generalized browder s theorem for tensor product and elementary operators abstract the transfer property for the generalized browder s theorem both of the tensor product and of the left right multiplication operator will be characterized in terms of the b weyl spectrum inclusion in addition the isolated points of these two classes of operators will be fully characterized

generalized browder s and weyl s theorems for left and right multiplication operators abstract the main objective of this work is to study generalized browder s and weyl s theorems for the multiplication operators l a and r b and for the elementary operator a b l a r b

the largest singular value of e sup x a sub 0 e sup x is convex on convex sets of commuting matrices a short and direct proof of the convexity property is given it is shown that the theorem applies to any convex commuting set of matrices in r sup nxn where a sub 0 in r sup nxn is fixed it is also shown that the result does not hold if x is permitted to be a general square matrix a counterexample is supplied for noncommuting matrices

the fractional representation approach to synthesis problems an algebraic analysis viewpoint part i weakly doubly coprime factorizations in this paper we show how to reformulate the fractional representation approach to analysis and synthesis problems within an algebraic analysis framework in terms of modules we give necessary and sufficient conditions so that a system admits weakly left right doubly coprime factorizations moreover we explicitly characterize the integral domains a such that every plant defined by means of a transfer matrix whose entries belong to the quotient field of a admits weakly doubly coprime factorizations finally we show that this algebraic analysis approach allows us to recover on the one hand the approach developed in m c smith ieee trans automat control 34 1989 pp 1005 1007 and on the other hand the ones developed in k mori and k abe siam j control optim 39 2001 pp 1952 1973 v r sule siam j control optim 32 1994 pp 1675 1695 and 36 1998 pp 2194 2195 m vidyasagar h schneider and b a francis ieee trans automat control 27 1982 pp 880 894 m vidyasagar control system synthesis a factorization approach mit press cambridge ma 1985

time varying versus time invariant compensation for rejection of persistent bounded disturbances and robust stabilization it is shown that time varying compensation does not improve the optimal rejection of persistent bounded disturbances this result is obtained by exploiting a key observation that any time varying compensator which yields a given degree of disturbance rejection must do so uniformly over time thereby removing any advantage of time variation this key observation is exploited to show that time varying compensation does not improve the optimal rejection of disturbances regardless of the norm used to measure the disturbances thus absolutely summable finite energy or persistent bounded disturbances may be treated in the same manner it is shown that time varying compensation does not help in the bounded input bounded output robust stabilization of time invariant plants with unstructured uncertainty in doing so it is also shown that the small gain theorem is both necessary and sufficient for the bounded input bounded output stability of certain linear time varying plants subject to unstructured linear time varying perturbations

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

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

an optimal control approach to robust tracking of linear systems in our early work we show that one way to solve a robust control problem of an uncertain system is to translate the robust control problem into an optimal control problem if the system is linear then the optimal control problem becomes a linear quadratic regulator lqr problem which can be solved by solving an algebraic riccati equation in this article we extend the optimal control approach to robust tracking of linear systems we assume that the control objective is not simply to drive the state to zero but rather to track a non zero reference signal we assume that the reference signal to be tracked is a polynomial function of time we first investigated the tracking problem under the conditions that all state variables are available for feedback and show that the robust tracking problem can be solved by solving an algebraic riccati equation because the state feedback is not always available in practice we also investigated the output feedback we show that if we place the poles of the observer

optimal h robustness and the relationship to lqg design problems a solution to the hx optimal control problem is obtained by employing a standard lqg polynomial based solution procedure the derivation is straightforward and avoids many of the more abstract mathematical results usually employed the relationship between the lqg and hm design methods becomes transparent and the similarities and differences can be identified

lqg control with an h sup infinity performance bound a riccati equation approach an lqg linear quadratic gaussian control design problem involving a constraint on h sup infinity disturbance attenuation is considered the h sup infinity performance constraint is embedded within the optimization process by replacing the covariance lyapunov equation by a riccati equation whose solution leads to an upper bound on l sub 2 performance in contrast to the pair of separated riccati equations of standard lqg theory the h sup infinity constrained gains are given by a coupled system of three modified riccati equations the coupling illustrates the breakdown of the separation principle for the h sup infinity constrained problem both full and reduced order design problems are considered with an h sup infinity attenuation constraint involving both state and control variables an algorithm is developed for the full order design problem and illustrative numerical results are given

a family of nonlinear h sup spl infin output feedback controllers state space formulas are derived for a family of controllers solving the nonlinear h sup spl infin output feedback control problem the formulas given are expressed in terms of the solutions to two hamilton jacobi inequalities in n independent variables these controllers are obtained by interconnecting the central controller with an asymptotically stable free system having l sup 2 gain spl les spl gamma all proofs given are simple and clear and provide deeper insight in the synthesis of the corresponding linear h sup spl infin controllers

robust stabilization of linear multivariable systems relations to approximation the stabilization of a linear system modelled as g a where g is a known rational transfer function and a is a perturbation satisfying w a2 g1 or equivalently there does not exist g with fewer poles in c than g such that w1 g g w2 a simple characterization of all robustly stabilizing controllers is then derived and state space formulae for maximally robust controllers are given finally reduced order controllers are considered

brief the weighted incremental norm approach from linear to nonlinear h control weighted induced norms can be used to handle robustness and sensitivity requirements in the linear context as shown by zames in its seminal work this paper deals with an attempt to extend linear h control concepts to the nonlinear context making use of a weighted incremental norm it is shown how these concepts make it possible to handle basic requirements such as robust stability disturbance attenuation and steady state behaviors

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

b weyl spectrum and poles of the resolvent abstract let t be a bounded linear operator acting on a banach space and let bw t c such that t i is not a b fredholm operator of index 0 be the b weyl spectrum of t define also e t to be the set of all isolated eigenvalues in the spectrum t of t and t to be the set of the poles of the resolvent of t in this paper two new generalized versions of the classical weyl s theorem are considered more precisely we seek for conditions under which an operator t satisfies the generalized weyl s theorem bw t t e t or the version ii of the generalized weyl s theorem bw t t t

optimal disturbance reduction in linear multivariable systems this paper presents a computational solution to an important optimization problem arising in optimal sensitivity theory the approach is to treat the multivariable problem exactly as the scalar problem in that stability constraints are handled via interpolation the resulting computations are easily implemented using existing methods

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

a non iterative method for computing the infimum in h optimization this paper presents a simple and non iterative procedure for the computation of the exact value of the infimum in the singular h optimization problem and is an extension of our earlier work the problem formulation is general and does not place any restriction on the direct feedthrough terms between the control input and the controlled output variables and between the disturbance input and the measurement output variables our method is applicable to a class of singular h optimization problems for which the transfer functions from the control input to the controlled output and from the disturbance input to the measurement output have no invariant zeros on the jw axis and also satisfy certain geometric conditions the computation of the infimum in our method involves solving two well defined riccati and two lyapunov equations

efficient robust optimization for robust control with constraints this paper proposes an efficient computational technique for the optimal control of linear discrete time systems subject to bounded disturbances with mixed linear constraints on the states and inputs the problem of computing an optimal state feedback control policy given the current state is non convex a recent breakthrough has been the application of robust optimization techniques to reparameterize this problem as a convex program while the reparameterized problem is theoretically tractable the number of variables is quadratic in the number of stages or horizon length n and has no apparent exploitable structure leading to computational time of mathcal o n 6 per iteration of an interior point method we focus on the case when the disturbance set is norm bounded or the linear map of a hypercube and the cost function involves the minimization of a quadratic cost here we make use of state variables to regain a sparse problem structure that is related to the structure of the original problem that is the policy optimization problem may be decomposed into a set of coupled finite horizon control problems this decomposition can then be formulated as a highly structured quadratic program solvable by primal dual interior point methods in which each iteration requires mathcal o n 3 time this cubic iteration time can be guaranteed using a riccati based block factorization technique which is standard in discrete time optimal control numerical results are presented using a standard sparse primal dual interior point solver that illustrate the efficiency of this approach

feedback minimax sensitivity and optimal robustness in this paper we look for feedbacks that minimize the sensitivity function of a linear single variable feedback system represented by its frequency responses sensitivity to disturbances and robustness under plant perturbations are measured in a weighted h infty norm in an earlier paper zames proposed an approach to feedback design involving the measurement of sensitivity by multiplicative seminorms which have certain advantages over the widely used quadratic norm in problems where there is plant uncertainty or where signal power spectra are not fixed but belong to sets the problem was studied in a general setting and some h infty examples were solved here a detailed study of the single variable case is undertaken the results are extended to unstable plants and explicit formulas for the general situation of a finite number of right half plane rhp plant zeros or poles are provided the q or approximate inverse parametrization of feedbacks that maintain closed loop stability is extended to the ease of unstable plants the h infty and wiener hopf approaches are compared

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

minimax frequency domain performance and robustness optimization of linear feedback systems it is shown that feedback system design objectives such as disturbance attenuation and rejection power and bandwidth limitation and robustness may be expressed in terms of required bounds of the sensitivity function and its complement on the imaginary axis this leads to a minimax frequency domain optimization problem whose solution is reduced to the solution of a polynomial equation

optimal model matching control for time delay systems model matching control for time delay systems is formulated as an optimization problem in h2 norm space as well as h norm space wiener hopf techniques and the theory of optimal interpolation are successfully employed to treat this model matching problem including perfect and imperfect model matching cases prior to this work for time delay systems only the problem of perfect model matching control had been considered and no attempt had been made to examine that of imperfect model matching control with the optimization approach presented here an optimal model matching control scheme is constructed for the imperfect model matching case to produce perfect model matching results whenever possible

a new coherent paired state with possible applications to fluctuation dissipation phenomena the authors extend their previous work on coherent paired states associated with the lie group su 1 1 whereas the earlier states were defined with respect to a single type of canonical boson or linear quantum harmonic oscillator the new states are defined in terms of two distinct types of bosons or oscillators the new coherent states may again on the one hand be viewed as ordinary glauber coherent states in the two boson hilbert space spanned by arbitrary numbers of two distinct bogoliubov quasiparticles associated with the original bosons via a generalised bogoliubov transformation alternatively expressed wholly in terms of the original bosons these new coherent states are reached from the ordinary coherent states via a unitary pairing transformation which is shown to be associated with the entire so called discrete series of representations of the group su 1 1 as an important illustration of the use of these states and transformations they study in detail a rather general class of quantum lagrangians which includes the damped linear harmonic oscillator they thereby illustrate their possible usefulness in applications to quantum many body or field theoretic processes involving fluctuation dissipation phenomena in general

elementary operators finite ascent range closure and compactness abstract given banach space operators a i b i b x 1 i 2 let ab b b x denote the elementary operator ab x a 1 x b 1 a 2 x b 2 then ab has finite ascent 1 for a number of fairly general choices of the operators a i and b i this information is applied to prove some necessary and sufficient conditions for the range of ab to be closed and in deciding conditions on the tuples a 1 a 2 and b 1 b 2 so that ab n x compact for some integer n 1 and operator x implies ab x compact this generalizes some well known results of anderson and foias 4 and yosun 25 also considered is the question what is a necessary and sufficient condition on the tuples a 1 a 2 b 1 b 2 and ab for ab n to be compact for some integer n 1

coherent mixed states and a generalised p representation the pure glauber harmonic oscillator coherent states provide a very useful basis for many purposes they are complete in the sense that an arbitrary state in the hilbert space may be expanded in terms of them furthermore the well known p representation provides a diagonal expansion of an arbitrary operator in the hilbert space in terms of the projection operators onto the coherent states the authors study the extensions of these results to the analogous mixed states which describe comparable harmonic oscillator systems in thermodynamic equilibrium at non zero temperatures their results are given for the general density operator which describes the mixed squeezed coherent states of the displaced and squeezed harmonic oscillator they show how these squeezed coherent mixed states similarly provide a very convenient complete description of a hilbert space in particular they show how the usual p and q representations of operators in terms of pure states may be extended to finite temperatures with the corresponding mixed states and various relations between them are demonstrated the question of the existence of the generalised p representation for an arbitrary operator is further examined and some pertinent theorems are proven they also show how their results relate to the glauber lachs formalism in quantum optics for mixtures of coherent and incoherent radiation particular attention is focused both on the interplay between the quantum mechanical and thermodynamical uncertainties and on the entropy associated with such mixed states

sampled data control of continuous time systems with an h optimality criterion abstract the synthesis of sampled data controllers for continuous time systems is studied in the context of h control theory the sampled data h optimal control problem consists of finding a sampled data controller which minimizes the l2 induced norm from the continuous time disturbance to the continuous time output the problem is here solved via a discrete system representation this is achieved by describing the closed loop system as a discrete system with finite state dimension and with a representation of the disturbance and the output signals of the discrete system as sequences of elements of certain infinite dimensional spaces via finite rank approximations of a compact integral operator associated with the system the sampled data h control problem can be solved to any degree of accuracy from the structure of the problem it follows that irrespective of the order of the approximation a feasible sampled data controller can be found which has the same order as the system

paper adaptive controllers for discrete time systems with arbitrary zeros an overview the paper presents a comprehensive review of the problem of designing discrete time adaptive controllers for systems with arbitrary zeros minimum or non minimum phase systems being particular cases the authors have attempted to give a unified presentation of the various existing schemes by stripping away the technical details and highlighting their close interrelation through the use of a common framework both algorithmic and theoretical aspects are discussed with particular emphasis on the stability of the schemes

on decoupling the h optimal sensitivity problem for products of plants abstract in this note we show how to solve the h optimal sensitivity problem for a siso plant p s p 1 s p 2 s given the solutions for p 1 s p 2 s this allows us to solve the problem for systems of the form e hs p 0 s where p 0 s is the transfer function of a stable lti finite dimensional system

h norm computation for ltv systems using nonlinear optimal control algorithms we discuss the computation for 2 induced norms of finite horizon linear time varying systems it is well known that 2 induced norm calculating problems can be converted into optimal control problems so with the help of optimal control algorithms we propose a numerical technique for 2 induced norm computation our approach is as follows first we present a new computational method for nonlinear optimal control problems that is numerically reliable stable and robust analysing the convergence property under some assumptions next by modifying the presented method we propose an algorithm for the 2 induced norm calculation of finite horizon linear time varying systems which is essentially different from all other existing methods finally the effectiveness of our algorithm is demonstrated by simulation

global parametrization of feedback systems with nonlinear plants for unity feedback systems with a plant p and controller c we show that given a nonlinear stable p the class of all controller stabilizing the feedback system is globally parametrized by a stable map q we also show how q is useful in studying the effects of modeling errors on the stability

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

weyl s theorem tensor products and multiplication operators abstract the transmission of weyl s theorem from operators on banach spaces to their tensor products and also to their associated multiplication operators is deconstructed

new approach to general nonlinear discrete time stochastic h_ infty control in this paper a new approach based on convex analysis is introduced to solve the h infty problem for discrete time nonlinear stochastic systems first by using the disintegration property of the conditional expectation sufficient necessary conditions are given for the internal stability of the concerned systems second in order to separate the unknown exogenous disturbance from the state variables the properties of convex functions are applied to prove the stochastic version of bounded real lemma brl third the state feedback h infty control problem is studied and an h infty control is designed based on the brl finally two numerical examples and one real world regulation control example of synthetic genetic circuit are presented to show the effectiveness of our developed theory

robust end point optimizing feedback for nonlinear dynamic processes the paper suggests two novel approaches to the synthesis of robust end point optimizing feedback for nonlinear dynamic processes classically end point optimization is performed only for the nominal process model using optimal control methods and the question of performance robustness to disturbances and model plant mismatch remains unaddressed the present contribution addresses the end point optimization problem for nonlinear affine systems with fixed final time through robust optimal feedback methods in the first approach a nonlinear state feedback is derived that robustly optimizes the final process state this solution is obtained through series expansion of the hamilton jacobi bellman pde with an active opponent disturbance as reliable measurements or estimates of all states may not always be available the second approach also robustly optimizes the process end point but uses output rather than state information this direct use of measurement information is preferred since the choice of a st

generalized browder s and weyl s theorems for banach space operators abstract we find necessary and sufficient conditions for a banach space operator t to satisfy the generalized browder s theorem we also prove that the spectral mapping theorem holds for the drazin spectrum and for analytic functions on an open neighborhood of t as applications we show that if t is algebraically m hyponormal or if t is algebraically paranormal then the generalized weyl s theorem holds for f t where f h t the space of functions analytic on an open neighborhood of t we also show that if t is reduced by each of its eigenspaces then the generalized browder s theorem holds for f t for each f h t

duality theory for mimo robust disturbance rejection banach space duality theory is used to characterize the solutions of a nonstandard h sup infinity optimization problem which is shown to be allpass in general and unique in the single input single output siso ease the theory leads to a numerical solution of duality and convex optimization which is applied to an example for a limiting case of sharp cutoff filters an explicit solution of the optimal robust disturbance attenuation problem ordap resembling the two arc theorem of complex analysis is derived

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

the general distance problem in h optimal control theory abstract the general distance problem that arises in h optimal control theory is considered transformations that reduce the general h problem to the general distance problem are reviewed and an iterative scheme called iteration is considered in which the general distance problem is reduced to a standard best approximation problem the iteration is viewed as a problem of finding the zero crossing of a function this function is shown to be continuous monotonically decreasing convex and bounded by some very simple functions these properties make it possible to obtain very rapid convergence of the iterative process the issues of model reduction and approximation in h synthesis will also be addressed

intermittent redesign of analog controllers via the youla parameter the paper studies digital redesign of linear time invariant analog controllers under intermittent sampling the sampling pattern is only assumed to be uniformly bounded but otherwise irregular and unknown a priori the contribution of the paper is twofold first it proposes a constructive algorithm to redesign any analog stabilizing controller so that the closed loop stability is preserved second it is shown that when applied to sub optimal boldsymbol h 2 and boldsymbol h infty controllers the algorithm produces sub optimal sampled data solutions under any a priori unknown sampling pattern the proposed solutions are analytic computationally simple implementable and transparent transparency pays off in showing the optimality under a fixed sampling density of uniform sampling for both performance measures studied

weight selection in feedback design with degree constraints we present an approach for feedback design which is based on recent developments in analytic interpolation with a degree constraint performance is cast as an interpolation problem with bounded analytic functions minimizers of a certain weighted entropy functional provide interpolants having degree less than the number of constraints the choice of weight parameterizes all such bounded degree solutions however the relationship between the weights and the shape of corresponding transfer functions is not direct thus in this paper we develop a formalism that guides weight selection

output feedback decentralized control of large scale systems using weighted sensitivity functions minimization this paper considers the problem of achieving stability and desired dynamical transient behavior for linear large scale systems by decentralized control it can be done by making the effects of the interconnections between the subsystems arbitrarily small sufficient conditions for stability and diagonal dominance of the closed loop system are introduced these conditions are in terms of decentralized subsystems and directly make a constructive h control design possible a mixed h pole region placement is suggested such that by assigning the closed loop eigenvalues of the isolated subsystems appropriately the eigenvalues of the overall closed loop system are assigned in desirable range the designs are illustrated by an example

h control with transients in h infty or uniformly optimal control problems it is usually assumed that the system initial conditions are zero in this paper an h infty like control problem that incorporates uncertainty in initial conditions is formulated this is done by defining a worst case performance measure both finite and infinite horizon problems are considered necessary and sufficient conditions are derived for the existence of controllers that yield a closed loop system for which the above mentioned performance measure is less than a prespecified value state space formulae for the controllers are also presented

h optimal feedback controllers for linear multivariable systems this paper treats the problem of designing for a linear multivariable plant a feedback controller which minimizes the h infty norm of a weighted sensitivity matrix there exists a family of optimal improper feedbacks this family is determined by application of a theory of ball and helton a method for computing optimal feedbacks is described in detail and a numerical example is included it is shown that an optimal improper feedback can be approximated by a proper one under certain conditions on the weighting matrices

on h optimal sensitivity theory for siso feedback systems this paper deals with the design of feedback controllers which minimize the h infty norm of the sensitivity function suitably weighted this approach to the theory of feedback design was introduced by zames 1 and developed by zames and francis 2 in this paper the theory of sarason 3 is applied to the determination of the optimal weighted sensitivity function and an upper bound on its norm the problem of achieving small sensitivity over a specified frequency band is studied and the effect of nonminimum phase is elucidated finally a method is introduced for handling plant poles and zeros on the imaginary axis

brief paper reliable stabilization using a multi controller configuration given a plant p we consider the problem of designing a pair of controllers c 1 and c 2 such that their sum stabilizes p and in addition each of them also stabilizes p should the other one fail this is referred to as the reliable stabilization problem it is shown that every strongly stabilizable plant can be reliably stabilized moreover one of the two controllers can be specified arbitrarily subject only to the constraint that it should be stable the stabilization technique is extended to reliable regulation

robust h sub spl infin filtering for nonlinear stochastic systems this paper describes the robust h sub spl infin filtering analysis and synthesis of nonlinear stochastic systems with state and exogenous disturbance dependent noise we assume that the state and measurement are corrupted by stochastic uncertain exogenous disturbance and that the system dynamic is modeled by ito spl circ type stochastic differential equations for general nonlinear stochastic systems the h sub spl infin filter can be obtained by solving second order nonlinear hamilton jacobi inequalities when the worst case disturbance is considered in the design procedure a mixed h sub 2 h sub spl infin filtering problem is also solved by minimizing the total estimation error energy it is found that for a class of special nonlinear stochastic systems the h sub spl infin filtering design can be given via solving several linear matrix inequalities instead of hamilton jacobi inequalities a few examples show that the proposed methods are effective

stabilization and sensitivity for eventually time invariant systems abstract this paper studies the stability and minimal sensitivity of a time varying system which as time passes becomes more and more time invariant

conjugation interpolation and model matching in h abstract the notion of conjugation is introduced to give a unified framework of algebraic operations connected to the analytic aspects of transfer functions two classes of conjugations are shown to be important one is the lossless conjugation that performs inner outer factorization and the other is the j lossless conjugation that solves the model matching problem related to h 8 control the solution of the model matching problem obtained by j lossless conjugation is regarded as the state space version of the directional interpolation theory and is much simpler than those obtained via the nehari approximation

min max kalman filtering abstract the problem of h optimal state estimation of linear continuous time systems that are measured with an additive white noise is addressed the relevant cost function is the expected value of the standard h performance index with respect to the measurement noise statistics the solution is obtained by applying the matrix version of the maximum principle to the solution of the min max problem in which the estimator tries to minimize the mean square estimation error and the exogenous disturbance tries to maximize it while being penalized for its energy the solution is given in terms of two coupled riccati difference equations from which the filter gains are derived in the case where an infinite penalty is imposed on the energy of the exogenous disturbance the celebrated kalman filter is recovered in the stationary case where all the signals are stationary an upper bound on the solutions of the coupled riccati equations is obtained via a solution of coupled linear matrix inequalities the resulting filter then guarantees a bound on the estimation error covariance matrix an illustrative example is given where the velocity of a maneuvering target has to be estimated utilizing noisy measurements of the position

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

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

robust parameter adjustment with nonparametric weighted ball in h sup infinity uncertainty using a parameter adjustment law robust monotonic parameter error reduction is proved for a linear parameter estimation problem in which the signal is corrupted by the presence of nonparametric dynamical uncertainty the nonparameterized dynamics are characterized by a known frequency domain magnitude bound a nonconservative bound on the finite time energy of the nonparametric dynamics output is constructed in real time when the error single energy is small enough to be due to nonparametric dynamics alone the parameter adjustment is shut off to avoid misadjustment the approach is an extension of existing adaptive control error dead zone ideas to the case of frequency domain bounded uncertainty

h2 optimal control of linear systems with tracking disturbance rejection constraints this paper considers the design of two degree of freedom linear multivariable controllers for four block matrix transfer functions with objectives pertaining to both random and deterministic signals the former are taken into account in the cost functional whereas the latter entail that the controller is constrained to solve a servomechanism problem involving a class of reference signals and two classes of disturbances the cost functional is set up by adding quadratic terms to the usual lqg wiener hopf cost in such a way as to ensure the existence of unique solutions even when the signals to be tracked rejected as well as models have poles on the imaginary axis the solutions of the servomechanism problem are explicitly parametrized by free stable rational and proper parameters under two sets of assumptions one which is necessary and sufficient for the existence of solutions and another stronger set which avoids kronecker products and the associated higher dimensional factorizations involved in

two properties of l sub 1 optimal controllers the solution of the l sub 1 sensitivity minimization problem is shown to have two properties which contrast markedly with properties of the solutions to the better known h infinity sensitivity minimization problem or the lqg linear quadratic gaussian problem is given of a one parameter family of first order plants where the order of the l sub 1 optimal compensator can be arbitrarily large and thus it is impossible to bound the order of an l sub 1 optimal compensator in terms of the order of the plant a plant is considered which has a continuous one parameter family of l sub 1 optimal compensators and thus l sub 1 optimal compensators need not be unique the author s two examples are considered to answer two questions left open by m a daleh and j b pearson ibid vol 32 p 314 23 1987

robustness with simultaneous pole and zero movement across the jω axis in this paper we present some new tests for assessing the robustness of the stability property of a feedback design the tests offer a wider scope of application than those previously described in the literature the main result is a robustness test which can be used to assess closed loop stability when the nominal plant model and perturbed plant model representing uncertainty have different numbers of poles and zeros in the right half plane this test also indicates over which frequencies good sensitivity and robust stability are compatible objectives and over which frequencies a compromise has to be made contrary to popular belief if there are no uncertain zeros which can move across the imaginary axis then good sensitivity and robust stability are compatible objectives over the same frequencies

thermalized displaced and squeezed number states in the coordinate representation within the framework of thermofield dynamics the wavefunctions of the thermalized displaced number and squeezed number states are given in the coordinate representation furthermore the time evolution of these wavefunctions is considered by introducing a thermal coordinate representation and we also calculate the corresponding probability densities average values and variances of the position coordinate some special cases of which are consistent with results in the literature

on the minimization of sensitivity to additive disturbances for linear distributed parameter mimo feedback systems the problem of minimizing the sensitivity of the output or input of a plant under a feedback control with respect to external additive disturbances is considered the mathematical model assumed for the plant is a rectangular matrix of transfer functions which can arise from both lumped parameter and distributed parameter systems including time delays the feedback controller is assumed to guarantee the closed loop stability it is shown that an arbitrarily small sensitivity function on a compact set of complex frequencies can be obtained by a suitable choice of feedback transfer function provided no zero of the plant transfer function is in this set additionally the sensitivity function can be designed to be bounded on an unbounded region containing the imaginary axis if there are no zeros of the plant in the right half plane and a certain hypothesis holds multiplicative delays are not excluded if there are zeros in the right half plane the latter design is not realizable since minimizing the sen

linear feedback systems and the graph topology it is established for general linear systems that the gap metric induces the coarsest topology with respect to which both closed loop stability and closed loop performance are robust properties in earlier works similar topological results were obtained by exploiting the existence of particular coprime factor system representations not known to exist in general by contrast the development here does not rely on any specific system representations systems are simply characterized as subspaces of norm bounded input output pairs and the analysis hinges on the underlying geometric structure of the feedback stabilization problem unlike other work developed within such a framework fundamental issues concerning the causality of feedback interconnections are discussed explicitly the key result of this paper concerns the difference between linear feedback interconnections with identical controllers in terms of a performance robustness related closed loop mapping upper and lower bounds on the induced norm of this difference are derived allowing for possibly infinite dimensional input output spaces and time varying behavior the bounds are both proportional to the gap metric distance between the plants which clearly demonstrates the gap to be an appropriate measure of the difference between open loop systems from the perspective of closed loop behavior to conclude an example is presented to show that bounds of the form derived here for linear systems do not hold in a general nonlinear setting

delay dependent stability and h control for a class of fuzzy descriptor systems with time delay this paper addresses issues of stability analysis and h control for a class of fuzzy time delay descriptor systems described by an extended takagi sugeno t s fuzzy model a delay dependent stability criterion is first presented and a h control synthesis approach is then developed both the stability criterion and the h controller synthesis are based on the solutions of linear matrix inequalities lmis finally two examples are given to illustrate the effectiveness of the approaches proposed in this paper

infinite horizon h control for nonlinear stochastic markov jump systems with x u v dependent noise via fuzzy approach in this paper the h control problem is studied for nonlinear stochastic markov jump systems with state control and external disturbance dependent noise x u v dependent noise for short a sufficient condition is derived for the infinite horizon h control of such systems in terms of a set of coupled second order hamilton jacobi inequalities hjis in general it is difficult to solve these coupled hjis by using fuzzy approach the infinite horizon h control design for nonlinear stochastic markov jump systems is developed via solving a set of linear matrix inequalities lmis instead of hjis two numerical examples are presented to illustrate the effectiveness of the proposed design methods

further results on limitations to the capability of feedback in this paper we study the limitations of the capability of the feedback mechanism in dealing with structural uncertainty for a class of pth order non linear autoregressive control systems the size of the structural uncertainty is characterized by multiple lipschitz constants say l 1 l 2 lp of the uncertain non linear function it is shown that if l 1 l 2 lp satisfy certain algebraic conditions then there exists no globally stabilizing feedback control law for the corresponding uncertain systems the result here is an extension of the previous result obtained by zhang and guo 2002

browder weyl theorems tensor products and multiplications article history received 1 september 2008 available online 10 june 2009 submitted by r curto a banach space operator t b x satisfies browder s theorem if the complement of the weyl spectrum w t of t in t equals the set of riesz points of t t is polaroid if the isolated points of t are poles no restriction on rank of the resolvent of t l et t denote the set of fredholm points of t browder s theorem transfers from a b b x to s l a r b resp s a b i f and only if a and b resp a and b h ave svep at points a and b for which w s i f a and b are finitely polaroid then the polaroid property transfers from a b x and b b y to l a r b again restricting ourselves to the completion of x y in the projective topology if a and b are finitely polaroid then the polaroid property transfers from a b x and b b y to a b

optimal sensitivity theory for multivariate distributed plants in this note we generalized the results of foias and tannenbaum 1986 a and foias et al 1987 on optimal sensitivity minimization to a much broader class of distributed plants we relate our results to certain kinds of hankel norm approximation problems that frequently appear in control engineering applications

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

the robust optimal control of uncertain systems state space method the robust optimal control of linear uncertain systems with l sub 2 bounded uncertainties is addressed the optimal state feedback control law in the so called worst disturbance case is derived the lql method minimax state estimation maximin state feedback closed loop control scheme lql means linear quadratic l sub 2 bounded uncertainties is proposed in a sense the lql method uses the idea of the linear quadratic gaussian lqg method to treat h sub infinity optimizing design problems

an approach to h control of a class of nonlinear systems this paper considers robust h control of a class of nonlinear systems that can be represented by a heterogeneous model a new controller design method using both state feedback is proposed based on quadratic stability theory a number of necessary and sufficient conditions are derived for quadratic stabilizability with disturbance attenuation of the control system concerned an example is also given to demonstrate the performance of the proposed controller design methods

synthesis of dissipative systems using quadratic differential forms part ii the problem discussed is that of designing a controller for a linear system that renders a quadratic functional nonnegative our formulation and solution of this problem is completely representation free the system dynamics are specified by a differential behavior and the performance is specified through a quadratic differential form we view control as interconnection a controller constrains a distinguished set of system variables the control variables the resulting behavior of the to be controlled variables is called the controlled behavior the constraint that the controller acts through the control variables only can be succinctly expressed by requiring that the controlled behavior should be wedged in between the hidden behavior obtained by setting the control variables equal to zero and the plant behavior obtained by leaving the control variables unconstrained the main result is a set of necessary and sufficient conditions for the existence of a controlled behavior that meets the performance specifications the essential requirement is a coupling condition an inequality that combines the storage functions of the hidden behavior and the orthogonal complement of the plant behavior

some applications of small gain theorem to interconnected systems based on small gain theorem decentralized control of interconnected systems is studied linear matrix inequality conditions are given for stabilizability of interconnected systems interconnections can be used to improve system robustness in the sense of small gain theorem an example is given to illustrate the results finally interconnected systems composed of symmetric subsystems are studied

the use of representations of h in placing classical control problems in a modern control framework abstract motivated by zames work on optimal sensitivity it is shown that there exists in functional analysis a foundation based upon the theory of representations of h upon which results may be developed to enable a range of classical control problems to be placed in a modern control framework

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

entropy minimizing and risk sensitive control rules abstract glover and doyle 4 have shown that the entropy minimising criterion is identical with the infinite horizon form of the leqg optimality criterion leqg linear exponential of quadratic gaussian whittle and kuhn 11 have given a hamiltonian formulation which provides a natural treatment of leqg optimisation this paper contains a direct proof that the control yielded by these hamiltonian methods is entropy minimising the conclusion would follow from the two assertions above but a simple adjoining of the proofs of these two assertions yields an argument so circuitous that a direct proof is of interest

drazin spectra of banach space operators and banach algebra elements abstract given a banach algebra a and a a several relations among the drazin spectrum of a and the drazin spectra of the multiplication operators l a and r a will be stated the banach space operator case will be also examined furthermore a characterization of the drazin spectrum will be considered

frequency response algorithms for h sub infinity optimization with time domain constraints a very broad framework for control system design is considered that encompasses frequency response methodologies for h sub infinity optimization that solve various aspects of the control design problem and that are less well known that state space methods the focus is on linear programming lawson s algorithm and trefethen s algorithm a modified lawson s algorithm is proposed and related to trefethan s method the modified algorithm is shown to be significantly faster than linear programming and lawson s algorithm it is also shown how to extend the modified lawson s algorithm so as to handle time domain constraints in addition to frequency domain specifications which distinguishes it from other h sub infinity optimization methods some steps are taken toward dealing with time domain constraints within an h sub infinity optimization framework

coherent states on the m sheeted complex plane coherent states on the m sheeted complex plane are introduced and properties like overcompleteness and resolution of the identity are studied operators am am which create and annihilate clusters of m particles are considered and it is shown that the properties of our coherent states with respect to them are analogous to the properties of the ordinary glauber coherent states with respect to a a an extended bargmann representation in the m sheeted complex plane and p and q representations with respect to our coherent states are studied applications of this formalism in the study of hamiltonians that describe m particle clustering are also considered