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

robust identification from band limited data consider the problem of identifying a scalar bounded input bounded output stable transfer function from pointwise measurements at frequencies within a bandwidth we propose an algorithm which consists of building a sequence of maps from data to models converging uniformly to the transfer function on the bandwidth when the number of measurements goes to infinity the noise level to zero and asymptotically meeting some gauge constraint outside error bounds are derived and the procedure is illustrated by numerical experiments

on uniform asymptotic stability of time varying parameterized discrete time cascades recently a framework for controller design of sampled data nonlinear systems via their approximate discrete time models has been proposed in the literature in this paper we develop novel tools that can be used within this framework and that are useful for tracking problems in particular results for stability analysis of parameterized time varying discrete time cascaded systems are given this class of models arises naturally when one uses an approximate discrete time model to design a stabilizing or tracking controller for a sampled data plant while some of our results parallel their continuous time counterparts the stability properties that are considered the conditions that are imposed and the the proof techniques that are used are tailored for approximate discrete time systems and are technically different from those in the continuous time context a result on constructing strict lyapunov functions from nonstrict ones that is of independent interest is also presented we illustrate the utility of our results in the case study of the tracking control of a mobile robot this application is fairly illustrative of the technical differences and obstacles encountered in the analysis of discrete time parameterized systems

a nearly interpolatory algorithm for h sub spl infin identification with mixed time and frequency response data a new algorithm is proposed for h sub spl infin identification with mixed time and frequency response data this algorithm shares similar properties to interpolatory algorithms and is nearly interpolatory meanwhile it greatly reduces the computational complexity of the interpolatory algorithm worst case identification error bounds are also characterized and derived for this nearly interpolatory algorithm a numerical example is used to illustrate the proposed identification algorithm

all optimal hankel norm approximations of linear multivariable systems and their l error bounds the problem of approximating a multivariable transfer function g s of mcmillan degree n by s of mcmillan degree k is considered a complete characterization of all approximations that minimize the hankel norm is derived the solution involves a characterization of all rational functions s f s that minimize where s has mcmillan degree k and f s is anticavisal the solution to the latter problem is via results on balanced realizations all pass functions and the inertia of matrices all in terms of the solutions to lyapunov equations it is then shown that where k 1 g s is the k l st hankel singular value of g s and for one class of optimal hankel norm approximations the method is not computationally demanding and is applied to a 12 state model

robust identification and interpolation in h abstract we consider system identification in h in the framework proposed by helmicki jacobson and nett an algorithm using the jackson polynomials is proposed that achieves an exponential convergence rate for exponentially stable systems it is shown that this and similar identification algorithms can be successfully combined with a model reduction procedure to produce low order models connections with the nevanlinna pick interpolation problem are explored and an algorithm is given in which the identified model interpolates the given noisy data some numerical results are provided for illustration finally the case of unbounded random noise is discussed and it is shown that one can still obtain convergence with probability 1 under natural assumptions

worst case system identification in l 1 ne equation 1 optimal algorithms and error bounds abstract this paper analyses worst case identification of linear shift invariant systems with an l 1 error criterion the assumed apriori information on the plant are bounds on the system gain and decay rate of the impulse response of the unknown system and the experimental data is assumed to be a finite set of corrupted impulse or step response samples of the system

robust identification in the disc algebra using rational wavelets and orthonormal basis functions worst case identification using corrupted frequency response measurements is considered in a framework which allows one to consider a variety of rational basis functions these include laguerre models kautz models and rational wavelets conditions are given for model sets to be complete in the disc algebra and examples are given explicit error bounds are provided for approximate modelling using specific basis functions

a globally optimal minimax solution for spectral overbounding and factorization in this paper an algorithm is introduced to find a minimum phase transfer function of specified order whose magnitude tightly overbounds a specified real valued nonparametric function of frequency this method has direct application to transforming nonparametric uncertainty bounds available from system identification experiments and or plant modeling into parametric representations required for modern robust control design software i e a minimum phase transfer function multiplied by a norm bounded perturbation

coprime fractional representations and stability of non linear feedback systems fractional representation methods for finite dimensional linear time invariant fdlti systems have been developed in an algebraic framework of principal ideal domains the concept of coprime fractional representations has been generalized to discrete time non linear systems by hammer 1985 1987 by viewing the bezout identity as the basis of the definition of coprimeness in this paper we view the role of right coprime fractional representation in representing the set of all stable input output pairs of a possibly unstable non linear system this notion of coprimeness is shown to be equivalent to that of fdlti systems using coprime fractional representations of non linear systems we obtain a characterization of the stability of non linear feedback systems this characterization is similar to that obtained in the case of fdlti systems we also obtain a parameterization of all non linear and time varying stabilizing controllers for linear possibly time varying plants which has the same form as t

the model matching problem for a class of polynomial non linear discrete input output systems with cross products an algorithmic approach in this paper we examine the model matching problem that concerns non linear input output discrete systems containing products among delays of input and output signals through a special factorisation the algebraic framework of e operators and the star product that we adopt describe these systems moreover a certain procedure that resembles the euclidean division allows us to discover the linear factors of those systems with respect to the above mentioned operations the entire approach is symbolically algorithmic and involves the use of suitable software

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

time domain identification for robust control abstract in this paper we are concerned with a problem of robust control oriented system identification in the time domain based on the well known schur takagi aak theorem we propose a linear algorithm to obtain the nominal model of the plant to be identified and the minimal bound of the uncertainty of the nominal model error which is measured by h norm it is also shown that in the model set defined by the nominal model and the uncertainty bound there exists at least one model which matches the prescribed input output data given in the time domain

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

hard frequency domain model error bounds from least squares like identification techniques the problem of deriving so called hard error bounds for estimated transfer functions is addressed a hard bound is one that is sure to be satisfied i e the true system nyquist plot will be confined with certainty to a given region provided that the underlying assumptions are satisfied by blending a priori knowledge and information obtained from measured data it is shown how the uncertainty of transfer function estimates can be quantified the emphasis is on errors due to model mismatch the effects of unmodeled dynamics can be considered as bounded disturbances hence techniques from set membership identification can be applied to this problem the approach taken corresponds to weighted least squares estimation and provides hard frequency domain transfer function error bounds the main assumptions used in the current contribution are that the measurement errors are bounded that the true system is indeed linear with a certain degree of stability and that there is some knowledge about the shape of the true frequency response

subspace based multivariable system identification from frequency response data two noniterative subspace based algorithms which identify linear time invariant mimo multi input multioutput systems from frequency response data are presented the algorithms are related to the recent time domain subspace identification techniques the first algorithm uses equidistantly in frequency spaced data and is strongly consistent under weak noise assumptions the second algorithm uses arbitrary frequency spacing and is strongly consistent under more restrictive noise assumptions promising results are obtained when the algorithms are applied to real frequency data originating from a large flexible structure

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

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

feedback bounded stabilisation of certain discrete volterra systems abstractthroughout this paper we present a method for designing feedback laws which stabilise nonlinear discrete volterra systems our method is based on a factorisation algorithm which decomposes the original system as a composition of a polynomial and a linear series

reversible linear and nonlinear discrete time dynamics it is shown that any causal discrete time system can be realized with a reversible or invertible state variable representation this bridges the gap with continuous time where flows always satisfy this property and settles an old debate on the relevance of invertibility in nonlinear discrete dynamics as the result applies to the linear case it is shown how the classic approach should be slightly modified and several examples of the benefits that can be obtained are given difference algebra is employed for nonlinear systems it leads to a most elegant characterization of causality which enables reversibility to be demonstrated module theory suffices in the linear case which is supplemented by elementary matrix manipulations

finite input output representative of a class of volterra polynomial systems n this paper the problem of finite input output representation of a special class of nonlinear volterra polynomial systems is studied via the notion of linear factorization of d series this is an algebraic method based mainly on the star product operation and on a related euclidean type algorithm

a new factorization of special nonlinear discrete systems and their applications in this paper a new factorization of discrete nonlinear systems is presented this factorization is based on the star product an operation corresponding to the linear connection of systems a relevant algorithm is developed based on a theoretical background finally some applications to system analysis bibo stability and feedback design are provided

fraction representations of non linear systems a simplified approach abstract a non linear system has a right fraction representation if it can be represented as pq 1 where p and q are stable systems and it has a left fraction representation if it can be represented as g 1 t where g and t are stable systems we develop here a theory of right and of left fraction representations for discrete time non linear systems with bounded input sequences we indicate the connection between fraction representations and the stabilization problem for non linear systems

a generalized kaplan meier estimator in the theory of competing risks the nonparametric kaplan meier estimator plays an important role in this paper a bivariate nonparametric estimator for the competing risk problem is given which for the special case of independent causes gives the kaplan meier estimator this paper also introduces a matrix w through which dependent models for the competing problem can be studied these results also indicate the special case on the matrix w for which the bounds to the survival function given by peterson 4 can be obtained

algorithms for identification in h with unequally spaced function measurements abstract worst case identification in h is considered in the situation in which corrupted frequency response measurements are available at an arbitrary set of frequencies two new classes of algorithms are presented one yields polynomial models directly the other is a two stage algorithm producing rational models each has improved convergence rates for the class of exponentially stable discrete time systems with errors typically o e o r for arbitrarily large r where e is the noise level and is the maximum spacing between identification points a numerical example is given

identification of rational approximate models in h sup spl infin using generalized orthonormal basis robust identification with fir models fails to be successful when the number of coefficients to be estimated becomes large caused by lightly damped modes of the plant or poles very close to the unit circle the paper presents a two stage algorithm to obtain a low order approximate model in frequency domain in a generalized orthonormal basis with guaranteed h sup spl infin error bound for deterministic linear time invariant stable systems the first stage being an l sup spl infin rational approximation and the second nonlinear step being an h sup spl infin rational approximation

the model matching problem for a certain class of nonlinear systems sufficient conditions are described for the solution of the model matching problem for a specific class of nonlinear systems represented by the so called polynomials and linear desired models an algorithm yielding causal controllers and a bibo closed loop connection is supplied

h sub spl infin identification and model quality evaluation set membership h sub spl infin identification is investigated using time domain data and mixed parametric and nonparametric models as well as supposing power bounded measurement errors the problem of optimally estimating the unknown parameters and evaluating the minimal worst case identification error called radius of information is solved for classes of models affine in the parameters the radius of information is obtained as function of the h sub spl infin norm of the unmodeled dynamics a method is given for estimating this norm from the available data and some general a priori information on the unmodeled dynamics thus allowing the actual evaluation of the radius of information the radius represents a measure of the predictive ability of the considered class of models and it is then used for comparing the quality of different classes of models and for the order selection of their parametric part the effectiveness of the proposed procedure is tested on some numerical examples and compared with standard statistical criteria for model order selection

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

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