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.
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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.
Stable SISO Plant
Let
where
General SISO Plant
Consider a general plant with a transfer function
Now, solve the Bézout's identity of the form
where the variables to be found (X(s), Y(s)) must be also proper and stable.
After proper and stable X, Y were found, we can define one stabilizing controller that is of the form
General MIMO plant
In a multiple-input multiple-output (MIMO) system, consider a transfer matrix
After finding
where
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 Q such that the desired criterion is met.