Prescribed time disturbance observer based on periodic delayed feedback does not estimate in prescribed settling time.

Hi, I hope you are feeling well. I have designed a prescribed time disturbance observer based on the idea that was used to design a prescribed time controller. First, I have apllied the controller to a first order and then to a third order system.

This is the link of the previous question that I have asked here:

https://ch.mathworks.com/matlabcentral/answers/2095136-a-nonlinear-system-and-its-control-input-in-simulink?s_tid=srchtitle

The problem I have is that the disturbance observer does not work in a prescribed time, I change the settling time, but it still works in a fixed settling time that is around 1.5 seconds. I have attached the simulink model here. I really appreciate your help if you can tell me how I can solve the problem.

[Sam Chak]

Hi @controlEE

The model appears incomplete and I cannot view it in full. I suggest first running the simulation in MATLAB to verify that the governing equations of the prescribed‑time disturbance observer are correct before constructing the model in Simulink.

[controlEE]

Hi @Sam Chak

Here is the simulation in MATLAB. As it can be seen the distubance observer always estimates the disturbance within 2 seconds no matter what I assign the value of h (settling time). I have played and changed the gains, but unfortunately I wasn't able to achieve desired results.

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)

●

[controlEE]

Hi @SamChak Were you able to identify the problem?

[Sam Chak]

Hi @controlEE

I am still investigating the issue because it took considerable time to convert the unannotated, index‑style code into a mathematically interpretable form. Several parameters and equations in the code remain unclear to me. To facilitate the analysis, could you provide the equations that lead to the derivation of the rate of change of the disturbance error, , where ?

That analysis should allow us to find out why the estimated disturbance consistently converges between 1.0 s and 1.4 s as tau varies from 0 to 1, regardless of the observer’s prescribed‑time setting h.

Note that the effect of the prescribed‑time observer becomes insignificant for large values of tau, but becomes noticeable when tau is very small (approaching zero).

% Sam: related to simulation time
% ------------------------------
t0      = 0;            % start time
tf      = 10;           % final time
dt      = 0.001;        % step size
t       = t0:dt:tf;     % time vector

% Sam: related to the control and estimation parameters
% ------------------------------
a       =  0.01;
% tau     =  6.65/7;    % system is still stable in the range 0 < tau < 1
tau     =  0;
h       =  3;           % prescribed settling time of the observer
delay   =  h/dt; 
Tfc     =  5;           % prescribed settling time of the controller
eps1    = -2;
eps2    =  3;

% Sam: related to external disturbance
% ------------------------------
% d     = 2*sin(0.2*pi*t) + 0.15*sin(2*pi*t);   % time-varying disturbance
d       = 1*ones(1, length(t));                 % constant disturbance (easy to analyze)

% Sam: related to initial condition
% ------------------------------
D1      = 0.7*pi;
x10     = 1;            % initial value of x1
x20     = -1;           % initial value of x2
d_hat(1)= -2;
x1(1)   = x10;
x2(1)   = x20;

% Sam: related to delayed terms
% ------------------------------
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;

% Sam: related to prescribed-time
% ------------------------------
W       = @(s) exp(a*s).*(s - h).^5.*(2*h - s).^5;
Wc      = 1/(integral(W, h, 2*h));

% Sam: Main code
% ------------------------------
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
    
    % Controller
    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));
    
    % the bell-shaped curve
    K(i)    = Rh(i)*Wc*exp(- a*(h - 2*t(i)));
    
    % discretized differential equation for state x1
    dx1(i)  = x2(i);
    x1(i+1) = x1(i) + dx1(i)*(t(i+1) - t(i));
    
    % discretized differential equation for state x2
    dx2(i)  = eps1*x1(i) + eps2*(1 - x1(i)^2)*x2(i) + d(i) + u(i);
    x2(i+1) = x2(i) + dx2(i)*(t(i+1) - t(i));
    
    % discretized differential equation for state phi
    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;
    
    % Sam: Most probably a sliding variable related to the disturbance error?
    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

figure
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');
title('Window 1'), 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');
title('Window 2'), 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');
title('Window 3'), 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');
title('Window 4'), xlabel('t(s)'), ylabel('Sd(t)')

leg = [leg1 leg2 leg3 leg4];
set(leg, 'FontSize', 12)

figure
plot(t(1:end-1), K), grid minor, ylim([-0.5, 1.5]), 
title('Prescribed-time observer gain')
xlabel('t'), ylabel('K')

●

figure
plot(t, d_hat); grid minor
axis([1.0, 1.4, 0.9, 1.1])
title('Convergence of the Estimated disturbance')
xlabel('t'), ylabel({'$\hat{d}(t)$'}, 'Interpreter', 'latex')

figure
plot(t, d_hat); grid minor
axis([2.5, 5, 0.9, 1.1])
title('Anomaly in the Estimated disturbance (see Window 3)')
xlabel('t'), ylabel({'$\hat{d}(t)$'}, 'Interpreter', 'latex')