solution of ode's with inputs using ode45

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

Answer 1

[@Johannes]

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

[savvas]

The modification you did still gives the same plot from both scripts, which is wrong for the nonlinear system. What nonlinear approach do you suggest to use?

Thanks

[@Johannes]

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

[savvas]

Yes I know that there is a small difference but still the response is not the expected one for the nonlinear system. I have the plot of the nonlinear system using a much different approach to get it and it is the correct one. Thanks for your input.