L2-optimal model reduction for unstable systems using iterative rational
Krylov algorithm
Caleb Magruder, Christopher Beattie, and Serkan Gugercin
Abstract— Unstable dynamical systems can be viewed from
a variety of perspectives. We discuss the potential of an inputoutput map associated with an unstable system to represent a
bounded map from L2 (R) to itself and then develop criteria
for optimal reduced order approximations to the original (unstable) system with respect to an L2 -induced Hilbert-Schmidt
norm. Our optimality criteria extend the Meier-Luenberger
interpolation conditions for optimal H2 approximation of stable
dynamical systems. Based on this interpolation framework, we
describe an iteratively corrected rational Krylov algorithm for
L2 model reduction. A numerical example involving a hardto-approximate full-order model illustrates the effectiveness of
the proposed approach.
I. INTRODUCTION
We consider single input/single output (SISO) linear dynamical systems described via a state space representation,
(
ẋ(t) = Ax(t) + bu(t)
H:
(1)
y(t) = cT x(t),
where A ∈ Rn×n , b, c ∈ Rn . We consider
R ∞ input functions
u(t) defined on the entire real line with −∞ |u(t)|2 dt < ∞.
To avoid trivialities, we assume, without loss of generality
to our results, that the representation in (1) is a minimal
realization of H. We have particular interest in cases where
the system order, n, is very large and that the system may be
unstable, i.e., A may have some eigenvalues with positive
real parts. Our goal will be to find, for any order r n, a
reduced-order model represented as
(
ẋr (t) = Ar xr (t) + br u(t)
Hr :
(2)
yr (t) = cTr xr (t),
that optimally approximates the full order system (1) with
respect to an error measure described in Section II-C that
extends in a natural way the usual H2 system norm.
Most model reduction methodologies, such as balanced
truncation [10], [11], Hankel norm approximation [7], or optimal H2 approximation [14], were developed originally for
asymptotically stable dynamical systems — systems having
all their poles in the left half-plane. However, there exist
prominent applications where model reduction of unstable
systems becomes a vital tool. One such case is controller
reduction. Controllers are designed to drive a plant into
desirable and robust performance settings, see [15]. However,
many controller design techniques, such as LQG and H∞
This work was supported in part by NSF under DMS-0645347.
C. Magruder, C. Beattie, and S. Gugercin are with the Department of Mathematics, Virginia Tech, Blacksburg, VA, 24061-0123, USA
{calebm, beattie, gugercin}@vt.edu
methods, lead to controllers that have the same order as
the plant to be controlled; see, for example, [13], [15]
and references therein for more details. Large order plants
lead to large order controllers. High-order controllers are
problematic for real-time applications due to the potential for
degraded numerical accuracy, computational lags that may
be difficult to compensate for, and the complex supporting
hardware that becomes necessary. Hence, one would like to
replace the original high-order controller with a low order
but high-fidelity approximation. Since controllers are usually
unstable systems [12], the controller reduction problem leads
directly to a model reduction problem involving unstable
systems. We refer the reader to [4], [5], [12], [15], [17] for
some recent works on model reduction of unstable dynamical
systems. We note that most of these works are based on
balanced truncation, unlike the framework we pursue here
which uses rational Krylov methods and an interpolatory
framework for model reduction.
II. BACKGROUND
A. L2 Systems
Denote bynL2(R) the set of functions
o with finite “energy”:
R ∞
L2 (R) = f −∞ |f (t)|2 dt < ∞ and by Ln2 (R) the
corresponding set of vector-valued functions on R:
Z ∞
2
n
n
kx(t)k dt < ∞ .
L2 (R) = x(t) ∈ R −∞
Define an operator A : Ln2 (R) 7→ Ln2 (R) as A x = ẋ − Ax
defined on all vector-valued functions x(t) ∈ Ln2 (R) having
components that are absolutely continuous, with derivative
ẋ ∈ Ln2 (R) as well. A is densely defined in Ln2 (R) and
is a 1-1 map of its domain onto Ln2 (R) as long as the
matrix A has no imaginary eigenvalues. If A has imaginary
eigenvalues then there will be f ∈ Ln2 (R) such that Ax = f
has no solution in Ln2 (R). Indeed, if A has purely imaginary
eigenvalues at ±ıω0 and Az0 = ıω0 z0 then
cos(ω0 t)
sin(ω0 t)
<e(z0 ) + √
=m(z0 ) ∈ Ln2 (R)
f (t) = √
2
1+t
1 + t2
but every solution to Ax = f grows asymptotically like
kx(t)k ∼ |arcsinh(t)| which grows logarithmically at ±∞;
thus, there can be no solution in Ln2 (R).
Conversely, if A has no imaginary eigenvalues, then for
every f ∈ Ln2 (R) we have an explicit representation of the
unique x ∈ Ln2 (R) that solves Ax = f . In the context of
the dynamical system given in (1), the resulting input-output
map H : u 7→ y then appears as a convolution operator
Z ∞
h(t − τ ) u(τ ) dτ :
y(t) = [cT A−1 b u](t) =
•
−∞
•
if A is stable, i.e., all eigenvalues of A have strictly
negative real part, then variation Rof parameters gives
t
immediately, [A−1 f ](t) = x(t) = −∞ eA(t−τ ) f (τ ) dτ ,
Z ∞
−1
T
h(t − τ ) u(τ ) dτ
y(t) =[c A b u](t) =
if A is unstable, then for any f ∈ Ln2 (R), calculate
[A−1 f ](t) = x(t) = Π+ x(t) + Π− x(t)
Z t
Z ∞
A(t−τ ) +
=
e
Π f (τ ) dτ −
eA(t−τ ) Π− f (τ ) dτ.
−∞
t
The system map H : u 7→ y appears explicitly as
Z ∞
y(t) = [cT A−1 b u](t) =
h(t − τ ) u(τ ) dτ
−∞
−∞
and h(t) =
•
cT eAt b
0
for t ≥ 0
for t < 0
with h(t) =
(3)
if A is antistable, i.e., eigenvalues
R ∞have positive real
parts, then [A−1 f ](t) = x(t) = − t eA(t−τ ) f (τ ) dτ ,
Z ∞
y(t) =[cT A−1 b u](t) =
h(t − τ ) u(τ ) dτ
cT eAt Π+ b
−cT eAt Π− b
for t ≥ 0
for t < 0
(7)
H maps functions u ∈ L2 (R) to y ∈ L2 (R), as before.
However, values of y(t) now depend on both past and future
values of u; h(t) is noncausal. Notice that (7) reduces to
(3) in the stable (causal) case and to (4) in the antistable
(anticausal) case.
−∞
and h(t) =
0
−cT eAt b
for t ≥ 0
for t < 0
B. Frequency Domain Representation
(4)
Note that in (3) the value of y(t) is independent of future
values of u; the corresponding h(t) is “causal”. For (4),
the value of y(t) is independent of past values of u; the
corresponding h(t) is “anticausal”.
The case where A is a general unstable matrix without
purely imaginary eigenvalues, i.e., some eigenvalues have
strictly positive real parts and the remaining have strictly
negative parts, is only slightly more involved. Let U +
and U − be maximal invariant subspaces of A associated
with stable and antistable eigenvalues, respectively. This
means that if X+ and X− are matrices having columns
that constitute bases for U + and U − : U + = Ran(X+ ) and
U − = Ran(X− ), then dim(U + ) + dim(U − ) = n, the n × n
matrix X = [X+ X− ] is invertible, and there are square
matrices M+ and M− such that
M+
0
+
−
+
−
A[X X ] = [X X ]
(5)
0
M−
with M+ stable and M− antistable (meaning that −M− is
stable). Let Y = (X−1 )T be partitioned as Y = [Y+ Y− ]
to conform with the partitioning of X. Define the stable and
antistable spectral projectors for A, respectively, as
Π+ = X+ (Y+ )T
and
Π− = X− (Y− )T .
To determine Ln2 (R) solutions to Ax = f , we use stable
and antistable spectral projectors to separate (1) into stable
and antistable subsystems with x+ = Π+ x and x− = Π− x:
 +
+
+

 ẋ (t) = Ax (t) + Π bu(t)
−
−
H : ẋ (t) = Ax (t) + Π− bu(t)
(6)


y(t) = cT x+ (t) + x− (t) ,
An explicit representation of Ln2 (R) solutions to Ax = f
can be found via variation of parameters as before for each
of the subsystems. So, finally
Laplace transforms are the usual approach to obtaining a
frequency domain representation of systems having the form
of (1). Although the usual (unilateral) Laplace transform
extends in a natural way to the bilateral Laplace transform:
Z ∞
L[u](s) =
u(t) e−st dt,
−∞
L[u](s) will not exist for <e(s) 6= 0 unless |u(t)| has a
sufficiently rapid decay at either ±∞; u ∈ L2 (R) does not
by itself imply that L[u](s) exists.
Instead, let ŷ(ω) and û(ω) denote Fourier transforms of
y(t) and u(t), respectively. Applying a Fourier transform to
(7) yields after some manipulation:
ŷ(ω) = cT (ıωI − A)−1 Π+ b û(ω)
+ cT (ıωI − A)−1 Π− b û(ω)
= cT (ıωI − A)−1 b û(ω)
= H (ıω) û(ω) + H − (ıω) û(ω) = H(ıω) û(ω),
+
introducing transfer functions H + (s) = cT (sI−A)−1 Π+ b,
H − (s) = cT (sI − A)−1 Π− b, and H(s) = cT (sI − A)−1 b.
Notice that the total transfer function, H(s), splits naturally
into the sum of a stable transfer function, H + (s), and an
antistable transfer function, H − (s) corresponding to the
splitting of A into stable and antistable components.
+
−
Notice also
that the+evident
realizations
for H −andH ,
A
Π
b
A
Π
b
i.e., H + :=
and H − :=
, are
cT
0
cT
0
nonminimal — both systems have the same apparent order as
H itself. Minimal realizations are immediate from the block
decomposition in (5):
H + (s) = cT X+ (sI − M+ )−1 Y+T b :=
H − (s) = cT X− (sI − M− )−1 Y−T b :=
»
M+
c+T
b+
0
–
»
M−
c−T
b−
0
–
where we use c±T = cT X± and b± = (Y± )T b.
(8)
C. The L2 (iR) norm
Definition 2.1: Let L2 (iR)
R ∞ denote the set of meromorphic
functions, G(s) such that −∞ |G(ıω)|2 dω < ∞.
L2 (iR) is a Hilbert space and is relevant here because
transfer functions of the systems we consider are elements
of L2 (iR). If G(s) and H(s) are elements of L2 (iR) that
are real-valued on R (i.e., if they represent real dynamical
systems), then their inner product is defined as
Z ∞
G(ıω)H(ıω)dω
hG, HiL2 =
−∞
Z ∞
=
G(−ıω)H(ıω)dω
(9)
Using the residue theorem, we obtain a form of the L2
inner product that can evaluated as a finite sum. Denote by
res[F (s), µ] the residue of F at µ.
Theorem 2.5: Let G(s), H(s) be in L2 (R). Suppose
H(s) has poles at µ1 , µ2 , . . . , µm labeled so that the first
k poles are stable, {µ1 , . . . , µk } ⊂ C− , and the last m − k
poles are antistable {µk+1 , . . . , µm } ⊂ C+ . Decompose G
and H into stable and antistable parts G = G+ + G− and
H = H + + H − . Then,
hG, HiL2 =
k
X
G− (−µ` ) res[H − (s), µ` ]
`=k+1
Proof:
(10)
−∞
+
Definition 2.2: Let H2 (C ) denote the set of functions,
G(s) that are analytic for s in the open right half plane,
C+ = {s|Re(s) > 0}, such that
Z ∞
sup
|G(x + ıy)|2 dy < ∞.
(11)
−∞
−
Similarly, let H2 (C ) denote the set of functions analytic in
the open left half plane, C− = {s|Re(s) < 0}, such that
Z ∞
sup
|G(x + ıy)|2 dy < ∞
(12)
x<0
m
X
+
and the L2 (iR)-norm of H is
1/2
Z ∞
2
|H(ıω)| dω
kHkL2 =
(13)
i=1
−∞
x>0
G+ (−µi ) res[H + (s), µi ]
−∞
The following well-known result describes an orthogonal
direct sum decomposition of L2 (iR).
Theorem 2.3: L2 (ıR) is an orthogonal direct sum of
H2 (C− ) and H2 (C+ ): L2 (ıR) = H2 (C− ) ⊕ H2 (C+ ).
Proof: Let H ∈ L2 (iR) and suppose that H has poles
in each of the left and right half-planes of C. There exist
H + ∈ H2 (C+ ) and H − ∈ H2 (C− ) so that H = H + + H − .
Notice that the product H + (−s)H − (s) is analytic in the left
half plane. For any R > 0, define a semicircular contour in
the left half-plane:
ΓR = {z|z = iω with ω ∈ [−R, R]}
π 3π
iθ
∪ z|z = Re with θ ∈
,
2 2
Using standard arguments and the Cauchy-Goursat theorem,
Z ∞
hH + , H − iL2 =
H + (−iω)H − (iω)dω
−∞
Z
1
= lim
H + (−s)H − (s)ds = 0
R→∞ 2πı Γ
R
Corollary 2.4: Given L2 systems H, Hr as in (1) and (2).
kH − Hr k2L2 = kH + − Hr+ k2H2 (C+ ) + kH − − Hr− k2H2 (C− )
Thus the quality of the approximation, Hr ≈ H is determined by how well the stable and antistable components of
Hr approximate the corresponding components of H.
hG, HiL2 = hG+ + G− , H + + H − iL2
= hG+ , H + iL2 + hG− , H − iL2
=
k
X
res[G+ (−s)H + (s), µi ]
i=1
m
X
+
res[G− (−s)H − (s), µ` ]
`=k+1
The last equality is an application of the residue theorem. The
conclusion then follows from the observation that G+ (−s)
is analytic in the left half-plane and that G− (−s) is analytic
in the right half-plane. See also Lemma 2.4 of [8].
III. O PTIMAL L2 MODEL REDUCTION .
Given an L2 system, H as in (1), we consider reduced
order systems, Hr of order r as in (2), which are best
approximations to H with respect to the L2 norm:
kH − Hr kL2 =
min
dim(H̃r )=r
kH − H̃r kL2
(14)
For stable systems, the L2 minimization problem (14) reduces to H2 minimization. For the analysis of that special
case, see [8], [14], [9] and the references therein.
Note that the model reduction problem we consider here is
different from the finite-horizon model reduction approaches
[4], [17] used for unstable dynamical systems where the fullorder model H remains causal but as a consequence is not
a mapping from L2 (R) to L2 (R). Since the output y(t) can
grow without bound, only a finite-horizon time window can
be considered.
The set of transfer functions associated with rth-order dynamical systems is not convex, so the optimal approximation
problem (14) allows for multiple minimizers. Indeed there
may be local minimizers that do not solve (14).
Definition 3.1: A reduced order system, Hr , is a local
minimizer for (14) if, for all > 0 sufficiently small,
kH − Hr kL2 ≤ kH − H̃r() kL2
()
(15)
()
for all dynamical systems H̃r with dim(H̃r ) = r and
()
kHr − H̃r kL2 ≤ C for some constant C.
Theorem 3.2: Suppose H ∈ L2 and Hr is a local L2 minimizer to H in the sense of (15). Suppose further that
Hr has simple poles: {λ̃1 , . . . , λ̃r }. Then
hH − Hr , s−1λ̃ iL2 = 0
and
`
Hr , (s−1λ̃ )2 iL2
`
Next, note that H + (−s) − Hr+ (−s) is analytic at s = λ̃` :
H + (−s) − Hr+ (−s) H + (−λ̃` ) − Hr+ (−λ̃` )
=
+
(s − λ̃` )2
(s − λ̃` )2
H +0 (−λ̃` ) − Hr+0 (−λ̃` )
+ ...
s − λ̃`
(16)
hH −
=0
The proof follows closely the pattern of proof described
in Theorem 5 of [3] (for stable systems), and is only
summarized here.
Proof: The definition (15) leads to
so we also have
0 =hH − Hr ,
kH − Hr k2L2 ≤kH − H̃r() k2L2
=
≤kH − Hr + Hr − H̃r() k2L2
≤kH −
Hr k2L2
+ kHr −
+ 2hH − Hr , Hr −
k
X
j=1
H̃r() k2L2
H̃r() iL2
1
iL2
(s − λ̃` )2
res[
H + (−s) − Hr+ (−s)
, λ̃j ]
(s − λ̃` )2
= H +0 (−λ̃` ) − Hr+0 (−λ̃` )
.
Similar arguments hold for k + 1 ≤ ` ≤ r.
1
So, 0 ≤ hH − Hr , Hr −
+ kHr − H̃r() k2L2 .
2
()
By choosing H̃r appropriately one can obtain (16): pick
H̃r() iL2
Hr −H̃r() =
±
± res[Hr , λ̃` ]
;
and Hr −H̃r() =
s − λ̃`
(s − λ̃` )(s − λ̃` ± )
then let → 0.
A. Interpolation-based optimality conditions
We now offer new interpolatory L2 optimality conditions
that extend the interpolatory optimal H2 conditions [9], [8]
from stable dynamical system settings to unstable ones.
Theorem 3.3: Given an L2 -system, H(s), as described in
(1), let Hr (s) be a local minimizer of dimension r for the
optimal L2 model reduction problem (14). Suppose further
that Hr (s) has simple poles, {λ̃i }r1 , ordered in such a way
that the first k poles are stable and the last r − k poles are
antistable: {λ̃1 , . . . , λ̃k } ⊂ C− and {λ̃k+1 , . . . , λ̃r } ⊂ C+ .
Then
Hr+ (−λ̃i ) = H + (−λ̃i ) and
˛
˛
d H + ˛˛
d Hr+ ˛˛
=
ds ˛s=−λ̃i
ds ˛s=−λ̃i
for i = 1, . . . , k
(17)
Hr− (−λ̃j ) = H − (−λ̃j ) and
=
Hr+ (σi )
d H + d Hr+ =
ds s=σi
ds s=σi
+
= H (σi ) and
for i = 1, . . . , k
Hr− (σj ) = H − (σj ) and
d Hr− ”
H + (−λ̃j ) − Hr+ (−λ̃j ) res[
1
, λ̃j ]
(s − λ̃` )
r
“
”
X
H − (−λ̃j ) − Hr− (−λ̃j ) res[
j=k+1
= H + (−λ̃` ) − Hr+ (−λ̃` ).
ds s=σj
(18)
dH =
.
ds s=σj
−
for j = k + 1, . . . , r
Notice that
1
iL2
s − λ̃`
j=1
+
Consider the system H described by A, b, c as in (1),
with associated stable and antistable quantities M± , b± , and
c± as described in (8). The interpolatory model reduction
problem involves finding a system (2) so that Hr (s) interpolates H(s) (perhaps also derivative values), at selected
interpolation points that are designated by “shifts”, {σi }ri=1 .
The conditions for optimal L2 approximation described in
(17) also involve an additional feature: Hermite interpolation
is necessary for both stable and antistable subsystems.
Toward this end, suppose two sets of distinct shifts are
given: {σi }ki=1 ⊂ C+ and {σi }ri=k+1 ⊂ C− , that are each
closed under conjugation (i.e. so that shifts within each set
are either real or occur in conjugate pairs). We wish to find
a reduced order system Hr (s) with stable and antistable
components, Hr+ (s) and Hr− (s), respectively, so that
1
if j = `
Proof: Evidently,
=
.
0 otherwise
Pick an index 1 ≤ ` ≤ k. From (13) and (16), we find
res[ s−1λ̃ , λ̃j ]
`
k “
X
A. The Interpolation Problem
˛
˛
d Hr− ˛˛
d H − ˛˛
=
ds ˛s=−λ̃j
ds ˛s=−λ̃j
for j = k + 1, . . . , r
0 =hH − Hr ,
IV. I TERATED INTERPOLATION .
1
, λ̃j ]
(s − λ̃` )
H ± (s) = cT (sI − A)−1 Π± b = c±T (sI − M± )−1 b±
d H±
= −cT (sI − A)−2 Π± b = −c±T (sI − M± )−2 b± ,
ds
so we may form reduced order interpolants to the stable and
antistable components H ± (s) independently based on the
stable/antistable components of A.
−
−
Define matrices Vk+ , Wk+ , Vr−k
and Wr−k
as
ˆ
˜
Vk+ = (σ1 I − M+ )−1 b+ , . . . , (σk I − M+ )−1 b+ R+
2 +T
3
c (σ1 I − M+ )−1
6
7
..
Wk+T = ST+ 4
5
.
+T
+ −1
c (σk I − M )
ˆ
˜
−
Vr−k
= (σk+1 I − M− )−1 b− , . . . , (σr I − M− )−1 b− R−
2 −T
3
c (σk+1 I − M− )−1
6
7
−T
..
Wr−k
= ST− 4
(19)
5.
.
−T
− −1
c (σr I − M )
S± and R± represent (invertible) change-of-bases matrices.
Since the shifts are distinct and closed under conjugation,
−T
−
Wk+T Vk+ and Wr−k
Vr−k
are invertible and S± and R±
−
−
can be chosen so that Vk+ , Wk+ , Vr−k
and Wr−k
are real
+T +
−T
−
matrices, Wk Vk = Ik , and Wr−k Vr−k = Ir−k .
Corollary 4.1: Suppose distinct shifts {σi }ri=1 are given
as described above and suppose real matrices Vk+ , Wk+ ,
−
−
Vr−k
and Wr−k
are computed as described in (19).
Define Hr (s) = Hr+ (s) + Hr− (s)
−1
with Hr+ (s) =c+T Vk+ (sI − M+
Wk+ b+
k)
−
−
−1
and Hr− (s) = c−T Vr−k
(sI − M−
Wr−k
b−
r−k )
+T
−
−T
− −
+ +
where M+
k = Wk M Vk and Mr−k = Wr−k M Vr−k .
Then Hr satisfies the interpolation conditions in (18).
The proof is omitted but follows directly from the related
interpolation properties true for rational Krylov subspaces
used to reduce stable systems.
B. Proposed Algorithm (L2-IRKA)
The L2 optimality conditions (17) reveal that Hr+ and Hr−
are Hermite interpolants to H + and H − at mirror images of
the poles of Hr+ and Hr− , respectively. Hence, as in the case
of the H2 problem, the optimal interpolation points depend
on a reduced system yet to be computed and are not known
a priori. The strategy we propose iteratively corrects the
interpolation points until the necessary conditions are met.
The resulting Algorithm 4.2 outlined here is inspired by the
Iterative Rational Krylov Algorithm (IRKA) of [8].
Algorithm 4.2: I TERATIVE RATIONAL K RYLOV ALGORITHM
FOR
L2 - OPTIMAL MODEL REDUCTION (L2-IRKA).
update k to be the total number of shifts in C+ ;
relabel the shifts so that {σ1 , . . . , σk } ⊂ C+ and
{σk+1 , . . . , σr } ⊂ C− .
−
−
c) Compute and update Vk+ , Wk+ , Vr−k
and Wr−k
according to (19).
+T
−
−T
+ +
− −
5) M+
k = Wk M Vk , Mr−k = Wr−k M Vr−k ,
−
−T
+T +
−
b+
k = Wk b , br−k = Wr−k b ,
−
+T
+
−T
ck = c+T Vk , and cr−k = c−T Vr−k
.
Upon convergence Hr = Hr+ + Hr− , will satisfy the L2
optimality conditions (17). We observe convergence behavior
similar to that of IRKA; alternative stopping criteria continue
to be studied. For example, careful use of system error norms
as opposed to relative shift change may be advantageous.
The final reduced order model is given as

+
+ +
+

 ẋk (t) = Mk xk (t) + bk u(t)
−
−
−
Hr : ẋ−
(20)
r−k (t) = Mr−k xr−k (t) + br−k u(t)


+T +
−T −
yr (t) = ck xk (t) + cr−k xr−k (t),
Unlike the Iterative Rational Krylov Algorithm of [8],
our proposed method iterates on two sets of interpolation
points, originating at each step from stable and antistable
reduced order poles. A key feature of Algorithm 4.2 is that
it adjusts the number of stable poles (k) and unstable poles
(r − k) during the iteration so that the user does not need
to determine this beforehand. This is similar to the balanced
truncation method of [15] where, for a given r, dimensions
of Hr+ and Hr− are chosen according to the Hankel singular
values of H + and H − . One need not specify the orders of
Hr+ and Hr− ; they are chosen automatically by the algorithm.
In the form we have presented, Algorithm 4.2 will carry
a practical restriction on system size due to the computation
of the block decomposition (5) in Step 1. The balanced truncation method of [15] carries a similar limitation. Even so,
system orders of a few thousand will present little difficulty
and circumstances are even more favorable if either the stable
or antistable invariant subspace is of modest dimension and
does not grow with overall system order — note that (5) does
not require a full eigendecomposition for A. Modifications to
Algorithm 4.2 that can take advantage of these circumstances
will be evaluated in future work.
V. A NUMERICAL EXAMPLE
1)
2)
3)
4)
A b
Decompose the full order system H :=
cT 0
into minimal
stable and
antistable subsystems:
M+ b+
M− b−
+
−
H :=
and H :=
.
c+T
0
c−T
0
Make an initial selection of σi for i = 1, . . . , r that is
closed under conjugation and ordered in such a way
that {σ1 , . . . , σk } ⊂ C+ and {σk+1 , . . . , σr } ⊂ C− .
Fix a convergence tolerance.
−
−
Compute Vk+ , Wk+ , Vr−k
and Wr−k
according to (19).
while (relative change in σi > tol)
+T
−
−T
+ +
− −
a) M+
k = Wk M Vk , Mr−k = Wr−k M Vr−k
b) Update the shifts:
−
{σ1 , . . . , σr } = {−λ(M+
k )} ∪ {−λ(Mr−k )};
We illustrate the performance of our proposed method
on an unstable model having 80 stable and 20 antistable
poles. The pole distribution, for both the stable and antistable
poles, is chosen to reflect a condenser distribution, making
the system very hard to reduce (see [6] for more details).
The normalized Hankel singular values, σk /σ1 , defined in
[16] for unstable systems, are depicted in Figure 1. The
slow decay of the singular values confirms that the system is
hard to approximate. Indeed, only near k ≈ 50 does the
normalized Hankel singular value, σk /σ1 , pass the 10−3
level (recall the full system order is 100).
We reduce the order of the system for r = 2 up through
r = 30, in increments of 2, using our Algorithm 4.2 and
compare with the balanced truncation method of [16]. The
)
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,
"
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(
ω∈R
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kHkL∞ = sup | H(ıω) | .
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resulting L2 error for every r is plotted in Figure 2. As the
figure illustrates, our proposed method consistently yields
better L2 performance than balanced truncation.
Even though our proposed method is geared towards L2
model reduction, we also investigate its performance in terms
of the the L∞ norm, which is defined as
Figure 3 depicts the resulting relative L∞ errors for both
balanced truncation and our proposed method as r varies
from 2 to 30. Even though the balanced truncation method
of [15] is precisely intended for L∞ -based approximation,
our proposed L2 -based method performs as well as balanced
truncation even in terms of the L∞ norm. Indeed, Algorithm
4.2 outperformed balanced truncation with respect to the L∞
error measure in 11 out of 15 cases. Balanced truncation was
better only in four cases: r = 4, r = 10, r = 14 and r =
18. This further supports the effectiveness of our proposed
approach for model reduction of unstable systems.
The Bode plots of the full-order model H, and the two
reduced-order models for r = 30 are plotted in Figure 4.
Both reduced models show a good match with the original
model, especially for lower frequencies. The Bode plots of
the two error systems are shown in Figure 5 and illustrate that
our proposed method can yield better L∞ error performance
– notice that balanced truncation produces a higher peak.
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Relative L2 error as r varies
Fig. 2.
This naturally extends the Meier-Luenberger interpolation
conditions for optimal H2 approximation. Based on these
interpolatory L2 optimality conditions, we developed an iteratively corrected rational Krylov algorithm that successively
adjusts the interpolation points until the necessary optimality
conditions are reached.
VI. CONCLUSIONS
R EFERENCES
By representing an unstable dynamical system as a noncausal bounded input-output map from L2 (R) to itself, we
are able to derive necessary conditions for optimal model reduction of the original system with respect to an L2 -induced
Hilbert-Schmidt norm. The optimality conditions reveal that
stable and antistable components of the optimal reducedorder model must be Hermite interpolants to the corresponding components of the original model at the mirror images
of the stable and antistable reduced-order poles, respectively.
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!
./01234/),!)5((6(
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)
&!
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7108)9(:;<8
,"!=.>?
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)**)+)!)+()**, )-)**)+)**,
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#!
$!
%!
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'!
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)!
*!
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Fig. 1.
Decay of Hankel singular values
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+
Fig. 3.
Relative L∞ error as r varies
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67.)+897:0+7'+:;)+'<99!7(.)(+-=.+().<1).!7(.)(+>7.)90
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+
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+3+4,5!2+3+
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Fig. 4.
Bode plots of H and the reduced models for r = 30
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67.)+897:0+7'+:;)+)((7(+<7.)90
+
6-9=+>(?@1=
A%!BCDE
!
+3+4,5!2+!+4(,5+!2+3+
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"
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!%
!" +
!&
!"
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!$
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!%
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!!
!"
"
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'()*+,(-./0)12
Fig. 5.
Bode plots of the error models for r = 30
!
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