stability of perturbed delay differential equations and stabilization of nonlinear cascade systems in this paper the effect of bounded input perturbations on the stability of nonlinear globally asymptotically stable delay differential equations is analyzed we investigate under which conditions global stability is preserved and if not whether semiglobal stabilization is possible by controlling the size or shape of the perturbation these results are used to study the stabilization of partially linear cascade systems with partial state feedback

continuous pole placement for delay equations in this paper we describe a stabilization method for linear time delay systems which extends the classical pole placement method for ordinary differential equations unlike methods based on finite spectrum assignment our method does not render the closed loop system finite dimensional but consists of controlling the rightmost eigenvalues because these are moved to the left half plane in a quasi continuous way we refer to our method as continuous pole placement we explain the method by means of the stabilization of a linear finite dimensional system in the presence of an input delay and illustrate its applicability to more general stabilization problems

slow peaking and low gain designs for global stabilization of nonlinear systems this paper presents an analysis of the slow peaking phenomenon a pitfall of low gain designs that imposes basic limitations to large regions of attraction in nonlinear control systems the phenomenon is best understood on a chain of integrators perturbed by a vector field up x u that satisfies p x 0 0 peaking may cause a loss of global controllability unless severe growth restrictions are imposed on p x u these growth restrictions are expressed as a higher order condition with respect to a particular weighted dilation related to the peaking exponents of the nominal system when this higher order condition is satisfied an explicit control law is derived that achieves global asymptotic stability of x 0 this stabilization result is extended to more general cascade nonlinear systems in which the perturbation p x spl upsi spl upsi spl upsi spl xi u sup t contains the state spl xi and the control u of a stabilizable subsystem spl xi spl dot a spl xi u as an illustration a control law is derived that achieves global stabilization of the frictionless ball and beam model

phaselocking in a reaction diffusion system with a linear frequency gradient a simple spatially inhomogeneous oscillatory reaction diffusion equation is studied existence of phaselocked waves is proven with a shooting argument stability is determined numerically applications to intestinal motility and a chemical experiment are described

nonexistence of abelian difference sets lander s conjecture for prime power orders in 1963 ryser conjectured that there are no circulant hadamard matrices of order 4 and no cyclic difference sets whose order is not coprime to the group order these conjectures are special cases of lander s conjecture which asserts that there is no abelian group with a cyclic sylow p subgroup containing a difference set of order divisible by p we verify lander s conjecture for all difference sets whose order is a power of a prime greater than 3

delay induced stability of vector second order systems via simple lyapunov functionals it is well known that some important classes of systems e g inverted pendulums oscillators double integrators that cannot be stabilized by a static output feedback may be stabilized by inserting an artificial time delay in the feedback static output feedback controllers have advantages over observer based controllers in the presence of uncertainties in the system matrices and or uncertain input output delays where the observer based design becomes complicated the existing lyapunov based methods that may treat the case of stabilizing delays and that lead to stability conditions in terms of linear matrix inequalities lmis suffer from high dimensionality of the resulting lmis with a large number of decision variables in this paper we suggest simple lyapunov functionals for vector second order systems with stabilizing delays that lead to reduced order lmis with a small number of decision variables moreover differently from the existing methods we show that the presented lmis are always feasible for small enough delays

stabilization by using artificial delays an lmi approach abstract static output feedback stabilization for the n th order vector differential equations by using artificial multiple delays is considered under assumption of the stabilizability of the system by a static feedback that depends on the output and its derivatives up to the order n 1 a delayed static output feedback is found that stabilizes the system the conditions for the stability analysis of the resulting closed loop system are given in terms of simple lmis it is shown that the lmis are always feasible for appropriately chosen gains and small enough delays robust stability analysis in the presence of uncertain time varying delays and stochastic perturbation of the system coefficients is provided numerical examples including chains of three and four integrators that are stabilized by static output feedbacks with multiple delays illustrate the efficiency of the method

cyclotomic integers and finite geometry the most powerful method for the study of finite geometries with regular or quasiregular automorphism groups g is to translate their definition into an equation over the integral group ring 2 g and to investigate this equation by applying complex representations of g for the definitions and the basic facts see section 2 if g is abelian this approach boils down to proving or disproving the existence of elements of 2 g with 0 1 coefficients whose character values are cyclotomic integers of certain prescribed absolute values up to now there have been two general methods to tackle this problem namely hall s multiplier concept and turyn s selfconjugacy approach however both methods need severe technical assumptions and thus are not applicable to many classes of problems despite many efforts over a period of more than 30 years no general method has been found to overcome these difficulties in this paper we present a new approach to the study of combinatorial structures via group ring equations which works without any restrictive assumptions in order to understand the method of the present paper it will be instructive to briefly discuss the self conjugacy concept first turyn 51 demonstrated that the character method for the study of group ring equations works very nicely under the so called self conjugacy condition an integer n is called self conjugate modulo m if all prime ideals above n in the mth cyclotomic field q m are invariant under complex conjugation under this condition it is possible to find all cyclotomic integers in q m of absolute value nt 2 for any positive integer t it is the complete knowledge of the cyclotomic integers of prescribed absolute value which makes the character method work so well under the self conjugacy condition since turyn s fundamental work 51 there have been dozens of papers extending and refining his approach however all these results are restricted to the case of self conjugacy and that is a very severe restriction indeed namely the probability that n is selfconjugate modulo m decreases exponentially fast in the number of distinct prime divisors of n and m see remark 2 2 this means that the self conjugacy method fails in almost all cases one may ask if it is possible to extend turyn s method in order to get rid of the self conjugacy assumption it turns out that in general this is impossible at least with present day methods the required complete knowledge of the cyclotomic integers of prescribed absolute value would yield an

proportional delayed controllers design for lti systems a geometric approach abstractthis paper focuses on the design of p controllers for single input single output linear time invariant systems the basis of this work is a geometric approach allowing to partitioning the parameter space in regions with constant number of unstable roots this methodology defines the hyper planes separating the aforementioned regions and characterises the way in which the number of unstable roots changes when crossing such a hyper plane the main contribution of the paper is that it provides an explicit tool to find p gains ensuring the stability of the closed loop system in addition the proposed methodology allows to design a non fragile controller with a desired exponential decay rate several numerical examples illustrate the results and a haptic experimental set up shows the effectiveness of p controllers

output feedback stabilisation of single input single output linear systems with i o network induced delays an eigenvalue based approach this work addresses the output feedback stabilisation problem for a class of linear single input single output systems subject to i o network delays more precisely we are interested in the characterisation of the set of delay and gain parameters guaranteeing the stability of the closed loop system to perform such an analysis we adopt an eigenvalue perturbation based approach various illustrative numerical examples complete the presentation

robust stabilization by linear output delay feedback the main result of this paper establishes that if a controller c comprising of a linear feedback of the output and its derivatives globally stabilizes a nonlinear plant p then global stabilization of p can also be achieved by an output feedback controller c h where the output derivatives in c are replaced by an euler approximation with sufficiently small delay h 0 this is proved within the conceptual framework of the nonlinear gap metric approach to robust stability the main result is then applied to finite dimensional linear minimum phase systems with unknown coefficients but known relative degree and known sign of the high frequency gain results are also given for systems with nonzero initial conditions

tracking the algebraic multiplicity of crossing imaginary roots for generic quasipolynomials a vandermonde based approach a standard approach in analyzing dynamical systems consists in identifying and understanding the eigenvalues bifurcations when crossing the imaginary axis efficient methods for crossing imaginary roots identification exist however to the best of the author s knowledge the multiplicity of such roots was not deeply investigated in recent papers by the authors 1 2 it is emphasized that the multiplicity of the zero spectral value can exceed the number of the coupled scalar delay differential equations and a constructive approach vandermonde based allowing to an adaptive bound for such a multiplicity is provided namely it is shown that the zero spectral value multiplicity depends on the system structure number of delays and number of non zero coefficients of the associated quasipolynomial rather than the degree of the associated quasipolynomial 3 this technical note extends the constructive approach in investigating the multiplicity of crossing imaginary roots j omega where omega neq 0 and establishes a link with a new class of functional confluent vandermonde matrices a symbolic algorithm for computing the lu factorization for such matrices is provided as a byproduct of the proposed approach a bound sharper than the polya szego generic bound arising from the principle argument is established

time delayed feedback stabilisation of nonlinear potential systems mechanical systems with nonlinear potential forces and delayed feedback are studied it is assumed that in the absence of control the trivial equilibrium positions of considered systems are stable but they are not attracting ones an approach for the constructing of nonlinear controllers providing the asymptotic stability of the equilibrium positions is proposed by the use of the lyapunov direct method and the razumikhin approach it is proved that for the corresponding closed loop systems the asymptotic stability can be guaranteed even in the cases when delay is unknown and time varying moreover estimates for solutions of closed loop systems are found an example and the results of a computer simulation are presented to demonstrate the effectiveness of the proposed approach

instabilities of front patterns in reaction diffusion systems abstract recent experiments in chemical reaction diffusion systems with externally imposed concentration gradients may provide access to a host of spatio temporal pattern formation phenomena these systems tend to form steep reaction fronts in response to the external gradient we use singular perturbation techniques normal form calculations and numerical simulations to investigate the existence and the stability of such sustained fronts in one dimensional systems the theoretical predictions are found in quantitative agreement with direct simulations of the hopf bifurcation from steady to periodically oscillating front structures observed in the couette flow reactor also conditions are found under which oscillations of the spatial structure become chaotic in two dimensional systems we address the issue of realizing an experimental situation hitherto unattained a one dimensional chain of coupled oscillators at the onset of the hopf destabilization of the front structure we point out the intimate relationship between the frequency of oscillation of the homogeneous front pattern and the characteristic wavelength of the turing pattern that can develop along the front 2 d 1 2 we comment on subsequent bifurcations that may result from the nonlinear interaction between hopf and turing instabilities as the precursors to spatio temporal chaos in conclusion we emphasize the possibility of probing the transition to mediated defect turbulence in thin film gel reactors

bifurcation to spatially induced chaos in a reaction diffusion system a one dimensional reaction diffusion equation is used to model a class of experimental open chemical systems in which spatiotemporal patterns can be sustained indefinitely numerical simulations for the model in parameter regimes corresponding to experiment reveal only low dimensional behavior the observed bifurcation sequence leads from steady state concentration profiles to temporally chaotic patterns the physical mechanism that causes this behavior is deduced from analysis of the local dynamics localized oscillators appear at each end of the reactor and one of the oscillators acts as a periodic forcing for the other the numerical results are in good qualitative agreement with the experiments

inverted pendulum stabilization characterization of codimension three triple zero bifurcation via multiple delayed proportional gains the paper considers the problem of stabilization of systems possessing a multiple zero eigenvalue at the origin the controller that we propose uses multiple delayed measurements instead of derivative terms doing so we increase the performances of the closed loop in presence of system uncertainties and or noisy measurements the problem formulation and the analysis is presented through a classical engineering problem which is the stabilization of an inverted pendulum on a cart moving horizontally on one hand we perform a nonlinear analysis of the center dynamics described by a three dimensional system of ordinary differential equations with a codimension three triple zero bifurcation on the other hand we present the complementary stability analysis of the corresponding linear time invariant system with two delays describing the behavior around the equilibrium the aim of this analysis is to characterize the possible local bifurcations finally the proposed control scheme is numerically illustrated and discussed

static output feedback stabilization necessary conditions for multiple delay controllers this note focuses on the static output feedback stabilization problem for a class of single input single output systems when the control law includes multiple distinct delays we are interested in giving necessary conditions for the existence of such stabilizing controllers illustrative examples second order system chain of integrators or chain of oscillators are presented and discussed

brief paper pid controller synthesis for a class of unstable mimo plants with i o delays conditions are presented for closed loop stabilizability of linear time invariant lti multi input multi output mimo plants with i o delays time delays in the input and or output channels using pid proportional integral derivative controllers we show that systems with at most two unstable poles can be stabilized by pid controllers provided a small gain condition is satisfied for systems with only one unstable pole this condition is equivalent to having sufficiently small delay unstable pole product our method of synthesis of such controllers identify some free parameters that can be used to satisfy further design criteria than stability

brief paper extraction of 3d stability switching hypersurfaces of a time delay system with multiple fixed delays stability switching hypersurfaces ssh of a general class of linear time invariant lti multiple time delay systems mtds are extracted the number of delays and system order n 0 in this problem are given but they are arbitrarily large hence the extraction of ssh becomes non trivial a novel and efficient procedure reveals the 3d projections of dimensional ssh without the need for finding the ssh in achieving this computation times required remain in the same order of magnitude independently of and n 0 case studies demonstrate the effectiveness of the approach

fundamental limits on uncertain delays when is a delay system stabilizable by lti controllers this paper concerns the stabilization of linear time invariant lti systems subject to uncertain possibly time varying delays the fundamental issue under investigation referred to as the delay margin problem addresses the question what is the largest range of delay such that there exists a single lti feedback controller capable of stabilizing all the plants for delays within that range drawing upon analytic interpolation and rational approximation techniques we derive fundamental bounds on the delay margin within which the delay plant is guaranteed to be stabilizable by a certain lti output feedback controller our contribution is threefold first for single input single output siso systems with an arbitrary number of plant unstable poles and nonminimum phase zeros we provide an explicit computationally efficient bound on the delay margin which requires computing only the largest real eigenvalue of a constant matrix second for multi input multi output mimo systems we show that estimates on the variation ranges of multiple delays can be obtained by solving lmi problems and further by finding bounds on the radius of delay variations third we show that these bounds and estimates can be extended to systems subject to time varying delays when specialized to more specific cases e g to plants with one unstable pole but possibly multiple nonminimum phase zeros our results give rise to analytical expressions exhibiting explicit dependence of the bounds and estimates on the pole and zeros thus demonstrating how fundamentally unstable poles and nonminimum phase zeros may limit the range of delays over which a plant can be stabilized by a lti controller

use of multiple time delays as controllers in imc schemes abstract multiple time delays with adjustable gains are proposed as controllers of imc schemes such a controller is composed of a finite number of time delays which are connected in parallel given a random disturbance with known power spectral density gains of the time delays are determined so as to minimize the output variance of the system the optimization is performed by means of orthogonal projection in a finite dimensional space in the case of a plant with minimum phase the minimum variance can be made arbitrarily small by increasing the number of time delays

asymptotic stabilization of an input delayed chain of integrators with nonlinearity in this paper we consider a stabilization problem of an input delayed chain of integrators with nonlinearity in this system the nonlinear part cannot be directly canceled by the input due to time delay we design a stabilizing controller and show that the system stability is determined by the types of nonlinear functions

global asymptotic stabilization for chains of integrators with a delay in the input the problem of the global uniform asymptotic stabilization by bounded feedback of a chain of integrators with a delay in the input is solved no limitation on the size of the delay is imposed to validate the approach a third order example is presented

homoclinic chaos in chemical systems abstract we first focus on complex dynamical phenomena generated by chemical kinetics in homogeneous media we comment on the alternating sequences of periodic and chaotic states observed in some experiments on the belousov zhabotinsky reaction when conducted in a continuously stirred tank reactor we present numerical results obtained with a seven variable oregonator model which reproduce most of the features of the experimental sequences we discuss the origin and the nature of the chemical chaos encountered along these sequences in terms of the chaotic orbits which exist on the way to homoclinicity we construct poincare map models which reduce to either multi humped or multi branched 1d maps in the strong area contraction limit we report on a recent experiment which provides evidence that the iteration scheme deduced from the time series of spiral type strange attractors satistifies the symbolic dynamics predicted by sil nikov theory of homoclinic chaos we then investigate spatio temporal pattern forming phenomena in a one dimensional reaction diffusion system with equal diffusion coefficients when imposing a concentration gradient through the system this model mimics the sustained stationary and periodically oscillating front structures observed in recent experiments conducted in an open couette flow reactor we emphasize the possibility that the oscillations of the spatial structure become chaotic we report on numerical simulations of a diffusion induced intermittent bursting phenomenon that is likely to be observed in bench experiments we elaborate on the interpretation of this intermittent occurrence of spatially localized structures in an extended system in terms of sil nikov s homoclinic chaos

spatiotemporal pattern formation and chaos in passive optical systems abstract the interaction and competition of transverse cavity modes in passive optical systems can lead to the formation of spatial field structures whose symmetry differs from that of the empty cavity while the time behaviour can become irregular after having introduced dynamical equations in which the spatial dependence of the electric field is described by frequency degenerate gauss laguerre gl modes the stationary solutions corresponding to a single or many modes are calculated and their stability is discussed a scan of the parameter space allows for the identification of several interesting dynamical regimes an extensive study of the chaotic states is performed in order to understand the relation between gl mode excitation and the relevant active degrees of freedom based on this analysis the conditions for the onset of spatiotemporal chaos for its characterisation and experimental observation are finally discussed

minimal vanishing sums of roots of unity with large coefficients a vanishing sum where n is a primitive nth root of unity and the a i s are non negative integers is called minimal if the coefficient vector a 0 a n 1 does not properly dominate the coefficient vector of any other such non zero sum we show that for every c there is a minimal vanishing sum of nth roots of unity with its greatest coefficient equal to c where n is of the form 3pq for odd primes p q this solves an open problem posed by lenstra jr

sampled data implementation of derivative dependent control using artificial delays we study a sampled data implementation of linear controllers that depend on the output and its derivatives first we consider an lti system of relative degree r geq 2 that can be stabilized using r 1 output derivatives then we consider pid control of a second order system in both cases the euler approximation is used for the derivatives giving rise to a delayed sampled data controller given a derivative dependent controller that stabilizes the system we show how to choose the parameters of the delayed sampled data controller that preserves the stability under fast enough sampling the maximum sampling period is obtained from lmis that are derived using taylor s expansion of the delayed terms with the remainders compensated by appropriate lyapunov krasovskii functionals finally we introduce the event triggering mechanism that may reduce the amount of sampled control signals used for stabilization

an improved time delay implementation of derivative dependent feedback abstract we consider an lti system of relative degree r 2 that can be stabilized using r 1 output derivatives the derivatives are approximated by finite differences leading to a time delayed feedback we present a new method of designing and analyzing such feedback under continuous time and sampled measurements this method admits essentially larger time delay sampling period compared to the existing results and for the first time allows to use consecutively sampled measurements in the sampled data case the main idea is to present the difference between the derivative and its approximation in a convenient integral form the kernel of this integral is hard to express explicitly but we show that it satisfies certain properties these properties are employed to construct the lyapunov krasovskii functional that leads to lmi based stability conditions if the derivative dependent control exponentially stabilizes the system then its time delayed approximation stabilizes the system with the same decay rate provided the time delay for continuous time measurements or the sampling period for sampled measurements are small enough

stabilizing a chain of integrators using multiple delays this note addresses the output feedback stabilization problem of a chain of integrators using multiple delays we shall prove that either n distinct delays or a proportional delay compensator with n 1 distinct delays are sufficient to stabilize a chain including n integrators we present two different approaches both are constructive and rely on frequency domain techniques on a derivative feedback approximation idea and a pole placement idea respectively an illustrative example triple integrator is presented

on vanishing sums of roots of unity an unsolved problem in number theory asked the following for a given natural number m what are the possible integers n for which there exist mth roots of unity 1 n c such that 1 n 0 we show in this paper that the set of all possible n s is exactly the collection of n combinations of the prime divisors of m where n denotes the set of all non negative integers the proof is long and involves a subtle analysis of minimal vanishing sums of mth roots of unity couched in the setting of integral group rings of finite cyclic groups our techniques also recovered with ease some of the classical results on vanishing sums of roots of unity such as those of redei de bruijn and schoenberg

sensitivity to infinitesimal delays in neutral equations the stability of a steady state solution of a neutral functional differential equation can be sensitive to infinitesimal changes in the delays this phenomenon is caused by the behavior of the essential spectrum and is determined by the roots of an exponential polynomial avellar and hale j math anal appl 73 1980 pp 434 452 have considered the case of multiple fixed and nonzero delays in the first part of this paper their results are illustrated by means of spectral plots in the second part we extend the theory of avellar and hale to the limit case whereby some of the delays are brought to zero which may lead to characteristic roots with arbitrarily large real part necessary and sufficient conditions are provided using these results we show that the ratio of the delays plays a crucial role when several delays tend to zero simultaneously as an illustration of the theory we analyze the robustness of a boundary controlled pde in the presence of a small feedback delay

technical communique on delay robustness analysis of a simple control algorithm in high speed networks this paper focuses on the robustness analysis of a simple second order control algorithm for data transfer in high speed networks the model under consideration derived using a fluid approximation technique can be found in izmailov siam j contr optimiz 34 1996 1767 the novelty of the approach lies in characterizing the stability of the scheme in the delay parameter space where the delays correspond to the control time interval and to the round trip time respectively thus we shall compute some delay insensitive measures for the algorithm which give upper and lower bounds for the uncertainty in the knowledge of the round trip times

sufficient conditions for a conjecture of ryser about hadamard circulant matrices let h be a hadamard circulant matrix of order n 4h2 where h 1 is an odd positive integer with at least two prime divisors such that the exponents of the prime numbers that divide h are big enough and such that the nonzero coefficients of the cyclotomic polynomial n t are bounded by a constant independent of n then for all the n n th primitive roots w of 1 p w n is not an algebraic integer in the cyclotomic field k q w where p t is the representer polynomial of h and is the euler function this implies that p w is not a real number

determining lyapunov exponents from a time series we present the first algorithms that allow the estimation of non negative lyapunov exponents from an experimental time series lyapunov exponents which provide a qualitative and quantitative characterization of dynamical behavior are related to the exponentially fast divergence or convergence of nearby orbits in phase space a system with one or more positive lyapunov exponents is defined to be chaotic our method is rooted conceptually in a previously developed technique that could only be applied to analytically defined model systems we monitor the long term growth rate of small volume elements in an attractor the method is tested on model systems with known lyapunov spectra and applied to data for the belousov zhabotinskii reaction and couette taylor flow