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

Abstract

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

Time-invariant uncertain systems: A necessary and sufficient condition for stability and instability via homogeneous parameter-dependent quadratic Lyapunov functions

“…The pairs (u, y) in R q 0 × R n 0 satisfying (15) can be found with the technique in [19,12] which amounts to finding the roots of a polynomial obtained via pivoting. The vectors c 1 , .…”

“…With m = 0 Theorem 2 proves that A is unstable, in particular ξ(u) = (0.3172, 0.6828, 0.0000, 0.0000) ′ spc(A(ξ(u))) = {0.0910, −0.6655, −1.0560} . Some details are: β ∈ R 5 ; α ∈ R 255 ; u is found from (15) with the technique in [19,12] by finding the roots of a quartic polynomial.…”

Quantifying the unstable in linearized nonlinear systems

“…, n} and φ ∈ Z j,k . For any admissible values of j and k, this can be done by computing Z j,k (e.g., through linear algebra operations, see for instance Chesi et al, 2003 and references therein), and then by trivial substitution of φ into (41).…”

“…Unfortunately, solving a system of polynomial equations is a difficult task and no reliable method does exist for this. Indeed, symbolic methods like resultants suffer of the problem that the univariate polynomial generated can have huge degree, while numerical methods like homotopy methods suffer of the problem that solutions might be lost, see Chesi, Garulli, Tesi, and Vicino (2003) and Mora (2005) and references therein. The third issue is that the computational time can be very large depending on the number of samples used.…”

Zero-convex functions, perturbation resilience, and subgradient projections for feasibility-seeking methods

“…A related example is Example 4 below. Zero-convex functions can help to analyze systems of (multivariate polynomial) equations, much like convex optimization helps doing so in other cases [39]. They can help in the analysis of (quasiconvex) quadratic functions which appear in the context of economics [5,Chapter 6], [28,Chapter 6].…”