technical communique h optimal terminal control for linear systems with delayed states and controls in this paper we consider a linear time varying system with state and control delay terms both lumped and distributed subject to unknown square integrable disturbances using a differential games approach we provide a necessary and sufficient condition for the existence of an optimal control law minimizing the h norm of the operator mapping the disturbance to the weighted control and terminal state

h sub infinity optimal control with state feedback a h sub infinity optimal control problem in which the measured outputs are the states of the plant is considered the main result shows that the infimum of the norm of the closed loop transfer function using linear static state feedback equals the infimum of the norm of the closed loop transfer function over all stabilizing dynamic even nonlinear time varying state feedback controllers

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

an algebraic riccati equation approach to h optimization abstract this paper considers the general so called four block h optimal control problem with the assumption that system states are available for feedback it is shown that infimization of the h norm of the closed loop transfer function over all linear constant i e nondynamic stabilizing state feedback laws can be completely characterized via an algebraic riccati equation it is further shown that the optimal norm is not improved by allowing feedback to be dynamic thus the general state feedback h optimal control problem can be solved by iteratively solving one are and the controller can be chosen to be static gain

loop gain optimization of unstable non minimum phase feedback systems the paper presents a design method for optimizing the loop gain of unstable non minimum phase feedback systems because obvious gain bandwidth limitations exist in such systems the proposed best loop gain optimization goal is maximization of the stability gain and phase margins which are inherently low in such feedback systems the design procedure is based on coprime factorization of the plant and controller and the bezout identity an optimization process varies iteratively the observer based controller parameters until maximization of the gain and phase margins is achieved the same design technique also allows one to solve the mixed sensitivity h standard control problem for linear time invariant feedback systems

percolation conductivity of penrose tiling by the transfer matrix monte carlo method abstract a generalization of the derrida and vannimenus transfer matrix monte carlo method has been applied to calculations of the percolation conductivity in a penrose tiling strips with a length 104 and widths from 3 to 19 have been used disregarding the differences for smaller strip widths up to 7 the results show that the percolative conductivity of a penrose tiling has a value very close to that of a square lattice the estimate for the percolation transport exponent once more confirms the universality conjecture for the 0 1 distribution of resistors

detecting unequal marginal scales in a bivariate population abstract a conditional test is proposed that uses the null hypothesis of bivariate symmetry to detect unequal marginal scales in a bivariate population the asymptotic normality of the test statistic and the consistency of the test are established asymptotic relative efficiency comparisons are made with the standard parametric test in both the bivariate normal case and in a trivariate reduction model case and with two of sen s 1967 tests in the bivariate normal case the good performance of the test is also illustrated via a monte carlo study

fast computation of achievable feedback performance in mixed sensitivity h design the computational bottleneck of the h infty design has been recognized to be the e iteration a computationally demanding direct search of the minimum achievable h infty performance verma and jonckheere showed that the optimal h infty performance can be characterized as the spectral radius of the so called toeplitz plus hankel operator even before the appearance of the toeplitz plus hankel operator in the h infty setting the same operator had already been shown to play a crucial role in the spectral theory of the linear quadratic problem developed by jonckheere and silverman in this paper we exploit this common toeplitz plus hankel operator structure shared by the seemingly unrelated linear quadratic and h infty problems and we construct fast state space algorithms for evaluating the spectral radius of the toeplitz plus hankel operator the salient feature of the algorithm is that the spectral radius can be evaluated with an accuracy predicted by an identifiable error bound from the antistabilizing solution of the algebraic riccati equation of the linear quadratic problem associated with the h infty design

logarithmic integrals interpolation bounds and performance limitations in mimo feedback systems we study performance limitation issues found in linear multivariable feedback systems our main contributions include bode and poisson type integral inequalities and performance limits for the sensitivity and complementary sensitivity functions these results characterize and quantify explicitly how open loop unstable poles and nonminimum phase zeros may impose inherent limitations on feedback design and fundamental limits on the best achievable performance the role of time delay is also studied in this context most notably we show that the performance and design limitations in multivariable systems intrinsically depend on the locations as well as directions of unstable poles and nonminimum phase zeros and in particular on how pole and zero directions are aligned the latter is characterized by angles measuring the mutual orientation between zero and pole directions and it is shown to play a crucial role in multivariable system design

a practical approach for evaluating population and individual bioequivalence pharmacokinetic measurements provided by subjects to each of two formulations of a drug have a joint distribution that can be characterized by parameters reflecting scale and correlation as well as location the bioavailability of the formulations can be expressed in terms of the means of the marginal distributions their means and variances or the marginal means and variances and the joint correlation these expressions correspond respectively to average population and individual bioequivalence when the joint distribution of the measurements is bivariate normal current proposals for assessing the degree of bioequivalence of two formulations are based on statistics that are composites of variance components and squares of expected mean differences from a mixed linear model there are technical and practical issues associated with these proposals particularly that they require more complicated designs than the familiar 2 x 2 cross over this paper describes an alternative approach that can be applied with standard 2 x 2 cross over designs and that provides evaluations of population and individual bioequivalence that should be adequate for all practical clinical purposes the approach is based on easily computed correlation and regression coefficients whose statistical properties under normality are well known and for which non parametric and robust alternatives exist when normality cannot be assumed the approach yields conclusions consistent with those obtained by the current proposals when applied to data sets supplied by the fda in the cases where the conclusions do not match the new approach appears to be more consistent with the data

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

linear control theory with an h 0e optimality criterion this expository paper sets out the principal results in bf h infty control theory in the context of continuous time linear systems the focus is on the mathematical theory rather than computational methods

a duality based approach to the multiobjective h 2 h optimisation problem in this article a duality approach to multiobjective h 2 h problems is pursued in which real rational para hermitian multipliers and real valued ones are associated to h and as usual h 2 constraints respectively it is shown that the maximisation of a dual functional over all such multipliers yields the optimal value of the original multiobjective h 2 h problem to compute lower bounds on the latter and the corresponding approximate solutions to the original problem the maximisation of the dual functional over linearly parameterised finite dimensional classes of real rational multipliers is shown to be equivalent to semi definite linear programming problems once the optimal multipliers in such a class are obtained the corresponding approximate solutions can be computed from an unconstrained h 2 problem iterative modification of such classes is discussed to obtain increasing sequences of lower bounds on the optimal value of the original problem this is done on the basis of locally incr

on a reasonable disaggregate criterion of population bioequivalence admitting of resampling free testing procedures the aggregate criterion of population bioequivalence pbe recommended in the new fda guidance for in vivo bioavailability studies is criticized for using the same distributional parameter as a reference for scaling the squared distance between the averages and the difference in variability in order to circumvent this and other difficulties inherent in the approach to pbe having been recommended thus far we introduce a disaggregate criterion instead which requires of population bioequivalent drug formulations that both the difference in means scaled by the pooled intrasubject standard deviation be sufficiently small in absolute value and the total variability be not substantially increased under the test as compared to the reference formulation for testing the first of the statistical hypotheses associated with this combined criterion an exact optimal procedure based on the ordinary two sample t statistic is presented as a test for equivalence with respect to total variability a modification of liu and chow s one sided test procedure for the assessment of intrasubject variability is recommended both subtests are combined by means of the intersection union principle explained here in general terms to form an overall test for disaggregate pbe maintaining any specified significance level the power of the combined procedure is investigated by means of monte carlo simulation

a mixed effects model for the analysis of repeated measures cross over studies a mixed effects model is developed for cross over trials in which the response is measured repeatedly within each time period relative to previous work on repeated measures cross overs the methodology synthesizes two important features first our procedure eliminates preliminary testing for carry over defined loosely as the component of a response that is due to treatment in the preceding period this is achieved by generalizing the methodology to cross over designs in which preliminary testing for carry over is unnecessary we focus largely on simple carry over that is carry over that lasts for exactly one period and is independent of the treatment administered in the period in which the carry over occurs however we also illustrate a modification of the procedure for a repeated measures cross over design which uses a more complicated model of carry over second the model allows both the between and within subject variance to differ among treatments conditions are described wherein closed form cf solutions to the variance components as well as closed form hypothesis tests of the treatment differences exist flexibility in the model is illustrated with an example in which inference based on the cf likelihood based estimates of the variance and estimates formed using an iterative routine proc mixed are compared

the subject by formulation interaction as a criterion of interchangeability of drugs summary the primary purpose of a bioequivalence study is often to meet the regulatory requirement of similar average bioavailabilities this is usually done by estimating the mean difference between formulations in a crossover design however a small difference as well as a rather narrow confidence interval might be obtained also if the subjects respond differently to the formulations i e if there is a nonnegligible subject by formulation interaction hence it can be argued that this interaction should be part of the criterion of interchangeability to be able to study this interaction the ordinary simple crossover design is not appropriate instead a design where each formulation is administered at least twice to each subject is desirable such a design is obtained by repeating a simple crossover layout to evaluate the impact of the subject by formulation interaction the ratio of the variance components of the interaction and the error is estimated this approach is illustrated by an example two different formulations of a drug are considered bioequivalent if their average bioavailabilities are similar the assessment of this similarity is usually based on an interval estimate of the mean difference between the formulations however this focus on means is an insufficient criterion since a small difference as well as a rather narrow confidence interval might be obtained also if the formulation differences show a considerable variation among subjects the impact on drug interchangeability of this so called subject byformulation interaction is discussed for example by hwang et al 1978 in order to estimate this interaction they suggest a design where each formulation is administered at least twice to each subject in this paper we briefly discuss how to design and analyze a bioequivalence study in order to evaluate the impact of the subject by formulation interaction as a measure of the magnitude of the interaction we propose the ratio of the variance components of the interaction and the error the ideas of the paper are illustrated with data from a comparative study of two generic formulations of furosemide

exact inference about the within subject variability in 2 2 crossover trials abstract the comparison of primary interest in a 2 2 crossover trial typically concerns the effect of the treatments say a and b on the mean response level this article deals with another important aspect namely the within subject response variability under a and b differences in drug formulation and or administration may lead to considerable differences in within subject variability whatever the difference in terms of mean level consideration of both these aspects may therefore be of considerable importance for the judgement of the treatments it is shown that although there are no within subject treatment replications it is possible to make various exact inferences about the a b ratio of within subject variances and about the a b difference in mean level simultaneously and marginally these inferences are semiparametric in that no distributional assumption is made about the between subject variability whereas a normality assumption is used for the within subject variability the infere

optimal disturbance attenuation for discrete time switched and markovian jump linear systems an exact condition for uniform stabilization and disturbance attenuation for switched linear systems is given in the discrete time domain via the union of an increasing family of linear matrix inequality conditions associated with each markovian jump linear system is a switched linear system so we obtain a necessary and sufficient condition for almost sure uniform stabilization and disturbance attenuation for markovian jump linear systems as well the results lead to semidefinite programming based controller synthesis techniques from which optimal finite path dependent linear dynamic output feedback controllers arise naturally in particular under the notion of path by path optimal disturbance attenuation finite path dependent controllers can outperform the usual mode dependent ones

a transfer matrix approach to random resistor networks introduces a transfer matrix formulation to compute the conductance of random resistor networks the authors apply this method to strips of width up to 40 and use finite size scaling arguments to obtain an estimate for the conductivity critical exponent in two dimensions t 1 28 or 0 03

feedback and optimal sensitivity model reference transformations multiplicative seminorms and approximate inverses in this paper the problem of sensitivity reduction by feedback is formulated as an optimization problem and separated from the problem of stabilization stable feedback schemes obtainable from a given plant are parameterized salient properties of sensitivity reducing schemes are derived and it is shown that plant uncertainty reduces the ability of feedback to reduce sensitivity the theory is developed for input output systems in a general setting of banach algebras and then specialized to a class of multivariable time invariant systems characterized by n times n matrices of h infty frequency response functions either with or without zeros in the right half plane the approach is based on the use of a weighted seminorm on the algebra of operators to measure sensitivity and on the concept of an approximate inverse approximate invertibility of the plant is shown to be a necessary and sufficient condition for sensitivity reduction an indicator of approximate invertibility called a measure of singularity is introduced the measure of singularity of a linear time invariant plant is shown to be determined by the location of its right half plane zeros in the absence of plant uncertainty the sensitivity to output disturbances can be reduced to an optimal value approaching the singularity measure in particular if there are no right half plane zeros sensitivity can be made arbitrarily small the feedback schemes used in the optimization of sensitivity resemble the lead lag networks of classical control design some of their properties and methods of constructing them in special cases are presented

sensitivity integrals and transformation techniques a new perspective we provide alternative proofs for the classical bode and poisson type sensitivity integrals and their extensions for both continuous time and discrete time systems our derivation uses the well known properties of laplace and z transformations this derivation helps establish a connection between bode and poisson type sensitivity integrals and laplace and z transforms hence providing an alternative perspective in interpreting these fundamental integral results

right half plane poles and zeros and design tradeoffs in feedback systems this paper expresses limitations imposed by right half plane poles and zeros of the open loop system directly in terms of the sensitivity and complementary sensitivity functions of the closed loop system the limitations are determined by integral relationships which must be satisfied by these functions the integral relationships are interpreted in the context of feedback design

derivation of the maximum entropy h controller and a state space formula for its entropy the problem of maximizing the entropy of a stabilized closed loop system is solved the solution exploits the parametrization of all closed loop systems that meet an h norm bound the central solution of this set is shown to maximize the entropy at infinity and for this case a particularly simple formula is derived for the maximum value of the entropy

state space formulae for all stabilizing controllers that satisfy and h norm bound and relations to risk sensitivity abstract given a linear system all stabilizing controllers such that a specified closed loop transfer function has h norm less than a given scalar are parametrized this characterization involves the solution to two algebraic riccati equations each with the same order as the system and further gives gives feasible controllers also with this order the relationship to the risk sensitive lqg stochastic control problem is established giving an equivalence between robust and stochastic control

numerical study of transport properties in continuum percolation the authors present numerical simulations of ac conductance for a random resistor capacitor network the conductance obeys a probability density function p g varies as g alpha 0 alpha 1 they use a highly efficient propagation algorithm to calculate the effective conductance of a long strip of a lattice at low frequencies they find that for the concentration p of conducting bonds less than the percolation threshold pc the imaginary part of conductance is proportional to frequency im geff approximately omega and the real part of conductance shows an anomalous frequency dependence re geff approximately omega 2 alpha the results of simulations in such a continuum system are in agreement with the predictions of the effective medium and the maxwell garnett approximation they also calculate the non universal dc conductivity exponents in continuum percolation the results are consistent with earlier theoretical predictions and numerical calculations

sensitivity minimization in an h norm parametrization of all suboptimal solutions a number of problems in the control of linear feedback systems can be reduced to the following we are given three stable rational matrix functions k of sizes p 1 x q 1 p 1 x q 2 and p 2 x q 1 respectively and seek a stable rational q 2 x p 2 matrix function s so as to minimize k s we assume that p 1 q 2 p 2 q 1 and that and have maximal rank q 2 and p 2 respectively on the j axis given a tolerance level sufficiently large we obtain a linear fractional map g f 0 11 g 0 12 0 13 0 21 g 0 22 0 23 1 such that f k s with s stable and f if and only if g is a stable q 2 x p 2 matrix function with g 1 the computation of 0 0 ij 1 i 2 1 j 3 reduces to solving a pair of symmetric wiener hopf factorization problems for the special case where i q2 0 t i p2 0 and k not necessarily stable to which the general case can be reduced we provide explicit state space formulae for 0 in terms of a state space realization of k and the

a polynomial approach to minimax frequency domain optimization of multivariable feedback systems the properties of a feedback system where the plant has rational transfer matrix h and the compensator has transfer matrix g can be characterized through the system functions s i hg and t g i hg 1 good disturbance attenuation robustness limited bandwidth and compensator roll off may be obtained by minimizing a criterion of the form z where z v s t w1 wls t w2 w2t v with respect to the compensator transfer matrix g here v w1 and w2 are suitable rational weighting matrices the solution of the problem can be reduced to a pair of matrix polynomial equations since the optimal solution is highly non unique special solutions with additional optimality properties are considered as well the paper includes a discussion of the numerical solution of the polynomial equations and of the question how to choose the weighting matrices to ensure required feedback system properties an example illustrates the results

wiener hopf design of the general two degree of freedom controllers and the connection to state space formulas h 2 design of the two degree of freedom controller configuration is treated for the generalized plant in both the frequency and state space domains a frequency domain wiener hopf formula is derived first under a set of compact assumptions that gives the entire class of controllers for which the feedback control system is stable and a quadratic cost functional is finite the h 2 design method suggested in this paper allows the reference inputs to include the persistent signals such as step or ramp functions and the class of persistent reference inputs that can be tracked with zero steady state error is characterized explicitly to circumvent the computational difficulties of the wiener hopf formulas the corresponding state space domain formulas are derived by rationalizing the polynomial fractional descriptions it is shown that the well known standard time domain assumptions for the h 2 design problem have the corresponding equivalent frequency domain conditions and this equivalence relationship is show

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

a controller degree bound for h 0e control problems of the second kind this paper is a continuation of our work on mathcal h infty optimal control problems which may be embedded in the linear fractional configuration of fig 1 in two previous articles 19 20 a controller degree bound was established for problems in which both p 12 s and p 21 s are square problems of the first kind if the mcmillan degree of p s is n it was shown that there exist mathcal h infty optimal controllers with mcmillan degree no greater than n 1 here we switch our attention to problems of the second kind that is we allow p 12 s to have more rows than columns with p 21 s square or alternatively we allow p 21 s to have more columns than rows with p 12 s square our main result shows that the degree bound derived previously for problems of the first kind carries over to problems of the second kind without change in addition to the controller degree bound our analysis suggests a number of modifications which are easily made to cur

disturbance attenuation and h optimization a design method based on the algebraic riccati equation this note presents a method for designing a state feedback control law to reduce the effect of disturbances on the output of a given linear system the problem under consideration involves attenuating the disturbances to a prespecified level the construction of the state feedback control law requires the solution of a certain algebraic riccati equation by applying the procedure with successively smaller values of the prespecifled disturbance level the result also provides an approach to the h infty optimization problem for the state feedback case

an analysis of the pole zero cancellations in h optimal control problems of the first kind the aim of this paper is to study the pole zero cancellations which occur in a class of h infty optimal control problems which may be embedded in the configuration of fig 1 h infty control problems are said to be of the first kind if both p 12 s and p 21 s are square but not necessarily of the same size it is primarily this class of problems which will concern us here a general bound on the mcmillan degree of all controllers which are stabilizing and lead to a closed loop which satisfies bf mathcal r s infty leqq rho rho need not be optimal in the l infty norm sense is derived as illustrated in fig 1 bf mathcal r s is the transfer function relating y 1 s to u 1 s if the mcmillan degree of p s in fig 1 is n we show that in the single loop siso case the corresponding unique h infty optimal controller never requires more than n 1 states in the multivariable case there is a continuum of optimal controllers whose mcmil

a simple method for deriving j spectral factors a simple method is presented for deriving the j spectral factor of a transfer function matrix explicitly in terms of the parameters of this matrix this method provides closed form expressions for the j spectral factor and its inverse which only require a solution of a single linear sylvester equation this method can be easily applied in the solution of the h sub infinity optimal regulation problem and it provides useful geometric insight into the structure of the optimal regulator return difference matrix