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

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

3 Develop a control schema (to set the plasma/antennas position) to minimize the reflected power during the LH pulse (extremum seeking algorithm). 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. The problem: University of Rome “Tor Vergata” Goal: Introduction Transmitted Power Reflected Power Plasma

4 Solutions implemented: University of Rome “Tor Vergata” Introduction 1. 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.

5 is unknown 1. Naïve approach. 2. Extremum seeking: the control schema in [M.Krstíc and H.H.Wang ’00]… Solutions implemented: University of Rome “Tor Vergata” Introduction The probing signal System with “fast” dynamics Extremum Seeking Controller X

6 1. Naïve approach. 2. 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]. Solutions implemented (cont’d): University of Rome “Tor Vergata” The previous extremum seeking algorithm Preprogrammed Reference Extremum Seeking Controller the probing signal Controlled plasma dynamics LH antennas: the sensing cells

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

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

9 Issues related to the previous extremum seeking algorithm: University of Rome “Tor Vergata” The previous extremum seeking algorithm 1. Overshoots; 2. Convergence (‘regularity’ of the noise d 1 ); 3. No formal proof. Shot # 26722

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

11 University of Rome “Tor Vergata” The proposed controller 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 ( )

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

13 University of Rome “Tor Vergata” The proposed controller The proposed controller: properties of d 1 Assumption 2: The disturbance is bounded and has bounded time derivative, namely there exist positive numbers and such that for all t ≥ 0. Assumption 3: The disturbance is such that there exist and satisfying

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

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

16 University of Rome “Tor Vergata” The proposed controller 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. 1)Both and are bounded, 2)the set is eventually forward invariant and 3)If in addition Assumption 3 holds and for some constant, then the set A is attractive and

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

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

19 University of Rome “Tor Vergata” Conclusions and future works Conclusions: The new extremum seeking algorithm: 1.Avoids overshoots and then copes with actuator’s rate limits; 2.Mild requirement (persistency of excitation like), increased performances; 3.Formal proof for the ideal case. Future works: 1.Experimental tests on FTU; 2.Formal proof with filters and noise ; 3.Generalization of the new approach;