1. The Swedish Fusion Association EURATOM – VR April 26-28, 2010EXTRAP T2R active control 1 Overview of active MHD control development on EXTRAP T2R and Output tracking control for RWMS J.R. Drake, K.E.J. Olofsson, L. Frassinetti, P.R. Brunsell, Association Euratom-VR, Alfvén Laboratory, Royal Inst. of Technology KTH, Stockholm e-mail contact: james.drake@ee.kth.sejames.drake@ee.kth.se

2. The Swedish Fusion Association EURATOM – VR April 26-28, 2010EXTRAP T2R active control 2 Research Unit VR activity in Topical Group MHD EFDA Work Programme 2010. WP10-MHD-03-03-xx-01/VR Stability at high Beta: Development of resistive wall mode control methods Development of advanced RWM magnetic feedback control methods for tokamak and RFP in collaboration with ENEA-RFX and IPP. Modelling, mode identification, input design, implementation, and conduction of physics experiments. Experimental verification of controllers on EXTRAP T2R device.

3. The Swedish Fusion Association EURATOM – VR April 26-28, 2010EXTRAP T2R active control 3 Overview of active MHD control development on EXTRAP T2R J.R. Drake, K.E.J. Olofsson, L. Frassinetti, P.R. Brunsell, Association Euratom-VR, Alfvén Laboratory, Royal Inst. of Technology KTH, Stockholm e-mail contact: james.drake@ee.kth.sejames.drake@ee.kth.se

4. The Swedish Fusion Association EURATOM – VR April 26-28, 2010EXTRAP T2R active control 4 Generic mode control strategy MHD (automatic) control system (the Plant): –Plasma, including the actual 3D boundary structure (wall, vessel, etc.) –Arrays of magnetic sensor coils. –Arrays of active saddle coils (actuators). –Controller hardware plus software. Advanced Control theory Feedback is implemented by a real-time controller which uses, eventually in a well- defined optimal way, the admissible actuators for MHD control in the form of currents in the active coils to constrain the MHD mode evolution at a specified reference state by responding to measured sensor voltages. The digital controller was originally developed by Consorzio RFX and implemented in a collaborative effort on both the EXTRAP T2R and RFX-mod experiments. New software incorporating advanced control theory has been developed for EXTRAP T2R and then installed and tested.

5. The Swedish Fusion Association EURATOM – VR April 26-28, 2010EXTRAP T2R active control 5 Theme Going Beyond simply stabilising Resistive Wall Modes or providing a smooth magnetic boundary using a controller with a feedback law for control of Fourier harmonic dynamics. ►Successful stabilisation of RWMs in RFPs is well known Important to optimize the control systems for future implementation. (Erik Olofsson). ►Use the actuators and the actual sensors for system identification which becomes the basis for controller synthesize. ►Enable the possibility for accessing, via the control actions, “non- zero” reference states. ►Develop a multiple input-multiple output (MIMO) controller using an appropriate state-space model to enable use of the very well developed automatic control tool box.

6. The Swedish Fusion Association EURATOM – VR April 26-28, 2010EXTRAP T2R active control 6 Benefits What is the benefit? ►Use modern control theory, specifically including system identification. There are many very powerful methods to synthesize controllers using only the measured plant features (active coil currents and sensor coil voltages). ►Through system identification, one can measure and account for non- axisymmetric features in the wall  mn. ►Develop control to track an “arbitrary” reference state. ►Optimise: robust controller stability and acceptable power requirements. Use the controller for generic MHD studies. ►Measure growth rates for “dry” and “wet” plants (i.e.with and without plasma) ►RMP effect on a rotating tearing mode.

7. The Swedish Fusion Association EURATOM – VR April 26-28, 2010EXTRAP T2R active control 7 limitations What is the disadvantage? ►The plant must be in operation. ►The configuration of the actuators and sensors must be based on a model. ►The initial plant operation must be based on a feedback controller that is synthesized using a model.

8. The Swedish Fusion Association EURATOM – VR April 26-28, 2010EXTRAP T2R active control 8 EXTRAP T2R experiment Sensor coils Active coils shell R/a=1.24m/0.18m I p  150kA, n=5x10 18 m -3, T i = 500eV, T e = 300 eV  wall =6ms  pulse = up to 90ms 4 poloidal x 32 toroidal active and sensor saddle coils (m=1 connected)

9. The Swedish Fusion Association EURATOM – VR April 26-28, 2010EXTRAP T2R active control 9 C 1 is the feedback control function. r(t) is a reference. Typical routing of the signals in the closed control loop u DAC (t) represents the actuator channels’ control voltage. P o, P 1 represent a time delay and a composite active coil and power amplifier. u sys (t) represents the active coil currents. P 2 represents the front- end wall/plasma dynamics. v 1 (t) is an exogenous signal representing the field errors and MHD noise. v 2 (t) is white noise. y sys (t) represent the time-integrated sensor voltages. v2v2 

10. The Swedish Fusion Association EURATOM – VR April 26-28, 2010EXTRAP T2R active control 10 Successful stabilisation of RWMs and acvhievment of a smooth magnetic boundary In T2R, RWM stabilization is achieved using both the Intelligent Shell (IS) and Fourier harmonic Mode Control (MC) feedback strategies.

11. The Swedish Fusion Association EURATOM – VR April 26-28, 2010EXTRAP T2R active control 11 System Identification by Dithering Experiments using vector dithering to concurrently excite both spatial and temporal dynamics of multiple RWMs motivates and justifies the use of direct multi-input multi- output (MIMO) linear plant estimation. During dithering, the IS controller is functioning so the baseline state is a smooth magnetic boundary.

12. The Swedish Fusion Association EURATOM – VR April 26-28, 2010EXTRAP T2R active control 12 Plucking the strings that are the plasma dynamics short current pulses to the 64 different active coil pairs measured signals on the 64 different sensor coil pairs

13. The Swedish Fusion Association EURATOM – VR April 26-28, 2010EXTRAP T2R active control 13 RWM Basics (Physicist’s point of view) A RWM radial field perturbation characterised by (m,n) measured at the wall has a growth rate, and the dynamics is described by, where is the wall penetration time for the mode and is that part of the resonant perturbation measured at the wall by the sensor coils that includes the field produced by the active coils, field errors, MHD noise,...).

14. The Swedish Fusion Association EURATOM – VR April 26-28, 2010EXTRAP T2R active control 14 Multiple-input multiple-output where the harmonic is the state variable The system dynamics for the MIMO model are expressed in state-space form; where: x is a vector of MHD modes (i.e. the state of the MHD fluid) u sys (coil currents) and y sys (sensor signals) are defined in the previous slide. A, B and C are system matrices defined by the parameters and geometry of the wall, actuators and sensors. Nv 1 is a source term which bundles effects of field-errors and MHD noise. v 2 is a white noise signal z is an optional vector expressing the desired performance M is a key factor for implementing process control and relates the MHD harmonics x to the merit vector z. (defines the control system objective).

15. The Swedish Fusion Association EURATOM – VR April 26-28, 2010EXTRAP T2R active control 15 MIMO autoregressiv exogenous (ARX) model structure Following equations are clipped from: “Closed-loop MIMO ARX estimation of concurrent external eigenmodes in magnetic confinement fusion” Erik Olofsson, et al Submitted to IEEE Conference on Decision and Control, Dec 15-17, 2010. (ARX = Autoregressive exogeneous) In time series modeling, an autoregressive exogenous model is an autoregressive model which has exogenous inputs. This means that the model relates the present value of the time series to both past values of the same series and present and past values of the driving (exogenous) series.time series autoregressiveexogenous

16. The Swedish Fusion Association EURATOM – VR April 26-28, 2010EXTRAP T2R active control 16 State space vector for the system based on actual measured quantities suitable for setting up a predictor. y(k) represents the sensor signal vector (n=64). k is the discrete time (k+1 represents time step k+T s ). u(k) represents the active coil current vector (m=64)

17. The Swedish Fusion Association EURATOM – VR April 26-28, 2010EXTRAP T2R active control 17 State space equation for the system based on actual measured quantities. y(k) represents the discrete time sensor signal vector (n=64). u(k) represents the discrete time  active coil current vector (m=64)

18. The Swedish Fusion Association EURATOM – VR April 26-28, 2010EXTRAP T2R active control 18 What is x? y(k) represents the discrete time sensor signal vector (n a represents number of time steps). u(k) represents the discrete time  active coil current vector (n b represents number of time steps).

19. The Swedish Fusion Association EURATOM – VR April 26-28, 2010EXTRAP T2R active control 19 Question: What are the matrices in the state space representation? p < nn a +(m-1)n b (i.e. spatial array dimension times number of time steps for both the sensors and the actuators) The matrix A has dimension p × p The matrix B has as the dimension p × m The matrix C  has the dimension n × p

20. The Swedish Fusion Association EURATOM – VR April 26-28, 2010EXTRAP T2R active control 20 Experiments Automated insight in the joint electromagnetic and MHD response is acquired by applying the robust numerics of linear algebra to data-sets from vacuum and plasma experimental data, respectively. Fully dense matrices for discrete-time MIMO ARX models are determined, in the prediction-error setting, for both vacuum and plasma. The dithering data is used.

21. The Swedish Fusion Association EURATOM – VR April 26-28, 2010EXTRAP T2R active control 21 The state space model is now set up. Two goals: 1.We are interested in the eigenvectors and eigenvalues of matrix A. 2.Synthesize an optimal, robust controller. Results describe the dynamics of the state of the sensors and active coils. This is related to the resistive shell geometry of T2R and to MHD normal mode stability. In order to decrypt the information in the state space model we can resort to the familiar spatial harmonics we are familiar with from MHD stability theory. The eigenvectors of A are sorted in terms of input-output amplification, stability, and output geometry.

22. The Swedish Fusion Association EURATOM – VR April 26-28, 2010EXTRAP T2R active control 22 The stability-periodogram* visualization of the results is something we recognize. T2R external plasma response eigenmodes from generic MIMO ARX linear estimation. n is a toroidal spatial spectral mode number and the poloidal spatial mode number is implicitly m=1. * Periodogram is an estimate of the spectral density of a signal.

23. The Swedish Fusion Association EURATOM – VR April 26-28, 2010EXTRAP T2R active control 23 Standard linear MHD calculation of RWM growth rates computed for T2R for m=1,3, 5 and -32≤n≤+31 for some reasonable equilibrium. γ m,n τ w n n n = -11

24. The Swedish Fusion Association EURATOM – VR April 26-28, 2010EXTRAP T2R active control 24 The visualisation of the dry plant results (without plasma) n

25. The Swedish Fusion Association EURATOM – VR April 26-28, 2010EXTRAP T2R active control 25 Mode wall diffusion times ( τ m,n / τ w ) computed for T2R for m=1,3,5 and -32 ≤ n ≤ +31. Measured dry plant shell magnetic diffusion eigenmodes for -15 ≤ n ≤ +16 n τ m,n /τ w 

26. The Swedish Fusion Association EURATOM – VR April 26-28, 2010EXTRAP T2R active control 26 Toroidal angle Poloidal angle The real-space eigenmode for n=-11 which is the well-known most unstable RWM in T2R

27. The Swedish Fusion Association EURATOM – VR April 26-28, 2010EXTRAP T2R active control 27 Two goals. Two goals: 1.We are interested in the eigenvectors and eigenvalues of matrix A. 2.Synthesize an optimal, robust controller. The first is of MHD physics interest (previous periodograms). The second involves using the “new” state space information (sensors and active coils) to synthesize a FFT decoupled controller. See reference: “Synthesis and operation of an FFT-decoupled fixed-order RFP plasma control system based on system identification” Erik Olofsson, et al Very recently published in Plasma Physics and Controlled Fusion

28. The Swedish Fusion Association EURATOM – VR April 26-28, 2010EXTRAP T2R active control 28 Results using the FFT-decoupled controller “First plasma operation with the digital controller developed here went smoothly. T2R was stabilized and we can declare a first- pass system success in this respect.” The design ran “out of the box.” A special feature of the controller is that it can simulate both an Intelligent Shell control system and a Clean Mode Control system. (see paper). Another special feature is the reference tracking capability of the controller.

29. The Swedish Fusion Association EURATOM – VR April 26-28, 2010EXTRAP T2R active control 29 Output tracking control for RWMS

30. The Swedish Fusion Association EURATOM – VR April 26-28, 2010EXTRAP T2R active control 30 Results using the FFT-decoupled controller Modes that the controller tracks. He modes are static-phase. Mode amplitude

31. The Swedish Fusion Association EURATOM – VR April 26-28, 2010EXTRAP T2R active control 31 Results using the FFT-decoupled controller Two modes rotating in opposite direction Real and Imaginary parts of two modes that the controller tracks. The real and imaginary parts. n=1 Im Re n=5 Re Im

32. The Swedish Fusion Association EURATOM – VR April 26-28, 2010EXTRAP T2R active control 32 Results using the FFT-decoupled controller One node in the other direction Five nodes in one direction

33. The Swedish Fusion Association EURATOM – VR April 26-28, 2010EXTRAP T2R active control 33 Comment on mode rigidity. This example of controlled amplitude and phase of two RWMs simultaneously must say something about mode rigidity.

34. The Swedish Fusion Association EURATOM – VR April 26-28, 2010EXTRAP T2R active control 34 Famous reference spectrum with simultaneous multiple harmonic amplitudes Note that some features of the reference spectrum (seen in sensor signal) can be recognized in the spectrum of the actuator coil currents, but it is clear that the controller provides a “broader” spectrum to the actuator in order to reproduce the sharp reference spectrum Time (ms) n, (m = -1) | U n | (spectrum from coil currents) | Y n | (spectrum from sensor signal)

35. The Swedish Fusion Association EURATOM – VR April 26-28, 2010EXTRAP T2R active control 35 Comment on selection of wall penetration time. What is the limit? Look at an earlier paper.

36. The Swedish Fusion Association EURATOM – VR April 26-28, 2010EXTRAP T2R active control 36 Normalized wall penetration time Normalized growth rate Latency time in  s

37. The Swedish Fusion Association EURATOM – VR April 26-28, 2010EXTRAP T2R active control 37 Future studies Output-tracking experiments. –Lorenzo Frassinetti will give examples concerning application of Resonant Magnetic Perturbations (RMP) in his presentation to come later in this workshop. –Tearing mode control using external RMPs. Maybe some more “QSH control” studies. RWM physics studies. –T2R experimental measurement of RWM eigenvalues for different plasma flows. Controlled slow rotation of non-resonant RWM with a rotating plasma fluid. Controlled plasma flow with Resonant Magnetic Perturbations (Lorenzo’s presentation). –Quantitative mode rigidity studies. –Discussion with ASDEX U to determine what capabilities should be incorporated in the ASDEX controller.

38. The Swedish Fusion Association EURATOM – VR April 26-28, 2010EXTRAP T2R active control 38 Future studies Edge Transport –Since the RMP affects flow, perhaps there is an effect on the coherent electrostatic structures observed in the edge region with probes. –Do the same edge electrostatic turbulence studies as done previously but with different plasma flow profiles. –Consider this an invitation.