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.
[1] The engineering significance of the YK formula is that if one wants to find a stabilizing controller that meets some additional criterion, one can adjust the parameter Q such that the desired criterion is met.
For ease of understanding and as suggested by Kučera it is best described for three increasingly general kinds of plant.
be a transfer function of a stable single-input single-output system (SISO) system.
be a set of stable and proper functions of
Then, the set of all proper stabilizing controllers for the plant
is an arbitrary proper and stable function of s. It can be said, that
parametrizes all stabilizing controllers for the plant
Consider a general plant with a transfer function
Further, the transfer function can be factorized as Now, solve the Bézout's identity of the form where the variables to be found
are found, we can define one stabilizing controller that is of the form
The set of all stabilizing controllers is defined as In a multiple-input multiple-output (MIMO) system, consider a transfer matrix
The factors must be proper, stable and doubly coprime, which ensures that the system
This can be written by Bézout identity of the form: After finding
that are stable and proper, we can define the set of all stabilizing controllers
using left or right factor, provided having negative feedback.
is an arbitrary stable and proper parameter.
Let their right coprime factorizations be: then all stabilizing controllers can be written as where