來源：https://github.com/aerojunkie/control-tools/blob/master/ufo_rotate.m

一個簡單的LQR例子

close all% Initial Conditions x0 = [3; % 3 radians0]; % 0 rad/s% System Dynamics A = [0 1; 0.01 0]; B = [0; 1]; C = [1 0]; D = 0;% Control Law Q = [1 0; % Penalize angular error0 1]; % Penalize angular rate R = 1; % Penalize thruster effort K = lqr(A,B,Q,R);% Closed loop system sys = ss((A - B*K), B, C, D);% Run response to initial condition t = 0:0.005:30; [y,t,x] = initial(sys, x0, t);%% Everything below this is just for the visualization% Patch drawings ufo_data = [0.54 .12;0.48 0.24;0.42 0.31;0.3 0.4;0.18 0.45;0.06 0.48;-0.06 0.48;-0.18 0.45;-0.3 0.4;-0.42 0.31;-0.48 0.24;-0.54 .12;0.54 .12;0.54 0.06;0.6 0.06;0.78 0.03; 0.91 0.01;0.91 0.03;1.01 0.03;1.01 -0.01; 1.1 -0.03; 1.26 -0.12; 1.36 -0.18; 1.40 -0.21;1.38 -0.24;1.26 -0.27;1.1 -0.3; 1.01 -0.32;1.01 -0.34;0.91 -0.34;0.91 -0.32; 0.72 -0.34;0.48 -0.35;0.72 -0.6;0.62 -0.6;0.38 -0.35; 0.3 -0.35; -0.3 -0.35;-0.38 -0.35;-0.62 -0.6;-0.72 -0.6;-0.48 -0.35;-0.91 -0.32;-0.91 -0.34;-1.01 -0.34;-1.01 -0.32;-1.1 -0.3;-1.26 -0.27;-1.38 -0.24;-1.40 -0.21;-1.36 -0.18;-1.26 -0.12;-1.1 -0.03;-1.01 -0.01;-1.01 0.03;-0.91 0.03;-0.91 0.01;-0.78 0.03;-0.6 0.06;-0.54 0.06;0.54 0.06; ];thrust = [0.05 0;-.05 0;-.06 -0.1;-.05 -0.1;-.06 -0.2;0 -0.25;0.06 -0.2;0.05 -0.1;0.06 -0.1;0.05 0];% yoffset moves the patch up from 0, 0 yoffset = 1;%% Create Animation% Rotate to inital condition ufo_rot = rot(ufo_data, x0(1)); r_thrust_rot = rot(thrust, x0(1)); l_thrust_rot = rot(thrust, x0(1));% Set up the figure and the subplots fig = figure('Position', [100 80 670 800]); h3 = subplot(3, 1, 3); h3.OuterPosition = [0 0.01 1 .19]; h3a = area(0, 0); hold on h3p = plot(0, 0, 'r'); h3.YGrid = 'on'; h3.XGrid = 'on'; h3.LineWidth = 1; h3.Color = [0 0 0]; h3p.LineWidth = 4; axis([0 12 -3 3]); h3.GridColor = 'w'; h3.GridAlpha = 0.5; text(6.1, 1.5, 'Acceleration (fuel)', 'Color', [1 0 0], 'FontSize', 24)h2 = subplot(3, 1, 2); h2.OuterPosition = [0 .2 1 .2]; h2p = plot(0, 0, 'g'); h2p.LineWidth = 4; h2.YGrid = 'on'; h2.XGrid = 'on'; h2.LineWidth = 1; h2.GridColor = 'w'; h2.GridAlpha = 0.5; axis([0 12 -.5 3.5]); h2.Color = [0 0 0]; text(6.1, 2.5, 'Angular Error (time)', 'Color', [0 1 0], 'FontSize', 24)h1 = subplot(3, 1, 1); h1.OuterPosition = [0 .4 1 0.63]; h1.XTick = []; h1.YTick = []; patch([-3 3 3 -3], [-1.7 -1.7 3.3 3.3], 'k'); axis([-3 3 -1.7 3.3]); line([0 0.28], [1 2.99], 'Color', [0.5 0.3 0.3], 'LineWidth', 1); line([0 0], [1 -1], 'Color', [0.3 0.5 0.3], 'LineWidth', 1); H = patch(ufo_rot(:, 1), ufo_rot(:, 2) + yoffset, [0.7 0.7 0.7]); RBT = patch(r_thrust_rot(:, 1), r_thrust_rot(:, 2), 'r'); LBT = patch(l_thrust_rot(:, 1), l_thrust_rot(:, 2), 'r'); RTT = patch(r_thrust_rot(:, 1), r_thrust_rot(:, 2), 'r'); LTT = patch(l_thrust_rot(:, 1), l_thrust_rot(:, 2), 'r'); RBT.FaceAlpha = 0; LBT.FaceAlpha = 0; RTT.FaceAlpha = 0; LTT.FaceAlpha = 0; RBT.EdgeColor = 'none'; LBT.EdgeColor = 'none'; RTT.EdgeColor = 'none'; LTT.EdgeColor = 'none'; hold on% Initial time and fuel values current_time = 0; fuel = 0;% Write time and fuel to the screen text(-2.5, 3, 'Time: sec', 'Color', [1 1 1], 'FontSize', 24,...'HorizontalAlignment', 'left'); text(-2.5, 2.7, 'Fuel: units', 'Color', [1 1 1], 'FontSize', 24,...'HorizontalAlignment', 'left');TME = text(-1.1, 3, num2str(current_time, '%.1f'), 'Color', [0 1 0], ... 'FontSize', 24, 'HorizontalAlignment', 'right'); FUL = text(-1.1, 2.7, num2str(fuel, '%.0f'), 'Color', [1 0 0], ... 'FontSize', 24, 'HorizontalAlignment', 'right');accel_true = diff(x(:, 2))/(t(2)-t(1));pause(2)% Update Animationfor i = 1:5:length(x)-1% Simulation ending conditionif and(abs(x(i,1)) < 0.011, abs(x(i,2)) < 0.011)x(i,1) = 0;pause(1);% Turn on tractor beamG = patch([0.1 0.8 -0.8 -0.1], [-0.35 -2.3 -2.3 -0.35] + yoffset, 'y');G.FaceAlpha = 0.6;returnend% Scale acceleration to make thruster size look better fuel units% just a few digits (asthetics)accel_scaled = (x(i+1, 2) - x(i, 2));fuel = fuel + abs(accel_scaled);accel_scaled = min(2, max(-2, accel_scaled * 1000));% Scale thruster based on accleration and directionif accel_scaled > 0r_scaled(:, 1) = thrust(:, 1) * accel_scaled * 0.5;r_scaled(:, 2) = thrust(:, 2) * accel_scaled;l_scaled(:, 1) = thrust(:, 1);l_scaled(:, 2) = thrust(:, 2);elseif accel_scaled < 0r_scaled(:, 1) = thrust(:, 1);r_scaled(:, 2) = thrust(:, 2);l_scaled(:, 1) = thrust(:, 1) * -accel_scaled * 0.5;l_scaled(:, 2) = thrust(:, 2) * -accel_scaled;elser_scaled(:, 1) = thrust(:, 1);r_scaled(:, 2) = thrust(:, 2);l_scaled(:, 1) = thrust(:, 1);l_scaled(:, 2) = thrust(:, 2);end% Offset thrusters to place correctly on UFOrb_thrust = r_scaled + [0.96 -0.34];lb_thrust = l_scaled + [-0.96 -0.34];rt_thrust = l_scaled * [1 0; 0 -1] + [0.96, 0.03];lt_thrust = r_scaled * [1 0; 0 -1] + [-0.96, 0.03];% Rotate thrusters with UFOrb_thrust_rot = rot(rb_thrust, x(i,1)) + [0 yoffset];lb_thrust_rot = rot(lb_thrust, x(i,1)) + [0 yoffset];rt_thrust_rot = rot(rt_thrust, x(i,1)) + [0 yoffset];lt_thrust_rot = rot(lt_thrust, x(i,1)) + [0 yoffset];% Update thruster data RBT.XData = rb_thrust_rot(:, 1);RBT.YData = rb_thrust_rot(:, 2);LBT.XData = lb_thrust_rot(:, 1);LBT.YData = lb_thrust_rot(:, 2);RTT.XData = rt_thrust_rot(:, 1);RTT.YData = rt_thrust_rot(:, 2);LTT.XData = lt_thrust_rot(:, 1);LTT.YData = lt_thrust_rot(:, 2);% Turn thrusters off if acceleration is low if accel_scaled > 2e-1RBT.FaceAlpha = 1;LBT.FaceAlpha = 0;RTT.FaceAlpha = 0;LTT.FaceAlpha = 1;elseif accel_scaled < -2e-1RBT.FaceAlpha = 0;LBT.FaceAlpha = 1;RTT.FaceAlpha = 1;LTT.FaceAlpha = 0;elseRBT.FaceAlpha = 0;LBT.FaceAlpha = 0;RTT.FaceAlpha = 0;LTT.FaceAlpha = 0;end% Rotate ufoufo_rot = rot(ufo_data, x(i,1));H.XData = ufo_rot(:, 1);H.YData = ufo_rot(:, 2) + yoffset;h2p.XData = t(1:i);h2p.YData = x(1:i, 1);h3p.XData = t(1:i);h3p.YData = accel_true(1:i);h3a.XData = t(1:i);h3a.YData = accel_true(1:i);current_time = num2str(t(i), '%.1f');% Update time and fuelTME.String = current_time;FUL.String = num2str(fuel*30, '%.0f');pause(0.0001); endfunction output = rot(data, angle) output = size(data); Q = [cos(angle) -sin(angle); sin(angle) cos(angle)]; for i = 1:length(data)output(i, :) = Q * data(i, :)'; end end

總結

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