On the role of homogeneous forms in robustness analysis of control systems

This paper considers the problem of positivity (or nonnegativity) of one-parameter families of homogeneous polynomial forms, i.e. of forms whose coefficients depend on a scalar parameter which is allowed to vary over either a continuous or a discrete set. The main contribution of the paper is an efficiently computable sufficient condition for positivity (or nonnegativity), which requires the solution of a Linear Matrix Inequalities (LMI) optimization problem for each value of the scalar parameter. Moreover, necessity of such a condition is investigated and proven to hold for some families of homogeneous forms. The paper also shows that several important problems in the analysis of control systems can be recast as positivity (or nonnegativity) of suitable one-parameter families of homogeneous forms. In particular, two robustness problems involving linear systems and the problem of computing all the equilibria of polynomial nonlinear systems axe discussed in detail.