

(

_dt

d

)

stable

⇐⇒

det

(

R

)

Hurwitz

Theorem

:

B

asymptotically stable iff

exists

Q

Φ

such that

Q

Φ

B

≥

0 and

Q

•

Φ

B

Example

B

=

ker

d

2

dt

2

+

3

d

dt

+

2

r

(

ξ

) =

ξ

2

+

3

ξ

+

2

Choose

Ψ(

ζ, η

)

s.t.

Q

Ψ

B

<

0, e.g.

Ψ(

ζ, η

) =

−

ζη

;

Find

Φ(

ζ, η

)

s.t.

_dt

d

Q

Φ

(

w

) =

Q

Ψ

(

w

)

for all

w

∈

B

:

(

ζ

+

η

)Φ(

ζ, η

) = Ψ(

ζ, η

) +

r

(

ζ

)

x

(

η

) +

x

(

ζ

)

r

(

η

)

|

{z

}

=0

on

B

Φ(

ζ, η

) =

−

ζη

+ (

ζ

2

₊

₃

_ζ

₊

₂

₎

1

6

η

+

1

6

ζ

(

η

2

₊

₃

_η

₊

₂

₎

ζ

+

η

=

1

Example

B

=

ker

d

2

dt

2

+

3

d

dt

+

2

r

(

ξ

) =

ξ

2

+

3

ξ

+

2

Choose

Ψ(

ζ, η

)

s.t.

Q

Ψ

B

<

0, e.g.

Ψ(

ζ, η

) =

−

ζη

;

Find

Φ(

ζ, η

)

s.t.

_dt

d

Q

Φ

(

w

) =

Q

Ψ

(

w

)

for all

w

∈

B

:

(

ζ

+

η

)Φ(

ζ, η

) = Ψ(

ζ, η

) +

r

(

ζ

)

x

(

η

) +

x

(

ζ

)

r

(

η

)

|

{z

}

=0

on

B

Φ(

ζ, η

) =

−

ζη

+ (

ζ

2

₊

₃

_ζ

₊

₂

₎

1

6

η

+

1

6

ζ

(

η

2

₊

₃

_η

₊

₂

₎

ζ

+

η

=

1

Example

B

=

ker

d

2

dt

2

+

3

d

dt

+

2

r

(

ξ

) =

ξ

2

+

3

ξ

+

2

Choose

Ψ(

ζ, η

)

s.t.

Q

Ψ

B

<

0, e.g.

Ψ(

ζ, η

) =

−

ζη

;

Find

Φ(

ζ, η

)

s.t.

_dt

d

Q

Φ

(

w

) =

Q

Ψ

(

w

)

for all

w

∈

B

:

(

ζ

+

η

)Φ(

ζ, η

) = Ψ(

ζ, η

) +

r

(

ζ

)

x

(

η

) +

x

(

ζ

)

r

(

η

)

|

{z

}

=0

on

B

Φ(

ζ, η

) =

−

ζη

+ (

ζ

2

₊

₃

_ζ

₊

₂

₎

1

6

η

+

1

6

ζ

(

η

2

₊

₃

_η

₊

₂

₎

ζ

+

η

=

1

Example

B

=

ker

d

2

dt

2

+

3

d

dt

+

2

r

(

ξ

) =

ξ

2

+

3

ξ

+

2

Choose

Ψ(

ζ, η

)

s.t.

Q

Ψ

B

<

0, e.g.

Ψ(

ζ, η

) =

−

ζη

;

Find

Φ(

ζ, η

)

s.t.

_dt

d

Q

Φ

(

w

) =

Q

Ψ

(

w

)

for all

w

∈

B

:

(

ζ

+

η

)Φ(

ζ, η

) = Ψ(

ζ, η

) +

r

(

ζ

)

x

(

η

) +

x

(

ζ

)

r

(

η

)

|

{z

}

=0

on

B

d

dt

Q

Φ

(

w

) =

Q

Ψ

(

w

)

for all

w

∈

B

Φ(

ζ, η

) =

−

ζη

+ (

ζ

2

₊

₃

_ζ

₊

₂

₎

1

6

η

+

1

6

ζ

(

η

2

₊

₃

_η

₊

₂

₎

ζ

+

η

=

1

Example

B

=

ker

d

2

dt

2

+

3

d

dt

+

2

r

(

ξ

) =

ξ

2

+

3

ξ

+

2

Choose

Ψ(

ζ, η

)

s.t.

Q

Ψ

B

<

0, e.g.

Ψ(

ζ, η

) =

−

ζη

;

Find

Φ(

ζ, η

)

s.t.

_dt

d

Q

Φ

(

w

) =

Q

Ψ

(

w

)

for all

w

∈

B

:

(

ζ

+

η

)Φ(

ζ, η

) = Ψ(

ζ, η

) +

r

(

ζ

)

x

(

η

) +

x

(

ζ

)

r

(

η

)

|

{z

}

=0

on

B

Equivalent to solving

polynomial Lyapunov equation

0

= Ψ(

−

ξ, ξ

)

ξ

2

+

r

(

−

ξ

)

ξ

2

₋

₃

_ξ

₊₂

x

(

ξ

) +

x

(

−

ξ

)

r

(

ξ

)

ξ

2

₊₃

_ξ

₊₂

;

x

(

ξ

) =

1

₆

ξ

Φ(

ζ, η

) =

−

ζη

+ (

ζ

2

₊

₃

_ζ

₊

₂

₎

1

6

η

+

1

6

ζ

(

η

2

₊

₃

_η

₊

₂

₎

ζ

+

η

=

1

Example

B

=

ker

d

2

dt

2

+

3

d

dt

+

2

r

(

ξ

) =

ξ

2

+

3

ξ

+

2

Choose

Ψ(

ζ, η

)

s.t.

Q

Ψ

B

<

0, e.g.

Ψ(

ζ, η

) =

−

ζη

;

Find

Φ(

ζ, η

)

s.t.

_dt

d

Q

Φ

(

w

) =

Q

Ψ

(

w

)

for all

w

∈

B

:

(

ζ

+

η

)Φ(

ζ, η

) = Ψ(

ζ, η

) +

r

(

ζ

)

x

(

η

) +

x

(

ζ

)

r

(

η

)

|

{z

}

=0

on

B

Φ(

ζ, η

) =

−

ζη

+ (

ζ

2

₊

₃

_ζ

₊

₂

₎

1

6

η

+

1

6

ζ

(

η

2

₊

₃

_η

₊

₂

₎

ζ

+

η

=

1

State-space case

_d

dt

I

x

−

A

x

=

0

;

R

(

ξ

) =

ξ

I

x

−

A

•

Choose

Q

<

0;

•

Solve polynomial Lyapunov equation

(

ξ

I

x

−

A

)

>

_P

+

P

(

ξ

I

x

−

A

) =

−A

>

_P

₋

_PA

=

Q

equivalent with

matrix

Lyapunov equation!

•

Lyapunov functional is

Outline

Lyapunov theory

Dissipativity theory

Dissipativity theory

supply

SYSTEM

Power is

supplied

;

energy is

stored

RLC circuits

Power

V

>

I

Storage in capacitors and inductors

Mechanical system

Power

F

>

v

+ (

_dt

d

ϑ

)

>

T

Setting the stage

Controllable system

w

=

M

(

_dt

d

)

`

;

M

(

ξ

)

Power (‘supply rate’)

Q

Φ

(

w

)

;

Φ(

ζ, η

)

Q

Φ

(

w

) =

Q

Φ

(

M

(

_dt

d

)

`

)

Φ

0

(

ζ, η

) :=

M

(

ζ

)

>

Φ(

ζ, η

)

M

(

η

)

Setting the stage

Controllable system

w

=

M

(

_dt

d

)

`

;

M

(

ξ

)

Power (‘supply rate’)

Q

Φ

(

w

)

;

Φ(

ζ, η

)

Q

Φ

(

w

) =

Q

Φ

(

M

(

_dt

d

)

`

)

Φ

0

(

ζ, η

) :=

M

(

ζ

)

>

Φ(

ζ, η

)

M

(

η

)

Setting the stage

Controllable system

w

=

M

(

_dt

d

)

`

;

M

(

ξ

)

Power (‘supply rate’)

Q

Φ

(

w

)

;

Φ(

ζ, η

)

Q

Φ

(

w

) =

Q

Φ

(

M

(

_dt

d

)

`

)

Φ

0

(

ζ, η

) :=

M

(

ζ

)

>

Φ(

ζ, η

)

M

(

η

)

Dissipation inequality

Q

Ψ

is

storage function

for the supply

Q

Φ

if

d

dt

Q

Ψ

≤

Q

Φ

Rate of storage increase

≤

supply

Q

∆

is

dissipation function

for

Q

Φ

if

Q

∆

≥

0 and

R

Q

∆

dt

=

R

Q

Φ

dt

DISSIPATION

SUPPLY

Dissipation inequality

Q

Ψ

is

storage function

for the supply

Q

Φ

if

d

dt

Q

Ψ

≤

Q

Φ

Rate of storage increase

≤

supply

Q

∆

is

dissipation function

for

Q

Φ

if

Q

∆

≥

0 and

R

Q

∆

dt

=

R

Q

Φ

dt

DISSIPATION

SUPPLY

Dissipation inequality

Q

Ψ

is

storage function

for the supply

Q

Φ

if

d

dt

Q

Ψ

≤

Q

Φ

Rate of storage increase

≤

supply

Q

∆

is

dissipation function

for

Q

Φ

if

Q

∆

≥

0 and

R

Q

∆

dt

=

R

Q

Φ

dt

_DISSIPATION

SUPPLY

Characterizations of dissipativity

Theorem:

The following conditions are equivalent:

•

R

_−∞

+

∞

Q

Φ

(

`

)

dt

≥

0 for all

C

∞

compact-support

`

;

•

Q

Φ

admits a storage function;

•

Q

Φ

admits a dissipation function

Also, storage and dissipation functions are one-one:

d

dt

Q

Ψ

=

Q

Φ

−

Q

∆

Example: mechanical systems

M

_dt

d

2

2

q

+

D

d

dt

q

+

Kq

=

F

F

q

=

M

_dt

d

2

2

+

D

d

dt

+

K

I

3

`

Φ(

ζ, η

) =

1

₂

(

M

ζ

2

₊

_D

_ζ

₊

_K

₎

>

_η

₊

1

2

ζ

(

M

η

2

₊

_D

_η

₊

_K

₎

∆(

ζ, η

) =

1

₂

(

D

>

₊

_D

₎

_ζη

Storage function

Ψ(

ζ, η

) =

Φ(

ζ, η

)

−

∆(

ζ, η

)

ζ

+

η

=

1

2

M

ζη

+

1

2

K

Example: mechanical systems

M

_dt

d

2

2

q

+

D

d

dt

q

+

Kq

=

F

F

q

=

M

_dt

d

2

2

+

D

d

dt

+

K

I

3

`

Supply rate: power

F

>

_d

dt

q

=

M

d

2

dt

2

`

+

D

d

dt

`

+

K

`

>

_d

dt

`

corresponding to

Φ(

ζ, η

) =

1

2

(

M

ζ

2

₊

_D

_ζ

₊

_K

₎

>

_η

₊

1

2

ζ

(

M

η

2

₊

_D

_η

₊

_K

₎

Φ(

ζ, η

) =

1

₂

(

M

ζ

2

+

D

ζ

+

K

)

>

η

+

1

₂

ζ

(

M

η

2

+

D

η

+

K

)

∆(

ζ, η

) =

1

₂

(

D

>

+

D

)

ζη

Storage function

Ψ(

ζ, η

) =

Φ(

ζ, η

)

−

∆(

ζ, η

)

ζ

+

η

=

1

2

M

ζη

+

1

2

K

Example: mechanical systems

M

_dt

d

2

2

q

+

D

d

dt

q

+

Kq

=

F

F

q

=

M

_dt

d

2

2

+

D

d

dt

+

K

I

3

`

Φ(

ζ, η

) =

1

₂

(

M

ζ

2

₊

_D

_ζ

₊

_K

₎

>

_η

₊

1

2

ζ

(

M

η

2

₊

_D

_η

₊

_K

₎

∆(

ζ, η

) =

1

₂

(

D

>

₊

_D

₎

_ζη

Storage function

Ψ(

ζ, η

) =

Φ(

ζ, η

)

−

∆(

ζ, η

)

ζ

+

η

=

1

2

M

ζη

+

1

2

K

Example: mechanical systems

M

_dt

d

2

2

q

+

D

d

dt

q

+

Kq

=

F

F

q

=

M

_dt

d

2

2

+

D

d

dt

+

K

I

3

`

Φ(

ζ, η

) =

1

₂

(

M

ζ

2

₊

_D

_ζ

₊

_K

₎

>

_η

₊

1

2

ζ

(

M

η

2

₊

_D

_η

₊

_K

₎

If dissipation inequality

Φ(

ζ, η

) = (

ζ

+

η

)Ψ(

ζ, η

) + ∆(

ζ, η

)

holds, then

Φ(

−

ξ, ξ

) =

−

1

2

ξ

2

(

D

>

+

D

) = ∆(

−

ξ, ξ

)

=

⇒

∆(

ζ, η

) =

1

2

(

D

>

+

D

)

ζη

Spectral factorization

of

Φ(

−

ξ, ξ

)

is key

∆(

ζ, η

) =

1

₂

(

D

>

+

D

)

ζη

Storage function

Ψ(

ζ, η

) =

Φ(

ζ, η

)

−

∆(

ζ, η

)

ζ

+

η

=

1

2

M

ζη

+

1

2

K

Example: mechanical systems

M

_dt

d

2

2

q

+

D

d

dt

q

+

Kq

=

F

F

q

=

M

_dt

d

2

2

+

D

d

dt

+

K

I

3

`

Φ(

ζ, η

) =

1

₂

(

M

ζ

2

₊

_D

_ζ

₊

_K

₎

>

_η

₊

1

2

ζ

(

M

η

2

₊

_D

_η

₊

_K

₎

∆(

ζ, η

) =

1

₂

(

D

>

₊

_D

₎

_ζη

Storage function

Ψ(

ζ, η

) =

Φ(

ζ, η

)

−

∆(

ζ, η

)

ζ

+

η

=

1

2

M

ζη

+

1

2

K

Example: mechanical systems

M

_dt

d

2

2

q

+

D

d

dt

q

+

Kq

=

F

F

q

=

M

_dt

d

2

2

+

D

d

dt

+

K

I

3

`

Φ(

ζ, η

) =

1

₂

(

M

ζ

2

₊

_D

_ζ

₊

_K

₎

>

_η

₊

1

2

ζ

(

M

η

2

₊

_D

_η

₊

_K

₎

∆(

ζ, η

) =

1

₂

(

D

>

₊

_D

₎

_ζη

Storage function

Ψ(

ζ, η

) =

Φ(

ζ, η

)

−

∆(

ζ, η

)

ζ

+

η

=

1

2

M

ζη

+

1

2

K

Outline

Lyapunov theory

Dissipativity theory