12/01/2004 E.Lazzaro Plasmi04, Arcetri Real-time tracking and feedback control strategies for rotating magnetic islands E. Lazzaro with the contribution of J.O.Berrino 1,G.D’Antona 2,S.Cirant 1 1 IFP “P.Caldirola”, Euratom-ENEA-CNR Association, Milano, Italy 2 Dip.Ingegneria Elettrotecnica, Politecnico di Milano, Italy

12/01/2004 E.Lazzaro Plasmi04, Arcetri Outline Introduction Key questions on observed processes Relation between measured signals and state variables Methods and models of control theory

12/01/2004 E.Lazzaro Plasmi04, Arcetri Introduction In the design of a realistic tokamak reactor device, the problem of reaching the desired performance targets and the problem of controlling various MHD instabilities have objectively a status of equal importance, scientifical and technical. Automatic control theory is highly developed and practically successful: in plasma physics there is ample scope of application of its principles and methods. The problem of control involves the basic aspects the plasma physics as well the need of a specific approach that requires a clear statement of the task, that includes: Development of the (simplest) models of the process to be controlled. Definition (selection )of the appropriate state variables {x i } and admissible control variables {u i } (i=1,…N), measurable and controllable with definite error bounds. Selection of the control policy,as a subset of the control variables. Choice of the decisional algorithms, implementation into a detection/ control device, and a suitable actuator.

12/01/2004 E.Lazzaro Plasmi04, Arcetri Key Questions on Resistive Modes in Tokamaks Theory predicts tearing of nested magnetic surface with rational (q=m/n ) with formation of growing and rotating magnetic “islands” that deteriorate the confinement properties and may cause a number of dire events.Several conditions for appearance of these instabilities have been predicted and confirmed. But what are the actual, observed aspects of the phenomenon that we may decide to control ? These instabilities are mainly observed as magnetic signals, picked up externally, with a certain frequency and amplitude (generally growing ). What are the footprints of these modes on the plasma properties, that suggest (or demand ) control? The most impressive footprints are the rapid, localized fluctuations on the background electron temperature profile measured by ECE radiation, in the density profile by reflectometry and sometimes in the plasma rotation profile measured by CXS. Other footprints are visible in soft -X-ray signals and the decrease of the signal of monitoring the thermal energy content.

12/01/2004 E.Lazzaro Plasmi04, Arcetri R Z R0R0 X Point O Point TEARING MODES (classical or neoclassical) Current perturbations alter the topology of magnetic confinement (isobaric) surface Safety factor q is rational On this surface the force line are closed after m toroidal loops and n poloidal loops

12/01/2004 E.Lazzaro Plasmi04, Arcetri Relation between measured signals and state variables For a single (m,n) mode the “theoretical“ state variables {x i } are the island width W(t) and rotation frequency  t  governed by the equations: The control variables {u i } are embodied in The control task would be to reduce to zero W in minimal time; RF current drive aimed on the “O” point could provide the suppression (“stabilizing” )effect Attempts of stabilization have been done with external control fields with (m,n ) helical pitch and recently with ECCD an, LHCD and ECRH applied the calculated q=m/n surface, without island tracking.

12/01/2004 E.Lazzaro Plasmi04, Arcetri Questions on the localization & modulation of RF power driving current in NTM rotating islands balanceWithin  J ECCD is spread ergodically on island flux tubes  1/W 2 scaling adequate to balance the J BOOT destabilizing term inadequate to balanceOutside  J ECCD is spread ergodically on flux surface  1/W scaling inadequate to balance J BOOT term Synchronizing RF pulses to keep a constant  ~0 favors the 1/W 2 scaling, but it‘s very complicated. The RF power depends on what is needed to balance the J BOOT term, not on the RF pulse strategy  ” ECCD ~1/W  ” ECCD ~1/W 2 A constant RF source in the lab frame appears oscillating in the moving island frame !  RF Steering the RF deposition profile on the correct radius is the most important task!

12/01/2004 E.Lazzaro Plasmi04, Arcetri Relation between measured signals and state variables We change the point of view and choose as state variables the ECE fluctuations (radiative Temperature) just as they are measured in amplitude, phase and frequency. The typical ECE signal is the superposition of a slowly varying component related to the equilibrium electron temperature, plus noise and coherent fluctuations due for instance to the magnetic islands The temperature fluctuation associated with a finite size island is generally expected to be negative on the inner island edge and positive on the outer edge Therefore the fluctuation amplitude should be  T e =0 on the island “O” and “X” stagnation points. The relative phase of two neighbouring ECE channels changes smoothly except if between them a rational surface q=m/n with an island is located: here a phase jump close to  occurs.

12/01/2004 E.Lazzaro Plasmi04, Arcetri TrTr 0 r Effects of the MHD perturbations on the Temperature profile q=m/n

12/01/2004 E.Lazzaro Plasmi04, Arcetri First step of the control problem: mode location & tracking We concentrate here in the development of an island tracking device based on ECE. Eventually the control variables {u i } shall be ECW steering angle, ECW power, pulse timing The rotating island frequency and phase are identified by a special Digital phase-locked loop that captures the ECE perturbation frequency within few cycles The island radial location is tracked by a fast algorithm that identifies the maximum phase change between of two neighbouring ECE channels Second step: designing optimal control strategy The basic “power cost” of the “stabilizing” action is due to the low J ECCD efficiency Formal control theory helps also understanding the physics of the controlled system and the best strategies can then be found

12/01/2004 E.Lazzaro Plasmi04, Arcetri A robust control technique for active MHD control with EC waves requires the real-time measurement of the location of both the island and the EC deposition layer. The most appropriate signals related to the state variables come from a multichannel EC emission (ECE) diagnostic.  T e oscillations caused by periodic flux perturbations reveal rotating modes. The location is identified by a channel number. The algorithm described can be used to localize Tearing Modes, sawteeth and the EC deposition layer. The algorithm has been tested off-line on FTU 12-channels polychromator data, on a variety of different MHD combinations. The algorithm is implemented on DSP modules for on-line action. DSP modules will be used for data acquisition, data processing and feedback control (ECH/ECCD is the actuator).

12/01/2004 E.Lazzaro Plasmi04, Arcetri 4 GYROTRONS 400 KW 140 GHz R BB R0R0 ECE CONTROL EXPERIMENT SCHEME Microwave heterodyne radiometers

12/01/2004 E.Lazzaro Plasmi04, Arcetri Mirror Gyrotron  (B) dep. R It is possible to change R dep by changing the the angle of the launcher mirror

12/01/2004 E.Lazzaro Plasmi04, Arcetri Sampling frequency = 20 KHz (0.05 msec) T1 = 2 x sec 500Hz T2 = 4 x sec 250Hz T3 = 20 x sec 50Hz Equivalent to cross-correlation: Radial Identification Algorithm: creating the sensitive parameter

12/01/2004 E.Lazzaro Plasmi04, Arcetri First steps in signal processing Low-pass filter ( ) Oscillation (o i ) and modulus (A i )

12/01/2004 E.Lazzaro Plasmi04, Arcetri Decisional algorithm P ij ≈ 1 if both i and j are on the same side with respect to the island O-point. P ij ≈ -1 if on opposite sides. A positive concavity in the P ij sequence locates the island. channels 1 0 P i j

12/01/2004 E.Lazzaro Plasmi04, Arcetri Island Found Between ch i and ch i+1 N Y Y N Next time step

12/01/2004 E.Lazzaro Plasmi04, Arcetri HARDWARE DEVELOPED BY IFP FOR FTU

12/01/2004 E.Lazzaro Plasmi04, Arcetri Digital Phase Locked Loop to track rotating island phase and frequency Implemented on DSP B Sampling rate 25 KHz

12/01/2004 E.Lazzaro Plasmi04, Arcetri Identification of locking island: the red DPLL trace tracks the physical signal DPLL performance: identification of mode frequency Shot ch7

12/01/2004 E.Lazzaro Plasmi04, Arcetri T e profile during one cycle correlation between nearby channels Case1 -ECRH boosts TMs if r dep < r island T e,EC E Mirnov coil

12/01/2004 E.Lazzaro Plasmi04, Arcetri oscillation frequency at minimum P ij m=1,n=1 axis minimum correlation (negative maximum) = island position (1,1) and (2,1) are coupled (2,1)

12/01/2004 E.Lazzaro Plasmi04, Arcetri TM modes with different m - order coexist T e &  T e profiles, fast ECE, showing presence of even and odd TM. All modes are tightly coupled same frequency at all radii Even mode:m=2, n=1. IAEA, Sorrento, October 4-10, 2000 ASSOCIAZIONE EURATOM/ENEA/CNR

12/01/2004 E.Lazzaro Plasmi04, Arcetri (2,1) stabilized (1,1) evolves in s.t. sawteeth are stabilized Case2- ECRH suppresses MHD if r dep ≈ r island and/or r dep ≥ r s.t. inversion (2,1) & (1,1) T e profile during one cycle correlation between nearby channels

12/01/2004 E.Lazzaro Plasmi04, Arcetri oscillation frequency at minimum P ij m=1,n=1 axis inversion radius (2,1) stabilized frequency jump of dominant oscillation: from (1,1) mode to (1,1) reconnection (s.t. crash) Sawteeth and TM dynamics simultaneously detected s.t. stabilized

12/01/2004 E.Lazzaro Plasmi04, Arcetri axis sawteeth (2,1 ) (2,1) locks (1,1 ) (3,2) (1,1) & (2,1) & (3,2) are coupled Coupled TMs simultaneously detected

12/01/2004 E.Lazzaro Plasmi04, Arcetri Formal aspects of the control problem The physical objective is to reduce the ECE fluctuation to zero in minimal time using ECRH /ECCD on the position q=m/n identified by the phase jump method The TM control problem in the extended Rutherfprd form, belongs to a general class known in in the theory of multistage decision processes [*]. In a linearized form the governing equation for the state variable x(t) is with the initial condition x(0)=x0, and a control variable (steering function) u(t). The formal problem consists in to reducing the state x(t) to zero in minimal time by a suitable choice of the steering function u(t) A number of interesting properties of this problem have been studied [*] [*] J.P. LaSalle, Proc. Nat. Acad. Of Sciences 45, (1959); R.Bellman,I. Glicksberg O.Gross, “On the bang-bang control problem” Q. Appl. Math (1956)

12/01/2004 E.Lazzaro Plasmi04, Arcetri Formal aspects of the control problem An admissible (piecewise measurable in a set Ω ) steering function u* is optimal if for some t*>0 x(t*,u*) =0 and if x(t,u)≠0 for 0<t<t* for all u(t)  Ω It can be proved [*] that : “ Anything that can be done by an admissible steering function can also be done by a bang-bang function” This leads to the theorem: “If for the control problem there is a steering function u(t)  Ω such that x(t,u)=0, for t>0, then there is an optimal steering function u* in Ω. Moreover all optimal steering functions u* are of the bang-bang form” Thus the only way of reaching the objective in minimum time is by using properly all the power available Steering times can be chosen testing ||x(t|| <  u(t) t

12/01/2004 E.Lazzaro Plasmi04, Arcetri CONCLUSIONS âAn algorithm performing real-time data analysis for locating the island has been developed and tested. âA multichannel EC emission diagnostic is used for generating input signals to the diagnostic/control unit. âThe algorithm described can be used to localize Tearing Modes, sawteeth and surface where q is rational. âTearing mode control based on ECE+ECCD/ECRH is possible with: âA high space resolution (many channels) ECE polychromator âPower requirements depend essentially on ratio J ECCD /J BOOT, and the radial location of RF profile âIf the basic physical process is efficient, formal control theory helps finding the best strategy (on-off switching of full RF power at selected times is most likely )

12/01/2004 E.Lazzaro Plasmi04, Arcetri THE END

12/01/2004 E.Lazzaro Plasmi04, Arcetri PC DSP A DSP B VGX M66 (DIGITAL IO) Internal Clock (ANALOG INPUT) Parameters Measurements Measurements/Status Service Commands off 4 Cutoff Freq. Cut Freq. Init. Parameters Measurem. GATE PRERUN FTU CLOCK PRERUN GATE FSC 4 On/off 4 GYROTRON msg B-PC msg A-PC 12 ECE signals 1

12/01/2004 E.Lazzaro Plasmi04, Arcetri axis inversion radius sawteeth are stabilized s.t inversion radius = minimum correlation (negative maximum)

12/01/2004 E.Lazzaro Plasmi04, Arcetri Case 4 ECRH affects mode coupling

12/01/2004 E.Lazzaro Plasmi04, Arcetri Case 3 - ECRH suppresses s.t. if r dep ≥ r s.t. inversion profile evolution during one s.t. cycle

12/01/2004 E.Lazzaro Plasmi04, Arcetri Formal aspects of the control problem The physical objective is to reduce the ECE fluctuation to zero in minimal time using ECRH /ECCD on the position q=m/n identified by the phase jump method The TM control problem in the extended Rutherford form, belongs to a general class known in in the theory of multistage decision processes [*]. In a linearized form the governing equation for the state variable x(t) is The formal problem consists in to reducing the state x(t) to zero in minimal time by a suitable choice of the steering function u(t). If  (x 0 ) represents the minimum time as function of the initial state, the optimality principle [**] stipulates that At each point in phase space of the state vectors x 0 the control vector u is chosen to maximize g.grad  Optimization of with respect to u at each x,