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.

• 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).