Finsler's Lemma for matrix polynomials

Abstract

Abstract. Finsler's Lemma charactrizes all pairs of symmetric n × n real matrices A and B which satisfy the property that v T Av > 0 for every nonzero v ∈ R n such that v T Bv = 0. We extend this characterization to all symmetric matrices of real multivariate polynomials, but we need an additional assumption that B is negative semidefinite outside some ball. We also give two applications of this result to Noncommutative Real Algebraic Geometry which for n = 1 reduce to the usual characterizations of positive polynomials on varieties and on compact sets.

Robust H2 and H∞ Filter Design for Continuous‐time Uncertain Linear Fractional Transformation Systems via LMI

“…In this paper, we are about to model the parametric uncertainty of the system via an LFT structure shown in Fig. 1, so for clarification of LFT representation, let M be a complex matrix partitioned as [4]…”

“…In this paper, we are about to model the parametric uncertainty of the system via an LFT structure shown in Fig. , so for clarification of LFT representation, let M be a complex matrix partitioned as and let be another complex matrix. Then we can formally define an LFT with respect to Δ as the map provided that the inverse ( I − M 11 Δ) −1 exists (if exists, the LFT is said to be well defined.).…”

Existence of Continuous or Constant Finsler's Variables for Parameter-Dependent Systems

“…Remembering that if µ (s) is a solution to (F2) then X (s) = − 1 2 µ (s) B T (s) is a solution to (F3), it is easy to apply the results of Section III-A to give some sufficient conditions that allow simple functional dependence like continuity or polynomial dependence on s for the variable X (s) in (F3) without loss of generality. Among these possible extensions, one may point out the next theorem which deals with a case closely related to [10] and [20]. Theorem 6.…”

“…Relaxation is a good resort when the solution space is little known. Another approach considered in the literature is to find some properties on the parameter set S and on the matrices of the system in order to reduce the search space of the solutions [10], [20]. In fact, if S is a compact set and the matrices of the PD-LMI depend continuously on the parameter, in [10] it is proved that if the PD-LMI has a solution, then one can restrict the search for a solution in the set of polynomial functions.…”

“…These results are important since, with the knowledge that a continuous function can be approximated by a polynomial, they form the theoretical justification for the polynomial relaxation approach (where at least a continuous solution is supposed to exist). If the parameter space is instead all the space and the matrices of the PD-LMI depends polynomially on the parameter, then [20] shows that one can restrict the search space and look for a rational solution.…”