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Licensed Unlicensed Requires Authentication 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, Benedikt Bartenschlager, Christoph Auenmüller and Sebastian Seitz

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.


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


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.

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Received: 2021-09-04
Accepted: 2022-02-15
Published Online: 2022-03-11
Published in Print: 2022-03-28

© 2022 Walter de Gruyter GmbH, Berlin/Boston

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