Frame 1 of 17

A brief overview of the state/signal approachto infinite-dimensional systems theory

Mikael Kurula

http://users.abo.fi/mkurula/

OSE Seminar – 3rd November 2010

Mikael Kurula A brief overview of the state/signal approach

Frame 2 of 17

Approximate contents

1 What are state/signal systems in discrete time?

2 Some examples of continuous-time systems

3 Well-posed, passive and conservative continuous s/s systems

4 My current research interests (if time permits)

5 Conclusions and references

The theory of state/signal systemshas chiefly been developed by Arovand Staffans in a series of approx. 10(long!) journal papers, publishedstarting from 2005.

Mikael Kurula A brief overview of the state/signal approach

Frame 2 of 17

Approximate contents

1 What are state/signal systems in discrete time?

2 Some examples of continuous-time systems

3 Well-posed, passive and conservative continuous s/s systems

4 My current research interests (if time permits)

5 Conclusions and references

The theory of state/signal systemshas chiefly been developed by Arovand Staffans in a series of approx. 10(long!) journal papers, publishedstarting from 2005.

Mikael Kurula A brief overview of the state/signal approach

Frame 2 of 17

Approximate contents

1 What are state/signal systems in discrete time?

2 Some examples of continuous-time systems

3 Well-posed, passive and conservative continuous s/s systems

4 My current research interests (if time permits)

5 Conclusions and references

The theory of state/signal systemshas chiefly been developed by Arovand Staffans in a series of approx. 10(long!) journal papers, publishedstarting from 2005.

Mikael Kurula A brief overview of the state/signal approach

Frame 2 of 17

Approximate contents

1 What are state/signal systems in discrete time?

2 Some examples of continuous-time systems

3 Well-posed, passive and conservative continuous s/s systems

4 My current research interests (if time permits)

5 Conclusions and references

The theory of state/signal systemshas chiefly been developed by Arovand Staffans in a series of approx. 10(long!) journal papers, publishedstarting from 2005.

Mikael Kurula A brief overview of the state/signal approach

Frame 3 of 17

What are state/signal systems? [AS1]

The classic state-space model of a discrete-time-invariant systemwith input u, state x , and output y is

Σi/s/o :[x(n+1)y(n)

]=[A BC D

] [ x(n)u(n)

], n ∈ Z+, (1)

where[A BC D

]is bounded on the Hilbert spaces

[ XU]→[ XY].

This can be turned into a state/signal system by settingw(n) := u(n) u y(n) and writing (1) equivalently as

Σs/s :

[x(n+1)x(n)w(n)

]∈[

Ax ′+Bu′

x ′

Cx ′+Du′uu′

] ∣∣ [ x ′u′

]∈[ XU]

=: V , n ∈ Z+.

Main idea: make minimal distinction between input u and output y .

Useful for unifying i/s/o theory [S1] and systems interconnection!

Mikael Kurula A brief overview of the state/signal approach

Frame 3 of 17

What are state/signal systems? [AS1]

The classic state-space model of a discrete-time-invariant systemwith input u, state x , and output y is

Σi/s/o :[x(n+1)y(n)

]=[A BC D

] [ x(n)u(n)

], n ∈ Z+, (1)

where[A BC D

]is bounded on the Hilbert spaces

[ XU]→[ XY].

This can be turned into a state/signal system by settingw(n) := u(n) u y(n) and writing (1) equivalently as

Σs/s :

[x(n+1)x(n)w(n)

]∈[

Ax ′+Bu′

x ′

Cx ′+Du′uu′

] ∣∣ [ x ′u′

]∈[ XU]

=: V , n ∈ Z+.

Main idea: make minimal distinction between input u and output y .

Useful for unifying i/s/o theory [S1] and systems interconnection!

Mikael Kurula A brief overview of the state/signal approach

Frame 3 of 17

What are state/signal systems? [AS1]

The classic state-space model of a discrete-time-invariant systemwith input u, state x , and output y is

Σi/s/o :[x(n+1)y(n)

]=[A BC D

] [ x(n)u(n)

], n ∈ Z+, (1)

where[A BC D

]is bounded on the Hilbert spaces

[ XU]→[ XY].

This can be turned into a state/signal system by settingw(n) := u(n) u y(n) and writing (1) equivalently as

Σs/s :

[x(n+1)x(n)w(n)

]∈[

Ax ′+Bu′

x ′

Cx ′+Du′uu′

] ∣∣ [ x ′u′

]∈[ XU]

=: V , n ∈ Z+.

Main idea: make minimal distinction between input u and output y .

Useful for unifying i/s/o theory [S1] and systems interconnection!

Mikael Kurula A brief overview of the state/signal approach

Frame 4 of 17

What are state/signal systems? [AS1] (cont.)

The subspace V =[A B1 0C Du1

] [ XU]

has the following properties:

1 V is closed in[ XXW

], where W = U u Y

2 If[z00

]∈ V then z = 0

3 The set D :=

[ xw ] |

[zxw

]∈ V

is closed in

[ XW]

4 For every x ∈ X there are some x and w such that[

zxw

]∈ V

Definition

We call any subspace V ⊂[ XXW

], where X ,W are Hilbert spaces,

with properties (1) – (4) a discrete-time state/signal system.

The sequence[

x(n)w(n)

]is a trajectory of V if

[x(n+1)x(w)w(n)

]∈ V , n ∈ Z+.

Mikael Kurula A brief overview of the state/signal approach

Frame 4 of 17

What are state/signal systems? [AS1] (cont.)

The subspace V =[A B1 0C Du1

] [ XU]

has the following properties:

1 V is closed in[ XXW

], where W = U u Y

2 If[z00

]∈ V then z = 0

3 The set D :=

[ xw ] |

[zxw

]∈ V

is closed in

[ XW]

4 For every x ∈ X there are some x and w such that[

zxw

]∈ V

Definition

We call any subspace V ⊂[ XXW

], where X ,W are Hilbert spaces,

with properties (1) – (4) a discrete-time state/signal system.

The sequence[

x(n)w(n)

]is a trajectory of V if

[x(n+1)x(w)w(n)

]∈ V , n ∈ Z+.

Mikael Kurula A brief overview of the state/signal approach

Frame 4 of 17

What are state/signal systems? [AS1] (cont.)

The subspace V =[A B1 0C Du1

] [ XU]

has the following properties:

1 V is closed in[ XXW

], where W = U u Y

2 If[z00

]∈ V then z = 0

3 The set D :=

[ xw ] |

[zxw

]∈ V

is closed in

[ XW]

4 For every x ∈ X there are some x and w such that[

zxw

]∈ V

Definition

We call any subspace V ⊂[ XXW

], where X ,W are Hilbert spaces,

with properties (1) – (4) a discrete-time state/signal system.

The sequence[

x(n)w(n)

]is a trajectory of V if

[x(n+1)x(w)w(n)

]∈ V , n ∈ Z+.

Mikael Kurula A brief overview of the state/signal approach

Frame 4 of 17

What are state/signal systems? [AS1] (cont.)

The subspace V =[A B1 0C Du1

] [ XU]

has the following properties:

1 V is closed in[ XXW

], where W = U u Y

2 If[z00

]∈ V then z = 0

3 The set D :=

[ xw ] |

[zxw

]∈ V

is closed in

[ XW]

4 For every x ∈ X there are some x and w such that[

zxw

]∈ V

Definition

We call any subspace V ⊂[ XXW

], where X ,W are Hilbert spaces,

with properties (1) – (4) a discrete-time state/signal system.

The sequence[

x(n)w(n)

]is a trajectory of V if

[x(n+1)x(w)w(n)

]∈ V , n ∈ Z+.

Mikael Kurula A brief overview of the state/signal approach

Frame 5 of 17

Input/state/output representations [AS1]

Definition

Let V be a s/s system on (X ,W).

A decomposition W = U u Y is admissible for V if there exists a

bounded operator[A BC D

]:[ XU]→[ XY]

s.t. V =[A B1 0C Du1

] [ XU].

In this case we call[A BC D

]an i/s/o representation of V .

Thus: If the i/o decomposition W = U u Y is admissible for thes/s system V , then

V =

[Ax ′+Bu′

x ′

Cx ′+Du′uu′

] ∣∣ [ x ′u′

]∈[ XU]

,

which means that [ xw ] is a trajectory generated by V if and only if[

x(n+1)y(n)

]=[A BC D

] [ x(n)u(n)

], n ∈ Z+.

Mikael Kurula A brief overview of the state/signal approach

Frame 5 of 17

Input/state/output representations [AS1]

Definition

Let V be a s/s system on (X ,W).

A decomposition W = U u Y is admissible for V if there exists a

bounded operator[A BC D

]:[ XU]→[ XY]

s.t. V =[A B1 0C Du1

] [ XU].

In this case we call[A BC D

]an i/s/o representation of V .

Thus: If the i/o decomposition W = U u Y is admissible for thes/s system V , then

V =

[Ax ′+Bu′

x ′

Cx ′+Du′uu′

] ∣∣ [ x ′u′

]∈[ XU]

,

which means that [ xw ] is a trajectory generated by V if and only if[

x(n+1)y(n)

]=[A BC D

] [ x(n)u(n)

], n ∈ Z+.

Mikael Kurula A brief overview of the state/signal approach

Frame 5 of 17

Input/state/output representations [AS1]

Definition

Let V be a s/s system on (X ,W).

A decomposition W = U u Y is admissible for V if there exists a

bounded operator[A BC D

]:[ XU]→[ XY]

s.t. V =[A B1 0C Du1

] [ XU].

In this case we call[A BC D

]an i/s/o representation of V .

Thus: If the i/o decomposition W = U u Y is admissible for thes/s system V , then

V =

[Ax ′+Bu′

x ′

Cx ′+Du′uu′

] ∣∣ [ x ′u′

]∈[ XU]

,

which means that [ xw ] is a trajectory generated by V if and only if[

x(n+1)y(n)

]=[A BC D

] [ x(n)u(n)

], n ∈ Z+.

Mikael Kurula A brief overview of the state/signal approach

Frame 5 of 17

Input/state/output representations [AS1]

Definition

Let V be a s/s system on (X ,W).

A decomposition W = U u Y is admissible for V if there exists a

bounded operator[A BC D

]:[ XU]→[ XY]

s.t. V =[A B1 0C Du1

] [ XU].

In this case we call[A BC D

]an i/s/o representation of V .

Thus: If the i/o decomposition W = U u Y is admissible for thes/s system V , then

V =

[Ax ′+Bu′

x ′

Cx ′+Du′uu′

] ∣∣ [ x ′u′

]∈[ XU]

,

which means that [ xw ] is a trajectory generated by V if and only if[

x(n+1)y(n)

]=[A BC D

] [ x(n)u(n)

], n ∈ Z+.

Mikael Kurula A brief overview of the state/signal approach

Frame 5 of 17

Input/state/output representations [AS1]

Definition

Let V be a s/s system on (X ,W).

A decomposition W = U u Y is admissible for V if there exists a

bounded operator[A BC D

]:[ XU]→[ XY]

s.t. V =[A B1 0C Du1

] [ XU].

In this case we call[A BC D

]an i/s/o representation of V .

Thus: If the i/o decomposition W = U u Y is admissible for thes/s system V , then

V =

[Ax ′+Bu′

x ′

Cx ′+Du′uu′

] ∣∣ [ x ′u′

]∈[ XU]

,

which means that [ xw ] is a trajectory generated by V if and only if[

x(n+1)y(n)

]=[A BC D

] [ x(n)u(n)

], n ∈ Z+.

Mikael Kurula A brief overview of the state/signal approach

Frame 6 of 17

Semi-canonical i/s/o representations [AS1]

A general s/s system V does not a priori have an i/s/o represent.

However, we can always construct one by choosing the canonical

input space U0 :=

w |[

z0w

]∈ V

and letting the output space Y

be an arbitrary complement to U0: W = U0 u Y.(The output space is not canonical.)

Then W = U0 u Y is an admissible i/o decomposition. Thecorresponding i/s/o representation of V is given by the map[

A BC D

]: [ x

u0 ] 7→ [ zy ]∣∣ [ z

xyuu0

]∈ V , y ∈ Y, u0 ∈ U0.

This is very useful,because it allows us to use the well-developed i/s/o theory.

Mikael Kurula A brief overview of the state/signal approach

Frame 6 of 17

Semi-canonical i/s/o representations [AS1]

A general s/s system V does not a priori have an i/s/o represent.

However, we can always construct one by choosing the canonical

input space U0 :=

w |[

z0w

]∈ V

and letting the output space Y

be an arbitrary complement to U0: W = U0 u Y.(The output space is not canonical.)

Then W = U0 u Y is an admissible i/o decomposition. Thecorresponding i/s/o representation of V is given by the map[

A BC D

]: [ x

u0 ] 7→ [ zy ]∣∣ [ z

xyuu0

]∈ V , y ∈ Y, u0 ∈ U0.

This is very useful,because it allows us to use the well-developed i/s/o theory.

Mikael Kurula A brief overview of the state/signal approach

Frame 6 of 17

Semi-canonical i/s/o representations [AS1]

A general s/s system V does not a priori have an i/s/o represent.

However, we can always construct one by choosing the canonical

input space U0 :=

w |[

z0w

]∈ V

and letting the output space Y

be an arbitrary complement to U0: W = U0 u Y.(The output space is not canonical.)

Then W = U0 u Y is an admissible i/o decomposition. Thecorresponding i/s/o representation of V is given by the map[

A BC D

]: [ x

u0 ] 7→ [ zy ]∣∣ [ z

xyuu0

]∈ V , y ∈ Y, u0 ∈ U0.

This is very useful,because it allows us to use the well-developed i/s/o theory.

Mikael Kurula A brief overview of the state/signal approach

Frame 6 of 17

Semi-canonical i/s/o representations [AS1]

A general s/s system V does not a priori have an i/s/o represent.

However, we can always construct one by choosing the canonical

input space U0 :=

w |[

z0w

]∈ V

and letting the output space Y

be an arbitrary complement to U0: W = U0 u Y.(The output space is not canonical.)

Then W = U0 u Y is an admissible i/o decomposition. Thecorresponding i/s/o representation of V is given by the map[

A BC D

]: [ x

u0 ] 7→ [ zy ]∣∣ [ z

xyuu0

]∈ V , y ∈ Y, u0 ∈ U0.

This is very useful,because it allows us to use the well-developed i/s/o theory.

Mikael Kurula A brief overview of the state/signal approach

Frame 6 of 17

Semi-canonical i/s/o representations [AS1]

A general s/s system V does not a priori have an i/s/o represent.

However, we can always construct one by choosing the canonical

input space U0 :=

w |[

z0w

]∈ V

and letting the output space Y

be an arbitrary complement to U0: W = U0 u Y.(The output space is not canonical.)

Then W = U0 u Y is an admissible i/o decomposition. Thecorresponding i/s/o representation of V is given by the map[

A BC D

]: [ x

u0 ] 7→ [ zy ]∣∣ [ z

xyuu0

]∈ V , y ∈ Y, u0 ∈ U0.

This is very useful,because it allows us to use the well-developed i/s/o theory.

Mikael Kurula A brief overview of the state/signal approach

Frame 7 of 17

Continuous-time boundary-control examples

Standard example: The transmission line [AKS2]

∂

∂ti(t, ξ) = −

1

L(ξ)

∂

∂ξv(t, ξ)

∂

∂tv(t, ξ) = −

1

C(ξ)

∂

∂ξi(t, ξ)

x(t) =[

i(t,·)v(t,·)

], x(0) given,

(t, ξ) ∈ R+ × [0, `].

The external signal is w(t) = (i(t, 0), v(t, 0), i(t, `), v(t, `))>.

Much more demanding: n-D spatial domains

The wave equation on a 2-D spatial domain Ω:

∂2

∂t2x(t, ξ, η) = c2

(∂2

∂ξ2+

∂2

∂η2

)x(t, ξ, η), (ξ, η) ∈ Ω.

We need Sobolev-space machinery: W is ∞-dimensional.

Mikael Kurula A brief overview of the state/signal approach

Frame 7 of 17

Continuous-time boundary-control examples

Standard example: The transmission line [AKS2]

∂

∂ti(t, ξ) = −

1

L(ξ)

∂

∂ξv(t, ξ)

∂

∂tv(t, ξ) = −

1

C(ξ)

∂

∂ξi(t, ξ)

x(t) =[

i(t,·)v(t,·)

], x(0) given,

(t, ξ) ∈ R+ × [0, `].

The external signal is w(t) = (i(t, 0), v(t, 0), i(t, `), v(t, `))>.

Much more demanding: n-D spatial domains

The wave equation on a 2-D spatial domain Ω:

∂2

∂t2x(t, ξ, η) = c2

(∂2

∂ξ2+

∂2

∂η2

)x(t, ξ, η), (ξ, η) ∈ Ω.

We need Sobolev-space machinery: W is ∞-dimensional.

Mikael Kurula A brief overview of the state/signal approach

Frame 8 of 17

Approaching continuous-time s/s systems

In discrete time[x(n+1)y(n)

]=[A BC D

] [ x(n)u(n)

], n ∈ Z+,

was turned into[x(n+1)x(n)w(n)

]∈[A B1 0C Du1

] [ XU]

=: V , n ∈ Z+.

In continuous time we have[x(t)y(t)

]=[A&BC&D

] [ x(t)u(t)

], t ∈ R+,

which similarly can be turned into[x(t)x(t)w(t)

]∈ V , t ∈ R+,

but here V has much more complicated structure. . .Mikael Kurula A brief overview of the state/signal approach

Frame 8 of 17

Approaching continuous-time s/s systems

In discrete time[x(n+1)y(n)

]=[A BC D

] [ x(n)u(n)

], n ∈ Z+,

was turned into[x(n+1)x(n)w(n)

]∈[A B1 0C Du1

] [ XU]

=: V , n ∈ Z+.

In continuous time we have[x(t)y(t)

]=[A&BC&D

] [ x(t)u(t)

], t ∈ R+,

which similarly can be turned into[x(t)x(t)w(t)

]∈ V , t ∈ R+,

but here V has much more complicated structure. . .Mikael Kurula A brief overview of the state/signal approach

Frame 8 of 17

Approaching continuous-time s/s systems

In discrete time[x(n+1)y(n)

]=[A BC D

] [ x(n)u(n)

], n ∈ Z+,

was turned into[x(n+1)x(n)w(n)

]∈[A B1 0C Du1

] [ XU]

=: V , n ∈ Z+.

In continuous time we have[x(t)y(t)

]=[A&BC&D

] [ x(t)u(t)

], t ∈ R+,

which similarly can be turned into[x(t)x(t)w(t)

]∈ V , t ∈ R+,

but here V has much more complicated structure. . .Mikael Kurula A brief overview of the state/signal approach

Frame 9 of 17

Some problems with continuous time [KS1]

A discrete-time state/signal system V ⊂[ XXW

]satisfies:

1 V is closed in[ XXW

]2 If

[z00

]∈ V then z = 0

3 The set D :=

[ xw ] |

[zxw

]∈ V

is closed in

[ XW]

4 For every x ∈ X there are some x and w such that[

zxw

]∈ V

Conditions (i) – (iii) mean that V is the graph of a bounded

operator F with domain D: V =[

zxw

] ∣∣ [ xw ] ∈ D, z = F [ x

w ]

.

Continuous time: For all PDE:s F is unbounded!

Even worse: in general there is no canonical input space.

Mikael Kurula A brief overview of the state/signal approach

Frame 9 of 17

Some problems with continuous time [KS1]

A discrete-time state/signal system V ⊂[ XXW

]satisfies:

1 V is closed in[ XXW

]2 If

[z00

]∈ V then z = 0

3 The set D :=

[ xw ] |

[zxw

]∈ V

is closed in

[ XW]

4 For every x ∈ X there are some x and w such that[

zxw

]∈ V

Conditions (i) – (iii) mean that V is the graph of a bounded

operator F with domain D: V =[

zxw

] ∣∣ [ xw ] ∈ D, z = F [ x

w ]

.

Continuous time: For all PDE:s F is unbounded!

Even worse: in general there is no canonical input space.

Mikael Kurula A brief overview of the state/signal approach

Frame 9 of 17

Some problems with continuous time [KS1]

A discrete-time state/signal system V ⊂[ XXW

]satisfies:

1 V is closed in[ XXW

]2 If

[z00

]∈ V then z = 0

3 The set D :=

[ xw ] |

[zxw

]∈ V

is closed in

[ XW]

4 For every x ∈ X there are some x and w such that[

zxw

]∈ V

Conditions (i) – (iii) mean that V is the graph of a bounded

operator F with domain D: V =[

zxw

] ∣∣ [ xw ] ∈ D, z = F [ x

w ]

.

Continuous time: For all PDE:s F is unbounded!

Even worse: in general there is no canonical input space.

Mikael Kurula A brief overview of the state/signal approach

Frame 9 of 17

Some problems with continuous time [KS1]

A discrete-time state/signal system V ⊂[ XXW

]satisfies:

1 V is closed in[ XXW

]2 If

[z00

]∈ V then z = 0

3 The set D :=

[ xw ] |

[zxw

]∈ V

is closed in

[ XW]

4 For every x ∈ X there are some x and w such that[

zxw

]∈ V

Conditions (i) – (iii) mean that V is the graph of a bounded

operator F with domain D: V =[

zxw

] ∣∣ [ xw ] ∈ D, z = F [ x

w ]

.

Continuous time: For all PDE:s F is unbounded!

Even worse: in general there is no canonical input space.

Mikael Kurula A brief overview of the state/signal approach

Frame 10 of 17

State/signal systems in continuous time [KS1, KS2, AKS2]

Definition (Continuous-time state/signal system)

Let V ⊂[ XXW

]be closed. Then [ x

w ] ∈[C1(R+;X )C(R+;W)

]is a classical

trajectory generated by V if

[x(t)x(t)w(t)

]∈ V for all t > 0.

V is a continuous-time state/signal system if:

1

[z00

]∈ V =⇒ z = 0

2 For every[

z0x0w0

]∈ V there exists a classical trajectory [ x

w ]

generated by V that satisfies

[x(0)x(0)w(0)

]=[

z0w0w0

].

Well-posed s/s systems by definition have some admissible i/odecomposition. However, there’s no way of constructing it. [KS1]

All discrete-time s/s systems are well-posed, because of U0.

Mikael Kurula A brief overview of the state/signal approach

Frame 10 of 17

State/signal systems in continuous time [KS1, KS2, AKS2]

Definition (Continuous-time state/signal system)

Let V ⊂[ XXW

]be closed. Then [ x

w ] ∈[C1(R+;X )C(R+;W)

]is a classical

trajectory generated by V if

[x(t)x(t)w(t)

]∈ V for all t > 0.

V is a continuous-time state/signal system if:

1

[z00

]∈ V =⇒ z = 0

2 For every[

z0x0w0

]∈ V there exists a classical trajectory [ x

w ]

generated by V that satisfies

[x(0)x(0)w(0)

]=[

z0w0w0

].

Well-posed s/s systems by definition have some admissible i/odecomposition. However, there’s no way of constructing it. [KS1]

All discrete-time s/s systems are well-posed, because of U0.

Mikael Kurula A brief overview of the state/signal approach

Frame 10 of 17

State/signal systems in continuous time [KS1, KS2, AKS2]

Definition (Continuous-time state/signal system)

Let V ⊂[ XXW

]be closed. Then [ x

w ] ∈[C1(R+;X )C(R+;W)

]is a classical

trajectory generated by V if

[x(t)x(t)w(t)

]∈ V for all t > 0.

V is a continuous-time state/signal system if:

1

[z00

]∈ V =⇒ z = 0

2 For every[

z0x0w0

]∈ V there exists a classical trajectory [ x

w ]

generated by V that satisfies

[x(0)x(0)w(0)

]=[

z0w0w0

].

Well-posed s/s systems by definition have some admissible i/odecomposition. However, there’s no way of constructing it. [KS1]

All discrete-time s/s systems are well-posed, because of U0.

Mikael Kurula A brief overview of the state/signal approach

Frame 10 of 17

State/signal systems in continuous time [KS1, KS2, AKS2]

Definition (Continuous-time state/signal system)

Let V ⊂[ XXW

]be closed. Then [ x

w ] ∈[C1(R+;X )C(R+;W)

]is a classical

trajectory generated by V if

[x(t)x(t)w(t)

]∈ V for all t > 0.

V is a continuous-time state/signal system if:

1

[z00

]∈ V =⇒ z = 0

2 For every[

z0x0w0

]∈ V there exists a classical trajectory [ x

w ]

generated by V that satisfies

[x(0)x(0)w(0)

]=[

z0w0w0

].

Well-posed s/s systems by definition have some admissible i/odecomposition. However, there’s no way of constructing it. [KS1]

All discrete-time s/s systems are well-posed, because of U0.

Mikael Kurula A brief overview of the state/signal approach

Frame 11 of 17

Passive state/signal systems [AS2, AS3, KS2, K1, AKS1]

Intuitively: A passive system has no internal energy sources.A conservative system is passive and dissipates no energy.

Mathematically: (For passive systems W is a Kreın space.)

1 The system has “enough” trajectories [ xw ].

2 Every trajectory satisfies (passive):

‖x(t)‖2X ≤ ‖x(0)‖2

X +

∫ t

0[w(s),w(s)]W ds, t ≥ 0. (2)

“Enough” means more or less that (2) holds for the dual system.

Passivity yields surprisingly useful additional structure:much of the discrete theory can be transferred to continuous time.

Passive systems have easily found admissible i/o decompositions:we know how to split the external signal w into [ yu ].

Conservative ⇒ passive ⇒ well-posed.Mikael Kurula A brief overview of the state/signal approach

Frame 11 of 17

Passive state/signal systems [AS2, AS3, KS2, K1, AKS1]

Intuitively: A passive system has no internal energy sources.A conservative system is passive and dissipates no energy.

Mathematically: (For passive systems W is a Kreın space.)

1 The system has “enough” trajectories [ xw ].

2 Every trajectory satisfies (passive):

‖x(t)‖2X ≤ ‖x(0)‖2

X +

∫ t

0[w(s),w(s)]W ds, t ≥ 0. (2)

“Enough” means more or less that (2) holds for the dual system.

Passivity yields surprisingly useful additional structure:much of the discrete theory can be transferred to continuous time.

Passive systems have easily found admissible i/o decompositions:we know how to split the external signal w into [ yu ].

Conservative ⇒ passive ⇒ well-posed.Mikael Kurula A brief overview of the state/signal approach

Frame 11 of 17

Passive state/signal systems [AS2, AS3, KS2, K1, AKS1]

Intuitively: A passive system has no internal energy sources.A conservative system is passive and dissipates no energy.

Mathematically: (For passive systems W is a Kreın space.)

1 The system has “enough” trajectories [ xw ].

2 Every trajectory satisfies (conservative):

‖x(t)‖2X = ‖x(0)‖2

X +

∫ t

0[w(s),w(s)]W ds, t ≥ 0. (2)

“Enough” means more or less that (2) holds for the dual system.

Passivity yields surprisingly useful additional structure:much of the discrete theory can be transferred to continuous time.

Passive systems have easily found admissible i/o decompositions:we know how to split the external signal w into [ yu ].

Conservative ⇒ passive ⇒ well-posed.Mikael Kurula A brief overview of the state/signal approach

Frame 11 of 17

Passive state/signal systems [AS2, AS3, KS2, K1, AKS1]

Intuitively: A passive system has no internal energy sources.A conservative system is passive and dissipates no energy.

Mathematically: (For passive systems W is a Kreın space.)

1 The system has “enough” trajectories [ xw ].

2 Every trajectory satisfies (passive):

‖x(t)‖2X ≤ ‖x(0)‖2

X +

∫ t

0[w(s),w(s)]W ds, t ≥ 0. (2)

“Enough” means more or less that (2) holds for the dual system.

Passivity yields surprisingly useful additional structure:much of the discrete theory can be transferred to continuous time.

Passive systems have easily found admissible i/o decompositions:we know how to split the external signal w into [ yu ].

Conservative ⇒ passive ⇒ well-posed.Mikael Kurula A brief overview of the state/signal approach

Frame 11 of 17

Passive state/signal systems [AS2, AS3, KS2, K1, AKS1]

Intuitively: A passive system has no internal energy sources.A conservative system is passive and dissipates no energy.

Mathematically: (For passive systems W is a Kreın space.)

1 The system has “enough” trajectories [ xw ].

2 Every trajectory satisfies (passive):

‖x(t)‖2X ≤ ‖x(0)‖2

X +

∫ t

0[w(s),w(s)]W ds, t ≥ 0. (2)

“Enough” means more or less that (2) holds for the dual system.

Passivity yields surprisingly useful additional structure:much of the discrete theory can be transferred to continuous time.

Passive systems have easily found admissible i/o decompositions:we know how to split the external signal w into [ yu ].

Conservative ⇒ passive ⇒ well-posed.Mikael Kurula A brief overview of the state/signal approach

Frame 11 of 17

Passive state/signal systems [AS2, AS3, KS2, K1, AKS1]

Intuitively: A passive system has no internal energy sources.A conservative system is passive and dissipates no energy.

Mathematically: (For passive systems W is a Kreın space.)

1 The system has “enough” trajectories [ xw ].

2 Every trajectory satisfies (passive):

‖x(t)‖2X ≤ ‖x(0)‖2

X +

∫ t

0[w(s),w(s)]W ds, t ≥ 0. (2)

“Enough” means more or less that (2) holds for the dual system.

Passivity yields surprisingly useful additional structure:much of the discrete theory can be transferred to continuous time.

Passive systems have easily found admissible i/o decompositions:we know how to split the external signal w into [ yu ].

Conservative ⇒ passive ⇒ well-posed.Mikael Kurula A brief overview of the state/signal approach

Frame 12 of 17

Time-domain behaviour, discrete time

Definition

The time-domain behaviour of a discrete s/s system V is the set

W =

w∣∣ [ x

w ] is a trajectory of V with x(0) = 0.

The behaviour of a passive s/s system is a passive behaviour:

1 W is invariant under right shift with zero padding: S∗W ⊂W

2 W is a maximal nonnegative subspace of the Kreın space`2

+(W):∑∞

k=0[w(k),w(k)]W ≥ 0 for all w ∈W.

The realisation problem

Given a passive time-domain behaviour W, find a passive s/ssystem V whose time-domain behaviour coincides with W.

Canonical realisations: Under what additional assumptions is therealisation V uniquely determined by W, and in what sense?

Mikael Kurula A brief overview of the state/signal approach

Frame 12 of 17

Time-domain behaviour, discrete time

Definition

The time-domain behaviour of a discrete s/s system V is the set

W =

w∣∣ [ x

w ] is a trajectory of V with x(0) = 0.

The behaviour of a passive s/s system is a passive behaviour:

1 W is invariant under right shift with zero padding: S∗W ⊂W

2 W is a maximal nonnegative subspace of the Kreın space`2

+(W):∑∞

k=0[w(k),w(k)]W ≥ 0 for all w ∈W.

The realisation problem

Given a passive time-domain behaviour W, find a passive s/ssystem V whose time-domain behaviour coincides with W.

Canonical realisations: Under what additional assumptions is therealisation V uniquely determined by W, and in what sense?

Mikael Kurula A brief overview of the state/signal approach

Frame 12 of 17

Time-domain behaviour, discrete time

Definition

The time-domain behaviour of a discrete s/s system V is the set

W =

w∣∣ [ x

w ] is a trajectory of V with x(0) = 0.

The behaviour of a passive s/s system is a passive behaviour:

1 W is invariant under right shift with zero padding: S∗W ⊂W

2 W is a maximal nonnegative subspace of the Kreın space`2

+(W):∑∞

k=0[w(k),w(k)]W ≥ 0 for all w ∈W.

The realisation problem

Given a passive time-domain behaviour W, find a passive s/ssystem V whose time-domain behaviour coincides with W.

Canonical realisations: Under what additional assumptions is therealisation V uniquely determined by W, and in what sense?

Mikael Kurula A brief overview of the state/signal approach

Frame 12 of 17

Time-domain behaviour, discrete time

Definition

The time-domain behaviour of a discrete s/s system V is the set

W =

w∣∣ [ x

w ] is a trajectory of V with x(0) = 0.

The behaviour of a passive s/s system is a passive behaviour:

1 W is invariant under right shift with zero padding: S∗W ⊂W

2 W is a maximal nonnegative subspace of the Kreın space`2

+(W):∑∞

k=0[w(k),w(k)]W ≥ 0 for all w ∈W.

The realisation problem

Given a passive time-domain behaviour W, find a passive s/ssystem V whose time-domain behaviour coincides with W.

Canonical realisations: Under what additional assumptions is therealisation V uniquely determined by W, and in what sense?

Mikael Kurula A brief overview of the state/signal approach

Frame 13 of 17

Current research topic: Functional models for s/s systems

Canonical realisations of arbitrary given passive behaviors weredeveloped in [AS5, AS6] (discrete time) and [AKS1] (cont. time).

These articles are very long and technical!

⇒ I am looking for (noncanonical) realisations in the frequencydomain using reproducing kernel Hilbert space techniques –hopefully this turns out simpler.

Mikael Kurula A brief overview of the state/signal approach

Frame 13 of 17

Current research topic: Functional models for s/s systems

Canonical realisations of arbitrary given passive behaviors weredeveloped in [AS5, AS6] (discrete time) and [AKS1] (cont. time).

These articles are very long and technical!

⇒ I am looking for (noncanonical) realisations in the frequencydomain using reproducing kernel Hilbert space techniques –hopefully this turns out simpler.

Mikael Kurula A brief overview of the state/signal approach

Frame 14 of 17

Conclusions

Main idea is minimal distinction between inputs and outputs.

Some formulations simplify, but some proofs are unintuitive.

Discrete-time theory simpler and much more developed.

Hot topics are realisation theory and interconnection:

I see this as the beginning of avery promising research field.

Thank you for your attention!Any questions?

mkurula@abo.fihttp://users.abo.fi/mkurula/

Mikael Kurula A brief overview of the state/signal approach

Frame 14 of 17

Conclusions

Main idea is minimal distinction between inputs and outputs.

Some formulations simplify, but some proofs are unintuitive.

Discrete-time theory simpler and much more developed.

Hot topics are realisation theory and interconnection:

I see this as the beginning of avery promising research field.

Thank you for your attention!Any questions?

mkurula@abo.fihttp://users.abo.fi/mkurula/

Mikael Kurula A brief overview of the state/signal approach

Frame 14 of 17

Conclusions

Main idea is minimal distinction between inputs and outputs.

Some formulations simplify, but some proofs are unintuitive.

Discrete-time theory simpler and much more developed.

Hot topics are realisation theory and interconnection:

I see this as the beginning of avery promising research field.

Thank you for your attention!Any questions?

mkurula@abo.fihttp://users.abo.fi/mkurula/

Mikael Kurula A brief overview of the state/signal approach

Frame 14 of 17

Conclusions

Main idea is minimal distinction between inputs and outputs.

Some formulations simplify, but some proofs are unintuitive.

Discrete-time theory simpler and much more developed.

Hot topics are realisation theory and interconnection:

I see this as the beginning of avery promising research field.

Thank you for your attention!Any questions?

mkurula@abo.fihttp://users.abo.fi/mkurula/

Mikael Kurula A brief overview of the state/signal approach

Frame 14 of 17

Conclusions

Main idea is minimal distinction between inputs and outputs.

Some formulations simplify, but some proofs are unintuitive.

Discrete-time theory simpler and much more developed.

Hot topics are realisation theory and interconnection:

I see this as the beginning of avery promising research field.

Thank you for your attention!Any questions?

mkurula@abo.fihttp://users.abo.fi/mkurula/

Mikael Kurula A brief overview of the state/signal approach

Frame 15 of 17

Selected references

AS1 D.Z. Arov and O.J. Staffans, State/signal lineartime-invariant systems theory. Part I: Discrete timesystems, The State Space Method, Generalizationsand Applications, Operator Theory: Advances andApplications, vol. 161, Birkhauser-Verlag, 2005,pp. 115–177.

AS2 , State/signal linear time-invariant systemstheory. Part II: Passive discrete time systems,Internat. J. Robust Nonlinear Control 17 (2007),497–548.

AS3 , State/signal linear time-invariant systemstheory. Part III: Transmission and impedancerepresentations of discrete time systems, OperatorTheory, Structured Matrices, and Dilations, TiberiuConstantinescu Memorial Volume, Theta Foundation,2007, available from AMS, pp. 101–140.

Mikael Kurula A brief overview of the state/signal approach

Frame 16 of 17

Selected references (cont.)

AS4 , State/signal linear time-invariant systemstheory. Part IV: Affine representations of discretetime systems, Complex Anal. Oper. Theory 1 (2007),457–521.

AS5 , Two canonical passive state/signal shiftrealizations of passive discrete time behaviors, J.Funct. Anal. 257 (2009), 2573–2634.

AS6 , Canonical conservative state/signal shiftrealizations of passive discrete time behaviors, J.Funct. Anal. 259 (2010), no. 12, 3265–3327.

KS1 M. Kurula and O.J. Staffans, Well-posed state/signalsystems in continuous time, Complex Anal. Oper.Theory 4 (2009), 319–390.

KS2 , Connections between smooth andgeneralized trajectories of a state/signal system,submitted, manuscript available athttp://users.abo.fi/mkurula/, 2010.

Mikael Kurula A brief overview of the state/signal approach

Frame 17 of 17

Selected references (cont.)

AKS1 D.Z. Arov, M. Kurula, and O.J. Staffans, Canonicalstate/signal shift realizations of passive continuoustime behaviors, submitted, manuscript athttp://users.abo.fi/mkurula/, 2010.

AKS2 , Continuous-time state/signal systems andboundary relations, Two submitted book chapters,2009, 42 pages, manuscript athttp://users.abo.fi/mkurula/, 2010.

K1 M. Kurula, On passive and conservative state/signalsystems in continuous time, Integral EquationsOperator Theory 67 (2010), 377–424, 449.

S1 O.J. Staffans, Passive linear discrete time-invariantsystems, Proceedings of the International Congress ofMathematicians, Madrid, 2006, pp. 1367–1388.

Mikael Kurula A brief overview of the state/signal approach