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.

Details

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]

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

Stable SISO Plant

Let 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 P(s) can be defined as

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

where Q(s) is an arbitrary proper and stable function of s. It can be said, that Q(s) parametrizes all stabilizing controllers for the plant P(s).

General SISO Plant

Consider a general plant with a transfer function P(s). Further, the transfer function can be factorized as

P(s)=\frac{N(s)}{M(s)}, where M(s), N(s) are stable and proper functions of s.

Now, solve the Bézout's identity of the form

 \mathbf{N(s)X(s)} + \mathbf{M(s)Y(s)} = \mathbf{1} ,

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 C(s)=\frac{X(s)}{Y(s)}. After we have one stabilizing controller at hand, we can define all stabilizing controllers using a parameter Q(s) that is proper and stable. The set of all stabilizing controllers is defined as

\left\{ \frac{X(s)+M(s)Q(s)}{Y(s) - N(s)Q(s)}, Q(s) \in \Omega \right\},

General MIMO plant

In a multiple-input multiple-output (MIMO) system, consider a transfer matrix \mathbf{P(s)}. It can be factorized using right coprime factors \mathbf{P(s)=N(s)D^{-1}(s)} or left factors \mathbf{P(s)=\tilde{D}^{-1}(s)\tilde{N}(s)}. The factors must be proper, stable and doubly coprime, which ensures that the system P(s) is controllable and observable. This can be written by Bézout identity of the form


\left[ \begin{matrix}
   \mathbf{X} & \mathbf{Y}  \\
   -\mathbf{\tilde{N}} & {\mathbf{\tilde{D}}}  \\
\end{matrix} \right]\left[ \begin{matrix}
   \mathbf{D} & -\mathbf{\tilde{Y}}  \\
   \mathbf{N} & {\mathbf{\tilde{X}}}  \\
\end{matrix} \right]=\left[ \begin{matrix}
   \mathbf{I} & 0  \\
   0 & \mathbf{I}  \\
\end{matrix} \right]
.

After finding \mathbf{X, Y, \tilde{X}, \tilde{Y}} that are stable and proper, we can define the set of all stabilizing controllers K(s) using left or right factor, provided having negative feedback.


\begin{align}
  & \mathbf{K(s)}={{\left( \mathbf{X}-\mathbf{\Delta\tilde{N}} \right)}^{-1}}\left( \mathbf{Y}+\mathbf{\Delta\tilde{D}} \right) \\ 
 & =\left( \mathbf{\tilde{Y}}+\mathbf{D\Delta} \right){{\left( \mathbf{\tilde{X}}-\mathbf{N\Delta} \right)}^{-1}}  
\end{align}

where  \Delta is an arbitrary stable and proper parameter.


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.

References

  1. V. Kučera. A Method to Teach the Parameterization of All Stabilizing Controllers. 18th IFAC World Congress. Italy, Milan, 2011.
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