Characterizing the solution set of polynomial systems in terms of homogeneous forms: an LMI approach

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.

[1]  David A. Cox,et al.  Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra, 3/e (Undergraduate Texts in Mathematics) , 2007 .

[2]  A. S. Hodel,et al.  Homotopy methods for the solution of general modified algebraic Riccati equations , 1993 .

[3]  Graziano Chesi,et al.  An LMI-based approach for characterizing the solution set of polynomial systems , 2000, Proceedings of the 39th IEEE Conference on Decision and Control (Cat. No.00CH37187).

[4]  Alberto Tesi,et al.  Robust absolute stability of Lur'e control systems in parameter space , 1991, Autom..

[5]  D. Seborg,et al.  On the number of solutions of multivariable polynomial systems , 1983 .

[6]  Uri Shaked,et al.  Minimum entropy static output-feedback control with an H∞-norm performance bound , 1997, IEEE Trans. Autom. Control..

[7]  Ioan Doré Landau,et al.  Reduced order bilinear models for distillation columns , 1978, Autom..

[8]  A. Tarski.  A Decision Method for Elementary Algebra and Geometry , 2023 .

[9]  Dale E. Seborg,et al.  Characterization of equilibrium sets for bilinear systems with feedback control , 1983, Autom..

[10]  B. Anderson,et al.  Output feedback stabilization—Solution by algebraic geometry methods , 1977, Proceedings of the IEEE.

[11]  N. Bose.  Applied multidimensional systems theory , 1982 .

[12]  Stephen A. Vavasis,et al.  Accurate solution of polynomial equations using Macaulay resultant matrices , 2004, Math. Comput..

[13]  David A. Cox,et al.  Using Algebraic Geometry , 1998 .

[14]  Dinesh Manocha,et al.  SOLVING SYSTEMS OF POLYNOMIAL EQUATIONS , 2002 .

[15]  Uri Shaked.  The effect of system zeros on the achievable H∞ estimation level , 1997, IEEE Trans. Autom. Control..

[16]  N. Jacobson.  Lectures In Abstract Algebra , 1951 .

[17]  A. Tesi,et al.  Robust stability of state-space models with structured uncertainties , 1990 .

[18]  Stephen P. Boyd,et al.  Linear Matrix Inequalities in Systems and Control Theory , 1994 .

[19]  A. Morgan.  Solving Polynomial Systems Using Continuation for Engineering and Scientific Problems , 1987 .

[20]  Graziano Chesi,et al.  A convex approach to a class of minimum norm problems , 1998, Robustness in Identification and Control.

[21]  A. Garulli,et al.  On the characterization of the solution set of polynomial systems via LMI techniques , 2001, 2001 European Control Conference (ECC).