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

on stability of lms methods and characteristic roots of delay differential equations we investigate the use of linear multistep lms methods for computing characteristic roots of systems of linear delay differential equations ddes with multiple fixed discrete delays these roots are important in the context of stability and bifurcation analysis we prove convergence orders for the characteristic root approximations and analyze under what condition for the steplength the discrete integration scheme retains certain delay independent stability properties of the original equations unlike existing results we concentrate on the recovery of both stability and instability we illustrate our findings with a number of numerical test results

new results on the synthesis of pid controllers this paper considers the problem of stabilizing a first order plant with dead time using a proportional integral derivative pid controller using a version of the hermite biehler theorem that is applicable to quasi polynomials the complete set of stabilizing pid parameters is determined for both open loop stable and unstable plants the range of admissible proportional gains is first determined in closed form for each proportional gain in this range the stabilizing set in the space of the integral and derivative gains is shown to be either a trapezoid a triangle or a quadrilateral for the case of an open loop unstable plant a necessary and sufficient condition on the time delay is determined for the existence of stabilizing pid controllers

a new perspective in the stability assessment of neutral systems with multiple and cross talking delays the stability of neutral systems with two cross talking delays is investigated using the method of cluster treatment of characteristic roots ctcr there are two main outcomes of this study a we create the strong stabilizability also called the delay stabilizability of the system directly from the ctcr procedure this is achieved by a small delay stability treatment while performing the steps of the ctcr for the delay stabilizable systems we also arrive at the exact bounds of the stability regions in the domain of the delays b we deploy a point wise algorithm which computes the rightmost roots of the characteristic quasi polynomial for cross verification of these stability regions several examples are presented the correspondence between the two methods for all of them is shown to be very strong

brief paper all stabilizing pid controllers for time delay systems this paper presents a method to compute the entire set of stabilizing pid controller parameters for an arbitrary including unstable linear time delay system the main contribution is to handle the infinite number of stability boundaries in the k d k i plane for a fixed proportional gain k p for retarded open loops it is shown that the stable region in the k d k i plane consists of convex polygons concerning neutral loops a new phenomenon is introduced for certain systems and certain k p the exact stable region in the k d k i plane can be described by the limit of a sequence of polygons with an infinite number of vertices this sequence may be well approximated by convex polygons moreover the paper describes a necessary condition for k p intervals potentially having a stable region in the k d k i plane thus the set of stabilizing controller parameters can be calculated after gridding k p in these intervals a matlab tool implementing the presented method is available

on a generalized max type difference equation from automatic control theory abstract this paper studies the boundedness character of positive solutions of the following generalization of a max type difference equation from automatic control theory see mishkis 1977 18 and popov 1966 21 x n 1 max a x n p x n 1 q x n 2 r n n 0 where the parameters a p q and r are positive numbers in the study of the equation we also introduce a new method called oachkatzlschwoif

assigning frequencies via determinantal equations new counterexamples and invariants the problem of arbitrary pole zero assignment by fixed structure compensators has been addressed so far in terms of either necessary or sufficient conditions the strongest existing condition is based on the rank of the differential of a related map on a degenerate controller and holds true generically when the degrees of freedom of the compensator exceed the number of frequencies to be assigned mp n for the output feedback problem the complete nongeneric solvability of the problem is still open even when complex controllers are considered a simple necessary solvability condition is that the linear map appearing as a factor of the main determinantal map defining the assignment problem is onto here we examine the special problem of assignment of matrix pencil zeros via diagonal perturbations and we present a new necessary and sufficient condition for complex solvability in terms of a new invariant involving the minors of this linear map based on this result we demonstrate that for the important case where the degrees of freedom of the controller are equal to the number of frequencies to be assigned the surjectivity of this linear map although it is necessary it is not sufficient for the solvability of the problem

stabilizing a class of time delay systems using the hermite biehler theorem abstract in this paper we use the hermite biehler theorem to establish results for the design of fixed order controllers for a class of time delay systems we extend results of the polynomial case to quasipolynomials using the property of interlacing in high frequencies of the class of time delay systems considered

new result on pid controller design of lti systems via dominant eigenvalue assignment this note considers the problem of assigning the dominant eigenvalues of a linear time invariant lti system to the desired positions by using proportional integral derivative pid controllers the procedure is based on first setting some rightmost eigenvalues of the system at the desired positions and then guaranteeing the dominance of those rightmost eigenvalues by using the generalization of the hermite biehler theorem it is worth pointing out that this work aims to ascertain the gains of pid controllers in a straightforwardly computational way which plays an important role in practical applications

on a system of two nonlinear difference equations abstract in this paper we study the oscillatory behavior the boundedness of the solutions and the global asymptotic stability of the positive equilibrium of the system of two nonlinear difference equationsxn 1 a yn xn p yn 1 a xn yn q n 0 1 p qare positive integers

composition constants for raising the orders of unconventional schemes for ordinary differential equations many models of physical and chemical processes give rise to ordinary differential equations with special structural properties that go unexploited by general purpose software designed to solve numerically a wide range of differential equations if those properties are to be exploited fully for the sake of better numerical stability accuracy and or speed the differential equations may have to be solved by unconventional methods this short paper is to publish composition constants obtained by the authors to increase efficiency of a family of mostly unconventional methods called reflexive

frequency domain synthesis of hereditary control systems via anisochronic state space a method of control feedback design for systems with hereditary properties is presented which is based on the concept of anisochronic state space a new kind of state feedback control is proposed using as state variables the system accumulation attributes only while the other delay dynamics are represented by functional terms assigning the control parameters is approached as a placement of characteristic function zeros however owing to the infinite set of these zeros in the hereditary system only the dominant zeros are placed for this purpose a new technique applying the conformal mapping of transcendental characteristic function of the hereditary system has been developed

global attractivity of a family of nonautonomous max type difference equations abstract this paper studies a family of nonautonomous max type difference equations with several delays using some transformations we prove global attractivity of positive solutions to these equations under some conditions some of our results considerably extend related ones in the literature two examples are given to illustrate the main results

on some solvable systems of difference equations abstract we show that the following systems of difference equations x n 1 u n 1 v n y n 1 w n 1 s n n n 0 where u n v n w n s n are some of the sequences x n or y n with real initial values x 0 and y 0 are solvable in fourteen out of sixteen possible cases two cases are left unsolved probably the most interesting is the result in the case u n x n v n x n w n x n s n y n where a fascinating formula is obtained in an elegant way by using some ad hoc ideas

low order stabilization of lti systems with time delay this paper considers the problem of stabilizing a single input single output siso linear time invariant lti plant with known time delay using a low order controller such as a proportional p a proportional integral pi or a proportional integral derivative pid controller for the siso lti system with time delay the closed loop characteristic function is a quasipolynomial that possesses the following features all its infinite roots are located on the left of certain vertical line of the complex plane and the number of its unstable roots is finite necessary and sufficient conditions for the stability of lti systems with time delay are first presented by employing an extended hermite biehler theorem applicable to quasi polynomials based on the conditions analytical algorithms are then proposed to compute the stabilizing sets of p pi and pid controllers the resulting characterizations of the stabilizing sets for p pi and pid controllers are analogous to the youla parameterization of all stabilizing controllers for plants without time delay numerical examples are provided to illustrate the proposed algorithm

the boundary constraint method for constructing vortex surface fields we develop a boundary constraint method for constructing the vortex surface field vsf in a three dimensional fluid velocityvorticity field the isosurface of the vsf is a vortex surface consisting of vortex lines which can be used to characterize the evolution of vortical structures in a lagrangian sense the evolution equation with pseudo time is solved under the vsf boundary constraint to obtain a numerical solution of the vsf compared with the existing two time method the boundary constraint method constructs the vsf from a single velocity dataset at a given time instead of a time series of velocity fields starting from a simple condition this improvement significantly increases the applicability of the vsf method and reduces the demanding computational cost and required velocity data size using the boundary constraint method we construct the vsfs in taylorgreen flow and transitional channel flow in addition the uniqueness of the vsf solution is discussed and the convergence of the error in the calculation of vsfs is analyzed

passivity based pi control of first order systems with i o communication delays a frequency domain analysis abstractin this paper the proportional integral pi control of first order linear passive systems through a delayed communication channel is revisited in light of the relative stability concept called stability treating the delayed communication channel as a transport partial differential equation pde the passivity of the overall control loop is guaranteed resulting in a closed loop system of neutral nature spectral methods are then applied to the system to obtain a complete stability map in particular we perform the d subdivision method to declare the exact stability regions in the space of pi parameters this framework is then utilised to analytically determine the maximum achievable exponential decay rate d of the system while achieving the pi tuning as explicit function of d and system parameters

on the stabilizing pid controllers for integral processes in this note we show how to determine the set of stabilizing parameters of a proportional integral derivative controller for an integrator plus dead time process in particular by exploiting a version of the hermite biehler theorem applicable to quasipolynomals the admissible range of the proportional gain is computed first then for each value of the proportional gain in that range the set of stabilizing values of the integral and derivative gain are found it is shown that the procedure is greatly simplified with respect to the case of a second order integral process with dead time

the pole placement problem via pi feedback controllers in this paper we study the pole placement problem using generalized pi controllers of a fixed lag k as compensators we derive a new strong sufficiency condition which guarantees the arbitrary pole assignability of a given system having m inputs p outputs and mcmillan degree n this sufficiency condition misses the theoretical best possible necessary condition by one degree of freedom the proof of the main result can be used to derive a numerical procedure this numerical procedure is however very sensitive in the parameters and further research is required to investigate the conditions under which sensitivity occurs and whether the solution can be made more robust in this regard

levy statistics in a hamiltonian system enhanced diffusion in a hamiltonian system is studied in terms of the continuous time random walk formulation for l evy walks the previous l evy walk scheme is extended i to include interruptions by periods of temporal localization and ii to describe motion in two dimensions we analyze a case of conservative motion in a two dimensional periodic potential numerical calculations of the mean squared displacements and the propagators for intermediate energies are consistent with the l evy walk description

on a max difference equation abstract in this paper we study the existence of generalized invariants and the periodicity of the positive solutions of max equations x n 1 max a n n i n k 1 x i b n n i n k x i n 0 1 where an bn are sequences of positive numbers x k x k 1 x0 0 and k 2 3

practical symplectic partitioned runge kutta and runge kutta nystrom methods we present new symmetric fourth and sixth order symplectic partitioned runge kutta and runge kutta nystrom methods we studied compositions using several extra stages optimising the efficiency an effective error ef is defined and an extensive search is carried out using the extra parameters the new methods have smaller values of ef than other methods found in the literature when applied to several examples they perform up to two orders of magnitude better than previously known method which is in very good agreement with the values of ef

brief pi stabilization of first order systems with time delay this paper considers the problem of stabilizing a first order plant with dead time using a pi controller using a version of the hermite biehler theorem applicable to quasipolynomials a complete and constructive characterization of all stabilizing pi gain values is obtained

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

on the difference equation xn xn 2 bn cnxn 1xn 2 abstract we prove that the difference equation x n x n 2 b n c n x n 1 x n 2 n n 0 where b n n n 0 and c n n n 0 are sequences periodic with period two and the initial values x 2 x 1 are real numbers can be solved explicitly some applications of obtained formulae are given

global stability of a difference equation with maximum we prove that every positive solution to the difference equationx n maxa 1x n 1 a 1 a 2x n 2 a 2 a kx n k a k email protected n 0 where email protected n a i 0 a email protected 0 1 i 1 k converges to the following quantity maxa 1 1 a 1 1 a k 1 a k 1 confirming a quite recent conjecture of interest we also prove another result on global convergence which concerns some cases when not all a i i 1 k belong to the interval 0 1

an eigenvalue based approach for the stabilization of linear time delay systems of neutral type an eigenvalue based approach for the stabilization of linear neutral functional differential equations is presented which extends the recently developed continuous pole placement method for delay equations of retarded type the approach consists of two steps first the stability of the associated difference equation is determined and a procedure is applied to compute the supremum of the real parts of its characteristic roots which corresponds to computing the radius of the essential spectrum of the solution operator of the neutral equation no restrictions are made on the dimension of the system and the number of delays also the effect of small delay perturbations is explicitly taken into account as a result of this first step the stabilization problem of the neutral equation is reduced to a problem involving only a finite number of characteristic roots as a second step stabilization is achieved by shifting the rightmost or unstable characteristic roots to the left half plane in a quasi continuous way by applying small changes to the controller parameters and meanwhile monitoring other characteristic roots with a large real part a numerical example is presented

implementation of high order spline interpolations for tracking test particles in discretized fields a systematic approach for constructing high order spline interpolation methods is proposed for fields known on regular rectangular grids these interpolation methods are tested in tracking trajectories of particles submitted to a force that derives from a potential known on a grid the interplay between the time advancement scheme and the spatial interpolation is studied in detail and it is shown how the order of the trajectory solver is directly affected by the order of the spline interpolation it is also shown how an interpolation method that preserves topological properties of physical fields can be better exploited with these higher order spline approximations

on the difference equation xn xn kb cxn 1 xn k abstract we describe the behavior of well defined solutions of the difference equation x n x n k b cx n 1 x n k n n 0 where k n parameters b c and the initial values x k x 1 are real numbers these results among others improve and extend our results in s stevic more on a rational recurrence relation appl math e notes 4 2004 80 85

on the periodic nature of the solutions of the reciprocal difference equation with maximum abstract we prove that every positive solution of the third order difference equation x n max a x n 1 b x n 2 c x n 3 n 0 1 is eventually periodic of not necessarily prime period t which is explicitly determined by the coefficients a b and c

study of the asymptotic behavior of the solutions of three systems of difference equations of exponential form in this paper we study the boundedness the persistence and the asymptotic behavior of the positive solutions of the following systems of two difference equations of exponential form xn 1 a be yn c yn 1 y n 1 d e xn f xn 1

tricubic interpolation in three dimensions the purpose of this paper is to give a local tricubic interpolation scheme in three dimensions that is both c 1 and isotropic the algorithm is based on a specific 64 64 matrix that gives the relationship between the derivatives at the corners of the elements and the coefficients of the tricubic interpolant for this element in contrast with global interpolation where the interpolated function usually depends on the whole data set our tricubic local interpolation only uses data in a neighbourhood of an element we show that the resulting interpolated function and its three first derivatives are continuous if one uses cubic interpolants the implementation of the interpolator can be downloaded as a static and dynamic library for most platforms the major difference between this work and current local interpolation schemes is that we do not separate the problem into three one dimensional problems this allows for a much easier and accurate computation of higher derivatives of the extrapolated field applications to the computation of lagrangian coherent structures in ocean data are briefly discussed

decentralized feedback pole placement of linear systems a projective product spaces model is used to analyse decentralized systems the degree of the pole placement map is computed the conditions under which the degree is odd are also given twin lift systems are studied it is proved that the poles of a twin lift system can be assigned to any values by local static and local dynamic feedback laws if and only if the system is jointly controllable communicated by professor a isidori

boundedness character of a class of difference equations abstract a complete picture regarding the boundedness character of positive solutions to the following difference equation x n max a x n 1 p x n k p n n 0 where k 2 and the parameters a and p are positive real numbers is given in particular for the case p k 1 0 k k k 1 k 1 we prove that all solutions to the equation are bounded we also present corresponding results concerning the following closely related difference equation x n a x n 1 p x n k p n n 0

technical communique synthesis of pid controllers for a class of time delay systems in this paper we use the hermite biehler theorem to establish results on the design of proportional plus integral plus derivative pid controllers for a class of time delay systems using the property of interlacing at high frequencies of the class of systems considered and linear programming we obtain the set of all stabilizing pid controllers as far as we know previous results on the synthesis of pid controllers rely on the solution of transcendental equations this paper also extends previous results on the synthesis of proportional controllers for a class of delay systems of retarded type to a larger class of delay systems

on a difference equation with maximum abstract this paper studies the asymptotic behavior of positive solutions to the difference equation x n max 1 x n 1 a x n 2 n 0 1 where 0 0 we prove that every positive solution of this equation approaches x 1 or is eventually periodic with period 4 according as a 1 or a 1

solutions of a max type system of difference equations we show in an elegant way that the general solution to the following max type system of difference equations xn 1 maxaxn ynxn yn 1 maxayn xnyn n n0 for the case y0 x0 a 0 y0 x0 max a 1 a is given by xn afk n 1 n x0fk n y0fk n 1 n n n y1 x0 y0 and yn y0fk n 1 1afk n 1 n 1x0fk n 1 1 1 n n 2 where k n n for n even k n n 1 for n odd n n0 n n mod2 for n n0 and where the sequence fn n n is the fibonacci sequence correcting and giving a proof of a wrong result in the paper d simsek b demir c cinar on the solutions of the system of difference equations xn 1 max a xn yn xn yn 1 max a yn xn yn discrete dyn nat soc 2009 2009 11 pp article id 325296

controller design for delay systems via eigenvalue assignment on a new result in the distribution of quasi polynomial roots this paper considers the eigenvalue distribution of a linear time invariant lti system with time delays and its application to some controllers design for a delay plant via eigenvalue assignment first a new result on the root distribution for a class of quasi polynomials is developed based on the extension of the hermite biehler theorem then such result is applied to proportional integral pi controller parameter design for a first order plant with time delay through pole placement the complete region of pi gains can be obtained so that the rightmost eigenvalues in the infinite eigenspectrum of the closed loop system with delay plant are assigned to desired positions in the complex plane furthermore on the basis of the previous result this paper also extended the pi control to the proportional integral derivative pid control it is worth pointing out that this work aims to improve the performance of the closed loop system on the premise of guaranteeing the stability

generalizations of the hermite biehler theorem abstract the hermite biehler theorem gives necessary and sufficient conditions for the hurwitz stability of a polynomial in terms of certain interlacing conditions in this paper we generalize the hermite biehler theorem to situations where the test polynomial is not necessarily hurwitz the generalization is given in terms of an analytical expression for the difference between the numbers of roots of the polynomial in the open left half and open right half planes the result can be used to solve important stabilization problems in control theory and is therefore of both academic as well as practical interest

on some periodic systems of max type difference equations we show that all positive solutions to the system of max type difference equations xn 1 max1 i m1f1ixn ki 1 1 1 xn ki 2 1 2 xn ki l 1 l n xn s 1 xn 2 max1 i m2f2ixn ki 1 2 1 xn ki 2 2 2 xn ki l 2 l n xn s 2 xn l max1 i mlflixn ki 1 l 1 xn ki 2 l 2 xn ki l l l n xn s l n n0 where s l mj ki t j n j t 1 l and for a fixed j i 1 mj and where the functions fji 0 l n0 0 j 1 l i 1 mj satisfy some conditions are eventually periodic with not necessarily prime period s a related result for the corresponding system of min type difference equations is also proved

on a symmetric system of max type difference equations we study behavior of positive solutions of the following max type system of difference equationsx n 1 maxc y n px n 1 p y n 1 maxc x n py n 1 p email protected n 0 where p email protected 0 extending some results in the literature among other results we prove that if p email protected 0 1 then every positive solution of the system converges to 1 1

on the system xn 1 ynx n k yn k 1 an bnynxn k yn 1 xnyn k xn k 1 cn dnxnyn k it is shown that the system of difference equationsx n 1 y nx n ky n k 1 a n b ny nx n k y n 1 x ny n kx n k 1 c n d nx ny n k email protected n 0 where email protected n the sequences a n b n c n d n email protected n 0 and the initial values x i y i i 0 email protected are real numbers can be solved in closed form asymptotic behavior of well defined solutions of the system for the case when all the sequences a n b n c n and d n are constant is studied in detail

on the dynamics of two exponential type systems of difference equations in this paper we study the asymptotic behavior of the positive solutions of the systems of the two difference equations i x n 1 a by n 1e x n y n 1 c dx n 1e y n ii x n 1 a by n 1e y n y n 1 c dx n 1e x n where the constants a b c d are positive real numbers and the initial values x 1 x 0 y 1 y 0 are also positive real numbers

strong stability of neutral equations with an arbitrary delay dependency structure the stability theory for linear neutral equations subjected to delay perturbations is addressed it is assumed that the delays cannot necessarily vary independently of each other but depend on a possibly smaller number of independent parameters as a main result necessary and sufficient conditions for strong stability are derived along with bounds on the spectrum which take into account the precise dependency structure of the delays in the derivation of the stability theory results from realization theory and determinantal representations of multivariable polynomials play an important role the observations and results obtained in the paper are first illustrated and validated with a numerical example next the effects of small feedback delays on the stability of a boundary controlled hyperbolic partial differential equation and of a control system involving state derivative feedback are analyzed

new results on eigenvalue distribution and controller design for time delay systems this paper considers the eigenvalue distribution of a linear time invariant lti system with commensurate time delays and its application to proportional integral derivative pid controller design for a delay plant via dominant eigenvalue assignment a new result on the root distribution of a quasi polynomial is first produced by applying part of pontryagin s conclusions this result which gives a necessary and sufficient condition can be directly used to judge the number of the right half plane eigenvalues of the characteristic equation of a time delay system based on the proposed result necessary and sufficient conditions on dominant eigenvalue assignment for pid control of time delay systems are presented and an algorithm is then provided to determine the pid controller gains the proposed approaches can assign some one or two eigenvalues to the desired positions and all the other eigenvalues to the left of a given line to guarantee the dominance of the assigned ones which enables us to design the controller according to the desired performance indexes for a standard first order system or a standard second order system in addition to stability the method is effective for the closed loop characteristic equation being retarded type or neutral type the controller gains to achieve the control objective can be characterized by a straightforward computation further a result on degradation to proportional integral pi control of time delay systems is given

on the system of difference equations xn cnyn 3 an bnyn 1xn 2y n 3 yn γnxn 3 αn βnxn 1yn 2 xn 3 we show that the system of difference equationsx n c ny n 3a n b ny n 1x n 2y n 3 y email protected nx n email protected email protected nx n 1y n 2x n 3 email protected n 0 where the sequences a n b n c n a n b n c n email protected n 0 and initial values x i y i email protected 1 2 3 are real numbers such that c n 0 c n 0 email protected n 0 can be solved in closed form and for the case when all the sequences a n b n c n a n b n and c n are constant we describe the asymptotic behavior of well defined solutions of the system

on the boundedness of positive solutions of the reciprocal max type difference equation xn max a1n 1 xn 1 a2n 1 xn 2 atn 1 xn t with periodic parameters we investigate the boundedness of positive solutions of the reciprocal max type difference equation xn max a1n 1 xn 1 a2n 1 xn 2 atn 1 xn t where for each value of i the sequence ain n 0 of positive numbers is periodic with period pi we give both sufficient conditions on the pi s for the boundedness of all solutions and sufficient conditions for all solutions to be unbounded this work essentially complements the work by bidwell and franke who showed that as long as every positive solution of our equation is bounded then every positive solution is eventually periodic thereby leaving open the question as to when solutions are bounded

eventually constant solutions of a rational difference equation we describe all the solutions of a rational difference equation from putnam s mathematical competition which are eventually equal to its positive equilibrium email protected 1 as a consequence we give a new elegant and short proof of the fact that the equation has a positive solution which is not eventually equal to one moreover we show that almost all solutions of the equation are not eventually equal to one

on a class of third order nonlinear difference equations this paper studies the boundedness character of the positive solutions of the difference equation x n 1 a x n px n 1 qx n 2 r n n 0 where the parameters a p q and r are positive numbers

further results on the synthesis of pid controllers this note focuses on a graphical approach to determine the stability region in the parameter space of pid controlled second order plants with dead time this work is the extension of and further results on the same topic of the previous paper in the literature on first order plants with dead time the main characterization of the new method is the utilization of a graphical stability criterion in the parameter space which releases a necessary and sufficient stability condition once the range of proportional gain is known the complete stabilizing set in the plain of integral derivative gains can be drawn and identified immediately not to be computed mathematically a practical algorithm of drawing the stabilizing boundaries in the integral derivative space is proposed

dynamic similarity approach to control system design delayed pid control loop abstractthe paper deals with a novel approach to dimensional analysis based design of control systems the case of delayed pid control loop is used to show the benefits of the dynamic similarity concept in providing the control with response resulting from a constrained integral absolute error iae optimisation the particular aim is to achieve this response in disturbance rejection ability of the control loop to reach a generalised description in evaluating the control loop dynamics a dimensionless model is applied resulting from the dimensional analysis of the control loop parameters by means of originally introduced similarity numbers of laggardness and swingability explicit relationships between the dominant pole coordinates and the pid controller gains are obtained the proposed optimum setting of control loop constrained in damping is referred to the ultimate oscillations of the plant

mapping based algorithm for large scale computation of quasi polynomial zeros a method for computing all zeros of a retarded quasi polynomial that are located in a large region of the complex plane is presented the method is based on mapping the quasi polynomial and on utilizing asymptotic properties of the chains of zeros first the asymptotic exponentials of the chains are determined based on the distribution diagram of the quasi polynomial secondly large regions free of zeros are defined finally the zeros are located as the intersection points of the zero level curves of the real and imaginary parts of the quasi polynomial which are evaluated over the areas of the region outside those free of zeros

robust stability of time delay systems there are two fundamental results available when we study stability of a polynomial family that is described by convex polytope in the coefficient space the edge theorem and the theory based on the concept of convex directions many known results can be explained with these two results this paper deals with a generalization of this line of research to the case of quasipolynomials that are entire functions which include both degree of the independent variable and exponential functions the main objects of the paper are the developing of the concept of convex directions for quasipolynomials and exploiting this concept for construction of testing sets for quasipolynomial families one of the primary sources of motivation for the class of problems considered in this paper is derived from process control a typical problem formulation almost always includes a delay element in each subsystem process block when we interconnect a number of such blocks in a feedback system the study of robust stability involves quasipolynomials of the sort considered in this paper

on a nonlinear generalized max type difference equation abstract the boundedness character of positive solutions of the following max type difference equation x n max a x n 1 p x n k r n n 0 where k n 1 the parameters a and r are positive and p is a nonnegative real number is studied in this paper our main results considerably improve results appearing in the literature

on a third order system of difference equations abstract we show that the system of difference equations x n 1 a 1 x n 2 b 1 y n z n 1 x n 2 c 1 y n 1 a 2 y n 2 b 2 z n x n 1 y n 2 c 2 z n 1 a 3 z n 2 b 3 x n y n 1 z n 2 c 3 n n 0 where the parameters a i b i c i i 1 2 3 and initial values x j y j z j j 0 1 2 are real numbers can be solved developing further the results in the literature

global stability of a max type difference equation we show that every positive solution to the difference equationx n maxa 1x n p 1 a 1 a 2x n p 2 a 2 a kx n p k a k n n 0 where p i i 1 k are natural numbers such that 1 0 a i 1 1 i 1 k converges to max 1 i ka i 1 a i 1 this result improves and complements the main result in our recent note s stevic global stability of a difference equation with maximum appl math comput 210 2009 525 529 since it also considers the case when a i 1 0 i 1 k

the boundedness character of two stevic type fourth order difference equations abstract the boundedness character of positive solutions of two nonlinear fourth order difference equations which are particular cases of two large classes of difference equations by stevic are studied in this paper

geometric integrators for odes geometric integration is the numerical integration of a differential equation while preserving one or more of its geometric properties exactly i e to within round off error many of these geometric properties are of crucial importance in physical applications preservation of energy momentum angular momentum phase space volume symmetries time reversal symmetry symplectic structure and dissipation are examples in this paper we present a survey of geometric numerical integration methods for ordinary differential equations our aim has been to make the review of use for both the novice and the more experienced practitioner interested in the new developments and directions of the past decade to this end the reader who is interested in reading up on detailed technicalities will be provided with numerous signposts to the relevant literature

on stability crossing curves for general systems with two delays for the general linear scalar time delay systems of arbitrary order with two delays this article provides a detailed study on the stability crossing curves consisting of all the delays such that the characteristic quasipolynomial has at least one imaginary zero the crossing set consisting of all the frequencies corresponding to all the points in the stability crossing curves are expressed in terms of simple inequality constraints and can be easily identified from the gain response curves of the coefficient transfer functions of the delay terms this crossing set forms a finite number of intervals of finite length the corresponding stability crossing curves form a series of smooth curves except at the points corresponding to multiple zeros and a number of other degenerate cases these curves may be closed curves open ended curves and spiral like curves oriented horizontally vertically or diagonally the category of curves are determined by which constraints are violated at the two ends of the corresponding intervals of the crossing set the directions in which the zeros cross the imaginary axis are explicitly expressed an algorithm may be devised to calculate the maximum delay deviation without changing the number of right half plane zeros of the characteristic quasipolynomial and preservation of stability as a special case

periodicity of a class of nonautonomous max type difference equations abstract in this note we prove several general and easily applicable results regarding the periodicity of positive solutions of the following difference equations x n max f 1 x n k 1 n f m x n k m n x n s n n 0 and x n min f 1 x n k 1 n f m x n k m n x n s n n 0 where 1 k 1 k m m s n and where the functions f i 0 n 0 0 1 i m satisfy some additional conditions these results considerably extend and unify numerous results on max type difference equations existing in the literature

on a system of difference equations abstract we show that the system of difference equations x n 1 ax n 1 by n x n 1 c y n 1 y n 1 x n y n 1 n n 0 where the parameters a b c and initial values x 1 x 0 y 1 y 0 are real numbers can be solved considerably improving the results in the literature

on the solutions of a system of difference equations with maximum in this paper we study the following max type system of difference equations x n max 1 x n m min 1 a y n r y n max 1 y n m min 1 b x n t n n 0 where a b 0 m r t 1 2 with r m and t m we show that every solution of this system with the initial values x d y d x d 1 y d 1 x 1 y 1 0 is eventually periodic with period 2m where d max m r t

eigenvalue assignment by static output feedback on a new solvability condition and the computation of low gain feedback matrices in this article the static output feedback problem for linear time invariant systems is considered for arbitrary assignability of the roots of the characteristic polynomial by static output feedback a new necessary and sufficient condition is derived although the proof is based on simple analysis the known sufficient conditions derived by techniques of algebraic geometry are directly covered furthermore an algorithm for the calculation of feedback matrices assigning a desired set of eigenvalues is proposed this algorithm does not require the desired eigenvalues to be distinct and it explicitly exploits the available degrees of freedom for reducing the feedback gain the presented approach is illustrated on computational examples

on the nonautonomous difference equation xn 1 an xn 1pxnq abstract in this paper we study the asymptotic behavior and the periodicity of the positive solutions of the nonautonomous difference equation x n 1 a n x n 1 p x n q n 0 1 where an is a positive bounded sequence p q 0 and x 1 x0 are positive numbers

long term behavior of positive solutions of a system of max type difference equations abstract some sufficient conditions which guaranty the attractivity of all positive solutions of the next system of difference equations x n max a y n 1 p j 2 k x n j q j y n max a x n 1 p j 2 k y n j q j n n 0 where a p 0 and q j j 2 k are nonnegative numbers are given the boundedness character of positive solutions of the system is also studied