Non-identifier based adaptive control in mechatronics

Transcription

1 Non-identifier based adaptive control in mechatronics 2.5 Application: Speed and current control of electrical drives Christoph Hackl Munich School of Engineering (MSE) Research group Control of renewable energy systems (CRES) Lecture & tutorial C. Hackl Non-identifier based adaptive control in mechatronics 1/43

2 Introduction Schedule (tentative) Date Content Introduction and 2. Non-identifier based speed control (relative-degree-one case) 2.1 High-gain adaptive stabilization High-gain adaptive tracking (using internal models) and 2.3 Adaptive λ-tracking control and funnel control Tutorials for Lectures canceled (Christi Himmelfahrt) Practical course (relative-degree-one case) Practical course (relative-degree-one case) [contd.] Practical course (relative-degree-one case) [contd.] & 2.3 Proofs of high-gain adaptive stabilization & funnel control Applications: Speed control of electrical drives (and some new results) 3. Non-identifier based adaptive position control (relative-degree-two case) High-gain adaptive stabilization and & 3.2 High-gain adaptive tracking Adaptive λ-tracking control and funnel control (with derivative feedback) Practical course (relative-degree-two case) Application: Position funnel control of servo-systems & industrial robots Conclusions & exam revision C. Hackl Non-identifier based adaptive control in mechatronics 2/43

3 Outline 2 Non-identifier based adaptive speed control 2.5 Applications C. Hackl Non-identifier based adaptive control in mechatronics 3/43

4 Outline 2 Non-identifier based adaptive speed control 2.5 Applications Speed funnel control of one-mass systems (1MS) Speed funnel control of 1MS with input saturation Speed funnel control of two-mass systems [1] Speed funnel control of wind turbine systems Current PI-funnel control with anti-windup of reluctance synchronous machines (RSMs) C. Hackl Non-identifier based adaptive control in mechatronics 3/43

5 Outline 2 Non-identifier based adaptive speed control 2.5 Applications Speed funnel control of one-mass systems (1MS) Speed funnel control of 1MS with input saturation Speed funnel control of two-mass systems [1] Speed funnel control of wind turbine systems Current PI-funnel control with anti-windup of reluctance synchronous machines (RSMs) C. Hackl Non-identifier based adaptive control in mechatronics 3/43

6 Speed funnel control of one-mass systems Stiff servo-system drive (m m ) load (m l ) Θ hkkkkkkkkkikkkkkkkkkj machine torque m m p q P CpR ą ; Rq rnms (control input) load torque m l p q P L 8 pr ą ; Rq rnms (disturbance) inertia Θ ą kg m 2ı state vector x pφ, ωq J : position φ rrads, speed ω rrad{ss friction (on machine and load side) gear ratio g r P Rztu r1s measured signals: angle φ and/or speed ω 9 φ (deteriorated by n m p q and/or 9n m p q) C. Hackl Non-identifier based adaptive control in mechatronics 4/43

7 Speed funnel control of one-mass systems Nonlinear model of stiff servo-systems [2, 3] 1{g r m L ν 2 ω g r ` F 2 ω g r 1MS ν 1 ω ` F 1 ω u actuator ω φ 9 k A 1{Θ 1{g m M r satûa 9ω φ φ{g r u A ω m φ m 9n m sensor(s) n m 9xptq Axptq ` b sat pua `mm ptq ` u a ptq pf 1 ωqptq ` b L`ml ptq ` pf 2 ω g r qptq yptq c J xptq, xpq pφ, ω q J where k a ą, pu a ą, u a P L 8 pr ą ; Rq, g r P Rztu, Θ ą, ν 1, ν 2 ą, and for all i P t1, 2u: F i P T, M Fi : sup t pf i βqptq t ě, βp q P CpR ą, Rquă8 and + (1MS) C. Hackl Non-identifier based adaptive control in mechatronics 5/43

8 Speed funnel control of one-mass systems Implementation [2, 3] implementation laboratory setup W 1,8 y ref ω ref e speed controller m m ω y ω ` 9n m 9n m Standard PI controller PI-Funnel controller ż t m m ptq k P eptq ` k I epτq dτ ż t 1 m m ptq kptq eptq ` k I kpτqepτq dτ with kptq ψptq eptq C. Hackl Non-identifier based adaptive control in mechatronics 6/43

9 Speed funnel control of one-mass systems Measurement results [2, 3] Set-point tracking ω + ṅm [rad/s] ω ref ( ) ω ref ( ) ± ψ( ) time t[s] PI-Funnel controller PI controller Reference tracking ω + ṅm [rad/s] e [rad/s] k, kp [Nms/rad] mm [Nm] ω ref ( ) ±ψ( ) 2 15 m L ( ) time t[s] C. Hackl Non-identifier based adaptive control in mechatronics 7/43

10 Outline 2 Non-identifier based adaptive speed control 2.5 Applications Speed funnel control of one-mass systems (1MS) Speed funnel control of 1MS with input saturation Speed funnel control of two-mass systems [1] Speed funnel control of wind turbine systems Current PI-funnel control with anti-windup of reluctance synchronous machines (RSMs) C. Hackl Non-identifier based adaptive control in mechatronics 7/43

11 Speed funnel control of 1MS with input saturation Funnel controller with input saturation [4, 5] Implementierung (xpc target) y ref e 1 ψ e e u m L satpua p q Laboraufbau ω ω ` 9n m 9n m pu a ě pu feas pu feas py ref, ψ, system,... q t ě : ψptq eptq ě ε! ) λ ε ď min, ψpq epq, λ 2 2ppu a` m L 8 q pu feas conservative! (for laboratory setup pu feas «63 rnms and û A 22 rnms) C. Hackl Non-identifier based adaptive control in mechatronics 8/43

12 Speed funnel control of 1MS with input saturation Measurement results ω+ṅm [rad/s] e [rad/s] k [Nms/rad] mm [Nm] y ref ( ) ±ψ( ) 5 m L( ) û A time t [s] Computed feasibility number: pu feas rnms C. Hackl Non-identifier based adaptive control in mechatronics 9/43

13 Speed funnel control of 1MS with input saturation PI-funnel control with input saturation Implementierung (xpc target) y ref e 1 ψ e e v? 9x I k I v u k P v `x I u m L sat pua p q Laboraufbau ω ω ` 9n m 9n m Tracking with prescribed transient accuracy guaranteed, t ě : eptq ă ψptq? C. Hackl Non-identifier based adaptive control in mechatronics 1/43

14 Speed funnel control of 1MS with input saturation PI controller with anti-windup ( conditional integration ) v k P u sat pua p q u sat k I 9x I x I f aw p q (PI aw ) pu A System theoretic interpretation: Lemma (see vp q P CpR ą ; Rq t ě : x I ptq ď maxtpu a, x I pqu : x max I ùñ Anti-windup: x I p q acts as bounded input disturbance C. Hackl Non-identifier based adaptive control in mechatronics 11/43

15 Speed funnel control of 1MS with input saturation PI-funnel control with saturation [6] Implementierung (xpc target) y ref e 1 ψ e e v 9x I f aw puqk I v u k P v `x I u m L sat pua p q Laboraufbau ω ω ` 9n m 9n m pu a ě pu feas pu feas py ref, ψ, system,... q t ě : ψptq eptq ě ε! ) λ ε ď min, ψpq epq, k p λ 2 pu feas «63 rnms 2ppu a`x max I ` m L 8 q C. Hackl Non-identifier based adaptive control in mechatronics 12/43

16 Speed funnel control of 1MS with input saturation Measurement results: (FC)+(PI), (FC)+(PI aw ) xi [Nm] k [Nms/rad] e [rad/s] ω+nm [rad/s y ref ( ) y ref ( )±ψ( ) ±ψ( ) max x I (t) u [Nm] 2 2 m L ( ) Zeit t [s] C. Hackl Non-identifier based adaptive control in mechatronics 13/43 ±û A

17 Outline 2 Non-identifier based adaptive speed control 2.5 Applications Speed funnel control of one-mass systems (1MS) Speed funnel control of 1MS with input saturation Speed funnel control of two-mass systems [1] Speed funnel control of wind turbine systems Current PI-funnel control with anti-windup of reluctance synchronous machines (RSMs) C. Hackl Non-identifier based adaptive control in mechatronics 13/43

18 Speed funnel control of two-mass systems Problem statement C. Hackl Non-identifier based adaptive control in mechatronics 14/43

19 Speed funnel control of two-mass systems Control objectives load (m L ) Θ 2, ω 2 Θ 1, ω 1 c S,d S drive (m M ) Control objective 1: Reference tracking ω 2 Ñ ω 2,ref (or ω 1 Ñ ω 1,ref ) Control objective 2: Active damping of resonant frequency d c ω s pθ 1 ` Θ 2 q «61 rrad{ss ùñ f Θ 1 Θ «9.7 rhzs 2 Constraint: only ω 1 is measured (available for feedback) C. Hackl Non-identifier based adaptive control in mechatronics 15/43

20 Speed funnel control of two-mass systems Two-mass system model d dt x Sptq A S x S ptq ` b S`uptq ` ua ptq ` B S yptq c J S x S ptq, x S pq x S P R 3 where x : `ω 1, φ s, ω 2 J, A S» c S 1, ds`g 2 r ν 1 g 2 r Θ 1 cs g r Θ 1 d s g r Θ 1 1 g r 1 d s g r Θ 2 c s Θ 2 ds`ν 2 Θ 2 fi ffi ffi fl, b S k a Θ 1 pf 1 ω 1 qptq m l ptq ` pf 2 ω 2 qptq», B S, 1 Θ 1 1 Θ 2 Θ 1, Θ 2 ą, c s, d s ą, g r P Rztu, ν 1, ν 2 ą, k a ą, u a p q, m l p q P L 8 pr ą ; Rq i P t1, 2u: F i P T (see Def. 1 in [7]) ^ M Fi : sup pf i ξqptq ˇˇ t ě, ξp q P CpRą, Rq ( ă8., /. /- (1) (2) C. Hackl Non-identifier based adaptive control in mechatronics 16/43 fi fl,, /. /-

21 Speed funnel control of two-mass systems Speed funnel control of two-mass systems with filter and state feedback [8] implementation in xpc target y ref pc 1 `c 2 qω 2,ref e W 1,8 PI-funnel controller u 2MS with sensors ω 1 ω 2 y filter & aug. output 9x F k f `xf ` S pφ y c 1 pω 1 `c 2φS p `c 3 pω 2 `c 2 x F pω 1 ω 1 `n m pφ S phid_s n m pω 2 ω 2 `n m2 n m2 2MS P S 1 ðù c 1 ą ^ c2 g r ě ^ c3 g r ą c 1 ^ k f ą. ùñ For active damping ω 1 and ω 2 must be fed back! C. Hackl Non-identifier based adaptive control in mechatronics 17/43

22 Speed funnel control of two-mass systems Two-mass system with disturbance observer (DO) [9] mech. system 1{g r m l ω ν 1 ω 1 `F 1 ω 1 d s ν 2 ω 2 `F 2 g 2 r g r u actuator ω 1 k rm A 1{Θ 1 1{g r c s 1{Θ 2 u m m φ S u A ω 2 1 1`sT do sθ p 1 { k p a 1`sT do pω 1 n msensor pω 1 m do 1 k do observer m do psq rmpsq k a u a psq (if p k a k a, p Θ 1 Θ 1, k do T do n m ) C. Hackl Non-identifier based adaptive control in mechatronics 18/43

23 Speed funnel control of two-mass systems Implementation: Funnel controller with disturbance observer [1] Implementierung (xpc target) y ref g r ω 2,ref e C 1 Funnel Regler u u 2MS mit Sensorik ω 1 ω 2 pω 1 ω 1 `n m m do Störgrößenbeobachter n m Design of disturbance observer k do ^ T do.3 s ˆ Hz ą f T 9.7Hz do C. Hackl Non-identifier based adaptive control in mechatronics 19/43

24 Speed funnel control of two-mass systems Measurement results: (FC)+(PI), (FC)+(DO) ω2 +nm [rad/s] 1 y ref ( ) 5 1 y ref ( )±ψ E( ) e [rad/s] 1 ±ψ E( ) m L( ) time t [s] C. Hackl Non-identifier based adaptive control in mechatronics 2/43 mm [Nm] k [Nms/rad]

25 Speed funnel control of two-mass systems Measurement results (Zoom): (FC)+(PI), (FC)+(DO).4 zoom φs [rad] ,2 zoom time t [s] C. Hackl Non-identifier based adaptive control in mechatronics 21/43

26 Outline 2 Non-identifier based adaptive speed control 2.5 Applications Speed funnel control of one-mass systems (1MS) Speed funnel control of 1MS with input saturation Speed funnel control of two-mass systems [1] Speed funnel control of wind turbine systems Current PI-funnel control with anti-windup of reluctance synchronous machines (RSMs) C. Hackl Non-identifier based adaptive control in mechatronics 21/43

27 Speed funnel control of wind turbine systems Problem statement [1] v w ω t A G γ r t ω t r t ω m gr A T r t β Operation regions p t,nom I II III IV p t rws v cut in v nom v cut out Simplified turbine model d dt ω m 1 ˆpvw, β, ω m q ` m Θ g m,ref r with Θ : Θ m ` Θ t {g 2 r v w m s C. Hackl Non-identifier based adaptive control in mechatronics 22/43

28 Speed funnel control of wind turbine systems Power coefficient, turbine torque and control objective c p pλ, βq λ p t c p pv w, β, ω m q 1 2 ϱr2 t πvw 3 g r c p pv w, β, ω m qp w ùñ m T pv w, β, ω m q loooomoooon ω m : p w Control objective in region II: Maximum power point tracking, i.e. λ r tω m g r v w Ñ λ c p p, βq C. Hackl Non-identifier based adaptive control in mechatronics 23/43 λ

29 Speed funnel control of wind turbine systems Nonlinear controller design ω m,ref Controller? m m,ref 3 2 n pψ pm i q s,ref 1 «1 T q ers,i s i q s m t pv w,β,ω m q g r n pψ pm m m ` 1 Θ ω m Standard controller [11]: m m,ref ptq k pω m ptq 2 with k p 1 2 ϱr2 t π c ppβ, λ q pλ q 3 Funnel controller: ςptq m m,ref ptq `ωm,ref ptq ω loooooomoooooon ψptq eptq looooooooooomooooooooooon m ptq with ω m,ref ptq g rλ v r w ptq t :kptq :eptq (WTC) (FC WT 1 ) C. Hackl Non-identifier based adaptive control in mechatronics 24/43

30 Speed funnel control of wind turbine systems Simulation results: 6 (FC WT 1 ), (WTC) vw / m s s λ v w λ cp / rad E / kwh.4.2 c p,betz =16/ c p time t / s C. Hackl Non-identifier based adaptive control in mechatronics 25/43

31 s s Speed funnel control of wind turbine systems Simulation results: 1.5 (FC WT 1 ), (WTC) ωm / rad 1.5 ω m,ref( ).2 ±ψ( ) e / rad k / 1 7 Nms rad time t / s C. Hackl Non-identifier based adaptive control in mechatronics 26/43

32 Outline 2 Non-identifier based adaptive speed control 2.5 Applications Speed funnel control of one-mass systems (1MS) Speed funnel control of 1MS with input saturation Speed funnel control of two-mass systems [1] Speed funnel control of wind turbine systems Current PI-funnel control with anti-windup of reluctance synchronous machines (RSMs) C. Hackl Non-identifier based adaptive control in mechatronics 26/43

33 Current PI-funnel control with anti-windup of RSMs Motivation and problem formulation: Are RSMs a viable alternative? advantages [12 14]: (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) C. Hackl Non-identifier based adaptive control in mechatronics 27/43

34 Current PI-funnel control with anti-windup of RSMs Look inside a RSM C. Hackl Non-identifier based adaptive control mechatronics (cut version/duration: :26 min) 28/43

35 Current PI-funnel control with anti-windup of RSMs Flux linkage in the RSM q d ψ d s ψ d s / Wb B 5 i q s / A i d s / A ψ q s.5 ψ q s / Wb.5 5 B i q s / A i d s / A C. Hackl Non-identifier based adaptive control in mechatronics 29/43

36 Current PI-funnel control with anti-windup of RSMs 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, (3) d dt ω kptq Θ p 3 2 p ik s ptq J Jψ k s `ik s ptq m l ptq loooooooooooomoooooooooooon lomon ψ d s / Wb i q s / A 5 5 machine torque B d-flux linkage i d s / A 5 ψ q s / Wb loads :J `νω k ptq ` pfω k qptq loooooooooooomoooooooooooon i q s / A B 5 q-flux linkage friction 5 5 i d s / A ı, (4) C. Hackl Non-identifier based adaptive control in mechatronics 3/43

37 Current PI-funnel control with anti-windup of RSMs Structural system properties of the RSM: Derivation of the nonlinear current dynamics To be derived: Nonlinear current dynamics d dt ik s ptq... (5) 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 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 (6) C. Hackl Non-identifier based adaptive control in mechatronics 31/43

38 Current PI-funnel control with anti-windup of RSMs Structural system properties of the RSM: Nonlinear Inductance matrix Definition 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 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 (7) symmetric 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 (8) positive definite 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 (9) C. Hackl Non-identifier based adaptive control in mechatronics 32/43

39 Current PI-funnel control with anti-windup of RSMs 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 (7) L k s pi k s ptqq d dt ik s ptq (1) Inserting (1) into (3) yields nonlinear current dynamics: 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 with system properties (see Remark IV.1): loads friction 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 [16], i.e. u k s ptq ď pu : u dc 2 for all t ě (11) C. Hackl Non-identifier based adaptive control in mechatronics 33/43

40 Current PI-funnel control with anti-windup of RSMs Multiple-input multiple-output funnel control: Tracking with prescribed transient accuracy ψpq epq λ ψptq eptq performance funnel F ψ ψp q BF ψ 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 2 C. Hackl Non-identifier based adaptive control in mechatronics 34/43

41 Current PI-funnel control with anti-windup of RSMs PI-funnel control with anti-windup e (3) u fc k p u k s,ref satû u k s k i 9ξ ξ f 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 ) C. Hackl Non-identifier based adaptive control in mechatronics 35/43

42 Current PI-funnel control with anti-windup of RSMs Control-loop and implementation i k s,ref PI-funnel controller (3)+ (4) u k s,ref Park/Clarke transform. dq αβ u s s,ref Clarke trafo. αβ abc u abc s,ref PWM VSI u dc power elect. u abc s nonlinear SM rotor stator ω k φ k ş ω k dt i k s dq i s s αβ i abc s pi a s,ib s,ic s qj 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 C. Hackl Non-identifier based adaptive control in mechatronics 36/43

43 Current PI-funnel control with anti-windup of 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 2 i q s [A] 2 i q s,ref time t [s] C. Hackl Non-identifier based adaptive control in mechatronics 37/43

44 Current PI-funnel control with anti-windup of 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] C. Hackl Non-identifier based adaptive control in mechatronics 38/43

45 Current PI-funnel control with anti-windup for RSMs Measurement results for 9,6 kw RSM e / A κ / V A i d s / A i d s,ref Λ i q s / A 2 i q s,ref 2 u k s,ref / V 4 2 û ωk / rad s 2 2 npωm,rated time t / s C. Hackl Non-identifier based adaptive control in mechatronics 39/43

46 Current PI-funnel control with anti-windup for RSMs Measurement results for 9,6 kw RSM (Zoom) e / A κ / V A i d s / A Λ i d s,ref i q s / A 2 i q s,ref 2 u k s,ref / V 4 2 û ωk / rad s time t / s npωm,rated C. Hackl Non-identifier based adaptive control in mechatronics 4/43

47 References I [1] C. M. Hackl, Funnel control with disturbance observer for two-mass systems, in Proceedings of the 52nd IEEE Conference on Decision and Control, pp , 213. [2] C. M. Hackl, A. G. Hofmann, R. W. De Doncker, and R. M. Kennel, Funnel control for speed & position control of electrical drives: A survey, in Proceedings of the 19th Mediterranean Conference on Control and Automation, pp , 211. [3] C. M. Hackl, A. G. Hofmann, and R. M. Kennel, Funnel control in mechatronics: An overview, in Proceedings of the 5th IEEE Conference on Decision and Control and European Control Conference, pp. 8 87, 211. [4] N. Hopfe, A. Ilchmann, and E. P. Ryan, Funnel control with saturation: Nonlinear SISO systems, IEEE Transactions on Automatic Control, vol. 55, no. 9, pp , 21. [5] C. M. Hackl, N. Hopfe, A. Ilchmann, M. Mueller, and S. Trenn, Funnel control for systems with relative degree two, SIAM Journal on Control and Optimization, vol. 51, no. 2, pp , 213. [6] C. 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, pp , 213. C. Hackl Non-identifier based adaptive control in mechatronics 41/43

48 References II [7] A. Ilchmann, E. P. Ryan, and C. J. Sangwin, Tracking with prescribed transient behaviour, ESAIM: Control, Optimisation and Calculus of Variations, vol. 7, pp , 22. [8] A. Ilchmann and H. Schuster, PI-funnel control for two mass systems, IEEE Transactions on Automatic Control, vol. 54, no. 4, pp , 29. [9] Y. Hori, H. Sawada, and Y. Chun, Slow resonance ratio control for vibration suppression and disturbance rejection in torsional system, IEEE Transactions on Industrial Electronics, vol. 46, no. 1, pp , [1] C. M. Hackl, Funnel control for wind turbine systems, in Proceedings of the 214 IEEE International Conference on Control Applications, pp , 214. [11] L. Y. Pao and K. E. Johnson, Control of wind turbines: Approaches, challenges, and recent developments, IEEE Control Systems Magazine, vol. 31, no. 2, pp , 211. [12] M. J. Kamper, F. van der Merwe, and S. Williamson, Direct finite element design optimisation of the cageless reluctance synchronous machine, IEEE Transactions on Power Conversion, vol. 11, no. 3, pp , C. Hackl Non-identifier based adaptive control in mechatronics 42/43

49 References III [13] T. A. Lipo, Synchronous reluctance machines A viable alternative for AC drives?, Electric Machines & Power Systems, vol. 19, no. 6, pp , [14] 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, pp. 1 13, [15] M. T. Ivrlač, Lecture notes on Circuit Theory and Communications, lecture notes (version 2.1), Institute for Circuit Theory and Signal Processing, Technische Universität München, 213. [16] N. Hopfe, A. Ilchmann, and E. P. Ryan, Funnel control with saturation: Linear MIMO systems, IEEE Transactions on Automatic Control, vol. 55, no. 2, pp , 21. C. Hackl Non-identifier based adaptive control in mechatronics 43/43