Brief PaperController tuning freedom under plant identification uncertainty: double Youla beats gap in robust stability☆
Section snippets
Problem set-up
We consider linear time-invariant finite dimensional systems and controllers, in a feedback configuration depicted in Fig. 1, denoted by H(G0,C), where G0 is the plant to be (modelled and) controlled, and C a present and known controller.
The closed-loop dynamics of H(G0,C) are described by the transfer matrixwhich maps the vector of variables col(r1,r2) into col(y,u).1 The closed-loop system is stable if and only if .2
Preliminaries
A coprime factor framework will be used to represent plants and controllers, employing both right and left coprime factorizations:where (N,D) and (Nc,Dc) are right coprime factorizations (rcf) and and are left coprime factorizations (lcf) over (Vidyasagar, 1985). The coprime factorizations are normalized (nrcf), (nlcf) if they additionally satisfy and , where denotes complex
Robust stability results for double-Youla representations
Uncertainty on a model Gx can be described in very many different ways. In a norm-bounded formulation, there are options for additive, multiplicative, coprime-factor, gap-metric uncertainties, all having their particular robust stability tests. See e.g. de Callafon, Van den Hof, and Bongers (1996) for an overview in a rather uniform (coprime factor) framework.
When considering robust performance tests on norm-bounded uncertainty sets, it has been motivated in de Callafon and Van den Hof (1997)
Gap metric results
When considering the gap metric as a measure for bounding plant uncertainty a similar analysis can be given as presented in the previous section. The gap metric distance between two systems Gx, GΔ is defined bywhere the directed gap is:where and are nrcf’s of Gx and GΔ. The stability result that is applicable to our problem set-up is the following. Proposition 3 LetH(Gx,C) be stable. ThenH(GΔ,CΔ) is stable ifGeorgiou and Smith, 1990
Comparison of the two uncertainty structures
Theorem 1 Given a set of plantsand a set of controllerssatisfying the gap stability condition of Corollary 1. Then for the sets of Proposition 2 withQ=Qc=I, it holds that , with , with , i.e. the two sets satisfy the stability condition of Proposition 2.
Proof
The proof is provided in Appendix A.
The result of this theorem implies that even when embedding the gap
Extension to ν-gap and Λ-gap
The analysis as presented in this paper so far can readily be extended to other uncertainty structures as well, as e.g. the ν-gap and the Λ-gap. The Vinnicombe or ν-gap metric is defined as (Vinnicombe, 1993):where W(g) denotes the winding number about the origin of g(s) as s follows the standard Nyquist D-contour.
The Λ-gap between two plants Gx and GΔ is defined as (Bongers, 1991; Bongers, 1994
Example
An example is considered in which robust stability is guaranteed by the condition of Proposition 2, but not by the gap-metric conditions of Corollary 1, Corollary 3, Corollary 4. We consider a (physical model) of a rotating drive system G0 and an identified model Ĝ based on input and output data. Fig. 3 shows that the model Ĝ is a fairly good model, although the error is quite large near the resonance frequencies. A controller C is designed, on the basis of the model Ĝ, which stabilizes
Concluding remarks
The use of different uncertainty structures for specifying model uncertainty leads to different sets of robustly stabilizing controllers. These controller sets generally intersect, and therefore a “best” choice cannot be made. However, when restricting attention to controller sets that can be described as a metric-bounded perturbation of a nominal/present controller, the use of a double Youla parametrization for representing plant and controller uncertainty is shown to be less conservative than
Acknowledgements
The authors would like to thank Ruud Schrama, Peter Bongers and Xavier Bombois for fruitful discussions on the topic of this paper.
Sippe G. Douma was born in Naaldwijk, The Netherlands, in 1974. He obtained the M.Sc. degree from the Department of Mechanical Engineering, Delft University of Technology, The Netherlands, 1999. He is currently working for his Ph.D. degree at the Department of Applied Physics, Delft University of Technology. His main research interests are in issues of identification for control.
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2005, AutomaticaCitation Excerpt :While extensive literature exists dealing with characteristics of each uncertainty structure, answering the posed question requires a thorough comparison and a bridging of the gap between identification and robust control that goes beyond the present state of the art. A first attempt with limited scope only directed towards robust stability issues was made in Douma, Van den Hof, and Bosgra (2003). Computational tools for calculating robust performance over uncertainty sets are available for many structures (Zhou et al., 1996; Vinnicombe, 2001).
Sippe G. Douma was born in Naaldwijk, The Netherlands, in 1974. He obtained the M.Sc. degree from the Department of Mechanical Engineering, Delft University of Technology, The Netherlands, 1999. He is currently working for his Ph.D. degree at the Department of Applied Physics, Delft University of Technology. His main research interests are in issues of identification for control.
Paul Van den Hof was born in 1957 in Maastricht, The Netherlands. He received the M.Sc. and Ph.D. degrees both from the Department of Electrical Engineering, Eindhoven University of Technology, The Netherlands in 1982 and 1989, respectively. From 1986 to 1999 he was an assistant and associate professor in the Mechanical Engineering Systems and Control Group of Delft University of Technology, The Netherlands. Since 1999 he is a full professor in the Signals, Systems and Control Group of the Department of Applied Physics at Delft University of Technology, and since 2003 Co-Director of the Delft Center for Systems and Control. He is acting as General Chair of the 13th IFAC Symposium on System Identification, to be held in Rotterdam, the Netherlands in 2003.
Paul Van den Hof's research interests are in issues of system identification, parametrization, signal processing and (robust) control design, with applications in mechanical servo systems, physical measurement systems, and industrial process control systems.
He is a member of the IFAC Council and Automatica Editor for Rapid Publications.
Okko H. Bosgra was born in Groningen, The Netherlands in 1944. Since 1986 he is a full professor of Systems and Control in the Department of Mechanical Engineering, Delft University of Technology, The Netherlands. His research interests are in the area where systems and control theory interacts with applications to mechatronics and process systems.
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A preliminary version of this paper was presented at the 15th IFAC World Congress, July 2002, Barcelona Spain. This paper was recommended for publication in revised form by Associate Editor Toshiharu Sugie under the direction of Editor Roberto Tempo. This work is part of the research program of the ‘Stichting voor Fundamenteel Onderzoek der Materie (FOM)’, which is financially supported by the ‘Nederlandse Organisatie voor Wetenschappelijk Onderzoek (NWO)’.