an input normal form homotopy for the l sup 2 optimal model order reduction problem in control system analysis and design finding a reduced order model optimal in the l sup 2 sense to a given system model is a fundamental problem the problem is very difficult without the global convergence of homotopy methods and a homotopy based approach has been proposed the issues are the number of degrees of freedom the well posedness of the finite dimensional optimization problem and the numerical robustness of the resulting homotopy algorithm a homotopy algorithm based on the input normal form characterization of the reduced order model is developed here and is compared with the homotopy algorithms based on hyland and bernstein s optimal projection equations the main conclusions are that the input normal form algorithm can be very efficient but can also be very ill conditioned or even fail

optimal h model reduction via linear matrix inequalities continuous and discrete time cases necessary and sufficient conditions are derived for the existence of a solution to the continuous time and discrete time h model reduction problems these conditions are expressed in terms of linear matrix inequalities lmis and a coupling non convex rank constraint set in addition an explicit parametrization of all reduced order models that correspond to a feasible solution is provided in terms of a contractive matrix these results follow from the recent solution of the h control design problem using lmis particularly simple conditions and a simple parametrization of all solutions are obtained for the zeroth order h approximation problem and the convexity of this problem is demonstrated computational issues are discussed and an iterative procedure is proposed to solve the h model reduction problem using alternating projections although global convergence of the algorithm is not guaranteed

numerical method for non linear wave and diffusion equations by the variational iteration method this paper applies he s variational iteration to the wave equations in an infinite one dimensional medium and some non linear diffusion equations a suitable choice of an initial solution can lead to the needed exact solution by a few iterations copyright 2007 john wiley sons ltd

frequency weighted spl lscr sub spl infin norm and optimal hankel norm model reduction a new relative error model reduction method is proposed using frequency weighted balanced realization and explicit spl lscr sub spl infin norm error bounds are also derived for the relative error and multiplicative error the method only needs to solve two lyapunov equations it is further shown that this method is equivalent to the balanced stochastic truncation if the plant is square and minimum phase this paper also gives a complete solution to the frequency weighted hankel norm approximation with antistable weighting these results are then applied to spl lscr sub spl infin norm model reduction and several numerically effective algorithms are proposed it is shown through many numerical examples that these algorithms work very well and in many cases produce almost optimal solutions

variational iteration method for autonomous ordinary differential systems in this paper a new iteration technique is proposed to solve autonomous ordinary differential systems in this method general lagrange multipliers are introduced to construct correction functionals for the systems the multipliers in the functionals can be identified by the variational theory the initial approximations can be freely chosen with possible unknown constants which can be determined by imposing the boundary initial conditions some examples are given the results reveal that the method is very effective and convenient

balanced parametrization of classes of linear systems canonical forms and parametrizations are presented for several sets of minimal systems of given dimension asymptotically stable systems allpass systems bounded real systems positive real systems minimum phase systems and the class of all minimal systems the approach is based on balancing techniques for these classes of systems applications of these results to hankel operators and model reduction are discussed

lqg control with an h sup infinity performance bound a riccati equation approach an lqg linear quadratic gaussian control design problem involving a constraint on h sup infinity disturbance attenuation is considered the h sup infinity performance constraint is embedded within the optimization process by replacing the covariance lyapunov equation by a riccati equation whose solution leads to an upper bound on l sub 2 performance in contrast to the pair of separated riccati equations of standard lqg theory the h sup infinity constrained gains are given by a coupled system of three modified riccati equations the coupling illustrates the breakdown of the separation principle for the h sup infinity constrained problem both full and reduced order design problems are considered with an h sup infinity attenuation constraint involving both state and control variables an algorithm is developed for the full order design problem and illustrative numerical results are given

numerical simulation of equal width wave equation by means of the variational iteration method the solution of the equal width wave equation is numerically obtained and compared with that obtained by the adomian decomposition method numerical results show that the variational iteration method can readily be implemented with excellent accuracy to nonlinear equations

on direct versus indirect methods for reduced order controller design comparisons are obtained between direct parameter optimization and indirect controller reduction methods in reduced order controller design in particular the authors compare the controller canonical correlation coefficients method of c de villemagne and r e skelton 1988 to the optimal projection theory of d c hyland and d s bernstein 1984 it is noted that suboptimal methods may in certain cases be simpler to implement than optimal methods thus the choice of direct versus indirect methods can be viewed as a tradeoff between design simplicity and actual control system performance

a review of the decomposition method in applied mathematics abstract the decomposition method can be an effective procedure for analytical solution of a wide class of dynamical systems without linearization or weak nonlinearity assumptions closure approximations perturbation theory or restrictive assumptions on stochasticitiy

balanced forms canonicity and parameterization we show that balanced forms are canonical and continuously parameterize the set of systems of a given order which are asymptotically stable minimal and have distinct singular values the advantage of the balanced canonical form over other classical canonical forms lies precisely in the guaranteed stability and minimality of the parameterization we also derive formulas allowing the synthesis of systems using gradient methods in balanced form

generalized riccati equations for the full and reduced order mixed norm h 2 h abstract this paper considers the mixed norm h 2 h standard problem specifically an lqg control design problem involving a constraint on h disturbance attenuation is addressed it is shown that the h 2 h dynamic compensator gains are completely characterized via coupled riccati lyapunov equations the principal result involves sufficient conditions for characterizing full and reduced order controlleers that satisfy bounds on both h 2 and h performance costs as a special case of this unified result we obtain the full order h solution to the standard control problem the pure reduced order h solution with no h 2 contribution further extensions include nonstrictly proper dynamics a direct transmission term from disturbances to h performance variables cross weighting and sensor noise plant disturbance correlation and a treatment of the pure reduced order h control problem

a homotopy for solving general polynomial systems that respects m homogeneous structures methods for finding all solutions to a polynomial system f z 0 of n equations in n unknowns using a generic homotopy have been presented by several researchers over the last ten years in this paper we describe a new method for defining a homotophy to find all solutions to a polynomial system a major feature of this new approach is that the homotopy need not to be generic or random the practical consequences are that the homotopy may be chosen to more closely resemble the system being solved with a consequent potential for reducing the arc length of paths and exploiting special structures in the original system in particular if the system is naturally m homogeneous then the number of homotopy paths that need to be tracked is reduced since many systems that arise in applications have such a structure this represents a significant advance over previous homotopies we also provide a proof that the homotopy parameter is strictly increasing on homotopy paths as a function of arc length a new result which has implications for the choice of numerical path tracking algorithms

a comparison of homotopies for alternative formulations of the l 2 optimal model order reduction problem abstract a fundamental problem in control systems theory is finding a reduced order model that is optimal in the l 2 sense to a given full order system model the numerical solution of this problem is challenging and the global convergence properties of homotopy methods are advantageous a number of homotopy based approaches have been developed the primary numerical issues are the number of degrees of freedom in the homotopy parameter vector the well posedness of the finite dimensional optimization problem and the numerical robustness of the resulting homotopy algorithm this paper develops two new homotopy algorithms for optimal model reduction and uses several examples to compare their performance with the performance of two previous algorithms the results show that the numerical well conditioning is inversely related to the algorithmic efficiency and that the relative performance of a given algorithm is problem dependent

piecewise homotopy perturbation method for solving linear and nonlinear weakly singular vie of second kind the purpose of this paper is to obtain the approximation solution of linear and strong nonlinear weakly singular volterra integral equation of the second kind especially for such a situation that the equation is of nonsmooth solution and the situation that the problem is a strong nonlinear problem for this purpose we firstly make a transform to the equation such that the solution of the new equation is as smooth as we like through modifying homotopy perturbation method an algorithm is successfully established to solve the linear and nonlinear weakly singular volterra integral equation of the second kind and the convergence of the algorithm is proved strictly comparisons are made between our method and other methods and the results reveal that the modified homotopy perturbation is effective

all optimal hankel norm approximations of linear multivariable systems and their l error bounds the problem of approximating a multivariable transfer function g s of mcmillan degree n by s of mcmillan degree k is considered a complete characterization of all approximations that minimize the hankel norm is derived the solution involves a characterization of all rational functions s f s that minimize where s has mcmillan degree k and f s is anticavisal the solution to the latter problem is via results on balanced realizations all pass functions and the inertia of matrices all in terms of the solutions to lyapunov equations it is then shown that where k 1 g s is the k l st hankel singular value of g s and for one class of optimal hankel norm approximations the method is not computationally demanding and is applied to a 12 state model

numerical linear algebra aspects of globally convergent homotopy methods probability one homotopy algorithms are a class of methods for solving nonlinear systems of equations that are globally convergent with probability one these methods are theoretically powerful and if constructed and implemented properly are robust numerically stable accurate and practical the concomitant numerical linear algebra problems deal with rectangular matrices and good algorithms require a delicate balance not always achieved of accuracy robustness and efficiency in both space and time the author s experience with globally convergent homotopy algorithms is surveyed here and some of the linear algebra difficulties for dense and sparse problems are discussed

application of he s homotopy perturbation method for nth order integro differential equations abstract in this paper he s homotopy perturbation method is proposed to solve n th order integro differential equations the results reveal that the method is very effective and simple

a numerical method for solving systems of linear and nonlinear integral equations of the second kind by hat basis functions abstract in this paper we use hat basis functions to solve the system of fredholm integral equations sfies and the system of volterra integral equations svies of the second kind this method converts the system of integral equations into a linear or nonlinear system of algebraic equations also we consider the order of convergence of the method and show that it is o h 2 application of the method on some examples show its accuracy and efficiency

steady state kalman filtering with an h error bound abstract an estimator design problem is considered which involves both l 2 least squares and h worst case frequency domain aspects specifically the goal of the problem is to minimize an l 2 state estimation error criterion subject to a prespecified h constraint on the state estimation error the h estimation error constraint is embedded within the optimization process by replacing the covariance lyapunov equation by a riccati equation whose solution leads to an upper bound on the l 2 state estimation error the principal result is a sufficient condition for characterizing fixed order i e full and reduced order estimators with bounded l 2 and h estimation error the sufficient condition involves a system of modified riccati equations coupled by an oblique projection i e idempotent matrix when the h constraint is absent the sufficient condition specializes to the l 2 state estimation result given in 2

probability one homotopy algorithms for solving the coupled lyapunov equations arising in reduced order h2 h modeling estimation and control optimal reduced order modeling estimation and control with respect to combined h 2 h criteria give rise to coupled lyapunov and riccati equations to develop reliable numerical algorithms for these problems this paper focuses on the coupled lyapunov equations which appear as a subset of the synthesis equations in particular this paper systematically examines the requirements of probability one homotopy algorithms to guarantee global convergence homotopy algorithms for nonlinear systems of equations construct a continuous family of systems and solve the given system by tracking the continuous curve of solutions to the family the main emphasis is on guaranteeing transversality for several homotopy maps based upon the pseudogramian formulation of the coupled lyapunov equations and variations based upon canonical forms these results are essential to the probability one homotopy approach by guaranteeing good numerical properties in the computational implementation of the homotopy algorithms

principal component analysis in linear systems controllability observability and model reduction kalman s minimal realization theory involves geometric objects controllable unobservable subspaces which are subject to structural instability specifically arbitrarily small perturbations in a model may cause a change in the dimensions of the associated subspaces this situation is manifested in computational difficulties which arise in attempts to apply textbook algorithms for computing a minimal realization structural instability associated with geometric theories is not unique to control it arises in the theory of linear equations as well in this setting the computational problems have been studied for decades and excellent tools have been developed for coping with the situation one of the main goals of this paper is to call attention to principal component analysis hotelling 1933 and an algorithm golub and reinsch 1970 for computing the singular value decompositon of a matrix together they form a powerful tool for coping with structural instability in dynamic systems as developed in this paper principal component analysis is a technique for analyzing signals singular value decomposition provides the computational machinery for this reason kalman s minimal realization theory is recast in terms of responses to injected signals application of the signal analysis to controllability and observability leads to a coordinate system in which the internally balanced model has special properties for asymptotically stable systems this yields working approximations of x c x bar o the controllable and unobservable subspaces it is proposed that a natural first step in model reduction is to apply the mechanics of minimal realization using these working subspaces

a computational method for solving nonlinear stochastic volterra integral equations in this paper an efficient method for solving nonlinear stochastic ito volterra integral equations nsivies is proposed by using new adjustment of hat basis functions and their stochastic operational matrix of integration the nsivie is reduced to a nonlinear system of algebraic equations also the error analysis of the proposed method is investigated and proved that the order of convergence is o h 4 finally numerical examples affirm the efficiency of the proposed method

numerical solution of second kind fredholm integral equations system by using a taylor series expansion method in this paper we present a taylor series expansion method for a second kind fredholm integral equations system with smooth or weakly singular kernels this method reduce the system of integral equations to a linear system of ordinary differential equation after constructing boundary conditions this system reduce to a system of equations that can be solved easily with any of the usual methods finally we show the efficiency of the method using some numerical examples

a numerical solution for solving system of fredholm integral equations by using homotopy perturbation method abstract homotopy perturbation method is applied to the numerical solution for solving system of fredholm integral equations comparison of the result obtained by the present method with that obtained by taylor series expansion method k maleknejad n aghazade m rabbani numerical solution of second kind fredholm integral equations system by using a taylor series expansion method appl math comput 175 2006 1229 1234 reveals that the present method is very effective and convenient

the optimal projection equations for model reduction and the relationships among the methods of wilson skelton and moore first order necessary conditions for quadratically optimal reduced order modeling of linear time invariant systems are derived in the form of a pair of modified lyapunov equations coupled by an oblique projection which determines the optimal reduced order model this form of the necessary conditions considerably simplifies previous results of wilson 1 and clearly demonstrates the quadratic extremality and nonoptimality of the balancing method of moore 2 the possible existence of multiple solutions of the optimal projection equations is demonstrated and a relaxation type algorithm is proposed for computing these local extrema a component cost analysis of the model error criterion similar to the approach of skelton 3 is utilized at each iteration to direct the algorithm to the global minimum

numerical solution of a biological population model using he s variational iteration method this paper presents numerical solution of a degenerate parabolic equation arising in the spatial diffusion of biological populations the variational iteration method and adomian decomposition method are used for solving this equation and then numerical results are compared with each other showing that the variational iteration method leads to more accurate results furthermore variational iteration method overcomes the difficulty arising in calculating the adomian s polynomials which is an important advantage over the adomian decomposition method

contragredient transformations applied to the optimal projection equations the optimal projection approach to solving the h2 reduced order model problem produces two coupled highly nonlinear matrix equations with rank conditions as constraints it is not obvious from their original form how they can be differentiated and how some algorithm for solving nonlinear equations can be applied to them a contragredient transformation a transformation which simultaneously diagonalizes two symmetric positive semi definite matrices is used to transform the equations into forms suitable for algorithms for solving nonlinear problems three different forms of the equations obtained using contragredient transformations are given an svd based algorithm for the contragredient transformation and a homotopy algorithm for the transformed equations are given together with a numerical example

new iterative methods for nonlinear equations by modified hpm a numerical method based on modified homotopy perturbation method hpm is proposed for solving nonlinear algebric equations it is shown that the proposed method has third order convergence to assess its validity and accuracy the method is applied to solve several test problems

a third order newton type method for nonlinear equations based on modified homotopy perturbation method in this paper we present a new iterative method for solving nonlinear algebric equations by using modified homotopy perturbation method we also discuss the convergence criteria of the present method to assess its validity and accuracy the method is applied to solve several test problems

a functional approach to lqg balancing the paper has as its theme a circle of problems related to lqg balancing with a special emphasis on the related problems of model reduction and robust stabilization the aim is to present a unified point of view to both previously described problems the unification is achieved by focusing on the study of three functions and the relationships between them and the various operators that are associated with these functions with an arbitrary transfer function g one can associate canonically two objects the conjugate inner function which is based on the nrcf and the r which is associated with the lqg controller of the function g the approach that is taken is functional emphasizing operators a balanced realization of a stable g arises as a matrix representation of the shift realization with respect to a basis made out of suitably normalized hankel singular vectors a similar result holds for lqg balanced realizations here the underlying hankel operator we study is h r where r u m v n and u v s

variational iteration method for solving nonlinear boundary value problems abstract in this paper he s variational iteration method is implemented to give approximate and analytical solutions for a class of boundary value problems the variational iteration method which produces the solutions in terms of convergent series requiring no linearization or small perturbation numerical examples are given and comparisons are made with the adomian decomposition method the fact that this method solves nonlinear equations without using adomian polynomials can be considered as an advantage of this method over adomian decomposition method

iterated he s homotopy perturbation method for quadratic riccati differential equation abstract in this paper iterated he s homotopy perturbation method is proposed to solving quadratic riccati differential equation comparisons are made between adomian s decomposition method adm and the exact solution and the proposed method the results reveal that the method is very effective and simple

the homotopy perturbation method for nonlinear oscillators with discontinuities in the paper the homotopy perturbation method is applied to the nonlinear oscillators with discontinuities only one iteration leads to high accuracy of the solutions

comment subjective probability and the theory of games comments on kadane and larkey s paper the normative solution concepts of game theory try to provide a clear mathematical characterization of what it means to act rationally in a game where all players expect each other to act rationally kadane and larkey reject the use of these normative solution concepts yet this amounts to throwing away an important piece of information to the effect that the players are rational and expect each other to be rational even in situations where the players do not expect each other to act with complete rationality normative game theory can help them heuristically to formulate reasonable expectations about the other players behavior

newton like iterative methods for solving system of non linear equations abstract homotopy perturbation method hpm is applied to construct a new family of newton like iterative methods for solving system of non linear equations comparison of the result obtained by the present method with newton raphson method reveals that the accuracy and fast convergence of the new method

asymptotology by homotopy perturbation method in the paper an heuristical example is given to illustrate the basic idea of the homotopy perturbation method so that homotopy perturbation method has made all that is necessary simple and all that is complex unnecessary

mixed spl hscr sub 2 and spl hscr sub spl infin performance objectives ii optimal control this paper considers the analysis and synthesis of control systems subject to two types of disturbance signals white signals and signals with bounded power the resulting control problem involves minimizing a mixed spl hscr sub 2 and spl hscr sub spl infin norm of the system it is shown that the controller shares a separation property similar to those of pure spl hscr sub 2 or spl hscr sub spl infin controllers necessary conditions and sufficient conditions are obtained for the existence of a solution to the mixed problem explicit state space formulas are also given for the optimal controller

coefficient parameter polynomial continuation we establish the theory for a very general approach to constructing homotopies for computing all geometrically isolated solutions to polynomial systems we assume that the coefficients for a polynomial system are given by parameters in engineering applications these would be derived from physical parameters since the coefficients themselves can be the parameters this is completely general we show that by considering the space of parameters instead of the space of coefficients we can exploit any special structure the coefficient parameter formulas impose on the solution set in particular we prove that we can ignore generic positive dimensional solution sets we also show that any side conditions additional equations can be used to reduce the number of paths these results unify and generalize all current approaches to polynimial continuation the effect of these new results is to greatly increase the efficiency of polynomial continuation in several engineering areas for example we establish a 36 path homotopy for a electrocardiographic application that needed 193 paths under the previous most appropriate theory also we show that a 16 path homotopy suffices for the inverse kinematics problem for 6 revolute joint manipulators a problem previously requiring 32 paths

a new family of iterative methods for solving system of nonlinear algebric equations abstract homotopy perturbation method hpm is applied to construct a new iterative method for solving system of nonlinear algebric equations comparison of the result obtained by the present method with that obtained by revised adomian decomposition method hossein jafari varsha daftardar gejji appl math comput 175 2006 1 7 reveals that the accuracy and fast convergence of the new method

a numerical simulation and explicit solutions of the generalized burgers fisher equation we consider solitary wave solutions of the generalized burgers fisher bf equation u t pu ru x u x x qu 1 u r 0 in this paper by considering the decomposition scheme we first obtain the exact solutions of the generalized bf equation for the initial condition without using any classical transformations and then its numerical solutions are constructed without using any discretization technique the numerical solutions are compared with the known analytical solutions its remarkable accuracy is finally demonstrated in the study of some values r 2 of the generalized bf equation

a fractional variational iteration method for solving fractional nonlinear differential equations recently fractional differential equations have been investigated by employing the famous variational iteration method however all the previous works avoid the fractional order term and only handle it as a restricted variation a fractional variational iteration method was first proposed in g c wu e w m lee fractional variational iteration method and its application phys lett a 374 2010 2506 2509 and gave a generalized lagrange multiplier in this paper two fractional differential equations are approximately solved with the fractional variational iteration method

games with incomplete information played by bayesian players i iii this article originally appeared in management science november 1967 volume 14 number 3 pp 159 182 published by the institute of management sciences the paper develops a new theory for the analysis of games with incomplete information where the players are uncertain about some important parameters of the game situation such as the payoff functions the strategies available to various players the information other players have about the game etc however each player has a subjective probability distribution over the alternative possibilities in most of the paper it is assumed that these probability distributions entertained by the different players are mutually consistent in the sense that they can be regarded as conditional probability distributions derived from a certain basic probability distribution over the parameters unknown to the various players but later the theory is extended also to cases where the different players subjective probability distributions fail to satisfy this consistency assumption in cases where the consistency assumption holds the original game can be replaced by a game where nature first conducts a lottery in accordance with the basic probability distribution and the outcome of this lottery will decide which particular subgame will be played i e what the actual values of the relevant parameters will be in the game yet each player will receive only partial information about the outcome of the lottery and about the values of these parameters however every player will know the basic probability distribution governing the lottery thus technically the resulting game will be a game with complete information it is called the bayes equivalent of the original game part i of the paper describes the basic model and discusses various intuitive interpretations for the latter part ii shows that the nash equilibrium points of the bayes equivalent game yield bayesian equilibrium points for the original game finally part iii considers the main properties of the basic probability distribution

mixed h sub 2 h sub infinity control a convex optimization approach the problem of finding an internally stabilizing controller that minimizes a mixed h sub 2 h sub infinity performance measure subject to an inequality constraint on the h sub infinity norm of another closed loop transfer function is considered this problem can be interpreted and motivated as a problem of optimal nominal performance subject to a robust stability constraint both the state feedback and output feedback problems are considered it is shown that in the state feedback case one can come arbitrarily close to the optimal even over full information controllers mixed h sub 2 h sub infinity performance measure using constant gain state feedback moreover the state feedback problem can be converted into a convex optimization problem over a bounded subset of n n and n q where n and q are respectively the state and input dimensions real matrices using the central h sub infinity estimator it is shown that the output feedback problem can be reduced to a state feedback problem in this case the dimension of the resulting controller does not exceed the dimension of the generalized plant

combined l2 h model reduction abstract a model reduction problem is considered which involves both l 2 quadratic and h worst case frequency domain aspects specifically the goal of the problem is to minimize an l 2 model reduction criterion subject to a prespecified h constraint on the model reduction error the principal result is a sufficient condition for characterizing reduced order models with bounded l 2 and h approximation error the sufficient condition involves a system of modified riccati equations coupled by an oblique projection i e idempotent matrix when the h constraint is absent the sufficient condition specializes to the l 2 model reduction result given by hyland and bernstein 1985

meshfree approach for solving multi dimensional systems of fredholm integral equations via barycentric lagrange interpolation abstract a highly accurate meshfree approach based on the barycentric lagrange basis functions is reported for solving the linear and nonlinear multi dimensional systems of fredholm integral equations fies of the second kind the method is an improved lagrange interpolation technique which possesses high precision and inexpensive procedure the systems of fies are handled with the barycentric formula and transformed into the corresponding linear and nonlinear systems of algebraic equations the stability of the solved integral equation systems is guaranteed by the numerical stability of the barycentric lagrange formula which is dominated by barycentric weights the convergence analysis and error estimation are included some computational results indicate that the method is simple and efficient

numerical complexiton solutions for the complex kdv equation by the homotopy perturbation method in this paper the homotopy perturbation method is extended to investigate the numerical complexiton solutions of the complex kdv equation by constructing special forms of initial conditions three new types of realistic numerical solutions are obtained numerical positon solution expressed by the trigonometric functions numerical negaton solution expressed by the hyperbolic functions and particularly the numerical analytical complexiton solutions expressed by combinations of the two kinds of functions all these numerical solutions obtained can rapidly converge to the exact solutions obtained by lou et al illustrative numerical figures are exhibited the efficiency of the proposed method

subjective probability and the theory of games this paper explores some of the consequences of adopting a modern subjective view of probability for game theory the consequences are substantial the subjective view of probability clarifies the important distinction between normative and positive theorizing about behavior in games a distinction that is often lost in the search for solution concepts which largely characterizes game theory since the work of von neumann and morgenstern many of the distinctions that appear important in conventional game theory two person versus n person zero sum versus variable sum appear unimportant in the subjective formulation other distinctions such as single play versus repetitive play games appear to be more important in the subjective formulation than in the conventional formulation

the optimal projection equations for fixed order dynamic compensation first order necessary conditions for quadratically optimal steady state fixed order dynamic compensation of a linear time invariant plant in the presence of disturbance and observation noise are derived in a new and highly simplified form in contrast to the pair of matrix riccati equations for the full order lqg case the optimal steady state fixed order dynamic compensator is characterized by four matrix equations two modified riccati equations and two modified lyapunov equations coupled by a projection whose rank is precisely equal to the order of the compensator and which determines the optimal compensator gains the coupling represents a graphic portrayal of the demise of the classical separation principle for the reduced order controller case

balanced gains and their significance for l 2 model reduction the singular values do not give enough information for model reduction in the l2sense other invariants are introduced which help describe the contribution of each state to the l2magnitude of the impulse response this leads to another suboptimal balanced truncation criterion

numerical solutions of highly oscillatory integrals abstract in this paper an application of he s homotopy perturbation method hpm is proposed to numerical solution of highly oscillatory integrals to apply the hpm to the oscillatory integrals we assume that the oscillatory function has not critical point at the endpoints of integration region the results reveal that the method is very effective and simple

approximate analytical solution for seepage flow with fractional derivatives in porous media abstract in this paper a new and more exact model for seepage flow in porous media with fractional derivatives has been proposed which has modified the well known darcy law and overcome the continuity assumption of seepage flow a new kind of analytical method of nonlinear problem called the variational iteration method is described and used to give approximate solutions of the problem the results show that the proposed iteration method requiring no linearization or small perturbation is very effective and convenient

identification and rational l 2 approximation a gradient algorithm this paper deals with the identification of linear constant dynamical systems when formalized as a rational approximation problem the criterion is the l2 norm of the transfer function which is of interest in a stochastic context the problem can be expressed as nonlinear optimization in a hilbert space but standard algorithms are usually not well adapted we present a generic recursive procedure to find a local optimum of the criterion in the case of scalar systems our methods are borrowed from differential theory mixed with a bit of classical complex analysis to our knowledge the algorithm described in this paper is the first that ensures convergence to a local minimum

application of he s methods to nonlinear chemistry problems in this paper he s variational iteration method vim and homotopy perturbation method hpm are implemented for solving analytically two systems of ordinary differential equations that often appear in chemical applications in both methods the initial approximations can be freely chosen with possible unknown constants which can be determined by imposing the boundary and initial conditions the solution procedure is of utter simplicity and the obtained solution is of high accuracy

convergence and error estimation of homotopy perturbation method for fredholm and volterra integral equations abstract in this paper an application of the homotopy perturbation method for solving fredholm and volterra integral equations of the second kind is presented discussed method consists in constructing the functional series sum of which determines the function giving the solution of considered problem conditions under which the constructed series is convergent are formulated and proved in the paper in general the composed series is rather fast convergent thanks to which calculation of few first terms ensures a very good approximation of the sought solution estimation of errors of approximate solution obtained by taking the partial sum of the series is also elaborated in the paper

iterative methods to nonlinear equations homotopy perturbation method is applied to construct a new iterative method for solving nonlinear algebraic equations the convergence of the new scheme is proposed an at least the cubic order of convergence is established several examples are presented and compared to other methods showing the accuracy and fast convergence of the new method

errata computing all solutions to polynomial systems using homotopy continuation in a previous paper we described a new method for defining homotopies for finding all solutions to polynomial systems a major feature of this new approach is that the start system for the homotopy need not be a random or generic system also homotopy paths are strictly increasing in the homotopy parameter in this paper we establish some principles of implementation and report on the performance of programs that use the new homotopies a feature of our implementation is that we eliminate divergent paths entirely we include performance statistics for homotopies derived from more traditional approaches for comparison generally the new approach is faster and more reliable

approximate solution of nonlinear differential equations with convolution product nonlinearities in this paper a new iteration method is proposed to solve nonlinear problems special attention is paid to nonlinear differential equations with convolution product nonlinearities the results reveal the approximations obtained by the proposed method are uniformly valid for both small and large parameters in nonlinear problems furthermore the first order of approximations are more accurate than perturbation solutions at high order of approximation

mixed h2 h control in a stochastic framework abstract this paper deals with a mixture of h 2 and h in the following way we have two inputs and one output one input signal is a white noise stochastic process and represents errors e g resulting from measurement noise the other input has a more deterministic character if one has a reference signal e g a step as input one cannot model this as white noise but it fits nicely into this new class of inputs the objective is to minimize the effect of these signals on the output of the system we define a cost function which enables us to combine these two exogenous inputs even though they are structurally different the analysis of this function leads to a standard h riccati equation we motivate this cost function by looking at two theoretical applications the derivation of robust performance bounds and a tracking problem