Numeric-Symbolic Solution for Satellite Trajectory Control by Pole Placement

Control design of satellites based on pole placement method results in determined or underdetermined multivariable polynomial systems. Since only the real solutions can be considered for hardware implementation, we are looking for exclusively these solutions. In this study we suggest a numeric-symbolic approach to compute only the real solutions directly. Employing computer algebra (Dixon or reduced Gröbner basis) a condition can be formulated for the free variables as parameters of the underdetermined system, which ensures only real solutions. Numerical example illustrates the procedure and the effectivity of the control law

CLASSIFICATION OF TIME SERIES VIA WAVELET SUBBAND ANALYSIS USING SUPPORT VECTOR MACHINE CLASSIFIER

An improved feature extraction method has been developed for classification and identification of time series, in case of the number of the experiments are considerably less than that of the samples in time series. The method based on the subband analysis of the wavelet transformation of the time signals, provides lower dimension feature vectors as well as much more robust kernel-based classifier than the traditional wavelet-based feature extraction method does. The application of this technique is illustrated by the classification of cerebral blood flow oscillation using support vector classifier with Gaussian kernel. The computations were carried out with Mathematica 5.1 and its Wavelet Application

A Numeric-Symbolic Solution of GNSS Phase Ambiguity

Solution of the Global Navigation Satellite Systems (GNSS) phase ambiguity is considered as a global quadratic mixed integer programming task, which can be transformed into a pure integer problem with a given digit of accuracy. In this paper, three alter-native algorithms are suggested. Two of them are based on local and global linearization via McCormic Envelopes, respectively. These algorithms can be effective in case of simple configuration and relatively modest number of satellites. The third method is a locally nonlinear, iterative algorithm handling the problem as {-1, 0, 1} programming and also lets compute the next best integer solution easily. However, it should keep in mind that the algorithm is a heuristic one, which does not guarantee to find the global integer optimum always exactly. The procedure is very powerful utilizing the ability of the numeric-symbolic abilities of a computer algebraic system, like Wolfram Mathematica and it is properly fast for minimum 4 satellites with normal configuration, which means the Geometric Dilution of Precision (GDOP) should be between 1 and 8. Wolfram Alpha and Wolfram Clouds Apps give possibility to run the suggested code even via cell phones. All of these algorithms are illustrated with numerical examples. The result of the third one was successfully compared with the LAMBDA method, in case of ten satellites sending signals on two carrier frequencies (L1 and L2) with weighting matrix used to weight the GNSS observation and computed as the inverse of the corresponding covariance matrix

APPLICATION OF COMPUTER ALGEBRA FOR GLUCOSE-INSULIN CONTROL IN H2/Hinf SPACE USING MATHEMATICA

In this case study, an optimal control in H2/Hinf space is presented for glucose- insulin system of diabetic patients under intensive care. The analysis is based on a modified two-compartment Bergman model. 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 symbolic solution of the modified Riccati equation. The numeric evaluation of the symbolic computation for γ > γ crit leads to two different solutions, but the norms of the vectors λ1, λ2 formed by the eigenvalues of the pair of the gain matrices are the same. The numerical results are in good agreement with that of the μ-Toolbox of MATLAB. One of the gain matrices with increasing γ , approaches the gain matrix computed with the traditional LQ optimal control design. The symbolic and numerical computations were carried out with \textitMathematica 5, and with the CSPS Application 2 as well as with MATLAB 6.5

SUPPORT VECTOR CLASSIFIER VIA MATHEMATICA

In this case study a Support Vector Classifier function has been developed in Mathematica. Starting with a brief summary of support vector classification method, the step by step implementation of the classification algorithm in Mathematica is presented and explained. To check our function, two test problems, learning a chess board and classification of two intertwined spirals are solved. In addition, an application to filtering of airborne digital land image by pixel classification is demonstrated using a new SVM kernel family, the KMOD, a kernel with moderate decreasing

SYMBOLIC SOLUTION OF BOUNDARY VALUE PROBLEM VIA MATHEMATICA

Symbolic computation has been applied to Runge-Kutta technique in order to solve a two-point boundary value problem. The unknown boundary values are considered as symbolic variables, therefore they will appear in a system of algebraic equations, after the integration of the ordinary differential equations. Then this algebraic equation system can be solved for the unknown initial values and substituted into the solution. Consequently, only one integration pass is enough to solve the problem instead of using an iteration technique like shooting method. This procedure is illustrated by solving the boundary value problem of the mechanical analysis of a liquid storage tank. Computations were carried out by the MATHEMATICA symbolic system

FAILURE RECOGNITION IN WASTE-WATER TREATMENT PROCESS

A failure recognition method based on the inverse solution of a linear dynamical model was applied to detect malfunctions of waste-water plant operations. Kalman filter could be sucessfully employed to eliminate process noise as well as modelling errors

Solving Robust Glucose-Insulin Control by Dixon Resultant Computations

We present a symbolic approach towards solving the Bergman three-state minimal patient model of glucose metabolism. Our work first translates the Bergman three-state minimal patient model into the modified control algebraic Riccati equation. Next, the modified control algebraic Ricatti equation is reduced to a system of polynomial equations, and an optimal (minimal) solution of these polynomials is computed using Dixon resultants. We demonstrate the use of our method by reporting on three case studies over glucose metabolism