An introduction to internal stabilization of infinite-dimensional linear systems

In these notes, we give a short introduction to the fractional representation approach to analysis and synthesis problems [12], [14], [17], [28], [29], [50], [71], [77], [78]. In particular, using algebraic analysis (commutative algebra, module theory, homological algebra, Banach algebras), we shall give necessary and sufficient conditions for a plant to be internally stabilizable or to admit (weakly) left/right/doubly coprime factorizations. Moreover, we shall explicitely characterize all the rings A of SISO stable plants such that every plant − defined by means of a transfer matrix with entries in the quotient field K = Q(A) of A − satisfies one of the previous properties (e.g. internal stabilization, (weakly) doubly coprime factorizations). Using the previous results, we shall show how to parametrize all stabilizing controllers of an internally stabilizable plants which does not necessarily admits a doubly coprime factorization. Finally, we shall give some necessary and sufficient conditions so that a plant is strongly stabilizable (i.e. stabilizable by a stable controller) and prove that every internally stabilizable MIMO plant over A = H∞(C+) is strongly stabilizable.