Systems, Structure and Control 2012 Part 7

Tham khảo tài liệu 'systems, structure and control 2012 part 7', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | Stability Analysis of Polynomials with Polynomic Uncertainty 113 procedure uses suitable properties of the Bernstein form of a multivariate polynomial and test stability by successive subdivision of the original parameter domain and checking positivity of a multivariate polynomial. It can be used in both algebraic checking positivity of Hurwitz determinant or geometric testing the value set approaches. Conceptually the same approach is adopted by Siljak and Stipanovic 1999 . They check robust stability by positivity test of the magnitude of frequency plot by searching minorizing polynomials and using Bernstein expansion. Methods of interval arithmetic are employed in Malan et al. 1997 . Solution of the problem using soft computing methods is presented in Murdoch et al. 1991 . 3. Backgrounds At first let us introduce the basic terms and general results used in robust stability analysis of linear systems with parametric uncertainty. Definition 1 Fixed polynomial A polynomial p s is said to be fixed polynomial of degree n if n p s TaJsJ ansn ---- a1s a0. 1 Definition 2 Uncertain parameter An -dimensional column vector q q1 . qt 7 e Q represents uncertain parameter. Q is called the uncertainty bounding set. In the whole work Q q eW q- q q for i 1 2 . l 2 where q- q i 1 2 . l are the specified bounds for the i-th component qr of q. Such a Q is called a box. Definition 3 Uncertain polynomial A polynomial n p s q T aj q sJ an q sn - ai q s a0 q q e Q. 3 is called an uncertain polynomial. Definition 4 Polynomic uncertainty structure An uncertain polynomial 3 is said to have a polynomic uncertainty structure if each coefficient function a J q J 0 . n is a multivariate polynomial in the components of q. Definition 5 Stability Hurwitz stability A fixed polynomial p s is said to be stable if all its roots lie in the strict left half plane. Definition 6 Robust stability A given family of polynomials P p - q q e Q is said to be robustly stable if for all q e Q p s q is stable .

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