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Observer_design_nonlinear.m
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Observer_design_nonlinear.m
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clear all
%% System Parameters %%
syms M m1 m2 l1 l2
% Parameters
g = 10; % gravity
M = 1000;
m1 = 100;
m2 = 100;
l1 = 20;
l2 = 10;
%% Nonlinear System Model %%
% Nx1_dot = x2;
% Nx2_dot = (-m1*g*sin(x3)*cos(x3)-m1*l1*x4*x4*sin(x3) - -m2*g*sin(x5)*cos(x5)-m2*l2*x6*x6*sin(x5)) / (m1*sin(x3)*sin(x3) + m2*sin(x5)*sin(x5)+M);
% Nx3_dot = x4;
% Nx4_dot = (-m1*g*sin(x3)*cos(x3)-m1*l1*x4*x4*sin(x3) - -m2*g*sin(x5)*cos(x5)-m2*l2*x6*x6*sin(x5)) /(l1 * (m1*sin(x3)*sin(x3) + m2*sin(x5)*sin(x5)+M)) - g*sin(x3)/l1;
% Nx5_dot = x6;
% Nx6_dot = (-m1*g*sin(x3)*cos(x3)-m1*l1*x4*x4*sin(x3) - -m2*g*sin(x5)*cos(x5)-m2*l2*x6*x6*sin(x5)) /(l2 * (m1*sin(x3)*sin(x3) + m2*sin(x5)*sin(x5)+M)) - g*sin(x5)/l2;
%% State Spaee Model %%
A = [0 1 0 0 0 0;
0 0 -m1*g/M 0 -m2*g/M 0;
0 0 0 1 0 0;
0 0 (-m1*g-M*g)/(M*l1) 0 -m2*g/(M*l1) 0;
0 0 0 0 0 1;
0 0 -m1*g/(M*l2) 0 (-m2*g-M*g)/(M*l2) 0];
B = [0; 1/M; 0; 1/(M*l1); 0; 1/(M*l2)];
C = [];
D = [];
%% LQR controller %%
R = 10;
Q = [100000 0 0 0 0 0;
0 100000 0 0 0 0
0 0 10 0 0 0;
0 0 0 10 0 0;
0 0 0 0 10 0;
0 0 0 0 0 10];
%% Observer design%%
C1 = [1 0 0 0 0 0];
C2 = [0 0 1 0 0 0;0 0 0 0 1 0];
C3 = [1 0 0 0 0 0;0 0 0 0 1 0];
C4 = [1 0 0 0 0 0;0 0 1 0 0 0;0 0 0 0 1 0];
L_T1 = acker(A',C1',[-1 -1 -1 -1 -1 -1]);
L_T3 = place(A',C3',[-1 -2 -3 -4 -5 -6]);
L_T4 = place(A',C4',[-0.61 -0.62 -0.63 -0.64 -0.65 -0.66]);
L1 = L_T1';
L3 = L_T3';
L4 = L_T4';
num = 1;
%% Observer %%
x = [0.01; 0.01; 0.01; 0.01; 0.01; 0.01];
non_x = x;
x_hat = [0; 0; 0; 0; 0; 0]; % Initial Condition
dt = 0.001;
Times = 1000/dt; %
record_state_non_x = zeros(6,Times);
record_state_x = zeros(6,Times);
record_state_x_hat = zeros(6,Times);
record_times = zeros(1,Times);
K = lqr(A,B,Q,R);
for t = 0.0 : dt : Times*dt
%% Nonlinear %%
non_x_A = [non_x(2);
(-m1*g*sin(non_x(3))*cos(non_x(3))-m1*l1*non_x(4)*non_x(4)*sin(non_x(3)) - m2*g*sin(non_x(5))*cos(non_x(5)) - m2*l2*non_x(6)*non_x(6)*sin(non_x(5))) / (m1*sin(non_x(3))*sin(non_x(3)) + m2*sin(non_x(5))*sin(non_x(5))+M);
non_x(4);
(-m1*g*sin(non_x(3))*cos(non_x(3))-m1*l1*non_x(4)*non_x(4)*sin(non_x(3)) - m2*g*sin(non_x(5))*cos(non_x(5)) - m2*l2*non_x(6)*non_x(6)*sin(non_x(5))) * cos(non_x(3)) / (l1 * (m1*sin(non_x(3))*sin(non_x(3)) + m2*sin(non_x(5))*sin(non_x(5))+M)) - g*sin(non_x(3))/l1;
non_x(6);
(-m1*g*sin(non_x(3))*cos(non_x(3))-m1*l1*non_x(4)*non_x(4)*sin(non_x(3)) - m2*g*sin(non_x(5))*cos(non_x(5)) - m2*l2*non_x(6)*non_x(6)*sin(non_x(5))) * cos(non_x(5)) / (l2 * (m1*sin(non_x(3))*sin(non_x(3)) + m2*sin(non_x(5))*sin(non_x(5))+M)) - g*sin(non_x(5))/l2];
non_x_B = [0;
1/(M + m1 * sin(non_x(3)) * sin(non_x(3)));
0;
cos(non_x(3))/(l1*(M + m1 * sin(non_x(3)) * sin(non_x(3)) + m2 * sin(non_x(5)) * sin(non_x(5))));
0;
cos(non_x(5))/(l2*(M + m1 * sin(non_x(3)) * sin(non_x(3)) + m2 * sin(non_x(5)) * sin(non_x(5))))];
%% Observer %%
% Observer 1
% y = C1 * non_x;
% y_hat = C1 * x_hat;
% non_x_dot = non_x_A - B * K * x_hat;
%
% x_hat_dot = A*x_hat - B*K*x_hat + L1*(y - y_hat);
% non_x = non_x + non_x_dot * dt;
% x_hat = x_hat + x_hat_dot * dt;
% Observer 3
% y = C3 * non_x;
% y_hat = C3 * x_hat;
% non_x_dot = non_x_A - B * K * x_hat;
%
% x_hat_dot = A*x_hat - B*K*x_hat + L3*(y - y_hat);
% non_x = non_x + non_x_dot * dt;
% x_hat = x_hat + x_hat_dot * dt;
%
% Observer 4
% y = C4 * non_x;
% y_hat = C4 * x_hat;
% non_x_dot = non_x_A - B * K * x_hat;
%
% x_hat_dot = A*x_hat - B*K*x_hat + L4*(y - y_hat);
% non_x = non_x + non_x_dot * dt;
% x_hat = x_hat + x_hat_dot * dt;
%% Unit Step Input to Closeloop%%
F = 1; % Unit Input
% % Observer 1
% y = C1 * non_x;
% y_hat = C1 * x_hat;
% non_x_dot = non_x_A - B * K * x_hat + B*F;
%
% x_hat_dot = A*x_hat - B*K*x_hat + B*F + L1*(y - y_hat);
% non_x = non_x + non_x_dot * dt;
% x_hat = x_hat + x_hat_dot * dt;
% Observer 3
% y = C3 * non_x;
% y_hat = C3 * x_hat;
% non_x_dot = non_x_A - B * K * x_hat + B*F;
%
% x_hat_dot = A*x_hat - B*K*x_hat + B*F + L3*(y - y_hat);
% non_x = non_x + non_x_dot * dt;
% x_hat = x_hat + x_hat_dot * dt;
%
% % Observer 4
%
% y = C4 * non_x;
% y_hat = C4 * x_hat;
% non_x_dot = non_x_A - B * K * x_hat + B*F;
%
% x_hat_dot = A*x_hat - B*K*x_hat + B*F + L4*(y - y_hat);
% non_x = non_x + non_x_dot * dt;
% x_hat = x_hat + x_hat_dot * dt;
%% Unit Step Input to Openloop%%
F = 1; % Unit Input
% % Observer 1
% y = C1 * non_x;
% y_hat = C1 * x_hat;
% non_x_dot = non_x_A + B*F;
%
% x_hat_dot = A*x_hat + L1*(y - y_hat) + B*F;
% non_x = non_x + non_x_dot * dt;
% x_hat = x_hat + x_hat_dot * dt;
% Observer 3
%
% y = C3 * non_x;
% y_hat = C3 * x_hat;
% non_x_dot = non_x_A + B*F;
%
% x_hat_dot = A*x_hat + B*F + L3*(y - y_hat);
% non_x = non_x + non_x_dot * dt;
% x_hat = x_hat + x_hat_dot * dt;
% % Observer 4
y = C4 * non_x;
y_hat = C4 * x_hat;
non_x_dot = non_x_A + B*F;
x_hat_dot = A*x_hat + B*F + L4*(y - y_hat);
non_x = non_x + non_x_dot * dt;
x_hat = x_hat + x_hat_dot * dt;
%% Record
record_state_x_hat(1,num) = x_hat(1);
record_state_x_hat(2,num) = x_hat(2);
record_state_x_hat(3,num) = x_hat(3);
record_state_x_hat(4,num) = x_hat(4);
record_state_x_hat(5,num) = x_hat(5);
record_state_x_hat(6,num) = x_hat(6);
record_state_non_x(1,num) = non_x(1);
record_state_non_x(2,num) = non_x(2);
record_state_non_x(3,num) = non_x(3);
record_state_non_x(4,num) = non_x(4);
record_state_non_x(5,num) = non_x(5);
record_state_non_x(6,num) = non_x(6);
record_times(num) = t;
num = num + 1;
end
%% Linear Figures %%
%% Observer Figure %%
%
subplot(2,3,1);
plot(record_times,record_state_x_hat(1,:),'b')
title('State 1: x')
hold on;
subplot(2,3,4);
plot(record_times,record_state_x_hat(2,:),'b')
title('State 2: x dot')
hold on;
subplot(2,3,2);
plot(record_times,record_state_x_hat(3,:),'b')
title('State 3: theta1')
hold on;
subplot(2,3,5);
plot(record_times,record_state_x_hat(4,:),'b')
title('State 4: theta1 dot')
hold on;
subplot(2,3,3);
plot(record_times,record_state_x_hat(5,:),'b')
title('State 5: theta2')
hold on;
subplot(2,3,6);
plot(record_times,record_state_x_hat(6,:),'b')
title('State 6: theta2 dot')
hold on;
%% Nonlinear Figures %%
subplot(2,3,1);
plot(record_times,record_state_non_x(1,:),'r')
legend('Observer','State')
title('State 1: x')
subplot(2,3,4);
plot(record_times,record_state_non_x(2,:),'r')
legend('Observer','State')
title('State 2: x dot')
subplot(2,3,2);
plot(record_times,record_state_non_x(3,:),'r')
legend('Observer','State')
title('State 3: theta1')
subplot(2,3,5);
plot(record_times,record_state_non_x(4,:),'r')
legend('Observer','State')
title('State 4: theta1 dot')
subplot(2,3,3);
plot(record_times,record_state_non_x(5,:),'r')
legend('Observer','State')
title('State 5: theta2')
subplot(2,3,6);
plot(record_times,record_state_non_x(6,:),'r')
legend('Observer','State')
title('State 6: theta2 dot')