Need help programming a 2-sliding mode differentiator by Arie Levant in matlab mfile.

Question

Nitish Mulye 2018-3-1

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https://ww2.mathworks.cn/matlabcentral/answers/385691-need-help-programming-a-2-sliding-mode-differentiator-by-arie-levant-in-matlab-mfile

编辑： Mostafa Sallam 2018-9-14

I am working on a model free iPID control which uses the derivative of the system output. I need to apply the 2-sliding mode differentiator designed by Arie Levant http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.218.6376&rep=rep1&type=pdf. The main three equations for the differentiator algoritm is as follows:

I have already made a small program using these equations. I am working with discrete data that updates the values of the parameters each Iterations. Miu and lambda are user defined, and all initial values are 0.

for k=1:200
  y(k)=k;     %Desired Trajectories (a linear lineas ás an input)
end
    % code
for k=2:200
    x(k) = x(k-1)+d_y(k-1);       %Integration of X (internal variable) using Euler backward method (x-dot= u)
    u_1(k) = (-miu)*sign(x(k)-y(k-1)); % Formula of u_1 dot in the algorithm 
    U_1(k) = U_1(k-1)+u_1(k);  % Integration of u_1 dot to obtain u_1 using Euler forward method 
    d_y(k) = U_1(k) - lamda*sqrt(abs(x(k)-y(k-1)))*sign(x(k)-y(k-1)); % derivative of y (U in the algorithm)
    dy(k)=y(k)-y(k-1); % Derivative of Y using differece formula
end
  end

My main question is that how do i integrate the x-dot and u1-dot (x(k) and U_1(k) in the program). I am using the Euler backward and forward method that are used by the discrete integrator block in simulink. I am geting the following result when i give a linear line as an input. Can anyone please check my program and suggest how to smooth the curve