robust stability of uncertain linear systems with saturating actuators we consider the robust stability of uncertain linear systems with saturating actuators and present two sufficient conditions to guarantee the stability of the closed loop system we assume that the bound of the initial state is known and that the uncertainty is unstructured the obtained results meet our intuition that the lower bound of initial state and the higher bound of saturation level make it easy to design a controller finally we show the usefulness by an example

quantum algorithmic integrability the metaphor of classical polygonal billiards we study the algorithmic complexity of motions in classical polygonal billiards which as the number of sides increases tend to curved billiards both regular and chaotic this study unveils the equivalence of this problem to the procedure of quantization the average complexity of symbolic trajectories in polygonal billiards features the same scaling relations with respect to the number of sides that govern quantum systems when a semiclassical parameter is varied two cases the polygonal approximations of the circle and of the stadium are examined in detail and are presented as paradigms of quantization of integrable and chaotic systems

asymptotic distribution of global errors in the numerical computations of dynamical systems abstract we propose an analysis of the effects introduced by finite accuracy and round off arithmetic on numerical computations of discrete dynamical systems our method which uses the statistical tool of the decay of fidelity computes the error by directly comparing the numerical orbit with the exact one or more precisely with another numerical orbit computed with a much higher accuracy furthermore as a model of the effects of round off arithmetic on the map we also consider a random perturbation of the exact orbit with an additive noise for which exact results can be obtained for some prototype maps we investigate the decay laws of fidelity and their relationship with the error probability distribution for regular and chaotic maps for both additive and numerical noise in particular for regular maps we find an exponential decay for additive noise and a power law decay for numerical noise for chaotic maps numerical noise is equivalent to additive noise and our method is suitable for identifying a threshold for the reliability of numerical results i e the number of iterations below which global errors can be ignored this threshold grows linearly with the number of bits used to represent real numbers

finite size lyapunov exponent review on applications in dynamical systems the growth of infinitesimal perturbations is well characterized by the lyapunov exponents in many situations of interest however important phenomena involve finite amplitude perturbations which are ruled by nonlinear dynamics out of tangent space and thus cannot be captured by the standard lyapunov exponents we review the application of the finite size lyapunov exponent fsle for the characterization of non infinitesimal perturbations in a variety of systems in particular we illustrate their usage in the context of predictability of systems with multiple spatio temporal scales of geophysical relevance in the characterization of nonlinear instabilities and in some aspects of transport in fluid flows we also discuss the application of the fsle to more general aspects such as chaos noise detection and coarse grained descriptions of signals this article is part of a special issue of journal of physics a mathematical and theoretical devoted to lyapunov analysis from dynamical systems theory to applications

the classification of normalizing groups abstract let x be a finite set such that x n let t n and s n denote the transformation monoid and the symmetric group on n points respectively given a t n s n we say that a group g s n is a normalizing if a g g g 1 a g g g where a g and g 1 a g g g denote the subsemigroups of t n generated by the sets a g and g 1 a g g g respectively if g is a normalizing for all a t n s n then we say that g is normalizing the goal of this paper is to classify the normalizing groups and hence answer a question of levi mcalister and mcfadden the paper ends with a number of problems for experts in groups semigroups and matrix theory

stability analysis for a class of nonlinear discrete time control systems subject to disturbances and to actuator saturation this paper addresses the stability characterisation problem for a class of nonlinear discrete time control systems subject to actuator saturation and energy bounded disturbance inputs the considered class of systems covers all nonlinear discrete time systems that can be modelled by rational difference equations based on quadratic and piecewise quadratic lyapunov functions conditions based on linear matrix inequalities are proposed to analyse the asymptotic stability internal stability and the l2 input to state stability external stability of the closed loop system the proposed conditions are then incorporated into convex optimisation problems to either maximise an estimate of the region of attraction or a bound on the admissible l2 disturbances and also to obtain an estimate of the system l2 gain for an admissible set of disturbances

the largest subsemilattices of the endomorphism monoid of an independence algebra abstract an algebra a is said to be an independence algebra if it is a matroid algebra and every map x a defined on a basis x of a can be extended to an endomorphism of a these algebras are particularly well behaved generalizations of vector spaces and hence they naturally appear in several branches of mathematics such as model theory group theory and semigroup theory it is well known that matroid algebras have a well defined notion of dimension let a be any independence algebra of finite dimension n with at least two elements denote by end a the monoid of endomorphisms of a we prove that a largest subsemilattice of end a has either 2 n 1 elements if the clone of a does not contain any constant operations or 2 n elements if the clone of a contains constant operations as corollaries we obtain formulas for the size of the largest subsemilattices of some variants of the monoid of linear operators of a finite dimensional vector space the monoid of full transformations on a finite set x the monoid of partial transformations on x the monoid of endomorphisms of a free g set with a finite set of free generators among others the paper ends with a relatively large number of problems that might attract attention of experts in linear algebra ring theory extremal combinatorics group theory semigroup theory universal algebraic geometry and universal algebra

technical communique lmi relaxations for robust stability of linear systems with saturating controls a method is proposed for studying robust stability of uncertain linear systems subject to actuator saturation it hinges upon successive lmi relaxations and therefore can readily be implemented using widely available numerical tools the technique is illustrated on two examples and is shown to significantly improve previously published results

primitive permutation groups and strongly factorizable transformation semigroups abstract let be a finite set and t be the full transformation monoid on the rank of a transformation t t is the natural number t given a t denote by a the semigroup generated by a let k be a fixed natural number such that 2 k in the first part of this paper we almost classify the permutation groups g on such that for all rank k transformations t t every element in s t g t can be written as a product eg where e 2 e s t and g g in the second part we prove among other results that if s t and g is the normalizer of s in the symmetric group on then the semigroup sg is regular if and only if s is regular recall that a semigroup s is regular if for all s s there exists s s such that s s s s the paper ends with a list of problems

augmented lyapunov function approach to l2 gain analysis for discrete time systems with saturation nonlinearities abstract this paper presents a method for l 2 gain analysis of discrete time systems with saturation nonlinearity by using an augmented lyapunov functional by using sector and slope properties of the saturation nonlinearity the l 2 gain performance criterion is obtained via lmi formulation the proposed lyapunov function have augmented quadratic form with states as well as the sector bounded nonlinear function numerical examples show that the proposed method is less conservative than the existing ones

anti windup based dynamic controller synthesis for nonlinear systems under input saturation this paper describes the design of dynamic controller and static anti windup compensator awc for lipschitz nonlinear systems under input saturation global and local awc based control schemes for stabilization of the nonlinear systems are proposed and necessary conditions for feasibility of the control approaches are investigated a one step approach for simultaneous design of h controller and awc by means of linear matrix inequalities lmis is presented herein which supports multi objective synthesis to attain stabilization or tracking robustness against disturbance and noise and penalization of large and high frequency control signals this multi objective synthesis can be accomplished by incorporating design weights as commonly used in the standard h control theory to design a performance oriented anti windup based control scheme lmis for the global control of the nonlinear systems subject to input saturation are derived by application of a quadratic lyapunov function the lipschitz condition the global sector condition l 2 gain reduction substantial matrix algebra and variable transformation in order to cope with unstable and oscillatory nonlinear systems lmi based local results are established using a local sector condition additional conditions are derived by incorporating properties of the saturation function to ensure well posedness of the controller two simulation examples are provided to show the effectiveness of the proposed control schemes for control of stable and chaotic nonlinear systems under input saturation

slow relaxation in weakly open rational polygons the interplay between the regular piecewise linear and irregular vertex angle boundary effects in nonintegrable rational polygonal billiards of m equal sides is discussed decay dynamics in polygons of perimeter p m and small opening delta is analyzed through the late time survival probability s m approximately equal t delta two distinct slow relaxation channels are established the primary universal channel exhibits relaxation of regular sliding orbits with delta 1 the secondary channel is given by delta 1 and becomes open when m p m delta it originates from vertex order disorder dual effects and is due to relaxation of chaoticlike excitations

simultaneous quadratic stabilization for a class of non linear systems with input saturation using dynamic surface control in this paper a new method to estimate the initial condition set which guarantees the quadratic stability for a class of non linear systems via dynamic surface control dsc in the presence of input saturation has been proposed the estimated set is enlarged to allow some degree of input saturation while achieving the simultaneous quadratic stability of the system in the domain a convex optimization problem to obtain ellipsoidal initial condition sets via a linear matrix inequality lmi approach as well as an illustrative example will be presented

properties making a chaotic system a good pseudo random number generator we discuss the properties making a deterministic algorithm suitable to generate a pseudo random sequence of numbers high value of kolmogorov sinai entropy high dimensionality of the parent dynamical system and very large period of the generated sequence we propose the multidimensional anosov symplectic cat map as a pseudo random number generator we show what chaotic features of this map are useful for generating pseudo random numbers and investigate numerically which of them survive in the discrete state version of the map testing and comparisons with other generators are performed

saturated linear quadratic regulation of uncertain linear systems stability region estimation and controller design this paper considers the problems of estimating the stability region domain of attraction and controller design for uncertain linear continuous time systems with input saturation when linear quadratic lq optimal controller is used by exploiting the structure of the lq controller and the property of saturation functions it is established that the estimation of stability region can be obtained by solving linear matrix inequality lmi problems moreover an iterative lmi ilmi algorithm is presented to design an lq controller such that the largest estimated stability region can be obtained two examples are given to compare our results with existing ones

groups that together with any transformation generate regular semigroups or idempotent generated semigroups let a be a non invertible transformation of a nite set and let g be a group of permutations on that same set thenhg aing is a subsemigroup consisting of all non invertible transformations in the semigroup generated by g and a likewise the conjugates a g g 1 ag of a by elements g2 g generate a semigroup denotedha g j g2 gi we classify the nite permutation groups g on a nite set x such that the semigroupshg ai hg aing andha g j g2 gi are regular for all transformations of x we also classify the permutation groups g on a nite set x such that the semigroupshg aing andha g j g2 gi are generated by their idempotents for all non invertible transformations of x

products of idempotent matrices over integral domains abstract we say that a ring r has the idempotent matrices property if every square singular matrix over r is a product of idempotent matrices it is known that every field and more generally every euclidean domain has the idempotent matrices property in this paper we show that not every integral domain has the idempotent matrices property and that if a projective free ring has the idempotent matrices property then it must be a bezout domain we also show that a principal ideal domain has the idempotent matrices property if and only if every fraction a b with b 0 has a finite continued fraction expansion new proofs are also provided for the results that every field and every euclidean domain have the idempotent matrices property

primitive groups graph endomorphisms and synchronization the third author has been partially supported by the fundacao para a ciencia e a tecnologia through the project cemat ciencias uid multi 04621 2013

commuting matrices equilibrium points for control systems with single saturated input in this paper commutative matrices of multiple input multiple output mimo linear systems are considered the existence of the feedback matrices of a commutative state matrix set in the mimo closed loops is reduced to the existence of an invariant subspace of a matrix a the existence of feedback matrices in systems in open loop is equivalent to the existence of the solution of matrix equations denoted by kronecker products by defining new equilibrium points the relationship between equilibrium points are discussed for a linear system with a single saturated input four criteria for equilibrium points are outlined for such linear systems finally four interesting examples including their corresponding simulink plots are shown to illustrate the above results

groups and actions in transformation semigroups let s be a transformation semigroup of degree n to each element s in s we associate a permutation group g r s acting on the image of s and we find a natural generating set for this group it turns out that the mathcal r class of s is a disjoint union of certain sets each having size equal to the size of g r s as a consequence we show that two mathcal r classes containing elements with equal images have the same size even if they do not belong to the same mathcal d class by a certain duality process we associate to s another permutation group g l s on the image of s and prove analogous results for the mathcal l class of s finally we prove that the schutzenberger group of the mathcal h class of s is isomorphic to the intersection of g r s and g l s the results of this paper can also be applied in new algorithms for investigating transformation semigroups which will be described in a forthcoming paper