linear and nonlinear algorithms for identification in h sub infinity with error bounds a linear algorithm and a nonlinear algorithm for the problem of system identification in h sub infinity posed by helmicki et al 1990 for discrete time systems are presented the authors derive some error bounds for the linear algorithm which indicate that it is not robustly convergent however the worst case identification error is shown to grow as log n where n is the model order a robustly convergent nonlinear algorithm is derived and bounds on the worst case identification error in the h sub infinity norm are obtained

worst case input output identification we consider worst case l1 identification of causal linear shift invariant systems from time series many results are given on general aspects of identification algorithm performance existence of optimal algorithms robust convergence and input experiment design the identification methodology studied here is compatible with the modelling requirements of modern robust control design

worst case identification of nonlinear fading memory systems abstract in this paper the problem of asymptotic identification for fading memory systems in the presence of bounded noise is studied for any experiment the worst case error is characterized in terms of the diameter of the worst case uncertainty set optimal inputs that minimize the radius of uncertainty are studied and characterized finally a convergent algorithm that does not require knowledge of the noise upper bound is furnished the algorithm is based on interpolating data with spline functions which are shown to be well suited for identification in the presence of bounded noise more so than other basis functions such as polynomials

on system identification of complex systems from finite data we introduce a new principle for identification based on choosing a model from the model parameterization which best approximates the unknown real system belonging to a more complex space of systems that do not lend themselves to a finite parameterization the principle is particularly effective for robust control as it leads to a precise notion of parametric and nonparametric error and the identification problem can be equivalently perceived as that of robust convergence of the parameters in the face of unmodeled errors the main difficulty in its application stems from the interplay of noise and unmodeled dynamics and requires developing novel two step algorithms that amount to annihilation of the unmodeled error followed by averaging out the noise the paper establishes robust convergence for a large class of systems topologies and unmodeled errors sample path based finite time polynomial rate of convergence and annihilation correlation algorithms for linearly parameterized model structures

nonparametric control algorithms for nonlinear fading memory systems we develop an algorithm to control an unknown nonlinear fading memory discrete time system our approach is based on nonparametric regression techniques rather than traditional feedback control we discuss a procedure that given a desired periodic output and a tolerance produces an acceptable output from any system in a wide class the algorithm uses as data past inputs which are selected by the algorithm and corresponding possibly noisy output observations and needs no extra parametric knowledge or restrictions on the system we also present an algorithm that produces an output that converges to the desired output when stricter conditions are imposed on the system our approach in its current form however cannot be used to control open loop unstable systems

control oriented system identification a worst case deterministic approach in h sub infinity the authors formulate and solve two related control oriented system identification problems for stable linear shift invariant distributed parameter plants in each of these problems the assumed a priori information is minimal consisting only of a lower bound on the relative stability of the plant an upper bound on a certain gain associated with the plant and an upper bound on the noise level the first of these problems involves identification of a point sample of the plant frequency response from a noisy finite output time series obtained in response to an applied sinusoidal input with frequency corresponding to the frequency point of interest this problem leads naturally to the second problem which involves identification of the plant transfer function in h sub infinity from a finite number of noisy point samples of the plant frequency response concrete plans for identification algorithms are provided for each of these two problems

modelling of uncertain systems via linear programming abstract real plants are in general time varying and uncertain yet most industrial control loops are designed using linear time invariant lti plant models the present work formalizes the problem of best approximate lti modelling of bibo stable linear time varying ltv systems in a way that is compatible with the notion of induced l norm used in robust l 1 control this setup is closely associated with certain identification methods and controlmotivated model validation invalidation procedures that can be efficiently implemented with special purpose linear programming lp techniques results are given for approximate modelling of bibo stable ltv and lti systems using lti models especially bibo stable fixed pole state space model parametrizations it is shown that such parametrizations are satisfactory from an approximate modelling point of view and can be used in output error type lp identification techniques and in lp model validation procedures various useful constraints on model parameters etc can be included as long as they are linear in the unknown parameters and both insensitive statistically robust and sensitive criteria can be used in identification and model validation simulation examples are included to illustrate that such techniques indeed give good results and can be used to solve problems of realistic size

identification of probabilistic system uncertainty regions by explicit evaluation of bias and variance errors a procedure is developed for identification of probabilistic system uncertainty regions for a linear time invariant system with unknown dynamics on the basis of time sequences of input and output data the classical framework is handled in which the system output is contaminated by a realization of a stationary stochastic process given minor and verifiable prior information on the system and the noise process frequency response pulse response and step response confidence regions are constructed by explicitly evaluating the bias and variance errors of a linear regression estimate in the model parametrizations use is made of general forms of basis functions conservatism of the uncertainty regions is limited by focusing on direct computational solutions rather than on closed form expressions using an instrumental variable method for identification the procedure is suitable also for input output data obtained from closed loop experiments

the caratheodory fejer problem and h sub spl infin l sub 1 identification a time domain approach in this paper we study a worse case robust control oriented identification problem this problem is in the framework of h sub spl infin identification but the formulation here is more general the available a priori information in our problem consists of a lower bound on the relative stability of the plant an upper bound on a certain gain associated with the plant and an upper bound on the noise level the plant to be identified is assumed to lie in a certain subset in the space of h sub spl infin characterized by the assumed a priori information the available experimental information consists of a corrupt finite output time series obtained in response to a known nonzero but otherwise arbitrary input the proposed algorithm is in the class of interpolatory algorithms which are known to possess desirable optimality properties in reducing the identification error this algorithm is obtained by solving an extended caratheodory fejer problem via standard convex programming methods both the algorithm and error bounds ran be applied to l sub 1 identification problems as well

sequential approximation of feasible parameter sets for identification with set membership uncertainty in this paper the problem of approximating the feasible parameter set for identification of a system in a set membership setting is considered the system model is linear in the unknown parameters a recursive procedure providing an approximation of the parameter set of interest through parallelotopes is presented and an efficient algorithm is proposed its computational complexity is similar to that of the commonly used ellipsoidal approximation schemes numerical results are also reported on some simulation experiments conducted to assess the performance of the proposed algorithm

experiment design in a bounded error context comparison with d optimality abstract the problem of optimal experiment design for linear regression models is addressed the measurement errors are supposed to be bounded with no other hypothesis on their distribution contrary to the classical statistical approach an experiment is considered here as optimal if it minimizes the volume of the set of all parameters that are consistent with the data the model structure and the noise bounds this new design policy is compared to the classical d optimal design used in a statistical context when the number of measurements is equal to the number of parameters to be estimated the new optimality criterion although relying on quite different hypotheses on the noise distribution is shown to lead to the same optimal policy as a d optimality criterion this is no longer true when more measurements are to be made depending on the shape of the admissible experimental domain it is sometimes possible to design optimal experiments without any calculations an algorithmic procedure is suggested to be used when this is not so

on the time complexity of worst case system identification in this paper we treat a general worst case system identification problem this problem is worst case with respect to both noise and system modeling uncertainty we consider this problem under various a priori information structures we determine bounds on the minimum duration identification experiment that must be run to identify the plant to within a specified guaranteed worst case error bound our results are algorithm independent we show that this minimum duration is prohibitively long based on our results we suggest that worst case with respect to noise system identification requires unrealistic amounts of experimental data

on the value of information in system identification bounded noise case assuming instantaneous bounds on the noise system parameter identification is formulated as membership set estimation problem sequential algorithms are constructed to estimate the membership sets of the parameters which are consistent with the measurements and the noise constraints the important new feature of the proposed algorithms is their ability to ignore redundant data the efficient data extraction property of the new algorithms is achieved with small computational effort and with improved performance when compared to the least square algorithm the convergence properties and the notion of identifiability in the set theoretic context are also studied

robust identification and galois sequences abstract worst case l1 identification is studied for bibo stable linear shift invariant systems it is shown that the chebyshev identification method when used with galois input designs satisfies a certain robust convergence property and provides l1 model error bounds in worst case identification of bibo stable systems with a uniformly bounded noise set up the robust identification methodology developed is compatible with the modelling requirements of modern robust control design

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

nonparametric output prediction for nonlinear fading memory systems the authors construct a class of elementary nonparametric output predictors of an unknown discrete time nonlinear fading memory system their algorithms predict asymptotically well for every bounded input sequence every disturbance sequence in certain classes and every linear or nonlinear system that is continuous and asymptotically time invariant causal and with fading memory the predictor is based on k sub n nearest neighbor estimators from nonparametric statistics it uses only previous input and noisy output data of the system without any knowledge of the structure of the unknown system the bounds on the input or the properties of noise under additional smoothness conditions the authors provide rates of convergence for the time average errors of their scheme finally they apply their results to the special case of stable linear time invariant lti systems

intermittency in the presence of noise we analyse iterations of maps on an interval with an added noise term in the neighbourhood of an intermittency threshold we rigorously derive a universal scaling function for the laminar time expressed as a function of the distance from the threshold and the variance of the noise

identification and application of bounded parameter models abstract the place of parameter bounding algorithms in identification methodology is discussed such algorithms provide a radical alternative to the computation of parameter point estimates and covariances instead of p d f or mean and covariance for the noise and prior parameter estimates they require bounds in an essentially deterministic model formulation the paper is a preliminary review of some standard tasks namely experiment design testing for outliers toleranced prediction and worst case control design in the context of parameter bounding identification computational requirements of these tasks are examined and the limitations of existing parameter bounding algorithms are noted

necessary and sufficient conditions for robust identification of uncertain lti systems abstract robust identification of uncertain systems arises whenever a chosen family of models does not completely describe reality in these situations the issue of unmodeled dynamics gains significance in addition to random measurement noise to deal with such mixed stochastic deterministic settings we introduce a novel notion for robust consistency which requires that the expectation with respect to noise of the worst case with respect to unmodeled dynamics identification error asymptotically approach zero it turns out that this notion leads to transparent necessary and sufficient conditions we show that robust consistency holds if and only if there is an instrument input pair capable of annihilating the residual error as well as stochastic noise an extension of this result to the well known bounded but unknown noise model shows that if we were to remove a set of lebesgue measure zero the error bound asymptotically approaches zero

input design in worst case system identification using binary sensors this technical note addresses system identification using binary valued sensors in a worst case set membership setting the main contribution is the solution of the optimal input design problem for identification of scalar gains which is instrumental to the construction of suboptimal input signals for identification of fir models of arbitrary order two different cost functions are considered for input design the maximum parametric identification error and the relative uncertainty reduction with respect to the minimum achievable error it is shown that in the latter case the solution enjoys the property of being independent of the length of the identification experiment and as such it can be implemented as an optimal recursive procedure over a time interval of arbitrary length

optimal algorithms theory for robust estimation and prediction this paper deals with the theory of optimal algorithms for problems which cannot be solved exactly the theory developed allows for the derivation of new and interesting results in parameter estimation and in time series prediction in situations where no reliable statistical hypothesis can be made on the functions and modeling errors involved but only a bound on them is known in particular the derivation of computationally simple optimal algorithms for these two problems is investigated the practical effectiveness of the algorithms obtained is illustrated by several numerical examples

the least squares identification of fir systems subject to worst case noise the least squares identification of fir systems is analyzed assuming that the noise is a bounded signal and the input signal is a pseudo random binary sequence a lower bound on the worst case transfer function error shows that the least square estimate of the transfer function diverges as the order of the fir system is increased this implies that in the presence of the worst case noise the trade off between the estimation error due to the disturbance and the bias error due to unmodeled dynamics is significantly different from the corresponding trade off in the random error case with a worst case formulation the model complexity should not increase indefinitely as the size of the data set increases

a class of algorithms for identification in h abs traet in this paper a class of algorithms for the problem of system identification in are investigated these algorithms are characterized by a two stage structure and involve a class of window functions some conditions in terms of properties of the window functions are derived which guarantee robust convergence of the algorithms identification errors are analyzed for several common window functions this leads to some insights into the trade off between the error induced by approximation and that due to noise

some iterative methods free from second derivatives for nonlinear equations in a recent paper noor m aslam noor new classes of iterative methods for nonlinear equations appl math comput 2007 doi 10 1016 j amc 2007 suggested and analyzed a generalized one parameter halley method for solving nonlinear equations using in this paper we modified this method which has fourth order convergence as special cases we obtain a family of third order iterative methods for appropriate and suitable choice of the parameter we have compared this modified noor method with some other iterative methods which shows that this new iterative method is robust and efficient one several examples are given to illustrate the efficiency and the performance of this new method

worst case conditional system identification in a general class of norms we consider optimality of identification of linear systems by reduced complexity models which belong to a subspace of possibly low dimension of a system space identification is based on the a priori knowledge that an unknown system belongs to a certain class and a posteriori information given by its nonexact output measurements measurement errors are assumed unknown but norm bounded the analysis is not restricted to specific norms in system and measurement spaces generic norms are considered we derive tight upper and lower bounds on the minimal local worst case identification error and define an algorithm whose error is within the derived bounds the results are specified for product norms which include l p norms email protected email protected in the system space the bounds are expressed in terms of the model error which measures a quality of representing systems by a model subspace and the diameter of information reflecting the size of a model uncertainty set the model error is further splitted into truncated model error and residual error which controls tail properties of the unknown system examples of model and input selection in some special cases are given and numerical experiments showing the behavior of the estimates reported

a generic intermittency model and its 1 d meta map power laws invariants and the succession of laminar sequences abstract intermittent behaviour has been found in many systems able to switch between two different dynamic states e g between long laminar phases and short chaotic bursts despite the apparently high dimensional complexity certain one dimensional 1 d maps are known to mimic properties of such dynamics to these belongs the iterative map x n 1 i x n i x n i z mod 1 giving rise to long laminar lengths the statistics of the laminar lengths are of special interest starting from this map we are interested in the values of x 0 i which arise after passing through the modulo operation these determine the laminar lengths uniquely a 1 d meta map x 0 i f x 0 i 1 is derived heuristically it is used to calculate statistical properties of the laminar phases our results show an improvement in the behaviour of short and very long laminar phases as compared to earlier analytical results introducing the concept of the generic starting value we find laminar phases not to be strictly independent of their predecessors

optimal robustness in the gap metric the application of the gap metric to robust stabilization of feedback systems is considered in particular a solution to the problem of robustness optimization in the gap metric is presented the problem of robust stabilization under simultaneous plant controller perturbations is addressed and the least amount of combined plant controller uncertainty measured by the gap metric that can cause instability of a nominally stable feedback system is determined included are a detailed summary of the main properties of the gap metric and the introduction of a dual metric called the t gap metric a key contribution of this study is to show that the problem of robustness optimization in the gap metric is equivalent to robustness optimization for normalized coprime factor perturbations this settles the question as to whether maximizing allowable coprime factor uncertainty corresponds to tolerating the largest ball of uncertainty in a well defined metric

worst case system identification in h sub spl infin validation of a priori information essentially optimal algorithms and error bounds resolves several important open issues pertaining to a worst case control oriented system identification problem known as identification in h sub spl infin first a method is presented for developing confidence that certain a priori information available for identification is not invalid this method requires the solution of a certain nondifferentiable convex program second an essentially optimal identification algorithm is constructed this algorithm is worst case strongly optimal to within a factor of two finally new upper and lower bounds on the optimal identification error are derived and used to estimate the identification error associated with the given algorithm interestingly the development of each of these results draws heavily upon the classical nevanlinna pick interpolation theory as such the authors results establish a clear link between the areas of system identification and optimal interpolation theory both the formulation and techniques in this paper are applicable to problems where the frequency data available for identification may essentially be arbitrarily distributed

rational basis functions for robust identification from frequency and time domain measurements this paper investigates the use of general bases with fixed poles for the purposes of robust estimation these bases which generalise the common fir laguerre and two parameter kautz ones are shown to be fundamental in the disc algebra provided a very mild condition on the choice of poles is satisfied it is also shown that by using a min max criterion these bases lead to robust estimators for which error bounds in different norms can be explicitly quantified the key idea facilitating this analysis is to re parameterise the chosen model structures into a new one with equivalent fixed poles but for which the basis functions are orthonormal in h 2 d

persistent identification of systems with unmodeled dynamics and exogenous disturbances in this paper a novel framework of system identification is introduced to capture the hybrid features of systems subject to both deterministic unmodeled dynamics and stochastic observation disturbances using the concepts of persistent identification control oriented system modeling and stochastic analysis we investigate the central issues of irreducible identification errors and time complexity in such identification problems upper and lower bounds on errors and speed of persistent identification are obtained the error bounds are expressed as functions of observation lengths sizes of unmodeled dynamics and probability distributions of disturbances asymptotic normality and complexity lower bounds are investigated when periodic inputs and ls estimation are applied generic features of asymptotic normality are further explored to extend the asymptotic lower bounds to a wider range of signals and identification mappings

the cumulant approach for investigating the noise influence on mode locking bifurcations the influence of noise on mode locking bifurcations is investigated for the circle map and for coupled logistic maps the technique based on the cumulant expansion is used for the bifurcation analysis of these systems it is shown that this cumulant analysis in gaussian approximation provides a suitable description of the influence of weak noise we find that the universal scaling properties for the circle map in the critical point for the golden mean sequence are the same as those obtained analytically using the path integral technique we find the same scaling behaviour in the case of weak multiplicative noise

optimal nonparametric identification from arbitrary corrupt finite time series formulates and solves a worst case system identification problem for single input single output linear shift invariant distributed parameter plants the available a priori information in this problem consists of time dependent upper and lower bounds on the plant impulse response and the additive output noise the available a posteriori information consists of a corrupt finite output time series obtained in response to a known nonzero but otherwise arbitrary input signal the authors present a novel identification method for this problem this method maps the available a priori and a posteriori information into an uncertain model of the plant which comprises a nominal plant model a bounded additive output noise and a bounded additive model uncertainty the upper bound on the model uncertainty is explicit and expressed in terms of both the l sub 1 and h sub spl infin system norms the identification method and the nominal model possess certain well defined optimality properties and are computationally simple requiring only the solution of a single linear programming problem

optimal asymptotic identification under bounded disturbances the intrinsic limitation of worst case identification of linear time invariant systems using data corrupted by bounded disturbances when the unknown plant is known to belong to a given model set is studied this is done by analyzing the optimal worst case asymptotic error achievable by performing experiments using any bounded input and estimating the plant using any identification algorithm it is shown that under some topological conditions on the model set there is an identification algorithm which is asymptotically optimal for any input and the optimal asymptotic error is characterized as a function of the inputs these results which hold for any error metric and disturbance norm are applied to three specific identification problems identification of stable systems in the l sub 1 norm identification of stable rational systems in the h sub infinity norm and identification of unstable rational systems in the gap metric for each of these problems the general characterization of optimal asymptotic error is used to find near optimal inputs to minimize the error

performance robustness of discrete time systems with structured uncertainty given an interconnection of a nominal discrete time plant and a stabilizing controller together with structured norm bounded nonlinear time varying perturbations necessary and sufficient conditions for robust stability and performance of the system are provided it is shown that performance robustness is equivalent to stability robustness in the sense that both problems can be dealt with in the framework of a general stability robustness problem the resulting stability robustness problem is shown to be equivalent to a simple algebraic one the solution of which provides the desired necessary and sufficient conditions for performance stability robustness these conditions provide an effective tool for robustness analysis and can be applied to a large class of problems in particular it is shown that some known results can be obtained immediately as special cases of these conditions

robust convergence of two stage nonlinear algorithms for identification in h two stage nonlinear algorithms for the problem of identification in are analyzed in terms of the fourier transform of the window function conditions characterizing the robust convergence of the two stage nonlinear algorithm are derived the sufficiency of the conditions holds for a general class of window functions while the necessity is proved for robust convergence of the first stage of the algorithm under the mild additional restriction that the window function is even symmetric some improved upper bounds for the worst case identification error are also obtained

fast identification n widths and uncertainty principles for lti and slowly varying systems the optimal worst case uncertainty that can be achieved by identification depends on the observation time in the first part of the paper this dependence is evaluated for selected linear time invariant systems in the l sup 1 and h sup spl infin norms and shown to be derivable from a monotonicity principle the minimal time required is shown to depend on the metric complexity of the a priori information set two notions of n width or metric dimension are introduced to characterize this complexity in the second part of the paper the results are applied to systems in which the law governing the evolution of the uncertain elements is not time invariant such systems cannot be identified accurately the inherent uncertainty is bounded in the case of slow time variation the n widths and related optimal inputs provide benchmarks for the evaluation of actual inputs occurring in adaptive feedback systems

pseudo regular oscillations induced by external noise an examination of the effect of noise on a general system at a saddle node bifurcation has revealed that in the limit of weak noise the probability density of the time to pass through the saddle node has a universal shape the specific kinetics of the particular system serving only to set the time scale this probability density is displayed and its salient features are explicated in the case of a saddle node bifurcation leading to relaxation oscillations this analysis leads to the prediction of the existence of noise induced oscillations which appear much less random than might at first be expected the period of these oscillations has a well defined nonzero most probable value the inverse of which is a noise induced frequency this frequency can be detected as a peak in power spectra from numerical simulations of such a system this is the first case of the prediction and detection of a noise induced frequency of which the authors are aware

h sub spl infin identification of multivariable systems by tangential interpolation methods the purpose of this paper is to present an extension to some of the current work on worst case identification problems to multivariable systems we consider an h sub spl infin identification problem for a class of linear shift invariant multi input multi output systems our main results are an interpolatory algorithm and a number of bounds on the identification error this algorithm operates on available input and output data in the time domain and is constructed by solving an extended matrix tangential caratheodory fejer problem similar to its counterpart for scalar systems this interpolatory algorithm possesses certain desirable optimality properties and can be obtained via standard convex programming methods

persistent identification of time varying systems identification of time varying systems especially slowly time varying systems is of importance in the development of a comprehensive theory of adaptation the persistent identification measures employed in this paper capture a main characterization in such identification problems namely one input signal must be used for identification of all possible observation windows this paper establishes several essential features in persistent identification problems which highlight their potential utility in adaptation 1 they have computable upper and lower bounds for typical classes of prior uncertainty sets 2 any full rank n periodic signals are optimal and the simple least squares estimates are optimal identification algorithms 3 optimal probing inputs can be approximately generated in a closed loop configuration when the plant and the controller are slowly time varying and 4 n periodic signals are asymptotically optimal for slowly time varying systems the main results of this paper have been successfully combined with a certain slow h sup spl infin design to derive an adaptive stabilization scheme

least squares type algorithms for identification in the presence of modeling uncertainty the celebrated least squares and lms least mean squares are system identification approaches that are easily implementable need minimal a priori assumptions and have very nice identification properties when the uncertainty in measurements is only due to noises and not due to unmodeled behavior of the system when there is uncertainty present due to an unmodeled part of the system as well however the performance of these algorithms can be poor here the authors propose a modified weighted least squares algorithm that is geared toward identification in the presence of both unmodeled dynamics and measurement disturbances the algorithm uses very little a priori information and is easily implementable in a recursive fashion through an example the authors demonstrate the improved performance of the proposed approach motivated by a certain worst case property of the lms algorithm an h sub spl infin estimation algorithm is also proposed for the same objective of identification in the presence of modeling uncertainty

modeling of normalized coprime factors with spl nu metric uncertainty modeling of uncertain systems with normalized coprime factor description is investigated where the experimental data is given by a finite set of frequency response measurement samples of the open loop plant that is linear time invariant and possibly infinite dimensional the objective is not only to identify the nominal model but also to quantify the modeling error with sup norm bounds in frequency domain the uncertainty to be identified and quantified is chosen as the spl nu metric proposed by vinnicombe 1993 because of its compatibility with h sub spl infin based robust control an algorithm is developed to model the normalized coprime factors of the given plant using techniques of discrete fourier analysis and balanced stochastic truncation and is shown to be robust in the presence of the worst case noise upper bounds are derived for the associated modeling error based on the minimum a priori information of the underlying model set and of the noise level in the measurement data a simulation example is used to illustrate the effectiveness of the proposed algorithm

the gap metric robustness of stabilization of feedback systems in this paper we introduce the gap metric to study the robustness of the stability of feedback systems which may employ not necessarily stable open loop systems we elaborate on the computational aspects of the gap metric and provide upper bounds to the gap in cases where the exact formulas do not apply by admissible uncertainties we mean those which preserve closed loop stability and a specified small tolerance on the i o behavior of a feedback system we show that admissible uncertainties are precisely those which are constrained in the gap finally we conclude that any metric which preserves a continuous relationship between open loop systems and the corresponding stable feedback interconnections must have the topology of the gap metric

input design in worst case system identification with quantized measurements this paper addresses the problem of set membership system identification with quantized measurements following the work developed for binary measurements the problem of optimal input design with multiple sensor thresholds is tackled for a fir model of order n the problem is decomposed into n static gain problems the one step optimal input problem is solved both for equispaced and generic sensor threshold distribution moreover the n step optimal input problem for the case of equispaced thresholds is addressed and a solution is provided under a suitable assumption on the sensor range and resolution the obtained results allow us to construct an upper bound on the time complexity of the fir identification problem for the case of equispaced thresholds numerical application examples are reported to show the effectiveness of the proposed algorithms

a convex analytic approach to system identification this paper introduces a new concept for system identification in order to account for random and nonrandom deterministic set membership uncertainties while random stochastic models are natural for modeling measurement errors nonrandom uncertainties are well suited for modeling parametric and nonparametric components the new concept introduced is distinct from earlier concepts in many respects first inspired by the concept of uniform convergence of empirical means developed in machine learning theory we seek a stronger notion of convergence in that the objective is to obtain probabilistic uniform convergence of model estimates to the minimum possible radius of uncertainty second the formulation lends itself to convex analysis leading to description of optimal algorithms which turn out to be well known instrument variable methods for many of the problems third we characterize conditions on inputs in terms of second order sample path properties required to achieve the minimum radius of uncertainty finally we present fundamental bounds and optimal algorithms for system identification for a wide variety of standard as well as nonstandard problems that include special structures such as unmodeled dynamics positive real conditions bounded sets and linear fractional maps

a paradigm for class identification problems the following problem arises in many applications involving classification identification and inference there is a set of objects x and a particular x spl isin x is chosen unknown to us based on information obtained about x in a sequential manner one wishes to decide whether x belongs to one class of objects a sub 0 or a different class of objects a sub 1 the authors study a general paradigm applicable to a broad range of problems of this type which they refer to as problems of class identification or discernibility they consider various types of information sequences and various success criteria including discernibility in the limit discernibility with a stopping criterion uniform discernibility and discernibility in the cesaro sense they consider decision rules both with and without memory necessary and sufficient conditions for discernibility are provided for each case in terms of separability conditions on the sets a sub 0 and a sub 1 they then show that for any sets a sub 0 and a sub 1 various types of separability can be achieved by allowing failure on appropriate sets of small measure applications to problems in language identification system identification and discrete geometry are discussed

a time domain approach to model validation in this paper we offer a novel approach to control oriented model validation problems the problem is to decide whether a postulated nominal model with bounded uncertainty is consistent with measured input output data our approach directly uses time domain input output data to validate uncertainty models the algorithms we develop are computationally tractable and reduce to generally nondifferentiable convex feasibility problems or to linear programming problems in special cases we give analytical solutions to these problems

reformulation of the parameter identification problem for systems with bounded disturbances abstract the problem of deriving parameter identification algorithms that incorporate prior knowledge of a bound on the disturbance is studied particular emphasis is given to the following aspects firstly to motivate from geometrical and optimization considerations the modifications dead zones required to cope with the disturbances secondly to guarantee the convergence properties which are essential in the establishment of stability of identification based adaptive control systems projection and least squares algorithms are studied in both cases an interpretation is given to the new dead zone that has proven to be a key ingredient in recent robustness studies e g goodwin et al 1986 automatica 22 199 207 kreisselmeier and anderson 1985 ieee trans aut control 31 127 133 and lozano leal and ortega 1986 2nd ifac workshop on adapt syst in control and signal proc lund sweden

estimation of model quality this paper provides an introduction to recent work on the problem of quantifying errors in the estimation of models for dynamic systems this is a very large field we therefore concentrate on approaches that have been motivated by the need for reliable models for control system design this will involve a discussion of efforts that go under the titles of estimation in th worst case estimation estimation in l1 and stochastic embedding of undermodelling a central theme of this survey is to examine these new methods with reference to the classic bias variance tradeoff in model structure selection

optimal rejection of persistent disturbances robust stability and mixed sensitivity minimization the problem of optimal disturbance rejection of bounded persistent disturbances is solved in the general nonsquare case the minimum value of the objective function can be obtained by solving a semi infinite linear programming problem and an iterative procedure for obtaining approximate solutions is introduced application of the l sup 1 optimal problem to robustness is discussed a mixed sensitivity problem is formulated and shown to guarantee good disturbance rejection in the presence of plant perturbations

the sample complexity of worst case identification of fir linear systems we consider the problem of identification of linear systems in the presence of measurement noise which is unknown but bounded in magnitude by some 0 we focus on the case of linear systems with a finite impulse response it is known that the optimal identification error is related within a factor of 2 to the diameter of a so called uncertainty set and that the latter diameter is upper bounded by 2 if a sufficiently long identification experiment is performed we establish that for any k 1 the minimal length of an identification experiment that is guaranteed to lead to a diameter bounded by 2k behaves like 2nf 1k when n is large where n is the length of the impulse response and is a positive function known in closed form while the framework is entirely deterministic our results are proved using probabilistic tools

lethargy results in lti system modelling modelling of linear time invariant lti systems is studied in an approximate modelling framework motivated by modern robust control theory as convolution type lti systems are represented by their unit impulse responses i e through a sequence of real complex numbers this paper provides also system theoretic results on modelling of sequences and their associated transfer functions the modelling results derived in this paper are mostly of lethargy type i e they establish limitations in approximate modelling of lti systems non existence of coprime factorizations is established for a number of different types of lti systems this leads to the somewhat surprising conclusion that many lti systems are not even stabilizable and that an even larger class of lti systems cannot be well approximated by rational transfer function models in addition it is discussed that often the rate of rational approximation of lti systems is not essentially better than the rate of approximation in some basis such as finite impulse response fir models furthermore different definitions of the important finite power signal setup are studied and new results given

time complexity and model complexity of fast identification of continuous time lti systems the problem of fast identification of continuous time systems is formulated in the metric complexity theory setting it is shown that the two key steps to achieving fast identification i e optimal input design and optimal model selection can be carried out independently when the true system belongs to a general a priori set these two optimization problems can be reduced to standard gel fand and kolmogorov n width problems in metric complexity theory it is shown that although arbitrarily accurate identification can be achieved on a small time interval by reducing the noise signal ratio and designing the input carefully identification speed is limited by the metric complexity of the a priori uncertainty set when the noise signal ratio is fixed

intermittent diffusion in the presence of noise abstract the diffusion coefficient for intermittent diffusion in the presence of weak additive gaussian white noise is calculated the recent result by tsimring physica d 63 1993 41 is shown to be correct only in a certain limit of the parameter values

critical diffusion behavior of a climbing sine map near intermittency threshold abstract the diffusion in a climbing sine map near intermittency threshold is considered a simple description is proposed for particles dispersion at small and moderate times when the model reveals a crossover from anomalous to normal diffusion and shows oscillations in the time dependence of low moments of the probability distribution function for larger times when flights of particles decorrelate a different approach is employed that allows one also to take into account the external noise a universal scaling function is found describing the critical behaviour of the diffusion constant under external noise near the intermittency threshold the results of numerical simulations are in agreement with the analytical formulas

estimation theory and uncertainty intervals evaluation in presence of unknown but bounded errors linear families of models and estimators the problem of parameter estimation and of the evaluation of related uncertainty intervals is considered in the case that a probabilistic description of noise and errors is not available of suitable but only a bound on them is known in the present paper only linear parametrizations and estimators are considered very simple and computationally feasible algorithms are derived for evaluating two different types of uncertainty intervals estimates uncertainty intervals parameter uncertainty intervals the relationships between the eui s and the pui s are established and the solution to the problem of the minimum uncertainty intervals estimator is given the latter can be obtained by means of a simple linear programming algorithm

robust approximate modelling of stable linear systems robust approximation and worst case approximate modelling of stable shift invariant systems from corrupted transfer function estimates are studied in the h sense connections between the problem formulations of the present work and certain problems of worst case system identification notably the helmicki jacobson nett problem formulation for identification in h are established issues of model set selection are addressed using the n width concept a concrete result establishes a priori knowledge for which a certain rational model set is optimal in then width sense a notion of robust convergence is defined so that any untuned approximation method satisfying it has a generic well posedness property for systems in the disk algebra the existence of robustly convergent approximation methods based on any complete model set in the disk algebra is shown in a constructive way a framework is given in which approximate models can be obtained as stable perturbations of the true system these can be combined

a necessary and sufficient condition for robust bibo stability abstract this paper is concerned with obtaining necessary and sufficient conditions for robustly stabilizing a class of plants characterized by a known linear shift invariant plant with additive perturbations of the form of possibly nonlinear time varying l stable operators we will show that in some sense the small gain theorem is necessary and sufficient even when the perturbations are restiricted to a class of linear operators

a framework for robust parametric set membership identification this paper proposes a new framework for studying robust parametric set membership identification the authors derive some new results on the fundamental limitations of algorithms in this framework given a particular model structure the new idea is to quantify uncertainty only with respect to the finite dimensional parametric part of the model and not the fixed size unmodeled dynamics thus the measure of uncertainty is different from the measures used in previous robust identification work where system norms are used to quantify uncertainty as an example the results are used to assess the fidelity of a certain approximate robust parametric set membership identification algorithm