In
control theory
Control theory is a field of control engineering and applied mathematics that deals with the control system, control of dynamical systems in engineered processes and machines. The objective is to develop a model or algorithm governing the applic ...
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. 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.
Stable SISO plant
Let
be a transfer function of a stable
single-input single-output system
In control engineering, a single-input and single-output (SISO) system is a simple single-Variable (computer science), variable control system with one input and one output. In radio, it is the use of only one antenna (radio), antenna both in the ...
(SISO) system. Further, let
be a set of stable and proper functions of ''
''. Then, the set of all proper stabilizing controllers for the plant
can be defined as
:
,
where
is an arbitrary proper and stable function of ''s''. It can be said, that
parametrizes all stabilizing controllers for the plant
.
General SISO plant
Consider a general plant with a transfer function
. Further, the transfer function can be factorized as
:
, where
,
are stable and proper functions of ''s''.
Now, solve the
Bézout's identity
In mathematics, Bézout's identity (also called Bézout's lemma), named after Étienne Bézout who proved it for polynomials, is the following theorem:
Here the greatest common divisor of and is taken to be . The integers and are called B� ...
of the form
:
,
where the variables to be found
must be also proper and stable.
After proper and stable
are found, we can define one stabilizing controller that is of the form
. After we have one stabilizing controller at hand, we can define all stabilizing controllers using a parameter
that is proper and stable. The set of all stabilizing controllers is defined as
:
.
General MIMO plant
In a multiple-input multiple-output (MIMO) system, consider a transfer matrix
. It can be factorized using right coprime factors
or left factors
. The factors must be proper, stable and doubly coprime, which ensures that the system
is controllable and observable. This can be written by Bézout identity of the form:
:
.
After finding
that are stable and proper, we can define the set of all stabilizing controllers
using left or right factor, provided having negative feedback.
:
where
is an arbitrary stable and proper parameter.
Let
be the transfer function of the plant and let
be a stabilizing controller. Let their right coprime factorizations be:
:
:
then all stabilizing controllers can be written as
:
where
is stable and proper.
Cellier: Lecture Notes on Numerical Methods for control, Ch. 24
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References
*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)
{{DEFAULTSORT:Youla-Kucera parametrization
Control theory