$D$-decomposition technique state-of-the-art

It is a survey of recent extensions and new applications for the classical $D$-decomposition technique. We investigate the structure of the parameter space decomposition into root invariant regions for single-input single-output systems linear depending on the parameters. The $D$-decomposition for uncertain polynomials is considered as well as the problem of describing all stabilizing controllers of the certain structure (for instance, PID-controllers) that satisfy given $H_\infty$. It is shown that the $D$-decomposition technique can be naturally linked with $M-\Delta$ framework (a general scheme for analysis of uncertain systems) and it is applicable for describing feasible sets for linear matrix inequalities. The problem of robust synthesis for linear systems can be also treated via $D$-decomposition technique.