Béla Paláncz

Algebraic Geodesy and Geoinformatics

While preparing and teaching Introduction to Geodesy I and II to undergraduate students at Stuttgart University, we noticed a gap which motivated the writing of the present book: Almost every topic that we taught required some skills in algebra, and in particular, computer algebra! From positioning to transformation problems inherent in geodesy and geoinformatics, knowledge of algebra and application of computer algebra software were required. In preparing this book therefore, we have attempted to put together basic concepts of abstract algebra which underpin the techniques for solving algebraic problems. Algebraic computational algorithms useful for solving problems which require exact solutions to nonlinear systems of equations are presented and tested on various problems. Though the present book focuses mainly on the two ?elds, the concepts and techniques presented herein are nonetheless applicable to other ?elds where algebraic computational problems might be encountered. In Engineering for example, network densi?cation and robotics apply resection and intersection techniques which require algebraic solutions. Solution of nonlinear systems of equations is an indispensable task in almost all geosciences such as geodesy, geoinformatics, geophysics (just to mention but a few) as well as robotics. These equations which require exact solutions underpin the operations of ranging, resection, intersection and other techniques that are normally used. Examples of problems that require exact solutions include; three-dimensional resection problem for determining positions and orientation of sensors, e. g. , camera, theodolites, robots, scanners etc.

Glucose-insulin control of type1 diabetic patients in H 2 /H ∞ space via computer algebra

This article presents the H2/H∞ control (disturbance rejection LQ method) of the Bergman minimal model [2] for Type1 diabetic patients under intensive care using computer algebra. To design the optimal controller, the disturbance rejection LQ method based on the minimax differential game is applied. The critical, minimax value of the scaling parameter γcrit is determined by using the Modified Riccati Control Algebraic (MCARE) equation employing reduced Grobner basis solution on rational field. The numerical results are in good agreement with those of the Control Toolbox of MATLAB. It turned out, that in order to get positive definite solution stabilizing the closed loop, γ should be greater than γcrit. The obtained results are compared with the classical LQ technique on the original non-linear system, using a standard meal disturbance situation. It is also demonstrated that for γ ≫ γcrit, the gain matrix approaches the traditional LQ optimal control design solution. The symbolic and numerical computations were carried out with Mathematica 5.2, and with the CSPS Application 2, as well as with MATLAB 6.5.

Application of pareto optimality to linear models with errors-in-all-variables

In some geodetic and geoinformatic parametric modeling, the objectives to be minimized are often expressed in different forms, resulting in different parametric values for the estimated parameters at non-zero residuals. Sometimes, these objectives may compete in a Pareto sense, namely a small change in the parameters results in the increase of one objective and a decrease of the other, as frequently occurs in multiobjective problems. Such is the case with errors-in-all-variables (EIV) models, e.g., in the geodetic and photogrammetric coordinate transformation problems often solved using total least squares solution (TLS) as opposed to ordinary least squares solution (OLS). In this contribution, the application of Pareto optimality to solving parameter estimation for linear models with EIV is presented. The method is tested to solve two well-known geodetic problems of linear regression and linear conformal coordinate transformation. The results are compared with those from OLS, Reduced Major Axis Regression (TLS solution), and the least geometric mean deviation (GMD) approach. It is shown that the TLS and GMD solutions applied to the EIV models are just special cases of the Pareto optimal solution, since both of them belong to the Pareto-set of the problems. The Pareto balanced optimum (PBO) solution as a member of this Pareto optimal solution set has special features and is numerically equal to the GMD solution.

Robust Blood-Glucose Control using Mathematica

A robust control design on frequency domain using Mathematica is presented for regularization of glucose level in type I diabetes persons under intensive care. The method originally proposed under Mathematica by Helton and Merino, now with an improved disturbance rejection constraint inequality - is employed, using a three-state minimal patient model. The robustness of the resulted high-order linear controller is demonstrated by nonlinear closed loop simulation in state-space, in case of standard meal disturbances and is compared with Hinfin design implemented with the mu-toolbox of Matlab. The controller designed with model parameters represented the most favorable plant dynamics from the point of view of control purposes, can operate properly even in case of parameter values of the worst-case scenario

CLASSICAL AND MODERN CONTROL STRATEGIES IN GLUCOSE-INSULIN STABILIZATION

Abstract This paper presents an analysis of classical and modern control methods for blood glucose control of diabetic patients under intensive care. Employing a modified two- compartment model proposed by Bergman, et al. (1979), linear feedback control law leads to a fully symbolic solution. In case of nonlinear approach, considering glucose injection as the only control variable and using Pontryagins maximum principle, a symbolic-numeric solution has been achieved. As modern control strategy, optimal glucose-insulin control in the H 2 /H ∞ -space is presented using LQ and disturbance rejection LQ methods, which result in a numerical solution to the control problem.

Pareto optimality solution of the Gauss-Helmert model

The Pareto optimality method is applied to the parameter estimation of the Gauss-Helmert weighted 2D similarity transformation assuming that there are measurement errors and/or modeling inconsistencies.In some cases of parametric modeling, the residuals to be minimized can be expressed in different forms resulting in different values for the estimated parameters. Sometimes these objectives may compete in the Pareto sense, namely a small change in the parameters can result in an increase in one of the objectives on the one hand, and a decrease of the other objective on the other hand. In this study, the Pareto optimality approach was employed to find the optimal trade-off solution between the conflicting objectives and the results compared to those from ordinary least squares (OLS), total least squares (TLS) techniques and the least geometric mean deviation (LGMD) approach.The results indicate that the Pareto optimality can be considered as their generalization since the Pareto optimal solution produces a set of optimal parameters represented by the Pareto-set containing the solutions of these techniques (error models). From the Pareto-set, a single optimal solution can be selected on the basis of the decision maker’s criteria. The application of Pareto optimality needs nonlinear multi-objective optimization, which can be easily achieved in parallel via hybrid genetic algorithms built-in engineering software systems such as Matlab. A real-word problem is investigated to illustrate the effectiveness of this approach.

Design of Luenberger Observer for Glucose-Insulin Control via Mathematica

Many articles dealing with insulin-glucose control have been published in the last decades, and they mostly assumed that all the system state variables are available for feedback. However, this is not usually the case, or they are not so cheap in practice as blood glucose measurements are. In this paper the use of the reduced-order estimator (also known as the Luenberger observer) is considered in symbolic form employing polynomial control system application of mathematica for the three-state minimal Bergman model, as this can be used to reconstruct those state variables that are hard to be recovered directly from the system outputs: remote compartment insulin and plasma insulin. Nonlinear closed loop simulations with H2/Hinfin control (disturbance rejection LQ method) showed that the observer, which is faster than the system itself, can provide a very good state recovery performance.

Classification of Time Series Using Singular Values and Wavelet Subband Analysis with ANN and SVM Classifiers

Oscillation of the cerebral blood flow (CBF) is a common feature in several physiological or pathophysiological states of the brain. It is promising to identify the disorders of the cerebral circulation based on the classification of CBF signals. In order to distinguish between different physiological states, two different classification methods have been developed; a Radial Basis Function based Neural Network and a Support Vector Classifier with Gaussian kernel. In order to describe the temporal blood flow patterns, two feature extraction procedures were applied; spectral matrix and wavelet subband analysis. These feature extraction and classification methods are evaluated and their efficiencies are compared. The computations were carried out with Mathematica 5.1 and its Wavelet Application.

Characterization of the temporal pattern of cerebral blood flow oscillations

Oscillation of the cerebral blood flow (CBF) is a common feature in several physiological or pathophysiological states of the brain. It is a promising opportunity to identify the state of the brain based on the classification of CBF signals. In order to carry out classification of the time signals, a feature vector has been extracted to characterize the signals. Unsupervised classification showed that the extracted feature vector is an acceptable representation of the time signals. It also turned out that the difference between normal signal and a signal indicating drug injection effect is significant, and much more dominant than the difference between signals of the right and left brain sides. For the signal classification an artificial neural network (ANN) model based on supervised backpropagation network has been developed and successfully applied.

Optimal Glucose-Insulin Control in H2 Space

In this case study optimal glucose-insulin control in the Hardy /spl Hscr//sub 2/-space is presented for diabetic patients under intensive care. The analysis is based on a modified two-compartment model. First the classical LQ optimal control design method is considered, and then its extension the so called disturbance rejection LQR (LQ rejection) method, based on the MINIMAX differential game is applied to control design. To demonstrate the results of these two methods, the simulation of the dynamical performance of the non-linear closed loop system in case of food (sugar) intake has been carried out. For the symbolic and numeric computations Mathematica and Matlab-Simulink are used.