Youla–Kucera parametrization

In control theory the Youla–Kučera parametrization (also simply known as Youla parametrization) is a formula that describes all possible stabilizing feedback controllers for a given plant P, as function of a single parameter Q.

The YK parametrization is a general result. It is a fundamental result of control theory and launched an entirely new area of research and found application, among others, in optimal and robust control.

For ease of understanding and as suggested by Kučera it is best described for three increasingly general kinds of plant.

Let ${\displaystyle P(s)}$ be a transfer function of a stable Single-input single-output system (SISO) system. Further, let Ω be a set of stable and proper functions of s. Then, the set of all proper stabilizing controllers for the plant ${\displaystyle P(s)}$ can be defined as

${\displaystyle \left\{{\frac {Q(s)}{1-P(s)Q(s)}},Q(s)\in \Omega \right\}}$,

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