output feedback direct adaptive fuzzy controller based on frequency domain methods this paper describes a design method that guarantees the stability of an adaptive fuzzy control system the system consists of an unstable plant with an irrational transfer function a takagi sugeno fuzzy adaptive controller a nonlinear reference model and an adaptation mechanism the gradient based adaptation mechanism changes the consequents of the controller rules in such a way that the closed loop system behaves like the reference model the proposed design procedure utilizes frequency domain methods in the forms of the nyquist and circle criteria it is assumed that the function of the fuzzy controller is a nonlinearity described by a sector condition this condition means that the controller function lies between two lines passing through the origin in this method the function of the fuzzy controller is verified during online adaptation therefore it stays in the allowed sector this method is illustrated by an example of a control system containing an unstable plant with a transport delay the main contribution of this paper is to use frequency methods combined with adaptive fuzzy control these methods have previously been used for nonadaptive fuzzy control but in this paper they are used for the first time for adaptive fuzzy control

on the noetherianity of some associative finitely presented algebras abstract we consider finitely generated associative algebras over a fixed field k of arbitrary characteristic for such an algebra a we impose some structural restrictions we call a strictly ordered we are interested in the implication of strict order on a for its noetherian properties in particular we prove that if a is a graded standard finitely presented strictly ordered algebra then a is left noetherian if and only if it is almost commutative in this case a has polynomial growth

unicity of minimal rank completions for tri diagonal partial block matrices abstract in this paper we establish necessary and sufficient conditions for a minimal rank completion of a tri diagonal partial block matrix to be unique the case of 4 4 block matrices with four consecutive diagonals is also considered along with its application to the problem of completing a matrix and its inverse

technical communique h optimal terminal control for linear systems with delayed states and controls in this paper we consider a linear time varying system with state and control delay terms both lumped and distributed subject to unknown square integrable disturbances using a differential games approach we provide a necessary and sufficient condition for the existence of an optimal control law minimizing the h norm of the operator mapping the disturbance to the weighted control and terminal state

generalized l sub 2 synthesis introduces a general framework for control design which can be used to solve several open problems in robust and optimal control these include systems subject to independently norm bounded disturbances robust stability for systems subject to element by element bounded structured uncertainty certain classes of robust performance problems and systems subject to deterministic white noise disturbances this is achieved by combining concepts such as h infinity optimization linear matrix inequalities and integral quadratic constraints necessary and sufficient conditions for the general problem to have a solution are in terms of a computationally attractive finite dimensional linear matrix inequality

h sub infinity optimal control with state feedback a h sub infinity optimal control problem in which the measured outputs are the states of the plant is considered the main result shows that the infimum of the norm of the closed loop transfer function using linear static state feedback equals the infimum of the norm of the closed loop transfer function over all stabilizing dynamic even nonlinear time varying state feedback controllers

nonlinear riemann hilbert problem for bordered riemann surfaces let be a bordered riemann surface with genus g and m boundary components let z z be a smooth family of smooth jordan curves in inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 01i which all contain the point 0 in their interior then there exists a holomorphic function f z on smooth up to the boundary with at most 2 g m 1 zeros on such that f z z for every z

formulas on preview and delayed h infty control a generalized hinfin control problem which covers preview and delayed control strategies is discussed based on a state space approach by introducing a hamiltonian matrix which is associated with a delay free generalized plant the analytic solution to the corresponding operator riccati equation is newly established based on the result obtained here the hinfin control problem is solved and for typical control problems e g hinfin and linear quadratic lq control for multiple input delay systems hinfin preview control some interpretations are provided on the resulting control system

an algebraic riccati equation approach to h optimization abstract this paper considers the general so called four block h optimal control problem with the assumption that system states are available for feedback it is shown that infimization of the h norm of the closed loop transfer function over all linear constant i e nondynamic stabilizing state feedback laws can be completely characterized via an algebraic riccati equation it is further shown that the optimal norm is not improved by allowing feedback to be dynamic thus the general state feedback h optimal control problem can be solved by iteratively solving one are and the controller can be chosen to be static gain

robust self learning fuzzy controller design for a class of nonlinear mimo systems a new learning paradigm that can be applied for the design of a fuzzy logic based learning controller that is robust to external signals such as disturbances and set point changes is proposed in the paper it is well known that the self organizing fuzzy controller proposed by procyk and mamdani is sensitive to external signals such a phenomenon may be observed in other types of direct fuzzy logic based learning controllers that utilize an adaptation scheme in which the locational information of the current error state vector determines the degree of adaptation that should be made in the learning controller to resolve the problem of sensitivity to external signals it is proposed that learning and modification of the controller be made in consideration of the motional trend of the error state vector as well as its locational information this paradigm is adopted in the design of a new robust self learning fuzzy controller for a class of nonlinear mimo systems the well known techniques of sliding mode control and the fuzzy decision making method are utilized to implement the proposed learning scheme in the fuzzy learning controller via simulation study and experimental results the proposed learning controller is verified to be robust in the presence of external disturbances

fast computation of achievable feedback performance in mixed sensitivity h design the computational bottleneck of the h infty design has been recognized to be the e iteration a computationally demanding direct search of the minimum achievable h infty performance verma and jonckheere showed that the optimal h infty performance can be characterized as the spectral radius of the so called toeplitz plus hankel operator even before the appearance of the toeplitz plus hankel operator in the h infty setting the same operator had already been shown to play a crucial role in the spectral theory of the linear quadratic problem developed by jonckheere and silverman in this paper we exploit this common toeplitz plus hankel operator structure shared by the seemingly unrelated linear quadratic and h infty problems and we construct fast state space algorithms for evaluating the spectral radius of the toeplitz plus hankel operator the salient feature of the algorithm is that the spectral radius can be evaluated with an accuracy predicted by an identifiable error bound from the antistabilizing solution of the algebraic riccati equation of the linear quadratic problem associated with the h infty design

positive definite completions of partial hermitian matrices abstract the question of which partial hermitian matrices some entries specified some free may be completed to positive definite matrices is addressed it is shown that if the diagonal entries are specified and principal minors composed of specified entries are positive then if the undirected graph of the specified entries is chordal a positive definite completion necessarily exists furthermore if this graph is not chordal then examples exist without positive definite completions in case a positive definite completion exists there is a unique matrix in the class of all positive definite completions whose determinant is maximal and this matrix is the unique one whose inverse has zeros in those positions corresponding to unspecified entries in the original partial hermitian matrix additional observations regarding positive definite completions are made

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

linear control theory with an h 0e optimality criterion this expository paper sets out the principal results in bf h infty control theory in the context of continuous time linear systems the focus is on the mathematical theory rather than computational methods

convergent presentations and polygraphic resolutions of associative algebras several constructive homological methods based on noncommutative grobner bases are known to compute free resolutions of associative algebras in particular these methods relate the koszul property for an associative algebra to the existence of a quadratic grobner basis of its ideal of relations in this article using a higher dimensional rewriting theory approach we give several improvements of these methods we define polygraphs for associative algebras as higher dimensional linear rewriting systems that generalise the notion of noncommutative grobner bases and allow more possibilities of termination orders than those associated to monomial orders we introduce polygraphic resolutions of associative algebras giving a categorical description of higher dimensional syzygies for presentations of algebras we show how to compute polygraphic resolutions starting from a convergent presentation and how these resolutions can be linked with the koszul property

geometry of pure spinor formalism in superstring theory abstract projective embedding of an isotropic grassmannian ogr 5 10 into projective space of spinor representation s can be characterized with a help of matrices by equations i 0 a polynomial function of degree n with values in s defines a map to ogr 5 10 if its coefficients satisfy a 2 n 1 quadratic equations algebra generated by coefficients of such polynomials is a coordinate ring of the quantum isotropic grassmannian we show that this ring is based on a lattice its defining relations satisfy straightened law this enables us to compute the poincare series of the ring

computer assistance for discovering formulas in system engineering and operator theory abstract the objective of this paper is two fold first we present a methodology for using a combination of computer assistance and human intervention to discover highly algebraic theorems in operator matrix and linear systems engineering theory since the methodology allows limited human intervention it is slightly less rigid than an algorithm we call it a strategy the second objective is to illustrate the methodology by deriving four theorems the presentation of the methodology is carried out in three steps the first step is introducing an abstraction of the methodology which we call an idealized strategy this abstraction facilitates a high level discussion of the ideas involved idealized strategies cannot be implemented on a computer the second and third steps introduce approximations of these abstractions which we call prestrategy and strategy respectively a strategy is more general than a prestrategy and in fact every prestrategy is a strategy the above mentioned approximations are implemented on a computer we stress that since there is a computer implementation the reader can use these techniques to attack their own algebra problems thus the paper might be of both practical and theoretical interest to analysts engineers and algebraists now we give the idea of a prestrategy a prestrategy relies almost entirely on two commands which we call ncprocess1 and ncprocess2 these two commands are sufficiently powerful so that in many cases when one applies them repeatedly to a complicated collection of equations they transform the collection of equations into an equivalent but substantially simpler collection of equations a loose description of a prestrategy applied to a list of equations is emsp 1 declare which variables are known and which are unknown at the beginning of a prestrategy the order in which the equations are listed is not important since ncprocess1 and ncprocess2 will reorder them so that the simplest ones appear first 2 apply ncprocess1 to the equations the output is a set of equations usually some in fewer unknowns than before carefully partitioned based upon which unknowns they contain 3 the user must select important equations especially any which solve for an unknown sayx when an equation is declared to be important or a variable is switched from being an unknown to being a known then the way in which ncprocess1 and ncprocess2 reorder the equations is modified 4 switchxto being known rather than unknown go to 2 above or stop when this procedure stops it hopefully gives the canonical necessary conditions for the original equations to have a solution as a final step we run ncprocess2 which aggressively eliminates redundant equations and partitions the output equations in a way which facilitates proving that the necessary conditions are also sufficient many classical theorems in analysis can be viewed in terms of solving a collection of equations we have found that this procedure actually discovers the classic theorem in a modest collection of classic cases involving factorization of engineering systems and matrix completion problems one might regard the question of which classical theorems in analysis can be proven with a strategy as an analog of classical euclidean geometry where a major question was what can be constructed with a compass and ruler here the goal is to determine which theorems in systems and operator theory could be discovered by repeatedly applying ncprocess1 and ncprocess2 or their successors and the human selection of equations which are important the major practical challenge addressed here is finding operations which when implemented in software present the user with crucial algebraic information about his problem while not overwhelming him with too much redundant information this paper consists of two parts a description of strategies a high level description of the algorithms a description of the applications to operator matrix and linear system engineering theory and a description of how one would use a strategy to discover four different theorems are presented in the first part of the paper thus one who seeks a conventional viewpoint for this rather unconventional paper might think of this as providing a unified proof of four different theorems the theorems were selected for their diverse proofs and because they are widely known so that many readers should be familiar with at least one of them the ncprocess commands use noncommutative grobner basis algorithms which have emerged in the last decade together with algorithms for removing redundant equations and a method for assisting a mathematician in writing a noncommutative polynomial as a composition of polynomials the reader needs to knownothingabout grobner basis to understand the first part of this paper descriptions involving the theory of grobner basis appear in the second part of the paper

a duality based approach to the multiobjective h 2 h optimisation problem in this article a duality approach to multiobjective h 2 h problems is pursued in which real rational para hermitian multipliers and real valued ones are associated to h and as usual h 2 constraints respectively it is shown that the maximisation of a dual functional over all such multipliers yields the optimal value of the original multiobjective h 2 h problem to compute lower bounds on the latter and the corresponding approximate solutions to the original problem the maximisation of the dual functional over linearly parameterised finite dimensional classes of real rational multipliers is shown to be equivalent to semi definite linear programming problems once the optimal multipliers in such a class are obtained the corresponding approximate solutions can be computed from an unconstrained h 2 problem iterative modification of such classes is discussed to obtain increasing sequences of lower bounds on the optimal value of the original problem this is done on the basis of locally incr

multiobjective h sub 2 h sub spl infin control for a linear time invariant system with several disturbance inputs and controlled outputs we show how to minimize the nominal h sub 2 norm performance in one channel while keeping bounds on the h sub 2 norm or h sub spl infin norm performance implying robust stability in the other channels this multiobjective h sub 2 h sub spl infin problem in an infinite dimensional space is reduced to sequences of finite dimensional convex optimization problems we show how to compute the optimal value and how to numerically detect the existence of a rational optimal controller if it exists we reveal how the novel trick of optimizing the trace norm of the youla parameter over certain convex constraints allows one to design a nearly optimal controller whose youla parameter is of the same order as the optimal one

stable adaptive fuzzy control of nonlinear systems a direct adaptive fuzzy controller that does not require an accurate mathematical model of the system under control is capable of incorporating fuzzy if then control rules directly into the controllers and guarantees the global stability of the resulting closed loop system in the sense that all signals involved are uniformly bounded is developed the specific formula for the bounds is provided so that controller designers can determine the bounds based on their requirements the direct adaptive fuzzy controller is used to regulate an unstable system to the origin and to control the duffing chaotic system to track a trajectory the simulation results show that the controller worked without using any fuzzy control rules and that after fuzzy control rules were incorporated the adaptation speed became much faster it is shown explicitly how the supervisory control forces the state to remain within the constraint set and how the adaptive fuzzy controller learns to regain control

how much uncertainty can be dealt with by feedback feedback is used primarily for reducing the effects of the plant uncertainty on the performance of control systems and as such understanding the following questions is of fundamental importance how much uncertainty can be dealt with by feedback what are the limitations of feedback how does the feedback performance depend quantitatively on the system uncertainty how can the capability of feedback be enhanced if a priori information about the system structure is available as a starting point toward answering these questions a typical class of first order discrete time dynamical control systems with both unknown nonlinear structure and unknown disturbances is selected for our investigation and some concrete answers are obtained in the paper in particular we find that in the space of unknown nonlinear functions the generalized lipschitz norm is a suitable measure for characterizing the size of the structure uncertainty and that the maximum uncertainty that can be dealt with by the feedback mechanism is described by a ball with radius 3 2 spl radic 2 in this normed function space

convergent relaxations of polynomial optimization problems with noncommuting variables we consider optimization problems with polynomial inequality constraints in noncommuting variables these noncommuting variables are viewed as bounded operators on a hilbert space whose dimension is not fixed and the associated polynomial inequalities as semidefinite positivity constraints such problems arise naturally in quantum theory and quantum information science to solve them we introduce a hierarchy of semidefinite programming relaxations which generates a monotone sequence of lower bounds that converges to the optimal solution we also introduce a criterion to detect whether the global optimum is reached at a given relaxation step and show how to extract a global optimizer from the solution of the corresponding semidefinite programming problem

sampled data h optimal control of time varying systems the sampled data h optimal control problem is studied for time varying systems the problem consists of finding all sampled data controllers with a piece wise constant control signal for which the l2 induced norm from the continuous time disturbance to the continuous time output is less than a specified bound a game theoretic approach is used to obtain a complete solution to the problem in the finite horizon time varying case the solution involves a mixed discrete continuous riccati equation and a variable transformation which takes the problem to a standard discrete finite dimensional h filtering problem

disturbance attenuation and h optimization a design method based on the algebraic riccati equation this note presents a method for designing a state feedback control law to reduce the effect of disturbances on the output of a given linear system the problem under consideration involves attenuating the disturbances to a prespecified level the construction of the state feedback control law requires the solution of a certain algebraic riccati equation by applying the procedure with successively smaller values of the prespecifled disturbance level the result also provides an approach to the h infty optimization problem for the state feedback case

completing a block diagonal matrix with a partially prescribed inverse abstract a complete solution of the matrix completion problem a b 1 c d is obtained in terms of solutions of a riccati type matrix equation several special cases are described in detail

noetherian properties of skew polynomial rings with binomial relations in this work we study standard finitely presented associative algebras over a fixed field k a restricted class of skew polynomial rings with quadratic relations considered in an earlier work of m artin and w schelter will be studied we call them binomial skew polynomial algebras we establish necessary and sufficient conditions for such an algebra to be a noetherian domain

an introduction to commutative and noncommutative gro bner bases abstract in 1965 buchberger introduced the notion of grobner bases for a polynomial ideal and an algorithm buchberger algorithm for their computation since the end of the seventies grobner bases have been an essential tool in the development of computational techniques for the symbolic solution of polynomial systems of equations and in the development of effective methods in algebraic geometry and commutative algebra moreover grobner bases have been also generalized to free noncommutative algebra and to various noncommutative algebras of interest in differential algebra e g weyl algebras enveloping algebras of lie algebras the aim of this paper is to give an introduction as elementary as i was able to make it to both commutative and noncommutative algebras grobner bases are in a sense a finite model of an infinite linear gauss reduced basis of an ideal viewed as a vector space and buchberger algorithm is the corresponding generalization of the gaussian elimination algorithm moreover the paper contains a survey of some applications of buchberger theory to noncommutative algebras together with these results surveyed this paper contains some minor new points e g the useless pair criteria in the noncommutative case and the final result on the existence and computability of grobner bases for two sided ideals in any finitely presented algebra

reduction of permutation invariant polynomials let r be a commutative ring with 1 let r x1 xn i be the polynomial algebra in the n 4 noncommuting variables x1 xn over r modulo the set of commutator relations i x1 xn xi xi x1 xn 1 i n furthermore let g be an arbitrary group of permutations operating on the indeterminates x1 xn and let r x1 xn ig be the r algebra of g invariant polynomials in r x1 xn i the first part of this paper is about an algorithm which computes a representation for any f r x1 xn ig as a polynomial in multilinear g invariant polynomials i e the maximal variable degree of the generators of r x1 xn ig is at most 1 the algorithm works for any ring r and for any permutation group g in addition we present a bound for the number of necessary generators for the representation of all g invariant polynomials in r x1 xn ig with a total degree of at most d the second part contains a first but promising analysis of g invariant polynomials of solvable polynomial rings

adaptive fuzzy control with guaranteed convergence of optimal approximation error with no a priori knowledge of plant boundary functions a novel direct adaptive fuzzy controller afc for a class of single input single output siso uncertain affine nonlinear systems is developed in this paper based on the theory of fuzzy logic systems flss with variable universes of discourse uds sufficient conditions that guarantee that the optimal fuzzy approximation error fae is locally convergent are given by the use of the output tracking error and its derivatives as input variables and by the selection of suitable adjusting parameters a variable ud fls with an optimal fae local convergence is constructed and its parameter adaptive law is derived by virtue of the lyapunov stability theorem under the assumption that the optimal fae is bounded it is proved that the closed loop system is asymptotically stable in the sense that all variables are uniformly ultimately bounded and that the tracking errors converge to zero the proposed approach eliminates the influence of the fae on the tracking errors by means of the inherent mechanism of the variable ud fls thus it has the potential to achieve high control performance without additional compensation under only a few fuzzy rules simulation studies demonstrate the superiority of the proposed afc in terms of the settling time tracking accuracy smoothness of the control input and robustness against external disturbances and parameter variations

groups and lie algebras corresponding to the yang baxter equations for a positive integer n we introduce quadratic lie algebras tr n qtr n and finitely discrete groups tr n qtr n naturally associated with the classical and quantum yang baxter equation respectively we prove that the universal enveloping algebras of the lie algebras tr n qtr n are koszul and compute their hilbert series we also compute the cohomology rings for these lie algebras which by koszulity are the quadratic duals of the enveloping algebras finally we construct a basis of u tr n we construct cell complexes which are classifying spaces of the groups tr n and qtr n and show that the boundary maps in them are zero which allows us to compute the integral cohomology of these groups we show that the lie algebras tr n qtr n map onto the associated graded algebras of the malcev lie algebras of the groups tr n qtr n respectively in the case of tr n we use quantization theory of lie bialgebras to show that this map is actually an isomorphism at the same time we show that the groups tr n and qtr n are not formal for n 4

adaptive control of a class of nonlinear systems with fuzzy logic an adaptive tracking control architecture is proposed for a class of continuous time nonlinear dynamic systems for which an explicit linear parameterization of the uncertainty in the dynamics is either unknown or impossible the architecture employs fuzzy systems which are expressed as a series expansion of basis functions to adaptively compensate for the plant nonlinearities global asymptotic stability of the algorithm is established in the lyapunov sense with tracking errors converging to a neighborhood of zero simulation results for an unstable nonlinear plant are included to demonstrate that incorporating the linguistic fuzzy information from human experts results in superior tracking performance

optimal disturbance attenuation for discrete time switched and markovian jump linear systems an exact condition for uniform stabilization and disturbance attenuation for switched linear systems is given in the discrete time domain via the union of an increasing family of linear matrix inequality conditions associated with each markovian jump linear system is a switched linear system so we obtain a necessary and sufficient condition for almost sure uniform stabilization and disturbance attenuation for markovian jump linear systems as well the results lead to semidefinite programming based controller synthesis techniques from which optimal finite path dependent linear dynamic output feedback controllers arise naturally in particular under the notion of path by path optimal disturbance attenuation finite path dependent controllers can outperform the usual mode dependent ones

an inertia formula for hermitian matrices with sparse inverses abstract let a mn be a nonsingular hermitian matrix let g be a chordal graph on vertices 1 n and suppose that g a 1 g then the inertia of a may be expressed in a simple way in terms of the inertias of those principal submatrices of a corresponding to certain readily identifiable sets of vertices of g specifically we prove that the inertia i a satisfies the identity i a c i a s m i a in which c denotes the collection of maximal cliques of g s denotes the collection of minimal vertex separators of g and m is the multiplicity of a separator

elimination of variables in linear solvable polynomial algebras and holonomicity abstract let k be a field of characteristic 0 based on the gelfand kirillov dimension computation of modules over solvable polynomial k algebras where solvable polynomial algebras are in the sense of a kandri rody and v weispfenning 1990 j symbolic comput 9 1 26 we prove that the elimination lemma obtained from d zeilberger 1990 j comput appl math 32 321 368 by using holonomic modules over the weyl algebra a n k and used in the automatic proving of special function identities holds for a class of solvable polynomial k algebras without any holonomicity restriction this opens a way to the solution of the extension contraction problem stemming from the automatic proving of multivariate identities with respect to the finiteness in the sense of f chyzak and b salvy 1998 j symbolic comput 26 187 227 it also yields a holonomicity so that automatic proving of multivariate identities may be dealt with by manipulating polynomial function coefficients instead of rational functions

feedback and optimal sensitivity model reference transformations multiplicative seminorms and approximate inverses in this paper the problem of sensitivity reduction by feedback is formulated as an optimization problem and separated from the problem of stabilization stable feedback schemes obtainable from a given plant are parameterized salient properties of sensitivity reducing schemes are derived and it is shown that plant uncertainty reduces the ability of feedback to reduce sensitivity the theory is developed for input output systems in a general setting of banach algebras and then specialized to a class of multivariable time invariant systems characterized by n times n matrices of h infty frequency response functions either with or without zeros in the right half plane the approach is based on the use of a weighted seminorm on the algebra of operators to measure sensitivity and on the concept of an approximate inverse approximate invertibility of the plant is shown to be a necessary and sufficient condition for sensitivity reduction an indicator of approximate invertibility called a measure of singularity is introduced the measure of singularity of a linear time invariant plant is shown to be determined by the location of its right half plane zeros in the absence of plant uncertainty the sensitivity to output disturbances can be reduced to an optimal value approaching the singularity measure in particular if there are no right half plane zeros sensitivity can be made arbitrarily small the feedback schemes used in the optimization of sensitivity resemble the lead lag networks of classical control design some of their properties and methods of constructing them in special cases are presented

quasilinear equation on bordered riemann surfaces we consider the existence of solutions of a nonlinear riemann hilbert problem for a quasilinear open image in new window equation on a bordered riemann surface

beurling s boundary differential relations on multiply connected domains abstract beurling s boundary differential relations for holomorphic functions on a multiply connected domain d in c are considered let k 3 the existence result is proved for the boundary differential relations of the form f f d where is a positive c k function on c moreover the existence of holomorphic solutions is proved for f f d where is a c k 1 defining function for a family of jordan curves in c containing the point 0 in its interior and is a positive c k bounded function on d c

one sided noncommutative grobner bases with applications to computing green s relations abstract standard noncommutative grobner basis procedures are used for computing ideals of free noncommutative polynomial rings over fields this paper describes grobner basis procedures for one sided ideals in finitely presented noncommutative algebras over fields the polynomials defining a k algebra a as a quotient of a free k algebra are combined with the polynomials defining a one sided ideal i of a by using a tagging notation whose essential effect is to forbid left multiplication standard noncommutative grobner basis techniques can then be applied to the mixed set of polynomials thus calculating a i whilst working in a free structure avoiding the complication of computing in a the paper concludes by showing how the results can be applied to completable presentations of semigroups and so enable calculations of green s relations

test for leibniz n algebra structure abstract for a given basis of a vector space l over a field k and a multiplication table which is defined by a multilinear map l n l we present an algorithm and develop a computer program on mathematica in order to test if the given multiplication table corresponds to a lie n algebra or a non lie leibniz n algebra or neither the algorithm is based on the computation of a grobner basis of the ideal which is used in the construction of the universal enveloping algebra of the leibniz algebra provided by the daletskii takhtajan functor by means of the ncalgebra package which permits the construction of grobner bases in noncommutative associative algebras

inverses of 2 2 block matrices abstract in this paper the authors give explicit inverse formulae for 2 2 block matrices with three different partitions then these results are applied to obtain inverses of block triangular matrices and various structured matrices such as hamiltonian per hermitian and centro hermitian matrices

computer algebra tailored to matrix inequalities in control a major advance in linear systems theory over the last decade has been a formalism for converting systems problems to matrix inequalities in this tutorial paper we describe computer algebra algorithms methodology and implementation which allows users to convert many systems problems to linear matrix inequalities lmis we shall focus on computer algebra methodology which can assist the user in producing lmis for control design we provide a step by step computer derivation of lmi formulas for the design of linear time invariant dynamic controllers that achieve a prespecified performance measured by the h norm of a certain closed loop transfer function

fuzzy basis functions universal approximation and orthogonal least squares learning fuzzy systems are represented as series expansions of fuzzy basis functions which are algebraic superpositions of fuzzy membership functions using the stone weierstrass theorem it is proved that linear combinations of the fuzzy basis functions are capable of uniformly approximating any real continuous function on a compact set to arbitrary accuracy based on the fuzzy basis function representations an orthogonal least squares ols learning algorithm is developed for designing fuzzy systems based on given input output pairs then the ols algorithm is used to select significant fuzzy basis functions which are used to construct the final fuzzy system the fuzzy basis function expansion is used to approximate a controller for the nonlinear ball and beam system and the simulation results show that the control performance is improved by incorporating some common sense fuzzy control rules

derivation of the maximum entropy h controller and a state space formula for its entropy the problem of maximizing the entropy of a stabilized closed loop system is solved the solution exploits the parametrization of all closed loop systems that meet an h norm bound the central solution of this set is shown to maximize the entropy at infinity and for this case a particularly simple formula is derived for the maximum value of the entropy

state space formulae for all stabilizing controllers that satisfy and h norm bound and relations to risk sensitivity abstract given a linear system all stabilizing controllers such that a specified closed loop transfer function has h norm less than a given scalar are parametrized this characterization involves the solution to two algebraic riccati equations each with the same order as the system and further gives gives feasible controllers also with this order the relationship to the risk sensitive lqg stochastic control problem is established giving an equivalence between robust and stochastic control

determinantal formulae for matrix completions associated with chordal graphs abstract let a be a partial positive definite hermitian matrix and let g a be the undirected graph of the specified off diagonal entries of a when g a is a chordal graph we present an explicit determinantal formula for the unique determinant maximizing positive definite completion maximum entropy completion of a in order to do this we first give a characterization of a chordal graph in terms of the existence of a proper spanning tree t of the intersection graph of its set of maximal cliques the formula for the maximum determinant is then a quotient of products of specified principal minors of a in which each minor in the numerator corresponds to a maximal clique of g a while each minor in the denominator corresponds to the intersection of two maximal cliques associated with an edge of t we next show that the denominator terms can be identified graph theoretically with the minimal vertex separators of g a and that these terms are independent of the particular proper spanning tree t then an alternative formula is given for the maximum determinant as an inclusion exclusion like ratio of principal minors associated with the maximal cliques of g a finally we show that the entries of the determinant maximizing completion may be calculated by solving a sequence of simple one variable maximization problems

sensitivity minimization in an h norm parametrization of all suboptimal solutions a number of problems in the control of linear feedback systems can be reduced to the following we are given three stable rational matrix functions k of sizes p 1 x q 1 p 1 x q 2 and p 2 x q 1 respectively and seek a stable rational q 2 x p 2 matrix function s so as to minimize k s we assume that p 1 q 2 p 2 q 1 and that and have maximal rank q 2 and p 2 respectively on the j axis given a tolerance level sufficiently large we obtain a linear fractional map g f 0 11 g 0 12 0 13 0 21 g 0 22 0 23 1 such that f k s with s stable and f if and only if g is a stable q 2 x p 2 matrix function with g 1 the computation of 0 0 ij 1 i 2 1 j 3 reduces to solving a pair of symmetric wiener hopf factorization problems for the special case where i q2 0 t i p2 0 and k not necessarily stable to which the general case can be reduced we provide explicit state space formulae for 0 in terms of a state space realization of k and the

low order controller design for siso systems using coprime factors and lmi this paper develops a low order controller design method for linear continuous time invariant single input single output systems requiring only the solution of a convex optimization problem the technique integrates several well known results in control theory an important step is the use of coprime factors so that based on strictly positive real functions feedback stabilization using low order controllers becomes a zero placement problem which is convex from this result we develop algorithms to solve two optimal control problems

the inverse of any two by two nonsingular partitioned matrix and three matrix inverse completion problems a formula for the inverse of any nonsingular matrix partitioned into two by two blocks is derived through a decomposition of the matrix itself and generalized inverses of the submatrices in the matrix the formula is then applied to three matrix inverse completion problems to obtain their complete solutions

a polynomial approach to minimax frequency domain optimization of multivariable feedback systems the properties of a feedback system where the plant has rational transfer matrix h and the compensator has transfer matrix g can be characterized through the system functions s i hg and t g i hg 1 good disturbance attenuation robustness limited bandwidth and compensator roll off may be obtained by minimizing a criterion of the form z where z v s t w1 wls t w2 w2t v with respect to the compensator transfer matrix g here v w1 and w2 are suitable rational weighting matrices the solution of the problem can be reduced to a pair of matrix polynomial equations since the optimal solution is highly non unique special solutions with additional optimality properties are considered as well the paper includes a discussion of the numerical solution of the polynomial equations and of the question how to choose the weighting matrices to ensure required feedback system properties an example illustrates the results

wiener hopf design of the general two degree of freedom controllers and the connection to state space formulas h 2 design of the two degree of freedom controller configuration is treated for the generalized plant in both the frequency and state space domains a frequency domain wiener hopf formula is derived first under a set of compact assumptions that gives the entire class of controllers for which the feedback control system is stable and a quadratic cost functional is finite the h 2 design method suggested in this paper allows the reference inputs to include the persistent signals such as step or ramp functions and the class of persistent reference inputs that can be tracked with zero steady state error is characterized explicitly to circumvent the computational difficulties of the wiener hopf formulas the corresponding state space domain formulas are derived by rationalizing the polynomial fractional descriptions it is shown that the well known standard time domain assumptions for the h 2 design problem have the corresponding equivalent frequency domain conditions and this equivalence relationship is show

approximation theory of fuzzy systems spl minus siso case in this paper the approximation properties of mimo fuzzy systems generated by the product inference are discussed we first give an analysis of fuzzy basic functions fbf s and present several properties of fbf s based on these properties of fbf s we obtain several basic approximation properties of fuzzy systems 1 basic approximation property which reveals the basic approximation mechanism of fuzzy systems 2 uniform approximation bounds which give the uniform approximation bounds between the desired control or decision functions and fuzzy systems 3 uniform convergent property which shows that fuzzy systems with defined approximation accuracy can always be obtained by dividing the input space into finer fuzzy regions and 4 universal approximation property which shows that fuzzy systems are universal approximators and extends some previous results on this aspect the similarity between fuzzy systems and mathematical approximation is discussed and an idea to improve approximation accuracy is suggested based on uniform approximation bounds

two mode indirect adaptive control approach for the synchronization of uncertain chaotic systems by the use of a hierarchical interval type 2 fuzzy neural network two mode adaptive controllers have two phases of operation a learning phase and an operation phase this paper presents a two mode indirect adaptive control approach for the synchronization of chaotic systems using a hierarchical interval type 2 fuzzy neural network ht2fnn its contribution to the existing literature is the adaptation laws derived for the parameters of the membership functions based on lyapunov stability analysis since in hierarchical case each t2fnn has only two inputs the computing of the derivatives is much simpler than the case in classical interval type 2 fnn moreover a novel approach is presented for the compensation of the approximation error the tuning of the parameters of the membership functions mf and the use of an interval type 2 fnn ensures that the estimation error is very small so that it can be negligible furthermore the number of mf required is seen to be less than that needed with type 1 fuzzy sets the simulation results confirm the efficacy of the proposed scheme in the synchronization of a uncertain nonidentical chaotic systems

generalized ahlfors functions let be a bordered riemann surface with genus g and m boundary components let z z be a smooth family of smooth jordan curves in c which all contain the point 0 in their interior let p and let f be the family of all bounded holomorphic functions f on s such that f p 0 and f z z for almost every z then there exists a smooth up to the boundary holomorphic function f 0 f with at most 2g m 1 zeros on s so that f 0 z z for every z and such that f 0 p f p for every f f if in addition all the curves z z are strictly convex then f 0 is unique among all the functions from the family f

gaussian markov distributions over finite graphs on s interesse aux modeles de selection de covariance introduites par dempster 1972 et etudies par wermuth 1976

robust adaptive fuzzy control in the presence of external disturbance and approximation error robust direct and indirect adaptive fuzzy controllers are proposed for tracking control of a class of nonlinear siso systems the behavior of the adaptive law is investigated in the error state space when the perturbations are introduced in the adaptive fuzzy control system a new adaptation scheme that can eliminate the effect of perturbations on the parameter adaptation is proposed it is proved that all the signals in the closed loop system are bounded in the presence of external disturbances and approximation error of the fuzzy logic system the concept of persistent excitation is utilized to guarantee the convergence and the boundedness of adaptation parameters it has also been observed that the convergence of parameters and tracking error could be smoother and faster with the proposed methods compared with the conventional adaptive fuzzy controllers two simulations have been conducted to demonstrate the effectiveness of the proposed controllers

regularity and k0 group of quadric solvable polynomial algebras abstract concerning solvable polynomial algebras in the sense of kandri rody and weispfenning j symbolic comput 9 1990 1 26 it is shown how to recognize and construct quadric solvable polynomial algebras in an algorithmic way if a k a1 an is a quadric solvable polynomial algebra it is proved that gl dima n and k 0 a z if a is a tame quadric solvable polynomial algebra it is shown that a is completely constructable and auslander regular

state space solutions to standard h sub 2 and h sub infinity control problems simple state space formulas are derived for all controllers solving the following standard h sub infinity problem for a given number gamma 0 find all controllers such that the h sub infinity norm of the closed loop transfer function is strictly less than gamma it is known that a controller exists if and only if the unique stabilizing solutions to two algebraic riccati equations are positive definite and the spectral radius of their product is less than gamma sup 2 under these conditions a parameterization of all controllers solving the problem is given as a linear fractional transformation lft on a contractive stable free parameter the state dimension of the coefficient matrix for the lft constructed using the two riccati solutions equals that of the plant and has a separation structure reminiscent of classical lqg i e h sub 2 theory this paper is intended to be of tutorial value so a standard h sub 2 solution is developed in parallel

a computed torque controller for uncertain robotic manipulator systems fuzzy approach computed torque control ctc is an effective motion control strategy for robotic manipulator systems which can ensure globally asymptotic stability however ctc scheme requires precise dynamical models of robotic manipulators to handle this impossibility in this paper a new approach combing ctc and fuzzy control fc is developed for trajectory tracking problems of robotic manipulators with structured uncertainty and or unstructured uncertainty fuzzy part with a set of tunable parameters is employed to approximate lumped uncertainty due to parameters variations unmodeled dynamics and so on in robotic manipulators based on lyapunov stability theorem it is shown that the proposed controller can guarantee stability of closed loop systems and satisfactory tracking performances the proposed approach indicates that ctc method is also valid for controlling uncertain robotic manipulators as long as compensative controller is appropriately designed finally computer simulation results on a two link elbow planar robotic manipulator are presented to show tracking capability and effectiveness of the proposed scheme

adaptive fuzzy control compensation design for direct adaptive fuzzy control for direct adaptive fuzzy control of perturbed uncertain nonlinear systems proportional integral derivative control compensation usually has to be used to guarantee h infty tracking performance or l 2 gain property of the closed loop system one common problem is that the control compensation often introduces unwanted high gain at the control input this paper proves that classical direct adaptive fuzzy control with adaptive fuzzy control compensation not only is capable of solving the problem of high gain but also can guarantee l 2 gain property of the closed loop system a sliding surface based adaptive fuzzy control is designed for control compensation by using a class of triangular membership functions nearest to the origin and a simple adaptive law the adaptive fuzzy control term is converted to an equivalence control term which is used to facilitate stability analysis moreover the system output can track not only the constant period periodic signals but also the constant signal compared with previous direct adaptive fuzzy control approaches that can guarantee h infty tracking performance or l 2 gain property in addition to the l 2 gain property of common tracking error rather than modified tracking error the major advantages of our approach are that the assumptions on control gain function are relaxed and a less conservative control design on robustness is achieved simulation results are given to demonstrate the effectiveness of the proposed approach

the structure of connected graded hopf algebras abstract in this paper we establish a structure theorem for connected graded hopf algebras over a field of characteristic 0 by claiming the existence of a family of homogeneous generators and a total order on the index set that satisfy some desirable conditions the approach to the structure theorem is constructive based on the combinatorial properties of lyndon words and the standard bracketing on words as a surprising consequence of the structure theorem we show that connected graded hopf algebras of finite gelfand kirillov dimension over a field of characteristic 0 are all iterated hopf ore extensions of the base field in addition some keystone facts of connected hopf algebras over a field of characteristic 0 are observed as corollaries of the structure theorem without the assumptions of having finite gelfand kirillov dimension or affineness on hopf algebras or of that the base field is algebraically closed

a controller degree bound for h 0e control problems of the second kind this paper is a continuation of our work on mathcal h infty optimal control problems which may be embedded in the linear fractional configuration of fig 1 in two previous articles 19 20 a controller degree bound was established for problems in which both p 12 s and p 21 s are square problems of the first kind if the mcmillan degree of p s is n it was shown that there exist mathcal h infty optimal controllers with mcmillan degree no greater than n 1 here we switch our attention to problems of the second kind that is we allow p 12 s to have more rows than columns with p 21 s square or alternatively we allow p 21 s to have more columns than rows with p 12 s square our main result shows that the degree bound derived previously for problems of the first kind carries over to problems of the second kind without change in addition to the controller degree bound our analysis suggests a number of modifications which are easily made to cur

completing a matrix when certain entries of its inverse are specified abstract suppose a a 11 a 12 a 21 a 22 is an n n matrix where a 11 has order p q if a is nonsingular let a 1 have the transposed partitioning b 11 b 12 b 21 b 22 first we show that the nullities of a 11 and b 22 are equal the remainder of our paper deals with the following completion problem suppose m 1 m 2 n 1 n 2 are nonnegative integers such that m 1 m 2 n 1 n 2 n 0

topological rewriting systems applied to standard bases and syntactic algebras abstract we introduce topological rewriting systems as a generalisation of abstract rewriting systems where we replace the set of terms by a topological space abstract rewriting systems correspond to topological rewriting systems for the discrete topology we introduce the topological confluence property as an approximation of the confluence property using a representation of linear topological rewriting systems with continuous reduction operators we show that the topological confluence property is characterised by lattice operations using this characterisation we show that standard bases induce topologically confluent rewriting systems on formal power series finally we investigate duality for reduction operators that we relate to series representations and syntactic algebras in particular we use duality for proving that an algebra is syntactic or not

an analysis of the pole zero cancellations in h optimal control problems of the first kind the aim of this paper is to study the pole zero cancellations which occur in a class of h infty optimal control problems which may be embedded in the configuration of fig 1 h infty control problems are said to be of the first kind if both p 12 s and p 21 s are square but not necessarily of the same size it is primarily this class of problems which will concern us here a general bound on the mcmillan degree of all controllers which are stabilizing and lead to a closed loop which satisfies bf mathcal r s infty leqq rho rho need not be optimal in the l infty norm sense is derived as illustrated in fig 1 bf mathcal r s is the transfer function relating y 1 s to u 1 s if the mcmillan degree of p s in fig 1 is n we show that in the single loop siso case the corresponding unique h infty optimal controller never requires more than n 1 states in the multivariable case there is a continuum of optimal controllers whose mcmil

a simple method for deriving j spectral factors a simple method is presented for deriving the j spectral factor of a transfer function matrix explicitly in terms of the parameters of this matrix this method provides closed form expressions for the j spectral factor and its inverse which only require a solution of a single linear sylvester equation this method can be easily applied in the solution of the h sub infinity optimal regulation problem and it provides useful geometric insight into the structure of the optimal regulator return difference matrix

using noncommutative grobner bases in solving partially prescribed matrix inverse completion problems abstract we investigate the use of noncommutative grobner bases in solving partially prescribed matrix inverse completion problems the types of problems considered here are similar to those in linear algebra appl 223 224 1995 73 there the authors gave necessary and sufficient conditions for the solution of a 2 2 block matrix completion problem our approach is quite different from theirs and relies on symbolic computer algebra here we describe a general method by which all block matrix completion problems of this type may be analyzed if sufficient computational power is available we also demonstrate our method with an analysis of all 3 3 block matrix inverse completion problems with 11 blocks known and 7 unknown we discover that the solutions to all such problems are of a relatively simple form we then perform a more detailed analysis of a particular problem from the 31 824 3 3 block matrix completion problems with 11 blocks known and 7 unknown a solution to this problem of the form derived in the above mentioned reference is presented not only do we give a proof of our detailed result but we describe the strategy used in discovering our theorem and proof since it is somewhat unusual for these types of problems