musings around the geometry of interaction and coherence we introduce the danos regnier category dr m of a linear inverse monoid m as a categorical description of geometries of interaction goi inspired from the weight algebra the natural setting for goi is that of a so called weakly cantorian linear inverse monoid in which case dr m is a kind of symmetrized version of the classical abramsky haghverdi scott construction of a weak linear category from a goi situation it is well known that goi is perfectly suited to describe the multiplicative fragment of linear logic and indeed dr m will be a autonomous category in this case it is also well known that the categorical interpretation of the other linear connectives conflicts with goi interpretations we make this precise and show that dr m has no terminal object no cartesian product of any two objects and no exponential whatever m is unless m is trivial however a form of coherence completion of dr m a la hu joyal which for additives resembles a layered approach a la hughes van glabbeek provides a model of full classical linear logic as soon as m is weakly cantorian one finally notes that girard s notion of coherence is pervasive and instrumental in every aspect of this work

robust stabilisation of nonlinear plants via left coprime factorizations abstract in this paper we take steps towards the development of a robust stabilization theory for nonlinear plants an approach using the left coprime factorizations of the plant and controller under certain differential boundedness assumptions is used we first focus attention on a characterization of the class of all stabilizing nonlinear controllers k q for a nonlinear plant g parameterized in terms of an arbitrary stable nonlinear operator q also we consider the dual class of all plants g s stabilized by a given nonlinear controller k and parameterized in terms of an arbitrary stable nonlinear operator s we show that a necessary and sufficient condition for k q to stabilize g s with q s not necessarily stable is that s stabilizes q this robust stabilization result is of interest for the solution of problems in the areas of nonlinear adaptive control and simultaneous stabilization it specializes to known results for linear operators

on the computation of proper feedforward internally stabilizing and pole placing compensators by combining two important results from the polynomial and fractional representation approaches we derive a state space formula for the direct computation of a proper feedforward internally stabilizing and pole placing compensator c s

a connection between state space and doubly coprime fractional representations explicit formulas for a doubly coprime fractional representation of the transfer function of a lumped lti system are given in terms of a stabilizable and detectable state space realization of the transfer function these formulas allow existing computational algorithms to be utilized for the purpose of computing doubly coprime factorizations several additional implications of this result are briefly discussed as are some immediate extensions and future research directions

a frequency domain approach to control of singular systems in this paper singular control systems are studied from the frequency domain point of view the proper stable factorization of a singular system is derived the stability of the closed loop system consisting of a singular system and controller is addressed most significantly all proper stabilizing controllers are parameterized for a given singular system

automatic recognition of tractability in inference relations a procedure is given for recognizing sets of inference rules that generate polynomial time decidable inference relations the procedure can automatically recognize the tractability of the inference rules underlying congruence closure the recognition of tractability for that particular rule set constitutes mechanical verification of a theorem originally proved independently by kozen and shostak the procedure is algorithmic rather than heuristic and the class of automatically recognizable tractable rule sets can be precisely characterized a series of examples of rule sets whose tractability is nontrivial yet machine recognizable is also given the technical framework developed here is viewed as a first step toward a general theory of tractable inference relations

construction of the left coprime fractional representation for a class of nonlinear control systems abstract a constructive approach to the left coprime fractional representations for a class of stable and unstable nonlinear control systems described by certain multi input multi output nonlinear ordinary differential equations is proposed the method works for the coprimeness defined by verma 15

on improving control loop robustness of model matching controllers this note first characterizes the class of all stabilizing controllers for a two degree of freedom control system which achieve a prescribed achievable transfer function the char acterization is in terms of an arbitrary proper stable transfer function with this characterization robust model matching is formulated as a standard hm optimization problem this means that standard controller designs for a nominal plant such as linear quadratic gaussian ones can be enhanced to give improved robustness properties using hm design tech niques

state feedback confinement and stabilization for non linear continuous time systems a theory of static state feedback for multivariable continuous time non linear systems is formulated the theory applies to systems described by differential equations of the form xdot t x t u t the basic objective is to design static feedback compensators which achieve the following properties a the state space trajectory of the closed loop system is confined within a specified subspace and b the closed loop system is internally stable an explicit method for designing such compensators is developed the construction of the compensators involves only quantities directly derived from the given function

robust right coprime factorization and robust stabilization of nonlinear feedback control systems robust right coprime factorization and robust stabilization of nonlinear feedback control systems are studied the concept of robust right coprime factorization of nonlinear operators for feedback control systems is introduced some conditions for the robustness of a right coprime factorization of a nonlinear plant under unknown but bounded perturbations are derived an example with a closed form solution is included to illustrate the general theory and a step by step construction of the robust factorization and the robust stabilization

from algol to polymorphic linear lambda calculus in a linearly typed functional language one can define functions that consume their arguments in the process of computing their results this is reminiscent of state transformations in imperative languages where execition of an assignment statement alters the contents of the store we explore this connection by translating two variations on algol 60 into a purely functional language with polymorphic linear types on the one hand the translations lead to a semantic analysis of algol like programs in terms of a model of the linear language on the other hand they demonstrate that a linearly typed functional language can be at least as expressive as algol

algebraic aspects of linear control system stability after formulating the problem of linear control system stability in very general terms the problem is resolved by 1 establishing a necessary and sufficient condition for stability 2 exploring in some detail the nature of this condition and 3 developing a parametric characterization of all controllers which stabilize a given plant the problem formulation is based on a ring theoretic setting for linear systems and is distinguished by the use of a novel control system configuration any interconnection of two distinct systems can be represented in terms of this configuration while preserving the distinction between the two systems furthermore no other configuration has this property several benefits and extensions are accrued as a result of using this configuration first a completely general yet physically meaningful problem formulation results consequently state space and operator theoretic approaches to linear control system stability are thoroughly reconciled this reconciliation embraces continuous time as well as discrete time systems lumped as well as distributed systems and 1 d as well as n d systems second the controller parametrization affords four degrees of freedom two more than any existing parametrization the additional degrees of freedom make explicit the design opportunity associated with controller implementation and thus determine several important control system characteristics this result is expected to provide a theoretical basis for the development of comprehensive control system design methodologies wherein design considerations related to controller implementation are addressed

preservation of stability under system perturbations in nonlinear control a simple characterization abstract a simple characterization of the perturbations a nonlinear closed loop control system can tolerate before losing its stability is derived the characterization is given in purely algebraic terms with the topological aspects being automatically incorporated through the theory of fraction representations of nonlinear systems the implications of system perturbations on internal stability are also discussed the presentation is for the case of discrete time nonlinear systems

internally stable nonlinear systems with disturbances a parameterization this paper deals with the control of a nonlinear system whose output is subjected to an additive disturbance the main result is a simple parameterization of the set of all system responses that can be obtained through internally stable control of the given system the parameterization provides a clear indication of the effects of the disturbance on the response of the stabilized closed loop system the class of achievable responses is determined by the numerator of a right coprime fraction representation of the system being controlled

on the youla kucera parametrization for nonlinear systems abstract we consider the problem of stabilizing a nonlinear plant through the use of a possibly unstable pre compensator and a stable feedback compensator and parameterizing a class of such stabilizers in terms of a bibo stable map q in the linear case by means of superposition the pre and feedback compensator can be combined in such a way that the bibo map q occurs in just one feedback loop within the controller feeding back output residuals prediction errors to the control inputs the main contribution of this paper is to develop a nonlinear generalization of this property building on earlier work in forming the youla kucera parametrization for nonlinear systems we show the equivalence of the class of all bounded input stabilizing nonlinear pre and feedback compensators to a class of possibly unstable feedback controllers in which a map q s is present in only the one feedback loop we then show that necessary and sufficient condition to achieve stability of the system is that q s be bibo stable one advantage of the new formulation is that differential boundedness assumptions do not involve the parametrization q s in any way just as the linear versions of our results have applications in the areas of optimal control and adaptive control of linear systems it is conjectured that the present results will underlie more general results for adaptive control and nonlinear systems

convergent ritz approximations of the set of stabilizing controllers abstract a necessary and sufficient condition for linear exponential sums to be dense in l p is derived under this condition a ritz approximation converges which has been proposed to solve linear control problems state space representations of orthonormal basis functions for multivariable systems are given

coprime fractional representations and stability of non linear feedback systems fractional representation methods for finite dimensional linear time invariant fdlti systems have been developed in an algebraic framework of principal ideal domains the concept of coprime fractional representations has been generalized to discrete time non linear systems by hammer 1985 1987 by viewing the bezout identity as the basis of the definition of coprimeness in this paper we view the role of right coprime fractional representation in representing the set of all stable input output pairs of a possibly unstable non linear system this notion of coprimeness is shown to be equivalent to that of fdlti systems using coprime fractional representations of non linear systems we obtain a characterization of the stability of non linear feedback systems this characterization is similar to that obtained in the case of fdlti systems we also obtain a parameterization of all non linear and time varying stabilizing controllers for linear possibly time varying plants which has the same form as t

left coprime factorizations and a class of stabilizing controllers for non linear systems abstract the problem of representing a class of non linear systems by both left and right bounded input bounded output bibo coprime factors is studied based on the coprime factorizations the class of all stabilizing controllers of a particular structure for the set of plants under consideration is then characterized in term of a bibo stable map q the results specialize to the youla kucera parametrization in the linear case results giving the various conditions for a non linear feedback system to be well posed are also presented

technical communique doubly coprime factorizations related to any stabilizing controllers in state space given a plant and an arbitrary stabilizing controller we present state space formulas for doubly coprime factorizations related to the given controller then we determine parametrizations of all stabilizing controllers centered on that controller

on the parametrization of nonlinear finite gain over set stabilizing controllers the concepts of gain and incremental gain over set have been introduced to alleviate some of the limitations of the conventional notions of gain and incremental gain in this paper the class of all finite gain over set stabilizing controllers is parametrized in terms of a finite gain over set stable map q provided that the plant p is finite incremental gain over set stable an alternate theorem for finite gain over ball stability of the closed loop is also developed which provides a superset and a subset of the set of all finite gain over set stabilizing compensators for the feedback system

towards a link between the factorization approach and mimo robust adaptive control based on a doubly coprime factorization theorem nett et al 1984 an adaptive controller for discrete time mimo systems consisting of a not necessarily minimum phase modelled part and fraction type unmodelled dynamics which are not necessarily stable non linear and time varying is developed then by virtue of a stability result which is itself of interest it is shown that the adaptive closed loop system is bibo stable even in the presence of small but not necessarily stable unmodelled dynamics and arbitrarily bounded initial conditions consequently an initial but important step for applying the factorization approach to mimo robust adaptive control is completely established

objects and classes in algol like languages many object oriented languages used in practice descend from algol with this motivation we study the theoretical issues underlying such languages via the theory of algol like languages it is shown that the basic framework of this theory extends cleanly and elegantly to the concepts of objects and classes moreover a clear correspondence emerges between classes and abstract data types whose theory corresponds to that of existential types equational and hoare like reasoning methods and relational parametricity provide powerful formal tools for reasoning about algol like object oriented programs 2002 elsevier science usa

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

right factorization of a class of time varying nonlinear systems a class of nonlinear continuous time time varying plants with a state space description which has a uniformly completely controllable linear part is considered for this class a right factorization is obtained for the case in which the state is available for feedback a normalized right coprime factorization is realized

on non linear systems additive feedback and rationality abstract the problem of stabilizing a nonlinear system by the application of additive output feedback is considered it is shown that if an infective recursive system can be so stabilized then it must be rational namely it must be possible to express as a ratio y pq l of two stable and recursive systems p and q

the graph metric for unstable plants and robustness estimates for feedback stability in this paper a graph metric is defined that provides a measure of the distance between unstable multivariable plants the graph metric induces a graph topology on unstable plants which is the weakest possible topology in which feedback stability is robust using the graph metric it is possible to derive estimates for the robustness of feedback stability without assuming that the perturbed and unperturbed plants have the same number of rhp poles if the perturbed and unperturbed systems have the same rhp poles then it is possible to obtain necessary and sufficient conditions for robustness with respect to a given class of perturbations as an application of these results the design of stabilizing controllers for unstable singularly perturbed systems is studied finally the relationship of the graph metric to the gap metric introduced by zames and el sakkary is studied in detail in particular it is shown that the robustness results of zames and el sakkary do not enable one to conclude the causality of the perturbed system whereas the present results do

stabilization and robustness of non linear unity feedback system factorization approach this paper is a self contained discussion of a right factorization approach in the stability analysis of the non linear continuous time or discrete time time invariant or time varying well posed unity feedback system s1 p c we show that a well posed stable feedback system s1 p c implies that p and c have right factorizations in the case where c is stable p has a normalized right coprime factorization the factorization approach is used in stabilization and simultaneous stabilization results

the construction of special coprime factorizations in discrete time given a rational discrete time transfer matrix it is shown how to construct two special kinds of coprime factorizations for it normalized and all pass denominator this is done in two ways first by carrying the existing continuous time construction results to discrete time via a bilinear transformation this is shown to be messy and computationally burdensome second direct constructions are derived in discrete time the derivations point out interesting differences between discrete and continuous time for these constructions

state feedback for non linear control systems abstract a theory of static state feedback for non linear discrete time systems is developed the theory applies to non linear systems possessing a recursive representation of the form x k 1 x k uk where is a continuous function and it deals with the construction of continuous state feedback functions that internally stabilize a given system the theory yields an explicit method for the computation of stabilizing feedback functions and several examples of the computation of such functions are provided

conditions for abstract nonlinear regulation this paper studies the general problem of the existence of output dynamic stabilizers for a control system the controller is not assumed to have any special structure necessary and sufficient conditions are given in terms of new notions which generalize the usual ones of stabilizability and detectability in the particular cases of analytic and bilinear systems these conditions are considerably simplified

fraction representations of non linear systems and non additive state feedback abstract the problem of constructing right coprime fraction representations is considered for non linear discrete time systems possessing a continuous realization it is shown that given a static possibly non additive state feedback that stabilizes the input state part of the realization a right coprime fraction representation of the entire system can be constructed the resulting coprime fraction representation has a particularly simple factorization space and can be used to derive stabilizing controllers for the original system without state access the construction of appropriate static state feedbacks is described in a companion paper hammer 1989 b

brief paper algebraic and topological aspects of the regulator problem for lumped linear systems the regulator problem with robustness is solved for systems modelled by rational transfer matrices a topology for possibly unstable plants is presented a necessary and sufficient condition is derived for the existence of a proper controller which provides internal stability and output regulation throughout an open neighbourhood of the plant a characterization of all such controllers is determined

parametrization of all stabilizing controllers of nonlinear systems this paper develops a new approach to the parametrization of all asymptotically stabilizing controllers of nonlinear systems including time varying ones both for state feedback and output feedback assumptions of stabilizability and a kind of detectability are required the approach is very direct and simple and is not based on the so called factorization and on the h framework of nonlinear systems which are used in previous papers a key of the derivation of the parametrization in this paper is to find an explicit state space representation of free functions such that the parametrized controller expresses all asymptotically stabilizing controllers of nonlinear systems

deciding knowledge in security protocols under equational theories the analysis of security protocols requires precise formulations of the knowledge of protocol participants and attackers in formal approaches this knowledge is often treated in terms of message deducibility and indistinguishability relations in this paper we study the decidability of these two relations the messages in question may employ functions encryption decryption etc axiomatized in an equational theory one of our main positive results says that deducibility and indistinguishability are both decidable in polynomial time for a large class of equational theories this class of equational theories is defined syntactically and includes for example theories for encryption decryption and digital signatures we also establish general decidability theorems for an even larger class of theories these theorems require only loose abstract conditions and apply to many other useful theories for example with blind digital signatures homomorphic encryption xor and other associative commutative functions

strictly doubly coprime factorizations and all stabilizing compensators related to reduced order observers abstract doubly coprime factorizations dcf play an important role in the design of compensators for lumped linear time invariant systems they are defined as stable proper rational matrices and they not only constitute a starting point for the fractional approach to all stabilizing compensators but they are also closely related to the frequency domain design of optimal state feedback and estimation and of the resulting observer based compensators when defining dcfs related to reduced order observers some rational matrices become improper this can be prevented by the introduction of artificial stable dynamics the identity elements which cancel in the system and in the compensator transfer matrices such pole zero cancellations however are contrary to the usual notion of coprimeness it is shown here that these additional dynamics which have no meaning in an observer based compensator scheme are superfluous and that the parametrization of all stabilizing compensators is feasible on the basis of possibly nonproper factorizations not containing identity elements for such factorizations the new definition of strictly doubly coprime factorizations sdcf is proposed

proper stable transfer matrix factorizations and internal system descriptions the exact relations between coprime proper stable factorizations of p s and coprime polynomial matrix factorizations are derived and they directly lead to relations with internal descriptions of the plant in differential operator or state space form it is shown that obtaining any right or left proper stable coprime factorization is equivalent to state feedback stabilization or to designing a full order full state observer respectively solving the diophantine equation is shown to be equivalent to designing a full or reduced order observer of a linear functional of the state and to designing a stable inverse system and this suggests new computational methods to solve the diophantine

a calculus for cryptographic protocols we introduce the spi calculus an extension of the pi calculus designed for describing and analyzing cryptographic protocols we show how to use the spi calculus particularly for studying authentication protocols the pi calculus without extension suffices for some abstract protocols the spi calculus enables us to consider cryptographic issues in more detail we represent protocols as processes in the spi calculus and state their security properties in terms of coarse grained notions of protocol equivalence

state feedback control of nonlinear systems a simple approach a simple methodology for the design of state feedback controllers for nonlinear continuous time systems is introduced the objective is to develop controllers that drive a nonlinear system from an initial condition into an assigned region of state space it is shown that such state feedback controllers can be derived by solving a set of inequalities obtained directly from quantities given in the controlled system s equation the results are applied to the design of state feedback controllers that achieve robust asymptotic stabilisation for a rather general class of nonlinear systems

numerically stable algorithm for transfer function matrix evaluation a numerically stable algorithm for the evaluation of the transfer function matrix corresponding to a state space description is proposed the algorithm performs orthogonal similarity transformations on the system matrices eigenvalue computations are used for pole and zero determinations

a new approach for computing the state feedback gains of multivariable systems this note presents some new results in linear systems first the relationship between the polynomial matrix description and the state space representation of multivariable systems is clarified then we show that once such a relationship is determined the coprime matrix fraction description can be easily computed and we can further develop a closed form formula to solve the pole assignment problem of multivariable systems such formula can be thought of as an extension of the ackermann s formula for mimo systems thus this note potentially gives us a clearer insight into linear systems from the theoretical viewpoint

right coprime factorizations and stabilization for nonlinear systems the problem of constructing right coprime factorizations which are based on the graph of an input output map instead of the bezout identity of nonlinear input output maps is considered the map is assumed to arise from a state variable realization with a fixed initial state the main result is that the existence of a stabilizing state feedback implies the existence of a right coprime factorization for the map the technique is illustrated by application to nonlinear systems which are affine in the control and have a controllable linear past and nonlinear systems which are feedback linearizable a notion of input output stability that requires a bound on the magnitude of the input signals is introduced methods of constructing such bounds are developed for a locally feedback linearizable system the problem of input bounds is transferred to the equivalent linear system this leads to a technique that allows state and input constraints for a feedback linearizable system to be mapped to the equivalent linear system

design of observer based compensators in the frequency domain the discrete time case for nth order systems with m outputs the design of compensators using state observers of order n x with 0 x m is presented in the z domain the design is carried out in equivalence with the usual time domain approach and the relations between the time and the frequency domain results are presented the state feedback and the observer are characterized by polynomial matrices in the frequency domain and the corresponding compensator matrix fraction descriptions are obtained from the solution of a linear diophantine equation in view of the optimal state estimation problem the use of an updated state estimate is of interest it is shown how this problem can also be solved in the frequency domain

comparing object encodings recent years have seen the development of several foundational models for statically typed object oriented programming but despite their intuitive similarity differences in the technical machinery used to formulate the various proposals have made them difficult to compare using the typed lambda calculus f as a common basis we now offer a detailed comparison of four models 1 a recursive record encoding similar to the ones used by cardelli car84 reddy red88 kr94 cook coo89 chc90 and others 2 hofmann pierce and turner s existential encoding pt94 hp95 3 bruce s model based on existential and recursive types bru94 and 4 abadi cardelli and viswanathan s type theoretic encoding acv96 of a calculus of primitive objects

perturbed nonlinear systems control using extended robust right coprime factorisation abstractin this paper extended operator based robust right coprime factorisation is investigated for dealing with a class of nonlinear systems with unknown bounded perturbations first a new kind of operator is introduced by which operator based right coprime factorisation approach is extended to consider the perturbed nonlinear systems by regulating the exponent of the proposed operator a broader class of nonlinear systems can be handled using the extended right coprime factorisation approach second for guaranteeing robust stability of the perturbed nonlinear systems feasible design schemes based on some sufficient conditions are discussed which can reduce complicated calculation in control processing using a different unimodular operator finally a simulation example is involved to illustrate the proposed design scheme for confirming the effectiveness of the proposed method

normalized coprime factorizations and balancing for unstable nonlinear systems in this paper we first study the normalized right and left coprime factorizations of a nonlinear system from a state space point of view in order to do so we introduce the notion of inner and co inner nonlinear systems secondly we deal with balancing for unstable nonlinear systems for linear systems there are several ways to approach this problem and we generalize the ideas of balancing the normalized coprime factorization and of lqg balancing to the nonlinear case lqg balancing can be seen as measuring the influence of a state component on a certain future and past energy function we extend this interpretation to the nonlinear case furthermore we use the normalized coprime factorization to give another way of balancing the unstable nonlinear system we give the relation between these two methods of balancing which is similar to the relation in the linear case

intruder deduction for the equational theory of abelian groups with distributive encryption cryptographic protocols are small programs which involve a high level of concurrency and which are difficult to analyze by hand the most successful methods to verify such protocols are based on rewriting techniques and automated deduction in order to implement or mimic the process calculus describing the execution of a protocol we are interested in the intruder deduction problem that is vulnerability to passive attacks in presence of equational theories which model the protocol specification and properties of the cryptographic operators in the present paper we consider the case where the encryption distributes over the operator of an abelian group or over an exclusive or operator we prove decidability of the intruder deduction problem in both cases we obtain a ptime decision procedure in a restricted case the so called binary case these decision procedures are based on a careful analysis of the proof system modeling the deductive power of the intruder taking into account the algebraic properties of the equational theories under consideration the analysis of the deduction rules interacting with the equational theory relies on the manipulation of z modules in the general case and on results from prefix rewriting in the binary case

a new approach for calculating doubly coprime matrix fraction descriptions using the concepts of state feedback in generalized state space explicit formulas for calculating doubly coprime matrix fraction descriptions mfds of the transfer matrix of a regular state space system and the polynomial generalized bezout identity elements corresponding to those coprime mfds are presented they can be easily calculated with the help of existing computational algorithms and software packages

an undecidability result for agh we present an undecidability result for the verification of security protocols since the perfect cryptography assumption is unrealistic for cryptographic primitives with visible algebraic properties several recent works relax this assumption allowing the intruder to exploit these properties we are interested in the abelian groups theory in combination with the homomorphism axiom we show that the security problem for a bounded number of sessions expressed by satisfiability of symbolic deductibility constraints is undecidable obtaining in this way the first undecidability result concerning a theory for which unification is known to be decidable f baader unification in commutative theories hilbert s basis theorem and grobner bases j acm 40 3 1993 477 503

2dof controller parametrization for systems with a single i o delay this note puts forward a parametrization of all stabilizing two degrees of freedom controllers for possibly unstable processes with dead time the proposed parametrization is based on a doubly coprime factorization of the plant and takes the form of a generalized smith predictor dead time compensator feedback part and a finite dimensional feedforward part prefilter some alternative dead time compensation schemes and disturbance attenuation limitations are also discussed