Approximating properties of linear models for input output descriptions

Pade approximations for linear and dynamic programming using Markov processes and new algorithm

Régularisation de problèmes inverses linéaires avec opérateur inconnu

Dans cette thèse, nous étudions des méthodes de résolution pour différents types de problèmes inverses linéaires. L'objectif est d'estimer un paramètre de dimension infinie (typiquement une fonction ou une mesure) à partir de l'observation bruitée de son image par un opérateur linéaire. Nous nous intéressons plus précisément à des problèmes inverses dits discrets, pour lesquels l'opérateur est à valeurs dans un espace de dimension finie. Pour ce genre de problème, la non­injectivité de l'opérateur rend impossible l'identification du paramètre à partir de l'observation. Un aspect de la régularisation consiste alors à déterminer un critère de sélection d'une solution parmi un ensemble de valeurs possibles. Nous étudions en particulier des applications de la méthode du maximum d'entropie sur la moyenne, qui est une méthode Bayésienne de régularisation permettant de définir un critère de sélection à partir d'information a priori. Nous traitons également des questions de stabilité en problèmes inverses sous des hypothèses de compacité de l'opérateur, dans un problème de régression non-paramétrique avec observations indirectes.We study regularization methods for different kinds of linear inverse problems. The objective is to estimate an infinite dimensional parameter (typically a function or a measure) from the noisy observation of its image through a linear operator. We are interested more specifically to discret inverse problems, for which the operator takes values in a finite dimensional space. For this kind of problems, the non-injectivity of the operator makes impossible the identification of the parameter from the observation. An aspect of the regularization is then to determine a criterion to select a solution among a set of possible values. We study in particular some applications of the maximum entropy on the mean method, which is a Bayesian regularization method that allows to choose a solution from prior informations. We also treat stability issues in inverse problems under compacity assumptions on the operator, in a general nonparametric regression framework with indirect observations

Estimation of an unknown disturbance of a waste shredder using an L-delay inverse model

In this work, a model-based system is developed for off-line estimation of an unknown disturbance of a waste shredder, utilizing the dynamic model of the shredder’s rotor and experimental data obtained from the shredder. During shredding, waste material generates an unknown resisting torque on the shredder’s rotor, which can be interpreted as an unknown disturbance input to the system. The application target is TANA shark 440-series industrial waste shredder of Finnish environmental technology company Tana Oy. The data used in the estimation includes pressure and rotational speed measurements, and reference values of hydraulic motors’ displacements. To perform the estimation, a linear state-space model of the hydraulic motors, gearboxes and rotor is constructed. The torques of hydraulic motors are inputs of the state-space model, and their values are computed based on pressure and rotational speed measurement, along with the reference values of the motors’ displacements. For the actual unknown disturbance estimation, a 1-delay inverse state-space model is derived from the rotor’s state-space model. This inverse model estimates the unknown disturbance input to the shredder using rotor speed measurement and calculated hydraulic motor torques. The unknown disturbance estimation result is validated using test data collected without material feed to the shredder, resulting in a zero disturbance input. The validation reveals an issue with the calculation of the hydraulic motor torques: the pressure without waste load is too low to determine mechanical efficiencies, leading to highly inaccurate torque calculations. An estimation conducted with a rough linear extrapolation of efficiency indicates that, on a large scale, estimating zero disturbance is moderately successful. However, due to inaccurate input signals, significant deviations from zero are observed in the result, especially during the transient phase of the rotor’s rotation. It is also concluded that identifying possible rotor model parameter changes due to operating conditions is challenging, due to uncertainty in the model’s input signals. In the future, tests performed with a test bench machine allow better evaluation of the accuracy of calculated hydraulic motor torques and the accuracy of non-zero unknown torque estimation. The sensitivity of the estimation result to changes in measurement data and rotor model parameters is investigated using simulation-based sensitivity analysis. The sensitivity analysis yields a clear outcome: the estimation result is several times more sensitive to changes in measurements used for calculating hydraulic motor torques than to changes in the rotor dynamic model parameters. Based on the sensitivity analysis, it is concluded that errors in the estimation result are most likely due to inaccuracies in the pressure measurements. The work proposes that placing the pressure sensors directly on the input and output sides of the hydraulic motors would enhance the accuracy of torque computation and thereby would also improve the accuracy of unknown disturbance estimation.Tässä työssä kehitetään mallipohjainen järjestelmä teollisen jäterepijän tuntemattoman häiriön off-line estimointiin käyttäen repijän roottorin dynaamista mallia sekä repijästä saatavaa kokeellista dataa. Revinnän aikana jäte aiheuttaa tuntemattoman suuruisen, repijän roottorin pyörintää vastustavan väännön, joka voidaan tulkita systeemin tuntemattomaksi häiriöinputiksi. Sovelluskohteena on suomalaisen ympäristöteknologiayritys Tana Oy:n TANA Shark 440-sarjan teollinen jäterepijä. Estimoinnissa repijästä hyödynnettävä data sisältää paine- ja pyörimisnopeusmittaukset sekä hydraulimoottorien kierrostilavuuksien asetusarvot. Estimointia varten työssä muodostetaan repijän hydraulimoottoreista, vaihdelaatikoista sekä roottorista lineaarinen tilamalli. Hydraulimoottorien vääntömomentit ovat tilamallin inputeina, joiden arvot lasketaan perustuen paine- ja pyörimisnopeusmittauksiin sekä moottorien kierrostilavuuksien asetusarvoon. Varsinaista tuntemattoman häiriön estimointia varten työssä muodostetaan roottorin tilamallista 1-viiveellinen inverssi tilamalli, joka estimoi repijän tuntemattoman häiriöinputin roottorin nopeusmittauksen sekä hydraulimoottorien laskettujen vääntömomenttien perusteella. Tuntemattoman häiriön estimointitulosta validoidaan käyttämällä testidataa, joka on kerätty ilman materiaalisyötettä, eli häiriöinputin arvo on nolla. Validoinnissa huomataan ongelma hydraulimoottorien vääntömomentin laskennassa: paine jää ilman jätekuormitusta niin alhaiselle tasolle, ettei moottorien mekaanisen hyötysuhteen arvoja ole määritelty. Tämän seurauksena vääntömomentin laskenta on hyvin epätarkkaa. Karkealla lineaarisella hyötysuhteen ekstrapoloinnilla suoritetun estimoinnin tulos osoittaa, että suuressa kuvassa nollahäiriön estimointi onnistuu kohtalaisesti, mutta epätarkkojen inputsignaalien johdosta tuloksessa esiintyy merkittäviä poikkamia nollasta, varsinkin roottorin pyörimisen transienttivaiheissa. Työssä todetaan, että mahdollisten toimintapisteestä johtuvien roottorimallin parametrien muutosten identifiointi on hankalaa, sillä myös mallin inputsignaalit sisältävät epävarmuutta. Tulevaisuudessa testipenkkilaitteella tehtävillä kokeilla kyetään paremmin arvioimaan hydraulimoottorien vääntömomentin laskennan tarkkuutta sekä nollasta poikkeavan tuntemattoman vastusväännön estimoinnin tarkkuutta. Estimointituloksen herkkyyttä mittausdatojen sekä roottorimallin parametrien muutoksille tutkitaan simulointipohjaisella herkkyysanalyysillä. Herkkyysanalyysi tuottaa selvän tuloksen: estimointitulos on moninkertaisesti herkempi muutoksille hydraulimoottorien vääntömomentin laskennassa käytettävissä mittausdatoissa kuin roottorin dynaamisen mallin parametreissa. Herkkyysanalyysin tuloksien perusteella todetaan, että estimointituloksen virheet ovat suurimmalla todennäköisyydellä peräisin painemittauksissa esiintyvistä epätarkkuuksista. Työssä todetaan, että paineantureiden sijoittaminen suoraan hydraulimoottoreiden tulo- ja lähtöpuolelle parantaisi vääntömomenttien laskennan ja sitä kautta myös tuntemattoman häiriön estimoinnin tarkkuutta

Coupled Electromagnetic Field/Circuit Simulation: Modeling and Numerical Analysis

Today's most common circuit models increasingly tend to loose their validity in circuit simulation due to the rapid technological developments, miniaturization and higher complexity of integrated circuits. This has motivated the idea of combining circuit simulation directly with distributed device models to refine critical circuit parts. In this thesis we consider a model, which couples partial differential equations for electromagnetic devices - modeled by Maxwell's equations -, to differential-algebraic equations, which describe basic circuit elements including memristors and the circuit's topology. We analyze the coupled system after spatial discretization of Maxwell's equations in a potential formulation using the finite integration technique, which is often used in practice. The resulting system is formulated as a differential-algebraic equation with a properly stated leading term. We present the topological and modeling conditions that guarantee the tractability index of these differential-algebraic equations to be no greater than two. It shows that the tractability index depends on the chosen gauge condition for Maxwell's equations. For successful numerical integration of differential-algebraic equations the index characterization plays a crucial role. The index can be seen as a measure of the equation's sensitivity to perturbations of the input functions and numerical difficulties such as the computation of consistent initial values for time integration. We generalize index reduction techniques for a general class of differential-algebraic equations by using the tractability index concept. Utilizing the index reduction we deduce local solvability and perturbation results for differential-algebraic equations having tractability index-2 and we derive an algorithm to calculate consistent initializations for the spatial discretized coupled system. Finally, we demonstrate our results by numerical experiments

Studies in Experimental Design and Data Analysis with Special Reference to Computational Problems

An experimenter, designing or analysing an experiment, frequently finds that not all can be done according to the text-book. Considerations outside his control may have to be taken into account that affect the design; he may not be interested in all his treatments equally but perhaps in some unusual contrasts between them. Again, unforseen circumstances and happenings can afterwards force him to discard some of his results, so upsetting the balance or orthogonality of the design. The aim of this thesis is to help an experimenter in such a situation. The first part is concerned with analysing a block experiment that is in general unbalanced and non-orthogonal. 'Two different methods, one iterative, one non-iterative, are derived for obtaining the analysis, each with its own advantages. The non-iterative method basically is derived from the actual design and produces matrices, which can then operate on any suitable data supplied. The iterative method, however, found in appendix A, is applied directly to the data from the start, to produce the treatment effects directly. Although the iterative method is easier to apply and can also be used with a wider class of design than can the non-iterative method, the inter-block analysis and the analysis of the dual become easier using the non-iterative method. Certain contrasts are related to the design in special ways, and, if known, make the analysis of the design easier. The implications are discussed in chapter 2, which is also concerned with finding the contribution to the sum of squares for these and other, more general, contrasts of interest. The dual of a design is defined as that design formed from the original design when treatments and blocks classifications are interchanged, i.e. treatments become blocks and vice-versa in the new design. It is useful for studying block differences eliminating those due to treatments, which may, for example, be required if the blocking system arose from the possible residual effects of treatments from some previous experiment on the same material. The second part of the thesis, in chapter 3> is concerned with the analysis of the dual. It is shown that there is no need to start again from the beginning when analysing the dual if the original design had already been analysed, because the analysis can provide information about that of the dual. The method is especially easy when the non-iterative method of analysing block designs, discussed in chapter 2, has been used for the original design. The experimenter will often be more interested in some contrasts between treatments than in others and a design can be selected to give more precise information about these contrasts. The construction of such designs is discussed in the third part of the thesis. Various measures can be used to judge which design is best as regards contrasts of interest. Algorithms for finding the optimal design according to these measures are derived and discussed in chapter 4. Listings and flowcharts of a program to carry out the non-iterative analysis of chapter 2 and of a program to construct optimal block designs appear in appendices B and C

Algorithms Seminar, 2001-2002

These seminar notes constitute the proceedings of a seminar devoted to the analysis of algorithms and related topics. The subjects covered include combinatorics, symbolic computation, asymptotic analysis, number theory, as well as the analysis of algorithms, data structures, and network protocols

Matrix Polynomials and their Lower Rank Approximations

This thesis is a wide ranging work on computing a “lower-rank” approximation of a matrix polynomial using second-order non-linear optimization techniques. Two notions of rank are investigated. The first is the rank as the number of linearly independent rows or columns, which is the classical definition. The other notion considered is the lowest rank of a matrix polynomial when evaluated at a complex number, or the McCoy rank. Together, these two notions of rank allow one to compute a nearby matrix polynomial where the structure of both the left and right kernels is prescribed, along with the structure of both the infinite and finite eigenvalues. The computational theory of the calculus of matrix polynomial valued functions is developed and used in optimization algorithms based on second-order approximations. Special functions studied with a detailed error analysis are the determinant and adjoint of matrix polynomials. The unstructured and structured variants of matrix polynomials are studied in a very general setting in the context of an equality constrained optimization problem. The most general instances of these optimization problems are NP hard to approximate solutions to in a global setting. In most instances we are able to prove that solutions to our optimization problems exist (possibly at infinity) and discuss techniques in conjunction with an implementation to compute local minimizers to the problem. Most of the analysis of these problems is local and done through the Karush-Kuhn-Tucker optimality conditions for constrained optimization problems. We show that most formulations of the problems studied satisfy regularity conditions and admit Lagrange multipliers. Furthermore, we show that under some formulations that the second-order sufficient condition holds for instances of interest of the optimization problems in question. When Lagrange multipliers do not exist, we discuss why, and if it is reasonable to do so, how to regularize the problem. In several instances closed form expressions for the derivatives of matrix polynomial valued functions are derived to assist in analysis of the optimality conditions around a solution. From this analysis it is shown that variants of Newton’s method will have a local rate of convergence that is quadratic with a suitable initial guess for many problems. The implementations are demonstrated on some examples from the literature and several examples are cross-validated with different optimization formulations of the same mathematical problem. We conclude with a special application of the theory developed in this thesis is computing a nearby pair of differential polynomials with a non-trivial greatest common divisor, a non-commutative symbolic-numeric computation problem. We formulate this problem as finding a nearby structured matrix polynomial that is rank deficient in the classical sense