Figure 4.7:

Closed-loop performance of combined nonlinear MHE/MPC with no disturbances. First column shows system states, and second column shows estimation error. Dashed line shows concentration setpoint. Vertical lines indicate times of setpoint changes.

Code for Figure 4.7

Text of the GNU GPL.

main.m

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% Control of a nonlinear CSTR using nonlinear MPC and MHE.


wmaxes = [0, 0.001, 0.001, 0.01, 0.01, 0.025];
vmaxes = [0, 0.1, 1, 1, 10, 10];

data = cell(size(wmaxes));

% Define helper functions.



% Simulate with varying disturbance sizes.
fig = figure();
ax = zeros(length(wmaxes));
for i = 1:length(ax)
    ax(i) = subplot(2, 3, i);
end
for i = 1:length(wmaxes)
    data{i} = runcstr(wmaxes(i), vmaxes(i));
    axes(ax(i));
    plot(data{i}.x(1,:), data{i}.x(2,:), '-ok');
    title(sprintf('|w| < %g, |v| < %g', data{i}.wmax, data{i}.vmax));
    xlabel('c');
    ylabel('T', 'rotation', 0);
    axis([0.7, 1, 322, 338]);
end

cstrode.m

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function rhs = cstrode(x, u, pars)
    % Nonlinear ODE model for reactor.
    c = x(1);
    T = x(2);
    h = pars.h;

    Tc = u(1);
    F = pars.F;

    F0 = pars.F;

    k = pars.k0*exp(-pars.E/T);
    rate = k*c;

    dcdt = F0*(pars.c0 - c)/(pars.A*h) - rate;
    dTdt = F0*(pars.T0 - T)/(pars.A*h) ...
           - pars.DeltaH/pars.rhoCp*rate ...
           + 2*pars.U/(pars.r*pars.rhoCp)*(Tc - T);
    rhs = [dcdt; dTdt];

stagecost.m

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function cost = stagecost(x, u, Deltau, xsp, usp, Q, R, S)
    dx = x - xsp;
    du = u - usp;
    cost = dx'*Q*dx + du'*R*du + Deltau'*S*Deltau;

runcstr.m

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function data = runcstr(wmax, vmax)
% data = runcstr(wmax, vmax)
%
% Runs a closed-loop simulation of the CSTR with the given disturbance sizes.
narginchk(2, 2);
mpc = import_mpctools();

% Sizes for the nonlinear system.
Delta = 0.5;
Nx = 2;
Nu = 1;
Ny = 1;
Nw = 1;
Nv = Ny;
small = 1e-5; % Small number.

% Parameters.
pars = struct();
pars.T0 = 350; % K
pars.c0 =  1;  % kmol/m^3
pars.r = 0.219; % m
pars.k0 = 7.2e10; % min^-1
pars.E =  8750; % K
pars.U =  54.94; % kJ/(min m^2 K)
pars.rho = 1e3;   % kg/m^3
pars.Cp = 0.239;  % kJ/(kg K)
pars.DeltaH = -5e4; % kJ/kmol
pars.A = pi()*pars.r.^2;
pars.rhoCp = pars.rho*pars.Cp;
pars.h = 0.659;
pars.F = 0.1;

ode = @(x, u) cstrode(x, u, pars);

cstrsim = mpc.getCasadiIntegrator(ode, Delta, [Nx, Nu], ...
                                  {'x', 'u'}, {'cstr'});

% Steady-state values.
cs = 0.878; % kmol/m^3
Ts = 324.5; % K
Tcs = 300; % K

xs = [cs; Ts];
us = [Tcs];

xlb = [0.1; 250];
xub = [1; 350];

% Simulate a few steps so we actually get dx/dt = 0 at steady state.
for i = 1:10
    xs = full(cstrsim(xs, us));
end
cs = xs(1);
Ts = xs(2);
C = [0, 1]; % Can't measure concentration.
ys = C*xs;
CVs = [1]; % Concentration is controlled variable.
G = [1; 0]; % State noise only enters at concentration.

% Bounds.
umax = 0.25*Tcs;
ulb = us - umax;
uub = us + umax;

% Get Kalman Filter weight as a prior.
[A, B] = mpc.getLinearizedModel(ode, {xs, us}, 'Delta', Delta, 'deal', true());
Qw = 0.05;
Rv = 50;

% Make functions for controller, target finder, and estimator.
fprintf('Building controllers.\n');

fmpc = mpc.getCasadiFunc(ode, [Nx, Nu], {'x', 'u'}, ...
                         'rk4', true(), 'Delta', Delta, 'M', 2, ...
                         'funcname', 'fnonlin');
fmhe = mpc.getCasadiFunc(@(x, u, w) fmpc(x, u) + G*w, [Nx, Nu, Nw], ...
                         {'x', 'u', 'w'}, 'funcname', 'fnonlin_w');
fsstarg = mpc.getCasadiFunc(ode, [Nx, Nu], {'x', 'u'}, 'funcname', 'fsstarg');

% Make stage costs.
lcontroller = mpc.getCasadiFunc(@stagecost, ...
                                {Nx, Nu, Nu, Nx, Nu, [Nx, Nx], [Nu, Nu], [Nu, Nu]}, ...
                                {'x', 'u', 'Du', 'xsp', 'usp', 'Q', 'R', 'S'}, ...
                                'funcname', 'l');

Vf = mpc.getCasadiFunc(@(x, xsp, P) (x - xsp)'*P*(x - xsp), ...
                       {Nx, Nx, [Nx, Nx]}, {'x', 'xsp', 'P'}, 'funcname', 'Vf');

lmhe = mpc.getCasadiFunc(@(w, v, Qinv, Rinv) w'*Qinv*w + v'*Rinv*v, ...
                         {Nw, Nv, [Nw, Nw], [Nv, Nv]}, ...
                         {'w', 'v', 'Qinv', 'Rinv'}, 'funcname', 'l');

h = mpc.getCasadiFunc(@(x) C*x, [Nx], {'x'}, 'funcname', 'h');

% Assemble controller, target finder, and estimator.
Ncontroller = 10;
Nmhe = 10;
Q = 0.01*diag(xs.^-2);
R = diag(us.^-2);
[~, P] = dlqr(A, B, Q, R); % For terminal penalty.
S = 10*R; % Rate-of-change penalty.

N = struct('x', Nx, 'u', Nu, 't', Ncontroller);
guess = struct('x', repmat(xs, 1, N.t + 1), 'u', repmat(us, 1, N.t));
lb = struct('x', xlb, 'u', ulb);
ub = struct('x', xub, 'u', uub);
par = struct('xsp', xs, 'usp', us, 'Q', Q, 'R', R, 'P', P, 'S', S, 'uprev', us);
controller = mpc.nmpc('f', fmpc, 'l', lcontroller, 'Vf', Vf, 'N', N, ...
                      'lb', lb, 'ub', ub, 'guess', guess, 'par', par, ...
                      'verbosity', 0);

N = struct('x', Nx, 'u', Nu, 'y', Ny);
guess = struct('x', xs, 'u', us, 'y', ys);
lb = struct('x', xlb, 'u', ulb);
ub = struct('x', xub, 'u', uub);
par = struct();
sstarg = mpc.sstarg('f', fsstarg, 'h', h, 'N', N, ...
                    'lb', lb, 'ub', ub, 'guess', guess, 'par', par, ...
                    'discretef', false(), 'verbosity', 0);

N = struct('x', Nx, 'u', Nu, 'w', Nw, 'y', Ny, 't', Nmhe);
guess = struct('x', repmat(xs, 1, N.t + 1));
lb = struct('x', xlb);
ub = struct('x', xub);
par = struct('y', repmat(ys, 1, N.t + 1), 'u', repmat(us, 1, N.t), ...
             'Qinv', mpc.spdinv(Qw), 'Rinv', mpc.spdinv(Rv));

mhe = mpc.nmhe('f', fmhe, 'h', h, 'l', lmhe, 'N', N, ...
               'guess', guess, 'lb', lb, 'ub', ub, 'par', par, ...
               'verbosity', 0);

% Closed-loop simulation.
Nsim = 200;
x = NaN(Nx, Nsim + 1);
x(:,1) = xs;
y = NaN(Ny, Nsim + 1);
u = NaN(Nu, Nsim);

rand('state', -1);
urand = @(n, m) 2*rand(n, m) - 1; % Uniform on [-1, 1].
v = vmax*urand(Ny, Nsim + 1);
w = wmax*urand(Nw, Nsim);
mhe.par.y = mhe.par.y + vmax*urand(Ny, Nmhe + 1);

xhat = NaN(Nx, Nsim + 1);

xtarg = NaN(Nx, Nsim);
utarg = NaN(Nu, Nsim);

% Disturbance and setpoint.
xsp = NaN(Nx, Nsim + 1);
t = 0:Nsim;
Nchange = 100;
xsp(CVs,:) = xs(CVs)*(1 - 0.1*(mod(t, Nchange) >= Nchange/2));
xsp(:,end) = xsp(:,end - 1);

% Start loop.
for i = 1:(Nsim + 1)
    fprintf('(%3d) ', i);

    % Use steady-state target selector.
    sstarg.fixvar('x', 1, xsp(CVs,i), CVs);
    sstarg.solve();
    fprintf('Target: %s, ', sstarg.status);
    if ~isequal(sstarg.status, 'Solve_Succeeded')
        fprintf('\n');
        warning('sstarg failed at time %d!', i);
        break
    end
    xtarg(:,i) = sstarg.var.x;
    utarg(:,i) = sstarg.var.u;

    % Take measurement and run mhe.
    y(:,i) = C*x(:,i) + v(:,i);
    mhe.newmeasurement(y(:,i), controller.par.uprev);
    mhe.solve();
    fprintf('Estimator: %s, ', mhe.status);
    if ~isequal(mhe.status, 'Solve_Succeeded')
        fprintf('\n');
        warning('mhe failed at time %d!', i);
        break
    end
    xhat(:,i) = mhe.var.x(:,end);
    mhe.saveguess();

    % Stop if at last time.
    if i == Nsim + 1
        fprintf('Done\n');
        break
    end

    % Apply control law.
    controller.fixvar('x', 1, xhat(:,i));
    controller.par.xsp = xtarg(:,i);
    controller.par.usp = utarg(:,i);
    controller.solve();
    fprintf('Controller: %s, ', controller.status);
    if ~isequal(controller.status, 'Solve_Succeeded')
        fprintf('\n');
        warning('controller failed at time %d', i);
        break
    end
    u(:,i) = controller.var.u(:,1);
    controller.saveguess();
    controller.par.uprev = u(:,i); % Save previous u.

    % Evolve plant.
    x(:,i + 1) = full(cstrsim(x(:,i), u(:,i))) + G*w(:,i);

    fprintf('\n');
end

% Make a plot.
t = Delta*(0:Nsim);
plottitle = sprintf('|w| < %g, |v| < %g', wmax, vmax);
style = struct('fig', figure(), 'marker', '', 't', t);
mpc.mpcplot('x', x, 'u', u, 'xnames', {'c', 'T'}, ...
            'unames', {'T_c'}, 'title', plottitle, 'legend', 'Actual', ...
            '**', style);
mpc.mpcplot('x', xhat, 'u', NaN(Nu, Nsim), 'color', 'b', 'linestyle', '--', ...
            'legend', 'Estimated', '**', style);
mpc.mpcplot('x', xsp, 'u', NaN(Nu, Nsim), 'color', 'r', 'linestyle', ':', ...
            'legend', 'Setpoint', '**', style);
mpc.mpcplot('x', [NaN(1, Nsim + 1); y], 'u', NaN(size(u)), 'color', 'g', ...
            'legend', 'Measurement', '**', style);

% Store data.
data = struct('x', x, 'u', u, 'y', y, 'xhat', xhat, 'xsp', xsp, 't', t, ...
              'v', v, 'w', w, 'vmax', vmax, 'wmax', wmax);