from time series to linear system part i finite dimensional linear time invariant systems abstract in the first part of this paper the definition of a dynamical system as simply consisting of a family of time series will be developed in this context the notions of linearity time invariance and finite dimensionality will be introduced it will be shown that a given family of time series may be represented by a system of ar equations riw t l rl 1w t l 1 r0w t 0 or equivalently by a finite dimensional linear time invariant system x t 1 ax t bu t y t cx t du t w u y if and only if this family is linear shift invariant and complete or as is equivalent closed in the topology of pointwise convergence this yields a very high level and elegant set of axioms which characterize these familiar objects it is emphasized however that no a priori choice is made as to which components of w are inputs and which are outputs such a separation always exists in any specific linear time invariant model starting from these definitions the structural indices of such systems are introduced and it is shown how an ar representation of a system having a given behaviour can be constructed these results will be used in a modelling context in part ii of the paper

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

on flat systems behaviors and observable image representations in this short paper we propose a definition of flatness for systems not necessarily given in input state output representation a flat system is a system for which there exists a mapping such that the manifest system behavior is equal to the image of this mapping and such that the latent variable appearing in this image representation can be written as a function of the manifest variable and its derivatives up to some order for linear differential systems flatness is equivalent to controllability we will generalize the main theorem of levine and nguyen systems control lett 48 2003 69 to general linear differential systems

technical communique the pbh test for multidimensional ltid systems in this article the classical popov belevitch hautus controllability test is generalized to higher dimensions

maximally orthogonalized higher order bases over generalized wires quadrilaterals and hexahedra this paper presents a general theory of maximally orthogonalized div and curl conforming higher order basis functions hobfs over generalized wires quadrilaterals and hexahedra in particular all elements of such bases necessary for fast and easy implementation are listed up to order n 8 numerical results given for div conforming bases applied in an iterative method of moments solution of integral equations show that the condition number and the number of iterations are a much lower than in the case of other hobfs of polynomial type and b practically not dependent on the applied expansion order

from time series to linear system part iii approximate modelling abstract in the third part of this paper the problem of finding a linear time invariant complete system which models an observed time series will be continued however in this part it will be assumed that the model is required to explain the observations only approximately this raises the question of what is meant by an approximate model and how one should judge a model against data which in a strict sense falsify the model the initial sections of this paper introduce the methodology either the maximal admissible complexity is fixed and the best fitting model in the model class is sought or the maximally tolerated misfit is fixed and the least complex model in the model class is sought this is illustrated by means of finite dimensional models where natural complexity and misfit functions suggest themselves subsequently what is meant by the complexity of a linear time invariant complete system is discussed a concrete complexity function is proposed which is defined in terms of the richness of the behaviour on finite time intervals and is in one to one correspondence with the number of inputs the number of states and the observability indices of the underlying minimal i s o system the next definition is that of the misfit between an observed time series and a system a concrete misfit function is proposed which takes into consideration the lag structure of the system and in how far the observations fail to corroborate the ar equations which define the system s behaviour how the misfit may be computed by means of the correlation function of the data is then shown these concepts are subsequently used in order to set up two algorithms for approximate modelling of an observed q dimensional time series the first algorithm fixes the maximal complexity in a sense this comes down to fixing the number and the lags of the required ar equations the second algorithm fixes the tolerated misfit it is adaptive in nature in the sense that the number and the structure of the resulting ar equations will depend on the observations these ideas and algorithms are illustrated by means of two simulated examples

moving target search a real time search for changing goals considers the case of heuristic search where the goal may change during the course of the search for example the goal may be a target that actively avoids the problem solver the authors present a moving target search algorithm mts to solve this problem the authors prove that if the average speed of the target is slower than that of the problem solver then the problem solver is guaranteed to eventually reach the target in a connected problem space the original mts algorithm was constructed with the minimum operations necessary to guarantee its completeness and hence is not very efficient to improve its efficiency the authors introduce ideas from the area of resource bounded planning into mts including 1 commitment to goals and 2 deliberation for selecting plans experimental results demonstrate that the improved mts is 10 to 20 times more efficient than the original mts in uncertain situations

continuous varying linear systems abstract we discuss implicit systems of ordinary linear differential equations with time variable coefficients their solutions in the signal space of hyperfunctions according to sato and their solution spaces called time varying linear systems or behaviours from the system theoretic point of view the basic result inspired by an analogous one for multidimensional constant linear systems is a duality theorem which establishes a categorical one one correspondence between time varying linear systems or behaviours and finitely generated modules over a suitable skew polynomial ring of differential operators this theorem is false for the signal spaces of infinitely often differentiable functions or of meromorphic hyper functions or of distributions on r it is used to obtain various results on key notions of linear system theory several new algorithms for modules over rings of differential operators and in particular new grobner basis algorithms due to insa and pauer make the system theoretic results effective

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

some basic structural properties of generalized linear systems abstract this paper shows how to give a kalman state variable realization to generalized linear systems simple characterizations of controllability observability of their duality and of a minimal realization are proposed our techniques based on elementary module theory are applicable to time varying systems as well as constant systems the note ends with a brief discussion of asymptotic observers in the constant case

surveillance search for a moving target the surveillance search problem is a generalization of the detection search problem studied by brown and stone among others in the surveillance version of the problem the search terminates only if the time and location of a detection satisfy certain problem specific conditions the objective is to maximize the expected value of a payoff received during and at the end of the search for instance we may wish to maximize the probability of finding the target in a specified subset of the state space or at a specified time consequently we might detect the target several times before the search terminates a search strategy i e an allocation of the available search effort must account for this possibility in this paper we formulate the surveillance search problem for applications in which the target moves according to a not necessarily time homogeneous markov process we then derive a set of necessary conditions for the optimality of a search strategy and develop algorithms based on the algorithm of brown for the detection search problem to find strategies that satisfy these necessary conditions for problems in which the search effort is infinitely divisible

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

algebraic characterization of approximate controllability of behaviours of spatially invariant systems we give algebraic characterizations of the properties of autonomy and of controllability of behaviours of spatially invariant dynamical systems consisting of distributional solutions w that are periodic in the spatial variables to a system of partial differential equations corresponding to a polynomial matrix m g c 1 d mxn

construction of nearly orthogonal nedelec bases for rapid convergence with multilevel preconditioned solvers this paper presents a systematic approach to constructing high order tangential vector basis functions for the multilevel finite element solution of electromagnetic wave problems the new bases allow easy computation of a preconditioner to eliminate or at least weaken the indefiniteness of the system matrix and thus reduce the condition number of the system matrix when these bases are used in multilevel solutions where the multilevels correspond to the order of the basis functions the resulting p multilevel ilu preconditioned conjugate gradient method mpcg provides an optimal rate of convergence we first derive an admissible set of vectors of order p and decompose this set into two subspaces rotational and irrotational gradient we then reduce the number of vectors by making them orthogonal to all previously constructed lower order bases the remaining vectors are made mutually orthogonal in both the vector space and in the range space of the curl operator the resulting vector basis functions provide maximum orthogonality while maintaining tangential continuity of the field the zeroth order space is further decomposed using a scalar vector formulation to eliminate convergence problems at extremely low frequencies numerical experiments show that number of iterations needed for the solution by mpcg is basically constant regardless of the order of the basis or of the matrix size computational speed is improved by several orders of magnitude due to the fast matrix solution of mpcg and to the high accuracy of the higher order bases

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

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

de rham diagram for hp finite element spaces abstract we prove that the hp finite elements for h curl spaces introduced in 1 fit into a general de rham diagram involving hp approximations the corresponding interpolation operators generalize the notion of hp interpolation introduced in 2 and are different from the classical operators of nedelec and raviart thomas

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

curl conforming hierarchical vector bases for triangles and tetrahedra a new family of hierarchical vector bases is proposed for triangles and tetrahedra these functions span the curl conforming reduced gradient spaces of nedelec the bases are constructed from orthogonal scalar polynomials to enhance their linear independence which is a simpler process than an orthogonalization applied to the final vector functions specific functions are tabulated to order 6 5 preliminary results confirm that the new bases produce reasonably well conditioned matrices

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

causal input output representation of 2d systems in the behavioral approach in this paper the concept of causal input output representation of two dimensional 2d systems in the behavioral approach is introduced these representations provide an interesting connection between classical 2d systems theory and 2d systems theory in the behavioral approach some characterizations of such representations are presented finally a technique that allows us to obtain causal input output representations is proposed

an optimal branch and bound procedure for the constrained path moving target search problem a searcher and target move among a finite set of cells c 1 2 n in discrete time at the beginning of each time period one cell is searched if the target is in the selected cell j it is detected with probability qj if the target is not in the cell searched it cannot be detected during the current time period after each search a target in cell j moves to cell k with probability pjk the target transition matrix p pjk is known to the searcher the searcher s path is constrained in that if the searcher is currently in cell j the next search cell must be selected from a set of neighboring cells cj the object of the search is to minimize the probability of not detecting the target in t searches

lyapunov stability of n d strongly autonomous systems in this article we look into stability properties of strongly autonomous n d systems i e systems having finite dimensional behaviour these systems are known to have a first order representation akin to 1 d state space representation we consider our systems to be already in this form throughout we first define restriction of an n d system to a 1 d subspace using this we define stability with respect to a given half line and then stability with respect to collections of such half lines proper cones then we show how stability with respect to a half line for the strongly autonomous case reduces to a linear combination of the state representation matrices being hurwitz we first relate the eigenvalues of this linear combination with those of the individual matrices with this we give an equivalent geometric criterion in terms of the real part of the characteristic variety of the system for half line stability then we extend this geometric criterion to the case of stability with respect to a proper

real time heuristic search abstract we apply the two player game assumptions of limited search horizon and commitment to moves in constant time to single agent heuristic search problems we present a variation of minimax lookahead search and an analog to alpha beta pruning that significantly improves the efficiency of the algorithm paradoxically the search horizon reachable with this algorithm increases with increasing branching factor in addition we present a new algorithm called real time a for interleaving planning and execution we prove that the algorithm makes locally optimal decisions and is guaranteed to find a solution we also present a learning version of this algorithm that improves its performance over successive problem solving trials by learning more accurate heuristic values and prove that the learned values converge to their exact values along every optimal path these algorithms effectively solve significantly larger problems than have previously been solvable using heuristic evaluation functions

efficient scalable parallel higher order direct mom sie method with hierarchically semiseparable structures for 3 d scattering a novel fast scalable parallel algorithm is proposed for the solution of large 3 d scattering problems based on 1 the double geometrical and current approximation higher order dho method of moments mom in the surface integral equation sie formulation and 2 a direct solver for dense linear systems utilizing hierarchically semiseparable hss structures namely an hss matrix representation is used for compression factorization and solution of the system matrix in addition a rank revealing qr decomposition for memory compression is used with a stopping criterion in terms of the relative rank tolerance value a method for geometrical preprocessing of the scatterers based on the cobblestone distance sorting technique is employed in order to enhance the hss algorithm accuracy and parallelization numerical examples show how the accuracy of the dho hss mom sie method is easily controllable by using the relative tolerance for the matrix compression moreover the examples demonstrate low memory consumption as well as much faster simulation time when compared to the direct lu decomposition the method enables dramatically faster monostatic scattering computations than iterative solvers and reduced number of unknowns when compared to low order discretizations finally great scalability of the algorithm is demonstrated on more than one thousand processes

factoring and decomposing a class of linear functional systems abstract within a constructive homological algebra approach we study the factorization and decomposition problems for a class of linear functional determined over determined under determined systems using the concept of ore algebras of functional operators e g ordinary partial differential operators shift operators time delay operators we first concentrate on the computation of morphisms from a finitely presented left module m over an ore algebra to another one m where m resp m is a module intrinsically associated with the linear functional system ry 0 resp r z 0 these morphisms define applications sending solutions of the system r z 0 to solutions of r y 0 we explicitly characterize the kernel image cokernel and coimage of a general morphism we then show that the existence of a non injective endomorphism of the module m is equivalent to the existence of a non trivial factorization r r 2 r 1 of the system matrix r the corresponding system can then be integrated in cascade under certain conditions we also show that the system ry 0 is equivalent to a system r z 0 where r is a block triangular matrix of the same size as r we show that the existence of idempotents of the endomorphism ring of the module m allows us to reduce the integration of the system ry 0 to the integration of two independent systems r 1 y 1 0 and r 2 y 2 0 furthermore we prove that under certain conditions idempotents provide decompositions of the system ry 0 i e they allow us to compute an equivalent system r z 0 where r is a block diagonal matrix of the same size as r applications of these results in mathematical physics and control theory are given finally the different algorithms of the paper are implemented in the maple package m orphisms based on the library oremodules

une interpretation algebrique de la transformation de laplace et des matrices de transfert abstract the tensor product of the module of a linear system with the quotient field of the ring of linear differential operators is a vector space where even in the time varying case a formal laplace transform and the transfer matrix are most naturally defined several classic problems are examined in this algebraic setting the relationship between left right coprime matrix decomposition and controllability observability the state variable canonical realization the transfer algebra with respect to parallel and series connections the input output inversion and model matching

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

hierarchal triangular elements using orthogonal polynomials hierarchal elements are finite elements which have the useful property that elements with different polynomial orders can be used together in the same mesh without causing discontinuities this paper introduces a new hierarchal triangular element in which the basis functions are constructed from orthogonal polynomials jacobi polynomials the resulting element is shown to be better conditioned than the earlier hierarchal element of rossow and katz 1 recursive formulas allow the complete set of basis functions for an element to be efficiently evaluated at a given point in addition the formulas can be used to generate pre computed universal matrices examples are given of universal matrices up to order 4 for the generalized helmholtz equation an electromagnetic problem involving a length of transmission line is used to show the usefulness of the new elements

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

duality in time varying linear systems a module theoretic approach abstract using elementary module theory an intrinsic definition of the dual or adjoint of a generalized time varying linear system is given with this the duality of controllability and observability is recovered from their intrinsic module theoretical definitions the duality of state feedback and output injection is discussed both in the static case and for its quasistatic generalization related brunovsky type canonical forms are derived in the quasistatic case the corresponding controllability indices and their duals the observability indices are defined intrinsically

higher order interpolatory vector bases for computational electromagnetics low order vector basis functions compatible with the nedelec 1980 representations are widely used for electromagnetic field problems higher order functions are receiving wider application but their development is hampered by the complex procedures used to generate them and lack of a consistent notation for both elements and bases in this paper fully interpolatory higher order vector basis functions of the nedelec type are defined in a unified and consistent manner for the most common element shapes it is shown that these functions can be obtained as the product of zeroth order nedelec representations and interpolatory polynomials with specially arranged arrays of interpolation points the completeness properties of the vector functions are discussed and expressions for the vector functions of arbitrary polynomial order are presented sample numerical results confirm the faster convergence of the higher order functions

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

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

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 construction of noetherian operators abstract we give an elementary and constructive purely algebraic proof for the existence of noetherian differential operators for primary submodules of finite dimensional free modules over polynomial algebras by means of these operators the submodule can be described by differential conditions on the associated characteristic variety this important result and the terminology are due to v p palamodov however his l ehrenpreis and later j e bjork s proofs of the existence theorem use complicated algebraic and analytic techniques and are not constructive as far as we see the idea to characterize primary ideals by their associated differential operators is due to w grobner but m noether s fundamentalsatz is based on similar ideas and is obviously the origin of palamodov s terminology

optimal sequential search a bayesian approach to the classical model of searching for one object out of n we add uncertainty about the parameters of the distribution of the n objects among the m boxes adopting a bayesian approach we study the optimal sequential search strategy for the case n 1 we obtain a generalization of the fundamental result of blackwell the strategy which searches at each stage in the most inviting box is optimal this strategy is also optimal form 2 and arbitrary n however for n 1 the optimal strategy may be very di erent from that of the classical model even when the uncertainty about is very small ams 1980 subject classi cation primary 90b40

finite poles and zeros of linear systems an intrinsic approach several kinds of finite poles and zeros are defined and their relationships are studied in an intrinsic manner some of these definitions are more precise than the usual ones all kinds of multiplicities of an input output decoupling zero are defined here in addition we believe that our approach clarifies those notions and it makes it possible to very concisely establish fundamental relationships between various poles and zeros some new relationships are derived the notions of input output stability and transfer stability are also introduced several examples illustrate the theory

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

zero structures of n d systems a detailed definition of the zero structure of an n d polynomial matrix is given the nature of the invariance of this zero structure is determined in both the polynomial matrix and polynomial system matrix contexts some connections with previous work are made

on invariant zeros of linear systems of pdes in this paper we provide a behavioral framework in which to describe and extend the concept of linear dynamics introduced by fliess from the one dimensional 1d to the multidimensional nd framework we provide an alternative description of the invariant zeros of a system equivalent to the smith zero description in the 1d case and use this to generalize the concept and characterization of invariant zeros to the nd case in particular we show that the definitions are equivalent in the 1d case to those in the classical literature we provide new results on the structural relations of nd invariant and transmission zeros

new coprimeness over multivariable polynomial matrices and its application to control delay systems abstract this paper presents a new notion of coprimeness over multivariable polynomial matrices where a single variable is given priority over the remaining variables from a characterization through a set of common zeros of the minors it is clarified that the presented coprimeness is equivalent to weakly zero coprimeness in the particular variable an application of the presented coprimeness to control systems with non commensurate delays and finite spectrum assignment is also presented because the presented coprimeness is stronger than minor coprimeness non commensurate delays are difficult to deal with in algebraic control theory the rational ratio condition between delays which can reduce non commensurate delays to commensurate delays proves to be both powerful and practical concept in algebraic control theory for delay systems

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

on the extension of solutions of linear constant coefficient pde abstract we solve the following over determined boundary value problem the extension problem let r be a matrix whose entries are linear partial differential operators with constant coefficients let be a non empty open bounded convex set we consider the homogeneous system r f 0 in a neighborhood of subject to the boundary condition f g in a neighborhood of for a given g we give a criterion for the unique existence of a smooth solution f to this problem there are two obvious necessary conditions g is smooth and r g 0 in a neighborhood of we characterize the class of differential operators r for which the problem is solvable for any g satisfying the necessary conditions finally in the case where the solution is non unique we consider the possibility of obtaining uniqueness by fixing several components of the desired solution

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

a parametrisation for dissipative behaviours the matrix case we study linear time invariant dynamical systems that are dissipative with respect to a generalised power defined by a quadratic differential form we address several cases in an increasing order of complexity and show how dissipative systems can be parametrised in this process we also establish a number of results for polynomial matrices that are of independent interest the present article is a generalisation of our earlier work that dealt with single input single output dissipative systems pendharkar i and pillai h k 2004 a parametrization for dissipative behaviours systems and control letters 51 123 132

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

the input output and controllable behavioural structures of a state space representation given the matrices of a linear state space representation we find an expression for a universal left annihilator of the matrix ctwovec zi a c and hence derive kernel representations for the input output behaviour and by duality the controllable part more generally in the discrete case we derive representations for the l completion for different values of l and the sub behaviours of trajectories reachable in a given time interval the representations are in certain trim canonical forms which are intimately connected with structure indices as a by product of the state elimination procedure we obtain a minimal state space representation for a given behaviour in terms of an arbitrary one

hierarchic hp edge element families for maxwell s equations on hybrid quadrilateral triangular meshes we construct and study a set of hierarchic basis functions for the galerkin discretisation of the space h curl 169 suitable for hybrid meshes containing both quadrilateral and triangular elements with arbitrary non uniform order polynomial approximation we investigate the conditioning and dispersive behaviour of the elements in addition numerical examples are shown which demonstrate the accuracy of the space for computing solutions of the time harmonic maxwell s equations that have singularities

module theoretic zero structures for system matrices the foundation for a coordinate free theory of the poles of a linear dynamical system was laid in 1965 by the module theoretic work of kalman however although the theoretical and application importance of transfer function zeros for feedback system design was widely known for single input single output systems by 1955 it remained for rosenbrock in 1970 to propose the ideas of transmission zeros input decoupling zeros and output decoupling zeros for multi input multi output systems by means of matrix theoretic methods a coordinate free module theoretic treatment of transmission zeros for a multi input mufti output transfer function was given in 1981 by wyman and sain this paper extends these coordinate free module theoretic studies to include systems which need not be controllable or observable interpretation of the rosenbrock system matrix is given on three levels rational finitely generated free modular and torsion divisible on the second level an omega zero module z omega is de

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

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

direct methods for primary decomposition leti be an ideal in a polynomial ring over a perfect field we given new methods for computing the equidimensional parts and radical ofi for localizingi with respect to another ideal and thus for finding the primary decomposition ofi our methods rest on modern ideas from commutative algebra and are direct in the sense that they avoid the generic projections used by hermann 1926 and all others until now

real time bidirectional search coordinated problem solving in uncertain situations this paper investigates real time bidirectional search rtbs algorithms where two problem solvers starting from the initial and goal states physically move toward each other to evaluate the rtbs performance two kinds of algorithms are proposed and are compared to real time unidirectional search one is called centralized rtbs where a supervisor always selects the best action from all possible moves of the two problem solvers the other is called decoupled rtbs where no supervisor exists and the two problem solvers independently select their next moves experiments on mazes and n puzzles show that 1 in clear situations decoupled rtbs performs better while in uncertain situations centralized rtbs becomes more efficient and 2 rtbs is more efficient than real time unidirectional search for 15 and 24 puzzles but not for randomly generated mazes it is shown that the selection of the problem solving organization is the selection of the problem space which determines the baseline of the organizational efficiency once a difficult problem space is selected the local coordination among problem solvers hardly overcome the deficit

controlling the learning process of real time heuristic search real time search provides an attractive framework for intelligent autonomous agents as it allows us to model an agent s ability to improve its performance through experience however the behavior of real time search agents is far from rational during the learning convergence process in that they fail to balance the efforts to achieve a short term goal i e to safely arrive at a goal state in the present problem solving trial and a long term goal to find better solutions through repeated trials as a remedy we introduce two techniques for controlling the amount of exploration both overall and per trial the weighted real time search reduces the overall amount of exploration and accelerates convergence it sacrifices admissibility but provides a nontrivial bound on the converged solution cost the real time search with upper bounds insures solution quality in each trial when the state space is undirected these techniques result in a convergence process more stable compared with that of the learning real time a algorithm

strong controlled invariance of behavioural nd systems abstractin this paper we give a stronger version of the notion of behavioral controlled invariance introduced in pereira rocha 2017 in the context of regular partial interconnections in such interconnections the variables are divided into two sets the variables to be controlled and the variables on which it is allowed to enforce restrictions control variables moreover regularity means that the restrictions of the controller do not overlap with the ones already implied by the laws of the original behavior a complete characterization of strong controlled invariance for nd behaviors is derived making use of a special controller behavior known as the canonical controller

control systems analysis for the fornasini marchesini 2d systems model progress after four decades the development of a control systems theory for multidimensional linear systems i e systems with more than one independent variable is an area where a very significant volume of research is based

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

l2 stability the cases of infinite dimensional discrete autonomous systems and 2 d autonomous systems abstract in this paper we analyze l 2 stability of infinite dimensional discrete autonomous systems given in a state space form with state transition matrix being a laurent polynomial matrix a 1 in the shift operator we give sufficient conditions and necessary conditions for l 2 stability of such systems we then use the theory of l 2 stability thus developed to analyze l 2 stability of discrete 2 d autonomous systems we achieve this by showing how a discrete 2 d autonomous system can be converted to an equivalent infinite dimensional state space discrete autonomous system where the state transition matrix turns out to be a laurent polynomial matrix in the shift operator finally we provide some easy to check numerical tests for l 2 stability of the above mentioned type of systems

the p and h p versions of the finite element method basic principles and properties in the classical form of the finite element method called the hversion piecewise polynomials of fixed degree p are used and the mesh size h is decreased for accuracy in this paper we discuss the fundamental theoretical ideas behind the relatively recent p version and h p version in the p version a fixed mesh is used and p is allowed to increase the h p version combines both approaches the authors describe and explain the basic properties and characteristics of these newer versions especially in areas where their behavior is significantly different from that of the h version simplified proofs of key concepts are included and computational illustrations of several results are provided a benchmark comparison between the various versions in included

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

a new computational search technique for ai based on the cantor set abstract this paper discusses the development of a new search algorithm for artificial intelligence systems the search technique is based on the mathematical theory of the cantor set the search algorithm will be efficient for specialized search domains where the distribution of the data elements to be searched is approximately normal the approach uses the iterative procedure of deleted middle thirds this facilitates quick pruning of a search space the new search technique has potential applications in computational systems with large input output data handling extensive experiments comparing the cantor set search css to binary search indicate that the new search technique holds good promise

higher order hierarchical legendre basis functions for electromagnetic modeling this paper presents a new hierarchical basis of arbitrary order for integral equations solved with the method of moments mom the basis is derived from orthogonal legendre polynomials which are modified to impose continuity of vector quantities between neighboring elements while maintaining most of their desirable features expressions are presented for wire surface and volume elements but emphasis is given to the surface elements in this case the new hierarchical basis leads to a near orthogonal expansion of the unknown surface current and implicitly an orthogonal expansion of the surface charge in addition all higher order terms in the expansion have two vanishing moments in contrast to existing formulations these properties allow the use of very high order basis functions without introducing ill conditioning of the resulting mom matrix numerical results confirm that the condition number of the mom matrix obtained with this new basis is much lower than existing higher order interpolatory and hierarchical basis functions as a consequence of the excellent condition numbers we demonstrate that even very high order mom systems e g tenth order can be solved efficiently with an iterative solver in relatively few iterations

strong practical stability and stabilization of differential linear repetitive processes abstract differential linear repetitive processes evolve over a subset of the upper right quadrant of the 2d plane where the unique feature is a series of sweeps or passes through a set of dynamics governed by the solution of a linear matrix differential equation over a finite duration t 0 where is termed the pass length or duration on each pass an output termed the pass profile is produced which acts as a forcing function on and hence contributes to the dynamics of the next pass profile the result can be oscillations in the pass to pass direction that cannot be controlled by direct application of standard or 1d linear systems theory the existing stability theory demands a bounded input bounded output property uniformly which in the case of the along the pass dynamics means for t 0 and for k t 0 0 0 0 where the integer k 0 denotes the pass number or index the pass length is always finite however and hence this stability theory could well be too strong in many cases and in particular impose very strong conditions in terms of control law design this paper develops an alternative in such cases by relaxing the requirement for the bounded input bounded output property to hold when k and t simultaneously provides an explanation of the implications of this in the frequency domain and then develops control law design algorithms

maximally orthogonal high order basis functions have a well conditioned gram matrix recently a novel high order finite element space for wires quadrilaterals and hexahedrons was presented 1 numerical results have shown a very favorable behavior of the condition number of the gram matrix of this finite element space as a function of the polynomial degree in this paper this high order finite element space is recognized to be expressible in terms of jacobi polynomials which can be easily computed using a three term recurrence in addition the condition number of the gram matrix of the one dimensional finite element space is rigorously analyzed for the general case of a piecewise smooth possibly curved geometry an explicit upper bound for the condition number in terms of the mesh quality is proved this bound implies that the one dimensional finite element space is stable for arbitrarily high polynomial degree numerical results corroborate the theoretical results and show that the basis can be used to perform hp refinement leading to an accurate handling of both large smooth regions and corners

maximally orthogonalized higher order basis functions in large domain finite element modeling in electromagnetics curl conforming max ortho basis functions mobfs are coupled with higher order large domain curved finite elements fes the performance of the functions is compared with that of the classical and near ortho basis functions through numerical experiments it is shown that max ortho fes yield highly orthogonal mass matrices for practically arbitrarily high orders of polynomial field approximations this facilitates the usage of iterative solvers and it significantly increases their efficiency accurate and fast computation of mobfs of arbitrarily high orders is enabled by the proposed two term recurrent formula

two dimensional proper rational matrices and causal input output representations of two dimensional behavioral systems the concept of proper rational matrix is strictly connected with the representation of causal transfer matrices in the two dimensional 2d case there is much freedom in defining proper rational matrices this freedom is connected to the fact that past and future in the 2d case can be determined by a 2d cone in this way the concept of rational matrix which is proper with respect to a cone can be introduced moreover an algorithm that checks the properness of a rational matrix is proposed finally this algorithm is used for determining all possible causal input output i o representations of a behavior given by a kernel representation

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

research on system zeros a survey this report provides a review of notable publications regarding the topic of zeros of multivariable systems in the period from 1970 into 1989 particular attention is paid to relationships between definitions of such zeros and their theoretical development transmission invariant system module infinite and finite zeros are addressed the latter subdivided into a hierarchical structure with respect to information considered in the definition process

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

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