Current PI-funnel control with anti-windup for reluctance synchronous machines (RSMs)

Transcription

1 Current PI-funnel control with anti-windup for reluctance synchronous machines (RSMs) Christoph Hackl Colloquium on the occasion of the 6th birthday of Achim Ilchmann Christoph Hackl: Current PI-funnel control with anti-windup for RSMs Seite 1/23

2 Outline 1 Funnel control 2 Current PI-funnel control with anti-windup for RSMs [1] 3 Conclusion Current PI-funnel control with anti-windup for RSMs, C. Hackl 2/23

3 Outline 1 Funnel control Principle idea The beginning Successful implementations in mechatronics Current PI-funnel control with anti-windup for RSMs, C. Hackl 2/23

4 Outline 1 Funnel control Principle idea The beginning Successful implementations in mechatronics Current PI-funnel control with anti-windup for RSMs, C. Hackl 2/23

5 Funnel control Tracking with prescribed transient behaviour Λpq epq Λptq eptq Λp q ep q λ t time t rss (performance) funnel Λpq uptq kptqeptq with kptq 1 Λptq eptq Current PI-funnel control with anti-windup for RSMs, C. Hackl 3/23

6 Outline 1 Funnel control Principle idea The beginning Successful implementations in mechatronics Current PI-funnel control with anti-windup for RSMs, C. Hackl 3/23

7 Funnel control The beginning... (at least for me in 24) ESAIM: Control, Optimisation and Calculus of Variations July 22, Vol. 7, URL: DOI: 1.151/cocv:2264 TRACKING WITH PRESCRIBED TRANSIENT BEHAVIOUR Achim Ilchmann 1,E.P.Ryan 2 and C.J. Sangwin 3 Abstract. Universal tracking control is investigated in the context of a class S of M-input, M-output dynamical systems modelled by functional differential equations. The class encompasses a wide variety of nonlinear and infinite-dimensional systems and contains as a prototype subclass all finite-dimensional linear single-input single-output minimum-phase systems with positive high-frequency gain. The control objective is to ensure that, for an arbitrary R M -valued reference signal r of class W 1, (absolutely continuous and bounded with essentially bounded derivative) and every system of class S, the tracking error e between plant output and reference signal evolves within a prespecified performance envelope or funnel in the sense that ϕ(t) e(t) < 1 for all t, where ϕ a prescribed real-valued function of class W 1, with the property that ϕ(s) > for all s>and lim infs ϕ(s) >. A simple (neither adaptive nor dynamic) error feedback control of the form u(t) = α(ϕ(t) e(t) )e(t) is introduced which achieves the objective whilst maintaining boundedness of the control and of the scalar gain α(ϕ( ) e( ) ). Mathematics Subject Classification. 93D15, 93C3, 34K2. Received February 7, 22. Revised April 17, Introduction In 1991, Miller and Davison [4] posed the problem of forcing this error (between plant output and reference signal) Current to be lesspi-funnel than an (arbitrary control with small) anti-windup prespecified for RSMs, constant C. after Hackl an (arbitrarily short) prespecified period 4/23

8 Outline 1 Funnel control Principle idea The beginning Successful implementations in mechatronics Current PI-funnel control with anti-windup for RSMs, C. Hackl 4/23

9 CRES Control of renewable energy systems Funnel control Successful implementations in mechatronics (since 24, [1 33]) Industrial drives Wind turbine systems Industrial robots y ref,e ye position control speed & position control Elastic drive trains speed control Electrical machines Machine tools ω motor gear long shaft (elastic) bearing load (applicator roll) speed & position control + damping mounting of blade y x speed & position control Current PI-funnel control with anti-windup for RSMs, C. Hackl current control 5/23

10 Outline 2 Current PI-funnel control with anti-windup for RSMs [1] Motivation and problem formulation Structural system properties of the RSM PI-funnel control with anti-windup Implementation and results Current PI-funnel control with anti-windup for RSMs, C. Hackl 5/23

11 Outline 2 Current PI-funnel control with anti-windup for RSMs [1] Motivation and problem formulation Structural system properties of the RSM PI-funnel control with anti-windup Implementation and results Current PI-funnel control with anti-windup for RSMs, C. Hackl 5/23

12 Motivation and problem formulation Reluctance synchronous machine (RSM): A viable alternative? advantages [34 36]: (very) cheap high efficiency feasible (IE4) high power density and torque-per-volume ratio sensorless control feasible disadvantages: power factor highly nonlinear inverter necessary (nonlinear control and good system knowledge necessary) Current PI-funnel control with anti-windup for RSMs, C. Hackl 6/23

13 Motivation and problem formulation Reluctance synchronous machine: A look inside (cut version/duration: :26 min) Current PI-funnel control with anti-windup for RSMs, C. Hackl 7/23

14 Motivation and problem formulation Flux linkage in the RSM q d ψ d s ψ d s [Wb] i q s [A] 5 5 B i d s [A] 5 ψ q s.5 ψ q s [Wb].5 5 B i q s [A] i d s [A] Current PI-funnel control with anti-windup for RSMs, C. Hackl 8/23

15 Motivation and problem formulation Control objective: Current reference tracking with prescribed transient accuracy j u k s ptq R s i k 1 s ptq ` d dt ψk s `ik s ptq ` ω k ptq ψ k looomooon 1 s `ik s ptq, (1) d dt ω kptq Θ p 3 2 p ik s ptq J Jψ k s `ik s ptq m l ptq loooooooooooomoooooooooooon lomon machine torque loads :J `νω k ptq ` pfω k qptq loooooooooooomoooooooooooon friction ı, (2) ψ d s [Wb] i q s [A] 5 5 B d-flux linkage i d s [A] 5 ψ q s [Wb] i q s [A] B 5 q-flux linkage 5 5 i d s [A] Current PI-funnel control with anti-windup for RSMs, C. Hackl 9/23

16 Outline 2 Current PI-funnel control with anti-windup for RSMs [1] Motivation and problem formulation Structural system properties of the RSM PI-funnel control with anti-windup Implementation and results Current PI-funnel control with anti-windup for RSMs, C. Hackl 9/23

17 Structural system properties of the RSM Derivation of the nonlinear current dynamics To be derived: Nonlinear current dynamics d dt ik s ptq... (3) Given: Stator voltage equation u k s ptq R s i k s ptq ` d dt ψk s `ik s ptq ` ω k ptqjψ k s `ik s ptq, ψ k s `ik s pq ψ k s, Assumption 1 Flux linkage ψ k s rwbs 2 is a continuously differentiable function of the stator currents i k s ras 2, i.e. ψ k s : R 2 Ñ R 2, i k s ÞÑ ψ k s pi k s q ^ ψ k s p q P C 1 pr 2 ; R 2 q (4) Current PI-funnel control with anti-windup for RSMs, C. Hackl 1/23

18 Structural system properties of the RSM Inductance matrix Definition 1 Inductance matrix L k s rhs 2ˆ2 is defined by» L k s pi k L dd s pi k s q L dq s pi k j s q s q : L qd s pi k s q L qq s pi k : s q symmetric [37]: Bψ d s pi k s q Bi d s Bψ q s pi k s q Bi d s Bψ d s pi k s q Bi q s Bψ q s pi k s q Bi q s fi fl ˇ ˇ i k s Bψk s pi k s q Bi k s ˇ ˇik s (5) positive definite k s P R 2 : L k s pi k s q L k s pi k s q J ô L dq s pi k s q L qd s pi k s q k s P R 2 : L k s pi k s q ą ô L dd s pi k s q ą, L qq s pi k s q ą, detpl k s pi k s qq ą (7) Current PI-funnel control with anti-windup for RSMs, C. Hackl 11/23

19 Structural system properties of the RSM Generic model with nonlinear current dynamics Time derivatives of the flux linkage are given by: d dt ψk s pi k s ptqq Bψk s pi k s ptqq Bi k s ptq d dt ik s ptq (5) Inserting (8) into (1) yields nonlinear current dynamics: L k s pi k s ptqq d dt ik s ptq (8) d dt ik s ptq L k s `ik s ptq 1 sat pu `uk s ptq R s i k s ptq ω k ptqjψ k s `ik s ptq ı d dt ω kptq p 3 Θ 2 p ik s ptq J Jψ k s `ik loooooooooooomoooooooooooon s ptq lomon m l ptq `νω ı loooooooooooomoooooooooooon k ptq ` pfω k qptq machine torque loads friction (9) with system properties (see Remark IV.1): 2-input 2-output system with vector relative degree one! BIBO stable internal dynamics, i.e. i k s p q P L 8 ^ m l p q P L 8 ñ ω k p q P L 8 Euclidean input saturation [38], i.e. u k s ptq ď pu : u dc 2 for all t ě Current PI-funnel control with anti-windup for RSMs, C. Hackl 12/23

20 Outline 2 Current PI-funnel control with anti-windup for RSMs [1] Motivation and problem formulation Structural system properties of the RSM PI-funnel control with anti-windup Implementation and results Current PI-funnel control with anti-windup for RSMs, C. Hackl 12/23

21 Current PI-funnel control with anti-windup for RSMs Funnel control: Tracking with prescribed transient accuracy Λpq epq λ Λptq eptq performance funnel F Λ Λp q BF Λ p q ep q t time rss Controller (applicable if feasibility condition pu ě pu feas holds, see Theorem IV.2): u fc ptq kptqeptq ùñ eptq ă Λptq (FC k i ) s where eptq pe d sptq, e q sptqq J i k s,refptq i k s ptq 1 kptq Λptq eptq ě 1 and eptq Λptq b e d sptq 2 ` e q sptq Current PI-funnel control with anti-windup for RSMs, C. Hackl 13/23

22 Current PI-funnel control with anti-windup for RSMs PI-funnel control with anti-windup k p sat pu e (FCi k ) s u fc u k s,ref u k s k i 9ξ ξ f pu awp q 1 d dt ξptq k i faw pu ξptq ` k p u fc ptq u fc ptq, ξpq ξ P R n u k s,refptq ξptq ` k p u fc ptq, where k p ą, k i ě. ùñ ξp q acts as bounded input disturbance (see Lemma III.1) ùñ PI-like extension for steady state accuracy + (PI aw ) Current PI-funnel control with anti-windup for RSMs, C. Hackl 14/23

23 Outline 2 Current PI-funnel control with anti-windup for RSMs [1] Motivation and problem formulation Structural system properties of the RSM PI-funnel control with anti-windup Implementation and results Current PI-funnel control with anti-windup for RSMs, C. Hackl 14/23

24 Current PI-funnel control with anti-windup for RSMs Control-loop and implementation i k s,ref PI-funnel controller u k s,ref Park trafo. dq u s s,ref Clarke trafo. αβ u abc s,ref PWM VSI power elect. u abc s nonlinear SM stator rotor ω m αβ abc u dc φ k ş pω m dt ` φ k pq i k s dq i s s αβ i abc s pi a s,i b s,i c s q J controller implementation αβ abc real world Realistic modeling: three-phase signals (machine, inverter, modulation,... ) pulse width modulation (PWM) and switching behavior of voltage source inverter (VSI) Park-/Clarke transformation Current PI-funnel control with anti-windup for RSMs, C. Hackl 15/23

25 Current PI-funnel control with anti-windup for RSMs Simulation results (1.1 kw RSM): FC, FC+PI and FC+PI aw e [A] 4 2 Λ k [V/A] i d s [A] i d s,ref i q s [A] time t [s] i q s,ref Current PI-funnel control with anti-windup for RSMs, C. Hackl 16/23

26 Current PI-funnel control with anti-windup for RSMs Simulation results (1.1 kw RSM): FC, FC+PI and FC+PI aw 4 i k s [A] 2 15 î max udc 2 u k s,ref [V] udc 2 ξ [V] time t [s] Current PI-funnel control with anti-windup for RSMs, C. Hackl 17/23

27 Outline 3 Conclusion Current PI-funnel control with anti-windup for RSMs, C. Hackl 17/23

28 Conclusion Funnel control: Facts simple and robust controller Tracking with prescribed transient behavior (Ñ funnel design) Λpq epq Λptq eptq Λp q ep q λ t time t rss (performance) funnel Λpq asymptotic accuracy possible (Ñ internal model design, e.g. PI) Happy Birthday, Achim and thank you for all your time and support! Current PI-funnel control with anti-windup for RSMs, C. Hackl 18/23

29 References I [1] Christoph M. Hackl. Current PI-funnel control with anti-windup for synchronous machines. In Proceedings of the 54th IEEE Conference on Decision and Control, pages , 215. [2] Hans Schuster, Christian Westermaier, and Dierk Schröder. High-gain control of systems with arbitrary relative degree: Speed control for a two mass flexible servo system. In Proceedings of the 8th IEEE International Conference on Intelligent Engineering Systems, pages , Cluj-Napoca, Romania, 24. [3] Hans Schuster, Christoph M. Hackl, Christian Westermaier, and Dierk Schröder. Funnel-control for electrical drives with uncertain parameters. In Proceedings of the 7th International Power Engineering Conference, Singapore, 25. [4] Hans Schuster, Christian Westermaier, and Dierk Schröder. Non-identifier-based adaptive control for a mechatronic system achieving stability and steady state accuracy. In Proceedings of the Joint CCA, ISIC and CACSD, pages , Munich, Germany, 26. [5] Hans Schuster, Christian Westermaier, and Dierk Schröder. Non-identifier-based adaptive speed control for a two-mass flexible servo system: Consideration of stability and steady state accuracy. In Proceedings of the 14th Mediterranean Conference on Control and Automation, pages 1 6, 26. [6] Hans Schuster, Christian Westermaier, and Dierk Schröder. Non-identifier-based adaptive tracking control for a two-mass system. In Proceedings of the International Conference on Intelligent Engineering Systems, pages , 26. [7] Christoph M. Hackl and Dierk Schröder. Funnel-control for nonlinear multi-mass flexible systems. In Proceedings of the 32nd Annual Conference on IEEE Industrial Electronics, pages , 26. [8] Christoph M. Hackl, Hans Schuster, Christian Westermaier, and Dierk Schröder. Funnel-control with integrating prefilter for nonlinear, time-varying two-mass flexible servo systems. In Proceedings of the 9th International Workshop on Advanced Motion Control, pages , Istanbul, Turkey, Current PI-funnel control with anti-windup for RSMs, C. Hackl 19/23

30 References II [9] Christoph M. Hackl, Yang Ji, and Dierk Schröder. Enhanced funnel-control with improved performance. In Proceedings of the 15th Mediterranean Conference on Control and Automation, pages Paper T1 16, 27. [1] Christoph M. Hackl, Yang Ji, and Dierk Schröder. Funnel-control with constrained control input compensation. In Proceedings of the 9th IASTED International Conference CONTROL AND APPLICATIONS, pages , 27. [11] Christoph M. Hackl and Dierk Schröder. Funnel-control with online foresight. In Proceedings of the 26th IASTED International Conference MODELLING, IDENTIFICATION AND CONTROL, pages , 27. [12] Christoph M. Hackl, Yang Ji, and Dierk Schröder. Nonidentifier-based adaptive control with saturated control input compensation. In Proceedings of the 15th Mediterranean Conference on Control and Automation, pages Paper T1 15, 27. [13] Christoph M. Hackl, Christian Endisch, and Dierk Schröder. Applicability of funnel-control in robotics. In Proceedings of the 5th International Conference on Computational Intelligence, Robotics and Autonomous Systems, pages 65 7, 28. [14] Christoph M. Hackl, Christian Endisch, and Dierk Schröder. Error reference control of nonlinear two-mass flexible servo systems. In Proceedings of the 16th Mediterranean Conference on Control and Automation, pages , 28. [15] Christoph M. Hackl, Christian Endisch, and Dierk Schröder. Funnel-control in robotics: An introduction. In Proceedings of the 16th Mediterranean Conference on Control and Automation, pages , 28. [16] Christoph M. Hackl, Christian Endisch, and Dierk Schröder. Specially designed funnel-control in mechatronics. In Proceedings of the 5th International Conference on Computational Intelligence, Robotics and Autonomous Systems, pages 59 64, Current PI-funnel control with anti-windup for RSMs, C. Hackl 2/23

31 References III [17] Christoph M. Hackl and Dierk Schröder. Non-identifier based adaptive control in robotics: An introduction. International Journal of Factory Automation, Robotics and Soft Computing, 4:9 99, 28. [18] Christoph M. Hackl, Christian Endisch, and Dierk Schröder. Contributions to non-identifier based adaptive control in mechatronics. Robotics and Autonomous Systems, Elsevier, 57(1):996 15, 29. [19] Hans Schuster. Hochverstärkungsbasierte Regelung nichtlinearer Antriebssysteme. PhD thesis, Lehrstuhl für Elektrische Antriebssysteme, (TUM), Germany, 29. [2] Christoph M. Hackl and Dierk Schröder. Non-identifier based adaptive control in robotics: An introduction. In Salvatore Pennacchio, editor, Recent adavances in Control Systems, Robotics and Automation, volume 1. INTERNATIONALSAR International Society for Advanced Research, Palermo, Italy, third edition, 29. [21] Christoph M. Hackl. Funnel Control: Implementierung, Erweiterung und Anwendung. In Dierk Schröder, editor, Intelligente Verfahren: Systemidentifikation und Regelung nichtlinearer Systeme, pages Springer-Verlag, Berlin, 21. [22] Achim Ilchmann and Hans Schuster. PI-funnel control for two mass systems. IEEE Transactions on Automatic Control, 54(4): , 29. [23] Christoph M. Hackl, Andreas G. Hofmann, Rik W. De Doncker, and Ralph M. Kennel. Funnel control for speed & position control of electrical drives: A survey. In Proceedings of the 19th Mediterranean Conference on Control and Automation, pages , 211. [24] Christoph M. Hackl, Andreas G. Hofmann, and Ralph M. Kennel. Funnel control in mechatronics: An overview. In Proceedings of the 5th IEEE Conference on Decision and Control and European Control Conference, pages 8 87, Current PI-funnel control with anti-windup for RSMs, C. Hackl 21/23

32 References IV [25] Christoph M. Hackl. Funnel control zur Positionsregelung von starren Drehgelenkrobotern mit grob bekannter Trägheitsmatrix. In Tagungsband des GMA-Fachausschuss 1.4, pages 32 31, 212. [26] Christoph M. Hackl and Ralph M. Kennel. Position funnel control for rigid revolute joint robotic manipulators with known inertia matrix. In Proceedings of the 2th Mediterranean Conference on Control and Automation, pages , 212. [27] Christoph M. Hackl and Ralph M. Kennel. Position funnel control with linear internal model. In Proceedings of 212 IEEE International Conference on Control Applications, pages , 212. [28] Christoph M. Hackl, Norman Hopfe, Achim Ilchmann, Markus Mueller, and Stephan Trenn. Funnel control for systems with relative degree two. SIAM Journal on Control and Optimization, 51(2): , 213. [29] Christoph M. Hackl. Funnel control with disturbance observer for two-mass systems. In Proceedings of the 52nd IEEE Conference on Decision and Control, pages , 213. [3] Christoph M. Hackl. PI-funnel control with Anti-windup and its application for speed control of electrical drives. In Proceedings of the 52nd IEEE Conference on Decision and Control, pages , 213. [31] Christoph M. Hackl. Funnel control for wind turbine systems. In Proceedings of the 214 IEEE International Conference on Control Applications, pages , 214. [32] Christoph M. Hackl. Speed funnel control with disturbance observer for wind turbine systems with elastic shaft. In Proceedings of the 54th IEEE Conference on Decision and Control, pages , 215. [33] Christoph M. Hackl. Non-identifier based adaptive control in mechatronics: Theory and Application. Lecture Notes in Control and Information Sciences. Springer-Verlag, Berlin, 215 (accepted; final version in preparation) Current PI-funnel control with anti-windup for RSMs, C. Hackl 22/23

33 References V [34] Marten J. Kamper, F.S. van der Merwe, and S. Williamson. Direct finite element design optimisation of the cageless reluctance synchronous machine. IEEE Transactions on Power Conversion, 11(3): , [35] Thomas A. Lipo. Synchronous reluctance machines A viable alternative for AC drives? Electric Machines & Power Systems, 19(6): , [36] A. Vagati. The synchronous reluctance solution: A new alternative in AC drives. In Proceedings of the 2th International Conference on Industrial Electronics, Control and Instrumentation, pages 1 13, [37] Michael T. Ivrlač. Lecture notes on Circuit Theory and Communications. Lecture notes (version 2.1), Institute for Circuit Theory and Signal Processing,, 213. [38] Norman Hopfe, Achim Ilchmann, and Eugene P. Ryan. Funnel control with saturation: Linear MIMO systems. IEEE Transactions on Automatic Control, 55(2): , Current PI-funnel control with anti-windup for RSMs, C. Hackl 23/23

34 Outline 4 Appendix Current PI-funnel control with anti-windup for RSMs, C. Hackl /1

35 Structural system properties of the RSM Nonlinear inductances L d s : Bψd s, L q Bi d s : Bψq s Bi q, L dq s : Bψd s s s Bi q s Bψq s Bi d s : L qd s L d s [H] i q s [A] i d s [A] L q s [H].2 5 i q s [A] i d s [A] L dq s [H] L qd s [H] i q s [A] i d s [A] 5 i q s [A] i d s [A] Current PI-funnel control with anti-windup for RSMs, C. Hackl 1/1