A new approach to the robust stability analysis and to the robust controller design is proposed via reflection coefficients of characteristic polynomials of discrete-time systems. The reason of using reflection coefficients instead of roots of the characteristic polynomial is that the mapping between reflection coefficients and polynomial coefficients is very simple, multilinear. So we can easily find Schur stable line segments in the polynomial coefficient space by varying a single reflection coefficient. The more serious tasks are: how to find the stability domain in the system parameter space and how to design a stabilizing fixed-order controller for an unstable plant?

We propose a randomized/deterministic approach for generating the (non-convex) stability domains both in the closed-loop system parameter space and in the controller parameter space via stable Schur segment bunches. The task of generation of stable line segments via reflection coefficients is simple, efficient and numerically stable. The robust controller design problem is formulated as a stability margin maximization task over the set of Schur segment bunches.

Future research will focus on the robust control of continuous-time systems. We have introduced the reduced Routh parameters and Routh rays of characteristic polynomials of continuous-time systems. We are looking forward to randomized/deterministic synthesis of stabilizing fixed-order controllers for unstable continuous-time plants via stable bunches of Routh rays.