the robust h sub 2 control problem a worst case design the worst case effect of a disturbance system on the h sub 2 norm of the system is analyzed an explicit expression is given for the worst case h sub 2 norm when the disturbance system is allowed to vary over all nonlinear time varying and possibly noncausal systems with bounded l sub 2 induced operator norm an upper bound for this measure which is equal to the worst case h sub 2 norm if the exogeneous input is scalar is defined some further analysis of this upper bound is done and a method to design controllers which minimize this upper bound over all robustly stabilizing controllers is given the latter is done by relating this upper bound to a parameterized version of the auxiliary cost function studied in the literature

an algorithm for robust performance tests based on frequency dependent lmis in this paper an algorithm is introduced for feasibility problems associated with frequency dependent linear matrix inequalities these problems correspond to h 2 or h robust performance tests for a given controller in the face of feedback perturbations to this effect an auxiliary h 2 cost functional is introduced and a sequence of h 2 problems with a single linear constraint is considered in which the h 2 cost functional is kept unchanged while the linear constraints are iteratively modified it is shown that in each step the auxiliary optimal solution moves closer in the sense of a weighted quadratic norm to the target feasible set conditions are also established under which the sequence of solutions to the auxiliary problems yields a solution to the original feasibility problem the method is illustrated by a numerical example corresponding to a h robust performance test for a multivariable system

entropy and global markov properties we extend refine and give simple proofs of some recent results on the validity of global markov properties for classical spin systems one of the new results is that there is a global markov property that is satisfied by equilibrium states in general the proof of this establishes formulas for the entropy and free energy that show that these quantities are ford dimensional systems given in terms of d 1 dimensional systems furthermore we show that global markov properties imply the absence of some types of symmetry breaking

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

uniqueness and global markov property for euclidean fields the case of general polynomial interactions we give a general method for proving uniqueness and global markov property for euclidean quantum fields the method is based on uniform continuity of local specifications proved by using potential theoretical tools and exploitation of a suitable fkg order structure we apply this method to give a proof of uniqueness and global markov property for the gibbs states and to study extremality of gibbs states also in the case of non uniqueness in particular we prove extremality for 2 4 also in the case of non uniqueness and global markov property for weak coupling 2 4 which solves a long standing problem uniqueness and extremality holds also at any point of differentiability of the pressure with respect to the external magnetic field

robust synthesis of feedforward compensators the design of a feedforward compensator for robust spl hscr sub spl infin or spl hscr sub 2 performance under structured uncertainty is considered for linear time invariant uncertainty a convex method based on linear matrix inequalities lmis across the frequency variable is given for nonlinear or time varying perturbations and spl hscr sub spl infin performance the design problem is reduced exactly to a state space lmi and extensions to spl hscr sub 2 performance are discussed an example illustrates the application of these techniques to two degree of freedom control design

neural network based approximate reasoning principles and implementation instead of seeking a structure mapping from a fuzzy reasoning system to a neural network this paper is intended to find a functional mapping from a fuzzy logic based algorithm to the network based approach by viewing the given rule base as defining a global linguistic association constrained by fuzzy sets approximate reasoning is implemented here by a backpropagation neural network bnn with the aid of fuzzy set theory by paying particular attention to the generalization capability of the bnn the underlying principles have been examined in detail using two examples a small demonstration at the linguistic level and a more realistic problem of multivariable fuzzy control of blood pressure the simulation results not only indicate the feasibility of the bnn based approach but also reveal some deeper similarities which exist in the two methods which may have some important implications for future studies into fuzzy control in addition this work may be considered as another application example of t

a hybrid neuro fuzzy pid controller a hybrid neuro fuzzy control strategy and its corresponding rule generating approach is proposed according to this approach the fuzzy control rules can be generated automatically via fuzzy inputs and then the appropriate control action can be deduced efficiently by a simplified fuzzy inference engine by combining the use of an incremental pi algorithm and a positional pd algorithm a pid fuzzy control strategy can be implemented simply from two input variables it results in the number of control rules being significantly reduced without decreasing the control performance the control parameters can be self tuned by introducing a single neuron together with a modified back propagation learning algorithm simulation results show that the proposed fuzzy controller is able to control unknown processes and provide good performance compared to traditional self organising and neural network based fuzzy controllers this method has simpler control algorithms and less computational burden

a set based approach for white noise modeling this paper provides a new framework for analyzing white noise disturbances in linear systems rather than the usual stochastic approach noise signals are described as elements in sets and their effect is analyzed from a worst case perspective the paper studies how these sets must be chosen to have adequate properties for system response in the worst case statistics consistent with the stochastic point of view and simple descriptions that allow for tractable worst case analysis the method is demonstrated by considering its implications in two problems rejection of white noise signals in the presence of system uncertainty and worst case system identification

some examples concerning the global markov property we present examples of interactions of classical lattice systems whose extremal gibbs states fail to have the global markov property one of the examples is translation invariant

analysis of robust h 2 performance using multiplier theory in this paper the problem of determining the worst case h2 performance of a control system subject to linear time invariant uncertainties is considered a set of upper bounds on the performance is derived based on the theory of stability multipliers and the solution of an original optimal control problem the numerical issues raised by the resulting computational problems are discussed in particular newly developed interior point convex optimization methods combined with linear matrix inequalities apply very well to the fast and accurate solution of these problems the new results compare favorably with prior ones the method can be extended to other types of perturbations

a hybrid neural network based self organizing controller this paper presents a novel and systematic approach to constructing a self organizing and self learning multivariable controller the proposed controller is built on a hybrid neural network consisting of a variable structure competitive network and a standard back propagation neural network bnn we develop the corresponding self organizing and learning algorithms the controller shares certain features with traditional fuzzy controllers in terms of using error based input forms and emphasizing the rule based paradigms however knowledge representation and reasoning are here carried out by the network structure and computing instead of logical inference on the other hand compared with ordinary bnn based control paradigms the proposed controller possesses a distinctive characteristic that is while having the advantages of computational efficiency and trainable capability of the network paradigm it maintains the clarity of the rule based paradigm by self organizing a rule base thereby providing an

on upper bounds on worst case h 2 performance obtained with dynamic multipliers in this note an iterative procedure is presented for obtaining a decreasing sequence of upper bounds on the worst case h 2 performance of a given stabilizing controller in the presence of normalized coprime factor perturbations to obtain such bounds a descent procedure is introduced for a dual lagrangean functional which gives upper bounds on the worst case h 2 performance index and is defined on the set of real rational and non negative functions dynamic multipliers specific ways are presented for selecting feasible and descent directions in this set and lower bounds are derived for the corresponding decreases on the dual functional at any step the dynamics of the current multiplier gives rise to a linear class over which the optimization of the dual functional is shown to be equivalent to linear optimization subject to linear matrix inequalities lmi this allows for a combination of function space and lmi techniques in the process of obtaining increasingly tighter upper bounds on worst case h

characterization and efficient computation of the structured singular value the concept of structured singular value was recently introduced by doyle 1 as a tool for the analysis and synthesis of feedback systems with structured uncertainties it is a key to the design of control systems under joint robustness and performance specifications and it very nicely complements the h infty approach to control system design in this paper it is shown that the structured singular value can be obtained as the solution of several smooth optimization problems properties of these optimization problems are exhibited leading to a fast algorithm that always yields the structured singular value for block structures of size no larger than 3 and often does for block structures of larger size

shannon mcmillan theorems for discrete random fields along curves and lower bounds for surface order large deviations the notion of a surface order specific entropy hc p of a two dimensional discrete random field p along a curve c is introduced as the limit of rescaled entropies along lattice approximations of the blowups of c existence is shown by proving a corresponding shannon mcmillan theorem we obtain a representation of hc p as a mixture of specific entropies along the tangent lines of c as an application the specific entropy along curves is used to refine follmer and ort s lower bound for the large deviations of the empirical field of an attractive gibbs measure from its ergodic behaviour in the phase transition regime

mixed spl hscr sub 2 and spl hscr sub spl infin performance objectives ii optimal control this paper considers the analysis and synthesis of control systems subject to two types of disturbance signals white signals and signals with bounded power the resulting control problem involves minimizing a mixed spl hscr sub 2 and spl hscr sub spl infin norm of the system it is shown that the controller shares a separation property similar to those of pure spl hscr sub 2 or spl hscr sub spl infin controllers necessary conditions and sufficient conditions are obtained for the existence of a solution to the mixed problem explicit state space formulas are also given for the optimal controller

robust performance against time varying structured perturbations in this paper we consider the problem of robust performance analysis for some nominal system m z against bounded linear time varying structured feedback perturbations we introduce a natural input output notion of rate of variation for a linear time varying operator we then exhibit upper and lower bounds on the maximum rate of variation of these perturbations against which robust performance is achievable using these bounds we show that the existence of frequency dependent d scales that render the norm of m z less than one is necessary and sufficient for robust performance against arbitrarily slowly varying structured linear perturbations of norm less than one this result suggests that it is natural and well justified to deal with the frequency dependent upper bound for the spl mu norm rather than spl mu itself this is reassuring given that the upper bound is easily and reliably computable while computation of complex spl mu appears difficult

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

stochastic competitive learning competitive learning systems are examined as stochastic dynamical systems this includes continuous and discrete formulations of unsupervised supervised and differential competitive learning systems these systems estimate an unknown probability density function from random pattern samples and behave as adaptive vector quantizers synaptic vectors in feedforward competitive neural networks quantize the pattern space and converge to pattern class centroids or local probability maxima a stochastic lyapunov argument shows that competitive synaptic vectors converge to centroids exponentially quickly and reduces competitive learning to stochastic gradient descent convergence does not depend on a specific dynamical model of how neuronal activations change these results extend to competitive estimation of local covariances and higher order statistics

a unified real time approximate reasoning approach for use in intelligent control part 1 theoretical development taking account of the fuzzy nature of human decision making processes and real time properties this paper establishes a unified approximate reasoning model based on possibility theory rather than on relational matrix computation both fuzzy and random uncertainties can be coped with in the model in the case of sensor based situations a simpler reasoning scheme is derived by introducing the concepts of matching measures the proposed models may provide another possibility for on line reasoning in real time expert system applications in part 1 the theoretical foundations are provided while in part 2 their application to multivariate blood pressure control is described

the lqg ltr procedure for multivariable feedback control design this paper provides a tutorial overview of the lqg ltr design procedure for linear multivariable feedback systems lqg ltr is interpreted as the solution of a specific weighted h2 tradeoff between transfer functions in the frequency domain properties of this solution are examined for both minimum phase and nonminimum phase systems this leads to a formal weight augmentation procedure for the minimum phase case which permits essentially arbitrary specification of system sensitivity functions in terms of the weights while such arbitrary specifications are not possible for nonminimum phase problems a direct relationship between weights and sensitivities is developed for nonminimum phase siso and certain nonminimum phase mimo cases which guides the weight selection process

robust stability with time varying structured uncertainty considers the problem of assessing robust stability in the presence of block diagonally structured time varying dynamic uncertainty it is shown that robust stability holds only if there exist constant scalings which lead to a small gain condition the notion of stability here is finite gain stability over finite energy signals in sharp contrast to the case of time invariant dynamic uncertainty this result is not limited by the number uncertainty blocks these results parallel previous results regarding finite gain stability over persistent bounded signals

convex methods for robust h sub 2 analysis of continuous time systems treats the robust h sub 2 performance question of evaluating the rejection of continuous time white noise in the worst case over structured uncertainty in the system a frequency domain convex condition for robust h sub 2 analysis is presented with analogous properties as in the discrete time case in particular necessary and sufficient results are obtained introducing a continuous time version of the methodology of set descriptions of white noise in addition a state space test in terms of linear matrix inequalities is developed for the robust h sub 2 problem in the case of constant uncertainty scalings which apply to nonlinear or time varying uncertainty thus the problem is rendered finite dimensional in the same situation in which robust h sub spl infin analysis is finite dimensional

robust h 2 control based on dynamic multipliers in this article upper bounds on the worst case h 2 performance index relative to structured feedback perturbations are considered which are based on the minimisation of dual lagrangean functionals over linearly parametrised finite dimensional classes of dynamic multipliers it is shown that the minimisation problems in question can be recast as optimisation problems with linear cost functional and linear matrix inequality lmi constraints an iterative scheme is suggested to generate linearly parametrised classes of multipliers of increasing dynamic order so that progressively tighter upper bounds can be obtained as illustrated by two simple numerical examples finally with a view to synthesis procedures based on d k iterations relative to multipliers and controllers it is shown that the minimisation of the upper bounds corresponding to given multipliers with respect to linearly parametrised classes of youla parameters can also be cast as linear cost lmi constraint problems

the complex structured singular value a tutorial introduction to the complex structured singular value is presented with an emphasis on the mathematical aspects of the based methods discussed here have been useful for analysing the performance and robustness properties of linear feedback systems several tests for robust stability and performance with computable bounds for transfer functions and their state space realizations are compared and a simple synthesis problem is studied uncertain systems are represented using linear fractional transformations lfts which naturally unify the frequency domain and state space methods

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

mixed h sub 2 h sub infinity control a convex optimization approach the problem of finding an internally stabilizing controller that minimizes a mixed h sub 2 h sub infinity performance measure subject to an inequality constraint on the h sub infinity norm of another closed loop transfer function is considered this problem can be interpreted and motivated as a problem of optimal nominal performance subject to a robust stability constraint both the state feedback and output feedback problems are considered it is shown that in the state feedback case one can come arbitrarily close to the optimal even over full information controllers mixed h sub 2 h sub infinity performance measure using constant gain state feedback moreover the state feedback problem can be converted into a convex optimization problem over a bounded subset of n n and n q where n and q are respectively the state and input dimensions real matrices using the central h sub infinity estimator it is shown that the output feedback problem can be reduced to a state feedback problem in this case the dimension of the resulting controller does not exceed the dimension of the generalized plant

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

from pca s to equilibrium systems and back stationary measures for probabilistic cellular automata pca s ind dimensions give rise to space time histories whose statistics may naturally be described by gibbs states ind 1 dimensions for an interaction energy obtained from the pca in this note we study the converse question do all gibbs states for this correspond to statistical space time histories for the pca our main result states that the answer is yes at least for translation invariant or periodic gibbs states thus ergodicity questions for pca s can at least partially be formulated as questions of uniqueness of gibbs states

frequency domain conditions for robust h sub 2 performance presents frequency domain conditions to analyze h sub 2 performance in the worst case over structured uncertainty in the system these conditions are convex feasibility tests in terms of linear matrix inequalities across frequency of the same nature as tests available for robust stability or h sub spl infin performance by introducing a deterministic generalization of the h sub 2 criterion the different versions of this test are characterized as necessary and sufficient for robust performance under corresponding classes of dynamic uncertainty

the global markov property for equilibrium states which are determined by correlations in a strip consider an interaction round face potential on the lattice d which may include nearest neighbor and next nearest neighbor pair interactions as well as more general plaquette terms assuming some periodicity of the potential it is shown that under the condition that equilibrium states can be distinguished by expectation values of sufficiently local observables equilibrium states possess a global markov property the condition under which this markov property is shown to hold is met in particular in case the equilibrium state is unique or determined by the magnetization the proof is based on an application of the variational principle to states which are constructed like markov chains

mixed h2 h control in a stochastic framework abstract this paper deals with a mixture of h 2 and h in the following way we have two inputs and one output one input signal is a white noise stochastic process and represents errors e g resulting from measurement noise the other input has a more deterministic character if one has a reference signal e g a step as input one cannot model this as white noise but it fits nicely into this new class of inputs the objective is to minimize the effect of these signals on the output of the system we define a cost function which enables us to combine these two exogenous inputs even though they are structurally different the analysis of this function leads to a standard h riccati equation we motivate this cost function by looking at two theoretical applications the derivation of robust performance bounds and a tracking problem