Tham khảo tài liệu 'recent advances in robust control theory and applications in robotics and electromechanics part 2', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | Parametric Robust Stability 19 This is an usful and inetresting property but above all properties there are three outstanding properties it would be very useful if they were fulfilled in complex numbers they are as follows Addition f q g q Subtraction f q g q Product f q g q a q a f q ag q a q af q ag q a q af q ag q fi q fi f q fig q fi q fif q fig q 5 fi q fif q fig q Most be noted that the alpha componet of subtraction is correct with a q af q ag q it is an addition of alphas. It is also important to highlight the simplicity with which made the addition subtraction and product in alpha beta representation. Sign decomposition of the determinant Sign decomposition of the determinant was developed in Elizondo 1999 and it was presented an application in Elizondo 2001A 2002B by simplicity only will mention Elizondo 1999 . In parametric robust stability is not very useful the sign decomposition of the determinant but it is a part of sign decomposition. We can analyze robust stability by means of the Hurwitz criterion means the robust positivity of determinants but it is so much easier by means of criterion 2 see table 1 . Taking account that the reader could work in other areas where the nonsingularity of a matrix dependent in parameters is important then sign decomposition of the determinant is included in this chapter. The a fi representation of the determinant In order to achieve the procedure to determine the robust positivity in necessary and sufficient conditions of a determinant with real coefficients depending on parameters qi the following fact is presented. By means of the a fi properties 5 is obtained the following fact in the development of the determinant appears the alpha part and beta part as shown in the following fact. Fact 1. Elizondo 1999 Let M q be a 2 X 2 matrix with elements mqj q Ễ K with representation ai j q fiij q . Then the a fi representation of the determinant of the matrix M q is det M q a a1 1 q a2 2 q a2 1 q a1 2 q det M q fi
