on the minimal number of matrices which form a locally hypercyclic non hypercyclic tuple abstract in this paper we extend the notion of a locally hypercyclic operator to that of a locally hypercyclic tuple of operators we then show that the class of hypercyclic tuples of operators forms a proper subclass to that of locally hypercyclic tuples of operators what is rather remarkable is that in every finite dimensional vector space over r or c a pair of commuting matrices exists which forms a locally hypercyclic non hypercyclic tuple this comes in direct contrast to the case of hypercyclic tuples where the minimal number of matrices required for hypercyclicity is related to the dimension of the vector space in this direction we prove that the minimal number of diagonal matrices required to form a hypercyclic tuple on r n is n 1 thus complementing a recent result due to feldman

noetherian extensions of commutative rings abstract let a b be an extension of commutative rings with identity and t a resp t b be the total quotient ring of a resp b we say that a is a b noetherian ring if every ideal i of a with i b b is finitely generated and the extension a b is called a noetherian extension if a is b noetherian in this paper among other things we introduce the ring a g b the global transform of a in b we then show that if x a is b regular i e x b b and r is a ring between a and a g b then r x r is a noetherian a module and every ideal of r containing x is finitely generated we also study some conditions on a and b under which if a is b noetherian with dim b a 1 then any ring between a and b is b noetherian this is the krull akizuki theorem when a is an integral domain and b t a finally we prove that if t b is integral over t a and a is integrally closed in b then every ideal i of a with i b b is t invertible i e i i 1 t a

asymmetricalgebraicriccatiequation ahomeomorphicparametrizationofthesetof solutions abstract in this paper asymmetric algebraic riccati equations are analyzed in particular we derive a new parametrization of the set of solutions generalizing on the symmetric case the proposed parametrization is obtained in terms of pairs of invariant subspaces of two related feedback matrices moreover the connection is clarified between the new parametrization and the classical homeomorphic one based on graph invariant subspaces of the pseudo hamiltonian matrix associated with the equation we finally show that also the newly introduced parametrization is given by a homeomorphic map

on discrete algebraic riccati equations a rank characterization of solutions abstract we give a rank characterization of the solution set of discrete algebraic riccati equations dares involving real matrices our characterization involves the use of invariant subspaces of the system matrix we observe that a dare can be considered as a specific type of a non symmetric continuous time algebraic riccati equation we show how to extract the low rank solutions from the full rank solution of a homogeneous dare we also study the eigen structure of various feedback closed loop matrices associated with solutions of a dare we show as a consequence of our observations that a solution x of a dare for controllable systems and certain specific types of uncontrollable systems satisfies the inequality x m i n x x m a x we characterize conditions under which the solution set of dare is bounded unbounded we use the method of invariant subspaces to identify some invariant sets of a difference riccati equation

on mockor s question abstract for certain classes of prufer domains a we study the completion a t of a with respect to the supremum topology t sup t w w where is the family of nontrivial valuations on the quotient field which are nonnegative on a and t w is a topology induced by a valuation w it is shown that the concepts sft prufer domain and generalized dedekind domain are the same we show that if e is the ring of entire functions then e t is a bezout ring which is not a t prufer ring and if a is an sft prufer domain then a t is a prufer ring under a certain condition we also show that under the same conditions as above a t is a t prufer ring if and only if the number of independent valuation overrings of a is finite in particular if a is a dedekind domain resp h local prufer domain then a t is a t prufer ring if and only if a has only finitely many prime ideals resp maximal ideals these provide an answer to mockor s question

on the newtonkleinman method for strongly stabilizable infinite dimensional systems we consider the newtonkleinman method for strongly stabilizable infinite dimensional systems under certain assumptions the maximal self adjoint solution to the associated control algebraic riccati equation is constructed the constructed solution is also the maximal solution to the corresponding control algebraic riccati inequality

existence and comparison theorems for algebraic riccati equations for continuous and discrete time systems abstract we discuss two comparison theorems for algebraic riccati equations of the form xbr 1 b x x a br 1 c a br 1 c x q c r 1 c 0 simultaneously we give sufficient conditions to obtain the existence of the maximal hermitian solution of a riccati equation from the existence of a hermitian solution of a second riccati equation further similar results are given for discrete algebraic riccati equations

on polynomial matrix spectral factorization by symmetric extraction we revise the 1963 davis algorithm 2 for the spectral factorization of a para hermitian nonnegative polynomial matrix phi by symmetric factor extraction this algorithm is careless about zeros at infinity by introducing the notion of diagonal reducedness of phi we obtain an easy sufficient test for the absence of zeros at infinity we show then how to get phi diagonally reduced by diagonal excess reduction steps similar to the oono and yasuura steps removing all zeros at infinity and then how to remove synunetrically finite zeros while keeping el diagonally reduced hence free of zeros at infinity this results in a revised symmetric extraction spectral factorization algorithm with monotone degree control an example shows the didactical conceptual simplicity of the method appropriate symmetric extraction is discovered by revising and discovering important particular one sided factor extraction properties of polynomial matrices

topological generators of abelian lie groups and hypercyclic finitely generated abelian semigroups of matrices abstract in this paper we bring together results about the density of subsemigroups of abelian lie groups the minimal number of topological generators of abelian lie groups and a result about actions of algebraic groups we find the minimal number of generators of a finitely generated abelian semigroup or group of matrices with a dense or a somewhere dense orbit by computing the minimal number of generators of a dense subsemigroup or subgroup of the connected component of the identity of its zariski closure

spectral factorization via hermitian pencils abstract the spectral factorization problem is solved for state space systems via results on the canonical forms and inertia of hermitian matrix pencils these algebraic results then give a deflation method for spectral factorization

lattice properties of sets of semidefinite solutions of continuous time algebraic riccati equations the set of negative semidefinite solutions of the algebraic riccati equation a x xa xbb x c c 0 is studied under the most general hypothesis namely that a b is stabilizable modulo the undetectable subspace of a c if is a lattice then is isomorphic to 0 s where t0 is the direct sum of copies of the set of nonnegative real numbers and is a closure of which is order isomorphic to a system of well defined a invariant subspaces necessary and sufficient conditions are given such that or is a lattice

lur e equations and even matrix pencils abstract in this work we consider the so called lur e matrix equations that arise e g in model reduction and linear quadratic infinite time horizon optimal control we characterize the set of solutions in terms of deflating subspaces of even matrix pencils in particular it is shown that there exist solutions which are extremal in terms of definiteness it is shown how these special solutions can be constructed via deflating subspaces of even matrix pencils

all optimal hankel norm approximations of linear multivariable systems and their l error bounds the problem of approximating a multivariable transfer function g s of mcmillan degree n by s of mcmillan degree k is considered a complete characterization of all approximations that minimize the hankel norm is derived the solution involves a characterization of all rational functions s f s that minimize where s has mcmillan degree k and f s is anticavisal the solution to the latter problem is via results on balanced realizations all pass functions and the inertia of matrices all in terms of the solutions to lyapunov equations it is then shown that where k 1 g s is the k l st hankel singular value of g s and for one class of optimal hankel norm approximations the method is not computationally demanding and is applied to a 12 state model

the solution set of the algebraic riccati equation and the algebraic riccati inequality abstract we present a parametrization of the solution set of the algebraic riccati equation and the algebraic riccati inequality of optimal control where we assume only sign controllability of the underlying system furthermore we characterize the existence of maximal and minimal solutions of both the equation and the inequality by checkable conditions

the sft property does not imply finite dimension for power series rings abstract in this paper we give an example to show that if r is finite dimensional and has the sft property then r t is not necessarily sft nor finite dimensional

the extended j spectral factorization for descriptor systems in this paper we study an extended j spectral factorization problem i e j spectral factorization problem for descriptor systems which may not have full column normal rank and may exhibit poles and zeros on the extended imaginary axis we present necessary and sufficient solvability conditions and develop a numerically reliable method for the underlying problem our method is implemented by using only orthogonal transformations and has an acceptable computational complexity

yet another characterization of solutions of the algebraic riccati equation abstract this paper deals with a characterization of the solution set of algebraic riccati equation are over reals for both controllable and uncontrollable systems we characterize all solutions using simple linear algebraic arguments it turns out that solutions of are of maximal rank have lower rank solutions encoded within it we demonstrate how these lower rank solutions are encoded within the full rank solution and how one can retrieve the lower rank solutions from the maximal rank solution we characterize situations where there are no full rank solutions to the are we also characterize situations when the number of solutions to the are is finite when they are infinite and when they are bounded we also explore the poset structure on the solution set of are which in some specific cases turns out to be a lattice which is isomorphic to lattice of invariant subspaces of a certain matrix we provide several examples that bring out the essence of these results

generating semisimple groups by tori abstract let g be a simple real lie group with finite center we prove that given a non central element g in g there is an elliptic element h in g such that h and g h g 1 generate a dense subsemigroup of g

brief suboptimal markovian smoothing estimates based on continuous curves of solutions of the algebraic riccati inequality based on a result on continuous dependence of solutions of an algebraic riccati equation on the data matrices we construct continuous curves of solutions of an algebraic riccati inequality and derive suboptimal markovian estimates for the steady state smoothing problem

spectral factorization with imaginary axis zeros abstract we examine the problem of the existence and calculation of hermitian solutions p of a linear matrix inequality corresponding to the spectral factorization of a proper rational spectral density in particular zeros are permitted on the finite imaginary axis or at infinity as part of our construction of solutions we show explicitly that each such imaginary axis eigenvalue either finite or infinite defines a fixed part of all solutions p

subspace hypercyclic abelian linear semigroups abstract we consider the subspace hypercyclicity for abelian semigroups g of matrices on k n n 1 k r or c we say that g is subspace hypercyclic for a non zero subspace m of k n if there exists x k n such that g x m is dense in m we provide an effective method for checking that a given abelian semigroup is subspace hypercyclic we construct a nontrivial example of an abelian semigroup that has a dense orbit relative to a certain straight line but it is not subspace hypercyclic for this straight line for every n 2 we also construct an example of an abelian semigroup generated by two matrices where the subspace is hypercyclic but not hypercyclic we determine the minimal number of diagonalizable matrices over c that generate a subspace hypercyclic abelian semigroup on c n moreover we prove that g k is subspace hypercyclic for every k n 0 p whenever g is a subspace hypercyclic abelian semigroup of matrices on c n

observer based fault detection and isolation filter design for linear time invariant systems in this paper we consider a model based fault detection and isolation problem for linear time invariant dynamic systems subject to faults and disturbances we use a state observer scheme that cancels the system dynamics and defines a residual vector signal that is sensitive only to faults and disturbances we then design a stable fault detection and isolation filter such that the norm of the transfer matrix function from disturbances to the residual is minimised for fault detection subject to the constraint that the transfer matrix function from faults to residual is equal to a pre assigned diagonal transfer matrix for isolation of possibly simultaneous occurring faults our solution is given in the form of linear matrix inequalities using state space techniques as well as a model matching problem using matrix factorisation techniques a numerical example is given to illustrate the efficiency of the fault detection and isolation filter

the krull dimension of power series rings over almost dedekind domains abstract let d be an almost dedekind domain that is not dedekind m be a non invertible maximal ideal of d x be an indeterminate over d and d x be the power series ring over d we first construct an example of 1 sets and we then use this 1 set and m to give a simple proof of dim d x 2 1 we show that ht m x m d x 2 1 when d has only countably many non invertible maximal ideals or when m is countably generated we finally construct a simple example of almost dedekind domains that are not dedekind with given cardinal number of non invertible maximal ideals

the dual algebraic riccati equations and the set of all solutions of the discrete time riccati equation abstractin this paper two new pairs of dual continuous time algebraic riccati equations cares and dual discrete time algebraic riccati equations dares are proposed the dual dares are first studied with some nonsingularity assumptions on the system matrix and the parameter matrix then in the case of singular matrices a generalised inverse is introduced to deal with the dual dare problem these dual ares can easily lead us to an iterative procedure for finding the anti stabilising solutions especially to dare by means of that for the stabilising solutions furthermore we provide the counterpart results on the set of all solutions to dare inspired by the results for care two examples are presented to illustrate the theoretical results

krull dimension of power series rings over non sft domains abstract a ring d is called an sft ring if for each ideal i of d there exist a finitely generated ideal j of d with j i and a positive integer k such that a k j for all a i for a cardinal number and a ring d we say that dim d if d has a chain of prime ideals with length arnold showed that if d is a non sft ring then dim d x 0 let c be the class of non sft domains the class c includes the class of finite dimensional nondiscrete valuation domains the class of non noetherian almost dedekind domains the class of completely integrally closed domains that are not krull domains the class of integral domains with non noetherian prime spectrum and the class of integral domains with a nonzero proper idempotent ideal the ring of algebraic integers the ring of integer valued polynomials on z and the ring of entire functions are also members of the class c in this paper we prove that dim d x 2 1 for every d c and that under the continuum hypothesis 2 1 is the greatest lower bound of dim d x for d c on the other hand there exists a finite dimensional sft domain d such that dim d x 2 1

lq control of descriptor systems a spectral factorisation approach a spectral factorisation algorithm of improper matrices and existence results under controllability conditions are presented based on these results an algorithm for linear quadratic lq control of rectangular descriptor systems without the detectability and infinite observability conditions is presented the matrix pencil associated with the euler lagrange differential equations can be singular can have finite generalised eigenvalues on the imaginary axis and infinite generalised eigenvalues of any multiplicity in the class of optimal state feedback controls we find a control that renders the closed loop system marginally stable and impulse free also we find conditions on x 0 that guarantee the optimality it is shown how the disturbance attenuation problem of descriptor systems can be re formulated as an optimal state feedback feedforward control problem for which the results of this article can be applied when applied to state space systems the algorithm of this article which is based on

hypercyclic tuples of operators and somewhere dense orbits in this paper we prove that there are hypercyclic n 1 tuples of diagonal matrices on cn and that there are no hypercyclic n tuples of diagonalizable matrices on cn we use the last result to show that there are no hypercyclic subnormal tuples in infinite dimensions we then show that on real hilbert spaces there are tuples with somewhere dense orbits that are not dense but we also give sufficient conditions on a tuple to insure that a somewhere dense orbit on a real or complex space must be dense

krull dimension and generic fibres for mixed polynomial and power series integral domains for a ring r and variables x1 xn we let r x1 xn denote a mixed extension ring of r where each xi is fixed as either xi for polynomials in the variable xi or xi for power series in xi it is well known that if r is a noetherian ring of krull dimension m then r x1 xn has krull dimension m n we assume throughout that at least one xi is xi and we prove for a certain class of integral domains r of dimension m that the dimension of r x1 xn is mn 1 or mn n each of our integral domains r is close to being noetherian and has a canonically associated prufer overring t such that the contraction map spec t spec r is a homeomorphism a second result in this paper used in the proof of the above result appears not to have been noticed before for an extension k k of fields the extension k x1 xn k x1 xn is integral and the dimension of the generic fibre is 0 if the maximal separable extension k0 of k in k is a finite extension of k and k has finite exponent over k0 otherwise the generic fibre has dimension n 1

homogeneous riccati equations versus quadratic inequalities it is proved that under certain hypotheses the existence of hermitian solutions of a homogenized algebraic riccati equation of continuous type is guaranteed if the corresponding quadratic inequality admits a hermitian solution both finite dimensional matrices and infinite dimensional hilbert space operators riccati equations are considered

a matrix factorization solution to the h h fault detection problem in this paper we give a matrix factorization solution to the h h fault detection fd problem for linear time invariant dynamic systems an h h fd filter minimizes the sensitivity of the residual signal to disturbances while maintaining a minimum level of sensitivity to faults more specifically it minimizes the largest singular value over the extended imaginary axis of the transfer matrix from the disturbance to the residual signal vectors subject to the constraint that the singular values of the transfer matrix from the fault to the residual vectors are larger than or equal to 1 we show through the use of matrix function factorizations and completions that the problem reduces to a nonstandard model matching problem which is then solved under a mild assumption concerning the rank of a matrix function we also give an illustrative example

geometric aspects of the riccati difference equation in the nonsymmetric case abstract this paper provides a collection of geometric results concerning the nonsymmetric riccati difference equation it gives representation formulas which express all the solutions in terms of a suitable number of known ones further results include geometric properties of the difference of two solutions and the generalization of some of the author s earlier results to the nonsymmetric case

an infinite cardinal valued krull dimension for rings abstract we define and study two generalizations of the krull dimension for rings which can assume cardinal number values of arbitrary size the first which we call the cardinal krull dimension is the supremum of the cardinalities of chains of prime ideals in the ring the second which we call the strong cardinal krull dimension is a slight strengthening of the first our main objective is to address the following question for which cardinal pairs does there exist a ring of cardinality and strong cardinal krull dimension relying on results from the literature we answer this question completely in the case where we also give several constructions utilizing valuation rings polynomial rings and leavitt path algebras of rings having cardinality and strong cardinal krull dimension the exact values of and that occur in this situation depend on set theoretic assumptions

uncontrollable dissipative systems observability and embeddability the theory of dissipativity is well developed for controllable systems a more appropriate definition of dissipativity in the context of uncontrollable systems is in terms of the existence of a storage function namely a function such that along every system trajectory its rate of change at each time instant is at most the power supplied to the system at that time however even when the supplied power is expressible in terms of just the external variables the dissipativity property for uncontrollable systems crucially hinges on whether or not the storage function depends on variables unobservable hidden from the external variables this paper investigates the key aspects of both cases and also proposes another intuitive definition of dissipativity these three definitions are compared we show that drawbacks of one definition are addressed by another dealing first with observable storage functions under the conditions that no two uncontrollable poles add to zero and that dissipativity is strict as f

the algebraic riccati inequality parametrization of solutions tightest local frames and generalized feedback matrices abstract extending on previous work by faurre scherer and pavon a parametrization of the symmetric solutions of the algebraic riccati inequality is established this is then applied to derive new results on tightest local frames and on generalized feedback matrices that arise in stochastic realization theory

minimal representations of continuous time processes having spectral density with zeros in the extended imaginary axis given a process y with stationary increments and rational incremental spectral density s we parametrize minimal state space representations of dy we consider the case when s may have zeros on the imaginary axis including infinity and we analyze the zero structure of each state space representation

krull dimension of the power series ring over a nondiscrete valuation domain is uncountable abstract let v be a rank one nondiscrete valuation domain with maximal ideal m we prove that the krull dimension of v x v 0 is uncountable and hence the krull dimension of v x is uncountable this corresponds to the well known fact that the krull dimension of the ring of entire functions is uncountable in fact we construct an uncountable chain of prime ideals inside m x such that all the members contract to 0 in v our method provides a new proof that the krull dimension of the ring of entire functions is uncountable it is also shown that v x v 0 is not even a prufer domain while the ring of entire functions is a bezout domain these are answers to eakin and sathaye s questions applying the above results we show that the krull dimension of v x is uncountable if v is a nondiscrete valuation domain

on a hardy space approach to the analysis of spectral factors the paper presents a study of several problems related to spectral factorizations we assume only a very weak form of coercivity for the p x p spectral function phi and look at the set wm of all rectangular p x m spectral factors the main object is the arithmetization of the geometry of the set of minimal stable spectral factors by employing hardy space techniques and the arithmetic of inner functions particular attention is paid to the study of various partial orders associated with the set of spectral factors

krull dimension of a power series ring over a valuation domain abstract let v be a one dimensional nondiscrete valuation domain and let v v 0 we prove that krull dim v x v 2 1 which is an analogue of the fact that krull dim e 2 1 where e is the ring of entire functions the lower bound 2 1 is sharp in fact if v is countable then krull dim v x v 2 1 under the continuum hypothesis we construct a chain of prime ideals in v x with length 2 1 such that each prime ideal in the chain has height 2 1 and contracts to 0 in v we also show that for a finite dimensional valuation domain w either krull dim w x or krull dim w x 2 1

the regular indefinite linear quadratic optimal control problem stabilizable case this paper addresses an open problem in the area of linear quadratic optimal control we consider the regular infinite horizon stability modulo a subspace indefinite linear quadratic problem under the assumption that the dynamics are stabilizable our result generalizes previous works dealing with the same problem in the case of controllable dynamics we explicitly characterize the unique solution of the algebraic riccati equation that gives the optimal cost and optimal feedback control as well as necessary and sufficient conditions for the existence of optimal controls

constructing chains of primes in power series rings for an integral domain d of dimension n the dimension of the polynomial ring d x is known to be bounded by n 1 and 2n 1 while n 1 is a lower bound for the dimension of the power series ring d x it often happens that d x has infinite chains of primes for example such chains exist if d is either an almost dedekind domain that is not dedekind or a rank one nondiscrete valuation domain one concern here is developing schemes by which such chains can be constructed in d x when d is an almost dedekind domain a consequence of these constructions is that there are chains of primes similar to the set of 1 transfinite sequences of 0 s and 1 s ordered lexicographically

on hermitian solutions of the symmetric algebraic riccati equation the structure of the set of hermitian solutions of the matrix quadratic equation xdx xa a x c 0 is studied under the conditions that c c d is positive semidefinite and a d is stabilizable new features e g nonexistence of the minimal solution appear in contrast with the known case when a d is controllable

families of solutions of algebraic riccati equations abstract we consider homogeneous algebraic riccati equations in the general situation when the matrix of the dynamics can be mixed we show that in this case the equation may have infinitely many families of solutions an analysis of these families is carried over and an explicit formula is derived when the matrix of the dynamics is unmixed this formula provides an explicit parametrization of the whole set of the solutions we finally show that there are situations in which the equation has spurious solutions that do not belong to any of the families and derive sufficient conditions under which on the contrary the union of the families covers the whole set of solutions

on the lyapunov equation coinvariant subspaces and some problems related to spectral factorizations a geometric approach to stochastic realization theory and hence to spectral factorization problems has been developed by lindquist and picci 1985 1991 and lindquist et al 1995 most of this work was done abstractly fuhrmann and gombani 1998 adopted an entirely hardy space approach to this set of problems studying the set of rectangular spectral factors of given size for a weakly coercive spectral function the parametrization of spectral factors in terms of factorizations of related inner functions as developed in fuhrmann 1995 had to be generalized this led to a further understanding of the partial order introduced by lindquist and picci in the set of stable spectral factors in the present paper we study the geometry of finite dimensional coinvariant subspaces of a vectorial hardy space h 2 via realization theory emphasizing the role of the lyapunov equation in lifting the hardy space metric to the statespace domain we follow this by deriving state space formulas for rectangular spect

monotonicity of maximal solutions of algebraic riccati equations abstract a monotonicity result for the maximal solution of the equation xbb x a x xa q 0 q q a b stabilizable is proved

generalizing the algebraic riccati equation to higher order behavioral systems the well known kalman yakubovich popov lemma establishes an equivalence between dissipativity and the solvability of a linear matrix inequality see r a b van der geest and h l trentelman systems control lett 32 1997 pp 283 290 i masubuchi systems control lett 55 2006 pp 158 164 in this paper we strengthen this result by showing the equivalence of dissipativity to the solvability of the so called lur e equation t reis linear algebra appl 434 2011 pp 152 173 which mainly is a linear matrix inequality with a rank minimizing condition while the results about the solvability of the algebraic riccati equation j c willems ieee trans automat control 16 1971 pp 621 634 cannot be obtained from the kalman yakubovich popov lemma we will see that with the lur e equation this is possible

singular optimal control a geometric approach linear quadratic singular optimal control problem is solved for nonminimum phase and noninvertible systems a state space decomposition is obtained and a reduced order nonsingular subproblem is solved the optimal stabilizing input of the singular problem has been found when there are no transmission zeros on the imaginary axis

geometry of the algebraic riccati equation part ii we prove that the set of real symmetric solutions of the algebraic riccati equation is isomorphic to the algebraic variety of invariant subspaces of a related n times n matrix by characterizing the structure of this variety we obtain a detailed description of the geometric properties of the solution set of the algebraic riccati equation

the kalman yakubovich popov inequality for differential algebraic systems abstract in this paper we revisit the kalman yakubovich popov lemma for differential algebraic control systems this lemma relates the positive semi definiteness of the popov function on the imaginary axis to the solvability of a linear matrix inequality on a certain subspace further emphasis is placed on the lur e equation whose solution set consists loosely speaking of the rank minimizing solutions of the kalman yakubovich popov inequality we show that there is a correspondence between the solution set of the lur e equation and the deflating subspaces of certain even matrix pencils finally we show that under certain conditions the lur e equation admits stabilizing anti stabilizing and extremal solutions we note that for our results we neither assume impulse controllability nor make any assumptions on the index of the system

the algebraic riccati equation conditions for the existence and uniqueness of solutions abstract unmixed solutions of the matrix equation xdx xa ax c 0 d 0 are studied

matrix riccati inequality existence of solutions abstract we consider the various criteria including hamiltonian and frequency domain ones for solvability of the matrix riccati inequality arising from control problems some criteria are given in terms of symplectic algebra which is also used as an important technical tool

error bounds in the gap metric for dissipative balanced approximations abstract we derive an error bound in the gap metric for positive real balanced truncation and positive real singular perturbation approximation we prove these results by working in the context of dissipative driving variable systems as in behavioral and state signal systems theory in such a framework no prior distinction is made between inputs and outputs dissipativity preserving balanced truncation of dissipative driving variable systems is addressed and a gap metric error bound is obtained bounded real and positive real input state output systems are manifestations of a dissipative driving variable system through particular decompositions of the signal space under such decompositions the existing bounded real and positive real balanced truncation schemes can be seen as special cases of dissipative balanced truncation and the new positive real error bounds follow

positive real and bounded real balancing for model reduction of descriptor systems we present an extension of the positive real and bounded real balanced truncation model reduction methods to large scale descriptor systems these methods are based on balancing the solutions of the projected lur e matrix equations important properties of these methods are that respectively passivity and contractivity are preserved in the reduced order models and that there exist approximation error bounds we also discuss the numerical solution of the projected lur e equations numerical examples are given

zeros of spectral factors the geometry of splitting subspaces and the algebraic riccati inequality in this paper it is shown how the zero dynamics of not necessarily square spectral factors relate to the splitting subspace geometry of stationary stochastic models and to the corresponding algebraic riccati inequality the notion of output induced subspace of a minimal markovian splitting subspace which is the stochastic analogue of the supremal output nulling subspace in geometric control theory is introduced through this concept the analysis can be made coordinate free and straightforward geometric methods can be applied it is shown how the zero structure of the family of spectral factors relates to the geometric structure of the family of minimal markovian splitting subspaces in the sense that the relationship between the zeros of different spectral factors is reflected in the partial ordering of minimal splitting subspaces finally the well known characterization of the solutions of the algebraic riccati equation is generalized in terms of lagrangian subspaces invariant under the corresponding hamiltonian to the larger solution set of the algebraic riccati inequality

acausal models and balanced realizations of stationary processes abstract we study acausal realizations of stationary or stationary increment processes in particular we characterize the family of models whose corresponding spectral factors have a fixed zero structure acausal models with a fixed zero structure are related to each other by a certain group of state feedback transformations which is naturally parametrized by the solution set of a homogeneous algebraic riccati equation each feedback transformation reflects some of the eigenvalues of the generator matrix a of the representation to a mirror image with respect to the imaginary axis dually acausal models with a fixed pole structure are parametrized by a dual riccati equation and by a corresponding family of output injection transformations from a general standpoint the results of this study clarify the role played by dual pairs of riccati equations in spectral factorization and may be relevant to other problem areas than stochastic modelling one natural application of the concepts discussed in the paper is to stochastic balancing balancing of models with an essentially arbitrary eigenvalue location can be accomodated very naturally in this framework a balancing algorithm involving the solution of a dual pair of riccati equations is discussed

state space solutions to standard h sub 2 and h sub infinity control problems simple state space formulas are derived for all controllers solving the following standard h sub infinity problem for a given number gamma 0 find all controllers such that the h sub infinity norm of the closed loop transfer function is strictly less than gamma it is known that a controller exists if and only if the unique stabilizing solutions to two algebraic riccati equations are positive definite and the spectral radius of their product is less than gamma sup 2 under these conditions a parameterization of all controllers solving the problem is given as a linear fractional transformation lft on a contractive stable free parameter the state dimension of the coefficient matrix for the lft constructed using the two riccati solutions equals that of the plant and has a separation structure reminiscent of classical lqg i e h sub 2 theory this paper is intended to be of tutorial value so a standard h sub 2 solution is developed in parallel

krull dimension of power series rings abstract an ideal i of a commutative ring r with identity is called an sft strong finite type ideal if there exist a finitely generated ideal j of r with j i and a positive integer k such that a k j for each a i a ring r is called an sft ring if every ideal of r is an sft ideal arnold showed that if r is a non sft ring then the krull dimension of the power series ring r x is infinite let c be the class of non sft rings in this paper we show that dim r x 2 1 for every r c in other words if r is a non sft ring then there exists a chain of prime ideals in r x with length at least 2 1 we also prove that under the continuum hypothesis 2 1 is the greatest lower bound of dim r x for r c if m is a non sft maximal ideal of a ring r such that m is the radical of a countably generated ideal then we construct a chain p of prime ideals in r x with length at least 2 1 lying between m r x and m x i e m r x p m x for each the same result holds when m is the non sft maximal ideal of a zero dimensional quasi local ring or a one dimensional quasi local domain r

a parametrization of the nonnegative definite solutions of the algebraic riccati equation in this paper the set of the hermitian nonnegative definite solutions of the algebraic riccati equation is studied the main result shows that any solution in this set is the product of a greatest solution and a projection the analysis only assumes the existence of a greatest solution and leads to generalize classification theorems via projections known in the stabilizable case

algebraic riccati equation and symplectic algebra questions of existence uniqueness and the parametric dependence of solutions of the algebraic riccati equation are considered the different criteria for the solubility of this equation are obtained with the help of symplectic algebra

computation of general inner outer and spectral factorizations in this paper we solve two problems in linear systems theory the computation of the inner outer and spectral factorizations of a continuous time system considered in the most general setting we show that these factorization problems rely essentially on solving for the stabilizing solution a standard algebraic riccati equation of order usually much smaller than the mcmillan degree of the transfer function matrix of the system the proposed procedures are completely general being applicable for a polynomial proper improper system whose transfer function matrix could be rank deficient and could have poles zeros on the imaginary axis or at infinity as an application we discuss the extension to the case of rational matrices of the complete orthogonal decomposition of a constant matrix numerical refinements and examples illustrating the proposed approach are discussed in detail

the set of positive semidefinite solutions of the algebraic riccati equation of discrete time optimal control in this paper the algebraic riccati equation are of the discrete time linear quadratic lq optimal control problem and its set of positive semidefinite solutions is studied under the most general assumption which is output stabilizability with respect to an appropriate basis the discrete time algebraic riccati equation dare decomposes into a lyapunov equation and an irreducible riccati equation the focus is on the riccati part which amounts to studying a dare where all unimodular modes are controllable a bijection between positive semidefinite solutions and certain well defined sets of f invariant subspaces is established which together with its inverse is order reversing as an application issues concerning positive definite or strong solutions are clarified analogous results for negative semidefinite solutions are valid only under an additional assumption on the unobservable subspace