49,177 research outputs found
Computing control invariant sets is easy
In this paper we consider the problem of computing control invariant sets for
linear controlled systems with constraints on the input and on the states. We
focus in particular on the complexity of the computation of the N-step
operator, given by the Minkowski addition of sets, that is the basis of many of
the iterative procedures for obtaining control invariant sets. Set inclusions
conditions for control invariance are presented that involve the N-step sets
and are posed in form of linear programming problems. Such conditions are
employed in algorithms based on LP problems that allow to overcome the
complexity limitation inherent to the set addition and can be applied also to
high dimensional systems. The efficiency and scalability of the method are
illustrated by computing in less than two seconds an approximation of the
maximal control invariant set, based on the 15-step operator, for a system
whose state and input dimensions are 20 and 10 respectively
Computationally efficient approximations of the joint spectral radius
The joint spectral radius of a set of matrices is a measure of the maximal
asymptotic growth rate that can be obtained by forming long products of
matrices taken from the set. This quantity appears in a number of application
contexts but is notoriously difficult to compute and to approximate. We
introduce in this paper a procedure for approximating the joint spectral radius
of a finite set of matrices with arbitrary high accuracy. Our approximation
procedure is polynomial in the size of the matrices once the number of matrices
and the desired accuracy are fixed
How scaling of the disturbance set affects robust positively invariant sets for linear systems
This paper presents new results on robust positively invariant (RPI) sets for
linear discrete-time systems with additive disturbances. In particular, we
study how RPI sets change with scaling of the disturbance set. More precisely,
we show that many properties of RPI sets crucially depend on a unique scaling
factor which determines the transition from nonempty to empty RPI sets. We
characterize this critical scaling factor, present an efficient algorithm for
its computation, and analyze it for a number of examples from the literature
A Sums-of-Squares Extension of Policy Iterations
In order to address the imprecision often introduced by widening operators in
static analysis, policy iteration based on min-computations amounts to
considering the characterization of reachable value set of a program as an
iterative computation of policies, starting from a post-fixpoint. Computing
each policy and the associated invariant relies on a sequence of numerical
optimizations. While the early research efforts relied on linear programming
(LP) to address linear properties of linear programs, the current state of the
art is still limited to the analysis of linear programs with at most quadratic
invariants, relying on semidefinite programming (SDP) solvers to compute
policies, and LP solvers to refine invariants.
We propose here to extend the class of programs considered through the use of
Sums-of-Squares (SOS) based optimization. Our approach enables the precise
analysis of switched systems with polynomial updates and guards. The analysis
presented has been implemented in Matlab and applied on existing programs
coming from the system control literature, improving both the range of
analyzable systems and the precision of previously handled ones.Comment: 29 pages, 4 figure
Long MDS Codes for Optimal Repair Bandwidth
MDS codes are erasure-correcting codes that can
correct the maximum number of erasures given the number of
redundancy or parity symbols. If an MDS code has r parities
and no more than r erasures occur, then by transmitting all
the remaining data in the code one can recover the original
information. However, it was shown that in order to recover a
single symbol erasure, only a fraction of 1/r of the information
needs to be transmitted. This fraction is called the repair
bandwidth (fraction). Explicit code constructions were given in
previous works. If we view each symbol in the code as a vector
or a column, then the code forms a 2D array and such codes
are especially widely used in storage systems. In this paper, we
ask the following question: given the length of the column l, can
we construct high-rate MDS array codes with optimal repair
bandwidth of 1/r, whose code length is as long as possible? In
this paper, we give code constructions such that the code length
is (r + 1)log_r l
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