229 research outputs found
A reduction technique for Generalised Riccati Difference Equations
This paper proposes a reduction technique for the generalised Riccati
difference equation arising in optimal control and optimal filtering. This
technique relies on a study on the generalised discrete algebraic Riccati
equation. In particular, an analysis on the eigen- structure of the
corresponding extended symplectic pencil enables to identify a subspace in
which all the solutions of the generalised discrete algebraic Riccati equation
are coin- cident. This subspace is the key to derive a decomposition technique
for the generalised Riccati difference equation that isolates its nilpotent
part, which becomes constant in a number of steps equal to the nilpotency index
of the closed-loop, from another part that can be computed by iterating a
reduced-order generalised Riccati difference equation
On Minimal Spectral Factors with Zeroes and Poles lying on Prescribed Region
In this paper, we consider a general discrete-time spectral factorization
problem for rational matrix-valued functions. We build on a recent result
establishing existence of a spectral factor whose zeroes and poles lie in any
pair of prescribed regions of the complex plane featuring a geometry compatible
with symplectic symmetry. In this general setting, uniqueness of the spectral
factor is not guaranteed. It was, however, conjectured that if we further
impose stochastic minimality, uniqueness can be recovered. The main result of
his paper is a proof of this conjecture.Comment: 14 pages, no figures. Revised version with minor modifications. To
appear in IEEE Transactions of Automatic Contro
The extended symplectic pencil and the finite-horizon LQ problem with two-sided boundary conditions
This note introduces a new analytic approach to the solution of a very
general class of finite-horizon optimal control problems formulated for
discrete-time systems. This approach provides a parametric expression for the
optimal control sequences, as well as the corresponding optimal state
trajectories, by exploiting a new decomposition of the so-called extended
symplectic pencil. Importantly, the results established in this paper hold
under assumptions that are weaker than the ones considered in the literature so
far. Indeed, this approach does not require neither the regularity of the
symplectic pencil, nor the modulus controllability of the underlying system. In
the development of the approach presented in this paper, several ancillary
results of independent interest on generalised Riccati equations and on the
eigenstructure of the extended symplectic pencil will also be presented
Pairs of -step reachability and -step observability matrices
Let and be matrices of size and ,
respectively. A necessary and sufficient condition is given for the existence
of a triple such that a -step reachability matrix of
and an -step observability matrix of .Comment: 5 page
Factor Models with Real Data: a Robust Estimation of the Number of Factors
Factor models are a very efficient way to describe high dimensional vectors
of data in terms of a small number of common relevant factors. This problem,
which is of fundamental importance in many disciplines, is usually reformulated
in mathematical terms as follows. We are given the covariance matrix Sigma of
the available data. Sigma must be additively decomposed as the sum of two
positive semidefinite matrices D and L: D | that accounts for the idiosyncratic
noise affecting the knowledge of each component of the available vector of data
| must be diagonal and L must have the smallest possible rank in order to
describe the available data in terms of the smallest possible number of
independent factors.
In practice, however, the matrix Sigma is never known and therefore it must
be estimated from the data so that only an approximation of Sigma is actually
available. This paper discusses the issues that arise from this uncertainty and
provides a strategy to deal with the problem of robustly estimating the number
of factors.Comment: arXiv admin note: text overlap with arXiv:1708.0040
Factor analysis with finite data
Factor analysis aims to describe high dimensional random vectors by means of
a small number of unknown common factors. In mathematical terms, it is required
to decompose the covariance matrix of the random vector as the sum of
a diagonal matrix | accounting for the idiosyncratic noise in the data |
and a low rank matrix | accounting for the variance of the common factors |
in such a way that the rank of is as small as possible so that the number
of common factors is minimal. In practice, however, the matrix is
unknown and must be replaced by its estimate, i.e. the sample covariance, which
comes from a finite amount of data. This paper provides a strategy to account
for the uncertainty in the estimation of in the factor analysis
problem.Comment: Draft, the final version will appear in the 56th IEEE Conference on
Decision and Control, Melbourne, Australia, 201
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