26 research outputs found

    Observer design for systems with an energy-preserving non-linearity

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    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

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    Let RR be a real closed field. We consider basic semi-algebraic sets defined by nn-variate equations/inequalities of ss symmetric polynomials and an equivariant family of polynomials, all of them of degree bounded by 2d<n2d < n. 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 2d12d-1 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 ss polynomials of degree dd in time (sn)O(d)(sn)^{O(d)}. This improves the state-of-the-art which is exponential in nn. When the variables x1,,xnx_1, \ldots, x_n are quantified and the coefficients of the input system depend on parameters y1,,yty_1, \ldots, y_t, one also demonstrates that the corresponding one-block quantifier elimination problem can be solved in time (sn)O(dt)(sn)^{O(dt)}

    Infinitesimal Periodic Deformations and Quadrics

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    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

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    We prove a nearly optimal bound on the number of stable homotopy types occurring in a k-parameter semi-algebraic family of sets in R\R^\ell, each defined in terms of m quadratic inequalities. Our bound is exponential in k and m, but polynomial in \ell. More precisely, we prove the following. Let R\R be a real closed field and let P={P1,...,Pm}R[Y1,...,Y,X1,...,Xk], {\mathcal P} = \{P_1,...,P_m\} \subset \R[Y_1,...,Y_\ell,X_1,...,X_k], with degY(Pi)2,degX(Pi)d,1im{\rm deg}_Y(P_i) \leq 2, {\rm deg}_X(P_i) \leq d, 1 \leq i \leq m. Let SR+kS \subset \R^{\ell+k} be a semi-algebraic set, defined by a Boolean formula without negations, whose atoms are of the form, P0,P0,PPP \geq 0, P\leq 0, P \in {\mathcal P}. Let π:R+kRk\pi: \R^{\ell+k} \to \R^k 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 (2mkd)O(mk). (2^m\ell k d)^{O(mk)}. Comment: 27 pages, 1 figur
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