Youla-Kučera Parameterization: Theory and Applications

Vladimír Kučera

Czech Institute of Informatics, Robotics, and Cybernetics, Czech Technical University in Prague, Czech Republic

Keywords: Linear Systems, Feedback, Stabilizing Controllers, Controller Parameterization, Multitask Controllers.

Abstract: Youla-Kučera parameterization is the parameterization of all linear controllers that stabilize a given linear

plant.

EXTENDED ABSTRACT

Youla, Bongiorno, and Jabr (Polytechnic Institute of

New York University) and Kučera independently

discovered the parameterization formula in the late

seventies, see (Youla et al., 1976a), (Youla et al.,

1976b) and (Kučera, 1975), (Kučera, 1979).

Ten years later, Vidyasagar (1985) provided a

comprehensive account of the result. Quadrat (2003)

generalized the parameterization results from

lumped-parameter systems to a class of distributed-

parameter linear systems. The survey paper

(Anderson, 1988) summarized the first twenty years

of theoretical developments. In contrast, the recent

survey paper (Mahtout et al., 2020) collects the latest

developments and industrial applications and

provides an impressive list of references.

Parameterization is essential when control

systems are designed to be stable and meet additional

performance specifications (Kučera, 1993). The

specifications beyond stability are achieved by

selecting an appropriate parameter. There is a one-to-

one correspondence between the set of stabilizing

controllers and the set of parameters. Furthermore,

the parameter appears linearly in the closed-loop

system transfer function, whereas the controller

appears nonlinearly. Selecting the parameter instead

of the controller thus simplifies the design

significantly. The system is made stable first, and then

the additional specifications can be accommodated,

one at a time.

Performance specifications, such as optimality

and robustness, are often conflicting and challenging

to achieve using a single controller. In such a case,

https://orcid.org/0000-0001-9397-6931

parameterization allows the designer to reconfigure

the controller to reach satisfactory performance while

guaranteeing overall system stability.

The Youla-Kučera parameterization is a

fundamental result that launched an entirely new area

of research and has been used to solve many control

problems (Kučera, 2007), ranging from optimal

control, robust control, disturbance and noise

rejection, and vibration control to stable controller

switching and fault-tolerant control.

There is a dual parameterization, which describes

all linear systems stabilized by a given linear

controller (Anderson, 1988). The parameter can then

describe plant variations. This is useful for solving the

problem of closed-loop plant identification (Hansen,

et al., 1989). Open-loop identification is more

straightforward, but it is often prohibitive to

disconnect the plant. Identifying the dual parameter

instead of the plant is a linear problem like open-loop

identification.

This keynote presentation is a guided tour through

the theory and applications of the Youla-Kučera

parameterization. It explains the origins of the result,

the derivation of the parameterization formula using

the transfer functions, and the state-space

representation of all stabilizing controllers (Nett et al.,

1984), (Kučera, 2011). It also explains how to select

the parameter to satisfy specific design requirements.

New and exciting applications of the Youla–Kučera

parameterization are then discussed: stabilization

subject to input constraints (Henrion et al., 2001),

output overshoot reduction (Henrion et al., 2005a),

and fixed-order stabilizing controller design (Henrion

et al., 2003), (Henrion et al., 2005b). A selection of

applications in different control fields is presented,

cera, V.

Youla-Ku

cera Parameterization: Theory and Applications.

DOI: 10.5220/0013113700003822

Paper published under CC license (CC BY-NC-ND 4.0)

In Proceedings of the 21st International Conference on Informatics in Control, Automation and Robotics (ICINCO 2024) - Volume 2, pages 9-10

ISBN: 978-989-758-717-7; ISSN: 2184-2809

showing the efficiency of this approach in controlling

complex systems (Cifdaloz et al., 2008), (Trangbaek

and Bendtsen, 2009), and (Mahtout et al., 2018).

ACKNOWLEDGEMENTS

This work was co-funded by the European Union

under Project Robotics and Advanced Industrial

Production CZ.02.01.01/00/22_008/0004590.

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