Extremum seeking without external dithering and its application to plasma RF heating on FTU

1. University of Rome “Tor Vergata” Extremum seeking without external dithering and its application to plasma RF heating on FTU Luca Zaccarian, Daniele Carnevale, Alessandro Astolfi and Salvatore Podda Technische Universiteit Eindhoven 7-8 May, 2008 Control for Nuclear Fusion

2. University of Rome “Tor Vergata” • Outline: • Introduction to the problem; • The previous extremum seeking algorithm; • The proposed controller; • Simulation results; • Conclusions and future works.

3. Reflected Power Transmitted Power University of Rome “Tor Vergata” The problem: During the LH pulse the plasma reflects a percentage of the heating power injected by the Lower Hybrid (LH) antennas: the power reflection function (wrt the plasma position y) is unknown. Plasma Goal: Develop a control schema (to set the plasma/antennas position) to minimize the reflected power during the LH pulse (extremum seeking algorithm). Introduction

4. Naïve approach: the unknown function is assumed to be quadratic, , and was detected on-line processing the available measurements. Then, the plasma reference position was modified with a star case funtion such that . University of Rome “Tor Vergata” Solutions implemented: Introduction

5. System with “fast” dynamics Extremum Seeking Controller X The probing signal University of Rome “Tor Vergata” Solutions implemented: Naïve approach. Extremum seeking: the control schema in [M.Krstíc and H.H.Wang ’00]… is unknown Introduction

6. LH antennas: the sensing cells University of Rome “Tor Vergata” Solutions implemented (cont’d): Naïve approach. Extremum seeking: the control schema in [M.Krstíc and H.H.Wang ’00] has been modified and applied to the FTU plant [Centioli et all ’05]. the probing signal Preprogrammed Reference Controlled plasma dynamics Extremum Seeking Controller The previous extremum seeking algorithm

7. University of Rome “Tor Vergata” Solutions implemented (cont’d) : Extremum Seeking Approach Naïve Approach The previous extremum seeking algorithm

8. University of Rome “Tor Vergata” Issues related to the previous extremum seeking algorithm: Shot # 26718 • Overshoots; The previous extremum seeking algorithm

9. University of Rome “Tor Vergata” Issues related to the previous extremum seeking algorithm: Shot # 26722 • Overshoots; • Convergence (‘regularity’ of the noise d1); • No formal proof. The previous extremum seeking algorithm

10. University of Rome “Tor Vergata” • Wavelet analysis of the signal d1: • wavelet function db(8); • Ts = 5 ms (sampling time); • scale settings [1,1,512]. Approximatively 250 Hz (Band-pass filters F:150-350Hz) The previous extremum seeking algorithm

11. University of Rome “Tor Vergata” The proposed controller: standing assumption Assumption 1: The unknown map is locally Lipschitz, locally bounded and there exist a and a class function such that for almost all : • Note that may not to be differentiable ( ) The proposed controller

12. As an ideal case, we consider , and University of Rome “Tor Vergata” The proposed controller: static • y is assumed measurable; • the pre-programmed reference is constant during the LH pulse The proposed controller

13. Assumption 2: The disturbance is bounded and has bounded time derivative, namely there exist positive numbers and such that for all t ≥ 0. University of Rome “Tor Vergata” The proposed controller: properties of d1 Assumption 3: The disturbance is such that there exist and satisfying The proposed controller

14. Theorem 1: Assume that Assumptions 1 and 2 hold. Then for any positive and , the closed-loop system (1) satisfies the following properties. • Both and are bounded, • the set is forward invariant and • If in addition Assumption 3 holds, then the set A is attractive. University of Rome “Tor Vergata” The proposed controller: static The proposed controller

15. As an ideal case, we consider , and University of Rome “Tor Vergata” The proposed controller: dynamic • is bounded; • is not measured. The proposed controller

16. University of Rome “Tor Vergata” The proposed controller: dynamic • Theorem 1: Assume that Assumptions 1 and 2 hold. Then for any positive and , the closed-loop system (2) satisfies the following properties. • Both and are bounded, • the set is eventually forward invariant and • If in addition Assumption 3 holds and for some constant ,then the set A is attractive and The proposed controller

17. University of Rome “Tor Vergata” Application to FTU of thedynamic controller: to approximate the derivate, to filter . Approximation of the function used to simulate the new controller wrt experimental data Simulation results

18. Shot # 26725 Shot # 26722 Simulation results fit experimental data. The new filters Fhave larger band…. Parameters of the dynamic algorithm: University of Rome “Tor Vergata” Simulation of thedynamic controller: Simulation results fit experimental data. Simulation results

19. University of Rome “Tor Vergata” Conclusions: • The new extremum seeking algorithm: • Avoids overshoots and then copes with actuator’s rate limits; • Mild requirement (persistency of excitation like) , increased performances; • Formal proof for the ideal case. Future works: • Experimental tests on FTU; • Formal proof with filters and noise ; • Generalization of the new approach; Conclusions and future works