Systematic mitigation of gain scheduling induced windup phenomena

Klaus Kefferpütz; Benedikt Bartenschlager; Christoph Auenmüller; Sebastian Seitz

doi:10.1515/auto-2021-0127

Published by De Gruyter (O) March 11, 2022

Systematic mitigation of gain scheduling induced windup phenomena

Systematische Vermeidung von Gain Scheduling induzierten Windup Phänomenen

Klaus Kefferpütz

Klaus Kefferpütz received his Diploma in electrical engineering in 2008 and his Ph. D. degree in control theory in 2012 from the Technical University of Darmstadt, Germany. From 2012 to 2018 he worked as a system engineer at MBDA Germany GmbH, Schrobenhausen, Germany. Since 2018 he has been a professor at the University of Applied Sciences Augsburg, Germany. His current research interests are collaborative control, navigation and tracking in multi-agent systems.
, Benedikt Bartenschlager

Benedikt Bartenschlager received his Diploma in aerospace engineering in 2014 from the Technical University of Munich (TUM), Germany. Since 2015 he has been a development engineer at MBDA Deutschland GmbH, Schrobenhausen, Germany.
, Christoph Auenmüller

Christoph Auenmüller received his double degree in the Joint European Master program in Space Science and Technology in 2015 from the Luleå University of Technology in Kiruna, Sweden and 2016 from the Julius-Maximilians-University of Würzburg, Germany. From 2017 to 2020 he worked at Diehl Defence in Röthenbach, Germany, as a development engineer in control engineering. Since 2020 he works as a system engineer at MBDA Germany GmbH, Schrobenhausen, Germany, in the field of missile design and flight mechanics.
and Sebastian Seitz

Sebastian Seitz received his M. Sc. degree in automation engineering from RWTH Aachen University in 2019. As part of his bachelor studies, he completed an internship in the field of flight control at MBDA Germany, Schrobenhausen. Since 2019 he is working on his Ph. D. degree in flight guidance and safe airspace integration of unmanned (tilt-wing) aircraft at the Institute of Flight System Dynamics at RWTH Aachen University.

From the journal at - Automatisierungstechnik

https://doi.org/10.1515/auto-2021-0127

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Abstract

In this paper, we discuss the systematic design of gain scheduling controllers for nonlinear systems. We discuss control signal blending induced windup phenomena and employ an extended Model-Recovery Anti-Windup (MRAW) scheme to mitigate them. Combining control signal blending with MRAW allows to build gain scheduling controllers without imposing the common slow variation assumption regarding the scheduling vectors. In addition, the approach offers a higher flexibility in the choice of the sub-controllers compared to classical approaches and convex optimization allows for a very efficient design of the anti-windup networks.

Zusammenfassung

In diesem Beitrag wird eine systematische Methode zum Entwurf von Gain Scheduling Reglern für nichtlineare Systeme vorgestellt. Dazu wird ein Ansatz, basierend auf Stellsignal-Überblendung verfolgt, der jedoch anfällig für das Auftreten von Regler-Windup selbst bei Abwesenheit von Sättigungseffekten ist. Mit Hilfe der vorgestellten Erweiterung eines modellbasierten Anti-Windup Ansatzes können diese unerwünschten Effekte erfolgreich vermieden werden. Aufgrund des Einsatzes von Stellsignal-Überblendung weist der resultierende Gain Scheduling Regler eine hohe Flexibilität hinsichtlich der verwendbaren Teilregler auf. Durch die Berücksichtigung von Bedingungen zur Vermeidung sogenannter versteckter Koppelterme kann auch auf die weit verbreitete Annahme langsam veränderlicher Scheduling Variablen verzichtet werden.

Keywords: gain scheduling; hidden-couplings; control signal blending; windup; model-recovery anti-windup

Schlagwörter: Gain Scheduling Regelung; Versteckte Koppelterme; modellbasiertes Anti-Windup; Stellsignal-Überblendung

Dedicated to the 60th birthday of Prof. Dr.-Ing. Jürgen Adamy.

About the authors

Klaus Kefferpütz

Klaus Kefferpütz received his Diploma in electrical engineering in 2008 and his Ph. D. degree in control theory in 2012 from the Technical University of Darmstadt, Germany. From 2012 to 2018 he worked as a system engineer at MBDA Germany GmbH, Schrobenhausen, Germany. Since 2018 he has been a professor at the University of Applied Sciences Augsburg, Germany. His current research interests are collaborative control, navigation and tracking in multi-agent systems.

Benedikt Bartenschlager

Benedikt Bartenschlager received his Diploma in aerospace engineering in 2014 from the Technical University of Munich (TUM), Germany. Since 2015 he has been a development engineer at MBDA Deutschland GmbH, Schrobenhausen, Germany.

Christoph Auenmüller

Christoph Auenmüller received his double degree in the Joint European Master program in Space Science and Technology in 2015 from the Luleå University of Technology in Kiruna, Sweden and 2016 from the Julius-Maximilians-University of Würzburg, Germany. From 2017 to 2020 he worked at Diehl Defence in Röthenbach, Germany, as a development engineer in control engineering. Since 2020 he works as a system engineer at MBDA Germany GmbH, Schrobenhausen, Germany, in the field of missile design and flight mechanics.

Sebastian Seitz

Sebastian Seitz received his M. Sc. degree in automation engineering from RWTH Aachen University in 2019. As part of his bachelor studies, he completed an internship in the field of flight control at MBDA Germany, Schrobenhausen. Since 2019 he is working on his Ph. D. degree in flight guidance and safe airspace integration of unmanned (tilt-wing) aircraft at the Institute of Flight System Dynamics at RWTH Aachen University.

Appendix A Proof of Lemma 1

Employing the Lyapunov-function V ( ξ a w ) = ξ a w T P ξ a w and demanding

V ˙ ( ξ a w ) < γ 2 y c , j T y c , j + γ u 2 u ˜ T u ˜ − y a w , u , j T y a w , u , j ,

we have to ensure that ‖ y a w , u , j ‖ L 2 < γ ‖ y c , j ‖ L 2 + γ u ‖ u ˜ ‖ L 2 holds. This can be shown integrating the above inequality and considering ξ a w ( 0 ) = 0. Assuming a LPV dependency of A ( p ), B ( p ) and C ( p ), all trajectories of (19) are contained in the convex hull (20) denoted by co. This assumption introduces some degree of conservatism but in turn allows for arbitrary fast changing scheduling vectors p. Introducing some degree of conservatism, the above inequality is fulfilled, if (21) holds with A k = A ( p k ), B k = B ( p k ) and C k = C ( p k ). In order to derive linear matrix inequalities, we take advantage of the fact that the deadzone nonlinearity dz ( u k ) is locally contained in the sector S [ 0 , Φ ] with Φ = diag ( Φ 1 , … , Φ m ) for all u ∈ T ( Φ ) = u ∈ R m : | u i | ≤ u max 1 − Φ i , i = 1 , … , m . Now, for arguments u ˆ = y c , j + y a w , y , j + u ˜ ∈ Y ( Φ ), where Y ( Φ ) = { y c , j , u ˜ ∈ R m : u ˆ ∈ T ( Φ ) ∀ t ≥ 0 } holds, the local sector condition [17] can be employed. It states that y ˜ T 2 Ψ Φ ( K a w ξ a w − y c , j − u ˜ ) − y ˜ ≥ 0 holds for all y c , j , u ˜ ∈ Y ( Φ ) and the nonlinear function y ˜ = dz ( y c , j + y a w , y , j + u ˜ ) contained in the sector S [ 0 , Φ ]. Therefore, inequality (21) is fulfilled for all y ˜, y c , j , u ˜ if (22) holds. Introducing the state ξ ˜ = [ ξ a w , j T y ˜ T y c , j T u ˜ T ] T and A c l , k = A k − B k K a w , we rewrite (22) and obtain (11). Applying the S-procedure/Schur-Complement Lemma [1], multiplying the result from both sides with diag ( Q , Ψ ˆ , I , I , I ) with Q = P − 1 , Ψ ˆ = Ψ − 1 and employing the substitution L = K a w Q, we finally obtain the LMI (12) in which we employed the abbreviation He ( M ) = M T + M.

References

1. Boyd, S., L. E. Ghaoui, E. Feron and V. Balakrishnan. 1994. Linear matrix inequalities in system and control theory. SIAM.10.1137/1.9781611970777Search in Google Scholar

2. Buschek, H. 1999. Full envelope missile autopilot design using gain scheduled robust control. Journal of Guidance, Control and Dynamics 22(1): 115–122.10.2514/2.4357Search in Google Scholar

3. Chilali, M., C. Scherer and P. Gahinet. 1997. Multiobjective output-feedback control via LMI optimization. IEEE Transactions on Automatic Control.10.1109/9.599969Search in Google Scholar

4. Fleischmann S., S. Theodoulis, E. Laroche, E. Wallner and J.-P. Harcaut. 2017. Controller design point selection for linearized gain scheduling. In: Proc. of American Control Conference.10.23919/ACC.2017.7963177Search in Google Scholar

5. Gahinet, P., P. Apkarian, M. Chilali. 1996. Affine parameter-dependent Lyapunov functions and real parametric uncertainty. IEEE Transactions on Automatic Control 41(3): 436–442.10.1109/CDC.1994.411442Search in Google Scholar

6. Hippe, P. 2006. Windup in control. Springer.Search in Google Scholar

7. Kelly, J. H. and J. H. Evers. 1997. An interpolation strategy for scheduling dynamic compensators. American Institute of Aeronautics and Astronautics.10.2514/6.1997-3764Search in Google Scholar

8. Kelly J. H., J. H. Evers, R. A. Korn and D. A. Lawrence. 1999. Scheduling dynamik compensators via control signal interpolation. AIAA 176–186.10.2514/6.1999-3974Search in Google Scholar

9. Lawrence D. A., J. H. Kelly and J. H. Evers. 1998. Gain scheduled missile autopilot design using a control signal interpolation technique. AIAA 1394–1402.10.2514/6.1998-4418Search in Google Scholar

10. Leit D. J. and W. E. Leithead. 2010. Survey of gain-scheduling analysis & design. International Journal of Control 73: 1001–1025.10.1080/002071700411304Search in Google Scholar

11. Löfberg, J. 2004. Yalmip: A toolbox for modeling and optimization in MATLAB. In: Proc. CACSD Conf., Taipei, Taiwan. URL http://control.ee.ethz.ch/~joloef/yalmip.php.Search in Google Scholar

12. Peter, F., F. Holzapfel, E. Xargay and N. Hovakimyan. 2012. L1 adaptive augmentation of a missile autopilot. In: AIAA Guidance, Navigation, and Control Conference.10.2514/6.2012-4832Search in Google Scholar

13. Pfiffer H. and S. Hecker. 2010. LPV controller synthesis for a generic missile model. In: IEEE international conference on control applications, pp. 1838–1843.10.1109/CCA.2010.5611127Search in Google Scholar

14. Rugh J. R. and S. S. Shamma. 2000. Research on gain-scheduling. Automatica 36: 1401–1425.10.1016/S0005-1098(00)00058-3Search in Google Scholar

15. Samet, H. and R. E. Webber. 1985. Storing a collection of polygons using quadtrees, ACM Transactions on Graphics 4(3): 182–222.10.1145/282957.282966Search in Google Scholar

16. Sturm, J. F. 1999. Using SeDuMi 1.02, a Matlab toolbox for optimization over symmetric cones. Optimization Methods and Software 11(1-4): 625–653.10.1080/10556789908805766Search in Google Scholar

17. Tarbouriech, S., G. Garcia, J. M. Gomez da Silva Jr. and L. Quiennec. 2011. Stability and stabilization of linear systems with saturation actuators. Springer.10.1007/978-0-85729-941-3Search in Google Scholar

18. Tan, W., A. K. Packard and G. J. Balas. 2000. Quasi-LPV modeling and LPV control of a generic missile. In: Proceedings of the American control conference, pp. 3692–3696.Search in Google Scholar

19. Toh K. C., M. Todd and R. Tutuncu. 1999. Sdpt3 – a Matlab software package for semidefinite programming. Optimization Methods and Software 11: 545–581.10.1080/10556789908805762Search in Google Scholar

20. Turner, M. C., G. Hermann and I. Postlethwaite. 2007. Anti-windup compensation using a decoupling architecture. Lecture notes in control and information sciences, 346. Springer, pp. 121–171.10.1007/978-3-540-37010-9_5Search in Google Scholar

Received: 2021-09-04

Accepted: 2022-02-15

Published Online: 2022-03-11

Published in Print: 2022-03-28

Systematic mitigation of gain scheduling induced windup phenomena

Abstract

Zusammenfassung

About the authors

Appendix A Proof of Lemma 1

References

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