PI-funnel control with Anti-windup

Transcription

1 PI-funnel control with Anti-windup Christoph Hackl Technische Universität München Institute for Electrical Drive Systems and Power Electronics 52 nd IEEE Conference on Decision and Control, Florence 13 th December /12/213 Christoph Hackl: PI-funnel control with Anti-windup Page 1/15

2 Outline 1 PI-funnel control 2 PI-funnel control with Anti-windup 3 Conclusions and future work 13/12/213 PI-funnel control with Anti-windup, Christoph Hackl 2/15

3 Funnel control [Ilchmann et al., 22] ψpq epq ψptq eptq λ t Time t rss ψpq Funnel Tracking with prescribed transient accuracy by design of ψp q, ě : eptq y ref ptq yptq ă ψptq Controller uptq 1 ψptq eptq loooooomoooooon :kptq eptq 13/12/213 PI-funnel control with Anti-windup, Christoph Hackl 3/15

4 Admissible system class Structural knowledge sufficient: relative degree one (or known relative degree [Ilchmann et al., 27]) BIBO stable zero dynamics (or minimum phase for linear systems) known sign of the high-frequency gain Example (LTI SISO system in Byrnes-Isidori form) d dt ˆyptq zptq j a1 a J 2 ˆyptq a 3 A 4 zptq ˆ γ ` uptq, n 1 ˆypq zpq ˆy P R n z γ specpa 4 q P C ă signpγ q known 13/12/213 PI-funnel control with Anti-windup, Christoph Hackl 4/15

5 FC with internal model [Ilchmann and Ryan, 26]? y ref e 1 ψ e e v internal model u e.g. F R psq augmented system linear system e.g. F S psq y Example ( Augmented system in the frequency domain) PI controller: F R psq k P ` ki s kp s`ki s, k P, k I ą system: F S psq 1`s s 2 s`1 kp s`ki 1`s augmented system : F R psqf S psq s s 2 s`1 ùñ lim tñ8 eptq for y ref p q const. 13/12/213 PI-funnel control with Anti-windup, Christoph Hackl 5/15

6 PI-funnel control for electrical drives [Ilchmann and Schuster, 29] implementation (xpc target) m L laboratory setup y ref e 1 ψ e e v 9x I k I v u k P v `x I u ω ω ` 9n m 9n m Scenario 1: y ref p q 1 rrad{ss Scenario 2: y ref p q 1 rrad{ss 13/12/213 PI-funnel control with Anti-windup, Christoph Hackl 6/15

7 Measurement results for y ref p q 1: PI-FC ω+ṅm [rad/s] y ref ( ) e [rad/s] k [Nms/rad] 2 ±ψ( ) u [Nm] 2 m L ( ) time t[s] 13/12/213 PI-funnel control with Anti-windup, Christoph Hackl 7/15

8 Measurement results for y ref p q 1: PI-FC ω+ṅm [rad/s] e [rad/s] k [Nms/rad] y ref ( ) y ref ( )±ψ( ) ±ψ( ) u [Nm] 2 m L ( ) 2 ±û A time t [s] 13/12/213 PI-funnel control with Anti-windup, Christoph Hackl 8/15

9 PI-funnel control with saturation 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? 13/12/213 PI-funnel control with Anti-windup, Christoph Hackl 9/15

10 Funnel control with saturation [Hopfe et al., 21] 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) 13/12/213 PI-funnel control with Anti-windup, Christoph Hackl 1/15

11 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: 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 13/12/213 PI-funnel control with Anti-windup, Christoph Hackl 11/15

12 PI-funnel control with saturation 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 13/12/213 PI-funnel control with Anti-windup, Christoph Hackl 12/15

13 Results for y ref p q 1: PI-FC, PI aw -FC 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] 13/12/213 PI-funnel control with Anti-windup, Christoph Hackl 13/15

14 Conclusions and future work To take home: PI-funnel control: windup possible use of anti-windup ( conditional integration ) windup eliminated improved control performance (no saw tooth response ) integral control action acts as bounded input disturbance ùñ PI aw -funnel control admissible for systems with saturation Open questions: extension feasible?... PI-funnel control for systems with relative degree two?... funnel control with internal model?... different anti-windup strategies beneficial? 13/12/213 PI-funnel control with Anti-windup, Christoph Hackl 14/15

15 References N. Hopfe, A. Ilchmann, and E. P. Ryan. Funnel control with saturation: Nonlinear SISO systems. IEEE Transactions on Automatic Control, 55(9): , 21. A. Ilchmann and E. P. Ryan. Asymptotic tracking with prescribed transient behaviour for linear systems. International Journal of Control, 79(8):91 917, 26. A. Ilchmann and H. Schuster. PI-funnel control for two mass systems. IEEE Transactions on Automatic Control, 54(4): , 29. A. Ilchmann, E. P. Ryan, and C. J. Sangwin. Tracking with prescribed transient behaviour. ESAIM: Control, Optimisation and Calculus of Variations, 7: , 22. A. Ilchmann, E. Ryan, and P. Townsend. Tracking with prescribed transient behaviour for nonlinear systems of known relative degree. SIAM Journal on Control and Optimization, 46(1): 21 23, /12/213 PI-funnel control with Anti-windup, Christoph Hackl 15/15