26 research outputs found
Observer design for systems with an energy-preserving non-linearity
Observer design is considered for a class of non-linear systems whose
non-linear part is energy preserving. A strategy to construct convergent
observers for this class of non-linear system is presented. The approach has
the advantage that it is possible, via convex programming, to prove whether the
constructed observer converges, in contrast to several existing approaches to
observer design for non-linear systems. Finally, the developed methods are
applied to the Lorenz attractor and to a low order model for shear fluid flow
Real root finding for equivariant semi-algebraic systems
Let be a real closed field. We consider basic semi-algebraic sets defined
by -variate equations/inequalities of symmetric polynomials and an
equivariant family of polynomials, all of them of degree bounded by .
Such a semi-algebraic set is invariant by the action of the symmetric group. We
show that such a set is either empty or it contains a point with at most
distinct coordinates. Combining this geometric result with efficient algorithms
for real root finding (based on the critical point method), one can decide the
emptiness of basic semi-algebraic sets defined by polynomials of degree
in time . This improves the state-of-the-art which is exponential
in . When the variables are quantified and the
coefficients of the input system depend on parameters , one
also demonstrates that the corresponding one-block quantifier elimination
problem can be solved in time
Infinitesimal Periodic Deformations and Quadrics
We describe a correspondence between the infinitesimal deformations of a periodic bar-and-joint framework and periodic arrangements of quadrics. This intrinsic correlation provides useful geometric characteristics. A direct consequence is a method for detecting auxetic deformations, identified by a pattern consisting of homothetic ellipsoids. Examples include frameworks with higher crystallographic symmetry
Bounding the number of stable homotopy types of a parametrized family of semi-algebraic sets defined by quadratic inequalities
We prove a nearly optimal bound on the number of stable homotopy types
occurring in a k-parameter semi-algebraic family of sets in , each
defined in terms of m quadratic inequalities. Our bound is exponential in k and
m, but polynomial in . More precisely, we prove the following. Let
be a real closed field and let with . Let be a
semi-algebraic set, defined by a Boolean formula without negations, whose atoms
are of the form, . Let be the projection on the last k co-ordinates. Then, the number of
stable homotopy types amongst the fibers S_{\x} = \pi^{-1}(\x) \cap S is
bounded by Comment: 27 pages, 1 figur