Post on 07-Jul-2020
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