Non-identifier based adaptive control in mechatronics: An overview

Transcription

1 Non-identifier based adaptive control in mechatronics: An overview Christoph Hackl (TUM) Munich School of Engineering (MSE) IEEE Guest Lecture sponsored by IEEE IAS/IES/PELS Joint SA Chapter 25th April 214, University of Cape Town Non-identifier based adaptive control in mechatronics: An overview 25/4/214, Page 1/54

2 Outline 1 Problem statement and motivation Reference tracking High-gain control: An intuition 2 Funnel control Relative-degree-one systems Relative-degree-two systems Steady-state accuracy with internal models 3 Application Speed and position control of stiff servo-systems Funnel control with linear internal model Funnel control for robotic systems Funnel control with saturation PI-funnel control with Anti-windup Funnel control for elastic systems Funnel control for wind turbine systems 4 Conclusion christoph.hackl@tum.de Non-identifier based adaptive control in mechatronics: An overview 25/4/214, Page 2/54

3 Reference tracking under load (CNC turning machine) ω y x ( Problem statement reference pω ref,x ref,y ref q: R ě Ñ R 3 precise position and speed control, e.g. for prescribed λ ě t : eptq y ref ptq yptq ď λ. Challenges nonlinear effects (e.g. actuator saturation, friction) friction and loads (disturbances) unknown and varying system parameters (solely) roughly known christoph.hackl@tum.de Non-identifier based adaptive control in mechatronics: An overview 25/4/214, Page 3/54

4 High-gain control: An intuition controller k py `k D 9yq u F 2 psq system ps`5q ps1q 2 ps`1q y 9y k D structural properties of F 2 psq relative degree (pole excess): r positive high-frequency gain (lim sñ8 s 2 F 1 psq 1) minimum-phase (numerator is Hurwitz) imaginary axis Root-locus (ˆ poles, zeros) k D = k D = 1/ real axis christoph.hackl@tum.de Non-identifier based adaptive control in mechatronics: An overview 25/4/214, Page 4/54

5 Admissible system class S 1 (simplified) + 9xptq Axptq `b`uptq `u d ptq `B Tdptq ` ptxqptq yptq c J xptq, with structural properties : (i) γ : c J b and signpγ q known; j sin A b P C with Rpsq ě : det c J ; (iii) operator T is of class T (see Def. 1 in [14]) and M T : sup t ptξqptq t ě, ξp q P Cprh, 8q,R n quă8; (iv) u d p q P L 8 prh, 8q;Rq and dp q P L 8 prh, 8q;R m q. (1) Example (Linear system in the frequency domain) F 1 psq c J psi Aq1 b ps`4qps`5q ps1q 2 ps`1q christoph.hackl@tum.de Non-identifier based adaptive control in mechatronics: An overview 25/4/214, Page 5/54

6 Control objective and funnel controller for S 1 [14] objective: tracking with prescribed transient accuracy of reference y ref p q ψ pq epq ψ pq ψ ptq eptq ψ p q ep q λ t time t rss funnel controller: funnel where eptq y ref ptq yptq, y ref p q P CpR ě ;Rq, ψ p q P CpR ě ; rλ, 8qq with λ ą uptq k ptqeptq where k ptq ς ptq ψ ptq eptq (FC 1 ) with gain scaling ς p q P CpR ě ;R ą q (e.g. to fix minimal gain: k ptq ě ς ptq{ψ ptq for all t ě ) christoph.hackl@tum.de Non-identifier based adaptive control in mechatronics: An overview 25/4/214, Page 6/54

7 Admissible system class S 2 (simplified) + 9xptq Axptq `b`uptq `u d ptq `B Tdptq ` ptxqptq yptq c J xptq, with structural properties : (i) c J b, γ : c J Ab and signpγ q known; j sin A b P C with Rpsq ě : det c J ; (iii) operator T is of class T (see Def. 1 in [14]) and M T : sup t ptξqptq t ě, ξp q P Cprh, 8q,R n quă8; (iv) u d p q P L 8 prh, 8q;Rq and dp q P L 8 prh, 8q;R m q. (v) yp q and 9yp q are available for feedback. (2) Example (Linear system in the frequency domain) F 2 psq c J psi Aq1 b ps`5q ps1q 2 ps`1q christoph.hackl@tum.de Non-identifier based adaptive control in mechatronics: An overview 25/4/214, Page 7/54

8 Control objective and funnel controller for S 2 [9] objective: tracking with prescribed transient accuracy of y ref p q and 9y ref p q ψ pq epq ψ pq ψ ptq eptq ψ p q ep q ψ 1 pq t λ 9ep q funnel λ 1 9epq ψ 1 pq t ψ 1ptq ě d dt ψptq ` δ 9eptq ψ 1 p q time t rss funnel controller (with derivative feedback) ˆ uptq k ptq 2 eptq ` k1ptq k ptq 9eptq, eptq y ref ptq yptq (FC 2 ) ς ptq ς 1 ptq where k ptq and k 1 ptq ψ ptq eptq ψ 1 ptq 9eptq scaling functions ς i p q P CpR ě ;R ą q, i,1 christoph.hackl@tum.de Non-identifier based adaptive control in mechatronics: An overview 25/4/214, Page 8/54

9 Steady-state accuracy with internal models [1, 6, 12] asymptotic accuracy (i.e. lim tñ8 eptq ), cannot be guaranteed, since ψ ptq ě λ ą for all t ě typical solution: use of internal model (IM) to achieve asymptotic tracking augmented system e v u y (FC 1 ) or (FC 2 ) (IM) system if (IM) has relative degree zero positive high-frequency gain and is minimum-phase then augmented system has identical structural properties as system christoph.hackl@tum.de Non-identifier based adaptive control in mechatronics: An overview 25/4/214, Page 9/54

10 Example: PI-funnel control [12, 13]? y ref e 1 ψ e e v internal model u e.g. F C psq augmented system linear system e.g. F S psq y Example ( Augmented system in the frequency domain) PI controller: F C psq k P ` ki s kp s`ki s, k P, k I ą system: F S psq 1`s s 2s`1 kp s`ki 1`s augmented system : F C psqf S psq s s 2s`1 ùñ lim tñ8 eptq for y ref p q const. christoph.hackl@tum.de Non-identifier based adaptive control in mechatronics: An overview 25/4/214, Page 1/54

11 Stiff servo-systems drive (m M ) load (m L ) Θ hkkkkkkkkkikkkkkkkkkj drive torque m M p q P CpR ě ;Rq rnms (control input) load torque m L p q P L 8 pr ě ;Rq rnms (disturbance) inertia Θ ą kgm 2 state x pφ, ωq J : position φ rrads, speed ω rrad{ss friction (on drive & load side) gear with ratio g r P Rztu r1s (neglecting dynamics & backlash) signals available for feedback: position φ and/or speed ω 9 φ (deteriorated by n m p q and/or 9n m p q, resp.) christoph.hackl@tum.de Non-identifier based adaptive control in mechatronics: An overview 25/4/214, Page 11/54

12 Nonlinear model of stiff servo-systems [7, 8] 1{g r m L ν 2 ω gr `F 2 ω gr 1MS u ν 1ω `F 1 ω actuator ω φ 9 k A 1{Θ 1{g r u 9ω φ φ{g satûa r u A ω m φ m 9n m sensor(s) ω 9xptq Axptq `bsatûa `mm ptq `u A ptq pf 1 ωqptq `b L`mL ptq ` pf 2 qptq g r yptq c J xptq, xpq pφ, ω q J where k A ą, û A ą, u A P L 8 pr ě ;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 ě,rquă8 and «ff ˆ 1 ˆ A,b 1,b L 1 and e.g. c Θ g rθ ν1`ν 2 {g 2 r Θ n m + ˆ 1 g r. (1MS) christoph.hackl@tum.de Non-identifier based adaptive control in mechatronics: An overview 25/4/214, Page 12/54

13 Speed control implementation implementation in xpc target laboratory setup with sensor C y ref ω ref e speed controller m M ω ω m ω ` 9n m 9n m Standard PI controller PI-funnel controller ż t m M ptq k P eptq `k I epτqdτ ż t ς ptq m M ptq k ptqeptq `k I k pτqepτqdτ where k ptq ψ ptq eptq christoph.hackl@tum.de Non-identifier based adaptive control in mechatronics: An overview 25/4/214, Page 13/54

14 Speed control measurement results Set-point tracking ω + ṅm [rad/s] time t [s] PI-funnel controller PI controller ω ref ( ) ω ref ( ) ± ψ ( ) Reference tracking ω + ṅm [rad/s] e [rad/s] k, kp [Nms/rad] mm [Nm] ω ref ( ) time t [s] ±ψ( ) 2 ml( ) christoph.hackl@tum.de Non-identifier based adaptive control in mechatronics: An overview 25/4/214, Page 14/54

15 Position control implementation implementation in xpc target C 1 y ref φ ref 9y ref ω ref e 9e position controller m M laboratory setup with sensor(s) φ ω 9y ω ` 9n m y φ `n m 9n m n m Standard PID controller (with feedforward) ż t m M ptq k P eptq `k I epτqdτ `k D 9eptq `u F ptq PID-funnel controller ˆ m M ptq k ptq 2 eptq ` k1ptq ż t ˆ k ptq 9eptq `k I k pτq 2 epτq ` k1pτq k pτq 9epτq dτ where k i ptq ς iptq ψ iptq e for i,1. piq ptq christoph.hackl@tum.de Non-identifier based adaptive control in mechatronics: An overview 25/4/214, Page 15/54

16 Position control measurement results Set-point tracking φ + nm [πrad] time t [s] φ ref ( ) φ ref ( ) ± ψ ( ) PID-funnel controller PID controller (with feedforward) Reference tracking φ + nm [π rad] e [π rad] k 2 [Nm rad ] ė [rad/s] kk1, kd [ Nms rad ] mm [Nm] time t[s] φ ref ( ) ±ψ( ) ±ψ1( ) 2 ml( ) christoph.hackl@tum.de Non-identifier based adaptive control in mechatronics: An overview 25/4/214, Page 16/54

17 Funnel control with linear internal model [6] implementation m L test bench drive load y ref φ ref e ke 2 φ 9y ref ω ref 9e u m M k k 1 9e ω 9y ω ` 9n m y φ `n m 9n m n m [π rad], [π rad/s].5 φ ref p q, 1.5 ω ref p q time t [s] [Nm] m L p q time t [s] [π rad], [π rad/s] ψ p q, ψ 1 p q time t [s] christoph.hackl@tum.de Non-identifier based adaptive control in mechatronics: An overview 25/4/214, Page 17/54

18 Measurements: (FC 2 ) angle φ+n m [πrad] error e [πrad] P-gain k 2 [Nm/rad] torque m M [Nm] φ ref ( ) φ ref ( )±ψ ( ) ±ψ ( ) m L ( ) time t[s] christoph.hackl@tum.de Non-identifier based adaptive control in mechatronics: An overview 25/4/214, Page 18/54

19 Design and combination of internal models y ref e funnel v u 9y 9e internal model ref controller extended system of class S 2 system of class S 2 y 9y 9n m n m reference y ref ptq : # y, ď t ă t y 4 cosp2πf pt t qq ` 3y,,t 4 ď t ď t end design: (i) y y s ùñ F PI psq k P s`k I s combination and (ii) y P R rrads f ě r1{ss cosp2πf tq 1 s 2 ` p2πf q 2 ùñ F COS psq ps `z q 2 s 2 ` p2πf q 2, z,f ą. F IM psq upsq vpsq k P s `k I{k P s ps `z q 2 s 2 ` p2πf q 2, k P,k I,z,f ą. (IM) christoph.hackl@tum.de Non-identifier based adaptive control in mechatronics: An overview 25/4/214, Page 19/54

20 Measur.: (FC 2 ), (FC 2 )+(PI), (FC 2 )+(IM) angle φ+n m [πrad] error e [πrad] P-gain k 2 [Nm/rad] torque m M [Nm] φ ref ( ) φ ref ( )±ψ ( ) ±ψ ( ) m L ( ) time t[s] christoph.hackl@tum.de Non-identifier based adaptive control in mechatronics: An overview 25/4/214, Page 2/54

21 Funnel control for robotic systems [5] y E y ref,e Applications (spot-)welding painting/enameling mounting/assembling laser-beam cutting... Kuka KR 15-2 (Series 2) ( Goals: Accurate position control of end effector, i.e. for given λ ą ě t : e E ptq y ref,e ptq y E ptq ď λ. controller: simple and intuitive to tune christoph.hackl@tum.de Non-identifier based adaptive control in mechatronics: An overview 25/4/214, Page 21/54

22 Rigid-link revolute-joint robotic systems Model of robotic systems (in joint space) Mpyptqq:yptq `Cpyptq, 9yptqq 9yptq ` pf 9yqptq `gpyptqq `dptq uptq, + pypq, 9ypqq J py, y 1 q J P R 2n (ROBOT) Standard assumptions Dc M,c M P R n : ă c M I n ď Mpyq Mpyq J ď c M I n Dc C P R n : }Cpy,vqv} ď c C }v} 2 F: CpR ě;rq Ñ L 8 locpr ě;rq (causal friction operator) Dc g P R n : }gpyq} ď c g dp q P L 8 pr ě;r n q (disturbance) up q P CpR ě;r n q (control input) Measurements: yp q and 9yp q Structural properties (strict) relative degree r 2 positive (definite) high-frequency gain (matrix) no internal dynamics (minimum-phase) derivative feedback admissible christoph.hackl@tum.de Non-identifier based adaptive control in mechatronics: An overview 25/4/214, Page 22/54

23 Control objective Tracking with prescribed transient accuracy for each joint i P ě : e i ptq y ref,i ptq y i ptq ă ψ,i ptq and 9e i ptq ă ψ 1,i ptq ψ,i pq ψ,i ptq ψ 1,i pq e i pq e i ptq ψ,i p q λ 1,i e i p q λ,i t 9ei p q t ψ 1,i ptq 9e i ptq ψ 1,i p q time t rss ψ,i pq funnel 9e i pq ψ 1,i pq (joint-)reference y ref,i p q P W 2,8 pr ě ;Rq (joint-)boundary pψ,i p q, ψ 1,i p qq: e ipq ă ψ,ipq und 9e ipq ă ψ 1,ipq ψ,ip q, ψ 1,ip q P L 8 pr ě;r ąq and globally Lipschitz continuous Dδ i ě : ψ 1,iptq ě d ψ,iptq `δi dt christoph.hackl@tum.de Non-identifier based adaptive control in mechatronics: An overview 25/4/214, Page 23/54

24 MIMO funnel controller ψ,i pq e i pq ψ,i pq ψ,i ptq e i ptq ψ,i p q e i p q ψ 1,i pq t λ,i 9ei p q funnel uptq Mpyptqq x Klooooomooooon ptq 2 eptq P λ 1,i 9e i pq ψ 1,i pq `Klooooooooomooooooooon ptqk 1 ptq 9eptq D t ψ 1,iptq ě d dt ψ,iptq ` δi 9e i ptq `u FF ptq ψ 1,i p q TIME t rss (FC n ) K ptq diag k,1 ptq,..., k,n ptq ( und K 1 ptq diag k 1,1 ptq,..., k 1,n ptq P t1,...,nu: k,i ptq u FF p q P L 8 pr ě ;R n q 1 ψ,i ptq e i ptq and k 1,i ptq 1 ψ 1,i ptq 9e i ptq christoph.hackl@tum.de Non-identifier based adaptive control in mechatronics: An overview 25/4/214, Page 24/54

25 Properties of the closed-loop system MIMO funnel controller W 2,8 y ref 9y ref e 9e PD controller & feed forward K 2 e `K K 1 9e `u FF decoupling xm u (ROBOT) y 9y 9n m n m For x Mpyq Mpyq diag 1 pyq,..., n pyq ( u FF p q P L 8 pr ě ;R n q n m p q P W 2,8 pr ě ;R n q (measurement noise admissible) we P t1,...,nu Dε i ě : ψ,i ptq e i ptq ě ε i and ψ 1,i ptq 9e i ptq ě ε i K p q,k 1 p q P L 8 pr ě ;R n ąq up q P L 8 pr ě ;R n q christoph.hackl@tum.de Non-identifier based adaptive control in mechatronics: An overview 25/4/214, Page 25/54

26 Example: Planar robot (2DOF) d 1,F 1 9y 1,u 1 d 2,F 2 9y 2,u 2 l 1 y 1 m 2 l 2 y 2 m 1 (point-)masses m 1, m 2 rkgs length of joints l 1, l 2 rms ˆy1 joint angle y : rrads 2 y 2 ˆu1 control input u : rnms 2 u 2 ˆF1 9y friction F 9y : 1 rnms 2 F 2 9y 2 ˆd1 disturbance d : rnms 2 d 2 + Mpyptqq:yptq `Cpyptq, 9yptqq 9yptq ` pf 9yqptq `gpyptqq `dptq uptq, pypq, 9ypqq J p, q J P R 4 (ROBOT) christoph.hackl@tum.de Non-identifier based adaptive control in mechatronics: An overview 25/4/214, Page 26/54

27 Implementation: MIMO funnel controller W 2,8 y ref 9y ref e 9e MIMO funnel controller PD controller K 2 e `K K 1 9e decoupling xm u (ROBOT) y 9y uptq xmpyptqq hkkkkkkkkkkikkkkkkkkkkj j 5 Mpyptqq 2 j eptq ` k,1 ptq 2 k,2 ptq P t1,2u: k,i ptq j k,1 ptqk 1,1 ptq 9eptq k,2 ptqk 1,2 ptq ψ,i ptq ψ,i ptq e i ptq ě 1 & k 5ψ 1,i ptq 1,iptq ψ 1,i ptq 9e i ptq ě 5 christoph.hackl@tum.de Non-identifier based adaptive control in mechatronics: An overview 25/4/214, Page 27/54

28 Implementation: References, disturbance & boundary MIMO funnel controller W 2,8 y ref 9y ref e 9e PD controller K 2 e `K K 1 9e decoupling xm u (ROBOT) y 9y [rad],[rad/s] y ref,1 p q, y ref,2 p q [Nm] d 1p q d 2p q [rad],[rad/s] ψ,ip q, ψ 1,ip q Zeit t[s] Zeit t[s] Zeit t[s] christoph.hackl@tum.de Non-identifier based adaptive control in mechatronics: An overview 25/4/214, Page 28/54

29 Simulation results (joints: #1, #2) Winkel y 1,y 2 [rad] 4 y ref,1 ( ) y ref,2 ( ) 2 Geschwindigkeit ẏ 1,ẏ 2 [rad] Moment u 1,u 2 [Nm] Zeit t[s] ẏ ref,1 ( ) ẏ ref,2 ( ) d 1 ( )=d 2 ( ) christoph.hackl@tum.de Non-identifier based adaptive control in mechatronics: An overview 25/4/214, Page 29/54

30 Simulation results (joints: #1, #2) Positionsfehler e 1,e 2 [rad] P-Verstärkung k 2,1,k 2,2 [Nm/rad] Geschw.-Fehler ė 1,ė 2 [rad/s] D-Verstärkung k,1 k 1,1,k,2 k 1,2 [Nms/rad] Zeit t[s] ±ψ,1 ( )=±ψ,2 ( ) ±ψ 1,1 ( )=±ψ 1,2 ( ) christoph.hackl@tum.de Non-identifier based adaptive control in mechatronics: An overview 25/4/214, Page 3/54

31 Funnel control with internal model for robotic systems lim tñ8 e i ptq not guaranteed (since ψ,i ptq ě λ,i ą for all t ě ) application of (linear) internal model admissible, if relative degree zero positive high-frequency gain control- and observable (ùñ minimum-phase) Standard internal model in industry (SISO): * 9x I ptq uptq, x I pq k P s `k I vpsq vptq k P uptq `k I x I ptq s extended MIMO funnel controller PD controller & feed-forward internal model decoupling y ref e K 2 e `K K 1 9e v e.g. u W 2,8 9y 9e `u FF (PI n) xm ref upsq, k P,k I ą (PI) (ROBOT) y 9y 9n m n m christoph.hackl@tum.de Non-identifier based adaptive control in mechatronics: An overview 25/4/214, Page 31/54

32 Combination of internal models W 2,8 extended MIMO funnel controller PD controller & feed-forward internal model decoupling y ref e K 2 e `K K 1 9e v u 9y 9e (IM n) ref `u xm FF (ROBOT) y 9y 9n m n m References (for each joints) y ref,i : R ě Ñ R, t ÞÑ y ref,i ptq : a i sin `2πf i t `b i, a i ą, b i Internal models (SISO) F PI psq k P s `k I ps `γq2, k P,k I ą and F SIN psq s s 2 ` p2πf i q 2, γ,f i ą Serial interconnection F IM psq k P s `k I s ps `γq 2 s 2 ` p2πf i q 2, k P,k I,γ,f i ą (IM) christoph.hackl@tum.de Non-identifier based adaptive control in mechatronics: An overview 25/4/214, Page 32/54

33 Simulation results with (IM n ) ( #1, #2) Winkel y 1,y 2 [rad] 4 y ref,1 ( ) y ref,2 ( ) 2 Geschwindigkeit ẏ 1,ẏ 2 [rad] Moment u 1,u 2 [Nm] Zeit t[s] ẏ ref,1 ( ) ẏ ref,2 ( ) d 1 ( )=d 2 ( ) christoph.hackl@tum.de Non-identifier based adaptive control in mechatronics: An overview 25/4/214, Page 33/54

34 Simulation results with (IM n ) ( #1, #2) Positionsfehler e 1,e 2 [rad] P-Verstärkung k 2,1,k 2,2 [Nm/rad] Geschw.-Fehler ė 1,ė 2 [rad/s] D-Verstärkung k,1 k 1,1,k,2 k 1,2 [Nms/rad] Zeit t[s] ±ψ,1 ( )=±ψ,2 ( ) ±ψ 1,1 ( )=±ψ 1,2 ( ) christoph.hackl@tum.de Non-identifier based adaptive control in mechatronics: An overview 25/4/214, Page 34/54

35 Funnel control with saturation [9, 1] implementation (xpc target) m L satûa p q laboratory setup y ref e 1 ψ e e u ω ω ` 9n m 9n m û A ě u feas u feas py ref,ψ, system,...q ě : ψptq eptq ě ε! ) λ ε ď min 2, ψpq epq, λ 2pû A` m L 8q u feas conservative (for setup u feas «63 rnms and û A 22 rnms) christoph.hackl@tum.de Non-identifier based adaptive control in mechatronics: An overview 25/4/214, Page 35/54

36 PI-funnel control with saturation feasible? implementation (xpc target) y ref e 1 ψ e e v? 9x I k I v u k P v `x I u m L satûa p q laboratory setup ω ω ` 9n m 9n m Tracking with prescribed transient accuracy endangered, ě : eptq ă ψptq? christoph.hackl@tum.de Non-identifier based adaptive control in mechatronics: An overview 25/4/214, Page 36/54

37 PI controller with Anti-windup ( conditional integration ) v k P u satûa p q u sat k I 9x I x I f aw p q (PI aw ) û A System theoretical interpretation: Lemma (see q P CpR ě ;Rq ě : x I ptq ď maxtû A, x I pqu : x max I ùñ Anti-windup: x I p q acts as bounded input disturbance christoph.hackl@tum.de Non-identifier based adaptive control in mechatronics: An overview 25/4/214, Page 37/54

38 PI-funnel control with saturation [3] implementation (xpc target) y ref e 1 ψ e e v 9x I f awpuqk I v u k P v `x I u m L satûa p q laboratory setup ω ω ` 9n m 9n m û A ě u feas u feas py ref,ψ, system,...q ě : ψptq eptq ě ε! ) λ ε ď min 2, ψpq epq, k Pλ 2pû A`x max ` m L 8q u feas «63 rnms I christoph.hackl@tum.de Non-identifier based adaptive control in mechatronics: An overview 25/4/214, Page 38/54

39 Measur. results: (FC 1 )+(PI), (FC 1 )+(PI aw ) ω+nm [rad/s] e [rad/s] k [Nms/rad] y ref ( ) y ref ( )±ψ E ( ) ±ψ E ( ) u [Nm] 2 m L ( ) 2 ±û A time t [s] christoph.hackl@tum.de Non-identifier based adaptive control in mechatronics: An overview 25/4/214, Page 39/54

40 Funnel control for elastic systems [2] load (m L ) Θ 2, ω 2 Θ 1, ω 1 c S,d S drive (m M ) Goal 1: Reference tracking ω 2 Ñ ω 2,ref (or ω 1 Ñ ω 1,ref ) Goal 2: Active damping of natural frequency d c S pθ 1 `Θ 2 q ω «61 rrad{ss ùñ f «9.7 rhzs Θ 1 Θ 2 Constraint: ω 1 as feedback signal only christoph.hackl@tum.de Non-identifier based adaptive control in mechatronics: An overview 25/4/214, Page 4/54

41 Elastic system: Two-mass system model mech. system 1{g r m L ν 1 ω 1 `F 1 ω 1 d S ν 2 ω2 gr `F 2 ω2 gr actuator ω 1 u u A k A m M 1{Θ 1 1{g r 1{Θ 2 φ S c S ω 2 n m pω1 sensor, d dt x pf 1 ω 1 qptq Sptq A S x S ptq `b S`uptq `ua ptq `B S, /. m L ptq ` pf 2 ω 2 qptq /- (2MS) yptq p1,, qx S ptq, x S pq pω1, φ S,ω 2q P R 3 christoph.hackl@tum.de Non-identifier based adaptive control in mechatronics: An overview 25/4/214, Page 41/54

42 Funnel control for elastic systems with filter and state feedback [13] 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 9 S pφs n m pω 2 ω 2 `n m2 n m2 2MS P S 1 ðù c 1 ą ^ c2 ě ^ c3 ą c 1 ^ k F ą. g r g r For active damping: Feedback of ω 1 and ω 2 necessary! christoph.hackl@tum.de Non-identifier based adaptive control in mechatronics: An overview 25/4/214, Page 42/54

43 Elastic system with disturbance observer (DO) [11] mech. system 1{g r m L ν 1 ω 1 `F 1 ω 1 d S ν 2 ω2 gr `F 2 ω2 gr u u actuator k A m M u A ω 1 rm 1{Θ 1 1{g r c S 1{Θ 2 φ S ω 2 pω 1 1 1`sT DO m DO 1k DO sθ p 1{ k p A 1`sT DO observer pω 1 n m sensor m DO psq rmpsq k A u A psq (for p k A k A, p Θ 1 Θ 1, k DO T DO n m ) christoph.hackl@tum.de Non-identifier based adaptive control in mechatronics: An overview 25/4/214, Page 43/54

44 Elastic system with (DO) element of class S 1 Extended two-mass system with x pω 1, φ S, ω 2, x F q J, d dt xptq Axptq `b`u pf 1 ω 1 qptq /. ptq `u A ptq `B, m L ptq ` pf 2 ω 2 qptq /- (3) yptq c J xptq, xpq ppx S qj, q J P R 4 where A j AS, b kdo T DO Proposition (see [2]) ˆbS 1, B T DO j BS and c, J p1,,, q. For T DO ą, k DO P R, p k A ą and p Θ 1 ą, the extended two-mass system (3) is element of system class S 1. Proof by checking system properties (i) (iv) of system class S 1. christoph.hackl@tum.de Non-identifier based adaptive control in mechatronics: An overview 25/4/214, Page 44/54

45 Implementation: Funnel control with (DO) implementation in xpc target y ref g rω 2,ref e W 1,8 funnel controller u u 2MS with sensors ω 1 ω 2 pω 1 ω 1 `n m m DO disturbance observer n m Disturbance observer design k DO ^ T DO.3 rss ˆ rhzs ą f 9.7 rhzs T DO christoph.hackl@tum.de Non-identifier based adaptive control in mechatronics: An overview 25/4/214, Page 45/54

46 Measur. results: (FC 1 )+(PI), (FC 1 )+(DO) ω 2 +n m [rad/s] e [rad/s] 5 1 k [Nms/rad] m M [Nm] y ref ( ) y ref ( )±ψ E ( ) ±ψ E ( ) m L ( ) time t [s] christoph.hackl@tum.de Non-identifier based adaptive control in mechatronics: An overview 25/4/214, Page 46/54

47 Meas. res. (zoom): (FC 1 )+(PI), (FC 1 )+(DO),3, zoom φs [rad],1.5 1,2 zoom , time t [s] christoph.hackl@tum.de Non-identifier based adaptive control in mechatronics: An overview 25/4/214, Page 47/54

48 Funnel control for wind turbine systems [4] v W ω T ωg g r r T ω T r T ω G gr r T Operation regimes PT rws P N I II III IV v cutin v N v cutout v m W s Simple turbine model d dt ω G 1 ˆmT pv W,β,ω G q `m M,ref Θ g r with Θ : Θ G `Θ T {g 2 r christoph.hackl@tum.de Non-identifier based adaptive control in mechatronics: An overview 25/4/214, Page 48/54

49 Power coefficient, turbine torque and goal c P,1 pλ q P T c P pv W,β,ω G q 1 2 r2 Tπv 3 W loooomoooon : P W λ c P,1 p q ùñ m T pv W,β,ω G q g rc P pv W,β,ω G qp W ω G λ Goal for regime II: Maximum power point tracking for fixed β β ě, i.e. λ rtωg g rv W Ñ λ (or ω G Ñ ω G ) for all λ P pλ min,λ max q and v cutin ď v W ă v N christoph.hackl@tum.de Non-identifier based adaptive control in mechatronics: An overview 25/4/214, Page 49/54

50 Nonlinear controller design ω G,ref Controller? m M,ref 3 2 pψ PM i q s,ref 1 «1 m T pv W,β,ω Gq g r pψpm T ers,i q s Θ m G ` ω G i q s 1 Standard controller [15]: m G,ref ptq k Pω G ptq 2 with k P 1 2 r2 Tπ c P pβ,λ q pλ q 3 (WTC) Funnel controller: ςptq m G,ref ptq ω G,ref ptq ω G ptq loooooomoooooon ψptq eptq looooooooomooooooooon with ω G,ref ptq g rλ v W ptq r T :kptq :eptq (FC WT 1 ) christoph.hackl@tum.de Non-identifier based adaptive control in mechatronics: An overview 25/4/214, Page 5/54

51 Simulation results: (FC WT 1 ), (WTC) 6 v W [m/s] ω G [rad/s].5 v W ω G,ref ( ).2 ±ψ( ) e [rad/s] k [1 7 Nms/rad] time t [s] christoph.hackl@tum.de Non-identifier based adaptive control in mechatronics: An overview 25/4/214, Page 51/54

52 Simulation results: (FC WT 1 ), (WTC) 1 λ [1] 5 c P [rad/s] λ c P,Betz = 16/ c P E [kwh] time t [s] christoph.hackl@tum.de Non-identifier based adaptive control in mechatronics: An overview 25/4/214, Page 52/54

53 Conclusion To take home funnel control applicable for speed and position control of rigid systems elastic systems (active damping by disturbance observer) rigid-link revolute-joint robotic manipulators (if inertia matrix is roughly known) wind turbine systems is feasible (if rotor radius, optimal tip speed ratio, gear ratio and wind measurements are available) only structural properties must be checked (robustness) no system identification or parameter estimation necessary (mainly) time-varying gains (also decrease possible) tracking with prescribed transient accuracy steady state accuracy in conjunction with internal model (e.g. PI) funnel control in presence of actuator saturation is feasible (conservative feasibility condition must be satisfied) PI-funnel control with Anti-windup feasible (wind-up effect removed) Non-identifier based adaptive control in mechatronics: An overview 25/4/214, Page 53/54

54 References [1] C. M. Hackl. High-gain adaptive position control. International Journal of Control, 84(1): , 211. [2] C. 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] 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, pages , 213. [4] C. M. Hackl. Funnel control for wind turbine systems. In submitted to the 214 IEEE Multi-conference on Systems and Control, 214. [5] C. M. Hackl and R. 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. [6] C. M. Hackl and R. M. Kennel. Position funnel control with linear internal model. In Proceedings of 212 IEEE International Conference on Control Applications, pages , 212. [7] 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, pages , 211. [8] 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, pages 8 87, 211. [9] 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, 51(2): , 213. [1] N. Hopfe, A. Ilchmann, and E. P. Ryan. Funnel control with saturation: Nonlinear SISO systems. IEEE Transactions on Automatic Control, 55(9): , 21. [11] 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, 46(1): , [12] A. Ilchmann and E. P. Ryan. Asymptotic tracking with prescribed transient behaviour for linear systems. International Journal of Control, 79(8): , 26. [13] A. Ilchmann and H. Schuster. PI-funnel control for two mass systems. IEEE Transactions on Automatic Control, 54(4): , 29. [14] A. Ilchmann, E. P. Ryan, and C. J. Sangwin. Tracking with prescribed transient behaviour. ESAIM: Control, Optimisation and Calculus of Variations, 7: , 22. [15] L. Y. Pao and K. E. Johnson. Control of wind turbines: Approaches, challenges, and recent developments. IEEE Control Systems Magazine, 31(2): 44 62, 211. christoph.hackl@tum.de Non-identifier based adaptive control in mechatronics: An overview 25/4/214, Page 54/54