Application of Bernstein expansion to the solution of control problems
The author provides an informative survey of the use of Bernstein expansion in control theory. In particular, he describes reliable methods for solving stability problems. Consider a family of polynomials \[ f(s,p)= a_0(p) s^m+\cdots+ a_{m-1}(p) s+ a_m(p), \] where the coefficients \(a_k(p)\) depend polynomially on the parameter vector \(p= (p_1,p_2,\dots, p_n)\). The parameter vector \(p\) is known to belong to a given box \(B\subseteq \mathbb{R}^n\). For \(p\in B\), the polynomial \(f(.,p)\) is \({\mathcal D}\)-stable, if all zeros of \(f(.,p)\) are inside \({\mathcal D}\), where \({\mathcal D}\) is a subset of the complex plane. The family is robust \({\mathcal D}\)-stable, if \(f(.,p)\) is \({\mathcal D}\)-stable for every \(p\in B\). Special choices of \({\mathcal D}\) lead to Hurwitz-stability, to Schur-stability, and to damping. Also covered is a related topic: Solving systems of strict polynomial inequalities over a given box \(B\subseteq \mathbb{R}^n\) with Bernstein expansion. Numerical examples using interval arithmetic illustrate the described methods.
scientific article; zbMATH DE number 1513373