practical internal stability of n d discrete systems this paper deals with the problem of asymptotic stability for n d discrete systems in the practical sense that the system input and output signals are unbounded in at most one dimension a definition of practical internal stability is introduced and necessary and sufficient conditions are derived the obtained results show that practical internal stability is less restrictive and more relevant for practical applications than the conventional two dimensional 2 d internal stability

parametrization of all strictly causal stabilizing controllers within the framework of the factorization approach we present parametrizations of all strictly causal stabilizing controllers

design parametrization and pole placement of stabilizing output feedback compensators via injective cogenerator quotient signal modules control design belongs to the most important and difficult tasks of control engineering and has therefore been treated by many prominent researchers and in many textbooks the systems being generally described by their transfer matrices or by rosenbrock equations and more recently also as behaviors our approach to controller design uses in addition to the ideas of our predecessors on coprime factorizations of transfer matrices and on the parametrization of stabilizing compensators a new mathematical technique which enables simpler design and also new theorems in spite of the many outstanding results of the literature 1 we use an injective cogenerator signal module f over the polynomial algebra d f s f an infinite field a saturated multiplicatively closed set t of stable polynomials and its quotient ring dt of stable rational functions this enables the simultaneous treatment of continuous and discrete systems and of all notions of stability called t stability we investigate stabilizing control design by output feedback of input output io behaviors and study the full feedback io behavior especially its autonomous part and not only its transfer matrix 2 the new technique is characterized by the permanent application of the injective cogenerator quotient signal module dtft and of quotient behaviors bt of df behaviors b 3 for the control tasks of tracking disturbance rejection model matching and decoupling and not necessarily proper plants we derive necessary and sufficient conditions for the existence of proper stabilizing compensators with proper and stable closed loop behaviors parametrize all such compensators as io behaviors and not only their transfer matrices and give new algorithms for their construction moreover we solve the problem of pole placement or spectral assignability for the complete feedback behavior the properness of the full feedback behavior ensures the absence of impulsive solutions in the continuous case and that of the compensator enables its realization by kalman state space equations or elementary building blocks we note that every behavior admits an io decomposition with proper transfer matrix but that most of these decompositions do not have this property and therefore we do not assume the properness of the plant 4 the new technique can also be applied to more general control interconnections according to willems in particular to two parameter feedback compensators and to the recent tracking framework of fiaz takaba trentelman in contrast to these authors however we pay special attention to the properness of all constructed transfer matrices which requires more subtle algorithms

polynomial matrices with given determinant abstract let a be a q n polynomial matrix 1 q a if suitable polynomials 1 n exist such that one can write 1 a i 1 n a i n where ai 1 ai n is one row in a and 1 n is a unimodular row then a can be row bordered to an n n polynomial matrix m with det m some examples are discussed

functional t observers abstract this article is a contribution to behavioral observer theory which was started by valcher and willems in 1999 and which was recently exposed by fuhrmann in a comprehensive survey article it is also a further development of the article on t observers by oberst and the author for a given continuous or discrete time linear time invariant behavior we assume that a linear function of a trajectory e g some components can be measured we are interested in estimating another linear function of this trajectory we generalize the notions of t observability and t observers introduced by oberst and the author t denotes a multiplicatively closed subset of the ring of operators for different choices of t t observability coincides with observability reconstructibility trackability or detectability a t observer is an exact dead beat tracking or asymptotic observer we show the equivalence of t observability and the existence of t observers and give a constructive parametrization of all t observers corresponding results for proper t observers are also presented partial observation of the state of a kalman state space system compare e g fuhrmann s work is a special case of our setting and so are the observers of certain unknown components of a behavior studied by bisiacco valcher and willems the first result on functional observers in context with rosenbrock equations or polynomial matrix descriptions is due to wolovich 1974

a two variable approach to solve the polynomial lyapunov equation abstract a two variable polynomial approach to solve the one variable polynomial lyapunov equation is proposed lifting the problem from the one variable to the two variable context allows to use faddeev type recursions in order to solve the polynomial lyapunov equation in an iterative fashion the method is especially suitable for applications requiring exact or symbolic computation

feedback stabilization over commutative rings the matrix case this paper provides a solution of the feedback stabilization problem over commutative rings for matrix transfer functions stabilizability of a transfer matrix is realised as local stabilizability over the entire spectrum of the ring for stabilizable plants certain modules generated by its fractions and that of the stabilizing controller are shown to be projective compliments of each other in the case of rings with irreducible spectrum this geometric relationship shows that the plant is stabilizable if and only if the above modules of the plant are projective of ranks equal to the number of inputs and the outputs if the maxspectrum of the ring is noetherian and of zero krull dimension then this result shows that the stabilizable plants have doubly coprime fractions over unique factorization domains the above stabilizability condition is interpreted in terms of matrices formed by the fractions of the plant certain invariants of these matrices known as elementary factors are defined and it is shown that the plant is stabilizable if and only if these elementary factors generate the whole ring this condition thus provides a generalization of the coprime factorizability as a condition for stabilizabilty a formula for the class of all stabilizing controllers is then developed that generalizes the previous well known formula in factorization theory for multidimensional transfer functions these results provide concrete conditions for stabilizabilty finally it is shown that the class of polynomial rings over principal ideal domains is an additional class of rings over which stabilizable plants always have doubly coprime fractions

the local euler obstruction and topology of the stabilization of associated determinantal varieties this work has two complementary parts in the first part we compute the local euler obstruction of generic determinantal varieties and apply this result to compute the chern schwartz macpherson class of such varieties in the second part we compute the euler characteristic of the stabilization of an essentially isolated determinantal singularity eids the formula is given in terms of the local euler obstruction and gaffney s m d multiplicity

observer synthesis in the behavioral approach analyzes the observer design problem in the behavioral context observability and detectability notions are first introduced and fully characterized necessary and sufficient conditions for the existence of an observer possibly an asymptotic or an exact one are introduced and a complete parameterization of all admissible observers is given the problem of obtaining observers endowed with a strictly proper transfer matrix and the design of observer based controllers are later addressed and solved finally the above issues are particularized to the case of state space systems thus showing they naturally generalize well known theorems of traditional system theory

regular implementation in the space of compactly supported functions abstract this article extends results on regular implementability in p rocha canonical controllers and regular implementation of nd behaviors in proceedings of the 16th ifac world congress 2005 and h l trentelman d napp avelli on the regular implementability of nd systems systems control lett 56 4 2007 265 271 to the case when the signal space is not an injective cogenerator for instance the space d of compactly supported smooth functions on r n in this case the bijective correspondence between behaviors and modules fails to hold also projections and sums of behaviors need not in general be behaviors a more general version of implementability is introduced and necessary and sufficient conditions are established for the implementation and regular implementation of a given desired behavior

envelopes of holomorphy and holomorphic discs the envelope of holomorphy of an arbitrary domain in a two dimensional stein manifold is identified with a connected component of the set of equivalence classes of analytic discs immersed into the stein manifold with boundary in the domain this implies in particular that for each of its points the envelope of holomorphy contains an embedded non singular riemann surface and also an immersed analytic disc passing through this point with boundary contained in the natural embedding of the original domain into its envelope of holomorphy moreover it says that analytic continuation to a neighbourhood of an arbitrary point of the envelope of holomorphy can be performed by applying the continuity principle once another corollary concerns representation of certain elements of the fundamental group of the domain by boundaries of analytic discs a particular case is the following given a contact three manifold with stein filling any element of the fundamental group of the contact manifold whose representatives are contractible in the filling can be represented by the boundary of an immersed analytic disc

2d systems feedback compensation an approach based on commutative linear transformations abstract algebraic properties of a pair of commutative matrices associated with an ideal in r z 1 z 2 are exploited for characterizing the closed loop polynomial variety of a 2d system also algorithms are given to find under what constraints the closed loop variety can be assigned and to compute the mfd of a compensator

conditions for time controllability of behaviours this paper develops a behavioural framework to study controllability of systems whose dynamics are described by systems of linear constant coefficient partial differential equations questions about the notion of controllability are addressed with special importance given to the time evolution algebraic conditions concerning time controllability are proved

semi cancellable fractions in system theory in this paper we propose to define the transfer function of a system in a new way namely as a semi cancellable fraction scf i e a fraction in which unstable pole zero cancellations are not allowed within this new framework it is possible to be rigorous and coherent in dealing with the stabilizability of a system from an input output viewpoint not focusing on the stabilizability of infinite dimensional input output systems

a behavioral approach to classic polynomial matrix descriptions abstract in this note we aim to show how the well known definitions and results obtained for the classic polynomial matrix descriptions pmds of a dynamic system kailath linear systems 1980 rosenbrock state space and multivariable theory 1970 wolovich linear multivariable systems 1974 can be seen as particular instances of the analogous definitions and results obtained within the more general behavioral setting more precisely pmds of dynamic systems correspond to a class of linear time invariant dynamic systems that we will call extended state space models for these models internal stability as well as the structural properties can be defined and fully characterized by applying the definitions and criteria introduced within the behavioral approach as a result we will obtain the main results of the pmd s approach also the control problem and the observer design problem are here stated in behavioral terms and solved finally the introduction of the dual system of an extended state space model completes the analysis

zero time controllability and dead beat control of two dimensional behaviors in this paper two dimensional 2d discrete behaviors defined on the grid mathbb z times mathbb z and having the time as first independent variable are investigated for these behaviors by emphasizing the causality notion that is naturally associated with the time variable we introduce two new concepts of controllability algebraic characterizations of time controllability and of zero time controllability are provided and it is shown that behaviors endowed with these properties admit special decompositions next the dead beat control dbc problem and the concept of admissible dbc are introduced and related to the zero time controllability property differently from what happens with one dimensional behaviors zero time controllability does not ensure the existence of regular dbcs and stronger algebraic properties need to be imposed on the behavior finally necessary and sufficient conditions for the existence of a dbc that makes the resulting behavior both strongly autonomous and nilpot

time autonomy versus time controllability we study systems whose dynamics are described by systems of linear constant coefficient partial differential equations in a behavioural framework questions about the notion of autonomy and controllability are addressed with special importance given to the time evolution we investigate consequences of time autonomy for time controllability

notes on the definition of behavioural controllability we take another look at the behavioural definition of controllability for discrete 1d systems and its extension to multidimensional nd systems defined on z n or n n we suggest that the current definition for nd systems is inappropriate for systems defined on the domain n n and that care has to be taken even with the 1d definition we propose a new definition which applies to all standard classes of discrete nd systems

positive extensions fejer riesz factorization and autoregressive filters in two variables in this paper we treat the two variable positive extension problem for trigonometric polynomials where the extension is required to be the reciprocal of the absolute value squared of a stable polynomial this problem may also be interpreted as an autoregressive filter design problem for bivariate stochastic processes we show that the existence of a solution is equivalent to solving a finite positive definite matrix completion problem where the completion is required to satisfy an additional low rank condition as a corollary of the main result a necessary and sufficient condition for the existence of a spectral fejerriesz factorization of a strictly positive two variable trigonometric polynomial is given in terms of the fourier coefficients of its reciprocal tools in the proofs include a specific two variable kronecker theorem based on certain elements from algebraic geometry as well as a two variable christoffel darboux like formula the key ingredient is a matrix valued polynomial that appears in a parametrized version of the schur cohn test for stability the results also have consequences in the theory of two variable orthogonal polynomials where a spectral matching result is obtained as well as in the study of inverse formulas for doubly indexed toeplitz matrices finally numerical results are presented for both the autoregressive filter problem and the factorization problem

solving linear systems via pfaffians abstract an explicit solution is given for the system of linear equations e e t where are alternating matrices of pfaffian 1 which at most differ in their first row and column and e is of the form i 2 2 r e 1 i x i if v v w r r 1 with v w 1 v w then a sequence of cohn transforms with respect to the fixed w which takes v w to v w is prescribed

paradigms and puzzles in the theory of dynamical systems a self contained exposition is given of an approach to mathematical models in particular to the theory of dynamical systems the basic ingredients form a triptych with the behavior of a system in the center and behavioral equations with latent variables as side panels the author discusses a variety of representation and parametrization problems in particular questions related to input output and state models the proposed concept of a dynamical system leads to a new view of the notions of controllability and observability and of the interconnection of systems in particular to what constitutes a feedback control law the final issue addressed is that of system identification it is argued that exact system identification leads to the question of computing the most powerful unfalsified model

stabilisation and control design by partial output feedback and by partial interconnection we study stabilisation and control design by partial output feedback on the one hand and by partial interconnection according to willems on the other hand this article is based on the paper design parametrization and pole placement of stabilizing output feedback compensators via injective cogenerator quotient signal modules by i blumthaler and u oberst and the same technique is used both discrete and continuous behaviours and various notions of stability are treated simultaneously in the case of partial output feedback we require that both the compensator and the feedback behaviour have proper transfer matrices without presuming properness of the plant we obtain conditions ensuring the existence of such stabilising compensators which perform additional control tasks like for instance tracking and an algorithm for constructing a large class of such compensators in the case of stabilisation by interconnection no input output structures of plant or compensator are assumed but the controlled be

a behavioral approach to control of distributed systems this paper develops a theory of control for distributed systems i e those defined by systems of constant coefficient partial differential operators via the behavioral approach of willems the study here is algebraic in the sense that it relates behaviors of distributed systems to submodules of free modules over the polynomial ring in several indeterminates as in the lumped case behaviors of distributed arma systems can be reduced to ar behaviors this paper first studies the notion of ar controllable distributed systems following the corresponding definition for lumped systems due to willems it shows that as in the lumped case the class of controllable ar systems is precisely the class of ma systems it then shows that controllable 2 d distributed systems are necessarily given by free submodules whereas this is not the case for n d distributed systems n ge 3 this therefore points out an important difference between these two cases this paper then defines two notions of autonomous distributed systems which mimic different properties of lumped autonomous systems control is the process of restricting a behavior to a specific desirable autonomous subbehavior a notion of stability generalizing bounded input bounded output stability of lumped systems is proposed and the pole placement problem is defined for distributed systems this paper then solves this problem for a class of distributed behaviors

constrained traffic equilibrium in routing we study noncooperative routing in which each user is faced with a multicriterion optimization problem formulated as the minimization of one criterion subject to constraints on others we address the questions of existence and uniqueness of equilibrium we show that equilibria indeed exist but uniqueness may be destroyed due to the multicriteria nature of the problem we obtain uniqueness in some weaker sense under appropriate conditions we show that the link utilizations are uniquely determined at equilibrium we further study the normalized constrained equilibrium and apply it to pricing

on the strong stabilizability of mimo n dimensional linear systems a plant is strongly stabilizable if there exists a stable compensator to stabilize it based on some theorems in complex analysis of several variables proved in this paper we present necessary conditions for the strong stabilizability of complex and real n d multi input multi output mimo shift invariant linear plants for the real case the condition is a generalization of the parity interlacing property of youla bongiorno and lu automatica j ifac 10 1974 pp 159 173 for the strong stabilizability of a real one dimensional mimo plant these conditions are also sufficient for the cases of n d plants with a single output miso or with a single input simo for general n d mimo plants we do not know if the conditions are sufficient or not a useful sufficient but not necessary condition for the strong stabilizability of a class of n d n geq 2 mimo plants is given

controller parameterization of anantharam s example we consider the controller parameterization of anantharam s example 1985 it is shown that the parameterization is achieved by using only one parameter even though the plant does not have a coprime factorization

brief paper generalized eigenvalue based stability tests for 2 d linear systems necessary and sufficient conditions this paper studies the stability of 2 d dynamic systems we consider systems characterized by 2 d polynomials and 2 d state space descriptions for each description we derive necessary and sufficient stability conditions which all require only the computation of a constant matrix pencil the stability of the underlying system can then be determined by inspecting the generalized eigenvalues of the matrix pencil the results consequently yield 2 d stability tests that can be checked both efficiently and with high precision additionally frequency sweeping tests are also obtained which complement the matrix pencil tests and are likely to be more advantageous analytically

a behavioral approach to estimation and dead beat observer design with applications to state space models the observer design problem is investigated in the context of linear left shift invariant discrete behaviors whose trajectories have supports on z necessary and sufficient conditions for the existence of a dead beat observer of some relevant variables from some measured ones in the presence of some unmeasured and irrelevant variables are introduced and a complete parametrization of all dead beat observers is given equivalent conditions for the existence of causal dead beat observers are then derived finally several classical problems addressed for state space models like state estimation the design of unknown input observers or the design of fault detectors and identifiers possibly in the presence of disturbances are cast in this general framework and the aforementioned equivalent conditions and parametrizations are specialized to these cases

time relevant stability of 2d systems for many 2d systems one of the independent variables plays a distinct role in the evolution of the trajectories since often this special independent variable is time we call such systems time relevant in this paper we introduce a stability notion for time relevant systems described by higher order difference equations we give algebraic tests in terms of the location of the zeros of the determinant of a polynomial matrix describing the system we also give an lmi characterization of time relevant stability involving only constant matrices

comments on a robust mimo terminal sliding mode control scheme for rigid robotic manipulators the authors state that there are two errors in the above paper zhihong et al ibid vol 39 no 12 p 2464 9 1994 the purpose of this note is to give correct versions of remark 2 1 and remark 3 3

dead beat control in the behavioral approach in this paper the concepts of controllability and zero controllability of a variable w appearing either in a standard or in a latent variable description as manifest variable are introduced and characterized by assuming this perspective the dead beat control dbc problem is posed as the problem of designing a controller involving both w and the latent variable c such that for the resulting controlled behavior the variable w goes to zero in a finite number of steps in every trajectory zero controllability of w turns out to be a necessary and sufficient condition for the existence of admissible dbcs as well as for the existence of regular dbcs the class of minimal dbcs namely dbcs with the least possible number of rows is singled out and a parametrization of such controllers is provided finally a necessary and sufficient condition for the existence of dbcs that can be implemented via a feedback law for which w is the input and the latent variable c the corresponding output is provided

algebraic geometric aspects of feedback stabilization this paper develops a theory of feedback stabilization for siso transfer functions over a general integral domain which extends the well known coprime factorization approach to stabilization necessary and sufficient conditions for stabilizability of a transfer function in this general setting are obtained these conditions are then refined in the special cases of unique factorization domains ufds noetherian rings and rings of fractions it is shown that these conditions can be naturally interpreted geometrically in terms of the prime spectrum of the ring this interpretation provides a natural generalization to the classical notions of the poles and zeros of a plant the set of transfer functions is topologized so as to restrict to the graph topology of vidyasagar ieee trans automatic control ac 29 1984 pp 403 418 when the ring is a bezout domain it is shown that stability of a feedback system is robust in this topology when the ring is a ufd this theory is then applied to the problem of sta

on quadratic differential forms this paper develops a theory around the notion of quadratic differential forms in the context of linear differential systems in many applications we need to not only understand the behavior of the system variables but also the behavior of certain functionals of these variables the obvious cases where such functionals are important are in lyapunov theory and in lq and h infty optimal control with some exceptions these theories have almost invariably concentrated on first order models and state representations in this paper we develop a theory for linear time invariant differential systems and quadratic functionals we argue that in the context of systems described by one variable polynomial matrices the appropriate tool to express quadratic functionals of the system variables are two variable polynomial matrices the main achievement of this paper is a description of the interaction of one and two variable polynomial matrices for the analysis of functionals and for the application of higher order lyapunov functionals

a convex characterization of multidimensional linear systems subject to sqi constraints this technical note proposes a convex characterization of the set of all stable closed loop linear systems that are obtained from a given plant which may be multidimensional by interconnecting it in feedback with controllers that satisfy a certain pre selected constraint we take an approach that is particularly useful when a doubly coprime factorization of the plant is difficult to obtain and a stable stabilizing controller may not exist the plant is not strongly stabilizable or when one may be difficult to find most related work requires one of these we adopt the so called coordinate free approach which unlike youla s parametrization does not rely on a doubly coprime factorization of the plant we show that if constraints which satisfy a condition called strong quadratic invariance sqi are imposed on the controllers then the set of all stable closed loop multidimensional linear systems has a convex representation and norm optimal control problems can be cast in convex form although the sqi condition is in general slightly stronger than quadratic invariance qi which was developed in related work they are equivalent for common classes of problems arising in decentralized control

a generalized tracking and disturbance rejection problem for multidimensional behaviors in this paper we study a generalized tracking and disturbance rejection problem for multidimensional behaviors given a multidimensional plant our first goal is to design a compensator to be connected to the plant through regular partial interconnection in such a way that the overall controlled system is autonomous and stable when no exogenous signal acts on the system on the other hand when exogenous signals affect the controlled system evolution we want to impose that a suitable linear combination of the overall system trajectories is negligibile in a sense we will clarify within the paper this problem setup formalizes a number of classical control problems first of all tracking of some reference signal together with rejection of another disturbance signal the adopted approach is extremely general and it is based on the idea of describing all behavior trajectories as the sum of a transient signal and a steady state component a decomposition that relies on gabriel s localization theor

stability and stabilization of multidimensional input output systems in this paper we discuss stability and stabilization of continuous and discrete multidimensional input output io behaviors of dimension r which are described by linear systems of complex partial differential resp difference equations with constant coefficients where the signals are taken from various function spaces in particular from those of polynomial exponential functions stability is defined with respect to a disjoint decomposition of the r dimensional complex space into a stable and an unstable region with the standard stable region in the one dimensional continuous case being the set of complex numbers with negative real part a rational function is called stable if it has no poles in the unstable region an io behavior is called stable if the characteristic variety of its autonomous part has no points in the unstable region this is equivalent to the stability of its transfer matrix and an additional condition the system is called stabilizable if there is a compensator io system such that the output feedback system is well posed and stable we characterize stability and stabilizability and construct all stabilizing compensators of a stabilizable io system parametrization the theorems and proofs are new but essentially inspired and influenced by and related to the stabilization theorems concerning multidimensional io maps as developed for instance by bose guiver shankar sule xu lin ying zerz and quadrat and of course the seminal papers of vidyasagar youla and others in the one dimensional case in contrast to the existing literature the theorems and proofs of this paper do not need or employ the so called fractional representation approach i e various matrix fraction descriptions of the transfer matrix thus avoiding the often lengthy matrix computations and seeming to be of interest even for one dimensional systems at least to the author an important mathematical tool new in systems theory is gabriel s localization theory which only in the case of ideal convex shankar sule unstable regions coincides with the usual one algorithmic tests for stability stabilizability and ideal convexity and the algorithmic construction of stabilizing compensators are addressed but still encounter many difficulties see in particular the open problems listed by xu e

a new parametrization of linear observers the estimation of unknown variables of a system the plant based on a set of known or measured variables is a fundamental and widely studied problem of system theory we investigate observers that solve this task in the behavioral setting our framework includes the discrete and the continuous linear time invariant cases and various notions of when a signal is considered to be small we characterize the observers that guarantee a small estimation error as those containing the essential part of the plant behavior a constructive one to one parametrization of all such observers follows naturally from this result

time autonomy and time controllability of discrete multidimensional behaviours predecessors and starting point of the present paper on multidimensional discrete time autonomy and time controllability were three papers from 2002 2004 by sasane controneo thomas and willems on the continuous case a paper from 2011 by napp rapisarda and rocha on time relevant 2d behaviors and a submitted paper by bisiacco and valcher on 2d time controllability and dead beat control we also acknowledge interesting talks on this subject by bisiacco and valcher in innsbruck we develop a general framework for discrete multidimensional behaviors in which the discrete time variable in n plays a special part among all independent variables following the models of our predecessors we define time autonomy and time controllability of a behavior by properties of its trajectories goal of the paper is the characterization of these pr operties by constructive algebraic conditions which is fully achieved for time autonomy and for timecontrollability in dimension two and partially for time co ntrollability in higher dimensions than two

regular almost interconnection of multidimensional behaviors reading the doctoral thesis of napp avelli 2007 i realized that gabriel s localization theory which i applied in context with the stabilization of multidimensional input output behaviors can also be used for the construction of regular almost interconnections of behaviors in arbitrary dimensions and not only in two dimensions in this paper i expose this theory in the language of quotient modules and derive an algorithm for arbitrary dimensions which has however not yet been implemented regular interconnections were introduced and discussed by willems ieee trans automat control 36 1991 pp 259 294 42 1997 pp 458 472 for one dimensional behaviors their multidimensional counterparts have been treated by rocha wood shankar zerz lomadze napp avelli and others since 1998 two dimensional almost direct sum decompositions and regular almost interconnections have been considered by valcher and bisiacco since 2000 and are also the main subject of napp avelli s thesis and his recent submitted paper roughly two behaviors are almost equal if they differ by finite dimensional behaviors only so the latter are considered negligible in this context two dimensional behaviors have special properties which are not discussed in the present paper i also briefly discuss other variants of regular almost interconnections where only stable autonomous behaviors are considered negligible

feedback control of multidimensional behaviors abstract in this paper we consider the problem of controlling an nd system by means of a feedback connection first we characterize the system interconnections that can be implemented as feedback connections by means of a suitable choice of input output structures then we give a necessary and sufficient condition for the possibility of obtaining a desired control performance b d b from a given nd behavior b by a feedback connection with a suitable controller in case both the input output structures of b and b d are a priori imposed our results generalize those obtained in willems ieee trans automat control 42 4 1997 458 for the 1d case

statistics of interior current distributions in two dimensional open chaotic billiards the probability current statistics of two dimensional open chaotic ballistic billiards is studied both analytically and numerically assuming that the real and imaginary parts of the scattering wave function are both random gaussian fields we find a universal distribution function for the interior probability current as a by product we recover previous analytic forms for wave function statistics the expressions bridge the entire region from goe to gue type statistics our analytic expressions are verified numerically by explicit quantum mechanical calculations of transport through a bunimovich billiard

parameterization of stabilizing controllers with either right or left coprime factorization a parameterization method of stabilizing controllers that requires only one of right left coprime factorizations is presented as an illustrative model we will consider the multidimensional system with structural stability for which it is not yet known in general whether or not a doubly coprime factorization always exists a parameterization of stabilizing controllers for the model is presented

multidimensional discrete stability by serre categories and the construction and parametrization of observers via gabriel localizations immediate predecessors of this work were a paper on two dimensional deadbeat observers by bisiacco and valcher multidimens systems signal process 19 2008 pp 287 306 and one on one dimensional functional observers by blumthaler linear algebra appl 432 2010 pp 1560 1577 compare also fuhrmann s comprehensive paper linear algebra appl 428 2008 pp 44 136 the present paper extends blumthaler s results to continuous or discrete multidimensional behaviors i e constructs and parametrizes all controllable observers of a given multidimensional behavior and for this purpose also discusses the required multidimensional stability such an observer produces a signal that approximates or estimates a desired component of the behavior such that the signal difference is negligible in a suitable sense this definition thus presupposes that of negligible or stable autonomous systems in the standard one dimensional case these are the asymptotically stable behaviors we define and investigate

feedback stabilization over commutative rings further study of the coordinate free approach this paper is concerned with the coordinate free approach to control systems the coordinate free approach is a factorization approach but does not require the coprime factorizations of the plant we present two criteria for feedback stabilizability for multi input multi output mimo systems in which transfer functions belong to the total rings of fractions of commutative rings both of them are generalizations of sule s results in siam j control optim 32 1994 pp 1675 1695 the first criterion is expressed in terms of modules generated from a causal plant and does not require the plant to be strictly causal it shows that if the plant is stabilizable the modules are projective the other criterion is expressed in terms of ideals called generalized elementary factors this gives the stabilizability of a causal plant in terms of the coprimeness of the generalized elementary factors as an example a discrete finite time delay system is considered

stability of distributed 3 d systems implemented on grid sensor networks using the givone roesser and the fornasini marchesini state space models for m d systems a method for distributed information processing in rectangular grid based wireless sensor networks is presented it can be used for distributed implementation of any general linear system on a grid sensor network nonlinearities caused by finite precision number representation schemes used for in node computations and inter node communication are modeled stability of the system is analyzed with special consideration given to the influence of inter node communication on system dynamics a necessary and sufficient condition for global asymptotic stability under finite precision arithmetic is established simulation results are presented to illustrate the theoretical results

canonical forms for polynomial and quadratic differential operators we define the canonical representative for the equivalence class consistingof all polynomial and quadratic differential operators that take the same values on a given subspaceof c

on the parametrization of all regularly implementing and stabilizing controllers in this paper we deal with problems of controller parametrization in the context of behavioral systems given a full plant behavior a subbehavior of the manifest plant behavior is called regularly implementable if it can be achieved as the controlled behavior resulting from the interconnection of the full plant behavior with a suitable controller behavior in such a way that the controller does not impose restrictions that are already present in the plant we establish a parametrization of all controllers that regularly implement a given behavior we also obtain a parametrization of all stabilizing controllers

on a generalization of the youla kucera parametrization part i the fractional ideal approach to siso systems abstract in this paper we show how to use the theory of fractional ideals in order to study the fractional representation approach to analysis and synthesis problems for siso systems within this mathematical framework we give necessary and sufficient conditions so that a plant is internally strongly bistably stabilizable or admits a weak coprime factorization moreover we show how to generalize the youla kucera parametrization of the stabilizing controllers to any stabilizable plant which does not necessarily admit a coprime factorization this parametrization is generally affine in two free parameters

some remarks on the intrinsic measures of eisenman this paper studies the intrinsic measures on complex manifolds first introduced by eisenman in analogy with the intrinsic distances of kobayashi some standard conjectures together with several new ones are considered and partial or complete answers are provided most of the counterexamples come from a closer examination of unbounded domains in complex euclidean space in particular a large class of unbounded hyperbolic domains are exhibited those unbounded domains of finite euclidean volume are also singled out for discussion

elementary proof of controller parametrization without coprime factorizability recently a controller parametrization method without coprime factorizability was presented by mori in this note we present this method using elementary matrix operations unlike in the earlier paper good knowledge of commutative algebra and module theory is not required

relationship between standard control problem and model matching problem without coprime factorizability we present the equivalence between a standard control problem and a model matching problem without coprime factorizability of plants this equivalence is well known if every stabilizable plant admit a doubly coprime factorization the result of this note does not require the coprime factorizability of plants

a behavioral approach to the pole structure of one dimensional and multidimensional linear systems we use the tools of behavioral theory and commutative algebra to produce a new definition of a finite pole of a linear system this definition agrees with the classical one and allows a direct dynamical interpretation it also generalizes immediately to the case of a multidimensional nd system we make a natural division of the poles into controllable and uncontrollable poles when the behavior in question has latent variables we make a further division into observable and unobservable poles in the case of a one dimensional 1d state space model the uncontrollable and unobservable poles correspond respectively to the input and output decoupling zeros whereas the observable controllable poles are the transmission poles most of these definitions can be interpreted dynamically in both the 1d and nd cases and some can be connected to properties of kernel representations we also examine the connections between poles transfer matrices and their left and right matrix fraction descriptions mfds we find behavioral results which correspond to the concepts that a controllable system is precisely one with no input decoupling zeros and an observable system is precisely one with no output decoupling zeros we produce a decomposition of a behavior as the sum of subbehaviors associated with various poles this is related to the integral representation theorem which describes every system trajectory as a sum of integrals of polynomial exponential trajectories

distribution of the wave function inside chaotic partially open systems we demonstrate both theoretically and experimentally that the distribution of the wave function inside a partially open chaotic time reversal symmetric system displays significant deviations from the porter thomas distribution we give arguments which show that this distribution resembles the distribution which is expected to be found in closed chaotic systems with broken time reversal symmetry

lossless and dissipative distributed systems this paper deals with linear shift invariant distributed systems by this we mean systems described by constant coefficient linear partial differential equations we define dissipativity with respect to a quadratic differential form i e a quadratic functional in the system variables and their partial derivatives the main result states the equivalence of dissipativity and the existence of a storage function or a dissipation rate the proof of this result involves the construction of the dissipation rate we show that this problem can be reduced to hilbert s 17th problem on the representation of a nonnegative rational function as a sum of squares of rational functions

a constructive solution to interconnection and decomposition problems with multidimensional behaviors by a multidimensional behavior we mean the solution space of a linear constant coefficient system of partial difference or differential equations within the behavioral framework a natural concept of interconnection has been introduced by j c willems the regular interconnection problem can be formulated as follows given a behavior the plant find if possible another behavior a controller such that their interconnection is not redundant and is equal to a certain desired behavior we give a constructive solution to this problem which is in terms of polynomial matrix equations that can be solved using the theory of grobner bases we also study a dual problem concerning the existence of direct sum decompositions of multidimensional behaviors finally we present a unified framework for studying both problems as special cases of constructing relative complements in the lattice of behaviors

noninteracting control of 2 d systems necessary and sufficient conditions for the existence of a decoupling bicausal precompensator for multivariable 2 d systems are derived in state space and frequency domains in general the decoupling problem for 2 d systems can be solved by feedback compensators if suitable injectivity assumptions are introduced on the input state matrices the structure of dynamic compensators is derived for this case and the 2 d decoupling problem with stability is solved

causal and stable input output structures on multidimensional behaviors in this work we study multidimensional nd linear differential behaviors with a distinguished independent variable called time we define in a natural way causality and stability of input output structures with respect to this distinguished direction we make an extension of some results in the theory of partial differential equations demonstrating that causality is equivalent to a property of the transfer matrix which is essentially hyperbolicity of the p c operator defining the behavior cal b c 0 y we also quote results which in effect characterize time autonomy for the general systems case stability is likewise characterized by a property of the transfer matrix we prove this result for the 2d case and for the case of a single equation for the general case it requires solution of an open problem concerning the geometry of a particular set in mathbb c n in order to characterize input output stability we also develop new results on inclusions of kernels freeness of variables and closure with respect to cal s cal s prime and associated spaces which are of independent interest we also discuss stability of autonomous behaviors which we believe to be governed by a corresponding condition

the lattice structure of behaviors if a linear continuous shift invariant distributed system is considered as a dynamical system converting input signals to output signals then this information is encapsulated in the impulse response or the transfer function of the system the set of all transfer functions has the structure of a ring corresponding to the operations of parallel and cascade connections of two systems however in the behavioral theory of willems a system is not described in terms of its input output transformation property indeed the concept of a behavior does not even need the notions of inputs and outputs and is therefore more fundamental than the classical concept of a system given by its transfer function the question then arises as to what is the structure of the set of all behaviors this paper argues that the relevant structure here is that of a modular lattice

dead beat observer synthesis in the paper the observer design problem is investigated in the context of linear left shift invariant discrete behaviors whose trajectories have support on the positive axis observability and reconstructibility properties of certain manifest variables from certain others in the presence of latent variables are defined and fully characterized necessary and sufficient conditions for the existence of either a dead beat or an exact observer are introduced and a complete parametrization of all dead beat observers is given

state space realization of 2 d finite dimensional behaviours this paper deals with the state space realization of autonomous autoregressive two dimensional 2 d systems in the context of the behavioural approach an arbitrary autoregressive 2 d system sig

on interconnections control and feedback the purpose of this paper is to study interconnections and control of dynamical systems in a behavioral context we start with an extensive physical example which serves to illustrate that the familiar input output feedback loop structure is not as universal as we have been taught to believe this leads to a formulation of control problems in terms of interconnections subsequently we study interconnections of linear time invariant systems from this vantage point let us mention two of the results obtained the first one states that any polynomial can be achieved as the characteristic polynomial of the interconnection with a given plant provided the plant is not autonomous the second result states that any subsystem of a controllable system can be implemented by means of a singular feedback control law these results yield pole placement and stabilization of controllable plants as a special case these ideas are finally applied to the stabilization of a nonlinear system around an operating point

algebraic aspects of two dimensional convolutional codes two dimensional 2d codes are introduced as linear shift invariant spaces of admissible signals on the discrete plane convolutional and in particular basic codes are characterized both in terms of their internal properties and by means of their input output representations the algebraic structure of the class of all encoders that correspond to a given convolutional code is investigated and the possibility of obtaining 2d decoders free from catastrophic errors as veil as efficient syndrome decoders is considered some aspects of the state space implementation of 2d encoders and decoders via finite memory 2d system are discussed

a note on output feedback stabilizability of multivariable 2d systems abstract the output feedback stabilizability conditions of 2d systems are expressed in term of structural properties of a pair of commuting linear transformations an algorithm is given for obtaining a stable closed loop 2d characteristic polynomial

lyapunov stability of 2d finite dimensional behaviours in this article we investigate a lyapunov approach to the stability of finite dimensional 2d systems we use the behavioural framework and consider a notion of stability following the ideas in pillai and shankar h pillai and s shankar 1998 a behavioral approach to control of distributed systems siam journal of control and optimization 37 388 408 rocha p rocha 2008 stabilization of multidimensional behaviors multidimensional systems and signal processing 19 273 286 valcher m valcher 2000 characteristic cones and stability properties of two dimensional autonomous behaviors ieee transactions on circuits and systems part i cas 47 290 302 we characterise stability in terms of the existence of a quadratic lyapunov function and provide a constructive algorithm for the computation of all such lyapunov functions

dynamic regulation of 2d systems a state space approach abstract the possibilities of modifying the dynamical behavior of 2d state space models by output feedback compensation are investigated and a complete characterization of the closed loop polynomial varieties is given it turns out that plant hidden modes and rank singularities of the transfer function are the unique constraints we have to cope with in the compensator synthesis the proof of this result is based on algebraic manipulations of 2d mfds and on a coprime realization algorithm

discrete time average positivity and spectral factorization in a behavioral framework abstract quite recently willems and trentelman siam j control opt 36 1998 1703 1749 have proposed quadratic differential forms for system analysis and synthesis from the behavioral point of view in this paper we develop their discrete time version called quadratic difference forms and some related fundamental concepts with respect to dissipativeness for a dynamical system which interacts with its environment along square summable trajectories moreover we provide an algorithm for spectral factorization of polynomial matrices as one example of quadratic difference forms a numerical example will be given to show the validity of our result

holomorphic flexibility properties of complex manifolds we obtain results on approximation of holomorphic maps by algebraic maps the jet transversality theorem for holomorphic and algebraic maps between certain classes of manifolds and the homotopy principle for holomorphic submersions of stein manifolds to certain algebraic manifolds

the fractional representation approach to synthesis problems an algebraic analysis viewpoint part ii internal stabilization in this second part of the paper a quadrat siam j control optim 40 2003 pp 266 299 we show how to reformulate the fractional representation approach to synthesis problems within an algebraic analysis framework in terms of modules we give necessary and sufficient conditions for internal stabilizability moreover we characterize all the integral domains a of siso stable plants such that every mimo plant defined by means of a transfer matrix whose entries belong to the quotient field k q a of a is internally stabilizable finally we show that this algebraic analysis approach allows us to recover on the one hand the approach developed in m c smith ieee trans automat control 34 1989 pp 1005 1007 and on the other hand the ones developed in k mori and k abe siam j control optim 39 2001 pp 1952 1973 s shankar and v r sule siam j control optim 30 1992 pp 11 30 v r sule siam j control optim 32 1994 pp 1675 1695 m vidyasagar h schneider and b a francis ieee trans automat control 27 1982 pp 880 894 m vidyasagar control system synthesis a factorization approach mit press cambridge ma 1985

beyond the berry robnik regime a random matrix study of tunneling effects we consider the energy spectra of quantum hamilton systems whose classical dynamics is of mixed type i e regular for some initial conditions in the classical phase space and chaotic for the complementary initial conditions in the strict semiclassical limit when the effective planck constant eff is sufficiently small the berry robnik br statistics applies while at larger values of eff or smaller energies one sees deviations from br due to localization and tunneling effects we derive a two level random matrix model which describes these effects and can be treated analytically in a closed form the coupling between the regular and chaotic levels due to tunneling is assumed to be gaussian distributed this two level model describes most of the features of matrices of large dimensions here n 1000 which we treat numerically and is predicted to apply in mixed type systems at low energies the proposed analytical level spacing distribution function has two parameters the br parameter characterizing the classical phase space and the coupling antenna distortion or tunneling parameter between states localization effects so far are not included in our analysis except in the special case where can be replaced by some effective eff

controllability of 2 d systems the concept of controllability of 2 d systems is introduced in a behavioral framework this property is defined as a systems concept and is characterized in terms of system representations state space realizations for controllable 2 d systems are presented the structure of state space systems is analyzed and the notions of trimness reachability and observability are introduced and characterized these realizations do not depend on any assumptions concerning causality

the fractional representation approach to synthesis problems an algebraic analysis viewpoint part i weakly doubly coprime factorizations in this paper we show how to reformulate the fractional representation approach to analysis and synthesis problems within an algebraic analysis framework in terms of modules we give necessary and sufficient conditions so that a system admits weakly left right doubly coprime factorizations moreover we explicitly characterize the integral domains a such that every plant defined by means of a transfer matrix whose entries belong to the quotient field of a admits weakly doubly coprime factorizations finally we show that this algebraic analysis approach allows us to recover on the one hand the approach developed in m c smith ieee trans automat control 34 1989 pp 1005 1007 and on the other hand the ones developed in k mori and k abe siam j control optim 39 2001 pp 1952 1973 v r sule siam j control optim 32 1994 pp 1675 1695 and 36 1998 pp 2194 2195 m vidyasagar h schneider and b a francis ieee trans automat control 27 1982 pp 880 894 m vidyasagar control system synthesis a factorization approach mit press cambridge ma 1985

internal model principles for observers this paper deals with the observer problem for dynamical systems in a behavioral context we are given a dynamical system together with a partition of the system variables into a set of known or measured variables and a set of unknown to be estimated variables the observer problem is to find a system that produces an estimate of the unknown variables on the basis of the known or measured variables for a given plant and partition we establish a characterization of all error behaviors that can be achieved by interconnecting the plant with some observer the main result of this paper is a very general behavioral formulation of an internal model principle for observers we will show that a nonintrusive observer achieves a stable error behavior if and only if in addition to a detectability condition on the observer the observer behavior contains the anti stabilizable part of the plant behavior

primeness of multivariate polynomial matrices abstract different characterizations of primeness of polynomial matrices that are mutually equivalent in the one dimensional case of univiriate polynomials lead to various primeness concepts in the multivariate situation applied to systems theory this fact gives rise to a more refined view to observability and controllability of multidimensional systems algorithms for testing primeness properties are given based on computer algebraic techniques

trajectory control and interconnection of 1d and nd systems in this paper we examine the relationship between control viewed as concatenation of trajectories and control viewed as interconnection of systems we show that for one dimensional linear time invariant systems the ability to obtain a given subsystem by regular interconnection a prerequisite for any feedback type structure is equivalent to the ability to drive any trajectory into that subsystem however in the case of multidimensional systems the former is a stronger property than the latter trajectory controllability can however be expressed as a regular interconnection of behaviors in an extended variable space by introducing latent or auxiliary variables this leads as a by product to the notion of controlling a system by means of latent variables