MODEL REDUCTION

7/27/2019 MODEL REDUCTION

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CME 345: MODEL REDUCTION - Balanced Truncation

CME 345: MODEL REDUCTION

Balanced Truncation

David Amsallem & Charbel FarhatStanford University

[email protected]

These slides are based on the recommended textbook: A.C. Antoulas, Approximation of

Large-Scale Dynamical Systems, Advances in Design and Control, SIAM,ISBN-0-89871-529-6

David Amsallem & Charbel Farhat Stanford University [email protected] CME 345: MODEL REDUCTION - Balanced Truncation
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Outline

1 Reachability and Observability

2 Balancing

3 Balanced Truncation

4 Preservation of Stability

5 Error Bounds

6 The POD method revisited

7 Numerical Computation of Balanced Truncation Decomposition

8 Application

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Reachability and Observability

Systems Considered

The following stable LTI full-order systems are considered:

d

dtx(t) = Ax(t) + Bu(t)

y(t) = Cx(t)

x(0) = x0

x Rn : vector state variables

u Rp : vector of input variables, typically p n

y Rq : vector output variables, typically q n

Recall the solution x(t) of the linear ODE:

x(t) = (t,u; t0, x0) = eA(tt0)x(t0) +

tt0

eA(t)Bu()d, t t0

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Reachability, Controllability and Observability

Definition (1)

A state x Rn is reachable if there exists an input function u(.) of finiteenergy and a time T < such that under that input and zero initialcondition, the state of the system becomes x.

Definition (2)A state x Rn is controllable to the zero state if there exist an inputfunction u(.) and a time T < such that

(T,u; 0, x) = 0n.

Definition (3)

A state x Rn is unobservable if, for all t 0,

y(t) = C(t, 0; 0, x) = 0q.David Amsallem & Charbel Farhat Stanford University [email protected] CME 345: MODEL REDUCTION - Balanced Truncation
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Completely Controllable Dynamical System

Definition (4 - R.E. Kalman 1963)

A linear dynamical system (A,B,C) is completely controllable at timet0 if it is not equivalent, for all t t0 to a system of the type

dx(1)

dt= A(1,1)x(1) + A(1,2)x(2) + B(1)u

dx(2)

dt= A(2,2)x(2)

y(t) = C(1)x(1) + C(2)x(2)

Interpretation: it is not possible to find a coordinate system in whichthe state variables are separated into two groups x(1) and x(2) suchthat the second group is not affected either by the first group or bythe inputs to the system.

Controllability is only a property of (A,B)

This definition can be extended to linear time variant systemsDavid Amsallem & Charbel Farhat Stanford University [email protected] CME 345: MODEL REDUCTION - Balanced Truncation
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Completely Observable Dynamical System

Definition (5)

A linear dynamical system (A,B,C) is completely observable at time t0if it is not equivalent, for all t t0 to any system of the type

dx(1)

dt= A(1,1)x(1) + B(1)u

dx(2)

dt= A(2,1)x(1) + A(2,2)x(2) + B(2)u

y(t) = C(1)x(1)

Dual of the previous definitionInterpretation: it is not possible to find a coordinate system in whichthe state variables are separated into two groups x(1) and x(2) suchthat the second group is not affected either by the first group or theoutputs of the system.Observability is only a property of (A,C)This definition can be extended to linear time variant systems


C 3 O C ON
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Example: Simple RLC Circuit

Equation of the network in terms of the current x1 flowing throughthe inductor and the potential x2 across the capacitor (LC = R = 1).

dx1dt

= 1

Lx1 + u1

dx2dt

= 1

Cx2 + u1

y1 =

1

L x1 +

1

Cx2 + u1David Amsallem & Charbel Farhat Stanford University [email protected] CME 345: MODEL REDUCTION - Balanced Truncation

CME 345 MODEL REDUCTION B l d T i
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Example: Simple RLC Circuit

Change of variable x1 = 0.5(x1 + x2), x2 = 0.5(x1 x2). Then thesystem becomes

dx1

dt

= 1

L

x1 + u1

dx2dt

= 1

Lx2

y1 =2

Lx2 + u1

x1 is controllable but not observablex2 is observable but not controllable


CME 345 MODEL REDUCTION B l d T ti
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Canonical Structure Theorem

Theorem (Kalman 1961)

Consider a dynamical system (A,B,C). Then:(i) there is a state space coordinate system in which the components ofthe state vector can be decomposed into four parts:

x = [x(a), x(b), x(c), x(d)]T

(ii) the sizes na, nb, nc and nd of these vectors do not depend on thechoice of basis (iii) the systems matrices take the form

A =

A(a,a) A(a,b) A(a,c) A(a,d)

0 A(b,b)

0 A(b,d)

0 0 A(c,c) A(c,d)

0 0 0 A(d,d)

,B = B(a)

B(b)

00

,

C =

0 C(b) 0 C(d).


CME 345: MODEL REDUCTION Balanced Truncation
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Canonical Structure Theorem

The four parts can be interpreted as follows

Part (a) is completely controllable but unobservable

Part (b) is completely controllable and completely observablePart (c) is uncontrollable and unobservablePart (d) is uncontrollable and completely observable

This theorem can be extended to the linear time variant case


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Reachable and Controllable Subspaces

Definition (6)

The reachable subspace Xreach Rn of a system (A,B,C) is the setcontaining all the reachable states of the system.

R(A,B) = [B,AB, ,An1B, ]

is the reachability matrix of the system.

Definition (7)

The controllable subspace Xcontr Rn of a system (A,B,C) is the setcontaining all the controllable states of the system


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Reachable and Controllable Subspaces

Theorem

Consider the system (A,B,C). Then

Xreach = im R(A,B) (1)

Corollary

(i) AXreach Xreach

(ii) The system is completely reachable if and only if rank R(A,

B) = n(iii) Reachability is basis independent


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Reachability and Observability Gramians

DefinitionThe reachability (controllability) Gramian at time t< is defined asthe n-by-n symmetric positive semidefinite matrix

P(t) = t

0

eABBeA

d

Definition

The observability Gramian at time t< is the n-by-n symmetric

positive semidefinite matrix

Q(t) =

t0

eA

CCeAd


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Reachability and Observability Gramians

Proposition: The columns of P(t) span the reachability subspaceR(A,B).

Corollary

A system (A,B,C) is reachable if and only if P(t) is symmetric positivedefinite at some time t> 0.


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Equivalence Between Reachability and Controllability

Theorem

The notions of controllability and reachability are equivalent forcontinuous linear dynamical systems, that is

Xreach = Xcontr




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Unobservability Subspace

Definition

The unobservability subspace Xunobs Rn is the set of all unobservable

states of system.

O(C,A) = [C,AC, , (A)iC, ]

is the observability matrix of the system.


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Unobservability Subspace

Theorem

Consider the system (A,B,C). Then

Xunobs = kerO(C,A) (2)

Corollary

(i) AXunobs Xunobs(ii) The system is completely observable if and only if rank O(A,B) = n(iii) Observability is basis independent


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Infinite Gramians

Definition

The infinite reachability (controllability) Gramian is defined as then-by-n symmetric positive semidefinite matrix

P =

0 e

At

BB

e

At

dt

Definition

The infinite observability Gramian is the n-by-n symmetric positive

semidefinite matrixQ =

0

eAtCCeAtdt




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Infinite Gramians

Using the Parseval theorem, the two Gramians can be written in thefrequency domain:

The infinite reachability Gramian

P= 12

Z

(jIn A)1BB(jIn A

)1d

The infinite observability Gramian

Q =1

2Z

(jIn A)1CC(jIn A)

1d


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Infinite Gramians

The two infinite Gramians are solution of the following Lyapunovequations:

for the infinite reachability Gramian

AP+ PA + BB = 0n

for the infinite observability Gramian

AQ + QA + CC = 0n




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Energetic Interpretation

P and Q are respective bases for the reachable and observablesubspaces.

P1 and Q are semi-norms

The inner product based on P1 characterizes the minimal energy

required to steer the state from 0 to x as t

x2P1 = xTP1x.

The inner product based on Q indicates the maximal energyproduced by observing the output of the system corresponding to an

initial state x0 when no input is applied

x02Q = x

T0 Qx0.


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Balancing

Model Reduction Based on Balancing

Model reduction strategy: eliminate the states x that are at thesame time:

difficult to reach, ie require a large energy x2P1

to be reached

difficult to observe, ie produce a small observation energy x2Q

These notions are basis depend

We would like to place ourselves in a basis where both concepts areequivalent, ie where the system is balanced


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Balancing

The Effect of Basis Change on the Gramians

Balancing requires changing the basis for the state using atransformation T GL(n)

x = Tx

Then:

eAtB TeAtT1TB = TeAtBBeA

t BTTeAtT = BeA

tT

The reachability Gramian becomes

P= TPT

CeAt

CT1

TeAt

T1

= CeAt

T1

eAtC TeA

tTTC = TeAtC

The observability Gramian becomes

Q = TQT1


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Balancing

Balancing Transformation

The balancing transformations Tbal and T1bal can be computed as

follows:1 compute the Cholesky factorization P= UU

2 compute the eigenvalue decomposition of UQU

UQU = K2K

where the entries in are ordered decreasingly

3 compute the transformationsTbal =

12 KU1

T1bal = UK 1

2

One can then check that

TbalPTbal = T

bal QT

1bal =

Definition (Hankel Singular Values)

= diag(1, , n) contains the Hankel singular values of thesystem


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Balancing

Variational Interpretation

Computing the balancing transformation Tbal is equivalent tominimizing the following function:

minTGL(n)

f(T) = minTGL(n)

tr(TPT + TQT1)

For the optimal transformation Tbal, the function takes the value

f(Tbal) = 2tr() = 2n

i=1

i

where {i}ni=1 are the Hankel singular values.




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Balanced Truncation

Block Partitioning of the System

Applying the change of variable x = Tbalx, the dynamical system

becomes (A, B, C) where

TbalAT1bal = A, TbalB = B, CT

1bal = C

Let 1 k n. The system can be partitioned in blocks as

A =

A11 A12A21 A22

,

B = B1B2

,C =

C1 C2

The subscripts 1 and 2 denote respectively a dimension of size k andn k



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Balanced Truncation

Block Partitioning of the System

The blocks with subscript 1 correspond to the most observable andreachable states.

The blocks with subscript 2 correspond to the least observable andreachable states.

The reduced-order model of size k obtained by balancedtruncation is then the system

(Ar,Br,Cr) = (A11, B1, C1) Rkk Rkp Rqk

The left and right reduced-order bases are respectively

W = Tbal(:, 1 : k), V = Sbal(:, 1 : k)

where Sbal = T1bal .



P ti f St bilit
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Preservation of Stability

Theorem

Theorem (Stability Preservation)

Consider (Ar,Br,Cr) = (A11, B1, C1), a ROM obtained by balanced

truncation. Then:(i) Ar = A1 has no eigenvalues on the open right half plane.

(ii) Furthermore, if the systems (A11, B1, C1) and (A22, B2, C2) have noHankel singular values in common, Ar has no eigenvalues on theimaginary axis.



Error Bo nds
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Error Bounds

Theorem

Theorem (Error Bounds)

The Balanced Truncation procedure yields the following error bound forthe output of interest.Let {i}

nSVi=1 {i}

ni=1 denote the distinct Hankel singular values of the

system and {i}nSVi=nk+1

the ones that have been truncated. Then

y(t) yr(t)2 2nSV

i=nk+1

iu(t)2.

Equivalently, in terms of the H-norm of the error system

G(s) Gr(s)H 2

nSVi=nk+1

i.

where G(s) and Gr(s) are the full- and reduced-order transfer functions.Equality holds when nk+1 = nSV (all truncated singular values are

equal).David Amsallem & Charbel Farhat Stanford University [email protected] CME 345: MODEL REDUCTION - Balanced Truncation


Error Bounds
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Error Bounds

Theorem

Proof. The proof proceeds in 3 steps

1 Consider the error system E(s) = G(s) Gr(s).

2 Show that if all the truncated singular values are all equal to , then

E(s)H = 2

3 Use this result to show that in the general case

E(s)H 2nSV

i=nk+1

i



The POD method revisited
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POD

Recall the theorem giving the POD basis:Theorem

Let K denote the n-by-n real symmetric matrix

K =

T

0 x(t)x(t)

T

dt.

Let 1 2 n 0 be the eigenvalues of K andi R

n, i = 1, , n their associated eigenvectors

K

i =i

i,

i = 1,

,

n.

Assume k > k+1. The subspace V = span(V) minimizing J( ) is theinvariant subspace of K associated with the eigenvalues 1, , k



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POD for an Impulse Response

For an single impulse input with zero initial condition, the responsefor an LTI system is

x(t) = eAtB

Consequence:

K =

T0

x(t)x(t)Tdt =

T0

eAtBBTeATtdt = P

is the reachability gramian.

Unlike the Balanced Truncation method, the method does not takethe observability gramian into account to determine the ROM: veryobservable states may be truncated.



Numerical Computation of Balanced Truncation Decomposition
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Numerical Methods

Computing the transformations

Tbal = 12 KU1, T1bal = UK

12

is usually ill-advised for numerical stability (computation of inverses).

Instead, it is better advised to use the following procedure:1 start from Cholesky decompositions of the Gramians

P= UU, Q = ZZ

2 Compute the SVD

U

Z = WV

3 LetTbal = V

Z, T1bal = UW



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Cost and Limitations

Balanced Truncation leads to ROM having quality and stabilityguarantees, however:

Computation of the Gramians is expensive as it requires computingthe solution of a Lyapunov equation (O(n3) operations)

As such, Balanced Truncation is generally impractical for largesystems of size n 105



Application
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pp

CD Player System

Goal is to model the position of the lens controlled by a swing arm2-inputs/2-outputs system



Application
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pp

CD Player System

Bode plot of the Full-Order Model (n = 120)

Each column represents one input and each row a different output

200

100

0

100

From: In(1)

T

o:Out(1)

100

105

200

100

0

100

To

:Out(2)

From: In(2)

100

105

Bode Diagram

Frequency (rad/sec)

Magnitude(dB)



Application
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CD Player System

Bode plot of the balanced truncation-based reduced-order models

Each column represents one input and each row a different output

100

50

0

50

100

150From: In(1)

T

o:Out(1)

105

150

100

50

0

50

To:Out(2)

From: In(2)

105

Bode Diagram

Frequency (rad/sec)

Magnitude(dB)

ssFOM

ssBT_5

ssBT_10

ssBT_20

ssBT_30

ssBT_40

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MODEL REDUCTION