solution of ode's with inputs using ode45
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Why do these two pieces of script give the same output?
Linear Dynamics:
% This code calculates the solution to the Linear system
%-------------------------------------------------------%
function LinSys = LinSol()
tspan = [0 10]; % time span
iniCon = [-1; -1; 0; 0; 0; 0]; % initial conditions
[t,y] = ode45(@(t,x) sys(t,x),tspan,iniCon);
plot(t,y,'linewidth',2);
legend({'\eta_1';'\eta_2';'\eta_3';'\eta_4';'\eta_5';'\eta_6'})
title('Linear Response');
xlabel('Time (s)')
ylabel('Position (m)');
grid on;
% This function simulates the Linear System
%-----------------------------------------%
function dx = sys(t,x)
m = 4; % mass
J = 0.0475; % moment of inertia
r = 0.25; % distance from c.o.m to vector force
g = 9.8; % gravity
c = 0.05; % damping coefficient
A = [0 0 0 1 0 0
0 0 0 0 1 0
0 0 0 0 0 1
0 0 -g -c/m 0 0
0 0 0 0 -c/m 0
0 0 0 0 0 0];
B = [0 0
0 0
0 0
1/m 0
0 1/m
r/J 0];
K = findLQR;
u = -K*x; % control law
% Linear System
%-------------%
dx = A*x + B*u;
% This function calculates the gain matrix K
%------------------------------------------%
function K = findLQR()
p = 0.1;
Q=diag([1 1 1 1 1 1]); % state cost matrix
R=diag([p p]); % input cost matrix
[K,~,~]=lqr(A,B,Q,R); % optimal gain matrix
end
end
end
and Nonlinear Dynamics:
% This code calculates the solution to the Nonlinear system
%---------------------------------------------------------%
function LinSys = NonLinSol()
clearvars
tspan = [0:0.01:10]; % time span
iniCon = [-1; -1; 0; 0; 0; 0]; % initial conditions
[t,y] = ode45(@(t,x) sys(t,x),tspan,iniCon);
plot(t,y,'linewidth',2);
legend({'\eta_1';'\eta_2';'\eta_3';'\eta_4';'\eta_5';'\eta_6'})
title('Nonlinear Response');
xlabel('Time (s)')
ylabel('Position (m)');
grid on;
% This function simulates the Nonlinear System
%--------------------------------------------%
function dx = sys(t,x)
m = 4; % mass
J = 0.0475; % moment of inertia
r = 0.25; % distance from c.o.m to vector force
g = 9.8; % gravity
c = 0.05; % damping coefficient
A = [0 0 0 1 0 0
0 0 0 0 1 0
0 0 0 0 0 1
0 0 -g -c/m 0 0
0 0 0 0 -c/m 0
0 0 0 0 0 0];
B = [0 0
0 0
0 0
1/m 0
0 1/m
r/J 0];
K = findLQR;
dx = zeros(6,1);
u = -K*x; % control law
% Nonlinear Dynamics
%------------------%
dx(1) = x(4);
dx(2) = x(5);
dx(3) = x(6);
dx(4) = -g*sin(x(3))-(c/m)*x(4)+(u(1)/m)*cos(x(3))-(u(2)/m)*sin(x(3));
dx(5) = g*cos(x(3))-g-(c/m)*x(5)+(u(1)/m)*sin(x(3))+(u(2)/m)*cos(x(3));
dx(6) = (r/J)*u(1);
% This function calculates the gain matrix K
%------------------------------------------%
function K = findLQR()
p = 0.1;
Q=diag([1 1 1 1 1 1]); % state cost matrix
R=diag([p p]); % input cost matrix
[K,~,~]=lqr(A,B,Q,R); % optimal gain matrix
end
end
end
0 个评论
回答(1 个)
@Johannes
2015-10-23
编辑:@Johannes
2015-10-23
Hello,
i agree with your linear approach. But in your nonlinear script you are mixing linear control theory with a nonlinear system. You calculate the matrix K by the matrices A and B of the linear System. But you have to use a nonlinear approach to control the nonlinear system.
I have modified your code and there you will see that your results are not equal.
Best Regards, Johannes
3 个评论
@Johannes
2015-10-23
You suggested that both scripts give the same output. I just wanted to show you that this is not true. If you zoom into the plot you will see that there are differences between the lines. The x x x lines represent the linear system.
There are some nice files from the ETHZ about Nonlinear control. http://control.ee.ethz.ch/~apnoco/lectures.php
Unfortunately i can not recommend you an approach. This depends on what you are trying to achieve.
Best Regards,
Johannes
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