One of the main applications of semidefinite programming (SDP) lies in linear systems and control theory. Many problems in this subject, certainly the textbook classics, have matrices as variables, and the formulas naturally contain non-commutative polynomials in matrices. These polynomials depend only on the system layout and do not change with the size of the matrices involved, hence such problems are called ``dimension-free". Analyzing dimension-free problems has led to the development recently of free real algebraic geometry (RAG). The main branch of free RAG, free positivity and inequalities, is an analog of classical real algebraic geometry, a theory of polynomial inequalities embodied in algebraic formulas called Positivstellensätze; often free Positivstellensätze have cleaner statements than their commutative counterparts. In this talk we present some of the latest theoretical advances, with focus on algorithms and their implementations in our computer algebra system NCSOStools. The talk is based on joint works with the following co-authors: J. Povh, K. Cafuta, S. Burgdorf, J. W. Helton, S. McCullough, M. Schweighofer.