the lqg optimal regulation problem for systems with perfect measurements explicit solution properties and application to practical designs the continuous time stationary linear quadratic gaussian lqg optimal regulation problem is considered where the measurements are free of white noise components a simple direct solution in the s domain is derived for the optimal controller for general linear time variant right invertible or left invertible systems the explicit expressions that are found for the controller transference and for the regulator return ratio matrix can be used to obtain a practical suboptimal design in the s domain these expressions are applied to derive simple conditions for the precise recovery of the return ratio matrix of the lqg regulator

almost invariant subspaces an approach to high gain feedback design part ii almost conditionally invariant subspaces in part i of this paper almost controlled invariant subspaces were studied in this part we will consider their duals the almost conditionally invariant subspaces these concepts give immediately by dualization of the almost disturbance decoupling control by state feedback the solution of the almost disturbance decoupled estimation problem finally we consider the problem of approximate disturbance decoupling by measurement feedback and it is shown that this problem is solvable to any arbitrary degree of accuracy if and only if 1 almost disturbance decoupling by state feedback and 2 almost disturbance decoupled estimation of the to be controlled output are both possible

a condition for robust stabilizability a family of linear time invariant finite dimensional plants is called robustly stabilizable if there exists a single controller that stabilizes each plant in the family a condition for robust stabilizability is given that requires the plant transmission zeros to remain in the left half complex plane and moreover imposes a restriction on the high frequency behavior of the plant transfer matrix

on the existence of an optimal observer in singular measurement systems an essential stage in each of the existing methods for singular optimal estimation is to obtain a nonsingular measurement vector which is a linear combination of the original measurement vector and a finite number of its derivatives a necessary and sufficient condition for the existence of such a nonsingular measurement is presented if this condition is satisfied then the existence of an optimal observer is guaranteed the new condition is shown to be considerably simpler and of much wider applicability than a recently published condition which is just a sufficient one the new condition requires just a rank check of a single composite matrix consisting of the system and noise matrices unlike the other condition the existence of an optimal estimator is determined here without carrying out the complex procedures for singular optimal estimation

potential theory for hyperbolic spdes keywords potential theory nonlinear hyperbolic stochastic partial differential equations multiparameter process gaussian process hitting probability reference prob article 2004 005view record in web of science record created on 2008 12 01 modified on 2017 05 12

comments on two recent papers on reduced order optimal state estimation for linear systems with partially noise corrupted measurement two recent correspondence on the optimal singular state estimation problem are shown to be in error optimal linear least squares estimators based on an innovations approach 9 10 are presented an important outcome is the clarification and hopeful elimination of some current misconceptions in relation to stochastic observer theory

an explicit expression for the minimum phase image of transfer function matrices a simple expression is obtained for the transfer function matrix of the minimum phase image of a left invertible continuous time invariant system with zeros in the right half plane that may be of multiplicities greater than one this expression is obtained by multiplying the system transfer function matrix from the right by a special inner matrix and it is explicitly given in terms of the input zero directions that correspond to the zeros of the system in the right half plane the result provides a simple expression for the inner outer factorization of transfer function matrices and it can thus be used in h sub infinity optimal control

polarity of points for gaussian random fields we show that for a wide class of gaussian random fields points are polar in the critical dimension examples of such random fields include solutions of systems of linear stochastic partial differential equations with deterministic coefficients such as the stochastic heat equation or wave equation with space time white noise or colored noise in spatial dimensions k 1k 1 our approach builds on a delicate covering argument developed by m talagrand ann probab 23 1995 767 775 probab theory related fields 112 1998 545 563 for the study of fractional brownian motion and uses a harmonizable representation of the solutions of these stochastic pdes

minimal order wiener filter for a system with exact measurements necessary and sufficient conditions for the existence of an optimal steady state state estimator are derived under the assumption that this estimator is a linear functional of the measurements and a finite number of derivatives of the exact measurements our conditions are shown to be dual to the generalized legendre clebsch conditions of the dual optimal singular regulator a separation principle is derived and it is shown that as all process and measurement noise vanishes the error covariance of our filter converges to a null matrix

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

the design of optimal reduced order stochastic observers for discrete time linear systems the minimum variance state estimation of linear discrete time systems with random white noise input and partially noisy measurements is investigated an observer of minimal order is found which attains the minimum variance estimation error the structure of this observer is shown to depend strongly on the geometry of the system this geometry dictates the length of the delays that are applied on the measurements in order to obtain the optimal estimate the transmission properties of the observer are investigated for systems that are left invertible and free of measurement noise an explicit expression is found for the transfer function matrix of this observer from which a simple solution to the linear discrete time singular optimal filtering problem is obtained

criterion for the convergence of the solution of the riccati differential equation the optimal control problem for a linear system with a quadratic cost function leads to the matrix riccati differential equation the convergence of the solution of this equation for increasing time interval is investigated as a function of the final state penalty matrix a necessary and sufficient condition for convergence is derived for stabilizable systems even if the output in the cost function is not detectable an algorithm is developed to determine the limiting value of the solution which is one of the symmetric positive semidefinite solutions of the algebraic riccati equation examples for convergence and nonconvergence are given a discussion is also included of the convergence properties of the solution of the riccati differential equation to any real symmetric not necessarily positive semidefinite solution of the algebraic riccati equation

the effect of system zeros on the achievable h sub spl infin estimation level lower bounds on the achievable estimation accuracy of stationary linear processes in the presence of energy bounded exogenous signals are derived these bounds are determined by the zeros of the process transfer function matrix a significant difference is observed between the minimum phase and the nonminimum phase cases in the latter case a lower bound is derived also for the norm of the matrix that solves the corresponding riccati equation the difference between the cases of minimum and nonminimum phase is accentuated when the measurement noise intensity tends to zero it is shown that in the minimum phase case the corresponding filter gain is almost in the range of the process input matrix in the nonminimum phase case the columns of the corresponding gain tend to stay in the space spanned by the rows of the system output matrix

brief modified stochastic luenberger observers a modified stochastic luenberger observer mslo structure is proposed to recover the optimal performance of the coventional slo for obtaining full state estimates in linear discrete time stochastic systems the optimal mslo omslo which gives the mmse estimates is derived by using the general two stage kalman filter a reduced order form of the omslo is also proposed for systems having singular measurement noises the connection between the omslo and the optimal minimal order observer of leondes and novak is also shown

the asymptotic minimum variance estimate of stationary linear single output processes the problem of minimum error variance estimation of single output linear stationary processes in the presence of weak measurement noise is considered by applying s domain analysis to the case of single input systems and white observation noise explicit and simple expressions are obtained for the error covariance matrix of estimate and the optimal kalman gains both for minimum and nonminimum phase systems it is found that as the noise intensity approaches zero the error covariance matrix of estimating the output and its derivatives becomes insensitive to uncertainty in the system parameters this matrix depends only on the shape of the high frequency tail of the power density spectrum of the observation and thus it can be easily determined from the system transfer function the theory developed is extended to deal with white measurement noise in multiinput systems where an analog to the single input nonminimum phase case is established the results are also applied to colored observation noise problems and a simple method to derive the minimum error covariance matrices and the optimal filter transfer functions is introduced

on the asymptotic behaviour of optimal root loci optimal root loci have received some attention in the recent literature results concerning the state space characterization of optimal root loci however either deal with lower order behaviour only or apply to a restricted class of a b c state space models in the present paper we develop a simple algorithm which deals with the state space uhaialtcn7 ation of all orders of optimal root locus behaviour and which applies to any system a b c

intersection equivalence of brownian paths and certain branching processes we show that sample paths of brownian motion and other stable processes intersect the same sets as certain random cantor sets constructed by a branching process with this approach the classical result that two independent brownian paths in four dimensions do not intersect reduces to the dying out of a critical branching process and estimates due to lawler 1982 for the long range intersection probability of several random walk paths reduce to kolmogrov s 1938 law for the lifetime of a critical branching process extensions to random walks with long jumps and applications to hausdorff dimension are also derived

lqg ltr procedure using reduced order kalman filters abstractthis paper discusses a linear quadratic gaussian loop transfer recovery lqg ltr procedure using reduced order kalman filters by extending known exact recovery result the state space realisation commonly used for reduced order observer design is employed the zero structure intrinsic to the realisation is revealed asymptotic recovery is achieved using a non singular reduced order kalman filter with a parameterised set of covariance matrices the proposed procedure provides a systematic method for directly designing reduced order lqg controllers without additional coordinate transformations a numerical design example for a simple multivariable plant is presented to compare the proposed design with the standard lqg ltr design using a full order kalman filter

estimation variance and h sub infinity error minimization of stationary process with perfect measurements the singular estimation problem of linear continuous time stationary processes is solved both in the minimum error variance and the minimum h sub infinity norm sense simple explicit expressions for the resulting estimators and their corresponding minimum criteria are derived in terms of the system structure parameters these expressions enable an easy comparison between the two minimization procedures the theory is demonstrated by a simple example of fourth order which is solved by both minimization methods

design of reduced order optimal estimators directly in the frequency domain the frequency domain design of optimal reduced order estimators caiman filters for continuous time linear time invariant systems containing 0 k m noise free output measurements is considered in a direct frequency domain approach to state feedback and estimation the states of a system do not appear explicitly and therefore we here focus our attention mainly on the frequency domain characteristics of the optimal filter by suitably modifying known time domain results a simple polynomial matrix equation can be derived from which the optimal filter parameters are obtained by spectral factorization the equivalence between the time and the frequency domain results is established with the aid of a non minimal representation of reduced order observers

parameterization of reduced order compensator in the frequency domain after a brief survey of the relations between time and frequency domain representations of the observer based linear state feedback loop the parameterization of the compensator is discussed it is shown that the free parameters specified by the state feedback matrix k and the output error feedback matrix l in the time domain appear as coefficients in the polynomial matrices dtilde s and s characterizing the closed loop dynamics in the frequency domain these polynomial matrices can either be chosen arbitrarily pole placement problem or they may be derived as solutions from the corresponding optimal control or the optimal estimation problems respectively starting from the polynomial matrices dtilde s and s the doubly coprime fractional representations of the system and of the compensator transfer matrices can be computed directly without recurrence to the time domain results from these representations the right and left coprime compensator matrix fraction descriptions can easily be evalu

the properties of reduced order minimum variance filters for systems with partially perfect measurements the problem of the finite time reduced order minimum variance full state estimation of linear continuous time invariant systems is considered in cases where the output measurement is partially free of corrupting white noise components the structure of the optimal filter is obtained and a link between this structure and the structure of the system invariant zeros is established using expressions that are derived in closed form for the invariant zeros of the system simple sufficient conditions are obtained for the existence of the optimal filter in the stationary case the structure and the transmission properties of the stationary filter for general left invertible systems are investigated a direct relation between the optimal filter and a particular minimum order left inverse of the system is obtained a simple explicit expression for the filter transfer function matrix is also derived the expression provides an insight into the mechanism of the optimal estimation

all sl 2 tilings come from infinite triangulations abstract an sl 2 tiling is a bi infinite matrix of positive integers such that each adjacent 2 2 submatrix has determinant 1 such tilings are infinite analogues of conway coxeter friezes and they have strong links to cluster algebras combinatorics mathematical physics and representation theory we show that by means of so called conway coxeter counting every sl 2 tiling arises from a triangulation of the disc with two three or four accumulation points this improves earlier results which only discovered sl 2 tilings with infinitely many entries equal to 1 indeed our methods show that there are large classes of tilings with only finitely many entries equal to 1 including a class of tilings with no 1 s at all in the latter case we show that the minimal entry of a tiling is unique

explicit expressions for cascade factorization of general nonminimum phase systems explicit expressions for two different cascade factorizations of any detectable left invertible nonminimum phase systems are given the first one is a well known minimum phase all pass factorization by which all nonminimum phase zeros of a transfer function g s are collected into an all pass factor v s and g s is written g sub m s v where g sub m s is considered as a minimum phase image of g s the second one is a new cascade factorization by which g s is rewritten as g sub m s u s where u s collects all awkward zeros including all nonminimum phase zeros of g s both g sub m s and g sub m s retain the given infinite zero structure of g s further properties of g sub m s g sub m s and u s are discussed these factorizations are useful in several applications including loop transfer recovery

the non commutative a 1 t system and its positive laurent property we define a non commutative version of the a1 t system which underlies frieze patterns of the integer plane this system has discrete conserved quantities and has a particular reduction to the known non commutative q system for a1 we solve the system by generalizing the flat gl2 connection method used in the commuting case to a 2 2 flat matrix connection with non commutative entries this allows us to prove the non commutative positive laurent phenomenon for the solutions when expressed in terms of admissible initial data these are rephrased as partition functions of paths with non commutative weights on networks and alternatively of dimer configurations with non commutative weights on ladder graphs made of chains of squares and hexagons

nearly singular filtering for uniform and non uniform rank linear continuous systems abstract the problem of the minimum variance stationary estimation of linear time invariant continuous systems is considered in the case where the norm of tho measurement white noise intensity approaches zero solutions in closed form are obtained for the kalman gain and the corresponding minimum error covariance matrix of the filtered estimate in terms of the noise intensity norm and the system transfer function matrix both uniform and non uniform rank systems are considered and for the first time a solution is obtained for the case where the infinite zeros of the system tond to infinity with multiple rates

a geometric approach to the singular filtering problem we consider the least squares filtering problem for a stationary gaussian process when the observation is not fully corrupted by white noise the so called singular case an optimal estimator is constructed consisting of an integrating part which is as in the regular case computed from a spectral factorization or an equivalent matrix problem and a differentiating part whose parameters are computed from a single matrix equation this improves on older results which either work under restrictive assumptions or describe the solution only as the result of some nested algorithm

multiple points of the brownian sheet in critical dimensions it is well known that an n parameter d dimensional brownian sheet has no k multiple points when k 1 d 2kn and does have such points when k 1 d 2kn we complete the study of the existence of k multiple points by showing that in the critical cases where k 1 d 2kn there are a s no k multiple points

generalized output nulling subspaces riccati equation computation and applications four generalized output nulling subspaces ons are defined and it is shown that many subspaces important in the analysis of linear systems can be expressed as special cases of the ons on the other hand it is shown that the ons can be computed from the solutions to square root riccati equations applications to computation of subspaces contained in the state space and applications to system inversion are demonstrated the properties of the ons considered as time dependent subspace sequences are investigated

on the geometry of the inverse system the geometry of a particular reduced order state space realization of the inverse of a linear continuous time invariant strictly proper system is considered this geometry is characterized by the eigenstructure of the dynamic part of the inverse which is related to the invariant zero structure of the inverted system a method for deriving explicit expressions for the zeros and the zero directions is introduced this method is demonstrated in an example of a uniform rank system

brownian sheet and capacity the main goal of this paper is to present an explicit capacity estimate for hitting probabilities of the brownian sheet as applications we determine the escape rates of the brownian sheet and also obtain a local intersection equivalence between the brownian sheet and the additive brownian motion other applications concern quasisure properties in wiener space

the generalized eigenstructure problem in linear system theory the algebraic theory of linear time invariant systems has been studied in large detail during the past few decades and numerous computational algorithms have been developed to solve problems arising in this context in this paper the numerical aspects of a certain class of such algorithms dealing with what the author calls generalized eigenstructure problems are discussed some new and or modified algorithms are presented both the nmnerical stability of the algorithms and the conditioning of the problems they solve are analyzed using numerical criteria

noncommutative marked surfaces abstract the aim of the paper is to attach a noncommutative cluster like structure to each marked surface this is a noncommutative algebra a generated by noncommutative geodesics between marked points subject to certain triangle relations and noncommutative analogues of ptolemy plucker relations it turns out that the algebra a exhibits a noncommutative laurent phenomenon with respect to any triangulation of which confirms its cluster nature as a surprising byproduct we obtain a new topological invariant of which is a free or a 1 relator group easily computable in terms of any triangulation of another application is the proof of laurentness and positivity of certain discrete noncommutative integrable systems

on the internal structure of minimal stochastic realization abstract the internal structure of minimal state space representations of stationary vector processes is investigated a necessary and sufficient condition for order minimality is given along with an upper bound on the minimal order the correspondence between the all pass structure of a multivariable system and the minimality of the output representation is also specified

state estimation for linear systems with observations partially corrupted by noise abstract this paper exploits the fact that any row vector of the observability matrix applied for transforming the state converts the latter to the new state component in the form of some derivative of the output component using the same but appropriately chosen vectors for transforming the system with the observation not fully corrupted by white noise we can accurately determine some state components these vectors create the basis for the l dimensional subspace of transformation vectors to the new accurately determinable state components using this basis the state transformation is constructed which in one step converts the singular linear filtering problem to a nonsingular one with state dimension decreased by l

extended limiting forms of optimum observers and lqg regulators the general problem of optimal estimation in singular observation systems is considered by generalizing the limiting procedure of friedland explicit and closed expressions for the transfer matrices of the optimum observer as well as for the optimal control law in the singular measurement lqg problem are derived the number of integrators that are required for the implementation of the singular optimum observer is shown for square systems to be equal to the number of the system s zeros the closed form of the optimum observer given here is advantageous as compared with existing procedures structural properties of the singular observation problem are discussed and the role of the zeros of the system in such problems is clarified the dual results for the case of stochastic cheap control are also given

explicit solution to the unstable stationary filtering problem an expression in closed form for the constant kalman gain in stationary filtered estimation of linear unstable processes is obtained and bounds on the achievable accuracy of the estimate are derived this expression is used to find lower and upper bounds for the singular values of the solution of the algebraic matrix riccati equation and to derive the minimum phase image of linear systems that possess zeros in the right half plane

brief paper reduced order state estimation for linear systems with exact measurements necessary and sufficient conditions are derived for the existence of reduced order steady state optimal linear state estimators for continuous linear dynamical systems when some output measurements are noise free these conditions are shown to be dual to the generalized strengthened legendre clebsch conditions for the extremal controls of a dual singular regulator when the order of singularity is at most two the optimal estimators are designed without the use of luenberger observers

images of the brownian sheet an n parameter brownian sheet in r d maps a non random compact set f in r n to the random compact set b f in r d we prove two results on the image set b f 1 it has positive d dimensional lebesgue measure if and only if f has positive d 2 dimensional capacity this generalizes greatly the earlier works of j hawkes 1977 j p kahane 1985 and khoshnevisan 1999 2 if dim h f d 2 then with probability one we can find a finite number of points 1 m r d such that for any rotation matrix that leaves f in one of the i s is interior to b f in particular b f has interior points a s this verifies a conjecture of t s mountford 1989 this paper contains two novel ideas to prove 1 we introduce and analyze a family of bridged sheets item 2 is proved by developing a notion of sectorial local non determinism lnd both ideas may be of independent interest we showcase sectorial lnd further by exhibiting some arithmetic properties of standard brownian motion this completes the work initiated by mountford 1988

direct solution to the general reduced order stochastic observation problem the stochastic optimal state observation problem is considered for a general linear continuous time invariant system with unmeasurable stationary inputs and measurement outputs that may be at least in part perfect a general solution to the problem is obtained by processing the perfect measurements through a specific differentiation transformation scheme in order to extract the maximum accurate information on the system states using this information the original system is transformed to a new reduced order model whose measurements are corrupted by a white noise of non singular intensity matrix a minimum order full state estimator to the original system is then constructed by combining the outputs of any full order observer to the reduced order model and the perfect combinations of the system states that were derived by the differentiation transformation scheme a solution to the general singular kalman filtering problem is then obtained by minimizing the variance of the estimation error of the obser

a simple solution to the singular linear minimum variance estimation problem a simple expression is obtained for the transfer function matrix of the minimum variance estimator for the states of a linear stationary continuous time right invertible system whose output is measured perfectly using the s domain approach it is shown that the optimal estimator first finds the driving noise input that achieves when applied on the minimum phase image model of the system an output spectrum which is identical to the measurement spectrum this input is then applied on a state space representation of the minimum phase image to produce the optimal estimate

the explicit structure of inner matrices and its application in h sup infinity optimization an expression for inner matrices is obtained which allows a detailed investigation of their structure the expression is used to solve the h sup infinity interpolation problem in a simple way a solution to this problem is obtained explicitly in terms of interpolation parameters without solving any equation it is found that the resulting solution encounters a reduction in its order at the critical point and the characteristics of the resulting reduced order solution are investigated

the all pass property of optimal open loop tracking systems the structure of the optimal open loop linear model following system is investigated it is shown that if the given plant is asymptotically stable but has zeros in the right half plane the stable optimal system contains an all pass network whose transference possesses unity singular values on the imaginary axis in the special case of optimal tracking it is shown that the resulting optimal transfer function matrix of the system is equal to the all pass transfer function matrix which is normalized to be the identity matrix at the zero frequency

the structure of inner matrices that satisfy multiple directional interpolation requirements the structure of an inner matrix that satisfies directional interpolation requirements for nondistinct values at the right half plane is derived explicitly in closed form in terms of the interpolation directions the simple expression that is obtained for the required inner transfer function matrix has the same structure as the one that has already been found for distinct interpolation requirements as a result this expression provides a comprehensive treatment in the frequency domain of interpolation problems that arise in many areas of linear system optimization

optimal reduced order estimators in the frequency domain the discrete time case the discrete time stationary kalman filter problem is solved in the z domain for nth order systems with m outputs where 0 k m measurements are noise free the design equations are very similar to the continuous time cage and they can be solved by spectral factorization giving the polynomial matrix d z which parametrizes the reduced order optimal estimator in the frequency domain by solving a single linear equation the equivalent time domain representation for the optimal filter can also be obtained a simple example demonstrates the filter design in the frequency domain

theory of ltr for non minimum phase systems recoverable target loops and recovery in a subspace part 1 analysis a complete analysis of loop transfer recovery problem using full order observer based controllers for general not necessarily left invertible and not necessarily minimum phase systems is considered the analysis here while showing that neither exact nor asymptotic loop transfer recovery is in general possible focuses on three fundamental issues the first issue is concerned with what can and what canot be achieved for a given system and for an arbitrarily specified target loop transfer function while the second issue is concerned with the development of necessary and or sufficient conditions a target loop has to satisfy so that it can be either exactly or asymptotically be recovered for a given system the third issue deals with the development of method s to test whether recovery is possible in a given subspace of the control space or not i e to test whether projections of target and achievable sensitivity and complimentary sensitivity functions onto a given subspace match each other or not such an

a unified solution to the singular and nonsingular linear minimum variance estimation problem a unified solution is derived for the transfer function matrix of the minimum variance estimator of the state of a linear continuous time system with noisy or noise free measurements for the noisy measurement case the kalman bucy filter standard equations are used to give the general format of the transfer function the kalman bucy gain using the spectral factorization on the algebraic in the s plane is derived and substituted in the transfer function expression the singular case is then shown to be a special case of the noisy measurement estimator

on the optimal unbiased functional filtering presents a simple solution to the optimal unbiased reduced order or functional filtering problem for linear time varying systems necessary and sufficient conditions for the existence of the obtained filter are given stability and convergence conditions are developed for the time invariant systems both continuous and discrete time cases are considered

critical brownian sheet does not have double points we derive a decoupling formula for the brownian sheet which has the following ready consequence an n parameter brownian sheet in r d has double points if and only if d 4n in particular in the critical case where d 4n the brownian sheet does not have double points this answers an old problem in the folklore of the subject we also discuss some of the geometric consequences of the mentioned decoupling and establish a partial result concerning k multiple points in the critical case k d 2n d

on the stability robustness of the continuous time lqg optimal control the robustness properties of the lqg optimal controller that is obtained by the asymptotic recovery method are discussed it is shown that in spite of the good recovered stability margins the resulting closed loop design is inherently unrobust in the sense that small parameter variations in the plant and or small realization errors in the controller may lead to an instantaneous instability of the design

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