ScienceDirect IFAC-PapersOnLine 48-14 (2015) 186–191 Available online at www.sciencedirect.com 2405-8963 © 2015, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Peer review under responsibility of International Federation of Automatic Control. 10.1016/j.ifacol.2015.09.455 © 2015, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Z. Szab´ o * , J. Bokor *** and F. Schipp * * Institute for Computer Science and Control, Hungarian Academy of Sciences, Budapest, Kende u. 13-17, Hungary, *** Institute for Computer Science and Control, Hungarian Academy of Sciences, Hungary, MTA-BME Control Engineering Research Group. Abstract: In order to design efficient algorithms that work on the set of controllers that fulfill a given property, e.g., stability or a norm bound, it is important to have an operation that preserves that property, i.e., a suitable blending method. Concerning stability, a traditional approach is to use the Youla parametrization and the corresponding parameters as a starting point. While this method guarantees stabilizability as the invariant property for the fairly large class of strictly proper plants, there are also other solutions to the problem. The authors already provide a detailed analysis for feedback stability placing the controller blending problem in a general setting by pointing to the basic global geometric structures that are related to well-posedness and feedback stability. In this paper these efforts are continued and the group structure corresponding to performance problems, e.g., those related to a suboptimal H ∞ design, is presented. Besides its educative value the presentation provides a possible tool for the algorithmic development. Blending for H ∞ performance: a group theoretic approach Keywords: controller parametrisation; controller blending; performance guarantee 1. INTRODUCTION Felix Klein, in the late 1800s, developed an axiomatic basis for Euclidean geometry that started with the notion of an existing set of transformations and he proposed that geometry should be defined as the study of transformations (symmetries) and of the objects that transformations leave unchanged, or invariant. The set of symmetries of an object has a very nice algebraic structure: they form a group. By studying this algebraic structure, we can gain deeper insight into the geometry of the figures under consideration. A common tool in formulating robust feedback control problems is to use system interconnections that can be described as linear fractional transforms (LFTs), as a general framework to include the rational dependencies that occur. In this context a special role plays the intimate relationship between M¨ obius transformations and LFTs. It is known that the set of stabilizing controllers for a given plant and the set of all suboptimal H ∞ can be expressed by using certain M¨ obius transformations and LFTs, respectively. In Szab´ o et al. [2014] the authors emphasise Klein’s ap- proach to geometry and demonstrate that a natural frame- work to formulate different control problems is the world that contains as points equivalence classes determined by stabilizable plants and whose natural motions are the M¨ obius transforms. The observation that any geometric property of a configuration, which is invariant under an euclidean or hyperbolic motion, may be reliably investi- gated after the data has been moved into a convenient position in the model, facilitates considerably the solution of the problems. In contrast to traditional geometric control theory, see, e.g., Wonham [1985], Basile and Marro [2002] for the linear and Isidori [1989], Jurdjevic [1997], Agrachev and Sachkov [2004] for the nonlinear theory, which is centered on a local view, this approach provides a global view. While the former uses tools from differential geometry, Lie algebra, algebraic geometry, and treats system concepts like controllability, as geometric properties of the state space or its subspaces the latter focuses on an input- output – coordinate free – framework where different transformation groups which leave a given global property invariant play a fundamental role. In the first case the invariants are the so-called invariant or controlled invariant subspaces, and the suitable change of coordinates and system transforms (diffeomorphisms), see, e.g., the Kalman decomposition, reveal these properties. In contrast, our interest is in the transformation groups that leave a given global property, e.g., stability or H ∞ norm, invariant. One of the most important consequences of the approach is that through the analogous of the classical geometric constructions it not only might gave hints for efficient algorithms but the underlaying algebraic struc- ture, i.e., the given group operation, also provides tools for controller manipulations that preserves the property at hand, called controller blending. There are a lot of applications for controller blending: both in the LTI system framework, Niemann and Stoustrup [1999], Stoustrup [2009] and in the framework using gain- scheduling, LPV techniques, see Shin et al. [2002], Chang