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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.<ref>V. Kučera. A Method to Teach the Parameterization of All Stabilizing Controllers. 18th IFAC World Congress. Italy, Milan, 2011.[http://www.nt.ntnu.no/users/skoge/prost/proceedings/ifac11-proceedings/data/html/papers/1148.pdf]</ref>


For ease of understanding and as suggested by Kučera it is best described for three increasingly general kinds of plant.
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===Stable SISO Plant===
 
Let <math>P(s)</math> 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 <math>P(s)</math> can be defined as
 
<math>\left\{ \frac{Q(s)}{1 - P(s)Q(s)}, Q(s)\subset \Omega \right\}</math>,
 
where <math>Q(s)</math> is an arbitrary proper and stable function of ''s''. It can be said, that <math>Q(s)</math> parametrizes all stabilizing controllers for the plant <math>P(s)</math>.
 
===General SISO Plant===
 
Consider a general plant with a transfer function <math>P(s)</math>. Further, the transfer function can be factorized as
 
<math>P(s)=\frac{N(s)}{M(s)}</math>, where M(s), N(s) are stable and proper functions of ''s''.
 
Now, solve the [[Bézout's identity]] of the form
 
'''<math> \mathbf{N(s)X(s)} + \mathbf{M(s)Y(s)} = \mathbf{1} </math>''',
 
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 <math>C(s)=\frac{X(s)}{Y(s)}</math>. 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
 
<math>\left\{ \frac{X(s)+M(s)Q(s)}{Y(s) - N(s)Q(s)}, Q(s)\subset \Omega \right\}</math>,
 
===General MIMO plant===
 
In a multiple-input multiple-output (MIMO) system, consider a transfer matrix <math>\mathbf{P(s)}</math>. It can be factorized using right coprime factors <math>\mathbf{P(s)=N(s)D^{-1}(s)}</math>  or left factors <math>\mathbf{P(s)=\tilde{D}^{-1}(s)\tilde{N}(s)}</math>. 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
 
<math>
\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]
</math>.
 
After finding <math>\mathbf{X, Y, \tilde{X}, \tilde{Y}}</math> that are stable and proper, we can define the set of all stabilizing controllers '''H(s)''' using left or right factor, provided having negative feedback.
 
<math>
\begin{align}
  & \mathbf{H(s)}={{\left( \mathbf{X}-\mathbf{K\tilde{N}} \right)}^{-1}}\left( \mathbf{Y}-\mathbf{K\tilde{D}} \right) \\
& =\left( \mathbf{\tilde{Y}}+\mathbf{DK} \right){{\left( \mathbf{\tilde{X}}-\mathbf{NK} \right)}^{-1}} 
\end{align}
 
</math>
 
where '''K(s)''' is an arbitrary stable and proper parameter.
<!-- Please correct the form of this page, I am still learning wiki formatting -->
<!--
Let <math>P(s)</math> be the transfer function of the plant and let <math>K_0(s)</math> be a stabilizing controller. Let their right coprime factorizations be:
:<math>P = N M^{-1}</math>
:<math>K_0 = U V^{-1}</math>
then '''all''' stabilizing controllers can be written as
:<math>K = (U+M Q) (V+N Q)^{-1}</math>
where Q is stable and proper.<ref>[http://www.inf.ethz.ch/personal/cellier/Lect/NMC/Lect_nmc_index.html Cellier: Lecture Notes on Numerical Methods for control, Ch. 24]</ref>
-->
 
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==
{{reflist}}
*D. C. Youla, H. A. Jabri, J. J. Bongiorno: Modern Wiener-Hopf design of optimal controllers: part II, IEEE Trans. Automat. Contr., AC-21 (1976) pp319–338
*V. Kučera: Stability of discrete linear feedback systems. In: Proceedings of the 6th IFAC. World Congress, Boston, MA, USA, (1975).
*C. A. Desoer, R.-W. Liu, J. Murray, R. Saeks. Feedback system design: the fractional representation approach to analysis and synthesis. IEEE Trans. Automat. Contr., AC-25 (3), (1980) pp399–412
*John Doyle, Bruce Francis, Allen Tannenbau. Feedback control theory. (1990). [http://www.gest.unipd.it/~oboe/psc/testi/dft.pdf]
 
{{DEFAULTSORT:Youla-Kucera parametrization}}
[[Category:Control theory]]

Latest revision as of 23:44, 26 April 2014


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