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The method of finding points of intersection of two cubic Bezier curves using the Sylvester matrix

Biły B.

Silesian Journal of Pure and Applied Mathematics

2016

Vol. 6, iss. 1

155--176

Sylvester matrix is used to create a 9th degree polynomial from coeﬃcients of two cubic Bezier curves. The real roots of this polynomial allow to compute points of intersection of aforementioned curves. An additional constraint is used to indicate valid points. The “reverseinverse law” is presented in order to reduce the cost of calculation in this particular case. Also some limitations of the method as well as the ways to avoid them, if possible, are pointed out.

Self-tuning control with regularized RLS algorithm for vibration cancellation of a circular plate

Leniowska L., Kos P.

Archives of Acoustics

2009

Vol. 34, No. 4

613-624

The self-tuning control assumes that the vibrating system is unknown and the controller procedure has the ability to identify the process and to update the necessary control law. Such an algorithm provides the relevant regulator parameters according to the obtained parametric object model. The algorithm can be described as a combination of the following two procedures: the online identification and the computation of the controller parameters. Nearly all of the identification procedures are related to the Least Squares (LS) estimate of a model output. Classified as an ill-posed problem, it implies that the obtained solution is potentially very sensitive to the data perturbations. In order to avoid such problems, the regularized version of the RLS method has been considered in this paper. By solving the linear system of equations with a non-singular Sylvester matrix, the formulas for the unknown coefficients of the considered PID-type controller structure have been obtained. The results of the tests and simulations for the circular plate vibration cancellation have been also included.