Hi Jouila,

Could you provide us the details?

The sliding surface plot is just a "symptom" indicating an issue, probably with the control law. You need to uncover the root cause of the issue, and then solve it appropriately.

Hi Jouila:

My suggestion is nonlinear switching surface, which can include linear dynamics and fraction order of tracking error. More important, the controller design must follow the finite-time stability criterion. It often has the result of finite-time bounded tracking error for the system in the presence of uncertainties. It also possesses transient response and chattering signal. Under appropriate design, null tracking error in finite time for the uncertain system can be achieved without risk of transients and chattering signals.

Sincerely,

Chih-Lyang Hwang

Dear Sir,

I'm working with a non-singular fast-terminal sliding mode to control a rigid one-link robot arm.

I wrote the Matlab code, but I couldn't get rid of the chattering phenomenon

You will find enclosed the expression of both the non-singular fast terminal sliding mode surface and the system that I used.

Thank you

Hi Jouila,

Given θ'' = f(θ, θ') + u

where

f(θ, θ') = [- d*θ' - mgl*cos(θ)] / (ml2),

then, the sliding surface and NTSM control law are proposed as follows:

s = θ + k2*{|θ'|β}*sign(θ'),

u = (ml2)*[− f(θ, θ') − (1/k2)*(1/β)*{|θ'|(2 − β)}*sign(θ') − tanh(s/0.25)].

Codes:

clear all; clc

function xdot = f(x, t)

xdot = zeros(2, 1);

m = 0.5;

l = 1;

d = 0.1;

g = 10;

k2 = 1;

f = (- d*x(2) - m*g*l*cos(x(1)))/m;

ap = (abs(x(2))^(2 - 5/3))*sign(x(2));

s = x(1) + k2*(abs(x(2))^(5/3))*sign(x(2));

u = m*(- f - (1/k2)*(3/5)*ap - tanh(s/0.25));

xdot(1) = x(2);

xdot(2) = (- d*x(2) - m*g*l*cos(x(1)) + u)/m;

endfunction

x0 = [1; 0],

t = linspace (0, 10, 100);

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

plot(t, y)

title ("Time responses of the state variables");

xlabel ("t");

legend ("x_1 theta", "x_2 theta dot");

u = 0.5*(0.2*y(:, 2)+ 10*cos(y(:, 1)) - (3/5)*(abs(y(:, 2)).^(2 - 5/3)).*sign(y(:, 2)) - tanh((y(:, 1) + (abs(y(:, 2)).^(5/3)).*sign(y(:, 2)))/0.25));

plot(t, u)

title ("Non-singular terminal sliding mode control");

xlabel ("t");

ylabel ("u");

s = y(:, 1) + (abs(y(:, 2)).^(5/3)).*sign(y(:, 2));

plot(t, s)

title ("Sliding surface");

xlabel ("t");

ylabel ("s");

% phase plot

plot(y(:, 1), y(:, 2))

title ("Phase plot");

xlabel ("x_1");

ylabel ("x_2");

If you understand the dynamical behavior of the one-link robot, you will see that it can also be effectively stabilized by a simple PD control algorithm:

u = (ml2)*(− kd*θ' − kp*θ + 10).

The desired transient performance and settling time can be easily achieved by tuning kd and kp. However, kp must be selected such that kp > 7.24615.

Codes:

clear all; clc

function xdot = f(x, t)

xdot = zeros(2, 1);

kp = 4^2;

kd = (1.65*4) - 0.1/0.5;

u = 0.5*(- kd*x(2) - kp*x(1) + 10);

xdot(1) = x(2);

xdot(2) = (- 0.1*x(2) - 5*cos(x(1)) + u)/0.5;

endfunction

x0 = [1; 0],

t = linspace (0, 10, 100);

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

plot(t, y)

title ("Time responses of states");

xlabel ("t");

legend ("x_1 theta", "x_2 theta dot");

% control input

u = 0.5*(- 6.4*y(:, 2) - 16*y(:, 1) + 10);

plot(t, u)

title ("PD control");

xlabel ("t");

ylabel ("u");

% phase plot

plot(y(:, 1), y(:, 2))

title ("Phase plot");

xlabel ("x_1");

ylabel ("x_2");

Thank you, sir,

In the expression o the sliding surface, why did you choose x instead of the error (e)?

s = e + k2*{|x'|β}*sign(e'),

Best regards,

Oops! It should be θ instead of x. In my example, the error is defined to be e = θ − θd. If the output θ is to be regulated at θd = 0, then the error is simply e = θ. If a time-varying reference signal θd is to be tracked, then e should be used.

Dear Yew-Chung Chak

Could you send me your email address, please!

Hi:

In general, a finite-time regulation problem can be achieved; however, null finite-time tracking error is difficult. To deal with this situation, the switching gains aren't constant but nonlinear.

Chih-Lyang Hwang