the matching of a prescribed linear input output behavior in a nonlinear system in this paper we deal with the problem of finding a feedback under which the input output behavior of a nonlinear system becomes exactly the same as that of a specified linear model the solvability of this problem is shown to depend on a formal infinite zero structure associated with the system

generalized controller canonical form for linear and nonlinear dynamics a generalized controller canonical form for linear and nonlinear dynamics is proposed it is obtained using the theorem of the primitive element from differential algebra the derivation of the controller form does not apply to multivariable constant linear systems with the transformations all nonlinear dynamics can be exactly linearized by dynamic feedback the main departure from standard theory is that transformations may depend on input derivatives once differential algebraic tools are introduced the proofs of the results are easy

transfer equivalence and realization of nonlinear input output delta differential equations on homogeneous time scales nonlinear control systems on homogeneous time scales are studied first the concepts of reduction and irreducibility are extended to higher order delta differential input output equations subsequently a definition of system equivalence is introduced which generalizes the notion of transfer equivalence in the linear case finally the necessary and sufficient conditions are given for the existence of a state space realization of a nonlinear input output delta differential equation

finite volterra series realizations and input output approximations of non linear discrete time systems abstract it is shown that a separability property of a given finite family of stationary kernels turns out to be necessary and sufficient for the realization of the associated functional by means of a polynomial affine system i e a system that is polynomial in the input and affine in the state moreover the discrete volterra kernels associated with the input output map of a non linear analytic discrete lime system initialized at an equilibrium point are shown to possess a separability property on this basis we state an approximation result for the given input output map by considering the first kernels of the discrete volterra series two explicit constructions of the approximating polynomial affine systems are proposed

brief transfer equivalence and realization of nonlinear higher order input output difference equations two fundamental modelling problems in nonlinear discrete time control systems are studied using the language of differential forms the discrete time nonlinear single input single output systems to be studied are described by input output i o difference equations i e a high order difference equation relating the input the output and a finite number of their time shifts a new definition of equivalence is introduced which generalizes the notion of transfer equivalence well known for the linear case our definition is based upon the notion of an irreducible differential form of the system and was inspired by the analogous definition for continuous time systems the second problem to be addressed is the realization problem the i o difference equation is assumed to be in the irreducible form so that one can obtain an accessible and observable realization necessary and sufficient conditions are given for the existence of a local state space realization of the irreducible i o difference equation these conditions are formulated in terms of the integrability of certain subspaces of one forms classified according to their relative degree the sufficiency part of the proof gives a constructive procedure up to finding the integrating factors and integration of the set of one forms for obtaining a locally observable and accessible state space system if the system is not in the irreducible form one has first to apply the reduction procedure to transform the system into the irreducible form

automatic classification of protein structure by using gauss integrals we introduce a method of looking at analyzing and comparing protein structures the topology of a protein is captured by 30 numbers inspired by vassiliev knot invariants to illustrate the simplicity and power of this topological approach we construct a measure scaled gauss metric sgm of similarity of protein shapes under this metric protein chains naturally separate into fold clusters we use sgm to construct an automatic classification procedure for the cath2 4 database the method is very fast because it requires neither alignment of the chains nor any chain chain comparison it also has only one adjustable parameter we assign 95 51 of the chains into the proper c class a architecture t topology and h homologous superfamily fold find all new folds and detect no false geometric positives using the sgm we display a map of the space of folds projected onto two dimensions show the relative locations of the major structural classes and zoom into the space of proteins to show architecture topology and fold clusters the existence of a simple measure of a protein fold computed from the chain path will have a major impact on automatic fold classification

weakly convergent expansions of a plane wave and their use in fourier integrals the fourier transform of an irreducible spherical tensor is normally computed with the help of the rayleigh expansion of a plane wave in terms of spherical bessel functions and spherical harmonics the angular integrations are then trivial however the remaining radial integral containing a spherical bessel function may be so complicated that the applicability of fourier transformation is severely restricted as an alternative the use of weakly convergent expansions of a plane wave in terms of complete orthonormal sets of functions is suggested the weakly convergent expansions of a plane wave are constructed in such a way that their application in fourier integrals leads to expansions of the fourier or inverse fourier transform that converge with respect to the norm of either the hilbert space l2 r3 or the sobolev space w 1 2 r3 accordingly these weakly convergent expansions may be viewed as distributions that are defined on either l2 r3 or w 1 2 r3 the properties of some complete orthonormal se

invariant codistributions and the feedforward form for discrete time nonlinear systems the goal of this paper is two fold first given an arbitrary n dimensional discrete time nonlinear dynamical system necessary and sufficient conditions for the existence of a one dimensional invariant codistribution are obtained second it is shown that the previous conditions can be used iteratively to obtain a nested sequence of n invariant codistributions with the properties that each codistribution contains the previous one and the last one coincides with the cotangent bundle of the state manifold as a byproduct necessary and sufficient conditions are obtained for a discrete time nonlinear dynamical system to be equivalent to the so called feedforward form

brief analysis of nonlinear time delay systems using modules over non commutative rings the theory of non commutative rings is introduced to provide a basis for the study of nonlinear control systems with time delays the left ore ring of non commutative polynomials defined over the field of meromorphic function is suggested as the framework for such a study this approach is then generalized to a broader class of nonlinear systems with delays that are called generalized roesser systems finally the theory is applied to analyze nonlinear time delay systems a weak observability is defined and characterized generalizing the well known linear result properties of closed submodules are then developed to obtain a result on the accessibility of such systems

authors reply to comments on on the discrete time normal form kotta 2000 commented on the above paper by califano et al 1998 kotta made two main criticisms in this paper the original authors reply to them

immersion of discrete time nonlinear system by regular dynamic state feedback into a linear system necessary and sufficient conditions for local solvability of the title problem around a given trajectory are obtained the proposed conditions are less restrictive than those obtained by lee and marcus for the problem of immersion a nonlinear system into a linear system via static state feedback instrumental in the problem solution is the inversion structure algorithm for a discrete time nonlinear system the solvability conditions are expressed in terms of the inversion algorithm moreover the construction of both the dynamic state feedback and the immersion map relies on this algorithm finally it is shown that for locally right invertible systems the considered problem unlike the problem of immersion via static state feedback is always solvable

the linking number and the writhe of uniform random walks and polygons in confined spaces random walks and polygons are used to model polymers in this paper we consider the extension of the writhe self linking number and linking number to open chains we then study the average writhe self linking and linking number of random walks and polygons over the space of configurations as a function of their length we show that the mean squared linking number the mean squared writhe and the mean squared self linking number of oriented uniform random walks or polygons of length n in a convex confined space are of the form o n2 moreover for a fixed simple closed curve in a convex confined space we prove that the mean absolute value of the linking number between this curve and a uniform random walk or polygon of n edges is of the form our numerical studies confirm those results they also indicate that the mean absolute linking number between any two oriented uniform random walks or polygons of n edges each is of the form o n equilateral random walks and polygons are used to model polymers in conditions we use numerical simulations to investigate how the self linking and linking number of equilateral random walks scale with their length

on the realization of nonlinear discrete time systems abstract it is shown on the basis of a compact expression of the kernels of the discrete volterra series associated to a linear analytic discrete time system that the kernels satisfy a suitable property which enables their inductive characterization we also give a first result on the realization of a given family of functions by a linear analytic discrete time system with linear invertible drift term

linearization of discrete time systems the algebraic formalism developed in this paper unifies the study of the accessibility problem and various notions of feedback linearizability for discrete time nonlinear systems the accessibility problem for nonlinear discrete time systems is shown to be easy to tackle by means of standard linear algebraic tools whereas this is not the case for nonlinear continuous time systems in which case the most suitable approach is provided by differential geometry the feedback linearization problem for discrete time systems is recasted through the language of differential forms in the event that a system is not feedback linearizable the largest feedback linearizable subsystem is characterized within the same formalism using the notion of derived flag of a pfaffian system a discrete time system may be linearizable by dynamic state feedback though it is not linearizable by static state feedback necessary and sufficient conditions are given for the existence of a so called linearizing output which in turn is a sufficient condition for dynamic state feedback linearizability

the linking number in systems with periodic boundary conditions periodic boundary conditions pbc are often used for the simulation of complex physical systems using the gauss linking number we define the periodic linking number as a measure of entanglement for two oriented curves in a system employing pbc in the case of closed chains in pbc the periodic linking number is an integer topological invariant that depends on a finite number of components in the periodic system for open chains the periodic linking number is an infinite series that accounts for all the topological interactions in the periodic system in this paper we give a rigorous proof that the periodic linking number is defined for the infinite system i e that it converges for one two and three pbc models it gives a real number that varies continuously with the configuration and gives a global measure of the geometric complexity of the system of chains similarly for a single oriented chain we define the periodic self linking number and prove that it also is defined for open chains in addition we define the cell periodic linking and self linking numbers giving localizations of the periodic linking numbers these can be used to give good estimates of the periodic linking numbers in infinite systems we also define the local periodic linking number associated to chains in the immediate cell neighborhood of a chain in order to study local linking measures in contrast to the global linking measured by the periodic linking numbers finally we study and compare these measures when applied to a pbc model of polyethylene melts

adaptive output feedback design for a class of nonlinear discrete time systems we present adaptive output feedback schemes for a class of discrete time systems whose continuous time counterparts are the output feedback canonical form we first develop a systematic design procedure for the deterministic case global boundedness and convergence are achieved with no restriction imposed on the nonlinearities then we proceed to develop an analogous design procedure for the adaptive case depending on the characteristics of the nonlinearities global boundedness and convergence can be achieved with various update laws a rank condition is required to ensure such properties for nonlinearities which are not globally lipschitz

non linear non interacting control with stability in discrete time a geometric framework the aim of the paper is to set the non linear non interacting control problem with stability in discrete time in a correct geometric framework so providing a solution which recalls the continuous approach and to point out the peculiarities of discrete time dynamics geometric necessary and sufficient conditions for the solvability of the problem are given and a constructive method figures out the regular static state feedback solution it is shown that if the given dynamics is feedback equivalent to a decoupled system characterized by an invertible drift the computation of the distributions involved can be carried out in a simple and elegant way the main results are thus proved exhaustively in this case and then generalized to submersive systems

generalization of transfer equivalence for discrete time non linear systems comparison of two definitions the problem of system reduction is studied for discrete time non linear single input single output systems described by high order input output i o difference equations that is given the i o equation we can find a minimal representation which is equivalent to the original system with the order being as small as possible comparison of two definitions of reducibility is provided the reducibility concepts addressed are both generalizations of the well known notion of transfer equivalence to the case of non linear control systems two roles of transfer equivalence are covered by these extensions the first is that of identity of the outputs for any fixed control sequence under zero initial conditions the second property is that of pole zero cancellation that may occur in the transfer function of equivalent systems the relationship between two reducibility definitions related to two equivalence notions is examined and the reducibility definition which extends pole zero cancellation property is shown t

controllability of linear stochastic systems abstract we discuss several concepts of controllability for stochastic systems and derive sufficient and in most cases also necessary algebraic conditions for the controllability of the state and the output of white and colored noise linear systems these criteria reduce to the well known deterministic ones in the absence of noise

the model matching problem for a class of polynomial non linear discrete input output systems with cross products an algorithmic approach in this paper we examine the model matching problem that concerns non linear input output discrete systems containing products among delays of input and output signals through a special factorisation the algebraic framework of e operators and the star product that we adopt describe these systems moreover a certain procedure that resembles the euclidean division allows us to discover the linear factors of those systems with respect to the above mentioned operations the entire approach is symbolically algorithmic and involves the use of suitable software

on the sliding mode control of multivariable nonlinear systems sliding mode control of multivariable nonlinear systems is addressed from the perspective of linear differential algebra for a special but large class of linearizable systems known as differentially flat systems essential orders and differential flatness are shown to be a relevant concept and a sensible requirement respectively associated with the possibilities of designing static or dynamical sliding mode multivariable feedback regulators for nonlinear systems

further results on identifiability of discrete time nonlinear systems different notions of identifiability are available for discrete time nonlinear systems the most popular in the current literature is structural identifiability besides algebraic identifiability and identifiability with known initial conditions these three notions are fully characterized in this paper within a unique linear algebraic framework the relationships between the notions are shown as well academic examples as well as the model of a power amplifier borrowed from the area of digital telecommunications are used to illustrate the different notions and computations involved by our results

orders of input output differential equations and state space dimensions this paper deals with the orders of input output equations satisfied by nonlinear systems such equations represent differential or difference in the discrete time case relations between high order derivatives or shifts respectively of input and output signals it is shown that under analyticity assumptions there cannot exist equations of order less than the minimal dimension of any observable realization this generalizes the known situation in the classical linear case the results depend on new facts themselves of considerable interest in control theory regarding universal inputs for observability in the discrete case and observation spaces in both the discrete and continuous cases included in the paper is also a new and simple self contained proof of sussmann s universal input theorem for continuous time analytic systems

irreducibility reduction and transfer equivalence of nonlinear input output equations on homogeneous time scales the purpose of this paper is to present a necessary and sufficient condition for irreducibility of nonlinear input output delta differential equations the condition is presented in terms of the common left divisor of two differential polynomials describing the behaviour of the system defined on a homogenous time scale the concept of reduction is explained subsequently the definition of transfer equivalence based upon the notion of an irreducible differential form of the system is introduced inspired by the analogous definition for continuous time systems

non linear systems stability and rationality the problem of when a non linear system can be represented as a quotient of two stable non linear systems is considered attention is mainly directed toward nonlinear discrete time recursive systems where recursive means that the relationship between an input sequence and the corresponding output sequence can be expressed in terms of a finite number of recursive equations necessary and sufficient conditions are derived for the existence of a fraction representation of a recursive system where the numerator and the denominator are stable recursive systems the explicit construction of such a fraction representation is described

on the mean and variance of the writhe of random polygons we here address two problems concerning the writhe of random polygons first we study the behavior of the mean writhe as a function length second we study the variance of the writhe suppose that we are dealing with a set of random polygons with the same length and knot type which could be the model of some circular dna with the same topological property in general a simple way of detecting chirality of this knot type is to compute the mean writhe of the polygons if the mean writhe is non zero then the knot is chiral how accurate is this method for example if for a specific knot type k the mean writhe decreased to zero as the length of the polygons increased then this method would be limited in the case of long polygons furthermore we conjecture that the sign of the mean writhe is a topological invariant of chiral knots this sign appears to be the same as that of an ideal conformation of the knot we provide numerical evidence to support these claims and we propose a new nomenclature of knots based on the sign of their expected writhes this nomenclature can be of particular interest to applied scientists the second part of our study focuses on the variance of the writhe a problem that has not received much attention in the past in this case we focused on the equilateral random polygons we give numerical as well as analytical evidence to show that the variance of the writhe of equilateral random polygons of length n behaves as a linear function of the length of the equilateral random polygon

knots in self avoiding walks discusses the existence of knots in random self avoiding walks on a lattice using kesten s 1963 pattern theorem it is shown that almost all sufficiently long self avoiding walks on the three dimensional simple cubic lattice contain a knot

the writhe of polygons on the face centred cubic lattice for polygons on the simple cubic lattice there is an important theorem due to lacher and sumners which shows that the writhe of a polygon is the average of the linking numbers of the polygon and its pushoffs in four particular directions this implies that the writhe of a polygon on the simple cubic lattice is always rational we prove a related theorem for the face centred cubic fcc lattice but show that on this lattice the probability that a polygon of length n has irrational writhe tends to unity as n tends to infinity in addition we show that the expectation of the absolute value of the writhe increases at least as fast as n 1 2 for large n

some realizations of algorithms for nonlinear input output systems this paper develops an algorithm which given an input output differential equation produces under appropriate circumstances a parametrization of the observation space of the underlying state space system the paper extends previous results by the authors to a more general setting so that state space systems in which the controls enter nonlinearly can be accommodated and most importantly the paper deals with implicitly defined systems at both the input output and state space level fliess and hasler have demonstrated the importance of implicit input output models which yield only local state space descriptions various realizations are obtained depending on the properties of the derived candidate observation space examples of the algorithm are described with the aid of an implementation using the computer algebra system axiom

flatness and defect of non linear systems introductory theory and examples we introduce flat systems which are equivalent to linear ones via a special type of feedback called endogenous their physical properties are subsumed by a linearizing output and they might be regarded as providing another nonlinear extension of kalman s controllability the distance to flatness is measured by a non negative integer the defect we utilize differential algebra where flatness and defect are best defined without distinguishing between input state output and other variables many realistic classes of examples are flat we treat two popular ones the crane and the car with n trailers the motion planning of which is obtained via elementary properties of plane curves the three non flat examples the simple double and variable length pendulums are borrowed from non linear physics a high frequency control strategy is proposed such that the averaged systems become flat

reversible linear and nonlinear discrete time dynamics it is shown that any causal discrete time system can be realized with a reversible or invertible state variable representation this bridges the gap with continuous time where flows always satisfy this property and settles an old debate on the relevance of invertibility in nonlinear discrete dynamics as the result applies to the linear case it is shown how the classic approach should be slightly modified and several examples of the benefits that can be obtained are given difference algebra is employed for nonlinear systems it leads to a most elegant characterization of causality which enables reversibility to be demonstrated module theory suffices in the linear case which is supplemented by elementary matrix manipulations

on the finiteness of accessibility test for nonlinear discrete time systems it is shown that for two large subclasses of discrete time nonlinear systems analytic systems defined on a compact state space and rational systems the minimum length r for input sequences ca

a new factorization of special nonlinear discrete systems and their applications in this paper a new factorization of discrete nonlinear systems is presented this factorization is based on the star product an operation corresponding to the linear connection of systems a relevant algorithm is developed based on a theoretical background finally some applications to system analysis bibo stability and feedback design are provided

generalized controlled invariance for discrete time nonlinear systems with an application to the dynamic disturbance decoupling problem in analogy with the continuous time case a general notion of controlled invariance with respect to quasi static state feedback is introduced for discrete time nonlinear systems which incorporates the earlier definition of controlled invariance with respect to regular static state feedback this new notion is used to derive a geometric solution to the dynamic disturbance decoupling problem the proposed solution is a natural generalization of the geometric solution to the static disturbance decoupling problem

numerical evaluation of integrals containing a spherical bessel function by product integration a method is developed for numerical evaluation of integrals with k integration range from 0 to that contain a spherical bessel function jl kr explicitly the required quadrature weights are easily calculated and the rate of convergence is rapid only a relatively small number of quadrature points is needed for an accurate evaluation even when r is large the quadrature rule is obtained by the method of product integration with the abscissas chosen to be those of clenshaw curtis and the chebyshev polynomials as the interpolating polynomials quadrature weights are obtained that depend on the spherical bessel function an inhomogenous recurrence relation is derived from which the weights can be calculated without accumulation of roundoff error the procedure is summarized as an easily implementable algorithm questions of convergence are discussed and the rate of convergence demonstrated for several test integrals alternative procedures are given for generating the integration weights and an error analy

writhe and mutual entanglement combine to give the entanglement length we propose a method to estimate n e the entanglement length that incorporates both local and global topological characteristics of chains in a melt under equilibrium conditions this estimate uses the writhe of the chains the writhe of the primitive paths and the number of kinks in the chains in a melt an advantage of this method is that it works for both linear and ring chains works under all periodic boundary conditions does not require knowing the contour length of the primitive paths and does not rely on a smooth set of data we apply this method to linear finitely extendable nonlinear elastic chains and we observe that our estimates are consistent with those from other studies

a linear algebraic framework for the analysis of discrete time nonlinear systems a linear algebraic framework for the analysis of synthesis type problems for discrete time nonlinear systems is introduced this is an extension of a similar tool for continuous time systems that established important connections between many algorithms associated with right invertibility left invertibility and dynamic decoupling as well as between these algorithms and an approach based upon differential algebra a similar payoff is seen to be possible in the discrete time setting

linearization of discrete time systems by exogenous dynamic feedback it is shown that a discrete time system may be linearizable by exogenous dynamic feedback even if it cannot be linearized by endogenous feedback this property is completely unexpected and constitutes a fundamental difference with respect to the continuous time case the notion of exogenous linearizing output is introduced it is shown that existence of an exogenous linearizing output is a sufficient condition for dynamic linearizability necessary and sufficient conditions for the existence of an exogenous linearizing output are provided the results of the paper are obtained using transformal operator matrices the properties of such operators are studied the theory is applied to the exact discrete time model of a mobile robot showing that the above mentioned property concerns not only academic examples but also physical systems

the writhe of a self avoiding walk the writhe of a self avoiding walk in a three dimensional space is the average over all projections onto a plane of the sum of the signed crossings we compute this number using a monte carlo simulation our results suggest that the average of the absolute value of the writhe of self avoiding walks increases as nalpha where n is the length of the walks and alpha approximately 0 5 the mean crossing number of walks is also computed and found to have a power law dependence on the length of the walks in addition we consider the effects of solvent quality on the writhe and mean crossing number of walks

computing the writhe on lattices given a polygonal closed curve on a lattice or space group we describe a method for computing the writhe of the curve as the average of weighted projected writhing numbers of the polygon in a few directions these directions are determined by the lattice geometry the weights are determined by areas of regions on the unit 2 sphere and the regions are formed by the tangent indicatrix to the polygonal curve we give a new formula for the writhe of polygons on the face centred cubic lattice and prove that the writhe of polygons on the body centred cubic lattice the hexagonal simple lattice and the diamond space group is always a rational number and discuss applications to ring polymers

product integration over infinite intervals i rules based on the zeros of hermite polynomials the paper discusses both theoretical properties and practical implementation of product integration rules of the form n j k x f x dx wnlj xnj o i l where f is continuous k is absolutely integrable the nodes xn are roots of the hermite polynomials hn x and the weights wn are chosen so that the rule is exact if f is any polynomial of degree n convergence of the rule to the exact integral as n oo is proved for a wide class of functions f and k including singular or oscillatory functions k and rates of convergence are estimated the rules are shown to have the property of asymptotic positivity and as a consequence exhibit good numerical stability numerical calculations for some practical cases are presented which show the method to be computationally effective for integrands including highly oscillatory ones that decay suitably at infinity applications of the method to integration over 0 oo are also discussed

design of coupled decentralized output regulators decentralized static output regulators are designed via projective controls in cases where the controllers are directly coupled i e when the controls of other decision makers appear in the local outputs for two decision makers the solution is reduced to the problem of solving a general non square algebraic riccati equation the existence and uniqueness of the solution is proved and then determined analytically these results are then generalized to an arbitrary number of decision makers finally the possibilities of the sequential design of static and dynamic regulators for this type of decentralized information structure are investigated a numerical example illustrates the design procedure

the model matching problem for a certain class of nonlinear systems sufficient conditions are described for the solution of the model matching problem for a specific class of nonlinear systems represented by the so called polynomials and linear desired models an algorithm yielding causal controllers and a bibo closed loop connection is supplied

on the problem of feedback linearization the paper studies and solves in a geometric framework the problem of partial feedback linearization for discrete time dynamics an algorithm for computing the largest linearizable subsystem is proposed this approach can be considered as dual to the one already proposed in literature in an algebraic context

a polynomial approach to discrete time non linear system controllability this paper uses a polynomial approach to present a necessary and sufficient condition for local controllability of single input single output siso discrete time non linear systems the condition is presented in terms of common factors of a non commutative polynomial expression this extends the result on controllability of linear systems and the recent result on controllability of continuous time non linear systems to expose controllability of discrete time non linear systems in the input output framework the controllability condition yields a finite step computing procedure for examining controllability of discrete time input output non linear systems

the writhe of a self avoiding polygon we discuss the writhe of a self avoiding polygon on a lattice as a geometrical measure of its entanglement complexity we prove a rigorous result about the dependence of the absolute value of the writhe on the number n of edges in the polygon and use monte carlo methods to estimate the distribution of the writhe both for all polygons with n edges and for the subset of polygons that are trefoils

an algebraic framework generalizing the concept of transfer functions to nonlinear systems an algebraic point of view in nonlinear control systems built up by introducing a differential field of meromorphic functions and a vector space of differential forms is in this paper extended by introducing skew polynomials such polynomials act as operators over a vector space of differential forms the left skew polynomial ring defined over the field of meromorphic functions is embedded to its quotient field to provide a basis for a symbolic computation of nonlinear systems members of such a quotient field are suggested as the transfer functions of nonlinear systems whereby the concept of the transfer functions is generalized to nonlinear systems the theory is applied to nonlinear control systems and argues the invariance of introduced transfer functions of nonlinear systems to static state transformations the existence of input output descriptions for state space representations availability of transfer function algebra for nonlinear systems etc

feedback linearization and lattice theory abstract the tools of lattice theory are applied to readdress the static state feedback linearization problem for discrete time nonlinear control systems unlike the earlier results that are based on differential geometry the new tools are also applicable for nonsmooth systems in case of analytic systems close connections are established between the new results and those based on differential one forms the mathematica functions have been developed that implement the algorithms methods from this paper

design method of exact model matching control for finite volterra series systems this paper is concerned with exact model matching control emm for finite volterra series systems first we show a structure of a causal controller which can achieve the exact model matching to a reference model and clarify the relationship between the proposed method and the emm for linear systems second to analyse the stability of the proposed control system we present an input dependent small gain theorem for the system with an external input then extend it for the system with two external inputs with the help of this theorem we clarify the condition under which the control system is stable for the reference input magnitude within a certain range and is also robust for small disturbances finally the effectiveness of the proposed method is illustated through numerical simulations

comparison of lpv and nonlinear system theory a realization problem abstract the paper explores the utilization of the nonlinear realization theory to address the problem of transforming linear parameter varying input output lpv io equations into a state space form with static dependence on the so called scheduling parameter the necessary and sufficient solvability conditions are given and three additional subclasses of lpv io equations are suggested that are guaranteed to have a realization of the considered type

on the discrete time normal form necessary and sufficient conditions under which the discrete time normal form exists are set both in geometric and in algebraic frameworks leading to equivalent conditions the results are therefore extended to generic nonlinear nonaffine continuous time systems

the mean squared writhe of alternating random knot diagrams the writhe of a knot diagram is a simple geometric measure of the complexity of the knot diagram it plays an important role not only in knot theory itself but also in various applications of knot theory to fields such as molecular biology and polymer physics the mean squared writhe of any sample of knot diagrams with n crossings is n when for each diagram at each crossing one of the two strands is chosen as the overpass at random with probability one half however such a diagram is usually not minimal if we restrict ourselves to a minimal knot diagram then the choice of which strand is the over or under strand at each crossing is no longer independent of the neighboring crossings and a larger mean squared writhe is expected for minimal diagrams this paper explores the effect on the correlation between the mean squared writhe and the diagrams imposed by the condition that diagrams are minimal by studying the writhe of classes of reduced alternating knot diagrams we demonstrate that the behavior of the mean squared writhe heavily depends on the underlying space of diagram templates in particular this is true when the sample space contains only diagrams of a special structure when the sample space is large enough to contain not only diagrams of a special type then the mean squared writhe for n crossing diagrams tends to grow linearly with n but at a faster rate than n indicating an intrinsic property of alternating knot diagrams studying the mean squared writhe of alternating random knot diagrams also provides some insight into the properties of the diagram generating methods used which is an important area of study in the applications of random knot theory