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

closed loop model set validation under a stochastic framework this paper deals with probabilistic model set validation it is assumed that the dynamics of a multi input multi output mimo plant is described by a model set with unstructured uncertainties and identification experiments are performed in closed loop a necessary and sufficient condition has been derived for the consistency of the model set with both the stabilizing controller and closed loop frequency domain experimental data fded in this condition only the euclidean norm of a complex vector is involved and this complex vector depends linearly on both the disturbances and the measurement errors based on this condition an analytic formula has been derived for the sample unfalsified probability sup of the model set some of the asymptotic statistical properties of the sup have also been briefly discussed a numerical example is included to illustrate the efficiency of the suggested method in model set quality evaluation

on stabilization and the existence of coprime factorizations shows that any transfer function matrix whose elements belong to the quotient field of h sub infinity and which is stabilizable has a matrix fraction representation over h sub infinity which is coprime in the sense that a matrix bezout identity can be satisfied

robust identification in h abstract we consider system identification in the banach space h in the framework proposed by helmicki jacobson and nett it is shown that there is no robustly convergent linear algorithm for identifying exponentially stable systems in the presence of noise which is not tuned to prior information about the unknown system or noise various nonlinear algorithms some closely related to one of gu and khargonekar are analysed and results on trigonometric interpolation used to provide new error bounds an application of these techniques to approximation is given and finally some numerical results are provided for illustration

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

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

worst case analysis of the least squares method and related identification methods abstract we consider worst case analysis of system identification under less restrictive assumptions on the noise than the l bounded error condition it is shown that the least squares method has a robust convergence property in l 2 identification but lacks a corresponding property in l 1 identification as well as in all other non hilbert space settings the latter result is in stark contrast with typical results in asymptotic stochastic analysis of the least squares method furthermore it is shown that the khintchine inequality is useful in the analysis of least l p identification methods

on input design in l conditional set membership identification this paper deals with input design in conditional set membership identification the problem is how to choose the input signal in order to minimize the global worst case identification error a characterization of the identification error is provided showing that the optimal input is the one that minimizes the radius of a suitable set moreover sufficient conditions under which the impulse input is optimal are provided

optimal estimation theory for dynamic systems with set membership uncertainty an overview in many problems such as linear and nonlinear regressions parameter and state estimation of dynamic systems state space and time series prediction interpolation smoothing and functions approximation one has to evaluate some unknown variable using available data the data are always associated with some uncertainty and it is necessary to evaluate how this uncertainty affects the estimated variables typically the problem is approached assuming a probabilistic description of uncertainty and applying statistical estimation theory an interesting alternative approach referred to as set membership or unknown but bounded ubb error description has been investigated since the late 1960s in this approach uncertainty is described by an additive noise which is known only to have given integral typically l2 of l1 or componentwise l bounds in this paper we review the main results of this theory with special attention to the most recent advances obtained in the case of componentwise bounds

robust stabilization bibo stability distance notions and robustness optimization abstract this paper studies robust stabilization of both linear shift invariant causal systems in an lp setting and linear time invariant causal continuous time systems in an lp p 1 or setting two key technical results in the paper establish the existence of lp and lp stable normalized coprime factorizations for discrete time and continuous time systems respectively which have coprime factorizations as lp and lp stable operators several distance measures for systems are then introduced including the graph metric the function the gap between the graphs of the systems and the projection gap it is shown that these distance measures lead to the weakest convergence notions for systems for which closed loop stability is a robust property the function can be computed using the dahleh pearson theory for l1 l1 optimal control robustness optimization in a directed p function is shown to be closely related to robustness optimization for bibo stable normalized coprime factor perturbations this result connects the stability margin of dahleh for coprime factor perturbations to the function these considerations are further supported by a robustness result in terms of the projection gap

sample complexity of worst case h identification abstract we show that the sample complexity of qorst case h identification is of order n 2 by proving that the minimal length of a fractional h cover for c n regarded as the linear space of complex valued sequences of length n is of order n 2 a unit vector u in l is a fractional h cover for c n if for some 0 h h for all rh c n where h z j 0 n 1 h j z j is the z transform of h we also give similar results for real valued sequences

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

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

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

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

on geometrical properties of preconditioners in ipms for classes of block angular problems one of the most efficient interior point methods for some classes of block angular structured problems solves the normal equations by a combination of cholesky factorizations and preconditioned conjugate gradient for respectively the block and linking constraints in this work we show that the choice of a good preconditioner depends on geometrical properties of the constraint structure in particular the principal angles between the subspaces generated by the diagonal blocks and the linking constraints can be used to estimate ex ante the efficiency of the preconditioner numerical validation is provided with some generated optimization problems an application to the solution of multicommodity network flow problems with nodal capacities and equal flows of up to 64 million variables and up to 7 9 million constraints is also presented these computational results also show that predictor corrector directions combined with iterative system solves can be a competitive option for large instances

finite sample properties of linear model identification in this paper we consider the finite sample properties of prediction error methods using a quadratic criterion function for system identification the problem we pose is how many data points are required to guarantee with high probability that the expected value of the quadratic identification criterion is close to its empirical mean value the sample sizes are obtained using risk minimization theory which provides uniform probabilistic bounds on the difference between the expected value of the squared prediction error and its empirical mean evaluated on a finite number of data points the bounds are very general no assumption is made about the true system belonging to the model class and the noise sequence is not assumed to be uniformly bounded further analysis shows that in order to maintain a given bound on the deviation the number of data points needed grows no faster than quadratically with the number of parameters for fir and arx models

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

identification of stabilizable systems closed loop approximation approximate modelling and identification of linear shift invariant possibly unstable discrete time systems or plants is studied in a framework compatible with the so called robust stability concept for feedback systems this unified framework is based on approximate modelling of the plant in the gap and graph metrics which is achieved here through approximation of certain closed loop transfer functions by finite dimensional systems properties of this approximate inverse modelling approach are studied and concrete rate of approximation results are given furthermore a consistency result in the gap and graph metrics is given for a certain experimental estimate of the plant constructed from closed loop input output data in a stationary noise set up under mild conditions on the unknown plant

new preconditioners for kkt systems of network flow problems we propose a new set of preconditioners for the iterative solution via a preconditioned conjugate gradient pcg method of the kkt systems that must be solved at each iteration of an interior point ip algorithm for the solution of linear min cost flow mcf problems these preconditioners are based on the idea of extracting a proper triangulated subgraph of the original graph which strictly contains a spanning tree we define a new class of triangulated graphs called brother connected trees bcts and discuss some fast heuristics for finding bcts of large weight computational experience shows that the new preconditioners can complement tree preconditioners outperforming them both in iterations count and in running time on some classes of graphs

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

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

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

an investigation of interior point algorithms for the linear transportation problem recently resende and veiga it siam j optim 3 1993 pp 516 537 proposed an efficient implementation of the dual affine da interior point algorithm for the solution of linear transportation models with integer costs and right hand side coefficients this procedure incorporates a preconditioned conjugate gradient pcg method for solving the linear system that is required in each iteration of the da algorithm in this paper we introduce an incomplete qr decomposition iqrd preconditioning for the pcg algorithm computational experience shows that the iqrd preconditioning is appropriate in this instance and is more efficient than the preconditioning introduced by resende and veiga we also show that the primal dual pd and the predictor corrector pc interior point algorithms can also be implemented by using the same type of technique a comparison among these three algorithms is included and indicates that the pd and pc algorithms are more appropriate for the solution of transportation problems with well scaled cost and right hand side coefficients and assignment problems with poorly scaled cost coefficients on the other hand the da algorithm seems to be more efficient for assignment problems with well scaled cost coefficients and transportation problems whose cost coefficients are badly scaled

brief on robustness in system identification this paper studies robustness issues in system identification specifically a general framework for robust convergence analysis is given for identification of stable linear time invariant systems this is used to derive a rather complete set of results on robust convergence under average l p norms orlicz norms and cross correlation type noise semi norms the new theory is used to discuss and to study robustness issues in identification of linear time invariant models when the true plant is not exactly linear robustness issues in system identification against unmodelled nonlinear dynamics are important as real plants exhibit typically at best nearly linear dynamics the results show clearly that the effects of unmodelled dynamics are typically more serious than those of random noise

identification and adaptive control towards a complexity based general theory abstract two recent developments will be surveyed here which are pointing the way towards an input output theory of h l 1 adaptive feedback the solution of problems involving 1 feedback performance exact optimization under large plant uncertainty on the one hand the two disc problem of h and 2 optimally fast identification in h on the other taken together these are yielding adaptive algorithms for slowly varying data in h l 1 at a conceptual level these results motivate a general input output theory linking identification adaptation and control learning in such a theory the definition of adaptation is based on system performance under uncertainty and is independent of internal structure presence or absence of variable parameters or even feedback

graph topology and gap topology for unstable systems a reformation is provided of the graph topology and the gap topology for a general setting including lumped linear time invariant systems and distributed linear time invariant systems in the frequency domain some essential properties and their comparisons are clearly presented in the reformulation it is shown that the gap topology is suitable for general systems rather than square systems with unity feedback it is shown that whenever an unstable plant can be stabilized by feedback it is a closed operator mapping a subspace of the input space to the output space hence the gap topology can always be applied whenever the unstable plant can be stabilized the graph topology and the gap topology are suitable for different subsets of systems and have many similar characteristics if one confines them to the same subset they will be identical the definitions of the graph metric and the gap metric are discussed

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

an improved algorithm and its parallel implementation for solving a general blood tissue transport and metabolism model fast algorithms for simulating mathematical models of coupled blood tissue transport and metabolism are critical for the analysis of data on transport and reaction in tissues here by combining the method of characteristics with the standard grid discretization technique a novel algorithm is introduced for solving a general blood tissue transport and metabolism model governed by a large system of one dimensional semilinear first order partial differential equations the key part of the algorithm is to approximate the model as a group of independent ordinary differential equation ode systems such that each ode system has the same size as the model and can be integrated independently thus the method can be easily implemented in parallel on a large scale multiprocessor computer the accuracy of the algorithm is demonstrated for solving a simple blood tissue exchange model introduced by sangren and sheppard w c sangren c w sheppard a mathematical derivation of the exchange of a labeled substance between a liquid flowing in a vessel and an external compartment bull math biophys 15 1953 387 394 which has an analytical solution numerical experiments made on a distributed memory parallel computer an hp linux cluster and a shared memory parallel computer a sgi origin 2000 demonstrate the parallel efficiency of the algorithm

robust identification of strongly stabilizable systems for strongly stabilizable systems for which a strongly stabilizing controller is known approximately the authors consider system identification in the graph gap and chordal metrics using robust h sub infinity identification of the closed loop transfer function in the framework proposed by a j helmicki et al 1990 error bounds are derived showing that robust convergence is guaranteed and that the identification can be satisfactorily combined with a model reduction step two notions of robust identification of stable systems are compared and an alternative robust identification technique based on smoothing which can be used to yield polynomial models directly is developed

stiff differential equations solved by radau methods radau iia methods are successful algorithms for the numerical solution of stiff differential equations this article describes radau a new implementation of these methods with a variable order strategy the paper starts with a survey on the historical development of the methods and the discoveries of their theoretical properties numerical experiments illustrate the behaviour of the code

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

worst case identification in banach spaces a general formulation of worst case identification of linear systems in a banach space framework is discussed this enables one to prove generalizations of various existence theorems from the theory of robust l1 and h identification to a variety of other situations

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

analysis of linear methods for robust identification in e 1 we consider worst case analysis of system identification by means of the linear algorithms such as least squares we provide estimates for worst case and average errors showing that worst case robust convergence cannot occur in the l1 identification problem the case of periodic inputs is also analysed finally a pseudorandomness assumption is introduced that allows more powerful convergence results in a deterministic framework

simple linear time algorithms to test chordality of graphs test acyclicity of hypergraphs and selectively reduce acyclic hypergraphs chordal graphs arise naturally in the study of gaussian elimination on sparse symmetric matrices acyclic hypergraphs arise in the study of relational data bases rose tarjan and lueker siam j c

on the computation of the gap metric abstract this paper deals with the computation of the gap metric introduced by zames and el sakkary 17 it is shown that the gap between two systems p1 p2 is precisely the maximum of the two expressions inf q h d i n i d j n j q for i j equal to 1 2 and 2 1 and ni di being normalized right coprine factorizations of pi i 1 2 in the sense of vidyasagar 12 this expression is computable using well known techniques from interpolation theory

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

brief robustness in h identification we consider identification in both the time domain for suitable choices of inputs and the frequency domain using a deterministic formulation that is less conservative than the usual worst case approach in that it allows for a certain number of outliers two new identification methods the orlicz and the top k method respectively are proposed and analysed theoretical h error bounds for the identified models are provided and the proposed methods are illustrated by a numerical example

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

accelerated multigrid for graph laplacian operators we consider multigrid type techniques for the numerical solution of large linear systems whose coefficient matrices show the structure of weighted graph laplacian operators we combine ad hoc coarser grid operators with iterative techniques used as smoothers empirical tests suggest that the most effective smoothers have to be of krylov type with subgraph preconditioners while the projectors which define the coarser grid operators have to be designed for maintaining as much as possible the graph structure of the projected matrix at the inner levels the main theoretical contribution of the paper is the characterization of necessary and sufficient conditions for preserving the graph structure in this framework it is possible to explain why the classical projectors inherited from differential equations are good in the differential context and why they may behave unsatisfactorily for unstructured graphs furthermore we report and discuss several numerical experiments showing that our approach is effective even in very difficult cases where the known approaches are rather slow as a conclusion the main advantage of the proposed approach is the robustness since our multigrid type technique behaves uniformly well in all cases without requiring either the setting or the knowledge of critical parameters as it happens when using the best known preconditioned krylov methods

bibo stability robustness in the presence of coprime factor perturbations a necessary and sufficient condition for robustly stabilizing a family of plants described by perturbations of fixed coprime factors of a plant is given the computation of the largest stability margin is discussed via solving a nonsquare l sup 1 optimal control problem an algorithm for obtaining lower approximations of the minimum value of the optimization problem mu sub 0 is proposed this together with the standard algorithm which provides upper approximations allows mu sub 0 to be computed within any degree of accuracy

on the worst case divergence of the least squares algorithm in this paper we provide a h norm lower bound on the worst case identification error of least squares estimation when using fir model structures this bound increases as a logarithmic function of model complexity and is valid for a wide class of inputs characterized as being quasi stationary with covariance function falling off sufficiently quickly

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

improving an interior point approach for large block angular problems by hybrid preconditioners abstract the computational time required by interior point methods is often dominated by the solution of linear systems of equations an efficient specialized interior point algorithm for primal block angular problems has been used to solve these systems by combining cholesky factorizations for the block constraints and a conjugate gradient based on a power series preconditioner for the linking constraints in some problems this power series preconditioner resulted to be inefficient on the last interior point iterations when the systems became ill conditioned in this work this approach is combined with a splitting preconditioner based on lu factorization which works well for the last interior point iterations computational results are provided for three classes of problems multicommodity flows oriented and nonoriented minimum distance controlled tabular adjustment for statistical data protection and the minimum congestion problem the results show that in most cases the hybrid preconditioner improves the performance and robustness of the interior point solver in particular for some block angular problems the solution time is reduced by a factor of 10

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

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

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

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

an approach to identification for robust control given measured data generated by a discrete time linear system we propose a model consisting of a linear time invariant system affected by norm bounded perturbation under mild assumptions the plants belonging to the resulting uncertain family form a convex set the approach depends on two key parameters an a priori given bound of the perturbation and the input used to generate the data it turns out that the size of the uncertain family can be reduced by intersecting the model families obtained by making use of different inputs the model validation problem in this identification scheme is analyzed for a given energy level the invalidation problem yields the family of those models which can never be invalidated for any possible input of fixed energy and any possible perturbation this leads to the intersection of all uncertain families a consequence of the invalidation problem is that for finite length measurements not all models can be invalidated using fixed energy inputs

worst case identification in e 2 linear and nonlinear algorithms abstract we consider worst case identification in the l2 norm given an unknown system h l1 one wishes to choose bounded inputs u l such that given finitely many corrupted output measurements y k 0 k of y h u where is noise assumed small one can construct an approximation g with g h 2 0 as 0 and n it is shown that inputs can be chosen such that to identify a sequence of length n with an l2 error of o one requires only o n measurements a numerical example is inclu

some contributions to fixed distribution learning theory we consider some problems in learning with respect to a fixed distribution we introduce two new notions of learnability these are probably uniformly approximately correct puac learnability which is a stronger requirement than the widely studied pac learnability and minimal empirical risk mer learnability which is a stronger requirement than the previously defined notions of solid or potential learnability it is shown that although the motivations for defining these two notions of learnability are entirely different these two notions are in fact equivalent to each other and in turn equivalent to a property introduced here referred to as the shrinking width property it is further shown that if the function class to be learned has the property that empirical means converge uniformly to their true values then all of these learnability properties hold in the course of proving conditions for these forms of learnability we also obtain a new estimate for the vc dimension of a collection of sets obtained by performing boolean operations on a given collection this result is of independent interest we consider both the case in which there is an underlying target function as well as the case of model free or agnostic learning finally we consider the issue of representation of a collection of sets by its subcollection of equivalence classes it is shown by example that by suitably choosing representatives of each equivalence class it is possible to affect the property of uniform convergence of empirical probabilities

h and bibo stabilization of delay systems of neutral type frequency domain tests for the h and bibo stability of large classes of delay systems of neutral type are derived the results are applied to discuss the stabilizability of such systems by finite dimensional controllers

approximation of unstable infinite dimensional systems using coprime factors abstract the effectiveness of comprime factor techniques in l 2 and l model reduction of unstable linear systems is analysed asymptotic estimates are given of the achievable error in the stable and unstable parts of the approximate system measured in a number of different norms some involving the associated hankel operators the chordal metric is introduced as a measure of approximation and is shown to yield the graph topology finally it is deduced that asymptotically optimal l 2 and l convergence rates can be obtained for a large class of unstable systems

a numerical method for stability windows and unstable root locus calculation for linear fractional time delay systems this paper aims to provide a numerical algorithm able to locate all unstable poles and therefore the characterization of the stability as a function of the delay for a class of linear fractional order neutral systems with multiple commensurate delays we start by giving the asymptotic position of the chains of poles and the conditions for their stability for a small delay when these conditions are met the root continuity argument and some simple substitutions allow us to determine the locations where some roots cross the imaginary axis providing therefore the complete characterization of the stability windows the same method can be extended to provide the position of all unstable poles as a function of the delay

software and algorithms for sensitivity analysis of large scale differential algebraic systems sensitivity analysis for dae systems is important in many engineering and scientific applications the information contained in the sensitivity trajectories is useful for parameter estimation optimization model reduction and experimental design in this paper we present algorithms and software for sensitivity analysis of large scale dae systems of index up to two the new software provides for consistent initialization of the solutions and the sensitivities interfaces seamlessly with automatic differentiation for the accurate evaluation of the sensitivity equations and is capable via mpi of exploiting the natural parallelism of sensitivity analysis as well as providing an efficient solution in sequential computations

a specialized interior point algorithm for multicommodity network flows despite the efficiency shown by interior point methods in large scale linear programming they usually perform poorly when applied to multicommodity flow problems the new specialized interior point algorithm presented here overcomes this drawback this specialization uses both a preconditioned conjugate gradient solver and a sparse cholesky factorization to solve a linear system of equations at each iteration of the algorithm the ad hoc preconditioner developed by exploiting the structure of the problem is instrumental in ensuring the efficiency of the method an implementation of the algorithm is compared to state of the art packages for multicommodity flows the computational experiments were carried out using an extensive set of test problems with sizes of up to 700 000 variables and 150 000 constraints the results show the effectiveness of the algorithm

worst case analysis of identification bibo robustness for closed loop data this paper deals with the worst case analysis of identification of linear shift invariant possibly infinite dimensional systems a necessary and sufficient input richness condition for the existence of robustly convergent identification algorithms in l sup 1 is given a closed loop identification setting is studied to cover both stable and unstable but bibo stabilizable systems identification or modeling error is then measured by distance functions which lead to the weakest convergence notions for systems such that closed loop stability in the sense of bibo stability is a robust property worst case modeling error bounds in several distance functions are included

l 1 optimal feedback controllers for mimo discrete time systems the problem considered in this paper is the design of a closed loop system consisting of a mimo discrete time plant and compensator in such a way that the system is internally stable and optimally tracks all persistent bounded inputs the solution consists of two parts first the calculation of the minimum value of the performance index a weighted transfer function which is done by solving a linear programming problem and second construction of the optimal transfer function by solving a set of linear equations it is shown that in general the discrete time problem will have a rational solution and thus appears to be of considerable practical significance

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

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

bezout factors and l sup 1 optimal controllers for delay systems using a two parameter compensator scheme the authors consider the simultaneous problem of optimal robust stabilization and optimal tracking for single input single output siso systems in an l sup spl infin setting using a two parameter compensator scheme optimal robustness is linked to the work done by georgiou and smith 1992 in the l sup 2 setting optimal tracking involves the resolution of l sup 1 optimization problems the authors consider in particular the robust control of delay systems they determine explicit expressions of the bezout factors for general delay systems which are in the callier desoer class spl bscr spl circ 0 finally they solve several general l sup 1 optimization problems and give an algorithm to solve the optimal robust control problem for a large class of delay systems

a combinatorial interior point method for network flow problems for solving minimum cost flow problems we develop a combinatorial interior point method based on a variant of the algorithm of karmarkar described in gonzaga 3 4 gonzaga proposes search directions generated by projecting certain directions onto the nullspace ofa by the special combinatorial structure of networks any projection onto the nullspace ofa can be interpreted as a flow in the incremental graph ofg in particular to evaluate the new search direction it is sufficient to choose a negative circuit subject to costs on the arcs depending on the current solution that approach results in an o mn2l algorithm wherem denotes the number of vertices n denotes the number of arcs andl denotes the total length of the input data

optimal rejection of persistent bounded disturbances in this paper we formulate the problem of optimal disturbance rejection in the case where the disturbance is generated as the output of a stable system in response to an input which is assumed to be of unit amplitude but is otherwise arbitrary the objective is to choose a controller that minimizes the maximum amplitude of the plant output in response to such a disturbance mathematically this corresponds to requiring uniformly good disturbance rejection over all time since the problem of optimal tracking is equivalent to that of optimal disturbance rejection if a feedback controller is used see 7 sect 5 6 the theory presented here can also be used to design optimal controllers that achieve uniformly good tracking over all time rather than a tracking error whose l 2 norm is small as is the case with the currently popular h infty theory the present theory is a natural counterpart to the existing theory of optimal disturbance rejection the so called h infty theory which is based on the assumption that the disturbance to be rejected is generated by a stable system whose input is square integrable and has unit energy it is shown that the problem studied here has quite different features from its predecessor complete solutions to the problem are given in several important cases including those where the plant is minimum phase or when it has only a single unstable zero in other cases procedures are given for obtaining bounds on the solution and for obtaining suboptimal controllers

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 stabilization of normalized coprime factor plant descriptions with h sub infinity bounded uncertainty the problem of robustly stabilizing a family of linear systems is explicitly solved in the case where the family is characterized by h sub infinity bounded perturbations to the numerator and denominator of the normalized left coprime factorization of a nominal system this problem can be reduced to a nehari extension problem directly and gives an optimal stability margin all controllers satisfying a suboptimal stability margin are characterized and explicit state space formulas are given

the upper lower solution method for the coupled system of first order nonlinear pdes abstract this paper is concerned with a coupled system of first order nonlinear partial differential equations this system is but not limited in the extended case of the general blood tissue exchange model btex we use the solutions of a coupled system of first order ordinary differential equations to construct a pair of ordered lower and upper solutions for the nonlinear partial differential system by monotone iterative methods we show the existence and uniqueness of the solution of the coupled system of nonlinear partial differential equations the asymptotic behavior of the solutions to the coupled nonlinear partial differential system can be obtained by investing the asymptotic behavior of the solution to the coupled system of first order ordinary differential equations finally we apply these results to the mathematical models of general blood tissue exchange and the gas solid inter phase heat transfer for the fast igniting catalytic converter problems

approximation and stabilization of distributed systems by lumped systems in this note we study two questions regarding linear time invariant distributed systems namely 1 when can the given system be approximated by a lumped system 2 when can the given system be stabilized by a lumped system in each case we obtain a complete answer to the question in terms of the behaviour of the purely atomic parts of the stable numerator and denominator of the plant

asymptotic properties of black box identification of transfer functions the problem of estimating the transfer function of a linear stochastic system is considered the transfer function is parametrized as a black box and no given order is chosen a priori this means that the model orders may increase to infinity when the number of observed data tends to infinity the consistency and convergence properties of the resulting transfer function estimates are investigated asymptotic expressions for the variances and distributions of these estimates are also derived for the case that the model orders increase it is shown that the variance of the transfer function estimate at a certain frequency is asymptotically given by the noise to signal ratio at that frequency mulliplied by the model order to number of data points ratio