balanced truncation for stochastic di erential equations with ......3 concept of observability 4...

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

Introduction Concept of Reachability Concept of Observability Balanced Truncation

Content

1 Introduction

2 Concept of Reachability

3 Concept of Observability

4 Balanced Truncation

Content

1 Introduction

Content

1 Introduction

Content

1 Introduction

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

]

Levy processes

E [M(t)] = 0 and E[M(t)2

]

Levy processes

E [M(t)] = 0 and E[M(t)2

]

Levy processes

E [M(t)] = 0 and E[M(t)2

]

Levy processes

E [M(t)] = 0 and E[M(t)2

]

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

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

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.

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.

Concept of Reachability

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

].

We call Pt :=∫ t

0E[Φ(s)BBT ΦT (s)

P :=

∫ ∞0

E[Φ(s)BBT ΦT (s)

]ds.

Proposition

im Pt = im P.

One can show that

0 = BBT + P AT + A P + Ψ P ΨT E[M(1)2

].

We call Pt :=∫ t

0E[Φ(s)BBT ΦT (s)

P :=

∫ ∞0

E[Φ(s)BBT ΦT (s)

]ds.

Proposition

im Pt = im P.

One can show that

0 = BBT + P AT + A P + Ψ P ΨT E[M(1)2

].

We call Pt :=∫ t

0E[Φ(s)BBT ΦT (s)

P :=

∫ ∞0

E[Φ(s)BBT ΦT (s)

]ds.

Proposition

im Pt = im P.

One can show that

0 = BBT + P AT + A P + Ψ P ΨT E[M(1)2

].

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.

Definition

E [X (T , 0, u)] = x .

Proposition

Definition

E [X (T , 0, u)] = x .

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 .

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 .

Concept of Observability

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.

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.

Y(t) = CΦ(t)x0.

‖Y‖2L2 := E∫ ∞

0

= xT0 Qx0.

Y(t) = CΦ(t)x0.

‖Y‖2L2 := E∫ ∞

0

= xT0 Qx0.

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.

(Ã, B̃, Ψ̃, C̃ ) = (TAT−1,TB,TΨT−1,CT−1),

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 .

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)

).

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)

).

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 .

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 ) .

Example

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

).

P = Q = Σ =(

5.97 0 00 4.23 00 0 1.48

).

PR =(

5.46 −0.38−0.38 3.43

),QR = ( 5.88 0.060.06 4.16 ) ,HVR = (

5.683.76 ) .

Example

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

).

P = Q = Σ =(

5.97 0 00 4.23 00 0 1.48

).

PR =(

5.46 −0.38−0.38 3.43

),QR = ( 5.88 0.060.06 4.16 ) ,HVR = (

5.683.76 ) .

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

.

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

.

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

].

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

].

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).

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.

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).

Example

[M(1)2

]for all i = 1, . . . , n.

Y (t) = (1, . . . , 1)X (t).

Example

[M(1)2

]for all i = 1, . . . , n.

Y (t) = (1, . . . , 1)X (t).

Example

[M(1)2

]for all i = 1, . . . , n.

Y (t) = (1, . . . , 1)X (t).

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

.

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.

IntroductionConcept of ReachabilityConcept of ObservabilityBalanced Truncation

balanced truncation for stochastic di erential equations with ......3 concept of observability 4...

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