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CME 345: MODEL REDUCTION - Balanced Truncation
CME 345: MODEL REDUCTION
Balanced Truncation
David Amsallem & Charbel FarhatStanford University
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
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CME 345: MODEL REDUCTION - Balanced Truncation
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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CME 345: MODEL REDUCTION - Balanced Truncation
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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CME 345: MODEL REDUCTION - Balanced Truncation
Reachability and Observability
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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CME 345: MODEL REDUCTION - Balanced Truncation
Reachability and Observability
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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CME 345: MODEL REDUCTION - Balanced Truncation
Reachability and Observability
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
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C 3 O C ON
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CME 345: MODEL REDUCTION - Balanced Truncation
Reachability and Observability
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
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CME 345: MODEL REDUCTION - Balanced Truncation
Reachability and Observability
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
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CME 345: MODEL REDUCTION - Balanced Truncation
Reachability and Observability
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).
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CME 345: MODEL REDUCTION - Balanced Truncation
Reachability and Observability
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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CME 345: MODEL REDUCTION - Balanced Truncation
Reachability and Observability
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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CME 345: MODEL REDUCTION - Balanced Truncation
Reachability and Observability
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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CME 345: MODEL REDUCTION Balanced Truncation
Reachability and Observability
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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CME 345: MODEL REDUCTION Balanced Truncation
Reachability and Observability
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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CME 345: MODEL REDUCTION Balanced Truncation
Reachability and Observability
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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Reachability and Observability
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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Reachability and Observability
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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Reachability and Observability
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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Reachability and Observability
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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Reachability and Observability
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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Reachability and Observability
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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CME 345: MODEL REDUCTION - Balanced Truncation
B l d T i
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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 .
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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.
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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
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
David Amsallem & Charbel Farhat Stanford University [email protected] CME 345: MODEL REDUCTION - Balanced Truncation
CME 345: MODEL REDUCTION - Balanced Truncation
The POD method revisited
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The POD method revisited
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
David Amsallem & Charbel Farhat Stanford University [email protected] CME 345: MODEL REDUCTION - Balanced Truncation
CME 345: MODEL REDUCTION - Balanced Truncation
The POD method revisited
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The POD method revisited
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.
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CME 345: MODEL REDUCTION - Balanced Truncation
Numerical Computation of Balanced Truncation Decomposition
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Numerical Computation of Balanced Truncation Decomposition
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
David Amsallem & Charbel Farhat Stanford University [email protected] CME 345: MODEL REDUCTION - Balanced Truncation
CME 345: MODEL REDUCTION - Balanced Truncation
Numerical Computation of Balanced Truncation Decomposition
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Numerical Computation of Balanced Truncation Decomposition
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
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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
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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)
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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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