Figure 2.5:

Closed-loop economic MPC versus tracking MPC starting at x=(-8,8) with optimal steady state (8,4). Both controllers asymptotically stabilize the steady state. Dashed contours show cost functions for each controller.

Code for Figure 2.5

Text of the GNU GPL.

main.m


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% Compare economic vs. tracking mpc.

pkg ('load', 'control')

mpc = import_mpctools();
N = struct('x', 2, 'u', 1, 't', 50);

% Linear model.
A = [0.5, 1; 0, 0.75];
B = [0; 1];
f = mpc.getCasadiFunc(@(x,u) A*x + B*u, [N.x, N.u], {'x', 'u'}, {'F'});

% Objective function.
q = [-2; 2];
r = -10;
lecon = mpc.getCasadiFunc(@(x, u) q'*x + r'*u, [N.x, N.u], {'x', 'u'}, {'lecon'});

Q = [10,0; 0,10];
R = 1;
ltrack = mpc.getCasadiFunc(@(x,u,xs,us) (x - xs)'*Q*(x - xs) + (u - us)'*R*(u - us), ...
                           [N.x, N.u, N.x, N.u], {'x', 'u', 'xs', 'us'}, {'ltrack'});

% Bounds.
lb = struct('u', -1, 'x', [-10; -10]);
ub = struct('u', 1, 'x', [10; 10]);

% Target finder.
target = mpc.sstarg('l', lecon, 'f', f, 'lb', lb, 'ub', ub, 'N', N, ...
                    'verbosity', 0);
target.solve();
xs = target.var.x;
us = target.var.u;
ls = target.obj;

% Set up controllers.
par = struct('xs', xs, 'us', us);
econ = mpc.nmpc('l', lecon, 'f', f, 'lb', lb, 'ub', ub, 'N', N, 'xf', xs, ...
                'verbosity', 0);
track = mpc.nmpc('l', ltrack, 'f', f, 'lb', lb, 'ub', ub, 'par', par, 'N', N, ...
                 'xf', xs, 'verbosity', 0);

% Simulate both.
Nsim = 25;
x0 = [-8; 8];
controllers = {econ, track};
names = {'econ', 'track'};
data = struct();
for i = 1:length(controllers)
    fprintf('Simulating %s MPC\n', names{i});
    x = NaN(N.x, Nsim + 1);
    x(:,1) = x0; % Initial condition.
    u = NaN(N.u, Nsim);
    controller = controllers{i};
    for t = 1:Nsim
        controller.fixvar('x', 1, x(:,t));
        controller.solve();
        u(:,t) = controller.var.u(:,1);
        x(:,t + 1) = controller.var.x(:,2);
        controller.saveguess();
    end
    data.(names{i}) = struct('x', x, 'u', u);
end

% Get data for objective function contours.
x1 = linspace(-10, 15, 251);
x2 = linspace(0, 10, 101);
[xx1, xx2] = meshgrid(x1, x2);
dx1 = xx1 - track.par.xs(1); % Deviation variables.
dx2 = xx2 - track.par.xs(2);
Vecon = q(1)*dx1 + q(2)*dx2;
Vtrack = Q(1,1)*dx1.^2 + 2*Q(2,1)*dx1.*dx2 + Q(2,2)*dx2.^2;
data.contours = struct('x1', x1, 'x2', x2, 'Vecon', Vecon, 'Vtrack', Vtrack);

% Make plots.
figure();
hold('on');
contour(xx1, xx2, Vecon, 10, '-c');
contour(xx1, xx2, Vtrack, 10, '-m');
colors = {'blue', 'red'};
for i = 1:length(names)
    n = names{i};
    plot(data.(n).x(1,:), data.(n).x(2,:), '-o', 'color', colors{i});
end
xlabel('x_1');
ylabel('x_2', 'rotation', 0);

% Find rotated cost function.
mu = -(eye(N.x) - A)'\q;
alpha = 0.01;
Mu = dlyap(A', alpha*eye(N.x));
lam = @(x) mu'*(x - xs) + (x - xs)'*Mu*(x - xs);

lrot = @(x, u) q'*(x - xs) + r'*(u - us) + lam(x) - lam(A*x + B*u) ...
               - alpha*(x - xs)'*(x - xs);
[x1, x2, u] = meshgrid(linspace(lb.x(1), ub.x(1), 21), ...
                       linspace(lb.x(2), ub.x(2), 21), ...
                       linspace(lb.u, ub.u, 21));
l = arrayfun(@(x1, x2, u) lrot([x1; x2], u), x1, x2, u);
fprintf('Minimum of dissipation inequality: %g\n', min(l(:)));

% Simulate multiple economic MPC trajectories.
econ.reset();
Nx0 = 16;
colors = jet(Nx0);
theta = linspace(0, 2*pi(), Nx0 + 1);
theta = theta(2:end);
x0 = [8*cos(theta); 8*sin(theta)];
Nsim = 21;
x = NaN(N.x, Nsim, Nx0);
u = NaN(N.u, Nsim, Nx0);
V = NaN(Nsim, Nx0);
plotopts = struct('fig', figure(), 'unames', {{'u', 'V'}}, 'marker', '');
fprintf('Simulating more economic MPC');
for j = 1:Nx0
    x(:,1,j) = x0(:,j);
    for i = 1:Nsim
        econ.fixvar('x', 1, x(:,i,j));
        econ.solve();
        if ~isequal(econ.status, 'Solve_Succeeded')
            warning('%s: failure at time %d!', names{i}, t);
            break
        end
        u(:,i,j) = econ.var.u(:,1);
        V(i,j) = econ.obj + lam(x(:,i,j)) - N.t*ls;
        if i < Nsim
            x(:,i + 1,j) = econ.var.x(:,2);
        end
    end
    fprintf('.');
    mpc.mpcplot(x(:,:,j), [u(:,:,j); V(:,j)'], 'color', colors(j,:), ...
                '**', plotopts);
end
fprintf('\n');
data.phase = struct('x', x, 'u', u, 'V', V);