1,105 research outputs found
Thomas decompositions of parametric nonlinear control systems
This paper presents an algorithmic method to study structural properties of
nonlinear control systems in dependence of parameters. The result consists of a
description of parameter configurations which cause different control-theoretic
behaviour of the system (in terms of observability, flatness, etc.). The
constructive symbolic method is based on the differential Thomas decomposition
into disjoint simple systems, in particular its elimination properties
Thomas Decomposition of Algebraic and Differential Systems
In this paper we consider disjoint decomposition of algebraic and non-linear
partial differential systems of equations and inequations into so-called simple
subsystems. We exploit Thomas decomposition ideas and develop them into a new
algorithm. For algebraic systems simplicity means triangularity, squarefreeness
and non-vanishing initials. For differential systems the algorithm provides not
only algebraic simplicity but also involutivity. The algorithm has been
implemented in Maple
Algorithmic Thomas Decomposition of Algebraic and Differential Systems
In this paper, we consider systems of algebraic and non-linear partial
differential equations and inequations. We decompose these systems into
so-called simple subsystems and thereby partition the set of solutions. For
algebraic systems, simplicity means triangularity, square-freeness and
non-vanishing initials. Differential simplicity extends algebraic simplicity
with involutivity. We build upon the constructive ideas of J. M. Thomas and
develop them into a new algorithm for disjoint decomposition. The given paper
is a revised version of a previous paper and includes the proofs of correctness
and termination of our decomposition algorithm. In addition, we illustrate the
algorithm with further instructive examples and describe its Maple
implementation together with an experimental comparison to some other
triangular decomposition algorithms.Comment: arXiv admin note: substantial text overlap with arXiv:1008.376
Bounds on the Automata Size for Presburger Arithmetic
Automata provide a decision procedure for Presburger arithmetic. However,
until now only crude lower and upper bounds were known on the sizes of the
automata produced by this approach. In this paper, we prove an upper bound on
the the number of states of the minimal deterministic automaton for a
Presburger arithmetic formula. This bound depends on the length of the formula
and the quantifiers occurring in the formula. The upper bound is established by
comparing the automata for Presburger arithmetic formulas with the formulas
produced by a quantifier elimination method. We also show that our bound is
tight, even for nondeterministic automata. Moreover, we provide optimal
automata constructions for linear equations and inequations
Complexity of Resolution of Parametric Systems of Polynomial Equations and Inequations
Consider a system of n polynomial equations and r polynomial inequations in n
indeterminates of degree bounded by d with coefficients in a polynomial ring of
s parameters with rational coefficients of bit-size at most . From the
real viewpoint, solving such a system often means describing some
semi-algebraic sets in the parameter space over which the number of real
solutions of the considered parametric system is constant. Following the works
of Lazard and Rouillier, this can be done by the computation of a discriminant
variety. In this report we focus on the case where for a generic specialization
of the parameters the system of equations generates a radical zero-dimensional
ideal, which is usual in the applications. In this case, we provide a
deterministic method computing the minimal discriminant variety reducing the
problem to a problem of elimination. Moreover, we prove that the degree of the
computed minimal discriminant variety is bounded by and
that the complexity of our method is bit-operations on a deterministic Turing machine
On the determination of cusp points of 3-R\underline{P}R parallel manipulators
This paper investigates the cuspidal configurations of 3-RPR parallel
manipulators that may appear on their singular surfaces in the joint space.
Cusp points play an important role in the kinematic behavior of parallel
manipulators since they make possible a non-singular change of assembly mode.
In previous works, the cusp points were calculated in sections of the joint
space by solving a 24th-degree polynomial without any proof that this
polynomial was the only one that gives all solutions. The purpose of this study
is to propose a rigorous methodology to determine the cusp points of
3-R\underline{P}R manipulators and to certify that all cusp points are found.
This methodology uses the notion of discriminant varieties and resorts to
Gr\"obner bases for the solutions of systems of equations
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