David A. Singer

A new structural framework for parity equation-based failure detection and isolation

Abstract The paper describes a new framework for developing parity equations that prevent incorrect isolation decisions under marginal size failures in a decision process that tests each residual independently. Test thresholds that take the noise conditions into account are set high to reduce the occurrence of false alarms while maintaining the algorithms ability to detect and isolate larger failures. The method is applicable to additive failures on the measured input and output variables and to additive plant disturbances. A transformation algorithm provides a multitude of models that satisfy the isolability requirements. A search procedure utilizing this model redundancy integrates model robustness considerations into the design.

Stable Orbits and Bifurcation of Maps of the Interval

Differentiable maps of class at least

Lagrangian Aspects of the Kirchhoff Elastic Rod

C^3

Curve straightening and a minimax argument for closed elastic curves

from the unit interval to itself are shown to have a finite number of stable periodic orbits, each of which attracts the iterates of some critical point, assuming the hypothesis of everywhere negative Schwarzian derivative. An example illustrates the necessity of this hypothesis; it shows that an endomorphism of the unit interval with one critical point may possess more than one stable orbit.

KNOT TYPES, HOMOTOPIES AND STABILITY OF CLOSED ELASTIC RODS

The relation between Eulers planar elastic curves and vortex filaments evolving by the localized induction equation (LIE) of hydrodynamics was discovered by Hasimoto in 1971. Basic facts about (an integrable case of) Kirchhoff elastic rods are described here, which amplify the connection between the variational problem for rods and the soliton equation LIE. In particular, it is shown that the centerline of the Kirchhoff rod is an equilibrium for a linear combination of the first three conserved Hamiltonians in the LIE hierarchy.

Augmented Models for Statistical Fault Isolation in Complex Dynamic Systems

THE RESULTS obtained here have to do with the following problem. Imagine the ends of a straight length of springy wire are joined together smoothly and the wire is held in some configuration described by an immersion y of the circle into the plane or into R3. According to the Bernoulli-Euler theory of elastic rods the bending energy of the wire is proportional to the total squared curvature of y, which we will denote by F (7) = 5, k2 ds. Suppose now the wire is released and it moves so as to decrease its bending energy as efficiently as possible, i.e. following “the negative gradient of F” (so our dynamics are Aristotelian rather than Newtonian, and we are also making the physically unrealistic assumption that the wire can pass through itself freely). How does the wire evolve, and what will happen ultimately (as time goes to infinity)? Of course, one wants to know first that one can actually define such a flow on the space of immersed circles, that it exists for all time, and that one can sensibly speak of a limiting curve yrn for the trajectory through a given initial curve yo. It is shown here that this is indeed the case and that in fact the Palais-Smale condition holds for this flow. It is proved, moreover, that if y0 is a plane curve of rotation index one (e.g. if y0 is embedded) then the flow carries y0 to a circle. Our main result, however, pertains to the non-planar case, where the situation is more complicated. In a space form it is possible to integrate the equations for an elastica, i.e. for a critical point of F, and this enables one to prove, in particular, that there is a countably infinite family of (similarity classes of) closed elastic curves in R3 (see Theorem 0.1). Thus, not all wire loops in R3 will flow to a circle. On the other hand, this leaves open the possibility that ycu is a circle for almost any initial curve yO, and indeed, our concluding Theorem 3.2 states that the circle is the only stable closed elastica in R3. The proof of this theorem itself depends on the dynamical, i.e. gradient flow approach to the study of F (and avoids a detailed analysis of the Hessian of F, which is quite complicated for non-planar elastic curves). The idea is as follows. One considers a discrete group G of rotations of R3 and an associated pair of multiply covered circular elastic curves which are Gequivariantly regularly homotopic, and which are both local minima for the restriction of F to G-symmetric curves (though multiple circles are unstable with respect to general variations). An appeal to the minimax and symmetric criticality principles then enables one to conclude that there exists a non-circular elastica of “saddle type”. Comparision with the classification theorem shows that one can account in this way for all non-circular solutions, hence all are unstable. We remark that a similar critical structure occurs for “free” (length unconstrained) elastic curves in the standard two-sphere: it was shown in [S] (by an entirely different method) that all closed non-geodesic solutions in S2 are unstable and can be regarded as minimax critical points arising from symmetrical regular homotopies between certain multiple coverings of a prime geodesic (though the minimax argument in [SJ is made only heuristically). To the extent that a similar picture holds as well for manifolds of (non-constant) positive curvature one gains a new view of closed geodesics as the limits of almost all trajectories of -VF. The organization of the paper is as follows. Section 0 is a brief review of some basic facts concerning elastic curves in space forms and the classification of closed elastic curves in R3 (details can be found in [S], [6] ). Section 1 is devoted mostly to the proof ofcondition (C)for the curve straightening flow. We have included details and have attempted to keep the discussion as self-contained as possible. In Section 2 we derive a second variation formula

The existence of nontriangulable cut loci

The energy minimization problem associated to uniform, isotropic, linearly elastic rods leads to a geometric variational problem for the rod centerline, whose solutions include closed, knotted curves. We give a complete description of the space of closed and quasiperiodic solutions. The quasiperiodic curves are parametrized by a two-dimensional disc. The closed curves arise as a countable collection of one-parameter families, connecting the m-fold covered circle to the n-fold covered circle for any m,n relatively prime. Each family contains exactly one self-intersecting curve, one elastic curve, and one closed curve of constant torsion. Two torus knot types are represented in each family, and all torus knots are represented by elastic rod centerlines.

A new flow on starlike curves in ℝ

The equation error approach to fault isolation implies the statistical testing of balance equation errors. In this paper, some substantial extensions to the existing methodology are proposed, including - generalized linear dynamic models - the concept of statistical isolability - the idea of and an algorithm for model augmenting - fault sensitivity analysis and filtering

The location of critical points of finite Blaschke products

We continue in this note the description of deformation theorems for geodesic fields on a Riemannian manifold begun in [1] , restricting ourselves here to surfaces of revolution and to deformations of metric within this class. Using real and holomorphic Fourier transforms, we obtain in Theorem 2 an explicit formula for the deformation of metric corresponding to a prescribed deflection of geodesies. As an application, we turn again to the structure of the cut locus and prove

Diffeomorphisms of the circle and hyperbolic curvature

In this note we find a new evolution equation for starlike curves in R 3 . We study the evolution of the subaffine curvature and subaffine torsion under the flow and show that it is completely integrable. The solutions to the evolution which move without changing affine shape are subaffine elastic curves. We integrate the subaffine elastica by quadratures.