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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.

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

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Extremum seeking without external dithering and its application to plasma rf heating on ftu

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


Extremum seeking without external dithering and its application to plasma rf heating on ftu

University of Rome “Tor Vergata”

  • Outline:

  • Introduction to the problem;

  • The previous extremum seeking algorithm;

  • The proposed controller;

  • Simulation results;

  • Conclusions and future works.


Extremum seeking without external dithering and its application to plasma rf heating on ftu

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


Extremum seeking without external dithering and its application to plasma rf heating on ftu

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


Extremum seeking without external dithering and its application to plasma rf heating on ftu

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


Extremum seeking without external dithering and its application to plasma rf heating on ftu

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


Extremum seeking without external dithering and its application to plasma rf heating on ftu

University of Rome “Tor Vergata”

Solutions implemented (cont’d) :

Extremum Seeking

Approach

Naïve Approach

The previous extremum seeking algorithm


Extremum seeking without external dithering and its application to plasma rf heating on ftu

University of Rome “Tor Vergata”

Issues related to the previous extremum seeking algorithm:

Shot # 26718

  • Overshoots;

The previous extremum seeking algorithm


Extremum seeking without external dithering and its application to plasma rf heating on ftu

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


Extremum seeking without external dithering and its application to plasma rf heating on ftu

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


Extremum seeking without external dithering and its application to plasma rf heating on ftu

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


Extremum seeking without external dithering and its application to plasma rf heating on ftu

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


Extremum seeking without external dithering and its application to plasma rf heating on ftu

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


Extremum seeking without external dithering and its application to plasma rf heating on ftu

  • 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


Extremum seeking without external dithering and its application to plasma rf heating on ftu

As an ideal case, we consider , and

University of Rome “Tor Vergata”

The proposed controller: dynamic

  • is bounded;

  • is not measured.

The proposed controller


Extremum seeking without external dithering and its application to plasma rf heating on ftu

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


Extremum seeking without external dithering and its application to plasma rf heating on ftu

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


Extremum seeking without external dithering and its application to plasma rf heating on ftu

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


Extremum seeking without external dithering and its application to plasma rf heating on ftu

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


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