norm based robust control of state constrained discrete time linear systems the author presents a theoretical framework for analyzing the stability properties of constrained discrete time systems in the presence of uncertainty and it is shown that this formalism provides a unifying approach including as a particular case the well known technique of estimating robustness bounds from the solution of a lyapunov equation these results are applied to the problem of designing feedback controllers capable of stabilizing a family of systems while at the same time satisfying state space constraints

discretization and event triggered digital output feedback control of lpv systems abstract this paper investigates the problem of discretization and digital output feedback control design for continuous time linear parameter varying lpv systems subject to a time varying networked induced delay the proposed discretization procedure converts a continuous time lpv system into an equivalent discrete time lpv system based on an extension of the taylor series expansion and using an event based sampling the scheduling parameters are continuously measured and modeled as piecewise constant a new transmission of the measured output to the controller is triggered by significant changes in the parameters yielding time varying transmission intervals the obtained discretized model has matrices with polynomial dependence on the time varying parameters and an additive norm bounded term representing the discretization residual error a two step strategy based on linear matrix inequality conditions is then proposed to synthesize a digital static scheduled output feedback control law that stabilizes both the discretized and the lpv model the conditions can also be used to provide robust i e independent of the scheduling parameter static output feedback controllers the viability of the proposed design method is illustrated through numerical examples

levels of knotting of spatial handlebodies if h is a spatial handlebody i e a handlebody embedded in the 3 sphere a spine of h is a graph s3 such that h is a regular neighbourhood of usually h is said to be unknotted if it admits a planar spine this suggests that a handlebody should be considered not very knotted if it admits spines that enjoy suitable special properties building on this remark we define several levels of knotting of spatial handlebodies and we provide a complete description of the relationships between these levels focusing our attention on the case of genus 2 we also relate the knotting level of a spatial handlebody h to classical topological properties of its complement m s3 h such as its cut number more precisely we show that if h is not highly knotted then m admits special cut systems for m and we discuss the extent to which the converse implication holds along the way we construct obstructions that allow us to determine the knotting level of several families of spatial handlebodies these obstructions are based on recent quandle coloring invariants for spatial handlebodies on the extension to the context of spatial handlebodies of tools coming from the theory of homology boundary links on the analysis of appropriate coverings of handlebody complements and on the study of the classical alexander elementary ideals of their fundamental groups

robust stability of interval matrices a computational approach a simple computational procedure is given to determine the robust stability of matrices containing interval parameter uncertainty the computation consists of determining the maximal phase difference of the characteristic polynomial evaluated over the vertices of the uncertain parameter set at each point of the stability boundary and verifying that this maximal phase difference is less than radians this follows as a direct application of the mapping theorem which allows us effectively to treat rank one perturbations of matrices we discuss the conservatism of the method and show how the solution may be improved to any desired level of accuracy we give several numerical examples of systems with state space perturbations and compare our results with those obtained by other methods

stability of polynomials under coefficient perturbation let the real polynomial a 0 x n a 1 x n 1 a n 1 x a n be stable and let the real numbers b k c k geq 0 0 leq k leq n be given we present a simple determinant criterion for finding the largest t 0 geq 0 such that the polynomial alpha 0 x n alpha 1 x n 1 alpha n 1 x alpha n is stable for all alpha k in a k b k t 0 a k c k t 0 cup a k 0 leq k leq n several further observations allow us to reduce the computational cost considerably

a trust region spectral bundle method for nonconvex eigenvalue optimization we present a nonsmooth optimization technique for nonconvex maximum eigenvalue functions and for nonsmooth functions which are infinite maxima of eigenvalue functions we prove global convergence of our method in the sense that for an arbitrary starting point every accumulation point of the sequence of iterates is critical the method is tested on several problems in feedback control synthesis

lpv system analysis via quadratic separator for uncertain implicit systems this paper considers a class of linear systems containing time varying parameters whose behavior is not known exactly we assume that the parameters vary within known intervals and there are known bounds on their rates of variation our objective is to give a computationally verifiable condition that guarantees stability of the system for all possible parameter variations we first point out that the information on the rate bounds can be exploited by considering an augmented system described by an implicit model we then develop a tool called the quadratic separator for stability analysis of uncertain implicit systems using this tool a sufficient stability condition is obtained for the original linear parameter varying system moreover we show that the stability condition obtained is equivalent to the existence of a lyapunov function that depends on the parameters in a linear fractional manner finally the computational aspects of the proposed stability conditions are addressed

lp parameter perturbation and design of robust controllers for linear systems in this paper linear systems with characteristic equations that are linearly dependent on some uncertain parameters are considered a new procedure is proposed for the calculation of d stability margins in the parameter space in general lp norms the procedure is simple and computationally efficient for several important regions of interest in the complex plane e g left hand plane sectors and circles centred at the origin it requires the solution of only one equation based on this result a new method is presented for the design of robust controllers robust placement of the closed loop poles inside the regions of interest is addressed the method provides a simple and computationally tractable solution to the robust controller design problem it can be used to synthesize robust controllers with general rational transfer functions for plants modelled as the ratio of two uncertain polynomials subjected to lp parameter perturbation

a heuristic approach to solving a class of bilinear matrix inequality problems abstract many canonical and modern control problems can be recast into the problem of a group of matrix inequalities some of them are in the form of linear matrix inequalities lmis which can be solved very efficiently by the powerful lmi toolbox in matlab but some others are in the form of bilinear matrix inequalities the characteristic of this latter class of problems is that when the so called communicating variables are fixed the overall problem will be reduced to the problem in lmis thus how to find the communicating variables is the key to solve the whole problem in this paper an optimal estimate for the communicating variables is presented we will illustrate our method by completely solving the problems of overshoot bound control and reachable set analysis for uncertain systems numerical examples are provided to show the effectiveness of the proposed method

an invariant of regular isotopy this paper studies a two variable laurent polynomial invariant of regular isotopy for classical unoriented knots and links this invariant is denoted lk for a link k and it satisfies the axioms 1 regularly isotopic links receive the same polynomial 2 lo0 1 3 l al l a l 4 l small diagrams indicate otherwise identical parts of larger diagrams regular isotopy is the equivalence relation generated by the reidemeister moves of type ii and type iii invariants of ambient isotopy are obtained from l by writhe normalization

robust control with structure perturbations the problem of making a given stabilizing controller robust so that the closed loop system remains stable for prescribed ranges of variations of a set of physical parameters in the plant the problem is treated in the state space and transfer function domains in the state space domain a stability hypersphere is determined in the parameter space using lyapunov theory the radius of this hypersphere is iteratively increased by adjusting the controller parameters until the prescribed perturbation ranges are contained in the stability hypersphere in the transfer function domain a corresponding stability margin is defined and optimized on the basis of the recently introduced concept of the largest stability hypersphere in the space of coefficients of the closed loop characteristic polynomial the design algorithms are illustrated by examples

improved measures of stability robustness for linear state space models in this paper the aspect of stability robustness of linear systems is analyzed in the time domain a bound on the structured perturbation of an asymptotically stable linear system is obtained to maintain stability using a lyapunov matrix equation solution the resulting bound is shown to be an improved bound over the ones recently reported in the literature also special cases of the nominal system matrix are considered for which the bound is given in terms of the nominal matrix thereby avoiding the solution of the lyapunov matrix equation examples given include comparison of the proposed approach with the recently reported results

static output feedback controllers stability and convexity the main objective of this paper is to solve the following stabilizing output feedback control problem given matrices a b sub 2 c sub 2 with appropriate dimensions find if one exists a static output feedback gain l such that the closed loop matrix a b sub 2 lc sub 2 is asymptotically stable it is known that the existence of l is equivalent to the existence of a positive definite matrix belonging to a convex set such that its inverse belongs to another convex set conditions are provided for the convergence of an algorithm which decomposes the determination of the aforementioned matrix in a sequence of convex programs hence this paper provides a new sufficient but not necessary condition for the solvability of the above stabilizing output feedback control problem as a natural extension we also discuss a simple procedure for the determination of a stabilizing output feedback gain assuring good suboptimal performance with respect to a given quadratic index some examples borrowed from the literature are solved to illustrate the theoretical results

graph invariants of vassiliev type and application to 4d quantum gravity we consider graph invariants of vassiliev type extended by the quantum group link invariants when they are expanded byx whereq e x the expansion coefficients are known as the vassiliev invariants of finite type in the present paper we define tangle operators of graphs given by a functor from a category of colored and oriented graphs embedded into a 3 space to a category of representations of the quasi triangular ribbon hopf algebra extended byu q sl 2 c which are subject to a quantum group analog of the spinor identity in terms of them we obtain the graph invariants of vassiliev type expressed to be identified with chern simons vacuum expectation values of wilson loops including intersection points we also consider the 4d canonical quantum gravity of ashtekar it is verified that the graph invariants of vassiliev type satisfy constraints of the quantum gravity in the loop space representation of rovelli and smolin

nonquadratic lyapunov functions for robust stability analysis of linear uncertain systems in this note we derive conditions of existence of a homogeneous polynomial lyapunov function of an arbitrary even degree establishing global asymptotic stability of linear system with box bounded uncertainty verification of these conditions is reduced to solving a convex minimization problem we produce numerical examples that demonstrate significant improvement in estimates of admissible uncertainty bounds compared with estimates obtained via the most commonly used quadratic lyapunov functions

further results on the robust stability of linear systems with a single time delay abstract for linear systems with a single time varying delay time we propose a method to characterize how large the deviation of the delay time can be compared to the nominal zero value such that the system still preserves its asymptotic stability both cases of linear time delay systems with and without uncertainties are considered the criteria developed are seen to give less conservative results in simulation examples

strong law of large numbers and shannon mcmillan theorem for markov chain fields on trees we study the strong law of large numbers and the shannon mcmillan theorem for markov chain fields on trees first we prove the strong law of large numbers for the frequencies of occurrence of states and ordered couples of states for markov chain fields on trees then we prove the shannon mcmillan theorem with almost everywhere a e convergence for markov chain fields on trees we prove the results on a bethe tree and then just state the analogous results on a rooted cayley tree in the proof a new technique for establishing the strong limit theorem in probability theory is applied

robust root clustering for linear uncertain systems using generalized lyapunov theory in this paper the problem of matrix root clustering in sub regions of complex plane for linear state space models with real parameter uncertainty is considered the nominal matrix root clustering theory of gutman and jury 1981 ieee trans aut control ac 26 403 using generalized lyapunov equation is extended to the perturbed matrix case and bounds are derived on the perturbation to maintain root clustering inside a given region the theory allows us to get an explicit relationship between the parameters of the root clustering region and the uncertainty range of the parameter space the current literature available on perturbation bounds for robust stability becomes a special case of this unified theory

strong laws of large numbers for asymptotic even odd markov chains indexed by a homogeneous tree abstract in this paper we are going to study the strong laws of large numbers for asymptotic even odd markov chains indexed by a homogeneous tree first the definition of the asymptotic even odd markov chain is introduced then the strong limit theorem for asymptotic even odd markov chains indexed by a homogeneous tree is established next the strong laws of large numbers for the frequencies of occurrence of states and ordered couple of states for asymptotic even odd markov chains indexed by a homogeneous tree are obtained finally we prove the asymptotic equipartition property aep for these markov chains

perturbation bounds for robust stability in linear state space models abstract this paper addresses the aspect of stability robustness of linear systems in the time domain upper bounds on the linear perturbation of an asymptotically stable linear system are obtained to maintain stability both for structured as well as unstructured perturbations using the lyapunov approach for structured perturbation the resulting bound is such that it garners the structural information about the nominal as well as the perturbation matrix into a single unified expression in the case of unstructured perturbations special features of the nominally stable matrix are exploited resulting in simpler expressions for the bound without the need to solve the lyapunov equation improvement of the proposed measures is illustrated with the help of examples

characterization of allowable perturbations for robust stability the problem of determining stability for a closed loop linear multivariable process in the presence of ill defined process perturbations is considered it is shown that m matrices may be used to characterize a class of allowable perturbations

convex analysis of output feedback control problems robust stability and performance this paper addresses the problem of optimal h sub 2 control by output feedback necessary and sufficient conditions on the existence of a linear stabilizing output feedback gain are provided in terms of the intersection of a convex set and a set defined by a nonlinear real valued function the results can be easily extended to deal with linear uncertain systems where uncertainties are supposed to belong to convex bounded domains providing an h sub 2 guaranteed cost output feedback control thanks to the properties of the above mentioned function we show that under certain conditions convex programming tools can be used for numerical purposes examples illustrate the theoretical results

invariants of graphs in three space par association d une collection de n uds et d aretes a un graphe dans un espace tridimensionnel on obtient des invariants calculables du type de plongement du graphe on considere deux types d isotopie

computation of nonconservative stability perturbation bounds for systems with nonlinearly correlated uncertainties consideration is given to the problem of robust stability analysis of linear dynamic systems with uncertain physical parameters entering as polynomials in the state equation matrices a method is proposed giving necessary and sufficient conditions for computing the uncertain system stability margin in parameter space which provides a measure of maximal parameter perturbations preserving stability of the perturbed system around a known stable nominal system a globally convergent optimization algorithm that enables solutions to the problem to be obtained is presented using the polynomial structure of the problem the algorithm generates a convergent sequence of interval estimates of the global extremum these intervals provide a measure of the accuracy of the approximating solution achieved at each step of the iterative procedure some numerical examples are reported showing attractive features of the algorithm from the point of view of computational burden and convergence behavior

characterization of allowable perturbations of a closed loop system with pole placement controller the problem of determining stability of a closed loop transfer function in the presence of coefficient perturbations is considered the results of interval polynomial theory are used to characterize the allowable interval of variation

a first order lyapunov robustness method for linear systems with uncertain parameters a method for stability robustness analysis based on a quadratic lyapunov function that varies linearly with uncertainty parameters is derived linear time invariant systems with structured uncertainties are discussed the lyapunov function is optimized numerically to maximize the robustness region in parameter space numerical results are given for four examples in which the first order method is compared to previous lyapunov methods for robustness analysis while the zero order method is slightly better than the first order method for one example the first order method is clearly superior in the other three more realistic examples the first order method is especially superior for applications to active control of flexible structures where robustness with respect to unmodeled coupling between modeled modes are important issues for such applications the first order method is much better at detecting the increased robustness associated with increased separation between frequencies

isotopy invariants of graphs the development of oriented and semioriented algebraic invariants associated to a class of embeddings of regular four valent graphs is given these generalize the analogous invariants for classical knots and links can be determined from them by means of a weighted averaging process and define them by means of a new state model this development includes the elucidation of the elementary spatial equivalences generalizations of the classical reidemeister moves and the extension of fundamental concepts in classical knot theory such as the linking number to this class spatial graphs

bibo stability robustness in the presence of coprime factor perturbations a necessary and sufficient condition for robustly stabilizing a family of plants described by perturbations of fixed coprime factors of a plant is given the computation of the largest stability margin is discussed via solving a nonsquare l sup 1 optimal control problem an algorithm for obtaining lower approximations of the minimum value of the optimization problem mu sub 0 is proposed this together with the standard algorithm which provides upper approximations allows mu sub 0 to be computed within any degree of accuracy

a lyapunov robustness bound for linear systems with periodic uncertainties abstract for linear systems involving uncertain parameters with known constant nominal values and uncertain perturbations that vary sinusoidally with time lyapunov robustness anslysis is used to determine a stability bound or margin for the amplitudes of the parameter perturbations this bound is the size of a hypercube in parameter space for which asymptotic stability is guaranteed the bound which is based on a quadratic lyapunov function that depends linearly on parameter perturbations varies with the frequency of the uncertain parameter perturbations the bound is asymptotically proportional to the square root of this frequency as it becomes large

invariance of the strict hurwitz property for polynomials with perturbed coefficients given a strictly hurwitz polynomial f lambda lambda n a n 1 lambda n 1 a n 2 lambda n 2 a 1 lambda a 0 it is of interest to know how much the coefficients a i can be perturbed while simultaneously preserving the strict hurwitz property for systems with n leq 4 maximal intervals of the a i are given in a recent paper by guiver and bose 1 in this note a theorem of kharitonov is exploited to obtain a general result for polynomials of any degree

stability bounds of singularity perturbed systems a necessary and sufficient condition is developed for computing the upper bound in sub 0 of the parasitic parameter in in a singularly perturbed system which ensures stability of such a system if 0

technical communique static output feedback stabilization an ilmi approach in this note the static output feedback stabilization problem is addressed using the linear matrix inequality technique a necessary and sufficient condition for static output feedback stabilizability for linear time invariant systems is derived in the form of a matrix inequality the extension of the result to h control is studied an iterative lmi ilmi algorithm is proposed to compute the feedback gain numerical examples are employed to demonstrate the effectiveness and the convergence of the algorithm

entropic aspects of random fields on trees the existence of the entropy rate of shift invariant random fields on binary trees is proven alternative representations of and bounds for the entropy rate and surface entropy rate are obtained in terms of conditional entropy particular emphasis is placed on markov chain fields on trees explicit results are obtained some of which extend to a more complicated class of tree models

2 d physics and 3 d topology invariants of three dimensional manifolds and of framed oriented labeled links in them are rigorously defined using any solution to the moore seiberg axioms for a rational conformal field theory these invariants are generalizations of witten s chern simons path integrals connections are explored with supersymmetry four dimensional manifolds and quantum gravity

some results on robust stability of discrete time systems the theorem of kharitonov on the hurwitz property of interval families of polynomials cannot be extended in genera to obtain sufficient conditions for the stability of families of characteristic polynomials of discrete time systems necessary and sufficient conditions for this stability problem are given such conditions naturally give rise to a computationally efficient stability test which requires the solution of a one parameter optimization problem and which can be considered as a counterpart to the kharitonov test for continuous time systems at the same time the method used to derive the stability conditions provides a procedure for solving another stability robustness problem i e the estimation of the largest domain of stability with a rectangular box shape around given nominal values of the polynomial coefficients

robust stability analysis of linear continuous discrete time systems with output feedback controllers the robust stability of linear systems with output feedback controllers and time varying uncertain parameters is considered the robust stability bounds for time varying uncertain parameters are given using the lyapunov method when there are no uncertain parameters in the input and or output matrices it is shown that the result for continuous time systems is the same as that presented by k m zhou and p p khargonekar 1987 and the result for discrete time systems is better than that of s r kolla 1989 for the same example

robust pole assignment abstract this paper presents new theorems on the theory of interval matrix inequalities and the theory of polynomials with interval roots and applies them to the problem of robust pole placement we formulate optimization problems and derive convergent iterative algorithms which allow the designer to find controllers that place closed loop poles within desired intervals for plants with unknown but bounded parameter uncertainties the algorithms are computationally reasonable and provide a useful addition to currently existing control cad tools

the xy centring algorithm for the dual lmi problem a new approach to fixed order control design many fixed order suboptimal control problems with stability performance and robustness specifications can be reduced to a search for a matrix x 0 satisfying a linear matrix inequality lmi while x 1 satisfies another lmi this paper defines a certain class of these problems we shall call the dual lmi problem and a computational algorithm to solve our dual lmi problem is given properties and limitations of the algorithm are discussed in comparison with the existing algorithm the min max algorithm an extension to optimal control problems is provided numerical examples for the fixed order stabilization problem and the static output feedback linear quadratic optimal control problem demonstrate the applicability of the proposed algorithm

the strong law of large numbers and the entropy ergodic theorem for nonhomogeneous bifurcating markov chains indexed by a binary tree guyon guyon j limit theorems for bifurcating markov chains application to the detection of cellular aging ann appl probab 2007 17 1538 1569 introduced an important model for homogeneous bifurcating markov chains indexed by a binary tree taking values in general state space and studied their limit theorems the results were applied to detect cellular aging in this paper we define a discrete form of nonhomogeneous bifurcating markov chains indexed by a binary tree and discuss the equivalent properties for them the strong law of large numbers and the entropy ergodic theorem are studied for these markov chains with finite state space in contrast to previous work we use a new approach to prove the main results of this paper

robust optimal parametric lq control with a guaranteed cost bound and applications for an optimal parametric linear quadratic lq control problem a design objective is to determine a controller of constrained structure such that the closed loop system is asymptotically stable and an associated performance measure is optimized in the presence of system uncertainty the system via a parametric lq design is further required to be robust in terms of maintaining the closed loop stability with a guaranteed cost bound this problem is referred to as robust optimal parametric lq control with a guaranteed cost bound and is addressed in this work a new design method is proposed to find an optimal controller for simultaneously guaranteeing robust stability and performance over a specified range of parameter variations the results presented generalize some previous work in this area a versatile numerical algorithm is also given for computing the robust optimal gains the usefulness of the design method is demonstrated by numerical examples and a design of the robust control of a vtol helico

robust stability with structured real parameter perturbations this paper considers the problem of robust stabilization of a linear time invariant system subject to variations of a real parameter vector for a given controller the radius of the largest stability hypersphere in this parameter space is calculated this radius is a measure of the stability margin of the closed loop system the results developed are applicable to all systems where the closed loop characteristic polynomial coefficients are linear functions of the parameters of interest in particular this always occurs for single input multioutput or single output multiinput systems where the transfer function coefficients are linear or affine functions of the parameters many problems with transfer function coefficients which are nonlinear functions of physical parameters can be cast into this mathematical framework by suitable weighting and redefinition of functions of physical parameters as new parameters the largest stability hyperellipsoid for the case of weighted perturbations and a stability polytope in parameter space are also determined based on these calculations a design procedure is proposed to robustify a given stabilizing controller this algorithm iteratively enlarges the stability hypersphere or hyperellipsoid in parameter space and can be used to design a controller io stabilize a plant subject to given ranges of parameter excursions these results are illustrated by an example

the asymptotic equipartition property for nonhomogeneous markov chains indexed by a homogeneous tree in this correspondence we first establish a strong limit theorem for countable nonhomogeneous markov chains indexed by a homogeneous tree as corollaries we obtain some strong limit theorems for frequencies of occurrence of states and ordered couple of states for these markov chains finally we prove the strong law of large numbers and the asymptotic equipartition property aep for finite nonhomogeneous markov chains indexed by a homogeneous tree

on the robustness of lq regulators for discrete time systems a sufficient condition for quadratic stabilizability of uncertain linear discrete time systems using lq regulators is presented the true system is represented by a nominal model plus additive terms representing the most common types of uncertainties in the state and input matrices sufficient robustness bounds as well as a robustification procedure for discrete time lq regulators are presented and the conservativeness of the proposed condition is discussed

on the stability properties of polynomials with perturbed coefficients given a polynomial p c s s n t 1 s n 1 t n 0 which is hurwitz or p d z z n t 1 z n 1 t n 0 which has zeros only within or on the unit circle it is of interest to know how much the coefficients t j can be perturbed while preserving the stability properties in this note a method is presented for obtaining the largest hypersphere centered at t t t 1 t n containing only polynomials which are stable

controllability observability and the solution of ax xb c a closed form finite series representation of the unique solution x of the matrix equation ax xb c is developed using this representation the image kernel and rank of x are related to the controllable and unobservable subspaces of the a c and c b pairs respectively bounds on the rank of x are obtainedin terms of the dimensions of these subspaces in the case that c has unitary rank an exact calculation of rank x is made the generic rank of x with a b fixed and c generic is evaluated

pole assignment with robust stability this paper uses convex analysis for the pole assignment design of discrete time siso systems incorporating robust stability against norm bounded parametric perturbations in the plant transfer function the method involves designing an overparameterized pole assignment controller for the nominal plant with the overparameterization chosen to reduce the size of changes in the closed loop characteristic equation which result from plant perturbations sufficiency bounds on the l sub p norm of the perturbation guaranteeing stability are obtained

nonsmooth h_infty synthesis we develop nonsmooth optimization techniques to solve h infty synthesis problems under additional structural constraints on the controller our approach avoids the use of lyapunov variables and therefore leads to moderate size optimization programs even for very large systems the proposed framework is versatile and can accommodate a number of challenging design problems including static fixed order fixed structure decentralized control design of pid controllers and simultaneous design and stabilization problems our algorithmic strategy uses generalized gradients and bundling techniques suited for the h infty norm and other nonsmooth performance criteria we compute descent directions by solving quadratic programs and generate steps via line search convergence to a critical point from an arbitrary starting point is proved and numerical tests are included to validate our methods the proposed approach proves to be efficient even for systems with several hundreds of states

lyapunov stability robust analysis and robustness design for linear continuous time systems the linear continuous time systems to be discussed are described by state space models with structured time varying uncertainties first the explicit maximal perturbation bound for maintaining quadratic lyapunov stability of the closed loop systems is presented then a robust design method is proposed given an open loop nominal system with information of the uncertainty structure the proposed robust design method produces a linear constant output feedback control law which maximizes the perturbation bound for maintaining quadratic lyapunov stability both the optimal results for robust analysis and robustness design can be obtained by using an efficient numerical algorithm e g minimax of matlab with analytic gradients given in this paper examples are investigated in some detail to show the improvements and advantages of the proposed methods over previous ones