TECHNISCHE UNIVERSITÄT BERLIN
Robust control via the computation of permuted graph bases
Volker Mehrmann Federico Poloni
Preprint 2012/19
Preprint-Reihe des Instituts für Mathematik
Technische Universität Berlin
http://www.math.tu-berlin.de/preprints
Preprint 2012/19 June 2012
Robust control via the computation of
permuted graph bases
Volker Mehrmann∗Federico Poloni†
08.06.12
We present a new numerical method for the
γ
-iteration in robust control
based on the extended matrix pencil formulation of [
6
]. The new method
bases the
γ
iteration on the computation of special subspaces associated
with matrix pencils. We introduce a permuted graph representation of these
subspaces, which avoids the known difficulties that arise when the iteration
is based on the solution of algebraic Riccati equations but at the same time
makes use of the special symmetry structures that are present in the problems.
We show that the new method is applicable in many situations where the
conventional methods fail.
1 Introduction
The optimal infinite-horizon output (or measurement) feedback
H∞
control problem
is one of the central tasks in robust control, see, e. g., [
19
,
31
,
39
,
40
]. Despite recent
developments [
6
,
4
,
14
,
15
,
16
,
20
,
25
,
32
,
36
,
38
] some of which are incorporated into
software libraries like SLICOT
1
[
8
,
9
,
21
] or the Matlab Robust Control Toolbox [
2
],
the development of robust numerical methods for the
H∞
control remains a problem [
12
].
Consider the linear dynamical system
˙x=Ax +B1w+B2u, x(t0) = x0,
z=C1x+D11w+D12u, (1)
y=C2x+D21w+D22u,
∗
Institut für Mathematik, MA 4-5, TU Berlin, Straße des 17. Juni 136, D-10623 Berlin, Fed. Rep.
Germany.
. Partially supported by DFG Research Center Matheon
‘Mathematics for key technologies’ in Berlin.
†
Institut für Mathematik, MA 4-5, TU Berlin, Straße des 17. Juni 136, D-10623 Berlin, Fed. Rep.
1See http://www.slicot.org
1
where
A∈Rn×n
,
Bi∈Rn×mi
,
Ci∈Rpi,n
,
Dij ∈Rmi,pj
, with
p1≥m2
and
m1≥p2
.
Here,
x
(
t
),
u
(
t
)and
y
(
t
)are state, input and output vectors respectively,
w
(
t
)is an
additional input term representing noise and errors in the modeled dynamics, and
z
is an
estimation error.
The optimal infinite-horizon
H∞
control problem [
6
,
19
,
40
] consists in designing
a controller for
(1)
such that the closed-loop system minimizes the magnitude of the
response
z
(
t
)with respect to the disturbance
w
(
t
)in the worst case. Mathematically,
one achieves the desired design goal of stabilizing the system and using the freedom
in the design to minimize the influence of the disturbance in the transfer function
Tzw
from
w
to
z
in the
H∞
norm
kTzwk∞
:=
supω∈Rσmax
[
Tzw
(
iω
)], where
σmax
denotes the
maximum singular value.
In the following we study problems with real coefficient matrices, however, all the
results can be formulated in a similar way for complex matrices. Following [
6
], we shall
make the following assumptions.
A1
(
A, B2
)is stabilizable and (
A, C2
)is detectable, i. e.,
rank
[
sI −A, B2
] =
rank
[
sI −
AT, CT
2] = nfor all s∈Cwith nonnegative real part;
A2 D22 = 0, and D12,D21 have full rank;
A3
for each
ω∈R
,
"A−iωI B2
C1D12#
has full column rank and
"A−iωI B1
C2D21#
has full
row rank.
When these assumptions hold, then the following theorem presents a method to estimate
the H∞norm, see [6, 19, 40].
Criteria Theorem 1.
Consider system (1) satisfying assumptions A1–A3. Then for a fixed
γ >
0, there exists an internally stabilizing controller with
kTzwk∞< γ
if and only if the
following conditions hold.
1.
The value
γ
satisfies
γ > max
(
ˆγH,ˆγJ
), where
ˆγH
,
ˆγJ
are the largest real values
such that
RH(ˆγH) = "DT
11
DT
12#hD11 D12i−"ˆγ2
HIm10
0 0#,
RJ(ˆγJ) = "D11
D21#hDT
11 DT
21i−"ˆγ2
JIp10
0 0#,
respectively, are nonsingular (they are well-defined under our assumptions).
2.
There exist positive semidefinite solutions
XH
(
γ
)
, XJ
(
γ
)to the algebraic Riccati
equations
0 = H11XH+XHHT
11 +H21 −XHH12XH,
0 = J11XJ+XJJT
11 +J21 −XJH12XJ(2)
2
associated with the Hamiltonian matrices
H(γ) = "H11 H12
H21 −HT
11#
:= "A0
−CT
1C1−AT#−"B1B2
−CT
1D11 −CT
1D12#R−1
H(γ)"DT
11C1BT
1
DT
12C1BT
2#,
J(γ) = "J11 J12
J21 −JT
11#(3)
:= "AT0
−B1BT
1−A#−"CT
1CT
2
−B1DT
11 −B1DT
21#R−1
J(γ)"D11BT
1C1
D21BT
1C2#.
3. If by ρ(·)we denote the spectral radius of a matrix, then ρ(XHXJ)< γ2,.
Theorem 1 provides computationally feasible criteria to decide whether a value
γ
is smaller or larger than the optimal
γ
value
γopt
=
kTzwk∞
and one can then use a
bisection process to determine
γopt
. However, as such a process approaches
γopt
, the
failure of one of these conditions often means that either one of the two Riccati equations
becomes exceedingly ill-conditioned as it approaches a non-solvable one, or one of the
matrices
RH
,
RJ
becomes singular. Therefore, recently, different numerical methods
have been designed [
4
,
6
,
25
] which avoid the formation of the Hamiltonian matrices and
the solution of the Riccati equations by computing special deflating subspaces for certain
matrix pencils. We will briefly recall this approach in Section 2.
Despite the described difficulties, the approach via the solution of Riccati equations is
still quite popular, in particular, because the symmetry and definiteness of the solutions
can be nicely monitored and allows for low-rank representations [
7
,
18
]. To achieve a
compromise between these nice properties and the avoidance of ill-conditioning, recently,
in [
30
] a variation of the structured doubling algorithm (SDA), which was originally
introduced in [
13
] for the solution of algebraic Riccati equations, has been presented that
achieves improved stability and robustness by using a new representation of the relevant
Lagrangian invariant subspaces of the Hamiltonian matrices in (3) which at the same
time retains the Riccati solution properties. In this paper, we extend these ideas and
adapt them to produce a robust implementation of the
γ
-iteration for
H∞
control. This
requires, in particular, a nontrivial extension of the theory in [
30
] to deal with extended
Lagrangian pencils, and turning the main iteration into a different form, which is not a
variant of SDA but rather a version of the inverse-free sign iteration [3].
The paper is organized as follows. In Section 2 we recall some basic facts about
algebraic Riccati equations, even pencils and Lagrangian subspaces and in Section 3
we introduce permutated graph bases of deflating subspaces of matrix pencils. We
then consider structured versions of these graph bases for Lagrangian subspaces and
Hamiltonian pencils in Section 4, and we extend them to a more general structure (called
partial Lagrangian) in Section 5. In Section 6 we show how this approach applies to
some special even pencils, and show how to extract a suitable Hamiltonian subpencil.
In Section 7 we introduce the algorithm that we use to compute their stable deflating
3
subspaces, and in Section 8 we describe the full
γ
iteration in this setting. Successively,
we present in Section 9 an analysis of why a similar algorithm, the doubling algorithm,
leads to a lower accuracy in the solution of our primary benchmark example. Finally,
conclusions are presented.
2 Algebraic Riccati equations and even pencils
sec:algric
Algebraic Riccati equations are a classic tool in almost all areas of control theory. They
allow for a nice and simple formulation of several results, but using them in numerical
solution methods may be problematic because their solutions may be highly ill-conditioned
even though the problem to be solved is well-conditioned. This is the case in particular
for the
γ
-iteration, where frequently near the optimal
γ
the solution of one the two
Riccati equations in (2) becomes unbounded. Therefore, at least for small-scale dense
problems, it is nowadays common numerical practice to replace the computation of the
solutions to the Riccati equation by the computation of Lagrangian invariant subspaces
for the Hamiltonian matrices in (3), by exploiting the following well-known result, see
e. g. [29], formulated for H(γ), with an obvious analogue for J(γ).
riclag Lemma 2.
The algebraic Riccati equation with Hamiltonian matrix
H
(
γ
)has a sym-
metric positive semidefinite solution
XH
(
γ
)if and only if
"I
XH(γ)#
is the Lagrangian
(semi-)stable invariant subspace of H(γ).
Here we call an invariant subspace (semi-)stable if it is associated with the eigenvalues
in the (closed) left half plane. An
n
-dimensional subspace of
U ⊂ R2n
is called Lagrangian
if uTJ2nv= 0 for all u, v ∈ U, where
J2n:= "0In
−In0#.
It is an interesting observation that we do not actually need
XH
and
XJ
to verify the
last two conditions of Theorem 1, but rather only a basis for the invariant subspace.
Theorem 3.
[
6
] Consider the Hamiltonian matrices (3) and associated Lagrangian
thmY
invariant subspaces
"YH
ZH#,"YJ
ZJ#,(4) invsubs
with all blocks in Rn,n.
1. There exist symmetric matrices XH,XJsuch that
"In
XH#,"In
XJ#
span the same invariant subspaces as
(4)
(respectively) if and only if
YH, YJ
are
invertible. In this case,
XH
=
ZHY−1
H
and
XJ
=
ZJY−1
J
are symmetric solutions
4
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