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”

Luca Zaccarian, Daniele Carnevale, Alessandro Astolfi and Salvatore Podda

Technische Universiteit Eindhoven 7-8 May, 2008 Control for Nuclear Fusion

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Outline:

• Introduction to the problem;

• The previous extremum seeking algorithm;

• The proposed controller;

• Simulation results;

• Conclusions and future works.

University of Rome “Tor Vergata”

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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

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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 .

2)()( yyyg y

yy

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is unknown

1. Naïve approach.

2. Extremum seeking: the control schema in [M.Krstíc and H.H.Wang ’00]…

Solutions implemented:

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Introduction

),( uyfy )(ygyg

)sin( ta The probing signal

)(gy

gy

System with “fast” dynamics

uExtremum Seeking

Controller hs

s

X

s

k1

l

l

s

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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

ryy

Preprogrammed

Reference

Extremum SeekingController

r

1d

2d

y)(ygyg

gy

s

k F

F

the probing signal

Controlled plasma

dynamics

LH antennas: the sensing cells

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Solutions implemented (cont’d) :

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The previous extremum seeking algorithm

Naïve Approach

Extremum Seeking Approach

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Issues related to the previous extremum seeking algorithm:

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The previous extremum seeking algorithm

Shot # 26718

1. Overshoots;

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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 d1);

3. No formal proof.

Shot # 26722

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University of Rome “Tor Vergata”

The previous extremum seeking algorithm

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)

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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 :

RRg :)(Ry K 00:)( RR

Rs|)(|||))(( ysysyssg

• Note that may not to be differentiable ( )

)(g|| x

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University of Rome “Tor Vergata”

The proposed controller

The proposed controller: static

).(

),()(

,

2121

2

1

zzksatky

dyy

dygz

dyz

As an ideal case, we consider , and 0, 21 ddd

• y is assumed measurable;• the pre-programmed reference is constant during the LH pulse

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The proposed controller

The proposed controller: properties of d1

Assumption 2: The disturbance is bounded and has bounded time derivative, namely there exist positive numbers and such that for all t ≥ 0.

dd d ,|)(|,|)(| dtddtd

Assumption 3: The disturbance is such that there exist and satisfyingd 0T 0c

.0,|)(|

tcddTt

t

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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.

1k 2k

).(

),()(

,

,

2121

2

1

zzksatky

dyy

dygz

dyz

yy

)1(

y

dyyRyA :

,0,)(),(max)( 0

tygagygAa

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University of Rome “Tor Vergata”

The proposed controller

The proposed controller: dynamic

).(

),()(

,

,

2121

2

1

zzksatk

dyy

dygz

dyz

yy

As an ideal case, we consider , and 0, 21 ddd

• is bounded;• is not measured.y

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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

1k 2k

).(

),()(

,

,

2121

2

1

zzksatk

dyy

dygz

dyz

yy

)2(

y

dkyyRyA 12:

,0,)(),(max)( 0

tygagygAa

ddtd |)(| dd.2Ly

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Simulation results

Application to FTU of the dynamic controller:

,)1)(1(

)(21

ss

ssF to approximate the derivate, to filter .11 21

Approximation of the function used to simulate the new controller wrt experimental data

)(g

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Simulation results

Simulation of the dynamic controller:

Parameters of the dynamic algorithm:

.10

,10

,3.0

,5.0

22

41

2

1

k

k

)(g

Simulation results fit experimental data.

Shot # 26725Simulation results fit experimental data.The new filters F have larger band….

Shot # 26722

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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:

d

1. Experimental tests on FTU;

2. Formal proof with filters and noise ;

3. Generalization of the new approach;