i am trying to solve a nonholonomic system against a given controller using ode solver but cann't get the right results.

Question

samee baig 2016-5-30

0
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此问题的直接链接

https://ww2.mathworks.cn/matlabcentral/answers/286430-i-am-trying-to-solve-a-nonholonomic-system-against-a-given-controller-using-ode-solver-but-cann-t-ge

编辑： Walter Roberson 2016-5-30

the controller is designed using model decomposition.the system has six states.

the mathematics for the controller is sound and all states should go to zero against a given initial condition.for different initial condition the sign of d5 and d6 is altered in the controllers to get convergence.

I have tried all possible scenarios but can't get the convergence.i also need to know if the variable 't' {cos(2*pi*t) and sin(2*pi*t)} in my controller has any thing to do with the problem.

the complete code is as follow

function dx=lbracket(t,x)
 dx = zeros(6,1);
 % define state variables
 T=0.5;
 x1 = x(1);
 x2 = x(2);
 x3 = x(3);
 x4 = x(4);
 x5 = x(5);
 x6 = x(6);
v1=-x1-x2*x3-x4*x5;
v2=-x2;
v3=-x4;
v4=-x6;
v5=x3;
v6=x5;
   d5=3.54491*sign(v5)*sqrt(abs(v5)/T);
   d6=3.54491*sign(v6)*sqrt(abs(v6)/T);
   u1=v1+d5*sin((2*pi*t)/T)-d6*cos((2*pi*t)/T);
   u2=v2-d5*cos((2*pi*t)/T);
   u3=v3+d6*cos((2*pi*t)/T);
   u4=v4;
  % 
  % 
   dx(1)     =  u1;
   dx(2)     =  u2;
   dx(3)     =  x(2)*u1;
   dx(4)     =  u3;
   dx(5)     =  x(4)*u1;
   dx(6)     =  u4;
   end
% function lbracket called in another m-file
Tspan = [0 50]; 
IC = [2 2 2 2 2 2];
options = odeset('RelTol',1e-2,'AbsTol',[1e-2 1e-2 1e-2 1e-2 1e-2 1e-2]);
[T,Y] = ode45(@lbracket, Tspan,IC,options);
x1  =   Y(:,1);
x2  =   Y(:,2);
x3  =   Y(:,3);
x4  =   Y(:,4);
x5  =   Y(:,5);
x6  =   Y(:,6);
figure()
plot(T,x1,T,x2,T,x3,T,x4,T,x5,T,x6,'linewidth',2)
legend('x1','x2','x3','x4','x5','x6',6);
xlabel('t(s)')
ylabel('x_1,x_2,x_3,x_4,x_5,x_6')
grid on