Youla–Kucera parametrization

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.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] 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.

Let $P(s)$ be a transfer function of a stable single-input single-output system (SISO) system. Further, let $\backslash Omega$ 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

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

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 $P(s)$.

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

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

Now, solve the Bézout's identity of the form

:$\backslash mathbf\{N(s)Y(s)\}\; +\; \backslash mathbf\{M(s)X(s)\}\; =\; \backslash mathbf\{1\}$,

where the variables to be found $(X(s),\; Y(s))$ must be also proper and stable.

After proper and stable $X,\; Y$ are found, we can define one stabilizing controller that is of the form $C(s)=\backslash frac\{Y(s)\}\{X(s)\}$. 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

:$\backslash left\backslash \{\; \backslash frac\{Y(s)+M(s)Q(s)\}\{X(s)\; -\; N(s)Q(s)\},\; Q(s)\; \backslash in\; \backslash Omega\; \backslash right\backslash \}$.

In a multiple-input multiple-output (MIMO) system, consider a transfer matrix $\backslash mathbf\{P(s)\}$. It can be factorized using right coprime factors $\backslash mathbf\{P(s)=N(s)D^\{-1\}(s)\}$ or left factors $\backslash mathbf\{P(s)=\backslash tilde\{D\}^\{-1\}(s)\backslash tilde\{N\}(s)\}$. The factors must be proper, stable and doubly coprime, which ensures that the system $\backslash mathbf\{P(s)\}$ 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 $\backslash mathbf\{X,\; Y,\; \backslash tilde\{X\},\; \backslash tilde\{Y\}\}$ that are stable and proper, we can define the set of all stabilizing controllers $\backslash mathbf\{K(s)\}$ 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 $\backslash Delta$ is an arbitrary stable and proper parameter.

Let $P(s)$ be the transfer function of the plant and let $K\_0(s)$ be a stabilizing controller. Let their right coprime factorizations be:

:$\backslash mathbf\{P(s)\}=\; \backslash mathbf\{N\}\; \backslash mathbf\{M\}^\{-1\}$

:$\backslash mathbf\{K\_0(s)\}\; =\; \backslash mathbf\{U\}\; \backslash mathbf\{V\}^\{-1\}$

then all stabilizing controllers can be written as

:$\backslash mathbf\{K(s)\}\; =\; (\backslash mathbf\{U\}+\backslash mathbf\{M\}\; \backslash mathbf\{Q\})\; (\backslash mathbf\{V\}+\backslash mathbf\{N\}\; \backslash mathbf\{Q\})^\{-1\}$

where $Q$ is stable and proper.[http://www.inf.ethz.ch/personal/cellier/Lect/NMC/Lect_nmc_index.html Cellier: Lecture Notes on Numerical Methods for control, Ch. 24]

{{reflist}}

• D. C. Youla, H. A. Jabr, 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 Tannenbaum. Feedback control theory. (1990). [http://www.gest.unipd.it/~oboe/psc/testi/dft.pdf]

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Category:Control theory