a new example of the lavrentiev phenomenon this paper describes a new example of a calculus of variations problem with the lavrentiev phenomenon a problem with different value functions for different classes of admissible trajectories such problems provide examples of nonuniqueness for hamilton jacobi equations since each value function is a solution the authors describe these solutions and the corresponding optimal trajectories and focus on how a standard proof of optimality is adapted for these examples

rational approximation in linear systems and control abstract in this paper we want to describe some examples of the active interaction that takes place at the border of rational approximation theory and linear system theory these examples are mainly taken from the period 1950 1999 and are described only at a skindeep level in the simplest possible scalar case we give comments on generalizations of these problems and how they opened up new ranges of research that after a while lived their own lives we also describe some open problems and future work that will probably continue for some years after 2000

an algebraic method to solve the minimal partial realization problem for scalar sequences abstract we give an algebraic method to construct all minimal partial realizations of a finite sequence of scalar numbers we use the minimal number of parameters to characterize all these minimal partial realizations three examples are worked out to illustrate the method

on triality and automorphisms and derivations of composition algebras abstract the principle of triality for orthogonal transformations on eight dimensional spaces is used to obtain different expressions for the automorphisms derivations and orthogonal transformations of cayley dickson algebras and symmetric composition algebras over fields of arbitrary characteristic

model reduction by minimizing the weighted equation error a method for model reduction is proposed which minimizes the equation error and yields a stationary exact model with this model the desired state variables or their physically meaningful linear combinations are reproduced this approach to the reduction problem leads to linear equations thus no search routines are needed moreover a time varying weighting matrix is introduced so that even unstable linear time invariant models can be reduced especially the characteristics of very low order reduced models depend highly on the weighting matrix the reduction for a distillation column of the 45th order is carried out

a suboptimal reduction algorithm for linear systems with a time delay in this paper a suboptimal model reduction algorithm is presented for models with time delay this method is an extension to the present authors method for models without a time delay an objective function is defined to minimize the weighted integral squared value between the outputs of the reduced order model and the original model when they are subjected to the same input signal examples are given to show the advantages of using the reduced order modelling technique provided

a method for optimal linear model reduction abstract this paper presents a new method for generating a reduced order model of a linear time invariant siso stable system the simplification criterion is the integral of a linear combination of the transient term of the original system step response and of a suitable number of its derivatives the reduction procedure is computationally simpler than similar procedures considered in the literature and leads to reduced order models whose step response fits satisfactorily the original one

a state space formulation for optimal hankel norm approximations the optimal hankel norm approximation problem studied in 1 is reformulated in a state space setting the crucial extension theorem is reestablished in this framework the minimal degree optimal approximation is then derived in terms of state space parameters

an optimal controller reduction by covariance equivalent realizations based upon the theory of covariance equivalent realizations 1 this note presents a technique to obtain reduced order controllers via an optimal projection of full order lqg controllers this projection is optimal in the sense that it is a weighted least square approximation to a covariance equivalent projection of the full order optimal closed loop system the technique which is capable of extracting optimal minimal lqg controllers whenever they exist is also shown to yield controllers that are equivalent to those obtained by component cost analysis 12

matrix pade approximation definitions and properties abstract some proposals are made to give a general definition of matrix pade approximants depending on the normalization of the denominator we define type i constant term is the unit matrix and type ii by conditions on the leading coefficient approximants existence and uniqueness are considered determinant expressions are given and relations among types i ii and left right approximants are considered

on direct versus indirect methods for reduced order controller design comparisons are obtained between direct parameter optimization and indirect controller reduction methods in reduced order controller design in particular the authors compare the controller canonical correlation coefficients method of c de villemagne and r e skelton 1988 to the optimal projection theory of d c hyland and d s bernstein 1984 it is noted that suboptimal methods may in certain cases be simpler to implement than optimal methods thus the choice of direct versus indirect methods can be viewed as a tradeoff between design simplicity and actual control system performance

balanced forms canonicity and parameterization we show that balanced forms are canonical and continuously parameterize the set of systems of a given order which are asymptotically stable minimal and have distinct singular values the advantage of the balanced canonical form over other classical canonical forms lies precisely in the guaranteed stability and minimality of the parameterization we also derive formulas allowing the synthesis of systems using gradient methods in balanced form

reduced order compensators via balancing and central control design for a structural control problem real time control of a physical system necessitates controllers that are low order in this article we compare two balanced truncation methods as a means of designing low order controllers for a nonlinear cable mass system the first is the standard technique of balanced truncation the second linear quadratic gaussian lqg balanced truncation can be thought of as balancing based on the controller and states that are important from the perspective of control and filter design are retained the control design applied to each reduced order model is the central controller we provide an overview of the central controller and devote attention to the design of this controller in the presence of balancing also described in this article is a method for reducing computational time in solving algebraic riccati equations for the design of low order lqg balanced controllers

pade techniques for model reduction in linear system theory a survey abstract techniques of pade approximation and continued fractions have been used often in model reduction problems an extensive bibliography on this topic is given the ideas are explained for the simple situation of a scalar function where no singular blocks appear in the pade table extensions to the matrix case and the multivariable case are not explained in detail

recursive partial realization from the combined sequence of markov parameters and moments abstract in this paper we develop the recursive structure of partial realization from a two sided sequence of markov parameters and moments the specific feature of this recursive structure is that it accommodates sequences growing in two directions like in the two point pade approximation we present a recursive algorithm for the minimal partial realization from a sequence of markov parameters and moments that amounts to recursive lu factorization of a toeplitz behavior matrix leaping over nongeneric sections it turns out that this algorithm reveals a new continued fraction expansion which extends the classical cauer third form as a corollary various results related to toeplitz matrices two point pade approximations continued fractions etc are unified within this partial realization setting

on dual algebraic riccati equations a method is presented to solve dual algebraic riccati equations it is shown that only one schur decomposition of the hamiltonian matrix is necessary to solve the dual equations

a note on the model reduction problem a mixed method of model reduction is proposed it is based on the differentiation method suggested by gutman et al 1 and on the usual pade method more precisely the differentiation procedure is used to obtain the reduced order denominator thus ensuring stability preservation while the numerator is obtained by matching the first maclaurin expansion coefficients of the original system and model transfer functions the method is computationally very simple and may lead to satisfactory results as shown by an example where the original system transfer function is taken from the relevant literature

principal component analysis in linear systems controllability observability and model reduction kalman s minimal realization theory involves geometric objects controllable unobservable subspaces which are subject to structural instability specifically arbitrarily small perturbations in a model may cause a change in the dimensions of the associated subspaces this situation is manifested in computational difficulties which arise in attempts to apply textbook algorithms for computing a minimal realization structural instability associated with geometric theories is not unique to control it arises in the theory of linear equations as well in this setting the computational problems have been studied for decades and excellent tools have been developed for coping with the situation one of the main goals of this paper is to call attention to principal component analysis hotelling 1933 and an algorithm golub and reinsch 1970 for computing the singular value decompositon of a matrix together they form a powerful tool for coping with structural instability in dynamic systems as developed in this paper principal component analysis is a technique for analyzing signals singular value decomposition provides the computational machinery for this reason kalman s minimal realization theory is recast in terms of responses to injected signals application of the signal analysis to controllability and observability leads to a coordinate system in which the internally balanced model has special properties for asymptotically stable systems this yields working approximations of x c x bar o the controllable and unobservable subspaces it is proposed that a natural first step in model reduction is to apply the mechanics of minimal realization using these working subspaces

stability preservation and computational aspects of a newly proposed reduction method the stability of the reduced order models obtained using the method suggested by a lepschy et al syst contr lett vol 8 pp 405 10 may 1987 is analyzed the relation between this method and the routh approximation is pointed out some interesting properties related to the construction of the reduced order transfer functions and to the evaluation of certain quadratic integrals are derived an examples shows the practicality of the procedures

derivations and automorphisms of jordan algebras in characteristic two a jordan algebra j over a field k of characteristic 2 becomes a 2 lie algebra l j with lie product x y x y and squaring x 2 x 2 we determine the precise ideal structure of l j in case j is simple finite dimensional and k is algebraically closed we also decide which of these algebras have smooth automorphism groups finally we study the derivation algebra of a reduced albert algebra j h3 o k and show that der j has a unique proper nonzero ideal vj isomorphic to l j k 1j with quotient der j v j independent of o on the group level this gives rise to a special isogeny between the automorphism group of j and that of the split albert algebra whose kernel is the infinitesimal group determined by vj

on the bilinear transformation of lqg balanced realizations abstract this contribution shows that some useful properties of lqg balanced realizations of continuous time systems are preserved by applying bilinear transformations this fact suggests that lqg balancing of discrete systems and its approximation can be performed by using the typical schemes successfully adopted for continuous time systems in the paper the approximation of discrete linear time invariant systems by using the singular perturbation approximation approach for lqg balanced realizations is analyzed showing its usefulness in approximating discrete models including unstable ones for closed loop aims

fine gradings on simple exceptional jordan pairs and triple systems abstract we give a classification up to equivalence of the fine group gradings by abelian groups on the jordan pairs and triple systems of types bi cayley and albert under the assumption that the base field is algebraically closed of characteristic different from 2 the associated weyl groups are computed we also determine for each fine grading on the bi cayley and albert pairs the induced grading on the exceptional simple lie algebra given by the tits kantor koecher construction

state space design of low order stabilizers an algorithm is presented for stabilizing a linear multivariable system with a controller of fixed dynamic order this is an output feedback stabilization problem the algorithm attempts to solve this by a sequence of approximate pole assignment problems the approximation is driven by the optimization of an objective function consisting of a weighted sum of the condition number of the closed loop eigenvectors and the norm of the difference between the computed and actual equivalent state feedback controls

a unified derivation and critical review of modal approaches to model reduction six of the most commonly used modal reduction techniques are represented by means of signal flow diagrams connecting inputs and state variables in this representation the derivation and the cross comparison of the various methods are greatly facilitated each method is then analysed to determine key characteristics i e i steady state agreement ii initial value agreement iii independence of the reduced model on the choice of retained state variables and inputs and iv eigen vector orientation realistic large scale systems are used to evaluate the static and dynamic characteristics of each model reduction method numerically these studies reveal some important and often unrecognized weaknesses of several traditional techniques and confirm the strength of the newly developed technique of litz

model reduction using new optimal routh approximant technique an optimal routh approximant of a single input single output dynamic system is a reduced order transfer function of which the denominator is obtained by the routh approximation method while the numerator is determined by minimizing a time response integral squared error ise criterion in this paper a new elegant approach is presented for obtaining the optimal routh approximants for linear time invariant continuous time systems the approach is based on the routh canonical expansion which is a finite term orthogonal series of rational basis functions and minimization of the ise criterion a procedure for combining the above approach with the bilinear transformation is also presented in order to obtain the optimal bilinear routh approximants of linear time invariant discrete time systems the proposed technique is simple in formulation and is amenable to practical implementation

a functional approach to lqg balancing the paper has as its theme a circle of problems related to lqg balancing with a special emphasis on the related problems of model reduction and robust stabilization the aim is to present a unified point of view to both previously described problems the unification is achieved by focusing on the study of three functions and the relationships between them and the various operators that are associated with these functions with an arbitrary transfer function g one can associate canonically two objects the conjugate inner function which is based on the nrcf and the r which is associated with the lqg controller of the function g the approach that is taken is functional emphasizing operators a balanced realization of a stable g arises as a matrix representation of the shift realization with respect to a basis made out of suitably normalized hankel singular vectors a similar result holds for lqg balanced realizations here the underlying hankel operator we study is h r where r u m v n and u v s

discrete linearized least squares rational approximation on the unit circle abstract we already generalized the rutishauser gragg harrod reichel algorithm for discrete least squares polynomial approximation on the real axis to the rational case in this paper a new method for discrete least squares linearized rational approximation on the unit circle is presented it generalizes the algorithms of reichel ammar gragg for discrete least squares polynomial approximation on the unit circle to the rationale case the algorithm is fast in the sense that it requires order m computation time where m is the number of data points and is the degree of the approximant we describe how this algorithm can be implemented in parallel examples illustrate the numerical behavior of the algorithm

lqg balancing for continuous time infinite dimensional systems in this paper we study the existence of linear quadratic gaussian lqg balanced realizations for continuous time infinite dimensional systems lqg balanced realizations are those for which the optimal cost operator for the system and its dual system are equal and diagonal the class of systems we consider is that of distributional resolvent linear systems which includes well posed linear systems as a subclass we prove the existence of lqg balanced realizations under a finite cost condition for both the system and its dual system we also show that an lqg balanced realization of a well posed transfer function is well posed we further show that approximately controllable and observable lqg balanced realizations are unique up to a unitary state space transformation finally we show that the spectrum of the product of the optimal cost operator of a system and its dual system is independent of the particular realization our method of proof shows the connections with coprime factorizations lyapunov balanced realizations and discrete time systems the main reason for studying lqg balanced realizations is that truncated lqg balanced realizations provide a good approximation of the original system we show that under certain conditions this is also true in the infinite dimensional case by proving an error bound in the gap metric

composition algebras over algebraic curves of genus zero we rephrase the classical theory of composition algebras over fields particularly the cayley dickson doubling process and zorn s vector matrices in the setting of locally ringed spaces fixing an arbitrary base field we use these constructions to classify composition algebras over complete smooth curves of genus zero applications are given to composition algebras over function fields of genus zero and polynomial rings

a least squares approach to reduce stable discrete linear systems preserving their stability abstract a new stability preserving model reduction algorithm for discrete linear siso systems based on a least squares approach is proposed similar to the pade approximation an equation system for the markov parameters involving a high dimensional hankel matrix is considered it is proved that approximate solutions computed via the moore penrose pseudo inverse give rise to a stability preserving reduction scheme furthermore the proposed algorithm is compared to the balanced truncation method showing comparable performance of the reduced systems

regularity properties of solutions to the basic problem in the calculus of variations this paper concerns the basic problem in the calculus of variations minimize a functional j defined by j x j l t x t x t dt a over a class of arcs x whose values at a and b have been specified existence theory provides rather weak conditions under which the problem has a solution in the class of absolutely continuous arcs conditions which must be strengthened in order that the standard necessary conditions apply the question arises what necessary conditions hold merely under hypotheses of existence theory say the classical tonelli conditions it is shown that given a solution x there exists a relatively open subset q of a b of full measure on which x is locally lipschitz and satisfies a form of the euler lagrange equation the main theorem of which this is a corollary can also be used in conjunction with various classes of additional hypotheses to deduce the global smoothness of solutions three such classes are identified and results of bernstein tonelli and morrey are extended one of these classes is of a novel nature and its study implies the new result that when l is independent of t the solution has essentially bounded derivative

computational approach to the simplicity of f 4 o s in the characteristic two case we present in this paper a computational approach to the study of the simplicity of the derivation lie algebra of the quadratic jordan algebra h3 os denoted by f4 os when the characteristic of the base field is two we will show not only a collection of routines designed to find identities and construct principal ideals but also a philosophy of how to proceed studying the simplicity of a lie algebra we have first implemented the quadratic jordan structure of h3 os into the computer system mathematica computing the derivation lie algebra of the quadratic jordon algebra h3 os at any characteristic preprint 2001 and then determined the generic expression of an element of the lie algebra f4 os der h3 os see 41 once the structure of f4 os is completely described it is time to analyze the simplicity by using the strategy mentioned if the characteristic of the base field is not two the lie algebra is simple but if the characteristic is two the lie algebra is not simple and there exists only one proper nonzero ideal i which is 26 dimensional and simple as a lie algebra in order to prove this last affirmation we have used again the set of routines to show the simplicity of the ideal and that it is isomorphic to f4 i which is also a simple lie algebra this isomorphism is constructed from a computed cartan decomposition of both lie algebras

normalized h sub spl infin controller reduction with a priori error bounds this paper describes an approach to the reduction of controllers for the normalized coprime factor robustness problem as well as the normalized h sub spl infin problem it is shown that a relative error approximation of a coprime factor representation of any suboptimal controller leads to a stability guarantee and an upper bound on the performance degradation when the reduced order controller is implemented when the approximation is performed on the controller generator guaranteed a priori stability and performance bounds are obtained in terms of the synthesis riccati equation solutions of the normalized h sub spl infin control problems

fractional balanced reduction model reduction via fractional representation a method for the model reduction of finite dimensional linear time invariant fdlti plants is proposed which uses normalized fractional representations is proposed the method dubbed fractional balanced reduction applies balance and truncate to a special representation of the graph operator of the plant this operation yields the graph operator of a reduced order plant the method has such properties as existence of an a priori error bound in the graph metric and preservation of sector containment coupling fractional representations with principal component analysis gives a model reduction method that is capable of producing in a numerically stable way a good reduced order model using the whole full order model sector properties are also preserved these are useful for deciding stability when nonlinearities are in the loop

a note on hankel approximation of all pass functions abstract the paper extends the results obtained by fuhrmann and ober on hankel approximation with an arbitrary transfer function g we can associate using left and right coprime factorization two all pass functions u and u and also a function r called the characteristic function of g which is connected with the lqg controller of the function g we analyse the connections between the hankel operators h u h r and h u h r and also establish a one to one correspondence between the optimal hankel approximation of h r and that of h u with some additional property finally the duality results on nehari complements and hankel approximations are extended

a new algorithm for l 2 optimal model reduction in this paper the quadratically optimal model reduction problem for single input single output systems is considered the reduced order model is determined by minimizing the integral of the magnitude squared of the transfer function error it is shown that the numerator coefficients of the optimal approximant satisfy a weighted least squares problem and on this basis a two step iterative algorithm is developed combining a least squares solver with a gradient minimizer convergence of the proposed algorithm to stationary values of the quadratic cost function is proved the formulation is extended to handle the frequency weighted optimal model reduction problem three examples demonstrate the optimization algorithm

reduction of linear continuous time multivariable systems by matching first and second order information this paper considers the approximation of stable continuous time multivariable linear systems from a finite number of markov parameters and second order information indexes it is shown that by properly choosing these indexes it is possible to uniquely identify an input output model of given order from an equal number of first and second order data

the structure of strongly prime quadratic jordan algebras abstract we extend the structure theory for strongly prime jordan algebras of arbitrary dimension to quadratic algebras of characteristic 2 we show that such j are either 1 hermitian forms h 0 a j h q a lying between an ample subspace of hermitian elements in a prime associative algebra a and those in its martindale ring of quotients q a 2 clifford forms with a scalar extension j j which is an algebra j j q 1 of a nondegenerate quadratic from q with basepoint 1 or 3 albert forms with a scalar extension j j which is a 27 dimensional split albert algebra j h 3 k of 3 3 hermitian matrices over an 8 dimensional split octonion algebra k as a consequence the simple jordan algebras of arbitrary dimension are either hermitian h 0 a for simple a or clifford forms in j q 1 or are albert algebras in particular our methods give an idempotent free classification of all simple finite dimensional algebras which are i special the key to this characterization is the existence of zel manov polynomials in the free special jordan algebra which are both hermitian polynomials whose values always look like hermitian elements and at the same time clifford polynomials nontrivial on 3 3 hermitian matrices hence on special algebras with 3 interconnected idempotents if a zel manov polynomial does not vanish on j then j is of hermitian type and if it does vanish on j then j is of clifford type one example of such a polynomial is q48 p16 x1 y1 z1 w1 p16 x2 y2 z2 w2 p16 x3 y3 z3 w3 for p16 d2x y z 2 dx y w dx y w dx y z x y z thus the structure of jordan algebras depends very directly on polynomial identities and commutators

bivariate thiele type matrix valued rational interpolants a new method for the construction of bivariate matrix valued rational interpolants on a rectangular grid is introduced in this paper the rational interpolants are of the continued fraction form with scalar denominator in this respect the approach is essentially different from that of bose and basu 1980 where a rational matrix valued approximant with matrix valued numerator and denominator is used for the approximation of a bivariate matrix power series the matrix quotients are based on the generalized inverse for a matrix introduced by gu chuanqing and chen zhibing 1995 which is found to be effective in continued fraction interpolation a sufficient condition of existence is obtained some important conclusions such as characterisation and uniqueness are proven respectfully the inner connection between two type interpolating functions is investigated some examples are given so as to illustrate the results in the paper

interpretation of lavrentiev phenomenon by relaxation the higher order case we consider integral functionals of the calculus of variations of the form f u fj f x u u u n dx defined for u e wn 0 1 and we show that the relaxed functional f with respect to the weak woc 1 0 1 convergence can be written as i f u j xu u n dx l u where the additional term l u the lavrentiev gap is explicitly identified in terms of f

model reduction via balanced state space representations a model reduction procedure based on balanced state space representations is studied in this paper the reduced order model is examined from the point of view of stability controllability and observability both continuous time and discrete time systems are considered

approximation of delay systems a case study abstract the approximation of delay systems is studied a numerically well behaved method for computing hankel optima rational approximations for delay systems is presented based on certain properties of hankel matrices and the theory of laguerre polynomials the cf method of trefethen and certain pade approximations of delay systems are also considered the importance of a certain wiener algebra property in the analysis of the rational approximation of an important class of delay systems in the l norm and the hankel norm is shown completing certain results presented in the literature for systems satisfying certain nuclearity or absolute continuity conditions a case study and numerical comparison is presented for the approximation of the parametric family of first order stable delay systems g s exp s ts 1 numerical experience indicates that l optimized cf approximations based on short truncated maclaurin series give in general somewhat smaller l approximation errors than hankel optim

optimal angle reduction a behavioral approach to linear system approximation we investigate the problem of optimal state reduction under minimization of the angle between system behaviors the angle is defined in a worst case sense as the largest angle that can occur between a system trajectory and its optimal approximation in the reduced order model this problem is analyzed for linear time invariant finite dimensional systems in a behavioral l2 setting without reference to input output decompositions and stability considerations the notion of a weakest past future link is introduced and it is shown how this concept is applied for the purpose of model reduction a method that reduces the state dimension by one is presented and shown to be optimal specific algorithms are provided for the numerical implementation of the approximation method the concepts and results are explicitly translated to an input output setting and related to balancing hankel norm reduction and normalized doubly coprime factorizations

interpretation of the lavrentiev phenomenon by relaxation we consider functionals of the calculus of variations of the form f u 01 f x u u dx defined for u w1 0 1 and we show that the relaxed functional f with respect to weak w1 1 0 1 convergence can be written as f u 01 f x u u dx l u where the additional term l u called the lavrentiev term is explicitly identified in terms of f

a unified least squares approach to identify and to reduce continuous asymptotically stable systems abstract the stability preserving model reduction algorithm for the discrete time case s feldmann p lang a least squares approach to reduce stable discrete linear systems preserving their stability laa 381 2004 141 163 is extended to both the identification and reduction of continuous time systems the obtained reduction algorithm turns out to be c stability preserving

the approximation of higher order integrals of the calculus of variations and the lavrentiev phenomenon we prove the following approximation theorem given a function x a b to mathbb r n in the sobolev space bf w nu 1 1 nu geq 1 and epsilon 0 there exists a function x epsilon in bf w nu 1 infty such that begin gather int a b sum i 1 m l i x epsilon nu x epsilon nu 1 psi i t x epsilon x epsilon dotsc x epsilon nu int a b sum i 1 m l i x nu x nu 1 psi i t x x dotsc x nu epsilon begin aligned x epsilon a x a quad x epsilon b x b x epsilon a x a quad x epsilon b x b vdots x epsilon nu a x nu a quad x epsilon nu b x nu b end aligned end gather provided that for every i in 1 dotsc m l i psi i is continuous in a neighborhood of x l i is convex in its second variable and psi i evaluated along x has positive sign we discuss the optimality of our assumptions comparing them with an example of sarychev j dynam control systems 3 1997 pp 565 588 as a consequence we obtain the nonoccurrence of the lavrentiev phenomenon in particular the integral functional int a b l x nu x nu 1 does not exhibit the lavrentiev phenomenon for any given boundary values x a a x b b x a a x b b dotsc x nu a a nu x nu b b nu furthermore we prove the following necessary condition an action functional with lagrangian of the form sum i 1 m l i x nu x nu 1 psi i t x x dotsc x nu with nu geq 0 exhibiting the lavrentiev phenomenon takes the value infty in any neighborhood of a minimizer

parametric generalized singular perturbation approximation for model order reduction a new model reduction technique for the approximation of balanced realization is introduced the method involves a further generalization of the generalized singular perturbation approximation by adding several parameters that can be tuned according to a specified performance criterion an a priori bound can be computed guaranteeing the quality of the approximated model in the whole range of parameter variations two numerical examples conclude this paper

viability of methods for generating stable reduced order models various methods for generating stable reduced order models are shown to have a serious disadvantage in that the reduced model may approximate the nondominant poles of the system and hence lead to erroneous approximations an example is given to compare these methods with the standard pade approximation method for which this disadvantage does not exist

generalized inverse matrix pade approximation on the basis of scalar products a new type of generalized matrix inverse is used to define the generalized inverse matrix pade approximants gmpa gmpa is introduced on the basis of scalar product of matrices with the form of matrix numerator and scalar denominator it is different from the existing matrix pade approximants in that it does not need multiplication of matrices in the construction process some algebraic properties are discussed the representations of gmpa are provided with the following three forms i the explicit determinantal formulas for the denominator scalar polynomials and the numerator matrix polynomials ii e algorithm expression iii thiele type continued fraction expression the equivalence relations above three representations are proposed

some further observations on the differentiation method of modal reduction the differentiation method of model reduction is shown to be equivalent to forming successive ratios of multipoint taylor polynomial approximation of the numerator and denominator of the transfer function respectively this reformulation allows the method to be applied entirely through a routh type array structure strong links are seen to exist between the method and other stability preserving methods via this routh type structure

on model reduction by modified cauer form a simple algorithm for obtaining the continued fraction quotients in the modified cauer form mcf from the given system matrices in companion form is presented in the sequel the triple of all lower order models in companion form is directly obtained a matrix method of obtaining the time moments and markov parameters from the mcf quotients is also outlined finally it is shown that system reduction by matching a set of mcf quotients is equivalent to system reduction by matching a set of time moments and markov parameters

minimal state space realization in linear system theory an overview we give a survey of the results in connection with the minimal state space realization problem for linear time invariant systems we start with a brief historical overview and a short introduction to linear system theory next we present some of the basic algorithms for the reduction of nonminimal state space realizations and for the minimal state space realization of infinite or finite sequences of markov parameters of linear time invariant systems finally we discuss some extensions of this problem to other classes of systems and point out some related problems

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

stability of reduced order models obtained via the equation error method some observations are made on the asymptotic stability and controllability of reduced order models which are obtained via the minimization of the equation error method

singular minimizers for regular one dimensional problems in the calculus of variations i i f x u x u x dx ja where a b is a finite interval denotes d dx f f x u p is c 0 and fpp 0 regularity we consider the problem of minimizing in the set a of absolutely continuous functions u a b r satisfying u a a u b 3 where a and 5 are given constants as is well known if a minimizer u of i in a is lipschitz continuous then u is smooth and satisfies the euler lagrange equation

overlapping block balanced canonical forms for various classes of linear systems abstract through the use of balanced realizations it has been possible to derive parametrizations and canonical forms for various classes of minimal linear systems of given dimension a possible problem of these parametrizations is that they are not overlapping this could be a drawback for the application of balanced parametrizations in such areas as system identification model reduction and optimization it is the topic of this paper to derive overlapping parametrizations which are closely related to the existing balanced parametrizations we first introduce input normal canonical forms which are defined through a novel way of choosing nice selections of columns of the reachability matrix these canonical forms provide overlapping parametrizations in the sense that they form a real analytic atlas of the manifold of systems which are considered then we introduce so called block balanced input normal forms which use the previously constructed input normal forms as building blocks the classes of systems for which such parametrizations are given are the stable minimal systems positive real minimal systems bounded real minimal systems and the class of all minimal systems of given mcmillan degree the results include both the single input single output and the multivariable case in the single input single output case however the issue of choosing nice selections of columns does not occur therefore in this case the derivation and presentation of the results is considerably simplified

one dimensional variational problems whose minimizers do not satisfy the euler lagrange equation in this paper we consider the problem of minimizing i u text int a b text f x u x u x dx 1 1 in the set a of absolutely continuous function u a b satisfying the end conditions u left a right text alpha text quad u left b right text beta text 1 2 where and are given constant in 1 1 a b is a finite interval denotes frac d dx and the integrand f f x u p is assumed to be smooth nonnegative and to satisfy the regularity condition f pp text 0 text 1 3

a new set of invariants for linear systems application to reduced order compensator design a new set of invariants for linear systems weighting the contribution of each state component to the inherent closed loop lqg behavior of the system is presented together with applications to model order reduction and reduced order compensator design

linear quadratic gaussian balancing for discrete time infinite dimensional linear systems in this paper we study the existence of linear quadratic gaussian lqg balanced realizations for discrete time infinite dimensional systems lqg balanced realizations are those for which the smallest nonnegative self adjoint solutions of the control and filter riccati equations are equal we show that the control filter riccati equation has a nonnegative self adjoint solution if and only if the system is output input stabilizable our main result is that the transfer function of a discrete time linear system has an approximately controllable and observable lqg balanced realization iff it has an input and output stabilizable realization the corresponding control and filter riccati equations have unique nonnegative self adjoint solutions moreover approximately controllable and observable lqg balanced realizations are unique up to a unitary state space transformation finally we show that the spectrum of the product of the smallest nonnegative self adjoint solutions of the control and filter riccati equations is independent of the particular realization

generically algebraic jordan algebras over commutative rings abstract the theory of the generic minimum polynomial norm and trace is developed for quadratic jordan algebras which are finitely generated and projective modules over an arbitrary commutative base ring using scheme theoretic methods we recover with new proofs most of the classical theory over fields and also obtain a number of results which are new even in the classical setting

some integral properties for balanced realizations of scalar systems certain structural properties for balanced minimal realizations of scalar transfer functions are used to develop closed forms for the semi infinite integral of the impulse response and of the square of the impulse response

controller reduction using canonical interactions a conceptual approach to controller reduction is proposed that is based on closed loop considerations the method assesses the interactions between the plant and a high order controller as described by canonical correlation coefficients a free parameter in the algorithm of the reduced controller makes it possible to trade stability robustness versus performance several examples demonstrate the efficacy of the method

a method for linear model reduction using the input independent equation error the paper deals with the order reduction problem for single input single output linear dynamical systems combining the uniform approximation approach with the equation error formulation a simplification method is proposed which leads to input independent optimal models by a relatively simple calculation procedure

reduced order control based on approximate inertial manifolds a reduced order method based on approximate inertial manifolds is applied to optimal control problems in infinite dimensional state spaces a detailed analysis of the method is given for the linear quadratic regulator problem the method can also be applied to higher order control systems with an appropriate decomposition of the state space in terms of slow and fast exponential decay

diagonalization in jordan pairs as every student of linear algebra knows a rectangular matrix over a division ring can be diagonalized by elementary row and column operations similarly there are normal forms for alternating and hermitian matrices which can be achieved by simultaneous row and column operations since matrices of this kind form the main examples of jordan pairs it is natural to ask whether similar results hold in the jordan setting we show that this is in fact the case study the obstruction to diagonalizability the defect and also prove that nondegenerate jordan pairs admit a rank function sharing many properties with classical matrix rank let v v v be a jordan pair and s e e a set of orthogonal idempotents the s diagonal elements are those in xi 1 v e now suppose that v is nondegenerate and satisfies dcc on principal inner ideals an element x e v a i is called diagonalizabfe if x is s diagonal for some s consisting of division idempotents to see that for matrices this is the same as the usual notion of diagonalizability suppose that v has in addition act on principal inner ideals then v contains a frame that is a finite set f of orthogonal division idempotents such that v f 0 for rectangular or hermitian matrices over division rings f can be taken to consist of the diagonal matrix units e eii whereas for alternating matrices over a field e eli1 2ie zi the jordan analogues of elementary row and column operations are the inner automorphisms b v e v e and v e v e where e e f which generate the group of f elementary automorphisms of v 6 any set of orthogonal division idempotents can be transformed up to association into f by an f elementary automorphism 6 th 21 it follows that x is diagonalizable if and only if cp x is f diagonal for some elementary automorphism cp of v this shows the equivalence with the usual definition

feedback properties of multivariable systems the role and use of the return difference matrix for linear time invariant multivariable feedback systems the feedback properties of plant disturbance attenuation sensor noise response stability margins and sensitivity to plant and sensor variation are quantitatively related to the bode magnitude versus frequency plots of the singular values of the return difference matrix i l and of the associated inverse return difference matrix i l 1 implied fundamental limits of feedback performance are quantitatively described and design tradeoffs are discussed the penalty function in the stochastic linear quadratic gaussian lqg optimal control problem is found to be a weighted sum of the singular values with the weights determined by the quadratic cost and noise intensity matrices this enables systematic tuning of lqg cost and noise matrices so that the resulting optimal return difference and inversereturn difference meet inequality constraints derived from design specifications on feedback properties the theory has been used to synthesize a multivariable automatic controller for the longitudinal dynamics of an advanced fighter aircraft

on the solution of a nonlinear matrix equation for mimo symmetric realizations abstract a result concerning a particular solution of a nonlinear solution matrix equation reported in a previous paper 2 and interesting for studying siso systems is extended to mimo symmetric realizations for such a class of systems other important siso results can be generalized

hankel singular values and lqg characteristic values of discrete time linear systems in cascade with inner systems recent results have shown that for continuous time systems obtained by cascading an asymptotically stable system with an inner system the first n hankel singular values are greater than or equal to those of the original system similarly cascading a system in minimal form with an inner system the same property holds for the linear quadratic gaussian lqg characteristic values in this article we consider the discrete time case and demonstrate that the property also holds for these systems a very important consequence stemming from these results is that the hankel singular values and the lqg characteristic values of input delayed discrete systems are greater than or equal to those of their zero delay counterpart