Optimization Routines for Gain Scheduling - CENIIT Project

Background and Industrial Motivation

The objective of this research proposal is to make analysis and design of gain scheduled controllers feasible for the general control community, including industry. Today this is not the case because of the lack of reliable software tools. One of the reasons for this is partly due to the fact that to analyze and design gain scheduled controllers often very large-scale Semidefinite Programs (SDPs) have to be solved, and general purpose software for SDPs cannot handle these huge problems. Even if the software in case of design is able to produce a controller that in theory stabilizes the closed loop system, the controller is often of too high order to be practically usable. In most applications, if not all, lower-order controllers are favored, at least if t he performance degradation is small or negligible. Another reason why current research results on gain scheduling are not used in practice is conservatism, i.e. often analysis and design of gain scheduled controllers result in the conclusion that it is not possible to determine if the analyzed system is stable or not, or if a stabilizing controller can be designed. However, practitioners in industry still often can design and verify that the closed loop system behaves satisfactory in simulations, however at huge costs in terms of labor and simulation time. These problems have been recognized by the Group for Aeronautical Research and Technology in EURope (GARTEUR), in which both aerospace industry and research organizations have been involved. The aim of this research proposal is to continue to bridge this gap between theory and applicability.

Objectives

This project focuses on the theoretical aspects of efficient optimization algorithms for analysis and design of gain scheduled controllers. The following research topics have been, are or will be investigated:

• Work have been carried out to develop efficient Interior-Point algorithms for SDPs related to the KYP lemma.
• Very promising results have also been obtained for these SDPs using an analytical center cutting-plane method.
• For many approaches to gain scheduling it is desirable to solve optimization problems involving parameterized LMIs. Under certain assumptions the parameterized LMIs are equivalent to a finite number of LMIs involving the same symmetric matrix variable. The essential difference as compared to the SDPs related to the KYP lemma is that there is more than one constraint for this variable. We have investigated how our results on iterative solvers for computing search directions for IP methods can be extended to this case. The challenge has been to develop efficient preconditioners.
• We are currently investigating how to make use of the general matrix variable structure often present in SDPs originating from control application. This work also involves an interface to YALMIP.
• There are various parameterizations proposed for analysis of LPV systems and synthesis of gain scheduling controllers in the litterature. Here, a better understanding of which parameterization to choose in different situations is required for finding more efficient solvers. Initial investigations are being carried out in collaboration with University of Siena.

It is surprising how little work has been done to develop efficient solvers customized to SDPs for specific control applications, especially considering the tremendous efforts that has been put into reformulating different control problems in terms of SDPs. The aim of this project is to continue filling out this huge gap. It should also be stressed that the results that will come out of this research proposal will not only be applicable to gain scheduling but hopefully also directly or indirectly to many other analysis and design problems in control theory. This is the long term objective of the project, which we have recently started to work on.

Results Achieved

Results achieved within the project related to items 1, 2 and 3 have been published, see publications below. Initial results related to items 4 and 5 have not yet been published.

Industrial Collaboration

In order to test our research results on industrial problems two application projects have been launched. One of the projects have been carried out in collaboration with Saab AB and Saab Bofors Dynamics AB in which clearance of flight control laws for fighter aircraft and missiles have been investigated. This has been funded by VINNOVA. The other project is an EC-project, COFCLUO, co-ordinated by Anders Hansson and carried out in collaboration with AIRBUS, ONERA, DLR, FOI and University of Siena in which clearence of flight control laws for civil aircraft is investigate. Contact persons are Fredrik Karlsson at Saab AB, Thomas Svantesson at Saab Bofors Dynamics AB and Guilhem Puyou at AIRBUS.

Publications

• M. Enqvist, A. Hansson, G. Hendeby E. Geijer-Lundin, D. Lindgren, J. Sjöberg, H. Tidefelt, D. Törnqvist, and R. Wallin: Utilization of low-rank structure in control applications of semidefinite programming, In: SIAM Conference on Optimization, Stockholm, Sweden, May 2005.
• L. Vandenberghe, R. Balakrishnan, R. Wallin, A. Hansson, and T. Roh: Interior-point algorithms for semidefinite programming problems derived from the KYP lemma, In: A. Garulli and D. Henrion, editors, Positive Polynomials in Control, Lecture Notes in Control and Information Sciences, pages 193--237. Springer Verlag, New York, 2005.
• R. Wallin, C.-Y. Kao, and A. Hansson: A decomposition approach for solving KYP-SDPs, In: Proceedings of hte 16 th IFAC World Congress, Prague, Czech Republic, July 2005.
• J. Harju, R. Wallin, and A. Hansson: Utilizing low rank properties when solving KYP-SDPs, In: Proceedings of 2006 IEEE Confrerence on Decision and Control, San Diego, 2006.
• R. Wallin, C.-Y. Kao, and A. Hansson: A cutting-pane method for solving KYP-SDPs, Automatica, 2008.
• J. H. Johansson: A structure utilizing inexact primal-dual interior-point method for analysis of linear differential inclusions, Licentiate Thesis no. 1367, Department of Electrical Engineering, Linköping university, May 2008.
• J. H. Johansson and A. Hansson: Structure exploitation in Semi-Definite Programs for Systems Analysis, IFAC World Congress, Seoul, July 2008.
• J. H. Johansson and A. Hansson: A tailored inexact interior-point method for systems analysis, IEEE Conference on Decision and Control, Cancun, Mexico, December 2008.
• R. Wallin, A. Hansson, and J. H. Johansson: A structure-exploiting solver for semidefinite programs derived from the Kalman-Yakubovich-Popov lemma, IEEE Transactions on Automatic Control, 2008.
• R. Falkeborn and A. Hansson: A decomposition algortim for KYP-SDPs, European Control Conference, 2009.
• J. H. Johansson and A. Hansson: An Inexact Interior-Point Method for System Analysis, International Journal of Control, to appear.