999 research outputs found
Computing control invariant sets in high dimension is easy
In this paper we consider the problem of computing control invariant sets for
linear controlled high-dimensional systems with constraints on the input and on
the states. 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 allow to overcome the complexity limitation inherent to the set
addition and vertices enumeration and can be applied also to high dimensional
systems. The efficiency and scalability of the method are illustrated by
computing approximations of the maximal control invariant set, based on the
10-step operator, for a system whose state and input dimensions are 30 and 15,
respectively.Comment: arXiv admin note: substantial text overlap with arXiv:1708.0479
Reach Set Approximation through Decomposition with Low-dimensional Sets and High-dimensional Matrices
Approximating the set of reachable states of a dynamical system is an
algorithmic yet mathematically rigorous way to reason about its safety.
Although progress has been made in the development of efficient algorithms for
affine dynamical systems, available algorithms still lack scalability to ensure
their wide adoption in the industrial setting. While modern linear algebra
packages are efficient for matrices with tens of thousands of dimensions,
set-based image computations are limited to a few hundred. We propose to
decompose reach set computations such that set operations are performed in low
dimensions, while matrix operations like exponentiation are carried out in the
full dimension. Our method is applicable both in dense- and discrete-time
settings. For a set of standard benchmarks, it shows a speed-up of up to two
orders of magnitude compared to the respective state-of-the art tools, with
only modest losses in accuracy. For the dense-time case, we show an experiment
with more than 10.000 variables, roughly two orders of magnitude higher than
possible with previous approaches
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
Computation of Atomic Fibers of Z-Linear Maps
For given matrix , the set
describes the preimage or fiber of under the -linear map
, . The fiber is called atomic, if
implies or . In this paper we present a
novel algorithm to compute such atomic fibers. An algorithmic solution to
appearing subproblems, computational examples and applications are included as
well.Comment: 27 page
Computing the vertices of tropical polyhedra using directed hypergraphs
We establish a characterization of the vertices of a tropical polyhedron
defined as the intersection of finitely many half-spaces. We show that a point
is a vertex if, and only if, a directed hypergraph, constructed from the
subdifferentials of the active constraints at this point, admits a unique
strongly connected component that is maximal with respect to the reachability
relation (all the other strongly connected components have access to it). This
property can be checked in almost linear-time. This allows us to develop a
tropical analogue of the classical double description method, which computes a
minimal internal representation (in terms of vertices) of a polyhedron defined
externally (by half-spaces or hyperplanes). We provide theoretical worst case
complexity bounds and report extensive experimental tests performed using the
library TPLib, showing that this method outperforms the other existing
approaches.Comment: 29 pages (A4), 10 figures, 1 table; v2: Improved algorithm in section
5 (using directed hypergraphs), detailed appendix; v3: major revision of the
article (adding tropical hyperplanes, alternative method by arrangements,
etc); v4: minor revisio
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