Prescribed Controller for a nonlinear system

I'm working on implementing a Strong Prescribed Time (SPT) controller for a manipulator in MATLAB. I've successfully designed the controller for the first example provided in a research paper. However, I'm facing challenges with the second example. Designing the SPT controller for the second example's manipulator dynamics.

The paper (Ding et al., 2023) outlines the SPT controller design procedure and manipulator dynamics.

type Exmp2.m

h=5;
a = -0.15;
history = @(t)[-0.2;0.1;0.1];

% Define the Rh function based on the given conditions
Rh = @(t) (0 <= t & t < h) .* 0 + ...
          (h <= t & t <= 2*h) .* ((t-h).^5.*(2*h-t).^5) + ...
          (t > 2*h) .* 0;

% Define the PDF gain function based on Rh
W_c = integral(@(s) Rh(s) .* exp(2 * a * s), h, 2 * h);
W = 1 / W_c;
K_a_h = @(t) Rh(t) .* W .* exp(-a * (h - 2 * t));

% Define the delay differential equation
dde = @(t,x,Z)manipulator_dynamics(t,x,Z,K_a_h);

sol = dde23(dde, h, history, [0 40]);
% Plot results
figure;
subplot(3,1,1);
plot(sol.x, sol.y(1,:));
xlabel('Time (s)');
ylabel('x1 (error in q)');
title('State x1 vs Time');

subplot(3,1,2);
plot(sol.x, sol.y(2,:));
xlabel('Time (s)');
ylabel('x2 (error in dq)');
title('State x2 vs Time');

subplot(3,1,3);
plot(sol.x, sol.y(3,:));
xlabel('Time (s)');
ylabel('x3 (error in I)');
title('State x3 vs Time');

% DDE function

function dxdt = manipulator_dynamics(t, x, Z, K_a_h)
    x1_d = 0;
    x2_d = 0;
    x3_d = 0;
    a = -0.15; % Example value
    tau = 81/83; % Example value
    h = 5; % Example value
    J = 1.625e-3; % kg·m²
    m = 0.506; % kg
    L0 = 0.305; % m
    R0 = 0.023; % m
    B0 = 16.25e-3; % N·m·s/rad
    L = 25e-3; % H
    R = 5; % Ω
    K_tau = 0.9; % N·m/A
    G = 9.81; % m/s²
    M0 = 0.434; % kg
    % Derived parameters
    M = J / K_tau + m * L0^2 / (3 * K_tau) + M0 * L0^2 / K_tau + 2 * M0 * R0^2 / (5 * K_tau);
    N = m * L0 * G / (2 * K_tau) + M0 * L0 * G / K_tau;
    B = B0 / K_tau;
    K_B = 0.9;
    Vp = 0.1 * sin(50 * pi *t);  
    z1 = x(1) - x1_d;
    z2 = x(2) - x2_d;
    z3 = x(3) - x3_d;
    z3lag = Z(3,1) - x3_d;

    % Control input
    u = L * ((-a / (2 * (1 - tau))) * z3 + x3_d ...
        + R / L * x(3) + K_B / L * x(2) ...
        - 5 * sign(z3) ...
        - (K_a_h(t) / (2 * (1 - tau))) * (sig(z3 * (abs(z3lag)^(2 * (1 - tau)))))^(2 * tau - 1));

    % Desired trajectory
    q_d = (pi / 2) * sin(t) * (1 - exp(-0.1 * t^2));

    % Dynamics
    dxdt = [x(2);
            (N / M) * sin(q_d) - (N / M) * sin(x(1) + q_d) - (B / M) * x(2) + (1 / M) * x(3);
            -(R / L) * x(3) - (K_B / L) * x(2) + (1 / L) * u + Vp / L];
end


function sigx = sig(x)
tau = 81/83;
sigx = sign(x) .* abs(x).^(2 * tau - 1);
end

[Sam Chak]

Hi @controlEE

Your design workflow looks correct. Since the controller u depends on , you need to find first so that you can determine its time derivative later. However, also depends on . Therefore, you need to backstep and find so that you can find its time derivative . When you backstep to the end, you will need .

Just be careful when performing substitutions. Because there are multiple layers of substitutions, if you make a mistake at the beginning (such as ), it will create a chain of mistakes that will propagate forward to the controller u.

%% terms extracted from the original system
F1      = ...;
F2      = ...;
F3      = ...;
G1      = ...;
G2      = ...;
G3      = ...;

%% miscellaneous terms to be used in Eq.(23) and Eq.(26)
a       = ...;
h       = ...;
rho     = ...;  % ρ, bounded non-negative function derived to satisfy |ϕ(t,x)| ≤ ρ(t,x)
tau     = ...;  % τ
K1      = ...;
K2      = ...;
K3      = ...;
z1      = ...;
z2      = ...;
z3      = ...;
...             % others

%% terms derived from Eq.(23) and their time derivatives
x1d     = 0;
dotx1d  = 0;
x2d     = ...;
dotx2d  = ...;
x3d     = ...;
dotx3d  = ...;

%% controller, according to Eq.(26)
u       = ...;

The perturbing term is not a constant, but a bounded time-varying sine wave. This info is for designing , which is a bounded non-negative function created to satisfy the condition .

By the way, the authors performed coordinate transformation and the x-dot system is expressed in strict-feedback form.

[Sam Chak]

Hi @controlEE

I have briefly reviewed the latest code, focusing on some critical terms. It appears that the authors' backstepping formulas and the "divide-and-conquer" coding guidance I had previously provided are not being followed. For instance, I am unable to locate the terms F1, F2, F3, G1, G2, and G3 within the code. This poses a challenge for me to thoroughly evaluate your implementation.

The code itself is meaningless if the underlying "step-by-step" backstepping control design is flawed. Providing a detailed, annotated step-by-step approach is essential, as it allows your "Supervisor" or "Software Development Manager" to review the code and ensure adherence to standard coding practices.

Regarding the code, the term in x3_d (which is not directly shown in the provided code) appears to be incorrect. Without the detailed "step-by-step" design, it is difficult to trace the reasoning behind this specific implementation of F2. Additionally, the derivation of "xdot_2_desire" is not clearly shown, as the mathematical time derivative is not explicitly presented.

[Sam Chak]

Looks like you have retracted your comment for some unknown reasons. The code runs and returns some plots, but I believe that , , are incorrectly coded, and is not coded in the controller, u.

[controlEE]

Hi @Sam Chak , I calculated x3_dot , could you check the code and especially x3_dot and x2_dot . I tried to assign them accurately based on page 6 , Eq.23 , Eq.24 . I deleted the last comment because the code needs some modifications .

[controlEE]

Hi @Sam Chak , Did u find the reason why we don't get a more ideal result ?

[Sam Chak]

Hi @controlEE, I don't see any error so far.

[controlEE]

So why we don't get the ideal results ? Should we switch to another solver ?

[Sam Chak]

Hi @controlEE

The most probable explanation is that the equations in your code may differ from those used by the authors. Unfortunately, without access to the authors' exact control law design equations for Example 2, we are unable to verify this.

Have you thoroughly checked to ensure that the equations in your code precisely follow the design methodology outlined in the paper? You need to show your every design step as outlined in order to thoroughly check it. Yes, "thoroughly"...

If you lump everything and write a single line equation in the code, it is very dificult to trace the substitutions and check it. I always DIVIDE and CONQUER it...

[controlEE]

Hi @Sam Chak

Ok , here is the last implementation of the example ;

There is an error in Line 54 , Can you determine what the problem is ?

clear all
close all
clc

dt = 0.001;
tf = 5;
t = 0:dt:tf;
J = 1.625e-3;
m = 0.506;
M0 = 0.434;
L0 = 0.305;
R0 = 0.023;
B0 = 16.25e-3;
L = 25e-3;
R = 5e3;
G = 9.81;
Kt = 0.9;
Kb = Kt;
B = B0/Kt;
Vp = 0.1*sin(50*pi*t);
q = pi/2*sin(t).*(1-exp(-0.1*t.^2));
a = -0.15;
tau = 81/83;
h = 5;
x0 = [-0.2,-0.1,0.1];
x(1,:) = x0;
x1d = zeros(length(t),1);
x1dd = zeros(length(t),1);
dx1d = zeros(length(t),1);
M = J/Kt + m*L0^2/(3*Kt) + M0*L0^2/Kt + 2*M0*R0^2/(5*Kt);
N = m*L0*G/(2*Kt) + M0*L0*G/Kt;
B0 = B0/Kt;
delay = h/dt;
W = @(s) exp(a*s).*(s-h).^5.*(2*h-s).^5;
Wc = 1/(integral(W,h,2*h));
for i=2:length(t)
    k = 0;
    for j=1:length(x0)
        if (t(i-1)-2*(length(x0)-j)*h)>=k*h && (t(i-1)-2*(length(x0)-j)*h)<(2*k+1)*h
            Rh(i-1,j) = 0;
            dR(i-1,j) = 0;
            ddRh(i-1,j) = 0;
        elseif (t(i-1)-2*(length(x0)-j)*h)>= (2*k+1)*h && (t(i-1)-2*(length(x0)-j)*h)<2*(k+1)*h
            Rh(i-1,j) = (t(i-1)-2*(length(x0)-j)*h-h).^5.*(2*h-(t(i-1)-2*(length(x0)-j)*h));
            dRh(i-1,j) = 5*((t(i-1)-2*(length(x0)-j)*h)-h).^4.*(2*h-(t(i-1)-2*(length(x0)-j)*h)).^5-5*((t(i-1)-2*(length(x0)-j)*h)-h).^5.*(2*h-(t(i-1)-2*(length(x0)-j)*h)).^4;
            ddRh(i-1,j) = 20*((t(i-1)-2*(length(x0)-j)*h)-h).^3.*(2*h-(t(i-1)-2*(length(x0)-j)*h)).^3*((2*h-(t(i-1)-2*(length(x0)-j)*h)).^2+((t(i-1)-2*(length(x0)-j)*h)-h).^2)-50*((t(i-1)-2*(length(x0)-j)*h)-h).^4.*(2*h-(t(i-1)-2*(length(x0)-j)*h)).^4;
        end
        if t(i-1)<=2*(length(x0)-j)*h
            K(i-1,j) = 0;
            dK(i-1,j) = 0;
            ddK(i-1,j) = 0;
        elseif t(i-1)>=2*(length(x0)-j)*h
            K(i-1,j) = Rh(i-1,j)*Wc*exp(-a*(h-2*(t(i-1)-2*(length(x0)-j)*h)));
            dK(i-1,j) = exp(-a*(h-2*(t(i-1)-2*(length(x0)-j)*h)))*Wc*(2*a*Rh(i-1,j)+dRh(i-1,j));
            ddK(i-1,j) = exp(-a*(h-2*(t(i-1)-2*(length(x0)-j)*h)))*Wc*(2*a*(2*a*Rh(i-1,j)+dRh(i-1,j))+2*a*dRh(i-1,j)+ddRh(i-1,j));
        end
    end



        if(i-1-delay)<0
            xd(i-1,:) = x0;
            x1d_d(i-1) = x1d(1);
        else
            xd(i-1,:) = x(i-1-delay,:);
            x1d_d(i-1) = x1d(i-1-delay);

        end
        z1d(i-1) = xd(i-1,1)-x1d_d(i-1);

        if (i-1-2*delay)<-delay
            xdd(i-1,:) = 0;
            z1dd(i-1) = 0;
        else
            z1dd(i-1) = z1d(i-1);
            xdd(i-1,:) = xd(i-1,:);
        end

        z1(i-1) = x(i-1,1)-x1d(i-1);
        x2d(i-1) = -a*z1(i-1)-K(i-1,1)*z1d(i-1);
        z2(i-1) = x(i-1,2)-x2d(i-1);
        if(i-1-delay)<0
            x2d_d(i-1) = -a*z1d(i-1);
        else
            x2d_d(i-1) = x2d(i-1-delay);
        end
        z2d(i-1) = xd(i-1,2)-x2d_d(i-1);


        dz1d(i-1) = -a*z1d(i-1)+z2d(i-1);
        dz1(i-1) = -a*z1(i-1)-K(i-1,1)*z1d(i-1)+z2(i-1);
        z1(i) = z1(i-1)+(t(i)-t(i-1))*dz1(i-1);

        dx2d(i-1) = -a*(-a*z1(i-1)-K(i-1,1)*z1d(i-1)+z2(i-1))-dK(i-1,1)*z1d(i-1)-K(i-1,1)*(-a*z1d(i-1)+z2d(i-1));
        
        dq(i-1) = pi/2*cos(t(i-1))*(1-exp(-0.1*t(i-1)))+0.2*pi/2*sin(t(i-1))*t(i-1)*exp(-0.1*t(i-1));
        x3d(i-1) = M*(-a*z2(i-1)-K(i-1,2)*z2d(i-1)-N/M*sin(q(i-1))+N/M*sin(x(i-1,1)+q(i-1))+B/M*x(i-1,2)+dx2d(i-1));

        if(i-1-delay)<0
            dx2d_d(i-1) = -a*(-a*z1d(i-1)+z2d(i-1));
            qd(i-1) = q(1);
            x3d_d(i-1) = M*(-a*z2d(i-1)-N/M*sin(qd(i-1))+N/M*sin(xd(i-1,1)+qd(i-1))+B/M*xd(i-1,2)+dx2d_d(i-1));
        else
            dx2d_d(i-1) = dx2d(i-1-delay);
            x3d_d(i-1) = x3d(i-1-delay);
        end
        z3d(i-1) = xd(i-1,3)-x3d_d(i-1);
        z3(i-1) = x(i-1,3)-x3d(i-1);
        dz2(i-1) = -a*z2(i-1)-K(i-1,2)*z2d(i-1)+1/M*z3(i-1);
        dz2d(i-1) = -a*z2d(i-1)+1/M*z3d(i-1);
        z2(i) = z2(i-1)+(t(i)-t(i-1))*dz2(i-1);
        ddx2d(i-1) = -a*(-a*dz1(i-1)-dK(i-1,1)*z1d(i-1)-K(i-1,1)*dz1d(i-1)+dz2(i-1))-ddK(i-1,1)*z1d(i-1)-dK(i-1,1)*dz1d(i-1)-dK(i-1,1)*(-a*z1d(i-1)+z2d(i-1))-K(i-1,1)*(-a*dz1d(i-1)+dz2d(i-1));
        dx3d(i-1) = M*(-a*dz2(i-1)-dK(i-1,2)*z2d(i-1)-K(i-1,2)*dz2d(i-1)-N/M*dq(i-1)*cos(q(i-1))+(x(i-1,2)+dq(i-1))*cos(q(i-1)+x(i-1,1))+B/M*(-N/M*sin(x(i-1,1)+q(i-1))+N/M*sin(q(i-1))-B/M*x(i-1,2)+1/M*x(i-1,3))+ddx2d(i-1));
        dz3(i-1) = -a/(2*(1-tau))*z3(i-1)-1/(2*(1-tau))*K(i-1,3)*abs(z3(i-1))^(2*tau-1)*sign(z3(i-1))*abs(z3d(i-1))^(2*(1-tau))+Vp(i-1)/L-0.1/L*sign(z3d(i-1));
        z3(i) = z3(i-1)+(t(i)-t(i-1))*dz3(i-1);

        u(i-1) = L*(-a/(2*(1-tau))*z3(i-1)+dx3d(i-1)+R/L*x3d(i-1)+Kb/L*x(i-1,2)-0.1/L*sign(z3(i-1))-K(i-1,3)/(2*(1-tau))*abs(z3(i-1))^(2*tau-1)*sign(z3(i-1))*abs(z3d(i-1))^(2*(1-tau)));

        dx(i-1,1) = x(i-1,2);
        dx(i-1,2) = N/M*sin(q(i-1))-N/M*sin(x(i-1,1)+q(i-1))-B/M*x(i-1,2)+1/M*x(i-1,3);
        dx(i-1,3) = -R/L*x(i-1,3)-Kb/L*x(i-1,2)+1/L*u(i-1)+Vp(i-1)/L;

        for j=1:length(x0)
        x(i,j) = x(i-1,j)+(t(i)-t(i-1))*dx(i-1,j);
        end
end

Unrecognized function or variable 'dRh'.

[Sam Chak]

Hi @controlEE

The function or variable 'dRh' is not defined in the code.

[controlEE]

Hi @Sam Chak

as showned in the code we defined dRh , Besides we tried to implement it in SIMULINK . could you take a look at it and troubleshoot the model .

thanks .

[Sam Chak]

Hi @controlEE

I have attempted to run the simulation, but it has been quite challenging to troubleshoot the Simulink model at this stage. The primary reason for this difficulty is that I do not have access to the mathematical model of the Control Law, which would allow me to evaluate the code against a known standard.

In fact, the mathematical model of the Control Law itself should be cross-checked against the standard procedure of the Backstepping Control design method, as outlined by Professor Petar V. Kokotovic. This step is crucial to ensure the correctness of your control design.

Additionally, I am unfamiliar with the specific parts of your MATLAB code where the delayed states of the prescribed-time control scheme are implemented using the array indexing approach. Your way of implementing the delay differential equations (DDEs) appears to be quite different from the 1st example provided, and it requires further understanding and examination.

[controlEE]

Hi @Sam Chak , It's been about months we've been working on 2nd example , although the control effort is not provided in the article , we investigate calculation times . we need to know where the problem lies , what's your suggestion ?

[Sam Chak]

Hi @controlEE

Unlike the formulated PID controller, most sophisticated controllers must be designed from scratch in order to satisfy certain forms of stability criteria. If you were to directly use the control formula in Eq. (26), which the authors claimed was derived from the Backstepping procedure, then you would not be able to ascertain whether the control design is correct or not.

I am curious to know if, without restricting yourself to the use of the Strong Prescribed-time Stabilized Nonlinear Periodic-Delayed Feedback Controller (SPTSNPDFC), what other types of controllers you could potentially design from scratch?

[controlEE]

Hi @Sam Chak , I need to design the SPTSNPDFC controller for the robot manipulator in the first place . I'll proceed to focus on drawbacks that posiibly can be modified and improved in the corresopnding controller . although PID controllers are quite appropriate and are widely used , however I'm seeking for the new control efforts designed theoritically . besides I'd like to share our ideas and prospectives with each other .

[Sam Chak]

Yes @controlEE, you can design the SPTSNPDFC controller for the robot manipulator, thoroughly examine each term in the control equation to identify any potential issues regarding implementation, and ultimately propose hypothetical solutions to address specific challenges.

However, it is essential to understand that the SPTSNPDFC controller must be designed from scratch following Prof. Kokotovic's Backstepping procedure. The SPTSNPDFC controller cannot be developed simply by substituting values into the seemingly ready-to-implement Eq. (26).

Designing the controller from scratch requires constructing the SPTSNPDFC controller incrementally, ensuring that it satisfies the stability condition at every step. This involves demonstrating that each design step is valid and stable beyond reasonable doubt, in line with Prof. Kokotovic's Backstepping procedure.

A code snippet is provided here (note that the steps to verify the stability condition are not included in the code):

https://www.mathworks.com/matlabcentral/answers/2132131-design-of-strong-prescribed-time-spt-controller-for-manipulator-example-2#comment_3196911

[controlEE]

Hi @Sam Chak

I have successfully matched the states and control effort to those depicted in the article.

Moving forward, I plan to make improvements in the system's response and aim to generalize the method further. While the article mentions the control effort in the context of a SISO system, I am considering extending this approach to MIMO systems and other nonlinear systems. I appreciate your guidance as I explore these avenues.

[Sam Chak]

Hi @controlEE

Congratulations! It is excellent to know that your persistence and efforts have paid off. Would you consider posting the full MATLAB code and the Simulink model in the ANSWER section below so that interested researchers can study how you successfully implemented the Strong Prescribed-time Stabilized Nonlinear Periodic-Delayed Feedback Controller (SPTSNPDFC)?

[controlEE]

Hi @Sam Chak

thanks for the guidance from the 1st example ,Once I've finished analyzing the model ,I'll share the model and an explanation here .

[controlEE]

Hi @Sam Chak . Here's the model attached , I hope it helps all the interested researcher to acquire a good prospective in Control Theory.Besides, If you share with me suitable systems and in general opinions which I can apply my controller to .(Strict feedback systems) ,That would be great .