noncommutative differential geometry and new models of gauge theory the noncommutative differential geometry of the algebra c v mn c of smooth mn c valued functions on a manifold v is investigated for n 2 the analog of maxwell s theory is constructed and interpreted as a field theory on v it describes a u n yang mills field minimally coupled to a set of fields with values in the adjoint representation that interact among themselves through a quartic polynomial potential the euclidean action which is positive vanishes on exactly two distinct gauge orbits which are interpreted as two vacua of the theory in one of the corresponding vacuum sectors the su n part of the yang mills field is massive for the case n 2 analogies with the standard model of electroweak theory are pointed out finally a brief description is provided of what happens if one starts from the analog of a general yang mills theory instead of maxwell s theory which is a particular case

bispectrum estimation a parametric approach higher order spectra contain information about random processes that is not contained in the ordinary power spectrum such as the degree of nonlinearity and deviations from normality estimation of the bispectrum which is a third order spectrum has been applied in various fields to obtain information regarding quadratic phase coupling among harmonic components and non gaussianness of processes existing methods of bispectrum estimation are patterned after the conventional methods of power spectrum estimation which are known to possess certain limitations the paper proposes a parametric approach to bispectrum estimation based on a non gaussian white noise driven autoregressive ar model the ar parameter estimates are obtained by solving the third order recursion equations which may be toeplitz in form but not symmetric it is shown that the method provides bispectral estimates that are far superior to the conventional estimates in terms of bispectral fidelity and that in the case of detecting phase coupling among sinusoids the method provides significantly better resolution

heat kernel regularization of quantum fields we discuss some consequences of the existence of a heat kernel regularization hkr for quantum fields we demonstrate that hkr applies in certain examples using methods which should be useful more generally

distributed control for identical dynamically coupled systems a decomposition approach we consider the problem of designing distributed controllers for a class of systems which can be obtained from the interconnection of a number of identical subsystems if the state space matrices of these systems satisfy a certain structural property then it is possible to derive a procedure for designing a distributed controller which has the same interconnection pattern as the plant this procedure is basically a multiobjective optimization under linear matrix inequality constraints with system norms as performance indices the explicit expressions for computing these controllers are given for both h infin or h 2 performance and both for static state feedback and dynamic output feedback in discrete time at the end of the paper two application examples illustrate the effectiveness of the approach

dynamical evolution in non commutative discrete phase space and the derivation of classical kinetic equations by considering a lattice model of extended phase space and using techniques of non commutative differential geometry we are led to a the concept of vector fields as generators of motion and transition probability distributions on the lattice b the emergence of the time direction on the basis of the encoding of probabilities in the lattice structure c the general prescription for the evolution of the observables in analogy with classical dynamics we show that in the limit of a continuous description these results lead to the time evolution of observables in terms of the adjoint of generalized fokker planck equations having 1 a diffusion coefficient given by the limit of the correlation matrix of the lattice coordinates with respect to the probability distribution associated with the generator of motion 2 a drift term given by the microscopic average of the dynamical equations in the present context these results are applied to one and two dimensional problems specifically we derive i the equations of diffusion smoluchowski and fokker planck in velocity space thus indicating the way random walk models are incorporated in the present context ii kramers equation by further assuming that motion is deterministic in coordinate space

extended lmi characterizations for stability and performance of linear systems over the past ten years extensive research has been devoted to extended lmi characterizations for stability and performance of linear systems these characterizations constitute a valuable tool for reducing conservatism in hard problems like multi objective control and robust stability and performance analysis the present paper proposes a general projection lemma based methodology for deriving such extended lmis and hereby provides a straightforward and unified proof for all known literature results as well as some currently missing extended lmis

multiobjective output feedback control via lmi optimization this paper presents an overview of a linear matrix inequality lmi approach to the multiobjective synthesis of linear output feedback controllers the design objectives can be a mix of h sub spl infin performance h sub 2 performance passivity asymptotic disturbance rejection time domain constraints and constraints on the closed loop pole location in addition these objectives can be specified on different channels of the closed loop system when all objectives are formulated in terms of a common lyapunov function controller design amounts to solving a system of linear matrix inequalities the validity of this approach is illustrated by a realistic design example

noncommutative differential geometry of matrix algebras the noncommutative differential geometry of the algebra mn c of complex n n matrices is investigated the role of the algebra of differential forms is played by the graded differential algebra c sl n c mn c mn c sl n c sl n c acting by inner derivations on mn c a canonical symplectic structure is exhibited for mn c for which the poisson bracket is to within a factor i the commutator also a canonical riemannian structure is described for mn c finally the analog of the maxwell potential is constructed and it is pointed out that there is a potential with a vanishing curvature that is not a pure gauge

differential calculi on the quantum group glp q 2 the author presents a systematical study of left covariant differential calculus on the quantum group glp q 2 a two parameter deformation of the general linear group in particular the author explicitly constructs the most general bicovariant calculus it depends on a new parameter s the corresponding lie algebra of left coinvariant vector fields is a two parameter deformation of the classical lie algebra p and q appear only in the combination r pq in the limit p q to 1 one obtains non standard bicovariant differential calculi on the classical lie group

differential calculus on compact matrix pseudogroups quantum groups the paper deals with non commutative differential geometry the general theory of differential calculus on quantum groups is developed bicovariant bimodules as objects analogous to tensor bundles over lie groups are studied tensor algebra and external algebra constructions are described it is shown that any bicovariant first order differential calculus admits a natural lifting to the external algebra so the external derivative of higher order differential forms is well defined and obeys the usual properties the proper form of the cartan maurer formula is found the vector space dual to the space of left invariant differential forms is endowed with a bilinear operation playing the role of the lie bracket commutator generalized antisymmetry relation and jacobi identity are proved

distributed control design for systems interconnected over an arbitrary graph we consider the problem of synthesizing a distributed dynamic output feedback controller achieving h sub spl infin performance for a system composed of different interconnected sub units when the topology of the underlying graph is arbitrary first using tools inspired by dissipativity theory we derive sufficient conditions in the form of finite dimensional linear matrix inequalities when the interconnections are assumed to be ideal these inequalities are coupled in a way that reflects the spatial structure of the problem and can be exploited to design distributed synthesis algorithms we then investigate the case of lossy interconnection links and derive similar results for systems whose interconnection relations can be captured by a class of integral quadratic constraints that includes constant delays

parameter estimation of cyclostationary am time series with application to missing observations time series with systematic misses occur often in practice and can be modeled as amplitude modulated arma processes with this as a motivating application modeling of cyclostationary amplitude modulated time series is addressed in the paper assuming that the modulating sequence is almost periodic parameter estimation algorithms are developed based on second and higher order cumulants of the resulting cyclostationary observations which may be corrupted by any additive stationary noise of unknown covariance if unknown the modulating sequence can be recovered even in the presence of additive perhaps nonstationary and colored gaussian or any symmetrically distributed noise if the arma process is nongaussian cyclic cumulants of order greater than three can identify non causal and non minimum phase models from partial noisy data simulation experiments corroborate the theoretical results

continuous time analysis eigenstructure assignment and h sub 2 synthesis with enhanced linear matrix inequalities lmi characterizations this note describes a new framework for the analysis and synthesis of control systems which constitutes a genuine continuous time extension of results that are only available in discrete time in contrast to earlier results the proposed methods involve a specific transformation on the lyapunov variables and a reciprocal variant of the projection lemma in addition to the classical linearizing transformations on the controller data for a wide range of problems including robust analysis and synthesis multichannel h sub 2 stateand output feedback syntheses the approach leads to potentially less conservative linear matrix inequality lmi characterizations this comes from the fact that the technical restriction of using a single lyapunov function is to some extent ruled out in this new approach moreover the approach offers new potentials for problems that cannot be handled using earlier techniques an important instance is the eigenstructure assignment problem blended with lyapunov type constraints which is given a simple and tractable formulation

penalized least squares estimation of volterra filters and higher order statistics volterra filters vfs and higher order statistics hos are important tools for nonlinear analysis processing and modeling despite their highly desirable properties the transfer of vfs and hos to real world signal processing problems has been hindered by the requirement of very large data records needed to obtain reliable estimates the identification of vfs and the estimation of hos both fall into the category of ill posed estimation problems we develop penalized least squares pls estimation methods for vfs and hos it is shown that pls is a very effective way to incorporate prior information of the problem at hand without directly constraining the estimation procedure hence pls produces much more reliable estimates the main contributions of this paper are the development of appropriate penalizing functionals and cross validation procedures for pls based vf identification and hos estimation

improved lmi representations for the analysis and the design of continuous time systems with polytopic type uncertainty simple modifications to the representation of the bounded real lemma brl in the linear continuous time invariant case are introduced these modifications reduce the overdesign that occurs in the analysis and the design of systems with polytopic type uncertainties a different lyapunov function accompanies each of the vertices of the uncertainty polytope thus eliminating the need for quadratic stability or stabilizability the advantages of these new representations are demonstrated by way of two examples

noncommutative differential calculus and lattice gauge theory the authors study consistent deformations of the classical differential calculus on algebras of functions and more generally commutative algebras such that differentials and functions satisfy nontrivial commutation relations for a class of such calculi it is shown that the deformation parameters correspond to the spacings of a lattice these differential calculi generate a lattice on a space continuum the whole setting of a lattice theory can then be deduced from the continuum theory via deformation of the standard differential calculus in this framework one just has to express the lagrangian for the continuum theory in terms of differential forms this expression then also makes sense for the deformed differential calculus there is a natural integral associated with the latter integration of the lagrangian over a space continuum then produces the correct lattice action for a large class of theories this is explicitly shown for the scalar field action and the action for su m gauge theory

all controllers for the general h control problem lmi existence conditions and state space formulas this paper presents all controllers for the general h control problem with no assumptions on the plant matrices necessary and sufficient conditions for the existence of an h controller of any order are given in terms of three linear matrix inequalities lmis our existence conditions are equivalent to scherer s results but with a more elementary derivation furthermore we provide the set of all h controllers explicitly parametrized in the state space using the positive definite solutions to the lmis even under standard assumptions full rank etc our controller parametrization has an advantage over the q parametrization the freedom q a real rational stable transfer matrix with the h norm bounded above by a specified number is replaced by a constant matrix l of fixed dimension with a norm bound and the solutions x y to the lmis the inequality formulation converts the existence conditions to a convex feasibility problem and also the free matrix l and the pair x y define a finite dimensional design space as opposed to the infinite dimensional space associated with the q parametrization

h sub spl infin design with pole placement constraints an lmi approach this paper addresses the design of state or output feedback h sub spl infin controllers that satisfy additional constraints on the closed loop pole location sufficient conditions for feasibility are derived for a general class of convex regions of the complex plane these conditions are expressed in terms of linear matrix inequalities lmis and the authors formulation is therefore numerically tractable via lmi optimization in the state feedback case mixed h sub 2 h sub spl infin synthesis with regional pole placement is also discussed finally the validity and applicability of this approach are illustrated by a benchmark example

deformations of super kms functionals we investigate the stability of the super kms property under deformations we show that a family of continuous deformations of the super derivation in the quantum algebra yields a continuous family of deformed super kms functionals these functionals define a family of cohomologous entire cocycles in this paper we investigate the super kms skms property of functionals co on a quantum algebra our interest in skms functionals was inspired by work of kastler and by conversations with alain connes ci k jlo2 the skms construction relies on the existence of a super derivation d acting on a dense subalgebra of a c algebra d the square of d is the infinitesimal generator of a continuous one parameter automorphism group c of the quantum algebra the usual kms property relates the cyclicity of a state co to the analytic continuation of a group et of automorphisms the skms property also involves invariance under the super derivation d whose square generates the automorphism group er it is known that an skms functional on a quantum algebra defines an entire cyclic cocycle z this is just the chern character which jaffe lesniewski and osterwalder defined in the context ofsupertrace functionals on a quantum algebra jlo 1 the skms property ensures that the functional co and the cocycle z which is derived from it are invariant under this group action in this paper we study the stability of this structure under perturbations of d we study only bounded perturbations which arise from the graded super commutator with an odd element q of the algebra d we show that such perturbations dq of d can be used to define a deformation coq of co which satisfies the skms property furthermore the corresponding family of cocycles z are cohomologous of course more singular unbounded perturbations can lead to

extended h 2 and h norm characterizations and controller parametrizations for discrete time systems this paper presents new synthesis procedures for discrete time linear systems it is based on a recently developed stability condition which contains as particular cases both the celebrated lyapunov theorem for precisely known systems and the quadratic stability condition for systems with uncertain parameters these new synthesis conditions have some nice properties a they can be expressed in terms of lmi linear matrix inequalities and b the optimization variables associated with the controller parameters are independent of the symmetric matrix that defines a quadratic lyapunov function used to test stability this second feature is important for several reasons first structural constraints as those appearing in the decentralized and static output feedback control design can be addressed less conservatively second parameter dependent lyapunov function can be considered with a very positive impact on the design of robust h 2 and h x control problems third the design of controller with mixed

quantum mechanics as noncommutative symplectic geometry the authors construct consistent differential calculi on the algebra generated by operators satisfying the canonical commutation relation this leads to a mathematical framework in which quantum mechanics can be understood as noncommutative symplectic geometry they point out the possibility to describe interactions geometrically as deformations of the differential calculus

decomposition based distributed control for continuous time multi agent systems in this technical note we consider the problem of designing distributed controller for a continuous time system composed of a number of identical dynamically coupled subsystems the underlying mathematical derivation is based on kronecker product and a special similarity transformation constructed from interconnection pattern matrix that decomposes the system into modal subsystems this along with the result of extended linear matrix inequality lmi formulation for continuous time systems makes it possible to derive explicit expressions for computing the parameters of distributed controllers for both static state feedback and dynamic output feedback cases the main contribution of the technical note is the solution of distributed control problem for continuous time systems under h stability and h2 stability performances by solving a set of lmis with noncommon lyapunov variables the effectiveness of this method is demonstrated by means of a satellite formation example

the bispectrum of complex signals definitions and properties the symmetry properties and the relationship between all forms of third order cumulants of complex signals are investigated it is shown that all cumulants for different position of the complex conjugate are related by simple transformations autoregressive modeling of complex valued signals using third order cumulants is also investigated it is shown that modeling of complex valued signals requires a different approach from modeling of real valued signals

a new robust d stability condition for real convex polytopic uncertainty abstract the problem of robust d stability analysis with respect to real convex polytopic uncertainties is tackled a new lmi based sufficient condition for the existence of parameter dependent lyapunov functions is proposed this condition generalises previously published conditions numerical comparisons with quadratic stability results as well as previous results based on parameter dependent lyapunov functions illustrate the relevance of this new condition finally this result appears to be promising for robust multi objective performance analysis and control synthesis purposes

differential calculi on commutative algebras a differential calculus on an associative algebra a is an algebraic analogue of the calculus of differential forms on a smooth manifold it supplies a with a structure on which dynamics and field theory can to some extent be formulated in very much the same way we are used to in the geometrical arena underlying classical physical theories and models in previous work certain differential calculi on a commutative algebra exhibited relations with lattice structures stochastics and parametrized quantum theories this motivated the present systematic investigation of differential calculi on commutative and associative algebras various results about their structure are obtained in particular it is shown that there is a correspondence between first order differential calculi on such an algebra and commutative and associative products in the space of i forms an example of such a product is provided by the ito calculus of stochastic differentials for the case where the algebra a is freely generated by coordinates xi i 1 n we study calculi for which the differentials dxi constitute a basis of the space of 1 forms as a left a module these may be regarded as deformations of the ordinary differential calculus on rn for n or 3 a classification of all orbits under the general linear group of such calculi with constant structure functions is presented we analyse whether these calculi are reducible i e a skew tensor product of lower dimensional calculi or whether they are the extension as defined in this paper of a one dimension lower calculus furthermore generalizations to arbitrary n are obtained for all these calculi

a new discrete time robust stability condition a new robust stability condition for uncertain discrete time systems with convex polytopic uncertainty is given it enables to check stability using parameter dependent lyapunov functions which are derived from lmi conditions it is shown that this new condition provides better results than the classical quadratic stability besides the use of a parameter dependent lyapunov function this condition exhibits a kind of decoupling between the lyapunov and the system matrices which may be explored for control synthesis purposes a numerical example illustrates the results

selective hbox scr h _2 and hbox scr h _ infty stabilization of takagi sugeno fuzzy systems this paper presents new results concerning the stability analysis and design of state feedback controllers for continuous time takagi sugeno t s fuzzy systems via fuzzy lyapunov functions the membership functions of the t s fuzzy systems are modeled in a space that is defined by the cartesian product of simplexes called a multisimplex if the time derivatives of the membership functions are bounded the bounds are used to construct a polytope that models the space of the time derivatives of the membership functions linear matrix inequality lmi relaxations that are based on polynomial matrices are provided for stability analysis and controller design extensions for the design of control laws that minimize upper bounds to h2 and h norms are also given the main novelty of this method is that it allows one to synthesize control gains which depends only on some premise variables that are selected by the designer numerical experiments illustrate the flexibility and advantages of the proposed method

distributed control design for spatially interconnected systems this paper deals with analysis synthesis and implementation of distributed controllers designed for spatially interconnected systems we develop a state space framework for posing problems of this type and focus on systems whose model is spatially discrete in this paper analysis and synthesis results are developed for this class of systems using the l sub 2 induced norm as the performance criterion the results are stated in terms of linear matrix inequalities and are thus readily amenable to computation a special implementation of the resulting controllers is presented which is particularly attractive for distributed operation of the controller several examples are provided to further illustrate the application of the results

low rank estimation of higher order statistics low rank estimators for higher order statistics are considered in this paper the bias variance tradeoff is analyzed for low rank estimators of higher order statistics using a tensor product formulation for the moments and cumulants in general the low rank estimators have a larger bias and smaller variance than the corresponding full rank estimator and the mean squared error can be significantly smaller this makes the low rank estimators extremely useful for signal processing algorithms based on sample estimates of the higher order statistics the low rank estimators also offer considerable reductions in the computational complexity of such algorithms the design of subspaces to optimize the tradeoffs between bias variance and computation is discussed and a noisy input noisy output system identification problem is used to illustrate the results

technical communique new dilated lmi characterizations for continuous time multiobjective controller synthesis this paper provides new dilated linear matrix inequality lmi characterizations for continuous time controller synthesis the dilated lmis enable us to parametrize controllers without involving the lyapunov variables in the parametrizations taking advantage of this feature we can readily design multiobjective controllers with non common lyapunov variables whereas we are forced to employ a common one in the well known lyapunov shaping paradigm in particular it is shown that the proposed dilated lmi based approach to h 2 d stability synthesis encompasses the corresponding lyapunov shaping paradigm as a special case in this sense the results of this paper can be viewed as partial counterparts of those obtained in the discrete time setting

the renormalization flow spaces of two dimensional field theories and connes geometry we formulate a local renormalization flow using connes non commutative geometry this formulation allows a geometric description of the renormalization flow and an intrinsic characterization of the operator product expansion we define spaces for string theory in terms of a ring of correlation functions and a renormalization flow on this ring which are the analogues of the category of riemannian manifolds with metric for general relativity the beta function is related to a differential form of relative entropy between two renormalization flow trajectories

consensus of multiple nonholonomic chained form systems abstract consensus problems of multiple nonholonomic systems are considered in this paper this problem is simplified into consensus problems of two subsystems based on the cascaded structure of nonholonomic chained form systems continuous and hybrid distributed controllers have been constructed for these two subsystems respectively based on the theory of cascaded systems consensus of multiple nonholonomic chained form systems can be realized using the methodology proposed in this paper no matter whether the group reference signal is persistently exciting or not different to previous assumptions on group reference such as persistent excitation or converging to nonzero constant the condition on the group reference signal have been further relaxed in this paper simulation results using matlab have illustrated the effectiveness of the results presented in this paper

the commutative limit of a matrix geometry the noncommutative geometry of complex n n matrices is considered as an approximation to the geometry of the two sphere s2 and the differential forms defined over the matrices are mapped into the ordinary differential forms on su2 considered as a fiber bundle over s2 it is found that the extra gauge invariant vacuum studied in previous publications has as image a constant multiple of the connection on su2 that defines the fibration the dirac monopole with unit magnetic charge