RAPTOR tutorial: Feedback controllers
In this tutorial we define feedback controllers for some plasma quantities. This means some inputs are not known a-priori, but are determined on-line by the controllers. Instead, the inputs represent references to the system.
In this example we will control the central heating power to obtain a reference 
. The controller implemented in beta_controller.m for this purpose is a standard PI controller. Inspect this file for more details, and use it as a template for your own controller.
Since some other actuators (for example Ip) are NOT controlled in feedback, we also need to define an "open-loop" controller, that will pass the inputs directly to the actuator command.
Contents
Load and modify default parameters
close all hidden; clear; run ../RAPTOR_path.m; [config] = RAPTOR_config; config.grid.tgrid = [0:0.001:0.1]; config.grid.rhogrid = linspace(0,1,21); config.debug.iterdisp = 10; % plot every 10; % actuators config.echcd = echcd('echcd_gaussian'); % use default with 2 EC sources config.echcd.params.active = true; config.nbhcd = nbhcd('none'); % no NBH
recall that this gives two EC sources, one central ECH and one off-axis ECCD.
disp(config.echcd.params)
active: 1
rdep: [0 0.4000]
wdep: [0.3000 0.3500]
cd_eff: [0 1]
eta_eccd0: 7.0000e+14
cd_trap: 0
uindices: [2 3]
check: 0
note that the EC actuator indices are [2,3] respectively (1=Ip by default).
Define controllers
We need two controllers: Controllers are defined very similarly to heating¤t drive actuators. First, call RAPTOR_contollers with controller function handles as inputs:
config.controllers = RAPTOR_controllers({@openloop_controller,@beta_controller});
This loads the default parameters for two controllers. We now modify the controller parameters.
config.controllers.beta_controller.params.Ki = 2e9; % i gain config.controllers.beta_controller.params.Kp = 2e6; % p gain config.controllers.beta_controller.params.t_fb = 0.05; % switch-on time of feedback controller config.controllers.beta_controller.params.iterplot = 10; config.controllers.beta_controller.params.fname = ... % file to store internal controller data. fullfile(config.envopts.RAPTOR_path,'personal','beta_controller_data');
Now define mapping between various controllers and actuators For each controller, specify a 2x1 cell which contains the indices of commanded actuators respectively inputs (u) received by that controller. For example:
Open-loop controller: command actuators [1,3] (Ip and ECCD power) in open-loop from inputs [1,2]
map_openloop = {[1 3],[1 2]}; %
Beta feedback controller command actuator 2 (ECH power) by input 3 (the reference beta)
map_betafb = {2,3};
Combine them both into the controllers.mapping parameter
config.controllers.mapping = {map_openloop,map_betafb};
Now inspect the controllers structure
disp(config.controllers)
openloop_controller: [1x1 struct]
beta_controller: [1x1 struct]
mapping: {{1x2 cell} {1x2 cell}}
Prepare the model
[model,params,init,g,v,U] = build_RAPTOR_model(config); % Define inputs % The inputs are now defined, following the above controller mapping, as % |u=[Ip,P_{ECCD},|$\beta_{ref}$|]| % Ip (feedforward) U(1,:) = 250e3*ones(size(params.tgrid)); % input Ip trace: ramp from 80kA to 250kA U(1,params.tgrid<0.02) = linspace(80e3,250e3,sum(params.tgrid<0.02)); % Off-axis power (feedforward) U(2,:) = 1e6*ones(size(params.tgrid)); % off-axis power U(2,params.tgrid<0.05) = linspace(0,1e6,sum(params.tgrid<0.05)); % ramp % Beta reference: 1% (feedback) U(3,:) = 0.01*ones(size(params.tgrid)); % beta reference (%) % Initial conditions init.Ip0 = U(1,1); x0 = RAPTOR_initial_conditions(model,init,g(:,1),v(:,1)); % Run simulation simres = RAPTOR_predictive(x0,g,v,U,model,params);
it telaps newt res t[ms] dt[ms] Ip[kA] Icd[kA] Ibs[kA] Ioh[kA] qe qmin q0 Vl[V] Te0[keV] Ti0[keV] ne0[e19] f_ss
1 0.17 4 5.1e-10 0 1 80 0 1.41 78.6 15.4 5.59 5.59 3.4e+00 0.21 0.21 1.00 1.2e+01
11 1.3 3 8.3e-15 10 1 169 1.48 7.65 160 7.27 3.59 3.7 5.3e+00 0.65 0.65 1.00 2.6e+01
21 1.8 3 5.2e-12 20 1 250 7.96 18.1 224 4.93 2.39 3.3 3.4e+00 1.71 1.71 1.00 2.0e+01
31 2.3 2 2.8e-10 30 1 250 20.5 31.5 198 4.93 1.86 3.39 1.6e+00 3.37 3.37 1.00 1.0e+01
41 2.8 2 8.0e-12 40 1 250 36.6 43.1 170 4.93 1.62 3.68 9.5e-01 4.95 4.95 1.00 6.3e+00
51 3.4 4 3.4e-14 50 1 250 56.2 52.5 141 4.93 1.5 4.3 6.0e-01 13.09 13.09 1.00 4.5e+00
61 3.9 2 5.6e-09 60 1 250 96.6 115 38.4 4.93 1.42 5.98 3.3e-01 12.84 12.84 1.00 2.7e+00
71 4.5 2 2.2e-10 70 1 250 98.7 131 19.8 4.93 1.36 7.67 2.9e-01 15.21 15.21 1.00 2.5e+00
81 5 2 2.7e-12 80 1 250 96.1 130 24 4.93 1.31 8.48 2.7e-01 15.38 15.38 1.00 2.3e+00
91 5.5 2 6.5e-12 90 1 250 95.4 129 25.1 4.93 1.27 8.8 2.5e-01 15.36 15.36 1.00 2.1e+00
saved controller data to /afs/ipp-garching.mpg.de/home/f/ffelici/matlab/RAPTOR/personal/beta_controller_data
101 6 2 7.0e-14 100 1 250 95.2 130 24.8 4.93 1.23 9.78 2.2e-01 15.40 15.40 1.00 2.0e+00
Plot results
out = RAPTOR_out(simres,model,params); clf; subplot(321); plot(out.time,out.U(1,:)); title('Ip'); subplot(322); plot(out.time,out.U([2:3],:)); title('P_{aux}') subplot(323); plot(out.time,[out.beta;U(3,:)]); title('\beta and \beta_{ref}') subplot(324); plot(out.time,out.f_ss); title('f_{ss} (steady stateness)') subplot(325); [cs,h] = contour(out.time,out.rho,out.q,[1 2 3 4],'color','k'); clabel(cs,h,'labelspacing',72); title('rational q locations'); xlabel('t [s]'); ylabel('\rho'); subplot(326); [cs,h] = contour(out.time,out.rho,out.te/1e3,[0.5 1 2 4 8],'color','k'); clabel(cs,h,'labelspacing',72); title('T_e contours [keV]') xlabel('t [s]'); ylabel('\rho');
end of tutorial