almost invariant subspaces an approach to high gain feedback design part ii almost conditionally invariant subspaces in part i of this paper almost controlled invariant subspaces were studied in this part we will consider their duals the almost conditionally invariant subspaces these concepts give immediately by dualization of the almost disturbance decoupling control by state feedback the solution of the almost disturbance decoupled estimation problem finally we consider the problem of approximate disturbance decoupling by measurement feedback and it is shown that this problem is solvable to any arbitrary degree of accuracy if and only if 1 almost disturbance decoupling by state feedback and 2 almost disturbance decoupled estimation of the to be controlled output are both possible

kalman decomposition for implicit linear systems the kalman decomposition for a general class of implicit systems is introduced and studied the properties of strong reachability observability acceptance of all inputs and outer sequences output uniqueness uniqueness and trajectory observability of the original system are related to the properties of its subsystems obtained via the kalman decomposition the problems of transfer relation regularizability by feedback output feedback and output injection are studied by means of this decomposition

algebraic characterizations of almost invariance this paper stresses the algebraic as opposed to analytic nature of the concept of almost invarianoe which has been introduced by willems 1980 as a means of studying asymptotic aspects of linear systems based on an interpretation in terms of difference equations it is shown that the basic properties and a number of characterizations of tho four fundamental classes of subspaces can be derived in a relatively simple and purely algebraic fashion among other things this paper also extends the results of hautus on the frequency domain interpretation of ordinary a b invariant subspaces re derives the pencil characterization of jaffe and karcanias 1981 and obtains a new hybrid type of characterization the last result also loads to a rank test for almost invariance

feedforward control pid control laws and almost invariant subspaces we define feedforward control as a control policy in which the exogenous disturbances are known for all time at the moment when the control is applied it is shown that disturbance decoupling by feedforward control is possible iff it is possible by pid control or iff approximate disturbance decoupling by state feedback is possible

descriptor representations without direct feedthrough term abstract descriptor representations are considered that are given by e a b c d with d 0 minimality under external equivalence is characterized in terms of the matrices e a b and c also transformations are given by which minimal e a b c representations are related under external equivalence the transformations turn out to be more simple than in the d 0 case algorithms for rewriting an e a b c d representation in e a b c form are also given finally a realization procedure is presented for obtaining a minimal e a b c representation for a system that is given in polynomial matrix fractional form

a computational algorithm for pole assignment of linear single input systems an efficient computational algorithm for pole assignment of linear multiinput systems is proposed a preliminary stage of the algorithm is a reduction of the system matrices into orthogonal canonical form the gain matrix elements are then found by orthogonal transformation of the closed loop system matrix into upper quasi triangular form whose diagonal blocks correspond to the desired poles the algorithm is numerically stable the computed gain matrix being exact for a system with slightly perturbed matrices it works equally well with real and complex distinct and multiple poles and is applicable to ill conditioned and high order problems

feedback and pole placement in descriptor variable systems the effects and uses of applying linear feedback to continuous time descriptor systems are studied structural changes resulting from feeding back the slow and fast parts of the trajectory separately are characterized it is shown that under certain conditions related to controllability the poles of the slow subsystem may be shifted arbitrarily and the impulsive behaviour of the fast subsystem may be eliminated

on ternary quotients of cubic hecke algebras we prove that the quotient of the group algebra of the braid group introduced by funar commun math phys 173 513 558 1995 collapses in characteristic distinct from 2 in characteristic 2 we define several quotients of it which are connected to the classical hecke and birman wenzl murakami quotients but which admit in addition a symmetry of order 3 we also establish conditions on the possible markov traces factorizing through it

generalized state space realizations of non proper rational transfer functions in this paper we propose an algebraic realization theory for polynomial transfer functions based on classical module theory realizations are given by linear systems whose generalized state space has a natural module structure over the ring of formal power series k z 1 existence and uniqueness of minimal realizations are proved and their properties are described the dimension of the minimal realization is called generalized degree of the polynomial transfer function and it is proved to be equal to the rank of an associated generalized hankel matrix finally a correspondence between realizations and fractional representations over k z 1 of polynomial matrices is established and some properties of such representations are investigated

squares and rectangles in relation categories three cases semilattice distributive lattice and boolean non unitary in this paper we investigate rectangle morphisms of dedekind categories or semilattice relation categories or unitary allegories which appear to be a basic structure for relation categories in order to reconstruct the dedekind category generated by all the rectangle morphisms we are obliged to clarify the role of both the square morphisms and the functions s which assign to every square morphism c the ideal morphism cs 1c1 we translate this observations into adapted new objects that we call the mp families in this way the reconstruction problem becomes an adjunction problem to each dedekind category we associate a mp family to each functor between two dedekind categories we associate a morphism between two mp families this association is a functor this functor has a left adjoint this result is also true for other universes distributive lattices boolean algebras

complex reflection groups braid groups hecke algebras presentations a la coxeter are given for all irreducible nite com plex re ection groups they provide presentations for the corresponding generalized braid groups for all but six cases which allow us to generalize some of the known properties of nite coxeter groups and their associated braid groups such as the computation of the center of the braid group and the construction of deformations of the nite group algebra hecke algebras we introduce monodromy representations of the braid groups which factorize through the hecke algebras extending results of cherednik opdam kohno and others summary introduction complex re ection groups and their presentations a background from complex re ection groups b presentations braid groups and their diagrams a generalities about hyperplane complements b generalities about the braid groups c the braid diagrams proofs of the main theorems for the braid groups b de e r a notation and prerequisites b computation of b de e r and of its center for d c computation of b e e r and of its center hecke algebras a background from di erential equations and monodromy b a family of monodromy representations of the braid group c hecke algebras diagrams and tables appendix generators of the monodromy around a divisor appendix tables to mathematics subject classi cation primary g we thank jean michel peter orlik pierre vogel for useful conversations and the isaac newton institute for its hospitality while the last version of this manuscript was written up the second named author gratefully acknowledges nancial support by the fondation alexander von humboldt for his stays in paris michel brou e gunter malle raphael rouquier

robust pole assignment in singular control systems abstract necessary conditions are given for the problem of pole assignment by state feedback in singular linear systems descriptor systems to have a solution which is regular and nondefective for a robust solution such that the assigned closed loop poles are insensitive to perturbations in the system data the same conditions must hold it can be shown that these conditions are also sufficient for the existence of a feedback which assigns the maximum possible number of finite poles with regularity these results provide the basis of a procedure for constructing closed loop semistate systems with given poles guaranteed regularity and maximum robustness

a maximal cubic quotient of the braid algebra i abstract we study a quotient of the group algebra of the braid group in which the artin generators satisfy a cubic relation this quotient is maximal among the ones satisfying such a cubic relation we also investigate the proper quotients of it that appear in the realm of quantum groups and describe another maximal quotient related to the usual hecke algebras finally we describe the connection between this algebra and a quotient of the algebra of horizontal chord diagrams introduced by vogel we prove that these two are isomorphic for n 5

controllability of generalized linear time invariant systems this note deals with the problem of c controllability of generalized systems a new criterion for determining c controllability of generalized systems is established with controllability established it is shown that it is always possible to transfer the system from the initial position to any target by a control vector valued function whose elements are polynomials of bounded degree the polynomial coefficients are obtained as a solution to a set of linear equations

lattice extensions of hecke algebras abstract we investigate the extensions of the hecke algebras of finite complex reflection groups by lattices of reflection subgroups that we introduced for some of them in our previous work on the yokonuma hecke algebras and their connections with artin groups when the hecke algebra is attached to the symmetric group and the lattice contains all reflection subgroups then these algebras are the diagram algebras of braids and ties of aicardi and juyumaya we prove a structure theorem for these algebras generalizing a result of espinoza and ryom hansen from the case of the symmetric group to the general case we prove that these algebras are symmetric algebras at least when w is a coxeter group and in general under the trace conjecture of broue malle and michel

on closed form solutions to the generalized sylvester conjugate matrix equation abstract in this paper several explicit parametric solutions to the generalized sylvester conjugate matrix equation are given the proposed solutions can provide all the degrees of freedom which are represented by a free parameter matrix in addition the approaches adopted in this paper do not require that the coefficient matrices are in any canonical forms

a general realization theory for higher order linear differential equations in this note we show that the geometric quotient under a natural group action of the generalized state space systems recently considered by schumacher kuijper and geerts is algebraically isomorphic to the space of a homogeneous autoregressive system this result essentially follows from work of stromme published earlier in the algebraic geometry literature in particular these generalized state space systems represent a realization of the space of homogeneous autoregressive systems

disturbance decoupling of linear time varying singular systems the problem of disturbance decoupling by state feedback is defined for a linear time varying singular system it is required that the closed loop system has a unique impulse free solution and its output is not affected by disturbances an algorithm namely disturbance decoupling algorithm is proposed it is proved that the feasibility of the disturbance decoupling algorithm is invariant under any regular feedback control law based on the disturbance algorithm a constructive method is provided to design a disturbance decoupling feedback control law sufficient conditions for the solvability of the disturbance decoupling problem are derived it is proved that one of the sufficient conditions is also necessary provided that other conditions are satisfied

solutions to the output regulation problem of linear singular systems in this paper we revisit the output regulation problem for linear singular systems and identify an important case of the general regulation problem where the measurement output is identical to the vector to be regulated we derive a necessary and sufficient condition for the regulation problem to be solvable via either full information feedback or error feedback then we show how full information feedback and error feedback controllers can be constructed explicitly

a system of generalized sylvester quaternion matrix equations and its applications let h m n be the set of all m n matrices over the real quaternion algebra we call that a h n n is hermitian if a a i j k where i j k are the quaternion units denote a a in this paper we derive some necessary and sufficient conditions for the solvability to the system of generalized sylvester real quaternion matrix equations a i x i y i b i c i z d i e i i 1 2 and give an expression of the general solution to the above mentioned system as applications we give some solvability conditions and general solution for the generalized sylvester real quaternion matrix equation a 1 x y b 1 c 1 z d 1 e 1 where z is required to be hermitian we also present some solvability conditions and general solution for the system of real quaternion matrix equations involving hermicity a i x i a i x i b i y b i c i i 1 2 where y is required to be hermitian our results include some well known results as special cases

feedback design for regularizing descriptor systems this paper surveys numerical techniques for the regularization of descriptor generalized state space systems by proportional and derivative feedback we review generalizations of controllability and observability to descriptor systems along with definitions of regularity and index in terms of the weierstras canonical form three condensed forms display the controllability and observability properties of a descriptor system the condensed forms are obtained through orthogonal equivalence transformations and rank decisions so they may be computed by numerically stable algorithms in addition the condensed forms display whether a descriptor system is regularizable i e when the system pencil can be made to be regular by derivative and or proportional output feedback and if so what index can be achieved also included is a a new characterization of descriptor systems that can be made to be regular with index 1 by proportional and derivative output feedback

poles and zeros of matrices of rational functions abstract this expository paper considers the problem of defining poles and zeros with multiplicity including those at infinity for a matrix of rational functions over a field the underlying theme question for the paper is does the number of poles of a matrix equal the number of zeros and what structural meaning underlies such an assertion our approach is motivated by ideas from linear system theory and control engineering control theoretic ideas are not a prerequisite and we hope to encourage specialists in classical linear and commutative algebra to investigate this promising source of new algebra problems

solvability controllability and observability of continuous descriptor systems in this paper we investigate the properties of the continuous descriptor system e dot x t ax t bu t 0 leqt leqb where e a and b are complex and possibly singular matrices and u t is a complex function differentiable sufficiently many times the traditional approach to such systems is to separate the state equations from the algebraic equations however such algorithms usually destroy the natural physically based sparsity and structure of the original system therefore we consider descriptor systems in their original form such systems possess numerous properties not shared by the well known state variable systems first we relate classical theories of matrix pencils to the solvability of descriptor systems then we extend the concepts of reachability controllability and observability of state variable systems to descriptor systems and describe the set of reachable states for descriptor systems

disturbance decoupling with stability and impulse free response in singular systems abstract the problem of disturbance decoupling with stability and impulse free response in square singular systems e x ax bu gq with output y dx by applying a state feedback u fx is considered by employing van dooren s recursive algorithm for the computation of the subspace supremal a b invariant subspace in kernel d and using kalman s decomposition theorem the paper s approach presents a complete solution to the underlying problem regarding state space systems it is shown that the solution to the underlying problem is obtained almost directly from van dooren s recursive algorithm

disturbance decoupling for descriptor systems by state feedback we study the disturbance decoupling problem for linear time invariant descriptor systems we give necessary and sufficient conditions for the existence of a solution to the disturbance decoupling problem with or without stability via a proportional and or derivative feedback that also makes the resulting closed loop system regular and or of index at most one all results are proved constructively based on condensed forms that can be computed using orthogonal matrix transformations i e transformations that can be implemented in a numerically stable way

pole placement and compensator design of generalized systems abstract the system considered is a generalized system e ax bu y cx with matrix e singular when the impulsive modes of the system are both controllable and observable it is shown in this paper that these modes can be eliminated by almost any constant output feedback for a strongly controllable and observable generalized system after its impulsive modes are eliminated the poles of the system can be assigned arbitrarily by state feedback even if the states are not available the paper adopts an efficient method to design a compensator for the system

dynamical order assignment for linear descriptor systems this paper considers the problem of dynamical order assignment for linear descriptor systems by derivative feedback control the approach is based on an ability to adjust the number of infinite non dynamical modes of the system within a permissible range increasing the dynamical order enables the designer to implement specific otherwise unrealizable control strategies on the other hand dynamical order reduction simplifies the design procedures for a large scale system the solution is obtained by imbedding the descriptor space problem in an equivalent state space problem where a zero value eigenvalue with a prescribed geometric multiplicity is assigned to a linear state system by proportional feedback control

pushout complements and basic concepts of grammars in toposes abstract an existence theorem of pushout complements is given in an elementary topos by using category theory of binary relations called relational calculus and it is also shown more explicitly in the category of directed graphs which is a typical example of toposes as an application moreover an embedding theorem and church rosser theorem on grammars derivations in a topos are proved

matrix pencil characterization of almost a z invariant subspaces a classification of geometric concepts the equivalence between the algebraic matrix pencil characterization of the sub spaces of the extended geometric theory and their dynamic characterization 13 established as a result a complete classification of almost a b invariant a b invariant almost controllability and controllability subspaces is derived in terms of matrix pencil invariants the frequency propagation aspects of infinite spectrum a b invariant subspaces are investigated and it is shown that they are limits of closed loop eigenspaces with arbitrarily largo eigenvalues finally the importance of the infinite frequency subspaces in the study of the asymptotic behaviour of the closed loop eigenspaces and eigenvalues under scalar gain output feedback is discussed

on the reduced order causal observer for generalized control systems abstract this paper considers the problem of a causal state observer for a square singular system necessary and sufficient conditions for solvability of the problem have been established the approach of this study leads to a new insight concerning synthesis of a causal observer and establishes new results with regard to the structure and the dimension of a reduced order observer for singular systems

on almost invariant subspaces for implicit linear discrete time systems abstract the concept of almost invariant subspace for an implicit linear discrete time system is introduced and studied in detail it is shown also that for regular homogeneous implicit systems the so called deflating subspaces can be identified with almost invariant subspaces

disturbance and input output decoupling of singular systems the disturbance decoupling and the simultaneous disturbance and input output decoupling problems for singular systems are considered in the context of the matrix fraction description mfd of the system solvability conditions are obtained in terms of the composite matrix of a column reduced mfd of the system a characterisation of the fixed poles of both problems is given and it is shown that the remaining poles can be arbitrarily assigned

the disturbance decoupling problem for implicit linear discrete time systems the disturbance decoupling problem for implicit linear discrete time systems is studied in detail necessary and sufficient conditions for the problem to be solved are given the results are obtained with the aid of the concepts of almost invariant sliding and coasting subspaces for implicit systems

generalized state space realizations from matrix fraction descriptions the problem of the realization of a matrix fraction description mfd of a nonproper transfer function is considered an algorithm for the realization is developed and it is shown that it is a natural generalization of wolovich s method 1973 for the case of proper systems the resulting generalized state space system is shown to be reachable and the conditions for its minimality in terms of the coprimeness and column reducedness of the mfd are derived the relation between the controllability indexes of the realization and the forney order 1975 of the mfd is established

eigenstructure assignment in descriptor systems coordinate free conditions are given for pole assignment by feedback in linear descriptor singular systems which guarantee closed loop regularity these conditions are shown to be both necessary and sufficient for assignment of the maximum possible number of finite poles transformation to special coordinates are not used and the results provide a robust algorithm for the computation of the required feedback

technical communique regularizability of linear time invariant descriptor systems under decentralized control regularizability conditions for linear time invariant descriptor systems under decentralized output feedback are presented in this paper the regularizability is characterized by rank conditions that involve the original system matrices only new rank conditions for regularity of the system are also derived

invariant subspaces and invertibility properties for singular systems the general case abstract open loop definitions and properties of several subspaces for general singular systems are characterized by means of a fully algebraic distributional framework simple recursive algorithms for producing these spaces as well as related duality aspects turn out to follow directly from these definitions next we provide definitions and conditions for two notions of left right invertibility of a general singular system in terms of our distributions subspaces and rosenbrock s system matrix and we show which conditions represent the gap between our invertibility concepts finally we prove that in many cases left right invertibility is equivalent to left right invertibility of the system matrix

on the solution to the sylvester matrix equation av bw evf a simple direct complete and explicit parametric solution of the generalized sylvester matrix equation av bw evf is proposed for matrix f in the jordan form with arbitrary eigenvalues using the smith canonical form of the matrix a se b the obtained solution is in a clear neat form and may have important applications in descriptor linear system theory

a generalized state space for singular systems systems of the form e dot x ax bu y cx with e singular are studied of particular interest are the impulsive modes that may appear in the free response of such systems when arbitrary initial conditions are permitted modes that are associated with natural system frequencies at infinity a generalized definition of system order that incorporates these impulsive degrees of freedom is proposed and concepts of controllability and observability are defined for the impulsive modes allowable equivalence transformations of such singular systems are specified the present framework is shown to overcome several difficulties inherent in other treatments of singular systems and to extend in a natural and satisfying way many results previously known only for regular state space systems

a case study for the open question disturbance decoupling problem for singular systems by output feedback we study the disturbance decoupling problem for singular systems ex spl dot ax bu gd y cx z hx with left invertibility we give necessary and sufficient conditions for the existence of a solution to the disturbance decoupling problem with or without stability by output feedback that also makes the resulting closed loop system regular and or of index at most one all results are proved based on a condensed form that can be computed using orthogonal transformations i e can be computed in a numerically stable way

a tutorial on the geometric analysis of linear time invariant implicit systems abstract a background of recent work in nonregular and regular implicit systems of the form e x ax bu is provided then the output nulling unknown input and composite subspaces are defined for singular systems with recursions given for their computation inner subspaces that is those in the domain of e and a are distinguished from outer subspaces those in the codomain of e and a the notions of composite preimage and composite image are introduced the result is a framework that unifies some of the recent work in the geometric theory of implicit systems system properties are discussed for both regular and nonregular systems and duality is investigated

disturbance decoupling by behavioral feedback for linear differentialalgebraic systems we study disturbance decoupling for linear differentialalgebraic systems which are not necessarily regular compared to previous approaches where state feedback is used we use the concept of behavioral feedback which allows to study a larger class of systems we derive geometric characterizations for solvability of the disturbance decoupling problem following the classical approach exploiting the freedom in the choice of the behavioral feedback we show that whenever disturbance decoupling can be achieved by behavioral feedback we may additionally achieve autonomous zero dynamics finally we solve lebrets twenty year old open problem concerning disturbance decoupling with output uniqueness using behavioral feedback

disturbance decoupled observer design for descriptor systems in this paper we study the observer design problem for descriptor systems with partly unknown inputs we give necessary and sufficient conditions for the existence of a solution to the disturbance decoupled estimation problem with or without stable error spectrum requiring at the same time that the resulting combined system is regular and of index at most one all results are proved based on a condensed form that can be computed using orthogonal matrix transformations i e transformations that can be implemented in a numerically stable way

categorical representation theorems of fuzzy relations abstract this paper provides a notion of zadeh categories as a categorical structure formed by fuzzy relations with sup min composition and proves two representation theorems for dedekind categories relation categories with a unit object analogous to one point set and for zadeh categories without unit objects

decoupling of square singular systems via proportional state feedback necessary and sufficient conditions for the decoupling of a solvable square singular system ex t ax t bu t with output y t dx t through an admissible control law of the form u t kx t hr t where h is a square nonsingular matrix it has been shown that for a given singular system that satisfies these conditions a propagational state feedback exists for which the system s transfer function is a diagonal nonsingular and proper rational matrix the proofs of the main results are constructive and provide a procedure for computing an appropriate proportional state feedback

homomorphisms of modules associated with polynomial matrices with infinite elementary divisors abstract if the inverse of a nonsingular polynomial matrix l has a polynomial part then one can associate with l a module over the ring of proper rational functions which is related to the structure of l at infinity in this paper we characterize homomorphisms of such modules

the generalized eigenstructure problem in linear system theory the algebraic theory of linear time invariant systems has been studied in large detail during the past few decades and numerous computational algorithms have been developed to solve problems arising in this context in this paper the numerical aspects of a certain class of such algorithms dealing with what the author calls generalized eigenstructure problems are discussed some new and or modified algorithms are presented both the nmnerical stability of the algorithms and the conditioning of the problems they solve are analyzed using numerical criteria

the generalized ackermann s formula for singular systems abstract being an elegant algorithm for state feedback pole placement ackermann s 1972 formula had been widely quoted in control texts in this paper the formula is extended to solve the root assignment problem for singular systems without loss of generality it is both well known and convenient that any regular generalized system can be transformed into the standard form e x ax t bu t where e a i and is a real constant in the derivation of the generalized ackermann s formula the closed loop characteristic polynomial det se a bk is simplified due to the relationship of e and a if e is nonsingular the feedback gain k can be computed from the generalized ackermann s formula directly in this case only the desired closed loop characteristic polynomial is required if e is singular the feedback algorithm needs both closed loop and open loop characteristic polynomials two numerical examples are presented to demonstrate our algorithms

dedekind categories with cutoff operators dedekind categories provide a suitable categorical framework for lattice valued binary relations it is known that the notion of crispness cannot be described by the basic tools of this theory only in this paper we will study dedekind categories with a cutoff operator in order to circumvent this shortage we will introduce and investigate the properties of the operator and its relationship with other tools previously used for the same purpose the main result of this paper is a representation theorem for dedekind categories with a cutoff operator satisfying the point axiom

on the regularizability of singular systems the property of regularity of singular systems is not a feedback invariant to correct this deficiency a new property of regularizability is defined and geometric tests are given for it in terms of system matrices regularizability is shown to be the natural extension of regularity which is a condition on the homogeneous system to controlled singular systems definitions of controllability and reachability are modified depending on regularizability rather than regularity a brief comparison of proportional and proportional plus derivative feedback laws in the context of making the closed loop system regular regular and reachable and regular and controllable is also given dynamical interpretations of these properties are also presented

processing in relational structures fuzzy relational equations abstract the paper provides an overview of methodology for processing fuzzy information in relational structures completed in terms of fuzzy relational equations the origin and the central role of relational calculus in general and fuzzy sets in particular is explained with special emphasis paid to its role and representation capabilities it is pointed out that fuzzy relational equations play a significant role as a platform for a uniform development of techniques in fuzzy sets many of the problems and frameworks developed in fuzzy sets so far can be easily translated into the language of fuzzy relational equations which immediately takes a significant advantage of their solid well developed formalisms techniques and clarity of exposition we will study the role of approximate solutions to fuzzy relational equations as a convenient tool to handle probabilistic type of uncertainty moreover those solutions can easily explain the origin and generate a way in which fuzzy sets of higher generality such as for instance type 2 sets interval valued sets can be algorithmically determined finally we study the role of equations in diverse fields of applications making use of an ample framework of general systems theory

a solution to the disturbance decoupling problem in singular systems via analogy with state space systems abstract this paper presents an approach for solving the disturbance decoupling problem in a square linear time invariant singular system a simple criterion and an efficient algorithm for solving the underlying problem by applying a state feedback which preserves the uniqueness property of the closed loop singular system has been established the present algorithm is based on a well established approach borrowed from state space system theory

deflation in eigenvalue assignment of descriptor systems using state feedback it is shown how deflation by unitary equivalence transformations can be used to solve the eigenvalue assignment problem for descriptor systems using state feedback the system is first transformed to a form that reveals its controllability the uncontrollable part of the system if any is then discarded and the assignments are continued with the completely controllable part of the system to give insight into the approach a mathematical method for the solution is derived without considering numerical issues using the idea of deflation it is shown how this method can be modified to derive a numerically efficient algorithm the numerical stability of the algorithm is proved and a numerical example is presented

disturbance decoupling with closed loop modes stability in switched linear systems this work deals with the problem of rendering the output of a switched linear system insensitive to a disturbance input with the requirement that the dynamics of the closed loop system be exponentially stable for all switching signals with a sufficiently large dwell time a necessary and sufficient condition for the problem to have a solution is shown a complete computational procedure to check the solvability condition and to synthesize the compensation scheme is provided a worked out example is presented with the aim of showing the effectiveness of the devised method

robust pole assignment in linear state feedback numerical methods are described for determining robust or well conditioned solutions to the problem of pole assignment by state feedback the solutions obtained are such that the sensitivity of the assigned poles to perturbations in the system and gain matrices is minimized it is shown that for these solutions upper bounds on the norm of the feedback matrix and on the transient response are also minimized and a lower bound on the stability margin is maximized a measure is derived which indicates the optimal conditioning that may be expected for a particular system with a given set of closed loop poles and hence the suitability of the given poles for assignment

geometric aspects of reduced order compensators for disturbance rejection the disturbance decoupling problem with stability constraint and pole placement is examined in a geometric framework two types of reduced order compensators are proposed one based on the observation scheme the other on the dual observation scheme the novelty lies mainly in these points 1 the avoidance of eigenspace computations in the syntheses as they rely on the determination of the maximum controlled invariant contained in a given subspace and its dual i e the minimum conditioned invariant containing a given subspace and pole assignability and 2 the concerted use of controlled and conditioned invariants which leads to a straightforward synthesis and also allows a clear interpretation of the dual compensation

on disturbance decoupling in descriptor systems the disturbance decoupling problem for a linear multivariable control system is to determine when and how a state feedback may be chosen so that some class of disturbances has no effect on the system output the development of necessary and sufficient conditions for the solvability of this problem for state space systems by wonham and his co workers has profoundly influenced geometric control theory particularly by means of the concept of an a b invariant subspace this paper develops necessary and sufficient conditions for solvability of this problem for linear systems described by a mixture of algebraic and differential equations although the argument presented has the same structure as in the state space case and incorporates a generalisation of a b invariant subspaces to descriptor systems there are some additional issues demanding particularly careful attention is the ensuring of existence and uniqueness of classical solutions of the underlying differential equations which cannot be tak

pole assignment in a specified disk the problem of assigning all poles of a closed loop system in a specified disk by state feedback is considered for both continuous and discrete systems a state feedback control law is determined by using a discrete riccati equation this kind of pole assignment problem is named d pole assignment and its relation to the optimal control problem and its robustness properties are discussed the gain and phase margins for all closed loop poles to stay inside the specified disk d are determined for the proposed control

eigenvalue placement for generalized linear systems this paper deals with some aspects of eigenvalue placement by state feedback for generalized linear systems described by e ax bu where e is a singular map it is shown that controllability of the infinite eigenvalues of the pencil se a is equivalent to the existence of a state feedback map which assigns those eigenvalues to pre specified complex numbers a procedure for the assignment of all eigenvalues to distinct complex numbers is also discussed

a reduced order observer for descriptor systems abstract in this paper a reducer order observer is analyzed for so called generalized state space or descriptor systems based on the staircase form for generalized state space models a recursive algorithm is presented to construct a reduced order observer for a given observable descriptor system in the algorithm only the eigenfrequencies of the observer have to be specified

a cubic defining algebra for the links gould polynomial abstract we define a finite dimensional cubic quotient of the group algebra of the braid group endowed with a essentially unique markov trace which affords the links gould invariant of knots and links we investigate several of its properties and state several conjectures about its structure

singular systems a new approach in the time domain a new approach in the time domain is developed for the study of singular linear systems of the form e dot x ax bu y cx with e singular central to the approach is the fundamental triple alpha e a 1 e alphae a 1 b c with a real number satisfying det alpha e a neq 0 controllability and observability matrices are expressed in terms of the fundamental triple new tests for impulse controllability and impulse observability are also established feedback control problems including pole placement decoupling and disturbance localization are studied by use of a modified proportional and derivative feedback of the state in the form of u f alpha x dot x v

a geometric approach to eigenstructure assignment for singular systems moore s well known result on the feedback assignment of eigenstructure in linear state space systems 5 was generalized to descriptor systems in 9 in this note we make some useful extensions to fill out the theory to begin with we do not assume controllability of the plant it is shown that the maximum number of finite closed loop eigenvalues is equal to dim el where l is the supremal a e r b invariant subspace in order to relate the possible closed loop structure to the open loop plant structure we define a set of indexes for descriptor systems which are in many respects similar to the controllability indexes of state space systems in terms of these controllability indexes we give a systematic approach to selecting the closed loop eigenvectors a major contribution of this note is the fact that it is geometric in nature and hence avoids any decomposition of the system matrices into a special form e g weierstrass form 1 the geometric setting introduced here should constitute a basis for further research in generalizing the well known results in geometric system theory disturbance decoupling input output decoupling etc to singular systems

output feedback of forward backward periodic systems stability an n periodic forward backward system defined by a collection of nonproper rational matrices is considered in addition an n periodic collection of output feedbacks such that the closed loop system is n periodic and forward backward is constructed block ordered coprime factorizations of a rational matrix are used in the stabilization of that closed loop system

remarks on controllability of implicit linear discrete time systems abstract some results concerning controllability of implicit linear discrete time system of the form ex k 1 fx k gu k are given it is not assumed that the corresponding pencil se f is regular the concept of x i x f controllability is defined and it is shown that it generalizes various definitions of controllability used in the existing literature of the subject computable geometric conditions for an implicit system to be x 1 x f controllable are given some particular cases are discussed the existence of a universal time of controllability is also proven

solution to generalized sylvester matrix equations an explicit solution to the generalized sylvester matrix equation ax by exf is established this solution is expressed in terms of the r controllability matrix of e a b a generalized symmetric operator matrix and an observability matrix moreover based on this solution solutions to some other matrix equations are also derived the results may provide great convenience for the analysis and synthesis problems related to these equations

observers for singular systems the following three major aspects of the design of observers for linear time invariant singular systems are resolved the necessary and sufficient conditions for the problem to have a solution the order of the minimal observer and the properties of the closed loop system separation principle an algorithm is presented for the construction of the minimal order observer

descriptor systems decomposition into forward and backward subsystems it is shown that a decomposition of the descriptor space with certain properties induces a decomposition of the descriptor system into forward and backward subsystems different decompositions correspond to different apportionments of the required boundary conditions between initial and final times applications include several different but equivalent methods for solving the descriptor system and a clarification of the relations between some previously known methods of solution

proportional and derivative state feedback decoupling of linear systems we consider the row by row decoupling of linear time invariant systems by proportional and derivative state feedback our contribution with respect to previous results is that our procedure is based on simple operator say matrices manipulations without any need to use a canonical form the only assumptions for applying such a decoupling strategy are that the system is right invertible which is a necessary condition to ensure solvability and minimum phase an illustrative example is proposed

on the quotients of cubic hecke algebras between the rank 3 quotients of cubic hecke algebras there is essentially one of maximal dimension we prove it has a unique markov trace having values in a torsion module therefore the description of a markov trace on the cubic hecke algebra corresponding tox 3 1 and having the parameters 1 1 is derived thus we obtain a numerical link invariant of finite degree and define a whole sequence of 3rd order vassiliev invariants

observers for discrete singular systems the author considers the design of observers for the discrete singular system ex k 1 ax k bu k y k cx k placing special emphasis on the problems of state reconstruction and minimal time state reconstruction it is shown that for a singular system finite poles can be moved to infinity by state feedback and the state can be reconstructed by causal observers

realization and partial fractions abstract we discuss the relation between two intrinsically different proposals that have been made in the literature concerning the representation by constant matrices of rational matrices given in fractional form it turns out that the relation is most naturally studied in the framework of partial fraction decompositions we develop the realization theory for decompositions with respect to arbitrary complementary parts of the extended complex plane which may for instance correspond to stability and instability an isomorphism is obtained which connects the spaces used in the two methods and several identities relating to the mcmillan degree are derived in a direct way finally a new computational procedure is given to obtain the partial fraction decomposition of a rational matrix given in fractional form

an approach for pole assignment in singular systems pole assignment in a singular system edx dt ax bu is discussed it is shown that the problem of assigning the roots of det se a bf by applying a proportional feedback u fx r in a given singular system is equivalent to the problem of pole assignment of an appropriate regular system an immediate application of this result is that procedures and computational algorithms that were originally developed for assigning eigenvalues in regular systems become useful tools for pole assignment in singular systems the approach provides a useful tool for the combined problem of eliminating impulsive behavior and stabilizing a singular system

a simple procedure for the stabilization of a class of uncontrollable generalized systems this paper gives a simple procedure for the stabilization with a prescribed rate of convergence of the free motions of a class of possibly single input uncontrollable generalized continuous time time invariant linear systems preserving the impulse free property if any