Disturbance observer and prescribed time controller

Hi, I simulated a disturbance observer based prescribed time controller, but I'm facing two problems :

1- When I choose a really small Tfc in the follwoing code, a singularity is resulted in the second state.

2- I wanted to see the performance of the disturbance observer. Unfortunately, it doesn't change. No matter what I choose as its settling time.

clear all
close all
clc

t0 = 0;
tf = 10;
dt = 0.001;
t = t0:dt:tf;
a = 0.01;
tau = 6.65/7;
h = 3;
delay = h/dt; 
Tfc = 5;
eps1 = -2;
eps2 = 3;
 d = 2*sin(0.2*pi*t)+0.15*sin(2*pi*t);
%  d = 0*ones(1,length(t));
D1 = 0.7*pi;
x10 = 1;
x20 = -1;
d_hat(1) = -2;
x1(1) = x10;
x2(1) = x20;
mu(1) = 0.3;
mu0 = mu(1);
b1 = 2;
b2 = 3;
phi(1) = mu(1)+x2(1);
phi1(1) = -b1*(1-exp(-x1(1)))/(Tfc-t(1));


Sd0 = 8;
W = @(s) exp(a*s).*(s-h).^5.*(2*h-s).^5;
Wc = 1/(integral(W,h,2*h));
for i=1:length(t)-1
    if t(i)<h
        Rh(i) = 0;
    elseif t(i)>=h && t(i)<= 2*h
%         Rh(i) = sin(pi*t(i)/h)^2;
          Rh(i) = (t(i)-h).^5*(2*h-t(i)).^5;
    else
        Rh(i) = 0;
    end
    
    if (i-delay)<=0
        mud(i) = mu0;
        Sdd(i) = Sd0;
    else
        Sdd(i) = Sd(i-delay);
        mud(i) = mu(i-delay);
    end
    z2(i) = x2(i)-phi1(i);
    %% Predefined-time Backstepping controller
    if t(i)<Tfc-0.0001
        dphi1_dt(i) = -b1*(1-exp(-x1(i)))/(Tfc - t(i))^2-b1*x2(i)*exp(-x1(i))/(Tfc - t(i));
        phi1(i+1) = phi1(i)+dphi1_dt(i)*(t(i+1)-t(i));
        dphi1_dx1(i) = -b1*exp(-x1(i))/(Tfc - t(i));
        phi2(i) = -b2*1/(Tfc - t(i))*(1-exp(-z2(i)));
    else
        dphi1_dt(i) = 0;
        dphi1_dx1(i) = -1500;
        phi1(i+1) = 0;
        phi2(i) = 0;
        
    end
    
    u(i) = -(x1(i)+eps1*x1(i)+eps2*x2(i)*(1-(x1(i))^2)+d_hat(i)-dphi1_dt(i)-dphi1_dx1(i)*x2(i)-phi2(i));
    
    K(i) = Rh(i)*Wc*exp(-a*(h-2*t(i)));
    dx1(i) = x2(i);
    x1(i+1) = x1(i)+dx1(i)*(t(i+1)-t(i));
    dx2(i) = eps1*x1(i)+eps2*(1-x1(i)^2)*x2(i)+d(i)+u(i);
    x2(i+1) = x2(i)+(t(i+1)-t(i))*dx2(i);
    dphi(i) = eps1*x1(i)+eps2*(1-x1(i)^2)*x2(i)+u(i)+d_hat(i)-(a/(2*(1-tau))*mu(i)+K(i)/(2*(1-tau))*sign(mu(i))*(abs(mu(i)))^(2*tau-1)*abs(mud(i))^(2*(1-tau)));
    phi(i+1) = phi(i)+dphi(i)*(t(i+1)-t(i));
    mu(i+1) = phi(i+1)-x2(i+1);
    
    %% Arbitrary Time Sliding Mode Disturbance Observer
    dmu(i) = (mu(i+1)-mu(i))/dt;
    Sd(i) = dmu(i)+a/(2*(1-tau))*mu(i)+K(i)/(2*(1-tau))*sign(mu(i))*(abs(mu(i)))^(2*tau-1)*abs(mud(i))^(2*(1-tau));
    %dd_hat(i) = -(D1*sign(Sd(i))+a/(1-tau)*Sd(i)+K(i)/(1-tau)*abs(Sd(i))^(2*tau-1)*sign(Sd(i))*abs(Sdd(i))^(2*(1-tau)));
    dd_hat(i) = -(D1+a/(2*(1-tau))*abs(Sd(i))+K(i)/(2*(1-tau))*abs(Sd(i))^(2*tau-1)*sign(Sd(i))*abs((Sdd(i)))^(2*(1-tau)))*sign(Sd(i));

    d_hat(i+1) = d_hat(i)+(t(i+1)-t(i))*dd_hat(i);
end
set(gcf,'color','white')
subplot(2,2,1)
plot(t,x1,'r','LineWidth',1.2);grid minor
leg1 = legend({'$x_1(t)$'});
set(leg1,'Interpreter','latex');
xlabel('t(s)')
ylabel('x1')
ylim([-1 1])
subplot(2,2,2)
plot(t,x2,'b','LineWidth',1.2);grid minor
leg2 = legend({'$x_2(t)$'});
set(leg2,'Interpreter','latex');
xlabel('t(s)')
ylabel('x2')
ylim([-1 1])
subplot(2,2,3)
plot(t,d,'c',t,d_hat,'m','LineWidth',1.2);grid minor
leg3 = legend({'$d(t)$';'$\hat{d}(t)$'});
set(leg3,'Interpreter','latex');
xlabel('t(s)')
ylabel('d(t)')
subplot(2,2,4)
plot(t(1:end-1),Sd,'LineWidth',1.2);grid minor
leg4 = legend({'$S_d(t)$'});
set(leg4,'Interpreter','latex');
xlabel('t(s)')
ylabel('Sd(t)')
leg = [leg1 leg2 leg3 leg4];
set(leg, 'FontSize', 12)

Answer 1

[CHANDRA BABU GUTTIKONDA]

Prescribed-time terms 1/(Tfc−t)1/(T_{fc}-t)1/(Tfc​−t) blow up near TfcT_{fc}Tfc​ in discrete simulation, creating an impulsive control that drives x2x_2x2​ unstable.