j spectral factorization and equalizing vectors abstract this note presents some basic results on j spectral factorization and its connection with state space realizations the assumptions needed are mild and the proofs are given in detail

robust control in the gap a state space solution in the presence of a single input delay a solution is provided to the problem of sub optimal robust design in the gap for a system with a single input lag the problem is shown to reduce to two matrix algebraic riccati equations that solve an allied linear quadratic gaussian h sub 2 problem and a parameterized differential matrix riccati equation the analysis intertwines finite dimensional delay equations and distributed abstract evolution models the parameterized family of suboptimal compensators is presented in terms of a neutral equation model

on stabilization and the existence of coprime factorizations shows that any transfer function matrix whose elements belong to the quotient field of h sub infinity and which is stabilizable has a matrix fraction representation over h sub infinity which is coprime in the sense that a matrix bezout identity can be satisfied

on spl hscr sub spl infin control for dead time systems a mixed sensitivity spl hscr sub spl infin problem is solved for dead time systems it is shown that for a given bound on the spl hscr sub spl infin norm causal stabilizing controllers exist that achieve this bound if and only if a related finite dimensional riccati equation has a solution with a certain nonsingularity property in the case of zero time delay the riccati equation is a standard riccati equation and the nonsingularity condition is that the solution be nonnegative definite for nonzero time delay the nonsingularity condition is more involved but still allows us to obtain controllers all suboptimal controllers are parameterized and the central controller is shown to be a feedback interconnection of a finite dimensional system and a finite memory system both of which can be implemented some spl hscr sub spl infin problems are rewritten as pure rational spl hscr sub spl infin problems using a smith predictor parameterization of the controller

formulas on preview and delayed h infty control a generalized hinfin control problem which covers preview and delayed control strategies is discussed based on a state space approach by introducing a hamiltonian matrix which is associated with a delay free generalized plant the analytic solution to the corresponding operator riccati equation is newly established based on the result obtained here the hinfin control problem is solved and for typical control problems e g hinfin and linear quadratic lq control for multiple input delay systems hinfin preview control some interpretations are provided on the resulting control system

fluctuation dissipation theorem and the linear glauber model we obtain exact expressions for the two time autocorrelation and response functions of the dimensional linear glauber model although this linear model does not obey detailed balance in dimensions d or 2 we show that the usual form of the fluctuation dissipation ratio still holds in the stationary regime in the transient regime we show the occurrence of aging with a special limit of the fluctuation dissipation ratio x infinity 1 2 for a quench at the critical point

mixed sensitivity optimization for a class of unstable infinite dimensional systems abstract in this paper we solve the h mixed sensitivity minimization problem for a class of unstable distributed systems the solution is based on an extension of the skew toeplitz methodology the key mathematical fact used is that the skew toeplitz operators arising in the unstable case are finite rank perturbations of the classical skew toeplitz operators obtained from compressions of rational functions a system of linear equations the singular system is derived for this class of operators from which one can compute the associated singular values and vectors an example is included to illustrate the results

optimal linear glauber model contrary to the actual nonlinear glauber model the linear glauber model lgm is exactly solvable although the detailed balance condition is not generally satisfied this motivates us to address the issue of writing the transition rate w j in a best possible linear form such that the mean squared error in satisfying the detailed balance condition is least the advantage of this work is that by studying the lgm analytically we will be able to anticipate how the kinetic properties of an arbitrary ising system depend on the temperature and the coupling constants the analytical expressions for the optimal values of the parameters involved in the linear w j are obtained using a simple moore penrose pseudoinverse matrix this approach is quite general in principle applicable to any system and can reproduce the exact results for one dimensional ising system in the continuum limit we get a linear time dependent ginzburg landau equation from the glauber s microscopic model of non conservative dynamics we analyze the critical and dynamic properties of the model and show that most of the important results obtained in different studies can be reproduced by our new mathematical approach we will also show in this paper that the effect of magnetic field can easily be studied within our approach in particular we show that the inverse of relaxation time changes quadratically with weak magnetic field and that the fluctuation dissipation theorem is valid for our model

robust controller design for delay systems in the gap metric a robust stabilization problem in the gap metric is discussed for a system with delays in control the authors derive a design procedure of robust controllers based on finite dimensional ricatti equations and it is shown that the resulting controller has a structure of observer based predictive action by employing completing the square argument a game theoretic interpretation is also provided on the trade off between the initial uncertainties and attenuating the disturbance caused by plant uncertainties

a unified smith predictor based on the spectral decomposition of the plant we point out a numerical problem in the well known modified smith predictor and propose a unified smith predictor usp which overcomes this problem the proposed usp combines the classical smith predictor with the modified one after a spectral decomposition of the plant we then derive an equivalent representation of the original delay system together with the usp based on this representation we give a controller parameterization and we solve the standard h 2 problem

parameterization of suboptimal solutions of the nehari problem for infinite dimensional systems the nehari problem plays an important role in hinfin control theory it is well known that hinfin control problem can be reduced to solving this problem this note gives a parameterization of all suboptimal solutions of the nehari problem for a class of infinite dimensional systems many earlier solutions of this problem are seen to be special cases of this new parameterization it is also shown that for finite impulse response systems this parameterization takes a particularly simple form

a note on essential spectra and norms of mixed hankel toeplitz operators in this brief hankel toeplitz operators which occur in feedback theory e g in the minimization of mixed h sensitivity and complementary sensitivity will be considered a method of computing their spectra eigenvectors and norms will be presented for infinite dimensional systems subject to continuous weightings

weighted sensitivity minimization in systems with a single input delay a state space solution a differential algebraic matrix riccati equation based design method is developed the solution is reached via a parallel two way analysis of an underlying differential game in the framework of both the finite order delay system and a distributed parameter ordinary evolution model

inner outer factorization for nonlinear noninvertible systems this paper considers inner outer factorization of asymptotically stable nonlinear state space systems in continuous time that are noninvertible our approach will be via a nonlinear analogue of spectral factorization which concentrates on first finding the outer factor instead of the inner factor an application of the main result to control of nonminimum phase nonlinear systems is indicated

weighted sensitivity minimization general plants in h and rational weights abstract for general plants in h and rational weighting functions the problem of computing the weighted h optimal sensitivity is considered a general formula is derived for a function whose singularities determine the optimal sensitivity 6 a determinantal expression for the optimal sensitivity is then deduced and related to a conjecture of zhou and khargonekar

diluted banded random matrices scaling behavior of eigenfunction and spectral properties we demonstrate that the normalized localization length of the eigenfunctions of diluted sparse banded random matrices follows the scaling law the scaling parameter of the model is defined as where is the average number of non zero elements per matrix row n is the matrix size and additionally we show that also scales the spectral properties of the model up to certain sparsity characterized by the spacing distribution of eigenvalues

robust stabilization of infinite dimensional systems by finite dimensional controllers abstract the problem of robustly stabilizing an infinite dimensional system with transfer function g subject to an additive perturbation is considered it is assumed that g b 0 of systems introduced by callier and desoer 3 the perturbation satisfies w1 w2 c now write w1gw2 g1 g1 where g1 is rational and totally unstable and g2 is stable generalizing the finite dimensional results of glover 12 this family of perturbed systems is shown to be stabilizable if and only if min g 1 the smallest hankel singular value of g 1 a finite dimensional stabilizing controller is then given by k w 2 k 1 i g 2 k 1 1 w 1 where 2 is a rational approximation of g2 such that g 2 g 2 min g 1 and k1 robustly stabilizes g1 to margin the feedback system g k will then be stable if w1 w2

weighted sensitivity minimization for delay systems in this note we discuss the h infty sensitivity minimization problem for linear time invariant delay systems while the unweighted case reduces to simple nevanlinna pick interpolation the weighted case turns out to be much more complicated and demands certain functional analytic techniques for its solution

parametric dependent hamiltonians wave functions random matrix theory and quantal classical correspondence we study a classically chaotic system that is described by a hamiltonian mathcal h q p x where q p are the canonical coordinates of a particle in a two dimensional well and x is a parameter by changing x we can deform the shape of the well the quantum eigenstates of the system are n x we analyze numerically how the parametric kernel p n m n x m x 0 2 evolves as a function of ensuremath delta x ensuremath equiv x ensuremath x 0 this kernel regarded as a function of n ensuremath m characterizes the shape of the wave functions and it also can be interpreted as the local density of states the kernel p n m has a well defined classical limit and the study addresses the issue of quantum classical correspondence both the perturbative and the nonperturbative regimes are explored the limitations of the random matrix theory approach are demonstrated

a skew toeplitz approach to the h optimal control of multivariable distributed systems in this paper the problem of the h infty optimization of multivariable distributed systems in the four block setting is studied this work is based on several previous papers and employs the skew toeplitz framework developed in operator theory adv appl 32 1988 pp 21 43 operator theory adv appl 32 1988 pp 93 112 operator theory and integral equations 11 1988 pp 726 767 j functional anal 74 1987 pp 146 159 siam j math anal 19 1988 pp 1081 1091

a j spectral factorization approach to h control it is shown that necessary and sufficient conditions for the existence of sub optimal solutions to the model matching problem associated with h control are that two coupled j spectral factorizations exist the second j spectral factor provides a parameterization of all solutions to the model matching problem the existence of the j spectral factors is then shown to be equivalent to the existence of stabilizing non negative definite solutions to two algebraic riccati equations allowing a state space formula for a linear fractional representation of all controllers to be given

brief every stabilizing dead time controller has an observer predictor based structure this paper considers the stabilization problem for linear time invariant systems with a single delay h in the feedback loop two parametrizations of all stabilizing controllers are derived in terms of a state space realization of the rational part of the plant these parametrizations have simple structures and clear interpretations the first is in the form of a generalized dead time compensator smith predictor and the second is in the observer predictor form applications of the proposed parametrization to the h 2 optimization and the robust stabilization problems are discussed

non perturbative response chaos versus disorder quantized chaotic systems are generically characterized by two energy scales the mean level spacing and the bandwidth b this implies that with respect to driving such systems have an adiabatic a perturbative and a non perturbative regimes a strong nonlinearity in the response due to a quantal non perturbative effect is found for disordered systems that are described by random matrix theory models is there a similar effect for quantized chaotic systems theoretical arguments cannot exclude the existence of an analogous weak version of the above mentioned nonlinear response effect but our numerics demonstrates an unexpected degree of semiclassical correspondence

brief on the h fixed lag smoothing how to exploit the information preview the problem of the continuous time h fixed lag smoothing over the infinite horizon is studied the first solution to the problem is derived in terms of one algebraic riccati equation of the same dimension as in the filtering case and the mechanism by which the performance improvements with respect to the h filtering occur is clarified it is shown that the h smoother exploits the information preview in an h 2 manner

on h sub 2 control of systems with multiple i o delays the h2 optimal control problem of systems with multiple i o delays is solved the problem is tackled by first transforming it to what is called the two sided regulator problem the latter is solved using orthogonal projection arguments and spectral factorization the resulting controller is an interconnection of rational blocks nonrational blocks having finite impulse response and delay operators all of which are implementable

a j spectral factorization approach to control necessary and sufficient conditions for the existence of suboptimal solutions to the standard model matching problem associated with mathcal h infty control j control are derived using j spectral factorization theory the existence of solutions to the model matching problem is shown to be equivalent to the existence of solutions to two coupled j spectral factorization problems with the second factor providing a parametrization of all solutions to the model matching problem the existence of the j spectral factors is then shown to be equivalent to the existence of nonnegative definite stabilizing solutions to two indefinite algebraic riccati equations allowing a state space formula for a linear fractional representation of all controllers to be given a virtue of the approach is that a very general class of problems may be tackled within a conceptually simple framework and no additional auxiliary riccati equations are required

transfer functions of regular linear systems part i characterizations of regularity we recall the main facts about the representation of regular linear systems essentially that they can be described by equations of the form x t ax t bu t y t cx t du t like finite dimensional systems but now a b and c are in general unbounded operators regular linear systems are a subclass of abstract linear systems we define transfer functions of abstract linear systems via a generalization of a theorem of foures and segal we prove a formula for the transfer function of a regular linear system which is similar to the formula in finite dimensions the main result is a simple to state but hard to prove necessary and sufficient condition for an abstract linear system to be regular in terms of its transfer function other conditions equivalent to regularity are also obtained the main result is a consequence of a new tauberian theorem which is of independent interest

cal h _ infty control and estimation problems with delayed measurements state space solutions most physical processes exhibit transport delay in the measured output and it is well known that this can have disastrous effects on system stability and performance if it is not accounted for in this paper we give necessary and sufficient conditions for existence of estimators and controllers that achieve the desired cal h infty performance criterion when such a measurement delay is present we also give the complete characterization of all controllers and estimators that achieve the desired performance criterion the necessary and sufficient conditions are easy to check and are given in terms of the familiar pair of algebraic riccati equations that appear in the nondelay versions of the corresponding cal h infty problems along with an additional riccati differential equation explicit state space formulas for the controllers and estimators are also obtained they have a linear periodic structure and are easily implementable to obtain these results we first obtain state space results for a modified nehari problem which may be of independent interest see problem 5 in section 2

evaluating and approximating fir filters an approach based on functions of matrices most of the controllers designed for systems with timedelays involve finite impulse response fir filters in their structure in particular prediction based controllers for systems with input output delays we present theory algorithms and software to evaluate the frequency response to approximate the filters by stable linear timeinvariant systems and to compute their induced cal l 2 gain the approach is based on the theory of functions of matrices in the approach pole zero cancelations in frequency domain representations are explicitly taken into account

robust stabilization of delay systems by approximation of coprime factors given a delay system with transfer function g s h2 s h1 s where h1 s 01npi s e is and h2 s 02nqi s e is with 0 0 1 n1 0 0 n2 the pi being polynomials of degree i and i 0for i 0 and the qi polynomials of degree di 0 for each i the robust stabilization of a class of perturbed coprime factors of this system is considered asymptotic estimates are obtained based on recent results on the approximation and stabilization of normalized coprime factors an explicit formula is given for the normalized coprime factor stability margin for the case of a multivariable transfer function of the form g s e str s with r rational

synthesis of non rational controllers for linear delay systems this paper addresses the control of linear delay systems using non rational controllers the structure of the controller is chosen so as to copy the structure of the plant reproducing the delays in the state and in the output the resulting stabilization and performance design problems are entirely expressed as linear matrix inequalities although the design inequalities are based on delay independent stability conditions the overall design is delay dependent in the sense that the controller makes use of the delay parameter of the plant this parameter is assumed to be constant yet arbitrary using non rational controllers we overcome the main difficulty faced when designing rational controllers for linear delay systems which is to incorporate in the design problem the matrix multiplier used to prove stability with respect to the delayed part of the system we illustrate the paper with several examples and provide extensive comparisons with existent results

h2 optimal model approximation by input output delay structured reduced order models abstract in this paper the h 2 optimal approximation of a multi input multi output transfer function by a finite dimensional system including input output delays is addressed this problem is of particular interest when the full order model represents a transport phenomenon the contribution is twofold firstly based on the pole residue decomposition an h 2 inner product formula in the presence of input output delays is derived secondly the underlying h 2 optimality conditions of the approximation problem are obtained as an extension of the tangential interpolatory conditions it is also demonstrated that for fixed delay values this problem can be recast as a rational function approximation one an iterative algorithm is sketched out and numerical results assess the theoretical contributions

explicit formulas for optimally robust controllers for delay systems this paper considers single input single output systems whose transfer functions take the form of a strictly proper rational function times a delay a closed form expression is presented for the controller which is optimally robust with respect to perturbations measured in the gap metric the formula allows the h sub spl infin loop shaping procedure of glover mcfarlane to be carried out explicitly for this class of systems without the need to first find a rational approximation of the plant the form of the controller involves a certain algebra of pseudo derivation operators these operators and their matrix generalizations play a central role in the derivation of the controller a discussion of the main properties of these operators is given an example is presented of a controller design to achieve disturbance attenuation and robust set point following for a plant with two lightly damped poles and a nontrivial time delay the performance is compared and shown to be superior to that of a smith predictor

h sup spl infin optimal and suboptimal controllers for infinite dimensional siso plants this paper presents a simple formula for h sup spl infin optimal and suboptimal controllers for unstable siso distributed plants with rational weighting functions the controller is expressed in terms of i inner and outer parts of the plant ii a finite dimensional spectral factor obtained from the weighting functions and iii a rational function satisfying certain interpolation conditions under certain genericity assumptions this rational function is of dimension less than or equal to n sub 1 l 1 n sub 1 l in the suboptimal case where l is the number of unstable poles of the plant and n sub 1 is the order of the sensitivity weighting function there are 2 n sub 1 l 2 n sub 1 l 1 in the suboptimal case linear equations which determine this rational function these linear equations can be written directly from the structure of the controller

a chain scattering matrix approach to the h output feedback control for state delayed systems a state space coprime factorization approach to the h controllers design for state delayed systems is developed the coprime factors that are generated by the present factorization possess the same delayed state as that of the original system this property together with the well known youla parameterization and the chain scattering matrix description provides a relatively simple method to obtain a family of the internally stabilizing h output feedback controllers a numerical example is given to demonstrate the validity and the applicability of the proposed approach

fock space for fermion like lattices and the linear glauber model abstract the concept of fock space representation is developed to deal with stochastic spin lattices written in terms of fermion operators a density operator is introduced in order to follow in parallel the developments of the case of bosons in the literature some general conceptual quantities for spin lattices are then derived including the notion of generating function and path integral via grassmann variables the formalism is used to derive the liouvillian of the d dimensional linear glauber dynamics in the fock space representation then the time evolution equations for the magnetization and the two point correlation function are derived in terms of the number operator

the nehari problem in systems with distributed input delays is inherently finite dimensional the nehari problem is considered in the context of linear systems with distributed input or equivalently output lags it is shown that both the computation of the optimal value and the parametrization of suboptimal solutions reduce to matrix computations in an associated undelayed nehari problem the solution is then given in terms of a system of neutral type

on the weighted sensitivity minimization problem for delay systems we consider the h optimal sensitivity problem for delay systems in particular we consider computation of inf w q q h jr where w s is any function in rh jr and in h jr is any inner function we derive a new explicit solution in the pure delay case where e sh h 0

h 0e sensitivity minimization for delay systems this paper announces results on the problem of feedback compensator design for x norm weighted sensitivity minimization when the plant contains a delay in the input a complete solution is presented for the case of one pole zero weighting function and a single input single output plant for stable minimum phase rational part generalizations and proofs will be published elsewhere

nehari problems and equalizing vectors for infinite dimensional systems for a class of infinite dimensional systems we obtain a simple frequency domain solution for the suboptimal nehari extension problem the approach is via j spectral factorization and it uses the concept of an equalizing vector moreover the connection between the equalizing vectors and the nehari extension problem is given

another look at finite horizon h control problems for systems with input delays we discuss a finite horizon h control problem for time varying systems with input delays clarifying a relationship between two h control problems in input delay case and in measurement delay case we derive a solution in input delay case based on the known result for the h control problem in measurement delay case and show that the solution has the same predictor observer structure as the solution in measurement delay case has using this structural information on the solution we also present an elementary proof of the solution to the finite horizon h control problem for systems with input delays which is based only on completion of squares

survival probability and local density of states for one dimensional hamiltonian systems for chaotic systems there is a theory for the decay of the survival probability and for the parametric dependence of the local density of states this theory leads to the distinction between perturbative and non perturbative regimes and to the observation that semiclassical tools are useful in the latter case we discuss what is left from this theory in the case of one dimensional systems we demonstrate that the remarkably accurate uniform semiclassical approximation captures the physics of all the different regimes though it cannot take into account the effect of strong localization

perturbations and chaos in quantum maps the local density of states ldos is a distribution that characterizes the effects of perturbations on quantum systems recently a semiclassical theory was proposed for the ldos of chaotic billiards and maps this theory predicts that the ldos is a breit wigner distribution independent of the perturbation strength and also gives a semiclassical expression for the ldos width here we test the validity of such an approximation in quantum maps by varying the degree of chaoticity the region in phase space where the perturbation is applied and the intensity of the perturbation we show that for highly chaotic maps or strong perturbations the semiclassical theory of the ldos is accurate to describe the quantum distribution moreover the width of the ldos is also well represented for its semiclassical expression in the case of mixed classical dynamics

quantum decay into a non flat continuum we study the decay of a prepared state into non flat continuum we find that the survival probability p t might exhibit either stretched exponential or power law decay depending on non universal features of the model still there is a universal characteristic time t0 that does not depend on the functional form it is only for a flat continuum that we get a robust exponential decay that is insensitive to the nature of the intra continuum couplings the analysis highlights the co existence of perturbative and non perturbative features in the local density of states and the nonlinear dependence of 1 t0 on the strength of the coupling

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

analytic system problems and j lossless coprime factorization for infinite dimensional linear systems abstract this paper extends the coprime factorization approach to the synthesis of internally stabilizing controllers satisfying an h norm bound to a class of systems with irrational transfer matrices using the coprime factorization description the h control problem can be reduced to two stable analytic system problems such problems have solutions if and only if a certain j lossless factorization exists the full h synthesis problem is shown to be equivalent to the solution of two nested j lossless factorizations if the irrational transfer matrix has a state space realization then the known state space formulas for the h control problem may be recovered using the relationship between j lossless factorizations and solutions of riccati equations however the results derived here are valid for a larger class of infinite dimensional systems

finite rank criteria for h infty control of infinite dimensional systems this paper studies the hinfin control of infinite dimensional systems whose transfer matrices are expressible as a series connection of a rational transfer matrix and a scalar possibly irrational inner function this class of systems is adequate for describing many control problems in practice when weighting functions are rational and plants have at most finitely many unstable modes we show that this problem can be reduced to solving two matrix valued riccati equations and an additional rank condition furthermore the obtained controller structure is characterized by the inner function and controllers for the finite dimensional part this result provides us with a solution that is easily implementable without much prior knowledge on infinite dimensional control theory a numerical example is given to illustrate the result

optimal mixed sensitivity for siso distributed plants the authors extend previous results for the single input single output siso mixed sensitivity problem for distributed parameter systems to a more general setting in which the transfer function of the plant to be controlled has both irrational stable and irrational unstable parts these developments were originally motivated by the computation of the infimal h sup spl infin norm of the weighted mixed sensitivity for the irrational transfer function model of a damped flexible beam

2dof controller parametrization for systems with a single i o delay this note puts forward a parametrization of all stabilizing two degrees of freedom controllers for possibly unstable processes with dead time the proposed parametrization is based on a doubly coprime factorization of the plant and takes the form of a generalized smith predictor dead time compensator feedback part and a finite dimensional feedforward part prefilter some alternative dead time compensation schemes and disturbance attenuation limitations are also discussed