controlled invariance for discrete time nonlinear systems with an application to the disturbance decoupling problems invariant distributions are defined for discrete time nonlinear control systems and necessary and sufficient conditions are given for their controlled invariance this extends to discrete time systems the basic tool which has been so important in solving the various synthesis problems for continuous time systems to indicate their utility in the discrete time setting they are used to locally solve the disturbance decoupling problem

reflection group presentations arising from cluster algebras we give a presentation of a finite crystallographic reflection group in terms of an arbitrary seed in the corresponding cluster algebra of finite type and interpret the presentation in terms of companion bases in the associated root system

nonlinear decoupling via feedback a differential geometric approach the paper deals with the nonlinear decoupling and noninteracting control problems a complete solution to those problems is made possible via a suitable nonlinear generalization of several powerful geometric concepts already introduced in studying linear multivariable control systems the paper also includes algorithms concerned with the actual construction of the appropriate control laws

invariant subspaces for lpv systems and their applications the aim of this paper is to extend the notion of invariant subspaces known in the geometric control theory of the linear time invariant systems to the linear parameter varying lpv systems by introducing the concept of parameter varying invariant subspaces for lpv systems affine in their parameters algorithms are given to compute many parameter varying subspaces relevant in the solution of state feedback and observer design problems

cellular algebras and inverse semigroups abstract we show that under certain compatability assumptions the semigroup algebra of an inverse semigroup is cellular if the group algebras of its maximal subgroups are cellular motivating examples include the symmetric inverse semigroup also known as the rook monoid and the dual symmetric inverse semigroup

hecke algebras of finite type are cellular let mathcal h be the one parameter hecke algebra associated to a finite weyl group w defined over a ground ring in which bad primes for w are invertible using deep properties of the kazhdan lusztig basis of mathcal h and lusztig s a function we show that mathcal h has a natural cellular structure in the sense of graham and lehrer thus we obtain a general theory of specht modules for hecke algebras of finite type previously a general cellular structure was only known to exist in types an and bn

congruence lattices of finite diagram monoids abstract we give a complete description of the congruence lattices of the following finite diagram monoids the partition monoid the planar partition monoid the brauer monoid the jones monoid also known as the temperley lieb monoid the motzkin monoid and the partial brauer monoid all the congruences under discussion arise as special instances of a new construction involving an ideal i a retraction i m onto the minimal ideal a congruence on m and a normal subgroup of a maximal subgroup outside i

presentations of general products of monoids abstract conditions are found under which a general product of two finitely presented monoids is itself finitely presented presentations in terms of the presentations of the factors are given subject to these conditions

cellularity of diagram algebras as twisted semigroup algebras abstract the temperley lieb and brauer algebras and their cyclotomic analogues as well as the partition algebra are all examples of twisted semigroup algebras we prove a general theorem about the cellularity of twisted semigroup algebras of regular semigroups which allows us to easily reproduce the cellularity of these algebras this theorem generalizes a result of east about the cellularity of semigroup algebras of inverse semigroups

sensor fault reconstruction and observability for unknown inputs with an application to wastewater treatment plants in this article we propose a general methodology for identifying and reconstructing sensor faults on dynamical processes this methodology is issued from the general identification theory developed in the previous papers busvelle e and gauthier j p 2003 on determining unknown functions in differential systems with an application to biological reactor esaim control optimisation and calculus of variations 9 509 553 busvelle e and gauthier j p 2004 new results on identifiability of nonlinear systems in 2nd symposium on systems structure and control oaxaca mexico busvelle e and gauthier j p 2005 observation and identification tools for non linear systems application to a fluid catalytic cracker international journal of control 78 208 234 in fact this identification theory also provides a general framework for the problem of observability with unknown inputs indeed many problems of fault detection can be formulated as such observability problems the e

the inverse braid monoid abstract we introduce an inverse monoid which plays a similar role with respect to the symmetric inverse semigroup that the braid group plays with respect to the symmetric group

on groups generated by involutions of a semigroup abstract an involution on a semigroup s or any algebra with an underlying associative binary operation is a function s s that satisfies x y y x and x x for all x y s the set i s of all such involutions on s generates a subgroup c s i s of the symmetric group sym s on the set s we investigate the groups c s for certain classes of semigroups s and also consider the question of which groups are isomorphic to c s for a suitable semigroup s

unknown input observer for state affine systems a necessary and sufficient condition the problem of designing an unknown input observer for linear systems and its application to fault detection is widely studied in the literature for nonlinear systems only subclasses of nonlinear systems and sufficient conditions have been stated in this paper an unknown input observer design for state affine systems is considered based on the geometric approach a necessary and sufficient condition is given for the existence of an unknown input observer

a geometric approach to the synthesis of failure detection filters a geometric formulation of beard s failure detection filter problem is stated using the concepts of c a invariant and unobservability subspaces the notions of output separable and mutually detectable families of subspaces introduced by beard are also clarified it is shown that mutual detectability is a necessary and sufficient condition for the existence of a detection filter with arbitrarily assignable spectrum moreover it is shown that the failure detection falter problem has a computationally simple solution when the failure events satisfy some mild restrictions finally the complete duality between a generalization of beard s detection filter problem and the restricted control decoupling problem is illustrated

a geometric approach to the theory of 2 d systems the authors develop a geometric approach to the theory of 2 d two dimensional systems defining in a suitable way the notion of a sub 1 2 b sub 1 2 invariant subspaces and of controlled invariant subspaces such subspaces are shown to have good computational and feedback properties which make them useful in application in particular sufficient conditions and constructive procedures are obtained for the solutions of disturbance decoupling and model matching problems moreover it is shown that certain structural properties of a 2 d system can be described by means of a set of indexes defined in geometric terms and that such structural indexes can be used to reformulate the sufficient condition for model matching

observability of autonomous discrete time non linear systems a geometric approach in this paper the observability of autonomous discrete time systems is studied from a purely differential geometric point of view as with continuous time systems this approach leads to a local canonical form for an observable system a proposal for the generalization of an invariant subspace is made

on the observability codistributions of a nonlinear system abstract the purpose of this is to introduce the notion of observability codistribution for a nonlinear system which extends the dual of the notion of unobservability subspace then we study its properties which bear important similarities with a number of properties which render the notion of unobservability subspace powerful in the solution of certain design problems

partial symmetry reflection monoids and coxeter groups abstract this is the first of a series of papers in which we initiate and develop the theory of reflection monoids motivated by the theory of reflection groups the main results identify a number of important inverse semigroups as reflection monoids introduce new examples and determine their orders

on decomposition of nonlinear control systems in this paper the problem is studied when the system x i k 1 u t g t x x 0 x 0 defined on an n dimensional analytic manifold can be decomposed into either independent subsystems or a cascade of subsystems necessary and sufficient conditions for the existence of such local decompositions are given in terms of lie algebras generated by the system

idempotents and one sided units lattice invariants and a semigroup of functors on the category of monoids abstract for a monoid m we denote by g m the group of units e m the submonoid generated by the idempotents and g l m and g r m the submonoids consisting of all left or right units writing m for the monoidal category of monoids g e g l and g r are all monoidal functors m m there are other natural functors associated to submonoids generated by combinations of idempotents and one or two sided units the above functors generate a monoid with composition as its operation we show that this monoid has size 15 and describe its algebraic structure we also show how to associate certain lattice invariants to a monoid and classify the lattices that arise in this fashion a number of examples are discussed throughout some of which are essential for the proofs of the main theoretical results

detection filter design for lpv systems a geometric approach this paper investigates fault detection and isolation of linear parameter varying lpv systems by using parameter varying c a invariant subspace and parameter varying unobservability subspaces the so called detection filter approach formulated as the fundamental problem of residual generation fprg for linear time invariant lti systems is extended for a class of lpv systems the question of stability is addressed in the terms of lyapunov quadratic stability by using linear matrix inequalities the results are applied to the model of a generic small commercial aircraft

partial mirror symmetry lattice presentations and algebraic monoids this is the second in a series of papers that develops the theory of reflection monoids motivated by the theory of reflection groups reflection monoids were first introduced in everitt and fountain adv math 223 2010 1782 1814 in this paper we study their presentations as abstract monoids along the way we also find general presentations for certain join semilattices as monoids under join which we interpret for two special classes of examples the face lattices of convex polytopes and the geometric lattices particularly the intersection lattices of hyperplane arrangements another spin off is a general presentation for the renner monoid of an algebraic monoid which we illustrate in the special case of the classical algebraic monoids

the semigroup generated by the idempotents of a partition monoid we study the idempotent generated subsemigroup of the partition monoid in the finite case this subsemigroup consists of the identity and all the singular partitions in the infinite case the subsemigroup is described in terms of certain parameters that measure how far a partition is from being a permutation as one of several corollaries we deduce howie s description from 1966 of the semigroup generated by the idempotents of a full transformation semigroup

conditioned invariant and locally conditioned invariant distributions abstract we define the concepts of conditioned invariant and locally conditioned invariant distributions for nonlinear system and show how they can be used to solve the problem of tracking an output signal of a nonlinear system in spite of unknown disturbances

four notions of conjugacy for abstract semigroups there have been several attempts to extend the notion of conjugacy from groups to semigroups one notion which we will denote byp was originally introduced for free semigroups and popularized by lallement s book in a general semigroup p is reflexive and symmetric but not transitive however one may consider the transitive closure ofp which is an equivalence relation in any semigroup another notion of conjugacy which we will denote byo was introduced by otto for monoids presented by finite thue systems the relationo is an equivalence relation in any semigroup but it reduces to the universal relation if a semigroup has a zero since there is a precise sense in which almost all finite semigroups have a zero it follows thato is not useful for almost all finite semigroups three authors of the present paper introduced another notion of conjugacy denoted byc which is an equivalence relation in any semigroup in addition if a semigroup s does not have a zero thenc o in s butc does not reduce to the universal relation when a semigroup has a zero in order to decide which notion is the most satisfactory in the sense that it allows the extension of the classic conjugacy results in groups and hence leads to strong and elegant results with interconnections with other parts of mathematics it is necessary to carry out a deep study comparing and separating these three notions of conjugacy this is the aim of our paper first we study the decidability and independence of the conjugacy problems for certain classes of finitely presented monoids we show that there exists a monoid m defined by a finite complete rewriting system such that the c conjugacy problem for m is undecidable and that for finitely presented monoids the c conjugacy problem and the word problem are independent as are the c conjugacy and p conjugacy problems as an especially interesting test case we describec andp in the class of polycyclic monoids and show that in this class the p conjugacy is almost transitive and thatc is strictly included inp we also characterizec in the symmetric inverse monoid i x for a countable set x and comparec andp in that monoid we then generalize an outstanding theorem proved by kudryavtseva showing that relationp is transitive in a variety of semigroups that contains all completely regular semigroups and their variants althougho andc coincide for semigroups without zero they differ in the most extreme way in some semigroups with zero we present an infinite family of semigroups with zero divisors in whicho is universal whilec is the identity we prove that commutative semigroups are precisely those where p is the identity and commutative and cancellative semigroups are those whereo is the identity we also show that a semigroup in whichp is universal is necessarily simple moreover if a semigroup s has an idempotent we conclude that s is a rectangular band if and only ifp is the universal relation the paper ends with a fairly large section of open problems

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

controlled invariance for nonlinear systems necessary and sufficient conditions are derived for a b invariance called here controlled invariance for nonlinear systems dot x f x u the obtained results generalize and elucidate already known results about systems dot x a x sum min i 1 max m u i b i x a new and direct differential geometric interpretation of the concept of controlled invariance and the derived conditions is given

representations of the q rook monoid the q rook monoid in q is a semisimple algebra over c q that specializes when q 1 to c rn where rn is the monoid of n n matrices with entries from 0 1 and at most one nonzero entry in each row and column when q is specialized to a prime power in q is isomorphic to the iwahori algebra hc m b where m mn fq is the monoid of n n matrices with entries from a finite field having q elements and b m is the borel subgroup of invertible upper triangular matrices in this paper we i give a new presentation for in q on generators and relations and determine a set of standard words which form a basis ii explicitly construct a complete set of seminormal irreducible representations of in q and iii show that in q is the centralizer of the quantum general linear group uqgl r acting on the tensor product w v n where v is the fundamental uqgl r module and w is the trivial uqgl r module

controlled invariance by static output feedback for nonlinear systems the paper deals with the notion of static output feedback for nonlinear systems necessary and sufficient conditions are derived for c a b invariance here called measured controlled invariance for nonlinear control systems

quiver mutations and boolean reflection monoids abstract in 2010 everitt and fountain introduced the concept of reflection monoids the boolean reflection monoids form a family of reflection monoids symmetric inverse semigroups are boolean reflection monoids of type a in this paper we give a family of presentations of boolean reflection monoids and show how these presentations are compatible with quiver mutations of orientations of dynkin diagrams with frozen vertices our results recover the presentations of boolean reflection monoids given by everitt and fountain and the presentations of symmetric inverse semigroups given by popova respectively surprisingly inner by diagram automorphisms of irreducible weyl groups and boolean reflection monoids can be constructed by sequences of mutations preserving the same underlying diagrams besides we show that semigroup algebras of boolean reflection monoids are cellular algebras

characterization of allowable perturbations for robust decoupling of affine non linear systems for affine non linear systems we study the effects of uncertainties in systems vector fields on decoupled linear input output behaviour a characterization of an allowable class of perturbations which preserve the decoupling properties of the nominal system is given an immediate consequence of this characterization indicates that decoupling position control of rigid robotic manipulators is not robust under any perturbations in robot dynamics the result is applied to simple hamiltonian control systems to characterize the allowable perturbation of total energy function

generalized output regulation problem for a class of nonlinear systems with nonautonomous exosystem the problem of the generalized output regulation is considered here in contrast to the classical output regulation problem where the references to be tracked and disturbances to be rejected are generated by an autonomous exogeneous system for the generalized output regulation an exogenous system is driven by an unknown external reference signal in such a way the class of reference or disturbance signals is expanded presented result solves the full measurement feedback case and is based on the well known notion of the regulator equation combined with the classical concept of the invariant distribution illustrative example including matlab simulink simulations is presented to demonstrate the suggested approach

generators and relations for partition monoids and algebras abstract we investigate the manner in which the partition monoid p n and algebra p n may be presented by generators and relations making use of structural properties of p n as well as presentations for several key submonoids we obtain a number of presentations for p n including that given without a complete proof by halverson and ram in 2005 we then conclude by showing how each of these presentations gives rise to an algebra presentation for p n

a graph theoretic approach to fault detection and isolation for structured bilinear systems in this paper using a graph theoretic approach we address some issues related to the fault detection and isolation for structured bilinear systems considering a structured bilinear system submitted to faults and disturbances we give necessary and sufficient conditions to the solvability of the so called bilinear fundamental problem of residual generation we also treat the cases where the system is submitted to multiple failures occurring simultaneously or only one at a time one of the main advantages of the proposed analysis tool is that all the given conditions are easy to check because they deal with finding paths in a digraph this makes our approach well suited to studying large scale systems

on solvability of several affine system problems we examine seven problems and properties of single input single output affine control systems and classify the properties topologically as being structural generic or residual these properties are accessibility observability invertibility preservation of a given distribution and partial linearizability the mathematics involved relies upon the thorn transversality theorem

a geometric approach to nonlinear fault detection and isolation we present a differential geometric approach to the problem of fault detection and isolation for nonlinear systems a necessary condition for the problem to be solvable is derived in terms of an unobservability distribution which is computable by means of suitable algorithms the existence and regularity of such a distribution implies the existence of changes of coordinates in the state and in the output space which induce an observable quotient subsystem unaffected by all fault signals but one for this subsystem a fault detection filter is designed

a probabilistic approach to some asymptotics in noiseless communication renewal theory is a powerful tool in the analysis of source codes we use renewal theory to obtain some asymptotic properties of finite state noiseless channels we discuss the relationship between these results and earlier uses of renewal theory to analyze the lempel ziv 1977 1978 codes and the tunstall 1967 code as a new application of our results we provide the asymptotic performance of two of the perl garey and even 1975 prefix condition codes

the quasi partition algebra abstract we introduce the quasi partition algebra q p k n as a centralizer algebra of the symmetric group this algebra is a subalgebra of the partition algebra and inherits many similar combinatorial properties we construct a basis for q p k n give a formula for its dimension in terms of the bell numbers and describe a set of generators for q p k n as a complex algebra in addition we give the dimensions and indexing set of its irreducible representations we also provide the bratteli diagram for the tower of quasi partition algebras constructed by letting k range over the positive integers

disturbance decoupling by measurement feedback with stability or pole placement in this paper we solve the disturbance decoupling problem by measurement feedback and requiring stability or pole placement on the closed loop system the problem is attacked using the geometric approach through the concepts of a bmod mathcal b invariant and controllability subspaces and their duals a mid k invariant and complementary observability subspaces the solution of this problem has an interesting structure consisting of a feedback processor which decomposes into i a disturbance decoupling loop ii a disturbance input stabilization or pole placement loop and iii a controlled output stabilization or pole placement loop

on normal forms of nonlinear systems affine in control the nonlinear equivalences of both finite and infinite zero structures of linear systems have been well understood for single input single output systems and have found many applications in nonlinear control theory the extensions of these notions to multiple input multiple output systems have proven to be highly sophisticated in this paper we propose constructive algorithms for decomposing a nonlinear system that is affine in control these algorithms require modest assumptions on the system and apply to general multiple input multiple output systems that do not necessarily have the same number of inputs and outputs they lead to various normal form representations and reveal the structure at infinity the zero dynamics and the invertibility properties all of which represent nonlinear equivalences of relevant linear system structural properties

vines and partial transformations abstract we introduce the partial vine monoid pv n this monoid is related to the partial transformation semigroup pt n in the same way as the braid group b n is related to the symmetric group s n and contains both the vine monoid t g lavers the theory of vines comm algebra 25 4 1997 1257 1284 and the inverse braid monoid d easdown t g lavers the inverse braid monoid adv math 186 2 2004 438 455 we give a presentation for pv n in terms of generators and relations as well as a faithful representation in a monoid of endomorphisms of a free group we also derive a new presentation for pt n

a combinatorial approach to finding the capacity of the discrete noiseless channel shannon defined the capacity of the discrete noiseless channel dnc and considered a finite state model from which the capacity can be calculated alternatively the dnc may be and often is represented by a finite list of forbidden strings in this correspondence we demonstrate the application of combinatorial techniques to finding the shannon capacity of the dnc directly from the forbidden list in our derivations the case of noninteger symbol durations as introduced by csiszar is considered

global transformations of nonlinear systems recent results have established necessary and sufficient conditions for a nonlinear system of the form dot x t f x t u t g x t with f 0 0 to be locally equivalent in a neighborhood of the origin in rnto a controllable linear system we combine these results with several versions of the global inverse function theorem to prove sufficient conditions for the transformation of a nonlinear system to a linear system in doing so we introduce a technique for constructing a transformation under the assumptions that g ldot f dotg ad n 1 f ldotg span an n dimensional space and that g ldot f ldot g ad n 2 f ldotg is an involutive set

idempotents and one sided units in infinite partial brauer monoids abstract we study monoids generated by various combinations of idempotents and one or two sided units of an infinite partial brauer monoid this yields a total of eight such monoids each with a natural characterisation in terms of relationships between parameters associated to brauer graphs we calculate the relative ranks of each monoid modulo any other such monoid it may contain and then apply these results to determine the sierpinski rank of each monoid and ascertain which ones have the semigroup bergman property we also make some fundamental observations about idempotents and units in arbitrary monoids and prove some general results about relative ranks for submonoids generated by these sets

braids and partial permutations abstract we study the inverse braid monoid ib n introduced by easdown and lavers in 2004 we completely describe the factorizable structure of ib n and use this to give a new proof of the easdown lavers presentation we also derive several new presentations each of which gives rise to a new presentation of the symmetric inverse monoid we then define and study the pure inverse braid monoid ip n which is related to ib n in the same way that the pure braid group is related to the braid group

bisimilar control affine systems the notion of bisimulation plays a very important role in theoretical computer science where it provides several notions of equivalence between models of computation these equivalences are in turn used to simplify verification and synthesis for these models as well as to enable compositional reasoning in systems theory a similar notion is also of interest in order to develop modular verification and design tools for purely continuous or hybrid control systems in this paper we introduce two notions of bisimulation for nonlinear systems we present differential geometric characterizations of these notions and show that bisimilar systems of different dimensions are obtained by factoring out certain invariant distributions furthermore we also show that all bisimilar systems of different dimension are of this form

on nonlinear observers in this note we make some remarks about the general observer problem for nonlinear systems we show how the existence of a partial observer without stability requirements is locally equivalent to the existence of a conditioned invariant distribution we argue that by considering not only identity observers we can put the stability issue in a broader perspective

shannon capacity and codes for communicating with a chaotic laser we investigate a method of communicating with chaos described by hayes grebogi and ott 19913 which uses the idea of controlling chaos by small perturbations although allowing the electronics controlling the output signal to remain at the low power level this method unavoidably incurs a capacity loss which we calculate for a chaotic co sub 2 laser model we first determine an approximation to the laser language by means of a forbidden set representation and then find its shannon 1963 capacity the results indicate that since the capacity loss is not significant the tradeoff is worthwhile finally we point to simple finite state encoders that satisfy the constraints imposed by the language

new results on f g invariance abstract the existing result for a nonsingular and involutive distribution to be f g invariant has a priori assumption that g is nonsingular this paper treats the problem of f g invariance for the case where g is singular necessary and sufficient conditions for a nonsingular and involutive distribution to be f g invariant are presented the constructive proof provides an algorithm for the construction of the feedback law

geometric tools for solving the fdi problem for linear periodic discrete time systems this paper studies the problem of detecting and isolating faults in linear periodic discrete time systems the aim is to design an observer based residual generator where each residual is sensitive to one fault whilst remaining insensitive to the other faults that can affect the system making use of the geometric tools and in particular of the outer observable subspace notion the fault detection and isolation fdi problem is formulated and necessary and solvability conditions are given an algorithmic procedure is described to determine the solution of the fdi problem

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

motzkin monoids and partial brauer monoids we study the partial brauer monoid and its planar submonoid the motzkin monoid we conduct a thorough investigation of the structure of both monoids providing information on normal forms green s relations regularity ideals idempotent generation minimal idempotent generating sets and so on we obtain necessary and sufficient conditions under which the ideals of these monoids are idempotent generated we find formulae for the rank smallest size of a generating set of each ideal and for the idempotent rank smallest size of an idempotent generating set of the idempotent generated subsemigroup of each ideal in particular when an ideal is idempotent generated the rank and idempotent rank are equal along the way we obtain a number of results of independent interest and we demonstrate the utility of the semigroup theoretic approach by applying our results to obtain new proofs of some important representation theoretic results concerning the corresponding diagram algebras the partial or rook brauer algebra and motzkin algebra