Dear Amardeep,

I suggest you to see links and attached files in topic.

-Sliding mode control with adaptive fuzzy approximator for MIMO ...

folk.ntnu.no/skoge/prost/proceedings/ecc03/pdfs/011.pdf

-Adaptive Fuzzy Sliding Mode Control for Inverted ... - Semantic Scholar

https://pdfs.semanticscholar.org/.../406c32b8b05970328c3ff2ae9d...

Best regards

Hi Mr Lafifi, Your approach doesn't require G(x) to be positive definite right? @Mohamed-Mourad Lafifi

Hi Amardeep,

Are you referring to Prof. Mohamed-Mourad Lafifi's G(x) approach?

You said that you have tried the power-rate method, sigmoid function, and the adaptive fuzzy logic to estimate the gain "K" for the discontinuous signum term {−K*sign(s)} in the Sliding Mode Controller. Assuming that K is varying gain, could you tell us why you estimate "K" in the first place?

Hi yew! Yes yes, I was referring to Mr lafifi's work, I mean the paper he shared doesn't seem to use the fact that g(x) is positive definite.

I used sigmoid instead of pure signum because it is a well known cheap fix solution to reduce chattering. Also, my seniors have used power rate reaching laws to attenuate chattering. However in some other literature I found out that they use adaptive fuzzy to estimate the gain(K) associated with the discontinous term(sgn) or estimate entire term (-K*sgn(s)). In both of these, the input to the fuzzy system is sliding surface variable 's'.

Hi Mohamed-Mourad Lafifi , what are the inputs to your fuzzy system? is it the entire state vector or error vector of states of interest(output)? Kindly specify.

Hi Amardeep Mishra, let's take your first case. Are you saying that the chattering continues to exist despite the discontinuous signum function is replaced by a sigmoid, say the hyperbolic tangent function, −K*tanh(s/ε)? If so, have you identified the source of chattering?

Yew-Chung Chak How can I identify the source of chattering? I mean, I am merely trying to control-

1) Normal quadcoter

2) Thrust vectored quadcopter

In #1, out of 4 inputs, almost all of them are chattering. Mostly the controls associated with attitude control loop seems to have dominant chattering vis-a-vis position control loop. This chattering results in chattering in phi, theta and psi as well i.e they dont appear neatly.

Hi Amardeep Mishra, in the quadcopter case, the "chattering" disturbances probably originate from the rotors. The unbalance in rotors can cause rotor vibrations.

Hi yew! I am still unable to attenuate the chattering. I have a system as mentioned in pdf attached below. I used adaptive fuzzy controller similar to the Mr Lafifi's work and I am seeing chattering as can be seen in the eps files attached below. I have used inverse tangent instead of plain sgn function. Kindly suggest some means to reduce this kind of chattering. Thanks in advance Yew-Chung Chak

Kindly note, the desired value for state x1 is 1 and for x2 it is 0.8

Hi Amardeep Mishra, let's try to solve the coupled ODEs in CodeCogsEqn.pdf using a very simple controller by feedback linearization. You can extend the result to sliding mode controller. Here is the code:

clear all; clc

function xdot = f(x, t)

xdot = zeros(2, 1);

u1 = (-x(1)*sin(x(2))*(-(x(1)*x(2) + (x(2))^2) - (x(1) - 1)) + sin(x(1))*(-(cos(x(1))*cos(x(2))) - (x(2) - 0.8)))/(x(2)*sin(x(1)) - x(1)*sin(x(2))*cos(x(2)));

u2 = (x(2)*(-(x(1)*x(2) + (x(2))^2) - (x(1) - 1)) - cos(x(2))*(-(cos(x(1))*cos(x(2))) - (x(2) - 0.8)))/(x(2)*sin(x(1)) - x(1)*sin(x(2))*cos(x(2)));

xdot(1) = x(1)*x(2) + (x(2))^2 + cos(x(2))*u1 + sin(x(1))*u2;

xdot(2) = cos(x(1))*cos(x(2)) + x(2)*u1 + x(1)*sin(x(2))*u2;

endfunction

x0 = [0.1; 0.4],

t = linspace(0, 10, 1000);

y = lsode("f", x0, t);

plot(t, y)

title("Time responses of states");

xlabel("t");

legend("x_1", "x_2");

Hi Yew-Chung Chak ,

Yeah feedback liearization and SMC are already working! There was never the problem with either of those, the main problem arises in the case of adaptive fuzzy SMC. Here are my results for sliding mode control:

Note that both the states x_1 and x_2 are able to converge without any chattering.

Also I just figured out the reason why it was chattering so much and the reason was, high gain K, to make Vdot negative definite. Here are the results for modified AFSMC:

However, one problem still persists and that is, sometimes the error converges to zero, sometimes it doesnt, is it because of the initial values of the parameters that I start my integration with? In my case I am starting off with rand, i.e random numbers.

Hi, I think it is better to plot the vector field or phase portrait using the results that you obtained about the convergence from different initial conditions.

Yew-Chung Chak ,

Also the convergence of parameters is heavily dependent on

(i) Initial values of the parameters, theta_f, theta_g

(ii) The desired values of x_1 and x_2.

Is there a way to guarantee convergence all the time? Kindly reply.

Hi Amardeep Mishra, is there anything wrong with your adaptive fuzzy sliding mode control? A fuzzy system can approximate any smooth function over a compact set to arbitrary accuracy. Please perform the convergence analysis.