Adaptive Stochastic Feedback Control of Resistive Wall Modes in Tokamaks

11th Workshop on MHD Stability & Control“Active MHD Control in ITER ”

Princeton, New JerseyNov 6- Nov 8 2006

Zhipeng Sun, Amiya Sen and Richard Longman

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Outline

• Motivation • Part I: System identification • Part II: Adaptive output feedback control • Part III: Neural network control• Summary

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Motivation• Resistive Wall Mode (RWM) in tokamaks

Need for stabilization• Past research on the suppression of RWM

Deterministic models usedNo optimal control usedFixed gain controllers without adaptive structurePID control inadequate for multi unstable modes

• Sen, Nagashima and Longman’s work Stochastic model used, optimal state feedback

• New methods as in the outlineState feedback control, output feedback control and its neural netowrk (NN) implementation

Part I

Online System Identification

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Objectives

• A mathematical model should be estimated from experimental data

Should be able to estimate time-varying systems Convergence time should be shorter compared to the inverse of the growth rateComputational burden should be small

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System models of a single unstable RWM

• State-space system model of a single unstable RWM

)()()()()()()(

ttHIttDItButAItI

m

n

ψψ +=++=

•

6

6

0.474 502 53114.7 79.0 16500

100 1.40 1030 2.30 10

T

A B

D H−

−

− − − = = − −

× = = ×

For a RWM in DIII-D tokamak

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• Difference equation model

Sampling rate is 1ms, e(k) is system noise q is the forward shift operator

• Noise modeling Total noise, approximately ½ to 1 Gauss, is evenly divided between the measurement noise and the plant noise RMS value of the plant noise is about 10-4 WeberRMS value of the measurement noise about 10-4

Weber

( ) ( ) ( ) ( ) ( ) ( )A q k B q u k C q e kψ = +

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( )1

ˆ ˆ ˆ( ) ( 1) ( )( ( ) ( 1))

( ) ( ) ( ) ( 1) ( )( ( ) ( 1) ( ))( ) ( ( ) ( )) ( 1)

T

T

T

k k K k k k k

K k P k k P k k I k P k kP k I K k k P k

θ θ ψ ϕ θ

ϕ ϕ ϕ ϕ

ϕ

−

= − + − −

= = − + −

= − −

use to approximate system noise

{ } (n)}(k)...(1)...{ :SequenceOutput ,)...u(n)u(1)...u(k :SequenceInput ψψψ( )1 2 0 1 1 2 a a b b c cθ = ))1()()1()()1()(()( −−−−−= kekekukukkkT ψψϕ

( ) ( ) ( ) ( )1 1Tk k k kε ψ ϕ θ= − − − ( )e k

• Autoregressive system model( ) ( 1)Tk kψ ϕ θ= −

• Extended least square (ELS) method

is defined as the estimate of θθ̂

• Regression model setup

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Identification of the time-invariant modelEstimation of A(q) Estimation of B(q)

Estimation

Estimation

True value

True value

Estimation

True value

True value

Estimation

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ELS method for the time-varying system• Real plasma systems are time-varying

Use a forgetting factor λ, 0 < λ ≤ 1, The ELS method becomes

Relationship between λ and the evolution of the system

• Simulation of a time-varying system modelThe simulation starts with the original modelThe system matrix A takes step increase of 10% every 50ms:

A A*1.1 A*1.2 … A*2

( )^ ^ ^

1

( ) ( 1) ( )( ( ) ( 1))

( ) ( ) ( ) ( 1) ( )( ( ) ( 1) ( ))( ) ( ( ) ( )) ( 1) /

T

T

T

k k K k k k k

K k P k k P k k I k P k kP k I K k k P k

θ θ ψ ϕ θ

ϕ ϕ λ ϕ ϕ

ϕ λ

−

= − + − −

= = − + −

= − −

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Identification of the time-varying systemEstimation of A(q) Estimation of B(q)

Estimation

Estimation

True value

True value

Estimation

Estimation

True value

True value

Part II

Adaptive Output Feedback Control

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Objectives • Minimize the output (fluctuation) energy

and control energy

• Stabilization time is short

• Control design should be simple and fast

• Computation burden should be low

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Block diagram of the controlled plasma

ControlDesign

OnlineEstimation

PlasmaControllerInput Output

Process ParametersSpecifications

Control Parameters In

ψm

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• Quadratic cost function

• Control law

• R and S satisfy the Diophantine equation

• P is the solution to a spectral factorization problem

( ){ }2 2( )J E k uψ ρ= +

( ) ( ) ( ) ( )R q u q S q qψ= −

)()()()()()( qCqPqSqBqRqA =+

)()()()()()( 111 −−− += qBqBqAqAqPqrP ρ

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Control Signal (Volt)Magnetic Flux (Weber)

Output feedback control of the time-invariant system

Stabilization point

Stabilization point

Steady-state value: 4*10-4Weber

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Magnetic Flux (Weber) Control Signal (Volt)

Adaptive output feedback control of the time-varying system

Step changes Step changes

Steady-state value: 1.8*10-3Weber

Part III

Neural Network Control

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Objectives

• Develop a control algorithm for a Neural Network (NN)

• Implement the adaptive output feedback control with the algorithm

• Use a digital neural network hardware

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• Mathematical model for the neuron.are inputs. They are multiplied by

connection weights and summed. The sum is passed to a transfer function and the result is the output of the neuron.

• A neural network (NN) is a system composed of many neurons

Its function is determined by network structure, connection strengths, and transfer functionsThe transfer function is chosen to be a linear function in the study

• A Neural Network processor (NNP) made by Accurate Automation Corporation (AAC) has been debugged and software improved.

muuu ,, 21

mwww ,, 21

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Block Diagram of the NNP controlDiophantine eq.

Output feedback control design

Input

Output

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A generalized linear Hopfield network

xm

x2

x1

bm

b2

b1

um

u2

u1

w2k

w1k

wmk

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• The generalized linear Hopfield network can solve simultaneous linear equations, e.g., the Diophantine Eq.

The Diophantine Eq. should be rewritten as Stage 1: a feedforward layer with b as its inputs and ATas its weight matrixStage 2: a Linear Hopfield layer whose inputs are the outputs of the Stage 1 layer, and weight matrix is ,

where

The outputs of the second layer give the negative of the conjugate of the solution being sought.

bAx =

)( TAAIW α−=)(

10AAtrace T<< α

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Interface of the NN controller

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Stabilization of the time-invariant system

Steady-state value: 4.5*10-4Weber

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Stabilization of the time-varying system

Steady-state value: 1.9*10-3Weber

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Computation time

• Matrix inversion is used as an example.• Sequential algorithms

Lower-upper decomposition algorithm is used to do the inversionComplexity of this algorithm is O(N3)(C++ notation).

• Parallel (NN) algorithm LHN is the neural network used to invert the matrix. Complexity of this algorithm is either O(N1) or O(1)(C++ notation).

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Summary• The ELS method can give an accurate estimate of the

single mode RWM • Stochastic optimal output feedback control can stabilize

the single mode RWM, it is able to stabilize the RWM with a convergence time of three times the inverse of the growth rate.

• Neural Network Processor can be used to implement the adaptive stochastic optimal output feedback control of a RWM.

• Computation time of the neural network control is similar to the output feedback control. However, it will be much faster for high-order systems.