Sum-of-square-of-rational-function based representations of positive semidefinite polynomial matrices
This provides a theoretical and computational framework for polynomial matrix representations in optimization and control, but it is incremental as it builds on existing diagonalization and Artin's theorem methods.
The paper tackles the problem of representing positive semidefinite polynomial matrices on special sets like ℝⁿ, intervals, and strips by proving sum-of-square-of-rational-function based representations, and it presents a numerical method for computing these representations with tests in OCTAVE.
The paper proves sum-of-square-of-rational-function based representations (shortly, sosrf-based representations) of polynomial matrices that are positive semidefinite on some special sets: $\mathbb{R}^n;$ $\mathbb{R}$ and its intervals $[a,b]$, $[0,\infty)$; and the strips $[a,b] \times \mathbb{R} \subset \mathbb{R}^2.$ A method for numerically computing such representations is also presented. The methodology is divided into two stages: (S1) diagonalizing the initial polynomial matrix based on the Schmüdgen's procedure \cite{Schmudgen09}; (S2) for each diagonal element of the resulting matrix, find its low rank sosrf-representation satisfying the Artin's theorem solving the Hilbert's 17th problem. Some numerical tests and illustrations with \textsf{OCTAVE} are also presented for each type of polynomial matrices.