self interactions of strands and sheets physical strands or sheets that can be modelled as curves or surfaces embedded in three dimensions are ubiquitous in nature and are of fundamental importance in mathematics physics biology and engineering often the physical interpretation dictates that self avoidance should be enforced in the continuum model i e finite energy configurations should not self intersect current continuum models with self avoidance frequently employ pairwise repulsive potentials which are of necessity singular moreover the potentials do not have an intrinsic length scale appropriate for modelling the finite thickness of the physical systems here we develop a framework for modelling self avoiding strands and sheets which avoids singularities and which provides a way to introduce a thickness length scale in our approach pairwise interaction potentials are replaced by many body potentials involving three or more points and the radii of certain associated circles or spheres self interaction energies based on these many body potentials can be used to describe the statistical mechanics of self interacting strands and sheets of finite thickness

a general class of sliding surface for sliding mode control in the conventional sliding mode control the relative degree of the chosen sliding surface is usually one this paper addresses a general class of sliding surface whose relative degree is no longer restricted to one we show that when the relative degree is more than two it is inevitable that there will exist a limit cycle for the nonlinear sign function and for the case of relative degree two the system possesses poor phase margin however these disadvantages can be remedied remarkably by smoothing out the control nonlinearity in a suitable boundary layer with a saturation function that ensures asymptotic stability of the overall system

on the minimum ropelength of knots and links the ropelength of a knot is the quotient of its length by its thickness the radius of the largest embedded normal tube around the knot we prove existence and regularity for ropelength minimizers in any knot or link type these are c 1 1 curves but need not be smoother we improve the lower bound for the ropelength of a nontrivial knot and establish new ropelength bounds for small knots and links including some which are sharp

adaptive sampled data based linear quadratic optimal control of stochastic systems the problem of sampled data sd based adaptive linear quadratic lq optimal control is considered for linear stochastic continuous time systems with unknown parameters and disturbances to overcome the difficulties caused by the unknown parameters and incompleteness of the state information and to probe into the influence of sample size on system performance a cost biased parameter estimator and an adaptive control design method are presented under the assumption that the unknown parameter belongs to a known finite set some sufficient conditions ensuring the convergence of the parameter estimate are obtained it is shown that when the sample step size is small the sd based adaptive control is lq optimal for the corresponding discretized system and sub optimal compared with that of the case where the parameter is known and the information is complete

an output feedback stabilization scheme using the filtered coordinate transformation an output feedback stabilizing controller is proposed for a class of nonlinear systems that are diffeomorphic to nonlinear observer form which contains uncertainty in constructing the controller a filtered coordinate transformation is utilized so as to make the vector field with input constant some riccati type inequality criteria are derived for the exponential convergence of an observer the filter and the system

adaptive high gain λ tracking with variable sampling rate it is well known that proportional output feedback control can stabilize any relative degree one minimum phase system if the sign of the feedback is correct and the proportional gain is high enough moreover there exists simple adaptation laws for tuning the proportional gain the so called high gain adaptive controllers which are not based on system identification or plant parameter estimation algorithms or injection of probing signals if tracking of signals is desired then these simple controllers are also applicable without invoking an internal model if the tracking error is not necessarily supposed to converge to zero but towards a ball around zero of arbitrarily small but prespecified radius 0 in this note we consider a sampled version of the high gain adaptive tracking controller the motivation for sampling arises from the possibility that the output of a system may not be available continuously but only at discrete time instants the problem is that the stiffness of the system increases as the proportional gain is increased our result shows that adaptive sampling tracking is possible if the product hk of the decreasing sampling rate h and the increasing proportional gain k decreases at a rate proportional to 1 logk

brief paper output feedback variable structure control for linear systems with uncertainties and disturbances this paper proposes a dynamic output feedback variable structure controller for linear mimo systems with mismatched and matched norm bounded uncertainties and matched nonlinear disturbances the proposed controller consists of nonlinear and linear parts similar to the unit vector type of controller the nonlinear part of the new controller takes care only of matched uncertainties and disturbances and on the other hand the linear part with full dynamics completely handles mismatched uncertainties designing such a linear part leads to use a lyapunov function associated with the full states which achieves the global stability against the mismatched uncertainties the resulting criteria are furthermore converted into solvable ones by using the so called cone complementary linearization algorithm for bi convex problems

local stabilization of minimum phase nonlinear systems abstract in this note we briefly review the notion of a nonlinear minimum phase system i e a system which when constrained in such a way as to produce zero output evolves with an asymptotically stable dynamics the main purpose of the paper is to show that any minimum phase nonlinear system can always be locally stabilized by smooth state feedback

adaptive stabilization of not necessarily minimum phase plants abstract we show that a first order adaptive regulator can stabilize any linear time invariant plant lti whose transfer function has arbitrary relative degree and order and is not necessarily minimum phase provided the dominant slow part of the plant is minimum phase and of relative degree one and the parasitic fast part is stable the results are extended to r r multivariable systems whose dominant parts are minimum phase and the spectrum of their high frequency gains is either in re s s 0 and their parasitic fast parts are stable

self contact for rods on cylinders we study self contact phenomena in elastic rods that are constrained to lie on a cylinder by choosing a particular set of variables to describe the rod centerline the variational setting is made particularly simple the strain energy is a second order functional of a single scalar variable and the self contact constraint is written as an integral inequality using techniques from ordinary differential equation theory comparison principles and variational calculus cut and paste arguments we fully characterize the structure of constrained minimizers an important auxiliary result states that the set of self contact points is continuous a result that contrasts with known examples from contact problems in free rods

stability for time varying linear dynamic systems on time scales we study conditions under which the solutions of a time varying linear dynamic system of the form x t a t x t are stable on certain time scales we give sufficient conditions for various types of stability including lyapunov type stability criteria and eigenvalue conditions on slowly varying systems that ensure exponential stability finally perturbations of the unforced system are investigated and an instability criterion is also developed

technical communique a remark on passivity based and discontinuous control of uncertain nonlinear systems we address the problem of robust stabilisation of nonlinear systems affected by time varying uniformly bounded in time affine perturbations our approach relies on the combination of sliding mode techniques and passivity based control roughly speaking we show that under suitable conditions the sliding mode variable can be chosen as the passive output of the perturbed system then we show how to construct a controller which guarantees the global uniform convergence of the plant s outputs towards a time varying desired reference even in the presence of permanently exciting time varying disturbances we illustrate our result on the tracking control of the van der pol oscillator

shapes of tight composite knots we present new computations of tight shapes obtained using the constrained gradient descent code ridgerunner for 544 composite knots with 12 and fewer crossings expanding our dataset to 943 knots and links we use the new data set to analyze two outstanding conjectures about tight knots namely that the ropelengths of composite knots are at least 4 4 less than the sums of the prime factors and that the writhes of composite knots are the sums of the writhes of the prime factors our numerics support the connect sum conjecture and argue against the additivity of writhe conjecture we also present data on the number of configurations having straight segments and highly curved kinked regions

output feedback multivariable global variable gain super twisting algorithm a novel output feedback super twisting algorithm design is proposed for uncertain multivariable plants in order to allow global finite time exact tracking in the presence of a rather general class of matched nonlinear disturbances which may depend on measured and or unmeasured states the algorithm is designed with variable gains this represents a significant extension of a recently proposed nondecoupled multivariable super twisting algorithm based on state feedback with fixed gains to the case of output feedback with variable gains

role of flexibility in entanglement entanglement is essential to the function of many physical systems flexibility and length determine the extent to which the system can become entangled given a perfectly flexible unit radius tube several researchers have studied the minimum length needed to tie different types of knots can one obtain the same configurations with less flexible tubing does more flexibility always yield tighter knots we demonstrate a phase change in flexibility beyond which more flexibility adds very little entanglement this level of flexibility is surprisingly low and appears to have a global bound since tensile strength and flexibility act inversely this level of flexibility provides the maximal tensile strength for materials that need to pack tightly this is a basic design principle that should be observable in nature

topological aspects of universal adaptive stabilization in this paper we consider two problems in non identifier based universal adaptive control within the framework of m aa rtensson adaptive stabilization ph d thesis lund institute of technology 1986 in this framework any linear system stabilizable by constant linear output feedback is adaptively stabilized by an adaptive piecewise linear output feedback control law the essential feature we exploit is that of a piecewise linear output feedback which switches through a set of feedback matrices with switching controlled by an output driven differential equation for each initial condition the state of the system converges to zero and the time varying gain matrix converges to a limit gain in this setting we consider two related problems the first concerns the sensitivity of closed loop solutions under small perturbations of the initial data the second concerns generic properties with respect to the set of initial conditions of stabilization by the limit gain we adopt a topological approach based on a decomposition of the dynamics of the resultant nonlinear closed loop system into a sequence of homeo diffeomorphisms derived from the switching nature of the dynamics using this decomposition we show that the set of initial conditions for which solutions are stable under small perturbations and the limiting gain is stabilizing has full lebesgue measure and dense interior this latter result has been conjectured in the literature the results are illustrated by examples of planar control systems where the sets of initial conditions yielding nonstabilizing limit gains are computed

integral menger curvature for surfaces abstract we develop the concept of integral menger curvature for a large class of nonsmooth surfaces we prove uniform ahlfors regularity and a c 1 a priori bound for surfaces for which this functional is finite in fact it turns out that there is an explicit length scale r 0 which depends only on an upper bound e for the integral menger curvature m p and the integrability exponent p and not on the surface itself below that scale each surface with energy smaller than e looks like a nearly flat disc with the amount of bending controlled by the local m p energy moreover integral menger curvature can be defined a priori for surfaces with self intersections or branch points we prove that a posteriori all such singularities are excluded for surfaces with finite integral menger curvature by means of slicing and iterative arguments we bootstrap the holder exponent up to the optimal one 1 8 p thus establishing a new geometric morrey sobolev imbedding theorem as two of the various possible variational applications we prove the existence of surfaces in given isotopy classes minimizing integral menger curvature with a uniform bound on area and of area minimizing surfaces subjected to a uniform bound on integral menger curvature

adaptive sampling control of high gain stabilizable systems it is well known that proportional output feedback control can stabilize any relative degree one minimum phase system if the sign of the feedback is correct and the proportional gain is high enough moreover there exist simple adaptation laws for tuning the proportional gain so called high gain adaptive controllers which do not need to know the system and do not attempt to identify system parameters the authors consider sampled versions of the high gain adaptive controller the motivation for sampling arises from the possibility that the output of a system may not be available continuously but only at sampled times the main point of interest is the need to develop techniques for adapting the sampling rate since the stiffness of the system increases as the proportional gain is increased our main result shows that adaptive sampling stabilization is possible if the product hk of the decreasing sampling interval h and the increasing proportional gain k decreases at a rate proportional to 1 log k

on the groundstate energy of tight knots new results on the groundstate energy of tight magnetic knots are presented magnetic knots are defined as tubular embeddings of the magnetic field in an ideal perfectly conducting incompressible fluid an orthogonal curvilinear coordinate system is introduced and the magnetic energy is determined by the poloidal and toroidal components of the magnetic field standard minimization of the magnetic energy is carried out under the usual assumptions of volume and flux preserving flow with the additional constraints that the tube cross section remains circular and that the knot length ropelength is independent from internal field twist framing under these constraints the minimum energy is determined analytically by a new exact expression function of ropelength and framing groundstate energy levels of tight knots are determined from ropelength data obtained by the sono tightening algorithm results for torus knots are compared with previous work and the groundstate energy spectrum of the first prime knots up to 10 crossings is presented and analysed in detail these results demonstrate that ropelength and framing determine the spectrum of magnetic knots in tight configuration

euler lagrange equations for nonlinearly elastic rods with self contact we derive the euler lagrange equations for nonlinearly elastic rods with self contact the excluded volume constraint is formulated in terms of an upper bound on the global curvature of the centre line this condition is shown to guarantee the global injectivity of the deformation of the elastic rod topological constraints such as a prescribed knot and link class to model knotting and supercoiling phenomena as observed e g in dna molecules are included by using the notion of isotopy and gaussian linking number the bound on the global curvature as a nonsmooth side condition requires the use of clarke s generalized gradients to obtain the explicit structure of the contact forces which appear naturally as lagrange multipliers in the euler lagrange equations transversality conditions are discussed and higher regularity for the strains moments the centre line and the directors is shown

a general integral quadratic constraints theorem with applications to a class of sampled data systems in the framework of integral quadratic constraints a more general stability theorem is derived by relaxing the assumptions on the uncertainty in contrast to existing results input output relations are only required to hold on a subspace of the full signal space this result is employed to incorporate uncertainties exhibiting sampling behavior into the framework of integral quadratic constraints the arising infinite dimensional stability test is exactly reduced to a finite dimensional linear matrix inequality advantages of this approach are illustrated for the specific case of an interconnection including a pulse width modulator

special coordinate basis for multivariable linear systems finite and infinite zero structure squaring down and decoupling abstract a special coordinate basis for multivariable linear systems is introduced here several fundamental properties of linear systems regarding controllability stabiliza bility observability detectability invariant zeros decoupling zeros infinite zero structure effect of feedback on zero structure squaring down diagonal and triangular decoupling etc can be directly displayed in terms of the special coordinate basis moreover connections between the special coordinate basis and the important invariant subspaces of the geometric theory of linear systems are established

on the groundstate energy spectrum of magnetic knots and links by using analytical results for the constrained minimum energy of magnetic knots we determine the influence of internal twist on the minimum magnetic energy levels of knots and links and by using ropelength data from the ridgerunner tightening algorithm ashton et al 2011 exp math 20 57 90 we obtain the groundstate energy spectra of the first 250 prime knots and 130 prime links the two spectra are found to follow an almost identical logarithmic law by assuming that the number of knot types grows exponentially with the topological crossing number we show that this generic behavior can be justified by a general relationship between ropelength and crossing number which is in good agreement with former analytical estimates buck and simon 1999 topol appl 91 245 57 diao 2003 j knot theory ramifications 12 1 16 moreover by considering the ropelength averaged over a given knot family we establish a new connection between the averaged ropelength and the topological crossing number of magnetic knots

operator formalism for topology conserving crossing dynamics in planar knot diagrams we address here the topological equivalence of knots through the so called reidemeister moves these topology conserving manipulations are recast into dynamical rules on the crossings of knot diagrams this is presented in terms of a simple graphical representation related to the gauss code of knots drawing on techniques for reaction diffusion systems we then develop didactically an operator formalism wherein these rules for crossing dynamics are encoded the aim is to develop new tools for studying dynamical behaviour and regimes in the presence of topology conservation this necessitates the introduction of composite paulionic operators the formalism is applied to calculate some differential equations for the time evolution of densities and correlators of crossings subject to topology conserving stochastic dynamics we consider here the simplified situation of two dimensional knot projections however we hope that this is a first valuable step towards addressing a number of important questions regarding the role of topological constraints and specifically of topology conservation in dynamics through a variety of solution and approximation schemes further applicability arises in the context of the simulated annealing of knots the methods presented here depart significantly from the invariant based path integral descriptions often applied in polymer systems and in our view offer a fresh perspective on the conservation of topological states and topological equivalence in knots

the average crossing number of equilateral random polygons in this paper we study the average crossing number of equilateral random walks and polygons we show that the mean average crossing number acn of all equilateral random walks of length n is of the form a similar result holds for equilateral random polygons these results are confirmed by our numerical studies furthermore our numerical studies indicate that when random polygons of length n are divided into individual knot types the for each knot type can be described by a function of the form where a b and c are constants depending on and n0 is the minimal number of segments required to form the profiles diverge from each other with more complex knots showing higher than less complex knots moreover the profiles intersect with the acn profile of all closed walks these points of intersection define the equilibrium length of i e the chain length at which a statistical ensemble of configurations with given knot type upon cutting equilibration and reclosure to a new knot type does not show a tendency to increase or decrease this concept of equilibrium length seems to be universal and applies also to other length dependent observables for random knots such as the mean radius of gyration rg

helical tubes in crowded environments when placed in a crowded environment a semiflexible tube is forced to fold so as to make a more compact shape one compact shape that often arises in nature is the tight helix especially when the tube thickness is of comparable size to the tube length in this paper we use an excluded volume effect to model the effects of crowding this gives us a measure of compactness for configurations of the tube which we use to look at structures of the semiflexible tube that minimize the excluded volume we focus most of our attention on the helix and which helical geometries are most compact we found helices of specific pitch to radius ratio 2 512 to be optimally compact this is the same geometry that minimizes the global curvature of the curve defining the tube we further investigate the effects of adding a bending energy or multiple tubes to begin to explore the more complete space of possible geometries a tube could form

global curvature for rectifiable loops we study in detail the notion of global curvature defined on rectifiable closed curves a concept which has been successfully applied in existence and regularity investigations regarding elastic self contact problems in nonlinear elasticity a bound on this purely geometric quantity serves as an excluded volume constraint to prevent selfintersections of slender elastic bodies modeled as elastic rods moreover a finite global curvature characterizes simple closed curv es whose arc length parameterizations possess a lipschitz continuous tangent field the investigation of local and non local properties of global curvature motivates in particular an extended definition of local curvature at any point of a rectifiable loop finally we show how a bound on global curvature can be used to define and control topological constraints such as a given knot type for closed loops or a prescribed linking number for closed framed curves suitable to describe e g supercoiling phen omena of biomolecules

stability analysis of pulse width modulated feedback systems new and improved stability results are obtained for pwm feedback systems through the use of quadratic lyapunov functions and optimization procedures

indirect sampled data control with sampling period adaptation it is known that if a continuous time feedback system is exponentially stable then the corresponding sampled data system obtained by sample hold discretisation with constant sampling period is also exponentially stable provided that the sampling period 0 is sufficiently small in general it is difficult to estimate how small the sampling period has to be in order to achieve the stability of the sampled data system in this article we present an adaptive mechanism for adjusting the sampling period this mechanism has the properties that for every initial state i the adaptation of the sampling period terminates after finitely many time steps and ii the state of the adaptive sampled data system is integrable and converges to zero as time goes to infinity

discrete time model reference adaptive control for continuous time systems using generalized sampled data hold functions the generalized sampled data hold functions approach of p t kabamba 1987 is extended to the control of linear time invariant systems with unknown parameters the idea of kabamba s approach is to periodically sample the plant output and define the control as the sampled output plus a discrete time reference each multiplied by an individual modulating function such a control makes it possible to assign an arbitrary discrete time transfer function for the sampled closed loop system and does not make assumptions on the plant other than controllability and observability the authors propose an indirect adaptive controller which is based on this approach and estimates the modulating functions online in particular the control is modified so that persistent excitation of the continuous time plant is ensured without making an assumption on the reference signal and discrete time asymptotic model following is nevertheless obtained the only assumptions on the plant are minimality for the continuous and sampled plant and known order

ideal geometry of periodic entanglements three dimensional entanglements including knots knotted graphs periodic arrays of woven filaments and interpenetrating nets form an integral part of structure analysis because they influence va

nonlinear feedback regulator design for the cuk converter the method of extended linearization is used for designing stabilizing nonlinear proportional integral pi feedback controllers which regulate to a constant set point value either the average output inductor current the average input inductor current or the average transfer capacitor voltage of a pulse width modulation pwm controlled cuk converter the design is carried out on the basis of the frequency domain ziegler nichols method applied to a family of transfer function models of the linearized average pwm controlled circuit parametrized by constant operating equilibrium points

area inequalities for embedded disks spanning unknotted curves author s hass joel lagarias jeffrey c thurston william p abstract we show that a smooth unknotted curve in r 3 satisfies an isoperimetric inequality that bounds the area of an embedded disk spanning the curve in terms of two parameters the length l of the curve and the thickness r maximal radius of an embedded tubular neighborhood of the curve for fixed length the expression giving the upper bound on the area grows exponentially in 1 r 2 in the direction of lower bounds we give a sequence of length one curves with r approaching 0 for which the area of any spanning disk is bounded from below by a function that grows exponentially with 1 r in particular given any constant a there is a smooth unknotted length one curve for which the area of a smallest embedded spanning disk is greater than a

embeddedness of minimal surfaces with total boundary curvature at most 4 pi this paper proves that classical minimal surfaces of arbitrary topological type with total boundary curvature at most 47r must be smoothly embedded related results are proved for varifolds and for soap film surfaces

link lengths and their growth powers for a certain infinite family of knots or links we study the growth power ratios of their stick number lattice stick number minimum lattice length and minimum ropelength compared with their minimum crossing number c k for every it is known that the stick number and lattice stick number grow between the and linear power of the crossing number and minimum lattice length and minimum ropelength grow with at least the power of crossing number which is called the four thirds power law furthermore the minimal lattice length and minimum ropelength grow at most as o but it is unknown whether any family exhibits superlinear growth for any real number r between and 1 we give an infinite family of non splittable prime links in which the stick number and lattice stick number grow exactly as the rth power of crossing number furthermore for any real number r between and 1 we give another infinite family of non splittable prime links in which the minimum lattice length and minimum ropelength grow exactly as the rth power of crossing number

what are the longest ropes on the unit sphere we consider the variational problem of finding the longest closed curves of given minimal thickness on the unit sphere after establishing the existence of solutions for any given thickness between 0 and 1 we explicitly construct for each given thickness theta n rm sin pi 2n n in mathbb n exactly varphi n solutions where varphi is euler s totient function from number theory then we prove that these solutions are unique and also provide a complete characterisation of sphere filling curves on the unit sphere that is of those curves whose spherical tubular neighbourhood completely covers the surface area of the unit sphere exactly once all of these results carry over to open curves as well as indicated in the last section

a simple selftuning controller for stably invertible systems in this paper we construct a new trajectory following adaptive control algorithm for linear time invariant finite dimensional systems that possess a stable inverse the exponential stability of the control scheme is demonstrated

divergence free fields energy and asymptotic crossing number in more than one context the equations of classical physics lead to a closed 2 form co on 3 dimensional space which is a conserved quantity in the sense that it pulls back under time evolution in the physics literature co is written as a divergence free vector field w appendix a describes two settings in which a divergence free w is conserved or frozen one is the magnetic field in ideal magneto hydrodynamics and the other is curl velocity in incompressible inviscid fluid flow if ht denotes the transport diffeomorphisms of a moving fluid in which the field is imbedded with ho id wt denotes the field in the fluid at time t and x0 is the space variable at time t 0 then

numerical upper bounds on ropelengths of large physical knots numerical computations explored for this paper show that an upper bound on the ropelength of large knots with crossing number n grows as fast as nln2n the algorithms to randomly generate samples of such large knots and to determine an upper bound on the ropelength for each knot are described the numeric results are presented and compared to the smallest known theoretical upper bounds on ropelength

length of the tightest trefoil knot the physical sense of tight knots provided by the sono algorithm is discussed a method allowing one to predict their length is presented an upper bound for the minimum length of a smooth trefoil knot is determined

brief integral control by variable sampling based on steady state data in this paper we consider integral control algorithms with convergent adaptive sampling for multivariable infinite dimensional systems steady state gain information is used in choosing suitable integrator gains and we also consider robustness with respect to error in measuring the steady state gain

a geometric approach to pulse width modulated control in nonlinear dynamical systems the author demonstrates under the assumption of high frequency control switchings the existence of an ideal equivalence among sliding regimes of variable structure feedback control and pulse width modulated pwm control responses in nonlinear dynamical systems this equivalence constitutes the basis for a geometric approach approach to pwm control loops design an illustrate example of energy conversion in a lossless switched controlled bilinear network is presented

brief variable structure output feedback control design for a class of uncertain dynamic systems in this paper we consider the problem of designing a variable structure output feedback control vsofc law for a class of uncertain systems with mismatched uncertainty in the state matrix we derive existence conditions of linear sliding surfaces in terms of constrained linear matrix inequalities lmis and we give vsofc strategies we also give a simple design procedure and parameterizations of the linear switching surfaces together with solvability conditions for the given constrained lmis finally we give a design example in order to show the effectiveness of our method

on rectifiable curves with l p bounds on global curvature self avoidance regularity and minimizing knots we discuss the analytic properties of curves whose global curvature function g 1 is p integrable it turns out that the lp norm mathcal u p gamma rho g gamma 1 l p is an appropriate model for a self avoidance energy interpolating between soft knot energies in form of singular repulsive potentials and hard self obstacles such as a lower bound on the global radius of curvature introduced by gonzalez and maddocks we show in particular that for all p 1 finite mathcal u p energy is necessary and sufficient for w2 p regularity and embeddedness of the curve moreover compactness and lower semicontinuity theorems lead to the existence of mathcal u p minimizing curves in given isotopy classes there are obvious extensions to other variational problems for curves and nonlinearly elastic rods where one can introduce a bound on mathcal u p to preclude self intersections

on variable structure output feedback controllers for uncertain dynamic systems a class of variable structure output feedback controllers for uncertain dynamic systems with bounded uncertainties is proposed no statistical information about the uncertain elements is assumed a variable structure systems approach and a geometric approach to the analysis and synthesis of system zeros are employed in the synthesis of the proposed controllers

peaking free variable structure control of uncertain linear systems based on a high gain observer an output feedback model reference variable structure controller based on a high gain observer hgo is proposed and analyzed for single input single output siso linear plants with relative degree greater than one the control law is generated using the hgo signals only to drive the sign function of the variable structure control component while the sign function gain also called modulation as well as the other components of the control signal are generated using signals from state variable filters which do not require high gain and are free of peaking this scheme achieves global exponential stability with respect to a small residual set and does not generate peaking in the control signal

simple adaptive control for a class of non linear systems with application to robotics simple adaptive control techniques have recently been developed for continuous stationary and non stationary linear systems based on the fact that the output stabilizability property of systems can be used to realize almost passive configurations that may facilitate the implementation of simple robust adaptive controllers this paper extends these concepts to a class of non linear systems that are linear in the control this includes manipulators and similar systems since it has recently been shown that non linear systems that can be stabilized via static or dynamic output feedback become almost passive via some corresponding parallel feedforward a rigorous proof of the robust stability of the simple adaptive control of non stationary almost passive systems is presented

output feedback stabilization of uncertain plants a variable structure systems approach the problem of output stabilization for uncertain s1so systems is considered using structural transformations uncertain systems can change to the form convenient for output feedback design synthesis of observer based variable structure control for asymptotical stabilization or uniform ultimate boundedness of the closed loop system is provided examples are considered and simulation results are given

adaptive stabilization using a variable sampling rate it is observed that digital versions of globally stable adaptive stabilization algorithms are at best locally stable due to incompatibilities between the gain adaptation algorithms and the choice of sampling rate this fact suggests the problem of retrieving global asymptotic stability for such difficulties by modifications to the stabilization algorithms to include sampling rates as control variables and the inclusion of extra plant information to relate sampling rates to gain evolutions in this contribution the theoretical problem of adaptive stabilization of single input single output siso linear systems s a b c in rnis approached using the sampling rate as the basic adaptive mechanism a wide range of sample interval adaptation schemes is derived that guarantees global asymptotic stability of the sampled outputs from the plant for any minimum phase siso systems satisfying the relative degree one constraint cb 0 the control laws are universal in the sense that stabilization is achieved des

on variable structure output feedback controllers in a recent work by zak hui 1993 a sliding mode controller for mimo systems using static output feedback was proposed very nice geometric conditions for how to design sliding surfaces were given however there are two restrictive assumptions in it one is that the uncertainties in the system must be bounded by a known function of outputs which excludes some possible uncertainties in the a matrix if the system is described by the triple a b c the other one requires a matrix equality to be held which may also be very difficult to satisfy in many systems in this paper we propose a modification of the sliding mode controller for a class of single input single output siso systems which can eliminate the above mentioned limitations and under certain conditions guarantee global closed loop stability hence the range of applicability of zak hui s method can be greatly broadened

best packing of identical helices in this paper we prove the unique existence of a ropelength minimizing conformation of the spun double helix in a mathematically rigorous way and find the minimal ropelength where t is the unique solution in of the equation using this result the pitch angles of the standard triple and quadruple helices are around and respectively which are almost identical with the approximated pitch angles of the zero twist structures previously known by olsen and bohr we also find the ropelength of the standard n helix

multirate modeling and control design for switched mode power converters a class of pulse width modulated pwm systems that includes many switched mode power converters is considered a new multirate low frequency sampling pwm scheme is introduced that not only provides the possibility for new types of operation but also unifies the switching logic corresponding to established pwm schemes an exact nonlinear discrete time model is used to develop multirate nonlinear controllers and observers based on map inversion concepts this nonlinear design methodology is widely applicable requiring only that the continuous time small signal average model be controllable and observable

star shaped control vector sets of second order systems with pwm type input this paper deals with second order continuous time linear time invariant systems by means of a pulse width modulation pwm type input in general it is difficult to analyze and design a pwm system because of its nonlinearity in terms of the duty ratio in this paper it is shown that the nonlinearity appearing in the discrete time model at the sampling instants can be eliminated by varying the location of the pulse as well as the duty ratio

feedback control of linear systems by multirate pulse width modulation in this note sampled data methods for modeling and control design of pulse width modulated systems are examined both single rate and multirate approaches are considered local controllability and observability analyses of the sampled data models are performed and feedback linearizability conditions for the sampled data models are derived multirate nonlinear state and output feedback designs are presented that permit asymptotic regulation or tracking

a variational rod model with a singular nonlocal potential the classical theory of elastic rods does not account for the possibility that large deformations may involve distinct points along the rod occupying the same physical space we develop an elastic rod model with a pairwise repulsive potential such that if two non adjacent points along the rod are close in physical space there is an energy barrier which prevents contact while for points nearby along the rod the potential is describable classically this framework is developed to prove the existence of minimizers within each homotopy class where the idea of topological homotopy of a curve is generalized to elastic rods as framed curves finally the relevant first order optimality conditions are derived and used to investigate the regularity of minimizers

a transformed lure problem for sliding mode control and chattering reduction the sliding mode control problem is investigated with the frequency domain approach the authors show that the sliding mode control can be transformed into a lure problem with this formulation it is shown that the conventional sliding mode control with a nonlinear sign function can only guarantee to make the overall system stable but not asymptotically stable the authors also consider a practical situation where the sign nonlinearity may contain a hysteresis loop for this situation they show that even if the hysteresis loop is very small there still exists a limit cycle however chattering phenomenon caused by the limit cycle behavior can be eliminated by a certain class of nonlinearities

adaptive stabilization of a class of uncertain nonlinear systems a differential inclusion approach abstract within the framework of controlled differential inclusions a stabilizing strategy is developed for a class of uncertain nonlinear dynamical systems the strategy has a linear plus discontinuous output feedback structure with bounded adaptive scalar gain

output feedback stabilization using variable structure control we consider the stabilization of a class of multivariable nonlinear systems about an equilibrium point at the origin using variable structure output feedback control in particular the system can be transformed into a normal form with no zero dynamics a robust high grain observer is used to estimate the state variables while rejecting the effect of disturbance a globally bounded discontinuous variable structure controller is designed to compensate for modelling error we show that the controller can stabilize the closed loop system and does not suffer from the peaking phenomenon that exists in previous designs

minimum lattice length and ropelength of 2 bridge knots and links knots are commonly found in molecular chains such as dna and proteins and they have been considered to be useful models for structural analysis of these molecules one interested quantity is the minimum number of monomers necessary to realize a molecular knot the minimum lattice length len k of a knot k indicates the minimum length necessary to construct k in the cubic lattice another important quantity in physical knot theory is the ropelength which is one of the knot energies measuring the complexity of knot conformation the minimum ropelength rop k is the minimum length of an ideally flexible rope necessary to tie a given knot k much effort has been invested in the research project for finding upper bounds on both quantities in terms of the minimum crossing number c k of the knot it is known that len k and rop k lie between o c k 34 and o c k ln c k 5 but unknown yet whether any family of knots has superlinear growth in this paper we focus on 2 bridge knots and links linear growth

singular perturbations and time scale methods in control theory survey 1976 1983 recent progress in the use of singular perturbation and two time scale methods of modeling and design for control systems is reviewed over 350 references are organized into major problem areas representative issues and results are discussed with a view to outlining research directions and indicating potential areas of application the survey is aimed at engineers and applied mathematicians interested in model order reduction separation of time scales and allied simplified methods of control system analysis and design the exposition does not assume prior knowledge of singular perturbation methods