On the Geometry and Parametrization of Almost Invariant Subspaces and Observer Theory

206 Pages·2012·0.88 MB·English

On the Geometry and Parametrization of
Almost Invariant Subspaces and Observer
Theory
Dissertation zur Erlangung des
naturwissenschaftlichen Doktorgrades
der Bayerischen Julius-Maximilians-Universita˜t Wu˜rzburg
vorgelegt von
Jochen Trumpf
aus
Buchen
Wu˜rzburg 2002

Acknowledgements
I want to thank Prof. Dr. U. Helmke for introducing me to the exciting
fleld of geometric control theory as well as for his always open ear dur-
ing his supervision of the last flve years of my work in the fleld. Special
thanks go to Prof. Dr. P.A. Fuhrmann for our valuable discussions during
my stays in Beer Sheva and San Diego. I feel indebted to the people of the
Math Department of the University of Wu˜rzburg, who created a pleasant
and family like working atmosphere. Especially, I want to mention I. Bo˜hm,
Dr. G. Dirr, Prof. Dr. R. Go˜b, Dr. R. Greiner, Prof. Dr. T. Grundho˜fer,
Dr.K.Hu˜per, M.Kleinsteuber, Dr.L.Kramer, K.Krumpholz, Dr.N.Rosehr
and Dr. O. Roth.
This work has been partially supported by a grant from the German-Israeli
Foundation GIF I-526-034.06/97.

Contents
Contents
1 Introduction 7
2 Almost invariant subspaces 13
2.1 Almost (A;B)-invariant subspaces . . . . . . . . . . . . . . . 13
2.1.1 Subspace algorithms I . . . . . . . . . . . . . . . . . 18
2.1.2 Feedback transformations . . . . . . . . . . . . . . . 20
2.1.3 Restricted and corestricted systems I . . . . . . . . . 22
2.2 Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.3 Almost (C;A)-invariant subspaces . . . . . . . . . . . . . . . 28
2.3.1 Subspace algorithms II . . . . . . . . . . . . . . . . . 31
2.3.2 Output injections . . . . . . . . . . . . . . . . . . . . 32
2.3.3 Restricted and corestricted systems II . . . . . . . . . 35
3 Observers 39
3.1 Identity observers . . . . . . . . . . . . . . . . . . . . . . . . 39
3.2 Tracking observers . . . . . . . . . . . . . . . . . . . . . . . 41
3.2.1 Singular (partially) tracking observers . . . . . . . . . 45
3.2.2 Uniqueness vs. pole assignment . . . . . . . . . . . . 55
3.2.3 Tracking output observers . . . . . . . . . . . . . . . 62
3.2.4 Tracking observers with output . . . . . . . . . . . . 64
3.3 Asymptotic observers . . . . . . . . . . . . . . . . . . . . . . 66
3.3.1 Asymptotic output observers . . . . . . . . . . . . . . 77
4 Moduli spaces of linear systems 89
4.1 Controllable pairs . . . . . . . . . . . . . . . . . . . . . . . . 89
4.1.1 Topology of „-regular pairs . . . . . . . . . . . . . . 91
4.2 Controllable triples . . . . . . . . . . . . . . . . . . . . . . . 95
4.3 The manifold of tracking observers . . . . . . . . . . . . . . 98
5 Kernel representations 105
5.1 The tight case . . . . . . . . . . . . . . . . . . . . . . . . . . 105
5.2 The instantaneous case . . . . . . . . . . . . . . . . . . . . . 111
5
Contents
5.3 The O = Fn case . . . . . . . . . . . . . . . . . . . . . . . 119
⁄
5.4 The observability case . . . . . . . . . . . . . . . . . . . . . 124
5.5 The restriction pencil . . . . . . . . . . . . . . . . . . . . . . 126
5.6 Partial realizations . . . . . . . . . . . . . . . . . . . . . . . 128
5.7 Observers II . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
6 Image representations 147
6.1 Brunovsky and Kronecker strata . . . . . . . . . . . . . . . . 147
6.2 Orders on combinations . . . . . . . . . . . . . . . . . . . . 158
6.3 A cell decomposition . . . . . . . . . . . . . . . . . . . . . . 164
A Topology of orbit spaces 177
B On the difierentiability of vector bundles 181
Bibliography 187
Notation 195
Index 199
6
Chapter 1
Introduction
The concept of an invariant subspace is the basic tool in the structure theory
of linear dynamical systems. Consider for example a linear flnite-dimensional
time-invariant ODE
x_ = Ax ; (1.1)
where x 2 Rn and A 2 Rn£n. An invariant subspace of this system is a linear
subspace V ‰ Rn with the property that every trajectory starting in V will
stay in V in the future. Apparently the system can be restricted to such a
subspace. As is well known, the invariant subspaces for the system (1.1) are
exactly the invariant subspaces of the linear operator A : Rn ¡! Rn, i.e. the
subspaces V ‰ Rn with AV ‰ V.
Whatiscalledthegeometricapproachtolinearsystemstheoryistheattempt
to apply the same idea to linear control systems. Consider, for example, a
linear flnite-dimensional time-invariant control systems in state space form
x_ = Ax+Bu ;
(1.2)
y = Cx :
A subspace V of the state space is called (A;B)-invariant or controlled invari-
ant, if for every starting point in V there exists a control input such that the
corresponding trajectory stays in V in the future. One of the most important
results of linear systems theory says that such a control input can always be
choseninaspecialform, namelyconstantstatefeedbacku(t) := Fx(t). Inal-
gebraic terms this means that V is (A;B)-invariant if and only if there exists
F such that V is (A+BF)-invariant, i.e. (A+BF)V ‰ V holds. Applying
7
1 Introduction
constant state feedback makes the (A;B)-invariant subspace dynamically in-
variant, such that the control system can be restricted to the subspace. An
equivalent condition is AV ‰ V+ImB, i.e. that V is A-invariant up to ImB.
While (A;B)-invariant subspaces address the input side of a control system
an analogous formation of a concept on the output side results in the notion
of (C;A)-invariant or conditioned invariant subspaces. A subspace V of the
state space is called (C;A)-invariant, if A(V \ KerC) ‰ V holds. This is
the case if and only if there exists J such that V is (A ¡ JC)-invariant,
i.e. (A¡JC)V ‰ V holds. Apparently, a (C;A)-invariant subspace can be
made dynamically invariant by applying constant output injection. Then the
control system can be restricted to the subspace.
Fromthepointofviewoflinearalgebratheconceptsof(A;B)-invarianceand
(C;A)-invariance are dual to each other. A subspace V is (A;B)-invariant
(i.e. controlled invariant) if and only if its orthogonal complement V? with
respect to the standard inner product is (B⁄;A⁄)-invariant (i.e. conditioned
invariant). In system theoretic terms this means that a subspace is (A;B)-
invariant with respect to the system (1.2) if and only if its orthogonal com-
plement is (C;A)-invariant with respect to the dual system. In view of du-
ality, restrictions to (A;B)-invariant subspaces translate to corestrictions to
(C;A)-invariant subspaces, i.e. restrictions to the factor space with respect
to the given subspace.
A natural question now is what can be said about the set of all invariant sub-
spaces for a given flxed system. In the case of a linear ODE of the form (1.1),
i.e. in the case of a linear operator A, the geometry of the set of A-invariant
subspaces is well understood. It can easily be seen to be a compact algebraic
subvariety of the Grassmann manifold it is part of. Most of the literature
in the fleld treats the more general case of A-invariant partial (cid:176)ags, which
includes the case of A-invariant subspaces as the special case of (cid:176)ag length
one. Steinberg [Ste76] obtained some results about the dimension of the va-
riety and the number of irreducible components. His results imply that the
variety is connected when A is nilpotent. For the case of (cid:176)ag length one,
connectedness was already proved by Douglas and Pearcy [DP68]. Shimo-
mura [Shi80] constructed a stratiflcation of the variety into a flnite number
of a–ne spaces indexed by a Young diagram. In [Shi85] he studied the
irreducible components of these strata. In the case of (cid:176)ag length one Shay-
man [Sha82] constructed a stratiflcation of the variety in a flnite number of
smooth submanifolds of the Grassmannian. The strata consist of the sub-
spaces with flxed cyclic structure, i.e. flxed Jordan type of the restriction of
A to the subspace. Shayman showed that these strata in general are neither
8
1 Introduction
unions nor intersections of Schubert varieties. Helmke and Shayman [HS87]
generalized these results to arbitrary (cid:176)ag length. Furthermore, they showed
that each Jordan stratum has a bi(cid:176)ag manifold as a strong deformation re-
tract, which allows to identify the homology type of the strata as that of a
product of Grassmannians. In the case of (cid:176)ag length one, the bi(cid:176)ag manifold
is a generalized partial (cid:176)ag manifold. Helmke and Shayman used the Schu-
bert cell decomposition of that (cid:176)ag manifold to calculate the Betti numbers
of the Jordan strata in this case.
The structure of the set of all (C;A)-invariant subpaces for a given flxed con-
trol system is considerably more complicated and not fully understood, yet.
In a remarkable paper Hinrichsen, Mu˜nzner and Pra˜tzel-Wolters [HMP81]
associated to each (C;A)-invariant subspace a module of Laurent series and
parametrized these modules using their so-called Kronecker-Hermite bases.
This leads to a parametrization of (C;A)-invariant subspaces, which depends
ontherestrictionindices, i.e. theobservabilityindicesoftherestrictionofthe
pair(C;A)tothesubspace. Thispaperalsocontainsarudimentarystructure
theory for (C;A)-invariant subspaces. In unpublished work based on these
ideas Mu˜nzner constructed a stratiflcation of the set of (C;A)-invariant sub-
spaces into smooth submanifolds of the Grassmannian. He also constructed
cell decompositions of these strata.
Polynomial descriptions of (A;B)-invariant and (C;A)-invariant subspaces
have also been derived by Fuhrmann and Willems [FW79, FW80], Emre and
Hautus [EH80] and Fuhrmann [Fuh81], focussing rather on characterizations,
not so much on parametrization.
During the last flve years there has been a sort of a revival of interest in
parametrization issues in geometric control theory. Fuhrmann and Helmke
[FH97, FH00, FH01] constructed a smooth map from the manifold of similar-
ity classes of „-regular controllable pairs of the appropriate size onto the set
V (C;A) of all (C;A)-invariant subspaces of codimension k of a flxed control
k
system of the form (1.2). They showed that this map restricts to a difieomor-
phism between the submanifold of „-tight pairs and the subset T (C;A) of
k
tight subspaces. The map is given in terms of kernels of truncated and per-
muted reachability matrices (hence the name kernel representation), whose
entries are directly related to corestrictions of the given control system to the
(C;A)-invariant subspaces under consideration. The exact relation is stated
and proved in this thesis, see the discussion below Proposition 5.30. Fur-
thermore, analogous mappings between sets of restricted system equivalence
classesofcontrollabletriplesandsetsofalmost(C;A)-invariantsubspacesare
constructed (Sections 5.2 and 5.3). Almost (C;A)-invariant subspaces were
9
1 Introduction
introduced in a series of papers by Willems [Wil80, Wil81, Wil82]. They sat-
isfy the invariance condition only up to an arbitrarily small " (in the metric
of the state space).
Ferrer, F. Puerta and X. Puerta [FPP98, FPP00] constructed a stratiflcation
ofV (C;A)intosmoothsubmanifoldsoftheGrassmannmanifoldofcodimen-
k
sion k subspaces. These so called Brunovsky strata consist of the subspaces
with flxed restriction indices. F. Puerta, X. Puerta and Zaballa [PPZ00]
explicitely described coordinate atlases for the strata, in [PPZ01] they con-
structed cell decompositions. While kernel representations are related to
corestrictions, these are not used in [FPP98, FPP00, PPZ00, PPZ01]. In-
stead, the stratiflcation and cell decomposition results of Ferrer, F. Puerta,
X. Puerta and Zaballa make use of restrictions of control systems to (C;A)-
invariant subspaces. They lead to image representations, i.e. the subspaces
are parametrized via columnspans of block Toeplitz type matrices. Such de-
scriptions have also been derived by Fuhrmann and Helmke [FH00, FH01],
using submodules of polynomial models to describe (C;A)-invariant sub-
spaces.
X. Puerta and Helmke [PH00] showed that the Brunovsky strata have gen-
eralized (cid:176)ag manifolds as strong deformation retracts, thus deriving efiective
formulas for the Betti numbers of the strata. Inspired by this paper, the
stratiflcation and cell decomposition results of Ferrer, F. Puerta, X. Puerta
and Zaballa are re-derived and simplifled in this thesis in the special case
of tight subspaces using a difierent approach involving Kronecker forms and
unipotent transformations (Chapter 6). In this way the known cell decom-
position of T (C;A) is shown to be induced by a Bruhat decomposition of
k
a generalized (cid:176)ag manifold. Furthermore, the adherence order of the cell
decomposition is identifled.
One of the di–culties arising in the study of the geometry and the topology
of V (C;A) is that this set is in general not compact, i.e. not closed in
k
the Grassmannian. A flrst attempt at obtaining a compactiflcation might
be to extend the set by including all almost (C;A)-invariant subspaces. An
˜ ˜
example due to Ozveren, Verghese and Willsky [OVW88] shows that this
attempt fails by constructing a subspace in the closure of all (C;A)-invariant
subspaces which is not almost (C;A)-invariant. With a difierent approach it
is however possible to construct a compactiflcation, at least in some special
cases. The corresponding result is not part of this thesis, but will appear in
a joint paper with Helmke and Fuhrmann [THF02].
While one of the most important applications of (almost) (A;B)-invariant
subspaces is the solution of the (almost) disturbance decoupling problem, in
10