balanced truncation for stochastic di erential equations with ......3 concept of observability 4...
TRANSCRIPT
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MAX PLANCK INSTITUTE
FOR DYNAMICS OF COMPLEX
TECHNICAL SYSTEMS
MAGDEBURG
February 13, 2013
Balanced Truncation for StochasticDifferential Equations with Levy Noise
Martin Redmann
Computational Methods in Systems and Control TheoryMax Planck Institute for Dynamics of Complex Technical Systems
Magdeburg, Germany
Max Planck Institute Magdeburg M. Redmann, Balanced Truncation for Stochastic Differential Equations with Levy Noise 1/25
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Introduction Concept of Reachability Concept of Observability Balanced Truncation
Content
1 Introduction
2 Concept of Reachability
3 Concept of Observability
4 Balanced Truncation
Max Planck Institute Magdeburg M. Redmann, Balanced Truncation for Stochastic Differential Equations with Levy Noise 2/25
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Introduction Concept of Reachability Concept of Observability Balanced Truncation
Content
1 Introduction
2 Concept of Reachability
3 Concept of Observability
4 Balanced Truncation
Max Planck Institute Magdeburg M. Redmann, Balanced Truncation for Stochastic Differential Equations with Levy Noise 2/25
-
Introduction Concept of Reachability Concept of Observability Balanced Truncation
Content
1 Introduction
2 Concept of Reachability
3 Concept of Observability
4 Balanced Truncation
Max Planck Institute Magdeburg M. Redmann, Balanced Truncation for Stochastic Differential Equations with Levy Noise 2/25
-
Introduction Concept of Reachability Concept of Observability Balanced Truncation
Content
1 Introduction
2 Concept of Reachability
3 Concept of Observability
4 Balanced Truncation
Max Planck Institute Magdeburg M. Redmann, Balanced Truncation for Stochastic Differential Equations with Levy Noise 2/25
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Introduction Concept of Reachability Concept of Observability Balanced Truncation
Levy processes
Properties of Levy processes
Levy processes generalize Wiener processes.
The trajectories may have jumps but at most countably many.
The trajectories are right-continuous and have left limits.
We will focus on square integrable Levy processes (M(t))t≥0 with zeromean, which means that
E [M(t)] = 0 and E[M(t)2
]
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Introduction Concept of Reachability Concept of Observability Balanced Truncation
Levy processes
Properties of Levy processes
Levy processes generalize Wiener processes.
The trajectories may have jumps but at most countably many.
The trajectories are right-continuous and have left limits.
We will focus on square integrable Levy processes (M(t))t≥0 with zeromean, which means that
E [M(t)] = 0 and E[M(t)2
]
-
Introduction Concept of Reachability Concept of Observability Balanced Truncation
Levy processes
Properties of Levy processes
Levy processes generalize Wiener processes.
The trajectories may have jumps but at most countably many.
The trajectories are right-continuous and have left limits.
We will focus on square integrable Levy processes (M(t))t≥0 with zeromean, which means that
E [M(t)] = 0 and E[M(t)2
]
-
Introduction Concept of Reachability Concept of Observability Balanced Truncation
Levy processes
Properties of Levy processes
Levy processes generalize Wiener processes.
The trajectories may have jumps but at most countably many.
The trajectories are right-continuous and have left limits.
We will focus on square integrable Levy processes (M(t))t≥0 with zeromean, which means that
E [M(t)] = 0 and E[M(t)2
]
-
Introduction Concept of Reachability Concept of Observability Balanced Truncation
Levy processes
Properties of Levy processes
Levy processes generalize Wiener processes.
The trajectories may have jumps but at most countably many.
The trajectories are right-continuous and have left limits.
We will focus on square integrable Levy processes (M(t))t≥0 with zeromean, which means that
E [M(t)] = 0 and E[M(t)2
]
-
Introduction Concept of Reachability Concept of Observability Balanced Truncation
We consider the following equation for t ≥ 0:
dX (t) = [AX (t) + Bu(t)] dt + ΨX (t−)dM(t), (1)Y(t) = CX (t),
where X (0) = x0, A,Ψ ∈ Rn×n, B ∈ Rn×m, C ∈ Rp×n, M is areal-valued square integrable Levy process with zero mean and u is anadapted stochastic processes with
‖u‖2L2 := E∫ ∞
0
‖u(t)‖22 dt
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Introduction Concept of Reachability Concept of Observability Balanced Truncation
We consider the following equation for t ≥ 0:
dX (t) = [AX (t) + Bu(t)] dt + ΨX (t−)dM(t), (1)Y(t) = CX (t),
where X (0) = x0, A,Ψ ∈ Rn×n, B ∈ Rn×m, C ∈ Rp×n, M is areal-valued square integrable Levy process with zero mean and u is anadapted stochastic processes with
‖u‖2L2 := E∫ ∞
0
‖u(t)‖22 dt
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Introduction Concept of Reachability Concept of Observability Balanced Truncation
Fundamental Solution
We introduce the stochastic processes (Φ(t, τ))t≥τ with values in Rn×n,which satify
Φ(t, τ) = In +
∫ tτ
AΦ(s, τ)ds +
∫ tτ
ΨΦ(s−, τ)dM(s)
for t ≥ τ ≥ 0. We write Φ(t, 0) = Φ(t).
Proposition
The solution of equation (1) is given by
Φ(t)x0 +
∫ t0
Φ(t, s)Bu(s)ds, t ≥ 0.
Max Planck Institute Magdeburg M. Redmann, Balanced Truncation for Stochastic Differential Equations with Levy Noise 5/25
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Introduction Concept of Reachability Concept of Observability Balanced Truncation
Fundamental Solution
We introduce the stochastic processes (Φ(t, τ))t≥τ with values in Rn×n,which satify
Φ(t, τ) = In +
∫ tτ
AΦ(s, τ)ds +
∫ tτ
ΨΦ(s−, τ)dM(s)
for t ≥ τ ≥ 0. We write Φ(t, 0) = Φ(t).
Proposition
The solution of equation (1) is given by
Φ(t)x0 +
∫ t0
Φ(t, s)Bu(s)ds, t ≥ 0.
Max Planck Institute Magdeburg M. Redmann, Balanced Truncation for Stochastic Differential Equations with Levy Noise 5/25
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Introduction Concept of Reachability Concept of Observability Balanced Truncation
Concept of Reachability
Max Planck Institute Magdeburg M. Redmann, Balanced Truncation for Stochastic Differential Equations with Levy Noise 6/25
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Introduction Concept of Reachability Concept of Observability Balanced Truncation
Reachability Gramian [Benner, Damm ’11]
We call Pt :=∫ t
0E[Φ(s)BBT ΦT (s)
]ds finite reachability Gramian at
time t ≥ 0.The asymptotic stability in mean square also provides the existence of theinfinite reachability Gramian
P :=
∫ ∞0
E[Φ(s)BBT ΦT (s)
]ds.
Proposition
For all t ≥ 0 it holds
im Pt = im P.
One can show that
0 = BBT + P AT + A P + Ψ P ΨT E[M(1)2
].
Max Planck Institute Magdeburg M. Redmann, Balanced Truncation for Stochastic Differential Equations with Levy Noise 7/25
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Introduction Concept of Reachability Concept of Observability Balanced Truncation
Reachability Gramian [Benner, Damm ’11]
We call Pt :=∫ t
0E[Φ(s)BBT ΦT (s)
]ds finite reachability Gramian at
time t ≥ 0.The asymptotic stability in mean square also provides the existence of theinfinite reachability Gramian
P :=
∫ ∞0
E[Φ(s)BBT ΦT (s)
]ds.
Proposition
For all t ≥ 0 it holds
im Pt = im P.
One can show that
0 = BBT + P AT + A P + Ψ P ΨT E[M(1)2
].
Max Planck Institute Magdeburg M. Redmann, Balanced Truncation for Stochastic Differential Equations with Levy Noise 7/25
-
Introduction Concept of Reachability Concept of Observability Balanced Truncation
Reachability Gramian [Benner, Damm ’11]
We call Pt :=∫ t
0E[Φ(s)BBT ΦT (s)
]ds finite reachability Gramian at
time t ≥ 0.The asymptotic stability in mean square also provides the existence of theinfinite reachability Gramian
P :=
∫ ∞0
E[Φ(s)BBT ΦT (s)
]ds.
Proposition
For all t ≥ 0 it holds
im Pt = im P.
One can show that
0 = BBT + P AT + A P + Ψ P ΨT E[M(1)2
].
Max Planck Institute Magdeburg M. Redmann, Balanced Truncation for Stochastic Differential Equations with Levy Noise 7/25
-
Introduction Concept of Reachability Concept of Observability Balanced Truncation
Reachability Gramian [Benner, Damm ’11]
We call Pt :=∫ t
0E[Φ(s)BBT ΦT (s)
]ds finite reachability Gramian at
time t ≥ 0.The asymptotic stability in mean square also provides the existence of theinfinite reachability Gramian
P :=
∫ ∞0
E[Φ(s)BBT ΦT (s)
]ds.
Proposition
For all t ≥ 0 it holds
im Pt = im P.
One can show that
0 = BBT + P AT + A P + Ψ P ΨT E[M(1)2
].
Max Planck Institute Magdeburg M. Redmann, Balanced Truncation for Stochastic Differential Equations with Levy Noise 7/25
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Introduction Concept of Reachability Concept of Observability Balanced Truncation
By X (T , 0, u) we denote the solution of the inhomogeneous system (1)at time T with initial condition 0 for a given input u.
Definition
An avarage state x ∈ Rn is called reachable (from zero) if a time T > 0and a control function u ∈ L2T exist such that
E [X (T , 0, u)] = x .
Proposition
An avarage state x ∈ Rn is reachable (from zero) if and only if x ∈ im P.
Max Planck Institute Magdeburg M. Redmann, Balanced Truncation for Stochastic Differential Equations with Levy Noise 8/25
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Introduction Concept of Reachability Concept of Observability Balanced Truncation
By X (T , 0, u) we denote the solution of the inhomogeneous system (1)at time T with initial condition 0 for a given input u.
Definition
An avarage state x ∈ Rn is called reachable (from zero) if a time T > 0and a control function u ∈ L2T exist such that
E [X (T , 0, u)] = x .
Proposition
An avarage state x ∈ Rn is reachable (from zero) if and only if x ∈ im P.
Max Planck Institute Magdeburg M. Redmann, Balanced Truncation for Stochastic Differential Equations with Levy Noise 8/25
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Introduction Concept of Reachability Concept of Observability Balanced Truncation
By X (T , 0, u) we denote the solution of the inhomogeneous system (1)at time T with initial condition 0 for a given input u.
Definition
An avarage state x ∈ Rn is called reachable (from zero) if a time T > 0and a control function u ∈ L2T exist such that
E [X (T , 0, u)] = x .
Proposition
An avarage state x ∈ Rn is reachable (from zero) if and only if x ∈ im P.
Max Planck Institute Magdeburg M. Redmann, Balanced Truncation for Stochastic Differential Equations with Levy Noise 8/25
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Introduction Concept of Reachability Concept of Observability Balanced Truncation
Proposition
The control u with minimal energy to reach x ∈ im P at time T > 0 isgiven by
u(t) = BT ΦT (T , t)P#T x , t ∈ [0,T ],
with
‖u‖2L2T = xTP#T x .
Thus, the minimal energy that is needed to steer the system to x is givenby
infT>0
xTP#T x = xTP#x .
Max Planck Institute Magdeburg M. Redmann, Balanced Truncation for Stochastic Differential Equations with Levy Noise 9/25
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Introduction Concept of Reachability Concept of Observability Balanced Truncation
Proposition
The control u with minimal energy to reach x ∈ im P at time T > 0 isgiven by
u(t) = BT ΦT (T , t)P#T x , t ∈ [0,T ],
with
‖u‖2L2T = xTP#T x .
Thus, the minimal energy that is needed to steer the system to x is givenby
infT>0
xTP#T x = xTP#x .
Max Planck Institute Magdeburg M. Redmann, Balanced Truncation for Stochastic Differential Equations with Levy Noise 9/25
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Introduction Concept of Reachability Concept of Observability Balanced Truncation
Concept of Observability
Max Planck Institute Magdeburg M. Redmann, Balanced Truncation for Stochastic Differential Equations with Levy Noise 10/25
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Introduction Concept of Reachability Concept of Observability Balanced Truncation
Observability Gramian [Benner, Damm ’11]
Due to the asymptotic stability in mean square the observability Gramian
Q = E[∫ ∞
0
ΦT (s)CTCΦ(s)ds
]exists and it is the solution of
AT Q + Q A + ΨTQΨ E[M(1)2
]+ CTC = 0.
Max Planck Institute Magdeburg M. Redmann, Balanced Truncation for Stochastic Differential Equations with Levy Noise 11/25
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Introduction Concept of Reachability Concept of Observability Balanced Truncation
We consider an uncontrolled system (1) (u ≡ 0).Hence, X (t) = Φ(t)x0 and
Y(t) = CΦ(t)x0.
The observation energy produced by x0 is
‖Y‖2L2 := E∫ ∞
0
YT (t)Y(t)dt = xT0 E∫ ∞
0
ΦT (t)CTCΦ(t)dt x0
= xT0 Qx0.
DefinitionAn inital condition x0 is unobservable if x0 ∈ ker Q.
Max Planck Institute Magdeburg M. Redmann, Balanced Truncation for Stochastic Differential Equations with Levy Noise 12/25
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Introduction Concept of Reachability Concept of Observability Balanced Truncation
We consider an uncontrolled system (1) (u ≡ 0).Hence, X (t) = Φ(t)x0 and
Y(t) = CΦ(t)x0.
The observation energy produced by x0 is
‖Y‖2L2 := E∫ ∞
0
YT (t)Y(t)dt = xT0 E∫ ∞
0
ΦT (t)CTCΦ(t)dt x0
= xT0 Qx0.
DefinitionAn inital condition x0 is unobservable if x0 ∈ ker Q.
Max Planck Institute Magdeburg M. Redmann, Balanced Truncation for Stochastic Differential Equations with Levy Noise 12/25
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Introduction Concept of Reachability Concept of Observability Balanced Truncation
We consider an uncontrolled system (1) (u ≡ 0).Hence, X (t) = Φ(t)x0 and
Y(t) = CΦ(t)x0.
The observation energy produced by x0 is
‖Y‖2L2 := E∫ ∞
0
YT (t)Y(t)dt = xT0 E∫ ∞
0
ΦT (t)CTCΦ(t)dt x0
= xT0 Qx0.
DefinitionAn inital condition x0 is unobservable if x0 ∈ ker Q.
Max Planck Institute Magdeburg M. Redmann, Balanced Truncation for Stochastic Differential Equations with Levy Noise 12/25
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Introduction Concept of Reachability Concept of Observability Balanced Truncation
Balanced Truncation
Max Planck Institute Magdeburg M. Redmann, Balanced Truncation for Stochastic Differential Equations with Levy Noise 13/25
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Introduction Concept of Reachability Concept of Observability Balanced Truncation
We start with a system
dX (t) = [AX (t) + Bu(t)] dt + ΨX (t−)dM(t), (2)Y(t) = CX (t), t ≥ 0,
which is completely reachable and observable, which is equivalent to Pand Q are positive definite.
We transfer the states with a regular matrix T and obtain a system with
(Ã, B̃, Ψ̃, C̃ ) = (TAT−1,TB,TΨT−1,CT−1),
which has the same output as system (2).
The Gramians of the transformed system are given by
P̃ = TPTT and Q̃ = T−TQT−1.
Max Planck Institute Magdeburg M. Redmann, Balanced Truncation for Stochastic Differential Equations with Levy Noise 14/25
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Introduction Concept of Reachability Concept of Observability Balanced Truncation
We start with a system
dX (t) = [AX (t) + Bu(t)] dt + ΨX (t−)dM(t), (2)Y(t) = CX (t), t ≥ 0,
which is completely reachable and observable, which is equivalent to Pand Q are positive definite.
We transfer the states with a regular matrix T and obtain a system with
(Ã, B̃, Ψ̃, C̃ ) = (TAT−1,TB,TΨT−1,CT−1),
which has the same output as system (2).
The Gramians of the transformed system are given by
P̃ = TPTT and Q̃ = T−TQT−1.
Max Planck Institute Magdeburg M. Redmann, Balanced Truncation for Stochastic Differential Equations with Levy Noise 14/25
-
Introduction Concept of Reachability Concept of Observability Balanced Truncation
We start with a system
dX (t) = [AX (t) + Bu(t)] dt + ΨX (t−)dM(t), (2)Y(t) = CX (t), t ≥ 0,
which is completely reachable and observable, which is equivalent to Pand Q are positive definite.
We transfer the states with a regular matrix T and obtain a system with
(Ã, B̃, Ψ̃, C̃ ) = (TAT−1,TB,TΨT−1,CT−1),
which has the same output as system (2).
The Gramians of the transformed system are given by
P̃ = TPTT and Q̃ = T−TQT−1.
Max Planck Institute Magdeburg M. Redmann, Balanced Truncation for Stochastic Differential Equations with Levy Noise 14/25
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Introduction Concept of Reachability Concept of Observability Balanced Truncation
Balancing Transformation
We want to choose T , such that the Gramians are equal, which meansthat P̃ = Q̃ = Σ = diag(σ1, . . . , σn).
TheoremThe balancing transformation T and its inverse are given by
T = Σ12 KTU−1,
T−1 = UKΣ−12 ,
where P = UUT and UTQU = KΣ2KT .
Max Planck Institute Magdeburg M. Redmann, Balanced Truncation for Stochastic Differential Equations with Levy Noise 15/25
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Introduction Concept of Reachability Concept of Observability Balanced Truncation
We consider the following partitions:
T =
[W T
TT2
], T−1 =
[V T1
]and X̂ =
(X̃X1
),
where W T ∈ Rr×n,V ∈ Rn×r and X̃ ∈ Rr .
Hence,(dX̃ (t)dX1(t)
)=
[W TAV W TAT1TT2 AV T
T2 AT1
](X̃ (t)X1(t)
)dt +
[W TBTT2 B
]u(t)dt
+
[W T ΨV W T ΨT1TT2 ΨV T
T2 ΨT1
](X̃ (t−)X1(t−)
)dM(t)
and
Y(t) =[CV CT1
]( X̃ (t)X1(t)
).
Max Planck Institute Magdeburg M. Redmann, Balanced Truncation for Stochastic Differential Equations with Levy Noise 16/25
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Introduction Concept of Reachability Concept of Observability Balanced Truncation
We consider the following partitions:
T =
[W T
TT2
], T−1 =
[V T1
]and X̂ =
(X̃X1
),
where W T ∈ Rr×n,V ∈ Rn×r and X̃ ∈ Rr .
Hence,(dX̃ (t)dX1(t)
)=
[W TAV W TAT1TT2 AV T
T2 AT1
](X̃ (t)X1(t)
)dt +
[W TBTT2 B
]u(t)dt
+
[W T ΨV W T ΨT1TT2 ΨV T
T2 ΨT1
](X̃ (t−)X1(t−)
)dM(t)
and
Y(t) =[CV CT1
]( X̃ (t)X1(t)
).
Max Planck Institute Magdeburg M. Redmann, Balanced Truncation for Stochastic Differential Equations with Levy Noise 16/25
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Introduction Concept of Reachability Concept of Observability Balanced Truncation
Selecting the first r rows and neglecting the X1 terms provide thefollowing reduced order model:
dX̃ (t) = W TAV X̃ (t)dt + W TBu(t)dt + W T ΨV X̃ (t−)dM(t)Ỹ(t) = CV X̃ (t).
Open Questions1 Is the reduced order model stable in general?
deterministic case: The reduced order model is stable.
2 What is the structure of the Gramians of the reduced model?deterministic case: PR = QR = diag(σ1, . . . , σr )
3 What is the structure of the Hankel values of the reduced model?deterministic case: The Hankel values are σ1, . . . , σr .
Max Planck Institute Magdeburg M. Redmann, Balanced Truncation for Stochastic Differential Equations with Levy Noise 17/25
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Introduction Concept of Reachability Concept of Observability Balanced Truncation
Selecting the first r rows and neglecting the X1 terms provide thefollowing reduced order model:
dX̃ (t) = W TAV X̃ (t)dt + W TBu(t)dt + W T ΨV X̃ (t−)dM(t)Ỹ(t) = CV X̃ (t).
Open Questions1 Is the reduced order model stable in general?
deterministic case: The reduced order model is stable.
2 What is the structure of the Gramians of the reduced model?deterministic case: PR = QR = diag(σ1, . . . , σr )
3 What is the structure of the Hankel values of the reduced model?deterministic case: The Hankel values are σ1, . . . , σr .
Max Planck Institute Magdeburg M. Redmann, Balanced Truncation for Stochastic Differential Equations with Levy Noise 17/25
-
Introduction Concept of Reachability Concept of Observability Balanced Truncation
Selecting the first r rows and neglecting the X1 terms provide thefollowing reduced order model:
dX̃ (t) = W TAV X̃ (t)dt + W TBu(t)dt + W T ΨV X̃ (t−)dM(t)Ỹ(t) = CV X̃ (t).
Open Questions1 Is the reduced order model stable in general?
deterministic case: The reduced order model is stable.
2 What is the structure of the Gramians of the reduced model?deterministic case: PR = QR = diag(σ1, . . . , σr )
3 What is the structure of the Hankel values of the reduced model?deterministic case: The Hankel values are σ1, . . . , σr .
Max Planck Institute Magdeburg M. Redmann, Balanced Truncation for Stochastic Differential Equations with Levy Noise 17/25
-
Introduction Concept of Reachability Concept of Observability Balanced Truncation
Selecting the first r rows and neglecting the X1 terms provide thefollowing reduced order model:
dX̃ (t) = W TAV X̃ (t)dt + W TBu(t)dt + W T ΨV X̃ (t−)dM(t)Ỹ(t) = CV X̃ (t).
Open Questions1 Is the reduced order model stable in general?
deterministic case: The reduced order model is stable.
2 What is the structure of the Gramians of the reduced model?deterministic case: PR = QR = diag(σ1, . . . , σr )
3 What is the structure of the Hankel values of the reduced model?deterministic case: The Hankel values are σ1, . . . , σr .
Max Planck Institute Magdeburg M. Redmann, Balanced Truncation for Stochastic Differential Equations with Levy Noise 17/25
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Introduction Concept of Reachability Concept of Observability Balanced Truncation
Example
The following matrices provide a balanced system:
A =(−5.12 2.99 2.05
1.25 −4.86 0.961.26 0.07 −9.52
),B =
(−3.39 −1.19 −0.611.31 −4.60 −0.09−0.68 1.70 4.57
),
Ψ =(−2.20 2.09 −0.57
1.30 −0.67 −2.060.30 −0.57 −1.14
),C =
(−2.77 1.31 1.62−3.94 −2.18 0.74−1.31 −2.49 1.74
).
The Gramians are given by
P = Q = Σ =(
5.97 0 00 4.23 00 0 1.48
).
The reduced order model has the following properties:
PR =(
5.46 −0.38−0.38 3.43
),QR = ( 5.88 0.060.06 4.16 ) ,HVR = (
5.683.76 ) .
Max Planck Institute Magdeburg M. Redmann, Balanced Truncation for Stochastic Differential Equations with Levy Noise 18/25
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Introduction Concept of Reachability Concept of Observability Balanced Truncation
Example
The following matrices provide a balanced system:
A =(−5.12 2.99 2.05
1.25 −4.86 0.961.26 0.07 −9.52
),B =
(−3.39 −1.19 −0.611.31 −4.60 −0.09−0.68 1.70 4.57
),
Ψ =(−2.20 2.09 −0.57
1.30 −0.67 −2.060.30 −0.57 −1.14
),C =
(−2.77 1.31 1.62−3.94 −2.18 0.74−1.31 −2.49 1.74
).
The Gramians are given by
P = Q = Σ =(
5.97 0 00 4.23 00 0 1.48
).
The reduced order model has the following properties:
PR =(
5.46 −0.38−0.38 3.43
),QR = ( 5.88 0.060.06 4.16 ) ,HVR = (
5.683.76 ) .
Max Planck Institute Magdeburg M. Redmann, Balanced Truncation for Stochastic Differential Equations with Levy Noise 18/25
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Introduction Concept of Reachability Concept of Observability Balanced Truncation
Example
The following matrices provide a balanced system:
A =(−5.12 2.99 2.05
1.25 −4.86 0.961.26 0.07 −9.52
),B =
(−3.39 −1.19 −0.611.31 −4.60 −0.09−0.68 1.70 4.57
),
Ψ =(−2.20 2.09 −0.57
1.30 −0.67 −2.060.30 −0.57 −1.14
),C =
(−2.77 1.31 1.62−3.94 −2.18 0.74−1.31 −2.49 1.74
).
The Gramians are given by
P = Q = Σ =(
5.97 0 00 4.23 00 0 1.48
).
The reduced order model has the following properties:
PR =(
5.46 −0.38−0.38 3.43
),QR = ( 5.88 0.060.06 4.16 ) ,HVR = (
5.683.76 ) .
Max Planck Institute Magdeburg M. Redmann, Balanced Truncation for Stochastic Differential Equations with Levy Noise 18/25
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Introduction Concept of Reachability Concept of Observability Balanced Truncation
Error Bound
We introduce the following partitions of a balanced realization:
A =[
A11 A12A21 A22
], Ψ =
[Ψ11 Ψ12Ψ21 Ψ22
], B =
[B1B2
]and C = [ C1 C2 ] .
We write E = E∥∥Y(t)− Ỹ(t)∥∥
2and obtain
E = E∥∥∥∥C ∫ t
0
Φ(t, s)Bu(s)ds − C1∫ t
0
Φ̃(t, s)B1u(s)ds
∥∥∥∥2
≤ E∫ t
0
∥∥∥(CΦ(t, s)B − C1Φ̃(t, s)B1) u(s)∥∥∥2ds
≤ E∫ t
0
∥∥∥CΦ(t, s)B − C1Φ̃(t, s)B1∥∥∥F‖u(s)‖2 ds
≤(E∫ t
0
∥∥∥CΦ(t, s)B − C1Φ̃(t, s)B1∥∥∥2Fds
) 12(E∫ t
0
‖u(s)‖22 ds) 1
2
.
Max Planck Institute Magdeburg M. Redmann, Balanced Truncation for Stochastic Differential Equations with Levy Noise 19/25
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Introduction Concept of Reachability Concept of Observability Balanced Truncation
Error Bound
We introduce the following partitions of a balanced realization:
A =[
A11 A12A21 A22
], Ψ =
[Ψ11 Ψ12Ψ21 Ψ22
], B =
[B1B2
]and C = [ C1 C2 ] .
We write E = E∥∥Y(t)− Ỹ(t)∥∥
2and obtain
E = E∥∥∥∥C ∫ t
0
Φ(t, s)Bu(s)ds − C1∫ t
0
Φ̃(t, s)B1u(s)ds
∥∥∥∥2
≤ E∫ t
0
∥∥∥(CΦ(t, s)B − C1Φ̃(t, s)B1) u(s)∥∥∥2ds
≤ E∫ t
0
∥∥∥CΦ(t, s)B − C1Φ̃(t, s)B1∥∥∥F‖u(s)‖2 ds
≤(E∫ t
0
∥∥∥CΦ(t, s)B − C1Φ̃(t, s)B1∥∥∥2Fds
) 12(E∫ t
0
‖u(s)‖22 ds) 1
2
.
Max Planck Institute Magdeburg M. Redmann, Balanced Truncation for Stochastic Differential Equations with Levy Noise 19/25
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Introduction Concept of Reachability Concept of Observability Balanced Truncation
We have
supt≥0
E∥∥Y(t)− Ỹ(t)∥∥
2
≤(E∫ ∞
0
∥∥∥CΦ(t, s)B − C1Φ̃(t, s)B1∥∥∥2Fds
) 12
︸ ︷︷ ︸#
‖u‖L2 .
It holds
# =(tr(CΣCT
)+ tr
(C1PRC
T1
)− 2 tr
(CPMC
T1
)) 12 ,
where
0 = BBT1 + PMAT11 + APM + ΨPM Ψ
T11 E
[M(1)2
].
Max Planck Institute Magdeburg M. Redmann, Balanced Truncation for Stochastic Differential Equations with Levy Noise 20/25
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Introduction Concept of Reachability Concept of Observability Balanced Truncation
We have
supt≥0
E∥∥Y(t)− Ỹ(t)∥∥
2
≤(E∫ ∞
0
∥∥∥CΦ(t, s)B − C1Φ̃(t, s)B1∥∥∥2Fds
) 12
︸ ︷︷ ︸#
‖u‖L2 .
It holds
# =(tr(CΣCT
)+ tr
(C1PRC
T1
)− 2 tr
(CPMC
T1
)) 12 ,
where
0 = BBT1 + PMAT11 + APM + ΨPM Ψ
T11 E
[M(1)2
].
Max Planck Institute Magdeburg M. Redmann, Balanced Truncation for Stochastic Differential Equations with Levy Noise 20/25
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Introduction Concept of Reachability Concept of Observability Balanced Truncation
Special Case
We assume
Ψ =
[Ψ11 0
0 Ψ22
].
We additionally need the partitions
Σ =
[Σ1
Σ2
]and PM =
[PM,1PM,2
].
Then, PR = QR = Σ1 and
#2 = tr((CT2 C2 + 2PM,2AT21)Σ2).
Max Planck Institute Magdeburg M. Redmann, Balanced Truncation for Stochastic Differential Equations with Levy Noise 21/25
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Introduction Concept of Reachability Concept of Observability Balanced Truncation
How to check stability of the reduced order model?
Theorem see [Damm ’04]
Let Y be an arbitrary symmetric and negative definite matrix. Then, thereduced order model is asymptotically mean square stable, if and only ifthe solution X of
AT11X + XA11 + ΨT11XΨ11 E
[M(1)2
]= Y
is unique, symmetric and positive definite.
Max Planck Institute Magdeburg M. Redmann, Balanced Truncation for Stochastic Differential Equations with Levy Noise 22/25
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Introduction Concept of Reachability Concept of Observability Balanced Truncation
Example
We consider a company, which produces n goods.Xi represents the asset of the company corresponding to good i .
We assume that the vector of all assets satisfies
dX (t) = [u(t)− diag(λ1, . . . , λn)X (t)] dt + X (t−)dM(t)X (0) = x0,
where λi is the depreciation rate of Xi and u is the vector of investments.
We observe that the system is asymptotically mean square stable if andonly if λi > 0.5 E
[M(1)2
]for all i = 1, . . . , n.
The output equation we consider is given by
Y (t) = (1, . . . , 1)X (t).
Max Planck Institute Magdeburg M. Redmann, Balanced Truncation for Stochastic Differential Equations with Levy Noise 23/25
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Introduction Concept of Reachability Concept of Observability Balanced Truncation
Example
We consider a company, which produces n goods.Xi represents the asset of the company corresponding to good i .
We assume that the vector of all assets satisfies
dX (t) = [u(t)− diag(λ1, . . . , λn)X (t)] dt + X (t−)dM(t)X (0) = x0,
where λi is the depreciation rate of Xi and u is the vector of investments.
We observe that the system is asymptotically mean square stable if andonly if λi > 0.5 E
[M(1)2
]for all i = 1, . . . , n.
The output equation we consider is given by
Y (t) = (1, . . . , 1)X (t).
Max Planck Institute Magdeburg M. Redmann, Balanced Truncation for Stochastic Differential Equations with Levy Noise 23/25
-
Introduction Concept of Reachability Concept of Observability Balanced Truncation
Example
We consider a company, which produces n goods.Xi represents the asset of the company corresponding to good i .
We assume that the vector of all assets satisfies
dX (t) = [u(t)− diag(λ1, . . . , λn)X (t)] dt + X (t−)dM(t)X (0) = x0,
where λi is the depreciation rate of Xi and u is the vector of investments.
We observe that the system is asymptotically mean square stable if andonly if λi > 0.5 E
[M(1)2
]for all i = 1, . . . , n.
The output equation we consider is given by
Y (t) = (1, . . . , 1)X (t).
Max Planck Institute Magdeburg M. Redmann, Balanced Truncation for Stochastic Differential Equations with Levy Noise 23/25
-
Introduction Concept of Reachability Concept of Observability Balanced Truncation
Example
We consider a company, which produces n goods.Xi represents the asset of the company corresponding to good i .
We assume that the vector of all assets satisfies
dX (t) = [u(t)− diag(λ1, . . . , λn)X (t)] dt + X (t−)dM(t)X (0) = x0,
where λi is the depreciation rate of Xi and u is the vector of investments.
We observe that the system is asymptotically mean square stable if andonly if λi > 0.5 E
[M(1)2
]for all i = 1, . . . , n.
The output equation we consider is given by
Y (t) = (1, . . . , 1)X (t).
Max Planck Institute Magdeburg M. Redmann, Balanced Truncation for Stochastic Differential Equations with Levy Noise 23/25
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Introduction Concept of Reachability Concept of Observability Balanced Truncation
We assume that n = 80, E[M(1)2
]= 1 and λi ∈ (0.5, 1.5]:
Dimension of reduced order model #
40 7.2277 · 10−620 7.2237 · 10−610 0.00265 0.35172 4.5929
.
Max Planck Institute Magdeburg M. Redmann, Balanced Truncation for Stochastic Differential Equations with Levy Noise 24/25
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Introduction Concept of Reachability Concept of Observability Balanced Truncation
Thank you for your attention!
ReferencesAntoulas ’05: Approximation of large-scale dynamical systems,Advances in Design and Control 6. Philadelphia, PA: Society for Industrialand Applied Mathematics (SIAM).
Applebaum ’09: Lèvy Processes and Stochastic Calculus, CambridgeStudies in Advanced Mathematics 116. Cambridge: Cambridge UniversityPress.
Benner, Damm ’11: Lyapunov equations, energy functionals, and modelorder reduction of bilinear and stochastic systems, SIAM J. ControlOptim., 49(2):686-711, 2011.
Damm ’04: Rational Matrix Equations in Stochastic Control, LectureNotes in Control and Information Sciences 297. Berlin: Springer.
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