transfer matrix approach to the structural block diagonal decoupling problem the block diagonal decoupling problem for structured systems is treated using the transfer matrix approach an associated graph which displays the internal structure of the system invariant and interconnection is used to establish necessary and sufficient conditions for the solution of the problem these conditions are expressed in the graph in terms of shortest input output paths which are related to the infinite structure of the system

the solution of diagonal decoupling problem by dynamic output feedback and constant precompensator the general case in this paper the solution to diagonal decoupling problem via dynamic output feedback and constant precompensator is presented in the general case the literature contains a number of particular solutions for this problem these solutions are valid for instance for square transfer matrices or they employ restricted dynamic output feedback schemes here we start with a nonsquare transfer matrix and consider a dynamic output feedback scheme where there is a proper compensator in the feedback loop accompanied by a constant gain precompensator in the feedforward path in this respect the problem being considered is the dynamic output feedback version of the well known morgan s problem the approach taken involves the characterization of open loop diagonalizers which admit a dynamic output feedback realization the key concepts involved in the solution are the generalized version of diagonal causality degree dominance and the set of attainable infinite zero orders an example is also included to illustrate the main result

minimal structure in the block decoupling problem with stability abstract in this paper we consider the block decoupling problem with stability we characterize the minimal mcmillan degree achievable for the block decoupled system for this we introduce a new list of integers which are feedback invariant these integers are related in a simple way to the minimal number of unstable and infinite zeros of the block decoupled system

block decoupling for linear systems over rings abstract this paper studies the block decoupling problem for injective linear systems defined over unique factorization domains and necessary and sufficient conditions for its solvability are obtained using the algebraic properties of transfer matrices further it is shown that if the problem is solvable the decoupling can be achieved by a biproper compensator and such a compensator can be realized by a regular static state feedback finally an example is given to illustrate the results

complete curves in moduli spaces of stable bundles on surfaces in this paper we give a partial answer to a question posed by hirschowitz and hulek in hi hu for moduli spaces of stable rank two vector bundles on the projective plane they asked for the maximal dimension of a complete subvariety of such a moduli space in fact they showed that for the projective plane these moduli spaces contain complete curves in almost every case for many cases they even contain complete surfaces cp hu lp and hu s the main theorem we will prove here is the following cp 3 8

periodic solutions of the intermediate long wave equation a nonlinear superposition principle periodic stationary wave solutions of the intermediate long wave ilw equation are derived using the bilinear transformation method and a new expression for the dispersion relation is obtained the class of physically important real valued solutions is identified these solutions may be represented as an infinite superposition of solitary wave profiles a property shared by the related korteweg de vries kdv and benjamin ono bo equation this nonlinear superposition principle which has been the subject of various interpretations in the literature is discussed the ilw periodic solution approximates to a sinusoidal wave and a solitary wave in the limits of small and large amplitudes respectively for intermediate amplitudes the solution can be well approximated by either a sine wave or solitary wave in the shallow water kdv limit the ilw periodic solution leads to the familiar cnoidal wave whereas the deep water bo limit yields benjamin s periodic wave a previously unknown expression for the cnoidal wave dispersion relation in terms of theta functions is obtained

algebro geometric analogues of donaldson s polynomials donaldson 5 has introduced polynomials on the second homology of any simply connected closed four manifold m with b odd at least 3 which up to are invariant under orientation preserving diffeomorphisms for every c ibm there is a polynomial 7c m syma h2 s where d c 4c z2 b 1 to define it one considers j c the set which is in a natural way a manifold of gauge equivalence classes of connections on the su 2 bundle on m with c2 c antiself dual with respect to a generic metric on m there is a natural map h2 m h2 j c namely slant product with 88 where p is the universal so 3 principal bundle on m x j c one way of defining 7c m 10 is to construct a compactification of r xc carrying a fundamental class xc one proves that extends to a map h2 m h2 xc then one defines

linear decoupling controllers for constrained dynamic systems linear decoupling controllers of proportional derivative pd type are considered for natural and robotic systems operating under holonomic constraints the dynamic system must be decomposed into orthogonal subspaces of motion and constraint the incremental motion of the system along the two hyper planes must be stable decoupling controllers including both non interacting and disturbance decoupling types are parameterized using a single matrix parameter the proof of stability is based on lyapunov function approach

a transfer matrix approach to feedback invariants of linear systems application to decoupling abstract the transfer matrix approach allows one to exhibit fairly easily fundamental invariants of multivariable systems under the action of transformation groups these transformation groups include the classical state feedback state and input changes of coordinates then the solvability of numerous classical control problems e g decouplability disturbance rejection model following can be expressed directly in terms of these invariants in this expository paper we present in a unified way three lists of feedback invariants related to the infinite behavior of linear systems these invariants turn out to be crucial in studying feedback decoupling problems the main tools for this study are factorization of transfer matrices over the p i d of proper rational functions when dealing with stability we factor transfer matrices over the p i d of proper rational stable functions we illustrate the use of these invariants by studying some decoupling problems

structured systems within the transfer matrix approach application to decoupling abstract we consider a new approach to structured systems within the transfer matrix framework we associate with such a structured system a graph which concentrates both knowledge on structural properties of independent subsystems and on their interconnections the analysis of this associated graph allows to conclude on the decouplability of structured systems

failure detection and identification using the geometric concept of an unobservability subspace a solution is given to the problem of detecting and identifying control system component failures in linear time invariant systems conditions are developed for the existence of a causal linear time invariant processor that can detect and uniquely identify a component failure first for the case where components can fail simultaneously and then for the case where they fail only one at a time explicit design algorithms are provided when these conditions are satisfied in addition to time domain solvability conditions frequency domain interpretations of the results are given and connections are drawn with results already available in the literature

feedback representation of precompensators application to decoupling linear multi input multi output plants in this paper we examine the problem of internally stabilizing and simultaneously diagonally decoupling a linear multivariable system by two parameter compensation based on our earlier results boussaid and guerin 1996 we prove that any given plant p of full row rank can be decoupled with stability by the considered configuration computation of the decoupling compensators follows by easy constructions the decoupling problem with a minimum number of unstable zeros in the decoupled system is also formulated and solved in the generic case

decoupling with internal stability for unity output feedback systems abstract this paper presents for the first time a necessary and sufficient condition for solvability of the decoupling problem with internal stability for square plants by unit output feedback compensation it consists of some polynomial matrix coprimeness conditions and is concerned only with the greatest c divisor of the plant parameterizations of all decoupling compensators and achievable diagonal loop maps with internal stabilizability are given and the results are also generalized in a very natural way to the decoupling problem with arbitrary pole assignment examples are included for illustration

block decoupling with stability by unity output feedback solution and performance limitations abstract the block decoupling problem with stability using unity output feedback for possibly non square plants of full row rank is considered a necessary and sufficient condition for solvability of this problem is given and the set of all compensators solving the problem is characterized previous results on the diagonal decoupling problem for square plants are thus generalized moreover a class of plants is specified where decoupling does not impose additional performance limitations as compared with a synthesis without decoupling quite general performance measures involving the sensitivity function and the complementary sensitivity function are considered including weighted h and h 2 norms

complete decoupling of linear multivariable systems by means of linear static and differential state feedback the concept of complete decoupling is introduced in this paper we show in particular that for any linear controllable system with non singular transfer matrix there always exists a static and differential state feedback such that the closed loop system is completely decoupled

weakly non local solutions for capillary gravity waves fifth degree korteweg de vries equation hunter and scheurle have shown that capillary gravity water waves in the vicinity of bond number bo l 3 are consistently modelled by the korteweg de vries equation with the addition of a fifth derivative term this wave equation does not have strict soliton solutions for bo l 3 because the near solitons have oscillatory wings that extend indefinitely from the core of the wave however these solutions are arbitrarily small perturbations of solitary waves because the amplitude of the wings is exponentially small in the amplitude 6 of the core pomeau ramani and grammaticos have calculated the amplitude of the wings by applying matched asymptotics in the complex plane in the limit e 0 in this article we describe a mixed chebyshev radiation function pseudospectral method which is able to calculate the weakly non local solitons for all e we show that for fixed phase speed the solitons form a three parameter family because the linearized wave equation has three eigensolutions we also show that one may repeat the soliton with even spacing to create a three parameter of periodic solutions which we also compute because the amplitude of the wings is exponentially small these non local capillary gravity solitons are as interesting as the classical localized solitons that solve the korteweg de vries equation

two types of bounded traveling wave solutions of a two component camassa holm equation the camassa holm equation ch is a well known integrable equation with is used to describe the velocity dynamics of shallow water waves this equation exhibits spontaneous emergence of peakons from smooth initial conditions the ch equation has been recently extended to a two component integrable system ch2 which includes both velocity and density variables in dynamics in this paper the bifurcation method of dynamical systems and numerical approach of differential equations are employed to study ch2 equation two types of bounded traveling waves are found one of them is called compacton another is called generalized kink wave the planar graphs of compactons and generalized kink awaves are simulated by using software maple exact explicit parameter expressions of the compactons and implicit expressions of the generalized kink waves solutions are given

a unifying approach to the decoupling of linear multivariate systems a unifying approach to the decoupling problem diagonal triangular block decoupling is introduced the key element is the interactor a polynomial matrix of unique structure associated with the transfer matrix of the system necessary and sufficient conditions for block decoupling are derived and the stability questions are resolved using the block coupling zeros the results obtained for decoupling via state feedback state feedback plus input dynamics and constant output feedback are conceptually simple unify and expand those existing and easily lead to constructive decoupling algorithms

solution of the static state feedback decoupling problem for linear systems with two outputs we give for linear systems with two outputs a necessary and sufficient condition for the solvability of the restricted decoupling problem rdp in terms of some structural invariants a geometric characterization is also given which provides the least dynamical extension needed when rdp is not solvable

coupled camassa holm equations n peakons and infinitely many conservation laws abstract three coupled camassa holm equations are proposed which are associated with a 4 4 matrix spectral problem with two potentials with the aid of the zero curvature equation we derive a hierarchy of new nonlinear evolution equations and establish their bi hamiltonian structures the three coupled camassa holm equations are exactly negative flows in the hierarchy and exact solutions with n peakons and infinite sequences of conserved quantities are discussed

paper transfer matrix approach to the triangular block decoupling problem the triangular block decoupling problem for linear multivariable systems is studied via the transfer matrix approach this approach clearly separates the problem of admissible control laws from the one of desired decoupled system specifications necessary and sufficient triangular decoupling conditions are given for various control laws these conditions are expressed in a very simple way in terms of linear dependance among the transfer matrix rows it turns out that when the problem is solvable this can be done by static state feedback on a minimal realization of the system furthermore it is shown that whenever triangular block decoupling is possible it is also attainable with stability

new compact and noncompact solutions for two variants of a modified camassa holm equation in this paper exact solutions of two variants of a modified camassa holm equation are determined by using direct ansatze as a result compactons solutions solitons with the absence of infinite tails solitons nonlinear localized waves of infinite support solitary patterns solutions having infinite slopes or cusps and plane periodic solutions are established it is also found that the qualitative change in the physical structure of solutions depends mainly on whether the exponent of the wave function u x t and the expression 2k c are positive or negative

feedback controls for pole subset assignment a pole assignment problem for multi input linear systems is discussed in conventional pole assignment problems an input is concerned with all poles simultaneously or it assigns only a specified subset of poles here we propose a design method which makes each input assign an arbitrary subset of poles thus when we replace a pole by using this method it is enough to reconstruct only the corresponding input

exact travelling wave solutions of a generalized camassa holm equation using the integral bifurcation method abstract in this paper a generalized camassa holm equation is studied by using the integral bifurcation method many travelling waves such as peaked compacton compacton peaked solitary wave solitary wave and kink like wave are found in some parameter conditions exact parametric representations of these travelling waves in explicit form and implicit form are obtained

groupe de picard des varietes de modules de fibres semi stables sur les courbes algebriques soient x une courbe algebrique projective lisse de genre g 2 sur r d des entiers avec r 2 on note u r d la variete de modules des fibres algebriques semi stables sur x de rang r et de degre d us r d l ouvert de u r d correspondant aux fibres stables on sait que u r d est une variete algebrique projective irreductible et normale si r et d ne sont pas premiers entre eux u r d n est pas lisse sauf dans une exception le cas ou g 2 r 2 et d est pair on supposera dans cet article qu on n est pas dans ce cas on a alors codimu r d u r d us r d 2 et u r d us r d est le lieu des points singuliers de u r d si l est un fibre en droites de degre d sur x on note u r l la sous variete fermee de u r d correspondant aux fibres vectoriels de determinant isomorphe a l on etudie dans cet article les groupes de picard pic u r d et pic u r l on montre d abord que les varietes u r d et u r l sont localement factorielles on calcule ensuite pic u r d et pic u r l qui sont decrits au moyen des diviseurs thetas generalises

complete integrable particle methods and the recurrence of initial states for a nonlinear shallow water wave equation we propose an algorithm for an asymptotic model of shallow water wave dynamics in a periodic domain the algorithm is based on the hamiltonian structure of the equation and corresponds to a completely integrable particle lattice in particular periodic particles are introduced in the algorithm for waves travelling through the domain each periodic particle in this method travels along a characteristic curve of the shallow water wave model determined by solving a system of nonlinear integro differential equations we introduce a fast summation algorithm to reduce the computational cost from o n 2 to o n where n is the number of particles with the aim of providing a test of the algorithms we scale the shallow water wave equation to make it asymptotically equivalent to the kdv equation in the form studied by zabusky and kruskal in their seminal 1965 paper thereby also testing the equivalence of the two models derived under similar asymptotic approximations of shallow water wave dynamics by using the fast summation algorithm and the asymptotic scaling analysis we further test this equivalence by investigating the interaction of solitons and recurrence of initial states for the shallow water wave equation in periodic domains finally to illustrate the hyperbolic nature of the dynamics of the shallow water wave model we introduce a particle algorithm and its integral counterpart for the initial boundary value problem with homogeneous boundary conditions on finite intervals

on the periodic solution of the burgers equation a unified approach for the most part the various known expressions for the periodic solution to the burgers equation have been derived by a mixture of ad hoc argument and approximation methods these include the well known solutions due to r d fay and d f parker remarkably these solutions were obtained without recourse to the celebrated hopf cole transformation which linearizes the burgers equation to the heat conduction equation we present here an alternative approach to the periodic problem which uses the fact that the riemann theta functions solve the burgers equation via the hopf cole transformation and thereby provide a unifying framework in which all previous representations of the periodic solution find their natural place yet another expression for the periodic solution in the form of an amplitude modulated sine wave is given and would appear to be new the limiting forms of the periodic solution as t rightarrow 0 embryonic sawtooth profile and t rightarrow infty old age sinusoidal profile are examined in detail the representation due to d f parker which expresses the periodic wave as a superposition of spreading taylor shock profiles is discussed within the context of a nonlinear superposition principle the composite asymptotic expansion due to m b lesser and d g crighton is examined in the light of our results the curious behaviour of this expansion in the embryonic region t ll 1 whereby a first order approximation yields an exact periodic solution the fay solution is explained

analysis of lie symmetries with conservation laws and solutions for the generalized 3 1 dimensional time fractional camassa holm kadomtsev petviashvili equation abstract in this paper under investigated is a generalized 3 1 dimensional camassa holm kadomtsev petviashvili gch kp equation which describes the role of dispersion in the formation of patterns in liquid drops with the help of the semi inverse method the euler lagrange equation and agrawal s method the time fractional gch kp equation is derived in the sense of riemann liouville fractional derivatives further the symmetry of the 3 1 dimensional time fractional gch kp equation is studied by fractional order symmetry meanwhile based on the new conservation theorem the conservation laws of 3 1 dimensional time fractional gch kp equation are constructed finally the solutions to the equation are given via a bilinear method and the radial basis functions rbfs meshless approach

linear feedback decoupling transfer function analysis the problem of linear system decoupling is examined based on recent results on linear feedback new insight is obtained through which resolution of the decoupliug problem is accomplished by calculations performed directly on the given transfer matrix computation of the decoupling compensators follows by easy constructions the problem of feedback block decoupling with internal stability is also formulated and resolved

a state space technique for the evaluation of diagonalizing compensators this note develops a technique for designing diagonalizing compensators based on state space descriptions the relationship with stable decoupling invariants is also explored

model matching and decoupling for continuous and discrete time linear time varying systems solutions to the exact model matching and block decoupling problems for both continuous and discrete time linear time varying systems are presented the parametrisation of the whole class of proper solutions is given for the decoupling the minimal delay problem is also considered in a time varying setting the approach is algebraic and based on the smith macmillan form at infinity of a transfer matrix of a time varying system which has been recently introduced in systems theory this avoids the difficulties related to the inversion of the transfer matrices with entries in non commutative fields over which the determinants of dieudonne or ore type are much more complicated the solutions presented here involve only standard matrix computations excluding direct matrix inversions and are thus easy to implement in practice examples are treated in detail to illustrate the theoretical results and the way in which the computations are done and a physical example is also shown

observer based fault detection and isolation for structured systems fault detection and isolation fdi problems are here consid ered for linear systems with faults and disturbances we design a set of ob server based residuals in such a way that the transfer from the disturbances to the residuals is zero and the transfer from the faults to the residuals al lows fault isolation we are interested in obtaining a transfer function from faults to residuals with either a diagonal structure i e a dedicated struc tured residuals set or a triangular one we deal with this problem when the system under consideration is structured that is the entries of the system matrices are either fixed zeros or free parameters to a structured system one can associate in a natural way a directed graph we can then provide necessary and sufficient conditions under which the fdi problems have a solution for almost any value of the free parameters these conditions are simply expressed in terms of input output paths in the associated graph index terms fault detection and isolation graph theory model based approach structured residuals structured systems

peakons and coshoidal waves traveling wave solutions of the camassa holm equation abstract camassa and holm 1 have recently derived a new integrable wave equation for the special case 0 they showed it has solitary waves of the form c exp x ct which they named peakons in this work we derive a perturbation series for general which converges even at the peakon limit we also give three analytical representations for the spatially periodic generalization of the peakon the coshoidal wave the three representations are i a closed form analytical solution ii a fourier series with coefficients that are explicit rational functions and iii an imbricate fourier series which is the superposition of an infinite number of peakons each separated from its neighbors by distance p where p is the spatial period lastly we have numerically tested the soliton superposition principle although the camassa holm equation is integrable for general it appears that imbricating solitary waves generates an exact spatially periodic solution only for the special cases 0 c 1 2 however the imbricate soliton series is a very good approximate solution for general even when the spatial period is small and the solution resembles a sine wave more than a solitary wave

partial desingularisations of quotients of nonsingular varieties and their betti numbers when a reductive group g acts linearly on a nonsingular complex projective variety x one can define a projective quotient variety x g using mumford s geometric invariant theory if the condition that every semistable point be properly stable is satisfied this quotient is the ordinary topological quotient of an open subset xss of the variety by the group in k a formula is obtained for the rational cohomology of x g under this condition the formula involves the rational cohomology of x and various linear sections of x together with the rational cohomology of the classifying spaces of g and certain reductive subgroups of g in many interesting examples the condition required in k is not satisfied thus the question arises as to what information we can obtain in general the quotient x g can now have serious singularities in contrast to the good case considered in k where the only singularities are those caused by finite isotropy groups it will be shown in this paper that there is a systematic way of blowing up x along a sequence of nonsingular subvarieties to obtain a variety x with a linear action of g such that every semistable point of x is stable the only assumption that we have to make is that there exists at least one stable point of x then x g is almost a resolution of singularities of x1 g in the sense that the most serious singularities have been resolved moreover there is a formula for the rational cohomology of x g again involving the rational cohomology of x and certain linear sections of x together with the rational cohomology of the classifying spaces of g and some reductive subgroups of g see theorem 8 14 for convenience we shall assume throughout that g is connected however the construction of x works in general and it is straightforward to modify the cohomological formulas to apply to the general case the construction of x g can also be modified to apply in some cases when x has no stable points the layout of the paper is as follows section 1 is a review of the basic facts of geometric invariant theory which will be needed and section 2 describes the relationship of geometric invariant theory with symplectic geometry and the moment map in section 3 semistability and stability in a blow up of x along a nonsingular c invariant subvariety are related to semistability and stability in x

well posedness global existence and blowup phenomena for a periodic quasi linear hyperbolic equation we establish the local well posedness of a recently derived model for small amplitude shallow water waves for a large class of initial data we prove global existence of the corresponding solution criteria guaranteeing the development of singularities in finite time for strong solutions with smooth initial data are obtained and an existence and uniqueness result for a class of global weak solutions is also given c 1998 john wiley sons inc

invariant spaces at infinity of linear systems application to block decoupling the dynamic state feedback block decoupling problem in the general case i e with singular or nonsingular input transformations is solved it is proven that the problem is solvable if the number of inputs is large enough the main tools for this study are some invariant spaces associated with the columns infinite behavior of a rational matrix it turns out that when the problem is solvable it is also solvable with stability

block decoupling by precompensation revisited the block decoupling problem by admissible dynamic precompensation for lti systems is considered admissibility refers to the preservation of the class of controlled output trajectories i e functional output controllability is concerned which is more demanding than just pointwise output controllability this problem has been solved by hautus and heyman within a transfer function matrix approach different new equivalent solvability conditions in terms of controllability subspaces transfer function matrices or matrix pencils are given one of these conditions expressed in the input space is at the origin of new necessary and sufficient conditions for block decoupling by general precompensation possibly non admissible and nonsquare in the wider sense of basile and marro

decoupling by dynamic measurement feedback with stability necessary and sufficient conditions the noninteracting control problem of linear time invariant multivariable systems by dynamic measurement feedback is solved in the state space geometric approach the generalization of w m wonham s edp 1979 is considered the problem of the reduction of compensator s order is not treated but the solution derived is able to solve simultaneous problems like the decoupling and the disturbance decoupling problems

block decoupling and arbitrary pole assignment for a linear right invertible system by dynamic compensation abstract the block decoupling of a linear right invertible system by dynamic compensation is studied the system structure of the block decoupled system is clarified from the viewpoint of the closed loop poles a dynamic compensator that realizes block decoupling and arbitrary pole assignment is proposed two examples are given to illustrate the design procedure of the dynamic compensator and its effectiveness

dynamic input output decoupling of nonlinear control systems the problem of dynamic input output decoupling of nonlinear control systems is studied based on an analytic algorithm the authors obtain necessary and sufficient conditions for the solvability of this problem the solution of the problem is constructed by applying a series of simple precompensations and linking maps some interesting connections with other approaches in nonlinear control theory are discussed the authors also give a few simple examples to illustrate the methods used

transfer function matrix approach to decoupling problem with stability a simple necessary and sufficient condition for solvability of decoupling with internal stability for unity output feedback for non singular plants is proposed

parameterization of decoupling controllers in the generalized plant model a diagonal decoupling problem in linear multi variable systems is formulated for the most general setting a necessary and sufficient condition for the existence of a decoupling controller is presented and when a solution exists the class of all decoupling controllers is characterized in terms of free parameters of stable rational vectors

new feedback invariants and the block decoupling problem a new list of feedback invariant integers is introduced we give several equivalent characterizations of these invariants within both geometric and transfer matrix approaches when solving the block decoupling problem it turns out that when decoupling is possible these integers represent the minimal mcmillan degree achievable for the blocks of the decoupled system

moduli of high rank vector bundles over surfaces the purpose of this work is to apply the degeneration theory developed in gl to study the moduli space of stable vector bundles of arbitrary rank on any smooth algebraic surface over c we will show that most of the recent progress in understanding moduli of rank two vector bundles can be carried over to high rank cases after introducing the notion of stable vector bundles the first author constructed the moduli schemes of vector bundles on surfaces he showed that for any smooth algebraic surface x with ample divisor h and line bundle i on x there is a coarse moduli scheme m x i h parameterizing modulo equivalence relation the set of all h semistable rank r torsion free sheaves e on x with dete i and c2 e d since then many mathematicians have studied the geometry of this moduli space especially for the rank two case to cite a few maruyama taubes and the first author showed that the moduli space m x m r d x i h is non empty when d is large moduli spaces of vector bundles of some special surfaces have been studied also the deep understanding of m x for arbitrary x and r 2 begins with donaldson s generic smoothness result roughly speaking donaldson do later generalized by friedman fr and k zhu zh showed that when d is large enough then the singular locus sing m x of m x is a proper subset of m 2 d x and its codimension in m x increases linearly in d this theorem indicates that the moduli m x behaves as expected when the second chern class d is large later using general deformation theory the second author proved that m x is normal and has local complete intersection l c i singularities at stable sheaves provided d is large l2 he also showed that when x is a surface of general type satisfying some mild technical conditions then m x is of general type for d 0 l2 in our paper gl we also proved that m x is irreducible if d is large in this and subsequent papers we shall show that the geometry of m x and the geometry of m x r 3 is rather similar the main obstacle in doing so is the lack of an analogy of the generic smoothness result in high rank case in this paper we will use the degeneration of moduli developed in gl to establish the following main technical theorem

decoupling of square singular systems via proportional state feedback necessary and sufficient conditions for the decoupling of a solvable square singular system ex t ax t bu t with output y t dx t through an admissible control law of the form u t kx t hr t where h is a square nonsingular matrix it has been shown that for a given singular system that satisfies these conditions a propagational state feedback exists for which the system s transfer function is a diagonal nonsingular and proper rational matrix the proofs of the main results are constructive and provide a procedure for computing an appropriate proportional state feedback

causal factorization and linear feedback an algebraic framework for the investigation of linear dynamic output feedback is introduced pivotal in the present theory is the problem of causal factorization i e the problem of factoring two systems over each other through a causal factor the basic issues are resolved with the aid of the new concept of latency kernels

wiener hopf factorization indices and infinite structure of rational matrices the relationship between the infinite structure and the wiener hopf factorization indices of polynomial matrix representations of controllable systems is investigated on the way local wiener hopf factorization indices must be defined in addition the finite and infinite structure of a rational matrix is characterized given its wiener hopf factorization indices

noninteracting control of 2 d systems necessary and sufficient conditions for the existence of a decoupling bicausal precompensator for multivariable 2 d systems are derived in state space and frequency domains in general the decoupling problem for 2 d systems can be solved by feedback compensators if suitable injectivity assumptions are introduced on the input state matrices the structure of dynamic compensators is derived for this case and the 2 d decoupling problem with stability is solved

the arctan tan and kepler burgers mappings for periodic solutions with a shock front or internal boundary layer abstract many periodic solutions have internal regions of rapid change internal boundary layers shock waves and geophysical fronts are one class of examples a second class is composed of functions which decay rapidly away from a central peak or peaks spherical harmonics mathieu eigenfunctions prolate spheroidal wave functions and geophysical hough functions may all be locally approximated by hermite functions in the appropriate parameter range and decay exponentially fast outside a narrow subinterval similarly the large amplitude cnoidal waves of the korteweg devries equation are narrow isolated peaks which are well approximated by the sech 2 y form of the solitary wave in this article we show that a change of coordinate is a powerful tool for resolving such internal boundary layers in the first part we develop a general theory of mappings for the spherical harmonic cnoidal wave class of examples which decay rapidly away towards the edges of the spatial period the particular map y arc tan l tan x is a particularly effective choice four numerical examples show that this map and the fourier pseudospectral method are a good team in the second part we generalize the earlier theory to describe mappings which asymptote to a constant but non zero resolution at the ends of the periodicity interval we explain why the kepler burgers mapping is particularly suitable for shock and fronts

the camassa holm kp equations with compact and noncompact travelling wave solutions abstract in this paper two variants of camassa holm kp equations are investigated compactons solitons with the absence of infinite tails solitons nonlinear localized waves of infinite support solitary patterns having infinite slopes or cusps and plane periodic solutions are formally derived the work highlights the qualitative change in the physical structures of the obtained solutions

geometric aspects of reduced order compensators for disturbance rejection the disturbance decoupling problem with stability constraint and pole placement is examined in a geometric framework two types of reduced order compensators are proposed one based on the observation scheme the other on the dual observation scheme the novelty lies mainly in these points 1 the avoidance of eigenspace computations in the syntheses as they rely on the determination of the maximum controlled invariant contained in a given subspace and its dual i e the minimum conditioned invariant containing a given subspace and pole assignability and 2 the concerted use of controlled and conditioned invariants which leads to a straightforward synthesis and also allows a clear interpretation of the dual compensation

the nonlinear interactor and its application to input output decoupling the solvability of the input output decoupling problem via pure dynamic state feedback a special type of dynamic compensator without zeros at infinity is investigated a necessary and sufficient condition for the solvability of this problem is established in terms of a nonlinear interactor a new concept in the theory of nonlinear control

factorization of linear systems a generalized framework abstract a generalized module theoretic framework for the study of linear time invariant systems is developed crucial in the present discussion are two new notions the generalized order and the adapted bases these notions form generalizations of the classical concepts of order or degree and of proper bases employed in the theory of linear systems the resulting framework is then applied to obtain explicit conditions for system factorization and to study output feedback systems in which the feedback compensator is stable

static state feedback triangular block decoupling for arbitrary systems a state space method abstractin this paper the static state feedback triangular block decoupling problem for a general system described by a b c d quadruples is studied necessary and sufficient conditions which are numerically verifiable are obtained and a numerically reliable decoupling method is established in the state space two examples are solved by our method which show the effectiveness

diagonal decoupling of linear systems by static state feedback a solution is presented to the long standing open problem of linear systems theory diagonal decoupling by static state feedback the earliest known investigation of system decoupling dates back to 1934 a state space formulation of the problem appeared in 1964 and a solution for square and invertible systems followed in 1967 the case of right invertible systems however has withstood all past efforts to obtain a solution save for several special cases the present formulation avoids restrictive hypotheses concerning system and decoupling feedback the existence of a solution is shown to depend on the existence of three lists of integers conditioned solely by system invariants with respect to the permissible transformations

input output decoupling by output feedback with disturbance attenuation of h control for any square system the combined problem of diagonal decoupling by output feedback with disturbance attenuation performance for any square system is formulated a sufficient condition which is more general than the necessary and sufficient one of static state feedback for the existence of the controller is determined the necessary and sufficient conditions for the controller and decoupled system that are both stable are presented an analytical form and a numerical computable method for the controller are obtained for the first time when the condition is satisfied the correctness of our method is tested by simulations for an example

decoupling in the design and synthesis of singular systems abstract necessary and sufficient conditions for the decoupling of singular systems using p d proportional derivative feedback are determined given a system which satisfies these conditions the class of all feedback matrices which decouple the system is characterized this class is used to determine the number of closed loop pole zeros which can be specified for the decoupled system and to develop a synthesis technique for the realization of desired closed loop pole zero configurations

achieving diagonal interactor matrix for multivariable linear systems with uncertain parameters abstract the notion of interactor matrix or equivalently the hermite normal form is a generalization of relative degree to multivariable systems and is crucial in problems such as decoupling inverse dynamics and adaptive control in order for a system to be input output decoupled using static state feedback the existence of a diagonal interactor matrix must first be established for a multivariable linear system which does not have a diagonal interactor matrix dynamic precompensation or dynamic state feedback is required for achieving a diagonal interactor matrix for the compensated system such precompensation often depends on the parameters of system and is thus difficult to implement with accuracy when the system is subject to parameter uncertainty in this paper we characterize a class of linear systems which can be precompensated to achieve a diagonal interactor matrix without the exact knowledge of the system parameters more precisely we present necessary and sufficient conditions on the transfer matrix of the system under which there exists a diagonal dynamic precompensator such that the compensated system has a diagonal interactor matrix these conditions are associated with the so called non generic singularity of certain matrix related to the system structure but independent of the system parameters the result of this paper is expected to be useful in robust and adaptive designs

solitons from sine waves analytical and numerical methods for non integrable solitary and cnoidal waves abstract the fkdv equation u t uu x u xxxxx 0 is used as a testbed for a variety of analytical and numerical methods that can be applied to solitary waves and cnoidal waves of non integrable differential equations that is to say to equations which cannot be solved by the inverse scattering transform the basic tools are i pade approximants formed from power series in the amplitude ii a newton kantorovich pseudospectral fourier continuation numerical method iii singular perturbation theory for two interacting solitons of almost identical phase speed iv bifurcation and branch switching methods v the imbricate soliton series a number of new results for the fkdv equation are obtained including extensive numerical calculations of the spatially periodic solutions with one peak cnoidal wave and two peaks bicnoidal wave per period an analytical expression for the double peaked soliton bion calculation of both the limit and bifurcation points for the bicnoidal wave and finally the computation of accurate analytical approximations to the cnoidal wave for all amplitudes more important all of these analytical and numerical tools are highly effective for this equation in spite of the fact that it cannot be solved by the inverse scattering transform work now in progress will apply these methods to non integrable equations in two space dimensions

input output decoupling of hamiltonian systems the linear case in this note we give necessary and sufficient conditions for a linear hamiltonian system to be input output decouplable by hamiltonian feedback i e feedback that preserves the hamiltonian structure in a second paper we treat the same problem for nonlinear hamiltonian systems

a simple existence condition of one degree of freedom block decoupling controllers a block decoupling problem in linear multivariable systems is treated for one degree of freedom controller configuration with unity output feedback the plant transfer matrix which may be non square is assumed to have unstable simple poles and zeros that may coincide a simple existence condition of a block decoupling controller is obtained by directional interpolation approaches

decoupling with stability for linear periodic systems in this paper the classical row by row decoupling problem with stability through the use of a dynamic compensator is studied for linear periodic discrete time systems the necessary and sufficient conditions for the existence of a solution are found and a design procedure of the dynamic compensator is provided by the sufficiency proof and also for the case of a dead beat requirement for the state free response

a legendre pseudospectral method for computing travelling waves with corners slope discontinuities in one space dimension with application to whitham s equation family if the dispersion in a nonlinear hyperbolic wave equation is weak in the sense that the frequency k of cos kx is bounded as k it is common that i travelling waves exist up to a limiting amplitude with wave breaking for higher amplitudes and ii the limiting wave has a corner that is a discontinuity in slope because corner waves are not smooth standard numerical methods converge poorly as the number of grid points is increased however the corner wave is important because at least in some systems it is the attractor for all large amplitude initial conditions here we devise a legendre pseudospectral method which is uncorrupted by the singularity the symmetric u x u x wave can be computed on an interval spanning only half the spatial period since u is smooth on this domain which does not include the corner except as an endpoint all numerical difficulties are avoided a key step is to derive an extra boundary condition which uniquely identifies the corner wave with both the grid point values of u x and phase speed c as unknowns the discretized equations imposing three boundary conditions on a second order differential equation are solved by a newton raphson iteration although our method is illustrated by the so called whitham s equation ut uux du dx where d is a very general linear operator the ideas are widely applicable

block decoupling of generalized state space systems abstract the problem of input output block decoupling of generalized state space systems via a regular static state feedback law is studied for the first time the necessary and sufficient conditions for the problem to have a solution are established the necessary and sufficient conditions for block decoupling with simultaneous stabilizability and the necessary and sufficient conditions for block decoupling with simultaneous arbitrary assignment of the system poles are established 1997 elsevier science ltd

a class of nonlinear fourth order variant of a generalized camassa holm equation with compact and noncompact solutions abstract in this paper we study a class of nonlinear fourth order analogue of a generalized camassa holm equation it is shown that this class gives compactons conventional solitons solitary patterns and periodic solutions it is also found that the qualitative change in the physical structure of solutions depends mainly on the exponent of the wave function u x t positive or negative and on the coefficient of u n as well

the orbital stability of the solitary wave solutions of the generalized camassa holm equation abstract in this paper we consider the orbital stability of smooth solitary wave solutions of the generalized camassa holm equation by constructing the functional extremum problem and using the orbital stability theory presented by grillakis shatah strauss and bona and souganidis we show that the solitary wave solutions of the generalized camassa holm equation are orbitally stable or unstable as determined by the sign of a discriminant the conclusions presented by the previous authors such as hakkaev and kirchev constantin and strauss can be considered as a special case of our results