secant zeta functions abstract we study the series s z n 1 sec n z n s and prove that it converges under mild restrictions on z and s the function possesses a modular transformation property which allows us to evaluate s z explicitly at certain quadratic irrational values of z this supports our conjecture that k k j q whenever k and j are positive integers with k even we conclude with some speculations on the bernoulli numbers

a connection between state space and doubly coprime fractional representations explicit formulas for a doubly coprime fractional representation of the transfer function of a lumped lti system are given in terms of a stabilizable and detectable state space realization of the transfer function these formulas allow existing computational algorithms to be utilized for the purpose of computing doubly coprime factorizations several additional implications of this result are briefly discussed as are some immediate extensions and future research directions

design of multivariable feedback systems with stable plant this paper considers in a general algebraic framework the design of a unity feedback multivariable system with a stable plant the method is based on a simple parameterization of the four closed loop transfer functions in terms of p the plant transfer function and q h y 1 u 1 in particular the i o transfer function q h y 2 u 1 pq using the framework of rational transfer functions we show that the closed loop system will be exponentially stable if and only if q is exponentially stable furthermore if both p and q are strictly proper then the controller is also strictly proper the basic result is design theorem 2 an algorithm is given for obtaining strictly proper controllers such that the resulting i o map is decoupled all its poles can be chosen by the designer and the same holds for zeros except of course for the c zeros prescribed by the c zeros of the plant a discussion is included to temper these results by the constraints imposed by noise and plant saturation

brief paper some new results for system decoupling and pole assignment problems in a related article we derived a canonical decomposition of the right invertible system c a b and applied this canonical decomposition to study the smith form of the matrix pencil p s a sibc0 and findout the finite zeros and infinite zeros of p s the range of the ranks of p s for email protected c and the controllability of the right invertible system in this paper we will apply this canonical decomposition of the right invertible system c a b to deduce the triangular decouple upon to row permutation provide some new results of the row by row decoupling and associated pole assignment problems

a solution to the diagonalization problem by constant precompensator and dynamic output feedback a solution is presented for the problem of diagonalization row by row decoupling the problem is solved using a constant precompensator and a dynamic output feedback compensator of a p m linear time invariant system the solvability condition is compact and concerns the dimension of a single subspace defined via the concepts of essential rows and static kernels associated with the transfer matrix a characterization of the set of all solutions to the problem is also given in solving this dynamic feedback problem a complete solution to its state feedback counterpart namely the restricted state feedback problem of diagonalization is also presented

open loop decoupling of mimo plants the complete set of decoupled forms of a mimo plant p which might be non square and may have unstable pole zero coincidences achievable using open loop precompensation is obtained in terms of certain multiplicities of the poles and zeros of p this approach which also yields the precompensator directly is much simpler than the methods available

decoupling controller design for linear multivariable plants decoupling controller design for linear time invariant square multiple input multiple output mimo plants under the unity feedback configuration is discussed for plants with no coincidences of unstable poles and zeros a simplified necessary and sufficient condition for closed loop stability is given the simplified condition leads to a simple parameterization of the set of all achievable decoupled input output i o maps and an algorithm which allows the design of decoupling controllers to achieve preassigned closed loop poles

decoupling with internal stability for unity output feedback systems abstract this paper presents for the first time a necessary and sufficient condition for solvability of the decoupling problem with internal stability for square plants by unit output feedback compensation it consists of some polynomial matrix coprimeness conditions and is concerned only with the greatest c divisor of the plant parameterizations of all decoupling compensators and achievable diagonal loop maps with internal stabilizability are given and the results are also generalized in a very natural way to the decoupling problem with arbitrary pole assignment examples are included for illustration

a fast algorithm for the division of two polynomial matrices a modification of the algorithm shown by q g wang and c h zhou see ibid vol ac 31 p 165 6 1968 is presented the performance of their algorithm is improved by the use of convolutions and therefore of fft techniques the present method is based on the fast inversion of block triangular toeplitz matrices and it is amenable to parallel implementation

on doubly coprime factorizations abstract this paper proposes a new doubly coprime factorization technique which forms a basis of algebraic control theory this enables us to generalize a previous result on doubly coprime factorization furthermore it is shown that all stabilizing compensators possess a simple model reference structure

solution to morgan s problem a necessary and sufficient condition is presented for the solution of the row by row decoupling problem known as morgan s problem in the general case that is without any restrictive assumption added to the system to the feedback law u fx gy g may be noninvertible this is a structural condition in terms of invariant lists of integers which are easily computable from a given state realization of the system these integers are the infinite zero orders morse s list i sub 4 and the essential orders of the system which only depend on the input output behavior and morse s list i sub 2 of the system which depends on the choice of a particular state realization

a unifying approach to the decoupling of linear multivariate systems a unifying approach to the decoupling problem diagonal triangular block decoupling is introduced the key element is the interactor a polynomial matrix of unique structure associated with the transfer matrix of the system necessary and sufficient conditions for block decoupling are derived and the stability questions are resolved using the block coupling zeros the results obtained for decoupling via state feedback state feedback plus input dynamics and constant output feedback are conceptually simple unify and expand those existing and easily lead to constructive decoupling algorithms

a real space renormalization approach to the kubo greenwood formula in mirror fibonacci systems an exact real space renormalization method is developed to address the electronic transport in mirror fibonacci chains at a macroscopic scale by means of the kubo greenwood formula the results show that the mirror symmetry induces a large number of transparent states in the dc conductivity spectra contrary to the simple fibonacci case a length scaling analysis over ten orders of magnitude reveals the existence of critically localized states and their ac conduction spectra show a highly oscillating behaviour for multidimensional quasiperiodic systems a novel renormalization plus convolution method is proposed this combined renormalization convolution method has shown an extremely elevated computing efficiency being able to calculate electrical conductance of a three dimensional non crystalline solid with 10 30 atoms finally the dc and ac conductances of mirror fibonacci nanowires are also investigated where a quantized dc conductance variation with the fermi energy is found as observed in gold nanowires

technical communique wiener hopf approach to derivation of rational doubly coprime factorizations this paper presents a method to construct a stable rational doubly coprime factorization dcf of a transfer matrix proper or improper from a given polynomial dcf wiener hopf factor polynomials are used to rationalize the polynomial dcf matrices it is shown that the suggested construction procedure results in the stable rational dcf description that includes the normalized matrix fractional description mfd and inner coprime mfd as special cases

the design of linear controllers with symbolic algebra abstract modern transfer function methods use quite a lot of abstract algebra in the design of control systems in computing the final results symbolic algebra turns out to be an important tool in this paper symbolic computation algorithms will be developed to enable computations directly in the theoretically useful transfer function rings a symbolic software package called hinf will be developed and used both for educational and theoretical purposes

linear feedback decoupling transfer function analysis the problem of linear system decoupling is examined based on recent results on linear feedback new insight is obtained through which resolution of the decoupliug problem is accomplished by calculations performed directly on the given transfer matrix computation of the decoupling compensators follows by easy constructions the problem of feedback block decoupling with internal stability is also formulated and resolved

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

optimal realisations of fixed point implemented digital controller with the smallest dynamic range a novel approach is proposed to design optimal finite word length fwl realizations of digital controllers implemented in fixed point arithmetic a minimax based search procedure is first formulated to obtain an optimal controller realization that optimizes an fwl closed loop stability measure since this fwl closed loop stability measure is solely linked to the fractional part or precision of fixed point format the resulting realization may not have the smallest dynamic range a measure is then derived to indicate the dynamic range of fixed point implemented realization by choosing an appropriate orthogonal transformation of this dynamic range measure of the optimal precision controller realization a numerical optimization method is developed to make the controller realization having the smallest dynamic range without sacrificing fwl closed loop stability robustness the proposed approach is more efficient than a direct optimization of some combined fwl closed loop stability and dynamic range measure

parameterization of decoupling controllers in the generalized plant model a diagonal decoupling problem in linear multi variable systems is formulated for the most general setting a necessary and sufficient condition for the existence of a decoupling controller is presented and when a solution exists the class of all decoupling controllers is characterized in terms of free parameters of stable rational vectors

robust triangular decoupling with application to 4ws cars the necessary and sufficient conditions for the problem of robust triangular decoupling are established for the case of a linear time invariant system with a nonlinear uncertain structure the general analytic expressions of the feedback matrices and the robust triangularly decoupled closed loop system are derived for the solution of the problem of robust triangular decoupling with simultaneous hurwitz invariability sufficient conditions are established finally all of the above results are successfully applied to control four wheel steering 4ws cars

transfer function matrix approach to decoupling problem with stability a simple necessary and sufficient condition for solvability of decoupling with internal stability for unity output feedback for non singular plants is proposed

generalized collinearity diagnostics abstract working in the context of the linear model y x e we generalize the concept of variance inflation as a measure of collinearity to a subset of parameters in denoted by 1 with the associated columns of x given by x 1 the essential idea underlying this generalization is to examine the impact on the precision of estimation in particular the size of an ellipsoidal joint confidence region for 1 of less than optimal selection of other columns of the design matrix x 2 treating still other columns x 0 as unalterable even hypothetically in typical applications x 1 contains a set of dummy regressors coding categories of a qualitative variable or a set of polynomial regressors in a quantitative variable x 2 contains all other regressors in the model save the constant which is in x 0 if 2 v denotes the realized variance of and 2 u is the variance associated with an optimal selection of x 2 then the corresponding scaled dispersion ellipsoids to be compared are v x x v

characterization of allowable perturbations for robust decoupling of affine non linear systems for affine non linear systems we study the effects of uncertainties in systems vector fields on decoupled linear input output behaviour a characterization of an allowable class of perturbations which preserve the decoupling properties of the nominal system is given an immediate consequence of this characterization indicates that decoupling position control of rigid robotic manipulators is not robust under any perturbations in robot dynamics the result is applied to simple hamiltonian control systems to characterize the allowable perturbation of total energy function

generalized lambert series and arithmetic nature of odd zeta values it is pointed out that the generalized lambert series sum nolimits n 1 infty n n 2h e n nx 1 studied by kanemitsu tanigawa and yoshimoto can be found on page 332 of ramanujan s lost notebook in a slightly more general form we extend an important transformation of this series obtained by kanemitsu tanigawa and yoshimoto by removing restrictions on the parameters n and h that they impose from our extension we deduce a beautiful new generalization of ramanujan s famous formula for odd zeta values which for n odd and m 0 gives a relation between 2 m 1 and 2 nm 1 a result complementary to the aforementioned generalization is obtained for any even n and m it generalizes a transformation of wigert and can be regarded as a formula for 2 m 1 1 n applications of these transformations include a generalization of the transformation for the logarithm of dedekind eta function z zudilin and rivoal type results on transcendence of certain values and a transcendence criterion for euler s constant

on a secant dirichlet series and eichler integrals of eisenstein series we consider the secant dirichlet series psi s tau sum n 1 infty frac sec pi n tau n s recently introduced and studied by lalin rodrigue and rogers in particular we show as conjectured and partially proven by lalin rodrigue and rogers that the values psi 2 m sqrt r with r 0 rational are rational multiples of pi 2 m we then put the properties of the secant dirichlet series into context by showing that for even s they are eichler integrals of odd weight eisenstein series of level 4 this leads us to consider eichler integrals of general eisenstein series and to determine their period polynomials in the level 1 case these polynomials were recently shown by murty smyth and wang to have most of their roots on the unit circle we provide evidence that this phenomenon extends to the higher level case this observation complements recent results by conrey farmer and imamoglu as well as el guindy and raji on zeros of period polynomials of hecke eigenforms in the level 1 case finally we briefly revisit results of a similar type in the works of ramanujan

proper stable transfer matrix factorizations and internal system descriptions the exact relations between coprime proper stable factorizations of p s and coprime polynomial matrix factorizations are derived and they directly lead to relations with internal descriptions of the plant in differential operator or state space form it is shown that obtaining any right or left proper stable coprime factorization is equivalent to state feedback stabilization or to designing a full order full state observer respectively solving the diophantine equation is shown to be equivalent to designing a full or reduced order observer of a linear functional of the state and to designing a stable inverse system and this suggests new computational methods to solve the diophantine

input output triangular decoupling and data sensitivity the problem of input output triangular decoupling and data sensitivity is studied for the case of linear time invariant systems using a static feedback law the necessary and sufficient conditions for the problem to have a solution are established the conditions are derived in terms of the variations of the system matrices the general solution of the controller matrices is also derived the necessary and sufficient conditions for triangular decoupling and data sensitivity with simultaneous asymptotic stability are established the structural property of the perturbation disturbance equivalence is also established

a polynomial matrix method for computing stable rational doubly coprime factorizations this paper proposes a new method for computing stable rational doubly coprime factorizations from a given transfer matrix in contrast to the well known method which requires a state space representation the proposed method makes full use of polynomial matrices and the whole operation is carried out directly in the frequency domain furthermore the paper clarifies the meaning of the obtained factorization as a controller by showing that this factorization coincides with the one by the state space method for a suitable choice of feedback and observer gains

ill conditioning and multicollinearity abstract it is well known that unstability of solutions to small changes in inputs causes many problems in numerical computations existence uniqueness and stability of solutions are important features of mathematical problems problems that fail to satisfy these conditions are called ill posed the purpose of this study is to remind briefly some methods of solution to ill posed problems and to see the impacts or connections of these techniques to some statistical methods

an algebraic theory for design of controllers for linear multivariable systems part i structure matrices and feedforward design in this two part paper a theory is presented in which several control problems can be solved the theory is applicable to a general class of linear multivariable systems where the measured output does not have to be equal to the controlled output or the state the system may be affected by nonmeasurable disturbances only controllers which stabilize the system and have proper transfer functions are allowed i e the controllers have to be physically realizable it is shown that the solutions to control problems of servo type e g problems of model matching decoupling and invertibility are special cases of the solution to a more general problem analogously the solutions to problems of regulator type e g disturbance decoupling output regulation and pole placement are also speclal cases of the solution to a more general problem it is shown that problems of servo type and regulator type can be solved independently of each other in part i generalized polynomials are introduced as a mathematical framework structue matrices which describe how well a system can be controlled are defined finally problems of servo type are solved part ii mainly deals with problems of regulator type

decoupling through specified input output channels with internal stability abstract a system with two sets of inputs u 1 u 2 and two sets of outputs y 1 y 2 is considered the objective is to construct a dynamic feedback compensator between u 1 and y 1 so that the closed loop transfer matrix between u 2 and y 2 is diagonal and nonsingular internal stability of the closed loop system is also required this problem can be regarded as the general version of the standard diagonal decoupling problem where the output to be controlled decoupled and the measured output are not distinguished along these lines it is shown that the diagonal causality degree dominance dcdd which is the key concept in standard diagonal decoupling problems also arises in the present context in a modified form the first modification is due to the constraint of internal stability which gives rise to a new notion diagonal stability and causality degree dominance dscdd these two notions are investigated in the first part of the paper the second modification comes from the crosschannel transfer matrices between u 1 and y 2 and u 2 and y 1 this yields another new notion joint essential stability functions the solvability conditions of the problem are expressed in terms of the properties of the subrings of the ring of stable and proper rational functions and their quotient rings these subrings are generated by joint essential stability functions of the cross channel transfer matrices the degrees of the joint essential stability functions are called the joint essential stability orders which are extensions of the well known essential orders and essential stability orders to the problem considered here

modular forms and period polynomials we study the space of period polynomials associated with modular forms of integral weight for finite index subgroups of the modular group for the modular group this space is en dowed with a pairing corresponding to the petersson inner product on modular forms via a formula of haberland and with an action of hecke operators defined algebraically by zagier we generalize haberland s formula to not necessarily cuspidal modular forms for finite index subgroups and we show that it conceals two stronger formulas we extend the action of hecke operators to period polynomials of modular forms we show that the pairing on period polynomials appearing in haber land s formula is nondegenerate and we determine the adjoints of hecke operators with respect to it we give a few applications for 1 n an extension of the eichler shimura isomorphism to the entire space of modular forms the determination of the relations satisfied by the even and odd parts of period polynomials associated with cusp forms which are independent of the period rela tions and an explicit formula for fourier coefficients of hecke eigenforms in terms of their period polynomials generalizing the coefficients theorem of manin

conditions for triangular decoupling control the main purpose of this article is to explore the relationship of two existing conditions for the triangular decoupling problem the first one is the triangular diagonal dominance condition proposed by hung and anderson the second one is the stable coprime factorisation described condition proposed by gomez and goodwin which has been proven as a necessary and sufficient condition for the triangular decoupling problem this article proves that the two conditions are actually equivalent it also provides easy to use criteria for assessment of the solvability of the triangular decoupling problem

logic based switching controllers a stepwise safe switching approach the present work proposes a new logic based safe switching technique called here stepwise safe switching this technique is based on the description of the process with linear approximants which are valid in operating areas around a finite number of nominal operating points of the process switching takes place between common controllers for adjacent nominal operating points moreover the switching between the controllers is allowed to take place only when the process has reached an operating point the proposed technique guarantees stability and good performance for all transitions between operating points that lie in the neighborhoods of the nominal operating points of the process this way a simple to design and easy to implement solution that provides good performance despite the drawback of lack of knowledge of the process dynamics can be derived the proposed technique is successfully applied to control an isothermal cstr process described by three linear approximants around corresponding nominal operating points

modular implementation of robust supervisory controllers for discrete event systems in this paper the problem of designing robust supervisory controllers for uncertain discrete event systems is studied it is assumed that both the plant behavior and the design specifications can be represented by regular languages it is also assumed that the unknown model of the uncertain plant belongs to a finite set of possibilities to solve the problem a modular approach based on a general recursive robust control scheme is developed the contribution of the present paper consists in deriving a maximally permissive modular robust supervisory controller for the case of prefix closed languages having the advantage of linear complexity in the number of models as opposed to the exponential complexity of the worst case scenario in the direct implementation of standard supervisors for augmented systems

internal stabilization and decoupling in linear multivariable systems by unity output feedback compensation we consider the problem of internally stabilizing and simultaneously diagonally decoupling a linear multivariable system by unity output feedback compensation a sufficient condition is derived for the existence of a cascade proper compensator c s such that when employed in a unity feedback loop involving the proper transfer function matrix p o of a free of unstable hidden modes system sigma p o will not only internally stabilize the feedback closed loop system sigma p o c but will also give rise to a closed loop transfer function matrix h yr diag which is nonsingular diagonal and has desired poles based on this analysis an algorithmic procedure for the computation of such a compensator is presented

static state feedback triangular block decoupling for arbitrary systems a state space method abstractin this paper the static state feedback triangular block decoupling problem for a general system described by a b c d quadruples is studied necessary and sufficient conditions which are numerically verifiable are obtained and a numerically reliable decoupling method is established in the state space two examples are solved by our method which show the effectiveness

path weights in concentration graphs a graphical model provides a compact and efficient representation of the association structure of a multivariate distribution by means of a graph relevant features of the distribution are represented by vertices edges and other higher order graphical structures such as cliques or paths typically paths play a central role in these models because they determine the independence relationships among variables however while a theory of path coefficients is available in models for directed graphs little has been investigated about the strength of the association represented by a path in an undirected graph essentially it has been shown that the covariance between two variables can be decomposed into a sum of weights associated with each of the paths connecting the two variables in the corresponding concentration graph in this context we consider concentration graph models and provide an extensive analysis of the properties of path weights and their interpretation more specifically we give an interpretation of covariance weights through their factorisation into a partial covariance and an inflation factor we then extend the covariance decomposition over the paths of an undirected graph to other measures of association such as the marginal correlation coefficient and a quantity that we call the inflated correlation the usefulness of these findings is illustrated with an application to the analysis of dietary intake networks

input output decoupling by output feedback with disturbance attenuation of h control for any square system the combined problem of diagonal decoupling by output feedback with disturbance attenuation performance for any square system is formulated a sufficient condition which is more general than the necessary and sufficient one of static state feedback for the existence of the controller is determined the necessary and sufficient conditions for the controller and decoupled system that are both stable are presented an analytical form and a numerical computable method for the controller are obtained for the first time when the condition is satisfied the correctness of our method is tested by simulations for an example

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

an algorithm for inverting rational matrices abstract we propose an algorithm for computing the inverses of rational matrices and in particular the inverses of polynomial matrices the algorithm is based on minimal state space realizations of proper rational matrices and the matrix inverse lemma and is implemented as a matlab 1 function experiments show that the algorithm gives accurate results for typical rational matrices that arise in analysis and design of linear multivariable control systems illustrative examples are given

necessary and sufficient conditions for existence of decoupling controllers it is well known that if a linear time invariant plant is free from coincidences of poles and zeros in the right half plane then it can be decoupled with internal stability under unity feedback configuration we consider plants for which such coincidences do occur and give necessary and sufficient conditions under which stabilizing decoupling controllers exist the conditions derived based on transfer matrices and residues are simple and straightforward

technical communique fixed poles of simultaneous disturbance rejection and decoupling a geometric approach when the simultaneous disturbance rejection and regular block decoupling problem by static state feedback is solvable and under some unrestrictive minimality assumption all the poles of the closed loop system can be placed except the so called fixed poles which are present for any solution we present here a geometric characterization for this set of fixed poles as well as a constructive procedure based on geometric tools for designing a feedback solution of the problem which assigns the remaining poles to arbitrary prespecified locations

common i o triangular decoupling for multi model descriptor systems abstract the problem of common input output triangular decoupling td for multi model descriptor linear systems is solved using a static state feedback controller the necessary and sufficient conditions for the solvability of the problem are established the set of all state feedback controllers solving the problem under the solvability conditions is derived for the derivation of the above results the problem of the inversion of a multiparameter matrix pencil in a triangular matrix form and under linear inequality constraints is solved

numerical computation of a coprime factorization of a transfer function matrix abstract a new algorithm is presented for computing a coprime factorization of a transfer function matrix the method is based on reformulating the problem as a problem of constructing a minimal basis of the right nullspace of a matrix pencil b a due to the special structure and rank properties of the pencil obtained after transforming b a to generalized schur form the original problem can be easily solved in a numerically reliable way

decoupling linear multiinput multioutput plants by dynamic output feedback an algebraic theory this paper presents an algebraic theory for the design of a decoupling compensator for linear time invariant multiinput multioutput systems the design method uses a two input one output compensator which gives a convenient parametrization of all diagonal input output i o maps and all disturbance to output d o maps achievable by a stabilizing compensator for a given plant it is shown that this method has two degrees of freedom any achievable diagonal i o map and any achievable d o map can be realized simultaneously by a choice of an appropriate compensator the difference between all achievable diagonal and nondiagonal i o maps and the cost of decoupling is discussed for some particular algebraic settings

block decoupling of generalized state space systems abstract the problem of input output block decoupling of generalized state space systems via a regular static state feedback law is studied for the first time the necessary and sufficient conditions for the problem to have a solution are established the necessary and sufficient conditions for block decoupling with simultaneous stabilizability and the necessary and sufficient conditions for block decoupling with simultaneous arbitrary assignment of the system poles are established 1997 elsevier science ltd

a new approach for calculating doubly coprime matrix fraction descriptions using the concepts of state feedback in generalized state space explicit formulas for calculating doubly coprime matrix fraction descriptions mfds of the transfer matrix of a regular state space system and the polynomial generalized bezout identity elements corresponding to those coprime mfds are presented they can be easily calculated with the help of existing computational algorithms and software packages

comments on on the structure of digital controllers with finite word length consideration in the above mentioned paper by li ibid vol 43 1998 a condition is presented as necessary and sufficient to give the optimal realization of state space controllers for minimal sensitivity of all closed loop eigenvalues subject to finite word length effects the condition is shown here not to be sufficient in addition the condition for minimal sensitivity of a single closed loop eigenvalue presented in the above mentioned paper is not complete the complete conditions are presented in this paper