a maximum likelihood algorithm for the estimation and renormalization of exponential densities we present an algorithm based on maximum likelihood for the estimation and renormalization marginalization of exponential densities the moment matching problem resulting from the maximization of the likelihood is solved as an optimization problem using the levenberg marquardt algorithm in the case of renormalization the moments needed to set up the moment matching problem are evaluated using swendsen s renormalization method we focus on the renormalization version of the algorithm where we demonstrate its use by computing the critical temperature of the two dimensional ising model possible applications of the algorithm are discussed

the kadanoff lower bound variational renormalization group applied to an su 2 lattice spin model the authors apply the variational lower bound renormalization group transformation of kadanoff 1975 to an su 2 lattice spin model in two and three dimensions even in the one hypercube framework of this renormalization group transformation the present model is characterized by having an infinite basis of fundamental operators they investigate whether the lower bound variational renormalization group transformation yields results stable under different truncations of this operator basis their results show that for this particular spin model this is not the case

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

robust stabilization of systems with multiple real pole uncertainties the robust stabilization of systems with several uncertain real poles is discussed necessary and sufficient conditions of existence of a controller in terms is obtained of an interpolation problem are given the interpolating function is obtained directly for a particular case two examples are shown for the synthesis of a controller an explicit solution of the controller for plants without zeros in the right hand plane rhp and without known poles in the rhp is given this shows the practical utility of the method to the design of robust stabilizers

optimal prediction with memory abstract optimal prediction methods estimate the solution of nonlinear time dependent problems when that solution is too complex to be fully resolved or when data are missing the initial conditions for the unresolved components of the solution are drawn from a probability distribution and their effect on a small set of variables that are actually computed is evaluated via statistical projection the formalism resembles the projection methods of irreversible statistical mechanics supplemented by the systematic use of conditional expectations and new methods of solution for an auxiliary equation the orthogonal dynamics equation needed to evaluate a non markovian memory term the result of the computations is close to the best possible estimate that can be obtained given the partial data we present the constructions in detail together with several useful variants provide simple examples and point out the relation to the fluctuation dissipation formulas of statistical physics

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

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

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

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

brief a parametric optimization approach to h and h2 strong stabilization this paper presents an approach for designing stable mimo h and h 2 controllers by directly computing the norm constrained stable transfer matrices q in the h and h 2 suboptimal controller parameterizations this is done by first converting the h 2 and h strong stabilization problems into some nonlinear unconstrained optimization problems through explicit parameterization of the norm constrained q s for any fixed order then a two stage numerical search is carried out by using a combination of a genetic algorithm and a quasi newton algorithm in order to reach an optimal solution the effectiveness of the proposed algorithms is illustrated through some benchmark numerical examples

stochastic global optimization as a filtering problem we present a reformulation of stochastic global optimization as a filtering problem the motivation behind this reformulation comes from the fact that for many optimization problems we cannot evaluate exactly the objective function to be optimized similarly we may not be able to evaluate exactly the functions involved in iterative optimization algorithms for example we may only have access to noisy measurements of the functions or statistical estimates provided through monte carlo sampling this makes iterative optimization algorithms behave like stochastic maps naive global optimization amounts to evolving a collection of realizations of this stochastic map and picking the realization with the best properties this motivates the use of filtering techniques to allow focusing on realizations that are more promising than others in particular we present a filtering reformulation of global optimization in terms of a special case of sequential importance sampling methods called particle filters the increasing popularity of particle filters is based on the simplicity of their implementation and their flexibility we utilize the flexibility of particle filters to construct a stochastic global optimization algorithm which can converge to the optimal solution appreciably faster than naive global optimization several examples of parametric exponential density estimation are provided to demonstrate the efficiency of the approach

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

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

hierarchical models and chaotic spin glasses renormalization group studies in position space have led to the discovery of hierarchical models which are exactly solvable exhibiting nonclassical critical behavior at finite temperature position space renormalization group approximations that had been widely and successfully used are in fact alternatively applicable as exact solutions of hierarchical models this realizability guaranteeing important physical requirements for example a hierarchized version of the sierpiriski gasket is presented corresponding to a renormalization group approximation which has quantitatively yielded the multicritical phase diagrams of submonolayers on graphite hierarchical models are now being studied directly as a testing ground for new concepts for example with the introduction of frustration chaotic renormalization group trajectories were obtained for the first time thus strong and weak correlations are randomly intermingled at successive length scales and a new microscopic picture and mechanism for a spin glass emerges an upper critical dimension occurs via a boundary crisis mechanism in cluster hierarchical variants developed to have well behaved susceptibilities

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

robust stabilizability for a class of transfer functions this paper is concerned with the robust stabilizability for single input single output plants robust stabilizability means that a fixed controller can stabilize simultaneously all the plants in a given class which is characterized by a frequency dependent uncertainty band function around the transfer function of a nominal model a necessary and sufficient condition for robust stabilizability is derived based on the well known nevanlinna pick theory in classical analysis it is shown that the values of the uncertainty band function should be restricted within a certain range at the unstable poles of the nominal model in order for the class to be robustly stabilizable a procedure of synthesizing a robust stabilizer is given and the parametrization of all the robust stabilizers is also shown

multiobjective h sub 2 h sub spl infin control for a linear time invariant system with several disturbance inputs and controlled outputs we show how to minimize the nominal h sub 2 norm performance in one channel while keeping bounds on the h sub 2 norm or h sub spl infin norm performance implying robust stability in the other channels this multiobjective h sub 2 h sub spl infin problem in an infinite dimensional space is reduced to sequences of finite dimensional convex optimization problems we show how to compute the optimal value and how to numerically detect the existence of a rational optimal controller if it exists we reveal how the novel trick of optimizing the trace norm of the youla parameter over certain convex constraints allows one to design a nearly optimal controller whose youla parameter is of the same order as the optimal one

non euclidian metrics and the robust stabilization of systems with parameter uncertainty this paper considers from a complex function theoretic point of view certain kinds of robust synthesis problems in particular we use a certain kind of metric on the disk the hyperbolic metric which allows us to reduce the problem of robust stabilization of systems with many types of real and complex parameter variations to an easily solvable problem in non euclidian geometry it is shown that several apparently different problems can be treated in a unified general framework a new result on the gain margin problem for multivariable plants is also given finally we apply our methods to systems with real zero or pole variations

how much uncertainty can be dealt with by feedback feedback is used primarily for reducing the effects of the plant uncertainty on the performance of control systems and as such understanding the following questions is of fundamental importance how much uncertainty can be dealt with by feedback what are the limitations of feedback how does the feedback performance depend quantitatively on the system uncertainty how can the capability of feedback be enhanced if a priori information about the system structure is available as a starting point toward answering these questions a typical class of first order discrete time dynamical control systems with both unknown nonlinear structure and unknown disturbances is selected for our investigation and some concrete answers are obtained in the paper in particular we find that in the space of unknown nonlinear functions the generalized lipschitz norm is a suitable measure for characterizing the size of the structure uncertainty and that the maximum uncertainty that can be dealt with by the feedback mechanism is described by a ball with radius 3 2 spl radic 2 in this normed function space

brief robust h minimum entropy static output feedback control of singularly perturbed systems the problem of designing static output feedback e independent controllers for linear time invariant singularly perturbed systems is considered the controller is required to satisfy a prescribed h norm bound and to minimize the closed loop entropy at s for all small enough e the optimal controller gain is designed on the basis of generalized riccati and lyapunov equations with symmetric block 2 2 that are coupled via a projection this gain is either purely fast purely slow or a composite one depending on the structure of the output coupling matrix a well posed problem with a finite value of entropy for e 0 is obtained by assuming that the entropy of the fast subproblem is zero or by scaling the matrices of the system in the first case the optimal controller is the one that minimizes the entropy of the corresponding descriptor system

sampled data h optimal control of time varying systems the sampled data h optimal control problem is studied for time varying systems the problem consists of finding all sampled data controllers with a piece wise constant control signal for which the l2 induced norm from the continuous time disturbance to the continuous time output is less than a specified bound a game theoretic approach is used to obtain a complete solution to the problem in the finite horizon time varying case the solution involves a mixed discrete continuous riccati equation and a variable transformation which takes the problem to a standard discrete finite dimensional h filtering problem

the xy model and kadanoff s variational real space renormalization group abstract the xy model in d dimensions is studied by means of a variational real space renormalization group transformation contrary to an earlier computation in the same framework for the d 2 case we find that a low order operator basis truncation is highly unstable for certain values of the variational parameter p the renormalization group flow can display period doubling sequences towards a chaotic regime the behavior in the d 3 case is very similar

reduction of robust stabilization problems to standard h problems for classes of systems with structured uncertainty we consider the problem of robustly stabilizing a family of linear time invariant systems with a linear time invariant controller for classes of robust stabilization problems with unmodelled dynamics recent breakthroughs in h theory lead to solution via two riccati equations for systems with structured real uncertainty however a similar comprehensive theory does not exist the objective of this paper is to delineate classes of linear time invariant systems with structured uncertainty either real or complex for which a similar solution to the robust stabilization problem is possible the theory has the following salient features first for classes of uncertainty structures no conservatism is introduced for structured real parametric uncertainty we make a connection with h theory that does not amount to any crude overbounding via complex discs the stabilizability conditions we obtain are not only sufficient but also necessary the second salient feature in the theory is a special weighting function which we construct for example for the case when the structured uncertainty enters only in the denominator the robust stabilization problem is replaced by a standard h sensitivity minimization problem with our special weighting function a similar h complementary sensitivity problem is associated with numerator uncertainty

modified nevanlinna pick interpolation and feedback stabilization of linear plants with uncertainty in the gain factor abstract this paper is a sequel to an earlier work in which we applied classical novnnlinna pick interpolation to the stabilization of a single input output plant with uncertainly in the gain factor by means of an asymptotically stable control and feedback sensor in this present work we show that for the proper design we need a modification of the classical interpolation procedure a point not clearly brought out in our previous paper

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

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

h control for semilinear systems in hilbert spaces abstract this paper considers the h controlled with state feedback for semilinear infinite dimensional systems in a hilbert space necessary and sufficient conditions for the existence of a suboptimal h controller are derived in terms of a hamilton jacobi equation

adaptive h control in finite frequency domain for uncertain linear systems this paper is concerned with the problem of adaptive h controller design in finite frequency domains for uncertain linear systems the uncertainties are assumed to be time invariant unknown but bounded which appear affinely in the matrices of system models an adaptive mechanism is introduced to construct a novel finite frequency h controller with time varying gains by using lyapunov theory and parseval s theorem the controller design conditions are given in terms of a set of linear matrix inequalities lmis it is shown that the proposed finite frequency controller with time varying gains can achieve better h performance than the traditional ones with fixed gains finally a numerical example of the f 18 aircraft model is given to illustrate the presented theoretical results

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

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

low order controller design for siso systems using coprime factors and lmi this paper develops a low order controller design method for linear continuous time invariant single input single output systems requiring only the solution of a convex optimization problem the technique integrates several well known results in control theory an important step is the use of coprime factors so that based on strictly positive real functions feedback stabilization using low order controllers becomes a zero placement problem which is convex from this result we develop algorithms to solve two optimal control problems

generalized l sub 2 synthesis introduces a general framework for control design which can be used to solve several open problems in robust and optimal control these include systems subject to independently norm bounded disturbances robust stability for systems subject to element by element bounded structured uncertainty certain classes of robust performance problems and systems subject to deterministic white noise disturbances this is achieved by combining concepts such as h infinity optimization linear matrix inequalities and integral quadratic constraints necessary and sufficient conditions for the general problem to have a solution are in terms of a computationally attractive finite dimensional linear matrix inequality

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

robust stability and design of linear discrete time siso systems under l sub 1 uncertainties shows how value sets and the zero exclusion principle can be used to obtain results on the robust stability and design of controllers against a special infinite dimensional l sub 1 norm bounded parametric uncertainty in both numerator and denominator of scalar discrete time plants this parametric uncertainty has properties of both parametric and unstructured uncertainty and allows standard h sup spl infin design tools to be used without conservatism

a convex parameterization of robustly stabilizing controllers this paper treats synthesis of robust controllers for linear time invariant systems uncertain real parameters are assumed to appear linearly in the closed loop characteristic polynomial the main contribution is to give a convex parameterization of all controllers that simultaneously stabilize the system for all possible parameter combinations with the new parameterization certain robust performance problems can be stated in terms of quasi convex optimization

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

constrained monte carlo maximum likelihood for dependent data maximum likelihood estimates mles in autologistic models and other exponential family models for dependent data can be calculated with markov chain monte carlo methods the metropolis algorithm or the gibbs sampler which simulate ergodic markov chains having equilibrium distributions in the model from one realization of such a markov chain a monte carlo approximant to the whole likelihood function can be constructed the parameter value if any maximizing this function approximates the mle

fractional representation algebraic geometry and the simultaneous stabilization problem an explicit relationship between the fractional representation approach feedback system design and the algebro geometric approach to system theory is formulated and used to derive a global solution to the feedback system problem these techniques are then applied to the simultaneous stabilization problem yielding a natural geometric criterion for a set of plants to be simultaneously stabilized by a single compensator

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

robust stabilization some extensions of the gain margin maximization problem considers robust stabilization for siso systems with linear dependence on a real uncertain parameter which may occur either in the numerator or denominator of the plant transfer function the authors show necessary and sufficient conditions for the existence of robust controllers their structure and parameterization can be obtained from the corresponding nevanlinna pick interpolation problems this work extends the previously known case of an uncertain gain parameter