fuzzy control turns 50 in 2015 we celebrate the 50th anniversary of fuzzy sets ten years after the main milestones regarding its applications in fuzzy control in their 40th birthday were reviewed in fss see 1 ten years is at the same time a long period and short time thinking to the inner dynamics of research this paper presented for these 50 years of fuzzy sets is taking into account both thoughts a first part presents a quick recap of the history of fuzzy control from model free design based on human reasoning to quasi lpv linear parameter varying model based control design via some milestones and key applications the second part shows where we arrived and what the improvements are since the milestone of the first 40 years a last part is devoted to discussion and possible future research topics

convergence rates in stochastic adaptive tracking abstract for stochastic control systems described by the armax model with unknown matrix coefficients the stochastic adaptive control is designed so that the parameter estimates converge to the true values with a rate of convergence o log n log log n c n a with a 0 c 1 and the tracking error tends to its minimum value at a speed of o n 1 2e with e 0

notions of stabilizability and controllability for a class of uncertain linear systems abstract this paper investigates the relationship between stabilizability and certain notions of controllability for a class of uncertain linear systems containing time varying unknown but bounded uncertain parameters the paper introduces a notion of controllability invariance roughly speaking an uncertain system is controllability invariant if it is controllable in the usual sense for each fixed value of the uncertain parameters the paper describes a class of uncertain systems for which controllability invariance implies stabilizability

discrete time stochastic adaptive control this paper establishes global convergence of a stochastic adaptive control algorithm for discrete time linear systems it is shown that with probability one the algorithm will ensure the system inputs and outputs are sample mean square bounded and the conditional mean square output tracking error achieves its global minimum possible value for linear feedback control thus asymptotically the adaptive control algorithm achieves the same performance as could be achieved if the system parameters were known

a multiloop generalization of the circle criterion for stability margin analysis in order to provide a theoretical tool well suited for use in characterizing the stability margins e g gain and phase margins of multiloop feedback systems multiloop input output stability results generalizing the circle stability criterion are considered generalized conic sectors with centers and radii determined by linear dynamical operators are employed to enable an engineer to specify the stability margins which he desires as a frequency dependent convex set of modeling errors including nonlinearities gain variations and phase variations which the system must be able to tolerate in each feedback loop without instability the resulting stability criterion gives sufficient conditions for closed loop stability in the presence of such frequency dependent modeling errors even when the modeling errors occur simultaneously in all loops so for example stability is assured as loop gains and phases vary throughout a set of nonzero measure whose boundaries are frequency dependent the stability conditions yield an easily interpreted scalar measure of the amount by which a muitiloop system exceeds or falls short of its stability margin specifications

multiobjective output feedback control via lmi optimization this paper presents an overview of a linear matrix inequality lmi approach to the multiobjective synthesis of linear output feedback controllers the design objectives can be a mix of h sub spl infin performance h sub 2 performance passivity asymptotic disturbance rejection time domain constraints and constraints on the closed loop pole location in addition these objectives can be specified on different channels of the closed loop system when all objectives are formulated in terms of a common lyapunov function controller design amounts to solving a system of linear matrix inequalities the validity of this approach is illustrated by a realistic design example

computing invariant densities and metric entropy we present a method for accurately computing the metric entropy or equivalently the lyapunov exponent of the absolutely continuous invariant measure for a piecewise analytic expanding markov map t of the interval we construct atomic signed measures m supported on periodic orbits up to period m and prove that super exponentially fast we illustrate our method with several examples

new fuzzy control model and dynamic output feedback parallel distributed compensation a new fuzzy modeling based on fuzzy linear fractional transformations model is introduced this new representation is shown to be a flexible tool for handling complicated nonlinear models particularly the new fuzzy model provides an efficient and tractable way to handle the output feedback parallel distributed compensation problem we demonstrate that this problem can be given a linear matrix inequality characterization and hence is immediately solvable through available semidefinite programming codes the capabilities of the new fuzzy modeling is illustrated through numerical examples

adaptive control with recursive identification for stochastic linear systems this paper presents a stochastic adaptive control algorithm which is shown to possess the following properties when applied to a possibly unstable inverse stable linear stochastic system with unknown parameters whenever that system satisfies a certain positive real condition on its moving average noise dynamics 1 the adaptive control part of the algorithm stabilizes and asymptotically optimizes the behavior of the system in the sense that the limit of the sample mean square variation of the output around a given demand level equals that of a minimum variance control strategy implemented with known parameters this optimal behavior is subject to an offset 2where 2is the variance of a dither signal added to the control action in order to produce a continually disturbed control for mu 2 0 it is shown that the input output process satisfies a persistent excitation property and hence subject to a simple identifiability condition the next property holds 2 the observed input and output of the controlled system may be taken as inputs to an approximate maximum likelihood algorithm aml which generates strongly consistent estimates of the system s parameters results are presented for the scalar and multivariable cases

dynamic output feedback h control synthesis for discrete time t s fuzzy systems via switching fuzzy controllers this paper studies the problem of dynamic output feedback h control of discrete time takagi sugeno t s fuzzy systems via a switched dynamic parallel distributed compensation dpdc scheme the considered switched dpdc control structure is determined based on the values of membership functions some new properties of membership functions are obtained and exploited to derive sufficient conditions for designing switching dpdc controllers that guarantee the stability and h performances of closed loop nonlinear systems the design conditions are given in terms of solvability of a set of linear matrix inequalities lmis it is shown that the new approach provides results at least as good as the corresponding dpdc methods a numerical example is given to illustrate the effectiveness of the proposed approach

h control of linear time varying systems a state space approach in this paper the standard problem of h infty control theory for finite dimensional linear time varying continuous time plants is considered the problem is given a real number gamma 0 f

structured and simultaneous lyapunov functions for system stability problems abstract it is shown that many system stability and robustness problems can be reduced to the question of when there is a quadratic lyapunov function of a certain structure which establishes stability of x ax for some appropriate a the existence of such a lyapunov function can be determined by solving a convex program we present several numerical methods for these optimization problems a simple numerical example is given

existence of similarity transformation converting bmis to lmis under a mild condition we investigate the existence of a similarity transformation regarding robust hinfin stability analysis and observer controller synthesis problems the observer based controller is capable of disturbance rejection in the presence of unknown but bounded disturbance we present results which illustrate plainly the role of the similarity transformation in converting a bmi problem into an lmi problem in a unified fashion applicable to both continuous and discrete time problems with or without uncertainty finally the validity and applicability of the approach are illuminated by examples

orthonormal expansions of invariant densities for expanding maps abstract we give a novel way of constructing the density function for the absolutely continuous invariant measure of piecewise expanding c markov maps this is a classical problem with one of the standard approaches being ulam s method problems in modern mathematics interscience new york 1960 of phase space discretisation our method hinges instead on the expansion of the density function with respect to an l 2 orthonormal basis and the computation of the expansion coefficients in terms of the periodic orbits of the expanding map the efficiency of the method and its extension to c k expanding maps are also discussed

gain scheduling via linear fractional transformations abstract a linear finite dimensional plant with rational state space parameter dependence is controlled using a parameter dependent controller the parameters are known to take on values in a unit ball and are known in real time the goal of control is to stabilize the parameter dependent closed loop system and provide disturbance error attenuation as measured in induced l 2 norms the approach taken uses the optimally scaled small gain theorem and solves the control synthesis problem by reformulating the existence conditions into a finite dimensional convex optimization

globally optimized power allocation in multiple sensor fusion for linear and nonlinear networks the present paper is concerned with a sensor network where each sensor is modeled by either a linear or nonlinear sensing system these sensors team up in observing either static or dynamic random targets and transmit their observations through noisy communication channels to a fusion center fc for locating tracking the targets physically the network is limited by energy resource according to the available sum power budget we develop a novel technique for power allocation to the sensor nodes that enables the fc produce the best linear estimate in terms of the mean square error mse regardless of whether the sensor measurements are linear or nonlinear the targets are scalar or vectors static or dynamic the corresponding optimization problems are shown to be semidefinite programs sdps of tractable optimization and thus are globally and efficiently solved by any existing sdp solver in other words new tractably computational algorithms of distributed bayes filtering are derived with full multisensor diversity achieved intensive simulation shows that these algorithms clearly outperform previously known algorithms

stochastic adaptive pole zero assignment with convergence analysis abstract the adaptive control u n is designed for the stochastic system a z y n 1 b z u n c z w n 1 with unknown constant matrix coefficients in the polynomials a z b z and c z in the shift back operator with the purposes that 1 the unknown matrices are strongly consistently estimated and 2 the poles and zeros are replaced in such a way that the system itself is transferred to a 0 z y n 1 b 0 z u n 0 e n 1 with given a 0 z b 0 z and u n 0 so that the pole zero assignment error e n 1 is minimized the problem of adaptive pole zero assignment combined with tracking is also considered in this paper conditions used are imposed only on a z b z and c z

parameterized lmis in control theory a wide variety of problems in control system theory fall within the class of parameterized linear matrix inequalities lmis that is lmis whose coefficients are functions of a parameter confined to a compact set such problems though convex involve an infinite set of lmi constraints and hence are inherently difficult to solve numerically this paper investigates relaxations of parameterized lmi problems into standard lmi problems using techniques relying on directional convexity concepts an in depth discussion of the impact of the proposed techniques in quadratic programming lyapunov based stability and performance analysis mu analysis and linear parameter varying control is provided illustrative examples are given to demonstrate the usefulness and practicality of the approach

quasi least squares identification and its strong consistency simple identification algorithms are proposed for both continuous and discrete time systems with correlated noise by a combined method connecting the martingale method with the ordinary differential equation method quite simple conditions are obtained to guarantee strong consistency of the proposed algorithms which give estimates for the unknown coefficients in both the dynamic system and the noise model

the strong consistency of the stochastic gradient algorithm of adaptive control by use of a technique due to chen sufficient conditions are established for the strong consistency of the stochastic gradient sg algorithm for mimo armax stochastic systems without monitoring this result is then used in conjunction with the method of disturbed adaptive controls introduced in 1 2 hence it is shown that the sg algorithm generates strongly consistent parameter estimates while it is operating as a part of the sg algorithm of adaptive control of goodwin ramadge and caines

fuzzy regulators and fuzzy observers relaxed stability conditions and lmi based designs this paper presents new relaxed stability conditions and lmi linear matrix inequality based designs for both continuous and discrete fuzzy control systems they are applied to design problems of fuzzy regulators and fuzzy observers first takagi and sugeno s fuzzy models and some stability results are recalled to design fuzzy regulators and fuzzy observers nonlinear systems are represented by takagi sugeno s ts fuzzy models the concept of parallel distributed compensation is employed to design fuzzy regulators and fuzzy observers from the ts fuzzy models new stability conditions are obtained by relaxing the stability conditions derived in previous papers lmi based design procedures for fuzzy regulators and fuzzy observers are constructed using the parallel distributed compensation and the relaxed stability conditions other lmi s with respect to decay rate and constraints on control input and output are also derived and utilized in the design procedures design examples for nonlinear systems demonstrate the utility of the relaxed stability conditions and the lmi based design procedures

on the stabilization of uncertain linear systems via bound invariant lyapunov functions this paper considers the problem of stabilizing a class of systems with an arbitrarily large uncertainty bounding set in particular the stabilization of linear uncertain systems via a bound invariant lyapunov function is examined necessary and sufficient conditions for the existence of a bound invariant lyapunov function are derived for systems with block structured uncertainty and a class of systems with scalar linear uncertainty

a novel approach to h_ bm infty decentralized fuzzy observer based fuzzy control design for nonlinear interconnected systems in general due to the interactions among subsystems it is difficult to design an h infin decentralized output feedback controller for nonlinear interconnected systems this study introduces h infin decentralized fuzzy observer based fuzzy control design where the premise variables depend on the state variables estimated by a fuzzy observer for nonlinear interconnected systems via t s fuzzy models the fuzzy control design for this case is more flexible but much more complex than that for the case where the premise variables depend on the state variables only a novel decoupled method is proposed in this study to transform the non linear matrix inequality non lmi conditions into some lmi forms by the proposed decoupled method the problem of h infin decentralized fuzzy observer based fuzzy control design for nonlinear interconnected systems is characterized in terms of solving an eigenvalue problem evp with five prespecified scalars for each subsystem in general it is a difficult task to solve the evp with five prespecified scalars fortunately this special evp can be easily solved by using a genetic algorithm and an lmi based optimization method finally a simulation example is given to illustrate the design procedure and robust performance of the proposed methods

mixed h2 h multi channel linear parameter varying control in discrete time this paper develops a new method for the synthesis of linear parameter varying lpv controllers in discrete time lpv plants under consideration have a linear fractional transformation lft representation in contrast to earlier results which are restricted to single objective lpv problems the proposed method can handle a set of h2 h specifications that can be defined channel wise this practically attractive extension is derived by using specific transformations of both the lyapunov and scaling multiplier variables in tandem with appropriate linearizing transformations of the controller data and of the controller scheduling function it is shown that the controller gain scheduling function can be constructed as an affine matrix valued function in the polytopic coordinates of the scheduled parameter hence is easily implemented on line finally these manipulations give rise to a tractable and practical lmi formulation of the multi objective lpv control problem

observer based relaxed cal h _ infty control for fuzzy systems using a multiple lyapunov function this short paper proposes a method of designing a fuzzy observer based h infin controller for discrete time takagi sugeno t s fuzzy systems to enhance the applicability of the output feedback controller and improve its performance this short paper first builds a set of fuzzy control rules with premise variables different from those of the t s fuzzy system and sets the overall controller to be dependent on not only the current time but also the one step past information on the estimated fuzzy weighting functions then based on the fuzzy control rules this short paper establishes a less conservative h infin stabilization condition incorporated with a multiple lyapunov function dependent on the estimated fuzzy weighting functions through a two step design procedure the h infin stabilization condition is formulated in terms of parameterized linear matrix equalities plmis which are reconverted into lmis with the help of an efficient and effective relaxation scheme

parameterized linear matrix inequality techniques in fuzzy control system design this paper proposes different parameterized linear matrix inequality plmi characterizations for fuzzy control systems these plmi characterizations are in turn relaxed into pure lmi programs which provides tractable and effective techniques for the design of suboptimal fuzzy control systems the advantages of the proposed methods over earlier ones are then discussed and illustrated through numerical examples and simulations

persistence of excitation in extended least squares in least squares parameter estimation schemes persistency of excitation conditions on the plant states are required for consistent estimation in the case of extended least squares the persistency conditions are on the state estimates here these persistency of excitation conditions are translated into sufficiently rich conditions on the plant noise and inputs in the case of adaptive minimum variance control schemes the sufficiently rich conditions are on the noise and specified output trajectory with sufficiently rich input signals guaranteed convergence rates of prediction errors improve and it is conjectured that the algorithms are consequently more robust

strong consistency of parameter estimates in direct self tuning control algorithms based on stochastic approximation in this paper convergence of the regulator parameter estimates generated by direct self tuning control algorithms of stochastic approximation type is considered the process is supposed to be represented by an armax model with an arbitrary delay while a dither signal converging to zero in the mean square sense is superimposed on a deterministic reference the strong consistency of the regulator parameter estimates is proved under assumptions concerning the estimator complexity and the properties of the optimal regulator transfer function for a broad class of dither signals the impact of a priori information on the dimension of the estimator and the consistency of the parameter estimates is indicated it is demonstrated that the classical persistence of excitation condition is not satisfied for the analyzed algorithms

robust fuzzy observer based fuzzy control design for nonlinear systems with persistent bounded disturbances a novel decoupled approach unlike the l 2 gain h control case the l gain control problem is dealing with persistent bounded disturbances this study introduces l gain fuzzy observer based fuzzy control design where the premise variables depend on the state variables estimated by a fuzzy observer for nonlinear systems via t s fuzzy models the fuzzy control design for this case is more flexible but much more complex than that for the case where the premise variables do not depend on the state variables estimated by a fuzzy observer a novel decoupled method is proposed in this study to transform the non lmi conditions into some lmi forms by the proposed decoupled method and the genetic algorithm the l gain fuzzy observer based fuzzy control problem can be easily solved by an lmi based method

convergence rate of least squares identification and adaptive control for stochastic systems the strong consistency and the convergence rate of least squares identification for the multidimensional armax model are established under some decaying excitation conditions which are satisfied if both input and output do not grow too fast and the attenuating excitation technique is applied the parameter identification results are applied to adaptive control systems with a quadratic loss function the rate of convergence of the loss function to its minimum is also obtained

robust fuzzy observer based fuzzy control design for nonlinear discrete time systems with persistent bounded disturbances to date nonlinear l infin gain control problems have not been solved by the conventional control methods for nonlinear discrete time systems with persistent bounded disturbances this study introduces fuzzy observer based fuzzy control design where the premise variables depend on the state variables estimated by a fuzzy observer to deal with the nonlinear l infin gain control problem the fuzzy control design for this case is more flexible but much more complex than that for the case where the premise variables depend on the state variables first the takagi sugeno t s fuzzy model is employed to represent the nonlinear discrete time system next based on the fuzzy model a fuzzy observer based fuzzy controller is developed to minimize the upper bound of l infin gain of the closed loop system under some nonlinear matrix inequality non lmi constraints a novel decoupled method is proposed in this study to transform the non lmi conditions into some linear matrix inequality lmi forms by the proposed decoupled method and the genetic algorithm the l infin gain fuzzy observer based fuzzy control problem for the nonlinear discrete time systems can be easily solved by an lmi based optimization method the proposed methods which efficiently attenuate the peak of perturbation due to persistent bounded disturbances extend the l infin gain control problems from linear discrete time systems to nonlinear discrete time systems a simulation example is given to illustrate the design procedures and to confirm the l infin gain performance of the proposed method

convergence of a multivariable self tuning controller with integral action abstract in this note global convergence is established for the multivariable self tuning control algorithm presented in 11 the plant is assumed to be described by a carima model the control algorithm relies on the generalized minimum variance approach the convergence proof is carried out by resorting to an interlaced stochastic gradient identification algorithm

recursive system identification and adaptive control by use of the modified least squares algorithm in this paper we first use a weaker than persistent excitation condition to establish the strong consistency of mls for mimo stochastic systems without monitoring and give a comparison between various conditions for consistency then we show the global convergence for adaptive tracking based on mls and finally the adaptive tracking and strong consistency results are combined

how smooth is your wavelet wavelet regularity via thermodynamic formalism a popular wavelet reference w states that in theoretical and practical studies the notion of wavelet regularity has been increasing in importance not surprisingly the study of wavelet regularity is currently a major topic of investigation smoother wavelets provide sharper frequency resolution of functions also the iterative algorithms to construct wavelets converge faster for smoother wavelets the main goals of this paper are to extend refine and unify the thermodynamic approach to the regularity of wavelets and to devise a faster algorithm for estimating regularity the thermodynamic approach works equally well for compactly supported and non compactly supported wavelets and also applies to non analytic wavelet filters

a convex characterization of gain scheduled h sub spl infin controllers an important class of linear time varying systems consists of plants where the state space matrices are fixed functions of some time varying physical parameters spl theta small gain techniques can be applied to such systems to derive robust time invariant controllers yet this approach is often overly conservative when the parameters spl theta undergo large variations during system operation in general higher performance can be achieved by control laws that incorporate available measurements of spl theta and therefore adjust to the current plant dynamics this paper discusses extensions of h sub spl infin synthesis techniques to allow for controller dependence on time varying but measured parameters when this dependence is linear fractional the existence of such gain scheduled h sub spl infin controllers is fully characterized in terms of linear matrix inequalities the underlying synthesis problem is therefore a convex program for which efficient optimization techniques are available the formalism and derivation techniques developed here apply to both the continuous and discrete time problems existence conditions for robust time invariant controllers are recovered as a special case and extensions to gain scheduling in the face of parametric uncertainty are discussed in particular simple heuristics are proposed to compute such controllers

meromorphic extensions of generalised zeta functions in this paper we give a full description of the spectrum of the ruelle perron frobenius operator acting on the banach space of holder continuous functions on a subshift of finite type theorem 1 these results are then used to extend the meromorphic domain of generalised zeta functions theorem 2 the most important application of these results is to the domain of the smale zeta function for axiom a flows theorem 3 in the course of this paper we settle questions raised by ruelle and sunada

guaranteed properties of gain scheduled control for linear parameter varying plants abstract gain scheduling has proven to be a successful design methodology in many engineering applications however in the absence of a sound theoretical analysis these designs come with no guarantees on the robustness performance or even nominal stability of the overall gain scheduled design this paper presents such an analysis for one type of gain scheduled system namely a linear parameter varying plant scheduling on its exogenous parameters conditions are given which guarantee that the stability robustness and performance properties of the fixed operating point designs carry over to the global gain scheduled design these conditions confirm and formalize popular notions regarding gain scheduled design such as the scheduling variable should vary slowly

nonlinear bayesian filtering using the unscented linear fractional transformation model for nonlinear state space model involving random variables with arbitrary probability distributions the state estimation given a sequence of observations is based on an appropriate criterion such as the minimum mean square error mmse this leads to linear approximation in the state space of the extended kalman filter ekf and the unscented kalman filter ukf which work reasonably well only for mildly nonlinear systems we propose a bayesian filtering technique based on the mmse criterion in the framework of the virtual linear fractional transformation lft model which is characterized by a linear part and a simple nonlinear structure in the feedback loop lft is an exact representation for any differentiable nonlinear mapping so the virtual lft model is amenable to a wide range of nonlinear systems simulation results demonstrate that the proposed filtering technique gives better approximation and tracking performance than standard methods like the ukf furthermore for highly nonlinear systems where ukf diverges the lft model estimates the conditional mean with reasonable accuracy