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Finite-Time-Horizon LQR for LTI system implemented. Meets goals 1 and 2. Does not meet goal 3 probably because of linearization. Creates oscillations about the equilibrium that explode.
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namanvs
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219
Naman_LQR_Working.m
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219
Naman_LQR_Working.m
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% Clear workspace
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clear all; close all; clc;
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% Parameters source: https://sal.aalto.fi/publications/pdf-files/eluu11_public.pdf
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g = 9.81; m = 0.468; Ix = 4.856*10^-3;
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Iy = 4.856*10^-3; Iz = 8.801*10^-3;
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% States:
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% X1: x X4: x'
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% X2: y X5: y'
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% X3: z X6: z'
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% X7: Pitch angle (x-axis) X10: Pitch rate (x-axis)
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% X8: Roll angle (y-axis) X11: Roll rate (y-axis)
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% X9: Yaw angle (z-axis) X12: Yaw rate (z-axis)
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% Inputs: Outputs:
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% U1: Total Upward Force (along z-axis) Y1: Position along x axis
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% U2: Pitch Torque (about x-axis) Y2: Position along y axis
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% U3: Roll Torque (about y-axis) Y3: Position along z axis
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% U4: Yaw Torque (about z-axis) Y4: Pitch (about x-axis)
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% Y5: Roll (about y-axis)
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% Y6: Yaw (about z-axis)
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% State Space Source: https://arxiv.org/ftp/arxiv/papers/1908/1908.07401.pdf
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% X' = Ax + Bu
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% Y = Cx
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nStates = 12;
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nInputs = 4;
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nOutputs = 6;
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A = [0 0 0 1 0 0 0 0 0 0 0 0;
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0 0 0 0 1 0 0 0 0 0 0 0;
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0 0 0 0 0 1 0 0 0 0 0 0;
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0 0 0 0 0 0 0 -g 0 0 0 0;
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0 0 0 0 0 0 g 0 0 0 0 0;
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0 0 0 0 0 0 0 0 0 0 0 0;
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0 0 0 0 0 0 0 0 0 1 0 0;
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0 0 0 0 0 0 0 0 0 0 1 0;
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0 0 0 0 0 0 0 0 0 0 0 1;
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0 0 0 0 0 0 0 0 0 0 0 0;
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0 0 0 0 0 0 0 0 0 0 0 0;
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0 0 0 0 0 0 0 0 0 0 0 0];
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% Note: In paper, 1/m is in wrong spot
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B = [0 0 0 0;
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0 0 0 0;
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0 0 0 0;
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0 0 0 0;
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0 0 0 0;
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1/m 0 0 0;
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0 0 0 0;
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0 0 0 0;
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0 0 0 0;
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0 1/Ix 0 0;
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0 0 1/Iy 0;
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0 0 0 1/Iz];
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C = [1 0 0 0 0 0 0 0 0 0 0 0;
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0 1 0 0 0 0 0 0 0 0 0 0;
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0 0 1 0 0 0 0 0 0 0 0 0;
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0 0 0 0 0 0 1 0 0 0 0 0;
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0 0 0 0 0 0 0 1 0 0 0 0;
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0 0 0 0 0 0 0 0 1 0 0 0];
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D = zeros(6,4);
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continuous_system = ss(A, B, C, D);
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T_s = 0.05;
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discrete = c2d(continuous_system, T_s);
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%Check if this works
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impulse(discrete, 0:T_s:1)
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%We should see that U1 gets us only translation in z, U2 couples Y2 and Y4,
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%U3 couples Y1 and Y5, and U4 gets us Y6
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%% Define goals
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%Goal 1: settle at 1m height <2s
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x_0_up = [0, 0, -1, ...
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0, 0, 0, ...
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0, 0, 0, ...
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0, 0, 0]'; %Redefine origin!
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%Goal 2: Stabilize from a 10-degree roll and pitch with <3deg overshoot
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x_0_pitch = [0, 0, 0, ...
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0, 0, 0, ...
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10, 0, 0, ...
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0, 0, 0]'; %Pitch of 5 degrees
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x_0_roll = [0, 0, 0, ...
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0, 0, 0, ...
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0, 10, 0, ...
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0, 0, 0]'; %Roll of 5 degrees
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%Goal 3: Move from position (0,0,0) to within 5 cm of (1,1,1) within 5 seconds.
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x_0_trans = [-1, -1, -1, ...
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0, 0, 0, ...
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0, 0, 0, ...
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0, 0, 0]'; %Redefine origin!
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%Define Q and R for the cost function. Begin with nominal ones for all.
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Q = diag([1, 1, 1, ... % x, y, z
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1, 1, 1, ... % x', y', z'
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1, 1, 1, ... % roll, pitch, yaw
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1, 1, 1]); % roll', pitch', yaw'
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R = diag([10, 6000, 6000, 1]); % upward force, pitch torque, roll torque, yaw torque
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%% Finite-Time Horizon LQR for Goal 1
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%Calculate number of timesteps.
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tSpan = 0:T_s:2;
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nSteps = length(tSpan);
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%Determine gains
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[K, P] = LQR_LTI(discrete.A, discrete.B, Q, R, nSteps);
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%Propagate
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[ulqr, xlqr] = propagate(nInputs, nStates, nSteps, x_0_up, K, A, B);
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%Plot
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plot_states(xlqr, tSpan);
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%% Finite-Time Horizon LQR for Goal 2
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%Calculate number of timesteps.
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tSpan = 0:T_s:0.2;
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nSteps = length(tSpan);
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%Determine gains
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[K, P] = LQR_LTI(discrete.A, discrete.B, Q, R, nSteps);
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%Pitch Goal
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%Propagate
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[ulqr, xlqr] = propagate(nInputs, nStates, nSteps, x_0_pitch, K, A, B);
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%Plot
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plot_states(xlqr, tSpan);
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%Roll Goal
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%Propagate
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[ulqr, xlqr] = propagate(nInputs, nStates, nSteps, x_0_roll, K, A, B);
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%Plot
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plot_states(xlqr, tSpan);
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%% Finite-Time Horizon For Goal 3
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%Calculate number of timesteps.
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tSpan = 0:T_s:5;
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nSteps = length(tSpan);
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%Determine gains
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[K, P] = LQR_LTI(discrete.A, discrete.B, Q, R, nSteps);
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%Pitch Goal
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%Propagate
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[ulqr, xlqr] = propagate(nInputs, nStates, nSteps, x_0_trans, K, A, B);
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%Plot
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plot_states(xlqr, tSpan);
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%% Helper Functions
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function [K, P] = LQR_LTI(A, B, Q, R, nSteps)
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%Set P up
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P = zeros(size(Q, 1), size(Q, 2), nSteps);
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%Initial value of P
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P(:, :, nSteps) = 1/2 * Q;
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%Set K up, initial K is 0, so this is fine.
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K = zeros(length(R), length(Q), nSteps);
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for i = nSteps-1:-1:1
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P_ = P(:,:, i+1);
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K(:, :, i) = ( 1/2 * R + B' * P_ * B )^(-1) * B' * P_ * A;
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P(:, :, i) = A' * P_ * ( A - B * K(:, :, i) ) + Q * 1/2;
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end
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end
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function [ulqr, xlqr] = propagate(nInputs, nStates, nSteps, x_0, K, A, B)
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%Set up for propagation
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ulqr = zeros(nInputs, nSteps);
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xlqr = zeros(nStates, nSteps);
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xlqr(:, 1) = x_0;
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for i = 1:(nSteps - 1)
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ulqr(:,i) = K(:,:,i) * xlqr(:,i);
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xlqr(:,i+1) = (A*xlqr(:, i) - B*ulqr(:, i));
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end
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end
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function plot_states(xlqr, tSpan)
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figure();
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subplot(1, 2, 1);
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plot(tSpan, xlqr(1, :), '-r');
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hold on;
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plot(tSpan, xlqr(2, :), '-g');
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plot(tSpan, xlqr(3, :), '-b');
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plot(tSpan, xlqr(4, :), '--r');
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plot(tSpan, xlqr(5, :), '--g');
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plot(tSpan, xlqr(6, :), '--b');
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legend('x', 'y', 'z', 'x`', 'y`', 'z`');
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title("Translations(-) and Velocities (--)");
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xlabel("Time(s)");
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ylabel("Displacement (m)");
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subplot(1, 2, 2);
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plot(tSpan, xlqr(7, :), '-r');
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hold on;
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plot(tSpan, xlqr(8, :), '-g');
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plot(tSpan, xlqr(9, :), '-b');
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plot(tSpan, xlqr(10, :), '--r');
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plot(tSpan, xlqr(11, :), '--g');
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plot(tSpan, xlqr(12, :), '--b');
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legend('Pitch (about x)', 'Roll (about y)', 'Yaw (about z)', 'Pitch Rate', 'Roll Rate', 'Yaw Rate');
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title("Angular Displacements(-) and Velocities(--)");
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xlabel("Time(s)");
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ylabel("Displacement (deg)");
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end
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