Remarks on the Pole-shifting Problem over Rings

Problems that appear in trying to extend linear control results to systems over rings R have attracted considerable attention lately. This interest has been due mainly to applications-oriented motivations (in particular, dealing with delaydifferential equations), and partly to a purely algebraic interest. We shall not touch here on the (nonalgebraic) motivationsmany can be found in the various references given-save to note that interest in applications lies not with arbitrary rings R but with certain broad classes, such as polynomial rings over R or C (delay systems), integers and finite rings (digital systems, coding), rings of suitably smooth real or complex functions (parametrized families of systems), and group algebras with real or complex coefficients (discretized p.d.e.‘s on certain manifolds). In this note, we shall restrict our attention to the problem(s) of modifying the characteristic polynomial of a given system through the use of feedback. A system (with m inputs, of dimension n, over the commutative ring R) is just a pair of matrices (F, G) over R, where F is n by n, and G is n by m. A feedback (matrix) for this system is any m by n R-matrix K. The closed-loop system obtained applying feedback K to the system (F, G) is by definition the new system (F+ GK, G). We shall be interested in the characteristicpolynomial of the system (F, G), meaning just the characteristic polynomial of F, ch.p.(F) = det(zlF).