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Characterizing the solution set of polynomial systems in terms of homogeneous forms: an LMI approach
Author(s) -
Chesi G.,
Garulli A.,
Tesi A.,
Vicino A.
Publication year - 2003
Publication title -
international journal of robust and nonlinear control
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.361
H-Index - 106
eISSN - 1099-1239
pISSN - 1049-8923
DOI - 10.1002/rnc.839
Subject(s) - mathematics , polynomial matrix , homogeneous polynomial , polynomial , matrix polynomial , stable polynomial , kernel (algebra) , polynomial kernel , dimension (graph theory) , linear matrix inequality , matrix (chemical analysis) , eigenvalues and eigenvectors , solution set , maximization , set (abstract data type) , mathematical optimization , pure mathematics , alternating polynomial , kernel method , computer science , mathematical analysis , materials science , physics , quantum mechanics , artificial intelligence , support vector machine , composite material , programming language
This paper considers the problem of determining the solution set of polynomial systems, a well‐known problem in control system analysis and design. A novel approach is developed as a viable alternative to the commonly employed algebraic geometry and homotopy methods. The first result of the paper shows that the solution set of the polynomial system belongs to the kernel of a suitable symmetric matrix. Such a matrix is obtained via the solution of a linear matrix inequality (LMI) involving the maximization of the minimum eigenvalue of an affine family of symmetric matrices. The second result concerns the computation of the solution set from the kernel of the obtained matrix. For polynomial systems of degree m in n variables, a basic procedure is available if the kernel dimension does not exceed m +1, while an extended procedure can be applied if the kernel dimension is less than n ( m −1)+2. Finally, some application examples are illustrated to show the features of the approach and to make a brief comparison with polynomial resultant techniques. Copyright © 2003 John Wiley & Sons, Ltd.